Finite Elements Analysis of Structures

Glossary

1

Glossary Acknowledgement This contribution has been prepared in the frame of the project MSM0021630519 Progressive reliable and durable civil engineering structures. GLOSSARY ...........................................................................................................................................................7 1

PRINCIPLE OF FINITE ELEMENT METHOD ..................................................................................13 1.1 1.2 1.3 1.4 1.5 1.6

2

MATHEMATICAL DEFINITION OF FEM.................................................................................................13 DEFORMATION VARIANT OF FEM USED IN PRACTICAL STATICS ..........................................................14 MORE GENERAL FORM OF FEM...........................................................................................................15 MATHEMATICAL FORMULATION OF BOUNDARY ELEMENT METHOD – BEM........................................18 PARTIAL DISCRETISATION OF THE PROBLEM - FINITE LAYER METHOD ...............................................18 IMPACTS OF CURRENT DIVISION OF LABOUR ON FEM IN PRAKTICE .....................................................19

BASIC TERMS AND ALGORITHMS OF FINITE ELEMENT METHOD.......................................22 2.1 2.2 2.3 2.4 2.5 2.6 2.6.1 2.6.2

EXPLANATION OF USED TERMINOLOGY ...............................................................................................22 EXPLANATION OF THE PROCEDURE ON AN EXAMPLE DEFORMATION VARIANT OF FEM ......................23 INDIVIDUAL STEPS OF DEFORMATION VARIANT OF FEM .....................................................................27 SPECIFICATION OF SELECTED OPERATIONS THAT ARE USEFUL TO UNDERSTAND FEM TERMS .............29 PRINCIPLE OF VIRTUAL WORK APPLIED IN FEM PROGRAMS ................................................................36 MAIN OUTCOME FOR THE USERS OF FEM PROGRAMS..........................................................................41 Selecting the elements of the FEM analysis model ........................................................................41 Interpretation of FEM output data ................................................................................................45

3 PHYSICAL AXIOMS AND VARIATIONAL PRINCIPLES OF MORE COMPLEX FEM PROBLEMS.........................................................................................................................................................48 3.1 PRINCIPLES OF APPROXIMATION OF THE SOUGHT DISTRIBUTION OF A QUANTITY IN FEM ...................48 3.2 ELEMENTARY PRINCIPLES OF PHYSICAL NATURE OF FEM...................................................................55 3.3 VARIATIONAL PRINCIPLES OF MECHANICAL PROBLEMS OF FEM.........................................................59 3.3.1 Position of variational principles in mechanics ............................................................................59 3.3.2 Scalar, vector and tensor field in FEM inputs and outputs ...........................................................62 3.3.3 General variational principle ........................................................................................................69 3.3.4 Consequences for some variants of FEM procedures....................................................................73 3.3.5 Special configurations used in FEM..............................................................................................76 3.3.6 Example of evaluation of elements by means of the variational principle.....................................80 3.3.7 Bufler’s variational principles.......................................................................................................81 3.3.8 Inverse variational principles........................................................................................................84 4

FINITE ELEMENTS.................................................................................................................................85 4.1 GEOMETRIC PROPERTIES OF ELEMENTS ...............................................................................................85 4.1.1 Differential elements and finite elements.......................................................................................86 4.1.1.1

4.1.2 4.1.3 4.1.4

4.1.4.1 4.1.4.2 4.1.4.3

4.1.5

Hermite 1D polynomials in FEM.........................................................................................................93 The most known 1D elements ............................................................................................................100 Thin-walled beams of open cross section...........................................................................................102

2D elements .................................................................................................................................105

4.1.5.1 4.1.5.2 4.1.5.3 4.1.5.4 4.1.5.5

4.1.6

How many differential elements are there?..........................................................................................87

Advantages and disadvantages of finite elements..........................................................................89 How not to get lost in the collection of finite elements ..................................................................91 1D elements ...................................................................................................................................93

Triangular elements............................................................................................................................105 Triangular elements with polynomials in L1, L2, L3 ...........................................................................113 Quadrilateral elements with polynomials in x,y.................................................................................117 Iso-, hypo- and hyper-parametric elements ........................................................................................121 Surface elements recommended by the authors..................................................................................127

3D elements .................................................................................................................................147

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Glossary 4.1.6.1 4.1.6.2 4.1.6.3 4.1.6.4 4.1.6.5

Tetrahedron ........................................................................................................................................147 Bricks .................................................................................................................................................150 Toroid.................................................................................................................................................157 Special 3D elements...........................................................................................................................157 Solid elements recommended by the authors .....................................................................................158

4.2 PHYSICAL PROPERTIES OF ELEMENTSE ..............................................................................................174 4.2.1 Physical models of materials of elements ....................................................................................174 4.2.2 What is the effect of physical properties in FEM algorithms.......................................................180 4.2.2.1 4.2.2.2 4.2.2.3 4.2.2.4 4.2.2.5 4.2.2.6 4.2.2.7 4.2.2.8

5

3D constitutive laws...........................................................................................................................180 Reduction of the dimension of a problem ..........................................................................................182 Description of deformation in a reduced problem..............................................................................186 Components of deformation in a reduced problem ............................................................................193 Physical constants of 2D FEM elements ............................................................................................197 Physical constants of 1D FEM elements ............................................................................................202 Physical constants of toroids ..............................................................................................................208 Gas elements ......................................................................................................................................209

MODELLING OF STRUCTURES FOR FEM ANALYSIS................................................................211 5.1 INTRODUCTION TO THE THEORY AND PRACTICE OF CREATION OF FEM MODELS .............................211 5.1.1 Present-day Approach to Modelling of Structures and Soil Environment...................................211 5.1.1.1 5.1.1.2

5.1.2

Dimensions of the Model for FEM analysis.................................................................................221

5.1.2.1 5.1.2.2 5.1.2.3

5.1.3

Defective FEM Results Due to Arithmetics .......................................................................................250 Present-day Possibilities of Improving Arithmetics in FEM Calculations .........................................258

Modelling of Non-linear Behaviour of Structures by means of FEM Algorithms........................260

5.1.4.1 5.1.4.2 5.1.4.3 5.1.4.4 5.1.4.5 5.1.4.6

5.1.5

The 1D Models ..................................................................................................................................226 2D Models..........................................................................................................................................234 Systems Consisting of 1D and 2D Elements ......................................................................................241

Numerical Stability of the Calculation of FEM Models...............................................................250

5.1.3.1 5.1.3.2

5.1.4

Objects and Terms .............................................................................................................................211 The Selection of an Effective FEM Model in Practice.......................................................................218

User Approach to Non-linear FEM Problems ....................................................................................260 Assembly of Equation Systems in Non-linear FEM problems...........................................................263 User’s interventions into the execution of non-linear FEM programs................................................266 More Complicated Constitutive Relations and Projects Depending on the Path................................270 The Selection of the Number of Increments and the Course of the Equilibrium Iteration .................280 Newton-Raphson Method and its Modifications................................................................................281

Transformation of Physical Quantities........................................................................................291

5.1.5.1 5.1.5.2

Transformation of Tensors of Stress, Deformation and Physical Constants ......................................292 Design Stress and Internal Forces ......................................................................................................309

5.2 NOTES CONCERNING THE PROBLEMS OF MODELLING OF CERTAIN STRUCTURES IN THE ENGINEERING PRACTICE ........................................................................................................................................................311 5.2.1 Introductory note .........................................................................................................................311 5.2.2 Modelling of Stiffeners in Planar Structures ...............................................................................312 5.2.3 Modelling of Column-Supports of Floor Slabs ............................................................................314 5.2.4 Boundary Effects in Slab Models.................................................................................................317 5.2.5 Singularities in the Analyses of Structures ..................................................................................318 5.2.6 Modelling of the Interactions between Foundation Grids and Subsoil........................................319 5.2.7 The Density of the Mesh...............................................................................................................320 5.2.8 Modelling of a “Double Beam” Bridge with wide Beams ...........................................................322 5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES ...............................................................323 5.3.1 Purpose of the guide ....................................................................................................................323 5.3.2 Method .........................................................................................................................................323 5.3.3 Core principal of solution............................................................................................................323 5.3.4 Stress components in physically orthotropic plates .....................................................................325 5.3.5 Internal forces in physically orthotropic plates...........................................................................328 5.3.5.1 5.3.5.2

5.3.6

Technical theory of plates with the effect of transverse shear not taken into account........................328 Plates with the effect of transverse shear taken into account..............................................................331

Shape orthotropy of plates...........................................................................................................332

5.3.6.1 5.3.6.2 5.3.6.3

Main principles of the transformation into physical orthotropy .........................................................332 Simple types of orthotropic plates......................................................................................................333 Plates with the effect of transverse shear taken into account..............................................................348

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Glossary 5.3.6.4 5.3.6.5 5.3.6.6

6

Box-sections.......................................................................................................................................352 Multi-cell slabs with linear hinges in longitudinal direction ..............................................................356 Other plate types ................................................................................................................................358

MODELLING OF STRUCTURE-SOIL INTERACTION ..................................................................360 6.1 INTRODUCTION............................................................................................................................360 6.1.1 Origin and Development of the Efficient Subsoil Model .............................................................360 6.1.2 The Main Ideas of the Efficient Subsoil Model ............................................................................362 6.1.3 The Efficient Structure-Soil Interaction Model Assuming an Arbitrary Shape of Structure-Soil Interface .....................................................................................................................................................364 6.1.4 Some Remarks about Soil-Foundation-Structure Interaction......................................................366 6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL..371 6.2.1 Reduction of the Three-dimensional Model to the Two-dimensional Model ...............................371 6.2.1.1 6.2.1.2

6.2.2

One-dimensional Efficient Subsoil or Soil Medium Model..........................................................391

6.2.2.1 6.2.2.2 6.2.2.3

6.2.3

Three-dimensional Models in Geomechanics ....................................................................................371 Two-dimensional Efficient Subsoil Model.........................................................................................373 Introduction........................................................................................................................................391 An Example of the Relation Between the Constants of One- and Two-dimensional Models ............392 Basic One-dimensional Relations ......................................................................................................394

Three-dimensional Efficient Subsoil Model as an Improvement on the Two-dimensional Model396

6.2.3.1 6.2.3.2 6.2.3.3

Main Idea of the Improvement of the Two-dimensional Efficient Subsoil Model.............................396 Basic Geometrical and Physical Relations .........................................................................................397 Some Special Cases ...........................................................................................................................402

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL.................................................405 6.3.1 Introductory Comment.................................................................................................................405 6.3.2 Variational Problem of the Plates on Efficient Subsoil Model ....................................................406 6.3.2.1 6.3.2.2 6.3.2.3 6.3.2.4 6.3.2.5

6.3.3

Implementation of the Soil-Structure Interaction Model Using the Finite Element Technique ...420

6.3.3.1 6.3.3.2 6.3.3.3 6.3.3.4 6.3.3.5 6.3.3.6

6.3.4

Dimension and Compatibility of Finite Elements ..............................................................................420 Kirchhoff's Plate on the 2D-Efficient Model of the Subsoil...............................................................423 General Remarks on the Finite Element Technique ...........................................................................427 Conclusions for the Solution of the Plate-Subsoil Interaction............................................................431 Mindlin's Plate on the 2D-Efficient Model of the Subsoil.................................................................432 Mindlin's Plate on the 3D-Efficient Model of the Subsoil..................................................................435

Nonlinear Analysis of Structure-Soil Interaction using the 2D Efficient Subsoil Model .............448

6.3.4.1 6.3.4.2 6.3.4.3 6.3.4.4 6.3.4.5 6.3.4.6 6.3.4.7 6.3.4.8 6.3.4.9 6.3.4.10 6.3.4.11 6.3.4.12

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Total Virtual Work of the Structure-Soil System ...............................................................................406 Potential Energy of the Plate-Soil System..........................................................................................409 Potential Energy of the Improved Subsoil Model ..............................................................................415 Variational Principles of Structure-Soil Interaction ...........................................................................416 Advantages of Lagrange's Variational Principle and Principle of Total Virtual Work.......................418

Introduction........................................................................................................................................448 Stress in subsoil..................................................................................................................................448 Physical model of soil based on the formula stated in CSN 73 1001 .................................................449 Physical model of soil according to DIN 4019...................................................................................452 Physical model of soil according to Eurocode 7 ................................................................................455 Variability of subsoil input data .........................................................................................................455 Reduction of the dimension of the interactive problem......................................................................459 Surface model of subsoil....................................................................................................................459 The effect of subsoil outside of the structure .....................................................................................464 Implementation into SCIA•ESA PT system.......................................................................................465 Statistical analysis of the structure-soil interaction ............................................................................469 Conclusion .........................................................................................................................................470

NONLINEAR MECHANICS OF CONTINUA AND STRUCTURES ...............................................471 7.1 INTRODUCTION ..................................................................................................................................471 7.1.1 Selected Mathematical Concepts and Notations..........................................................................471 7.1.1.1 7.1.1.2 7.1.1.3 7.1.1.4 7.1.1.5

7.1.2 7.1.3

Index, tensor and matrix notations .....................................................................................................471 Voigt notation ....................................................................................................................................474 Voigt rule for higher order tensors .....................................................................................................476 Tensors...............................................................................................................................................477 Transformation of finite elements matrices........................................................................................484

Classification of Nonlinearity......................................................................................................489 Basic Equations, Eulerian and Lagrangean Elements ................................................................492

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Glossary 7.2 GEOMETRICAL NONLINEARITY...............................................................................................494 7.2.1 Foundational Concepts................................................................................................................494 7.2.1.1 7.2.1.2 7.2.1.3

7.2.2

Systems of coordinates in nonlinear mechanics .................................................................................494 Deformation gradient .........................................................................................................................494 Rate of deformation ...........................................................................................................................497

Strain Measures ...........................................................................................................................498

7.2.2.1 7.2.2.2

Green – Lagrange strain tensor E ....................................................................................................500 Euler - Almansi strain tensor ( e ) ......................................................................................................501

7.2.2.3

Logarithmic strain measure ( ε n )......................................................................................................502

7.2.2.4 7.2.2.5 7.2.2.6

ˆ ) .................................................................................................502 Infinitesimal strain tensors ( ε ), ( e Other strain measures.........................................................................................................................503 Comparison of strain tensors..............................................................................................................504

7.2.3

Stress Measures ...........................................................................................................................506

7.2.3.1

Cauchy stress ( σ ) .............................................................................................................................508

7.2.3.2

Nominal stress ( N ), First Piola – Kirchhoff stress ( P ) ..................................................................508

7.2.3.3

Second Piola – Kirchhoff stress ( S ) .................................................................................................509

7.2.3.4 7.2.3.5

ˆ ) ........................................................................................................................509 Corotation stress ( σ Kirchhoff stress ( τ ) ..........................................................................................................................509

7.2.3.6 7.2.3.7 7.2.3.8

Biot stress ( T )..................................................................................................................................510 Transformations between different types of stress .............................................................................510 Objective stress rate ...........................................................................................................................510

7.2.4 7.2.5

Energetically Conjugate Stress And Strain Measures .................................................................511 Two Formulations of Geometrical Nonlinearity in FEM ............................................................514

7.2.5.1 7.2.5.2

Formulation based on current configuration (updated Lagrangean)...................................................515 Formulation based on reference configuration (total Lagrangean).....................................................522

7.3 MATERIAL NONLINEARITY.......................................................................................................527 7.3.1 Uniaxial Stress.............................................................................................................................527 7.3.1.1

7.3.2

7.3.2.1 7.3.2.2 7.3.2.3

7.4

Uniaxial nonlinear elasticity...............................................................................................................531

General Stress..............................................................................................................................533 Saint-Venant – Kirchhoff material .....................................................................................................534 Hyper-elastic materials.......................................................................................................................535 Hypo-elastic Materials .......................................................................................................................536

SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS...................................537 7.4.1.1 7.4.1.2 7.4.1.3

Picard Iteration Method......................................................................................................................538 Newton – Raphson Iteration Method .................................................................................................539 Riks Method.......................................................................................................................................542

7.5 LINEAR AND NONLINEAR STABILITY, POST-CRITICAL ANALYSIS .................................547 7.5.1 Introduction .................................................................................................................................547 7.5.2 Linear stability.............................................................................................................................548 7.5.3 Nonlinear Stability.......................................................................................................................550 7.5.4 Post-critical Analysis...................................................................................................................551 8

LINEAR AND NONLINEAR DYNAMICS OF STRUCTURES ........................................................553 8.1 VARIATIONAL FORMULATION OF THE INERTIAL PROBLEM ...............................................................553 8.2 DYNAMICS OF FOUNDATION PLATES.................................................................................................555 8.2.1 Consistent Mass Matrix of the Plate on the 2D Subsoil Model ...................................................555 8.2.1.1 8.2.1.2 8.2.1.3

8.2.2

Consistent Mass Matrix of the Plate...................................................................................................555 Consistent Mass Matrix of the Subsoil...............................................................................................556 Resulting Consistent Mass Matrix of the Plate on the 2D Subsoil Model..........................................557

Consistent Mass Matrix of the Plate on the 3D Subsoil Model ...................................................558

8.2.2.1

Damping Properties of the Plate-Soil System ....................................................................................559

8.3 LINEAR SOLUTION OF STRUCTURES SUBJECTED TO VIBRATION .........................................................562 8.3.1 The decomposition into eigenmodes method ...............................................................................562 8.3.1.1

8.3.2 8.3.3 8.3.4 8.3.5

Calculation of seismic effects from response spectrum......................................................................563

Numerical methods of direct integration .....................................................................................565 Explicit methods...........................................................................................................................565 Method of central differences ......................................................................................................565 Implicit methods...........................................................................................................................566

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Glossary 8.3.5.1 8.3.5.2

Newmark method...............................................................................................................................566 Wilson method ...................................................................................................................................568

8.4

NUMERICAL METHODS FOR NONLINEAR SOLUTION OF STRUCTURE MODELS SUBJECTED TO DYNAMIC 569 8.4.1 Modification of relations in motion equations.............................................................................569 8.4.2 Modification of relations between displacement, velocity and acceleration vectors...................570 8.4.3 Variants of connection of methods...............................................................................................572

LOAD

8.4.3.1 8.4.3.2 8.4.3.3 8.4.3.4

9

Algorithm of linear solution...............................................................................................................573 Variant I .............................................................................................................................................574 Variant II............................................................................................................................................579 Variant III ..........................................................................................................................................580

BENCHMARKS AND ILLUSTRATIVE EXAMPLES .......................................................................585 9.1 9.1.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.4 9.5 9.6 9.7 9.8

BENDING WITH THE SHEAR DEFORMATION ........................................................................................585 General remarks ..........................................................................................................................585 GEOMETRIC NONLINEARITY ..............................................................................................................586 General remarks ..........................................................................................................................586 Axially and transversally loaded cantilever beam.......................................................................589 SUBSOIL ............................................................................................................................................594 General remarks ..........................................................................................................................594 CABLES .............................................................................................................................................605 MEMBRANES .....................................................................................................................................610 MECHANISMS ....................................................................................................................................612 STABILITY (BUCKLING).....................................................................................................................617 DYNAMICS ........................................................................................................................................618

LITERATURE...................................................................................................................................................621

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1.1 Mathematical definition of FEM

Glossary SYMBOLS f&

For a field, the superposed dot denotes the material time derivative, i.e. f& ( X, t ) = ∂f ( X, t ) ∂t ; for a function of time only, it is the ordinary time derivative, i.e. f& (t ) = df (t ) dt

f' x

Derivative with respect to the variable x ; when the comma is followed by an index, such as i, j , k , to s , it is the derivative with respect to the corresponding spatial coordinate, i.e. f 'i = ∂f ∂xi

⋅

As in a ⋅ b indicates contraction of inner indices; for vectors, a ⋅ b is the scalar product ai bi ; if one or more of the variables are tensors of second order or higher, the contraction is on the inner indices, i.e. A ⋅ B represents Aij B jk , A ⋅ a represents Aij a j

:

As in A : B indicates double contraction of inner indices: A : B is given by Aij Bij , C : D is Cijkl Dkl ; note the order of the indices! Note also that if A or B is symmetric A : B = Aij B ji

×

As in a × b indicates cross-product, or vector product; in indicial notation, a × b → eijk a j bk

⊗

As in a ⊗ b indicates matrix product of vectors, or Kronecker product of matrices; in indicial notation, a ⊗ b → ai b j ; in matrix notation, a ⊗ b → {a}{b}

T

∂

symbol of partial derivative

7


VARIABLES A, Ax , Ay , Az

Cross section area of a beam, Ax full area, Ay , Az shear area in the y and z directions

BI , B

Matrix of spatial derivatives of shape functions in Voigt notation arranged so that δ {e} = B ⋅ δ d or {D} = B ⋅ d& ; B is a I

rectangular matrix [ B1 , B 2 , K , B n ]

I

B

bandwiceth of the overall (global) stiffness matrix

BI , B

Matrix of material derivatives of shape functions in Voigt notation arranged so that {E} = B ⋅ d or {E& } = B I ⋅ d& I ; B is a rectangular matrix B1 , B 2 , K , B n 

C

SE C SE , Cijkl , C SE  τ Cτ , Cijkl , C τ 

Cauchy Green tensor, C = FT ⋅ F ; it is distinguished from the material response matrices which follow by absence of a superscript, or damping matrix & Material tangent moduli relating S& to E Material tangent moduli relating convected rate of Kirchhoff stress τ ∇c to D

Cσ J , CσijklJ , Cσ J 

Material tangent moduli relating Jaumann rate of Cauchy stress σ ∇J to D

 στ  Cστ , Cστ ijkl , C 

Material tangent moduli relating

Truesdell

rate of Cauchy

stress σ ∇τ to D CS , C1S , C2Sx , C2Sy

Stiffness of subsoil, CS subsoil stiffness matrix, C1S , C2Sx , C2Sy subsoil stiffness parameters

D, Dij , {D}

Rate of deformation, velocity strain, D = sym ( ∇v )

E, Eij

Green strain tensor, E =

F, Fij

Deformation gradient, Fij = ∂xi ∂X j

1 T (F ⋅ F − I) 2

8


J

Determinant of Jacobian between spatial and material coordinates, J = det ∂xi ∂X j 

K0

Linear stiffnes matrix

KT

Tangent stiffnes matrices

K M , Kσ

Material and geometric tangent stiffness, respectively

L, Lij

Spatial gradient of velocity field

M

Mass matrix

NI

Shape functions

P, Pij

Nominal stress (transpose of first Piola-Kirchhoff stress)

R, Rij

Rotation matrix (rotation tensor)

S, Sij

Second Piola-Kirchhoff (PK2) stress

T

Transformation matrix

Tε

Transformation matrix for rotation of strain vector

Tσ

Transformation matrix for rotation of stress vector

Td

Transformation matrix for deformation parameters

U, U ij

Right stretch tensor

W

Work

W virt

Virtual work

W int , W ext

Internal and external work

X , Xi

Material (Lagrangian) coordinates

X I , X iI

X I = [ X , Y , Z ] nodal material coordinates

b, bi

Body force

9

1.1 Mathematical definition of FEM d

Nodal displacements stored in Voigt form

e, eij

Euler – Almansi strain tensor e =

ei

e x , e y , e z  , base vectors of coordinates

f , f I , fiI

Nodal forces

f int , f Iint , f iIint

Internal nodal forces

f ext , f Iext , f iIext

External nodal forces

l

length, span i, j , k indices of the x, y and 2 axes respectively

n, ni , n 0 , ni0

Unit normal in current (deformed) and initial (reference, undeformed) configurations

q, qi

Heat flux, also collection of internal variables in constitutive models

t

time

t , ti

Surface tractions

u, ui

Displacement field

u , v, w

Displacements in the x, y and 2 directions respectively

u I , uiI

Matrix of components of displacement at node I

v I , viI

Matrix of components of velocity at node I

v, vi

Velocity field

w, w

Hyperelastic potential on reference and configurations respectively, e.g. S = ∂w ∂E

xIJ ≡ xI − xJ

Difference in nodal coordinates

x, xi

Spatial (Eulerian) coordinates

x I , xiI

x I = [ xI , yI , z I ] nodal spatial coordinates

Γ, Γ 0

Boundary of body in current (deformed) and initial (reference,

1 ( I − F −T ⋅ F −1 ) 2

intermediate

10

1.1 Mathematical definition of FEM undeformed) configurations

[ξ , η , ξ ]

ξ, ξ i

are natural coordinates (parent element coordinates), also used as curvilinear coordinates

ε, ε ij

Infinitesimal strain tensor

εn

logarithmic strain tensor

Π

Total potential energy

Π int e

potential energy of one element

Π int

potential energy of internal forces

Π ext

potential energy of external forces

ρ , ρ0

Current and original density

Σ

Stress matrix from the relation σ = Σ ⋅ d , in Section 6 stress tensor, where each component is multiplied by the unit diagonal matrix I

σ, σ ij , {σ}

Cauchy (physical) stress tensor

σˆ , σˆ ij , {σˆ }

Corotational stress tensor

τ, τ ij , {τ}

Kirchhoff stress tensor

Φ( X, t )

Mapping from the initial configuration Ω0 to the current or spatial configuration Ω

ˆ ( X, t ) Φ

ˆ to the spatial Mapping from the referential configuration Ω configuration Ω

Ω, Ω 0

Domain of current (deformed), initial (undeformed)

Ωe

Domain of one element

∂

Matrix of differential operators, its application to vector u yield tensor ε : {ε} = ∂ ⋅ u ; ∂ is matrix of the type ( s, d ) , where s is the number of components of the tensor {ε} in Voigt notation

and d is dimension of the model, i.e. number of the components of the displacement vector

11


12


1 Principle of Finite Element Method 1.1 Mathematical definition of FEM When speaking about a general concept of finite element methods – including FEM, BEM, finite layer method and strip method – the mathematical nature inheres in what is termed discretisation of the problem. The term discrete is the opposite of continuous. To be clear: searching for unknown functions in domain Ω with boundary Γ is replaced by searching for a finite number of values of these functions or parameters d from which an approximate solution can be formulated, as explained later. Older methods used the same approach: classical variational methods (W. Ritz, 1908) searched for coefficients of preselected functions with generally non-zero values across the whole domain Ω . The wellknown sieving method, known also as a differential method, replaced derivatives by differences or, more generally, by combinations of several function-values in the nodes of a mesh. The collocation method limited itself to the requirement of satisfying the given conditions (roughly) in several selected points of Ω and Γ . Formally, the analytical solution of differential equations was always transformed to the solution of systems of algebraic linear equations. The same applies to FEM. The improvement is in the way this transformation is carried out, or mathematically speaking, in the selection of base functions into which the sought functions are decomposed. The decomposition is closely related to the division of domain Ω (or Γ in BEM) into subdomains Ωe , briefly called finite elements, contrary to “infinitely small differentials” dΩ , dΓ of the exact analysis. In the beginning, mathematicians were not interested into this approach. The first approximation of the function of two variables by a combination of linear functions over triangular elements (R. Courant 1943, termed plated surface) remained completely unnoticed and even significant accomplishments of engineers in 1956-1965 who started to use polynomials of second, third and even higher degree over the elements, attracted no attention of analysts. Only after the first international conference where the FEM was presented (1st Conference on Matrix Methods in Engineering, Ohio, 1965) even mathematicians noticed what engineers had thought for long – that a qualitatively new mechanism was created and that it should be thoroughly researched. And in around 1968, a quite exact mathematical definition of FEM was already given. FEM is a generalised Ritz-Galerkin variational method that uses base functions defined in a small compact domains closely linked to the selected division of the whole analysed domain to finite elements. In the same period it was shown that FEM generates systems of linear equations that are numerically significantly better conditioned than the still commonly used sieving method; formally flawless definitions of many useful terms were presented, hierarchies of various 1D, 2D and 3D finite elements were established according to conditions of continuity, etc. For detailed information refer to [3] to [5]. Present-day engineers benefit from this extensive research in the following way: they do not have to doubt about the mathematical unobjectionability of FEM and can rely on the convergence towards the exact solution with gradual refinement of the mesh in FEM programs that employ what is termed correct or 13

1.2 Deformation variant of FEM used in practical statics compatible elements (brief overview in [5], Appendix). In order to understand the core principle of FEM, it is convenient to use the engineering explanation of its most often used variant: deformation method, which employs what is called Lagrangean finite elements. This explanation will be made in the following paragraph. It forms the basis of almost all (in 1995 estimated 90 to 95%) commercially successful FEM systems.

1.2 Deformation variant of FEM used in practical statics This method can be easily programmed and produces well conditioned equation systems. The simplicity lies in the energetic concept of the problem, generally in the variational formulation of the problem where we search for an extreme of an operator Π (a “functional” in mechanics) that is of additive nature. It means that its value is for the whole system (domain) equal to the sum of values in the parts or elements of the system (subdomains – finite elements). This nature is characteristic especially for all the quantities defined by means of any bounded integral in the domain. Thus, for example total potential energy Π = Π int + Π ext of internal and external forces in the body is – according to the Lagrange variational principle – minimal just for the real state of the body (u, ε, σ ) . The FEM equation can be in this particular situation obtained through the differentiation of Π with respect to individual deformation parameters d1 , d 2 ,K , d m ,K , d N . For example, the m –th equation is: ∂ ( Π int + Π ext ) ∂d m

= ∂Π int ∂d m + ∂Π ext ∂d m = K m d − f m = 0

(1.2.1)

and can be obtained through the versatile addition theorem, the principle of which lies in the additivity of energy as a scalar, i.e. in the additivity of energy derivations: P

∂Π int ∂d m =

∂ ∑ Π int e e =1

∂d m

P

= ∑ ∂Π int ∂d m = K d e e =1

(1, N ) ( N ,1)

(1.2.2)

P

∂Π ext ∂d m =

∂ ∑ Π ext e e =1

∂d m

P

= ∑ ∂Π ext ∂d m = − f m e

(1.2.3)

e =1

Notice that the geometric dimension of the elements e = 1, 2,K , P is not important. Only their stiffness matrices K e are involved together with vectors of load parameters fe that may be of different dimension for different elements of the same system (ne , ne ) , (ne ,1) . The elements may be one-, two, or three-dimensional with no negative impact on the principle of the solution. One to-be-solved system can generally include e.g. beam, plate, wall and/or shell elements, as well as 3D elements. Before they can be summed, all the matrices K e and fe must be formally transformed to the same dimension ( N , N ) and ( N ,1) , if N is the total 14

1.3 More general form of FEM number of deformation parameters. This applies to matrices K er , fer that are established when elements of matrices K e , fe are written to those positions in matrices ( N , N ) , ( N ,1) where they belong to according to their global code numbers that define the topology of the system. The remaining elements of matrices ( N , N ) , ( N ,1) are zero. The whole system of linear equations for the calculation of unknown parameters d = [d1 , d 2 ,K , d m ,K , d N ]T is formed in a simple way: K d = f

( N , N ) ( N ,1)

(1.2.4)

( N ,1)

P

P

Κ = ∑ Κ er

(N,N)

f = ∑ fer

( N ,1)

e =1 ( N , N )

(1.2.5)

e =1 ( N ,1)

As the global code numbers (list of element nodes) hold complete information about the position of the matrix elements in the ( N , N ) grid, it is not necessary to factually establish the extended matrices K er , fer , which would consume a considerable portion of the memory, but the algorithm employs only the original matrices K e , fe . And it is just this simplicity and universality together with the fact that the system of equations is well-conditioned, that represents the practical advantage of FEM in comparison with classical approaches and that is the main reason why it is so universally and widely used in the present advanced era of computer development. The generalisation through the orthogonalisation principle or through the weighed residua method is explained in the following text.

1.3 More general form of FEM After a more detailed investigation, we can easily find that the additivity of the functional Π = ∑ Πe e

or potentially of any bounded integral in domain Ω = ∑ Ωe e

Ι = ∫ K d Ω = ∑ ∫ K d Ωe Ω

(1.3.1)

e Ωe

is the only condition for the application of a powerful apparatus for establishing the system of equations (i.e. the calculation of coefficients of unknown and absolute elements) developed in FEM. Therefore, even in the early period of FEM development, it became used for problems where no Π -nature quantity could be defined, but we know differential equation L1 (u) = 0 in domain Ω and boundary conditions L 2 (u) = 0 on its boundary Γ that must be met by the sought function u . This may be a set of functions u = [u , v,K]T , shortly a vector of functions, 15

1.3 More general form of FEM or a “vector function”. In that situation we also have more equations L1 and conditions L 2 , which is indicated by the used matrix notation. Then, the problem may be parameterised by the substitution uˆ = Ua or directly by uˆ = Nd (more exact matrix notation will be discussed later) with unknown coefficients a or parameters d . After substituting a set of values a or d into the above-mentioned equations, they will not be fulfilled for obvious reasons and the right-hand side will not give identical zeroes, but certain functions in Ω and Γ that can be called “residuum ε1 and ε 2 ”; which, in general, represent vector functions: L1uˆ = ε1 ≠ 0 in Ω

(1.3.2)

L 2û = ε 2 ≠ 0 on Γ

(1.3.3)

Now, we can use the “orthogonalisation principle”, termed recently also the “principle of weighed residua”, that was in fact used already by Bubnov (1910) and Galerkin (1915) as the main improvement of the Ritz method (1908 - 1909), but without the modern parameterisation of the problem by a numerically suitable base functions with a small compact support ( uˆ = Nd ), in the classical Ritz form uˆ = Ua . From this point of view, FEM can be seen as a generalised Ritz - Galerkin method resulting in significantly better conditioned systems of linear equations. The fundamental idea is simple. First, it will be explained for one unknown function. For the exact solution of u , the residua (1.3.2), (1.3.3) are identical zeros, and consequently, also products ε1 g1 in Ω and ε 2 g 2 on Γ , for arbitrary “weight functions” g1 in Ω and g 2 on Γ , are identical zeroes. The bounded integral from an identical zero is exactly zero, and thus, the following must be satisfied for an exact solution of u and arbitrary “weights” g1 , g 2 : R = ∫ ε1 g1d Ω + ∫ ε 2 g 2 d Γ = 0 Ω

(1.3.4)

Γ

Condition (1.3.4) is not satisfied by an approximate solution. If we had enough time and could gradually substitute various sets d (in modern approaches) or a (in classical approaches) into (1.3.4), then, with pre-selected weights g1 , g 2 , we would obtain different values of the total “error R in comparison against zero”. For example, sets d1, d 2 , d3 ,K , d m would give

R = −6385.51, 800.75, − 1263.04, K , − 38.51 . And R would be absolutely smallest, let us say, for set d83615 , and it its value would be R = 0.0445996 . We might even come across a set that, for the given accuracy, would give R B 0 . This set of parameters can be then declared the best with respect to meeting condition (1.3.4) for the pre-selected g1 , g 2 and the corresponding approximate solution uˆ can be used in further steps. The procedure is clear, but completely useless in practice. Even if we (i) abstracted away from the continuous possibility of changes of individual parameters d j in the set d =  d1 , d 2 , d3 ,K , d j ,K , d N −1 , d N 

T

(1.3.5)

(ii) limited ourselves to certain nodal points of the number axis, (iii) eliminated technically irrational values d j , (iv) managed to get over the impact of the pre-selection of g1 , g 2 , etc., we would have such a huge number of sets (1.3.5), that even the fastest state-of-the-art computers would not find the best set in a reasonable time.

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1.3 More general form of FEM The way out of this is that after the substitution of (1.3.5) into (1.3.4) and after integration (over the elements Ωe , Γ e ), R becomes a function of N parameters d1 , d 2 ,K , d N : R(d1 , d 2 , d3 ,…, d N ) = 0

(1.3.6)

If equations (1.3.2), (1.3.3), respectively their operators L1 , L 2 , are linear, then also formula (1.3.4) is linear in parameters d1 , d 2 ,K , d N . It is therefore useful to establish for them N algebraic linear equations, which can be achieved if we write equation (1.3.4) N - times with N different weight functions ( g1 , g 2 ) j , j = 1, 2,K , N : R j (d1 , d 2 ,K , d N ) = 0

j = 1, 2,K , N

(1.3.7)

The sought parameters d1 , d 2 ,K , d N , can be found from system of N equation (1.3.7), which gives the approximate solution of the problem. If operators L1 , L 2 are not linear, also equations (1.3.7) are non-linear, which on one hand results in (i) well-known complications of solution algorithms (e.g. Newton-Raphson method), (ii) problem of ambiguity, etc., but, on the other hand, does not mean a principal limitation of the application of the approach. In problems of mechanics, this mathematical algorithm can be sometimes clearly mechanically interpreted. For example, already Bubnov (1910) and Galerkin (1915) considered it to be a generalised principle of virtual work, which was elaborated on by a number of other authors, best by V. Z. Vlasov in 1950 - 1960. Weight functions g are in this sense considered to be a generalised virtual “displacements” (states of deformation) of the system and L are considered to be a generalised “force” quantities. As most of the authors did not reproduce the original ideas correctly, we recommend that readers should study the original publications [10] to [14] and the literature cited in [3] to [5]. It is clear from the formal derivation of (1.3.7) that different weight functions g would have different corresponding equation systems (1.3.7) and thus different solutions d and subsequently different internal forces, stresses and deformation in the analysed structure. If the solution d was strongly dependent on the selection of g , it would not be reliable for technical applications. Bubnov, Galerkin and also Vlasov tried to clear this issue through the mechanical interpretation of g . We can briefly say: if we are not able to respect all the possible virtual displacements, let us respect at least N mutually independent and, for the given system and its connections, most characteristic displacements, i.e. functions g . Relation to other methods was scrutinised as well. For example, in mechanical problems with a potential energy, if we select the functions g to be gradually equal to all base functions of set U , and if we fulfil in advance boundary conditions L 2 = 0 on Γ , we obtain equations (1.3.7) identical to the Ritz method. If we fulfil in advance conditions L1 = 0 in the analysed domain, the procedure is identical to Trefftz method, which is the historically oldest “boundary method” preceding the nowadays BEM that will be briefly described in the following paragraph.

17

1.4 Mathematical formulation of boundary element method – BEM

1.4 Mathematical formulation of boundary element method – BEM The most concise description of BEM reads: We select such weight functions g1 , g 2 and perform such per partes domain integral integrations (Gauss - Ostrogradsky theorem), so that equation (1.3.4) contains only integrals in Ω and on Γ that can be numerically calculated from the input data and so that the only unknowns are the distributions of the quantities along boundary Γ . In general, this can be achieved in problems of mechanics with boundary Γ = Γ p + Γu (stress vector p is given on Γ p , displacement vector u on Γu ) through the selection of g1 = u ∗j ,

g 2 p = u ∗j ,

g 2u = p ∗j

(1.4.1)

where u ∗j is the source function for displacements and p ∗j its “reaction” on Γu . The stated distributions along the boundary are parameterised the usual way: we select a finite number of elements on boundary Γ , define finite boundary elements and their nodal parameters, in total N values for set (1.3.5). If we gradually select the source functions for individual boundary nodes to be the weight functions, we obtain such a number of algebraic equations that is just required to solve the parameters. This rather mnemonic and encapsulated overview should be elaborated at least in the following: If there is just one unknown function – e.g. the deflection of a membrane w , temperature T , torsion function F , etc.– then each boundary node has just one unknown parameter d = w, T , F etc. If there are two or three unknown functions – e.g. u, v, ( w) in a 2D (or 3D) elasticity problem – then each boundary node has two or three parameters d , marked locally d1 , d 2 , d3 , e.g. u, v, w in a pure deformation variant with reaction components px , p y , pz in the fixed nodes. Similarly to FEM, also other variants are possible. Also notations (1.3.2), (1.3.3), (1.3.4) then represent two or three equations, so that also the number of conditions for the solution increases correspondingly as well. For the weight functions are the used what is termed fundamental functions, under special conditions the exact source functions of individual displacement components u , v, w . These are, in accordance with the general influence principle, identical with the distribution of displacement components caused by singular loads Px = 1 , Py = 1 or Pz = 1 acting on the analysed system in boundary nodes. The method has developed from the older method of integral equations (Boundary Integral Equation Method, BIEM) and the present-day common international name is BEM (Boundary Element Method).

1.5 Partial discretisation of the problem - Finite Layer Method The discretisation of the problem – i.e. the substitution of unknown functions defined over the continuum of an domain and along its boundary by a countable, even finite, set of 18

1.6 Impacts of current division of labour on FEM in praktice parameters (“parameterisation of the problem”) – does not have to be complete. If we can in a certain direction, let us say in the x- direction, make a very good estimate of the character of the course of functions f ( x, y ) e.g. by means of trigonometric components, it is sufficient to divide just the y - interval (the front arch of a prismatic folded plate, support edge of a bridge deck, etc.) and we get elements of a “strip method” that reduces a 2D problem into a 1D task. The reduction in the dimension of the problem can also be made in advance through the following. Instead of unknown function fD of several variables in domain Ω D we introduce its projection to function f D − s of fewer variables in domain Ω D − s , where D is the dimension of the original domain and s is the reduced dimension – practically, s = 1 or 2 in common transformations (Fourier, Laplace, Hankel and others). It is an integral transformation with different “weight” functions g D , defined in the original domain Ω D . After bounded integral f D−s = ∫ f D g D d Ω

(1.5.1)

Ω

is introduced, the only remaining variables are those from domain Ω D − s . One of the oldest technical applications deals with a layered continuum (Bufler, Nikitin-Shapiro, Falk and others) and reduces a 2D symmetrical problem or a 3D general problem into a 1D problem within the interval 0 ≤ z ≤ H , divided into layers of thickness H i , i = 1, 2,K , n . The procedure is known as a finite layer method = FLM. The reduction in the dimension of type (1.5.1) is also intensively exploited for the reduction of time variable t , e.g. in viscoelasticity problems. The solution then deals with what is termed assigned elastic problem. The reduction in the dimension of a problem from a 3D construction subsoil massif into a 2D surface problem in the footing surface is the fundamental precondition for the effectiveness of FEM programs in the field of common foundation engineering; see [8, 9].

1.6 Impacts of current division of labour on FEM in praktice It is a well-known fact that in the current period the extent of practically useful scientific/technical knowledge doubles within 10 years, while the half-life of scientific knowledge (replacement by new, more accurate, more economic with regard to scientific thought, and more generally valid pieces of knowledge) is about 5 years. This gives rise to the question what part of the present-day information explosion is supposed to penetrate down to the engineer-designer, what part should be understood and used actively in their design practice, what part they are supposed to know as existing in order to be able to find the details, etc. This is not a simple task as the human brain competes with the most powerful computers as far as the structure is concerned, but it dramatically lags behind in terms of the (i) speed of performed operations, (ii) scanning of information (concentration about 6 bits per second), and (iii) time over which the information is stored (some data, even most of them, are erased immediately). An erudite specialist just before retirement holds in their brain, i.e. in their operational memory, approximately 109 - 1010 bits of information, unless they develop sclerosis (through bad diet and insufficient mental gymnastics) when they are about 40 that 19

1.6 Impacts of current division of labour on FEM in praktice keeps progressing and increases the natural handicap of hundreds of thousands of neurons dying every day. Even under optimal circumstances, it is illusionary to require that the engineer-designer fully understands everything they use for their work, e.g. that they have acquired comprehensive knowledge of methods of analysis, their numerical algorithms and programs, that they know everything about the work with computer and its peripheries, scanners, printers, plotters, hardcopy generators, digitizers, etc. Similarly, they can hardly be capable of citing (by heart) even a fraction of various standards and regulations. We have to accept the fact that also the division of labour has increased dramatically and that engineering and design institutions now have specialists to tackle this issue: mathematicians-analysts, programmes, electronics engineers, operators, specialists in technical fields, sometimes even documentalists of technical standards and their amendments, etc. What is thus left for the engineer, what burden cannot be taken from their shoulders and what cannot be done by anyone else? There is quite a bit of it, as the everyday practice of structural engineers, production boards, site engineers and other noninterchangeable roles proves. In this text, we will focus on structural engineering, and in this field it is the structural engineer (which includes a team of structural engineers in the case of larger projects) who must (i) define the analysis model, (ii) find all the related material in applicable standards or request a corresponding survey, (iii) prepare input data for the program that will be used to analyse the model, (iv) perform the solution at the computer terminal themselves or with the help of operators, (v) correctly interpret the results for further steps in the design process and (vi) during all phases carry out effective checks of all input data and outputs. The structural engineer has to safely sort all the obtained information relating to the designed structure into categories: geometrical data (lengths, angles, shapes, topology), statical (external impulses, loads, action conditions) and physical (in general, rules for the behaviour of substances, or Hooke’s law in the simplest terms), as they are sorted into these groups by software input interfaces. They need to have a clear idea about the conditions of continuity and equilibrium, both in the structure and at its boundary (support conditions). To sum up: the intrinsic task of the structural engineer is to possess the knowledge of mechanics to the extent that is required by the selected analysis model. As far as the calculation methods (that form the basis for the applied programs) are concerned, the structural engineer has to be familiar with their core principle to the extent relevant to their reliable application under standard circumstances. When FEM programs are used, it is necessary to know what elements are implemented, i.e. what approximations of functions are assumed on them. This must be taken into account when the density (size) of finite elements is being chosen. In addition, also accepted assumptions concerning the internal forces with the resulting limitations in the practical application must be known. Moreover, one must understand the style of expression used by manuals and become familiar with the terminology employed, in order to be able to communicate with the programmers when unavoidable errors in input and output have to be clarified, quickly located and corrected. One also needs to have a correct idea of the extent of application of the selected program or calculation method. With regard to complex problems that appear only occasionally in engineering practice, one must be aware of the programs and technical literature suitable for resolving the problem and must be able to study these material on one’s own. Basically, a civil engineer has been prepared for all these key tasks already at 20

1.6 Impacts of current division of labour on FEM in praktice university, even though to the extent that is proportional to the (i) possibilities of the curriculum, (ii) study plan, (iii) teaching staff, and (iv) university facilities. The present era makes life-long learning in practice vitally important. And the aim of this book is to provide a part of the knowledge required.

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2.1 Explanation of used terminology

2 Basic terms and algorithms of finite element method 2.1 Explanation of used terminology Element is a part of the whole that is either physically composed of the elements, or that is divided into the elements in our theorization. Infinitely small element, also infinitesimal, dx , dxdy , dxdydz etc. is a limit of a finite size element whose dimensions approach zero. It is used in differential and integral calculus and is commonly used in the technical practice as it has its place in a university curriculum. Finite element is an element of finite dimensions, contrary to the infinitely small element. For example, a rectangle a ⋅ b can be divided into 4 finite elements with dimensions a 2 ⋅ b 2 , or a triangle can be split into several triangles on condition that we create the additional vertices through a kind of triangulation, etc. A domain is a connected set of points (open, unless we consider also its boundary; closed including the points of the boundary). It can be simply or multiply connected (with openings). In technical practice, this means so-called bodies (3D) that are in fact three-dimensional in Euclidian space with coordinates x, y, z – in fact a set of points ( x, y, z ). And this is the way that they are handled in dams, thick-walled blocks, soil massifs, etc. where none of the dimensions is significantly smaller. Often, however, simpler analysis models are introduced: Two-dimensional domain (2D) of walls, plates, shells, box structures, etc., planar or spatial of more or less complex structure, the points of which are assigned certain physical properties including those depending on thickness, i.e. sectional dimension that does not exist in a twodimensional domain. One dimensional domain (1D) of beams, frames, truss girders, grids, netting, etc., in general in structures composed of what is termed beams (the beam is thus a one-dimensional model of a body that is in fact three-dimensional). Subdomain is a set of points of the domain that have the same properties, i.e. non-zero measure, in the corresponding dimension d = 3 or 2 or 1 in spatial, planar and linear, respective three-, two-, one-dimensional domains. From this we get the shortest definition: Finite element is a subdomain. A connected set of finite elements, which is usually treated as what is termed substructure or segment, can also be a subdomain. This is also related to the formation of more complex finite elements, see further in the text. Element is the English term that gave the name to the Finite Element Method. Subelement is an element that forms a part of a larger element. For example, four triangular subelements can form one quadrilateral element, or five tetrahedrons can form one hexahedron (called brick), etc.

22

2.2 Explanation of the procedure on an example deformation variant of FEM Superelement is an element created from two or more subelements. Substructure is a part of the structure that can be analysed separately usually with the aim to simplify the solution of a larger structure. Load is in FEM perceived in a generalised sense as all impacts (external and internal, gravitational, thermal, hygroscopic) that produce internal forces and displacement of the structure. In input they are sorted into force load (common forces and moments in points or distributed over a certain 3D, 2D, or 1D domain) and deformation load (similar deformation impulses) and, quite often, thermal and similar impacts (shrinkage – analogy to cooling, etc.) are extracted from the deformation ones. Support (support conditions) is a technical term for boundary geometric or, as the case may be, kinematical, conditions. Basically, it means the reduction of degrees of freedom of a 0D (point), 1D (linear) or 2D (planar support) figure. Inputs strictly distinguish (i) fixed supports in which all degrees of freedom, or more precisely deformation parameters, are a priori nullified, (ii) support with given values of deformation parameters (if zero values are input, this type coincides with the previous one) and (iii) flexible supports. Elastic support is the most frequent type of flexible support. It is assumed that the size of reactions depends only on the magnitude of deformation parameters in the state when the reactions are analysed. In general, the relation may be non-linear, but the term “elastic” eliminates possible dependence on other factors, especially on the history of the loading process. That means that the relation between reactions and deformation parameters is always unambiguous. Physically linear elastic support is a type of support where the relation between reactions and deformation parameters is approximated by a linear law of the following type: r = k d , which is similar to Hooke’s law. In general, the set of reactions can contain also parameters d of other nodes. Usually, however, we assume in one node a linear nature of this relation. In the simplest example with just one reaction r the matrix of various connections k is converted into what is termed a spring constant k in relation r = kd . The linkage between quantities r and d is that virtual work r on d is the full product rd (principle of net virtual work). Element node list is a sorted list of numbers of the element nodes.

2.2 Explanation of the procedure on an example deformation variant of FEM The finite element method is a method based on dividing the analysed domain into subdomains, or illustratively into finite elements. In the widest meaning of the word, this name can be used for any calculation that exploits finite elements. For example, already in the years before Christ, in order to determine the area of a planar figure U , ancient mathematicians divided it into finite elements U i of a simple shape (rectangles, sectors) and applied rule

23

2.2 Explanation of the procedure on an example deformation variant of FEM U = ∑U i

(2.2.1)

i

that is actually commonly used until now, and even for other quantities relating to planar figures, e.g. cross-sections of beams. In addition, it is sufficient that the quantity is additive in nature, i.e. that its magnitude for U is the sum of magnitudes for U i . This is for example the second moment of area (moment of inertia) about the same x -axis: J = ∫∫ y 2 dxdy Ω

J = ∑ Ji

J i = ∫∫ y 2 dxdy Ωi

(2.2.2)

i

It can be easily understood that this applies to any quantity defined by the value of a bounded integral over domain U . Whatever the dimension of this domain, the theorem on the calculation of bounded integral (only one integration symbol is used in the notation here, the type of the integral depends on the dimension of domain U ) whose value equals V holds for all continuous functions (this assumption can be even weakened):

∫ fdU = ∑ ∫ fdU U Ui

U

V = ∑ Vi

(2.2.3)

i

For example, in a one-dimensional domain 0 ≤ x ≤ L we have: After the domain is divided into finite elements - intervals ( x0 , x1 ) , ( x1 , x2 ) , ..., ( xn −1 , xn ) for x0 = 0 , xn = L in dividing points (termed “nodes”) x j , j = 1, 2,K , n − 1 , the mesh has 2 end-nodes and n − 1 internal nodes plus n finite elements and applies to any continuous function f ( x) : xj

L

n

0

j =1 x j −1

∫ f ( x)dx = ∑ ∫

f ( x)dx

(2.2.4)

Function f ( x) can be for example the density of potential energy of internal forces in a beam: 2 2 1  N 2 ( x) β y Qy ( x) β z Qz2 ( x) M x2 ( x) M y ( x) M z2 ( x)  + + + + + f ( x) =   2  EA GA GA GJ k EJ y EJ z 

(2.2.5)

where sectional characteristics and modules can even be – for beams of variable cross-section – functions of x . The additive nature of the energy, which is scalar, is evident: the energy accumulated in the beam as what is called deformation work of an elastic state is the sum of energies of its parts, i.e. finite elements into which the beam is divided. This applies to all, even multidimensional, elastic bodies (structures): Π int = ∑ Π intj

(2.2.6)

j

Here, j is the summation index. The potential energy of external forces, i.e. of the given load (index z ) and reactions, resulting usually from gravitation and other sources outside of the body, is marked Π z and is of the same additive nature, e.g. for a general body Ω with boundary Γ :

24

2.2 Explanation of the procedure on an example deformation variant of FEM Π ext = ∫∫∫ XT u d Ω + ∫∫ pT u d Γ Ω

(2.2.7)

Γ

Also the singular load by concentrated forces and moments is included into the integration, i.e. elements of type

∑ (P u k

k

+ M kϕ k )

(2.2.8)

k

and we sum over all the loaded points k . Using the oldest and most general principle of virtual works (used in simple machines already by Archytas of Tarentum and Archimedes in fourth and third century B.C.), we can derive the Lagrangean variational principle of the minimum (for details see [5]) total potential energy of the system (body + its load) and of the minimum of its internal and external forces. Π = Π int + Π

ext

= min .

(2.2.9)

This principle means that the algebraically lowest value of Π is achieved in a real state of the body – when related to the initial state in which the zero energetic levels of Π , Π int and Π ext are defined. Following from the Clapeyron theorem, Π int = − 12 Π ext in linearly elastic bodies, and therefore Π = −Π int (there is always a decline from the initial state). If we calculate the value of Π ext from any other than the real system of stresses and deformations, we always get – on condition that we meet all geometrical supporting and continuity conditions – algebraically higher value of Π . Instead of finding the stress-state and deformation of the body from the equilibrium and continuity conditions – which are in the form of differential equations with enormous demands on the smoothness of the function (continuity up to the third derivative) – we can find the solution directly from the above-mentioned condition of the minimum, without having to transform to Euler differential equations of this variational problem, that would lead us back to the equilibrium and continuity conditions. The finite element method in a narrower meaning of the word is the oldest form of FEM, only for elements Ωe of domain Ω . Methods that introduce finite elements on the boundary, finite strips and layers (see art. 1.5) have special names. In the period of early development between 1956-1965 FEM used to be usually connected with the problem of finding the extreme of an operator (in mechanics it is also called functional) Π . Contrary to classical variational methods that do not divide domain U and that narrow down the class of allowable functions (among which the extremal meeting the requirement of the variational problem is sought) to a linear combination f = ∑ ak g k = ga

(2.2.10)

k

where base functions g k are non-zero throughout almost the whole domain U (except the set of points of zero measure in which the “coordinate axes or planes are possibly intersected”), the finite element method uses the following new means: a) Base functions g k , closely related to the division of the domain into finite elements and non-zero only in the elements that contain one common node of the mesh (this set of elements is “a small compact support of function g k ”). These functions have a typical “pyramidal” character and can be named in a popular way as source functions of the sought function in terms of coefficients ak , as they define the distribution of 25

2.2 Explanation of the procedure on an example deformation variant of FEM function f in the case that only a1 = 1, a2 = a3 = K = 0 we have f = g1 ,

ak = 1

and

other

ak = 0 , e.g. for

b) Particular meaning of decomposition coefficients f (it is an analogy to the decomposition of a vector into components, hence the name “base” functions) in technical problems: the original decomposition of f within one element is replaced by the decomposition that uses as coefficients certain parameters of nodes in the domain mesh contained in this element and common to all adjacent elements. The relation between these parameters d and original coefficients a in matrix notation is d = Sa a = S −1d (2.2.11) The original decomposition of f within one element is thus transformed to f = ga = gS −1d = Nd N = gS −1 (2.2.12) with base functions N with which it is significantly easier to satisfy the continuity between elements. Once again, these are source functions, this time in terms of parameters d , and e.g. for d1 = 1 , other d = 0 we have f = N1 etc., which gives an illustrative and technically feasible meaning of FEM base functions. Expressed in popular terms, they can be compared to influence lines. c) with regard to the additive nature of the bounded integral (2.2.3), e.g. energy (2.2.9), the general algorithms for the calculation of coefficients of the equations with unknown nodal parameters can be based on the summation of corresponding expressions (integrals) over elements, which can be conveniently programmed using code numbers – as explained later. Therefore, FEM algorithms do not depend on the shape of the analysed domain, but on the shape and type of applied elements. Similarly, it is not necessary to distinguish between internal and boundary points of the domain and conditions (connections) for parameters can be prescribed the same way in internal and external nodes, i.e. various supporting conditions can be met, which was unmanageable in classical variational methods. d) The character of FEM base functions means that the equations have a sparse and under certain circumstances even band matrix of coefficients of left-hand sides and are well-conditioned for numerical solution. The most significant band character of the matrix would occur for mutually orthogonal base functions, i.e. for (2.2.13) ∫ qk qm d Ω = 0 k ≠ m Ω

∫N

k

Nmd Ω = 0 k ≠ m

(2.2.14)

Ω

which was used even in classical methods, e.g. in the analysis of plates using double series. In that case, only diagonal coefficients of the equations remain non-zero and the system splits into separate equations of one unknown. This ideal configuration can be achieved in classical methods only at the cost of a dramatic increase in the stringency of the conditions, e.g. for plates we have to limit ourselves to rectangular plates without openings that are simply supported around the whole circumference. In FEM, the base functions have the already mentioned pyramidal character and their small compact supports have non-zero intersection only for closely “neighbouring” functions, i.e. functions relating to the nodes that are common to one element. For a fine mesh the stated condition of orthogonality is satisfied almost for all pairs of base functions, and thus their system is almost orthogonal. How to achieve this favourable 26

2.3 Individual steps of deformation variant of FEM condition through a suitable numbering of nodes or through other means will be explained later. The aim is always the same: to get the narrowest band of non-zero coefficients around the diagonal. The state-of-the-art program systems perform this minimisation internally and automatically. In conclusion, let us emphasise the following: FEM is a method of analysis of a given model of reality the behaviour of which is described by a variational problem. The problem can be from the field of mechanics, electromagnetics, etc. FEM cannot improve the properties of the used model of reality. On the other hand, it allows for the input of models that are closer to the reality than the models that could be analysed by means of classical methods.

2.3 Individual steps of deformation variant of FEM For the sake of brevity and clarity, we limit ourselves mainly to (i) one area of FEM application – mechanical problems from the field of civil- and mechanical-engineering and to (ii) the most frequent solution of such problems – Lagrange variational principle. In this standard situation we can keep a unified approach according to the enclosed summary, which can be applied with small modifications even in other FEM applications.

2.

o. 4.

.

3.

Matrix of selected functions 6. d = number of unknown function in of coordinates U (mono-norms) the problem, d = 1 to 6 for FEM programs in type (d ,1) common structures

8.

Deformation parameters type (n,1)

9.

10. .

11. Vector of unknown coefficients – type (n,1)

13. .

14. Distribution of displacement components and possibly rotation 15. components – type (d ,1)

16. .

17. Matrix depending on element shape - type (n, n) 19.

20. .

Formulas:

5. .

7.

Quantity:

12.

18.

d = S a

( n ,1)

( n , n ) ( n ,1)

a = S −1 d

( n ,1)

( n , n ) ( n ,1)

u = U a = U S −1 d = N d

( d ,1)

( d , n ) ( n ,1)

( d , n ) ( n , n ) ( n ,1)

( d , n ) ( n ,1)

S

Note: 1 to 5 = selection of the finite element

21. Distribution of strain components or other strain characteristics – type ( s,1)

22.

ε = B a = B S −1d = B d

( s ,1)

( s ,n )

( s , n ) ( n ,1)

27

2.3 Individual steps of deformation variant of FEM 24. 23. .

Matrix derived

25.

G – through derivations

26.

( s ,d )

( s ,n )

ε = G u  ( s ,1) ( s ,d ) ( d ,1) 

27.

B = G U

( s ,d ) ( d , n )

from U 29. Distribution of stress components or internal forces type ( s,1) : 28. .

30.

Space:

s =6,

31.

plane:

s = 3,

32.

plate:

s = 5,

33.

shell:

s = 8 etc.

34.

35. .

36. Matrix of physical constants 37. - type ( s, s )

38. 0.

39.

41. 1.

Matrix of stresses – type ( s , n)

42. e, 43. 44.

Stiffness matrix of element Ω = analysed domain

52. 3.

( s , s ) ( s ,n ) ( n ,n )

∫S

∫B

Ωe

56.

−1T

( s , s ) ( s ,n )

B T C B S −1 d Ω e =

( n,n ) ( n, s ) ( s , s ) ( s ,n ) ( n,n )

Ωe

=

Ωe = element domain

53. Structure stiffness matrix, type ( N , N ) (created through addition 54.

Σ = C B S −1 = C B

( s ,n )

45.

50. Element stiffness matrix in global parameters, extended to 51. dimension ( N , N )

( s , n ) ( n ,1)

C

K =

48.

( s , s ) ( s , n ) ( n , n ) ( n ,1)

( s , s ) ( s ,1)

( n,n )

46. 47. Introduction of global 1a deformation parameters

49. 2.

40.

σ = C ε = C B S −1 d = Σ d

( s ,1)

C B d Ωe

T

( n, s ) ( s , s ) ( s ,n )

d e = Td eg d eg = TT d e K eg = TT K e T (N ,N )

K er = LT K eg L ( N ,n )

(N ,N )

( n,n )

(n,N )

K = ∑ K er

(N ,N )

e ( N ,N )

theorem)

55. 58. 57. 4.

Vector of load parameters (n,1) of element e.

59.

X volume load,

60.

p surface load,

61.

Pi concentrated impulses.

63.

Type X , p , P , is (d ,1)

64. fe = 65.

∫N

T

X dΩ +

Ωe

∫ N p dΓ + T

Ωe

+ ∑ N P + ∑ NϕTi M i T ui i

i

i

28

2.4 Specification of selected operations that are useful to understand FEM terms 62. M i moments – if u does not contain rotations ϕ , ϕ is differentiated from u

66. 67.

68. 5.

69. Matrix of influence functions

71. 6.

72. Vector of load parameters of element in global coordinates, 73. extended to dimension ( N ,1)

70.

Types: (n,1) = (n, d ) (d ,1) N = US −1

type (d , n)

B = GN = BS

−1

type ( s, n)

feg = TT fe ( n ,1)

fer = LT feg ( N ,n )

( N ,1)

( n ,1)

76. 74. 7.

75. Vector of right-hand sides of equations, type ( N ,1)

77.

f = ∑ fe e

78.

All types ( N ,1)

79. 80. 8.

81. System of linear equations 82. for unknown deformation parameters of the body d ( N ,1)

83. 9.

84. Calculation of stresses and displacements in elements once d 85. has been solved, type σ ( s,1) , type U (d ,1)

K d = f

( N , N ) ( N ,1)

( N ,1)

σ = Σ d

( s ,1)

( s , n ) ( n ,1)

86.

U = N d

( d ,1)

( d , n ) ( n ,1)

2.4 Specification of selected operations that are useful to understand FEM terms Detailed derivation of the procedure is given in common textbooks [1 to 5]. FEM is nowadays a part of the curriculum at civil engineering and mechanical engineering faculties. Consequently, the graduates are familiar with its fundamentals. Other users will however find it useful to describe some operations the understanding of which facilitates the handling of inputs and outputs in FEM practice. Also users familiar with the method may benefit from this brief summary. The deformation variant of FEM, used almost in all FEM programs, uses displacement components u , sometimes together with rotation components ϕ , as unknowns. In general, the unknowns are put together into one matrix vector d . For the simplest one-dimensional problem ( d = 1 ), it is a single deflection function w( x) of a beam in plane ( x, z ) . For bending of plates according to the Kirchhoff theory it is w( x, y ) in the direction of z –axis. For planar 29

2.4 Specification of selected operations that are useful to understand FEM terms elasticity problems u = [u , v]T : there are two functions ( x, y ) , dimension of the problem d = 2 . For 3D bodies we have three functions u , v, w with variables x, y, z etc. Normally, we deal at most with six components of vector d – displacements u , v, w and rotations ϕ x , ϕ y , ϕ z – for planar and beam elements of Mindlin type (for details see art. 4). If we can find u accurately enough (including required derivatives), calculation of other quantities necessary for the design practice is a routine work. The distribution of u across one element is approximated polynomially according to no. 4 of art. 2.3 with coefficients a , i.e. expression u = Ua , where U is matrix (no.1 of art. 2.3), containing just zeros (“0”), ones (“1”) and monomials, i.e. expressions of type xα , y β , z γ with integer non-negative α , β , γ . We select values u , or derivatives, exceptionally also linear combinations of components of u as deformation parameters d . And we do it in what is called nodal points or element nodes. These are in fact nodes of the mesh that represents the division of the domain into elements – called corner (or vertex) nodes – or sometimes also some boundary (mid-side) nodes or even inner nodes, e.g. in the centroid. If we introduce such parameters, band width BW of the global stiffness matrix K increases while time T = a ⋅ N ⋅ BW 2 necessary for the solution of the system Kd = f (no. 18 of art. 2.3) increases quadratically with BW . Therefore, we often eliminate these parameters through suitable procedures, which will be explained later in the text. However, it is always true that when substituting nodal coordinates into the expression u = Ua or corresponding derivatives of components u we can obtain relation (2.2.11) and (2.2.12) between the deformation parameters d and coefficients a in the form d = Sa (no.2, art. 2.3), i.e. also its inversion a = S −1d (no.3, art. 2.3.). Consequently, following from no. 4, art. 2.3, the distribution of u across the element can be expressed only through its deformation parameters: u = U S −1 d = N d

( d ,1)

( d , n ) ( n , n ) ( n ,1)

( d , n ) ( n ,1)

(2.4.1)

Also the second form of (2.4.1) is for many elements obtained directly. This introduces a slight shortening of the form of writing the formulas. For better understanding, we keep both forms of (2.4.1). The first one is indispensable for classical elements, which use polynomial coefficients that are only in the second part of the reasoning replaced by the deformation parameters. This represents what is termed parameterisation of the problem: instead of unknown functions, we search only for a finite number of parameters d . In the given variant, we would use equilibrium conditions for the classical solution, e.g. in the theory of elasticity the Lamé’s differential equations of second order. This approach is however applicable only to special homogenous and isotropic problems and due to its complexity was replaced in practical problems by the variational problem of finding the minimum of the total potential energy Π of internal forces ( Π int ) and external forces, i.e. loads ( Π ext ) acting on the body. This is the Lagrange variational principle (2.2.9) that is superior to all equilibrium conditions that prove to be only its Euler differential equations. We omit the mathematically crucial, but for the first explanation less important, consideration about the definition of the “solution” of the original and variational problem, the term “generalised solution” and relating questions concerning the function spaces, equivalence, and convergence. We present only the algorithm of the procedure. 30

2.4 Specification of selected operations that are useful to understand FEM terms The potential energy has in fact always the form of work, i.e. the product of force and distance. The original, non-deformed, state is chosen as the zero energetic level. External forces X in body Ω and p on its boundary Γ acting in the real final deformed state of the body have always negative potential energy when related to this level Π ext = − ∫ uT Xd Ω − ∫ uT pd Γ Ω

(2.4.2)

Γ

because they would not do any work during the return into the original state, on the contrary, they would consume it, it would have to be supplied to them, the loads would have to be lifted, etc. On the other hand, the internal forces would do some work, similarly to a wound spring in a watch that supplies energy to overcome friction and to move the hands. This energy is generated during the static deformation of the body gradually, following the increase in the action of external forces from zero to final values. Therefore, potential energy Π int of internal forces equals to a half of the product of internal forces σ and corresponding strains ε , generally in the form Π int =

1 T ε σd Ω 2 Ω∫

(2.4.3)

What we understand under internal forces σ and strains ε is briefly summarised under symbols σ and ε . See also matrix G in the procedure overview stated under no. 7, art. 2.3. In general, components σ and components ε are mutually linked through the requirement that their product represents the virtual work, or in the case of reversible problems that do not depend on the trajectory (conservative, scleronomic constraints, etc.) the potential energy arising from them. Thus, in terms of physics, everything is ready for the formulation of mathematical algorithms. It is enough to substitute formulas (2.4.1) and no. 6 to 8, art. 2.3 into (2.4.2) and (2.4.3): int Π e = Π ext e + Π e = − ∫ ( Nd ) X d Ω e − ∫ ( Nd ) p d Γ e + T

Ωe

T

Γe

T 1 BS −1d ) CBS −1d d Ω e ( ∫ 2 Ωe

For the transposition of the product we can apply the well-known rule

( BS d ) −1

T

( Nd )

T

= dT NT ,

= dT S −1BT , and thus once the parameters are factored out of the integrals

(parameters are constants, even though still unknown) and the notation according to no. 11 and 14, art. 2.3 is introduced, we get a lucid expression for the total potential energy Π e of one element: 1 Π e = −dTe fe + dTe K ed e 2

(2.4.4)

For clarity reason, we introduced element subscript e for all quantities in (2.4.4). In the whole body Ω = ∑ Ωe e

the potential energy is the sum of formulas (2.4.4) in individual elements. Before we can perform the summation, we have to swap from parameters d e , fe , defined in 31

2.4 Specification of selected operations that are useful to understand FEM terms element coordinates, to parameters d eg , feg in global coordinates that are common to all elements in the system. Practically, this represents a transformation displacement components of the same vector u and small rotation ϕ and components of force P and moment M in nodal points from the element coordinates into global ones. Usually, these are right-angle coordinates. Then we have a well-known operation of rotation of the coordinate-trihedral from the element position xe , ye , ze into the global position xg , y g , z g . For any vector, e.g. u , we can apply the projection formulas with direction cosines that are independent on the order of axes, as e.g. cos( xe xg ) = cos( xg xe ) , which follows from the identity cos α = cos(−α ) . In manuals for FEM programs the user can come across both directions of the transformation, one of which is chosen as the basic one, e.g. from global to element components: ue  cos( xe xg ) cos( xe y g ) v  = cos( y x ) cos( y y ) e g e g  e   we  cos( ze xg ) cos( ze yg ) 

cos( xe z g )  ueg     cos( ye z g )  ⋅ veg  cos( ze z g )   weg 

Brief matrix notation with matrix T3 (3,3) : u e = T3u eg This direction of transformation, the calculation of element components from the global ones, is usually applied at the end of calculation before the stresses in elements are determined. It can be simply proved that multiplication T3 × T3T results in unit matrix 1 , it is sufficient to apply common trigonometric theorems. Therefore, we have T3T = T3−1 and the opposite direction of the transformation is defined by the transposed matrix T3T (3,3) : u eg = T3T u e The same matrix T3 applies also to the transformation of vectors ϕ , P , M . The transformation is usually written for all parameters of both the element deformation d e and external load fe . Thus, for example, a tetrahedron with linear polynomials (art. 4.1.6.1., Fig. 4.19a) has in each of the four vertices 1, 2, 3, 4 three components u , v, w of vectors u1 , u 2 , u3 , u 4 . Its element-coordinates can follow some of the distinctive directions of the tetrahedron, generally different from the global axes. Then we can write  u1e  T3 u   0  2e  =  u3e   0    u 4 e   0

0

0

T3 0 0

0 T3 0

0   u1eg    0  u 2 eg  ⋅ 0  u 3eg     T3  u 4 eg 

in the matrix form d e = T12d eg d eg = T12T d e with transformation matrix T12 (12,12) : 32

2.4 Specification of selected operations that are useful to understand FEM terms T12 = DIAG [ T3 , T3 , T3 , T3 ] Similarly, a 1D beam element with two end-nodes 1 and 2 has in general 2 × 6 = 12 parameters in space combined in vector d e = [u1 , ϕ1 , u 2 , ϕ 2 ]T . Their components are transformed by means of a similar matrix T12 , from global axes xG , y G , z G into central ones xC , y C , z C and back by means of matrix T12T . Components of force parameters f e are transformed in the same way as d e . In general, the transformation can be even more complex, it can also include other geometrical operations, which is usually presented in theoretical manuals. The important thing is that it is always possible to write the matrix relation: d eg = TT d e

d e = Td eg

feg = TT fe

fe = Tfeg

(2.4.5)

What remains is numbering of all deformation and force parameters globally in the whole structure by numbers 1, 2,K , N , which is done by a special subprogram which remains hidden from the user. The aim is to sum properly formulas (2.4.4), i.e. just those items that belong to one particular parameter. They represent the contribution of elements sharing the same node and common value of one parameter in it. This is done through what is termed code number, which will be explained later. Formally, we can introduce what is called extended quantities from element dimension n to dimension N and place the non-zero elements to the positions where they belong by their code numbers. This can be written using localisation matrix L whose size is (n, N ) . In total, n elements of the matrix are equal to 1 and the remaining ones are zero. When the extended quantities are being abandoned in favour of the element ones, i.e. during narrowing from N to n , matrix L can be called the selection matrix: d eg = L d er ( n ,1)

f

eg ( n ,1)

( n , N ) ( N ,1)

= L fer

(2.4.6)

( n , N ) ( N ,1)

The element stiffness matrices K e are first defined in global components of the parameters: K eg = TT K e T

(2.4.7)

and then extended to dimension ( N , N ) : K er = LT K eg L

(2.4.8)

which will be described in detail in formula (2.4.14). That will allow for clear summation of potential energy (2.4.4) of all elements of the system. Π = ∑ Π e = −∑ dTe fe + e

e

1 dTe K ed e = min ∑ 2 e

(2.4.9)

Only this formula is minimal according to (2.2.9), not some partial items of (2.4.4). Further modification is based on simple reasoning: dimensions of the matrices in (2.4.9) are d e (n,1) , f e ( n,1) , K e (n, n) , where n is the number of deformation parameters of one element. In practical applications of FEM to beam structures n equals to 4 (planar truss girders) up to 12 33

2.4 Specification of selected operations that are useful to understand FEM terms (spatial frames), in structures with slabs n is 6 to 48, and even in 3D bodies it is always rather small number, e.g. in 3D bricks it is 24 (tri-linear), 60 (tri-quadratic), 96 (tri-cubic), mostly less then 100. On the other hand, the total number of all deformation parameters of the analysed system, structure or body is usually even for middle-size practical problems significantly larger, approximately N = 103 − 105 . These parameters are globally (comprehensively) numbered from 1 to N by what is termed global indices or code numbers 1 to N , see art. 2.1: d = [ d1 , d 2 , d3 ,K , d N −1 , d N ]

T

(2.4.10)

The definition of the state of each element e requires only n of these parameters that do not have to go in the sequence in the global numbering as, in general, the nodes of the domain mesh (from the numbers of which the global indices of the parameters are derived) cannot be numbered this way. However, we can always define a relation between the local indices 1, 2,K , n and global indices J1 , J 2 , K , J n belonging to the same deformation parameters, which are the first time perceived as a set of deformation parameters of element e , the second time as a part of all deformation parameters of the whole system. The information contained in selection matrix L in formula (2.4.6) can be simply coded in the following way: d eg = [ d1 , d 2 , d 3 ,K , d n −1 , d n ]e =  d J1 , d J 2 , d J3 ,K , d J n−1 , d J n  GLOB T

T

(2.4.11)

What is this primitive, almost entirely “administrative” commonplace in notation, good for? We will show that after a certain amendment or extension, we get the fundamental assumption for the success of FEM algorithms that gives the right to apply what is called addition theorem (assemblage), which is exceptionally suitable for computers. The extension consists of the following: the matrix vector (n,1) according to (2.4.11) is written into a grid ( N ,1) in such a way that positions to which no value is attributed are set to zero: d er = 0, 0,K , 0, d J1 , d J 2 , 0,K 0, d J3 , d J 4 , 0,K , 0, d J n−1 , d J n , 0,K , 0, 0 

T

(2.4.12)

In the presented example, the non-zero elements emerge in pairs, which is, for instance, the situation in planar elasticity problems with components u, v, but it is just a sample, not a rule. The appearance can be fully arbitrary. In practice, especially after the optimisation of the band width of global matrix K , the appearance of non-zero elements has cumulative nature – they form a kind of an island or a group of islands in the sea of zeros. Expression (2.4.12) is called extended or localised (positioned) vector of element deformation parameters d er . In addition to information (2.4.11) it holds also important topological information about the position of the element in the system. That is, however, already specified by n global indices J1 to J n written precisely in the order of local element indices 1, 2,K , n : J1 , J 2 , J 3 , K , J n −1 , J n

(2.4.13)

We briefly call expression (2.4.13) the global code number of the set of deformation parameters of element e . Each element has different number (2.4.13). Strictly speaking, it is not a “number”, but a “string” composed of n integers. Let us go back to formula (2.4.9). It is quite clear that vector of load parameters f e ( n,1) can 34

2.4 Specification of selected operations that are useful to understand FEM terms be similarly extended to f er ( N ,1) as well. Each d1 , d 2 ,K is virtually associated with just one parameter f1 , f 2 ,K on which the virtual or possibly the real work (potential energy Π z is work that must be done to return the load to the position in the original non-deformed state) is done. But also square matrix K eg (n, n) , see the overview, no. 11 to 13, art. 2.3., can be extended to the order ( N , N ) and marked as K er ( N , N ) , localised stiffness matrix of element e . It is sufficient to write its elements k in rows i = 1, 2,K , n and columns j = 1, 2,K , n symbolically into rows J1 , J 2 , K , J n and columns J1 , J 2 , K , J n of square matrix ( N , N ) : 1 1 M  J1   J2  M   K er = J 3  (N ,N ) M   M  Jn   M   N

K

J1

J2

K

J3

K

K

Jn

k11 k21

k12 k22

k13 k23

k1n k2 n

k31

k32

k33

k3 n

kn1

kn 2

kn 3

knn

K

N                 

(2.4.14)

What are all these extended notations good for? They enable us to write potential energy (2.4.4) of one element e first in a completely equivalent form 1 Π e = −dTer fer + dTer K er d er 2

(2.4.15)

from which there is only one small step towards the final form. That is obtained when d er is replaced by the vector (column matrix) of all deformation parameters of the analysed system d: 1 Π e = −dT fer + dT K er d 2

(2.4.16)

This replacement is possible for a simple reason: from the whole vector d only non-zero elements K er “act” in mentioned multiplication operations, while other elements of vector d are only multiplied by zeros, which does not influence the value of energy (2.4.16). The simplicity of this reasoning, easily verifiable in practical examples with a small number of parameters N , is the reason why it is not in fact explicitly mentioned in any FEM text. This, however, is annoying for curious readers as the whole vector d suddenly appears in the derivation of the procedure. Even the famous monographs [1], [2] use incorrect formulation such as " K is the sum of K e ", even though the summation operation is defined only for matrices of the same order, i.e. summation of matrices ( n, n) cannot result in matrix ( N , N ) ! 35

2.5 Principle of virtual work applied in FEM programs The detailed explanation that is important not only for programmers, but also for anybody who wants to understand principal advantages of FEM in comparison with other methods is given in books [4]. Adding the elements of the stiffness matrix into the appropriate positions in the global matrix was briefly named the addition theorem. Even though that this term penetrated into the international literature thanks to the translation of [5] already in 1975, the present-day FEM texts still use vague formulations such as “arithmetic sum of matrices”, or, most often, just the term “assemblage”. This applies to almost all theoretical manuals and user’s guides of FEM programs as well. Expression (2.4.16) is the final form of potential energy Π e = Π i + Π z of one element of the system. Next steps represent just a simple summation of the energies of all the elements as scalar values into total energy Π of the analysed system with common d factored out: 1   Π = ∑ Π e = −dT ∑ fer + dT  ∑ K e  d 2  e e 

(2.4.17)

marking of global stiffness matrix K as vector of load parameters f : K = ∑ K er e

f = ∑ fer

(2.4.18)

e

1 Π = −dT f + dT Kd 2

(2.4.19)

application of the condition of the minimum: ∂Π ∂Π = −f + Kd = 0

(2.4.20)

and rewriting of the obtained system of equations in usual form with vector of right-hand sides and matrix of coefficients K : Kd = f

(2.4.21)

This reasoning is in full applicable to arbitrary functionals (operators) Π that are additive in their nature Π = ∑ Πe e

2.5 Principle of virtual work applied in FEM programs First, let us focus on the most frequent FEM applications in technics, i.e. mechanical problems, that are in general non-linear but “finite”, which means that the solution is independent on the history, i.e. on the conditions that the analysed system was exposed to prior to the analysed state. Or more precisely: we will assume that the system of external and internal forces is conservative and that the geometrical constraints are scleronomous. The following text brings just brief information for the users of FEM programs. 36

2.5 Principle of virtual work applied in FEM programs The above-mentioned examples of mechanical problems allow for a simple application of the oldest known mechanical principle of virtual work (Archytas of Tarentum – 400, Stevin 1550, Galileo 1600, Newton 1678, Huygens 1700, Mach 1880). Its axiomatic extension to non-rigid bodies and systems (Navier 1830, Lamé 1860, Saint-Venant 1870, Mohr 1900) includes also the work of internal forces into the virtual work. Let's define a virtual work of external forces: δ WPvirt = ∑ δ WPivirt = ∑ Pi δ ri i

(2.5.1)

i

and internal forces δ WSvirt = ∑ δ WSvirtj = −∑ Pj s j j

(2.5.2)

j

where the sign “–” (minus) in (2.5.2) is introduced axiomatically, i.e. from the experience with the accumulation of a part of the work (energy) into the body. The principle of virtual work then axiomatically says (without any proof) that for every system of forces {Pi , S j } that are in equilibrium and for every virtual displacement {δ ri , s j } the total virtual work of external and internal forces equals to zero: δ W virt = δ WPvirt + δ WSvirt = ∑ Piδ ri − ∑ Pj s j = 0 i

(2.5.3)

j

Several popularisation formulations have been created with the aim to explain the principle in an intelligible way. E. Mach analyses them in a voluminous chapter of his famous “Mechanics” (7th edition [24]) and he sums up them in: ES GESCHIEHT NICHTS, WENN NICHTS GESCHEHEN KANN, i.e. if nothing can happen (i.e. there is no motion in any direction under which the impulse of force Pt could change into momentum mv, etc.), then nothing really happens (i.e. there is no motion, it is an equilibrium steady state). If we have distributed loads and internal forces, the sums from (2.5.3) become integrals, which is nothing important, just a formal modification of the notation of the principle of virtual work that may in general contain both sums and integrals depending on the type of loads (point, linear, surface, volume) and internal forces (1D, 2D, 3D models, singularity) and we may have a component, tensor or matrix notation. As the work is scalar, we can add together work done in parts Ωe (subdomains, elements) of domain Ω into the total work to which the principle applies. Should it happen that a system contains parts that have no connection to the remainder of the system, we can simply find out – through the application of a virtual displacement that is non-zero only in such parts – that the principle of virtual work applies to every such unconnected part separately, which is correct as the bodies (domains) are independent. Therefore, we can limit ourselves only to systems (domains) Ω where each part Ωe has at least one connection with another part Ω . There is an unlimited number of such connections in classical continuum Ω divided into elements Ωe . They are geometrically described by the Saint-Venant equations of compatibility. We will, however, consider more general systems consisting of elements of various dimension and non-classical continuum. The parameterisation of deformation degrees of freedom of such systems ensures that the analysis of their behaviour is integrated into the 37

2.5 Principle of virtual work applied in FEM programs standard procedure. The principle of virtual work (2.5.3) applies to arbitrary non-rigid bodies and was axiomatically extended also to elastic, or more generally deformable, continuum of 1D, 2D, 3D dimension. We employ symbols from the overview in art. 2.3. and from notation:

Pi :

External volume and surface forces

δ ri :

Displacement

Sj :

Internal forces

sj :

Strains on which σ work

b , p (d ,1)

u= N d

(d ,1)

σ= C B d

( s,1)

ε= B d

( s,1)

( d , N ) ( N ,1)

( s , s ) ( s , N ) ( N ,1)

( s , N ) ( N ,1)

All the terms are perceived as generalised in such a way that they contain not only the classical 3D continuum 3D ( d = 3, s = 6 ), but also e.g. planar 2D problems ( d = 2, s = 3 ), axially symmetrical problem ( d = 2, s = 4 ), Mindlin shells ( d = 6, s = 8 ), etc. Statical and also geometrical quantities are written in the form of matrix vectors, i.e. column matrices, the dimension of which is added in brackets. External forces b , p are given. Other quantities are assumed as depending only on N parameters d , similarly to the derivation of the FEM procedure, i.e. after the parameterisation of the system has been carried out through the selection of base functions that is closely related to the division into finite elements and to the chosen polynomial distribution over elements. We have already put down the global forms of these dependencies on d , that is for the whole analysed system, using matrices N , B extended to dimension (d , N ) , ( s, N ) . Let us remind that in dimension (a, b) the a is the number of lines and b the number of columns of the matrix. For linear problems it is sufficient to take the final state of the system after loads and deformations have been applied as the equilibrium system {b, p, σ} . The following N displacements of the system are sequentially chosen as the virtual displacement: (1)

d1 ≠ 0

di = 0

i ≠1

(2)

d2 ≠ 0

di = 0

i≠2

di = 0

i≠ j

M ( N ) d N ≠ 0 di = 0

i≠N

( j)

M dj ≠ 0

(2.5.4)

Let us call them “unit parametric states” as just one parameter is non-zero in these states. In textbooks we often see its size set to one, which is neither necessary nor illustrative, as it soon disappears from the derivation and the physical meaning is not apparent. The following virtual displacements and strains correspond to the above-mentioned parameters:

38

2.5 Principle of virtual work applied in FEM programs (1)

u1 = N1d1

ε 1 = B1d1

(2)

u 2 = N 2d2

ε 2 = B2d2

( j)

M u j = N jd j

(N )

M uN = N N d N

ε N = BN dN

( d ,1)

( s ,1)

(2.5.5)

ε j = B jd j

( d ,1) (1,1)

( s ,1) (1,1)

if we use subscript j to mark the row submatrices of which the global matrix shapes N are composed  N1  N   2  M  N =  (d ,N ) Nj   M     N N 

 B1  B   2  M  B =  (d ,N ) B j   M    B N 

(2.5.6)

With the selected notation, the principle of virtual work (2.5.3) now gets for the continuum in domain Ω with boundary Γ the form: δ V = ⌠ uTv b d Ω + ⌠ uTv p d Γ − ⌠ εTv σ d Ω (1,1) ⌡ (1,d ) ( d ,1) ⌡ (1, d ) ( d ,1) ⌡ (1,s ) ( s ,1) Ω

Γ

(2.5.7)

Ω

where u v is an arbitrary virtual displacement of the points of the continuum that may be subjected to volume and surface external forces b , p and ε v = ∂u v is the virtual deformation derived from it. If we choose the parametric state d i ≠ 0 as u v , principle (2.5.7) produces one equation with the defined notation:

∫ (N d )

T

j

Ω

j

b d Ω + ∫ ( N j d j ) p d Γ − ∫ ( B j d j ) CBd d Ω T

Γ

T

(2.5.8)

Ω

After the transposition of products in the form ( AC ) = CT AT and factoring of d out of the T

integrals, while for scalar d j (one number, matrix (1,1) ) we have d Tj = d j , we get:       NTj b d Ω + ⌠  NTj p d Γ  −  ⌠  BTj C B d Ω  d  = 0 d j  ⌠ ⌡ (1,d ) ( d ,1)   ⌡ (1, s ) ( s , s ) ( s , N )  ( N ,1)  ( d ,1)  ⌡ Γ  Ω   Ω (1, d ) 

(2.5.9)

This can be satisfied for non-zero d j only if the expression in the box brackets equals to zero. When comparing with no. 11, 14, art. 2.3 of the overview, it can be found that the expression in the first round brackets is the j -th parameter f j of vector f of all load parameters and the expression in the second round brackets is the j -th line K j of global stiffness matrix K of the system. Therefore, after rearrangement to both sides of the equals sign, equation (2.5.9) reads: 39

2.5 Principle of virtual work applied in FEM programs K jd = f j

(2.5.10)

If we gradually select as virtual displacements all N parametric states, i.e. j = 1, 2,K , N , we obtain N equation of type (2.5.10), in other words the whole system Kd = f

(2.5.11)

that is identical with system (2.4.21) derived through the Lagrangean variational principle in the previous art. 2.4. from the condition of Π = min . Why is the system of equations, i.e. also the solution, identical? There are two reasons for this: mechanical and mathematical. In terms of mechanics, the principle of virtual work is superior to all equilibrium conditions and it can generate an arbitrary number of such conditions as there exists an arbitrary (infinite) number of selectable virtual displacements. It is superior to the Lagrangean variational principle Π = min , or, as the case may be, δΠ = 0 that is one of its consequences. In terms of mathematics, we have limited in both situations the deformability of the system to states that are a linear combination of the same unit parametric states, which reduces the unlimited freedom of the continuum to “ N degrees of (deformation) freedom” that are the same in both procedures. Is there any advantage in the derivation through the principle of virtual work in comparison with the Lagrangean variational principle? Yes, and the advantage is significant, as during the derivation we do not have to limit ourselves just to problems of classical equilibrium that is independent on the trajectory (= history proceeding to the analysed state). The system {b, p, σ} must be in equilibrium in every time instant and thus also the increment at time dt , {db, dp, d σ} is in equilibrium and the principal of virtual work can be applied to it. This allows for the application of FEM – solution in steps, increments of load – using what is called an incremental method. The main problem is then how to add together increments dσ, dε, du in a way that is consistent with the laws of physics if deformations are large. Instead of details, let us give only brief notes that make the modelling of structures for inputs needed in FEM programs easier: 1. If body Ω is subjected to singular forces concentrated in figures of lower dimension than that of Ω (e.g. for 3D body Ω on 1D lines Γ or in 0D points "i"), then we can use directly the definition of such loads: the measure of the area on which it acts converges to zero but the resultant remains constant, as the intensity increases in indirect proportion to the measure. Consequently, instead of integrals in (2.5.9) se use directly the sums of products of the resultants with the virtual displacements (or moments with rotations), i.e. what is termed concentrated impact, see overview no. 15, art. 2.3. For linear loads we use line integrals, always with the aim to get the virtual work in the form of a sum or integral of products of the force and distance. It is thus possible to calculate load elements f , i.e. vector f of the right-hand sides of the set of equations Kd = f , for a very wide set of loads. FEM program are usually limited to : Loads distributed over elements, of general direction, decomposed into components px , p y , pz or bx , by , bz , Concentrated forces with components Fx , Fy , Fz , or also moments M x , M y , M z in mesh nodes with deformation parameters u , v, w , or also ϕ x , ϕ y , ϕ z , Linear, mainly straight-line, loads on element boundaries with components into the direction 40

2.6 Main outcome for the users of FEM programs of axes x, y , z . The state-of-the-art programs allow for the selection of various coordinates, e.g. global or planar (element) axes. The given load is approximated by this modelled load, meeting the requirement that the statical resultants must be identical. There should be a node in the point of action of the concentrated load or moment, which is taken into account already in the phase of division of the domain (generation of the mesh). 2. The global (localised, extended) forms of matrices N and B used in the derivation hold, apart from the information on element matrices N e , B e , only the topological data about the position of the element in the globally numbered mesh of nodes that is stored in element code numbers (2.4.13). These can be created from the sorted global numbers of its nodes with the in-advance-nullified deformation parameters taken into account, as explained in art. 2.4. This mass notation – allowing for mathematically correct summation of matrices of the same type – does not mean that the huge matrices are really created and stored in the memory. The program employs the “addition theorem” and sends the corresponding items of individual matrices of elements “ e ” where they really belong to according to their global code number and thus gradually creates 1st, 2nd, ..., to N -th equation. As we almost always have band matrices K with non-zero elements only in a band with the “width of BW ” on the left- and right-hand side of the diagonal, and as matrix K is symmetrical, only this band is stored in the memory. Some programs combine the assembly with the solution, or “pre-solution” or with the transformation into a triangular matrix, which means that no equation is stored in the original form. Other programs first assemble and store the whole system Kd = f . 3. Some programs do not use optimisation (almost minimisation) of the band width BW , but take advantage of a small density of matrix K (schwach besetze Matrix, sparse matrix) instead in order to speed up the solution of the system of equations. The present-day FEM programs enable the users to number the nodes by themselves or to renumber the nodes in such order that suits their particular needs. Any potential renumbering is performed just internally and the communication with the user is done in the user’s numbering.

2.6 Main outcome for the users of FEM programs 2.6.1 Selecting the elements of the FEM analysis model I. In each FEM program the corresponding analysis model is defined at the beginning of the manual. This is usually done in the description of the applied finite element. Each real object is three-dimensional, which means that 3D elements do not cause major problems. The most frequent type today is what is called bricks, or more precisely hexahedrons or dodecahedrons, with planar or curved sides and straight or 41

2.6 Main outcome for the users of FEM programs curved edges. Usually, they represent shapes that are formed through a transformation of a cube with sides equal to 2 with vertices (±1, ±1, ±1). Other spatial elements with details about the distribution of displacement components u ( x, y, z ) , v( x, y, z ) , w( x, y, z ) in various elements can be found e.g. in chapter 3 of title [5]. The main issue that need to be addressed in practice is the problem of accurate physical constants, unless we deal with elementary examples of homogenous and isotropic matter with two easily determinable constants E , ν . Physical constants for general configurations of non-homogenous and anisotropic bodies are specified by means of 21 functions d ik ( x, y, z ) . As FEM programs perform the integration within one element numerically, typically using the Gauss procedure, only values dik (in general any physical quantity) in the nodes of the integration mesh apply. For the simplest situation of one-point integration (in 3D space it is an analogy of a “staircase diagram”) only the value in the centroid of the element applies, which is the point where the origin (0,0,0) in the original cube is located. Depending on the required accuracy, various 3D bricks are nowadays used: especially what is termed brick 24, 60, and 96, art. 4.1.6.2, fig. 4.19, with tri-linear, tri-quadratic and tri-cubic distribution of displacement components u , v, w in one element (for details see formulas (4.1.65) (4.1.82)). II. Analysis models of FEM programs for what is termed planar structures (i.e. walls, plates and shells) are of 2D dimension. It means that one of the geometrical dimensions, thickness h, is missing in the geometry. This is the source of traditional troubles in the creation and selection of the analysis model. A whole hierarchy of options how to tackle the missing thickness h, or more precisely how to tackle the distribution of displacement components in the interval − h 2 ≤ z ≤ h 2 along h , has been established over the years. This includes the Reissner sequences that expand the displacement components u , v, w into Taylor’s series in variable z , see art. 4.1.6. Thus, a countable set of plate- or shell-models can be created, of which mainly two appear in the present-day FEM programs: e) The oldest known model is the Kirchhoff one. It assumes that component w along h is constant (common deflection of a rigid normal h ) and that components u , v use one member of the Reissner sequence, linear in z , which introduces a physically rigid mass normal h . Moreover, it is assumed that this normal remains normal even after the deformation of the mid-plane (plate) or mid-surface (shell) into a general deflection surface. Therefore, the deformation of an element is uniquely defined by means of a single function w( x, y ) . The difficulty in FEM programs is that the compatibility between elements (no breaks allowed) requires that not only function w , but also derivatives ∂w ∂x , ∂w ∂y are continuous. FEM terminology uses term class of function C1 (subscript 1 means continuity up to the 1st derivative). Hundreds of publications that deal with this problem for triangular and quadrilateral elements were presented between 1956 and 1976. They introduce what is called rational correction functions for correction of common polynomials, etc. An overview of the hierarchy of elements without correction can be found in [5]. For further reading about correction functions refer to [2], and the literature cited there. Some companies today treat these functions as a business secret, in order to hide them from competitors. Their importance significantly decreased as new elements come into play – see point (b). 42

2.6 Main outcome for the users of FEM programs What is important for users is that internal forces are differentiated out of function w( x, y ) using rule no. 19 from the overview in art. 2.3. According to [31] chapter 3, pp. 212 – 223, bending moments mx , my and torque mxy are determined by second derivatives, shear forces qx , q y and reactions r by third derivatives of w with respect to x, y . The lowest degree of polynomial w( x, y ) is thus equal to 3 if it is supposed to generate non-zero transverse shear. Classical compatible elements, as they were called, used degree 5 (21 coefficients) on a triangle and Ahlin’s bi-cubic polynomials on a quadrilateral. It was this disharmony between the 10 coefficients of a polynomial of third degree in x,y and 3 triangle vertices (which means that for the selection of parameters w , ∂w ∂x , ∂w ∂y there is only 9 parameters) that started explosion of publications in 1960 – 1970 where also correction functions were developed. As also these functions apply to derivatives and as they are rational, not every modification results in reasonable approximations of mx , my , mxy and especially qx , q y , r . After a decade of experience authors of these publications also admit that transverse shear is obtained more reliably when elements from paragraph (b) are applied. f) A newer model, the Mindlin one, seems to be only a small generalisation of the Kirchhoff’s model: It does not include the requirement that the rigid normal h remains normal in the deflection surface. The right angles between directions ( x, z ) and ( y, z ) can alter by slope γ xz , γ yz that are permanently zero in the Kirchhoff approach, which can be demonstrated using what is called pin model. Pins jabbed up to their centre point into a rubber mid-plane represent normals h , the heads z = − h 2 represent the top face and the peaks z = h 2 are the lower face of the element. In the Mindlin approach these normals have, in addition to deflection w , two more degrees of freedom ϕ x , ϕ y – components of rotation about the x –axis and y –axis that are considered a vector, which is possible only for very small rotations, i.e. in geometrically linear mechanics of small deformations, both displacements and rotations (classical linear theory). The disadvantage that three unknowns w( x, y ) , ϕ x ( x, y ) , ϕ y ( x, y ) are introduced is outweighed by the fact that instead of one differential equation of fourth order we get one equation of second order (equilibrium of forces in the zdirection) and two equations of first order (moment equilibrium about x –axis and y – axis). The impact on the FEM theory is that the requirement on the continuity of base functions is reduced by one order, i.e. from class C1 to class C0 , which represents formulas that are continuous only in function values (0th derivative = the function itself). This eliminates complications that are solved, among others, also through correction functions. Moreover, it is sufficient to choose all three functions as linear polynomials in the form a + bx + cy . As a result, we get 9 unknown coefficients on a triangle and a regular expression with 9 deformation parameters w , ϕ x , ϕ y in its vertices. This basic element can be then improved in various ways, e.g. through the rule of Kirchhoff-style permanence, with the tendency to have the polynomial for ϕ x and ϕ y one degree lower that the polynomial for w (as in the Kirchhoff approach we have ϕ x = ∂w ∂y , ϕ y = − ∂w ∂x ). The first improvement (still used in present-day 43

2.6 Main outcome for the users of FEM programs FEM programs) was presented already by Lynn and Dhillon at the 2nd Conference on Matrix Methods Japan - USA in Tokyo in 1971. Another substantial improvement relating to the extension to spatial shell systems was made in 1977 by Kolar and Nemec. Similar elements are now implemented in all prestigious FEM programs. The most important fact for users is that internal forces of the Mindlin model mx , my , mxy , qx , q y depend only on the first derivatives of functions w , ϕ x , ϕ y (at qx , q y we have in addition to ∂w ∂x , ∂w ∂y only the functions themselves ϕ x , ϕ y , i.e. not differentiated!). This ensures better numerical reliability, especially when no rational correction functions are necessary. For the above-mentioned elementary triangle all the moments in the element are constant and shear forces have linear distribution. It is an inverted ratio in comparison with Kirchhoff’s elements (without correction functions) where, primarily, the distribution of shear forces is one degree lower than the distribution of moments. III. Similarly to paragraph II, 1D beam elements can be defined in the “Kirchhoff style” (more precisely using the Bernoulli-Navier hypothesis about the preservation of planar sections in normal planes of deflection curves) – termed classical beams (classical 1D continuum)) – or in the “Mindlin style” – i.e. with the transverse shear taken into account. For the first approach, the distribution of all the three displacement components u , v, w in the body of the beam can be derived from four functions of one variable s on the (generally curved) beam axis. u ( s ), v( s ), w( s ), υ ( s ) and their derivatives with respect to s . In the second approach, we get, in terms of geometry, a Cosserat 1D continuum in which each point has 6 degrees of freedom u ( s ), v( s ), w( s ), ϕ x ( s ), ϕ y ( s ), ϕ z ( s ) . The users can see the following differences (see also art. 4.1.4.): In general, a classical straight beam element in space has in the central (principal centroidal) axes x, y , z at each end 6 parameters of type u , v, w, υ , dv dx , dw dx , in total 12 parameters, with corresponding end components of force and moment vectors Px , Py , Pz , M x , M y , M z . Axial effects u , υ (tension/compression and torsion) are distributed linearly over the element and transverse effects v, w (bending and shear) by polynomials of third degree. A similar Mindlin element has at its ends 6 parameters, independent degrees of freedom of end-sections u , v, w, ϕ x , ϕ y , ϕ z . The requirements of FEM applications would be satisfied if the six functions were selected in the form of linear polynomials. The element is however significantly improved if we take into account the wellproven Kirchhoff principle, i.e. if we select a one degree higher polynomial for v, w – the polynomial of second degree. In order to guarantee compatibility with 2D elements in space it is necessary to rise the degree of polynomial u accordingly. This results in an element with three nodes I ( x = 0) , J ( x = L) , K ( x = L 2) , six parameters in I , J and just three parameters u , v, w in K , in total 15 parameters. It can be reduced through energetic elimination of internal parameters in node K into an element with the same number and location of parameters at ends I , J as the classical 44

2.6 Main outcome for the users of FEM programs element. Practical applications have proven that, provided the mesh is of normal density, the compatibility between 1D and 2D elements only in nodes is sufficient. On the other hand, it is useful not to neglect the effect of transverse shear. Some FEM programs introduce the effect of shear in 1D beam elements through a systematic procedure. Slopes γ xy , γ xz due to shear forces Qy , Qz are considered to be 13th and 14th parameter added to the 12 common above-mentioned parameters. The energetic elimination then modifies the stiffness matrix into the classical dimension (12,12) and with elements that include the effect of shear in what is termed shear factors α , see also chapter 2. The important factor in modelling of a real structure is not the shear factor of elements, but of the whole structural members that can be, of course, divided into several elements (curved beams, beams with haunches, etc.). The same applies to plates. So-called thin and thick plates are considered depending on the ratio h L , where h is the thickness and L the characteristic dimension of the whole slab (diameter, shorter side of a rectangle, etc.), not the dimension of its element. The limit value is usually the ratio h L = 1 5 . Thinner plates can be analysed by FEM programs with Kirchhoff elements, which may be “thick” themselves if the mesh is dense enough, it may even happen that h > L . For thicker plates it is better to apply FEM analysis using Mindlin elements. Most such elements feature a built-in numerically-stabilising test that allows for convergence to a reasonable result even for thin plates. The feature is termed shear locking. A numerical limit value over which the shear no longer increases, must be introduced.

2.6.2 Interpretation of FEM output data In order to meet the needs of design practice, sizing, checking of resistance and assessment of serviceability, the outputs of the FEM procedure are further processed either manually or in numerical or graphical postprocessors. The following principle is valid: FEM programs with elements whose dimension is n D hold in the outputs only such an amount of information of dimension m D, m > n that is embedded into n D finite elements through hypotheses that reduce the dimension from m to n . In particular: FEM programs with 2D elements cannot be used for a detailed 3D stress-state analysis in such places of 2D structures (walls, plates, shells) that require such an analysis (the vicinity of column heads in flat slabs, bridge bearings, concentrated impulses acting on area smaller than h 2 , where h is the thickness of the planar structure, etc.). Even less applicable to the 3D analysis are the outputs from FEM programs that use 1D elements. For example, such programs are not able to tell on which particular part of the cross-section the load or reaction acts (top or bottom flange, etc.), as only the resultant over the cross-section is known. Therefore, it is not possible to perform an accurate analysis of the stress-state in e.g. fixed ends where the specific design of the fixing cannot be taken into account or in connections where the exact welding, bolting or gluing characteristics cannot be considered, etc. If the real design is modelled through a kind of abstraction, e.g. if a foundation strip is 45

2.6 Main outcome for the users of FEM programs modelled by a rigid 1D line, the singular outputs from the FEM analysis (that converges to exact singularities) cannot give an accurate idea about the detailed stress-state in the vicinity of the singularities. A typical example: If we define a rigid line support of a bridge deck, either perpendicular or skewed, we get non-designable moments in the corners of the deck, which is a correct trend towards the singularity of the corner. If we want to have realistic outputs, it is sufficient to specify the real stiffness of the supporting beam that can never be infinite, which is included in the definition of a rigid support with w ≡ 0 , moreover with a zero area measure σ z → ∞ . Even the lowest estimate of the stiffness of the line support or the lowest estimate of the spring constant k in older programs that did not allow for such a support type produces usable moments. Hundreds of similar situations can be summed up to express the following principle: If we want correct FEM results, it is sufficient to exploit the wide capabilities of FEM in order to express correctly the real design and realisation of the structure and its supports and the real distribution of loads. In this context, mistakes are made by both younger engineers (due to the classical tuition at faculties) and experienced specialists who were – during their long practice – forced to simplify everything (loads, supports, material properties, etc.) as program with the power of present-day FEM systems were missing. Many engineers learn about the effectiveness of FEM at various courses, but they doubt about the perplexity of input data for complex spatial systems consisting of thousands of 1D and 2D (and possibly also 3D) elements, which usually prevented the full utilisation of FEM in the previous era of development (in the Czech Republic approximately until 1980). The main idea of the division of a domain into finite elements is however much stronger and it includes also the following capabilities: g) The analysed structure is first divided into smaller substructures, which makes the division more understandable. The substructures may have technological and production functions, but may also be parts selected only on a formal basis, in order to improve the handling of the model. Sometimes, it is possible to combine (i) parts made of 1D elements and other parts composed of 2D elements, sometimes (ii) individual walls and ceilings, foundation slabs, pile foundations, etc. If we use this division just with the aim to make the handling of inputs more comfortable, it is better to talk about macroelements and sometimes only about segments. If the program creates for these parts separate global stiffness matrices K e and vectors of load parameters fe and if the addition theorem is applied to them (no. 13 and 17 of the overview in art. 2.3.), then we can refer to real substructures in terms of FEM. What is remarkable about this procedure is that the substructure can have much more “internal nodes” than “boundary nodes” through which it is connected to other substructures. Only the parameters of these joint nodes then appear in the global stiffness matrix K and in the vector of load parameters f of the whole structure. This reduces the number of unknowns N and also the band width BW at the cost of certain preparatory works – resembling pre-elimination – which is just the formation of matrices K e and fe . h) Creation of what is called superelements by assembling simpler elements, called subelements, is not in fact too different from (a), but it has slightly different technical meaning. One of the oldest superelements is a quadrilateral composed of four triangles 46

2.6 Main outcome for the users of FEM programs or a hexahedron (i.e. a brick with the shape of a cube, block, or parallelepiped with either planar or curved faces) composed of five tetrahedrons. The meaning is the following: we apply simple base functions U (no. 1 and 4 in the table in art. 2.3.) in the subelement. These functions usually impose a significant deformation restriction, physical strengthening of the element by means of fictive connections that impose prescribed displacements. This restriction is somewhat relieved for the superelement. Its internal parameters are regularly (in compliance with the energetic principle) eliminated and the result is an element that is “softer in terms of deformation” than it would be after a simple addition of the subelements. Let us present one illustrative example: if we use for a triangle linear functions U , i.e. monomials 1, x, y , or for a tetrahedron 1, x, y, z , then we force it to displace along the plane or hyperplane inserted into 3D space. The created quadrilateral can move along four partial planes (tetrahedron along five partial hyperplanes), which can already approximate more complex, out-of-plane deformation. The reduction of the number of unknowns N and band width BW through the eliminated parameters is also welcome. In addition to the above-mentioned elimination, also a simple interpolation of parameter values is sometimes used, mainly with the aim to exclude undesired nodes in the middle of the sides. On side 123 we thus set d 2 = ( d1 + d 3 ) 2 (similarly for more complex geometry) and linear or quadric interpolation is applied. The principle of compatibility between adjacent elements must be satisfied, i.e. the elimination must generate the same values of deformation parameters in both elements. The users are advised to pay attention to the definition of the element in the manual as it can make clear some characteristics of the element that may show up in the output – interpretation of the results. Let us describe a simple example: let us have d = w = slab deflection without other parameters (angles or derivatives). If d 2 is eliminated through a linear interpolation, then for supporting conditions d1 = 0 , d3 = 0 , side 123 is straight with all possible consequences for the internal forces in the element. If d 2 is eliminated energetically, its value may be different and the boundary of the element can be curved, which produces different internal forces. The situation is different if additional nodal parameters are defined. The requirement of d = 0 in nodes does not have to mean w ≡ 0 at the edge of the plate, i.e. exact linear support, but just “skewering” of the plate on the vertex nodes of the element. This has an impact on the output internal forces and must be interpreted appropriately with respect to real support conditions required, see chapter 5.

47

3.1 Principles of approximation of the sought distribution of a quantity in FEM

3 Physical axioms and variational principles of more complex FEM problems 3.1 Principles of approximation of the sought distribution of a quantity in FEM Every single day, professionals in engineering practice face the problem how to reliably forecast the behaviour of various objects under different conditions, i.e. “the response of the analysed system to external influences that cause changes the mechanical state of the system”, or simply: how to reliably forecast the behaviour of the object and its surroundings (soil environment, subsoil, etc.) in the production phase (construction, assembly), in service, in exceptional situations (accidents, failures) and in what is termed limit states (ultimate, serviceability, deformation, cracks, fatigue). With regard to the needs of practice, this is not a mathematical problem and no theory is required. The only necessary thing is to comply with all the clauses of valid standards that concentrate many-year experience and that reflect the production possibilities in a particular state. If this is feasible for common objects without any special calculations, i.e. if it is possible to prove that the design is safe, economical and even optimal and if it can be evidenced that the assessment is reliable without any additional analysis exploiting modern methods, then no engineer will do that. In practice, there is no time and there are no means for such “mathematical amusement”. In today’s competitive environment, it is becoming more and more a must for companies to be innovative and implement new things. Also increasing is the number of cases when it is necessary to perform calculations using such methods that are appropriate to the nature of these new things, if the organisation is supposed to do everything that can be reasonably required in order to prevent any damage. No methods are used in practice just for fun, but only due to the instinct of self preservation. It is a certain tool for the fight against nature, even though it is different from the famous stick of a primeval man, however, with the same purpose in the engineering field where only the must-be-solved, and not the can-be-solved, is solved (contrary to the realm of mathematics where, and it is right, the can-be-solved is solved). Art. 1. presented a brief overview of different FEM procedures that are available in nowadays practice for the analysis of various engineering problems. The typical feature of these problems is that we seek the distribution of certain quantities and their extreme values, which means that we work with functions. Briefly and in popular terms, we can say that the final aim of almost every numerical method is to represent the sought solution, in general a set of functions (e.g. u , v, w , etc.) of several variables (e.g. x, y, z , t , etc.), as a vector function, e.g. U ( x) = [ u ( x, y, z , t ), v( x, y, z , t ), K]

T

(3.1.1)

48

3.1 Principles of approximation of the sought distribution of a quantity in FEM approximately in the form of a linear combination (sums of products) of pre-selected functions ui ( x) , i = 1, 2, K , n with coefficients ai : n

u(x) = ∑ aiU i ( x)

(3.1.2)

i =1

which can be most concisely written in matrix notation according to no. 4 from the overview in art. 2.3: u = Ua

(3.1.3)

Functions u can have an arbitrary physical meaning, in mechanics they may represent any geometrical or statical quantities, in the deformation variant of FEM (used to illustrate finite elements methods in art.1.2. and 2.2.-4) usually the displacement components (or their derivatives) and small rotations. Their values in one point of the solution of the system are written in a column matrix, called also matrix vector. r r r r T The historically oldest vector is vector of displacement u = [ u , v , w] in Euclidian 3D space (D denotes the dimension of the space). Let us mark the unit vectors in the direction and r r r orientation of x -, y -, and z -axis by symbols u , v , w and the magnitude of the components r r r r r r by u = au u1 , v = av v1 , w = aw w1 We can apply the decomposition of the vector into three components, in other words the vector sum r r r r u = au u1 + av v1 + aw w1 (3.1.4) which can be symbolised by a diagonal in a parallelepiped with the sides equal to the components. This approach was used to add vectors already in antiquity, either axiomatically or with attempts for “proofs”. Only at the end of the 19th century, the correctness of this operation was consistently axiomatically analysed using the principles of independence and superposition (E. Mach: Die Mechanik, 7th edition. 1912), see art. 3.2. Analogically, formula (3.1.3) can be called the decomposition of function, or a set of functions (i.e. vector function, with an arbitrary number of components, it is a mathematical vector – column matrix, not a physical vector), into components U . If these components are known, or chosen, then function u is determined only by the set of coefficients a , which is practically always a finite number of numbers and even for n → ∞ it is a countable set of the lowest cardinality, no continuum. Through this the discretisation of the problem has been performed (discrete is the opposite of continuous, we could also say “point” or “isolated”). r Analogically to the base, or the base for the decomposition of vector u (3.1.4), i.e. to base r r r components u , v , w , functions in U can be called base function. The same way as the endpoints of vector u fill in a certain space (Euclidean 3D), it can be said that all possible functions u that can be written in the form (3) fill in a certain function space the “base” of which is represented by functions U . There are no physical concepts related to this reasoning. We have just mathematical terms that are very useful for further abstraction and mutual understanding. Seeking the unknown function is a significantly more complex problem then seeking a (even large) number of unknown values, because even for the simplest situation of one function of one variable it represents a problem in which we have to find function values assigned to the continuum of an independent variable. In popular terms: it is a problem of seeking the infinite number of values, and we do not deal with a countable infinity, but with 49

3.1 Principles of approximation of the sought distribution of a quantity in FEM greater cardinality. In a smallest possible interval, the cardinality of unknown function values assigned to the independent variable from this interval is greater than countable infinity. Therefore, if we (e.g. on a powerful computer) degrade the solution of a problem to the solution (for non-linear problems the repetitive solution) of a system of algebraic equations with 105 – 106 variables, it is really a significant degradation represented by such discretisation. After the degradation of a problem through its discretisation it becomes clear that the success of a numerical method depends primarily on the selection of base functions of decomposition (3.1.3). Even a trivial decomposition of vector (3.1.4), which gave the name to the “base functions”, could cause numerical difficulties if the selected base was not orthogonal, but oblique with a very small angle between a pair of the base, which could lead to the requirement of high precision of the function of the projected cosines, etc. Already the selection of the base functions could in the classical methods satisfy the boundary conditions on Γ (RITZ) or the differential equation in Ω (TREFFTZ) - see table (3.1.5) for the overview.

Differential equation

Boundary conditions

Type of method

(0) satisfied

satisfied

exact solution

(1) not satisfied

satisfied

original RITZ

(2) not satisfied

not satisfied

Bubnov-Galerkin

(3) satisfied

not satisfied

TREFFTZ

The original versions of corresponding methods were published in [10] to [14]. Unfortunately, the clear formulations were distorted by later interpreters. Another obstacle for normal users was that most interpretations focused on the mathematical side of the problem and completely neglected the physical and technical needs of practice that initiated the method. An illustrative example is given in [3, 5]. In our text, let us present this simple but rather exact explanation of mathematical advantages:

50

3.1 Principles of approximation of the sought distribution of a quantity in FEM

Figure 3.1: a) Analysed domain Ω and its boundary Γ. b) Elements Ωe of the domain and elements Γe of its boundary. c) Division of the analysed domain into layers (FLM). d) Division of the analysed domain into strips (strip method).

51

3.1 Principles of approximation of the sought distribution of a quantity in FEM The advantage of the modern variational FEM is in the “support” of the base function, i.e. in domain Ωb in which the base function has non-zero value, with the exception of points where their measure Ω0 = 0 – which are usually points in which the diagrams of base functions (e.g. trigonometric ones) intersect the basic coordinate planes. In the classical variational methods, the support is the whole analysed domain Ωb = Ω

(3.1.6)

which results in the well-known numerical difficulties (especially due to the fact that the equation systems are ill-conditioned), impossibility to adapt the base functions to the shape and modification of the analysed domain. Any opening or cut eliminates the possibility to apply e.g. Fourier solution of plates by means of trigonometric series, etc. Base functions in the finite element method are much more expedient. First, let us note that, for historical reasons, the term finite element method is nowadays used only for the solution based on the division of a domain with dimension nD (fig. 3.1a). It was established for the elements with dimension nD (fig. 3.1b) also at the national FEM course [6], even though all other methods that use the element-principle work with finite elements too - FEM - finite element method = only elements of a domain with dimension nD , BEM = boundary element method = elements only at the boundary of the domain, BIEM = boundary integral equation method, FLM = finite layer method, FSM = finite strip method. In FEM we use decomposition (3.1.3) in such a way that we define base functions U with a small compact support that is always created only by a few elements that share a certain node (vertex, midside). For example, we use pyramidal functions with the value equal to one in the given node. If the mesh is fine, these functions are equal to zero over the major part of the domain and create non-zero graphs only on the mentioned small compact support. Mathematics [3, 5] adopted term finite functions. The integral of the product of two such functions is equal to zero for most functions (as the integrand itself is zero), except when the intersection of the supports is not an empty set. Just this ensures that the analysed algebraic equations of FEM are well-conditioned. The equations have a distinctly band (in general sparse) matrix of the coefficients in contrast to the full matrix for BEM. Historically, a similar progress in numerical methods was made for the first time 140 years ago when Clapeyron was analysing a continuous beam and used as unknowns the support (hogging) moments M instead of (at those times) usual reactions R . The base functions of deflection on the basic system that represents a single long simple beam (removal of intermediate supports (constraints) of the continuous beam) use the whole analysed domain as the support and lead to the full matrices of the coefficients of the left-hand sides of the linear equations for unknown reactions, in contrast to well-known three-moment equations with the band width (half-wide, for symmetrical equations) equal to 2 regardless of the number of spans of the beam. The mechanical meaning of the present-day base functions in FEM can be better expressed if we, in addition to the discretisation, introduce also convenient parameterisation, i.e. if we choose as unknowns the (technically illustrative) quantities in nodes (displacements, rotations, moments, stress components, etc.). This illuminates the relation with source functions of the discretised system (influence lines of the parameters). The base functions in the boundary element method (BEM) (fig. 1a, b) become clearer if we realise that it is in fact a finite treatment of the Trefftz method, i.e. we choose such base functions that precisely satisfy the differential equations inside domain Ω and if we try to achieve that their linear combination (3.1.3) satisfies also, in a certain exactly defined sense 52

3.1 Principles of approximation of the sought distribution of a quantity in FEM (i.e. in what is termed “norm”, e.g. energetic, minimum of the sum of the squares of variations or just collocations), the boundary conditions. With this aim in mind we divide boundary Γ (= edge of the domain) into finite elements Γe and all operations are then carried out only on the boundary, which is a one-dimension-lower figure than domain ( Ω ... nD , Γ ... (n − 1)D , see fig. 1b). It is indisputable that this significantly reduces the number of unknown parameters and even though we get equations with full (non-band) matrices of the coefficients of the left-hand sides, which may result in saving of total-time and memory-size. Method called BIEM (boundary integral equation method) appeared already in the prehistory of this procedure. Hundreds of articles were describing the method in 1965 - 1980. Those who are seriously interested in BEM are recommended to study [34], original sources [15] to [22]. Similarly to FEM, a vast number of unoriginal authors explain the BEM inaccurately and usually in a completely intelligible way for engineers. They only lay the emphasis on the exact fundamental functions and used symbols that are all Greek to a normal engineer. However, the principle is in fact fairly simple: If we know the exact solution for a certain domain and given internal load obtained through influence functions and if this solution holds for certain boundary conditions (usually for infinitely large domain), then the sought solution can be obtained in the following way: we cut out the given domain and situate it into the given boundary conditions. Apparently, we intervene only on boundary Γ . This can be demonstrated on a rubber-made model, e.g. a rubber band with the dimensions of 500 × 500 × 2 mm , that serves the purpose of the “infinite domain”. Let us put the band on a table and draw, somewhere close to its centre, boundary Γ of the analysed domain, e.g. circle d = 80 mm , ring etc. Inside this domain we apply e.g. one horizontal force acting in a certain point (centre, etc.). The boundary deforms. Next, let us imagine that distributed loads are distributed along Γ and these loads cause that the boundary takes the original shape. The same happens with defined geometrical conditions. Or we can imagine that one section is defined on Γ and both of its banks are subjected to such compressive stresses (double-vectors, actions and reactions) that the given domain is subjected to the required stress component, which represents statical boundary conditions. There exist also mixed situations when we require a certain given displacement (e.g. zero) on one part of Γ and certain given stress components (e.g. zero) on another part of Γ . We may even require a certain mix of geometrical and statical conditions be met on a part of Γ , i.e. in total boundary Γ can be divided into three parts, symbolically (they do not have to be continuous): Γ = Γu + Γ p + Γup It is quite obvious that throughout the 30-year history of BEM its principles have continuously developed and nowadays conferences and publications with new pieces of knowledge are intended rather for professionals who are following this development. Practical engineers cannot usually understand these works, but they should be able to use the appropriate software. Generally, the above-mentioned popularisation, which is definitely not intended for mathematicians, is sufficient for them. The appearance of each new method is followed by a period of uncertainty in the engineering practice and doubting questions rise such as: “what is it in fact”, “does it have any negative impact on the old software package”, “has it been already verified”, “what are the advantages”, etc. All this is quite logical as the engineer has to add their signature to the design and is legally liable. Senior engineers certainly remember the confusion about FEM 53

3.1 Principles of approximation of the sought distribution of a quantity in FEM from the years 1967 – 1972. These difficulties still increase with the continuing development as (i) the structural engineers no longer play the roles of mathematicians – analysts and programmers (as was the case in the past) and (ii) long past is the time when new methods were presented to engineers in an understandable way with the subject matter of mechanics. This used to be common in the prehistory of BEM when crystal clear information was presented [14, 15 to 19]. It is beyond the capabilities of a practical engineer to read or even browse through these publications and, consequently, he/she has to rely on wide-spread rumours that are quite often commercially oriented. Let us add a few conclusions concerning both the theory and practice based on our own experience and drawn from international resources: i) Contrary to FEM, BEM is advantageous in simple problems dealing with homogenous domains in which we know the source function of the solution (e.g. influence surface of plate deflection, displacement of a wall – half-space, etc.). The method results in a lower number of equations with full matrices of coefficients. Another positive feature is that infinite domains can be handled without any special elements. j) In practice however, the analysed domain is often non-homogenous (e.g. fig. 3.1c), but it is possible to define not too many homogenous domains where, once again, a certain known “unit” solution holds, i.e. the source function is known. Then, the difficulties can be overcome by the introduction of additional, internal segments of the boundary ( Γ1− 2 , Γ 2−3 , etc.) and their “boundary” elements. On the other hand, this increases the number of unknowns in the nodes of all boundaries and, because matrices of coefficients are not band matrices, also the time needed for the solution of the equation system increases. Using the well-known formula for the determination of the time that is required to solve the system of equations: T (minutes) = a ⋅ N ⋅ BW 2 (3.1.7) where N is the number of unknowns and BW the band width, we can easily estimate under which configuration the advantage of BEM over FEM vanishes. It is rather soon, as BW is squared in (3.1.7). The time estimate (3.1.7) can be used approximately also for what is termed sparse matrices where the non-zero elements are concentrated in a band around the main diagonal. BW can be set equal to approximately a half of the average number of non-zero elements in one equation. Quite often, these elements form a kind of islands in the sea of zeros and are spread almost over the whole matrix, especially in large problems with 100,000 or more deformation parameters. The time estimates can never be performed accurately in advance, but numerous programs tell them to users as a tentative initial information after the input has been completed. Moreover, the progress of the solution is shown in the screen for individual phases of the solution and, therefore, the user can at any moment see the current status. Also the solution of the system of equations can be followed in the same way, even though it is usually more time demanding. k) Any change of physical properties, thickness, reinforcement, etc. prevents application of the source function of the homogenous configuration and there is no chance that it could be possible or useful to create source functions for such situations. Consequently, if an engineering office is supposed to be able to analyse anything, it must own (as a sine qua non) a certain software package. And this basis can be extended by a BEM package. As a result, homogenous domains can be analysed more effectively. In particular, parametric studies of the influence of the shape of a planar 54

3.2 Elementary principles of physical nature of FEM domain are effective in BEM, as only the boundary is changing. The reduction in the dimension of the problem can also be achieved through an integral transformation (Fourier, Hankel, Laplace, etc.), which is used in the finite layer method – fig. 1c. Or alternatively, it can be done using the assumption about the distribution of the solution along one of the coordinates, e.g. x (see fig. 3.1d) and discretisation of a cross-section, e.g. in the yz-plane. This strip method was introduced already in 1965 by Y. K. Cheung.

3.2 Elementary principles of physical nature of FEM The contents of this article belongs to the sphere of human thinking that always remains reserved to the human – engineer, as it represents learning, decision-making, intuition, invention and sensation processes and other functions of a human brain that cannot be turned to an algorithm and that cannot be reduced to the seven basic operations of a Turing machine. It means that as the performance capacity of computers is growing, also this component must increase otherwise the human would become an insignificant peripheral device of a computer. Unfortunately, this is in fact frequently happening due to lack of education and related ethics and aesthetics. And it is accompanied by the well-known impacts on the real standard of life, ecology of nature in whole, social catastrophes, etc. What was enough for the engineer equipped just by a hand-held calculator cannot be sufficient for the same engineer if they are fitted with a fourth-generation computer. The speed of numerical operations has grown (approximately) from 10-1 op./s in antiquity and 100 op./s in the era of Pascal’s calculator (called the Pascaline or the Arithmétique) in modern times up to 1010 op./s in present-day computers. Throughout this historical period, the “homo sapiens sapiens” (“double-wise” or “civilised” human according to the scientific terminology of the theory of species) remains all the same, with the maximum speed of the receipt of information at around 6 bit/s, maximum usable memory about 1 – 10MB, which is slower than a normal PC by a factor of 109, continuously forgetting most of the information received within several hours and doing in average 1 to 2 errors per 100 elementary operations (physiologically normal state). The human can compete with the machine only as its creator in the above mentioned processes that cannot be turned into algorithms. This includes e.g. a deep understanding and further development of principles of physics. Misapplication of these principles, if it leads to a logically possible machine, cannot be revealed by the machine. And it cannot be sensitive to fine ecological connections of the production process controlled by CAD - CAM - CAE - CIM algorithms. It means that an inaccurate but possible analysis FEM or BEM model cannot be corrected by a machine. The engineer meets in practice a lot of problems whose correct solution is based on physical principles. Principles of mechanics are prevailing and the most often used (even though sometimes only subconsciously) ones are summarised in fig. 3.2. The principles are of axiomatic nature, i.e. they cannot be proved by decomposition into simpler theorems or pieces of knowledge.

55

3.2 Elementary principles of physical nature of FEM

87.

PRINCIPLE OF ISOLATION

90.

PRINCIPLE OF RELEASE

93.

CLASSICAL BOLTZMANN:

100.

EFFECTS

91.

OBJECT BODIES

95.

94. 3 DEGREES OF FREEDOM IN A POINT 99.

88.

MODELS OF CONTINUUM

89.

PRINCIPLE OF SUPERPOSITION

92.

PRINCIPLE OF INTERACTION

96.

MULTIPOLAR MODELS

97. 98.

6 DEGREES OF FREEDOM

MODELS OF PARTICULATE MATERIALS DEFINITION OF MODEL BEHAVIOUR 103.

101.

GEOMETRY

COSSERAT:

102.

PHYSICS

STATICS

104.

DYNAMICS

105.

REOLOGY

Figure 3.2: Some principles and model used in the practical application of finite methods in the field of mechanics

The academic curriculum usually covers only the commonly used principle of superposition of the effects (fig. 3.3), but the principle of isolation, which must precede it, is rarely explained. In a real world, each object is under influence of many effects and, based on human experience, e.g. m effects may be important for that analysed behaviour of a structure. Let us assume a simple example shown at the top of fig. 3.3 where there are two effects: '1' e.g. gravity and the load following from it and '2' e.g. thermal changes in the vicinity of the object, magnetic field, etc. The human brain has just a limited number of neurons (approximately 120 billion) and their connections (each neuron is connected to roughly 8 000 of others) and is able to think about a “problem of limited capacity”, e.g. only about the effect '1' or only about the effect '2'. Therefore, we have to isolate individual effects and analyse them separately (as isolated). The philosophical background for this is the belief (i.e. unproven axiom) that this conduct has some purpose, that if we analyse the effect '1' we can abstract away from the effect '2' if we have no other choice due to the limited capacity of our brain. If we accept this axiomatically as the principle of isolation, it can be extended by the well-know axiom of the “superposition of effects” that is common in linear mechanics. In a non-linear mechanics we always have to carefully determine if these principles can be applied. It is usually an exception if it is possible, e.g. the Timoshenko’s beam under compression subjected to a constant force N and the superposition of transverse load. On the other hand, the effects of load P and thermal load in fig. 3.4 must be analysed together and 56

3.2 Elementary principles of physical nature of FEM cannot be separated. It goes similarly with the nowadays-popular principle of “interaction”. Already in the years B.C. people (Chinese mechanics, Greek Archytas of Tarentum, Archimedes, Heron, etc.) knew the preceding principle of “release”, without which the principle of interaction is meaningless. Let us have an object (fig. 3.5) and let us be interested in a part of it, e.g. D or E. We must require axiomatically, i.e. we have to believe, that – following from the human experience acquired so far – it is possible to “release” the analysed part and consider it separately on its own and that, at the same time, the impact of the neighbouring parts and other surroundings J can be somehow “replaced”. Mechanics uses this approach to introduce all the fundamental terms of what is called internal forces in beams, plate structures and, in general, stress in bodies. The action of the “surroundings” is replaced by forces acting on the boundary of the extracted part. Particularly topical is the application of this principle through the introduction of what is termed boundary connections. For example, the system allows for the solution of an arbitrary foundation limited to a small domain Ω1 with boundary Γ1 , which brings considerable savings of computer time. The surrounding of the foundation is modelled by energetically equivalent connections and the subsoil itself – from a certain depth H (within the extent of which there are 3D elements) – is replaced by a surface model. This approach would be impossible without the initial axiom that the foundation can be released. Both the classical and modern conception of forces and force fields is closely related to the above-mentioned four principles and today used models of continuum and particular materials (fig. 3.2). Before Isaac Newton, forces were known from the physiological analogy as tensile or compressive forces directly acting on objects. Newton opened the era of forces acting remotely. Even though he remained bothered throughout the whole life by how they can be transferred, he stuck to his formulation “hypotheses non fingo”, i.e. to the description of how it works without faking the hypotheses on why it is so. The remote force (Fernkraft) became so popular with later physicists that even contact forces were explained by remote forces, direct contacts were abandoned and were replaced by remote transfer, even though – of course – in the atomic or, at most, molecular scale. Today we know four groups of forces (gravitational, electromagnetic, of weak and strong interaction) with a problematic fifth group of what is termed antigravitational forces. However, any relation to these four fundamental physical forces is still missing in the mechanics in the explanation of the stress-state of different models of continuum and discontinuum. Those who are interested in these problems are recommended to study [24 to 31] containing both popularising and original formulations.

57

3.2 Elementary principles of physical nature of FEM

Figure 3.3: Principle of isolation. Principle of superposition of the results.

Figure 3.4: Example of a strut or suspension subjected to force load and thermal load when the principles of isolation and superposition cannot be applied. Force F is transferred by internal forces R. This behaviour can be expressed by a simple of compound rheological model (see art. 4.2.1, fig. 4.23 and 4.24). The picture shows the frequent Kelvin’s model, i.e. parallel connection of spring H a damper N.

58

3.3 Variational principles of mechanical problems of FEM

Figure 3.5: Principle of release. Principle of interaction. The problem of the size of the surroundings J that should be together with the analysed cut-out AB...H released from the objective reality. Subjective conception of the substitution of the action of removed parts by forces F.

3.3 Variational principles of mechanical problems of FEM 3.3.1 Position of variational principles in mechanics The elementary explanation of articles 1 and 2 is sufficient for a general idea about how the finite element method provides for the substitution of unknown functions for their approximate shapes – a linear combination of base functions – that are closely related to the division of the analysed body Ω (FEM) or Γ (BEM) into finite elements. To sum up: how the discretisation of the project and the related finite number of unknown parameters is carried out in FEM. The solution of the systems of algebraic equations is the most effective procedure the state-of-the-art computers can offer. Non-linear problems must be solved by the repetitive application of the same procedure too. The given explanation focused on the most often used variant called deformation variant of FEM, in which the unknown parameters d have the nature of geometrical quantities: components of displacement and small rotations, or their derivatives. In article 1.3 we showed that the application of the first-rate FEM apparatus is possible in all the situations in which the functional (whose extreme (minimum or maximum) is being sought) is of additive nature. We also explained that formulas (1.3.2) (1.3.4) clearly show how such a functional can be created in problems where we lack any physical definition for it – through the principle of weighed residua (1.3.4). This can be done also in non-mechanical, even non-physical, purely mathematical problems. It is clear that for engineers especially mechanical problems are of great importance. For these, the development of FEM has offered not only the deformation, but also force and mixed 59

3.3 Variational principles of mechanical problems of FEM variants – for problems of varying content and extent. Some FEM systems contain them in the set of implemented finite elements. Therefore, it is useful to be familiar with the variational principles that represent the basis of the particular solution in order to be able to evaluate their practical advantages and disadvantages. There exist many illustrious original resources dealing with these issues, e.g. the literature cited in [3-9]. For the first acquaintance with the subject let us mention certain “sine qua non” for the engineers. The principles of mechanics belong to the oldest principles of physics. This includes such statements that can be used to derive the equations of motion or under special conditions of statics the equations of mechanical systems, both in arbitrary coordinates. Depending on the mathematical formulation and mechanical meaning we can distinguish two types of principles: Differential principles examine an arbitrary instantaneous state of the system and compare it with the nearby state. Integral principles are in general variational principles that examine such states which the system passes during a limited time interval and compare them with certain nearby states. Some principles can exist in both formulations and are usually closely related to each other. The differential principles include e.g. Gauss’s principle of least constraint and Hertz’s principle of least curvature. The typical integral principles include Hamilton’s, EulerMaupertuis’s and Jacobi’s principles – called together the principle of least effect (effect = energy × time) – which differ from each other in the assignment of the comparative trajectories. D’Alembert’s principle and the principal of virtual work are not in fact extremal principles, but also represent statements from which the equations of motion or, as the case may be, equilibrium, can be derived. When dealing with problems in the field of statics of elastic bodies, we can start either with the system of differential equations of with energetic reasoning, which leads either to the classical boundary problem or to the variational problem from the theory of elasticity. There is no general answer to the question which formulation of the problem should be considered the primary one, unless we take into account some additional, e.g. methodological, positions. Each basic principle is actually an unprovable axiom and can be replaced only by another axiom (or derived from it). The basis for the finite element is however formed by the fundamental variational (integral) principles. The oldest principles of mechanics focus only on what is termed mechanical systems of mass points or rigid bodies that have finite number of degrees of freedom and such connections whose reactions do no work for any virtual displacement. On the other hand, the virtual work of stresses appears in elastic bodies. The most general principle is Gauss’s principle of least constraint: the real motion of the system is such that in every time instant the resistance Z is minimal: N

Z (a1 , a2 , K , an ) = ∑ i =1

1 ( Pi − mi ai ) 2 = min mi

(3.3.1)

on condition that we take into account all possible motion states of the system with the same positions and speeds but different accelerations ai ( i = 1, 2,K , N ) that meet the given conditions. Pi means external forces and mi the mass of N points of the system. Under special circumstances, this principle is equivalent to d'Alembert’s principle, but makes it 60

3.3 Variational principles of mechanical problems of FEM possible to take into account more general (non-linear, non-homogeneous) conditions. It is based on the method of the least sum of squares and for a free motion without forces ( Pi = 0 ) it can be harmonised with the Hertz’s principle of least curvature. The most important principle for the field of statics is the principle of virtual works, which can be traced back to Aristotle (383 – 322 B.C.) and which was generally formulated by Bernoulli in 1717 in a letter to Varignon who published it in 1725. In the original form it is applicable just to the systems of rigid bodies, non-compressible liquids without friction and non-expansible ropes and reads: a mechanical system is in equilibrium only if the virtual work V of given forces Pi ( i = 1, 2,K , N ) is for any virtual displacement δ ri ( i = 1, 2,K , N ) equal to zero N

V = ∑ Piδ ri = 0

(3.3.2)

i =1

A virtual displacement is an arbitrary infinitesimal change in the position of the mechanical system that is assumed in a certain time instant and that complies with the given geometrical connections. Forces Pi (contrary to the reactions that are subsequently determined by the analysis) must be defined in advance. They are of physical origin and, in general, depend on physical constants (gravitation, friction, electrical or magnetic forces, etc.). They can act both outside and inside of the system. The principle of virtual works can be extended to elastic continuum only axiomatically. All the mechanical (statical) conditions of equilibrium in the body and on its boundary are then based on it. On the other hand, if we accept these conditions as an axiom, the principle of virtual work can be derived from them. The statements are actually equivalent. In order to understand the wider context, let us add the following facts. We deal only with what is termed scleronomous mechanical systems with scleronomic constraints in which time t is not explicitly included (contrary to rheonomic constraints). In addition, we take into account only such forces P whose potential is independent on time or – more exactly – that do not follow the deformation of the bodies and keep their direction, so that their potential energy Π is unequivocally defined by the deformation of the body. We term such system shortly as conservative. The following statement applies to them: A scleronomous mechanical system with given conservative forces moves in such a way that the sum of kinetic ( K ) and potential ( Π ) energy remains constant during the full movement ( k is the energetic constant). K +Π = k

(3.3.3)

It must be stated that in technical practice we may encounter also rheonomic constraints and, in particular, given forces that are not conservative and their potential energy is not independent on time, e.g. external forces follow the deformation of the structure (fluid pressure according to Pascal’s law), are influenced by its vibration (various aerodynamic effects), etc. Similarly, internal forces may follow more complex constitutive law in the case of ideally elastic bodies. The system is then non-conservative (dissipative) and the mechanical energy is partially transformed into other forms of energy, e.g. thermal, etc.

61


3.3.2 Scalar, vector and tensor field in FEM inputs and outputs Texts dealing with FEM often use the term field as the opposite to the term single value. Users sometimes misunderstand it and consider it identical to the term function, which is known to be a relation that relates one set if independent variables to another set of dependant variables, regardless of whether the relation is analytical (formulas), numerical (tables) or graphical (graphs). A field is just a description of the state in all points x = ( x1 , x2 , K , xN ) of a body with dimension N D – in particular in FEM in points of beams and structures composed of them (1D), walls, plates, shells and their systems (2D), spatial figures (3D), and occasionally even 4D figures in what is called space-time elements in space ( x, y, z , t ) . Such elements were developed for certain dynamic problems and were applied in practice. In general, however, another discretisation of time factor t has proved useful. In free vibration problems it is the harmonic analysis, in forced vibration it is either modal analysis (decomposition into the free vibration shapes) or what is termed direct integration in time t , e.g. over reliably small differences ∆t , which allows for a good estimate of the response of the analysed system to the given dynamic excitation. Consequently, we stick to three dimensions N = 1, 2, 3 (as the configuration N = 1 and 2 can always be obtained from the basic configuration N = 3 ), because this is the way they are defined by means of various static and in particular geometric restrictions concerning the nature of quantities acting in real 3D bodies (other bodies do not exist in physical terms). Therefore, the explanation can be limited to 3D body Ω and its points x = ( x1 , x2 , x3 ) = ( x, y, z ) with the illustrative notation of coordinate axes xi (i = 1, 2, 3) using letters x, y, z . The subscript notation i (or j if two different axes play a certain role) is in FEM common also for other quantities than coordinates. For example, displacement components ui (i = 1, 2, 3) instead of u, v, w , components of volume forces in Ω X i instead of X , Y , Z , components of surface forces on Γ (boundary of Ω ) pi instead of px , p y , pz , components of stress tensor σ ij instead of σ 11 = σ x , σ 12 = σ 21 = τ xy = τ yx , σ 13 = σ 31 = τ xz = τ zx , σ 22 = σ y , σ 23 = σ 32 = τ yz = τ zy , σ 33 = σ z (in total 6 components of a symmetrical stress tensor of the classical Boltzmann continuum), components of strain tensor ε ij instead of ε 11 = ε x , ε 12 = ε 21 = γ xy 2 = γ yx 2 , etc. as for σ ij , γ is the change of the originally right angle between the directions specified by the subscripts. The fraction 1 2 has been introduced to ensure that quantity ε has the character of tensor, i.e. that it transforms during the rotation of the coordinate trihedral following the same rules as any other tensor, e.g. σ . In addition, this coefficient is important for the unification of the notation of the geometrical link between displacements ui and components of strain ε ij . The following notation is introduced in which the subscript following the comma denotes the partial derivative with respect to this index: ui , j =

∂ui ∂x j

for example : u1,2 =

∂u1 ∂u = ∂x2 ∂y

(3.3.4)

Then the well-known relation from textbooks dealing with the theory of elasticity (six

62

3.3 Variational principles of mechanical problems of FEM relations, three for ε , three for γ ) has the following form: ε ij =

1 (ui , j + u j ,i ) 2

i = 1, 2,3

j = 1, 2,3

(3.3.5)

1 ∂u u1,1 + u1,1 ) = u1,1 = ( 2 ∂x 1 1 1  ∂u ∂v  ε 12 = γ xy = ( u1,2 + u2,1 ) =  +  2 2 2  ∂y ∂x  ε 11 = ε x =

e.g.

Components of rotation that were earlier neglected for many reasons were implemented into modern FEM programs (approximately from 1980 onwards). Formally, they differ from the components of strain only by the minus sign in the general formula that is similar to (3.3.5): ωij =

1 ( u j , i − ui , j ) 2

i≠ j

(3.3.6)

In common engineering notation in detail: 1  ∂w ∂v  ω x = ω yz =  −  2  ∂y ∂z  1  ∂u ∂w  ω y = ω zx =  −  2  ∂z ∂x  1  ∂v ∂u  ω z = ω xy =  −  2  ∂x ∂y 

(3.3.7)

The notation with one subscript denotes the axis of rotation, the notation with two subscripts the plane of rotation. It represents the real physical rotation. If it is small enough, it is exactly satisfied for the limit approaching zero, then the rotation can be considered a vector and the three quantities (3.3.7) its components in the x –, y –, and z –axis. As a preliminary point it should be observed that they can be accepted as parameters of deformation in nodes of finite elements, which makes it possible to concentrate all the parameters into their vertices. This eliminates the intermediate side parameters that worsen the good numerical conditionality of the equation systems for the solution. They increase the band width BW or occupy more elements in matrix K . A scalar field is defined in each point of body Ω by a single value (e.g. by specific weight or density ρ , temperature T 0 etc.) as a function of coordinate xi , i.e. the position of the point. Whether we know the values (e.g. definition of the density of the body) or not (e.g. temperature T 0 ) plays no role in the nature of the field. On the other hand, a scalar field (i) can vary over time t , which is a common situation towards the rheological process, or (ii) can be constant over time t , e.g. in the case of stationary (stable) thermal flow through the body each point has permanently constant temperature. A vector field is in each point of body Ω by one vector, which is a quantity representable by an arrow the origin of which is in the analysed point, the length expresses the magnitude and the arrow defines the direction and orientation of the vector. The way of input is once again unimportant. A useful aid is an idea of a dense system of arrows each of which starts in one point of a body. 63

3.3 Variational principles of mechanical problems of FEM The way these arrows are documented is of no primary significance. In practice, it is always by means of what is termed vector components in the selected coordinate axes, of which there is a wide range of selections and which can be the global axes common to the whole body Ω ( x, y, z ; r , z , ϕ ; r , ϕ , ψ etc.), or local. For example, for shells it is possible to select one axis in the normal n to the mid-surface and the other two in the tangents to the normal sections that have the extreme, i.e. principle, radii of curvature (max. a min.). Users are recommended not to overlook in which axes the components of the vector field are defined. It is usually displayed in a help line on the status bar in the input graphical environment or sometimes only mentioned in the manual. This relates mainly to the following vector fields: (i) displacement components ui , both in the input of wanted displacements and in the output of sought displacements, etc; (ii) components of volume forces X i , usually defined in what is termed gravitational global system x, y, z , with the vertical + z –axis following the gravitation of the Earth, but sometimes also differently (centrifugal forces!); (iii) components of surface forces pi on boundary Γ , defined only for this 2D area; (iv) speed components often in the product with the density (scalar field) ρ , i.e. the quantities of shape vi = ρ ui* = ρ ∂ui ∂t

(3.3.8)

The same applies to components of acceleration that are often also in the product with the density for use in d’Alembert’s principle, i.e. what is termed volume fictitious forces (inertial forces) in the meaning of the second Newton’s law, or more precisely the density of these forces in a point: Pi = ρ ai =

ρ ∂ 2ui ∂t 2

vi* =

∂vi ∂t

(3.3.9)

Components ni of unit vector (length=1) of the normal to the boundary Γ , or mi of unit vector m of the normal to another surface, usually internal boundary between elements, form a vector field that is of purely geometrical nature in a 2D area. A tensor field is defined in every point of body Ω by one tensor, which is a physical quantity originally related to the perception of a general stress (tension) and later also general deformation of a small neighbourhood of a point. Today, it is formally defined in mathematical terms through the requirement on the transformation of its components during the rotation of the coordinate trihedral. In practice, in FEM we come across stress tensor σ and strain tensor ε . With the exception of Cosserat’s continuum and some other tensors in non-linear mechanics, we deal with symmetrical tensors, i.e. σ ij = σ ji , ε ij = ε ji , which reduces the number of independent components to six. As a default, the components of normal stress σ ii , i = 1, 2, 3 are marked σ x , σ y , σ z and components of shear stress σ ij (i ≠ j ) are denoted τ yz , τ zx , τ xy . Arranged in this order, they are often written in the form of singlerow matrices – always as column matrices – in order to save the notation as a transposed row with one subscript defining the position of the element: σ = σ x , σ y , σ z ,τ yz ,τ zx ,τ xy  = [σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 ] T

T

(3.3.10)

The matrix algebra calls the single-row matrices shortly row- or column vector, which is 64

3.3 Variational principles of mechanical problems of FEM justified by the generalisation from the original dimension of 3 in Euclidian space to the dimension of N in the N -dimensional space. This rather innocent choice of terms has caused considerable troubles to the users of FEM programs. In addition to notation (3.3.10), FEM algorithms and manuals use (inevitably) also the original matrix notation σ 11 σ 12 σ 13  σ x τ xy τ xz    σ ij = σ 21 σ 22 σ 23  = τ yx σ y τ yz  σ 31 σ 32 σ 33  τ zx τ zy σ z   

(3.3.11)

in all operations where the simplification (3.3.10) would result in nothing useful. The subscript operations in algorithms cause no difficulties in the current languages even if a larger number of indices are used. Similarly, also the strain tensor uses both the vector notation: γ γ γ   T ε = ε x , ε y , ε z , yz , zx , xy  = [ε1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 ] 2 2 2   T

(3.3.12)

and the original matrix notation:

ε11 ε12 ε ij = ε 21 ε 22 ε 31 ε 32

γ xy γ xz   ε x 2 2  ε13   γ yz  γ ε 23  =  yx ε y 2 2   ε 33    γ zx γ zy  εz   2 2  

(3.3.13)

Components of rotation (3.3.6), (3.3.7) form a skew-symmetric (antisymmetric) matrix 0 ω12 ω13  ωij = ω21 0 ω23  ω31 ω32 0 

ω ji = −ωij

(3.3.14)

The antisymmetry is due to the fact that the rotation is in the limit case of geometrical linearity (for very small rotations) an oriented vector. The rotation in a plane with the orientation ji is positive if the j -axis transforms into the i -axis. Apparently, it is the same vector k ⊥ ij , but of opposite direction. A plane has an oriented normal that together with the plane axes creates a right handed system. If axes i and j are swapped, k must change its orientation in order to keep the system right handed. The physical formula known as a generalised Hooke’s law can be thus written in the original form termed tensor notation 3

3

σ ij = ∑∑ Cijkl ε kl

(3.3.15)

k =1 l =1

where most of FEM manuals (for the reason of brevity) omit the summation symbols, which is what is called Einstein’s summation convention (the summation is carried out with respect to the repeating subscripts k , l ): 65

3.3 Variational principles of mechanical problems of FEM σ ij = Cijkl ε kl

(3.3.16)

or a typical FEM notation, called matrix notation, is used σ = C ε

(6,1)

(6,6) (6,1)

(3.3.17)

with matrix of physical constants C of type (6,6), which, thanks to the symmetry, contains only 21 different elements d , double subscripted cij , i, j = 1, 2, K , 6 . Let us note that the stress components depend only on the components of what is termed pure strain ε . The physical law contains no rotations. It can be proved that quantities ω are the exact average of rotations of all elementary line segments passing the investigated point, having all the possible directions, measured e.g. from a certain base direction with the angle ranging from 0o to 360o or from 0 to 2π. For better understanding, ω can be imagined to be a rotation of close surroundings of a point as a rigid compact whole, without giving rise to any stress. This nature of the rotation components is the root of the peculiar behaviour of finite elements with rotational parameters of deformation, which will be explained later in the appropriate paragraphs about elements. As a preliminary point it should be observed that in such elements there exist states when the elements are not deformed at all, even though they embody a certain set of non-zero rotational parameters. These states are in technical literature called “zero displacement modes”. A certain amount of work of deformation accumulates in the elements so that it represents one of the “zero energy modes”. Zero work of deformation can be accepted only when the element moves as a rigid whole – termed rigid body displacement, otherwise it is an unwanted situation that must be eliminated, usually through an artificial stiffening of the element. It is perceived as a kind of penalty for undue behaviour – hence the term “penalty stiffness” and “penalty functions” for the corresponding additional base functions. These terms are commonly used in the nowadays FEM manuals without detailed explanations which normal users may not be interested in. Important notice for all FEM users: l) A vector is not equivalent to its three components, i.e. a vector field is different from three scalar fields, because (i) there exist an infinite number of triplets of components for every vector, depending on the selection of coordinate axes, but namely (ii) due to physical reasons. There is an irreconcilable physical difference between a quantity defined by a single value (e.g. temperature T 0 ) and a vector representable by an arrow of a certain size and direction. This difference cannot be reduced to any scalar reasoning, it can only be documented in the form of scalars with a comprehensive explanation of what is going on. m) A tensor is not equivalent to its nine (non-symmetrical) or six (symmetrical) components neither to three vectors as it is often popularised, e.g. stress tensor = three components of principal stress σ 1 , σ 2 , σ 3 , etc. It must be somehow numerically documented, usually by six components (3.3.10) for stress and (3.3.12) for strain. The fact that we deal with a completely different physical quantity can be shown on what is termed uniaxial stress states σ 33 = σ z (remaining components equal to zero) or strain ε 33 = ε z (remaining components equal to zero) in the direction of the z -axis, e.g. gravitation, which is a state recommended by most standards for subsoil. Values 66

3.3 Variational principles of mechanical problems of FEM ε z can be formally treated as a scalar field. It is however enough to rotate the coordinate system into another orientation x* , y* , z * where no axis is neither parallel nor perpendicular to z . All six components of tensor ε are generally non-zero in these axes! it means that a pure compression of elementary element dxdydz in the z direction results in a general change of the element dx* dy* dz * into a parallelepiped, including sloping of right angles, what can be simply found after substitution into the well-known transformation formulas. The situation is similar for stress tensor σ . Both tensors describe a physical state in a point neighbourhood, which can be documented by means of certain values (scalars) with the explanation of what is documented, but such a phenomenon cannot be reduced to any reasoning that uses scalars or vectors. n) Internal forces in 1D and 2D FEM elements, i.e. six values for 1D elements of beams with a solid cross-section N x , Qy , Qz , M x , M y , M z (3.3.18) and 3 membrane and 5 bending components for 2D-elements of surfaces: nx , n y , qxy , mx , my , mxy , qx , q y (3.3.19) have the character of what is called double vectors. For beams these are force vectors with components N x , Qy , Qz and moments M x , M y , M z acting in the section across the 1D element on both banks of the section within the meaning of the principle of release (art. 3.2.). If it is a section x = x0 in an element 0 ≤ x ≤ L , then one set of vectors replaces the action of the part x > x0 on the part x < x0 and the other set of vectors the action of the part x < x0 on the part x > x0 . They have the same magnitude but opposite signs, which is expressed in classical definitions of summation by the well-known phrase “if we proceed from the other side, we change the sign”. The principle is the same for plated structures, but quantities (3.3.19) represent only local point intensities of forces and moments. If these intensities remained unchanged along a specified length of the section c, it would result in a similar situation as in beams. A planar element cx × c y , in planar coordinates cut by sections x p , x p + cx , y p , y p + c y , would be subjected to forces and moments (kN, kNm): c y nx , cx n y , c y qxy , cx q yx (qxy = q yx ) (3.3.20) c y mx , cx my , c y mxy , cx m yx , c y qx , cx q y (mxy = m yx ) Such a state is very rare in practice, it is what is termed elementary stress-state, and formulas (3.3.20) are applicable only to the situation c y = dy → 0 , cx = dx → 0 and only for distributed internal forces without singular external loads. If, in such situations, we need forces and moments over a certain length of sections cx , c y , in particular in the design of reinforcement in reinforced concrete structures where a unit length is usually chosen, the resultant must be obtained through the integration along the sections:

67

3.3 Variational principles of mechanical problems of FEM N x = ∫ nx dy

N y = ∫ n y dx

N xy = ∫ qxy dy

N yx = ∫ q yx dx

M x = ∫ mx dy

M y = ∫ my dx

M xy = ∫ mxy dy

M yx = ∫ myx dx

Qx = ∫ qx dy

Qy = ∫ q y dx

cy

cy

cy

cy

cy

cx

cx

(3.3.21)

cx

cx

cx

Let us mention the school definition of the meaning of the subscripts: the first one denotes the direction of the normal to the face against which the force is acting, the second one (if required as the symbol of the quantity itself is insufficient) determines the direction and orientation of the action. Therefore, the statement about the reciprocity of shear stresses τ xy = τ yx , resulting in the equalities qxy = q yx and mxy = myx , implies the equality of forces N xy = N yx and moments M xy = M yx only for identical lengths of sections cx = c y in (3.3.20). This is rare in a general configuration (3.3.21), as we deal with distributions in completely different points of a planar structure, which means that these distribution should be identical, which practically happens only for elementary stress-states (pure shear, pure torsion). o) It follows, among others, from formulas (3.3.21) that local point values of internal forces are not decisive for the strength of the structure, but their distribution over lengths cx , c y are. These lengths are always finite, even though not necessarily always equal to 1 meter (reinforced concrete), but e.g. values corresponding roughly to thickness h . If a FEM program prints in the output document singular data about internal forces (3.3.19), either enormously large numbers (e.g. 109 against 103) or symbols of computer-infinity XXXXX, i.e. overflow in the output format , there is no reason to worry about the safety of the structure. The output just gives a signal about proper convergence of FEM towards the exact solution of the problem. This “infinity” becomes more apparent with refinement of the mesh. The exact solution often gives some internal forces of “infinitely large magnitude”, or more precisely “increases beyond all limits”, e.g. under concentrated load P in a Kirchhoff plate the deflection w is finite, but the bending moments are “infinitely large”, similarly to a planar problem, where, however, already the term stress from elementary technical mechanics defined as ratio P σ= (3.3.22) A necessarily increases beyond all limits for A → 0 and a permanent concentrated force P . What can be derived from the stated above for FEM users is explained on an example of modelling of primary (load) and secondary (reactions) external forces in chapter 5. For the time being, let us only repeat the rule from art. 2.6 in a slightly modified wording: 68

3.3 Variational principles of mechanical problems of FEM 4. If we want that FEM gives correct results, we must model correctly both loads and reactions, i.e. on 2D shapes whose planar measure is not zero in the way they actually act on the structure. 5. If we apply the useful Saint-Venant principle and replace the real distribution of loads and reactions by singular abstractions (point or linear resultants) on area with size d, then the FEM outputs can be directly used only in the distance nd from the edge or from the action point of load or reaction. Strict authors admit only n =3 to 5 (e.g. A. Föppl even in classical calculations). In FEM applications with normally required accuracy of famous standard-defined two percent, the use of n =1 to 2 is sufficient. 6. If we use user-friendly inputs in the form of concentrated forces and line loads (nothing like this does not exist in the real world), then we have to interpret the outputs in domain d correctly within the meaning of formulas (3.3.21). Usually, we have limited integrals with acceptable (designable) magnitude. Graphically, it is demonstrated by “cutting off of infinite peaks” – as recommended already by A. Pucher in the evaluation of influence areas for moments in plates – or rounding of extreme values of the curves – which is in fact recommended in standards. 7. A question of a user: “what would really cause a concentrated P ” of linear q can be answered surprisingly simply: “Nothing at all!” It would pass through the atomic space between particles and with regard to the dimension of d → 0 there is even no chance that it would hit any of the particles. Just this strict explanation can discourage persistent users. This explanation is in fact in another dimension proven experimentally by tests made by C . Bach at the beginning of the 20th century in Dresden. He applied load to very thin glass plates (window panes) by means of gradually sharper spike. A special device ensured that the holder was stopped before hitting the pane. From a certain value of sharpness (defined by a contact area with the diameter lower than the thickness of the plate) the plate did not break but was punched. An elementary thought [47] can be used to find out when the failure due to shear on the surface of the pressed out little cylinder happens earlier than the rupture of the plate. This is now implemented also in regulations of technical standards dealing with what is called punching shear in flat slabs. These slabs require different arrangement of reinforcement. It is clear that the Bach’s spike is not a real concentrated load P , but something close to distributed load over a small circle and thus it represents a dimension far different from the theoretical limit d → 0 . It is however a valuable source of guidance. The present-day FEM would provide, on condition of appropriate modelling of the load and implementation of relevant strength characteristics, Bach’s results even without any experiment.

3.3.3 General variational principle Simple, though a bit formal is the derivation of the general variational principle in mechanics of solid the phase using orthogonalisations (1.3.2) to (1.3.4) art. 1.3, that will be extended to eight components. The left-hand sides of nullified equations of dynamic equilibrium, compatibility of body matter, continuity between adjacent elements (that may be broken in non-conformal elements) static and geometric boundary or support conditions will 69

3.3 Variational principles of mechanical problems of FEM be selected as residua ε 1 to ε 8 . Distributions of displacement, deformation, stress components, dislocations and step changes between adjacent elements, reactions or displacements of supports (or their variations) will be chosen for weight functions g1 to g8 . This is an approach used in FEM literature that is intended mainly for mathematicians and programmers. For engineers – users of FEM programs – another derivation will be more illustrative – the derivation to which a physical content can be assigned, which enables the engineers not to get lost in the vast number of offers by various companies that usually emphasize which conditions are perfectly satisfied, but at the same time keep quiet about what remains unsatisfied and to what extent. This lesson became topical already around 1970 when, in addition to improved Lagrangean elements (art. 2.2 to 2.6), Castigliano’s elements (“force method”) and Hellinger-Reissner’s elements (“mixed method”) appeared. Later, around 1980, rigid finite elements emerged – that concentrated all the elasticity or plasticity into their contact (“rigid element method”) – together with various transitional and semirigid elements, etc. The aim of these historical notes is to persuade the users of FEM programs that after twenty years of further development any attempt to prepare an overview of existing finite elements would be just a hopeless and useless, never-ending, task. Therefore, it is necessary to make the users familiar with something else: (i) with the overview of principles on which the elements are based and (ii) with what can be expected from them in engineering practice, why they are applicable to something and why in different situations they completely fail to meet the expectations promoted by dishonest advertisements appearing in the fierce competition of commercial companies. Suitable for this aim is the following formulation of the general variational principle: First, let us define the following fields (art. 3.3.2.): 8. Vector field of components of displacement, matrix notation u = [u1 , u2 , u3 ]T = [u , v, w]T , tensor notation ui (i = 1, 2,3) . 9. Tensor field of components of strain, matrix notation ε (3,3) see art. 3.3.2., tensor notation ε ij (i, j = 1, 2,3) , assumption: symmetry ε ij = ε ji , i.e. classical Boltzmann continuum with the axiom about the reciprocity of shear components ε ij i σ ij . 10. Tensor field of components of dislocation stresses, matrix notation σ (3,3) see art. 3.3.2., tensor notation σ ij (i, j = 1, 2,3) , the choice of subscript symbols is apparently not important, if required, it is possible to use e.g. σ kl , similarly for the previous field ε kl etc., if it is useful for overall understandability. Warning: Fields u , ε , σ are defined in a completely independent way. No relation is a priori expected among them. This is emphasised also by the adjective “dislocation” in field no. 3, as even meeting of the compatibility conditions of body Ω is not pre-assumed (Saint-Venant relations between u and ε ). 11. Vector field of the dislocation boundary load acting on part Γu of boundary Γ , i.e. in a 2D area of a 3D body on which the distribution of displacement components u is prescribed, e.g. prescribed settlements or displacements of points of support. Matrix notation p 0 = [ p10 , p20 , p30 ]T = [ px0 , p 0y , pz0 ]T , tensor notation pi0 (i = 1, 2,3) .

70

3.3 Variational principles of mechanical problems of FEM The following fields are derived from the four basic fields 1.-4.: 12. Tensor field d of components of deformation derived from vector field u :  ∂u ∂v ∂w ∂v ∂w ∂w ∂u ∂u ∂v  dT =  + + +  (3.3.23)  ∂x ∂y ∂z ∂z ∂y ∂x ∂z ∂y ∂x  A priori there is no relation between the derived field d and field ε , neither any relation between other two derived fields d ≠ ε , σ ≠ σ 0 , ε ≠ ε 0 . 13. Tensor field of stress components σ derived from the tensor field of deformation components through physical, in general non-linear, relations ∂W(ε ) ∂W(ε ) ∂W(ε ) σ= σx = σy = etc. (3.3.24) ∂ε ∂ε x ∂ε y where W is the density (intensity) of the potential energy of the elastic deformation of the body in point ( x, y, z ) . This scalar function of x, y, z can be in general of form W (ε, σ ) , but for the explicit expression of stress components it is necessary to obtain form W (ε) . The function must be determined through physical experiments. In physically linear theory W is defined by known quadratic expressions [5,31]. Consequently, formulas (3.3.24) are linear and represent general Hooke’s law σ = Cε (3.3.25) where C is the matrix of elastic constants (art. 3.3.2.) 14. Tensor field of components of strain ε 0 , derived from tensor field of dislocation stresses σ 0 through similar physical relations: ∂Φ(σ) ∂Φ(σ) ∂Φ(σ) ε= εx = εy = etc. (3.3.26) ∂σ ∂σ x ∂σ y where ε , σ is substituted by symbols ε 0 , σ 0 in order to induce field ε 0 . Φ is the density (intensity) of the complementary energy in point ( x, y, z ) of body Ω . It is once again a scalar function of x, y, z and is related to function W through the Legendre transformation: Φ + W = σ T ε ( = εT σ ) (3.3.27) which explains its name, as the product εT σ is the full virtual work of finite stress components σ over finite strain components ε in point ( x, y, z ) . The total energies in body Ω are then Π int = ∫ Wd Ω Π int ,* = ∫ Φd Ω (3.3.28) Ω

Ω

Formula (3.3.26) is linear in a physically linear theory and follows directly from Hooke’s law (3.3.25) ε = C−1σ (3.3.29)

The following simple quadratic formula holds for functions Φ and W 1 1 W = Φ = σ T ε = (σ x ε x + σ y ε y + K + τ xy γ xy ) 2 2

(3.3.30)

71

3.3 Variational principles of mechanical problems of FEM Using (3.3.29) or (3.3.25), formula (3.3.30) can be expressed just by means of quadratic expressions in σ or in ε . Certain boundary values are assigned to the derived fields if the point ( x, y, z ) of body Ω gets closer to boundary Γ . In general, no relation between these values and given p 0 , u 0 is required in advance. Similarly, the derived fields do not have to be necessarily in equilibrium, compatible, etc. Let us assign a number – functional – to body Ω with boundary Γ = Γ p + Γu and to 3 + 3 + 6 + 6 = 18 functions of x, y, z defined as vector fields u , p 0 and tensor fields ε , σ 0 :

Π = Π int + Π ext + Π D

(3.3.31)

by means of formulas: Π int = ∫ W (ε ) d Ω

(3.3.32)

Ω

Π ext = − ∫ bT u d Ω − Ω

∫ p u dΓ T

Γp

Π D = ∫ σ 0 ( d − ε ) d Ω + ∫ p0 ( u0 − u ) d Γ T

Ω

(3.3.33)

T

(3.3.34)

Γu

Π int is the total potential energy of the elastic deformation of the body, Π ext is the total potential energy of primary external forces of the body, i.e. specified loads b , p in body Ω and on its boundary Γ or, more precisely, on part Γ p of this boundary. The primary state of the body is always used as the comparison level. Π D is the total virtual work of components of dislocation stress σ 0 in dislocations d − ε , including the virtual work of dislocation load p 0 on boundary dislocations u 0 − u . The dislocation stress can be considered an external tensor field of force character that does virtual work over the difference between d and ε called dislocation. It is clear that if we assume that ε is an integrable and compact field derived from field u , then we get d ≡ ε , as following from (3.3.23) d is us just derived from u . Then d − ε = 0 and Π D = 0 , i.e. Π = Π int + Π ext , art. 2.2., (2.2.9). Π D can be briefly called dislocation potential energy. The extension of the definition of virtual work for continuum must be accepted axiomatically as proved by experience, as the transition from a finite or countable set of mass points to a continuum is too radical (in terms of both physics and mathematics). In this context we can say that functional Π according to (3.3.31) to (3.3.34) is the total virtual work of the system of forces (σ, b, p, σ 0 , p 0 ) acting on body Ω and its boundary Γ over virtual displacement (ε, u, d − ε, u 0 − u) . In general, any system of functions [σ, b, p, σ 0 , p 0 ] can be taken as a system of forces and any system of functions [ε, u] can be taken as a virtual displacement. In addition, d − ε and u 0 − u are derived from the latter in order to determine the value of functional Π . After such a choice, however, no physical law holds for functional Π . Let us assume that Ω + Γ is a continuous physical body in the state of statical equilibrium and that all four original fields

72

3.3 Variational principles of mechanical problems of FEM u, ε, σ 0 , p 0 have been changed to new fields u + ∂u ε + ∂ε σ 0 + ∂σ 0

in Ω

p 0 + ∂p 0

on Γ

(3.3.35)

At the same, the given load b in Ω , p on Γ also supporting conditions u 0 on Γu remain unchanged. This change (variation of fields u, ε, σ 0 , p 0 ) results in the change of the functional value Π to the value of Π + ∂Π . This change is ∂Π = ∂ ( Π int + Π ext + Π D ) = = ∫ ∂W (ε) d Ω − ∫ bT ∂u d Ω − Ω

Ω

∫ p ∂u d Γ + T

(3.3.36)

Γp

0 0 0 0 0 0 +⌠  ∂σ ( d − ε ) + σ ∂ ( d − ε )  d Ω + ∫ ∂p ( u − u ) + p ∂ ( u − u )  d Γ ⌡ Γu T

T

T

T

Ω

If we extend the principle of virtual work to a generalised virtual displacement ∂u, ∂ε, ∂σ 0 , ∂p 0 , then the general variational principle reads ∂Π = ∂ (Π int + Π ext + Π D ) = 0

(3.3.37)

It is in fact an extension of the Lagrangean variational principle by the dislocation potential, which can be viewed as an amendment that expresses the impact of secondary conditions – compatibility conditions in Ω and on Γ . These conditions or, more precisely the left-hand sides of nullified equations that express these conditions, are in (3.3.34) multiplied by Lagrangean coefficients σ 0 and p 0 . The classical Lagrangean principle assumes a priori that these conditions are met and thus expression (3.3.34) equals to zero.

3.3.4 Consequences for some variants of FEM procedures First of all, it can be proved that all the equilibrium conditions in body Ω and on boundary Γ p , geometrical relations between u and ε in Ω , geometrical boundary conditions on u and also physical relations follow from the principle (3.3.37). It is sufficient to select in turns such variations in which the only non-zero variation is ∂σ 0 , then ∂p 0 , ∂u and ∂ε [48]. These four simple variations do not consider all possible variations of four independent fields u, ε, σ 0 , p 0 and represent only the prove that the variational principle (3.3.37) is superior to all basic relations in the theory of physically generally non-linear elasticity. In linear elasticity the general formula (3.3.24) is replaced by Hooke’s law (3.3.25) and formula (3.3.30) holds for W . In geometrical terms, the whole reasoning is linearised by formulas (3.3.23) that hold only for small u and d , i.e. ε . For large displacements u but small deformations, it is enough to apply formulas (3.3.23) extended by quadratic terms, which practically exploits all technical applications except materials with a fractional modulus of elasticity, such as rubber, etc. Such problem is considerably complicated and those who are interested in more information are referred to [32].

73

3.3 Variational principles of mechanical problems of FEM The general variational principle in the form (3.3.37) was derived by Hu Hai - Chang already in 1955 as what is termed the principle of generalised potential energy, as it is a direct extension of the Lagrange-Dirichlet principle of dislocation potential Π D . Equation (3.3.36) is a generalisation of Lagrangean variational equation [5, 31]. We integrate the first term in (3.3.34), i.e. the product σ 0 d per partes according to the Gauss-Ostrogradsky formula over area Ω with boundary Γ , we divide the boundary integral over Γ into two parts ( Γ p and Γu ) and merge the boundary integral in (3.3.33) and (3.3.34): T

Π ext = ∫ ( bT − b* ) u d Ω − T

Ω

∫ (p

Γp

T

− p* ) u d Γ T

(3.3.38)

Π D = ∫ σ 0 ε d Ω + ∫ p0 u0 d Γ T

T

Ω

T

Γu

where in brief notation b* = bx* , b*y , bz*  * ∂σ x* ∂τ yx ∂τ zx* * bx = − − − ∂x ∂y ∂z T

b*y = −

∂τ xy* ∂x

−

∂σ *y ∂y

−

p* =  p*x , p*y , p*z  T

∂τ zy* ∂z

∂τ ∂τ ∂σ z* b =− − − ∂x ∂y ∂z * z

* xz

* yz

(3.3.39)

p*x = 1σ x* + mτ *yx + nτ zx* p*y = 1τ xy* + mσ *y + nτ zy* p*z = 1τ xz* + mτ *yz + nσ z* l , m, n are direction cosines of the outer normal to Γ . The meaning of the functionals is thus modified. Hu Hai - Chang called this principle (3.3.37) the principle of generalised complementary energy. If we follow the Legendre transformation and make substitution in functional (3.3.32) W = σT ε − Φ , then if σ ≡ σ 0 the element σT ε is deducted by the element - σT ε . If we add the volume integrals in (3.3.32) and (3.3.38) in the substitution into (3.3.32), the density of potential energy W in (3.3.31) is replaced by the formula for the density of complementary energy Φ . The practical consequence of both mentioned forms of the variational principles in FEM is: p) What causes certain problems in the application of the Lagrangean variational principle is the selection of the approximation of the deformation state of the body, as the principle assumes that all conditions of continuity (compatibility) are met and expects that only one field is under variation. Practically, it is directly vector field u , because if we select the vector field u as a system of three functions u ( x, y, z ) ,

74

3.3 Variational principles of mechanical problems of FEM v( x, y, z ) , w( x, y, z ) that are continuous throughout the whole body Ω , the unobjectionability of the application of the principle is granted. In classical variational methods it is necessary to choose the shape of functions u for the whole body Ω with parameters that are used throughout the whole body Ω . In simplified terms, only one formulation (formula) for substitute functions u holds for the whole body Ω . This fact has often been criticised as it is sometimes almost impossible to find a suitable formula for function u that holds throughout the whole Ω , and the task is even more complicated by the necessity of satisfying also all the geometrical boundary conditions. If, however, such a formulation for u is found, then the fact that all the geometrical requirements of the Lagrangean variational principle are satisfied is a big advantage. In the finite element method, this advantage does not apply to the mere selection of the formula for function u over elements Ω k . This factor was initially ignored due to the fact that in the years 1956-1964 the method was pursued mainly by engineers – technicians who intuitively assumed that the introduction of nodal parameters of deformation common to all elements with a contact in the node can for a refined mesh give a true picture of the non-nodal continuity. The convergence, however, does not have to be monotonous. The practical consequence is that even quite a fine mesh does not guarantee generally accurate results. We may express an intuitive theorem that the results obtained with incompatible elements are the poorer the greater is the proportion of dislocation potential Π D . Therefore, it is recommended to use compatible elements. q) A similar reasoning holds for the application of Castigliano’s variational principle in which we have to start only with the equilibrium states and in which we once again have just one field subjected to variation. Usually, it is tensor field σ for which we ensure in advance that it meets the requirements for any parameters of Cauchy’s equilibrium equations and statical boundary conditions. In classical variational methods we select the field using one formula that holds throughout the whole body Ω . On the other hand, in the finite element method an arbitrary selection of σ over elements (without further analysis) does not guarantee that the statical conditions of equilibrium on boundaries between elements are satisfied if we limit ourselves only to the continuity in mesh nodes. Noncorrect element may be created. An intuitive concept that refinement of the mesh increases the number of nodes in which the continuity of the stress is guaranteed in the parameters is not justified as, at the same time, the area of these discontinuities increases and the size of the border between elements grows. Consequently, the force methods required the same studious analysis as the deformation variant. r) The force variant is less frequently used due to clear practical advantages of the deformation variant. It is, however, impossible to overlook the considerable importance of the possibility to obtain solution of both types (a) and (b) for the same problem. It is a well-known fact that the compatible solution approaches: a) the exact solution from below in terms of energy, i.e. it gives the lower limit of the energetic functional. On the other hand, the equilibrium solution approaches b) the exact solution from the top in the same sense. The intuitive prove follows 75

3.3 Variational principles of mechanical problems of FEM already from the fact that in the deformation method we define just a finite number of parameters and thus introduce into body Ω many geometrical constraints that do not exits in reality, while in the force method we – with the same number of the parameters – release the constraints. Naturally, also exact proofs could be presented. Direct utilisation of this fact is possible if the body is subjected to just one load. Then, according to the Betti theorem, it follows that the solution ‘a’ (‘b’) gives the lower (upper) limit of the ordinates of influence surfaces, or functions. Using this we can – for common load configurations (such as e.g. full distributed load or periodic groups of forces) – make estimates about the lower or upper limit of the result, which is the fundamental issue of each approximation. s) The mixed variant of FEM was introduced into practice already around 1970 through what is termed hybrid elements. These elements define two independent fields of displacement (symbolically marked u ) and stresses ( o ). One of the oldest elements is Herrman’s plate element with independent deflections w and moments P . Both fields are linear over the triangular element and introduced are three parameters w in vertices and three parameters M n in the centres of sides with normals n . The popularity of hybrid elements in practice gradually faded as they a priori satisfy neither equilibrium conditions as in paragraph (a), nor the continuity conditions as in (b). The engineer can quite simply assess to what extent the solution satisfies the equilibrium conditions. Deviations can be clearly seen e.g. in step-changes in the distribution of internal forces at the boundaries between elements, in differences between given external nodal forces and those forces that correspond to the calculated parameters are printed by the programs like what is called “nodal loads” – we mean the real (IST in German) as against wanted (SOLL in German). The solution may produce certain unwanted forces (residua) on free unloaded edges. If these deviations are negligible in comparison with the quantities that are decisive for the design (e.g. 2% of the absolute extreme of similar values in the whole structure), technical standards accept even such calculation. On the other hand, it is hopeless in common practice to try to evaluate the admissibility of deviations in the continuity between elements or between the structure and its supports. Even a subtle dislocation may be a cause of fatal errors in the stress-state, which can be proved when we input such a dislocation as a deformation load – especially very stiff structures are sensitive. Therefore, purely force-variant Castigliano’s (b) elements and classical hybrid elements can be nowadays only rarely seen. In following years, finite elements based on more general Hellinger-Reissner’s principle with pure deformation parameters emerged. They guaranteed, when refined, a good convergence towards what is called weak solution of the given problem. They are used in the present-day practice, especially after deformation rotational parameters of type ω were introduced. See also the end of art. 3.3.5.

3.3.5 Special configurations used in FEM Hu Hai - Chang and K. Washizu formulated the principle in a general form similar to (3.3.37) in 1955 for both elastic and plastic zones. Our text deals only with elastic applications. It is possible to proceed from the classical variational principle to the general 76

3.3 Variational principles of mechanical problems of FEM principle in such a way that the condition of continuity (Lagrange-Dirichlet) or equilibrium (Castigliano - Menabrea) is not set a priori, but as a secondary condition that must be met by the functions minimising functional Π to the greatest possible extent. For example, when the Lagrangean variational principle is extended, the dislocation stresses and loads σ 0 , p 0 have the meaning of the Lagrange coefficients by which the left-hand sides of the nullified secondary conditions are multiplied in the element that includes the total value of functional Π . The other way round, both the classical principles (Lagrange and Castigliano) can be derived from the general principle, as well as other more up-to-date, or what we may call nonclassical, principles (Hellinger - Reissner, Veubeke, etc.). The derivation of these more specific principles only requires that we assume in the general principle that fields u, ε, σ 0 , p 0 are subject to variations that meet some of the conditions which otherwise follow from the general principle, such as Euler’s differential equations of the variational problem (3.3.37). As already said, there are in total five conditions: A) in body Ω 15. The physical relation between tensor fields σ and ε , which is generally in the form (3.3.24) or (3.3.26); for the material with a linear relation between σ and ε in the form (3.3.25) or (3.3.29); i.e. Hooke’s law. 16. The geometrical relation between fields u and ε . Tensor field ε can be derived from vector field u by means of well-known formulas for ε ≡ d . This relation can be also expressed through the Saint-Venant conditions of compatibility, on condition that we first calculate only with field ε . If they are met, then there also exists a vector field u from which the tensor field can be derived. 17. Cauchy’s equations of equilibrium, i.e. the well-known statical relation between tensor field σ and vector field of loads b . B) on boundary Γ = Γ p + Γu : 18. Geometrical boundary conditions for field u on Γu . 19. Statical boundary conditions for field σ ; see detailed notation (3.3.39), as the principle leads to the identity of fields σ 0 = σ . If we sort these conditions in the given order, then the general variational principle can be classified by symbol EEEEE, as all the conditions are Euler’s equations of the appropriate variational problem. For other less general principles we assume that one or more from the five presented conditions is satisfied a priori, which is marked by symbol a . These special situations sorted in the order following from [50] are summarised in table 3.1. It can be mathematically proved that what is termed correct finite elements that make it possible to satisfy the conditions (a priori) have the following physically favourable property: when the mesh is gradually refined, the solution converges to such a state of the structure that embodies the exact value of the appropriate energetic functional – briefly said: it converges in terms of energy. This is related to the basic fact that the classical solution using the differential equations of equilibrium (Cauchy, Lamé) and continuity (Saint-Venant, Beltrami) – very demanding on the continuity of functions (up to the third derivative of the 77

3.3 Variational principles of mechanical problems of FEM displacement components, inclusive!) – called strong solution, is in FEM replaced by a problem of seeking an extreme of some energy significantly less demanding on the continuity of functions over the analysed domain. It is a different solution (in the sense solution = result) and is called weak solution – see the end of art. 3.3.4. The functional space, in which the solution is sought, is thus dramatically limited. It is a certain small subspace from the space belonging to the strong solution. Let us symbolically mark a technically interesting function, e.g. deflection or stress, by symbol F (strong solution) and f (weak solution). The principal question for engineering applications is whether f differs from F and what the consequences on the design may be. Those who are interested in these convergence issues are kindly referred to the literature cited e.g. in [5]. This complicated issue can be briefly summarised as follows: t) The energetic convergence of FEM does not generally imply the convergence in all functional norms, i.e. in all defined remainders between the distribution of functions F , f , e.g. the integrals from the squares of remainders ( F − f ) . The same applies also to the derivative of functions F and f and naturally also to individual values Fi , f i in a certain point i . Only in exceptional situations it is possible to deduce from the energetic convergence the convergence in one point, on condition that such a value itself can be decisive for the energy. This typically involves what is termed influence lines with one concentrated load, e.g. a plate subjected to concentrated load Pi in point i has in linear mechanics – according to the Clapeyron theorem – the total potential energy of internal and external forces related to the primary state Π = − Pw i i 2. u) In FEM with correct Lagrangean elements energy Π int converges to the exact value, i.e. deflection wi under load P converges to the exact deflection too. But,

already for two loads Pi , Pk we have Π = − ( Pw i i + Pk wk ) 2 . The refinement of the mesh in a good program results in correct Π , but there exist infinitely large number of deformation pairs wi , wk that can give it. Of course, the engineer’s intuition tells that no “wild pair” is possible, but the computer has no intuition. It only works with formal, fully transmittable terms – see art. 5.1.1.1. Therefore, pure mathematical analyses of base functions of elements are welcome, as well as proofs that the solution converges even in functions f (even though it is still a weak solution) where it is not possible to make any statements about higher derivatives as in the case of strong solution F .

v) 108. 106.

107.

Variational principles

109. 111. hys

116.

117. .

123. I.

General (Hu Hai-Chang, K. 118. Washizu) vers126. 124. Hellinger- 125. ion a) Reissner: 131. vers132. ion b)

Conditions

in body 110.

on boundary

112. P 113. G 114. S 115. G eom tat eom tat

S

119. E

120. E

121. E

122. E

E

127. a

128. E

129. E

130. E

E

133. E

134. a

135. E

136. E

E 78

3.3 Variational principles of mechanical problems of FEM 137. 138. Lagrange-Dirichlet: 139. II. vers147. 144. 145. Castigliano- 146. ion a) V. Menabrea: 152. vers153. ion b) 158. 159. Boundary (Trefftz) 160. . 167. vers168. ion a) 173. vers174. ion b) 165. vers180. 166. Special [50] 179. I. ion c) 185. vers186. ion d) 191. vers192. ion e)

140. a

141. a

142. E

143. a

E

148. a

149. E

150. a

151. E

a

154. E

155. a

156. a

157. E

a

161. a

162. a

163. a

164. E

E

169. a

170. E

171. a

172. E

E

175. E

176. a

177. a

178. E

E

181. E

182. E

183. a

184. E

a

187. E

188. E

189. a

190. E

a

193. a

194. a

195. E

196. E

E

Table 3.1: The overview of a priori satisfied conditions (symbol a) in various variational principles. Other conditions are Euler equations of the problem (symbol E).

w) A common user quite often meets the consequences of the introduction of the weak solution in elements with rotational parameters of deformation that cannot generally satisfy the compatibility on common borders (Novozhilov’s effect, see later in the text), i.e. in general in elements resulting in dislocations that are minimised by means of the general Hellinger-Reissner principle. The obtained solution thus satisfies neither the conditions of equilibrium nor the conditions of continuity. Both improve with the refinement of the mesh, but in terms of energy, e.g. the dislocation potential, work done by internal forces on the dislocations between elements, reduces. In general, this does not imply the convergence in functions, respectively in norms of remainders of two functions, not mentioning the values assigned to a certain point. This usually escapes the attention of undemanding users due to the standard-defined 2% tolerance for numerical accuracy, because the “nature of the problem” does not lead to any substantial differences in comparison with the exact solution F obtained analytically in a simple test. x) In practice, no one knows the exact solution of their problem since absolutely no one, including analysts, knows it for more complex shapes, loads and supports of the analysed structure. The results of FEM are accepted in good faith (bona fide), i.e. the reputation of a software company is trusted, which is verified through independent tests. Or alternatively one relies on the fact that structures designed by means of FEM programs serve reliably their purpose for years and (even) were not too expensive, i.e. they are safe and economical! This is an intuitive factor in the trust in the FEM program that cannot be underestimated by the engineering science.

79


3.3.6 Example of evaluation of elements by means of the variational principle After the initial period between 1956-1965 when elements based on principle III from table 3.1 (Lagrange, deformation) were exclusively used, elements of type IV from table 3.1 were developed (Castigliano, force) and also elements of type II from table 1 (Reissner [51], mixed). In classical matrix analysis of structures composed of beam they correspond to deformation, force and mixed method, which means that the unknown parameters are of geometric or static nature or both. Close attention was paid to mixed elements and a considerable numerical effect was expected from them due to the minimum requirements concerning the grade of the base functions. Even today, users are still interested in offers of companies that contain what is termed Herrman’s plate element: triangular with parameters w1 , w2 , w3 (deflections) in nodes and m1 , m2 , m3 (bending moments in mid-sides opposite to the vertices 1,2,3), acting through compression or tension on faces in the direction of the normal of the side. Linear base functions are sufficient. Many similar mixed elements based on the Reissner principle have been developed, in particular version IIb, in which the physical relation between fields σ and ε is not fixed in advance (symbol "a" = a priori). Authors usually claimed that the classical deformation variant III was improved through the introduction of another free field σ added to the free (unknown) field u , which cause that the equilibrium conditions were better met (the geometrical conditions are satisfied a priori). However, the analysis [52] revealed the following: The best tensor field σ within the meaning of the Reissner principle is such that is derived from field u (and the corresponding field ε ) through physical formulas (3.3.24), for linear configuration through (3.3.25). This field can be obtained directly by means of “unimproved” Lagrangean variational principle! Other authors used version IIa, i.e. they did not meet a priori the geometrical constraints in body Ω , which represents the well-known Saint-Venant equations of compatibility. The common force variant IV was reportedly improved through the introduction of another free field u in addition to free field σ , the aim of which was to better meet the conditions of continuity in body Ω (statical conditions of equilibrium are satisfied a priori). With no particular difficulty it was however shown ([52]) that the following statement is true: The best tensor field σ within the meaning on the Reissner principle is identical with the best σ within the meaning of the Castigliano principle! No improvement of FEM with force elements is thus possible using the Reissner principle. With regard to the quality of field u , corresponding to the force solution, there is nothing definite to be said about it, as field σ corresponding to the sought field according to (3.3.26), in a linear configuration according to (3.3.29), does not have to be integrable into field u , because the Saint-Venant conditions of compatibility were not guaranteed. We have six differential relations between components of tensor ε that follow in fact from the requirement of continuity of components u , v, w of field u up to the third derivative inclusive (classical smoothness of displacement components).

80


3.3.7 Bufler’s variational principles The generalisation of the reasoning from art. 3.3.3.-3.3.6. was made already in 1969 by H. Bufler [53]. In particular, he assumed that all fields can vary over time t , which means that we deal with a problem of four variables x, y , z , t . The principle of virtual displacements then uses the variations of components of displacement ui in the analysed state in time t , which is emphasised by the name dynamic principle of virtual displacements. Similarly, the principle of complementary work (Castigliano) is transferred into the dynamic principle of virtual forces. This extends the reasoning from art. 3.3.3. to 3.3.6. (table 3.1) to problems of dynamics. What is however important for further (also the present-day) development of FEM is the introduction of a significantly more general dislocation potential on the boundaries between elements and the possibility that the impulse theorem can be met only approximately (Second Newton’s law, Ft = mv ) in the surroundings of each point of body Ω . Art. 3.3.3. established four independent fields u, ε, σ 0 , p 0 , in tensor notation ui , ε ij , σ ij0 , pi0 and (from them) derived fields (3.3.23) to (3.3.29). Now we add the following: Vector field vi = ρ ∂ui ∂t where ρ is the density of the matter of the body Ω and ∂ui ∂t is the speed of the motion of the point, or, more precisely, its components into axes x, y, z (i = 1, 2, 3) . The field is called briefly specific impulse. Dislocation border Γ d , called also the internal boundary, which is a set of all boundaries between finite elements that are located within body Ω , but not on its boundary Γ . This boundary is – as in art. 3.3.3. – divided into two (not inevitably continuous) parts Γ = Γu + Γ p on which the geometrical ( u ) or static ( p ) conditions are prescribed. These, however, now depend also on time t and thus we have kinematic ( u ) or dynamic ( p ) conditions. Discontinuity in the components of displacement is now allowed on the dislocation boundary (in the way that may really happen in FEM according to the nature of the FEM variant). Boundary Γ d divides in each point body Ω by a section on one bank of which one finite element is located and on the second bank of which another element exists, on which the displacement is marked as “prime” (e.g. u’ = u prime). In general, certain step-changes appear in the displacement components that are called: Vector field of dislocations, defined only on Γ d : d ui = ui − ui′

(i = 1, 2,3)

In FEM practice, this field has zero values in nodes of the mesh if the deformation FEM variant (art. 2.2.) is used for which components ui are in common parameters of all elements adjacent to one node. Dislocations commonly happen for incompatible base functions, e.g. the well-known rectangular plate element with twelve-member biharmonic polynomial w( x, y ) , which was so far used in older program systems. This is nowadays known even in FEM practice. What is less known is the fact that dislocations may happen in the force variant just for the reason that this variant provides no guarantees in advance concerning the continuity of the body. Step-changes of stress may appear on the dislocation boundary Γ d . These changes are called:

81

3.3 Variational principles of mechanical problems of FEM Tensor field of step-changes in stress, defined only on Γ d :

dσ ij = σ ij where two adjacent elements Ωe and Ω′e have their external normal m j and m′j defined by means of the components of its unit vector m1 , m2 , m3 and m1′ = −m1 , m2′ = − m2 , m3′ = − m3 , on condition that we establish that the local positive direction of the normal to the element boundary points from the element outwards. Now, we could generalise art. 3.3.3 without formidable obstacles, however, the auxiliary formulas and the final notation are formally complex. For the purpose of this overview of FEM principles let us mention the general variational equation that follows through independent variations from the formulation common to both options (variation of displacement or forces): ⌠1 − ∫ (σ ij , j + X i − vi∗ ) ∂ui d Ω +  ( vi − ρ ui∗ ) ∂vi d Ω + ∫ (σ ij n j − pi ) ∂ui d Γ + ⌡ρ Ω Γu Ω

+

1 1 ⌠  dσ ij ∂ (ui + ui′) d Γ −  ε ij + ( ui , j + u j ,i )  ∂σ ij d Ω − ∫ ( ui − ui∗ ) ∂pi∗ d Γ + ∫ 2 Γd 2 ⌡  Γu

(3.3.40)

Ω

+ ∫ (σ ij n j − pi∗ ) ∂ui d Γ − Γu

1 dui ∂ (σ ij + σ ij′ ) m j d Γ = 0 2 Γ∫d

Equation (3.3.40) has 8 members and with independent variations ∂ui , ∂vi , ∂σ ij , ∂pi∗ in integration domains Ω, Γu , Γ p , Γ d can be clearly seen that it can be satisfied identically only if the coefficients for these variations are equal to zero. Thus we get gradually 8 Euler’s equations of the analysed variational problem: 20. Dynamic equations of equilibrium in body Ω (Cauchy’s equations). 21. Definition of the specific impulse, see vector field vi . 22. Dynamic boundary conditions on boundary part Γ p , where vector of stress pi is specified. 23. Condition of continuity for tensor of stresses σ ij along the whole internal boundary Γ d , i.e. step-changes in stress must be zero. 24. Kinematic relation between vector field ui and tensor field ε ij in body Ω . 25. Kinematic boundary conditions on boundary part Γu , where vector of displacements ui∗ is specified. 26. Dynamic relation for vector of reactive stresses pi∗ on boundary part Γu , where displacements are prescribed. 27. Condition of continuity for displacement ui along the whole internal boundary Γ d .

82

3.3 Variational principles of mechanical problems of FEM The Bufler’s variational equation (3.3.40) holds generally and does not assume existence of the potential of external or internal forces. It can be thus used for the solution of problems with non-conservative external forces, with non-elastic material, etc. What remains unresolved is the question whether the Lagrangean multipliers of secondary conditions can be applied to variational equations as the existence of the corresponding functional is not clear. The equation can, however, be viewed also from the point of view of physics as an axiomatic extension of the principle of virtual work, which already several times proved correct during the development of mechanics. For the first time with the extension from rigid bodies to flexible ones and then with the invention of Galerkin method [10,11] and so on. Equation (3.3.40) does not look appealing to practical FEM users, but it presents in a concise way an explanation of the behaviour of a great many of new FEM elements offered today, with an explanation of what can be expected from these elements. The overview of today-already-classical elements (Lagrange, Castigliano, Reissner) in statical problems – to which the simpler variational equation (3.3.36), (3.3.37) applies – was provided in table 3.1, art. 3.3.4. The table contains in fact 12 “element families” I, IIa, IIb, III, ... VIe. Elements III have best stood the test of time, which will be further justified in the following passages. A similar overview of “element families” for the general variational equation would be too extensive. For the time being, most of them have not been developed at all and many of them would show that they improve nothing. As the present-day explosion of new FEM systems sometimes contains completely new features, we will present here at least an abridged amendment to table 3.1 in the following table 3.2 that focuses only on statical problems, which means that the specific impulse is omitted. From dozens of options we selected only two from table 3.1 and (in the last row) what is termed rigid elements that are offered by some companies as well.

197.

Order

198.

207. Domain

206.

199. .

200. 2 .

208.

3201. .

209. Ω

202. .

Γ210. p

. 211.

203.

6.204.

205. 7.

212.

Γ213. u

Γ214. u

216. 218. 220. S D222. 224. 226. K228. D230. TEP PEC. IN. YN. IN. YN. TEP Meanin YN. g 217. 219. 221. I B223. 225. 227. B229. B231. Q. MP. OUND EL. OUND OUND

215.

232.

General 233. ly

234.

235. E

E236.

237.

238.

E239.

E240.

241.

Tab. 242. 1, I.

243.

244. -

E245.

246.

247.

E 248.

a249.

250.

III. 251.

252.

253. -

E254.

255.

256.

a 257.

a258.

Rigid 260. elements

261.

262. -

E263.

264.

265.

E266.

E267.

259.

.

Table 3.2:

83

3.3 Variational principles of mechanical problems of FEM Overview of a priori satisfied conditions (symbol a=a priori) for some special situations of application of the general variational equation. Symbol E (Euler’s equation) denotes which condition should be approximately met by the solution.

The symbol E warns that the conditions – stated in the same order as in the numbered list in the text of art. 3.3.7 by numbers 1 to 8 – are satisfied only approximately. In the classical methods these would be Euler’s equations of the variational problem. The chosen example of rigid elements is interesting as it cannot be assigned to table 3.1 – it thus belongs among typical extensions within the meaning of Bufler’s variational equation (3.3.40). A finite element is considered a completely rigid, absolutely non-deformable body. Its actual elastic or plastic properties are concentrated in its edges, e.g. in the case of a triangular shell elements into their perimeters. A single variant thus offers an incredibly simple element with three parameters w1 , w2 , w3 in vertices (for the analysis of spatial shells), w is the deflection in the direction of the normal to the element. All the possible combinations of letters E and a in table 3.2 have not been considered so far, but most of them will not probably be a big asset. The only factor that deserves the attention of the users of FEM programs is the fact that various new elements can be reliably assigned to a certain category and that it is possible to evaluate whether their “weak point” (symbol E) is not in fact desirable for specific problems solved by them and whether fulfilment of a certain condition a priori (letter a) is not – for that particular group of problems – detrimental. The common conditions of continuity of fields σ and u on internal boundaries Γ d are strongly violated in the relatively new category of rigid elements suitable e.g. for the analysis of global behaviour of rockfil dams. The contact is not in fact continuous but it is realised by means of substantially smaller contact faces on which the contact stress reaches values that are larger (normally by a factor 100) that the values in continuum. This is related to the possibility to define what is termed transitional or contact elements of dimension (n − 1)D for domain nD that can be used for a successful modelling of various technical effects.

3.3.8 Inverse variational principles The name is not entirely accurate but it took hold thanks to pioneering publications [54], [55]. The simplest definition of an “inverse” problem is: all fields that define the stressstate of a domain are known. We are supposed to immerse body Ω whose shape (boundary) Γ is unknown into this domain. We seek such a body that is in a certain sense optimal. It is thus an opposite task to common problems in which we know body Ω and try to find its stress-state. This task and also a much wider range of problems can be solved if we generalise Bufler’s variational equation (3.3.40), or directly the original dynamic principle of virtual displacements ([5], first edition from 1972, pp. 303, formula (48)) for variable body Ω and boundary Γ . We thus accept the existence of the variation of Ω and Γ , which was – for fields that are continuous throughout the whole body Ω – introduced already by V. Horak [54], [55] in the secondary conditions of constant volume V of body Ω or surface U of boundary Γ (or both the conditions can be demanded simultaneously). More general formulation can be obtained if we: y) either (i) admit that also non-continuous fields can appear in Horak’s inverse 84

4.1 Geometric properties of elements variational principle and (ii) introduce dislocation boundary Γ d that can be possibly also subject to variations, z) or admit that Bufler’s variational principle may contains also variations of body Ω or boundary Γ or both ( Ω and Γ ). Moreover, if we even allow for variations of dislocation boundary Γ d , it may result in the definition of a principle that optimises the division of the domain for non-continuous fields [56]. This more general principle will be stated without proof using tensor notation. Ω* is the variable body, Γ* , Γ*d the variable boundary. Variation ∂* applies also to integration domains Ω* , Γ* , Γ*d . The inverse dynamic principle of virtual displacements:

∫σ

Ω*

ij

∂ε ij d Ω* −

∫ (X

Ω*

i

⌠1 − vi∗ ) ∂ui d Ω* +  ( vi − ρ ui∗ ) ∂vi d Ω* − ∫ pi ∂ui d Γ* − ⌡* ρ Γ*p Ω

  1 1   * ⌠ * ∗ ∗ * * −∂  σ ij ε ij − ( uij + u ji )  d Ω + ∫ pi ( ui − ui ) d Γ + ∫ (σ ij + σ ij′ ) m j dui d Γ  = 0 2 2 Γ*   ⌡  Γ*u d Ω*

(3.3.41)

The inverse principle of virtual forces can be modified the same way. In addition to adaptations leading to the variational equation (3.3.40), also all elements in the fixed limits of integrals Ω, Γ, Γ d must be expressed. This can be achieved using the Horak’s formulas for the variation of functionals caused by the variation of integration domains [54]. The meaning of variational principles was for a long time underestimated in FEM practice. One of the reasons was the inappropriately selected application example in [54]. The theoretically challenging text of the author’s thesis was followed by a single example: determination of an optimal height ( h ) to width ( b ) ratio of a rectangular cross-section of a beam subjected to bending on condition that the beam is made (cut) from a cylinder of a specified diameter. That problem had been solved in the distant past using a substantially simpler approach called “golden cut”. Later works by V. Horak and other authors presented a much wider range of problems related to the optimisation of the shape of a body. In the future we can expect further progress in the application of FEM to such problems.

4 Finite elements 4.1 Geometric properties of elements

85

4.1 Geometric properties of elements

4.1.1 Differential elements and finite elements Classical analysis employs differential elements, e.g. dx dy dz for bodies in right handed global coordinates X , Y , Z , or dr rdϕ dz in semi-polar coordinates r , ϕ , z , or – in the simplest configuration – ds for 1D beams with the coordinates s as the position of a cross-section measured from the chosen origin of the beam. The characteristic feature is that these are limit shapes, i.e. it is assumed that differentials dx, dy etc. approach zero without any limits, which is often expressed – inaccurately – in a popular way that the elements are infinitely small. What is in fact substantial for these elements is the variability in the vicinity of zero. No equation from the classical analysis could be derived if we imagine an element 0 × 0 × 0 or ε × ε × ε with constant ε . Even the fundamental term of the classical analysis, derivative, is defined by a limit transition, e.g. df ( x) f ( x + ε ) − f ( x) = lim ε →0 dx ε ∂f ( x, y ) f ( x + ε , y ) − f ( x, y ) = lim ε →0 ∂x ε 2 ∂ f ( x, y ) 1  ∂f ( x + ε , y ) ∂f ( x, y )  = lim  −  2 ε → 0 ∂x ε ∂x ∂x 

(4.1.1)

The above-mentioned formulas express the values of derivatives through limits of differences of values that are valid beyond the point x , i.e. for x + ε , and values that are valid in the point x . Therefore, these are forward derivatives. The formulas can be written also for backward derivatives – then we have to use differences f ( x) − f ( x − ε ) , or as middle derivative using a symmetrical form f ( x + ε 2) − f ( x − ε 2) etc. Engineering faculties devote quite a considerable part of mathematics curriculum to derivatives and therefore, here we only refer to the appropriate textbooks, books or guides and popular monographs [57], [58]. For the needs of the following text, it is useful to remind especially the term continuity of functions and derivatives. Function f is continuous in point B of domain 1D, 2D or 3D with coordinates x , ( x, y ) or ( x, y, z ) if: aa) it is in point B defined by certain value f 0 , bb) values f in the vicinity of point B approach this single common value f 0 , if the points of the vicinity approach point B from any side or direction. A similar definition applies to an arbitrary derivative f . If we use an illustrative concept of derivative f ( x) as a tangent (tan) of the angle between (i) the tangent to the graph of f and (ii) the x -axis, then the conception of a common tangent on the left- or right-hand side (alternatively limit middle secant) leads to the popular definition of continuity of the first derivative: the graph of the function is not allowed to have a peak in point B. This applies also to two variables if we consider sections y = const or x = const across the graph f ( x, y ) . If we admit the conception of n -dimensioned graphs embedded into (n + 1) dimensional space, it is possible to handle the term peak even in these graphs that cannot be displayed by means of any technical measures in a real 3D model of the space (Euclid, Newton). Similarly for the second derivative, it is possible under certain assumptions approach to the conception 86

4.1 Geometric properties of elements of graph curvature (1 variable) or three components of the curvature of a planar graph (2 variables) that must have – in the case of satisfied continuity – the same limit from all sides, etc.

4.1.1.1 How many differential elements are there? This provocative question can be perceived in different ways. If it is to mean how many different shapes of differential elements can be introduced in continuum, then the answer is: the same number as the number of all possible coordinate systems. In 1D continuum it is just one possible element ds , which is generally curved, but straight in a straight continuum ( dx ). In 2D and 3D continuum we can theoretically have an unlimited number of coordinate systems, even though only some are used in practice (right-angled, semi-polar, curved, etc.). Consequently, we get an unlimited number of shapes of differential elements. However, the question can be understood in different way: We have a given 1D, 2D, or 3D body, we define in it just one fixed system of coordinates and we divide the body into differential elements – all of the same shape. How many of these elements are there? This formulation of the question has a closer relation to the following text dealing with FEM and it is useful to think about it a bit more. Every FEM user, including students, would give the correct answer that their number is unlimited, as their dimension approaches, in limit, zero, or as students sometimes put it: they are infinitely small and therefore there are infinitely many of them. The basis for this concept is that the whole body Ω is composed of these dΩ = Ωe : Ω = ∑ Ωe

(4.1.2)

e

If all the elements are identical and their number is n , then: Ω = nΩ e

(4.1.3)

In limit: Ω = lim nΩe = lim" ∞ "*"0" n →∞ Ωe → 0

(4.1.4)

The popular idea of n as a sequence of natural number n = 1, 2,3,K could lead to a misconception that there are countable infinity of these elements, which is briefly marked ∞0 . The subscript 0 indicates the lowest possible cardinality of the set of the infinite number of elements, which is termed aleph-null. If we admit, within the same conception, that differential element Ωe has a zero measure, then the meaning of expression (4.1.4) is just a sum of countable (even though infinite) number of nulls (zeros), i.e. Ω = 0 ! Apparently, something is wrong. And painstaking students who had good teachers of mathematics report the same problem. A continuum of an arbitrary dimension (1D, 2D, etc.) and size (from the smallest possible domains to meta-galactic dimensions) is defined by unlimited divisibility. Each of its parts has again properties of a continuum. More illustrative example of 1D continuum: the interval between arbitrarily close points x , x + h is again a continuum, it thus have the same 87

4.1 Geometric properties of elements number of points as the continuum between −∞ and +∞ . The same must be understood in the meaning of the possibility to assign the elements of one set to the elements of a second set. In the case of infinite sets we cannot speak about a concrete number since the number is unlimited, but about cardinality which is identical for the two sets, if such assignment can be made and no elements remain unassigned in either set. What is not correct in the result Ω = 0 is apparently the wrong conception of a continuum that sticks to the common sense of a practical engineer who does not accept some strange higher infinity. It is sufficient to get over this useless backwardness and simply admit that the number of points in an arbitrary continuum is greater that countable infinity, i.e. that they cannot be assigned to natural numbers 1,2,3,... If we had an unlimited amount of time and started to assign natural numbers 1,2,3,... to the points of any howsoever small (e.g. ( x, x + h) ) or large (e.g. (0, ∞) ) continuum, which would represent a never-ending process, so that some infinitely long-lived creatures would have to carry on with it after the extinction of the universe, they would see with some astonishment that the continuum has not decreased at all, that it still has the same cardinality. After billions of years of assigning, e.g. with the speed of 1 point / second, still almost no point of the continuum would be marked, which is the mathematical formulation of the continual measure of a countable, even though infinite, set of points. We can imagine a differential element around each point of a continuum, so that in limit dx → 0 , dy → 0 , etc., we get a set of differential elements which is not only infinite, but which has much larger cardinality than countable infinity. Thus the number of differential elements can be marked e.g. ∞1 , with the subscript 1 indicating a greater cardinality aleph one. The extremely interesting mathematical considerations, disputes and proofs about whether there is something between ∞0 and ∞1 , the alternative up-to-date definition of the measure, etc., are not interesting for FEM users. On the other hand, it is very useful for the needs of modelling of inputs and interpretation of outputs of FEM programs to consider the principal capabilities of FEM in general, with no regard to advertising slogans printed on the glossy paper of world-renowned companies’ colourful marketing materials. These capabilities are ultimately limited by a simple factor: The number of finite elements in FEM is always finite, as it follows from the name of the method and the nature of the matter: division of body (4.1.2) to elements of the same dimension (FEM) and similarly for the boundary of the body (BEM) or partially discretised problems (FLM and strip method). The number of parameters d , i.e. geometric quantities in the deformation variant, or statical quantities in the force variant, or both in the mixed variant, bound to the nodes of the mesh is finite as well. Therefore, always a double degradation of the problem takes place: cc) Reduction of the cardinality of the continua aleph one to a countable set aleph-null, reduction of ∞1 to ∞0 . dd) Reduction of countable infinity ∞0 to finite number of N free (independent) parameters d . Formally it is manifested by: ee) Replacement of differential equations by the variational principles in which we handle countable sets of free fields (art. 3.3.2.).

88

4.1 Geometric properties of elements ff) Reduction of the system of ∞0 equations to systems of N equations, which can be processed by the current computers.

4.1.2 Advantages and disadvantages of finite elements The advantages prevail for both rational users of FEM, mathematicians and programmers. The biggest advantage can be explained through the oldest and nowadays most frequently used deformation variant of FEM in statical problems, art. 2.3. The selection of the finite element is only under numbers 1 to 5 of the overview. It covers the shape of the element, its base functions and deformation parameters. The whole procedure under numbers 6 to 19 is standard and applicable to all elements. It just has to be programmed with general number of base functions and deformation parameters as what is termed main program. It is extended by a continually updated library of finite elements, i.e. subprograms or routines according to points 1 to 5. This, to a certain extent, turns the situation against classical methods, in which it was necessary to deal with each shape of body Ω with boundary Γ separately and in which the differential element remained the same. The following principle applies to FEM: The algorithm of FEM solution is independent on the shape of body Ω and its boundary Γ , including possible openings, notches, unusual loads and supports etc. The essential part of the FEM algorithm is just the main program that realises the addition theorem following from the additivity of energetic or other (orthogonalising) functional, no. 13 and 17 of the overview in art. 2.3., together with a library of subprograms for various finite elements. It is difficult to explain the advantages of this concept today, when there are only a few engineers who had to calculate structures by hand using the classical methods and thus there is nothing to compare with. Hardly anybody would believe that only 60 years ago (1945) a unique theory was used for each shape of a shell and that all of them failed if one or more air-conditioning openings had to be designed. And it sounds like a fairy tale, even though it is a proven fact, that in 1955 an engineer was given 14-day vacation to analyse a frame with 16 unknowns and that after the return to work with the solved system of 16 unknowns for 6 right-hand sides he was celebrated as a super-hero of statics. These historical notes were included here for the reason that it is the most popular way to explain the connection of FEM and state-of-the-art computers. The state-of-the-art computer including the present-day top PCs have similar approach to the double degradation of the problem as FEM (art. 4.1.1.): a) They operate only with a countable set of numbers. b) They can take into account only a finite number of numbers from a countable set, e.g. in double precision of N numbers 10-17, 2.10-7, 3.10-17, ..., 1017 - 2.10-17, 1017 - 10-17, 1017, in arbitrary precision N < ∞ . Analytical operations (derivative, integration) are carried out by computers usually numerically as simple algebraic operations. Best suited for it are polynomial functions, analytically the simplest expressions of the type of rational functions (termed correction functions), etc. This leads us to another advantage of finite elements: 89

4.1 Geometric properties of elements Finite elements are an ideal means to exploit the capacity of the present-day computers that are not capable of working with a continuum, but that can handle exceptionally well (speed, accuracy) a finite set of numbers. Now we get the weak points of FEM that, if correctly accepted and known, cannot overweight the advantages in common engineering practice. They represent in fact the dark side of the advantages: c) FEM can express neither any singularities, which are typical for mechanics of a continuum, nor various limit terms of the exact analysis. FEM does not work with any differential formulas, equations or terms. It is based on variational principles of mechanics or on other formal principles of a general nature (orthogonalisation with weight functions = weighted residual principles, art. 1.3 and 3.3.3) and it obtains what is called weak solution through their approximate analysis using a finite number of base functions. This solution features lower (by a factor of ten or more) demands on continuity of sought functions in the analysed domain (structure). For this reason, FEM cannot give a true picture of higher derivatives of sought functions along internal boundaries Γ d between elements, as they are not defined there at all. If the users manage to live with the fact that it is not possible to extract from FEM and if they learn the correct technical interpretation of the outputs (taking into account the discussed fact), they will experience no disadvantage. However, if they try to obtain something that does not exist, without paying the attention to the core of the matter, then even the friendliest ever hot-line support cannot help them. All explanation given over the phone sound to them more like excuses of foxy authors and distributors of the software who are stealing the precious time of the engineers just when they need to (i) perform the design of a plate above a line support for the known bearing strength, (ii) design the reinforcement of a flat slab above a column modelled by means of a single simple fixed support, or when they need to (iii) design a similar plate subjected to a concentrated load for which, with gradually refined mesh, the output value approaches the absolutely useless Pucher singularity w = r 2 ln r 8π with an infinitely large moment, etc. In order to provide accurate information, we have to add that, today, there exist dozens of what is termed singular finite elements that can be inserted into required parts of the model in similar situations (acute-angle corners of openings or boundaries, concentrated force actions, etc.) and that allow for exact modelling of such singularities. On one hand, they are a rather inorganic element in common programs, but, on the other hand, they can be included into the addition theorem in a normal way. After a period of massive use of these improvements (around 1980), their application became rare, mainly due to market-related reasons. The users called and asked how to size the structure for these infinities. They learnt that the infinite values represent local point infinities with no meaning for the strength, as the decisive factors are what is termed integral strength factors, which, for the simplest configuration, are integrals from the distribution of quantities along a certain curve, the values that numerous programs printed in their output. What followed was a disillusion from the improved accuracy and the return to good old elements that themselves smoothed such singularities, with best results in the case that loads and supporting conditions were modelled in such a way that corresponded to technical practice where, after all, no point singularities cannot be created (and why, if 90

4.1 Geometric properties of elements they are so inconvenient). We should realise that FEM is not an academic issue but a useful tool for practice. The development of FEM rarely followed exact scientific demands, even though whole teams of mathematicians-analysts, who did not know what the engineers would model by those elements, were engaged in this process. This was a good thing, since otherwise we would have today seemingly purposeless hierarchies of elements, convergence criteria, error estimates and many other useful, but only formal pieces of knowledge without any technical content. d) The advantage stated above in (a) (i.e. the finite number of parameters and application of computers that can operate just with the finite number of digits) is linked to a wellknown disadvantage that is briefly called numerical instability. The simplest explanation is rather dramatic. With regard to the measure of the continuum of numbers, almost no number is stored in any numerical computer. Mathematics uses this diction to express the fact that the continual measure of displayed numbers equals to zero. It is a kind of a sieve where almost all numbers pass through. Double precision only mitigates this problem to some extent but it does not resolve it. It could be solved only through the introduction of interval arithmetic: either classical, i.e. number x is defined as interval x1 < x < x2 , or spherical, i.e. number x includes dispersion r , which for 1D problems equals interval ( x − r 2) < x < ( x + r 2) . In 2D and 3D we think about points in circular or, as the case may be, surroundings of point x , or generally in space nD about the appropriate super-sphere – hence the term spherical arithmetic. The problem is pursued by a special commission named GAMM (Gesellschaft für angewandte Mathematik und Mechanik), led by Professor K. Nickel, who regularly reports at annual GAMM conferences. In 1996 this conference took place for the second time (since the establishment in 1920) in Prague. The first time it was in 1968. We can extract from the published materials two facts that are interesting for FEM users: unfavourable fact that the calculation takes considerably longer time (e.g. 300% of time in common arithmetic) and favourable fact that all fuzzy tests and all numerical instabilities are not relevant any more.

4.1.3 How not to get lost in the collection of finite elements Let us begin with the same question as in art. 4.1.1 where we asked: how many finite elements are there? Once again, there are several conceptions available. If we assume that the question covers elements of different shapes and sizes, it has no practical meaning since it represents an infinite set of type ∞1 . Users, usually, raise the question that takes no account of particular lengths and angles, i.e. for example all triangles with three vertex nodes represent one element type, on condition that the sought quantities are distributed through the same functions. Thus the question “How many types of elements are there?” is meaningful. Within this book we limit to mechanical problems and we will formulate two question variants of different sophistication: e) How many types of elements are theoretically known today including the analysis of their mechanical properties and problems related to their application in different problems? There is a simple answer even to this question: once again, it is an infinite set that (not 91

4.1 Geometric properties of elements taking into account the particular lengths and angles) is of type ∞0 , which means a countable set. For various particular types of elements we today know countable infinite hierarchies (sequences, families) of elements the elements of which have gradually increasing numbers of parameters, i.e. degrees of freedom for their behaviour, which makes it possible for them to satisfy the ever growing demands on the smoothness of the results along the boundaries between elements. The term smoothness means the continuity in function values and in derivatives up to a certain degree d inclusive, which, for 1D, 2D and 3D problems, means formulas: d (α ) f ( x ) dx (α )

α = 0, 1, 2, K , d

(4.1.5)

∂ (α + β ) f ( x , y ) ∂xα ∂y β

α , β = 0, 1, 2, K , max (α + β ) = d

(4.1.6)

∂ (α + β + γ ) f ( x, y , z ) ∂xα ∂y β ∂z γ

α , β , γ = 0, 1, 2, K , max (α + β + γ ) = d

(4.1.7)

The function itself, in this notation, is considered the zero-th derivative, i.e. for example. f ( x) =

d (0) f ( x) dx (0)

f ( x, y ) =

d (0) f ( x, y ) ∂x 0 ∂y 0

(4.1.8)

Technical problems always require that certain minimal demands on smoothness are met. For example, we cannot allow any crack, hollows or cuts on the boundaries between elements. It is necessary to require that components of displacement u , v, w are continuous everywhere. Such continuity in the function value is the lowest demand on smoothness. The class of functions that meets this requirement inside body (analysed structure) Ω is called class C0 . The letter C stands for continuity and subscript 0 denotes zero-th derivative, the function itself. There are problems in which we have stricter demands on the smoothness. A typical example is the Kirchhoff theory of 2D plates where we introduce just one component of displacement in the mid-plane w( x, y ) and the other two are derived from the assumption of keeping the normal perpendicular. u ( x, y, z ) = z

∂w( x, y ) ∂w( x, y ) v ( x, y , z ) = − z ∂x ∂y

(4.1.9)

The continuity of u, v results in the requirement that the first derivative of function w( x, y ) is continuous. Therefore, we have to work with the class of functions C1 in 2D area of the plate. Technically speaking: no breaks in deflection surface w( x, y ) are allowed, which would mean step-changes in the first derivatives that manifest themselves as peaks in the graphs. The demands can be made stricter, e.g. we could require – in addition to continuity (4.1.9), that bending moments (depend on the second derivative) or shear forces (depend on the third derivative) are continuous as well. Then we could accept only function classes C2 or C3 . Other examples will be presented later. For the time being, let us emphasize the characteristic of the hierarchy of elements that is most important from the users’ point of view: higher elements generate function classes Cd of higher smoothness d . 92

4.1 Geometric properties of elements f) How many types of elements are included in the present-day assortment of FEM programs? If we focus on the prevailing deformation FEM variant for standard statical 1D, 2D and 3D problems, then the answer is not too satisfactory for curious users – we have thousands of element types the brief overview of which would fill in several volumes. Larger program systems include more than 100 types of elements, often in numerous variants depending on various keys reflecting the demands of the users. It is difficult for common users to find the identity of elements offered by various companies if they study just the user’s guides. It is inevitable to read also theoretical manuals that are published by reputable companies. However, some details are not included there either. Normal elements are extended by (i) transition elements that make it possible to connect two elements featuring different valence (see later in the text) and (ii) contact elements, featuring measure equal to zero in the analysed domain, that give a true picture of only special configurations of connections between two elements of connections of elements to the surroundings. The number of all today offered and commercially successful elements for the deformation variant of FEM can be roughly estimated to be about 1,000. The overview given in the following articles can help the engineers not to get lost.

4.1.4 1D elements 4.1.4.1 Hermite 1D polynomials in FEM All the statements in the following overview will be stated without proofs, which can be found in [5]. The same reference contains also formulations that are stricter in terms of mathematics. Here, we will use the formulation that is closer to engineers who use FEM programs (without causing unacceptable inaccuracies). Each 1D element (fibre, truss, part of a frame or grid beam, column, pile, etc.) has a very simple geometry. It is a line segment IJ or an arc IJ with end-points I ( x = 0) and J ( x = L) , for which we can introduce coordinates x with the origin in point I, so that an arbitrary point B has coordinate xB equal to the length of line segment or arc IB (fig. 4.1). 1D element forms a 1D continuum of points in the interval [0, L] . In practice, we need to know various functions in this interval, e.g. components of displacement or rotation of beam sections, internal forces N , Qy , Qz , M x , M y , M z , components of external force or moment loads, generally variable geometrical characteristics, etc.

93


Figure 4.1: a) Straight 1D element b) Curved 1D element c) Hermite interpolation polynomial, general configuration

In FEM we have to accept the fact that the exact results can only rarely be obtained for these functions (only for what is termed elementary situations). Practically, the sought function will be approximated by a polynomial with variable x , which is in fact a linear combination of (n + 1) monomials in x : x 0 = 1, x1 = x, x 2 ,

x3 ,

...,

xn

(4.1.10)

c3 ,

...,

cn

(4.1.11)

with coefficients c0 ,

c1 ,

c2 ,

i.e. common type of formula 94

4.1 Geometric properties of elements p( x) = c0 + c1 x + c2 x 2 + ... + cn x n

(4.1.12)

briefly n

p ( x) = ∑ ck x k

(4.1.13)

k =0

There are many ways to create such a polynomial if it is supposed to satisfy given conditions (Lagrange, Stirling, Bessel polynomials, etc., see [57, 58]. The most suitable for FEM proved to be Hermite polynomials, which follows from this favourable characteristic (fig. 4.1c): In every point 0, 1, 2, K , with coordinates x0 < x1 < x2 < ... < xs it is possible to define α 0 , α1 , α 2 ,..., α s conditions for p( x) , with the sum of these conditions equal to the sum of coefficients (4.1.11), i.e.: α 0 + α1 + α 2 + ... + α s = n + 1

(4.1.14)

In one point it is possible to define one (with the corresponding α = 1 ) or more (α > 1) conditions for p ( x) and derivatives p( x) with respect to x that are denoted by the order of the derivative in the brackets: d (a) p p ( x) = ( a ) dx

a = 1, 2, K

(a)

(4.1.15)

or by a number of primes at lower derivatives, e.g. p (1) ( x) = p (I) ( x),

p (2) ( x) = p (I I) ( x),

...

(4.1.16)

The conditions must be systematic, i.e. they must relate to value p (always) and all subsequent derivatives p I , p I I , K (for greater number of α ). No derivatives can be skipped, including the 0th derivative and the function itself. A set of such conditions is called Hermitean set. If we mark the given numbers by letter y , the Hermitean conditions are always in the form: p( x0 ) = y0 (0)

p′( x0 ) = y0 (1)

p( x1 ) = y1(0)

p′( x1 ) = y1(1)

M M

p( xs ) = ys (0)

K K

p (α 0 −1) ( x0 ) = y0 (α0 −1) p (α1 −1) ( x1 ) = y1(α1 −1) (4.1.17)

p′( xs ) = ys (1)

K

p (α s −1) ( xs ) = ys (α s −1)

These conditions are satisfied by the Hermitean interpolation polynomial, the general form of which can be explicitly written in the form of a rather unintelligible sum (see [5]). Its meaning is in the applicability to arbitrary conditions in the form (4.1.17). The formula is significantly simplified in the following practical applications: 28. All numbers α in (4.1.14) are equal to one, i.e. only the function values and no derivatives are prescribed in all the points. Apparently, n + 1 values must be prescribed for a polynomial of n- th degree. This is the well-known (both from school and from practice) (oldest) Lagrangean interpolation polynomial. Unfortunately, it 95

4.1 Geometric properties of elements is not suitable for the purposes of FEM, as it operates with many intermediate points. It is clear from the principle of FEM (art. 1. - 2.) that it is essential to try to concentrate the determining values to the ends of the interval. This is best satisfied by the second extreme configuration. 29. We select only two end-points x0 (for 1D element x0 = 0 ) and xs ( xs = L) as defining. We prescribe the same number of conditions (4.1.17) in each point, i.e. α 0 = α s = (n + 1) 2 . As the number of conditions must be integer, we have to limit ourselves to odd degrees n . Consequently, for n = 1 we obtain a linear Hermitean polynomial defined only by functional values at ends y0 , ys , which is a standard linear interpolation. For n = 3 we get a cubical Hermitean polynomial, defined at each end by the function value and its derivative. In FEM, this situation is most common for 1D elements. It is technically interpreted in various ways and is decomposed into what is termed unit parametrical states, see fig.4.2. For n = 5 we obtain a quintic polynomial degree 5. Three conditions for p, p′, p′′ , etc. are prescribed at each end. This special configuration can be characterised by table Chyba! Nenalezen zdroj odkazů.:

96


Figure 4.2: Decomposition of function w(x) of the deflection of a prismatic beam into base functions Vi(x). The picture shows the part between nodes 1 and 3, in which six parameters of deformation ∆ i fully defines the distribution of deflection w(x)

97


Figure 4.3: Most often used 1D-elements – beam type – for: a) planar truss girders, b) spatial truss girders, c) planar frames, d) planar grids, e) spatial frames. Shown are their parameters of deformation in global axes.

98


Figure 4.4: a) Example of a 1D-element with three nodes IJK compatible with a 2D-element of a Mindlin shell with 15 parameters of deformation. B) Example of the definition of the position ICJC of a physical 1D element in space by means of an auxiliary axial line segment IAJA and eccentricities eI,eJ.

99

4.1 Geometric properties of elements Special Hermite polynomials with two defining points at interval ends: Total number of Number of Polynomial conditions and conditions at Type of conditions degree polynomial one end members p 1 1 2 3

2

p, p ′

4

5

3

p, p′, p′′

6

n

(n + 1) 2

p, p′, p′′, K , p ( n −1) 2

n +1

(4.1.18)

The literature dealing with FEM almost always use the term Hermitean polynomial only for this special situation, which is a kind of an established slang term justified by the extreme endeavour to repel all the determining quantities to the end of the interval.

4.1.4.2 The most known 1D elements The most often used is a straight beam element IJ as in fig. 4.1a. FEM took this element from a classical matrix analysis of beam structures, in which the terms beam and element coincided. Fig. 4.3 presents five basic situations a), b), c), d), e) having 4, 6, 6, 6, 12 parameters of deformation d with the components u, v, w, ϕ x , ϕ y , ϕ z (no. 2 of the overview in art. 2.3). Components in element coordinates are used to work with the element – for 2D elements it is the centroidal axis x C and the principal centroidal axes of the cross-section y C , z C . The stiffness matrix and vector of load parameters are then defined in these coordinates – no. 11 and 14 of the overview in art. 2.3. To operate with the whole structure, all this is transformed into the global coordinates that are common to the whole structure – no. 12,13 and 16,17 in art. 2.3. Less frequent is the element with three nodes as in fig. 4.4a, which can have in general spatial configuration 2 × 6 = 18 parameters in nodes I, J, K (K is the centre of the element). Usually however, what is termed Kirchhoff’s principle is utilised, which is based on the expectation of a certain permanence of geometrical relations even for the Mindlin model of a beam (see art. 4.2), where the rotations of mass sections are independent of deflections. The article 4.2 indicates that it represents two independent components of rotation ϕ z (instead of dv dx ) and ϕ y (instead of − dw dx ). If we do not expect large distortion of the right angles between directions x, y and x, z and if we require that the Mindlin model with the increasing slenderness converges to the Kirchhoff one, the distribution of ϕ z and ϕ y should be expressed by means of a polynomial one degree lower than v and w. The simplest elements in fig. 4.4a assume that the distribution of u , v, w follows polynomials of degree 2 and ϕ x , ϕ y , ϕ z polynomials of degree 1. The centre K then introduces only parameters u , v, w , and, as a result, the whole element has 15 parameters of deformation. This element is often used as an element compatible with 2D elements (see e.g. 100

4.1 Geometric properties of elements fig. 4.6b, 4.7b and 4.8d in the following art. 4.1.5.1) with nodes in the middle of edges. Such elements, however, make the numerical processes more complicated and prolong the computation time. Therefore, modern FEM abandons these mid-nodes and, in general, all non-vertex nodes. It can be made on the level of one element through interpolation or preferably pre-elimination (termed static condensation). Even better is to eliminate such nodes from the element through the definition of other vertex parameters such as rotation ω , see art. 3.3.2, formulas (3.3.6), (3.3.7) and the corresponding text in the following art. 4.1.5. Unfortunately, this has been discovered only after nearly 20 years of fumbling influenced by the personality of B. M. Irons, who in 1980 in the famous title “The Finite Element Technique” still called such effort as vasting money. He was apparently strongly influenced by Lagrangian variational principle and who was annoyed by zero displacement modes – see art. 3.3.2. The introduction of ω –type rotational parameters in 2D and 3D elements allowed for a unified conception of these parameters for all 1D, 2D and 3D elements. These parameters are commonly used at the ends (“in the vertices of line segments”) of 1D elements since the oldest beam conceptions. Their physical meaning is rotation ϕ x , ϕ y , ϕ z of end cross-sections around central axes x, y, z . The geometry of a 1D-element can be defined directly by its ends I, J. The orientation of principal centroidal axes y C ⊥ z C is already a physical property, art. 4.2. For many structures it may be, however, suitable to define first an auxiliary grid of lines IAJA, which can be called axial configuration of the structure. The physical (real technical) elements are then attached with (in general) eccentricities eI , eJ (Fig. 4.4b) to the nodes of this configuration with respect to the actual design of joints. The eccentricities are position vectors IAIC and JAJC that can be defined through their global components in coordinates ( x, y, z )G common for the whole structure: exIG

eGyI

ezIG

exJG

eGyJ

ezJG

(4.1.19)

Or alternatively by the axial components in the auxiliary coordinates ( x, y, z ) A , introduced separately for each line segment JAJA according to the given fixed rules: exIA

e yIA

ezIA

exJA

eyJA

ezJA

(4.1.20)

1D elements defined in this way behave in FEM like polygonal tri-beams IA IC JC JA the endparts IAIC, JCJA of which are absolutely rigid (they do accumulate no deformation energy). Therefore, it represents a comfortable modelling of rigid joints. Elastic joints can be modelled by means of elastic joint beams eI , eJ , see chapter 5. Special situations when the elastic connection of the beam-end (or several beam-ends) must be taken into account can be handled with just a dimensionless contact element with a very simple stiffness matrix. On condition that the elastic connections do not depend on each other, all four main stiffness submatrices are diagonal. The elimination of the deformation parameters at the ends of the beam (similar to a general statical condensation of internal parameters of a substructure or superelement) can give a transformed stiffness matrix with respect to the elastic connections to the joints. They then enter the addition that is used to create the global stiffness matrix of the whole structure. The set of physical elements defined with eccentricities (4.1.19) or (4.1.20) from the axial configuration is called central configuration of the structure or also physical configuration, as it is formed by the axes of physical (real) beams. This terminology 101

4.1 Geometric properties of elements is useful for nonlinear tasks too.

4.1.4.3 Thin-walled beams of open cross section Beams, whose thickness of individual partial cross section part ti is small comparing to beam cross section dimensions (b, h) of we call thin-walled. Literature specifies rate app. 1/10. There are differentiated open and closed cross sections, according to fact whether centre line constitutes closed curve. Thereinafter we will deal only with open cross sections according to theory of Prof. Vlasov. There belong e.g. I, C, T, U sections. Theory is based on 2 assumptions: 30. Deformation of cross section outline in its plane does not exist. It follows that cross sections twist in their plane as rigid aggregates. 31. Elements of centre line surface originally rectangular, remain after deformation rectangular, as well, i.e. their angular deformation is nought. If the element is stressed only by torque, in the element could originate: gg) aaaa) only tangential stresses. This phenomenon originates at free torsion (SaintVenant), diamonding (deplanation) of cross section is not restrained, it could freely wage. hh) bbb) tangential stresses and normal stresses. Most frequent case – free torsion is restrained by boundary condition, loading, change of torque value along length of the beam, or change of cross section along length of the beam. This is a bounded torsion. We neglect it at massive cross sections, at thin-walled it play significant role on overall elasticity of the beam. Bounded torsion could arise also due to other types of loading, it need not be just a torque. It could be also shear force that does not pass through centre of shear, or axial force acting apart from centre of gravity of the cross section. Vector of deformation: u = u, v, w, ϕ x , ϕ y , ϕ z ,υ 

T

Differential equations of equivalence: dϕ x dx d 2ϕ x B = − EI w dx 2 d 3ϕ x M w = − EI w dx3 M sv = GIT

102

4.1 Geometric properties of elements where M sv is a free torsion (St.Venant), M w is a bounded torsion (Vlasov, warping), B is bending torsion bi-moment. Total torque is: M x = M sv + M w . After substitution from antecedent equations we receive differential equations of torsion: M x = GIT ϕ ′ − EI wϕ ′′′ For solid cross-sections we assume that the shear centre coincides with the centroid of the cross-section. This means that: - the calculation of static quantities is simplified, - in the evaluation of internal forces it is not necessary to distinguish to which point the forces are related. For thin-walled cross-sections, the shear centre is usually not identical with the centroid. The ideal situation when both points coincide can happen only for cross-sections that are symmetrical around both principal axes (I-section, X-section). For cross-sections that are symmetrical around one axis, both the centroid and shear centre are located on the axis of symmetry, but at a certain distance from each other. The calculation requires that the sectional characteristics be specified in the following way: sectional area Ax , moments of inertia I y , I z are related to the centroid, shear areas Ay , Az , torsional moment of inertia I x and sectorial moment of inertia Iω are related to the shear centre. A joint in a real structure may connect two beams with a different position of the shear centre (e.g. two different thin-walled cross-sections or a solid cross-section with a thin-walled crosssection). The assembly of stiffness matrices of individual beams can be performed in a usual way, but the different positions of the shear centres must be matched together – they must be related to a specific point of the cross-section. This point may be: the centroid of the cross-section (this point is unambiguous and is common to all beams connected to the joint), the shear centre of one specific beam with all the remaining beams connected to that beam in the joint being transformed into the LCS of that specific beam. The most suitable is the first option when we transform nodal parameters related to the shear centre, which is based on:

103

4.1 Geometric properties of elements addition of the torsional moment due to the eccentricity of shear forces and addition of bimoment B due to bending moments.

INITIAL STRESS Let us assume that a beam is initially subjected to forces ( N 0 , M y 0 , M z 0 , B0 ) . Using the formula in [23] we have: σ0 =

P0 M y 0 M z 0 B0 + − − A Iz Iy Iω

The geometric matrix (the effect of the stress-state on the stiffness) can be written as a linear function of the initial stress-state: K g = N 0 K 1 + M y 0 K 2 − M z 0 K 3 − B0 K 4 The final matrix K g is symmetrical and it dimension is (14 × 14) .

Thin-walled finite element Number of nodes: 2 Shape functions: u = a1 + a2 x v = b1 + b2 x + b3 x 2 + b4 x3 w = c1 + c2 x + c3 x 2 + c4 x3 ϕ = d1 + d 2 x + d 3 x 2 + d 4 x3 For thick cross-section is used only linear term: ϕ = d1 + d 2 x Example no. 1 Critical force of cantilever of unequal angle. Length L = 2 [m] , E = 210 [GPa] , G = 84 [GPa] . Cross section rectangular, free of slopes and radiuses, thickness of arms 10 [mm], length centreline arms 250 [mm] and 150 [mm]. Numeric solution and solution according to Vlasov approaches with increasing length of beam the Euler critical force.

104

4.1 Geometric properties of elements analytical Euler

analytical Vlasov

FEM 1D model

FEM 2D model

2250

650

632

658

Pcrit [kN]

Results.

4.1.5 2D elements

4.1.5.1 Triangular elements The geometrically simplest 2D element is a triangle. It is what is termed a simplex in 2D space – which is similar to a line segment in 1D, tetrahedron in 3D, super- pentahedron in 4D, etc. The theory of simplex is well-elaborated (simplex – simple, the simplest possible shape). With regard to FEM the interested readers are referred to texts in [5], where proofs can be found together with further references. As a brief introduction we will state some useful terms and formulas relating to common 2D triangular elements. In particular, it is clear that a continuous distribution of an arbitrary quantity over a triangle can be described by means of a polynomial of degree n with two variables: p ( x, y ) = a1 + a2 x + a3 y + a4 x 2 + a5 xy + a6 y 2 + + a7 x 3 + a8 x 2 y + ... + aN −1 xy n −1 + aN y n

(4.1.21)

where N = ( n + 1)( n + 2 ) 2

(4.1.22)

If we want to write the complete polynomial of degree n in a more compact form, let us rewrite coefficients a1 , K , an in (4.1.21) in such a way that the coefficient at product xi y i , where i + j = k , is written aij( k ) . Then, instead of (4.1.21) we can write n

p ( x, y ) = ∑

∑a

k =0 i + j = k

ij

(k ) i

x yj

(4.1.23)

where the symbol

∑

i + j =k

means that the sum is over all ordered pairs (i, j ) of non-negative integers the sum of which equals to k . It can be derived from (4.1.23) that the number of terms of a complete polynomial of degree n with two variables is given by formula (4.1.22). We try to use in FEM applications interpolation polynomials with the highest possible number of conditions determining the polynomial concentrated in the vertices of the triangle. The reason is simple: a condition prescribed for an edge is common at most to two triangles, but a condition prescribed in a vertex is common to all triangles that share this particular 105

4.1 Geometric properties of elements vertex. This is the reason why the total number of conditions in the same triangular mesh is lower if more conditions are concentrated to vertices. Let us explain it by the following example: A polynomial of third degree can be defined on a triangle unambiguously in two ways: 32. By function-values in vertices, centroid and points that divide sides into thirds (in total 10 values). 33. By function-values in vertices, centroid and by first derivatives in vertices (in total 10 values). If we divide a square by two diagonals to four triangles, then the first variant (1) gives the total number of prescribed values equal to 25, while the second option (2) results only in 19 values. The more triangles are in the mesh the greater the differences in the total number of values are. The problems with triangular elements in FEM are of analytical and topological nature. The topological problem is that the triangulation of domain Ω (wall, plate, section across a body of revolution, section across a prismatic volume) generates a vast number of elements and even their numbering does not help to improve orientation. Modern FEM program get over this weak point through the implementation of quadrilateral elements (art. 4.3) composed of four triangles with some inconvenient deformation parameters being eliminated or condensed (art 4.3.2-3). However, this does not settle the analytical problem which was – after years of groping – eventually solved by Czech mathematicians A. Zenisek and M. Zlamal [3, 5], which is still cited in the world literature. In essence, it is a seemingly simple problem: A function f ( x, y ) in 2D-domain Ω – e.g. plate deflection w( x, y ) , stress component in a planar problem σ x ( x, y ) , bending moment in a plate M x ( x, y ) , etc. – is to be substituted by a function that is prescribed separately for each element into which the domain Ω is divided by (fixedly selected) triangulation. This prescription should be a complete polynomial of degree n (4.1.21). Naturally, its coefficient may be different for every element. What is identical is just the degree n – the substitution should be so smooth to fit into class Cd with d being the required level of continuity (see art. 4.1.3, formulas (4.1.5) – (4.1.8) and the related text). For C0 – i.e. the lowest level of continuity solely in functional values (with allowed breaks in boundaries between elements) – there is in fact nothing to solve, as the requirement is met already by the age-old Courant plated surface from 1943. A plane a + bx + cy over every triangle defined by three ordinates f in its vertices is sufficient. Linear polynomials were in FEM actually the first applied base functions (fig. 4.5) and many FEM programs still have them in their element libraries for the analysis of planar problems. The question arises concerning the level of continuity that can be guaranteed by polynomials of degree 2, 3, etc., at the boundary between elements. The answer can be general for an arbitrary degree n and it will show that only the degrees 5, 9, 13, etc., can guarantee C1 , C2 , C3 , etc., continuity, i.e. up to the first, second, third, etc., derivative inclusive. Therefore, it is convenient to group the polynomial into foursomes: n = 4m + χ

m = 0,1, 2,...

χ = 1, 2,3, 4

(4.1.24)

According to (4.1.22) every polynomial has N = ( n + 1)( n + 2 ) 2 terms and, therefore, the 106

4.1 Geometric properties of elements same number N of parameters is required for its determination. They cannot be chosen arbitrarily if we want to satisfy the main principle of FEM: to concentrate the highest possible number of parameters into the vertex nodes of an element. We mark the points as in fig. 4.6. In order to obtain the maximum brevity and full generality, we write functions p ( x, y ) in point P or Q with coordinates ( x, y ) using the symbol p(P) or p (Q) and we indicate the point by an index. In mechanical FEM problems the function p is usually one of the components of a displacement or small rotation vector. The approximations of vector field over elements are created this way. Function p can be of course also e.g. a stress component or deformation component, torque function (Prandtl), Airy function, capillary pressure of liquids in twophase environments, etc. Partial derivatives of this function will be symbolically marked as follows: α = (α1 , α 2 )

α = α1 + α 2

Dα p =

α ∂ p ∂xα1 ∂yα 2

(4.1.25)

107


Figure 4.5: The simplest triangular FEM element for planar problem or deformation in walls and prismatic volumes. Displacement components u(x,y) are distributed linearly over the element. a) Six deformation parameters d1, d2, …, d6 has the meaning of displacement components ui, vi, …, vk in element vertices. b) Six parameters of external forces (load). c) Unit element e1 d) Distribution of six base functions V in one element is linear too.

108


Figure 4.6: Depiction of points in the triangle in which the parameters of deformation are defined. Vertices P1, P2, P3, centroid P0, points on edges Q with determining index: lower (subscript) starting with 1, 2, 3 in the middle of edges 12, 23, 31. If there are more points on one edge, the notation is the same and additional indices 1,2; 3,4; 5,6 are attached to the newly added points. Upper (superscript) in brackets (k) mean that the points divide the edge into k+1 intervals. Fig. shows only a few first configurations of FEM elements with regard to the following fig. 4.7. Deformation parameters are most often the displacement components, sometimes their derivatives and in vertices also the components of small rotation.

109


Figure 4.7: The first nine terms in the hierarchy of triangular FEM elements with the polynomial of degree n=1 to 9 having (n+1)(n+2)/2 coefficients. This corresponds to the same number of parameters that unequivocally define these polynomials, i.e. 3, 6, 10, … 55 parameters for a), b), c), etc., respectively. The small full circle in the figure indicates points where the parameter is the value of function p, the little arrow means the derivative dp/dn in the direction of the arrow n, the double arrow represents the second derivative d2p/dn2, the circle with number no (no.=1, 2, 3 or 4 in the figure) means value p and all partial derivatives up to the order no.

110


Figure 4.8: a) Rectangular or oblique coordinates in a rectangle or rhomboid. b) Continuity between rectangular or planar coordinates L1, L2, L3 in a triangle. c), d) e) Decomposition of function p (L1, L2, L3) into base functions of degree 1, 2 and 3, the parameters are just function values p in 3, 6 and 10 nodes according to fig. f) Assumed distribution of deformation of one side of the element following what is termed beam deflection in beams with rotation parameters. g) Element deformed only by three rotation parameters.

111

4.1 Geometric properties of elements For example the statement that values Dα p (P) are prescribed in point P, with α ≤ 5 , means that 21 values are prescribed in point P: function values and all partial derivatives up do degree 5 inclusive, i.e. two first derivatives, three second, …, six fifth derivatives. This general statement applies: Let P1, P2, P3 be vertices of triangle T, P0 its centroid and O1k , O k2 , K , O3k k points dividing sides of triangle T into k + 1 identical intervals (see fig. 4.6). Let m ≥ 0 and κ (1 ≤ κ ≤ 4) be integers. Then, there exist just one polynomial (4.1.21) of degree n = 4m + κ that reaches the given values Dα p(Pj )

( j = 1, 2,3)

D β p (P0 )

∂1 p(O(sk ) ) ∂v1

( s = 1, 2,K ,3k )

(4.1.26)

where ∂p ∂v denotes the derivative with respect to the normal and superscripts α , β , k , 1 are defined as follows: α ≤ 2m

β ≤ m−2

1 = k = 1,..., m

α ≤ 2m

β ≤ m −1

1 = k − 1, k = 1,K , m + 1

α ≤ 2m + 1

β ≤m

1 = k = 1,..., m

α ≤ 2m + 1

β ≤ m +1

1 = k − 1; k = 1,K , m + 1

(4.1.27)

This statement can be used to create an infinite number (n = 1, 2,K) of elements whose geometrical and continuity properties are known and can be directly used for the solution of FEM problems with specified technical requirements. A classical example is the element for bending of the Kirchhoff plate. Convergence to a weak solution (art. 3.3.) is guaranteed already by continuity C1 (first derivative), i.e. elimination of breaks between elements, which is satisfied by a polynomial of fifth degree. If we want also the C2 continuity (second derivatives, bending moments) a polynomial of degree 9 is required. For higher demands, e.g. C3 (third derivative, shear forces), we have to work with a polynomial of degree 13. Further, we will see that a Mindlin plate has lower demands on the degree of the polynomial at the cost of increase in the number of degrees of a point of a 2D model from 1 ( w ) to 3 ( w, ϕ x , ϕ y ). For better understanding, the first nine terms of a general hierarchy (4.1.26), (4.1.27) are drawn in fig. 4.7. In more up-to-date modifications of FEM we try to achieve even higher concentration of parameters into the vertices than permitted by the polynomial analysis. This can be achieved by eliminating or condensating non-vertex parameters (art. 4.3). Consequently, the element in fig 4.7e, which originally had 3 × 6 + 3 = 21 parameters, becomes an element with 3 × 6 = 18 vertex parameters of type w, wx , wy , wxx , wyy , wxy (in plates). Also interesting is the evolution of opinions on the useful polynomial degree n . Historically first was n = 1 (1943 R. Courant, 1956 Clough-Ohio), n = 2 (Veubecke, 1965) in planar problem up to n = 5 (1968) in plates, among others M. Zlamal. A complete hierarchy was created by A. Zenisek (1971). The application of higher polynomials collided with the capacity of computers. They became used only after 1985 thanks to what is termed 112

4.1 Geometric properties of elements p –refinement of results as an alternative to h –refinement, or the combination termed h − p or p − h process. The state-of-the-art FEM programs allow for both an automatic increase of the polynomial degree ( p –process) and refinement of the mesh, i.e. decrease of the h-norm of their size ( h –process) according to comfortably input demands of the user. It was contributed, among others, by excellent research performed by I. Babuska (living in the U.S. since 1962) concerning the adaptive form of FEM carried out between 1980-1995. The opinion on the required degree of the polynomial was also changed by the fact that the “Irons taboo” was overcome after 1980, i.e. components of small rotation, and possibly even the rotation fields independent on the displacement fields, were permitted as deformation parameters. See art. 3.3.2. and notes relating to this issue in art. 4.1.4.2.

4.1.5.2 Triangular elements with polynomials in L1, L2, L3 Let us notice the main difference between the rectangular (or oblique) coordinates x, y on a rectangle (or rhomboid) 0 ≤ x ≤ a , 0 ≤ y ≤ b and on a triangle. For a rectangle (or rhomboid) we can introduce dimensionless coordinates according to fig. 4.8a: ξ1 =

x1 A1x = a A

η1 =

y1 A1 y = b A

(4.1.28)

where A = ab is the area of the whole rectangle (or rhomboid) and A1x and possibly A1y , are parts of this area in the interval 0 ≤ ξ ≤ ξ1 or, as the case may be, 0 ≤ η ≤ η1 . Coordinates (4.1.28) are natural in a certain sense, apparently the best and most illustrative of all possible ones. Such a level of naturalness and illustrativeness cannot be achieved through any rectangular or oblique coordinates for a triangle (fig. 4.5). In FEM it was quite early discovered (already about 1960) that just the ratio of areas in (4.1.28) is typical and what is termed natural coordinates started to be used, called planar coordinates by numerous authors, which are applied in many present-day FEM programs. The definition is quite simple (fig. 4.8b). Each point P has three dimensionless coordinates L1 , L2 , L3 . These are ratios of triangle areas (P23), (P31), (P12) to the area of the whole element (123), symbolically: L1 =

( P 23) (123)

L2 =

( P31) (123)

L3 =

( P12) (123)

(4.1.29)

Even this definition makes it clear that L1 + L2 + L3 = 1

(4.1.30)

as the three partial triangles always exactly cover the whole element. In addition, a very simple expression, or equation of sides of the element, is clear. Each point on side 12, 23 or 31 of the element results in a zero triangle (P12), (P23) or (P31), therefore: L3 = 0 (side 12), L2 = 0 (side 31), L1 = 0 (side 23)

(4.1.31)

Moreover, vertex coordinates are obvious as in them always one partial triangle occupies the whole element. It is that triangle which is formed by a vertex with the opposite side. Thus, for a general marking of the point

113

4.1 Geometric properties of elements P( L1 , L2 , L3 ) = P( x, y )

(4.1.32)

the vertices P1, P2, P3 are: P1 (1, 0, 0) = P1 ( x1 , y1 ) P2 (0,1, 0) = P2 ( x2 , y2 )

(4.1.33)

P3 (0, 0,1) = P3 ( x3 , y3 ) There is a simple linear relation between planar and rectangular coordinates. If the planar coordinates L1 , L2 , L3 of point P are known, its rectangular coordinates x, y can be calculated from the coordinates of its vertices in the same (otherwise arbitrary) coordinates ( x1 , y1 ) , ( x2 , y2 ) , ( x3 , y3 ) : x = L1 x1 + L2 x2 + L3 x3 y = L1 y1 + L2 y2 + L3 y3 1 = L1 + L2 + L3

(4.1.34) (verification formula)

114

4.1 Geometric properties of elements The form of this formula is useful also for other FEM elements, see further in the text (quadrilateral elements). Planar coordinates L1 , L2 , L3 can be viewed as influence lines of point P in terms of coordinates of the vertices of the element. For example, L1 shows the influence of coordinate x1 and y1 of vertex P1 on coordinates x and y of point P. Let us go back to e.g. the unit element in fig. 4.5c) with vertex coordinates (0,0), (1,0), (0,1). Formulas (4.1.33) have the form following also directly from the ratio between triangle heights: x = L2 ⋅1,

y = L3 ⋅1

The multiplication by one (1m) is not omitted in order to keep the dimension of lengths x, y . The inversion of (4.1.34) can give us formulas to calculate planar coordinates L1 , L2 , L3 from known coordinates x, y of point P. The most lucid notation is: L1 = ( a1 + b1 x + c1 y ) 2 A

L2 = ( a2 + b2 x + c2 y ) 2 A

(4.1.35)

L3 = ( a3 + b3 x + c3 y ) 2 A

where 2 A means the double of the area of element 123. It can be found fast from the determinant: 1

x1

y1

2A = 1 1

x2 x3

y2 y3

(4.1.36)

Coefficients in formulas (4.1.35) have a common cyclic formula, where ijm = 123, 231 or 312: ai = x j ym − xm y j

bi = y j − ym

ci = xm − x j

(4.1.37)

As planar coordinates L1 , L2 , L3 are just linear transformation of rectangular ones x, y , their application does not change the polynomial degree on the triangle. Therefore, a similar hierarchy holds as in art. 4.1.5.1., with nodal polynomials of degree 1, 5, 9, 13, etc., for the generation of function class C0 , C1 , C2 , C3 , etc., in the analysed 2D structure, i.e. with the requirement on continuity both in the function ( C0 ) or up to partial derivatives of degree 1, 2, 3, etc., inclusive ( C1 , C2 , C3 , etc.). This fact was unknown to many authors for long time. Consequently, there was a certain period in the history of FEM when even reputable experts attempted to solve the known discrepancy between the number of coefficients of a polynomial of third degree (4.1.21) (which is according to (4.1.22) N = 4 ⋅ 5 2 = 10 ) and the number of vertices of a triangle multiplied by three (parameters w, wx , wy ), i.e. 3 × 3 = 9 . To add the centroidal value w as the tenth parameter according to fig. 4.7c) was numerically inconvenient as the elimination and condensation were not yet normally used and mathematicians warned about it. The reason was that it lowered the convergence by a factor of ten or hundred. A vast number of triangular FEM elements based on L1 , L2 , L3 coordinates were derived at those times, in particular of degree 3. A very useful integration formula – still used in FEM literature – found its 115

4.1 Geometric properties of elements application in the formation of stiffness matrices:

∫∫ L

a 1

Lb2 Lc3 dA =

A

a !b !c ! 2A ( a + b + c + 2)!

(4.1.38)

The continuity of derivatives could not be guaranteed by the polynomials of third degree. The class of functions was still C0 . Various sophisticated modifications of what is termed correction functions, which were not polynomials but rational functions, shifted them towards class C1 . Some of them were published (B. M. Irons, O.C. Zienkiewicz, [1,2]), others were kept secret and even today some companies treat them as confidential information as they reportedly represent the core of the success of specific elements. The correction functions lose their importance in present-day elements. On the other hand, the elements with rotational parameters now newly require new types of functions relating to the possibility of strain without energy (zero energy modes) or even zero displacements at a certain configuration of rotational parameters (zero displacement modes). The technical literature uses abbreviation ESF (extra shape functions) for functions that enlarge the class of allowable element deformation and RDOF (rotational degrees of freedom) through the rotational parameters. We mentioned the necessity to use a special kind of reinforcement in the elements (penalty energy, penalty stiffness) already in art. 3.3.2. As an example let us use the first three members in the hierarchy of triangular elements with polynomials in planar coordinates. Contrary to the hierarchy in rectangular coordinates (art. 4.1.5.1.), where – following from fig. 4.7 – also partial derivatives are used, we can do just with function values pi in points i = 1, 2,3,K of the triangle, mainly in its vertices and on its sides. Generally, the polynomial has then the form of a linear combination N

p( L1 , L2 , L3 ) = ∑ pi vi ( L1 , L2 , L3 )

(4.1.39)

i =1

where N = 3, 6, 10 , etc. for degree n = 1, 2,3 , etc. Base, or influence, functions – analogous to fig. 4.5d in linear configuration – are very simple (fig. 4.8c): V1 = L1

V2 = L2

V3 = L3

(4.1.40)

p( L1 , L2 , L3 ) = p1 L1 + p2 L2 + p3 L3 Quadratic polynomials (fig. 4.8d) require introduction of different base functions for vertices and for midpoints of sides:: V1 = ( 2 L1 − 1) L1

V2 = ( 2 L2 − 1) L2

V3 = ( 2 L3 − 1) L3

V4 = 4 L1 L2

V5 = 4 L2 L3

V6 = 4 L3 L1

(4.1.41)

p( L1 , L2 , L3 ) = p1 ( 2 L1 − 1) L1 + K + p6 4 L3 L1 It is similar for cubic polynomials (fig. 4.8e), where centroid 10 is added and each side has two nodes in the thirds of sides: Vi =

1 2

( 3Li − 1)( 3Li − 2 ) Li

V4 = 94 L1 L2 ( 3L1 − 1)

i = 1, 2,3 etc. for V5 and V9

(4.1.42)

116

4.1 Geometric properties of elements V10 = 27 L1 L2 L3 p( L1 , L2 , L3 ) = p1 12 ( 3L1 − 1)( 3L1 − 2 ) L1 + ... + p10 27 L1L2 L3 To conclude this article, let us mention at least one example of correction function for the distribution of deflection w of a Kirchhoff triangular plate element: w( L1 , L2 , L3 ) =

L1 L22 L23 ( L1 + L2 )( L2 + L3 )

(4.1.43)

Following from (4.1.31) it is clear that w reaches on all three sides of the element values 05 02 → 0 and a proof can be provided that also the derivatives ∂w ∂n are along two sides (12 and 13) equal to zero. ∂w ∂n is non-zero only along side 23, i.e. the mass normals h of the plate rotate. If we use cyclic substitution in (4.1.43), we can obtain two similar functions with nonzero ∂w ∂n only on side 12 or only on side 13. Linear combination w123 of these three functions with suitable selected parameters of type ∂w ∂n in the middle of the sides thus has two very useful features: ii) it does not break continuity of deflection w between elements, i.e. class C0 , jj) it can establish continuity ∂w ∂n , i.e. smooth the breaks in the deflection surface of the plate between elements (class C1 ), on condition that we add w123 to a common polynomial of degree 3, which in general does not guarantee this continuity – as we already know from art. 4.1.5.1., fig. 4.7. Without w123 it would have to be a polynomial of degree 5. It is possible to create an infinite number of correction functions of type (4.1.43) and really about 100 of them were made between 1960 and 1968. Universities in the U.S.A. in 1968 explicitly forbade submitting of new PhD theses dealing with this theme, as there was a danger of real explosion of complicated formulas. The present-day FEM programs usually have such elements in their libraries, as already stated. The users may find it useful to read the following note: correction functions are not polynomials and thus their integration and sometimes even derivation are performed numerically. Singularity-surprises are not eliminated for higher derivatives – used e.g. to obtain moments in plates (second derivative) and shear forces (third derivative). Formulas are more complex – see the rule about the derivation of rational functions ( f g )′ = ( f g′ − fg ′) g 2 , which was applied several times! If the element is not properly treated in terms of programming, its internal forces do not have to be reliable. Fortunately enough, we are now witnessing an overall tendency of swapping from Kirchhoff elements to Mindlin ones, for which function class C0 is sufficient.

4.1.5.3 Quadrilateral elements with polynomials in x,y This is the oldest known FEM element used already in 1956. The very oldest element is a rectangular one shown in fig. 4.9. The original authors based their approach on a naive idea – that unfortunately thrived for long in fields isolated from technical information: it is just enough to generalise polynomials from beams, i.e. 1D elements, to 2D space. This really gives an applicable element for planar elastic problems that are a kind of extension of tension 117

4.1 Geometric properties of elements and compression in both directions amended by planar shear to provide for full membrane (wall) stress-state. Tension or compression in 1D elements requires just a linear base displacement function u ( x) . For a 2D element in fig. 4.9a) two bilinear displacement functions are sufficient u ( x, y ) = a1 + a2 x + a3 y + a4 xy

(4.1.44)

v( x, y ) = a5 + a6 x + a7 y + a8 xy

together with eight deformation parameters in four vertices 1,2,3,4 in the following order d = [ d1 , d 2 , d 3 , d 4 , d5 , d 6 , d 7 , d8 ] = [u1 , v1 , u2 , v2 , u3 , v3 , u4 , v4 ] T

T

(4.1.45)

Base functions are of simple form (similar to fig. 4.5 for a triangle). At the same time, it is guaranteed that a set of these elements generates in the analysed 2D domain function class C0 , with continuous function values u,v in all points including the boundary between the elements. This is sufficient for a weak solution of the problem. The solution of a Kirchhoff plate requires that the deflection surface w( x, y ) is continuous in derivatives ∂w ∂x , ∂w ∂y , i.e. function class C1 . The generalisation from a beam used originally (for almost 9 years) a biharmonic polynomial with twelve terms w( x, y ) = a1 + a2 x + a3 y + a4 xy + a5 x 2 + a6 y 2 +

(4.1.46)

+ a7 xy 2 + a8 x 2 y + a9 x3 + a10 y 3 + a11 xy 3 + a12 x3 y which satisfies biharmonic equation (the subscript denotes partial derivative) wxxxx + 2wxxyy + wyyyy = 0

(4.1.47)

which means that plane load on the element p ( x, y ) ≡ 0 . This was originally considered a positive feature (everything is concentrated into nodes). Only on the occasion of the first declassification of FEM – which was in the meantime used for the analysis of many strategic structures including the preparation of the APOLLO module for landing on the Moon – at the legendary first FEM conference (1st Conference on Matrix Methods, WPAFB, Ohio, 1965), a warning was published that (4.1.46) does not generate function class C1 . The continuity of first derivatives is guaranteed only in nodes of rectangular mesh, not along their common sides where the plate is broken. The proof is self-evident. After a partial derivation with respect to x , only degree 3 is left in y , and thus ∂w ∂x goes along sides x = 0 and x = a (fig. 4.9a) following a cubical parabola in y . This inevitably requires four defining parameters, e.g. values ∂w ∂x and derivative with respect to y , i.e. ∂ ∂y ( ∂w ∂x ) = ∂ 2 w ∂x∂y in every side end, i.e. in nodes of the element. This means that the original number of 12 parameters – three in each node ( w, ∂w ∂x , ∂w ∂y ) – increases to 16 (derivatives are marked briefly by a subscript): d = [ d1 , d 2 , d 3 , d 4 , d 5 ,..., d16 ] =  w1 , wx1 , wy1 , wxy1 , w2 ,..., wxy 4  T

T

(4.1.48)

The polynomial with twelve terms (4.1.46) must be extended by additional four terms, in order to get a regular relation d = Sa (no. 2, art. 2.3.). These are a13 x 2 y 2 + a14 x 2 y 3 + a15 x3 y 2 + a16 x 3 y 3

(4.1.49) 118

4.1 Geometric properties of elements amending (4.1.46) to what is termed Ahlin’s bicubic polynomial with 16 terms, which exploits all possibilities of degree in variable x or y up to the degree 3. It is an incomplete polynomial of sixth degree, as the last term is of degree 6, but many terms of degree 4, 5 and 6 are missing (e.g. x 4 y, y 6 , etc.). Polynomial (4.1.46) with amendment (4.1.49) can be simply written as an Ahlin bicubic polynomial, if we use two subscripts for the notation of coefficients a meaning the power of x (1st subscript) and y (2nd subscript): 3

3

w( x, y ) = ∑∑ aij xi y j

(4.1.50)

i =0 j = 0

What can be successfully applied to FEM are Ahlin’s bi- n- th polynomials with odd n = 1, 3, 5, 7 , etc., with (n + 1) 2 coefficients including the absolute term a00 : n

n

w( x, y ) = ∑∑ aij xi y j

(4.1.51)

i =0 j = 0

It can be proved [5] that they ensure the continuity between elements up to (n-1)-th derivative inclusive, i.e. following from fig. 4.9b to 4.9d:

For n =

1

3

5

7

Continuity:

function

+ 1st derivative

1st and 2nd derivative

up to 3rd derivative

For Kirchhoff plate:

of deflection

rotation without breaks

curvature (moments)

shear forces

of displacement

deformation components

deformation components without breaks

SaintVenant equations

In planar elastic problem:

In our country, they were commercially used for the first time already in 1969-1973 in the oldest versions of NE01 and NE03 programs by I. Nemec. This revealed that the continuity of the first derivative can be sometimes unwelcome, e.g. in multi-cell slabs with linear hinges in the longitudinal direction it is necessary to admit the break in the transverse direction. This can be solved through doubling deformation parameters wy , wxy in all nodes (fig. 4.9e).

119


Figure 4.9: a) The oldest known rectangular element with four nodes in vertices. b) Generalisation to a rhomboid in oblique coordinates x,y. c) Four deformation parameters in a bilinear element. d) 16 deformation parameters in a bicubic element. e) Doubling of parameters to ensure discontinuity (break) in the y-direction. f) Character of the boundary of a region divided to rectangles, the region may contain openings (hatched). g) Character of the boundary of a region divided to rhomboids. h) Quadrilateral element composed of four triangular subelements. i) Effective element of NE14, NE15 programs. When two neighbouring nodes coincide, it transforms itself without any problems into a triangular element composed of three triangles.

120

4.1 Geometric properties of elements Ahlin’s elements offer a big numerical advantage in the concentration of all parameters into nodes = vertices. Polynomial (4.1.51) with (n + 1) 2 coefficients, if we admit only odd degrees n = 2m + 1 , require that just (m + 1) 2 parameters are introduced in the nodes. For m = 0 , n = 1 it is just one parameter (fig. 4.9c), for m = 1 , n = 3 there are four parameters (fig. 4.9d), etc. A practical disadvantage of Ahlin’s elements in the original form is that they are capable of covering only certain areas (fig. 4.9f and 4.9g). Their boundary must be composed of line segments that are parallel with axes x, y . This disadvantage can be overcome through the transformation of the shape into a general quadrilateral, see the following article. Another possibility is to compose the quadrilateral of four triangles as in fig. 4.9h. The internal node is located in the intersection of two diagonals defined by the mid-points of the sides. This guarantees geometrical regularity in the case that the element degenerates into a triangle due to the coincidence of two adjacent nodes. This, together with the introduction of rotation parameters of deformation for the planar stress-state, leads to a very effective modern node in fig. 4.9i. All deformation parameters, in total 4 × 6 = 24 , are concentrated into its four vertices.

4.1.5.4 Iso-, hypo- and hyper-parametric elements The basic operation for the definition of these elements is the projection of a square onto an arbitrarily curvilinear quadrilateral. An additional operation consists in the selection of base functions – most often components of displacement or rotation – on this quadrilateral. Usually, both operations use the same functions that are called in the first operation shape functions and in the second one base functions, which gives rise to isoparametric elements. Nodal coordinates (ξ v , ηv ) of the arbitrarily curvilinear quadrilateral can be considered to be the element shape parameters. The values of the unknown function in these nodes pv (ξ v , ηv ) are the parameters of deformation related to the base functions, most often components of displacement or rotation of these nodes (fig. 4.10). The oldest isoparametric element is in fact the triangle from art. 2. with six shape parameters (ξ v , ηv ) and six deformation parameters (uv , vv ) , v = 1, 2, 3 , linear shape and base functions, known already in 1956. Also its extension by nodes in the centres of sides (v = 4, 5, 6) , with twelve parameters of both shape and deformation (Veubecke, 1965) and quadratic functions, is an isoparametric element (fig. 4.7b). It is however possible to define a triangular element with straight sides (v = 1, 2, 3) over which the displacement components follow quadratic functions and in which other nodes in the centres of sides are utilised (v = 1 to 6). Such an element is in terms of base functions hyper-parametric and in terms of shape functions hypo-parametric. A similar example of a quadrilateral can be shown in fig. 4.10a (selection of the shape) with a higher base function with higher number of nodes, e.g. 4.10b,c, being selected for the distribution of the unknown function. If we lower base functions in the selected element, e.g. only linear functions with nodes 1, 3, 5, 7 for the element from fig. 4.10b) – and the values in nodes 2, 4, 6, 8 are determined only by these linear functions (interpolated) and are not among the unknown parameters of deformation – we obtain an element that is hypo-parametric in terms of base 121

4.1 Geometric properties of elements functions and hyper-parametric in terms of shape functions. In practice mainly isoparametric elements are used. Their analysis and properties are based on special functions called parametric functions p(ξ ,η ) , analysed in detail in Czech publication [5], pp. 148 to 196, accompanied with a great number of examples. The same publication also removes errors that commonly appeared in these functions between 1965 and 1978 even in contributions by prominent authors. The error usually resulted from an incorrect generalisation of simple bilinear functions to what is termed biquadratic, bicubic, etc., functions without verification of values of the coefficients and absolute terms. The procedure was strongly influenced by a very popular class of polynomials called by O.C. Zienkiewicz [2] serendipity. For more precise definition see [5]. Users of parametric functions should know just the main piece of knowledge that it is better to define these functions separately for every degree n = 1, 2, 3 , etc., as general expressions are almost always incorrect.

122


Figure 4.10: Projection of a unit square with vertices (±1, ±1) onto a general curvilinear quadrilateral, nodal points for degree: a) n=1 (only straight-line sides), c) n=3 (cubic parabolas).

123

4.1 Geometric properties of elements The nature of parametric functions can be illustrated on three examples from fig. 4.10. Let us apply the procedure by A. Zenisek from [5], which holds for an arbitrary degree n . Let us have a square with vertices (−1, −1) , (1, −1) , ( 1,1) , (−1, 1) in the Cartesian coordinate system ξ , η . Let us divide its sides to n identical intervals. There are in total 4(n − 1) dividing points and together with square vertices we get 4n nodal points. Depiction of these points for n = 1 to 3 is clear from fig. 4.10. Square vertices are now marked with symbols A1* = (−1, −1) , An*+1 = ( 1, −1) , A2*n +1 = ( 1, 1) , A3*n +1 = (−1, 1) . Let us assume polynomials in the form p(ξ ,η ) =

∑α

ab

ξ aη b

(4.1.52)

( a ,b )

where the summation is carried out over all integer pairs (a, b) having the following properties: 1. 0 ≤ a ≤ n , 0 ≤ b ≤ n , 2. only one of the numbers a,b can be greater than one. Set [ (a, b) ] of all these pairs is thus of the following form ( a, b )  = [

( 0, 0 ) , ( 0,1) , ..., ( 0, n ) , (1, 0 ) , (1,1) , ..., (1, n ) , ( 2, 0 ) , ( 2,1) , ( 3, 0 ) , ( 3,1) ,..., ( n, 0 ) , ( n,1) ]

(4.1.53)

and has 4n members. The set of all polynomials (4.1.52) forms 4n – dimensional linear function space. For our needs we seek such a base of this subspace (i.e. 4n base functions p1 (ξ ,η ) , p2 (ξ ,η ) , ..., p4 n (ξ ,η ) ) which has an advantageous feature for FEM: all base functions pi (i ≠ k ) have non-zero value in nodes Ak* , only function pk has the value equal to one in the circumstance. This can be written by means of Kronecker’s symbol δ ik , which is equal to zero for i ≠ k and equal to one for i = k , in a commonly used form: pi ( Ak* ) = δ ik

( i, k = 1, 2,..., 4n )

(4.1.54)

The literature cited in [5] correctly derives the most common situations according to fig. 4.10, which will be mentioned here explicitly for their significant importance in FEM: n = 1 (fig. 4.10a)

pv (ξ ,η ) =

1 4

(1 + ξvξ )(1 + ηvη )

( v = 1,..., 4 )

(4.1.55)

where ξ v , ηv are coordinates of vertices Ak* , i.e. (−1, −1) , (−1, 1) , ( 1,1) , (1, −1) . n = 2 (fig. 4.10b)

124


(1 + ξvξ )(1 + ηvη )(ξvξ + ηvη − 1) pk (ξ ,η ) = 12 (1 − ξ 2 ) (1 + ηkη ) pk (ξ ,η ) = 12 (1 + ξ k ξ ) (1 − η 2 ) pv (ξ ,η ) =

1 4

( v = 1,3,5, 7 ) ( k = 2, 6 ) ( k = 4,8)

(4.1.56)

( v = 1, 4, 7,10 ) ( k = 2,3,8,9 ) ( k = 5, 6,11,12 )

(4.1.57)

n = 3 (fig. 4.10c)

(1 + ξvξ )(1 + ηvη ) (ξ 2 + η 2 − 109 ) 27 pk (ξ ,η ) = 32 (1 − ξ 2 ) (1 + ηkη ) ( 13 + 3ξkξ ) 27 pk (ξ ,η ) = 32 (1 + ξk ξ ) (1 − η 2 ) ( 13 + 3ηkη ) pv (ξ ,η ) =

9 32

For n = 2 the nodal coordinates are, respectively, according to fig. 4.10b (−1, 1) , (0, −1) etc., up to (−1, 0) , for n = 3 (fig. 4.10c) the twelve node coordinates are (−1, −1) , (− 13 , −1) , ( 13 , −1) , (1, −1) , (1, − 13 ) , etc., up to (−1, − 13 ) . Property (4.1.54) can be easily verified through direct substitution. Experienced readers probably anticipate the influence principle that will be now explained. The most illustrative is the situation for n = 1 (fig. 4.10a). In terms of element shape, formula (4.1.55) can be applied as what is termed shape function, i.e. for the transformation of unit square (its sides are equal to 2) – with vertices according to the text following (4.1.55) and fig. 4.10a) left – to a general quadrilateral. Each point (ξ , η ) can be assigned unequivocally a point ( x, y ) according to a simple rule of transformation of a rectangular mesh (net) with identical meshes (elements) ∆ξ × ∆η (in fig. we selected ∆ξ = ∆η = 13 ) to a more general mesh specified by uniform division of sides of the quadrilateral to identical intervals (there are six them in fig. 4.10). It can be easily verified that coordinates of point ( x, y ) that is an image of point (ξ , η ) are 4

x = ∑ xv pv (ξ ,η ) 1

4

y = ∑ yv pv (ξ ,η )

(4.1.58)

1

where p are functions (4.1.55), ( xv , yv ) coordinates of four vertices v = 1, 2, 3, 4 . In particular for the vertices we get correct relations: x1 = 14 x1 (1 + ( −1)( −1) ) (1 + ( −1)( −1) ) + 0 + 0 + 0 = x1 x2 = 0 + 14 x2 (1 + (1)(1) ) (1 + ( −1)( −1) ) + 0 + 0 = x2 x3 = 0 + 0 + 14 x3 (1 + (1)(1) ) (1 + (1)(1) ) + 0 = x3

(4.1.59)

x4 = 0 + 0 + 0 + 14 x4 (1 + ( −1)( −1) ) (1 + ( −1)( −1) ) = x4 and similarly for y1 to y4 . For centre ξ = 0, η = 0 we obtain 1 1 4 x (1 + 0)(1 + 0) + x ⋅ 1 ⋅ 1 + x ⋅ 1 ⋅ 1 + x ⋅ 1 ⋅ 1 = [1 ] ∑ xv 2 3 4 4 4 1 1 1 4 yc = [ y1 (1 + 0)(1 + 0) + y2 ⋅1 ⋅1 + y3 ⋅1 ⋅1 + y4 ⋅1 ⋅1] = ∑ yv 4 4 1

xc =

(4.1.60)

125

4.1 Geometric properties of elements which are the coordinates of the centroid of the quadrilateral. Notice please that each vertex v = 1 to 4 or more precisely its coordinates xv , yv contribute to coordinate x, y just by value pv (ξ ,η ) , which can be perceived as an influence or source function for coordinate x or y . This is an analogy to the sum of effects of four loads P on a certain static quantity if we know its influence surface. What is important is the fact that formulas (4.1.58) and (4.1.55) assign (again) a straight line segment to the points of a certain straight line segment (that is a part of a rectangular mesh – fig. 4.10a left, e.g. ξ = ξ 0 = const . ). The reason is that only η changes – linearly – and thus (4.1.58) becomes an equation of a straight line segment with parameter η . Consequently, also sides ξ = −1 , ξ = 1 , η = −1 , η = 1 become straight line segments – sides of a quadrilateral. Similarly, it can be proved that formulas (4.1.56) and (4.1.57) transform a rectangular mesh ∆ξ , ∆η to a curvilinear mesh composed of parabolas of second and third order, which holds even for the sides of the element. This way we can obtain elements with curved sides. They are useful in applications where we need rather smooth substitution of the boundary of the analysed domain – one curved side on the boundary is sufficient, others may be straight. In general, the influence principle is still valid for an arbitrary n : 4n

x = ∑ xi pi (ξ ,η ) i =1

4n

y = ∑ yi pi (ξ ,η )

(4.1.61)

i =1

For n = 1 formulas (4.1.58) are with notation i = v n = 2, 3 , etc., the nodes are also located on sides of marked k and together with v are numbered i = 1 to etc. A general summation expression – proved in [5] formulas:

(nodes are only in vertices v ). For elements outside of the vertices, are 8 (fig. 4.10b), i = 1 to 12 (fig. 4.10c), – is useful for the verification of the

4n

∑ p (ξ ,η ) = 1 i =1

(4.1.62)

i

In terms of the distribution of displacement components u ( x, y ) , v( x, y ) within the extent of one element, we can use polynomials pi (ξ ,η ) as base functions. We use a formula similar to (4.1.61): 4n

u ( x, y ) = ∑ ui pi (ξ ,η ) i =1

4n

v( x, y ) = ∑ vi pi (ξ ,η )

(4.1.63)

i =1

However, instead of coordinates of nodes i = 1 to 4n , we now handle their displacement components ui , vi , which are in FEM used as deformation parameters. They are shared by elements that have a common node, regardless of whether a vertex v or side-point k . This safely ensures the required continuity of functions u , v over the whole domain. It is not necessary to select in (4.1.63) the same n as in (4.1.62). If we do so, we get an isoparametric element with exactly the same number of shape parameters (coordinates) as deformation parameters (displacement components). It is the most frequent element of this type. Otherwise we get a hypo-parametric or hyper-parametric element, as already stated at the beginning of this article, in terms of shape or distribution of base functions, which must be – for the needs of users – explicitly stated in the manuals.

126


4.1.5.5 Surface elements recommended by the authors Planar elasticity and deformation Geometry is defined in plane. It is possible to solve both cases of planar elasticity: - planar elasticity – loading, as well as reactions act in wall plane.

Vector of stress:

T

σ = σ x , σ y , 0,τ xy , 0,0  , after substitution to physical equations: 1 1 σ x − µ (σ y + σ z )  γ xy = τ xy  E G 1 1 ε y = σ y − µ (σ x + σ z )  γ yz = τ yz E G 1 1 ε z = σ z − µ (σ x + σ y )  γ zx = τ zx E G εx =

T

we will reach vector of strain: ε = ε x , ε y , ε z , γ xy , 0, 0  . Deformation w( x, y, z ) in direction upright to the plane of plate exists, but it is irrelevant. Wall in this direction freely change thickness, nothing prevent this. - planar deformation - layer with thickness h = 1 cut from the points that does not allow T

deformations in direction upright to the wall plane. Vector of strain: ε = ε x , ε y ,0, γ xy , 0, 0  . T

We will reach stress vector through inversion of physical equations: σ = σ x , σ y , σ z ,τ xy , 0, 0  . Wall try to deform also upright to its plane, what is prevented by adjacent layers. In next we work with reduced vectors of stresses and strains:  σx    σ =  σy  τ xy = τ yx    Missing components is possible to computed of physical equations. Thickness h within one element is constant. Vector of strain and corresponding geometric equations that comprise only linear part:  ∂u     ε x   ∂x     ∂v  ε = εy  =  ∂y  γ xy      ∂u ∂v   +   ∂y ∂x  In solution of geometrically non-linear tasks by the Neton-Raphson method is used 127

4.1 Geometric properties of elements quadratically Green vector of strain ε II that comprises non-linear terms that are in theory of small deformation neglected:  1  ∂u 2  ∂v 2      +      2  ∂x   ∂x      2 2  1  ∂u   ∂v    ε II =    +      2  ∂y   ∂y     ∂u ∂u ∂v ∂v    +  ∂x ∂y ∂x ∂y    Element could be physically isotropic, or orthotropic. Matrix of physical constant CM :  C11 C12 CM =  C22  sym.

C13  C23  C33 

In case of isotropy is valid: C13 = C23 = 0 planar elasticity:

planar deformation:

E 1− µ2 E C33 = 2. (1 + µ )

C11 = C22 =

C11 = C22 =

C12 =

C33 =

E.µ 1− µ2

E. (1 − μ ) (1 + μ ) . (1 − 2.μ )

E. (1 − μ ) 2. (1 + μ ) . (1 − 2.μ )

C12 =

μ.E. (1 − μ ) (1 + μ ) . (1 − 2.μ )

Material orthotropy is given by angle of rotation - angle β of orthotropy to planar co-ordinate element system. Vector of deformation in case thermal loading: ε 0 = [α T , α T , 0]T , where T is change of temperature from stadium of preparation (zero elasticity) after actual status 1 - fig. 4.11.

Figure 4.11: Uniform change of temperature.

128

4.1 Geometric properties of elements Result of solution are at:  nx    - planar elasticity of internal forces n (normal and shear):, n =  ny   qxy = q yx     σx    - planar deformation of stresses σ (normal and shear):. σ =  σ y  τ xy = τ yx   

Membrane isoparametric elements with rotational degrees of freedom (Serendipity family) Each vertex of element has 3 degrees of freedom - 2 translations in wall plane walls and rotation: u = [u, v, ω z ]T .

Numeric integration of matrices There is utilised Gauss numeric integration that converts integral to sum of products of weighted coefficients and functional values of integrated function. Number of integration points and their position is for each element different; their exact description is presented at description of each element. At composition of matrix of elements by numeric integration is not allowed to change its parameters, numeric integration is firmly bounded with each type of element and there is selected such number of integration points and their position that provides best results comparing to amount of machine instructions. Integration on unit square is: 1 1

∫∫

−1 −1

f (ξ ,η ) dξ dη = ∑∑ wi w j f (ξi ,η j ) m

m

i =1 j =1

where m is a number of integration points, w are weight coefficients, f (ξ ,η ) are functional of value of shape function in corresponding point.

For 3-node element is used analogical procedure, boundary of integral are 0,1 . Elimination of nodes in centre of sides Field of deformation parameters of 3-nodal and 4-nodal element is approximated by quadratic multinomial, original geometry is with node in centre of each side, fig. 4.12. On each side of the element is anticipated that is straight. This enables to eliminate central node, what is 129

4.1 Geometric properties of elements moreover favourable from aspect of width of the system of equations and thus consumption of machine time. Resulting geometry of the element is described freely of centre node – it is linear. Resulting element is than sub-parametrical.

Figure 4.12: Edge of element with centre node.

Original elements have nodes at the centres of sides [uk , vk ] , these nodes are condensed by transformation of deformation parameters to vertices by interpolation: uk =

y − yi 1 ui + u j ) + j ( (ωzj − ωzi ) 2 8

vk =

x −x 1 vi + v j ) + j i (ω zj − ω zi ) ( 2 8

There will arise element that is compatible with other finite elements 1D and 3D that have corresponding degrees freedom in rotation.

Spurious mode control For isoparametric elements is necessary to numerically stabilize elements, to avoid zeroenergy modes. In case of 3-nodal element there appears only equal rotation of spurious mode, in case of 4-nodal element there is also hourglass mode. Stabilisation consists in adding of small contribution to the potential energy to Π element. Size of the parameter α R and α H is not possible to be changed. 1/ Hourglass mode It occurs only at 4-nodal beams if there is used reduced integration. This mode vanishes, if there is used disordered net, or more elements, fig. 4.13. There is valid: 130

4.1 Geometric properties of elements ω z1 = −ω z 2 = ωz 3 = −ω z 4 Element has added energy EH : EH = α H ⋅ V ⋅ ωH ⋅ Gxy ⋅ ωH α H – parameter of energy ( 1 ⋅10−3 ) V – element volume

ωH =

1 4

( ω z1 − ω z 2 + ω z 3 − ω z 4 )

Gxy – modulus of elasticity in shear

Figure 4.13: Hourglass spurious mode.

2/ Equal rotation mode It occurs at 3-nodal and 4-nodal elements, fig. 4.14. There is valid: ω z1 = ω z 2 = ω z 3 = ω z 4 Element have added energy ER : ER = α R ⋅ V ⋅ ωR ⋅ Gxy ⋅ ωR α R – parameter of energy ( 1⋅10−6 ) V – element volume

ωR – relative rotation, computed in the centre of the C element:

131

4.1 Geometric properties of elements 1 n ωR = ωC − .∑ ω zi , n i =1 1  ∂v ∂u  where ωC = .  −  2  ∂xC ∂yC 

Figure 4.14: Equal rotation spurious mode.

4-nodal element:

Figure 4.15: 8-nodal origin 16 DOF membrane without RDOF. 4-nodal new 12 DOF membrane with RDOF.

Number of nodes: 8

Unit coordinates of nodes: {−1, −1}, {1, −1}, {1, 1}, {−1, 1} Shape functions:

132


(1 − ξ )(1 − η )( −ξ − η − 1) N 2 = 14 (1 + ξ )(1 − η )(ξ − η − 1) N 3 = 14 (1 + ξ )(1 + η )(ξ + η − 1) N 4 = 14 (1 − ξ )(1 + η )( −ξ + η − 1) N 5 = 12 (1 − ξ 2 ) (1 − η ) N 6 = 12 (1 + ξ ) (1 − η 2 ) N 7 = 12 (1 − ξ 2 ) (1 + η ) N8 = 12 (1 − ξ ) (1 − η 2 ) N1 =

1 4

Numeric integration: Reduced (2 × 2) integration points and weight multiplier: ξ =  −1 3 , 1 3 , 1 3 , − 1 η =  −1 3 , − 1 3 , 1 3 , 1 w = [1, 1, 1, 1]

3  3 

Evaluation: Function of surface on extrapolation of qualities of integration points to peaks: F = a1 + a2ξ + a3η + a4ξη Loading: Element could be loaded: – of node force (moment) in the node, – uniform loading on edge, – volume loading, – relative deformation of surface ε caused by uniform warming up, eventual uniform shrinkage.

3-nodal element: 133


Figure 4.16: 6-nodal origin 12 DOF membrane without RDOF. 3-nodal new 9 DOF membrane with RDOF.

Unit coordinates of nodes: {0, 0}, {1, 0}, {0, 1} Shape functions: N1 = 1 − 3ξ − 3η + 2ξ 2 + 4ξη + 2η 2 N 2 = −ξ + 2ξ 2 N 3 = −η + 2η 2

N 4 = 4 (ξ − ξ 2 − ξη ) N 5 = 4ξη

N 6 = 4 (η − ξη 2 ⋅η )

Numeric integration: Selective reduced (4-point), 3 for ε x , ε y , 1 for γ xy integration points and weight multiplier: ξ = [1 6, 2 3, 1 6, 1 3]

η = [1 3, 1 3, 2 3, 1 3] w = [1 3,1 3, 1 3, 1]

Evaluation: F = a1 + a2ξ + a3η

134

4.1 Geometric properties of elements Normal stresses are extrapolated of integration points to vertices according to the function F . Shear stress is constant, there is evaluation centre of gravity of the element.

Matrices of finite elements

Sttiffness matrix Dependency among vector field of deformation ε and unknown nodes by deformation parameters u is expressed by: ε = δ Nu = B Lu . Linear deformation matrix B L is obtained by derivation of shape functions N . Matrix of differential operators: ∂  ∂x δT =   0 

∂ ∂y   ∂ ∂x 

0 ∂ ∂y

Matrix of shape functions: N N= 1 0

0 N1

N2 0

0 K N 2 K

Of potential energy of internal forces Π i : Π i =

1 T ε σd Ω is obtained sttiffness matrix of 2 Ω∫

element K L .

Geometric matrix, Matrix of influence of initial stress on element stiffness This matrix is used for solution of problem of stability: K L − λKG

det

=0

or for compilation of total tangency matrix of element with influence of elasticity: KT = K L + K G Non-linear part B NL , taking into account influence of stress on overall stiffness of the element:

B NL

 ∂N1  ∂x =  ∂N1  ∂y

0 0

∂N 2 ∂x ∂N 2 ∂y

 0 K  0 K 

Cauchy stress matrix of previous iteration, in case of geometric non-linearity, or initial 135

4.1 Geometric properties of elements stresses – pre-stresses:  Nx τ% =   N xy

N xy  N y 

Than, geometric matrix of the element K G is: % NL d Ω K G = ∫ BTNL τB Ω

Plate elements Planar constructions, whose all points lay in one common plane. Thickness h is small, comparing to dimensions of the plate. At determination of characteristic dimension L of the plate (what is in case of rectangular plate a shorter dimension, in case of round plate diameter) it is possible to set up solutions of plate according to corresponding theory to: 34. Thin plate – ratio 1 50 < h L < 1 10 - technical theory of bend of thin plates, based on Kirchhoff’s assumptions. There belong almost all ceiling plates. 35. Thick plate - ratio 1 10 < h L < 1 5 – Reissner-Mindlin’s theory that respects shear of normal line slope due to shear deformation γ xz , γ yz . There could be used also thin plates, while it is necessary to eliminate shear of solving of cross section at small thicknesses. While the ratio is h L > 1 5 - construction should be already evaluate as 3D task, at ratio h L < 1 50 the construction is necessarily to be solved a membrane, what is possible in the shell model, or membrane force are accentuated and in overall energetic potential could not be neglected. Force loading acts upright to the plane of the plate, moment around axes of the plane plate. Normal line to the centreline plane of the element remains after deformation straight, it depends on theory of calculation, whether it remain perpendicular to the centreline plane (Kirchhoff’s theory), or not (Reissner-Mindlin’s theory). Each vertex of element has 3 degrees of freedom: deflexions and rotation: u = [ w, ϕ x , ϕ y ]T . Vector of deformation in case of thermal loading: ε 0 = [α∆T h , α∆T h , 0, 0, 0] where ∆T is difference of temperature on upper and bottom surface, fig. 4.17. T

136


Figure 4.17: Non-uniform change of temperature.

Result of solutions are internal moments and force:  mx   m  y   m =  mxy = myx     qx   qy    Thickness of element is constant.

Loading: Element could be loaded: – by node force (moment) in vertex, – by uniform force loading on edge, – by volume loading, – by curvature of surface κ caused by non-uniform warming up, eventually non-uniform shrinkage.

Application of affect of flexible subsoil. Flexible subsoil with shearing spreading of the Kolar-Nemec type could be easily added to the stiffness matrices. Stiffness matrix of subsoil is added to the stiffness matrix of the plate element. Potential energy of subsoil is given by: Πp =

1 C1S w2 + C2Sx wxx2 + C2Sy w2yy ) d Ω ( ∫ 2Ω

Thin plates Element is based on Kirchhoff theory of thin plate, where is neglected shear 137

4.1 Geometric properties of elements deformation - slope. Deformations of plate are described by 1 function of 2 variables w( x, y ) . Rotations are derivations of the deflexions. Out of assumptions of this theory there is valid ε = [ε x , ε y , 0, γ xy , 0, 0]T . Normal line to non-deformed surface of the plate remains a normal line also after its deformation, and under assumption of zero displacement u , v points of centreline surface there is valid: ∂w ∂x ∂w v = − zϕ y = − z ∂y

u = zϕ x = z

After substitution to the geometric equations: ∂u ∂u ∂v γ xy = + ∂x ∂y ∂x ∂v ∂v ∂w εy = γ yz = + ∂y ∂z ∂y ∂w ∂w ∂u + εz = γ zx = ∂z ∂x ∂z εx =

we will reach vector of deformation ε that vitiates linearly by high of cross section h . We describe this vector for short by the vector of curvature of deflexions surface κ , what is for plate more convenient inscription:   ∂u ∂2w =z 2   ∂ ∂ x x   εx  2   ∂ ∂ v w   ε = εy  =  = −z 2  ∂y ∂y  γ xy      ∂u ∂v 2 ∂ w = 2z  +  ∂x.∂x   ∂y ∂x

 ∂2w    2 ∂ x   κx  2   ∂ w   κ = κ y  =  − 2  ∂y  κ xy      ∂2w  2   ∂x.∂y 

Between both vectors there is valid relation: ε = z κ Normal stress σ z is comparing to stresses σ x and σ y negligible small, they are not considered. Vector of stress is σ = [σ x , σ y , 0, τ xy , 0, 0]T . Out of condition of zero slope there are zero also shear of component τ yz and τ zx . Discrepancy of assumptions with reality of this theory is in: 1/

εz =

1  − µ (σ x + σ y )  physical equations, but σ z = 0 .  E

138

4.1 Geometric properties of elements 1 τ yz = 0 G , 1 γ zx = τ zx = 0 G γ yz = 2/

but out of conditions of equilibrium there is valid that course of shearing stresses by high of cross section is parabolic. Number of nodes: 3

Coordinates of nodes: {1, 0, 0}, {0,1, 0}, {0, 0,1} in planar co-ordinates

Deflexions function of the plate: w = a1L1 + a2 L2 + a3 L3 + a4 L12 L2 + a5 L22 L3 + a6 L23 L1 + a7 L12 L3 + a8 L22 L1 + a9 L23 L2 + 2a10 L1L2 L3 Meaning of parameters a1−9 : deflexions w and rotations ω x , ω y in each node. Parameter a10 is linear combination of parameters: a10 = (a4 + a5 + a6 + a7 + a8 + a9 ) 4 . To this parameter appertain deflexions w in the centre of gravity of element, we work in calculation free of internal nodes, and that is why we eliminate this parameter. L1 , L2 , L3 are planar coordinates. At 3-nodal element it is more convenient to work in the system of planar co-ordinates, as in the Cartesian system, due to calculation of integrals. Each planar co-ordinate Li of the item P expresses ration of surfaces: Li =

A ( P, j , k ) A ( i, j , k )

Point P defined inside the element with planar co-ordinates [ L1 , L2 , L3 ] is transformed to the Cartesian system with co-ordinates [ x, y ] according to: xP = x1 L1 + x2 L2 + x3 L3 yP = y1 L1 + y2 L2 + y3 L3 1 = L1 + L2 + L3 It follows that point 1 of the element has planar coordinates of [1, 0, 0], point 2 [0, 1, 0], and point 3 [0, 0, 1], fig. 4.18.

139


Figure 4.18: Area coordinates on the triangle

Inverse relating to antecedent 3 equations represent equations for calculation of planar coordinates of the point P, defined in the Cartesian system: Li = ( ai + bi x + ci y ) ( 2 A ) , accordingly for j , k .

Area of the element A could be defined e.g. according to: 1 x1 1 A = det 1 x2 2 1 x3

y1 y2 y3

ai = x j yk − xk y j bi = y j − yk ci = xk − x j we will obtain expressions for a j , b j , K , ck by cyclic exchange of indexes. We derive each function defined in the system of planar co-ordinates f ( L1 , L2 , L3 ) for needs of stiffness matrix, vector of volume forces and internal forces as composite function according to: ∂f ( L1 , L2 , L3 ) ∂f ∂L1 ∂f ∂L2 ∂f ∂L3 1  ∂f ∂f ∂f  = + + = + b2 + b3  b1  ∂x ∂L1 ∂x ∂L2 ∂x ∂L3 ∂x 2∆  ∂L1 ∂L2 ∂L3  ∂f ( L1 , L2 , L3 ) ∂f ∂L1 ∂f ∂L2 ∂f ∂L3 1  ∂f ∂f ∂f  = + + = + c2 + c3  c1  ∂y ∂L1 ∂y ∂L2 ∂y ∂L3 ∂y 2.∆  ∂L1 ∂L2 ∂L3 

140

4.1 Geometric properties of elements Integration: Implicitly analytically. Calculation of integral is in this case simple, we advance according to Fellippe equation: p !q !r !

∫ L L L dA = ( p + q + r + 2 )! 2 A p q r 1 2 3

A

Thick plates Element is based on assumption of Reissner-Mindlin’s theory: – Normal lines to the centreline plane remain straight also after deformation, however, they are not upright to the centreline plane of the plate, but there is neglected deplanation of cross section, – Normal line of stress σz is comparing to stresses σx and σy negligible small, such as in the Kirchhoff theory. Of this condition their is also in this theory contradiction with ε z = 0 . This phenomenon originates by reduction of task dimension. Vector of deformation ε = [ε x , ε y , 0, γ xy , γ yz , γ zx ]T is composed of deflexion deformation of surface ε x , ε y , γ xy that are linearly changed by high of the cross section and constant slope by high of the cross section γ yz , γ zx .  ∂φ y    ∂ x   κx  ∂φx     κ = κ y  =  −  ∂y  κ xy      ∂φ ∂φ  y − x  ∂x   ∂y

 ∂w  +ϕy   γ  ∂x  γ =  yz  =  γ zx   ∂w − ϕ  x  ∂y 

While w, ϕ x and ϕ y are independent variable, contrary of the the Kirchoff's plate theory. The element could bet physically isotropic, as well as orthotropic, shape of physical matrix C B is:  C11 C12  C22  CB =     sym.

C13 C23

0 0

C33

0 C44

0  0  0   C45  C55 

In case of isotropy there is valid:

141

4.1 Geometric properties of elements C13 = C23 = C45 = 0 Eh3 C11 = C22 = 12 (1 − µ 2 ) Eh3 1− µ C33 = ⋅ 2 12 (1 − µ 2 ) µ Eh3 C12 = 12 (1 − µ 2 ) Eh π2 C44 = C55 = ⋅ 12 2 (1 + µ ) Material orthotropy is represented by angle of rotation of angle β orthotropy to the planar coordinate system of the element.

LYNN - DHILLON Element defined in the Cartesian co-ordinates with linear approximation functions with adding of quadratic beams for deflexion for improvement of convergence. Problem with shear locking at decreasing thickness is solved by introduction of numeric-stabilisation test:

( C44 , C55 ) ≤ 500 ( C11 , C22 )

A

where A is planar area of the element.

Number of nodes: 3 Coordinates of nodes: {0, 0}, {x2 , y2 }, {x3 , y3} Shape functions: w = a1 + a2 x + a3 y − 12 a8 x 2 + 12 ( a5 − a8 ) xy + 12 a6 y 2 ω x = a4 + a5 x + a6 y ω y = a7 + a8 x + a9 y Integration: Implicitly analytically

142

4.1 Geometric properties of elements MITC4 Isoparametric linear element according to [Bathe], elimination of shear locking at decreasing thickness is done by mixed interpolation of deflexion, rotation and slope. Qualities of the element: – element matrix is obtained by full Gauss’s numeric integration , – element has no zero mode of energy

Number of nodes: 4 Unit coordinates of nodes: {−1, −1}, {1, −1}, {1, 1}, {−1, 1} Shape functions:

(1 − ξ )(1 − η ) N 2 = 14 (1 + ξ )(1 − η ) N 3 = 14 (1 + ξ )(1 + η ) N 4 = 14 (1 − ξ )(1 + η ) N1 =

1 4

Numeric integration: Full (2 × 2) Integration of points and weight multiplier: ξ =  −1 3 , 1 3 , 1 3 , − 1 η =  −1 3 , − 1 3 , 1 3 , 1 w = [1, 1, 1, 1]

3  3 

Shell structures Facet-shell structure is an operating term for planar (2-dimensional) structures situated in 3dimensional space and loaded so that there is not possible in them to separate the plate impact from the wall impact. There is used an element from planar tension. Bending element depends on selected plate theory. The final stress vector: σ = [σ x , σ y , 0, τ xy , τ yz , τ zx ]T , strain vector: ε = [ε x , ε y , ε z , γ xy , γ yz , γ zx ]T . Internal forces are composed of a vector of σ membrane stresses and vector of m bending moments and shear forces. 143

4.1 Geometric properties of elements  mx   m  y   m =  mxy = myx     qx   qy   

 σx    σ =  σy  τ xy = τ yx   

The matrix of physical constants of the element C is composed of a bending part C B , membrane part CM . Material orthotropy is defined by the angle of the rotation of the angle β of orthotropy to planar co-ordinate system of the element, into which are because of so defined element transformed physical quantities. In case of physical non-linearity, (for example by origination of cracks along the height of a section) will come to moving of the position of bearing of centreline plane in comparing to the original one, which divided the width of the element into 2 equal parts. This effect respects a sub matrix C BM , it describes cohesion of wall and plate effects: C BM  C C= B   sym. CM 

Large deformations In solving of geometrically non-linear tasks by the Neton-Raphson method is used quadratic element of Green vector in deformation ε II , which includes non-linear terms that are neglected in the theory of small deformations:  1  ∂w  2        2  ∂x      2  1  ∂w    ε II =      2  ∂y     ∂w ∂w     ∂x ∂y    Subsoil Every finite element can have along the entire surface continuous contact with effective model of a subsoil of the Kolar-Nemec type, which is defined by five constants in planar coordinates [ x p , y p , z p ] : C1Sx

C1Sy

C1Sz

C2Sx

C2Sy

Relevant forces for deformations are membrane (ru , rv ) like and bending (rw , t x , t y ) like.

144

4.1 Geometric properties of elements rup = C1Sx u p rvp = C1Sy v p

∂w p ∂x p ∂w p t yp = C2Sy p ∂y t xp = C2Sx

rwp = C1Sz w p The constants C1Sx , C1Sy express the resistance against planar movements of centreline plane of the element (friction). The constants C1Sz , C2Sx , C2Sy are coefficients of relations expressing resistance of surrounding against the movement and angular rotation. The constant C1Sz responds to the Winkler model, C2Sx , C2Sy Pasternak model. In most cases is C1Sx = C1Sy and C2Sx = C2Sy .

Sandwich elements The element is along the height divided into a few (at least 2) isotropic, or orthotropic layers, ideally resistant linked together, so that there does not come to shearping. Every layer has its physical parameters, which are possible to solve so far only by physical linear elasticity. According to Newton-Raphson is possible to solve such elements by geometrical nonlinearity. The number of layers is unlimited. The thickness of every layer is constant. Load of an element is analogous, as well as with facet-shell elements. Alternate homogeneous cross-section with ideal static quantities, which are dealt with in calculation will be created from a composite cross-section. For specification we alternate resistance come out of equality of work of internal and external forces. Evaluation of internal forces comes off in optional point along the height of a cross-section, where on the basis of relevant equations of elasticity is finished calculation of final quantities. Example: Bimetallic strip - console, constant change of the temperature. E = 3 ⋅107 [Pa], µ = 0 , h1 = h2 = 0.05 [m], ∆T = 100 [K], α1 = 1⋅10−5 , α 2 = 2 ⋅10 −5 . The length of a beam L = 10 [m], the width 1 [m].

Figure E4.1: Geometry, cross-section of an element and results – process of stresses along the height of a cross-section.

145


Deformations and stresses

ux [m]

u z [m] σ 1I [Pa] σ 2I [Pa] σ 1II [Pa] σ 2I I [Pa]

numerically

0,150

0,750

–7500

15000

–15000

7500

analytically

0,150

0,750

–7500

15000

–15000

7500

Results.

146


4.1.6 3D elements 4.1.6.1 Tetrahedron The simplest and oldest 3D finite element is a tetrahedron as a natural extension of a sequence of what is termed simplex in nD space. A simplex is the simplest shape, a line segment in 1D, triangle in 2D, tetrahedron in 3D, super-pentahedron in 4D, etc. It is intercepted by the lowest possible number of shapes of dimension (n − 1)D , i.e. for n = 1 by two points (dimension 0D), for n = 2 by three line segments (dimension 1D), for n = 3 by four triangles (dimension 2D), for n = 4 by five tetrahedrons (dimension 3D), etc. Spaces nD or (n − 1)D can be curved, i.e. they can be immersed into spaces (n − 1)D or nD , which gives rise to curved elements. We will describe the most frequently used 3D tetrahedron intercepted by four planar triangles. It is rare for present-day programs to offer this element separately. It usually forms a sub-element of a brick-shaped element, which is a prism, or more generally a block or an arbitrary hexahedron, etc. – see art. 4.1.6.2. A whole hierarchy of polynomials in x, y, z of degree n = 1, 2, 3, K on a tetrahedron is known, with m coefficients, where M=

1 6

( n + 1)( n + 2 )( n + 3)

(4.1.64)

which for n = 1, 2, 3, K is in turns 4, 10, 20, ... coefficients with monomials xα y β z γ , 0 < (α + β + γ ) ≤ 1, 2, 3 , etc. This corresponds – for each of the three displacement components u, v, w – to the same number of m deformation parameters in m nodal points which must be selected in compliance with certain rules. The first three situations are shown in fig. 4.19a-c. The tetrahedron has in total 3m parameters of deformation. These include components of displacement u , v, w and possibly also their derivatives in nodes. For n = 1 and 2 only displacement parameteres are udes, in fig. 4.19a)b) marked by full circle. It represents 3 × 4 = 12 (n = 1) and 3 × 10 = 30 (n = 2) parameters. For n = 3 we have 3 × 20 = 60 parameters. The components and their first derivatives (marked by index) must be introduced in all vertices, i.e. in total 3 × 4 × 4 = 48 quantities (u , u x , u y , u z , v, K , wz ) . In

addition, also introduced must be 3 × 4 = 12 displacement components (u , v, w) in centroids of the sides (fig. 4.19c). A complete hierarchy was elaborated by A. Zenisek [5]. Up to n = 8 , these polynomials guarantee the continuity of u, v, w only in function values, i.e. function class C0 . Only the polynomial of degree 9 with 220 coefficients and 660 deformation parameters can generate function class C1 , and continuity also in the first derivatives, i.e. in components of strain ε . This element was normally unworkable in technical applications due to the performance capacity of present-day computers. It is now convenient to make a summary of features of polynomials in simplexes in space 1D, 2D, 3D, i.e. in a line segment, triangle and tetrahedron. If the dimension of the space is d (1,2,3), the continuity in function values (C0 ) can be achieved already by a polynomial of first degree (a + bx, a + bx + cy, a + bx + cy + dz ) with 2,3,4 coefficients. Continuity in the first derivatives (for d > 1 partial ones) requires a polynomial of degree 3, 5, 147

4.1 Geometric properties of elements 9 with 4, 21 and 220 coefficients. The continuity in second derivatives needs polynomials of degree 5, 9, 17 with 6, 55 and 1,140 coefficients, etc. The number of coefficients relates to just one approximation function, which means that the number of parameters of deformation is either the same (if we work with just one unknown function, e.g. deflection w( x) in a 1D problem) or higher, e.g. three times if we have a 3D problem with three unknown displacement components u , v, w . As the demand for what is termed p-version of FEM, which improves the solution for a rather coarse mesh and large elements through increasing the polynomial degree in elements, has increased in recent years, we will address this issue again in art. 4.3.3.

148


Figure 4.19: 3D-elements FEM: a-c) Tetrahedrons with polynomials of degree 1 - 3. d-f) Bricks with tri-linear, tri-quadratic and tri-cubic polynomials. Tetrahedron b) with quadratic polynomials and 3 x 10 = 30 parameters of deformation of type u, v, w can be modified to a significantly more effective element with 4 x 6 = 24 parameters of type u, v, w, ωx, ωy, ωz only in vertices with preserved favourable properties of quadratic approximation as explained in art. 4.1.6. Then there are no parameters in the centres of sides. Similar improvement can be applied to the brick in fig. e), again through parameters of rotation ω, which means that it has only 8 x 6 = 48 parameters, all in vertices, see art. 4.1.6.1 and 4.1.6.2 and cited literature. Merging of all upper nodes 3, 11, 4, 20, 8, 15, 7, 19 into a single node can give a useful element in the shape of a four-sided pyramid, which does not cause any mathematical difficulties.

149


4.1.6.2 Bricks The most often used 3D elements are elements termed bricks. This originally slang term has been widely used in FEM since 1960. It has been adopted in most languages and normally is not translated. Originally, it was a body with the shape of a cube, prism or block with six planar or curved sides, i.e. a hexahedron, or depending on the number of sides also dodecahedron with eight vertices. Later on, other shapes were developed too, e.g. pentahedrons (fig. 4.20). The main advantage against tetrahedrons from art. 4.1.6.1. is better orientation in the division of the domain and the possibility to introduce numerically convenient base functions. During the years of FEM development, several bricks were constructed as pure super-elements composed of five tetrahedrons, which, apart from better orientation, brought no other advantages. Progress was made only when what is termed isoparametric 3D elements was introduced. It is in fact a consistent extension of the idea of 2D isoparametric elements (art. 4.1.5.4.) to a 3D space. This extension was initially (1962-1968) made through a formally not well thought out addition of z -terms to x, y terms. Defect elements were created even in topranking FEM centres, e.g. B. M. Irons in Swansea, Wales, in 1975 honestly admitted errors in his base functions following the correspondence with authors of publication [5], where A. Zenisek published the first correct formulas for what is termed tri-quadratic and tri-cubic polynomials. The rule from 2D-domains that each polynomial degree must be defined separately and general formulas fail holds for 3D bricks as well. In order to illustrate the stated, let us present the correct formulas for the first three polynomial degrees called briefly tri- n -th, in the domain of a unit cube with the side equal to 2 and vertices as in fig. 4.19d-f, the centre of the cube is in the coordinate origin (0,0,0). Generally, the tri- n -th polynomials have the following form: p(ξ ,η , ς ) =

∑

aabcξ aη bς c

(4.1.65)

( a ,b , c )

where the summation is carried out over all integer triplets (a, b, c) having the following properties: 1.

0 ≤ a ≤ n, 0 ≤ b ≤ n, 0 ≤ c ≤ n ,

2.

only one of the numbers a, b, c can be greater than one.

It is clear that for n = 1 we have a polynomial with the highest term of degree 3 ξηζ , for n = 2 the highest terms are of degree 4 ξ 2ηζ , ξη 2ζ , ξηζ 2 , for n = 3 the highest terms have degree 5 ξ 3ηζ , ξη 3ζ , ξηζ 3 etc. Based on the highest power of one coordinate, a rather vague name tri- n -th polynomial is used, n = 1, 2, 3 , etc. The simplest ones are tri-linear polynomials (n = 1) , assigned in turns to individual vertices of the cube in fig. 4.19d with coordinates (ξ v , ηv , ζ v ) , v = 1 to 8, which reach only unit values +1 or –1:

150

4.1 Geometric properties of elements pv (ξ ,η , ζ ) = 1 8 (1 + ξvξ )(1 + ηvη )(1 + ζ vζ )

( v = 1, 2,...,8)

(4.1.66)

Features of polynomial (4.1.66) in 3D are identical to the features of polynomial (4.1.55) in a 2D problem, see art. 4.1.5.4. If we use it as the shape function in the meaning of formula (4.1.61) which is in 3D extended by coordinate z : 12 n − 4

∑ x p (ξ ,η , ζ )

x=

i =1

12 n − 4

∑

y=

i =1

z=

i

i

yi pi (ξ ,η , ζ )

− 1 ≤ ξ ,η , ζ ≤ 1

(4.1.67)

12 n − 4

∑ z p (ξ ,η , ζ ) i =1

i

i

then for n = 1 the sum deals with 12 − 4 = 8 terms assigned to 8 vertices i = v = 1 to 8. Rectangular mesh ∆ξ , ∆η , ∆ζ is transformed by (4.1.67) to the mesh of straight line segments, and also edges of the cube are transformed to straight line segments and sides of the cube becomes a warped line surface with the shape of a hyperbolic paraboloid, fig. 4.19d. We get a brick of degree 1. If we use (4.1.66) also as base functions in the meaning of formula (4.1.63), extended in 3D by displacement component w : 12 n − 4

∑ u p (ξ ,η , ζ )

u=

i =1

v=

i

i

12 n − 4

∑ v p (ξ ,η , ζ ) i =1

w=

i

i

− 1 ≤ ξ ,η , ζ ≤ 1

(4.1.68)

12 n − 4

∑ w p (ξ ,η , ζ ) i =1

i

i

then the displacement components u , v, w follow inside the 3D brick tri-linear polynomials ( n = 1 ). This creates the simplest isoparametric 3D brick. Its shape is defined by coordinates of eight vertices, its displacement components by eight triplets (u , v, w) in these vertices, in total 24 parameters ( x, y, z )v and deformation (u , v, w)i , i = 1 to 8. In a standard FEM terminology we call it brick 24, which is today included almost in all FEM programs. The next closest higher polynomial (4.1.65) for n = 2 , tri-quadratic, cannot be written using a collective formula for all 12 × 2 − 4 = 20 nodes (fig. 29e), which was in vain attempted until circa 1970. We will present here the correct form according to [5]. Eight polynomials relating to eight vertices (ξ v , ηv , ζ v ) are of the following form: pv (ξ ,η , ζ ) = 1 8 (1 + ξvξ )(1 + ηvη )(1 + ζ vζ )(ξ vξ + ηvη + ζ vζ − 2 )

(4.1.69)

four polynomials relating to nodal points (0, ηk , ζ k ) in centres of sides ξ : pk (ξ ,η , ζ ) = 1 4 (1 − ξ 2 ) (1 + η kη )(1 + ζ k ζ )

(4.1.70)

four polynomials relating to nodal points (ξ k , 0, ζ k ) in centres of sides η :

151

4.1 Geometric properties of elements pk (ξ ,η , ζ ) = 1 4 (1 + ξ k ξ ) (1 − η 2 ) (1 + ζ k ζ )

(4.1.71)

four polynomials relating to nodal points (ξ k , ηk , 0) in centres of sides ζ : pk (ξ ,η , ζ ) = 1 4 (1 + ξ k ξ )(1 + ηkη ) (1 − ζ 2 )

(4.1.72)

in total 8 + 3 × 4 = 20 polynomials, in domain −1 ≤ ξ ,η , ζ ≤ 1 . In the present times, the highest degree of polynomial (4.1.65) implemented in FEM programs is n = 3 . The correct form of these polynomials is determined by their relation to nodes of the element in fig. 4.19f. This represents 8 vertices and 12 × 2 = 27 nodes in the thirds of the sides, in total 32 nodes. Eight polynomials relating to eight vertices (ξ v , ηv , ζ v ) are of the following form: pv (ξ ,η , ζ ) = 1 64 (1 + ξvξ )(1 + ηvη )(1 + ζ vζ ) 9 (ξ 2 + η 2 + ζ 2 ) − 19 

(4.1.73)

four polynomials relating to nodal points (−1 3, ηk , ζ k ) in one third of sides ξ : pk (ξ ,η , ζ ) = 9 64 (1 − 3ξ ) (1 − ξ 2 ) (1 + η kη )(1 + ζ k ζ )

(4.1.74)

four polynomials relating to nodal points (+1 3, ηk , ζ k ) in two thirds of sides ξ : pk (ξ ,η , ζ ) = 9 64 (1 + 3ξ ) (1 − ξ 2 ) (1 + ηkη )(1 + ζ k ζ )

(4.1.75)

similarly eight polynomials relating to nodal points (ξ k , ±1 3, ζ k ) in the thirds of sides η : pk (ξ ,η , ζ ) = 9 64 (1 + ξ k ξ )(1 ± 3η ) (1 − η 2 ) (1 + ζ k ζ )

(4.1.76)

and eight polynomials relating to nodal points (ξ k , η k , ±1 3) in the thirds of sides ζ : pk (ξ ,η , ζ ) = 9 64 (1 + ξ k ξ )(1 + η kη )(1 ± 3ζ ) (1 − ζ 2 )

(4.1.77)

in total 8 + 3 × 8 = 32 polynomials in domain −1 ≤ ξ ,η , ζ ≤ 1 . Similarly to (4.1.62) in 2D problems, the summation statement must hold for all correctly defined polynomials (4.1.65): 12 n − 4

∑ i =1

pi (ξ ,η , ζ ) = 1

(4.1.78)

FEM practice, in addition to the already mentioned brick 24, commonly uses brick 60, an isoparametric element as in fig. 4.19e with the same tri-quadratic polynomials (4.1.69) (4.1.72) for element shape (4.1.67) and also for the distribution of displacement components (4.1.68). The edges of the element are generally formed by a parabola of second order passing through three nodes, i.e. the vertices and the mid-point of the edge. The sides of the element are generally curved. Program packages may also contain the isoparametric brick 96 as in fig. 4.19f, with the edges formed by parabola of third order with one inflection allowed. These elements are isoparametric, because they have the same number of shape and deformation parameters. The shape of brick 60 is defined by means of 3 × 20 = 60 coordinates of nodes ( x, y, z ) and the deformation is prescribed by 3 × 20 displacement components u, v, w in these nodes. For brick 96 we have 3 × 32 = 96 parameters of shape and deformation. 152

4.1 Geometric properties of elements Analogously to 2D problems discussed in art. 4.1.5.4. it is possible to define hyper– or hypo–parametric elements in terms of shape or deformation. It is applied to situations when we need the best representation of the boundary (smooth without breaks) and, therefore, we use higher polynomials for the shape functions. At the same time we limit ourselves to a coarse approximation of displacement components, i.e. to a lower number of unknown parameters of deformation, e.g. to a tri-linear distribution specified just by nodal values ( u, v, w ). For other nodes we content ourselves with the interpolation, in which we substitute the coordinates of non-vertex nodes into the tri-linear polynomials. This creates an element that is hyper-elastic in terms of shape and hypo-elastic in terms of base. Nowadays, however, this is not a typical approach as the computational capacity of present-day computers does not represent such a strong limiting factor for the allowable number of unknowns, as it was in the past. The present-day FEM programs offer tetrahedrons and bricks that concentrate all the parameters of deformation into the vertices. This is possible thanks to the rotation parameters of deformation – see the previous art. 3.3.2, form. (3.3.6), (3.3.7) and notes in subsequent paragraphs. The algorithms of such elements are not simple, they require a large number of transformations, additional functions (ESP = extra shape functions) and a special treatment of singular states (penalty energy, penalty stiffness) in order to eliminate zero displacement fields under certain configurations of vertex rotational parameters, which influences the stiffness matrix in a way similar to excessive release of the body. The basic principles were published in [93] and applied to [65, 73]. They can be briefly summarised as follows: They are based on quadratic fields of displacement components, i.e. for a tetrahedron on the element in fig. 4.19b) with 30 deformation parameters of displacement type and for a brick from the element in fig. 4.19e) with 60 parameters of deformation of the same type, i.e. u, v, w . All parameters relating to the mid-points of edges are eliminated through special interpolation according to fig. 4.8f), which can be popularised as a beam-like state, as linear interpolation is used for the effect of end-point displacements and cubic interpolation for the effect of rotational parameters that were added into all of the vertex nodes. Through standard modifications we then obtain a tetrahedron stiffness matrix of dimension (24, 24) and a brick stiffness matrix (48, 48). There are six parameters in every vertex (six degrees of freedom u, v, w, ϕ x , ϕ y , ϕ z ), i.e. for a tetrahedron in total 4 × 6 = 24 and for a brick 8 × 6 = 48 . The matrix created in this way is, however, singular, as certain constellations of rotational parameters lead to no (zero) displacement field. This can be avoided, similarly to normal singularity, if zero values are prescribed for some rotational parameters in the analysed structure, which is analogous to the necessity to define certain support conditions in order to prevent the rigid body motion. This approach was, however, abandoned (in around 1980) for many good reasons (subjectivity and the fact that the elements were still ill-conditioned). What proved useful was ordinary stiffening of elements with regard to these effects, which was applied from 1960 to eliminate equations of type 0 = 0 in complementary connections of two planar systems in space. In terms of physics, this represents an insertion of certain mechanisms into an element in which deformation energy forms even under the mentioned circumstances (penalty energy, mentioned earlier in the text). Moreover, the overall displacement field is amended (enriched) by special states ESP (extra shape functions) that are quantified by, usually three, additional formal parameters that are related to the whole element and can be imagined e.g. in the centroid. Publication [93] uses a purely Lagrangean conception of the element. There exist even other elements, e.g. with mutually independent displacement and rotation fields, the incompatibility of which decreases with the refinement 153

4.1 Geometric properties of elements of the mesh thanks to the application of more general variational principles [27, 49, 51, 53, 62]. A list of literature contributing to the problem of rotational parameters contains over 100 items. It is neither feasible nor purposeful to present all of them to ordinary FEM users. Therefore, those who are interested in this topical FEM issue are referred to only a few selected articles [89] to [107], listed chronologically, that are friendly to the engineering mentality and that make it possible to follow the interesting development from 1973 ([89], naive and partially incorrect approach) to 1991 ([107], very effective procedure, verified in ANSYS [65]). Bricks can be modified in various ways. For example, it is not necessary that the division of their edges is uniform, brick 96 does not have to have the nodes exactly in the thirds, just one (boundary) side can be curved and the other (internal) planar, etc. It is definitely useless to introduce curved sides of bricks inside of a 3D domain. Useful bricks are those with the shape of a nonagon (fig. 4.20), which are suitable to fill some corner subdomains of analysed volumes (blocks, cast volumes, concrete dams). Another name of this type of brick is pentahedron. It is possible to create a similar hierarchy of polynomials like bricks, each polynomial being of shape n

p(ξ ,η , ζ ) = ∑ f k (ξ ,η )ζ k

(4.1.79)

k =0

where f 0 (ξ ,η ) and f1 (ξ ,η ) are polynomials of n- th degree and f 2 (ξ ,η ), K , f n (ξ ,η ) are polynomials of degree 1. Polynomial (4.1.79) has generally d members, where d = ( n + 1)( n + 2 ) + 3 ( n − 1)

(4.1.80)

Let us consider a nonagon in fig. 4.20 with vertices (0,0,–1), (1,0,–1), (0,1,–1), (0,0,1), (1,0,1), (0,1,1). Nodal points of this nonagon are for n = 1 the vertices, for n = 2 the vertices and middle points on edges and fro n = 3 the vertices, points dividing the edges to thirds and the centroid of the triangular sides. The total number of nodes is for all configurations specified by expression (4.1.80). The nodal points are sorted according to fig. 4.20 and numbered 1, 2, K , d . The detailed analysis of the properties of polynomials (4.1.79) was made by A. Zenisek in [5]. For the sake of clarity, let us present at least the first two cases of degree n : n = 1 , six polynomials assigned to vertices 1 to 6, variables ξ , η within the scope of the unit triangle (0,0), (1,0), (0,1), third variable ζ within the interval −1 ≤ ζ ≤ 1 , fig. 4.20a):

(1 − ξ − η )(1 − ζ ) p3 (ξ ,η , ζ ) = 12 η (1 − ζ ) p5 (ξ ,η , ζ ) = 12 ξ (1 + ζ ) p1 (ξ ,η , ζ ) =

1 2

p2 (ξ ,η , ζ ) = 12 ξ (1 − ζ )

(1 − ξ − η )(1 + ζ ) p6 (ξ ,η , ζ ) = 12 η (1 + ζ ) p4 (ξ ,η , ζ ) =

1 2

(4.1.81)

n = 2 , d = 3 × 4 + 3 × 1 = 15 polynomials assigned to nodes 1 to 15 according to fig. 4.20b), the same scope of variables ξ , η , ζ :

(1 − ξ − η )(1 − ζ )( −2ξ − 2η − ζ ) p3 = 12 ξ ( 2ξ − ζ − 2 )(1 − ζ ) p5 = 12 η (1 − ζ )( 2η − ζ − 2 ) p1 =

1 2

p2 = 2ξ (1 − ξ − η )(1 − ζ ) p4 = 2ξη (1 − ζ )

p6 = 2η (1 − ξ − η )(1 − ζ )

154

4.1 Geometric properties of elements p7 = (1 − ξ − η ) (1 − ζ 2 )

p9 = η (1 − ζ 2 )

p11 = 2ξ (1 − ξ − η )(1 + ζ ) p13 = 2ξη (1 + ζ )

p15 = 2η (1 − ξ − η )(1 + ζ )

p8 = ξ (1 − ζ 2 )

(4.1.82)

(1 − ξ − η )(1 + ζ )( −2ξ − 2η + ζ ) p12 = 12 ξ (1 + ζ )( 2ξ + ζ − 2 ) p14 = 12 η (1 + ζ )( 2η + ζ − 2 )

p10 =

1 2

Nonagons can be continuously connected to bricks – simple example can be seen in fig. 4.20d). At the same time, it is possible to eliminate, in advance, the unknown deformation parameters (u, v, w) in the nodes of the nonagon that are not needed for the connection (art. 4.3.). Moreover, the effectiveness of the element in fig. 4.20 b1) can be increased similarly to the tetrahedron (fig. 4.19b) and brick (fig. 4.19e). Instead of the parameters in the centres of the edges, rotations in the vertices are introduced. They represent small rotations, i.e. a vector with three components. Each vertex has thus all the six degrees of freedom and the element has in total 6 × 6 = 36 deformation parameters.

155


Figure 4.20: Nonagon: a) n=1, b) n=2, c) n=3, d) connection of a nonagon to brick 60. Other name of this element is pentahedron. Case n=2 can be improved with the help of rotational parameters so that all parameters of deformation are just in vertices, in total 6×6=36, as explained in art. 4.1.6.2.

156


4.1.6.3 Toroid 3D elements include also elements created by rotation of a 2D element around a certain axis o that is common to the whole problem. If the 2D element were circular, we would get a torus. The original element is usually a quadrilateral or triangle and, therefore, we call the element annular (analogy to anulus) or simply toroidal. With reference to geometry and base functions, everything told about 2D elements in art. 4.1.5 holds. The difference is significant in terms of physical nature. Contrary to plane stress or plain strain described in fact by three components of tensor σ or ε , in a toroid we get a general stress-state with all six components of these tensors. In a special configuration of an axially symmetrical problem in axes r , z , ϕ , where all planes ϕ = const . form the plane of symmetry, the shear stress components τ rϕ , τ zϕ are eliminated and only four stress components τ r , σ z , σ ϕ , τ rz remain together with similar strain components ε r , ε z , ε ϕ , γ rz . This differs from 2D elements of planar problems with σ x , σ y , τ xy and ε x , ε y , γ xy only in the fourth component σ ϕ and ε ϕ . Component σ ϕ is a certain analogy to component σ z = v(σ x + σ y ) in a plane strain problem with ε z = 0 . Component ε ϕ is analogous to component ε z = v(σ x + σ y ) E in a plane stress problem with σ z = 0 . The difference is essential in the physical relation between tensor σ and ε where the impact of incurvation of element fibres into circles of various radii r takes hold.

4.1.6.4 Special 3D elements It is natural that various special technical problems initiated development of different finite elements that are more suitable for the particular task than standard versatile elements. About 100 of such problem-oriented elements were published just in the period between 1965 and 1985, and it is estimated that approximately 1,000 useful modifications of shape and base functions were not made public. These were made partially according to the procedure given in art. 4.3 and sometimes following the engineering intuition. The attempts to document all FEM elements miserably failed already around 1980. Famous Japanese centres admitted that they could trace about 8% of innovations at most, even when focusing only on literature searches of top-ranking publications and company prints. Manuals describe such elements only from the user’s perspective and if any theoretical manuals exist, they are usually not detailed enough to analyse the element in question (and possibly integrate into one’s own system). The user must rely on what is called numerical tests that are already about 20 years centrally gathered and analysed e.g. in NAFEMS publications. Authors’ tests are not accepted. The tests are made by independent professionals and the comparison of results is made for similar conditions (number of unknowns, computation time). The results are commonly discussed at specialised conferences, which proved useful and which represents a no-effect advertisement. The conclusions may be surprising and procedures and elements promoted for log time sometimes lose their glory. Around 1980 – 1985 this happened to hybrid elements when examples were presented in which the well-tried elements for thick shells (semiloof, Reissner), etc., failed. On the other hand, some procedures for the selection 157

4.1 Geometric properties of elements of base functions – not based on any mathematical derivation but following utterly from an engineering intuition or a lucky idea – proved their worth This includes in particular the wellknown serendipity family [2]. The name has its origins in the late 18th century. It is an artificial word created by Horace Walpole (1717-1797), 4th Earl of Orford, Member of the United Kingdom Parliament from 1741 to 1768. In his fairy tale he endowed the three princes of Serendip with the ability (serendipity) to make discoveries “by accident and sagacity”. Nowadays, this is considered a significant attribute of creative thinking and is supported with e.g. the discoveries of X-rays, radioactivity, radioastronomy (quasars, pulsars), etc. With regard to FEM, many elements and procedures developed from seemingly unimportant and secondary reasoning, just thanks to the fact that someone noticed that it could be utilised usefully in a different way, more generally, that something could be generalised from 1D and 2D and 3D, etc. This human ability is nowadays distinctly limited by powerful computers that induce a feeling that such typically human, almost emotional, processes of development of knowledge are not appropriate for the end of 2nd millennium, that they cannot be correct, that they cannot be proved, etc. This is, however, a false belief. It is a very desirable part of modelling in technical practice. If we carried out a survey into the history of FEM development starting from 1956, we could reveal that a vast majority of such suddenly appearing lucky ideas were later proved by mathematicians to be entirely correct. With regard to what is termed serendipity family, it represents an attempt to extend the polynomials used in 2D problems in art. 4.1.5.4. and bricks in art. 4.1.6.2. to general degree n through an analytical transformation from 1D problems (beams) to 2D problems (plates, walls, shells) and to 3D problems (volumes in Euclidian space). Only after a few years of practical application, the mathematical analysis was provided by A. Zenisek in [5]. He proved that in 3D problems the serendipity family is defined only for n ≤ 4 . This means the final stop for inventing of new members of this family of elements.

4.1.6.5 Solid elements recommended by the authors 3-D isoparametric elements with rotational degrees of freedom (Serendipity family) Every vertex of an element has 6 degrees of freedom: [u, v, w, ϕ x , ϕ y , ϕ z ] . Elements do not have nodal -point in the centre of sides, edges can be straight only. Elements are defined by geometry of nodes and isotropic material. In case of 6-nodal -point element and 8-nodal point is possible to use material orthotropy, that is used in contact elements.

158


Figure 4.21: 20-nodal -point original 60 DOF brick without rotational degrees of freedom. 8- nodal -point new 48 DOF brick with rotational degrees of freedom.

Is used numeric gauss integration. Every element has relevant number of integral points, while because of the fastness of calculation is used reduced integration. This results in origination of zero energy states, which are treated by additional resistances. We distinguish 2 types: equal rotations and hourglass. For every element ´s area is used analogical technique of additional resistance introduction into resistant matrix of an element so as for 2D membrane elements. For all isoparametrical elements is used equal principle of transformation of nodal point deformations in centres of sides into vertices so as it is in the case of 2-dimensional elements, but it is completed by the third dimension. Movements in centres of the sides are eliminated by mediation: y − yi z −z 1 ui + u j ) + j ω zj − ω zi ) + j i (ω yj − ω yi ) ( ( 2 8 8 z − z x −x 1 vk = ( vi + v j ) + j i (ω xj − ω xi ) + j i (ω zj − ω zi ) 2 8 8 x −x y − yi 1 wk = ( wi + w j ) + j i (ω yj − ω yi ) + j (ωxj − ωxi ) 2 8 8 uk =

Space elements with smaller number of vertices than 8 are solved in 2 ways: 36. by degeneration of 8-nodal -point element, fig. 4.22, 37. by defining of own shape functions and by position of integral points. Both techniques are implemented, for a user is available only the technique No.2. Hereinafter is in detail described only this technique. 8-nodal-point element - brick is derived from quadratic izoparametric element with 20 nodalpoints. Such an element allows using curved edges, its disadvantage is large width of a half belt and consequently large consumption of machine time. By above mentioned elimination of nodal-points in centres of sides of 20-nodal-point brick was achieved equations system 159

4.1 Geometric properties of elements with tighter half-bend, what lead to acceleration of calculation. From the original stiffness matrix of the 60×60 element will arise a matrix of the 48×48 size.

Figure 4.22: Scheme of brick degeneration.

Stiffness matrix Hereinafter presented shape functions N create together with matrix of differential operators of δ 6 geometrical equations: ε = δ N ⋅ u = BL ⋅ u while: ∂  ∂x   δT =  0   0 

0

0

∂ ∂y

0

0

∂ ∂z

∂ ∂y ∂ ∂x 0

0 ∂ ∂z ∂ ∂y

∂ ∂z   0  ∂  ∂x 

Transformation of derivations of non-dimensional co-ordinates to Cartesian co-ordinates is:

160

4.1 Geometric properties of elements  ∂   ∂x  ∂ξ   ∂ξ     ∂   ∂x  =  ∂η   ∂η  ∂   ∂x     ∂ζ   ∂ζ

∂y ∂ξ ∂y ∂η ∂y ∂ζ

∂z   ∂  ∂ξ   ∂x    ∂z   ∂  ⋅  ∂η   ∂y  ∂z   ∂    ∂ζ   ∂z 

Inverse relation of solution of equations:  ∂y ∂z ∂z ∂y ∂   ∂η ∂ζ − ∂η ∂ζ  ∂x      ∂  1  ∂z ∂x ∂x ∂z −  =   ∂y  J  ∂η ∂ζ ∂η ∂ζ ∂   ∂x ∂y ∂y ∂x −     ∂z   ∂η ∂ζ ∂η ∂ζ

∂z ∂y ∂y ∂z − ∂ξ ∂ζ ∂ξ ∂ζ ∂x ∂z ∂z ∂x − ∂ξ ∂ζ ∂ξ ∂ζ ∂y ∂x ∂x ∂y − ∂ξ ∂ζ ∂ξ ∂ζ

∂y ∂z ∂z ∂y   ∂  − ∂ξ ∂η ∂ξ ∂η   ∂ξ  ∂z ∂x ∂x ∂z   ∂  − ⋅  ∂ξ ∂η ∂ξ ∂η   ∂η  ∂x ∂y ∂y ∂x   ∂  −    ∂ξ ∂η ∂ξ ∂η   ∂ζ 

Where J is determinant of Jacobi matrix of transformation: ∂x ∂y ∂z ∂x ∂z ∂y ∂y ∂z ∂x − + − ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ ∂y ∂x ∂z ∂z ∂x ∂y ∂z ∂y ∂x − + − ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ

J = J (ξ ,η , ζ ) =

Evaluation of Jacobean in any point: M M M  ∂N J = ∑∑∑ xi y j zk ⋅  i i =1 j =1 k =1  ∂ξ ∂N j − ∂ξ

∂N j ∂N k ∂N i ∂N k ∂N j ∂N j ∂N k ∂N i − + − ∂η ∂ζ ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ ∂N i ∂N k ∂N k ∂N i ∂N j ∂N k ∂N j ∂N i  + −  ∂η ∂ζ ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ 

Loading vector Vector of volume forces: fv = − ∫ NbdV V

Vector of initial strain of temperature has in case of 3D shape: ε 0 = ∆T ⋅ α x α y α z

0 0 0 

T

fe = − ∫ BTL Cε 0 dV V

161

4.1 Geometric properties of elements Geometric matrix Linear part of the deformation matrix B L , is assemble on base of matrix of differential operators δ , introduced at previous text. Non-linear part B NL , necessary for solving of stability (or for calculation of tangent matrix) is of shape:

B% NL

 ∂N1  ∂x  ∂N =  1 ∂y   ∂N1  ∂z

0 0 0 0 0 0

∂N 2 ∂x ∂N 2 ∂y ∂N 2 ∂z

K K K

       

Cauchy’s matrix of stresses: σ x τ xy τ xz    τ% = τ xy σ y τ yz  τ xz τ yz σ z    On base of presented assumptions we are able to assemble geometric matrix K G , with which we could approach to solving of eigennumbers: K L − λKG

det

=0

Mass matrix This matrix is not diagonal in most of cases. However, it is symmetric, positively defined. We convert it to diagonal one to mass vector by thereinafter described algorithm. We adjust mass matrix in general: K M = ρ ⋅ ∫ NT NdV V

to shape of vector according to algorithm: 1/ we calculate N

M i = ∑ K M (i , j ) j =1

for i = 1, K , N

2/ set up K M (i , j ) = 0

for i ≠ j

K M ( i ,i ) = M i

for i = 1, K , N

162

4.1 Geometric properties of elements Stress Results - stresses – we will obtain directly in integrating Gauss points. For each element we will reach field of stresses: σ g = [σ x1 ,K , σ xN , σ y1 ,K , σ yN , σ z1 ,K , σ zN ,τ xy1 ,K ,τ xyN ,τ yz1 ,K ,τ yzN ,τ zx1 ,K ,τ zxN ] N is number of integration points. Each element has specific number of integration points and their position in dimensionless co-ordination system ξ , η , ζ .

Taking into account evaluation of results in 3D, it is more convienent to provide user results directly in vetices σ v , what we solve with their extrapolation to vertices of brick tranship of virtual hyperbolic function F that is different for each type of element. We will reach transformation matrix Tσ , that assure extrapolation of results to vertices, from solutions of equations of function F in corresponding integration points: σ v = Tσ ⋅ σ g For solving of stability we operate with resulting stress in brick that we obtain as an arithmetical mean of vertices.

163

4.1 Geometric properties of elements Tetrahedron Number of nodes: 10 Unit coordinates of nodes: {0, 0, 0}, {1, 0, 0}, {0,1, 0}, {0,0,1} Shape functions: N1 = (1 − ξ − η − ζ ) ⋅ (1 − 2ξ − 2η − 2ζ ) N 2 = ξ ( 2ξ − 1) N 3 = η ( 2η − 1)

N 4 = ζ ( 2ζ − 1)

N 5 = 4ξ (1 − ξ − η − ζ )

N 6 = 4η (1 − ξ − η − ζ )

N 7 = 4ζ (1 − ξ − η − ζ ) N8 = 4ξη N 9 = 4ηζ N10 = 4ζξ Numeric integration: Selective, reduced (6-point), 5 for ε x , ε y , ε z , 1 for γ xy , γ yz , γ zx integration points and weight multiplier: ξ =[ 1 6

η =[ 1 6

ζ =[ 1 6

12

16

16

14

16

12

16

14

16

16

12

14

w = [ 9 20 9 20 9 20 9 20 −4 5

] 14] 14] 1 ] 14

Evaluation: Function of surface for extrapolation of qualities of integration points to vertices: F = a1 + a2ξ + a3η + a4ζ

164

4.1 Geometric properties of elements Pyramid element Number of nodes 10

Unit coordinates of nodes: {−1, −1, 0}, {1, −1, 0}, {1,1,0}, {−1,1, 0}, {0, 0,1} Shape of the function

( −ξ − η − 1) ( (1 − ξ )(1 − η ) − ζ + (ξηζ ) (1 − ζ ) ) N 2 = 14 (ξ − η − 1) ( (1 + ξ )(1 − η ) − ζ − (ξηζ ) (1 − ζ ) ) N 3 = 14 (ξ + η − 1) ( (1 + ξ )(1 + η ) − ζ + (ξηζ ) (1 − ζ ) ) N 4 = 14 ( −ξ + η − 1) ( (1 − ξ )(1 + η ) − ζ − (ξηζ ) (1 − ζ ) ) N 5 = ζ ( 2ζ − 1) N 6 = (1 + ξ − ζ )(1 − ξ − ζ )(1 − η − ζ ) ( 2 (1 − ζ ) ) N 7 = (1 + η − ζ )(1 − η − ζ )(1 + ξ − ζ ) ( 2 (1 − ζ ) ) N8 = (1 + ξ − ζ )(1 − ξ − ζ )(1 + η − ζ ) ( 2 (1 − ζ ) ) N 9 = (1 + η − ζ )(1 − η − ζ )(1 − ξ − ζ ) ( 2 (1 − ζ ) ) N10 = ζ (1 − ξ − ζ )(1 − η − ζ ) (1 − ζ ) N11 = ζ (1 + ξ − ζ )(1 − η − ζ ) (1 − ζ ) N12 = ζ (1 + ξ − ζ )(1 + η − ζ ) (1 − ζ ) N13 = ζ (1 − ξ − ζ )(1 + η − ζ ) (1 − ζ ) N1 =

1 4

Numeric integration reduced 8 - point 2 × 2 × 2 integration of points and weight multiplier : ψ 1 = 0,50661630334978769

ψ 2 = 0, 26318405556971380

ψ 3 = 0,12251482265544100

ψ 4 = 0,54415184401122496

ψ 5 = 0, 23254745125350801

ψ 6 = 0,10078588207982500

ξ = [ −ψ 1 , ψ 1 , ψ 1 , −ψ 1 , −ψ 2 , ψ 2 , ψ 2 , −ψ 2 ]

η = [ −ψ 1 , −ψ 1 , ψ 1 , ψ 1 , −ψ 2 , −ψ 2 , ψ 2 , ψ 2 ] ζ = [ψ 3 , ψ 3 , ψ 3 , ψ 3 , ψ 4 , ψ 4 , ψ 4 , ψ 4 ]

w = [ψ 5 , ψ 5 , ψ 5 , ψ 5 , ψ 6 , ψ 6 , ψ 6 , ψ 6 ]

Evaluation: Function of surface for extrapolation of qualities of integration points to vertices: 165

4.1 Geometric properties of elements F = a1 + a2ξ + a3η + a4ζ + a5ξη + a6ηζ + a7ζξ + a8ξηζ Triangular prism (wedge) Number of nodes 15

Unit coordinates of nodes: {0, 0, −1}, {1, 0, −1}, {0,1, −1}, {0, 0,1}, {1, 0,1}, {0,1,1} Shape of the function

(1 − ζ )(1 − ξ − η )( −2ξ − 2η − ζ ) N 2 = 12 ξ (1 − ζ )( 2ξ − ζ − 2 ) N 3 = 12 η (1 − ζ )( 2η − ζ − 2 ) N 4 = 12 (1 + ζ )(1 − ξ − η )( −2ξ − 2η + ζ ) N 5 = 12 ξ (1 + ζ )( 2ξ + ζ − 2 ) N 6 = 12 η (1 + ζ )( 2η + ζ − 2 ) N 7 = 2ξ (1 − ζ )(1 − ξ − η ) N8 = 2ξη (1 − ζ ) N1 =

N 9 = 2η (1 − ζ )(1 − ξ − η )

1 2

N10 = 2ξ (1 + ζ )(1 − ξ − η ) N11 = 2ξη (1 + ζ )

N12 = 2η (1 + ζ )(1 − ξ − η ) N13 = (1 − ξ − η ) (1 − ζ 2 ) N14 = ξ (1 − ζ 2 ) N15 = η (1 − ζ 2 )

Numeric integration reduced 6 - point 3 × 2 integration points and weight multiplier : ξ = [1 6, 2 3, 1 6, 1 6, 2 3, 1 6]

η = [1 6, 1 6, 2 3, 1 6, 1 6, 2 3] ζ =  −1 3 , −1 3 , −1 3 , 1 w = [1 3, 1 3, 1 3, 1 3, 1 3, 1 3]

3,1

3,1

3 

Evaluation: Function of surface for extrapolation of qualities of integration points to vertices: Function of extrapolation is obtained by combination of triangle and line element:

( b1 + b2ξ + b3η ) , ( b4 + b5ζ ) , their multiplying and adjusting to shape: F = a1 + a2ξ + a3η + a4ζ + a5ξζ + a6ηζ

166


8-nodal brick Number of nodes 20

Unit coordinates of nodes: {−1, −1, −1}, {1, −1, −1}, {1,1, −1}, {−1,1, −1}, {−1, −1,1}, {1, −1,1}, {1,1,1}, {−1,1,1} Shape of the function N1 = 18 (1 − ξ )(1 − η )(1 − ζ )( −ξ − η − ζ − 2 )

N 2 = 18 (1 + ξ )(1 − η )(1 − ζ )( ξ − η − ζ − 2 ) N 3 = 18 (1 + ξ )(1 + η )(1 − ζ )( ξ + η − ζ − 2 )

N 4 = 18 (1 − ξ )(1 + η )(1 − ζ )( −ξ + η − ζ − 2 ) N 5 = 18 (1 − ξ )(1 − η )(1 + ζ )( −ξ − η + ζ − 2 )

N 6 = 18 (1 + ξ )(1 − η )(1 + ζ )( ξ − η + ζ − 2 )

N 7 = 81 (1 + ξ )(1 + η )(1 + ζ )( ξ + η + ζ − 2 ) N8 = 18 (1 − ξ )(1 + η )(1 + ζ )( −ξ + η + ζ − 2 ) N9 =

1 4

(1 − ξ ) (1 − η )(1 − ζ ) (1 + ξ ) (1 − η ) (1 − ζ ) (1 − ξ ) (1 + η )(1 − ζ ) (1 − ξ ) (1 − η ) (1 − ζ ) (1 − ξ ) (1 −η )(1 + ζ ) (1 + ξ ) (1 − η ) (1 + ζ )

N10 =

1 4

N11 =

1 4

N12 =

1 4

N13 =

1 4

N14 =

1 4

2

2

2

2

2

2

N15 =

1 4

N16 =

1 4

N17 =

1 4

N18 =

1 4

N19 =

1 4

N 20 =

1 4

(1 − ξ ) (1 + η )(1 + ζ ) (1 − ξ ) (1 − η ) (1 + ζ ) (1 − ξ )(1 − η ) (1 − ζ ) (1 + ξ )(1 − η ) (1 − ζ ) (1 + ξ )(1 + η ) (1 − ζ ) (1 − ξ )(1 + η ) (1 − ζ ) 2

2

2 2

2

2

Numeric integration reduced 8 - point reduced (2 × 2 × 2) integration points and weight multiplier: ξ =  −1 3 , 1 3 , 1 3 , −1 3 , − 1 3 ,1 3 , 1 3 , − 1 3  η =  −1 3 , −1 3 , 1 3 , 1 3 , − 1 3 , −1 3 , 1 3 , 1 3  ζ =  −1 3 , −1 3 , −1 3 , −1 3 , 1 3 , 1 3 , 1 3 , 1 3  w = [ 1,1,1,1,1,1,1,1]

167

4.1 Geometric properties of elements Evaluation: Function of surface for extrapolation of qualities from integration points to vertices: F = a1 + a2ξ + a3η + a4ζ + a5ξη + a6ηζ + a7ζξ + a8ξηζ Examples For comparison we have selected following examples. Example No. 1 Plate Fig. E4.2, a × b = 6 × 4 m , thickness h = 0, 2 m , E = 20 GPa , µ = 0,15 , ρ = 2300 kg ⋅ m -3 , supported along entire circumference fixing. We have solved own frequency and own shapes.

Figure E4.2: Plate fixed along circumference.

dividing

f1 [Hz] f 2 [Hz] f3 [Hz]

6 × 4 ×1

54.10

90.68

162.7

12 × 8 × 1

46.98

72.53

116.9

60 × 40 × 2

45.61

70.05

110.1

analytically 45.974 71.145

112.58

Results.

168


Figure E4.3: Dynamics - 1st own shape of oscillation.

Figure E4.4: Dynamics - 2nd own shape of oscillation.

Figure E4.5: Dynamics - 3rd own shape of oscillation.

169

4.1 Geometric properties of elements Example No. 2 Plate Fig. E4.6, a × b = 9 × 3 m , thickness h = 0,1 m , E = 32,5 GPa , µ = 0, 0 , supported along entire circumference articulated. Uniform membrane loading qn = −3, 33 MNm -2 , resultant N = −1 MN . We have solved critical loading and own shapes. Wall buckling in square valves - this shape represents minimal resistance against buckling.

Figure E4.6: Plate along circumference fulcrumed, axially loaded.

dividing

N cr [MN]

18 × 6 × 2 45 × 15 × 2 90 × 30 × 2 analytically

42.87 35.50 34.71 33.01

Results.

Figure E4.7: Stability - 1. own shape of buckling.

170

4.1 Geometric properties of elements Example No. 3 Plate of previous example - fig. E4.8. Loading uniform shear qt = 100 kNm -2 at circumference. We have solved critical loading and own shapes, fig. E4.9.

Figure E4.8: Plate by circumference fulcrumed, loaded by shear flow.

dividing

Qcr [MN.m-1]

18 × 6 × 2 45 × 15 × 2 analytically

24.26 17.76 17.46 Results.

Figure E4.9: Stability – 1st own shape of buckling.

171

4.1 Geometric properties of elements Example No.4 Rotation of the element as rigid body, fig. E4.10. Theory of large rotations. Brick supported by hinge in 1 node, loaded with force in oposite node, solved by geometric nonlinearity according to Newton-Raphson.

Figure E4.10: Geometry and deformed shape.

CONTACTS A connection of two rigid plates (2D elements) between which the relevant physical relations hold true is called a contact. A contact allows for a small mutual displacement of the two elements. Basic assumption is, that during increases of load will not come to change of contact space of 2 elements, which have a contact among themselves. From that follows necessity of knowledge of a contact region before calculation. By this assumption is limited number of tasks, which are possible to solve. In the next stage of development can come to further development in this issue on the basis of user response. The limitation of the scope of the problem follows from the introduction of an element that is geometrically bound to two contact surfaces. The displacement (translation) in the plane of the contact must be small. No such restriction is necessary for the normal direction. The stress-strain diagram in the normal direction can be: - full action in both tension and compression,

172

4.1 Geometric properties of elements - elimination of tension, - elimination of compression. The most frequent is the second situation when two rigid elements are in mutual contact and can be separated from each other. The stiffness in the normal direction is taken from stiffness of the basic material of the contact plates. The stress-strain diagram in the tangential direction can be: - full action in shear, - elastic Coulomb friction, - elastic friction with a limited maximum shear τ s , τ s = µ σ n - max. allowed stress in friction σ n normal stress on the contact plane µ frictional coefficient - rigid friction. The most common is the second situation when the shearing failure in the contact surfaces occurs only after the ultimate shear force, which depends on the pressing force and friction coefficient, has been exceeded. Process of defining the contact is necessary except of geometry of contact areas, to choose adequate physical law of contact. This one is defined especially for normal treatment on contact plane and tangentially in its plane (friction). Normal influence does not cause bigger problems in calculation. On the contrary, by introduction of friction conditions in plane of the contact and mainly, which is dependent on normal force to the surface (Coloumb) mostly in the course of calculation causes problems with convergence, whole calculation can be markedly slowed down..

Elimination of tension

Elastic Coulomb friction

173

4.2 Physical properties of elementse

4.2 Physical properties of elementse 4.2.1 Physical models of materials of elements An article dealing with physical properties of elements is not, strictly speaking, relevant to the FEM issue, as FEM does not apply any special physical laws, i.e. relations between static quantities (internal forces, components of stress tensor) and geometric quantities (components of displacement or rotation and their derivatives, components of strain tensor). These laws must be observed in any method: analytical and numerical, classical and modern. With regard to the variety of materials used in analysed buildings, structures and their environment or subsoil, there exists a separate and wide-ranging scientific and technological field based on an experimental basis. If a physical law is to be used in technical applications, it is not enough just to philosophise them out. It is essential to obtain for them reliable parameters as input data for calculations. Otherwise, any prognosis of behaviour of an object has no chance to find a purposeful application. FEM assumes that a reliable research of characteristics of all materials present in the analysed problem has been carried out and that stress-strain diagrams are available and valid for all configurations (states)) predictable in the structure (assembly, service, limit). These states are more analysed by means of constitutive laws, which can contain also the process of gradual loading and unloading, i.e. the path of the load or components of vector u a tensors σ , ε at time t . It is suitable to divide the problems analysed by means of FEM into two fundamentally different groups: a) Finite problems, in which it is not necessary to take into account the way how the analysed state of stress or deformation originated, i.e. the history of the analysed state. The most typical example is the analysis of an elastic system. In general, such a system is defined by uniquely corresponding (i) tensor stress field σ and (i) tensor strain field ε , without the necessity to know what was leading to the current state. For example, it is not important whether some components of σ were greater or lower before, etc. This general definition includes, as a special case, also the educational theorem saying that after unloading an elastic body returns to the original state. Naturally, it returns to this original state after loading too, on condition that it was subjected to lower load before, and if we consider the original state without stress and strain (σ = ε = 0) . Then, if all external loads disappear, the body returns to the original state as well. If there was some initial stress σ 0 , the body returns to this stress. It may seem superfluous to speak of returns, as the history is completely irrelevant and it is sufficient to analyse just a single particular state of the elastic body. In simpler situations the relation between components of σ and components of ε is linear, which can be written according to no. 8 and 9 tab., art. 2.3 and art. 3.3.2, formula. (3.3.15) (3.3.17) – preferably in matrix form to be as concise as possible: σ = Cε

(4.2.1)

with square matrix of physical constants C of dimension ( s, s ) , if s is the number of components of tensors σ and ε . According to Glossary, the dimension in ordinary elastic 174

4.2 Physical properties of elementse problems is s = 3 to 6. The same article contains examples where the matrix vector σ is composed of what is termed internal forces of 1D or 3D element. They represent in fact integrals of stress components and their moment over the cross-section of a beam or over the thickness of a slab, wall or shell, see the following art. 4.2.2. to 4. Most programs assume that for one element C is a matrix of constants, which means that non-homogenous shapes must be divided to so small elements so that the change of C could be modelled by a step-like diagram. On the other hand, there are programs that allow for modelling of changes of C inside the element, usually by means of a polynomial of low degree that is defined unequivocally by nodal values of C , similarly to shape and base functions (briefly physical functions). For example, the very useful element SHELL43-ANSYS of a Mindlin plate is a general quadrilateral, in the vertices of which different thickness h1 to h4 can be specified, which represents a physical term (not geometrical) for 2D space, as there is no geometrical dimension in the direction of the normal. Function h( x, y ) is inside the element a bi-linear polynomial of type (4.1.55). Generally, instead of (4.2.1) we can write: σ (x) = C(x) ε(x)

(4.2.2)

where x = x for 1D elements, x = ( x, y ) for 2D elements and x = ( x, y, z ) for 3D elements, alternatively with dimensionless coordinates ξ ; ξ ,η ; ξ ,η , ζ or cylindrical ones r , z , ϕ etc. It is typical for FEM that the variability of C does not lead to any significant numerical difficulties. According to the table in art. 2.3., no. 10, stress matrix Σ is simply varying as well, and if the integration in no. 11 at stiffness matrix K is carried out exclusively numerically using the Gauss formula, only the values of C in the integration nodes apply. The case of a nonlinear law, combining components of σ with components ε , is more complex. This includes in particular laws (3.3.24) and (3.3.26) that can be used to differentiate tensor σ and tensor ε out of the density of potential energy W or complementary energy Φ , which leads to a linear relation (4.2.1) or (4.2.2), with the later happening only occasionally if we deal with special quadratic functions W and Φ . Generally nonlinear relations with a vector function are obtained experimentally from tests with a controlled strain: σ = f (ε )

(4.2.3)

or with a controlled stress: ε = g (σ )

(4.2.4)

or the implicit dependence F (σ , ε ) = 0

(4.2.5)

The feature of the finite problem – i.e. the independence on the history preceding the analysed state – is preserved. It represents a nonlinear elasticity and formula (4.2.5) can be applied to some elastic-plastic substances only for the special configuration of initial monotonous loading, without the possibility to monitor the behaviour during subsequent unloading. b) Problems depending on the path include all problems excluding those stated under (a). A typical example is a viscous substance in which the magnitude of the deformation is determined among others by the speed of stress alteration over time t , over which the stress acts. The range of such problems is significantly broader than for (a). The finite problems are in fact a mere abstraction that is useful for the first (and in practice often the last as well) 175

4.2 Physical properties of elementse prognosis of the behaviour of a real object. Even a brief classification of problems under (b) goes beyond the scope of this text. Practically speaking, it always represents a nonlinear problem. In order to clearly demonstrate how the dependence on the path of the analysed process over time arises, let us combine what is termed elementary rheological models (fig. 4.23) into systems, in which two or more models are arranged one next to another (parallel arrangement) or one after another (serial arrangement). Three simple examples are shown in fig. 4.24 with the scheme of corresponding stress-strain diagrams for force P(t ) monotonously increasing over time t attached at the bottom. If P(t ) has other than monotonous distribution, i.e. if the system is alternately loaded and unloaded, the behaviour of rheological models can be very complex. Only systems with up to 3 – 4 parts have been analysed in detail so far. An illustrative example of the Kelvin two-part model is in fig. 4.25. In this text, let us emphasize just the fact following from the graph in fig. 4.25c): If we know the magnitude of force P that acts on the system at time t , we can determine neither the deformation, nor the percentage of force P transferred by model H (spring) and by N (damper). If the problem is to be defined at all, we must know the state of the system at time t = 0 , e.g. the initial state without any stress and deformation, and the complete history of the magnitude of P up to the analysed time instant, i.e. function P(t ) . In the case of such a definition we talk about loading by controlled force, as we assume that a certain mechanism introduces the force P(t ) into the system in this way. If the deformation is defined (elongation or shortening in the cited examples), we talk about controlled deformation and force P acting on the system is unknown. This force could be read from a dynamometer connected to jaws of a press used for the test carried out with the model. It could be simply demonstrated that force P depends on the complete history of the introduced displacement u (t ) prior to the instant of measurement. In particular, fig. 4.25c shows that even such a simple two-part model (spring and damper) completely fails to comply with the fundamental requirements of finite problems – the independence of the state on the path of load over time. It behaves fundamentally differently from an elastic body. In order to determine σ (in our example just two internal forces Pel , Pvis ), it is insufficient to input just ε (in our example just translation u ), similarly as the pure value u does not determine force P . If we study the model at a certain time instant t , in which there is a certain displacement u , we cannot determine the force in the spring and in the damper at all, until we know what has been happening since the initial state. Such behaviour can be seen quite often e.g. in foundation engineering where the settlement of a building strongly depends on the history that the subsoil went through not only in geological terms (history of geostatic stress-state and structural strength) but also in terms of the construction process (excavations, temporary unloading of the vertical geostatic component) and operation (alternate loading of adjacent objects, e.g. in a system of tanks for bulk material). More details can be found in [8, 9, 22].

176


Figure 4.23: Six basic rheological models marked by initials of authors (the last without author) and corresponding stressstrain diagrams for monotonous loading by force P causing elongation u of the model.

177


Figure 4.24: Three examples of complex rheological models and corresponding stress-strain diagrams with a brief wording of the constitutional law for the behaviour at time t under constant force P.

178


Figure 4.25: The Kelvin model according to fig. 4.24 K for cyclic loading by a constant force P alternated with complete unloading to the value P = 0. The dependence of the model state (i.e. its deformation and percentage of transferred force P in parts H and N) on the full history of loading starting from the initial state of the model.

179


4.2.2 What is the effect of physical properties in FEM algorithms

4.2.2.1 3D constitutive laws To find out the physical properties of materials is the problem for experiments and no FEM algorithm can substitute this kind of information. On the other hand, it is convenient if the author of the FEM program and the experimenter were not completely isolated from each other, as only a certain form of information is useful for a particular FEM methodology and any other result would have to be transformed in a complicated way. The complex constitutive laws in the implicit form (4.2.5) are usually implemented only in problem oriented – not versatile – programs, e.g. for the prognosis of behaviour of large soil structures (earth dams, caverns, tunnels). An ordinary user most often comes across the explicit form (4.2.3) that takes form (4.2.1) for linear problems. This form can be directly and without problems applied to FEM algorithms for 3D problems, in which the dimension of matrix σ and ε is 6 and symmetrical matrix C (6, 6) has 21 of elements that may be generally different, see art. Glossary, 2.3. and 3.3.3. The tensor notation clearly indicates the stress components σ ij (i, j = 1, 2, 3) , strain components ε kl (k , l = 1, 2, 3) and physical constants cijkl . The summation is made over indices k , l and the signs of these sums are omitted for the sake of brevity: σ ij = cijkl ε kl

(4.2.6)

Thus, a fourth-order tensor of physical constants c is introduced. It is transformed as a tensor during the rotation of the coordinate trihedral x, y, z . The matrix notation takes advantage of the symmetry of tensors σ and ε , i.e. σ ij = σ ji , ε kl = ε lk . Six generally different components are numbered using one index and are sorted into column matrices written, for the sake of brevity, as transposed rows: σ = [σ 11 , σ 22 , σ 33 , σ 23 , σ 31 , σ 12 ] = [σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 ] T

T

(4.2.7)

ε = [ε 11 , ε 22 , ε 33 , ε 23 , ε 31 , ε 12 ] = [ε1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 ] T

T

If we keep this cyclic order, tensor ε is in a geometrically linear mechanics derived from vector field u according to Glossary by means of the rule ε = Gu (the first matrix G in Glossary). The same rule applies to equilibrium conditions in the transposed form: G T σ + x = 0 . Matrix of physical constants C in the relation σ = Cε is a square matrix (6,6) with the elements marked with two subscripts according Glossary: C = [Cik ]

i, k = 1, 2,K , 6

(4.2.8)

Due to the symmetry Cik = Cki , the maximum number of different constants is 21 for general material anisotropy. Various types of orthotropy reduce this number and in the case of isotropy in which the physical properties do not depend on the analysed direction, all the 180

4.2 Physical properties of elementse constants can be derived from two fundamental characteristics. Usually, modulus of elasticity E and Poisson’s coefficient of transverse contraction ν are used. These two are related to shear modulus G through the formula G=

E 2 (1 + ν )

(4.2.9)

Optionally, another relation can be applied – see e.g. [31, 42-46]. Nonlinear problems, firstly, make the relation between strain tensor ε and displacement tensor u more complex (i.e. matrix of operators G ), and, secondly, the definition of stress σ must be done more accurately. There exist voluminous both theoretical and experimental studies the outcomes of which have been implemented into FEM practically always only in the simplified form (4.2.1) with constants (4.2.8), e.g. in incremental form in matrix notation: dσ = C dε

(4.2.10)

or in index notation dσ ij = cijkl d ε kl

(4.2.11)

Constants cijkl have in this for the nature of tangent modulus. They are part of input data for different levels of stress and deformation, in the form of a function, table or graph. The reason why the present-day exceptionally developed nonlinear constitutional relations cannot get into FEM algorithms in a different way is simple. Contemporary computers and their peripherals can operate with continuous inputs and outputs, the cooperation with analogous devices is not eliminated, etc., but the only tried and standing core or atom of every algorithm is the linear matrix algebra, and in FEM in particular the solution of a system of linear algebraic equations. Without exaggeration, we can say that all FEM algorithms conform to this fact. Several concessions must be made in nonlinear problems just with regard to the physical laws, so that we get linear systems, as even the simplest assumptions without these concessions result in systems of equations of degree 3 to 4, which are totally unusable for many reasons. Consequently, an ordinary FEM user comes into contact with the physical properties of materials in fact only in the form of a certain number of physical constants: at least two ( E , ν ) and at most the whole set of 21 constants for different levels of stress and deformation of an anisotropic matter. They can neither input them randomly nor assume that the FEM program derives them from nothing. They represent irreplaceable input data. Users obtain them from standards, from their own or ordered experiment, from accepted published sources, from databanks (e.g. geo-fund), etc. In addition to these constants that represent deformation properties of a substance, the physical constants include also strength characteristics, which are in fact certain limit values or relations between them. If these limit values are reached, one of the following happens: failure, plasticizing, change of structure of a multi-phase environment, bulk material, soil, etc. They represent technical limits of elasticity, plasticity, strength, structural stability and, in general, quantitative transition towards another physical model. In FEM algorithms they often appear in the solution of limit states, e.g. propagation of cracks in reinforced concrete structures initiated when the tensile strength of concrete is locally exceeded, which with increasing load results in continuous areas that all the internal forces are transferred solely by 181

4.2 Physical properties of elementse the steel reinforcement. After its plasticization, a certain collapse mechanism is created and a part or the whole structure collapses. Strength characteristics of soil are a typical example already for the simplest Coulomb law on limit shear stress τ nt , acting in the direction of t on the surface with normal n : τ nt = c + σ n tg ϕ

(4.2.12)

If there is no normal stress σ n , the limit shear stress equals to cohesion c , which for bulk materials equals to their cohesion, for continuous model it is the tensile strength that originates under pure tension τ nt in the direction that halves the right angle (n, t ) . For noncohesive soil (sand, gravel) c = 0 , and what remains is just the friction between individual grains with the coefficient f = tg ϕ , with ϕ representing the friction angle. This parameter, however, comes in play even in cohesive soil and, consequently, a FEM program for a strength problem of this type has two constants: c, ϕ . They may have different value for various construction and service stages. For example, at the beginning we have values cu , ϕu (subscript u means undrained), when consolidation of subsoil ended we get values cef , ϕef (subscript ef = effective, directly between soil grains). Deformation and strength constants defined in a real 3D material are theoretically clearly and unambiguously specified for the needs of FEM algorithms. They can be found experimentally and are constitutive in nature, i.e. hold the information on the real behaviour of material in physical processes and this information is not distorted by additional speculations about the reduction of the dimension of the problem. Only 3D objects exist in reality and their 3D physical models fully respect this fact.

4.2.2.2 Reduction of the dimension of a problem Problems that are prevailing in common practice model given objects by means of 1D and 2D. Therefore, the problem that emerges is what the physical properties of such elements can be if we know that only 3D bodies are real. A wide-ranging series of studies have been focusing on this topic since the very beginning of FEM. In fact, every users’ manual must tackle this issue and give instructions concerning the representation of physical properties of the material of the structure. Consequently, we limit ourselves to the main principles that usually remain unnoticed in the torrent of otherwise useful information. In the first place, it is convenient to empathise with the dimension of the element and consider what can be defined geometrically in this dimension. For 2D elements (plates, walls, shells) nothing extending into the third dimension can be defined this way, i.e. thickness h of elements. According to art. 4.1.5. an element is defined on a surface – let us call it planar (surface) – with coordinates x P , y P (fig. 4.26a). A spatially curved 2D element is defined in a similar way on a 2D surface, which adds neither complexity nor simplicity to our problem – the 2D space is just curved (inserted into a 3D space), but is still a 2D space (fig. 4.26b). A real physical element is a 3D body, filled continuously with mass normals to the middle-plane ( x P , y P ) of the element in such a way that we measure out h 2 in the direction + z P and

182

4.2 Physical properties of elementse − z P , which creates the positive face of the element z P = + h 2 and negative face z P = − h 2 . The physical volume of the element is located in between them. They form the physical boundary of the element in z P direction. Boundaries in x P and y P directions are defined by outer normals. If h is constant, we obtain an element of a constant thickness. In general, the thickness may be variable. This however changes nothing on the core of the following text. A simple fact is true: The thickness in a 2D element has a character of a physical constant. It is never used in any algorithm on its own without being multiplied by some physical constant, e.g. deformational constant E , gravitational constant g , etc. Only 3D stress-state and strain σ, ε can exist in a real physical element. In a 2D element we have to introduce some representatives of it. It follows from the nature of the problem that it must represent the distribution of σ, ε just inside the interval − h 2 ≤ z P ≤ + h 2 , which is not geometrically present in the 2D space, i.e. it must substitute functions σ( z P ) , ε( z P ) in this interval. As far as the functions of stress are concerned, the simplest approach – in terms of statics – is to work with their resultants or with the resultants of their moments. It is at the same time necessary in order to satisfy the elementary conditions of equilibrium. This is the way we define in planar coordinates what is termed membrane (wall) and bending (plate) internal forces in walls, plates and shells, as stated in Glossary: nx = ∫ σ x dz

n y = ∫ σ y dz

qxy = ∫ τ xy dz = q yx

mx = ∫ σ x z dz

m y = ∫ σ y z dz

mxy = ∫ τ xy z dz = myx

qx = ∫ τ xz dz

q y = ∫ τ yz dz

(4.2.13)

183


Figure 4.26: a) Planar 2D-element, b) Curved 2D-element, generally curvilinear planar coordinates, c) 2D internal forces and their positive direction, d) 1D-element, centroidal coordinates xC yC zC, e) 1D internal forces and their positive direction in a sign-consistent convention, f) end vectors f.

184

4.2 Physical properties of elementse The integration is carried out from − h 2 to h 2 . If no negative sign is introduced, we deal with a sign-consistent definition of internal forces, which features certain advantages in connection with practical outputs and their processing, determination of the location of the tensile face under bending, etc. The dimension of quantities n and q is [kN/m], that of m is [kNm/m=kN]. Technical names: n axial forces (tension +, compression –), q shear forces, m moments ( mx , m y bending moments, mxy torsional moment). The positive direction is clear from fig. 4.26c. Stress components σ z are neglected, as under common load they have no technical meaning in comparison with other components. A more dramatic reduction is made in the case of 1D beam elements. We can operate just with a straight line segment within the interval 0 ≤ xC ≤ L or with a part of an arc 0 ≤ s ≤ L for curved elements, which is not a significant difference. The missing two dimensions in y C and z C directions cannot be in 1D space considered to be geometrical quantities and they do not appear in algorithms separately without some physical constant. We use the centroidal (central) coordinate system xC (beam axis), y C , z C (principal centroidal axes) as in fig. 4.26d. All classical (Navier, Bernoulli) and also newer (Mindlin) theories of beams neglect stress components σ y , σ z , τ yz in comparison with σ x , τ xy , τ xz . For the sake of brevity we omit in x, y, z the index C of the central axes. The considered components must be represented through their resultants over what is called a cross-section of the beam, related to its centroid: the term cross-section Ax means a 2D area of a 3D beam volume, filled with points that are located in the normal plane ( y C z C ) , perpendicular to the xC axis. The whole volume of the beam is an infinitely rigid of its cross-sections with boundaries represented by its faces (endsections) and surface. Similarly, we could construe the physical volume corresponding to a 2D element as an infinitely rigid set of its mass normals, also known as the pin model. In 1D elements we have to do with a single point, centroid C of the cross-section, into which the resultants of three considered stress components in this cross-section are concentrated. In practice, we call them internal beam forces: N = ∫∫ σ x dAx

Qy = ∫∫ τ xy dAx

Qz = ∫∫ τ xz dAx

M x = ∫∫ (τ xz y − τ xy z )dAx

M y = ∫∫ σ x zdAx

M z = ∫∫ σ x ydAx

(4.2.14)

The integration is performed over the whole area Ax of the cross-section. The formulas contain just one minus sign at the torsional moment M x , which cannot be eliminated regardless of which coordinate system we have selected. There is no minus sign before the integral. This sign-consistent convention has its practical advantages in the design of beams subjected to bending (fig. 4.26d). Positive moments M y , M z result in tension at the positive side of the beam, which is the side with positive y C , z C coordinates. In the theory of beams it may be convenient to use the minus sign before the last integral (4.2.14), which results in vector-consistent convention. In this convention, the components of vectors P2 , M 2 acting at the right-hand end of the beam (generally at x = l ) have the same sign as the internal forces at this end. At the other end, components P1 , M1 have the opposite sign. Besides, also other conventions, usually traditional in nature, are used in technical practice. An ordinary user can learn about the convention applied in the purchased program from manuals and quite often it 185

4.2 Physical properties of elementse is not necessary to know it at all, if the program offers convenient graphical postprocessors that draw the diagram on the tensile side where the reinforcement must be inserted, etc. It is however useful to be familiar with the convention if one compares the output of reactions (vector components acting at bound ends of beams) with internal forces that are not simple vectors, but double-vectors. Each section of the beam at point x1 has two banks, one limits the part x < x1 before the section and the other one the part x > x1 after the section. The double-vector of internal forces is composed of the vector acting on the part before the section and the vector with the opposite direction of the components acting on the part after the section. This feature is caused by the fact that internal forces (4.2.14) are the integral representation of stress components that are not vectors, but tensors. After a part of the body is released in the section, the action of the released part manifests itself only through interrupted stress components that follow the principle of action and reaction on both parts of the body, in the opposite direction when expressed in the vector form. For the sake of clarity, let us state in detail the relation between end-vectors acting on a 1D element (fig. 4.26e) T

f = f1T , f 2T  =  Px1 Py1 Pz1 M x1 M y1 M z1 Px 2 Py 2 Pz 2 M x 2 M y 2 M z 2 

T

(4.2.15)

with internal forces at its ends T

S k = S1T , S T2  =  N x1 Qy1 Qz1 M xd 1 M yd 1 M zd 1 N x 2 Qy 2 Qz 2 M xd 2 M yd 2 M zd 2 

T

(4.2.16)

For all conventions we have only changes of the sign, which can be briefly written by using diagonal matrix DIA (12,12) , which have zeros in all elements except the main (descending) diagonal. The elements of this diagonal are equal to plus or minus one. Consequently, the following relation is always true: f = DIA × S k

S k = DIA × f

(4.2.17)

In the vector-consistent convention we have DIA = DIAG [ −1, −1, −1, −1, −1, −1;1, 1, 1, 1,1, 1]

(4.2.18)

Using the convenient sign-consistent convention (fig. 4.26e): DIA = DIAG [ −1, −1, −1, −1, −1, 1; 1, 1, 1, 1, 1, −1]

(4.2.19)

Program ESA uses convention that can be defined by following signs in DIAG: DIA = DIAG [ −1, 1, 1, −1, 1, −1;1, −1, −1, 1, −1, 1]

(4.2.20)

4.2.2.3 Description of deformation in a reduced problem In reality, only 3D volumes exist. Their overall deformation is completely described by means of three functions u , v, w of three variables x, y, z . From this we can derive the deformation in details (detailed description of geometrical changes), i.e. elongation, compression, skewing, volume and shape changes in an arbitrarily small vicinity of every 186

4.2 Physical properties of elementse point ( x, y, z ) . To be more precise, each vector field u = [u ( x, y, z ) v( x, y, z ) w( x, y, z ) ]

T

(4.2.21)

can be assigned a tensor field of strain ε through some geometrical operation G , which in mathematical terms represents mainly differentiation ε = Gu

(4.2.22)

The simplest operation G is in what is termed geometrically linear mechanics, where we have at most the first partial derivative with respect to x, y, z , in the first power (examples can be found in Glossary at symbol G where six components of classical deformation tensor are listed under symbol ε ). They represent three relative changes in length ε in x − , y − , z − direction and three slopes γ which represent the change of the originally right angles between directions ( y, z ) , ( z , x) and ( x, y ) . Formulas (4.2.24) can be simply derived through the summation of the contributions of individual displacement components to the total slope. Formulas (4.2.25) that are very similar apply to the components of small rotation defined already in art. 3.3.2, in which there are differences of these contributions. In the right handed system x, y, z , the sign of the contribution is defined by the unified convention of the positive rotation about the + x -axis (direction from + y towards + z ) and similarly around the + y -axis (from + z towards + x ) and about the + z -axis (from + x towards + y ). The components are the average from the rotation of all line segments passing the given point and they can be roughly interpreted as the rotation of a small vicinity of the point as a rigid unit. Consequently, in linear mechanics we get nine components of tensor ε and operation G has a simple form that can be explicitly written: ε x = ∂u ∂x

ε y = ∂v ∂y

ε z = ∂w ∂z

(4.2.23)

1 1 ε yz = γ yz = (∂w / ∂y + ∂v / ∂z ) 2 2 1 1 ε zx = γ zx = (∂u / ∂z + ∂w / ∂x) 2 2 1 1 ε xy = γ xy = (∂v / ∂x + ∂u / ∂y ) 2 2

(4.2.24)

1 (∂w / ∂y − ∂v / ∂z ) 2 1 ω y = ω zx = (∂u / ∂z − ∂w / ∂x) 2 1 ω z = ω xy = (∂v / ∂x − ∂u / ∂y ) 2

(4.2.25)

ω x = ω yz =

The components of slope are written in theoretical texts with subscripts indicating the plane of the slope. The quantity ε can be then transformed as a tensor if the position of the coordinatetrihedral x, y, z changes. Most FEM manuals, however, use technical components of γ that are equal to the whole change of the right angle between the original directions, i.e. full slope. The components of rotation (4.2.25) appear in FEM only from 1975 – 1980 and are marked by one index indicating the rotation axis or by two indices indicating the plane of rotation. 187

4.2 Physical properties of elementse There is no stress-state directly linked to these components as they do not appear in the physical relation σ = Cε . A statement by B. M. Irons – well-known co-author with O. C. Zienkiewicz – saying that any attempt to introduce the rotation into the deformation parameter is a waste of company money caused that really effective elements with rotations were developed only 20 years after FEM was born. After 2D elements also excellent 3D elements were created – see art. 4.1.6.2. Now we can exploit a whole set of 1D, 2D and 3D elements, including their problem-free connection as all the shared nodes have the same valence. We have six deformation parameters with a clear meaning of displacements (translations) and rotations, which fact is whole-heartedly welcome in engineering practice. It is not significant for the following reasoning that in geometrically nonlinear mechanics on condition of large displacements or deformations formula (4.2.22) can be more complex than (4.2.23) - (4.2.25), as it contains – in commonly used approximation – the derivatives in the quadratic terms and we strictly meet also irrational expressions under square roots. The main point is that relation (4.2.22) is fully sufficient to find out the tensor of deformation in a 3D problem with 3D elements. In the case of 2D elements, it is clear that it is not possible to define vector field u (4.2.21) directly in the 2D area of an element (e.g. plane ( x P , y P ) ) also for the reason that the variable z P is not positioned in this area. Therefore, it is necessary to start with values u( x P , y P ) and assume a certain analytical extension towards the z P direction. This is not typical only of FEM, but also of classical methods for the solution of 2D systems, walls, plates and shells. The general starting point is what is termed Reissner’s sequences [62], i.e. in fact the expansion into Taylor’s series in variable z , which can be written separately for each component u , v, w of vector u and we can even use different number of members in the series. Using an economical common notation omitting the superscript P for planar coordinates x, y, z of a 2D element: u ( x, y, z ) = u ( x, y, 0) +

x  ∂u ( x, y, z )    +… 1!  ∂z  z =0

z m  ∂ m u ( x, y, z )  …+   + Z m +1 m!  ∂z m  z =0

(4.2.26)

The higher the number m of terms, the lower the absolute value of the remainder Z m +1 – on condition that certain requirements mentioned in the cited texts dealing with the mathematical analysis are met. Standard FEM algorithms now use only two examples of expansion (4.2.26) that respect just the first two terms and that differ just in the assumption with regard to the second term. Manuals usually call them the Kirchhoff and Mindlin 2D elements. Historically younger and more topical is the Mindlin element, which is used practically in all present-day packages of FEM programs for both plates and shells. Its theory is based on a quite inconspicuous article [63] from 1951 that explained the difference between the behaviour of thin and thick plates subjected to bending vibration through the introduction of two rotations ϕ x , ϕ y of the mass normal of the plate which were independent on its deflection surface w . At the same time, it kept Kirchhoff’s assumption that the normal remains straight, its length (h) does not change and on condition of small slopes of the deflection surface it has everywhere the same displacement component w , so that just the identity remains at this component in series (4.2.26) 188

4.2 Physical properties of elementse w( x, y, z ) = w( x, y, 0) = w0 ( x, y )

(4.2.27)

In addition, let us neglect the membrane (wall) stress-state and deformation of the element that is in linear mechanics commonly superimposed to the bending ones and is analysed conveniently with components u , v in the plane of the 2D element using [43], or with all details using [31]. Under bending, in the middle plane of a physical element u ( x, y, 0) = v( x, y, 0) = 0

(4.2.28)

which eliminates the first term of (4.2.26). The value of the second term is given by the first derivative ∂u ∂z or ∂v ∂z at point z = 0 , which for small slopes represents the inclination angle of the mass normal around y C or xC , i.e. ϕ y or ϕ x . According to fig. 4.27 we introduce vector signs for the angles and thus the second terms of (4.2.26) get the following form in a right handed system: u ( x, y, z ) = zϕ ( x, y ) v( x, y, z ) = − zϕ x ( x, y )

(4.2.29)

189


Figure 4.27: a) Geometric assumption of Mindlin plate, b) Kirchhoff plate, c) Mindlin beam, d) Navier beam.

190

4.2 Physical properties of elementse This, together with (4.2.27), determines the complete vector field u of the physical element of a Mindlin plate by means of three functions defined only in its centroidal plane z = 0 : w0 ( x, y )

ϕ x ( x, y )

ϕ y ( x, y )

(4.2.30)

Notice, please, that the originally right angle between axes ( x P , z P ) and ( y P , z P ) changes after deformation due to sloping γ xy = ∂w0 ∂x + ϕ y

γ yz = ∂w0 ∂y − ϕ x

(4.2.31)

Kirchhoff element implemented in older FEM programs and used as an alternative for plates and shells of very small thickness even in the present-day programs is based on a radical assumption that no shear deformations (4.2.31) and thus: γ xz = 0

ϕ y = − ∂w0 ∂x

γ yz = 0

ϕ x = ∂w0 ∂y

(4.2.32)

As a result, instead of three functions (4.2.30) just a single function w0 ( x, y ) is sufficient to determine the complete vector field u in the physical element: w( x, y, z ) = w0 ( x, y ) u ( x, y, z ) = − z ∂w0 ( x, y )

(4.2.33)

v( x, y, z ) = − z ∂w0 ( x, y ) Modern FEM programs usually employ Mindlin elements and use a special numerical procedure – called briefly shear locking – to ensure that the reduction of the element thickness h does not lead to numerical instability and that the results are close to the Kirchhoff assumption, which is a theoretically and experimentally verified trend. Individual programs differ only in the magnitude of the parameter used for the shear locking and serious manuals state this value. A user who would obtain slightly different results from two welltried programs, can thus find the explanation and does not lose time by searching for nonexisting bugs. The situation is similar with 1D elements where we today use mainly Mindlin elements offering the slope and (as an alternative) less classical elements based on the Bernoulli – Navier assumption that are called Navier elements. A brief explanation is in fig. 4.27c,d. Instead of mass normals h we have in beams (non-warping) cross-sections Ax . Instead of the middle plane z P = 0 we have a centroidal axis xC of the beam where y C = z C = 0 . In terms of geometry we can consider the Mindlin beam a one-dimensional Cosserat continuum in which each point has all six degrees of freedom both in translation and rotation with regard to centroidal axes xC , y C , z C . For the sake of brevity, let us now omit superscript C and write the distribution of the degree of freedom along the x -axis in the form of six functions of one variable x : u ( x), v( x), w( x), ϕ x ( x), ϕ y ( x), ϕ z ( x)

(4.2.34)

If vector field (4.2.21) – defined through three functions u ( x, y, z ) , v( x, y, z ) – is to be completely defined by these six functions, we have to make some assumption concerning the distribution along y and z coordinates, i.e. across cross-section Ax . Theoretically, it is

191

4.2 Physical properties of elementse possible to start with the expansions similar to (4.2.26) and define the whole hierarchy of beam models. Practically, this is necessarily awkward, as technically accurate models can be obtained through simple assumptions concerning the behaviour of cross-section Ax . In principle, there are two possibilities: kk) Points ( x, y, z ) filling planar cross-section Ax in the initial state fill after deformation some general area Ax* , i.e. the cross-section warps, it is no longer planar. The mode of warping can be prescribed on the basis of theoretical analyses of the behaviour of real beams. The best known is the prescription of displacement component u A ( y, z ) perpendicular to the, generally inclined, plane of Ax according to Vlasov’s rule for sector areas of what is termed thin-walled beams. This introduces, in addition to (4.2.34), also another geometrical quantity – warping parameter ω ( x) . ll) Cross-section Ax moves during the deformation of a beam as a rigid planar unit. Therefore, it has just six degrees of freedom (4.2.34) defined in its centroid C . If there is no bond between them, it represents a more general instance of a beam – let us call it briefly the Mindlin beam (fig. 4.27c) following the analogy with the Mindlin model of a plate discussed earlier in the text. In order to get a better overview of the distribution of u , v, w in the 3D volume of such a beam, it proved useful in technical practice to introduce two groups of load cases and deformations called axial and transverse effects. If we select the x, y, z axes in the principal centroidal axes of the beam and its cross-section xC , y C , z C (and omit the superscript C for the sake of brevity), these effects are independent on each other, can be analysed separately and superimposed. The axial effects can be divided into translation (tension or compression), when each cross-section simply moves in the x -direction as a rigid unit, i.e. u ( x, y , z ) = u ( x ) v( x, y, z ) = w( x, y, z ) = 0 (4.2.35) and rotational (torsion) when the cross-section rotates about the x-axis and, therefore, for small rotations ϕ x we have: u ( x, y , z ) = 0 v( x, y, z ) = − zϕ x ( x) w( x, y, z ) = yϕ x ( x) (4.2.36)

The transverse effects can be divided into effects in the plane ( xy ) when component v applies as what is termed deflection and component ϕ z as the rotation of cross-section Ax . In geometrically linear reasoning we neglect the effect of the rotation and we consider v to be relating to all points of the cross-section. We get the following displacement field: u ( x, y, z ) = − yϕ z ( x)

v ( x, y , z ) = v ( x )

w( x, y, z ) = 0

(4.2.37)

The effects in the plane ( xz ) are characterised by deflection w and rotation of the crosssection ϕ y with this displacement field: u ( x, y, z ) = zϕ y ( x)

v ( x, y , z ) = 0

w( x, y, z ) = w( x)

(4.2.38)

This defines the whole displacement field using six functions of one variable (4.2.34), i.e. the reduction of a 3D problem to a 1D problem is carried out. The classical theory of beams 192

4.2 Physical properties of elementse uses another simplification and defines a beam without the slope (Bernouilli, Navier) that can be briefly called according to the analogy with plates a Navier beam. The added requirement is that the plane of cross-section Ax remains even after deformation the normal (perpendicular) plane to the bent beam axis. The last two functions (4.2.34) can be then derived from the first function through differentiation with the sign convention for angles as vectors (fig. 4.27d) respected: ϕ y ( x) = − dw( x) dx

ϕ z ( x) = dv( x) dx

(4.2.39)

The deformation of such a classical beam is described just by four functions of one variable u, v, w, ϕ x . The assumption of the perpendicularity of Ax to the beam axis can also be understood in the way that the change of the initially right angles between directions x, y and x, z is not permitted, i.e. there is no slope γ xy = 0

γ xz = 0

(4.2.40)

This physical assumption is disputable in reality, as it implies zero shear transverse stresses after multiplication by shear modulus G τ xy = Gγ xy = 0

τ xz = Gγ xz = 0

(4.2.41)

After the integration over cross-section Ax this results in zero shear forces: Qy = 0

Qz = 0

(4.2.42)

which is possible only for load by pure bending without any shear, i.e. constant moments M zd or M yd with a constant curvature of the beam. This is a very rare situation. Every other type of load leads to non-zero Q, τ , γ . This changes the right angles between the initial directions x, y and x, z . Formulas (4.2.39) no longer apply and must be extended by the effect of slope. It is convenient to write it with derivatives of w on the left-hand side and extend thus the idea about the behaviour of the Mindlin beam: dw( x) dx = −ϕ y ( x) + γ xz ( x) dv( x) dx = ϕ z ( x) + γ xy ( x)

(4.2.43)

A detailed analysis of the corollaries with regard to FEM is contained in [42], chapter 3. For the purpose of this text, formulas (4.2.34) - (4.2.43) are sufficient for further physical reasoning.

4.2.2.4 Components of deformation in a reduced problem If we know the complete displacement field u in a 3D body modelled by means of 2D or 1D elements, we can derive tensor field of strain ε without any reduction problems through the ordinary 3D rule in the simplest linear form ε = Gu (art. 2.3., no. 6 and 7), or using more complex operations in the case of a nonlinear problem. Then we can determine the tensor stress field σ from the constitutional physical relations in the form σ = Cε (art. 2.3., no. 8) or using more complex relations (art. 3.3.2), again, without any technical speculations. If we reduce the dimension of the problem to 2D or 1D, we do not work directly with tensor σ , but 193

4.2 Physical properties of elementse with its integral representatives, 2D internal forces (4.2.13) or 1D internal forces (4.2.14), fig. 4.26, which can be summarised in matrix vectors: s = [sTm , sTb ]

S = STa , STb 

s m = [nx , ny , qxy ]T

s b = [mx , my , mxy , qx , q y ]T

S a = [ N , M xd ]

T

Sb = STbxy , STbxz 

T

Sbxy = Qy , M zd 

T

(4.2.44)

T

Sbxz = Qz , M yd 

T

(4.2.45)

2D internal forces (4.2.44) were divided to membrane ones (wall) marked with index m and bending (plate) with index b . 1D internal forces (4.2.45), which are usually written in the following order S =  N , Qy , Qz , M xd , M yd , M zd 

T

(4.2.46)

were divided into axial effects S a and transverse effects, which are further divided to bending and shear in plane ( XY ) and ( XZ ) , i.e. Sbxy and Sbxz . This is convenient for physical relations into which we need to introduce – instead of tensor ε – components on which internal forces s or S do the virtual work. At the same time, the virtual works done by components of σ on components of ε corresponding to the applied reduction of the dimension from 3D to 2D or 1D equal, which can be formally written by the following equations:

∫∫∫ ε

T

3D

∫∫∫ ε 3D

σ dxdydz = ∫∫ εT2Ds dxdy

(4.2.47)

2D

T

σ dxdydz =

∫ε

T 1D

S dx

(4.2.48)

1D

The derivation of strain components ε 2D and ε1D of reduced problems is one of the main themes in the textbooks of theoretical mechanics. It is often done using various popularising forms, in which there is a risk of factual mistakes. Taking into account only more recent texts, we can recommend [2, 4, 31, 42, 43, 61]. The consistent conception of reduced problems in a way that it is always necessary to know the exact situation in the 3D body whose 2D or 1D model is analysed was introduced already in classical mechanics by G. Hamel, A. Nádai and in particular A. and L. Föppl [47]. The importance of this principle has not diminished with the development of numerical methods and is respected by serious authors. This was often underestimated in common practice and internal forces s , S and their strain virtual equivalents ε 2D , ε1D were unthinkingly accepted as separately existing and everything representing quantities without any regard to the fact through which reduction they originated. No attention is paid to the fact that all real bodies of any dimensions or proportions of these dimensions are only 3D bodies and that there is no other stress-state than 3D that is defined by tensors σ and ε . On the other hand, this does not prevent us from using resultants, whenever it is useful, e.g. if we need to balance the tension in a concrete structure with reinforcement. Users of FEM programs, vast majority of which employs 2D and 1D elements (i.e. the real 3D dimension is reduced), should be informed in the manual which type of reduction has been applied and what analysis the given model can be used for, see chapter 2. Any attempts to use such elements for the solution of typical 3D problems of stress 194

4.2 Physical properties of elementse concentration, etc., are pointless, as the contemporary FEM already offers both the required, even though demanding, software with 3D elements and advanced computers. Let us state here explicitly (according to manuals [42, 43]) the components of deformation assigned to internal forces s and S for the most frequent FEM problems. For 2D problems analysed by means of shell elements with internal forces (4.2.44) we can similarly divide the strain components to membrane ones (wall) ε 2D m and bending ones (plate) ε 2 Db : ε 2D = εT2 D m , εT2 D b 

ε 2 D m = ε 2D x , ε 2 D y , γ 2 D xy 

T

ε 2 D b = ϕ ′y , − ϕ x* , (ϕ *y − ϕ x′ ) , γ xz , γ yz 

T

(4.2.49)

Partial derivatives with respect to x and y are marked by a prime and a dot. Let us notice that the membrane stress-state can use standard components of deformation known from plane stress problems. The stress-state is constant along thickness h of the element. The bending stress-state is introduced for each of five internal forces by such a quantity that is present in the formula for virtual work (4.2.47) on a differential element dxdy . For bending moments mx and my with what we call direction or differentiation or reinforcement subscripts x, y (the direction of tension or reinforcement at the face of the plate), it represents mutual inclination of mass normals to the plate in sections x, x + dx and y, y + dy about the y and x axis. Similarly, for torsional moment mxy it is about the x - and y -axis, which is added together and both vector index i and signs of angles ϕ x and ϕ y are respected. For shear forces qx , q y it represents slope that can be expressed by means of three functions (4.2.30) according to formulas (4.2.31) which means that we have no additional unknown functions. The described more general Mindlin model of a plate can be for plates of a very small thickness replaced by the Kirchhoff model according to (4.2.32) – (4.2.33) with just one unknown function w0 ( x, y ) . This, however, results in a physical paradox that shear forces qx , q y are nullified due to zero slopes (4.2.32). Kirchhoff got around this through the derivation of these forces from the conditions of moment equilibrium of the dxdy element with regard to the x - and y -axis, which caused that they became independent on bending moments mx , my , mxy and they cannot be physically derived from the slope. What remains is the relation between moments and angle increments that are, according to (4.2.32), replaced by second derivatives of a single function w0 ( x, y ) – deflection surface of the plate: ϕ ′y = −∂ 2 w0 ∂x 2

−ϕ x* = −∂ 2 w0 ∂y 2

ϕ *y − ϕ x′ = −∂ 2 w0 ∂x∂y − ∂ 2 w0 ∂x∂y = −2∂ 2 w0 ∂x∂y

(4.2.50)

Let us notice that rotation components ω (art. 3.3.2) do not apply, which is in accordance with physical experience that stress arises only due to “pure deformation” and is not affected by the rotation of a small vicinity of the point as a rigid unit, or more precisely – by the average rotation of all directions passing through this point. In 1D beam problem, the six internal forces (4.2.45) are virtually assigned simple deformation quantities. Following the order of common notation and output (4.2.46), this is the following set:

195

4.2 Physical properties of elementse ε1D = ε x , γ xy , γ xz , ϑx , k y , k z 

T

(4.2.51)

Internal forces N , Qy , Qz work in the element dx on its relative elongation ε x and slope γ xy , γ xz . Internal moments M xd , M yd , M zd work on relative slope ϑx and on curvatures caused by the differences in the angles of adjacent cross-sections ϕ y , ϕ z . We can employ six degrees of freedom (4.2.34), which can be, according to art. 4.2.2.3, used to express values (4.2.51) in the following way: ε x = ∂u dx

γ xy = ∂vQ dx

γ xz = ∂wQ dx

ϑx = ∂ϕ x dx

k y = ∂ϕ y dx

k z = −∂ϕ z dx

(4.2.52)

We introduced the negative sign for curvature k z , which applies to a technically convenient definition of bending moments, see art. 4.2.2.2. and the text following formula (4.2.14). This minus sign disappears in the vector-consistent convention. Users of FEM programs do not have to know the convention, on condition that they use exclusively the graphical output of the bending moments that are drawn at the tensile face of the beam. On the other hand, in numerical outputs the sign plays an important role, is mentioned in manuals and numerous users find it on their own through a simple comparison with moment diagrams. The total (final) diagram of transverse displacement v, w can be put together from the part resulting from bending vM , wM and the part resulting from shear vQ , wQ (detailed explanation can be found in [42], chapter 3): v = vM + vQ

w = wM + wQ

(4.2.53)

At the same time, relation (4.2.43) is valid, which enables us eliminate slope γ from components (4.2.52) and deal just with the introduced six degrees of freedom (4.2.34) without decomposition (4.2.53): γ xy = dv dx − ϕ z

γ xz = dw dx + ϕ y

(4.2.54)

This gives the system of strain components (4.2.51) the form suitable for FEM: ε1D =  du dx , ( dv dx − ϕ z ) , ( dw dx + ϕ y ) , dϕ x dx , dϕ y dx , − dϕ z dx 

T

(4.2.55)

An extreme example is Navier slender beams, in which zero slope γ can be assumed, see (4.2.39) – (4.2.40). This situation is similar to the Kirchhoff 2D elements, i.e. shear forces Qy , Qz cannot be physically derived from zero γ , but only from the conditions of moment equilibrium of element dx . Instead of angles ϕ y , ϕ z , we can use the derivatives of w , because ϕ y = − dw dx , ϕ z = dv dx . Formulas (4.2.55) get simplified to the following form: ε1D =  du dx , 0, 0, dϕ x dx , − d 2 w dx 2 , − d 2 v dx 2 

T

(4.2.56)

used now only in the classical theory of beams, with the effect of transverse shear not taken into account. For 1D elements we do not encounter any more the problems related to the components of 196

4.2 Physical properties of elementse small rotation ω in 2D and 3D areas. Each point of a 1D element is the model of one crosssection of the beam with six degrees of freedom: three translations u , v, w and three rotations ϕ x , ϕ y , ϕ z . The question how to deal with them effectively comes to the fore only in geometrically nonlinear problems (art. 5.1.4) with large rotations that can no longer be treated as vectors. Some programs apply the conception according to which rotation is treated as a quaternion (four data: the direction of rotation axis α , β , γ and the magnitude of rotation ω ), see e.g. Argyris’s algorithms in [61, 65]. The incremental approach is more feasible if load increments are selected so small that they produce only small increments of rotation, with reference to the previous (reference) configuration of the structure, that may be considered vectors. However, it is always necessary to properly separate the pure strain of the beam from its rigid body motion, including its rotation around its centroidal axis.

4.2.2.5 Physical constants of 2D FEM elements In the previous paragraphs, static and geometric quantities of a reduced 2D problem were defined. These are: internal forces s (4.2.44) and strain components ε 2D (4.2.49), related to each other through the definition of virtual work (4.2.47), the density of which (the value related to a unit area) is specified by coefficient εT2D s . When a particular problem is to be analysed, it is necessary to know the physical relation between s and ε 2D . For the simplest configuration this relation can be linear and have the form of generalised Hooke’s law σ = Cε (4.2.1), (4.2.2), (4.2.6), for more complex situations incremental form is suitable (4.2.10), (4.2.11), etc. In 3D problems these relations are not burdened with any speculation, they express the physical essence or constitution of the matter – hence the name constitutional in art. 4.2.2.1. The reduction 3D→2D brings to the physical relations one geometric quantity – element thickness h – existing in the third dimension, that is not contained in the 2D area. It gets there only during the integration of σ , which gives rise to internal forces s (4.2.44). Furthermore, it comes into play when ε 2D is created from 3D tensor ε and 3D vector u . The aim is always to have the relation that is either (i) linear in all components: s = C2 ε 2 D

(4.2.57)

with the matrix of physical constants C2 (8,8), or (ii) more complex, incremental, but always – if possible – explicit, e.g. (4.2.3) etc. Matrix C2 is marked with subscript 2 corresponding to the dimension of the problem and is substantially different from matrix C of the non-reduced 3D problem, whose subscript 3 is not used, as it represents the fundamental constitutive quantities. The difference is that C2 is not determined just by the physical properties of the matter, but also by statical and geometrical assumptions that were made during the reduction of the problem to the 2D dimension. In FEM practice matrix C2 includes also what is termed shape orthotropy (models of ribbed or corrugated slabs), change of constitutive matrices C over thickness h (effect of steel bars in concrete, stratification) and similar effects. There exist a vast number of published (hundreds) and unpublished matrices C2 , starting with resources from 1900 – 1950 before FEM was developed, as this reasoning is independent on the method of solution of the structure itself. An older overview can be found in Czech publication [31], a newer FEM-oriented overview is given in [64]. Despite this, it sometimes 197

4.2 Physical properties of elementse happens that a FEM user who is to design a 2D structure cannot find what they would expect on the basis of their engineering feeling even in the new manuals [43], [65], etc. Should this happen, they can create the input data for C2 themselves. The starting point is the definitions of internal forces (4.2.13) in which stress components σ are substituted by strain components ε according to undisputed 3D constitutional relations that must be in explicit form, e.g. (4.2.1), (4.2.2) or (4.2.3). This gives rise to no speculation as to the reduction of the dimension. This only comes after the physically existing components ε are replaced by ε 2D of the 2D model. These are generally defined by energetic (virtual) equivalence (4.2.47). A FEM user practically always opts for the Mindlin model (4.2.49), occasionally with the Kirchhoff assumption (4.2.50), i.e. they get satisfied with a rigid mass normal h . The membrane components of the 2D model are thus transferred unchanged into the whole 3D body as its strain components. ε x ( x, y, z ) = ε 2D x ( x, y ) ε y ( x, y, z ) = ε 2D y ( x, y )

(4.2.58)

γ xy ( x, y, z ) = γ 2 D xy ( x, y ) The consequences of bending components of the 2D model (4.2.49) are more complex. They can be found from general formulas (4.2.23) - (4.2.25) in which the displacement components u, v, w are replaced by expressions (4.2.27), (4.2.29). We get: ε x = ∂u ∂x = zϕ ′y

γ xy = ∂v ∂x + ∂u ∂y = − zϕ x′ + zϕ y∗ = z (ϕ y∗ − ϕ x′ )

ε y = ∂v ∂y = − zϕ x∗

γ xz = ∂w ∂x + ϕ y

εz = 0

γ yz = ∂w ∂y − ϕ x

(4.2.59)

198

4.2 Physical properties of elementse For the sake of brevity, we omit the marking of variables x, y, z and index 0 at w , which is the shared deflection of the whole normal h (4.2.27). As a result, we can substitute into internal forces (4.2.13). If the constitutive 3D relation between σ and ε has the form of a general anisotropy, matrix C (6,6) in the formula σ = Cε is full and each stress component depends on all six strain components, e.g. σ x = c11ε x + c12ε y + c13ε z + c14γ yz + c15γ zx + c16γ xy

(4.2.60)

After integration of the first formula (4.2.13) we get the following relation between internal force nx and components ε 2D (4.2.49) of the strain of the 2D model: ⌠ c11 (ε 2D x + zϕ ′y ) + c12 (ε 2 D y − zϕ x∗ ) + c13 ⋅ 0 +  nx =    dz ∗   + c γ + c γ + c (γ ′ + z ( ϕ − ϕ ))  y x ⌡  14 yz 15 zx 16 2D xy

(4.2.61)

The integration is carried out over the thickness of the physical body of the element − h 2 ≤ z ≤ h 2 . After the integration we get the required formula that forms the first row in matrix (4.2.57): nx = C2,11ε 2D x + C2,12ε 2D y + C2,13γ 2 D xy + C2,14ϕ ′y2 Dx + +C2,15 (−ϕ x* ) + C2,16 (ϕ *y − ϕ x′ ) + C2,17γ xz + C2,18γ yz

(4.2.62)

where: C2,11 = ∫ c11dz

C2,12 = ∫ c12 dz

C2,13 = ∫ c16 dz

C2,14 = ∫ c11 z dz

C2,15 = ∫ c12 z dz

C2,16 = ∫ c16 z dz

C2,17 = ∫ c15 dz

C2,18 = ∫ c14 dz

(4.2.63)

Similarly, we get the 2nd to 8th line of matrix (4.2.57) from the 2nd to 8th formula (4.2.13). If we sort the internal forces (4.2.13) into a matrix vector s = sTm , sTb 

T

sb =  mx , my , mxy , qx , q y 

s m =  nx , n y , qxy  T

T

(4.2.64)

in a similar way we sorted the strain components (4.2.49), we can write the whole obtained physical relation (4.2.57). In general, matrix C2 is full, but always symmetrical, which means that we do not have 8 × 8 = 64 constants C2ij , but only 36 different constants. The material of the structure can be generally anisotropic, which results in the full integration of membrane and bending conditions. It is clear already from the dependence of membrane internal force nx on all, i.e. also bending, components of strain. Such complex modelling is rare in practice, even though the functioning of the FEM program itself is not affected. An ordinary user cannot, due to financial and time constraints, afford to order all the values that form constitutive matrix C (21 items), almost no data can be traced in data banks, and it can be considered a success if reliable C are obtained for a much simpler orthotropic configuration. In the case of orthotropy, matrix C splits into two separate submatrices Cn (3,3) for axial 199

4.2 Physical properties of elementse components and Cs (3,3) for shear components. On condition that the axes of orthotropy coincide with the planar axes of the element, Cs is diagonal, which reduces the number of independent physical constants of the 3D matter to 6 + 3 = 9 : 0  C (3,3) C(6,6) =  n Cs (3,3)   0  c11 c12 c13  Cn (3,3) = c12 c 22 c23   c13 c 23 c33 

c 44 Cs (3,3) =  0  0

0 c55 0

0 0  c66 

(4.2.65)

Axial components σ z , ε z (reasons and consequences can be found in [31], [43]) are neglected in walls, plates and shells, as they are considerably lower than σ x , σ y , ε x , ε y . It is then sufficient to know 3 physical constants c11 , c12 , c22 for Cn and 3 for Cs , in total 6 constants. It can be easily proved that this significantly simplifies matrix C2 too. For example, only the first two members with c11 , c12 apply in formulas (4.2.60) and (4.2.61), which means that four of eight constants (4.2.63) are not employed and what remains are two constants linked to membrane components ε 2D x , ε 2D y and two for bending components ϕ ′y , −ϕ x∗ . It is now convenient to add an important note that often escapes the attention of most users of FEM programs: Constitutive 3D constants cij after the integrals remain in formulas (4.2.63). Therefore, it is possible to model even structures with varying physical properties over the thickness, e.g. members with greater stiffness at one face, layered members (called sandwiches), etc. The behaviour of corresponding 2D members is then, however, more general than commonly known. If we relate all internal forces to their geometrically middle plane z = 0 , it, in general, results in the interaction of membrane and bending stress-state. This is not a hindrance in general shell (plate-wall) elements, on condition that the FEM program allows for such an input. However, the situation simplifies if this variation of physical properties is symmetrical with reference to plane z = 0 . This happens for most sandwich plates with both soft core (more rigid surface layers) and rigid core (symmetrical insulation cover layers). As a result, both c11 ( z ) and c12 ( z ) in formulas (4.2.63) are symmetrical functions of z , product zc11 ( z ) and zc12 ( z ) are asymmetrical functions and their integral over the interval − h 2, h 2 equals zero. Consequently, another two constants C2,14 , C2,15 are no longer employed and membrane forces nx , n y depend only on membrane components of strain ε 2D x , ε 2D y . If the axes of material orthotropy were different from the planar x- and y - axes of the 2D element (which is a frequent situation, as the axes for automatic mesh generation are assigned to individual elements by the program according to unified rules independently on the physical properties), then just the third member of (4.2.62) with membrane slope γ xy would be added, which does not affect the independence of the membrane conditions with regard to bending. A similar formula can be actually easily proved even for the next two components n y , qxy and, moreover, it can be shown that also the bending conditions are independent – not affected by the membrane components.

200

4.2 Physical properties of elementse The just derived independence is implemented into most (almost all) FEM programs in such a way that the user is required to provide (for the needs of the input) only very simplified matrix of physical constants of relation (4.2.57) that splits into two independent matrices: s = C2 m ε 2 D m

m (3,1)

(3,3)

s = C2 b ε 2 D b

b (5,1)

(3,1)

(4.2.66)

(5,5) (5,1)

Both matrices are symmetrical and, therefore, C2m has just six independent elements that must be input by the user of the FEM program in a given order: (1,1), (2, 2), (3,3), (1, 2), (1,3), (2,3)

(4.2.67)

In case of isotropic plate made of a material which is defined just by its modulus of elasticity, Poisson’s coefficient and thickness: E ,ν , h

(4.2.68)

Then the following constitutive equations are valid for the plane stress: σ x = E ( ε x + νε y ) (1 −ν 2 )

σ y = E ( ε y + νε y ) (1 −ν 2 )

(4.2.69)

τ xy = Gγ xy The coefficients of the membrane material stiffness matrix are then as follows: C2,11 = C2,22 = Eh (1 −ν 2 )

C2,33 = Gh

C2,12 = ν C2,11

C2,13 = C2,23 = 0

(4.2.70)

Matrix C2b for bending of the element is a little more complicated. On the grounds of symmetry, this does not involve 5 × 5 = 25 different numbers, but only 15 for a general anisotropy and 6 + 3 = 9 for the most frequent configuration, in which case we can assume mutual independence of the elementary bending and shear states. Actually, in that case, matrix C2b (5,5) further splits into two separate symmetrical matrices C2bo (3,3) , C2bs (2, 2) , on condition that we divide the corresponding internal forces sb (4.2.64) and strains ε 2 Db (4.2.49) to bending and shear: S = [ N , Qy , Qz , M xd , M yd , M zd ]T

(4.2.71)

sbo = C2 bo ε 2 D bo

(4.2.72)

sbs = C2bs ε 2D bs

Nine elements of matrices of physical constants must be specified: C11 , C22 , C33 , C44 , C55 , C12 , C13 , C23 , C45

(4.2.73)

The constitutive equations can be then written in the following expanded form:  mx   C11 m    y  C12  mxy  = C13     qx   0  q y   0  

C12 C22

C13 C23

0 0

C23 0 0

C33 0 0

0 C44 C45

0   ϕ ′y    0   −ϕ x*  * 0  ⋅ (ϕ y − ϕ x′ )     C45   γ xz  C55   γ yz 

(4.2.74)

201

4.2 Physical properties of elementse The simplest configuration is an isotropic substance that was already analysed in the section dealing with the membrane stress-state, see (4.2.68) – (4.2.70). For this situation, only three constants are input and they are stored in the first three positions. The program automatically detects from the zeros stored in the remaining positions that an isotropic matter is dealt with and calculates all nine constants. Formulas can be obtained through an elementary integration over the interval − h 2 ≤ z ≤ h 2 , similarly as it was shown for the membrane forces. As the integral definition contains at moments in (4.2.13) multiplication with the z -coordinate, the substitution of stress components that are linear in z produces integrals from the square of z , i.e. the order is h3 . The order for shear forces (constants in z ) is just h . We get: C11 = C22 = Eh3 (12 − 12ν 2 )

C33 = Gh3 12

C44 = C55 = Gh β

C12 = ν C11

(4.2.75)

C13 = C23 = C45 = 0 For a uniform distribution of shear stress τ xz , τ yz over thickness h we have β = 1 . Using the analogy with a beam of rectangular cross-section, it is possible to apply parabolic distribution and shear coefficient β = 1, 2 . Depending on the circumstances, also different values of β can be used for 2D models of 3D physical walls, plates and shell of various shapes. None of the physical relations contains components of rotation ω . This complies with perceiving a classical continuum (Boltzmann) as a thick set of non-oriented, i.e. nonpolarised, points in which it is “not possible to detect if they rotate". Each point has only three relevant degrees of freedom – displacements. In a 2D model, each of its points substitutes the whole mass normal of length h (the thickness of the modelled wall, plate or shell). As it follows from both the Kirchhoff (no slope) and Mindlin (with slope) hypothesis, the normal remains straight and its rotation models the rotation of a point of the 2D model. In terms of geometry we assign one point with additional three degrees of freedom – rotational. This is the case of the polarised (Cosserat) continuum, where “it is possible to discover that the point rotates”, as if we equip it with some suitable marks, e.g. like on a globe. Therefore, some FEM manuals contain information that the Cosserat model is used. This, however, does not apply in terms of physics. The Cosserat continuum has nine different components of stress and strain (non-symmetrical tensors, theorem of reciprocity does not apply) and in addition to common components σ , τ it contains another stress-state due to the mutual rotation of points – termed moment stress-state. In 2D models, for example, moments (4.2.74) are also linked to the difference of rotations of adjacent points, but physical constants C of this relation are obtained after the reduction of the 3D dimension to 2D by means of constants of a classical continuum.

4.2.2.6 Physical constants of 1D FEM elements The reduction of a 3D problem to a 1D one is much more dramatic and only six constants – on condition that we deal with an ordinary prismatic straight 1D element without warping cross-section – are left to express the physical properties of a 3D body that is 202

4.2 Physical properties of elementse considered to be a 1D beam. It features six internal forces (4.2.14) and six degrees of freedom (4.2.34) of point x that models cross-section Ax and six components of 1D strain (4.2.51), virtually assigned to the internal forces in such a way that each of them does virtual work only on its own component. The advantage is that, contrary to formula (4.2.57) for 2D problems, we now deal with a simpler, diagonal relation in the form S = C1ε1D

(4.2.76)

where according to (4.2.14) a (4.2.51) there is: S = [ N , Qy , Qz , M xd , M yd , M zd ]T

(4.2.77)

ε1D = [ex , γ xy , γ xz , ϑx , k y , − k z ]T

(4.2.78)

C1 = DIAG C1,11 , C1,22 , C1,33 , C1,44 , C1,55 , C1,66 

(4.2.79)

For 1D elements, we pay attention to only three stress components σ x , τ xy , τ xz and, for these components, we employ just three rows from the constitutive physical relation σ = Cε . An isotropic material is practically always assumed and we can apply the simplest formulas: σ x = Eε x

τ xy = Gγ xy

τ xz = Gγ xz

(4.2.80)

With regard to axial effects, tension and compression cause no problems and the integration of the first formula gets N = ∫∫ Eε x dAx = ε x ∫∫ E dAx thus C1,11 = ∫∫ E dAx

(4.2.81)

which, for the simplest configuration of a constant E over cross-section Ax , is a well known stiffness quantity C1,11 = EAx

(4.2.82)

Quite often we have two moduli, for example steel Ea and concrete Eb , in cross-section part Aa (not necessarily continuous, e.g. steel reinforcement bars) and Ab . The integration is carried out separately for Aa and Ab and we get C1,11 = Ea Aa + Eb Ab

(4.2.83)

Apparently, even more general configurations, e.g. laminated beams, are not complicated. It is usually convenient to factor out some constant modulus E0 and deal with the dimensionless ratio after the integral: ⌠⌠ E ( y, z ) C1,11 = E0  dAx ⌡⌡ E0

(4.2.84)

This form is suitable also for physically nonlinear substances with a tangent modulus 203

4.2 Physical properties of elementse E of the diagram σ x = f (ε x ) . Torsion as the second axial effect requires the integration of the fourth formula in (4.2.14), in which the substitution (4.2.80) introduces slopes γ that originate from relative twisting ϑx = dϕ x dx . Generally, such a strain is accompanied with warping of the cross-section, i.e. with formation of displacement components u ( y, z ) , which breaks the assumption that the cross-section behaves like a rigid unit (under all other five types of loading). If we insisted that no warping of the cross-section occurs, we would have to introduce residual axial stresses σ x ( y, z ) , which is in fact really done in what is termed restrained warping. Consequently, the seventh internal force – bimoment B – must be defined, whose resultant is zero, i.e. it results into a system stress σ x that is in equilibrium in the cross-section. This theory is applied in what is termed thin-walled beams, which means, in particular, various steel rolled cross-sections, etc. This configuration will be skipped here and the readers are referred to standard texts dealing with the theory of V. Z. Vlasov, A. Umansky, etc., contained in a directly applicable form in the textbooks of steel structures and corresponding standards. We limit ourselves just to what is termed thick-walled (solid) beams of ordinary rectangular, circular, T-shaped, etc. cross-section. For them we can admit, without any negative impact on technical accuracy, that free warping occurs under torsion, which means the state when σ x ( y, z ) = 0 , i.e. no additional axial stresses constrain the beam. Even this simplified problem cannot be solved elementary, but by means of either the strain method with the unknown function of warping u ( y, z ) , or force method with what is termed the Prandtl stress function Φ( y, z ) . It is related to shear components of the stress in the fourth formula in (4.2.14) through derivative formulas τ xy = ∂Φ ∂z

τ xz = −∂Φ ∂y

(4.2.85)

Therefore, function Φ is a certain potential, similarly to the classical Airy function known in planar elasticity problems or the Maxwell or Galerkin functions used in spatial problems. Many years of practice lead to the compilation of tables with results for the most frequent cross-section. These tables contain both torsional stiffness C1,44 from formula (4.2.76): M xd = C1,44ϑx

(4.2.86)

and formulas for extreme shear stress, usually in the form that is similar to the traditional formula for bending: max τ = M xd Wτ The selection of values for ordinary cross-sections can be found in tabular form in [43], pp. 63-65 – for isotropic shear modulus G where the following applies C1,44 = GI k

(4.2.87)

where I k [m4] is purely geometric cross-sectional characteristic with the same physical dimension [m4] as moments of inertia I x , I y or polar moment of inertia I p . This is a good place to mention one common mistake committed by users of FEM programs due to the fact that quantities I x , I y , I p , I k have the same dimension [m4]. This mistake results from the fact that for a circular cross-section (symmetrical hollow cross-section as well) with 204

4.2 Physical properties of elementse diameter D = 2 R we get the following equity: I k = I p = ∫∫ r 2 dAx =

π 4 π 4 R = D 2 32

(4.2.88)

This equity does not exist for any other cross-section, which means that I k is not a polar moment of inertia and, moreover, is not related to inertia (such as I x , I y ) at all. Values I p contained in numerous tables serve a completely different purpose and must be substituted for I k ! This can be actually clearly seen from the oldest approximate formula derived already by Saint-Venant for cross-sections close in shape to circle, e.g. polygons: I k = Ax4 40 I p

(4.2.89)

For a precise circle this formula produces

(π R )

2 4

Ik =

40π R 4 2

=

π 3R4 , 20

which, applying the notation π 3 = π 2 ⋅ π and approximation π 2 = 10 , corresponds with formula (4.2.88) variation defined by means of exact π 2 = 9,87 , i.e. 1.3%. Of all the formulas for individual cross-section shapes listed in [42], pp. 63-65, let us mentions here a rectangle with sides h ≥ b : h≥b

1

1.2

1.5

2

3

∞

Multiplier:

Ik

0.1404

0.1661

0.1957

0.2286

0.2633

0.3333

b3 h

The assumption of Saint-Venant torsion is strongly negatively affected for ratio h / b greater than 3. Often published approximate formula I k = b3 h 3 for slender rectangles is not recommended. It is better to model such beams using 2D shell elements. The cited publication [42], pp. 59-60, contains the analysis of the accuracy of the Grashof-Zhuravsky assumption for transverse shear (see the following text). For slender, flat cross-sections, e.g. a shallow ellipse, the extreme shear stress is burdened with an error that is largest for ν = 0 , when it even increases towards infinity with the growing shallowness of the ellipse. The error vanishes for ν = 0.5 . It is therefore recommended to abandon the 1D model for common cross-sections with ν = 0.2 to 0.3 and employ 2D elements. Bending and shear in ( XY ) plane are in a 1D beam model expressed through the relation between two internal forces (4.2.45) and two strain components (4.2.51) Qy = C1,22γ xy

M zd = C1,66 k z

(4.2.90)

Both slope and curvature can be expressed by means of formulas (4.2.52), (4.2.54), i.e. the form (4.2.55) is used instead of (4.2.51) for the strain components, in order to have only six degrees of freedom (4.2.34) in the FEM algorithm: Qy = C1,22 ( dv dx − ϕ z )

M zd = −C1,66 dϕ z dx

(4.2.91)

205

4.2 Physical properties of elementse The procedure to determine physical constants C1,22 and C1,66 is the same as for axial effects, i.e. it is based on the integral definition (4.2.14), where τ xy and σ x are substituted by strain components γ xy and ε x in their undisputed 3D meaning. Then we can proceed to 2D components of strain according to (4.2.52), (4.2.54) and calculate the integrals over the area of cross-section Ax . A problem is encountered in the case of τ xy – what distribution of τ xy ( y, z ) should be assumed, as the reduction principle 3D → 1D in art. 4.2.2.4. leaves this issue completely open. It could be e.g. possible to take τ xy ( y, z ) = τ xy = const . , which corresponds in physical terms to γ xy ( y, z ) = γ xy = τ xy G . The second formula in (4.2.14) then gives a very simple result: Qy = ∫∫ τ xy dAx = ∫∫ Gγ xy dAx = GAxγ xy C1,22 = GAx

(4.2.92)

Unfortunately, this would break not only the concordance with experimentally proven distribution of τ xy ( y, z ) , but also the fundamental principle such as the reciprocity theorem for shear stress components τ xy = τ yx at the face of the beam that is obviously without any shear stress when subjected to transverse axial load. Fortunately enough, already the classical theory of elasticity found a simple hypothesis (Grashof-Zhuravsky) that provides sufficient compliance with all requirements for common solid cross-sections, on condition that these are symmetrical around the bending XY plane, i.e. around the vertical principal centroidal axis Y C . The resultant of the stress components τ xy , τ xz is τ xn = τ xy2 + τ xz2

(4.2.93)

for which we assume that its ray intersects the Y C axis in point O that is for certain level of y shared by all coordinates. The point O is defined by the intersection of the tangent to the profil contour in the level y . Due to symmetry, both tangents in points ( y, z ) and ( y, − z ) intersect in the same point O. This overcomes the indeterminacy of the 1D model and the subsequent procedure is already straightforward. It is presented in detail in all elementary textbooks dealing with the theory of elasticity. FEM users should be aware of one dissimilarity in the recommended formulas: Less technical texts, university textbooks in particular, simplify the problem even more and in the energetic equivalence (4.2.47) completely neglect the impact of components τ xz in comparison with τ xy under bending and shear in the XY plane. Similarly, they neglect the influence of τ xy in comparison with τ xz for a problem in the XZ plane – they simply take into account only the components that are parallel with the plane of the problem. More elaborate texts respect both components. In order to provide for a practical application in FEM inputs, the result is always modified to the form analogous to (4.2.92). Most often, what is termed cross-sectional area Ay effective in shear is introduced in plane XY , and similarly Az in plane XZ : Ay = Ax β y

Az = Ax β z

(4.2.94)

206

4.2 Physical properties of elementse This introduces coefficients β y and β z expressing the influence of the distribution of shear stresses over the cross-section. Roughly, the rule applies that β > 1 , i.e. Ay < Ax , Az < Ax , which means that the assumption that Ax = Ay = Az stiffens the beam with regard to transverse shear and may be thus dangerous. For common cross-sections we can consider values β = 1.2 (rectangle), β = 1.18 (full circle), β = 1.11 (full hexagon), β = 2.0 (thin-walled circles - pipes), β = 2.1 to 2.8 (rolled cross-sections no. 8 to 45 under normal bending and shear in the plane of their web). The traditional form of formula (4.2.92) is modified according to (4.2.94) to the following form: C1,22 = GAy

C1,33 = GAz

(4.2.95)

The second physical constant of formula (4.2.91) for bending due to moment M zd is simpler with regard to both objectivity and calculation. The sixth formula of (4.2.14) is used, where σ x = Eε x = E du dx = E d ( yϕ z ) dx = Ey dϕ z dx It directly gives the sought after relation: M zd = ∫∫ σ x ydAx =

( ∫∫ Ey dA ) dϕ 2

x

z

dx

with physical constant C1,66 = ∫∫ Ey 2 dAx

(4.2.96)

If the modulus of elasticity is constant over cross-section Ax , E can be factored and the integral contains just the well-known expression for the moment of inertia of a cross-section about its principal centroidal z -axis: C1,66 = EI z

(4.2.97)

Similarly, for bending in plane XZ , we can obtain the relation between moment M yd and strain component dϕ y dx of the 1D model of a beam according to the fifth formula in (4.2.14): M yd = ∫∫ σ x zdAx =

( ∫∫ Ez dA ) dϕ 2

x

y

dx

with physical constant C1,55 = ∫∫ Ez 2 dAx

(4.2.98)

which is, for constant E over the cross-section, equal to an ordinary product: C1,55 = EI y

(4.2.99)

with the moment of inertia of the cross-section about its principal centroidal y -axis. For a homogenous cross section the pertinent cross section stiffness can be introduced in the following formula (4.2.100):

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Homogenous cross-section:

C1,11

C1,22

C1,33

C1,44

C1,55

C1,66

EAx

GAy

GAz

GI x

EI y

EI z (4.2.100)

For nonhomogenous cross sectio these stiffnesses must be calculated by pertinent numerical quadrature. For homogenous cross section the following values must be input: Ax , Ay , Az , I x , I y , I z , E , ν

(4.2.101)

and the program calculates the required G = E (2 + 2ν ) . The calculation of Ay , Az , I x is an easy task for users only in the case of very simple cross-sections, which are overviewed in [43]. A manual calculation is not possible for complex configurations. Therefore, it was completely automated, e.g. by program [94], in which the user defines just the shape of the cross-section. Coefficient β for the effective cross-sectional area in shear Ay = A β y , Az = A β z is determined by the program from the exact theoretical formula by means of a numerical integration. Torsional stiffness I k can be calculated by FEM that exploits a triangular element of T. Moan (1973, Trondheim, Norway) with three stress parameters in a node. It is a function of torsion F and its partial derivative with respect to y and z , which is equal to components of shear stress τ xy , τ xz , occurring under Saint-Venant torsion of a beam with free warping (unconstrained torsion of solid (thick-walled) beams).

4.2.2.7 Physical constants of toroids A special status in FEM belongs to rotationally symmetrical shapes divided to toroidal elements according to art. 4.1.6.3. Even if in the inputs we deal with a 2D area, in which the given 3D body is generated through the rotation about an axis, we do not use 2D physical relations of art. 4.2.2.5, but actually the non-reduced 3D relations σ = Cε , or more complex 3D relations between σ and ε , yet in cylindrical coordinates (r , z , ϕ ) . It is of no advantage to introduce here internal forces. FEM program can be divided to two groups. Most of them can be applied only to problems with a rotationally symmetrical load. It is actually the first term in the expansion of a general load with respect to variable ϕ , i.e. into a series with terms cos nϕ , n = 0, 1, 2, K , as cos(0) = 1 = const . Advanced programs allow for the input of an arbitrary load whose harmonic analysis with respect to cos nϕ provides a set of loading components, the effect of which is analysed separately, because it features different stiffness matrices, and therefore the left-hand sides of the equations Kd = f are also different. Most often, only the first term ( cosϕ ) is used in practice to approximately express the effect of wind (pressure and suction).

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4.2.2.8 Gas elements Special type of element that is possible to be used in case of validity of state equation of ideal gases. They are constructions that are closed, filling if constituted by gas that at the same time assures synergism. It could to be insulation glass, where among several solid layers of glass and case of glued foil is present of given production temperature and pressure. The potential energy of external forces causes a change in the potential energy of the gas that depends on the change of volume and pressure. Π i = p∆V

(4.2.102)

The dependence of the work on these quantities is shown in the diagram.

Figure 4.28 Diagram of isotermic action of an ideal gas.

Let us assume that there is no change of the temperature in the gas – let us consider an isometric action according to Boyle Mariotte law which says that for a fixed mass of ideal gas at fixed temperature, the product of pressure and volume is a constant. pV = const .

(4.2.103)

209


Figure E4.11: Air hall solidly supported along bottom edge, asymmetrically loaded by uniform force loading, geometry, deformed shape.

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5.1 Introduction to the Theory and Practice of Creation of FEM Models

5 Modelling of Structures for FEM Analysis 5.1 Introduction to the Theory and Practice of Creation of FEM Models 5.1.1 Present-day Approach to Modelling of Structures and Soil Environment 5.1.1.1 Objects and Terms When “New Generation Computing, An International Journal on Fifth Generation Computers” (University of Tokyo, editor T. Moto-oka) magazine was established in 1983 with the participation of reputable experts from all developed countries, there was a prognosis made in the editorial articles that the 5th generation of computers would be on commercial sale in about 1989-1990. Their expected parameters, based on the experienced theoretical studies of informatics, arose a considerable optimism as far as the possibilities of modelling of engineering tasks were concerned, in particular the full automation of this process liberating the user from demanding considerations. As we know today, the predictions were only partly met. Moreover, in about 1990 the theory of modelling was further advanced with active participation of engineers, mathematicians and physiologists and it became clear that leaving the engineering problems related to the modelling to an uncompromising algorithm was not desirable because it could potentially put an end to the whole technical progress. This is actually conditioned by the human ability of invention, which cannot be transformed into an algorithm. It is even possible to say that mere deduction does not lead to any truly new piece of knowledge. The outstanding physicist E. Mach [24] was right when he already in 1912 defended the statement that all pieces of theoretical knowledge are acquired only as useful concentrates of massed experience (empiriocriticism) with the specific human ability to create abstract terms playing its role. A user of a FEM program is, first of all, interested in a certain object that exists and should be evaluated or that does not exist yet and is to be designed. The object we deal with can be a real object that can be, or will be able to be, perceived, or it may be an abstract object of some internal spiritual activity, a schedule of ideas, etc. In any case, the object features four characteristics: 38. It is the subject of the user’s interest. 39. It may be defined, differentiated from other objects. 40. It represents a whole at a given level of consideration. 41. It is formulated as a concept, either in the existing standards, manuals, literature or it is created according to the symbolic rule [36] just for the needs of the problem in question: 211

5.1 Introduction to the Theory and Practice of Creation of FEM Models P( F , S , I , p) (5.1.1) F is the form, i.e. the name of the term, symbol, sentence, etc, S is its contents, communicable by communication means or by a language, verbally or graphically, I is the intuitive, non-communicable contents and p is the assignment to a particular object O . The identification term assigns the object to a broader class of objects {O} . The descriptive term expresses the properties of a real object: mm) lassification (qualitative)

c

nn) topological (typical in a system of FEM elements, comparative, it places the element into the right position within the system), oo) metrical (quantitative, the magnitude of lengths and angles of an element, structure, loading, etc). What is considered the least dangerous for users are formal terms in which the contents is 100% communicable, i.e. I = O . These are all purely mathematical terms, which by their contents remain in mathematics and have no physical contents. Computers can work with such terms only. All engineering considerations must be mathematically modelled through these terms, otherwise a computer cannot be used. On the other end of the spectrum, there are intuitive terms, whose contents is 100% non-communicable ( S = 0 ). And completely out of the spectrum there are vague terms, i.e. an unclear, fashionable, lax connection of words uttered without any idea and expressing nothing sound – even to the author, S = 0 , I = 0 . If they occur in manuals or even in textbooks, they are of little use and represent a waste of time for the user or company. On the other hand, they are invaluable and indispensable in art where the explanation of the work may be just intuitive, and in politics as the “art of the possible” with the necessity to avoid any formulations that are hard to change after the election and that make coalition agreements difficult to conclude. No belles-lettres, music, picture, statue, etc. or successful politics could be made of just formal terms. If we rightly eliminate the vague terms from FEM, we cannot do it to the full extent with the intuitive terms for a simple reason: We do not understand FEM just as a purely mathematical matter, but we take into account also the engineering component. If this component used just formal terms (100% S ), all technical progress would stop and everything would remain at the level communicated in the present-day resources through the mathematical form. The engineering component applies in the modelling of structures with the aim to treat them using the FEM methodology. Already the decision about the dimension of the model – 3D, 2D, 1D (art. 5.1.1.2) – cannot be unambiguously prescribed by any manual. Depending on the immediate need in the given stage of a project, the user must use the professionalism and decide intuitively whether a 1D beam model, a 2D shell model or even, for a detail near a bearing, a 3D model will be used. Similarly, the user must decide about the model for external forces, i.e. loading. They may feel e.g. that it is not enough to concentrate all the load in nodes using a primitive rule of a simple beam, although no formal exact mathematical analysis is at hand.. They select one support type of the structure from the types offered by the FEM program: point or linear, rigid or flexible – without exactly analyzing the effect of the flexibility on stress distribution. They feel that finding the real flexibility for the input data could be 212

5.1 Introduction to the Theory and Practice of Creation of FEM Models dramatically more complicated task than the analysed problem itself, although they make no formal analysis of this feeling. FEM program manuals must provide enough room for all these intuitive considerations. Simply said, in addition to formal terms that are necessary for the communication with PCs, manuals must use normal terms as well. Normal terms in technical literature have different ratio S : I , on average about 80% of S and 20% of I , but e.g. in soil mechanics, in foundation engineering, etc. there is sometimes even 60% of S and 40% of I , which is typical for piles. Without intuitive imagination cultivated in expert contacts (reading, conferences, seminars), many resources are not applicable because they silently assume this imagination. The user must simply know what is a pile, although e.g. a gravel-sand drainage pile is something completely different (local change of subsoil, stiffening, acceleration of primary consolidation). The best score of S : I is that of steel structures. Reinforced concrete is considerably much worse off due to the intuitive imagination about the way the steel reinforcement may take over the tensile stress of a 3D region if this is already affected by cracks in which there is no tensile stress at all, whether and how shear can arise there, etc. Obviously, theoretical and experimental studies have been elaborated dealing with these phenomena, with some simplified conclusions accessible to a common user mostly in standards, but they often come up with nothing applicable to the very problem in question. The situation in soil mechanics is the worst due to its subject matter. Nearly every introduction to a textbook contains a warning that only an experienced expert with about thirty-year practice can make a good design of a building foundation, which is in fact just before their retirement. Literally, it means that sufficient intuition can be obtained only over such a period of time, because no satisfactory abstraction and evolution of generalizing experiences into a communicable contents S have taken place yet. What is important is the differentiability of terms, i.e. the ability of the FEM user to recognize at a given level of consideration whether the term relates to the object. The level may be variably demanding with regard to the differentiability. It gradually grows from a generally technical area to a design, production and calculation area (absolute in the contact with a PC). It may be generally factual, more complex and area-related, objective in the given year 1997 and ultimately absolute in the mathematics. Roughly speaking, the differentiability has lower limit d and upper limit h with possible density of occurrence v in between – ultimately a continuum in the interval f (d , h) , e.g. in mathematics all real numbers between 0 and 1. Normally, the assignment may be much weaker, e.g. only a few instances may be included.. For example, the term typified prefabricated beam K67 and I73 for bridge structures included only four possible spans l = 21, 24, 27 and 30 m , which specified all its dimensions, reinforcement etc. according to typified drawings. The term large diameter pile according to CSN 73 1002 (1982) included all cases of diameter d larger than 0.6 m and lengths D greater than 4d , without any limitation from the top. The differentiability may be naturally different – individual – for each person. In a group of experts it increases to team differentiability and if the team is extended by other experts it may grow to details with sharp boundaries. There exists a certain effective differentiability that corresponds economically to the most advantageous division of work. The active age of one man is approximately 15 000 md (= man-days) with 30 000 working seconds per one man-day. The amount of information received in each second is about 2 b (bits) with the exception of at most 6 bits over continuous 30 minutes, which has no influence on the total amount of the received information: 213

5.1 Introduction to the Theory and Practice of Creation of FEM Models 15000 × 2 × 30 000 = 900 000 000 b = 900 Mb = 125 MB Up to 90% of this amount is sooner or later forgotten, which means that even the most experienced expert has not too much information in his operating memory (brain) – compare it with a disk of a common PC. In addition to a team work it is obviously necessary to engage a kind of an external memory, databank, Internet, etc. Only the most concentrated knowledge, i.e. the theory, should be left to the brain. An expert that is able to quote by heart the contents of one standard, but who knows nothing about the purpose of the principle of virtual works (which is extraordinarily useful in both linear and non-linear mechanics) waists his internal memory for useless things that can be easily obtained from external memories (standards). FEM programs relieved us of the drudgery of calculations, but they are simply not capable of performing typically human considerations. Also the following characterization of term P created for object O is related to human qualities: It is the objectivity of the term that can be evaluated by measure M 0 M0 =

n0 m

(5.1.2)

where m in the denominator is equal to the number of all subjects (persons, teams) that have created term P (5.1.1) for the given object O . The numerator n0 is the number of subjects used for the creation of terms P with identical communicable parts S . To be exact, formal mathematical terms have M 0 = 1 , vague terms have the tendency towards M 0 = 0 , common technical terms at various levels have M 0 in the interval (0,1) . Let us take an example from soil mechanics: participants of advanced training for structural engineers created for a specific foundation a term with measure M 0 = 0.6 up to 0.8. The lower ratio appears traditionally in a pile foundation, the higher one in a raft foundation. The measure M 0 was lower in a group of newly graduated participants who studied the subject, approximately 0.5. Generally, it was proved that the objectivity of terms is greater for steel structures and is high in the field of modelling for FEM calculations without the interaction with subsoil. The given example of the creation of a term to a particular object is a deductive process. The opposite is the inductive process, when we are looking for the object for a term created by abstraction. The engineering activity belongs to constructive processes, when we create artificially an object for our terms. It is connected with the rate of completeness M c of the delimitation of the term Mc =

nc m

(5.1.3)

The amount of all subjects creating object O on the basis of term P is marked m . The amount of all identically created objects is nc . It denotes the identity on the given differentiability (d , ν , h) , explained above. We take into account only the communicable part S of term P that was transferred to all m subjects by the primary subject that created term P for object O . Insubstantial details may differ in the intuitive part. In technical practice, the rate of completeness M c varies within quite broad limits according to the character of the subject. The average value is about 0.8, which means that the number of substantially different objects is roughly 0.2 (20%). The effort of the authors of standards is to 214

5.1 Introduction to the Theory and Practice of Creation of FEM Models increase the rate M c . It is traditionally low in the field of foundation engineering which has been already mentioned (raft foundations, deep foundations, pile foundations, micropiles, etc.). It is not a Czech speciality, it also occurs on an international scale. E.g. EUROCODE 7 does not contain any chapter about common large-size foundation slabs, it is necessary to check them according to the principles laid in the chapter Spread Foundation, which covers rather pad foundations, strip foundations or grillage foundations. Users of FEM programs who want to analyse foundation slabs, grillage foundations or piles [8, 9, 22], etc. must involve their intuition in the process of selection of the most suitable model at the given level of structural design. Design of a technical object O always assumes a sufficient understanding in the sense that the created object O satisfies all actual requirements during the construction and service life and, possibly, all limit states prescribed in standards. In addition to the limit bearing capacity (safety against collapse due to any possible reason), the limit deformation is critical in today’s practice more and more often, which means – in general terms – the applicability in operation, the tolerance of deflections, settlement, etc. The progressing development of FEM programs resulted in gradual disappearance of primitivism, which is practicism based on rough estimates. The development rather takes the trend to precisionism, which is the currently maximally reachable technical level of the project. It is clear from fig. 5.1 that both extremes are harmful, because they may result in increase of total costs on the realisation of the technical object. These must be increased by the costs of repairs of defects, adaptations, renovations, etc. that always make a threat if a primitive design was applied. What must be also included are the costs of a higher-quality investigation of the properties of materials and subsoil, the time and financial factor related to longer and higher-quality calculations, etc. Obviously – for various levels of design-readiness, i.e. designs for competitions, detail designs of objects of various importance and investment costs under various conditions, etc. – a certain economical level of definition of the term and design of the technical object exists in which the total costs are minimal (Fig. 5.1). The main principle for the creation of a model is that neither misunderstanding and misapprehension of the term and object nor excessive costs from both extremes may occur. Also this principle should be born in mind when users chose a suitable FEM program from today’s rich offer. It is certainly pointless to primitively estimate the tension in a certain structural detail if it can be easily calculated by FEM. On the other hand, it is also pointless to resolve the issue of a safe foundation of the building in the complex building-foundation-subsoil system using 3D brick elements for the soil-massive, which may lead to a system with 105 − 106 deformation parameters. Moreover, an advanced research of physical input data – which further increases the price of project-preparation – is adequate for such a complicated calculation.

215


Figure 5.1 Economical level of the definition of the term and and design of a technical object, according to [36].

216


Figure 5.2 Application of different models of structures and their subsoil for FEM analysis. Selection of an effective FEM model in practice.

217


Figure 5.3

A typical evolution of the creation of calculation models in three phases, according to [75]. The primitive phase I is characterised by a considerable unfamiliarity with real processes and estimates made by practitioners. The experience with the costs for repair of defects lead gradually to phase II that is exploited by real users – structural engineers – and scientific and research institutions oriented to the needs of the structural engineers and sponged on by a considerable amount of theoreticians. Their work is often sterile, it falls into a cemetery of non-useful ideas and is forgotten forever. The real benefit is brought by the transition to phase III where the complexity of the model is adequate to the necessity to guarantee both safety and economy of the product.

5.1.1.2 The Selection of an Effective FEM Model in Practice

218

5.1 Introduction to the Theory and Practice of Creation of FEM Models The term effective FEM model (hereinafter Pefm ) comprises the modelling of physical phenomena undergoing in the structure, the choice of the dimension of the calculation model n D, the selection of the type of finite elements including the degree of polynomial p and the mesh density h , the formulation of static and geometrical conditions (loads and supports), the interaction with neighbouring buildings, subsoil and soil environment, the consideration of how the outputs will be interpreted in the design office, in particular which limit states are decisive and which secondary, whether exact stresses are needed for the assessment of steel structures or whether just the resultants are adequate for the design of reinforcement of reinforced concrete structures, and many problem-oriented details. It is clear – even from this – that the communicable part of this term (5.1.1) (that may be expressed by a language of any sort verbally or graphically) does not characterise its contents to the full extent of 100%. A part of contents I is necessarily left intuitive. The objectivity M 0 according to (5.1.1) and also the rate of completeness M c according to (5.1.2) are lower than one. A pessimistic approach to Pefm considers it even to be vague, fashionable, changeable depending on the development and accessibility of PCs in practical engineering, variable according to prices of software and peripheries, etc., without any particular communicable S and intuitive I contents, as the present-day explosion of computer technologies and FEM programs gives no time to develop any desirable intuition. It is difficult to argue with this opinion, as there is much truth in it. It is, however, not useful for structural engineers at all. It gives no guidance as far as the choice is concerned. Let us try to perform a purely economic evaluation that can be understood in the market environment. Let us neglect the subjective approach to the analysis of structures, which comprises (i) the curiosity of the user who is interested to know more about things than is necessary to put the design into practice and (ii) the interest of the researcher who wants to collect a set of results at various h-p level and who uses FEM to play interesting computer games or to obtain annexes for thesis, etc. The economic evaluation must start from the total costs of all the FEM calculations relating to the given project in the phase of preparation and service of the analysed structure (Fig. 5.2). The user must realise that there is no sense in selecting a more complex model unless they order for it the corresponding physical input data or their investigation. This applies mainly to the subsoil of buildings, as each structure must be laid on a foundation (with the exception of space satellites). The high precision of the superstructure model with the foundation comes completely in vain in a thick liquid, which the Winkler´s subsoil in fact is. The reliability of a whole is decided by the weakest link of the chain. If the savings achieved in the calculation are 100%, i.e. we apply no FEM model and limit ourselves to an expert estimate, it represents a correct allowable procedure. It is sufficient if the project is properly signed, as calculations are required by Building Control departments only for more complex projects in order to prove that there is no danger of considerable public damages, threat to human lives or life interests of neighbours, etc. Consequently, both the structural engineer and investor take the risk of defects in the structure. These defects may appear already during the construction (which usually has just economic, not juridical consequences) or later during the service life of the building. Once a defect is discovered, it is usually advisable to perform the FEM calculation that determines both the origin of the fault and the responsible party, suggests the appropriate structural changes, etc. These calculations require the input data obtained through measurements performed on the structure, which is usually an expensive matter: the magnitude of absolute and relative displacements, the strength of specimens of materials taken, the photo documentation of the defect, the non-destructive defectoscopic 219

5.1 Introduction to the Theory and Practice of Creation of FEM Models monitoring (ultrasonic and radiation devices, verification of the position of reinforcement), etc. The threat of defects in the structure and considerable damage leads to the opposite extreme, analogous precisionism (Fig. 5.1) in the overall conception of the project, now in the field of FEM calculations (Fig. 5.2). What is typical for this attitude is the increase in the dimension of finite elements n D (art. 5.1.2). The users do not realise that the division of a flat slab laid on columns to 3D elements (art. 4.1.6) could make some details in stresses more precise only on condition that they input adequate input data about physical properties of concrete and reinforcement, including the detailed location of the reinforcement, which is not yet known. The modulus of elasticity of concrete Eb and its Poisson´s coefficient ν b have distinctly cumulative character and are not applicable to tensile regions with hair-like cracks. If the users manage to get more exact physical relations between σ and ε , it is just an additional expense of time or money for invoices paid to experts and laboratories. The result may be an unpleasant surprise: practically nothing has changed in the resultants that are needed for the design of reinforcement in comparison with the cheaper calculation using 1D + 2D elements. Analogously, also the application of an expensive expert program of the most up-to-date hp-version may result in disillusion, as the results practically do not differ from those obtained by a cheap program using professionally correct segmentation and choice of mesh size. It is necessary to realise that no one is interested in the FEM calculation itself, unless it is not the final goal, e.g. the research of the properties of an element, convergence issues, etc. – which is the problem of the authors of the program. FEM is just an intermediate link that is more or less useful in a certain phase of the design, and it may happen that the zero variant is the best option, i.e. there is no calculation at all. Mostly, however, there exists some effective model for which the total costs of all calculations made within the project (including possible additional corrections) are minimal (Fig. 5.2). It follows from the previous explanation that it is not possible to present any 100%-valid rule for this selection and many considerations must necessarily remain intuitive. This irritates inexperienced users of FEM programs and they complain either that the less complex model has not produced the expected results (usually in a certain point of stress concentration) or that the calculation with the most exact model took many hours and that – as they knew nothing about the physics of the materials but only the modulus of elasticity E – the result was a triviality that could have been estimated for free in advance. The most frequent complaints arise from the misunderstanding of the reduction of the dimension of the model – theoretically explained in art. 4.2.2.2., Fig. 4.26 4.27, formulas (4.2.1) - (4.2.100). Therefore, we will dedicate a few practical remarks to this question in the next article. To conclude the considerations about effective FEM models, it is suitable to point out that it is not a formal, intuitive or vague term (art. 5.1.1.1.), but a normal term with communicable ( S ) and non-communicable intuitive ( I ) contents that develops both over real time (history of technology), and over physiological time of the life of an individual with growing experience or existence and changes of a working team. This process was suggested with a kind of exaggeration as early as in 1976 by H. Duddeck [75] and is schematically indicated in Fig. 5.3. The phase II is nowadays typical for FEM models. Everyone has already understood that mere estimates are not sufficient to achieve success in competition, in tenders and price quotes. Therefore, FEM is used generally, but – similarly to any demand for technology – a great many people whose immediate interest is not an engineering work but

220

5.1 Introduction to the Theory and Practice of Creation of FEM Models FEM itself parasitize on it. This interest may be considerably useful. Without professional mathematicians we would hardly find our way in the finite elements - see the pioneering works of M. Zlamal and A. Zenisek. [3] to [5]. Also thousands and thousands of theses and conference presentations, the purpose of which was just to represent an item in the list of produced publications (required by the school or academy), belong to this group. Most of them brought nothing understandable for people in practice or, in the worst case, they frightened the readers by unproven statements about disadvantages of e.g. commonly used Lagrange elements, which were temporarily abandoned in favour of a vast number of elements based on other variational principles (art. 3), but which are once again used widely. Fig. 5.3 indicates also a completely sterile branch of theoreticians who are far away from practice and who doggedly follow more and more complicated models, which are sometimes not understood even by similarly obsessive theoreticians, which brings immense losses of spiritual potential due to duplicities and errors flooding hundreds of international conferences, bulletins and monographs. The approach of theoreticians who are working directly for practice and who are directly commercially interested in practical application of their theories is indicated in the figure by a thick line and follows the trend towards the engineering mastership characterised by a maximal possible simplicity for the required level of accurateness of the model. Due to the infinite diversity of possible applications, it is impossible to derive explicitly the “what is not substantial” and “what can be neglected”. The intuition, which can be trained only through (i) a continuous contact with practice, (ii) successes and failures, (iii) lessons taken from mistakes made by others and from one’s own slips, (iv) a continuous dealing with topical requests of hot-line users and (v) the principle to use exclusively true statements, plays an important role here, even though it is unpleasant to everyone including the author of the theory and model and even though it may be sometimes accompanied with temporary financial losses.

5.1.2 Dimensions of the Model for FEM analysis The selection of the dimension of the model for FEM calculation is extremely important both objectively and economically. Therefore, finite elements of various 1D to 3D dimensions were discussed in detail in theoretical chapter 4, art. 4.1.4 to 4.1.6, fig. 4.1 to 4.20. It was stressed in art. 4.2 that only 3D object-volumes exist in physical terms and that objects of lower dimensions 2D and 1D are abstract (art. 5.1.1.1) and the terms related to them must be understood in that way. The reduction of the dimension is of course very useful. It is only necessary to understand correctly the meaning of 2D internal forces and deformations of walls, slabs and shells and 1D internal forces and deformations of beam elements of beams, trusses, frames and grids, ribs and other stiffeners of planar structures. The information in art. 4.2.2.2 to 4.2.2.6, fig. 4.26 - 4.27 dealt with this issue. These pieces of knowledge will be illustrated here through a few practical examples of modelling for FEM calculations. In order to understand which considerations are in question, it is necessary to emphasise the following fact at the very beginning as it is unjustly omitted by many: Most users of FEM programs have no feeling of subjective happiness connected with the use of those programs. Briefly said, they do not enjoy themselves and FEM rather represents a required task (similar to performing some arithmetical operations on a calculator) that is necessary for the main goal: the prognosis of the behaviour of the structure. If one

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5.1 Introduction to the Theory and Practice of Creation of FEM Models needs to find out e.g. the result of the mathematical operation 15.73π / sin 10.5°, they type the numbers and functions on a keyboard and read the result or store it in the memory for subsequent use. Only very few individuals are interested in the number of decimal digits of π and the representation of sine function, whether the calculator uses in a subsequent operation just the digits displayed on the display or whether it employs the internally stored 16-digit value. Perhaps no one is interested at all in (i) which Taylor´s series was used to derive the sine, (ii) which test is used to check the number of elements in this series to reach 16 valid digits, (iii) how the decimal inputs are converted to binary, octal or hexadecimal forms accessible to the arithmetic unit of the calculator, etc. All this was solved by specialists in the prehistory of electronics. It may be found in various older resources and encyclopaedias. Nobody today asks mathematicians or company dealers about it. Using a hyperbole we may say that the corresponding experts have built for themselves an eternal monument of glory and, at the same time, they dug the grave for the interest in consulting service in this field, as it works without any mistake. A similar situation is today with the systems of programs based on FEM that reached such perfection and users-friendliness that the questions relating to FEM itself disappeared completely, even though they were quite frequent in our country in the years 1970 -1980. The interest in training courses dealing with FEM principles dropped, while the interest in the creation of models for FEM increased – however, for purely utilitarian reasons as an assistance to practice. This is given by the position of the user of FEM program for whom it represents just a part, even through an important part, of the whole work on a project according to Fig. 5.4, blocks 6 to 11. Even the present-day level of input and output preprocessors, graphical environment and automation of many tasks does not allow for removing the human factor from block 6 (preparation) and 11 (checking by professional feeling before releasing the results for the next project phase). In block 11 we may discover crucial factual mistakes in inputs (formal mistakes will be reported by the program in time and the calculation will not be performed), e.g. in loading, stiffnesses of elements and connections, units, etc. Such inputs define a formally correct problem, the program can process it, but it has nothing in common with the given structure – it represents a completely different problem. It is necessary to (i) come back to the node E or as far as to D of the flow chart in Fig. 5.4, (ii) correct the mistakes and (iii) repeat the calculation. Not even the most state-of-the-art graphical environments, e.g. WINDOWS 2000, can protect against factual mistakes, handling of inputs is just more comfortable and faster. If the transition to subsequent project phase reveals that the design is unrealistic (e.g. the bending moments cannot be transferred by means of any reasonable reinforcement), it is sometimes necessary to go back to the initial concept of the statical model. Repeating of the FEM calculation is sometimes enforced by factual mistakes of the user – some sources of which are given in Fig. 5.5. Many manuals produced by different companies are written for promotional reasons in such a way as if all users were ideal unmistakable beings, while the truth is just the opposite. There exists no FEM program, including the newest expert hp-version, which could prevent factual mistakes. Formally correct input data are processed by the program even if they represent a technical nonsense – e.g. a foundation slab made of thin rubber (error in E modulus, invalid dimension of thickness).

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Figure 5.4 An overview of the activities of a structural engineer – designer, from the initial concept of the statical model and model for FEM program (block 1) to the supervision during the realisation of the project (block 13).

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Figure 5.5 Main sources of mistakes made by the structural engineer – designer that result in the necessity to repeat the FEM solution using the same model but different input data, or using a changed model or applying more detailed model upgraded to a higher dimension from 1D to 2D, etc.

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Figure 5.6 Two basic types of 1D beam models used in common FEM programs. a) The newer Mindlin model taking into account the influence of transverse shear (slope), geometrically understood as Cosserat´s 1D continuum with six independent degrees of freedom in each point. b) A classical beam according to Bernoulli-Navier hypothesis with zero shear deformation. c) Explanation of the difference between (a) and (b) on a truss girder. d) An ordinary solid beam

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5.1.2.1 The 1D Models Practically all newer FEM programs use the model of a beam according to Fig. 5.6a. Exceptionally, classical beams neglecting the influence of deformation due to shear forces (Fig. 5.6b) may occur. The theory of reduction of 3D and 1D was explained in art. 4.1.4 and 4.2.2. The users should be aware of what they gain by the reduction and what they lose. If only six values – the resultants of three stress components σ x , τ xy , τ xz (another three are neglected in the whole body of the beam) – are available in one section of a beam, then an infinite number of stress-states correspond to this group of six numbers and they cannot be differentiated by the 1D model at all. Let us emphasize now just that for short beams the deformation may be caused mostly by shear and thus the calculation using a classical model gives a dangerously small deflection, even just 10% of the actual one, which is thus ten times larger. Also bending moments M and shear forces Q are commonly affected by the error which even relatively grows in limit situations (load located near supports) above all limits, yet for quantities that are small in absolute value. For quantities that are decisive for sizing the errors reach even tens of percent. Only very slim beams are not sensitive to the influence of shear deformation (slope) as most of deformation in them is due to bending. The influence of transverse shear can be illustratively demonstrated on a truss girder (Fig. 5.6c) where the upper and lower chords transfer bending moments (stress σ x is concentrated into the chords) and the diagonals and verticals (generally the beams between chords) transfer shear forces (shear stress τ xy concentrated into their axial forces). The model of a classical beam is a truss with absolutely stiff diagonals and verticals that do not allow for the change of the right angle of the diagonals. If we omit the diagonals and verticals, we get a limit configuration that models a beam with no shear resistance in which the total deformation is due to shear – due to the change of the shape of the diagonals and verticals – which ultimately results in a kinematically indeterminate structure. Real solid beams (Fig. 5.6d) behave rather in a classical way, but their shear stiffness (art. 4.2.2.6) is finite and corresponds to an intermediate configuration – Fig. 5.6c. Fig. 5.7 is a good example to clearly demonstrate what we lose by the reduction 3D – 2D – 1D. Each physical body, even a beam, is three-dimensional and no other stress-state σ then the 3D-stress-state in a continuum exists. If we use a 2D model, we lose the possibility to follow the details in the interval − h 2 < z < h 2 . An infinite number of stress distributions may be assigned to the same internal forces of a 2D model in the above mentioned interval – it is sufficient if they have identical resultants, more exactly identical integrals (4.2.13), art. 4.2.2.2, Fig. 4.26c or the same intensities in point ( x, y ) of the 2D model. The reduction to a 1D model is more dramatic, as the overall stress-state of a section of the beam – i.e. three functions of two variables σ x ( y, z ) , τ xy ( y, z ) , τ xz ( y, z ) – are characterised by six numbers N , Qy , Qz , M xd , M yd , M zd defined by integrals (4.2.14), art. 4.2.2.2., Fig. 4.26e. In a symbolic way, we deal with the degradation of information whose cardinality is 3× ∞ 2 into six numbers, which is true even if we admit just the countable infinite number of points ( y, z ) of the section – even though we deal with the infinity of a larger cardinality of a continuum. Practically said: the given six numbers that are called the internal forces in the section of a beam cannot characterise any differences in the stress-state 226

5.1 Introduction to the Theory and Practice of Creation of FEM Models σ x , τ xy ,τ xz if the resultants (4.2.14) are identical. It follows from the above that any system of forces with a zero resultant can be added to the load, because it does not change the total resultant (4.2.14). The example is in Fig. 5.7b (it is not possible to differentiate the load on the upper and lower flange), Fig. 5.7c (supports of the beam) and Fig. 5.7d (various stress-states resulting in the same moment M yd ). Following from the Saint Venant principle, the static effect of an equilibrium system of forces acting on a small domain Ω0 with the characteristic dimension h manifests itself just in this domain and its close vicinity up to the distance kh , where k = 2 (ordinary beams) and 4 (more complex configurations). This is in fact the main reason why we operate with the resultants (4.2.14). In a Saint Venant domain Ω0 and its close vicinity we cannot analyse the detailed stress-state by means of beam internal forces. Those who are interested in the stress in the vicinity of the beam end must define its support conditions more accurately than by mere definition of deflection w and rotation ϕ of the end section. On condition that we require even more detailed analysis that cannot be covered by such approach (e.g. the concentration of stress in the corners of flanges welded to the endplate or the stress-state of an anchoring concrete block) a 3D model may be necessary, as the general spatial stress-state cannot be simplified by any credible assumption. Common practice often applies 1D elements whose centroidal axes do not meet in the common point of intersection, which results in eccentric joints (Fig. 5.8). Older FEM programs modelled these

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Figure 5.7 a) The reduction of the dimension of the model. b) If we add a state with a zero resultant we may transfer the load from one flange to the other one, which cannot be covered by a 1D model. c) A 1D-model cannot differentiate whether the beam is supported on the upper or lower flange. d) Quite different distributions of stress σx along the section give the same bending moment Myd for the model with a 1D beam.

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Figure 5.8 Various structures composed of beams, modelled by means of 1D elements. a) Continuous beams with differently strengthened flanges. b) Concrete frames with different effective widths of slabs. c) Eccentric connections of beams. d) Eccentricity of 1D-element in relation to the basic 2D-model of a slab or wall. e) Eccentricity of the joints of beams in a truss girder. f) Eccentricity of the rib in relation to the middle plane of a shell. g) Eccentric joints in masts.

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Figure 5.9 Modelling of eccentric connections of beams to nodes of an analysed system – composed in general of 1D and 2D elements – in the configuration when the centroidal axis of the beam remains parallel with the line connecting the nodes and the eccentricity is defined just by two components in axial axes xA, yA.

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Figure 5.10 General eccentricity in which centres of gravity IC, JC of the end-sections of the beam are arbitrarily shifted from nodes IA, JA of the mesh of the analysed structure that consists generally of 1D and 2D elements. Vectors of eccentricity eI, eJ are generally different, therefore the centroidal axis xC = IC JC is not parallel with the axial line segment xA = IA JA.

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Figure 5.11 C

C

C

Main centroidal axes x , y , z of each 1D beam element determine its position in space through the line connecting the centres of gravity xC = IC AC that is defined in eccentric joints according to Fig. 5.10 and through the rotation of the coordinate axes yC ⊥ zC from the basic position (defined by the program) by a certain angle β0. To be more user-friendly, up-to-date FEM programs enable the user to define this position simply and in an illustrative way by an arbitrary point By located in plane yC ⊥ xC or Bz in plane zC ⊥ xC. We select a distinctive point e.g. on the surface of the beam, which enables the user to work in the interface with AutoCAD, see [80].

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eccentricities by means of what was termed joint-beams whose lengths were equal to the eccentricities and whose stiffness was either (i) infinitely large, practically e.g. 106 times larger than that of structural beams or (ii) finite – which made it possible to express the elasticity of the joint. This lead to an increased number of nodes of the calculation model and increased number of unknown Ns , notwithstanding the unpleasant inputs relating to the definition of the fictive beams. Newer programs treat the eccentric nodes in a smarter way without increasing the number of unknown Ns . As early as in 1980 the program system NEXX enabled the user to define a parallel connection of a beam, e.g. a rib of a slab, to the nodes of the system in a way shown in Fig. 5.9, the legend of which gives the explanation of the procedure. The possibility to define a general eccentricity according to Fig. 5.10 was implemented in the same system in 1993. This creates a new possibility to define a system composed of beams. It is sufficient to define the axial configuration as a system of lines connecting axial nodes I A , J A that (in general) do not have to be structural nodes. Moreover, even the whole axial configuration may be just a geometrical auxiliary system. In the next step, the central configuration consisting of centroidal axes I C , J C of structural, physical, beams is defined. This can be achieved through the definition of three components of the vector of eccentricity eI , eJ at each end-point of the beam, which can be done – depending on the view of the user – either in global axes X G , Y G , Z G that are common for the whole system, or in axial axes defined separately for each line segment I A , J A . This defines the position of a physical beam in space with the exception of one parameter that represents the position of the coordinate axes y C ⊥ z C . To do that, the structural engineer may define any point By in the main centroidal plane xC y C or Bz in the main centroidal plane xC z C . Such a point may be easily chosen for example on the surface of the beam (e.g. in beams made of rolled profiles in the intersection of the web axis and the face of the flange and it is similar in concrete T sections etc). All ambiguities are thus resolved. The input is always unambiguous, has no exceptions and may be easily connected to AutoCAD. To conclude, let us mention that the position of 1D element in space must be unambiguously defined including the position of central axes y C ⊥ z C , i.e. angle α x or point By or Bz also in the case of elements of solid circular and hollow circular profile, i.e. bars and tubes. Their axes y C ⊥ z C are not physically unambiguously determined – their position is arbitrary. If the user provides no input (default), the program assigns them a standard (termed basic) position with one of the axes situated in the basic projection plane. The calculation is performed without any problems and unless the user is interested in the position on the beam in which the extreme stress occurs, there are no difficulties. If, however, a reinforced concrete thick-walled tube with a distinct bending in one plane is to be analysed – which should happen even in a complex irregular distribution of longitudinal tensile reinforcement – then the user would have to be aware of the direction of moments M yd , M zd , i.e. they would have to know the position of axes y C ⊥ z C . The situation is similar with steel tubes if the position of the stress extreme is to be found. For such situations it is definitely sensible to select the position of axes y C ⊥ z C in advance and in such a way (i) that e.g. the axis z C is situated in the direction of the expected extreme or in the direction that is important for other structural reasons and (ii) that it makes the calculation model clear. 233


5.1.2.2 2D Models The reduction of the dimension of a problem to 2D was theoretically explained in art. 4.2.2.2-5, fig. 4.26 - 4.27 where we introduced 2D internal forces (4.2.13) and deformations (4.2.49) related with each other through physical formula (4.2.57) with the matrix of physical constants D2 . The elements of this matrix may be determined from constitutive (corresponding to 3D reality) constants dik by means of integrals (4.2.63) within the interval of the thickness of the 2D model (− h 2, h 2) . The decomposition of internal forces into three membrane (wall) forces nx , ny , qxy and five bending (slab) forces mx , my , mxy , qx , q y according to (4.2.64) is useful in technical terms. Using common assumptions of a linear mechanics and orthotropy of materials we may obtain two separate physical relations for the membrane and bending states (4.2.66). The membrane state is simpler – in fact the plane stress with matrix of physical constants D2 m (3,3) , for example see (4.2.72). What is more complicated is the bending state with matrix D2b (5,5) . Its decomposition into the bending part D2bo (3,3) and shear part D2 bs (2, 2) according to (4.2.75) is almost always sufficient in common practice. The decomposition of the required nine constants (4.2.73) is described in detail in (4.2.74) with an example of isotropy (4.2.75). Therefore, the reduction of a 3D body of a slab, wall or shell to a 2D model is theoretically clear. In this paper we will try to point out some useful remarks to FEM users concerning the technical meaning of 2D models whose finite elements were described in art. 4.1.5 There are no technical problems with models of walls with plane stress, because all FEM programs assume silently (without mentioning it in manuals) the existence of what is termed generalised plane stress. This is the state very close to reality with no normal stress σ z acting on faces of walls. The detailed theoretical analysis, differences against the theoretical plane stress and plane strain and the overall analysis of what is termed plane problem of elasticity can be found in Czech title [31]. Structural engineers can do with the following remark: Individual 2D models differ from each other just by their physical constants and, therefore, we may obtain every solution by means of a FEM program on condition that we input the corresponding constants. Only three of them must be input for the most frequent isotropic case: modulus of elasticity E , Poisson’s ratio ν and thickness h that (as we know from art. 4.2.2) belongs in 2D models to physical data, because the geometrical dimension in the direction of the normal does not exist. The algorithms also contain the thickness h only in the product Eh . Only in postprocessors, which determine the 3D stressstate, thickness h may occur independently, e.g. in section modulus W = h 2 6 related to width b = 1 of a section of the wall. Similarly, there are no troubles with the membrane stressstate of shells, especially if the FEM program uses planar shell finite elements, which is in fact applied in majority of FEM programs. If the finite element is a planar one, its membrane parameters define the same stress-state regardless whether it is a part of a wall or a shell. Certain difficulties arise of course in 2D models of plates or in the bending state of shells, which we explain on an example of a planar slab (Fig. 5.12). Many users of FEM 234

5.1 Introduction to the Theory and Practice of Creation of FEM Models programs have the tendency to require from the model more than it is able to offer and to absolutise its results up to absurd conclusions contradicting both the correct technical interpretation and the simple engineering intuition. A slab is in fact a very frequent model and may be of various shapes with corners, openings and cuts, may be exposed to continuous and concentrated loads, may have linear, column and elastic flexible supports, etc (Fig. 5.12a). Misunderstandings resulting from unfamiliarity with the theory sometimes occur in practical applications. For example, an ordinary rectangular bridge slab subjected to a uniformly distributed load should – according to the opinion of a naive structural engineer – behave as a simple beam (Fig. 5.12b), which really occurs for zero transverse contraction ν = 0 . If we input for concrete ν = 0.2 , the program draws and prints in the corners of the slab large concentrations of reactions r ( y ) . In the effort to remove them, the user applies finer mesh of finite elements, but the concentration increases and, naturally, its extent is reduced. Technical support of the program producer confirms it as a correct result which would be even more distinctive for larger values of ν, e.g. roughly 500% for ν = 0.5 . What can reassure the user is the print of verification results obtained by the prestige program ANSYS as well as the exact theoretical solution that gives in the corner an infinitely large q – the well known corner singularity. Moreover, it is possible to explain this effect in a popular way by cutting the slab into beams according to Fig. 5.12c. For ν = 0.2 and the slab subjected to bending the lower faces of the beams get narrower and the upper ones get broader, which is in a compact slab possible only through the effect of transverse bending in the y -direction accompanied by moments M y = 0.2 M x which in the y -direction act everywhere in an infinitely wide slab. If a finite bridge slab has a free edge, the condition of M y = 0 can be met by the removal of M y = 0.2 M x , which means by the application of moments 0.2 M x along this edge. This will necessarily produce in linear supports of the slab additional reactions r ( y ) that violate the constant r expected by the naive structural engineer. The system of removing moments 0.2 M x and corresponding reactions r ( y ) is in equilibrium and if the bridge slab is considerably wide, i.e. B ? L , it has the Saint Venant character. It affects just the parts near the free edges. In the middle of the slab the distribution of reactions r ( y ) is practically uniform, as an educated structural engineer familiar with beams can intuitively expect. The whole mistake is thus grounded on the incomprehension of the differences between a slab and a series of beams caused by the Poisson’s coefficient of transverse contraction. The 2D model is in practice often used to substitute the typical thin-walled box bridge structures, which is usually accurate enough, if vertical webs guarantee proper shear transfer between the horizontal slabs (Fig. 5.13a). The user must solve the problem of longitudinal overhanging strips bearing usually just the pavements but considerably interacting with the whole structure. Two extremes are possible: (i) to extend the neutral plane or axis o1 of the section as far as to the edge (Fig. 5.13d), or (ii) to put it into the middle plane o2 of the overhanging strip (Fig. 5.13b). The correct solution would be obviously represented by an intermediate position o3 (Fig. 5.13c). The problem was analysed in detail for both coarse (Fig. 5.13e) and fine (Fig. 5.13f) mesh including extensive photo-elastic tests performed on models. Also the transversal arrangement of the structure proved to be decisive. If densely distributed diaphragms are used, they provide for the rigidity of the shape of the cross-section. Consequently, axis o3 may be closer to o1 even for large projection of the overhanging part. These studies were of a considerable importance at the time when the 235

5.1 Introduction to the Theory and Practice of Creation of FEM Models capacity of common PCs made it possible to solve just systems of equations with 1000 to 10000 unknowns. Today, it is possible to include 2D shell elements into 3D space and model the box structure in 3D. Equations with 10 000 to 100 000 unknowns can be solved by the present-day PCs without any problems. FEM programs automatically assign certain planar coordinates xP, yP to 2D elements, e.g. applies the principle that xP is parallel with the intersection line of plane ρ of the element and the global coordinate plane xG yG and its positive direction is given by the requirement that the angle of axes (xP, yG) must be acute (smaller than 90°) while 0° is also allowed (Fig. 5.14). The zP axis is always the normal to the plane ρ of the element and its positive direction is defined by numbering of IJK or IJKL vertices in a triangular and quadrilateral element respectively. When viewed in the direction of the zP axis the numbering goes clockwise. The yP axis completes the xP and zP axes to create a right handed system xP, yP, zP. This rule causes a certain discomfort, because the internal forces are calculated, printed and drawn in these coordinates (fig. 5.14) and also all deformations are related to them (fig. 5.15). It may happen that for a general situation when a plated surface is used to model a complicated shell, e.g. a bucket of a turbine, a bucket of an excavator etc., each element can have different direction of xP, yP towards the global xG, yG, zG axes. Effective postprocessors may provide the directions of principal moments and there also exist postprocessors designing optimum reinforcement of reinforced concrete shells in a complex configuration of up to 2 x 3 layers in selected directions. Theoretically, it is necessary to satisfy all exceptional cases when the definition of the position of planar xP axis fails, because the above mentioned intersection line of ρ × (xG yG) does not exist. This is handled by the program and it will explained, for better clarity, on the most frequent example of a box structure with (i) horizontal walls parallel with (xG yG), (ii) frontal wall parallel with (yG zG) and (iii) side walls parallel with (xG zG), see Fig. 5.16 and 5.17. At the same time, it is obvious from these figures that the user may influence the positive direction of planar normals zP of all 2D elements in such a way, that e.g. the positive faces zP = h/2 lie completely inside the box, or on lower, back and right faces etc. Positive bending moments mx, my then produce tension on these faces, i.e. tensile reinforcement must be designed in reinforced concrete structures. Consequently, a certain systematic approach is very desirable.

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Figure 5.12 Some consequences of the 2D dimension of the model of the slab for the users of FEM programs: a) Shape, supporting and loading singularities. b) A bridge deck with free edges, concentration of reactions in corners with non-zero transverse contraction. c) Explanation of the origin of the corner effect. d) Fixing of the slab into a rigid column. e) An estimate of the distribution of moments in the slab above the real column. f) The state after the column is removed. g) The detail around the column head must be modelled by means of 3D elements.

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Figure 5.13 A box section of a bridge deck solved by a 2D model. a) Uncertainty of the position of the neutral area or axis o of the section of the overhanging part supporting usually the pavement. b) A rather pessimistic estimate of o2 in the centroidal plane of the overhanging part. c) The state as analysed by a more complex model with 2D shell elements inserted into 3D space. d) Technical estimate for a dense system of transverse diaphragms that guarantee the stability of the shape of the whole section of the bridge deck including the overhanging parts. e) Too coarse finite element mesh. f) Minimal fineness of the mesh for the estimate of real distribution of the quantities in the transverse direction.

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Figure 5.14 a) The principle by which the FEM program ESA PT assigns the planar coordinate system xP, yP, zP to finite elements. If the triangles are subelements of quadrilateral elements, the directions of the axes are the same for the whole quadrilateral, as far as it is a planar one. b) All internal forces are calculated in the planar coordinates. Then they may be further processed by a postprocessor that can determine the principal values, directions, etc.

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Figure 5.15 Internal forces according to Fig. 5.14 cause the deformations of the differential element dxP dyP of the finite element: a) membrane (wall), b) bending, c) torsional, d) transverse shear.

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5.1.2.3

Systems Consisting of 1D and 2D Elements

Many FEM programs allow for modelling of structure by means of 2D shell and 1D beam elements in one model. This covers in particular planar structures (walls, slabs, shells) with ribs, bracing, ties, side beams, etc. The theoretical issues relating to the mutual compatibility were indicated in art. 4.1. The user is most interested in the technical interpretation of the output data, for which we give here a few remarks illustrated by figures 5.18 to 5.22. They deal mainly with reinforced concrete structures. The present-day standards for the design of reinforcement are still affected by the “beam era” and provide instructions and formulas (verified by experience) for the reinforcement of T-sections in which the web is formed by a beam of a rectangular cross-section and the flange by what is called effective slab width. This originally exactly defined term (R. Chwall, 1944) for the special configuration was gradually generalised and became used in standards for both concrete and steel structures in connection with two conceptions of analysis of ribbed shells. The first one was the method of substituting framework which reduces the system to 1D beam elements equipped with the effective widths of the slab structure, i.e. it concentrates the physical body into a set of linear 1D elements. The other one was the method of substituting continuum which calculates just with 2D slab structure and which models the influence of beam ribs just through physical constants of this structure. This includes in particular shape orthotropy used not only in ribbed, but also otherwise shaped walls, slabs or shells (corrugated walls, boxes). The present-day level of FEM software and PCs gives an optimistic idea about the uselessness of the two presented conceptions, as the stresses in 1D and 2D elements are handled separately. For steel structures we come across the issue of continuity of stress in the connection of 1D and 2D elements, as the correct base functions of Lagrangean elements guarantee just the continuity of components of displacement, which is sufficient for the convergence of what is called weak solution. The step-like changes in the stress may be considered a consequence of not satisfying the equilibrium conditions, they appear also between the elements of the same 2D dimension. In the field of reinforced concrete structures the users come across an traditional problem of steel reinforcement that was already commented earlier. Outputs of FEM programs contain 1D internal forces of beam elements and 2D internal forces of shell elements. Nothing else is available for the design of the reinforcement using either a suitable postprocessor or manual calculation. However, these forces are in central coordinates xC , y C , z C of 1D elements and planar coordinates x P , y P , z P of 2D elements. But these coordinates are not generally parallel as can be seen in Fig. 5.18. There exist simple orthogonal systems where (as required) the components are parallel and can be easily summed (Fig. 5.19). In that case there is only one substantial discrepancy in the impossibility of a versatile coincidence of the signs of torsional moments in 2D and 1D elements. It may be formally removed. When bending moments (Fig. 5.20) are summed we have to respect the fact that for 2D elements we deal with the intensity of physical dimension (kNm/m), art. 4.2.2.2., formula (4.2.13). Under the simplest precondition that this intensity is constant in a certain effective width b , the values M + mb and analogously the normal forces N + nb are added together. It is similar with the summation the shear forces Q + qb and 241

5.1 Introduction to the Theory and Practice of Creation of FEM Models torsional moments M k + mk b , always with the subscripts of the corresponding axes (Fig. 5.21). In a non-orthogonal system (Fig. 5.22) we must first transform the internal forces into identical axes, in order to be able to sum the components. If the ribs are thinly distributed it is more suitable not to use the effective width b , but to perform the integration of internal forces of 2D elements over the whole width between the ribs. The actual distribution of the reinforcement depends on a set of other details and engineering invention, which is typical for reinforced concrete structures.

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Figure 5.16 a) An example of planes in a box structure. In all three planes the positive direction +n was selected to coincide with the positive direction of the global axes. In visible planes, +n goes from inside to outside of the box. Note: In quadrilateral elements, please, substitute the numbering of vertices IJK by IJKL. b) The orientation of the planar axes of the elements in face plane r of the box (more generally, in the planes parallel with global coordinate plane XG ZG ). Only the direction of the +XP axis can be determined unambiguously – it is parallel with the intersection of plane r and plane XG YG and forms with the +XG axis angle α in the interval 0 ≤ α < π/2, i.e. 0°. It is a general rule, not an exceptional plane. The positive direction of normal +n = +ZP may be selected independently for the front and back plane – into or out of the box – and vertices IJK should be marked accordingly.

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Figure 5.17 a) The orientation of the planar axes of the elements in horizontal planes of the box (more generally, the planes parallel with the global coordinate plane XG YG). Only the direction of the +XP axis ≡ +XL // +XG (exception from b) can be determined unambiguously. The positive direction of the normal +n = +ZP may be chosen as going into or out of the box and the local numbering of vertices IJK must be made accordingly. The +n direction can be chosen independently for the upper and lower plane, e.g. everywhere going out of the box. b) The orientation of the planar axes of the elements in the side planes of the box (more generally, the planes parallel with the global coordinate plane YG ZG). Only the direction of the +XP axis ≡ +YL // +YG (exception) can be determined unambiguously. The positive direction of normal +n = +ZP may be chosen independently for the lefthand and right-hand side – out of or into the box – and the local numbering of vertices IJK must be made accordingly.

244


Figure 5.18 a) The planar coordinates of a 2D element of a FEM program. b) The central coordinates of 1D elements adjacent to some sides of 2D elements for different order of coding of their ends – which determines the positive direction of the xC axis.

245


Figure 5.19 a) The simplest case of the interaction of 2D and 1D elements in FEM programs: perpendicular ribs in the XP and YP directions. b), c) Coinciding directions of axes XC // XP, XC // YP and coinciding position of relative sides z > 0. d), e), f), g) The effect of the coding of 1D elements on the direction of the positive XC axis. h ), i), j) Impossibility to get coincidence of the signs of torsional moments of 2D and 1D elements without taking their coding into account.

246


Figure 5.20 When bending moments of 2D and 1D elements are summed, it is necessary to respect the differences in their physical character. For 2D elements it is the intensity in a point. The meaning of the values printed by the FEM program is: if the vicinity of this point was subjected to bending f constantly along width b = 1m, the moment acting over this width would be numerically equal to this intensity. If we use another, e.g. the effective, width b, then moment m × b acts over this width in the 2D element. On condition that m is the assumed a constant intensity. Just this moment in (kNm) may be summed with the moment (kNm) in the 1D element.

247


Figure 5.21 Shear forces q of 2D elements printed by a FEM program are point intensities. If they acted constantly over width b = 1m of a section across a 2D element, then the area h × 1 (m2) would be subjected to the force (kN) numerically equal to the printed value q. If we use another, e.g. the effective, width b, the shear force on the area hb (m2) is equal to q × b (kN). Just this force can be physically summed with the shear force in the 1D element (kN). Similarly, the torsional moment in the 2D element mxy is also the point intensity and only the multiplication by length b, along which its constant intensity is assumed, produces the moment (kNm) that can be physically summed with the torsional moment Mx in the 1D element – only if its positive direction is taken into account correctly. Algebraic sums cannot be generally performed.

248


Figure 5.22 Note relating to the co-action of a 1D element with two adjacent 2D elements in FEM programs. Generally, the internal forces of all three elements are printed in different coordinates: in 2D planar and 1D central. Before the conversion of 2D quantities to physical dimension that can be summed with 1D quantities (i.e. force in (kN) and moments in (kNm)), it is necessary to transform the 2D quantities into the directions defined by the central axis of the 1D element. In addition, the distribution of the intensities of the internal forces in the 2D elements must be taken into account and it must be decided whether we sum their integrals (in the case of significant variation) or whether we simply multiply the intensities obtained from the transformation by some effective width b taken from standards or regulations. These standards extrapolate the term b considerably out of the scope of the theory of 2D+1D systems. It may happen that no formula for b is applicable for a general 3D configuration, in particular for ribs located in two directions, complex loading conditions and complex geometry of the whole system.

249


5.1.3 Numerical Stability of the Calculation of FEM Models

5.1.3.1

Defective FEM Results Due to Arithmetics

In practice, we may usually easily find out who is to blame for intuitively defective results of FEM analysis: whether the user (invalid inputs) or the author of the program (mishandled critical places of the algorithms). However, we may come across a situation that no one is to blame, because the input data are perfectly correct and the program for many years worked fine for similar problem. It is no surprise for mathematicians and there are specialists who deal just with such problems. The German Society of Applied Mathematics and Mechanics (GAMM) established as early as after 1980 a department headed by K. Nickel [76] the research of which is very useful even for engineers. Recently, a special magazine for mathematical modelling of systems [77] has been started where we may find interesting topics for all those who learnt in their practice that the difficulties of modelling of reality does not have to end with the mathematical formulation and creation of programming algorithms – even when FEM is applied. We will give here a few remarks for common users that may explain some failures and give advice about what to do in such situations. Let us start from the elementary example in Fig. 5.23a that is even statically determinate, which means that there will be no doubts concerning the exact solution. A vertical suspension 012 consists of two different ties 01, 12 with lengths L1 and L2 and tensile stiffnesses [kN/m] K1 = E1 A1 L1 K 2 = E2 A2 L2

(5.1.4)

It is subjected to vertical force P . Therefore, the same axial force N1 = P , N 2 = P is in both ties. The first tie elongates by length w1 , the other one by w2 , the node 1 drops by length ∆1 and node 2 by ∆ 2 , while the formulas of technical elasticity evidently apply: ∆1 = w1 = PL1 E1 A1 ∆ 2 = w1 + w2 = PL1 E1 A1 + PL2 E2 A2

(5.1.5)

When written using stiffnesses we get ∆1 = P1 K1 ∆ 2 = P K1 + P K 2 = P( K1 + K 2 ) K1 K 2

(5.1.6)

Practically all present-day FEM programs solve this problem by the deformation method with two unknowns ∆1 and ∆ 2 , for which they assemble two equations of vertical equilibrium as a very primitive special case of the procedure given in art. 2.3. The detailed form of the system of two equations K ∆ = f is:

250

5.1 Introduction to the Theory and Practice of Creation of FEM Models (1)

( K1 + K 2 )∆1 − K 2 ∆ 2 = 0

(2)

− K 2 ∆1 + K 2 ∆ 2 = P

(5.1.7)

From the second equation we get: ∆ 2 = ( P + K 2 ∆1 ) K 2 = P K 2 + ∆1

(5.1.8)

which, when substituted into the first equation, gives ∆1 from the equation with one unknown: ( K1 + K 2 )∆1 − K 2 ( P K 2 + ∆1 ) = 0

(5.1.9)

which, for exact arithmetics, is:

[ K1 + ( K 2 − K 2 )] ∆1 = P

(5.1.10)

which for K 2 − K 2 = 0 fully corresponds to the elementary solution (5.1.6). The real arithmetics of PCs with the finite number of valid digits applied in the solution of the system of equations (5.1.7) using the elimination does not have to lead to the exact equality K 2 − K 2 = 0 . Even the beginners in programming know that what is called zero tests must be extended to a small interval ε , otherwise they would not work. Therefore, the parentheses (5.1.10) may contain the result K 2 − K 2 = ε and we have an inaccurate relation:

[ K1 + ε ] ∆1 = P ∆1 = P ( K1 + ε )

(5.1.11)

This produces the inaccurate reaction of the whole tie: R = K1∆1 = K1 P ( K 1 +ε ) = P (1 + ε K1 )

(5.1.12)

In common practice, the differences in the size of the stiffnesses are not too big and thus the numerical error ε is so small in comparison with the value of K1 that nobody will spot it in the output. However, let us imagine the situation that K 2 ? K1 e.g. K 2 = 109 K1 , which may

occur for E2 ? E1 (steel 12, rubber 01) or A2 ? A1 (dramatically different cross-sections) or L2 ? L1 (considerably different lengths). When real (single precision) numbers are used, we may encounter error ε = 10−9 K 2 , which leads to ε K1 = 10 −9 K 2 10−9 K 2 = 1

(5.1.13)

Consequently, formula (5.1.12) produces not the correct reaction R = P , but the incorrect one R = P 2 , which even a beginner must notice in the output – it is a 50% error in equilibrium. We may provide even more drastic examples from real practice: In order to harmonise the free vibration of a very rigid foundation block of a turbo-generator, a long adjustable steel tie was used to connect it with the 60 m deep anchoring block. The stiffness of the tie was negligible in comparison with the stiffness of the blocks and the first solution failed. Let us give a possible correction: If we divide a very flexible structural element into a larger number of finite elements, then their stiffness will increase in indirect proportion to lengths L and the

251

5.1 Introduction to the Theory and Practice of Creation of FEM Models discussed effect will decrease or disappear at all. One of the oldest principles of FEM follows from that: The division of the structure into finite elements should never produce dramatic differences in stiffnesses of adjacent elements. The graphical representation of the sensitivity of the system of equations (5.1.7) can be seen in Fig. 5.23b. We have in fact equations of two lines in coordinate axes ∆1 , ∆ 2 . Both equations can be transformed into a practical tangent form (1)

∆ 2 = [ ( K1 + K 2 ) K 2 ] ∆1 = k ∆1

(2)

∆ 2 = ∆1 + P K 2 = ∆1 + p

(5.1.14)

The tangent of angle α between the first line and the ∆ 2 -axis is k , the second line forms with the ∆ 2 -axis angle 45°. The first line passes through the origin, the interval on the ∆ 2 -axis defined by the second line is p . What is important for the stability of the solution is obviously the value of the tangent of the line (1): k = ( K1 + K 2 ) K 2

(5.1.15)

If, for example, K1 = 0.732 K 2 , then k = 1.732 , α = 60° and the intersection of lines (1) and (2) in Fig. 5.23 is distinct. Its coordinates ∆1 , ∆ 2 represent the numerically stable solution of the system of equations (5.1.7) or (5.1.14). If K1 = 0.073K 2 , it leads to k = 1.073 , α = 47° , the lines intersect each other under a small angle 2°. The point of intersection is not distinct, the graphical solution would already be unreliable, but the PC’s arithmetics would master it without any problems. The problems occur only with K1 = K 2 – the lines are almost parallel. In the limit, k = 1 and the problem has no finite solution – the parallel lines intersect each other in the vanishing point ∆1 = ∆ 2 = ∞ . The situation is similar for three degrees of freedom (symbolically shown in Fig. 5.23c). We obtain three equations with three unknowns ∆1 , ∆ 2 , ∆3 analogous to (5.1.7) or (5.1.14), each of them being the equation of a plane in 3D space ∆1 , ∆ 2 , ∆3 . The solution is represented by the coordinates of the common point, i.e. the point of intersection of these three planes. If the planes are collinear, they have always one point of intersection – see e.g. the coordinate planes which have a common origin (0, 0, 0) . If a pair of planes or even three planes are

252


Figure 5.23 a) A suspension consisting of two different ties. b) Roots D1, D2 of the system of equations (5.1.7) are the coordinates of the intersection point of two lines (1), (2) from formula (5.1.14). For accuracy e we have strips the width of which is 2e. The point of intersection may lie in the intersection of these strips. The more the two lines get parallel, the bigger the intersection of the strips is. c) The example with 3 degrees of freedom, roots D1, D2, D3 are the coordinates of the point of intersection of the three planes. For accuracy e the possible positions of the point of intersection form a 3D body whose size grows with the decreasing angle of the two planes. d) The symbolic representation of N-dimensional point D and corresponding vector of right-hand sides f. For N=1 it is a common diagram f1 = f(D1), for a linear problem f1 = K1D1.

253

5.1 Introduction to the Theory and Practice of Creation of FEM Models nearly parallel, the point of intersection is not distinct. And for planes that are exactly parallel in he limit there exist no finite set ∆1 , ∆ 2 , ∆3 which would satisfy the system of three equations. Now we will discuss particular examples of FEM from practice where the system of equations K∆ = f has 10 4 to 106 unknown parameters ∆1 , ∆ 2 ,K , ∆ N . For N = 2 or 3 we were looking for the intersection of two straight lines or three planes. In general, we deal with N -dimensional linear bodies, inserted into ( N + 1) dimensional space that may be illustratively called N -superplanes starting with N ≥ 4 . The solver of the equations integrated into a FEM program has to find the common point of intersection of N superplanes, i.e. N -dimensional arithmetical point ∆1 , ∆ 2 ,K , ∆ N . We do not need a big imagination to realise how many nearly parallel couples may appear in today’s common case of 100 000 superplanes and which factors may influence the distinctness of the position of the point of intersection of such a quantity of linear bodies (of N -dimensional variants). Mathematics has payed a great attention to this and related problems for many decades. Important factors are (i) the method that is used to find the point of intersection or the roots of the system of equations, (ii) the possibilities for adaptations or transformations into what is called well conditioned systems and effective solution methods. As the solution of equations in non-linear problems is performed in one problem commonly 10 to 100 times, the solver of equations is literally the engine of the program and its effectiveness either increases or decreases the applicability of the program. That is the reason why we will return to this issue again in art. 5.1.4. In this chapter, let us give just the following useful consideration: Let us imagine that we have obtained some solution of the equation system (5.1.7) about which we know that its error is ε , i.e. we do not have exact roots ∆1 , ∆ 2 , but approximate values ∆1* , ∆*2 . We know neither the algorithm nor the precision of the arithmetics and, therefore, we can analyse neither the causes of the error nor its magnitude. Nevertheless, we want to evaluate whether the solution ∆1* , ∆*2 is useful. Let us substitute the approximate values into equations (5.1.7) and we get some other right-hand sides: (1)

( K1 + K 2 )∆1* − K 2 ∆*2 = P1 ≠ 0

(2)

− K 2 ∆1* + K 2 ∆*2 = P2 ≠ P

(5.1.16)

If P1 is very small in comparison with P , it represents, physically, some negligible additional loading of node 1 (Fig. 5.23a) and if P2 is nearly identical with P , the node 2 is nearly subjected to the required load and, as a result, the approximate solution may be accepted as the basis for further design. The rate of approximation may be, for example, the traditional difference of 2% in comparison with the required value P – that means that we admit P2 in the interval 0.98P to 1.02P and in node 1 we will tolerate the unwanted additional load of −0.02P to 0.02P . For more complex problems we can use, for example, norm (272) or tests (288), (289), etc. Let us generalise this indirect solution in which we try to find the input of right-hand sides f i for roots ∆i for an arbitrary number of unknowns N , that means for a system of N equations K∆ = f . For each set of unknowns ∆ ∗ we can easily find the right-hand sides f * 254

5.1 Introduction to the Theory and Practice of Creation of FEM Models and using the above-mentioned tests we can evaluate if they are satisfactory or if they require some correction. The interpretation of the roots as the coordinates of the common point of linear bodies in the N -dimensional space is possible only for N = 2 (Fig. 5.23b) and N = 3 (Fig. 5.23c). For bigger N we can use the symbolic Fig. 5.23d: The point on the horizontal axis represents the point (∆1 , ∆ 2 ,K , ∆ N ) , the point on the vertical axis represents the vector of right-hand sides ( f1 , f 2 , K , f N ) . For N = 1 we have an elementary graph f1 = f (∆1 ) , in a linear problem we get line f1 = K1∆1 . For larger N this is only symbolic. Of all solutions ∆ ∗

obtained by an indirect method, i.e. the solutions corresponding to some right-hand sides f * , we may be interested only in solution ∆ corresponding to an acceptable right-hand side f . The above-mentioned tests cover obviously also the inaccuracies of the arithmetics resulting from rounding of intermediate results to a given number of valid digits. Therefore, it may be difficult to distinguish them from the inaccuracies resulting from other sources. There are, however, situations when the source of the error is quite clear. If, for example, a program code working with real (single precision) numbers produces error (5.1.13) and the program prints the value of the reaction R = P 2 with the 50% error – even though we analyse a statically determined structure with evident R = P and the input value is correct – we must have come across a numerical instability, which may be overcome by transition to double precision code working with numbers with precision 10 −18 . As a result, instead of precise zero, we get according to (5.1.13) ε K1 = 10−9 , which has practically no effect on the accuracy of the result. The present-day FEM programs are offered with large capacities – e.g. 32 500 nodes with 6 degrees of freedom, which is N = 195000 equations – for common PCs available in the market. The solution represents millions of arithmetical operations during which anything can happen. The continuum of real numbers (−∞, ∞) is even in the most state-of-the-art digital computer represented after double substantial reduction: pp) The cardinality of the set of the continuum is aleph one, the cardinality of the set of all rational numbers is aleph zero, termed countable infinity, whose continual measure equals zero. Any large number of such numbers cannot fully fill even an arbitrarily small interval of the continuum. qq) The set of rational numbers is sparse. The interval between them is in the real (single precision) arithmetics equal to 10−9 , in the double precision it is 10 −17 , and in multiple arithmetics it may be even smaller, but there are always infinite amount of numbers of the continuum between two displayable numbers. It is possible to formulate a mathematical statement that almost no number is displayed from the set of the continuum of real numbers, which means that the continual measure of displayed numbers equals zero. No assignment can create enough numbers to fill continuously any small or large interval of the continuum. The consequences can be illustrated on elementary operations of subtraction and division. Expressions (a − b) (c − d ) are sensitive. If (a, b) and (c, d ) are almost identical, small deviations in numbers may cause even the change of the sign of the result. There exist even seemingly stable operations with surprising failures. One of the oldest algorithms was known already in the years long before Christ to Euclid who 255

5.1 Introduction to the Theory and Practice of Creation of FEM Models probably derived it. It is the decomposition of an arbitrary real number a into the form of a continued fraction containing only integers c0 , c1 , c2 , K – according to Fig. 5.24a, e.g. 3.1 = 3 + 1 10 , 3.14 = 3 + 1 (7 + 1 7) , etc. The fraction is finite for rational numbers, otherwise it is infinite. The set of numbers c0 , c1 , c2 , K always defines some number a . Numbers c0 , c1 , c2 , K for number a can be found comfortably through repeated division. The corresponding algorithm e.g. in ALGOL is: begin real a; integer c; start:c=entier(a);

(the whole part of a)

if a=c then goto stop; print (c);

(integers are printed here, one after another)

a:= 1/(a-c);

(core of the calculation)

go to start; stop: print (a); end

The computer thus prints a series of numbers c , which are in the Euclid’s fraction marked c0 , c1 , c2 , K , cn and which can be called the decomposition of a real number into integers. If a given number a is rational, the test a = c applies always, even though sometimes after a very long series of numbers, and the computer stops, otherwise n → ∞ . To perform the calculation, we do not inevitably have to use a PC, a usual calculator may be sufficient. As an example we use the already mentioned number 3.14 with the finite decomposition {3, 7, 7}. This is really the result obtained on acalculator. But for another calculator (the precision of numbers in the register is 10 −99 ) we get (starting from the third number) something completely different: {3, 7, 6, 1, 476190 476, 5, 3, 1, 4, 78125000,K} and the process does not end at all! K. Nickel [76] claims that he has obtained on an unspecified type of calculator with the accuracy 10 −10 this: {3, 7, 6, 1, 142857142, 1, 9, K} and the process did not end either. And 3.14 is a rational number. For the transcendental number π the fraction would be – correctly – infinite, but for the rational value shortened to 15 decimal numbers it would be naturally finate. Let us give the obtained decompositions according to Fig. 5.24a: Exactly: EL-514: K.Nickel:

π = 3.14159 26535 89793 π = {3, 7, 15, 292, 1, K} π = {3, 7, 15, 1, 292,1, 1, K} π = {3, 7, 15, 1, 293, 10, K} 256

5.1 Introduction to the Theory and Practice of Creation of FEM Models EL-5020:

π = {3, 7, 15,1, 292, 1, 1, 1, 1, 1, 15, 2,1, 3, K}

Figure 5.24 a) The decomposition of number a into a continued fraction. b) Babushka’s paradox for Kirchhoff slab: the solution of a regular polygon with n vertices converges for n→∞ to the deflection of the centre of the slab, which is only 0.615–multiple of the deflection of a circular slab. c) Zone c between the mesh density (number of elements) n1 and n2 in which most the common applications of FEM usually fluctuate and where the predominant error is caused by the decomposition of the exact function into base functions over elements, briefly by interpolation. The finer the mesh, the larger the number of equations N and the larger the numerical error. There exists division no in which the total error is minimal. If a complex application of FEM is used to extensive systems (N=105 to 106), the numerical error may predominate.

257

5.1 Introduction to the Theory and Practice of Creation of FEM Models Large numbers that appeared in the number 3.14 at the 5th position of the decomposition signal that something is probably wrong with the arithmetics. No excessively large number occurs for number π , but even there we cannot speak about unambiguity. We cannot even say that the multiple character of the arithmetics or the increase in the accuracy of displayed numbers has any generally positive effect. Errors caused by the arithmetics, which can be analysed or removed purely mathematically (art. 5.1.3.2) must be distinguished from the errors resulting from the physical nature of the model of the structure. For example, a simply supported Kirchhoff slab in Fig. 5.24b has the shape of a regular polygon with n vertices and was analysed in 1968 by the team of authors of [3] for growing n = 12, 36, 60 , etc. by means of a fully compatible triangular element and polynomials of 5th degree (art. 4.1.5.1) with the elimination of needlelike elements from the centre. The team expected the convergence towards deflection w0 of the centre of a solid circular slab. And really, for n = 60 the printed w0n only slightly differed from w0 . However, with n growing further, it started to decrease and for a large n the result was approaching the limit of 0.615w0 . At first, the co-author mathematician considered it to be a defect of the element. Nevertheless, it was a defect of the model and complex boundary condition w = 0 defined along he whole periphery. For such configuration the derivative ∂w ∂n = 0 along both edges that meet each other in one node, which implies ∂w ∂n = 0 in an arbitrary direction n , i.e. the nodal mass normal of the slab is fully fixed. This decreases the deflection of the slab, even though not up to the deflection of a fully fixed slab 0.200w0 . It would require that a set of boundary normals with the cardinality of the continuum be fully fixed, while for n → ∞ we still have just a countable set. This is covered by the mathematical theory and it represents another illustrative example of the difference in the cardinality of sets that was already mentioned in the passage dealing with the display of numbers in a computer. I. Babushka addressed this problem as early as in 1964 at the 1st Czechoslovak conference on mechanics in Smolenice and it is also published in documents [3,5]. The paradox disappears if we use the Mindlin model of a slab with independent rotation of mass normals with components ϕ x , ϕ y (art. 4.2.2). It disappears even in the Kirchhoff model, on condition that we require w = 0 only in the boundary nodes and not on the boundary sides. This eliminates the cause of point fixation. FEM users can learn from this that a seemingly less exact input of boundary conditions may improve the solution. It is an instruction that proper attention should be always paid to the model that is used in the particular FEM program and that is described in the manual. The input data should be adapted to correspond to the physical substance of the model including the boundary conditions – if the given structure should be reliably modelled.

5.1.3.2 Present-day Possibilities of Improving Arithmetics in FEM Calculations We will introduce just briefly three topics addressed by numerical mathematicians in the corresponding section of GAMM [76] that relate to the solution of the system of equations, which is the main arithmetical part of FEM. The first one is arithmetics of fractions in which we limit ourselves to rational numbers. They (except zero) can be always 258

5.1 Introduction to the Theory and Practice of Creation of FEM Models written as fraction m n , where m and n are integers. There are no rounding errors in this arithmetics. The systems of equations can be solved exactly and the roots can be presented in the form m/n as well. Various tests suggest that the solution takes about five times longer time. The drawback is the impossibility to express exactly irrational numbers, especially roots and transcendental numbers such as π , e , functions log, sin, etc. The solution was found at around 1970 in what is termed interval or spherical or (generally said) set arithmetics. It is based either on the interval in which the number a must lie, that is a ∈ (a1 , a2 ) , or on the form a ∈ (a1 − r , a1 + r ) , which once again represents two determining numbers – this time the centre a1 of the interval and the half of length r . For complex number, this second form leads to a circle in a plane, or generally, for numbers defined by more numbers to a sphere, or supersphere in the corresponding space – hence the name “spherical arithmetics”. The advantage is that the amount of numbers describing the intervals is reduced, because r may be common for all. The term single number does not exist in this arithmetics (Dedekind’s section on the number axis), there is only a set of numbers – hence the name “set arithmetics”. Simple procedures are derived for numerical operation (+, −, ×, ÷) and the result is again a set of numbers, i.e. the interval in which the result lies. The term of equality b = c must be understood as a complete equivalence of interval (a1 , a2 ) , or (a1 − r , a1 + r ) , etc. Moreover, a very useful term of inclusion substituting the equality is introduced for situations when it is sufficient that two sets representing two numbers b and c have a non-empty intersection. Roughly speaking, it is an approximate equality of the order r – it is possible to elaborate tests of programs for it, etc. If we take into account the fact that input data in technical practice are also intervals or sets with tolerance r , it is obvious why the interval arithmetics became established so quickly. In our country it was first reported by J. Kratochvil and F. Leitner already in 1972. The contemporary literature represents an endless series of several hundreds of titles. The advantages can be summarised into the fact that it is an apparatus that is capable of displaying the numerical continuum by means of the finite amount of numbers without discrepancies and unambiguously for all computers. Even if the set arithmetics overcomes most of the numerical troubles, it will not solve some of them on its own. As an example we may state a typical logical (decision, branching) statement, e.g. if a=b then c else d;

which for the equality a = b invokes jump to c , otherwise to d . But what is the equality in a digital computer? Can also inclusion be considered equality? Let us demonstrate it on an example: Let the numbers a, b are produced during the execution of the program in this way: a = 3 × 1 3 ; b = 1 . Let us have a usual n -multiple arithmetics, in which the result of the operation 1 3 = 0.333K 33 and the result a = 0.999K 99 , that means that a ≠ b for any precision and that command d is performed incorrectly instead of command c ! It may seem that this can be removed by the introduction of the set arithmetics. On the other hand, in the set arithmetics not just two, but three situations may occur, because the numbers a, b are described by sets (intervals) a ∈ A, b ∈ B : Case 1:

A = B identity of

Case 2:

A ∩ B = 0 zero inters

259

5.1 Introduction to the Theory and Practice of Creation of FEM Models Case 3:

A ∩ B ≠ 0 non-zero

For case 1 the jump to statement c is definitely correct. For case 2 the jump to statement d is correct. But for case 3 it is not decided what should be done. This case 3 must be added as the 3rd possible value of the logic expression: true, false, unknown. If the latter happens, it is necessary to start some additional decision-making procedure, so that the algorithm might proceed to the correct target. For example, the equality a = b in the test will be in this case substituted by the inclusion A ∈ b and, knowing the numerical precision in the computer, set A is created in such a way that A ∋ a , e.g. A = (0.999K 99, 1.000K 01) . As b = 1 , it must be true that b ∈ A and jump to statement c will be correctly performed. We presented a very simple example that would be handled by an experienced programmer by an adaptation of the test, supposing that they would come across any discrepancy during debugging. But the cases may be significantly more complex and an unusual exceptional case of a test can be met only after long years of flawless operation of the program. Consequently, it pays off to prevent these incidents in advance also in the procedures of FEM programs.

5.1.4 Modelling of Non-linear Behaviour of Structures by means of FEM Algorithms 5.1.4.1 User Approach to Non-linear FEM Problems The commonly used linear mechanics is from the physical point of view only a special limiting case of a general mechanics. The analysed structure does not change at all in terms of geometry. The components of displacement vector u are negligible with regard to the dimensions of elements. The components of the tensor of deformation ε and vector of rotation ω are near zero and, therefore, they may be neglected with regard to 1, i.e. to the dimensionless unit. Quantities u , ε and ω are considered to be only the source of stressstate σ that is attributed to the non-deformed body that is used to specify the conditions of equilibrium. In terms of physics, we assume that σ and ε are related to each other by a linear relation – by a generalised Hooke’s Law. Also the geometrical relations between ε and u are linear, they do not contain any expressions of 2nd and higher degree, roots, etc. Statically, we work just with external forces which do not depend on the deformation of the structure. We deal with a conservative system of primary (given loads) and secondary (unknown reactions) external forces that do a common virtual work over virtual displacements defined by the sum or by the integral of products of forces and components of displacement in their direction. With regard to the primary state u = ε = ω = 0 they have the potential energy of the position (art. 3.3). The support (boundary) conditions do not change during the process of deformation, they are scleronomous. The results of a linear problem do not depend on (i) how the given external forces reached their final size, (ii) how they are distributed over the structure, (iii) whether loading or unloading occurred – briefly said they do not depend on the loading path over time. Similarly, the Hooke’s Law does not contain any information about the path of stress and deformation over time. For a given σ or ε we may unambiguously 260

5.1 Introduction to the Theory and Practice of Creation of FEM Models find ε or σ without examining the history through which the material of the structure passed before the investigated state. The assumptions of the linear mechanics are so strong that in reality they are never fully satisfied and, therefore, it is the engineer who must decide whether they accept all of them and limit themselves to a linear version of FEM programs. In exceptional situations, e.g. if only slightly stressed and deformed steel structures subjected to monotonous static load with permanent connections in joints and bearings are analysed, there is no substantial doubt concerning the justification of the full linearization. Usually, however, some or all the given assumptions can be rightly criticised and their fulfilment is questionable. Slender steel structures may show deflections that evidently influence the arms of the forces, i.e. their moments related to the points of the deformed system. Partial plasticisation of certain parts in the analysed limit state can be expected and allowed, etc. The material of concrete structures is far from the assumptions of Hooke’s Law and prestressed systems are very sensitive to even relatively small deflections (fig. 24d). Each structure must be laid on some foundation and the subsoil (which has a significant impact on the behaviour of the structure) is always strongly physically non-linear. Moreover, its deformation depends on the history of the loading process, on consolidation, changes of water regime, etc. Membrane and cable roof structures are subjected to non-conservative forces, water pressure is a typical force that follows the deformation and remains permanently perpendicular to the load bearing surface, etc. Briefly summarized: There are substantially fewer structures for the analysis of which we can apply the linear mechanics without any objections than those which evidently do not meet the strict assumptions. In the EU countries the civil engineering design practice was strongly influenced in the years 1991 to 1997 by seven EUROCODES numbered temporarily EC1 to EC7. Majority of them was already accepted by Czech standards CSN P ENV 1991, etc. with corresponding symbols of the field including what is termed national application documents (NAD). Non-linear calculations are required if the expected error of the linear theory exceeds 10%. It is up to the structural engineer to decide who and how should anticipate this error.

These facts contradict with the fact that the present-day design practice of ordinary structures (estimated 95% of all static calculations) is fully linearised with the exception of foundation engineering where the physical non-linearity of subsoil is, at least roughly, taken into account. Otherwise, we would obtain completely unreliable settlements (dramatically larger than the real ones), non-economical foundations and invalid prognoses of non-uniform settlements that fail to meet technological tolerances. Besides, applicable standards EC7, CSN, DIN, ÖNORM, etc. would not be satisfied. As we deliberately skip this problem in this chapter, as it will be described in chapter 6 in more details, we may, even in this paper, limit ourselves to what is called superstructure. The weakest link in terms of the theory of reliability is today the foundation engineering and thus the limitation to pure structures is just a methodological issue. Such a limitation can be applied in practice only in exceptionally favourable foundation conditions of less deformable subsoil or in some types of statically determined foundations, e.g. bridges on three supports. The reasons why structural engineers in real practice use in 95% of situations linear FEM programs can be summarized into two groups: rr) Traditional lectures at universities, textbooks, technical guides, numerous standards and manuals are still usually based on the linear mechanics with an occasional 261

5.1 Introduction to the Theory and Practice of Creation of FEM Models reference to a certain non-linear effect that is sometimes considered unimportant. A typical example is the present-day approach to the calculation of internal forces in reinforced concrete structures where both EC2 and relating national standards admit the linear calculation, even though the design takes into account the physical nonlinearity and heterogeneity of material. Also the commonly offered FEM programs available for an affordable price for smaller companies have contained, until recently, only the linear statics. ss) The time and financial factor slow down the application of non-linear FEM programs. Large mechanical engineering companies, universities and scientific institutes can already utilise efficient systems, e.g. ANSYS [73], which are not accessible to entrepreneurs with limited financial resources. They can perform the required calculations, but on order and they have to pay for it. A calculation performed by your own program on your own PC has the following disadvantages: (i) it takes significantly longer time than the linear problem, (ii) it may require more complicated input data, e.g. the non-linear physical relation between bending moments and curvatures in cracked reinforced concrete, detailed geological profile of the subsoil, constants for conditions of plasticity, etc. The static calculation is just a small part of the whole project, see art. 5.1.2., Fig. 5.4 and 5.5. To pay a great attention to the project is reasonable only if the savings made during the realisation of the project exceed the expenditures spent on “obtaining more detailed input data” and “performing the calculation”, or if there is a chance to get another advantage, for example in a tender for a contract, or if it is not possible to guarantee the safety of the building in some ultimate or serviceability limit states, especially stability and deformation. The procedure applied in practice follows subconsciously the schedules given in art. 5.1.1., Fig. 5.1 to 5.3. Practice is not interested in the two extremes: primitivism and perfectionism, i.e. no calculation on the one side or unnecessarily complex non-linear analysis on the other side. It intuitively minimises the total time and financial demands on the construction and its project. Each case has its specific features. The abstract term “it is necessary to use a non-linear mechanics” has its communicable part S and noncommunicable intuitive part I , see art. 5.1.1.1, formula (5.1.1). The ratio of objectivity of this necessity M 0 (5.1.1) the ratio of completeness of the specification of the problem M c (5.1.2) is definitely smaller than one. In the set of subjects, persons and teams only a certain part n0 shares the same communicable form that (in some particular situations) the non-linear calculation is unconditionally necessary. In chart (282) they hold at least one positive answer YES to the questions about the expected extent of the change of the shape of the structure and about the influence of internal forces on the stiffness of elements. If both answers are NO, the linear calculation is justified on condition that there is no risk of shape instability that would require the application of determinant equation (286). Let us notice that none of the above mentioned positions is exactly quantified. That is where the intuitive part of decision about the necessity of application of a non-linear mechanics is. The experts simply conclude – on the grounds of his greater or smaller experience, concern about the future of the construction, subconscious reactions based on 262

5.1 Introduction to the Theory and Practice of Creation of FEM Models what they learnt from various seminars, conferences and other information sources – that a non–linear program should be used. They cannot give any exact reason for this conclusion. To be able to do so they would have to know the result. The intuition occasionally fails and a sophisticated program consuming ten-times more computer time produces almost the same result as the linear mechanics – the quantities decisive for the design differ e.g. just by 2% and large differences are only in non-substantial details. The engineering proficiency (Fig. 5.3) is, among others, related to a genius intuition and simplicity – the goal is a high-quality construction and not the calculation itself. An extraordinarily efficient aid that helps to achieve this target is FEM, the results of which either objectively confirm the prognosis of the behaviour of the structure or correct it and provide a welcome guidance for the future. Unlike the mathematicians who base their considerations just on exactly proved theorems and work with formal terms P (100% S , 0% I ), (popularly said: they solve what they are capable of), the engineers solves what must be solve and this solution is not the core or final aim of their effort. From time to time we may even come across an opinion that the engineer must at least roughly know in advance what the result will be. If we take into account the fact that they must input the analysed structure in a way that is not too far from the reality in terms of designed cross-sections of beams, thicknesses of slabs, reinforcement, etc. – which is the decisive factor in the design of statically indeterminate structures (and practically nearly all structures are statically indeterminate), this ability to foresee is really desirable, otherwise there is a risk that the calculation would have to be repeated even several times. Software companies offering FEM programs emphasise in the marketing materials printed in colours the user-friendliness of their products, efficient graphical preprocessors and outputs, quality implementation of all possible non-linear and time-dependent problems, which is after all true information. However, for tactical reasons, it is not mentioned how educated the user must be to exploit at least 5% from the corresponding program packages or not to remain just at the level of linear modelling. Similarly, time and financial demands relating to gathering of input data corresponding to the level of the program are not stated. All thee factors decide in practice about the total effectiveness of the resources spent on the preparation of the project, i.e. also about the fact to which extent the non-linear mechanics is employed.

5.1.4.2 Assembly of Equation Systems in Non-linear FEM problems FEM program systems are usually versatile and make it possible to adjust the required mode according to (282) or stability calculation according to (286) already during the declaration phase. In addition, it is possible to follow individual solution steps, to control their accuracy and final accuracy of the level and distribution of loads, to follow the convergence, etc. The solution can be performed also manually and can be compared with (i) the exact calculation (irrational expressions with roots) or with (ii) commonly applied cubic calculation that is not suitable for rubber materials. This approach clarifies all necessary terms of basic equation (280) for one step of the general algorithm, which solves the transition of the analysed structure from an arbitrarily loaded and stressed configuration 1 into a subsequent configuration 2. Structural engineers with an average expertise can skip elementary textbooks. For them, the following concise explanation that skips all the details that are interesting 263

5.1 Introduction to the Theory and Practice of Creation of FEM Models primarily for mathematicians and programmers. The present-day computers require the complete linearization of one step (of the atom of the solution), because the only applicable means is the solution of the system of linear algebraic equations. In algorithms of complex problems this may be implemented just as a sub-program of some more complicated non-linear procedure that is incorporated into a package of mathematical programs with attractive names. It does not alter the fact that, after all, our PCs are not capable of using any other means. Theoretical documents prefer the procedure from the most general definitions of mechanical quantities (requiring irrational geometrical relations with roots (Novozhilov)) to quadratic geometry (Green-Lagrange tensor of deformation and 2nd Piola-Kirchhoff tensor of stress) which, however, when applied in FEM without any adaptations, generates systems of equations of 4th degree. These can be relatively easily adapted to equations of 3rd degree. The reduction to 2nd degree is already bound by strict assumptions and in order to achieve complete linearization, it is necessary to make a kind of intervention into physical relations. The procedure requires various adaptations, symmetrisation, definition of quantities of tensor character, etc. The final effect can be explained to users of FEM programs in a popular way from below without stating anything from the whole theoretical base. It requires to limit ourselves to so-called reasonable or well conditioned problems and to make a sincere confession of what we are able to solve. We can analyse this problem: We have an arbitrary structure or body Ω with boundary Γ in state 1 in which the shape is Ω1 and boundary Γ1 and in which it shows, in comparison with the initial state, parameters of deformation ∆1 , stress-state σ 1 and in which it is subjected to external load represented by virtual nodal equivalents f1S that are in equilibrium with internal forces or stress σ 1 . We are interested in shape Ω 2 with boundary Γ 2 in state 2 in which it is subjected to external forces with virtual nodal equivalents f 2 . The unknowns are the parameters of deformation ∆ 2 and stress-state σ 2 in state 2. The deformation and force parameters are defined or measured in both states in the same coordinate system X G (let us call it global) that does not change during the whole process of solution. The stress-state is related to the same areas of sections across the physical body Ω , which means that we may perform decomposition of state 2 to the sum of known state 1 and unknown increment: ∆ 2 = ∆1 + ∆ σ 2 = σ1 + σ f 2 = f1S + f

(5.1.17) (5.1.18) (5.1.19)

We assume that only small rotations occur during the transition from state 1 to state 2, and that they can be handled as vector. The question how to obtain the total rotations after Ω passes through several states will be solved later using a special consideration in which we separate the pure deformation of elements from their motion as a rigid body. We are able to solve increments ∆ from the system of equations. The analysed structure or body Ω may consist of 1D elements with internal forces S1 , 2D elements with internal forces s1 and possibly also 3D elements with stress σ 1 in state 1. As both S1 and s1 are just integral representatives of real physical stresses σ 1 according to art. 4.2.2.2., formulas (4.2.13), (4.2.14), it is obviously enough to use the notation σ 1 in parenthesis, which points 264

5.1 Introduction to the Theory and Practice of Creation of FEM Models out the fact that matrix K σ depends on the stress-state of the body, i.e. the stress-state of its elements. It is after all contained also in its most correct name: matrix of the influence of the initial stress-state on the stiffness of the element. The most concise notation of the pertinent set of equation is: KT Δ = f

(5.1.20)

with the meaning:

[ K L + K σ (σ 1 )] Δ = f2 − f IS For the sake of further considerations it is advantageous to write in the parenthesis also the dependence of both matrices on parameters of deformation ∆1 in state 1. That is because the matrices are obtained through summation of element matrices transformed into global coordinates xG , yG , zG , where the increments ∆ are defined and solved. This transformation is written in art. 2.3. only once in no. 12 of the overview, because the transformation matrix T consists of constant cosines of the angles between the element and global axes ( xe , ye , ze ) , ( xG , yG , zG ) . This is explained in detail in the text following formula (2.4.4). Now, however, we take into account the change of the position of the element axes during the deformation of the structure and, therefore, axial trihedrals ( xe , ye , ze ) relating to every element form with the global axes different angles in different states. These angles (and subsequently also matrices T ) can be in state 1 determined from parameters of deformation ∆1 of this state, especially from those that are components of displacements u, v, w of the nodes of the structure. Naturally, also the components of rotation are important for the separation of the motion of the element as a rigid body from the total deformation or displacement of all points of the element. It is therefore convenient to indicate this dependence in the parenthesis after T and write symbol T(∆1 ) in geometrically non-linear mechanics. No 12 of the overview in art. 2.3 will thus gets a more general form K eg (∆1 ) = TT (∆1 )K e T(∆1 )

(5.1.21)

which is valid for physical linearity – with a constant stiffness matrix of the element K e that is independent on the level of the stress-state and deformation σ 1 , ε1 (Hooke’s Law) of state 1. The situation will not get more complicated if we omit this classical requirement and input some more complex physical law (art. 4.2.1), e.g. in an incremental form (4.2.10), (4.2.11) or even in the form depending on the history passed by the material of the element before state 1, which can be recorded by means of time factor t . It can represent a real time (if the duration of different stress-states is really important – see rheological models in Fig. 4.23 - 4.25) or just a formal time (if the duration is not important). Generally, it is possible to assume a variable stiffness matrix K e also in element coordinates, which can be written as K e (σ1 , ε1 , t ) , where factor t means the phase from the initial state of the material of the element to state 1. This gives a more detailed notation of the stiffness matrix of the element in the global coordinates: K eg (∆1 , σ 1 , ε1 , t ) = TT (∆1 )K e (σ1 , ε1 , t )T(∆1 )

(5.1.22)

If we apply the addition theorem (art. 2.3, no.13) to matrices (5.1.21), we get the stiffness

265

5.1 Introduction to the Theory and Practice of Creation of FEM Models matrix of the whole structure K L . In a general non-linearity, it obviously depends on the same quantities as K eg , which is indicated by symbol K L (∆1 , σ1 , ε1 , t ) . The matrix of the influence of the initial stress-state on the stiffness of the element K σ in state 1 depends in one element in its coordinates on its stress-state σ1 . After the transformation into the global axes analogous to (5.1.20) and after the application of the addition theorem, we obtain matrix K σ that depends not only on stress-state σ1 but also on ∆1 , because the transformation matrix T depends on ∆1 . This is marked by symbol K σ (∆1 , σ1 ) . The equilibrium equation can be then written in the following form: K T (∆1 , σ1 , ε1 , t )∆ = f

(5.1.23)

In more detail:

[ K L (∆1 , σ1 , ε1 , t ) + K σ (∆1 , σ1 )] ∆ = f2 − f1S

(5.1.24)

5.1.4.3 User’s interventions into the execution of non-linear FEM programs Now we may proceed to the explanation of interventions that the user may make during the execution of a versatile FEM program when solving a particular problem. We will give several variants of answers to questions (a), (b) and (c) concerning the physical nonlinearity including a possible time factor. For the time being we will omit questions relating to stability and post-critical behaviour. Possible answers are marked in table (5.1):

(5.1) a) geometrical nonlinearity b) influence of the stress-state on stiffness c) physical nonlinearity

1.

2.

3.

4.

5.

6.

7.

8.

NO

YES

NO

YES

NO

YES

NO

YES

NO

NO

YES

YES

NO

NO

YES

YES

NO

NO

NO

NO

YES

YES

YES

YES

Variant 1. It is a classical linear calculation in which matrix K σ does not apply and matrix K L is independent on all stated factors, which means that equation (5.1.22) gets a common form K L∆ = f The solution runs just once. The primary non-stressed state is taken as state 1, the required state is state 2 and neither increments nor iterations are needed. If the structure is in the initial state subjected to some tension, its influence on the stiffness is neglected and the calculated 266

5.1 Introduction to the Theory and Practice of Creation of FEM Models stress-state is simply added to it. Variant 2. This is a minor adaptation of variant 1 and can be briefly called adjustment of nodal coordinates with a symbol of summation of the components of displacement and the coordinates. x2 = x1 + u y2 = y1 + v

(5.1.25)

z2 = z1 + w To be more precise, it means that changes of the position of elements with regard to the global axes – i.e. the changes of transformation matrices T(∆1 ) in state 1 to T(∆ 2 ) in state 2 – are taken into account. The influence of the stress-state on the stiffness of elements is neglected. There exists an example that can be solved in this way with any required precision. It is a cantilever subjected on its free end to a bending moment. The axial and shear forces in the cantilever are zero and the bending moment is constant. Consequently, matrix K σ = 0 . The only inaccuracy of the solution is the fact that we assume that each 1D element remains straight after the deformation, even though its axis in this case is an exact circle. This inaccuracy can be arbitrarily reduced through the refinement of the mesh of 1D elements in the cantilever. The more the solved structure differs from this ideal configuration – especially the larger the axial forces are – the bigger the error is. The axial force N has the biggest impact of the six internal forces of beams. This may be compensated by what is termed ecorrection, which in fact corresponds to nodal balancing f 2 − f1S with the remaining five components of the deformation of the beam being neglected. Taking into account the current state of the development of the theory and computer technology, programs of this type are meaningful as a simpler approximate calculation of suitable configurations. The influence of K σ in equation (5.1.24) disappears and matrix K L depends only on the reference configuration of the system given by parameters ∆1 : K (∆1 )Δ = f 2 − f1S

(5.1.26)

The problem further simplifies if we omit the tests of the level and distribution of nodal force parameters f and if we simply divide the total load into n equal parts that we gradually add to the load the structure is subjected to. The number of equilibrium iterations in one incremental step is set to one, i.e. the system of equation (5.1.25) is solved only n -times. The size or the number of intervals n is specified by the user, i.e. the rotations in one increment must not be greater than e.g. 5°, which is recommended in the ANSYS [73] and the upper limit that can be tolerated is about 8°. Less complex programs trust the user and perform no test (292). Variant 3. This is a kind of contradiction to variant 2. The deformations of the structure are small, but the influence of internal forces on the stiffness of elements is big (Fig. 4.28d). This is typical for prestressed structures. The dependence of stiffness on parameters ∆1 disappears and what remains in equation (5.1.24) is the matrix of the influence of internal forces K σ :

267


[ K L + K σ (σ 1 )] ∆ = f2 − f1S

(5.1.27)

All transformations by means of matrices T(∆1 ) do not apply any more and matrix K L is constant during the whole solution. This may mean just one run through the block of the solution of equation (5.1.27) that has been prepared from found nodal parameters f1S that form with σ 1 an equilibrium system of state 1. It can be stored in the memory from one of previous calculations. The program calculates parameters ∆ and – to be safe – it verifies whether the selection of this variant by the user is justified. It performs all transformations that depend on ∆ as if the user opted for a more complex variant 4 and continues with tests of the given load. Variant 4 does not differ from variant 3 in the first step, on condition that the full value of load (full load) is input without any increments. However, the deformation is assumed (in advance) so large that the dependence of matrices K on ∆ . cannot be neglected. As a result, we use equation (5.1.24) in a physically linear form:

[ K L (∆1 ) + K σ (∆1 , σ 1 )] Δ = f2 − f1S

(5.1.28)

This variant enables the user to control the complexity of the calculation through tolerances specified in tests. Experienced users can intuitively guess from the character of the problem that there is no danger of divergence of ∆ in no phase of loading (i.e. that there is no risk of loss of stability of the shape), they do not insist on the accuracy of levels and distribution of load in partial incremental phases and divide the total load into n increments just to ensure that the tolerance for rotations in one step is not exceeded. A relatively small n can ensure it, e.g. n = 3 . On the other hand, this iteration is always necessary after the last increment, otherwise no one could guarantee what is the factual load of the structure producing the given graphical and numerical output data. The variants 5 to 8 differ considerably from the variants 1 to 4 by the fact that the physically non-linear behaviour of materials of elements of the structure is taken into account together with their connections – which will be omitted for the time being for the sake of simplicity (it may be substituted by bound or contact non-linear elements). The complexity of variants 5 to 8 changes according to the requirements of the user and according to the character of the problem. Firstly, we explain the simplest case of what is called finite problems that do not depend on the path over time. Consequently, both the real time t and the fictive development parameter t vanish from all calculations. In practice, it is in particular the case of monotonous incremental loading of the structure, if the stress and deformation increase in all elements without any discontinuity in stress-strain diagrams, e.g. elimination of tension after cracking and after similar step-like changes in σ − ε diagrams. This does eliminate the possibility of loss of elasticity and development of plastic, i.e. irreversible, deformations. As the phase of unloading is not solved and thus there is no way to say whether it is the case of non-linear elasticity (a possible return to the initial state) or elastic-plasticity (after the return a part of deformations is permanent). For these problems the program requires that the stressstrain diagrams of the material of the elements be input in the usual form obtained from tests that can be performed either with controlled deformation ε , (i.e. function in the form σ = σ (ε ) ) or with controlled stress ε = ε (σ ) . Implicit formulas of the type f (σ , ε ) = 0 are 268

5.1 Introduction to the Theory and Practice of Creation of FEM Models not suitable. For the simplest configurations we use the relation for uniaxial stress-state σ x = f (ε x ) , more precise Hooke’s Law with one cubic member σ x = E1ε x + E3ε x3 , bilinear law with two different moduli E1 , E2 varying at a certain level of σ x while the other modulus may be zero (plasticity without strengthening), similarly for G modulus in shear, etc. In each state 1 the program can find both deformation ε 1 of the element and stress-state σ 1 – depending on the achieved values of parameters ∆1 , because the history of the structure plays no role here. The variant 5 is then controlled by a considerably simplified equation (5.1.24) or by a slightly extended equation (5.1.26): K L (σ 1 )Δ = f2 − f1S = f

(5.1.29)

In the simplest situation, state 1 is the initial state with zero ∆1 , σ1 , f1S , the sought-after state 2 is final (full load state). Contrary to the linear Hooke’s Law σ = Dε (valid for any σ 1 and σ 2 , i.e. permanently during the transition from state 1 to state 2), the moduli or other constants of the physical relation between s and ε vary. The moduli usually decrease – the material softens, but they may also increase – the material hardens. It may happen that we determine some average values of the moduli between states 1 and 2 that have the following properties: The calculation using variant 1 gives, with a satisfactory precision, the same result as variant 5. These are what is called secant moduli. In the stress-strain diagram of a uniaxial stress-state σ x = f (ε x ) these are the tangents of angles between the secant of the graph and the ε x axis. Ideally, it would be enough to solve equation (5.1.29) only once. Let us write it in the form K L (σ 1 , σ 2 )Δ = f

(5.1.30)

After solving Δ and corresponding stress-state σ 2 we may determine the real load acting on the structure – i.e. to calculate f 2 from stress σ 2 . If the user-specified tolerance TOLF (288289) is satisfied, the calculation ends. If the tolerance is not met, the calculation must be repeated with a new estimate of the secant moduli. That, however, does not have to be a better one, if we consider that a structure today usually has 1000 – 10 000 elements and all of them, to a different extent, participate in the creation of factual load f 2 . Therefore, it is usually more reliable to perform the calculation using the tangent moduli with the load divided into n increments. In each of them the stress-strain diagram is substituted by the tangent in the point corresponding to state 1 that was reached in the previous step. For more complex physical laws we deal with tangential planes or superplanes. Matrix K L (σ1 ) is then a typical tangent matrix of stiffness – marked usually K T , see (280). This procedure solves equation (5.1.29) n -times, every time with a different left-hand side. It must be ensured that no systematic error occurs and that the whole process does not produce results that are permanently above or under the real ones and continuously move away from them. This can be ensured by at least occasional equilibrium iteration that has been already mentioned in the text about the secant modulus. It is sufficient to apply a simplified iteration reduced to just one step, in which the factual load of the structure is determined. This state is then used as the starting point for the next increment. There exist many practical adaptations of this procedure, see [1-7, 32, 61, 70, 73]. Also the averages of tangent moduli at the beginning and at the end of the increment 269

5.1 Introduction to the Theory and Practice of Creation of FEM Models proved useful, i.e. a new repetition of the calculation of one step, etc. Variants 6 to 8 have been derived from variants 2 to 4 through a similar adaptation that was used to create variant 5 from variant 1. The difference is that it is reasonable to insist on a certain reasonable accuracy of the level and distribution of the real load of the structure in comparison with the required state after the considered load increment. Subsequently, the tangential moduli of the stress-strain diagrams of the material of elements apply in this state. Another difference is that it is necessary to specify the term stress. It is usually sufficient to use an intelligible engineering conception and the stress is related to the same planes of elements in their element coordinates. These move together with the elements as rigid bodies and their pure deformation is separated. In 1D elements the internal forces simply relate always to the same cross-section, regardless of the changes of its position in space. Similarly, in 2D elements they relate to the same mass normal h . More details can be found in [3, 49, 61, 65, 69 to 74].

5.1.4.4

More Complicated Constitutive Relations and Projects Depending on the Path

The current state of FEM programs and computer technology allows for the input of physical properties of materials corresponding to their real behaviour that may be very complex. What became a problem is how to obtain these properties effectively from experiments and how to store this information for numerical calculations. This is the issue addressed by specialists who publish corresponding publications, e.g. [78]. A special group is represented by geomechanical data [79], which are not covered in this publication for the sake of brevity and for which we refer to documents [8, 9]. The physical relations are elaborated in more detail also in newer FEM texts, e.g. [80]. FEM also penetrated many non-mechanical fields and it becomes commonly applied method in problems dealing with the protection of the environment [81]. Non-traditional models [82] are newer. They abandon the classical conceptions [83, 84]. The current trend is to increase the reliability and applicability of FEM in modelling and optimisation in the design in all engineering fields, see the preparation of the congress [85]. From the enormous number of innovations of physical character, the problems depending on the path (this means either a formal history of the loading process or the real time factor t , i.e. also the duration of individual phases preceding the analysed state of the structure) are those that have the biggest chance to be applied in common civil engineering and mechanical practice. An example of the dependence on t was given in article 4.2.1, Fig. 4.23 - 4.25. The typical feature of problems depending on time is obvious already from the simple Kelvin´s rheological model (Fig. 4.24, 4.25): The input of the current load of the structure does not determine its stress-state and deformation. In general, an arbitrary number of stress-states and deformation-states correspond to such an instantaneous load-state depending on the previous history of loading and unloading. Also the speed of loading and unloading and the time period during which the load is constant have an impact on the result. This is in agreement with the examination of structures whose material or subsoil feature strongly rheological properties, e.g. reinforced concrete foundation slabs with the effect of creep, relaxation, cracking, consolidation of cohesive soils, change of water regime, etc. taken 270

5.1 Introduction to the Theory and Practice of Creation of FEM Models into account. Older FEM programs sometimes offered a special process to follow the relaxation, i.e. the drop of stress over time t under constant deformation, or creep, i.e. the growth of deformation under a constant stress. Newer programs have introduced the possibility to follow the relaxation in a more general way – for an arbitrarily controlled distribution of deformation ε = g1 (t ) with given function g1 (Fig. 5.25a,b), which in the special case of a constant distribution ε = ε1 in time t leads to a traditional relaxation. Similarly, it is possible to separately follow a generalised creep with the input of introduced stress σ = g3 (t ) , Fig. 5.25c. In real materials both phenomena can occur simultaneously according to more or less complex constitutive laws that can be described (in a popular way) as a variable plane F (σ , ε , t ) in Fig. 5.25d, even though the dependence is more complicated. In symbolic coordinates σ , ε , t the plane σ = σ 1 defines a section ε (t ) representing a common creep. Similarly, the dashed section σ (t ) across the plane ε = ε1 represents a normal relaxation. The points of dash-dotted line σ = f 0 (ε ) can be considered the representation of a physically nonlinear dependence of σ on ε in time t = 0 . Similarly, σ = f1 (ε ) in time t = t1 , where, however, the symbolism of (Fig.5.25d) fails, because function f1 does not generally depend only on ε and t1 , but also on the course and duration of individual phases of the development of ε (t ) over time 0 ≤ t ≤ t1 . Complex FEM programs require the cooperation of the structural engineer with an expert on the given material. When dealing with consolidating subsoil (where the tensor of stress is according to Terzaghi principle divided into the spherical tensor of hydrostatic or hydrodynamic pressure of water in pores of soils and the remaining general tensor of effective stress between individual grains of the soil), a consultation with an engineer-geologist is normally required. This two-phase particular environment can be accompanied by a third, gaseous, phase (in the pores not completely filled with water). The constitutive law is adapted according to the degree of saturation (saturation of pores with water). An aid that proved useful for the explanation of the difference between the linear and non-linear mechanics is the example of a strut or stay in Fig. 5.26 that can be solved scientifically or – with a various rate of simplification and with various numerical modifications – even manually, which contributes to the confidence of users in FEM programs that sometimes produce surprising results. Common structural engineers are, from both the faculty and practice, so soaked with the linearity and resulting principles of linearity (proportionality) and summation of effects, that they trust the results of a non-linear mechanics only when they can verify them themselves using their own effort and reasoning. For clarity reason, we will present here just a few typical results and conclusions. All of them relate to symmetrical stays or struts in Fig. 5.26 a-d, whose solution can be reduced to the analysis of behaviour of one beam subjected to tension or compression, on condition that we limit ourselves just to a vertical load in nodes. One beam is subjected to vertical force P . The condition of symmetry takes the effect in the right end of the beam with the vertical sliding support (Fig. 5.26e), that prevents horizontal displacement u , which means that the problem has just one unknown: sagging and deflection in the stay and strut, respectively. A classical solution follows from the conditions of equilibrium in the non-deformed state and from the Hooke’s Law (Fig. 5.26e):

271

5.1 Introduction to the Theory and Practice of Creation of FEM Models v = P ( D sin 2 α ) D = EA L

(5.1.31)

with a common tensile or compression stiffness D . It is advantageous to introduce dimensionless quantities according to the marks of length in Fig. 5.27e ν =v L v =ν L

β = sin α = b L b = βL

p = P EA P = EAp

(5.1.32)

Then, the classical solution has the form ν = p β2

p = β 2ν

(5.1.33)

and in coordinate axes ν , p it represents a straight line – indefinitely long for both the stay and strut, – passing trough the origin (ν , p ) = (0, 0) in Fig. 5.27a. For clarity reason, a particular example has been selected that can be for the stay easily demonstrated by a small model of thin rubber: EA = 2 [ N ], a = 0.05 m, b = 0.02 m,

L2 = 0.0029 m 2 , L = 0.05385165 m,

D = EA L = 37.13907 [ Nm −1 ], α = arctg (b a ) = 21.80141°, β = b L = sin α = 0.371391, β 2 = 0.137931. The equation of the line is (5.1.34): p = 0,137931ν

p = P EA

ν =v L

(5.1.34)

A thin rubber fibre can sustain even large elongations without failure (rupture), e.g. ν = 5 . In the limit we may imagine the state, in which both fibres of the stay are nearly vertical (v → ∞) and no geometrical change of the inclination of the fibre appears and the limit relation holds: p = ν − 0.628609

(5.1.35)

marked by the dashed straight line in Fig. 5.27a. The behaviour of the stay is given by the drawn curve with the tangent (5.1.34) and asymptote (5.1.35). Its equation can be easily derived by means of the exact Pythagorean theorem: −0.5 p = (ν + β ) 1 − (1 + 2 βν + ν 2 )   

(5.1.36)

The inclination of the tangent of the graph is in fact given by the derivative:

272


−0.5 −1.5 dp 2 = 1 − (1 + 2βν +ν 2 ) + (ν + β ) (1 + 2βν + ν 2 ) = KT dν

(5.1.37)

For very small sagging ν → 0 we get dp dν = β 2 = 0.137931 , like according to (5.1.34). For large sagging ν → ∞ we may use strong inequality 2 βν + ν 2 ? 1 and substitute the 1 in parenthesis by number β 2 from the interval 0 to 1, without changing considerably the value in the parenthesis. As a result, we can easily extract ( β 2 + 2 βν +ν 2 )0.5 = ( β + ν ) and we get (in agreement with (5.1.35)): p =ν + β −1

dp dν = 1

(5.1.38)

273

5.1 Introduction to the Theory and Practice of Creation of FEM Models Figure 5.25 Special cases of sections across a general constitutive plane F(s, e, t). a) The distribution of stress s = h1(t) in time t under controlled deformation e=g1(t) introduced into the element in the vicinity of the analysed point x, y, generalization of relaxation. b) The same for another distribution e=g2(t) of the introduced deformation. Generally different stresses s1, s2 correspond to the same parameters e1, t1 for different history e. c) The distribution of deformation e=h3(t) for controlled stress s=g3(t), generalization of creep, dependence of value e3 not only on parameters s3, t1, but on the whole history s. d) Illustration of relaxation and creep as parts of a general rheological process.

274


Figure 5.26 a) to d) Symmetrical stays, in which the solution of a single tie or beam 12 is sufficient. e) to g) Geometrical relations for large sagging. h), i) The limit case of the initial horizontal position of the elements of the stay.

275


Figure 5.27 a) The example of the stay used in the text. The solid line represents the exact solution, the dashed line shows the asymptote for an infinitely large sagging, the dash-dotted line means the linear technical solution for very small sagging. b) The limit case of the stay whose beams are horizontal in the initial state. In a geometrically linear mechanics, we deal with a kinematically indeterminate problem that cannot be solved. If we admit large sagging, the solution exists similarly to case b).

276


Figure 5.28 The behaviour of the strut from Fig. 5.26 b for all possible values of load P. The change of the strut into the stay (due to break through) happens in part “a” between deflections w1 and w2. Only the stay can transfer another increment of force P. This part can be experimentally examined only with controlled deformation w. If we attach a vertical beam or a spring with stiffness k to the strut, three cases may occur: a) The phenomenon of the breakthrough remains preserved in another interval a. b) The break-through does not happen, only a little indication of it. c) The system is permanently stable, which occurs for larger stiffnesses k .

277

5.1 Introduction to the Theory and Practice of Creation of FEM Models This means that the exact solution (5.1.36) includes also both limit cases. In addition, it includes also the case that cannot be solved by means of a technical linear static calculation (Fig 5.27b). If β = 0° , the initial position of the beams of the stay is horizontal. Linear programs for PCs report immediately after the input that the problem is kinematically indeterminate. An attempt to solve one equation for one unknown sagging ν discloses the fact that stiffness K is zero in comparison with ν . That means that the equation of type Kν = P gives an unusable result ν = ∞ . In Fig. 5.27b this is marked by the horizontal tangent to the graph in the origin. The exact formula (5.1.37) provides for β = 0° exactly such sagging ν that is necessary to transfer force P – in accord with the behaviour of the model! Large P or p = P EA is naturally accompanied with large ν or n = ν L . In limit ν → ∞ it holds that the asymptote p = n − 1 – see Fig. 5.27b where also the detail of behaviour with very small ν is shown. It leads us to an important general conclusion for users of non-linear FEM programs: In geometrically non-linear mechanics there exists no term of static or kinematic determinateness or indeterminateness. All problems can be in principle solved. If the degree of freedom is not limited in any way (by a claw or stop), the system deforms up to the state in which the static equilibrium of loads and reactions is reached, i.e. the equilibrium of external forces of the system. Only if such a case can be never reached, the Newton’s laws of motion are applied and the problem is transformed into a dynamic motion problem, e.g. forced damped vibration, etc. This is rather a rare exception for common static loads which then change their character according to the nature of the response of the structure, see chap. 5 art. 5.4. A drastic example: When we attempt to release the fixation of a horizontal cantilever with vertical load P at the free end, i.e. after the insertion of a hinge into the fixed end, the linear theory fails. Consistently geometrically non-linear theory finds (e.g. by the method of controlled deformation) some states of the broken cantilever, it assigns zero bearing capacity P to all of them until it reaches the vertical position of the cantilever when it correctly assigns normal force N = P . If a kind of stop was defined in the problem somewhere along the path of the cantilever (e.g. at some deflection ν K of its loaded end), then the motion of the cantilever would stop and force P would be transferred by the reaction R in the stop. The derived exact solution (5.1.36), (5.1.37) valid also for the strut, or for the “reversed stay”, in which the length b < 0 and, therefore, β = b L < 0 . The behaviour of the strut in the whole extent p is displayed in Fig. 5.28. Several intervals of variability of p can be noticed there. For all negative p , it is simply a strut subjected to load acting upwards whose behaviour is identical to the behaviour of the stay – it is stable in all phases. The response of the strut to positive p acting downwards is at the beginning stable according to (5.1.36), (5.1.37) with negative β . Quite simple algebraic operations can be applied to discover that for a certain deflection (better “displacement”) ν K the derivative (5.1.37) equals zero and that for ν = ν K it is negative. In technical terms, it means that load pK , corresponding to critical value ν K , is the maximum possible one. In an attempt to increase the load, the strut would break through and converts itself into the shape of the stay, where such increase is already possible. The process during the break-through is dynamic, as there exists an excess of the force, i.e. acceleration occurs according to the Newton’s law. The breakthrough may be easily followed on a model made of suitable bars. If we return to Fig. 5.26a and 5.26b we can imagine a rotationally symmetrical system of beams of the strut that 278

5.1 Introduction to the Theory and Practice of Creation of FEM Models approximately model a flat shell with membrane stress-state. The break-through is in every day practice known as the behaviour of the bottom of an oilcan under required compression and also as the behaviour of various switches, etc. – it is thus a common and actual phenomenon and the graph in Fig. 5.28 is no technical surprise. We will extend it further by this consideration: Let us place a vertical beam or spring with axial stiffness k (in the relation R = kv ) under the loaded node of the strut. For the beam k = EK AK LK , for the springs k is the known spring constant. It also transfers a part of the load (according to the ratio of its own stiffness k to the stiffness of the strut) that is in the geometrically non-linear process variable and is given by incremental form (5.1.37) by the formula dp = KT dv . The summation of both parts of the load produces the graph in Fig. 5.28, from which it is obvious that three situations may occur: tt) A small stiffness in comparison to KT , i.e. k < k K – the phenomenon of the breakthrough occurs again, only the magnitude of the load under which another increase in loading starts to be impossible changes and the strut becomes the stay strengthened by the spring (Fig. 5.28a). uu) Critical stiffness k K , equal to KT in a horizontal position that is passed during the break through when the strut becomes the stay. When corresponding load is reached, we can observe in the model a very easy increase of the deflection under a very small (in the limit none) increase of the load (Fig.5.28b). vv) Large stiffness k > k K . The graph in Fig. 5.28c is constantly growing. The system is permanently stable. No break-through occurs. The lesson learnt by the user of FEM programs: The systems solved in practice usually have 10 000 – 100 000 elements. Elements stressed differently in different loading phases meet in one node. If we limit ourselves just to their axial forces N (truss girders), we may say, in quite a popular way, that some beams may act in the node as “springs” that stabilise the “buckling behaviour” of other beams. The behaviour of a technical structure would be characterised by an N -dimensional graph analogous to Fig. 5.28 (where N = 1 ) and with N = 1000 − 10 000 with a vast number of inflection, “saddle”, etc. points. Numerically it demonstrates itself in co-action with other internal forces of 1D elements (similarly in 2D elements) by the fact that the program may during the non-linear calculation report a collapse of the system (local or even global). It is quite complicated to distinguish whether it is a real collapse for the given level and distribution of load (i.e. the bifurcation of equilibrium, shape instability) or just a collapse during some equilibrium iteration due to the influence of an auxiliary constellation of the system of forces. The user has practically just one option: to increase the number of increments, i.e. to reduce one level of loading undergoing the equilibrium iteration. If the system continues to collapse either locally or globally, it is very probable that the system is really unsuitable for transfer of the given load and some structural changes must be made. A mere change of cross-sections and thicknesses of elements usually has no or little effect. It is better to change the whole conception of the structure, i.e. to modify the geometry, introduce stiffeners, etc.

279


5.1.4.5 The Selection of the Number of Increments and the Course of the Equilibrium Iteration Most FEM programs for non-linear analysis require a kind of user-cooperation in the solution and expert versions of the programs provide some assistance. Only robust programs have chance for success in practice. Briefly said: they produce equilibrium results and the users are able to compare the load with their input values. However, they have practically no chance to compare the analysed shape of the deformed system (structure) with the real shape that can be determined through measurement of the real system or at least its model, or alternatively by means of very complex numerical analysis with such a large number of small elements that it surpasses the capacity of present-day PCs. What is the core of the approximation of the results? The complete linearization of one step of the is possible only under such drastic simplifications that any additional preconditions would completely kill the non-linearity of the problem. Let us summarise at least the following main sources of errors in one step of the solution from chapter 1: ww) W hen the influence of the increment, or the correcting load increment, on the reference configuration 1 is calculated, every 1D or 2D element passes into configuration 2 generally deformed (elongated, compressed, bent, skewed, twisted). The rigid body motion of the element is somehow separated from this deformation, which causes some problems in 1D elements due to the rotation around their axis. Of all components of deformation we admit just the axial (1D) or planar (2D) components and we consider that each element in configuration 2 is again straight (1D) or planar (2D) one. Otherwise, we could not apply any standard FEM procedure. As a result, the system is in each phase (including the final one) modelled by a plated surface (2D) and line segments (1D) although in fact these are complicated areas and curves. xx) During the transition from configuration 1 to 2 each element undergoes a certain rotation ω .Common programs treat it as a vector, which is admissible just for “infinitely small rotations” in the limit ω → 0 . Then, even the order of the components of the rotation is not important, the same transformation formulas apply here as for real physical vectors of force, moment or displacement. For large rotations their non-vector character is so obvious that it can be easily demonstrated in textbooks of physics. A mere change of order of the rotation by 90° around the x - and y -axis, the body gets to a completely different position. The magnitude of the rotation ω that can be still treated using vectors depends on the required accuracy of the result. Simple manual tests show that acceptable values can be obtained if the maximum admitted rotations ω is by angle “TOLA”, which is equal approximately to 5°, approximately 0.1 rad. For a larger ω the error quickly increases and the results become purposeless. The user sometimes has the possibility to opt for smaller tolerance “TOLA”. Nevertheless, each step from this source is still affected by an error. yy) The stress-state of each element found in configuration 1 during the step from 1 to 2 is assigned to the appropriate element (displaced or rotated) in configuration 2. Virtually, other components of nodal forces in global coordinates belong to it, which 280

5.1 Introduction to the Theory and Practice of Creation of FEM Models is balanced by the equilibrium iteration. However, the procedure contains a systematic error in the estimate of the stress-state and the larger the element and load increment (or rotation made by the element), the larger the error. It follows from the above-said that even in a physically linear analysis (when, theoretically, it should be sufficient to calculate with just the full load) it may be useful and sometimes even necessary to divide the load into several increments that are separately balanced by the equilibrium iteration. This is practical for complicated physical relations between stress and deformation in order to obtain problem-free intermediate equilibrium states with an overview of activated connections, areas where discontinuous relations apply, etc. This is absolutely necessary in problems depending on the path of loading over time (art. 5.1.4.4 and 4.2.1) even if the physical time, i.e. the duration t , does not play any role and the only important thing is the history of loading and unloading, etc. If also the duration t is important, e.g. in viscoelastic or viscoplastic materials, the increments relate also to time factor t . The users of non-linear FEM programs are provided with a tool that can make the prognosis of the behaviour of the structure significantly easier and more accurate, on condition that they fully exploit the technical parameters of the tool. Unprofessional use – e.g. the effort to get the shortest possible time of calculation through the application of the full load without increments and using a very coarse finite element mesh – may completely devalue the results, unless the whole solution collapses completely (in time and before the devalued results may get to the user) thanks to integrated tests. Beginners are strongly recommended to consult the problem with the technical support department of the program manufacturer already before the preparation of the input data.

5.1.4.6 Newton-Raphson Method and its Modifications Already in the beginnings of the development of FEM in around 1950-1960 certain analogy was found between the incremental method for the solution of non-linear problems designed by engineers and almost 300 year-old Newton’s method for finding the root ∆ E of the equation F (∆) = 0 that was later extended by Joseph Raphson to systems of equations with several unknowns. It was later proved that the derivative F ′ in the Newton’s method is an analogy to tangential stiffness KT of the system with one free parameter ∆ . Similarly, the Jacobian matrix of partial derivatives in the system of equations is an analogy to tangential matrix of stiffness K T in systems with several free parameters, i.e. with a matrix vector of unknowns ∆ . A clear proof for the exact solution of certain systems of 1D elements (ropes) was given in 1968 [88]. It can be easily extended to arbitrary systems. This gave mathematical credibility to intuitively derived engineering procedures. This positive side of the development resulted in 1970-1980 in a general awareness of engineering practice that everything is in order and that a reliable apparatus is available for non-linear problems. The size of problems grew with the development of computer technology, in particular the number of unknown parameters and complexity of non-linearities. However, problems started to arise 281

5.1 Introduction to the Theory and Practice of Creation of FEM Models in practice. In about 1990 it was clear to all informed experts that only an indirect method is available. Regardless of which approach is used to estimate the stress-state of the structure (in the deformation method using the estimate of parameters ∆ ), we must always assign them a certain equilibrium load and use iterations to try to improve the estimate until this load only slightly differs from the required one. Various algorithms (hundreds of them) differ just in the method used to obtain suitable estimates. Authors and companies usually claim that their approach is the best one, i.e. that it requires the least number of steps (increments and iterations), which they back up by examples that work well with the applied method. In other examples the user may find that a considerably larger number of steps is required and sometimes it is even not too difficult to come up with an example for which the program fails. The h-p version of FEM and expert programs brought a hope into practice that users have finally received what they need in non-linear problems. Nevertheless, we often witness disappointments that cannot be explained by an ordinary user. The reason is that the authors and companies do not mention in promotional materials things that would affect negatively the marketability of the program and in the euphoria about the fact that several examples were analysed without any problems they deliberately or unconsciously generalise this success to all design work. They omit two crucial facts: zz) No FEM program solves the exact equations of the given structure, provided that such equations can be written and documented at all, which is rather an exception, see [88]. The exact equation even for the simplest 1D problems contains unknown parameters D in an irrational form under the radical sign. Generally, the equations often contain angles in trigonometric and other transcendent function. Even the equations drastically simplified by means of Taylor series to equations of 3rd degree in ∆ (i.e. considerably weakened geometrical non-linearity) cannot be solved by any FEM program. The only thing which the present-day PCs are capable of is the solution of systems of linear algebraic equations. Through a sequence of such simplest systems in incremental or iteration steps the FEM programs try to mathematically approximate what is described by means of exact or simplified non-linear systems of equations. This produces an overwhelming difference between the mathematical and physical model of the structure even in the top-class programs. aaa) T o define of the second source of deviations of the results from the real behaviour of the structure let us abstract away from the first source (a) and imagine that we really solve exact systems of non-linear equations. Even this ideal case can be solved (i.e. it converges to the exact solution) only under very strict preconditions. It is not possible to make sure in general that these preconditions are met even for relatively small systems let alone the systems with 10 000 to 100 000 unknowns, which is a common number in today’s practice. This causes a grave uncertainty: • whether the whole process converges to any solution at all and • whether this solution is statically correct, because the systems of non-linear equations have no unambiguous solution, but an unbelievably large number of mathematically satisfying roots, even for low levels of non-linearity and even if we limit ourselves to real numbers. Two roots may be here arbitrarily close or distant. Each FEM program must start from an initial estimate, which is usually represented by the classical linear solution. If it is close to the statically invalid root, the program usually finds this root. No intuitive processes run in computers, all operations are purely formal. Only the final check performed with a professional feeling of the users, i.e. by their intuition 282

5.1 Introduction to the Theory and Practice of Creation of FEM Models and experience, may discover such an error. Most users have no useful intuition and experience in non-linear calculations and many correct results are surprising for them. Let us try to put some light on this issue and present here a kind of explanation that is not intended for experienced mathematicians, but that may help get the first idea about what is going on during the execution of a FEM program and what the relation to the real structure is. We base the explanation on the conditions under which the Newton’s method used to find root ∆ E (index E = exact) of non-linear equations F (∆) = 0 (Fig. 5.29) converges. First of all, it must be stressed that it is a local method that does not find all the roots but just one which is, in a sense, nearest to the first estimate ∆1 . In FEM programs it is usually the linear (classical, technical) solution of the problem. If it were too far from ∆ E , the method does not have to be successful. Already this may be a reason for the introduction of the incremental process, as dramatic changes of ∆ . are not assumed to happen during one increment Then, let us write the Newton’s formula that results directly from the graphical interpretation (Fig. 5.29) of the substitution of curve F (∆) by its tangent. It calculates the improved (k + 1) -th estimate ∆ k +1 from the k -th estimate of ∆ k : ∆ k +1 = ∆ k − f (∆ k ) f ′(∆ k )

(5.1.39)

In all textbooks and guides the mathematicians draw the attention to the conditions under which the procedure (5.1.39) converges to the exact root ∆ E . Firstly, at least one root must exist in some interval a, b . Secondly, the first derivative f ′ with respect to ∆ must have the same sign along this interval, i.e. function f has no extreme there. Also the second derivative f ′′ is not allowed to change the sign, i.e. no inflection is there. To meet the condition that the graph of f must at least once intersect the ∆ -axis, the product f (a ) ⋅ f (b) must be negative. The initial estimate of root ∆1 = a can be selected if f (a ) has the same sign as f ′′( x) . This sign must not change within a, b , i.e. it is the same also for f ′′(a) , f ′′(b) . If f (a ) has the opposite sign, it is necessary to opt for initial estimate ∆1 = f (b) . Thus the sign of f (b) coincides with the sign of f ′′( x) . The estimate of the error of the k -th improvement of root ∆ k depends on the flatness of curve f , whose measure is the minimum of the absolute value of the first derivative f ′ in the interval a, b , marked simply m : ∆ k − ∆ E ≤ f (∆ k ) m

(5.1.40)

Let us notice that the preconditions that are necessary for the success of the Newton’s method are very strict and only the cases according to Fig. 5.29a), b) have a chance to succeed. The users always intuitively expect such “reasonable behaviour” of the structure. But even the simplest example of the strut in art. 5.1.4.4 (Fig. 5.28) proved that “less reasonable” distributions according to Fig. 5.29 c), d) may occur and an invalid root ∆*E may be provided by the procedure in addition to the statically valid root ∆ E . Inflections occurred already there, which may lead to a complete collapse of the process as shown in Fig. 5.29a. Also the number of steps k = 1, 2, K that are required to obtain a sufficiently accurate estimate ∆ n is 283

5.1 Introduction to the Theory and Practice of Creation of FEM Models practically important. If a point with zero f ′ occurred in the interval a, b , then m = 0 and the estimate of the error (5.1.40) cannot be used – it simply says that the error is smaller than infinity, which we know anyway. The situation with f ′ = 0 is, however, eliminated already in the preconditions of the applicability of the procedure, because f ′ is not allowed to change its sign. On the other hand, the case of an arbitrarily small f ′ , i.e. of an arbitrarily large error ∆ k , is not eliminated.. However, in FEM application we do not have not just one unknown, but usually approximately 10 000 to 100 000 and even more. For more equations written briefly using matrix vectors as f (∆) = 0

(5.1.41)

J. Raphson extended the Newton’s procedure by a formula very similar to (5.1.39): ∆ k +1 = ∆ k − [ J (∆ k )] ⋅ f (∆ k ) −1

(5.1.42)

All first partial derivatives apply in the formula in what is called Jacobean matrix J , which we write, for the sake of clarity, using its components:  ∂f1 ∂∆1  ∂f ∂∆ 1 J (∆1 , ∆ 2 ,K , ∆ N ) =  2  M   ∂f N ∂∆1

∂f1 ∂∆ 2 ∂f 2 ∂∆ 2 M ∂f N ∂∆ 2

L L L

∂f1 ∂∆ N  ∂f 2 ∂∆ N   M  ∂f N ∂∆ N 

(5.1.43)

The conditions of convergence are similar to the Newton’s method, e.g. determinant J must not be zero in any point of the domain in which the iteration is performed, etc. The proximity of the first estimate ∆1 to the exact root ∆ E – which is now an arithmetical point in an N dimensional space (∆1 , ∆ 2 , K , ∆ N ) – can be expressed by the requirement that ∆1 must lie inside the N -dimensional sphere with the centre in ∆ E and a radius of convergence r . No one tests this condition in practice and all applications are performed by the users in good faith that the result will be usable. What the risk is will be demonstrated at least on an elementary example of two equations with two unknowns ∆1 , ∆ 2 , used already in the linear problem in art. 5.1.3.1 (Fig. 5.23) dealing with errors of arithmetics. The non-linear case is illustrated in Fig. 5.30. Instead of an intersection of two straight lines we now deal with the intersection of two curves in plane (∆1 , ∆ 2 ) . The first step of the Raphson´s procedure substitutes areas F1 , F2 by tangential planes in points B1 , B2 whose coordinates ∆11 , ∆12 are equal to the first estimate of the roots. These planes intersect the base plane (∆1 , ∆ 2 ) in two straight lines, which represents the linearization analogous to Fig. 5.23b). We expect that the intersection of these straight lines (∆12 , ∆ 22 ) is not too far from the exact solution, i.e. from the intersection of curves (∆1E , ∆ 2 E ) . We repeat the procedure using this improved estimate. The straight lines (tangents) of the Newton’s method (Fig. 5.29) are replaced by tangential planes, i.e. by 2-dimensional bodies, which corresponds to the number of non-linear equations being solved. Users–engineers will find in Fig. 5.31 a particular application: an asymmetrical stay 284

5.1 Introduction to the Theory and Practice of Creation of FEM Models made of two ties. The figure shows the common procedure with the importance of the partial derivatives in matrix (5.1.43) for N = 2 . These are the tangents of angles formed by the characteristic tangents in points B1 , B2 of surfaces F1 , F2 and the coordinate-plane (∆1 , ∆ 2 ) . In FEM programs these partial derivatives are represented by members K1,1 , K1,2 , K 2,1 , K 2,2 of the tangential stiffness matrix K T (2, 2) of the system that is valid for the currently performed step of the solution. In general, matrix J is the tangential matrix K T for an arbitrary N . The complexity of the Raphson´s procedure and the risks of failure are apparent already from Fig. 5.30-5.31 for N = 2 . The situation is so unclear for large N that the success of the solution cannot be generally predicted and it is necessary to limit the predictions to problemoriented ones. The root is the arithmetic point in an N dimensional space with base (∆1 , ∆ 2 , K , ∆ N ) . The point is defined as the common intersection of N superplanes inserted into an ( N + 1) -dimensional space (∆1 , ∆ 2 , K , ∆ N , F ) . For common N = 10 000 − 100000 , we can hardly estimate the character of these planes in the vicinity of the root, their vertices, “valleys” and saddle points, mutual inclination influencing the variance of the intersection for the given arithmetics, i.e. non-zero thickness of these planes analogous to Fig. 5.23b, etc. The users of FEM programs can do nothing but rely on the promotional materials of the companies that usually stress the robustness of the program. This can be in fact guaranteed only if the application is limited to a particular problem, e.g. to a certain type of structure, a kind of non-linearity, etc., which is usually soon discovered in practice.

285


Figure 5.29 Newton’s method that finds root DE of one equation F (D) = 0 with one unknown D. The first and second derivatives F with respect to D are marked F’ and F”. They must fulfil the conditions (given in the text of art. 5.1.4.6) that are rather strict and correspond to the situation illustrated in Fig. 5.29a) b). Cases c), d) already does not guarantee that the method finds the required statically valid root DE and that an invalid D*E is not found instead. In case e) the method does not have to converge to any solution at all.

286


Figure 5.30 Raphson´s extension of Newton’s method from Fig. 5.29 to the method searching for root (D1E, D2E) of two equations (1),(2) with two unknowns. a) Initial estimate D11., D21. corresponds to points B1, B2 on surfaces F1, F2. b) The tangential planes in these points approximate the distribution of functions F1, F2 in the vicinity of the exact root that is situated in the intersection of exact curves F1 = 0, F2 = 0. These are substituted by straight lines F1.1 = 0, F1.2 = 0. Their intersection is the second estimate of the root, and the procedure is repeated again and again. When certain conditions are met, the solution converges to the exact root.

287


Figure 5.31 Two characteristic tangents a) to surface F1 in point B*1, b) to surface F2 in point B*2 marked a) F1,1, F1,2 b) F2,1, F2,2. Their inclinations (partial derivatives written in the figure) represent members K1,1, K1,2, K2,1, K2,2 of tangential stiffness matrix K of the system with two free parameters, e.g. D1 = u, D2 = w for an asymmetrical stay.

288

5.1 Introduction to the Theory and Practice of Creation of FEM Models To conclude this article, let us give another piece of information that is important for the users. Nearly all non-linear FEM programs make it possible to exploit the fact that the solution of linear equations K∆ = f 2 can be very quickly obtained if the solution of the system K∆ = f1 with the same left-hand side K and an arbitrary right-hand side f1 has been performed before. It is sufficient to keep in the memory either what is termed the back substitution or the decomposed matrix K * (triangular matrix). For large N the time savings are enormous. This resulted – especially in the times of inefficient PCs – in modifications of the Newton-Raphson´s procedure that are presented in Fig. 5.32. The original procedure changes the tangential stiffness matrix K1 in each step k , i.e. in every iteration of every increment, according to the reached configuration of the system and its stress. The change of stress results in the change of matrix K NL of the initial stress-state. Then, tangential matrix K T = K L + K NL of each element is transformed into the global coordinates for new direction cosines between the element and global coordinates (Fig. 5.32a). This guarantees a very fast convergence of the second order, but in exchange for the need to repeatedly perform a new solution of the equations with new left-hand sides. If we are quite close to the root for one increment f1 , we may try, starting from the j -th iteration (e.g. j = 3 ), to continue with the calculation with unchanged K T (Fig. 5.32b). This, on the one hand, requires more iteration steps, but, on the other hand, each of them is substantially shorter (measured in time). Only the back substitution is performed, the decomposed K T is kept in the memory. If, for example, one such iteration is ten times shorter, but we need five times more iteration steps, we can save, starting from the j-th iteration, 50% of the computer time. Many FEM programs offer the possibility not to change matrix K T during one increment at all (Fig. 5.32c) and even to calculate the whole problem with the initial matrix K T – which is termed “initial stiffness matrix procedure” (Fig. 5.32d). The users meet these terms just at the beginning of the input process and are asked on the screen to answer the following question: When do you want to change the tangential matrix K T ? Answers according to Fig. 5.32a) to d) are offered: bbb) onstantly.

C

ccc) I n first j iterations of each increment, default value is usually 2 to 3, number j may be selected. ddd) ust in the transition to next increment.

J

eee) ever.

N

fff) I leave this question to the program. This may be possible mainly in expert programs that make some suggestion that can be possibly changed during the increment, which can be approved or altered by the user’s own decision. Sometimes, such a program takes its suggestion back during the execution of the solution and reports something 289

5.1 Introduction to the Theory and Practice of Creation of FEM Models like: “I made a mistake in determining the character of the problem” or “unfortunately, my estimate was not optimal”, which may have an impact on the total computer time.

290


Figure 5.32 a) The original non-modified Newton-Raphson´s procedure changes the tangential matrix KT in each step, i.e. the left-hand sides of the solved systems of linear equations, which guarantees a good convergence of the second order. b) In order to save the computer time, we may perform a few last iterations of each increment without changing matrix KT, which leads to a weaker convergence of the first order, but one step lasts considerably shorter time as only the back substitution must be run. c) The change of KT only in the transition to the next increment (incremental stiffness procedure). d) Only the initial matrix KT is used (initial stiffness procedure).

Warning: The points through which the solution passes do not lie on the curve corresponding to the exact solution. This is due to the influence of many approximations that are necessary to completely linearize one step ( 1.2.1.4.6) Fig. a) – d) demonstrate a frequent case of monotonous convergence from below. In practice we may witness the convergence that is (i) monotonous from above (the points lie above the reached level), (ii) sometimes oscillating at the beginning (Fig. e) and (iii) variously combined for various parameters.

5.1.5 Transformation of Physical Quantities

291


5.1.5.1 Transformation of Tensors of Stress, Deformation and Physical Constants 5.1.5.1.1 3D Bodies Stress may occur just in real physical bodies which are always three dimensional. Nothing changes due the fact that some of the dimensions, e.g. the thickness of a wall, may be very small in comparison with the other two dimensions. This dimension cannot vanish to zero in physical terms. If that happened, the whole body would disappear, because even the smallest particle of mass are three-dimensional, regardless whether we consider that the smallest one is (according to a particular purpose) a grain, crystal, molecule, atom, baryon and lepton, etc. FEM programs for calculations of structures and technical bodies assume that the material may be modelled by a physical continuum, which proved to be a useful concept. Even if this assumption is met, each particle, including the famous elementary hexahedron dxdydz , is three-dimensional and its physical properties are assumed not to change if we arbitrarily decrease its volume. This idea is absolutely wrong, it fails completely even with dimensions of approximately 10−10 m (molecular and atomic level) and it makes problems in the analysis of singular loads, supports, shapes, etc. The only thing which keeps it alive in practice is its usefulness according to the Mach’s Principle of Economy of Thinking. In FEM algorithms we commonly use limiting operations such as derivatives and integrals, for which dxdydz → 0 , which is a paradox in terms of the real structure of mass, but this approach is applied for about 300 years in the whole field of mechanics to a full satisfaction of the users of both the algorithms and realised constructions – popularly said “nothing collapses due to this fact”. In this term, the term stress as it was given in art. 3.3.2 can be accepted as a useful concept, naturally only for 3D bodies. What must be done with 2D– models of walls, slabs and shells and 1D models of beams was explained in art. 4.1.4 and 4.1.5. Any reduction of the dimension of the problem from 3D to 2D and 1D must guarantee that the three dimensional character of the stress does not vanish. The established internal forces must be always understood as resultant forces (integrals) of stress and their moments. As soon as any doubts or uncertainties occur, it is always necessary to recall the 3D character of the stress. Consequently, all difficulties usually disappear in a simple and natural way without any wild speculations. These must be abandoned anyway after some useless time is spent on a hopeless idea that they could be of any help in practice without leading us into an endless mud of nonsense, absolutely inadequate to the accuracy of present-day FEM programs. The above mentioned reasoning applies also to various transformations, which is a short name for rules for the calculation of the components of vectors and tensors in various coordinate systems. The term “transformation” is not too appropriate, because neither vector nor tensor is transformed during this operation, it is the same all the time – e.g. it is always the same force, the same moment, the same displacement, the same stress-state, the same deformation and the same physical property of the material. The only thing that changes is the coordinates in which the quantity is recorded, because we cannot describe them numerically otherwise than using their components. Unfortunately, this is related to a wide spread mistake – that appears frequently even in popular textbooks and texts – that a vector is identical with its three components, or even that a tensor is identical with three vectors, etc. 292

5.1 Introduction to the Theory and Practice of Creation of FEM Models But we deal with physical objects of exactly defined character that share in terms of mathematics just the rules for the transformation of the components into various coordinates, which is a subjective matter of the observer or documentalist. Manuals of FEM programs sometimes require that the user must define some properties of the structure in precisely defined coordinates, e.g. element coordinates x P , y P , z P (superscript P ). It may be considerably

293


Figure 5.33 a) to d) The transformation of stress components in 3D space, e) to h) The transformation in a 2D-model, see also Fig. 5.34.

294

5.1 Introduction to the Theory and Practice of Creation of FEM Models difficult especially in 3D problems, as most 3D elements use what is termed natural coordinates that correspond to the coordinate grid of basic unit-elements on which all base functions are primarily defined. But it may be impractical even for 2D problems as we will see in art. 5.1.5.2. User-friendly programs (which are today practically all commercially successful systems) offer the users the option to select any directions of the coordinates that are suitable and automatically transform the data into these directions. The procedures may be rather complex. Firstly, the directions of element axes x P , y P , z P must be determined (in general, each element in one defined macro-element can have different axes) and, secondly, the angles to the users axes must be calculated. Only a few simple formulas from the mechanics of continuum are needed to understand what is going on. They can be briefly summarized using Fig. 5.33 in which we indicate the user-axes x, y, z (the user does not come in contact with other axes). From the material of the structure we extract an element dxdydz in these axes (Fig. 5.33a), in which the stress is described by six components according to art. 3.3.2., i.e σ = [σ x , σ y , σ z , τ yz , τ zx , τ xy ]T . If we want to analyse (a little) area An defined by normal n , it is advantageous to extract an element of the shape of a tetrahedron according to Fig. 5.33a), in which An is dashed. Three conditions of component equilibrium in space immediately provide the first useful formulas for the transformation of components px , p y , pz of the result stress on area An into the user-axes x, y, z : px = σ x cos(nx) + τ xy cos(ny ) + τ zx cos( nz ) p y = τ xy cos(nx) + σ y cos(ny ) + τ yz cos( nz )

(5.1.44)

pz = τ zx cos(nx) + τ yz cos(ny ) + σ z cos( nz ) The advantage of the cosine function is the equality cos α = cos(−α ) , which makes it possible to write in parenthesis the axes that form angle α in an arbitrary order, e.g. cos( xn) = cos(nx) , even if we consider the angles to be oriented vectors. If we measure the angles from the first indicated axis to the other one and the angle is positive if the second axis is formed through a positive rotation of the first axis. The rotation is positive if it transforms the first coordinate axis in the plane of rotation into the second axis, which is in a right handed system just the positive direction of the third coordinate axis that is perpendicular to the plane of rotation. When viewed in this direction, the positive rotation is the clock-wise rotation which may be verified in Fig. 5.33a). This property holds in turns for all the three possibilities of the order of axes 123, 231 and 312. The components (5.1.44) are generally inclined to area An , which is not illustrative for physical reasoning about the phenomena happening there (tension, compression, shear). Therefore, it is suitable to introduce other components according to Fig. 5.33b) d), in particular component σ n which is perpendicular to An , i.e. it has the direction of positive normal n . Using projection we may obtain an unambiguous formula that once again contains only the problem-free cosine functions and stress components in a correct cyclic order: σ n = σ x cos 2 (nx) + σ y cos 2 (ny ) + σ z cos 2 (nz ) + +2 τ yz cos(ny ) cos(nz ) + τ zx cos( nz ) cos( nx) + τ xy cos( nx) cos( ny) 

(5.1.45)

295

5.1 Introduction to the Theory and Practice of Creation of FEM Models A certain component τ n situated in An (Fig. 5.33b) corresponds to it. Added together, components σ n , τ n give resultant σ r that is generally inclined in space. To find τ n the Pythagorean Theorem is sufficient: σ r2 = ( px2 + p y2 + p z2 ) τ n2 = σ r2 − σ n2

(5.1.46)

It is practical to decompose shear component τ n into two components τ nt , τ ns according to Fig.5.33d), with the directions t and s chosen as perpendicular and forming a right handed system together with normal n(n, t , s ) . In FEM procedure this system is usually represented by what is called element axes ( x P , y P , z P ) , see Fig. 5.33c. The contributions of individual components to τ nt may be found through projection into the t -direction. When summation is performed we get a formula similar to (5.1.45): τ nt = σ x cos(nx) cos(tx) + σ y cos(ny ) cos(ty ) + σ z cos( nz ) cos(tz ) + +τ yz [ cos(ny ) cos(tz ) + cos( nz ) cos(ty ) ] + +τ zx [ cos(nx) cos(tz ) + cos( nz ) cos(tx) ] +

(5.1.47)

+τ xy [ cos(nx) cos(ty ) + cos( ny ) cos(tx) ]

The formula for the second shear component τ ns (Fig. 5.33d) is obtained from formula (5.1.47) through a simple substitution of t by s . This provides an unambiguous and problemfree relation between the components of the same tensor of stress in different axes. This relation may represent the last (reliable) resort that can be used in case of uncertainty arising from unclear instructions of some FEM program manuals. It contains only the + signs, cosine functions and it has a sort of cyclic structure. Let us apply it now to the most frequent practical problem. The user inputs a structure or body in macroelements, in which he selects some technically most suitable axes x, y, z , which are called for the sake of brevity simply “user-axes´”. The program itself divides the structure into finite element according to the specified requirements. In each element the program (for the needs of its algorithms) defines unambiguously “element” axes x P , y P , z P using a uniform rule, e.g. in 3D elements in what is termed natural directions that correspond to directions ξ , η , ζ of the unit element (art. 4.1.6, Fig. 4.19). In general, the trihedral x P ⊥ y P ⊥ z P has in each element different orientation to the trihedral x ⊥ y ⊥ z that is shared by the whole macro-element or even by the whole structure, which can be expected in simpler configurations of planar structures with 2D elements, i.e. walls and slabs. Anyway, the program must reliably describe the quantities whose components have been input or stored in the database in the x, y, z -axes using the components in the element axes x P , y P , z P , because the element algorithms are capable of processing just these ones. It is an internal need of the program, and the users do not have to care about it in user-friendly programs. On the other hand, there still exist programs that require the input data in element axes, which may considerably complicate the preparation of input data. The components of the stress tensor in element axes x P , y P , z P are: 296

5.1 Introduction to the Theory and Practice of Creation of FEM Models σ P = σ xP , σ yP , σ zP , σ yzP , σ zxP , σ xyP 

T

(5.1.48)

The formulas for the transformation between components σ P and σ , i.e. what is termed transformation rule, is in fact already available: It is sufficient to substitute the n in formula (5.1.45) by the directions x P , y P , z P , which gives σ xP , σ yP , σ zP . Then we substitute (nt ) in formula (5.1.47) by direction y P z P or z P y P , which gives the same result τ yzP = τ zyP . Subsequently, (nt ) is replaced by direction z P x P or x P z P , which gets τ zxP = τ xzP . Finally, we replace (nt ) by direction x P y P or y P x P and get τ xyP = τ yxP . The result is six formulas that can be written in a clear form using the transformation matrix Tσ of the dimension (6, 6) : σ P = Tσ σ

(5.1.49)

The explicit notation of matrix Tσ is only seldom mentioned in the technical literature, although it is very useful. A clear notation can be obtained by the introduction of brief indication of direction cosines using symbol c with two indexes: i, k . The indexes are more important than the symbol c itself. Therefore, it is suitable to write them in big letters directly behind the symbol, i.e. in the form cik . The first index i = 1, 2 or 3 means the element axis x P , y P or z P – in physical terms it is the normal to the area subjected to the searched stress component, which is its first (in normal components the only) index. The second index k means the x, y or z axis of another arbitrary right-handed system, e.g. that of the “usersystem” in which the same tensor is written with components that appear in formula (5.1.49) as the given quantities. The importance of symbols cik can be best seen in a common transformation of a vector with three components from form v to form v P by means of matrix of rotation R vxP   c11 c12   v P = v yP  = c21 c22  P  c c vz   31 32

c13   vx    c23   v y  = Rv c33   vz 

(5.1.50)

c11 = cos( x P x),

c12 = cos( x P y ),

c13 = cos( x P z )

c21 = cos( y P x),

c22 = cos( y P y ),

c23 = cos( y P z )

c31 = cos( z P x),

c32 = cos( z P y ),

c33 = cos( z P z )

(5.1.51)

The orthogonality demonstrates itself through the fact that we do not have nine independent constants, but in fact just three values from which the remaining six ones can be determined by means of the (i) rule of orthogonality and (ii) trigonometric “one” (1): 3

∑c K =1

ik

3

∑c k =1

2 ik

c jk = 0 =1

i≠ j for i = 1, 2 or 3

i, j = 1, 2 or 2, 3 or 3, 1

(5.1.52) (5.1.53)

Matrix R is not symmetrical, which is obvious from (5.1.51). It has however one favourable

297

5.1 Introduction to the Theory and Practice of Creation of FEM Models feature: its inverse matrix R −1 equals its transposed matrix RT , i.e. the matrix is orthogonal. Therefore, the inverse formula to (5.1.50) can be easily found v = R −1 v P = R T v P

(5.1.54)

Using notation (5.1.51), matrix Tσ (6,6) has in transformation (5.1.49) the following form (written as a table to save some space):     Tσ =     

c112 2 c21 c312 c21c31 c31c11 c11c21

c122 2 c22 c322 c22 c32 c32c12 c12 c22

c132 2 c23 2 c33 c23c33 c33c13 c13c23

2c12 c13 2c22 c23 2c32 c33 c22 c33 + c32c23 c32c13 + c12c33 c12c23 + c22 c13

2c13c11 2c11c12   2c23c21 2c21c22  2c33c31 2c31c32   (5.1.55) c23c31 + c33c21 c21c32 + c31c22  c33c11 + c13c31 c31c12 + c11c32   c13c21 + c23c11 c11c22 + c21c12 

Similarly to matrix R , matrix Tσ is not symmetrical. Moreover, it is not orthogonal either, which means that Tσ−1 = TσT is not valid. However, its inverse matrix, needed for the inverse relation to relation (5.1.49): σ = Tσ−1σ P

(5.1.56)

can be obtained easily and directly from its definition. It in fact again combines the stress components according to general formulas (5.1.45), (5.1.47), but the right-hand side contains the element components and the left-hand side collects general or “users´” components. So it is enough to swap the order of indexes ik in the nine constants (5.1.51), i.e. also in (5.1.55). The only constants not affected by this change are c11 , c22 , c33 , the other six will change. For example, instead of c12 = cos( x P y ) representing the contribution of the y-component to component x P , the inverse matrix contains cos( xy P ) = cos( y P x) = c21 , which expresses the contribution of component y P to component x . And again, we exploit the favourable feature of the cosine function cos α = cos(−α ) , regardless of the direction of angle α . Therefore, no special table is necessary for the inversion. Let us point out that diagonal members Tii , i = 1, 2, K , 6 of this table will not be changed by the inversion. For i = 4 to 6 it is contributed also by the commutativity of multiplication. The transformation of components of deformations follows the laws as the components of stress, on condition that the tensor of deformation is properly defined. It is necessary to keep the same cyclic order as for the stress, i.e. ε = ε x , ε y , ε z , ε yz , ε zx , ε xy 

T

ε P = ε xP , ε yP , ε zP , ε yzP , ε zxP , ε xyP 

T

(5.1.57)

and to take the shear components only by half values of the full technical slopes, i.e. of the changes in right angles that were between the directions defined by the indexes, see art. 3.3.2 and clear Fig 5.33f) – for one slope γ xy : 298

5.1 Introduction to the Theory and Practice of Creation of FEM Models 1 ε ik = γ ik 2

i≠k

(5.1.58)

Then, formulas analogous to (5.1.49) and (5.1.56) apply: ε P = Tσ ε ε = Tσ−1ε P

(5.1.59)

with matrix Tσ according (5.1.55) and Tσ−1 according to the same table with swapped order of symbols ik in elements cik . The physical relation between the components of stress and deformation is in practice most often modelled by a generalised Hooke’s Law in some coordinates that are suitable for the given structure of material, lamination, important directions, etc. In general, we may deal with a real or what is called technical or shape anisotropy or various special orthotropies (wood, fibreglass, consolidated soil, reinforced or composite materials, etc.). In the matrix formula σ = Cε

(5.1.60)

the matrix of physical constants is symmetrical in the vicinity of one point of the structure. Its dimension is (6, 6) , therefore it has at most 21 independent constants Cij , i, j = 1 to 6 , with Cij = C ji . If full slopes γ were used in the test to obtain Cij , there is no problem to swap to half-slopes v according to (5.1.58). For example, instead of the relation τ ik = Gγ ik we write τ ik = 2Gε ik . This considerably simplifies the problem that must be solved by the program in every element of the analysed body. The physical relation (5.1.60) must be written in element coordinates or components, because the FEM algorithm requires the following form σ P = CP ε P

(5.1.61)

The matrix of physical constants C P has again 21 independent constants CijP . Their relation to constants Cij (specified by the user during the input phase explicitly or implicitly from a material database) can be obtained from energetic equivalence and in general from the equivalence of virtual work for non-linear processes. It can be easily demonstrated on the equivalence of the density of the work of deformation, in fact on its double (art. 2) that cannot be dependent on a subjective choice of coordinates: 2 ∏ = εT σ = εT Cε = (Tσ−1ε P )T CTσ−1ε P = = (ε P )T (Tσ−1 )T CTσ−1ε P = (ε P )T C P ε P

(5.1.62)

The following formula clearly follows from the above: C P = (Tσ−1 )T CTσ−1

(5.1.63)

It makes it possible to swap from the input matrix of physical constants specified in arbitrary axes x, y, z (“the user-axes”) to the matrix of physical constants in the element coordinates x P , y P , z P . Matrix Tσ is defined (5.1.55) and matrix Tσ−1 by the table in which we change the order of indexes IK in elements cIK . As a result, for example, the first row begins with (c11)2 , (c 21) 2 , (c31)2 , 2 ⋅ c 21 ⋅ c31 , etc. Matrix (Tσ−1 )T is then a mere transposition of Tσ−1 , i.e. 299

5.1 Introduction to the Theory and Practice of Creation of FEM Models the lines of matrix Tσ−1 are written as columns in matrix (Tσ−1 )T . Practically less important is the inverse procedure when we need to obtain – from physical formula (5.1.61) – relation (5.1.60) in other than element axes. Using the energetic equivalence of the density of work of deformation 2 ∏ = (ε P )T σ P = (ε P )T C P ε P = (Tσ ε )T C P Tσ ε = εT TσT C P Tσ ε = εT Cε we obtain C = TσT C P Tσ

(5.1.64)

It contains directly matrix Tσ according (5.1.55) and its transposition.

5.1.5.1.2 2D Models of Walls, Slabs and Shells Transformation relations are considerably simplified in these models, which can be demonstrated on the example of plane-stress, i.e. on the well-known problem of the theory of elasticity. In this problem we work with three-component quantities that have the character of a tensor in a 2D continuum that is located in the xy -plane and that models a wall of thickness h according to Fig. 5.33e). The whole problem is symmetrical around this plane, e.g. the stress must be a symmetrical function of z in the interval (−h 2, h 2) . We introduce its average value from this interval in arbitrary x, y axes or in element axes x P , y P : σ = σ x , σ y ,τ xy 

T

σ = σ , σ ,τ  P

P x

P y

P xy

T

(5.1.65)

This stress is physically related to deformation ε = ε x , ε y , ε xy  ε xy =

T

γ xy 2

ε P = ε xP , ε yP , ε xyP  γ xyP P ε xy = 2

T

(5.1.66)

through the generalised Hooke’s Law with the matrices of physical constants of dimension (3 × 3) : σ = Cε σ P = CP ε P

(5.1.67)

Relations (5.1.49), (5.1.56), (5.1.59) between the components of the same tensor of stress and deformation in different axes apply also to 2D models:

300

5.1 Introduction to the Theory and Practice of Creation of FEM Models σ P = Tσ σ = T −1σ P

(5.1.68)

ε P = Tε ε = T −1ε P

Only the first, second and sixth column and the row for three components (5.1.65), (5.1.66) of stress and deformation of the 2D model apply in transformation matrix T according (5.1.55), components σ z , τ yz , τ zx are not defined in it. The dimension of matrix T in (5.1.68) is just 3 × 3 and, for the sake of clarity, it can be written in this way explicitly. Such a small matrix requires only little space and thus we may use a more detailed notation (5.1.51) from which the meaning of the elements of the matrix is clear

 cos 2 ( x P x)    T= cos 2 ( y P x)    P P  cos( x x) cos( y x) 

cos 2 ( x P y ) cos 2 ( y P y ) cos( x P x) cos( y P y )

 2 cos( x P x) cos( x P y )    P P 2 cos( y x) cos( y y )    cos( x P x) cos( y P y ) +  + cos( y P x) cos( x P y ) 

(5.1.69)

We may also write the inverse matrix according to the explanation following formula (5.1.56): It is sufficient to swap the order of indexes ik in (5.1.55), which is demonstrated by changed order of the axes in the parentheses. This complies with the fact that the inverse matrix performs the transformation in the opposite direction: we known the components in axes x P , y P and we search for the components in axes x, y :  cos 2 ( xx P )    −1 T = cos 2 ( yx P )    2 P P  cos ( xx ) cos( yx ) 

cos 2 ( xy P ) cos 2 ( yy P ) cos( xy P ) cos( yy P )

 2 cos( xx P ) cos( xy P )    P P  2 cos( yx ) cos( yy )   cos( xx P ) cos( yy P ) +  + cos( yx P ) cos( xy P ) 

(5.1.70)

The advantage of formulas (5.1.69), (5.1.70) is the cosine function. When the formulas are applied we do not have to care about the direction of the angles between the axes, cos α = cos(−α ) , e.g. cos( xy P ) = cos( x p y ) . Practical engineers appreciate that it is possible to work with the absolute values of angles instead of taking care about their sign. They, however, together with programmer raise an objection that all the angles in the parentheses are, in the case of a 2D problem, determined by a single value! It is the inclination of coordinate axes x ⊥ y from x P ⊥ y P , which, however, must be already provided with the sign according to a common rule: rotation α is positive if it follows the rotation from the + x -axis to the + y -axis, and equally from the + x P -axis to the + y P -axis. When viewed in the

301

5.1 Introduction to the Theory and Practice of Creation of FEM Models direction of the axis + z = z P it is a clock-wise rotation (Fig. 5.33e). Another notation then depends only on the definition of angle α . There are two possibilities. We can measure it from the x -axis (the user-axis) to the x P -axis (element axis), i.e. to choose ( xx P ) = α P

(5.1.71)

or from the x P -axis to the x -axis, which is usually more practical: ( x P x) = α

(5.1.72)

The relation between the two created formulas is very simple, because ( xx P ) = −( x P x)

i.e. α = −α P

(5.1.73)

therefore, it is only the change of the sign of angle α . This has no effect in cosine function cosα , but it results in a change in sine function sin α = − sin(−α ) = − sin α P

(5.1.74)

Let us use Fig. 5.33e) to express all the angles by means of the definition angle ( xx P ) = α , i.e. by means of (5.1.71). For the sake of brevity we omit the angle-parentheses: xx P = α

(5.1.75)

yy P = α x P y = 90° + α

y P x = α − 90°

(5.1.76)

Next, let us apply the well-known properties of trigonometric functions: cos(−α ) = cos α cos(90° − α ) = sin α

cos(α + 90°) = − sin α cos(α − 90°) = sin α cos(−α − 90°) = cos(90° + α ) = − sin α

(5.1.77)

If we substitute into (5.1.69) and (5.1.70), we obtain the program algorithms in which we write shortly: c = cos α

s = sin α

(5.1.78)

Transformation matrix T for the transformation from the user-components in x, y to the element components x P , y P (which are, based on a 2D model, called planar ones) and inverse matrix T−1 for reverse transformation according to (5.1.68) (positive angle α is measured from the x -axis to the x P -axis (5.1.71)) have the form:

T

c 2 s 2 −2sc   2  2 = s c 2 sc   sc − sc (c 2 − s 2 )   

T −1

 c2 s2 2 sc   2  2 = s c −2 sc   − sc sc (c 2 − s 2 )   

(5.1.79)

Let us notice that T−1 differs from T just in the elements with function s = sin α in the first

302

5.1 Introduction to the Theory and Practice of Creation of FEM Models power, which means only four elements T13 , T23 , T31 , T32 . The matrices are neither symmetrical nor orthogonal, i.e. the symmetry of elements Tik = Tki or the transposition rule T−1 = TT does not apply here – they apply only to the matrix of rotation R in the transformation of the components of vectors (5.1.50). In planar problem it is reduced to the dimension (2, 2) and for two-component vectors v, v P it looks like this: v P = Rv R R −1

v = R −1 v P = RT v P

 cos( x P x) cos( x P y )  cos α = = P P cos( y x) cos( y y )   sin α  c s =  − s c 

(5.1.80) − sin α  c − s  = cos α   s c 

Physical relations (5.1.60) to (5.1.64) are valid also for 2D models – the only change is that the dimension of the matrices decreases from 6 to 3. In order to simplify the formulas, it is advantageous to introduce inverse matrices: t = T −1

(5.1.81)

t −1 = T and work with the following formulas: σ = Cε

σ P = CP ε P

C = TT C P T

C P = (T−1 )T CT−1 = tT Ct

(5.1.82)

Let us remind here that this assumes a correct order and definition of components in tensors of stress and deformation (5.1.65) and (5.1.66). The analysis of load bearing walls and generally the analysis of plane-stress of bodies (wheels, discs, etc.) modelled by means of a 2D-continuum (with thickness h assigned as a physical property (art. 4.2.2)) in practice usually uses what is called internal forces, e.g. in [kN/m], according to Fig. 5.33h): n =  nx , n y , qxy  nx = hσ x

T

ny = hσ y

qxy = hτ xy

(5.1.83)

These are intensities of fictitious internal forces of the 2D-body. Such a body cannot exist in terms of physics, it is a mathematical model. The practical meaning of intensities (5.1.82) was explained in a detailed way in art. 4.2.2.2., see (4.2.13), the relation to deformations was presented in art. 4.2.2.4, see (4.2.44) - (4.2.47), the physical laws were addressed in art. 4.2.2.5, see (4.2.57) - (4.2.66), practical application was discussed in art. 5.1.2.2, Fig. 5.125.17. As internal forces (5.1.82) differ from the components of stress only in the multiple h , the same transformation formulas apply to them, i.e.

303

5.1 Introduction to the Theory and Practice of Creation of FEM Models n P = Tn

n = T−1n P

= Cn ε

n P = CnP ε P

n

Cn = TT CnP T

(5.1.84)

CnP = (T −1 )T Cn T −1 = tT Cn t

For physically anisotropic and orthotropic walls the matrix of physical constants Cn differs from matrix C only in the h-multiple, e.g. Cn = hC

CnP = hC P

(5.1.85)

For shape orthotropic (e.g. ribbed, corrugated, etc. walls (Fig. 5.34e)) modelled by a physically orthotropic 2D-model, matrix Cn is determined independently and both physical constants of the material and the principles of energetic, or virtual, equivalence apply – see art. 5.1.2.2. Let us come back to the remark following (5.1.82). Full slopes γ = 2ε are often used in practice. Also matrices C are related to them. Consequently, also the transformation matrices must be modified in members that combine changes in angles and lengths. Two elements in positions (1,3) and (2,3) are half-size elements, two elements in positions (3,1) and (3,2) are double-size elements. The transformation of C into C P then uses matrix:  c2 s2 sc   2  2 t2 =  s c − sc   −2 sc 2sc (c 2 − s 2 )   

(5.1.86)

Five components of stress (5.1.48) act in slabs subjected to bending with the influence of transverse shear taken into account. Only component σ zP is neglected – it is vertical to the middle-plane x p y p and is small (with the exception of mathematical singularities of point loads) in comparison with the other five components. Handling of these singularities was described in art. 5.2.5. As a result, the third row and column do not apply in the versatile transformation matrix (5.1.55), on condition that it is limited to transformations in which the axis z P = z and the coordinate axes x ⊥ y rotate just around this axis, the same as in the above mentioned walls, i.e. σ z = σ zP is still negligible. In technical terms: We are interested just in the elements of the type dxdyh that are extracted from the whole slab (Fig. 5.33g, Fig.5.34) by planes that are parallel with the axis z P = z , i.e. not in elements dxdydz that are oriented generally in space, which naturally feature a complete system of six generally nonzero stress components even for σ zP = 0 . Similarly, we are not interested in the tensors of deformation of elements dxdydz , but in what is termed 2D components of deformation corresponding to 2D internal forces (art. 4.2.2.2 to 5). Therefore, we work with what is called plate-quantities (4.2.64) to (4.2.74). Subscript b ( b =bending) is extended by subscript o (pure bending and torsion), s (transverse shear). The positive direction of the components is displayed in Fig. 5.33g, 5.34a-d:

304

5.1 Introduction to the Theory and Practice of Creation of FEM Models sb = s boT , sbsT 

T

sbo =  mx , m y , mxy  εb = ε boT , ε bsT 

T

sbs =  qx , q y 

(5.1.87)

T

T

εbo = ∂ϕ y ∂x , ∂ϕ x ∂y , ( ∂ϕ y ∂y − ∂ϕ x ∂x )  ε bs = γ xz , γ yz  sbo = Cbo εbo

Cbo = [Coik ]

sbs = Cbs ε bs i, k = 1, 2, 3

Cbs = [Csik ]

i, k = 1, 2

T

(5.1.88)

(5.1.89)

The dimension of matrices of physical constants C is 3 × 3 for bending and 2 × 2 for transverse shear. The precondition of all linear plate theories is mutual independence of constants for bending and transverse shear, which is obvious also from art. 4.2.2.5., formula (4.2.74). It is not too strict limitation in technical terms, on condition that we analyse slabs that are symmetrical around their middle-plane z = 0 , which must be satisfied, among others, also in what is termed shape-orthotropic slabs. Some of the examples in Fig. 5.34e do not satisfy this condition (one-side ribs). The wooden slab in Fig. 5.34f is not even physically symmetrical around plane z = 0 and a general 3D stress-state arises. We have already mentioned the mutual interaction of the bending and membrane state under large deflections (Fig. 5.34g) in art. 5.1.4. First, let us focus on pure bending and torsion in Fig. 5.34a,b. Let us take the (i) definition of moment intensities mx , my , mxy as integrals σ x , σ y , τ xy in interval − h 2 ≤ z ≤ h 2 and (ii) linear distribution of stress along thickness h . Now we can derive the well-known relations between the extreme stress σ x* , σ *y , τ xy* on the positive face of the slab z = h 2 and moment intensities: mx =

σ x*h 2 6

my =

σ *y h 2 6

mxy =

τ xy* h 2 6

With regard to the common multiple h 2 6 in one point of the 2D model of the slab, it is obvious that the same rules apply to the transformation of moment components (Fig. 5.34c) and to the stress components in a planar problem (5.1.82) and to the internal force in walls (5.1.84). Using a brief matrix notation for (5.1.87)–(5.1.89) and technical demonstration in Fig. 5.34a-c we get: P sbo = mP

sbo = m

m = Tm

m = T−1m P

P P m P = Dbo ε

m = Cbo ε

P Cbo = TT Cbo T

Cbo = (T−1 )T Cbo T−1 = tT Cbo t

P

(5.1.90)

If we operate with technical full slopes, it will affect the 2D components of deformation of the plate model in the same way as it was mentioned in the text following formula (5.1.85). It means that the calculation of C P from C uses matrix t 2 (5.1.86).

305


Figure 5.34 Bending and transverse shear of a plate. a) Positive bending moments. b) Positive torsional moments. c) Transformation of moment components. d) Transformation of components of transverse forces. e) Shape orthotropy in axes x, y. f) Asymmetry around plane z=0. g) Large deflections.

306

5.1 Introduction to the Theory and Practice of Creation of FEM Models The case of transverse shear (Fig. 5.33g) (that is neglected or even misinterpreted in most manuals) requires, first of all, the return to a general 3D transformation with matrix T (6, 6) according (5.1.55) and with formulas (5.1.51) for direction cosines. Our example is considerably simpler. Only the following applies: the fourth row and column for component τ yz (which, when integrated, produces shear force q y ), and the fifth line and column for component τ zx = τ xz – basis for qx (Fig. 5.33g). Moreover, we require just the transformation that alters neither the direction nor the orientation of the axis z P = z , which means that angles x P z , y P z , z P x, z P y are always 90° and their cosines c13, c 23, c31, c32 are equal to zero. Angle z P z is 0°, i.e. c33 = 1 . The remaining four constants (5.1.51) depend just on one quantity α – the angle of rotation of the coordinate axes x ⊥ y into the position x P ⊥ y P , as already found in formulas (5.1.75) and briefly written using symbols (5.1.78): c11 = c, c 21 = s, c12 = − s, c 22 = c that were used also in the matrix of rotation (5.1.80). The fourth and fifth rows (5.1.55) have thus the following form: 0 0 0 c s 0 0 0 0 −s c 0 As a result, we obtain a simple formula for the transformation of transverse shear forces (Fig. 5.33g, Fig. 5.34d): τ yzP = cτ yz + sτ zx

→

q yP = cq y + sqx

τ zxP = − sτ yz + cτ zx

→

qxP = − sq y + cqx

We write it in alphabeticl order and we mark the matrix of transformation of the dimension (2 × 2) as Ts : qxP = cqx − sq y

(5.1.91)

q yP = sqx + cq y q P =  qxP , q yP 

T

q =  qx , q y 

T

q P = Ts q

q = Ts−1q P = t s q P

c − s  Ts =   s c 

 c s Ts−1 = t s =   −s c 

(5.1.92)

(5.1.93)

The minus sign in matrices (5.1.87) resulted formally from the general spatial transformation of (5.1.55). They can be easily verified if we perform the transformation according to Fig. 5.34d for angle α = 90° , where c = 0, s = 1 , and, as a result, qxP = − q y , q yP = qx . The definition of the positive direction is in Fig. 5.33g. Fig. 5.34d marks the positive qx , q y on positive faces by a filled (coloured) circle. The negative faces (with lower x, y coordinates) are subjected to positive transverse forces if these act against the orientation of the z P -axis, which is marked by empty circles. The effect of the forces on the element remains naturally the same for the same stress-state in the examined point of the 2D 307

5.1 Introduction to the Theory and Practice of Creation of FEM Models model of the slab. It is completely independent on our subjective choice of coordinates. For angle α = 90° neither the position of the element nor the position of the filled and empty circles changes. The y P -axis follows the direction and orientation of the x -axis, but the x P axis has opposite orientation than the y-axis, i.e. x P = − y . It means that the same forces qx must be (with regard to this y P -axis) registered correctly according to the definition of positive transverse forces, which follows even from transformation (5.1.93). Using the technical notation for transverse forces sbs = q (Fig.5.34d) and transverse slopes ε bs = γ , we can write the physical relations (5.1.89) in the usual engineering form: q P = CbsP γ P

q = Cbs γ

(5.1.94)

with the matrix of physical constants according to (4.2.74) of the dimension (2 × 2) , which is usually input by users in the x, y axes that suit their needs C Cbs =  44 C54

C45  C55 

(5.1.95)

The matrix of constants for planar coordinates x P , y P will be then determined using matrices (5.1.93) by the transformation similar to (5.1.90) (but with a smaller dimension of the matrices): CbsP = (Ts−1 )T Cbs Ts−1 = tTs Cbs t s

(5.1.96)

If it is necessary to obtain the constants for another (e.g. the original user) system of coordinates x, y, the following transformation is used Cbs = TT CbsP T

(5.1.97)

The transformation matrices combine here just the shear slopes and are thus valid for both tensor and full slopes, i.e. for their doubles. All the derived formulas apply also to shell (plate-wall) elements (i.e. planar 2D elements of arbitrary shells), on condition that the linear mechanics assumes that their membrane stress-state is independent on bending and transverse shear (art. 5.1.2.1.). This breaks the matrix of physical constants (8 × 8) into two independent matrices (3 × 3) and (5 × 5) and the letter then into two matrices (3 × 3) and (2 × 2) – separately for bending with torsion and transverse shear. Except these matrices, the whole matrix (8 × 8) has only zero elements, which characterises the principle of mutual orthogonality or the principle that the three mentioned stress-states do not affect each other. Notes relating to the geometrical non-linearity are given in art. 5.1.4 where the bending and membrane stress-state interact. This must be expressed by additional matrices in a form that can be easily transformed in an algorithm. Most often used are the matrices of the effect of the initial stress-state on the change of flexural stiffness in the incremental form, which allows for use of the matrices of physical constants similar to the linear mechanics. In physical terms, however, the problem is considerably more complicated. It is necessary to opt either for the idea of secant or tangential moduli and in problems that depend on the path of loading we must know the complete constitutive laws of the given material of the analysed structure.

308


5.1.5.2 Design Stress and Internal Forces Once the analysis of the structure by a FEM program has been performed, it is necessary to exploit the obtained results for a safe and economic design of the structural elements (thickness, profiles, strength characteristics of the material and connections, etc.) or for checking of the whole structure. It may be necessary to repeat both tasks if a considerable difference is discovered between the input data and the values resulting from the engineering considerations that can be called by the term sizing. The deviation of decisive results for linear problems is admitted to reach up to 2%, for non-linear problems it is usually larger – up to about 10%. The problem is that the magnitude of the deviation cannot be determined accurately unless we know the exact solution, which is never the case in complex practical problems. A possible way-around is to perform FEM analyses with two gradually finer meshes that show only very small differences in the results. It proves nothing in mathematical terms, which may be demonstrated on the example of simple infinite series, e.g. on the wellknown series 1 n with natural numbers n, the sum of which differs only slightly for a large number of members N and N + 1 . Nevertheless, the sum is completely different for another distant N and it diverges for the infinite N . Engineers intuitively assume that their problem does not behave like that. This iterative character that is typical for the whole design process – continuous refinement of intentions and prognosis of the behaviour – was partially described in art. 5.1.1. In this chapter we focus only on the problem of sizing as a one-step phase of the design process. From the point of view of the philosophy of practical applications of mechanics, the simplest problem is sizing in 3D problems. It uses directly the calculated tensors of stress and deformation from which the size and direction of principal stresses and deformations are calculated and substituted into the criteria of plasticity or strength of various materials (usually of energetic nature), etc. We often deal with very complicated and time-dependent constitutive relations, which can be, e.g. for soil, affected by the presence of more phases of the mass (solid, liquid and gaseous). The applications often require the participation of experts and experimenters and use of highly expert and problem-oriented programs. However, no reduction of the dimension of the problem is made. In common situations, e.g. for steel structures, the design stress can be replaced by a certain equivalent stress, under which the same work of deformation is accumulated in the sample of material in the case of a simple stress-state. Elsewhere, e.g. in problems with the concentration of stress in the vicinity of notches and cuts, a suitable integral around the singular point is used to remove the “nonsizable infinity”, which means that we operate with a certain volume or planar area defined by the stress diagram. The corresponding procedures have nothing in common with the finite element method. They were known and used long time before the FEM came to life in 1956, some of them even in the 19th century. A considerable progress was made in 1930 – 1940. The famous work by F. Neuber (Krebsspannungslehre) was published as early as in 1937. Today we can select from numerous specialised literature for all possible materials including fibreglass, plastics, soil and rock and, naturally, concrete of dam blocks and other 3D bodies. What is considerably more difficult in terms of mechanical principles is the sizing in 2D models of walls, plates and shells. These are in fact physical 3D bodies – the physical tensor of stress and deformation cannot arise in other than physical bodies. Moreover, only a correct reduction of the dimension of the problem according to art. 5.2.2 can assign some internal 309

5.1 Introduction to the Theory and Practice of Creation of FEM Models forces to the 2D model. This numerically required simplification of the idea about the action of the structure has an unpleasant consequence at the end of the whole process of the prognosis of the behaviour of the structure. We must apologetically return back to the 3D reality, which may be accompanied with minor difficulties in homogeneous bodies for which it is normally sufficient to find the extreme stress components at faces according to formulas (5.1.83), (5.1.90), e.g. including maximum shear in the middle plane: σ x* =

nx 6mx ± 2 h h

τ xz* =

1.5qx h

(5.1.98)

etc. These components may be then handled as in 3D bodies. We can search for the values and directions of principal stresses, use yield strength and ultimate strength, apply energetic criteria, etc. Postprocessors of FEM programs may arrange for various problem-oriented outputs, e.g. for steel shells or places with stress concentrations. On the other hand, nonhomogenous bodies cause considerably larger problems. Approximately since 1920 – i.e. long before the origin of FEM and at times of just manual calculations made by means of the force method and at beginnings of the sieving method (with manually assembled and solved equations) – an explosive development of reinforced concrete structures has started, and this type of structure has been representing an important, even predominant, part of construction industry. Therefore, top class experts have been trying for already 80 years to find out whether and how it is possible to apply the 2D models even here. At the beginnings, 1920-1934, experts like H. Marcus (director of Huetta-Werke, essays 1924 – 1928 dealing with slabs analysed by means of sieving method, grid, or truss and beam analogy that is even today popular among the majority of structural engineers), H. Leitz (1923-1926, plates reinforced in two perpendicular directions, effect of torsional moments) and E. Suenson (1922, oblique reinforcement) devoted their time to this issue. This period culminated in the book by W. Fluegge “Statik und Dynamik der Schalen” written in 1934 that contains already a complete theory of design forces in reinforcement and fictitious concrete struts including many details and detailed formulas (chapter 4) for reinforcement arranged arbitrarily in two and three directions. The book was translated into English and other languages. Many ideas may be today subjected to criticism, nevertheless, one idea is extremely valuable: the separation of the stress-state in steel and in concrete, its description by special Mohr’s circles and differentiation of the first and second state (today we would call it “limit” states) before and after cracking. The present-day idea of a composite two-component material can be traced there: a 2D continuum that is filled continuously by two materials, which can be advantageously expressed using the FEM method by means of two elements with different properties in the limit states that are placed into the same location (one on another). The research that was interrupted by the World War II (1939 – 1945) soon recovered. Outstanding works made between 1945 and 1960 were completed by H. Ruesch (1958), A. Pucher (1949), K. Bayer (1948), R. H. Wood (1961). These are followed by A. Sawczuk (1963) and especially F. Ebner whose theses, supervised by F. Leonhardt and defended at Technical University in Karlsruhe in 1963, contains a detailed documentation of experiments made with reinforced concrete slabs. Another big contribution was the dissertation of Th. Baumann defended with the active participation of Professor H. Ruesch in 1972 in Stuttgart and published in brief in the Der Bauingenieur in 1972 and printed at his own costs in full detail later in 1976. It contains practically a complete evaluation of all design theories up to 1972 and the justification of personal contribution – the concept that the problem of the 310

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice determination of design forces in reinforcement and in concrete is addressed as a statically indeterminate problem that can be solved with a help of various additional conditions added to the equations of equilibrium. In particular, the condition of the minimum of the whole reinforcement in a wall or slab or shell is practically important. What is valuable in terms of mechanics is (i) the systematic procedure starting with the principal stress components sorted according to algebraic inequality and (ii) the definition of critical inclination of the reinforcement from the larger principal stress according to the ratio of both principal stresses. The today’s recommendation in EC2 (Eurocode 2), accepted also in the Czech Republic in CSN P ENV 1992-1 /731201 from 1994 and accompanied by the national application document, defines the critical limit between two stress-states more simply by means of the absolute value of shear component τ xy in the right-angled x, y axes of the reinforcement. This leads to the relation between nx and − | qxy | for walls and the relation between mx , m y and − | mxy | for bending of plates. Unfortunately, it may be applied only to the reinforcement arranged in two perpendicular directions even though these may be arbitrarily inclined from the principal stress axes. Consequently, another arrangement of reinforcement requires the use of (i) the recommendations made by F. Ebner or H. Ruesch that proved their value in motorway bridges in Germany or (ii) a more precise theory by F. Baumann. It depends which limit state we are interested in. It may be the serviceability limit state (also known as the second limit state of deformation or cracks evaluating the structure under service load) in which no continuous large cracks in concrete are admitted. Or the ultimate limit state, called also the first limit state that arises in the instant of the collapse of the structure that is subjected to the limit load (collapse or other catastrophic deformations). In this case it is suitable to transform the problem already at the very beginning to the analysis of two states of planar stress-states in walls of a certain thickness located at the positive and negative face of the plate or shell. In principle, it is a substitution of moments by pairs of opposing forces. What remains a problem is the arm and the thickness of the “model-walls”. The fact that these stress-states may in the limit state affect favourably each other in the z direction along the thickness h may be handled by suitable experimentally verified coefficients.

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice 5.2.1 Introductory note Static calculations of structures always make the engineers face the problem of creation of the calculation model of the structure. In general, we can say: the simpler the model in terms of calculation is, the greater number of speculations must be included into the model by the structural engineer. A more complex model can more accurately correspond to 311

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice reality. The complexity of the model is always a kind of compromise between the following factors: the endeavour to get the most accurate solution, capabilities of the program and computer, accuracy of the data delivered, laboriousness of the input and other aspects which are taken into account by the structural engineers when they choose the model. There exist no exact solutions of real structures – neither now nor in the future. On the other hand, the technical progress will produce more and more accurate models and results. For example, the analysis of a foundation slab requires the user to input the properties of the subsoil (e.g. parameters C1 and C2 of the two-parametric model of the subsoil). These are, however, distinctly nonlinear and depend on the stress level in the foundation surface. Moreover, this stress depends on the applied load and on the deformation of the foundation slab. The distribution of the load of the foundation slab (which is mainly the effect of the self-weight and external load of the building) depends on the stiffness of the whole structure and also on the time course of the settlement. In other words, everything is interlinked with everything and, therefore, the structural engineers more and more often analyse the model of the whole structure including its subsoil. The performance capacity of programs and computers already allows for such approach and the development definitely follows this trend. As already said, the development does not stop here and, in the future, the calculations of structures will become even more accurate.

5.2.2 Modelling of Stiffeners in Planar Structures These problems occur very often with various types of structures. The method of modelling of ribs depends on the model of the problem. The idea of some authors of FEM programs, which is also the idea of some users, that one model of a structure can be analysed arbitrarily as a planar problem or plate or as a shell is wrong in principle. The creation of the calculation model of the analysed structure depends on whether the structure will be solved e.g. as a plate or as a shell – which applies in general and not only to the modelling of ribs. Let us demonstrate the difference e.g. on the example of a simple rectangular rib of a floor slab. If we calculate the floor slab according to the plate theory (Kirchhoff or Mindlin) which reduces the 3D body of the slab into its middle-plane and uses only one function of displacement ( w) , we must take this precondition into account when we determine the stiffness of the rib that is modelled by a beam located in the middle-plane of the slab. In general, when we determine the corresponding stiffness (flexural, shear or torsional), it is necessary to calculate the corresponding stiffness of the T-profile with a suitable effective width of the slab (e.g. according to the appropriate standard for concrete structures) and subtract the part of stiffness that is already included in the effective width of the slab. The obtained difference is assigned to the rib. Let us demonstrate the above-mentioned procedure on the example of the flexural stiffness, which is the most important one here. Let us use IT to mark the moment of inertia of the T-profile around the axis parallel with the plane of the slab o passing through the centroid of the T-profile IT . Then, the following formula is recommended to determine the moment of inertia of the beam representing the rib located in 312

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice the middle-plane of the slab: I = IT −

bh3 12

There are other possible ways to model the rib in a slab, but – in our opinion – the abovementioned procedure is suitable for common applications in design practice. Its advantage is that it gives a very good picture of the influence of the rib on the behaviour of both the slab and the whole structure. The disadvantage is that, in order to design the reinforced concrete T-profile, we cannot take just the internal forces in the beam, but that it is also necessary to add the values corresponding to the effective width of the slab, which causes problems in design algorithms. If we calculate the ribbed slab using a program for the analysis of shells, the preparation of the calculation model is simpler and more realistic, however, at the cost of the need to solve a considerably bigger problem. It is

Figure 5.35 Transverse section across a slab with a strengthening beam.

possible to model the rib by means of 2D elements. This approach, however, leads to the increase of the size of the problem, requires additional effort of the user and is reasonable for very high ribs only. In most cases it is better to model the rib using a beam eccentrically connected to the slab (see Fig. 5.35). The parameters of the deformation of the ends of the beams are transformed into the nodes in the middle-plane of the slab under the precondition that the mass normals remain straight. This approach does not increase the number of nodes in the mesh and the size of the problem does not increase in comparison with the solution of the slab without any ribs. The determination of the stiffness of the rib is simpler than for the plate-problem. The input data include the sectional characteristics of a part of the crosssection that sticks out of the slab. The moments of inertia are related to the centroidal axes of the sticking-out part. The procedure described for the plate-model would be more accurate only for the torsional stiffness. For example, the sectional characteristics of a rectangular rib can be written: effective area in compression

( Ax ) = b0 h0 313

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice effective area in shear moments of inertia

5 ( Ay , Az ) = b0 h0 6 b0 h03 Iy = 12 bh3 Iz = 0 0 12

It is more problematic to determine the torsional moment of inertia. To meet the needs of practice, we recommend calculating it for a rectangle with dimension b0 and h + h0 . If the design requires the internal forces corresponding to the whole T-profile, they must be composed of all contributing parts. Let us write, as an example, moment M y : M y = N 0 e + M y 0 + mb where e is the eccentricity of the connection of the beam, i.e. the distance of the centroid of the beam from the middle-plane of the shell N0

is the axial force in the beam

M y0

is the bending moment around the y -axis of the beam,

m

is the mean value of the moment in the slab in the direction of the rib.

The above mentioned procedure is a problem-oriented transformation of FEM outputs (which are not affected at all) for the needs of a design program.

5.2.3 Modelling of Column-Supports of Floor Slabs Column-supports of a floor slab are usually modelled as point supports in the centroid of the column. As the increase of calculation accuracy through finer mesh causes the internal forces in the vicinity of the point support to rise quickly and converge to the theoretical value for the point support (the infinity), the users more and more often face the problem what to do with the astronomic values that cannot be handled in the design. Moreover, the sectional dimensions of the columns are not small enough to allow the user to neglect them and replace the columns by point supports. Let us show here a few approaches that may improve the model of the column-support ggg) o increase the stiffness of the slab in the intersection with the column.

T

This method gives a true picture of the effect of the real column on the overall behaviour of the slab. The traditional approach to modelling produces the largest

314

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice values of the curvature of the slab just in the area of the intersection (connection) of the column and the slab. Such a curvature is, however, far from reality. If we base our reasoning on the idea that a part of the mass of the column can be considered a haunch of the slab, we can tell that the stiffness in the area around the intersection of the column with the slab corresponds to the stiffness of the corresponding haunch. The increase of the stiffness by one order of magnitude gives good results. Similarly to other similar situations, it is not possible to establish an unambiguous rule specifying the stiffness that should be assigned to the area where the slab and column intersect each other. With regard to the fact that even large differences, even multiples, in the stiffness lead to a relatively small difference in deformations and internal forces in the slab, a rough estimate made by the user is fully satisfactory. As stated above, this methodology gives a true picture of the effect of the column on the behaviour of the slab, but it does not solve the problem of sizing in the intersection of the slab and the column. It is more likely based on the idea that the intersection of the slab and the column is a 3D-body in which the term “internal forces in a slab” lose its meaning and the decision about the design of the reinforcement in the area around the intersection is left to the user or to the design-making program. Also an excessive concentration of the internal forces in the corners of the columns occurs here – but it is realistic and must be taken into account. hhh) o support the slab by an elastic environment.

T

This variant assumes that the supporting column is in the area of intersection replaced by an elastic environment, that is described by parameters C1 and C2 (see [8] ). However, this kind of support must be in regions of the slab distant from the columns equivalent to the real column. The effect of the column on the slab is characterised by the stiffness of the column head in both compression and bending. Let us introduce the following notation: CN

the stiffness characterising the resistance of the support against displacement w stiffnesses representing the resistance of the support against rotation ϕ x , ϕ y

CM x , CM y A Ix, Iy

cross-sectional area of the column head in contact with lower face of the slab. moments of inertia in the section across the column head related to the centroidal axes

C1 , C2 x , C2 y

parameters describing the elastic support of area A .

The condition of the equivalence of the resistance against displacement can be written:

∫ C dA = C 1

N

(5.2.1)

A

315

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice The conditions of the equivalence of the resistance against rotation are formulated by the following relations:

∫ C x dA + ∫ C 2

1

A

2x

dA = CM y

A

(5.2.2)

∫ C1 y dA + ∫ C2 y dA = CM x 2

A

A

If we consider that the following is true

∫ dA = A A

∫ x dA = I

∫ y dA = I

2

2

y

A

(5.2.3)

x

A

we can write the following formulas for the parameters of subsoil: C1 =

CN A

C2 x =

CM y − C1 I y A

C2 y =

CM x − C1 I x A

(5.2.4)

The advantage of this approach is that the continuous elastic support automatically smoothens the singularities above the point support without the necessity to introduce unjustified speculations. On the other hand, the disadvantage is the need to obtain exact values of reactions that are necessary for the design of columns and the impossibility to specify any settlement of the supports. iii) To combine variants (a) and (b).¨ It seems that the optimal model of a column-support of the slab would be represented by a combination of the above mentioned approaches. Let us present the scheme of this combination in Fig. 5.36.

Figure 5.36 A possible method for the modelling of slabs supported by a column.

316

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice Parameters C1 , C2 , CM x , CM y can be again obtained from the conditions of equivalence. However, it is necessary to consider that C N and C1 act as springs connected in a series and that the resistance against rotation acts in a similar way. This model is, however, relatively very complex and causes considerable problems during the design of the algorithm.

5.2.4 Boundary Effects in Slab Models The analysis of plates or shells using the Mindlin bending theory (which is today probably the most often used method in FEM programs) gives often rise to user questions concerning the distribution of internal forces in the vicinity of a free and simply supported edge – w = 0 . Contrary to the Kirchhoff bending theory, the Mindlin theory contains independent rotations of the normal to the middle-plane. This makes it possible to meet the condition M nt = 0 at a free or simply supported edge (where n is the direction of the normal and t is the direction of the tangent to the edge). The consequence is that the gradient of torsional moment mnt in the vicinity of an edge (free or simply supported) is steep. As the condition of the vertical equilibrium must be fulfilled, shear force qt = −∂mm ∂n occurs around this gradient. The gradient ∂mm ∂n depends on the density of the mesh, because the transition from moment mnt that is not influenced by the boundary defect takes place in a strip along the edge of the slab. The width of this strip depends on the used elements. This phenomenon occurs practically in one row of elements along the edge. Therefore, the size of the shear force cannot be determined, because it increases with the refinement of the mesh, which reduces the width of the strip along the edge affected by this defect. It is however possible to determine resultant Qt of shear force qt that results from zeroing of the torsional moment along the edge. d

Qt = ∫ qt dn

(5.2.5)

0

where the upper boundary d is the distance from the edge that is big enough to eliminate the boundary (edge) defect. The origin of the n coordinate is at the edge of the slab. If we substitute qt = −∂mm ∂n , we may perform the integration. d

∂m d Qt = ⌠  nt dn = [ mnt ]0 = mnt (d ) ⌡ ∂n

(5.2.6)

0

If the condition for mnt is met at the edge where τ nt = 0 , it produces line shear force Qt whose magnitude is equal to the size of torsional moment mnt at the edge not effected by the boundary defect. The stated boundary defect is the consequence of the reduction of a 3D problem to a 2D one, which cannot be performed by a theoretically flawless finite process, i.e. 317

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice by means of the finite number of operation, which is, however, the principle of all FEM programs.

Figure 5.37 The distribution of the magnitude of torsional moment in the slab with the effect of shear taken into account (Timoshenko, Mindlin) in the vicinity of a free edge.

5.2.5 Singularities in the Analyses of Structures The growing performance capacity of both computers and programs for the analysis of structures brings new problems which many users working in the past with coarse mesh were not aware of. The problem is the singular points in the model of the analysed structure, i.e. the points in which the theoretical solution gives infinite internal forces or stress and in which the numerical solution converges to infinity with the refinement of the mesh. Some users even think that it is a bug in the program if it gives too large values in these points while competitive programs gives smaller results. This attitude is based on the fact that the stress in these points is greater than the allowable one or that a concrete structure cannot be designed at all. In general, the only way out of this situation is through the fact that even if the internal forces or stress converge in the singular points to infinity, any integral of internal forces along a straight line (or an integral of the stress over an area) passing through this point gives a finite value. As the material in the vicinity of the singular point plasticises and the internal forces are redistributed, it is possible to perform required checks or design using the average value of the corresponding internal force over a straight line of a given length or over a given area (on condition that the applicable standard allows for it). Nevertheless, the linear solution of the problem does not make it possible to handle the problem in another way. Let us state a few typical singularities which occur in the models of structures. For planar structures (slab, wall, shell) these are in particular: • points of action of concentrated forces or moments, • point supports – both rigid and flexible,

318

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice • end-points of line supports or points where the line support changes its direction, • concave turns of edges of plates, e.g. corners of openings. For 3D bodies there are some more: line loads or line supports and non-rounded corners of the surface.

5.2.6 Modelling of the Interactions between Foundation Grids and Subsoil The application of a two-parametric model of the interaction between a foundation slab and its subsoil has become common in the engineering practice. Parameters of interaction C1 and C2 can be obtained either by means of approximate formulas from literature or more accurately by means of the corresponding program (SOILIN) that determines these parameters for the given geological conditions and loads acting on the footing surface. Consequently, when a strip foundation is being analysed, the structural engineer faces the problem of determining the parameters of interaction for beams with known surface parameters of interaction C1 and C2 . Parameters C1*z , C2*z , C1*ϕ , C2*ϕ have practical meaning for a strip foundation. The physical meaning of the first two parameters is the resistance of the subsoil against vertical displacement (C1*z ) and its first derivative in the direction of the axis of the beam (C2*z ) . The physical meaning of the other two parameters is the resistance of the subsoil against the rotation and its first derivative in the direction of the axis of the beam. The requirement of the equivalence of virtual work gives the following formulas for 1D and 2D parameters of the interaction between the building and subsoil. C1*z = C1b + 2 C1C2 C2*z = C2b + C2

C2 C1

C1b3 b2 C = + C1C2 ⋅ + C2b 12 2 3 Cb C2 b 4 C2*ϕ = 2 + C2 ⋅ C1 4 12 * 1ϕ

(5.2.7)

where b is the width of the foundation strip. A similar procedure can be applied to obtain the relations between parameters C1 and C2 and discrete parameters K z* , Kϕ* that must be introduced at the ends of the foundation strip:

319

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice K z* = C2 + b C1C2 b 2C2 bC2 K = + 4 2 * ϕ

C2 b 3 + C1C2 C1 12

(5.2.8)

It is not necessary to emphasise that – similarly to the modelling of other parts of the structure – also here the described model is a result of certain hypotheses and its validity is only relative and that it is suitable to use the formulas only if the structural engineer is not able to determine the parameters in another, more accurate way.

5.2.7 The Density of the Mesh Generally, it holds that the higher the density of the mesh is, the closer the results are to the theoretical values. More accurate results can be obtained at the cost of longer calculation time and higher demands on the capacity of the disk memory of the computer. The above-mentioned statement is, however, only relative, see Fig 5.38. As the size of the problem grows, also the numerical error due to rounding of numbers in the computer increases. The sum of the errors due to the approximation of functions and numerical error has its minimum for a certain density the mesh. Further refinement of the mesh would deteriorate the results. However, for double accuracy (ca 15 digits in the mantissa) the density of the mesh producing the minimum error is usually substantially higher than the densities commonly used. Therefore, the users are usually not forced to take the numerical error into account when they define the density of the mesh. On the other hand, they should determine the size of the numerical error once the calculation has been performed, e.g. through the comparison of (i) the sum of load components and (ii) the sum of reaction components. Moreover, an extremely refined mesh may be sometimes contra-productive also for another reason. The refinement of the mesh around singularities causes that the results converge to the theoretical value, i.e. to infinity, which may lead to certain problems for the users of the program.

320

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice

Figure 5.38 The influence of the density of the mesh on the size of the error of solution.

An experienced user chooses reasonable density of the mesh, i.e. the density that is adequate to the program and computer used for the solution of the problem, and, in particular, that corresponds to the demands on the results. For instance, if the aim of the calculation is to determine the settlement and total deformation of the building, rather coarse mesh is sufficient. But if the results of the solution are supposed to be used for the design (e.g. the design of a foundation or floor slab) the mesh in the corresponding part of the structure must be refined accordingly. The required density depends on the program used – i.e. on the element employed, but, generally, it can be said that if the results are to be used for design, it is necessary to have at least 8-10 elements between supports. The input in state-of-the-art programs is independent on the generation of the finite element mesh. Consequently, the user may use the created model of the structure for various purposes by a simple change of the mesh density in individual parts. If the building is modelled as a whole, it is not feasible to have the mesh in all parts so refined that it could be used for the design of all parts. The system of the equations would be so excessive that it would be impossible to solve it on available computers. Another possibility for the solution of the problem is the application of what is termed a substructure method. Various parts of the structure are considered substructures (or also superelements) in which the program eliminates the parameters of deformation in all the nodes that are not in contact with any other substructures. The final solution thus contains only the parameters of deformations of the nodes in contact of superelements (master nodes). Instead of the need to solve a huge system of equations a set of smaller systems must be solved. The internal parameters are then calculated only in those substructures in which it is necessary. This method is extremely effective for the solution of large structures that could not be analysed otherwise.

321

5.2 Notes Concerning the Problems of Modelling of Certain Structures in the Engineering Practice

5.2.8 Modelling of a “Double Beam” Bridge with wide Beams If the beams are rather massive (Fig. 5.39) it may seem reasonable to solve the problem using 3D elements. If this approach is applied to concrete structures, it has – in addition to the enormous increase in the size of the problem – another substantial disadvantage. The user obtains the distribution of stresses throughout the structure, but – for the needs of the design – he needs to know the internal forces. It would be arduous to obtain them through the integration over the cross-section of the beam. A specialised program focussing on such structures that would perform the integration automatically could help. Consequently, for the reasons stated above, the modelling by means of 3D elements need not be suitable. Therefore, let us discuss the possibility to model such a structure using a shell with beams. Unless the width of the beam is big, the procedure given in the paragraph for beam structures could be used. However, if the width is large, the beams have strong influence on the transverse stiffness of the structure and the effect of the cross-section of the beam on the transverse stiffness of the structure must be taken into account. Therefore, we propose e.g. the following model. The slab outside the beam is modelled by means of shell elements using the common method. The part of the slab above the beams is modelled using orthotropic shell elements. Their stiffness – in comparison with the orthotropic shell with the thickness equal to the thickness of the slab outside the beams – differs in flexural and membrane stiffness in the direction of the width of the beam where the values are derived from the thickness of the shell corresponding to the height of the beam. This represents the influence of the beam on the transverse stiffness of the bridge. The longitudinal stiffness of the beam is modelled by a beam – the area of which is equal to the area of the beam under the slab – in the centroid of the area of the beam outside the slab. The stiffness of the beam in torsion is determined for the total height of the beam including the slab. The moment for the design of the beam is obtained as a sum of the (i) moment in the beam, (ii) product of the axial force in the beam and its eccentricity, and (iii) product of the moment in the slab in the point of connection of the beam and the effective width of the slab.

Figure 5.39 Cross-section of a “double beam” bridge

322

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES 5.3.1 Purpose of the guide The purpose of this guide is to provide instructions for the calculation of those input data for orthotropic plates that characterise their physical properties, i.e. flexural, torsional and shear stiffness. In the program, these properties are described by the matrix of physical constants D = [ Dik ] . It follows from the said that that elements Dik of this matrix are the very data that must be calculated manually.

5.3.2 Method It is assumed that the solution is obtained by means of the finite element method in its most often used deformation variant with unknown deformation parameters of geometric nature. Matrix D combines (i) mutually determined static quantities σ (stress components or, in case of plates, their resultant over a section – i.e. internal forces in the plate) with (ii) corresponding geometric quantities ε (deformation components or, in case of plates, those derivatives of deflection area w on which deformation components depend): σ = Dε

(5.3.1)

Matrix D can be defined separately for (i) each element or (ii) a specific group of elements. This allows for the analysis of plates that are non-homogenous over the elements or that change their shape over the elements. Consequently, it is possible to express, rather approximately, even the gradual change of geometry in a haunch or in a similar detail. As the text should serve as a fast and easy-to-read reference, it is practically without any literature references.

5.3.3 Core principal of solution In plate model we need to introduce the following functions of the x, y variables needed because of reduction of the general 3D problem by one dimension. In the plate theory we have 3 generalized displacement components, the displacement w and the rotations of the “mass” normal around the x and y axes denoted as ϕ x and ϕ y respectively. In the case of the Kirchhoff theory the mass normal remains perpendicular to the plate surface and in the Mindlin theory the rotations ϕ x and ϕ y are independent of the plate surface.

323

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES The plate deformation is in the Kirchhoff theory defined by the curvatures κ x , κ y and twisting ϑ by the following kinematic equations: κ x = − w, xx

κ y = − w, yy

ϑ = w, xy

For the plates with the transverse shear effect let us introduce the average shear deformations along the plate normal denoted as γ x and γ y , the γ x being the average of the γ xz and the γ y being the average of the γ yz . The vector of deformation is defined by the following kinematic relations: ϕ y , x , − ϕ x , y , ϕ y , y − ϕ x , x , γ x , γ y 

T

where γ x = w,x + ϕ y and γ y = w,y − ϕ x .

The plate internal forces are defined as the following stress resultants: M x = ∫ σ x z dz

M y = ∫ σ y z dz

Tx = ∫ τ xz dz

Ty = ∫ τ yz dz

M xy = M yx = ∫ τ xy z dz

The constitutive relations between the internal forces and deformations are for the Kirchhoff plates in (5.3.23) and for the Mindlin plates in (5.3.40). Present-day programs and solution methods for orthotropic plates deal, in fact, only with what we call physically orthotropic planar (two-dimensional) continuum filled with points in the mid-plane ( x, y ) of the plate, i.e. in the plane z = 0 . The body of a real slab is limited by the upper and lower surface z = ± h 2 , where h is the thickness of the plate. Alternatively, the height may be variable, i.e. h( x, y ) . It is assumed that a normal to the midplane (e.g. points with coordinates ( x1 , y1 , z ), − h 2 ≤ z ≤ h 2 remains straight, i.e. undistorted, even in the deformed plate. In the standard Kirchhoff theory, it remains even perpendicular to the deflected surface of the plate w( x, y ) . If we take into account the effect of transverse shear τ xz ,τ yz on angular changes γ xz , γ yz , this normal is generally rotated around x - and y axis by angles ϕ x ( x, y ) and ϕ y ( x, y ) . Even after such a generalisation, we have only three functions of two variables to describe the deformation of the plate continuum (Boltzmann continuum for Kirchhoff plates and Cosserat continuum for Mindlin plates) w( x, y ) ϕ x ( x, y )

ϕ y ( x, y )

(5.3.2)

In a real three-dimensional body of a given plated structure, e.g. box-section, the displacement can be fully described by means of three functions of three variables u ( x, y , z )

v ( x, y , z )

w( x, y, z )

(5.3.3) 324

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES from which the full description of its stress-state σ can be derived. In order to be able to solve such a body as a plate, we must define certain geometric assumptions that make it possible to apply functions (5.3.2) to find functions (5.3.3). For example, the simplest Kirchhoff assumption mentioned above is of the following form (Fig. 1): w( x, y, z ) → w( x, y ) = w( x, y, 0) ϕ x ( x, y ) = ∂w( x, y ) ∂x = w, x ϕ y ( x, y )

= −∂w( x, y ) ∂y = w, y

(5.3.4)

u ( x, y, z ) = − z ∂w( x, y ) ∂x = − zw, x v( x, y, z ) = − z ∂w( x, y ) ∂y = − zw, y

Fig. 1

Iust one function of two variables w( x, y ) is sufficient to describe functions (5.3.3). This is the basis of the core principal of the solution, i.e. the transformation of the structure into a plate, and, conversely, the utilisation of the results obtained from the platecalculation in the design and checks of the analysed structure. An unambiguous relation must exist between functions (5.3.2) and (5.3.3) - with the simplest one being of type (5.3.4). Under such conditions, it is no more problematic to establish similar relations between internal forces or stress in (i) the structure and (ii) its plated-model. In general, we thus reduce the three-dimensional problem into a two-dimensional one.

5.3.4 Stress components in physically orthotropic plates For brevity, we will limit ourselves to the most frequent examples with the coordinate axes ( x and y ) put into the axes of orthotropy. This approach is generally recommended, as it simplifies the notation. The approach is based on general relations between (i) deformation components and (ii) stress components in the primary form ε = D0−1σ :

325

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES  ε x   a11 ε    y   a21  ε z   a31  = γ yz   γ zx      γ xy  

a12 a22

a13 a23

a32

a33

0 a44

0

a55

 σ x   σ   y  σ z     τ yz   τ zx    a66  τ xy 

(5.3.5)

Let us define the following constants known as technical constants: • Three (Young) moduli of elasticity: E1 =

1 a11

E2 =

1 a22

E3 =

1 a33

(5.3.6)

G12 =

1 a66

(5.3.7)

• Three (Lamé) shear moduli of elasticity: G23 =

1 a44

G31 =

1 a55

• Six (Poisson) coefficients of lateral contraction µik using the following formulas: a12 = −ν 12 E1 ≡ a21 = −ν 21 E2 a13 = −ν 13 E1 ≡ a31 = −ν 31 E3

(5.3.8)

a23 = −ν 23 E2 ≡ a32 = −ν 32 E3 The identities (≡) follow from the symmetry aik = aki , which means that from the nine E and µ constants, only six are independent. Thus, together with three G constants, we get nine constants E , G and µ that are necessary to describe the physical properties of the analysed type of an orthotropic substance. Axes x, y and z are marked with subscripts 1, 2 and 3. The coefficient µik equals to the relative lateral contraction in the i -direction with tension σ k = E2 in the k -direction. The subscripts of shear moduli can be swapped: Gik = Gki . The physical law (5.3.5) with technical constants can be, for clarity, re-written separately for normal and shear components (this decomposition appears only in orthotropy):    1 ε x   E    1    ν 21 ε y  =  − E    2 ε   ν 31  z   − E  3

−

ν 12 E1

1 E2 −

ν 32 E3

ν 13     σ E1   x    ν 23    −  σ y  E2    1  σ   z E3    −

ε = Dε−1σ

(5.3.9)

Each line of matrix Dε−1 contains the same E modulus. The matrix is symmetrical and, therefore, it remains identical after transposition. Consequently, we can write the formula 326

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES with the same E moduli in columns:    1 ε x   E    1    ν 12 ε y  =  − E    1 ε   ν 13  z   − E  3

−

ν 21 E2

1 E2 −

ν 23 E3

ν 31     σ E3   x    ν 32    −  σ y  E3    1  σ  E  z E3    −

ε = Dε−T σ

(5.3.10)

In considerations and in calculations we always use the form that is more suitable under the given circumstances. The form (5.3.9) is more frequent. The matrix of physical constants is diagonal for shear components:    1 γ yz   G    23    γ zy  =  0    γ    xy   0 

0 1 G31 0

 0  τ yz      0  τ zx    1    τ xy G12   

γ = Dγ−1τ

(5.3.11)

0  0  G12 

(5.3.12)

It is thus simple to write a reverse formula: G23 Dγ =  0  0

τ = Dγ γ

0 G31 0

The reverse formula to (5.3.9) is less clear in terms of technical constants. It is however significantly simpler for orthotropic plates, as it is based on the main static assumption of the theory of plates: σ z ( x, y , z ) ≡ 0

(5.3.13)

Here, the formula (5.3.9) splits into two simpler formulas:    1 ε x   E  = 1 ε   − ν 21  y   E  1

ν 12    σ E1   x  =  1    σy E2   

(5.3.14)

ν 32  σ x    E3  σ y 

(5.3.15)

−

and  ν ε z  =  − 31  E3

−

Component ε z can be considered insignificant and can be omitted in further considerations, which applies also to orthotropy. Contrary to isotropy, we can even define a 327

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES material with non-zero lateral contraction in the ( x, y ) plane, but with zero ν 31 , ν 32 or with E3 → ∞ , and meet the condition that ε z ≡ 0 at σ z ≡ 0 . It is however of no practical meaning. The formula (5.3.14) can be easily inverted, and thus we get to the full relation σ = Dε of type (5,1) = (5,5)(5,1) that takes form (5.3.16) for plates, if we arrange the components in a way that is practical for plates:    E1 σ x  1 −ν 12ν 21    σ   ν 12 E2  y  1 −ν 12ν 21    τ   0  xy  =     τ   0  xz      τ   0  yz   

ν 21 E1 1 −ν 12ν 21

0

0

0

E2 1 −ν 12ν 21

0

0

0

0

G12

0

0

0

0

G13

0

0

0

0

G23

   εx       εy       γ xy       γ xz       γ yz   

(5.3.16)

5.3.5 Internal forces in physically orthotropic plates

5.3.5.1 Technical theory of plates with the effect of transverse shear not taken into account This is what is termed as the Kirchhoff theory of thin plates, based on formulas (5.3.4) and valid approximately within the range of wm ≤ h≤

L c

C ≈ 100

L 5

(5.3.17) (5.3.18)

where: wm - the maximal plate deflection, h - the plate thickness, L - the characteristic plate dimension in plan-view, L = diameter of a circular plate, L = the shorter side of a rectangular or rhomboid plate, etc. The consequences of failing to comply with (5.3.17) are: Stress of significant intensity appears in the mid-plane of the plate, plane stress is 328

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES added to the plate stress. We talk about geometrically non-linear plates with large deflections. Only tensile stress appears in the limit point h → 0 , the plate becomes a membrane under tension (film of pneumatic structures, three-dimensional geometrically non-linear problems). The consequences of failing to comply with (5.3.18) are: For h > L 5 , we get thick plates with a significant influence of transverse shear on the overall energy, deformation and stress-state of the plate, see paragraphs 5.2 and 6.3. When conditions (5.3.17) and (5.3.18) are met, we define the following internal forces: Bending moment (subscript = the direction) of reinforcement: M x = ∫ σ x z dz

M y = ∫ σ y z dz

(5.3.19)

Twisting moment: M xy = M yx = ∫ τ xy z dz

(5.3.20)

Shear forces: Tx = ∫ τ xz dz

Ty = ∫ τ yz dz

(5.3.21)

They are integrated over the thickness of the plate in the interval − h 2 ≤ z ≤ h 2 . They are taken into account in the equilibrium relationships, however not as for deformation. Considering the hypothesis (5.3.4), geometric conditions ε x = ∂u ∂x

ε y = ∂v ∂y

γ xy = ∂u ∂y + ∂v ∂z

(5.3.22)

physical relation (5.3.16) and conditions of moment equilibrium of a plate element around x and y - axis, we get formula (5.3.1) in the following form:

 M x   D11 M    y   D21  M xy  =  0     Tx   0    Ty   0

D12 D22

0 0

0 0

0 0

0 0

0 0 0

D33 0 0

0 D11 0

0 D3 0

0 0 D22

 w, xx    0   w, yy  0   2 w, xy    0   w, xxx   0   w, yyx    D3   w, yyy  w   , xxy 

(5.3.23)

with the following elements of stiffness matrix D : • Flexural stiffness: D11 =

E1h3 12 (1 −ν 12ν 21 )

D22 =

E2 h3 12 (1 −ν 12ν 21 )

(5.3.24)

• Contraction stiffness: D12 = ν 21 D11 ≡ D21 = ν 12 D22

(5.3.25) 329

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES • Torsional stiffness: D33 = G12

h3 12

(5.3.26)

• Mixed stiffness: D3 = 2 D33 + D12 = 2 D33 + D21

(5.3.27)

In total, we need five entries for matrix D : h, E1 , E2 , G12 , ν 12 . The second coefficient of lateral contraction in the x -direction with the elongation in the y -direction is E2 D = ν 12 22 E1 D11

ν 21 = ν 12

(5.3.28)

This number of entries can be reduced to four if we accept the assumption that a relationship similar to that valid for isotropy can be applied to shear modulus G12 , even though for geometric averages (another derivation was already presented by M. T. Huber): G12 =

(

E1 E2

2 1 + ν 12ν 21

(5.3.29)

)

Therefore, torsional stiffness D33 is no longer an independent constant, but, following from (5.3.26) and (5.3.24), it can be expressed as D33 =

(

1 1 − ν 12ν 21 2

)

D11D22

(5.3.30)

which can be written using (5.3.28) in the form: 1 D33 = 1 −ν 12 2 1 D33 = 1 −ν 21 2

  D11 D22  D11   D11 D22 D22 

D22 D11

(5.3.31)

Formula (5.3.27) represents a coefficient at mixed derivation in the main equation for plates D11w, xxxx + 2 D3 w, xxyy + D22 w, yyyy = p

(5.3.32)

and is decisive for the type of orthotropy defined by constant κ=

D3 D11 D22

(5.3.33)

In the civil engineering practice, the type (0 ≤ κ < 1) is usual. Sometimes we can meet the second type (κ = 1) that can be reduced to an isotropic solution. The third type (κ ≥ 1) can appear only rarely in steel plates with closed ribs that are rigid in torsion.

330

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES Orthotropic constant κ can be, in certain types of plates, considered a primary piece of data that has been verified in practice and that represents the mixed stiffness of the plate D3 = κ D11 D22 Thus, using (5.3.27) and (5.3.30), we can obtain the contraction stiffness in the form

(

D12 = D3 − 2 D33 = κ − 1 + ν 12ν 21

)

D11 D22

(5.3.34)

Usually, for κ = 1 we get D12 = ν 12ν 21 D11 D22

(isothropy D12 = ν D11 )

(5.3.35)

Plate reactions Qx , Qy are equal to shear forces Tx , Ty (similarly to beams) only if twisting moments M xy at the edge are equal to zero (e.g. fully fixed edge). Generally, e.g. in a simply supported edge, the complement due to torsion must be added: Qm = Tm + ∂M mn ∂ n

(5.3.36)

where m ⊥ n is either x ⊥ y or y ⊥ x or any other direction of the edge n with normal m . If the reactions are not calculated, they can be obtained from presented results with the derivatives calculated approximately by means of two adIacent values, e.g. in a sequence of equidistant border points 1, 2, 3 with step d : Qx (2) = Tx (2) +  M xy (3) − M xy (1)  2d

(5.3.37)

5.3.5.2 Plates with the effect of transverse shear taken into account In the classical theory of plates that was discussed in the previous paragraph, shear moduli G13 , G23 from (5.3.16) have no effect as the shear forces (5.3.21) are determined from the condition of moment equilibrium of the element, which leads to the last two lines of matrix (5.3.23). In thick plates and within the range of approximately L 5
(5.3.38)

the right angle between the normal and the mid-plane of the plate is skewed by γ due to transverse shear τ xz , τ yz and, following from (5.3.4), the second and third line are eliminated. Then, according to (5.3.2) we have three independent functions w, ϕ x , ϕ y . Under such conditions, the following formulas are considered, see figure 9 below γ x = w, x + ϕ y γ y = w, y − ϕ x

(5.3.39)

together with the fourth and fifth line of matrix (5.3.16). Therefore, instead of matrix (5.3.23) 331

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES we get the matrix:  M x   D11 M    y   D21  M xy  =  0     Tx   0  Ty   0  

D12 D22

0 0

0 0

0 0 0

D33

0 D44

0 0

0

0   ϕ y,x    0   −ϕ x , y  0  ϕ y , y − ϕ x , x    0   w, x + ϕ y  D55   w, y − ϕ x 

(5.3.40)

with new elements that, for the simplest configuration of a constant transverse shear across the plate thickness h , are: D44 = G13h

D55 = G23h

(5.3.41)

and consequently, the five entries h, E1 , E2 , G12 , ν 12 are extended by two more : G13 , G23 , which means in total seven entries for the calculation of input data for D .

5.3.6 Shape orthotropy of plates

5.3.6.1 Main principles of the transformation into physical orthotropy Certain bridge, floor, foundation and other structures are similar to plates in terms of the hypothesis (5.3.18), i.e. their total “thickness” h is small in comparison with plandimensions L . On the other hand, they represent a body of a more general shape, e.g. ribbed plates, hollow core slabs, plates with both weak and stiff reinforcement, e.g. with I-beams embedded in the concrete, corrugated plates, double-layer braced plates, etc. Only occasionally is the total flexural stiffness of such shapes identical in both (i) longitudinal x direction and (ii) transverse y -direction. Usually, the stiffness in the two directions can differ even by a factor of ten. This is not due to different modulus E1 and E2 , but due to a varying section in planes x =constant. It is therefore the shape (not physical) orthotropy of the plate in terms of shape or technical orthotropy. Global behaviour of such plates (without a detailed stress analysis in the vicinity of statically, geometrically or physically singular points and without other irregularities due to the real shape of the body) can be analysed by means of methods and programs developed for physically orthotropic plates, as long as the following assumptions are taken into account: jjj) Displacement components u, v, w (5.3.3) must be unequivocally derivable from the plate deflection surface w (or from the three functions (5.3.2) in case of plates with the effect of shear taken into account) across the whole analysed body, which means also its deformation ε and stress σ components, or the internal forces acting in the section across any part of the body. This requires establishment of suitable geometrical hypotheses, such as (5.3.4), verified in terms of accurateness through 332

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES various reasoning, experiments, practical experience, etc. kkk) B ased on the hypotheses (a), the following must be unequivocally specified for every type of shape orthotropy in plates: b1) INPUT:Constants Dik in the matrix of physical constants D for physically orthotropic plate that is going to substitute the analysed plate in the calculation. b2) OUTPUT: Further utilisation of outputs of internal forces in the substitute physically orthotropic plate for the needs of design and checking of the analysed plate with shape orthotropy, i.e. what internal forces or what stresses arise in the analysed plate. An accurate analysis of meeting the assumptions (a) and accurateness or technical applicability of results would require a comparison with the exact three-dimensional solution of a real structure at least in several characteristic or limit states, or with reliable experiments and tests. This is available only for certain examples, e.g. for steel ribbed plates, etc. Then, more precise data about the composition of the slab section appear in the input (b1). Mostly however, such analysis can not be performed and only an approximate method can be used for the comparison, e.g. for box-section plates, which, however, is not a proof, as the variation from the exact solution is not known. In view of numerous factors that influence the properties of civil engineering structures, some well-tried formulas – extended by up-todate knowledge about, in particular, torsional stiffness – can be accepted for the approximate calculations.

5.3.6.2 Simple types of orthotropic plates In section 6.2 we will discuss plates in which the effect of transverse shear can be neglected and in which the classical Kirchhoff hypothesis (5.3.4) is satisfied within the whole range. Practically speaking, these are common, not-too-thick plates that are ribbed, corrugated or stiffened in two directions x ⊥ y identical with the selected direction of coordinates.

5.3.6.2.1 Energetic equivalence of sectional characteristics The principle of potential energy equivalence of internal forces in the real and substitute body must be observed in the transformation into a physically orthotropic plate, i.e. in the calculation of constants Dik in the stiffness matrix D (5.3.23). For the plates in question, it is the following formula

333

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES Πi =

1  M x (− w, xx ) + M y (− w, yy ) + M xy (− w, xy ) + M yx (− w, yx )  dxdy 2 ∫∫ 

(5.3.42)

which is the integral of the products that represent only work of moments (similarly to a slim beam subIected to bending and torsion) on curvatures that can be, for small deflections w , expressed by second derivatives. In the calculation of the substitute plate by means of the finite element method, the second mixed derivative will be continuous everywhere (or at least everywhere with the exception of shapes with zero surface area) and the following condition of equivalence is met w, xy = w, yx

(5.3.43)

In shape orthotropic plates in general M xy will not be equal to M yx . We can define the average moment M xyF , which is the substitute physical orthotropic twisting moment M xyF = ( M yx + M xy ) 2

(5.3.44)

and thus (5.3.42) can be simplified to: Πi =

1  M x (− w, xx ) + M y (− w, yy ) + M xyF (−2w, xy )  dxdy ∫∫ 2

(5.3.45)

Fig. 2

The positive direction of all quantities is shown in Fig. 2. The comparative level (Π i = 0) is the primary non-deformed shape. Π i is always positive. Let us have a system of two beam skeletons that are parallel to x - and y - axes and have flexural stiffness ( E ′I ) x , ( E ′I ) y , torsional stiffness (GI k ) x , (GI k ) y , flexural curvatures κ x = − w, xx , κ y = − w, yy and relative twisting ϑx = −∂ϕ x ∂x = − w, yx , ϑy = ∂ϕ y ∂y = − w, xy . The potential energy of bending and twisting moments of such a system is 334

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES Πi =

1  ( E ′I ) x κ x2 + ( E ′I ) y κ y2 + (GI k ) x ϑx2 + (GI k ) y ϑ y2  ds ∑ ∫ 2

(5.3.46)

Comparing (5.3.46) with (5.3.42), we can get – only formally at the moment – the following formulas for plate strips of unit width d x = 1 or d y = 1 : M x = −( E ′I ) x w, xx

M y = −( E ′I ) y w, yy

M xy = M kx = −(GI k ) x w, yx

M yx = M ky = −(GI k ) y w, xy

(5.3.47)

Fig. 3

As the lateral contraction and elongation of element sections cannot occur freely in a compact plate (Fig. 3), the plate strips are rather stiffer in bending than the beams. This can be included into the modulus of elasticity, and thus e.g. in the case of an isotropic plate we have the module E′ =

E 1 −ν 2

(5.3.48)

which follows from the exact relation between σ and ε . In addition, it can be seen in Fig. 3 that the lateral contraction of the section is prevented in plates by certain transverse moments M ′ . E.g., the transverse moments occurring in isotropic plates subjected to bending moment M y and deformed to a cylindrical surface w( y ) are M ′ = M x =ν M y

(5.3.49)

and similarly, in plates subjected to bending moment M x and bent to the surface w( x) we have: M ′ = M y =ν M x

(5.3.50)

This effect vanishes in materials without lateral contraction (ν = 0) . Let us examine the consequences of formal equations (5.3.47) to (5.3.50) in isotropic plates with the thickness h = 1 , so that 335

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES Ix = Iy =

1 3 h 12

I kx = I ky =

1 3 h 6

(5.3.51)

There is a difference between the twisting moment of inertia of beam with the height identical with the thickness of plate and with the substantially larger width and that of a plate strip cut from a continuous plate with the same thickness. The 1 3 factor holds for a thin strip in which both horizontal shear stresses and vertical shear stresses occur if the strip is considered as a beam. The vertical shear stresses at both ends of the thin strip do not occur if we consider a unit width of a plate. Only horizontal shear stresses occur then (see Fig. 4). The horizontal shear stresses result in half the twisting moment in the strip and the vertical stresses in the other half. Thus the factor reduces to half the value, being 1 6 .

Fig. 4

Let us employ the well-known relation for shear modulus G=

E 2 (1 + ν )

(5.3.52)

and let us name the known plate-constant as Eh3 D= 12 (1 −ν 2 )

(5.3.53)

We get formulas defining the relation between moments and curvature M x = − D ( w, xx + ν w, yy )

M y = − D ( w, yy + ν w, xx )

(5.3.54)

336

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES that are in full compliance with formulas derived from the hypothesis (5.3.4). In addition, using Fig. 5, let us consider that the total twisting ϑ (more precisely the torsional curvature) of the plate element is influenced by both pairs of twisting moments, that are mutually dependent, because the relative twisting ϑ must be the same for both directions.

Fig. 5

In plates the continuous mixed derivative w, xy = w, yx = ϑ (see also Fig. 2, the element −2 ϑ xy in d local coordinates x, y with the origin in the centre of the element, so that in the corners d d x=± , y=± we get w = ± w0 . A general formula then follows from (5.3.47) 2 2 is deformed into a warped line surface of a hyperbolic paraboloid type w( x, y ) =

ϑ = ϑx = ϑy = w, xy M xy = w, xy (GI k ) x

(5.3.55) M yx = w, yx (GI k ) y

(5.3.56)

and in the case of isotropic plates it is M kx = M ky = M xy = M yx w, xy =

M xy GI k

(GI k ) x ≡ (GI k ) y = GI k M xy = GI k w, xy

After substitution from (5.3.51) and (5.3.52), multiplication by one in the form of

(5.3.57) 1 −ν and 1 −ν

use of

337


(1 +ν )(1 −ν ) = (1 −ν 2 ) M xy = −

we obtain

(1 −ν ) = − Eh3 1 −ν w = − D 1 −ν w E h3 w, xy ( ) , xy ( ) , xy 2 (1 +ν ) 6 (1 −ν ) 12 (1 −ν 2 )

(5.3.58)

which is the same formula that can be derived by an accurate procedure (integration of τ xy ) from hypothesis (5.3.4). Therefore, the beam-based theorization about physical constants of a plate leads to the correct stiffness matrix D of an isotropic plate. Using (5.3.23) or (5.3.40), (5.3.54) and (5.3.58), we can write it in the form usual in FEM:   1  Mx      My  = D ν  M xy      0 

ν 1 0

 0    − w, xx    0   − w, yy    −2 w, xy   1 −ν   2 

(5.3.59)

It can be therefore expected that such a theorization will not be principally (i.e. as far as equilibrium and continuity conditions are considered) defective even in simple plates with shape orthotropy, which is also supported by the existing experience obtained in experiments and in real practice.

5.3.6.2.2 Bending and twisting moments The dimension of these quantities is force, or more clearly, force × length per unit width of a plate section. Using the main SI units, it is N (Newton) or Nm/m (Newton meter per 1 meter of width). Conversion to previously used units: 1 kp = 9,80665 N = 10 N 1 Mp = 104 N = 10 kN

All sectional characteristics I x , I y , I kx , I ky must be calculated either (i) for a unit width of a section or (ii) for another width of the section b , e.g. for the distance between ribs or the size of the finite element and then the result must be divided by this b . The dimension of I is therefore m 3 .

Plates without lateral contraction

338

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES This group includes practically all ribbed plates with open ribs (not hollow core slabs), because the flexural stiffness of such plates is derived mainly from ribs that do not influence each other in transverse direction as there is no continuous contraction in this direction. The value termed as “effective value of ν ” is very small (e.g. 0.02) also in reinforced concrete plates with thin ribs, especially after the formation of cracks in the tensile concrete, and calculations of such plates for ν = 0 are accurate. The matrix of physical constants is diagonal as D12 = D21 = 0 , see (5.3.35). What remains is to determine D11 , D22 and D33 . The first two bending constants are quite clear and they can be calculated using (5.3.47) from D11 = ( EI ) x ,

D22 = ( EI ) y

(5.3.60)

The formulas include also possible diverseness Ex ≠ E y . I x and I y are related to the unit width of plate sections in planes x = constant and y = constant. For the most common situation of Ex = E y = E and for the calculation of I xb , I yb for ribs with an effective width of the plate bx , by we have D11 =

EI xb bx

D22 =

EI yb

(5.3.61)

by

If the plate of the height h is ribbed only in its x -direction, then D22 =

Eh3 12

(5.3.62)

For the needs of this calculation, in common concrete plates with ribs in both x - and y direction, the full distance shown in Fig. 6a can be taken as the effective width bx , by .

Fig. 6

Only for thin plates, e.g. steel orthotropic deck slabs, the reduction of bx , by according to technical standards is more significant (Fig. 6b). It is however necessary to consider the real loading conditions of the plate and other circumstances and not only apply the formula given in the standard, as it is valid for stress rather than for the substitute flexural stiffness. When bx , by are uncertain, we recommend a consultation with the author of this guide. The third constant, torsional D33 , is more problematic, but in most situation we can get satisfactory result using the formula that follows from (5.3.31) for ν = 0 . 339

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES D33 =

1 D11 D22 2

(5.3.63)

For isotropy and ν = 0 , this formula transforms into the correct value of 1 2 D , see (5.3.59). This is then represented by one twisting moment M xyF = M yxF of the substitute physically orthotropic slab. M xyF = M yxF = − D33 2w, xy = − D11 D22 w, xy

(5.3.64)

The equality M xyF = M yxF follows from the theorem of reciprocity of shear stresses τ xy = τ yx (Fig. 7a) that must be valid on the vertical edge of the plate element also in plates with shape orthotropy, but that does not imply the equality of twisting moments, which clearly follows from Fig. 7b. Twisting moments can be obtained by the integration of the shear stress flow over the section through planes x = constant and y = constant: M xy = ∫∫ τ xn r dFx = − D33 x 2w, xy

(5.3.65)

M yx = ∫∫ τ yn r dFy = − D33 y 2 w, yx

(5.3.66)

Fx

Fy

In order to apply formulas (5.3.64) and (5.3.65) accurately, we would have to know the exact distribution of shear flow over the sections of the given plate with shape orthotropy. It is quite a difficult three-dimensional problem that would require rather challenging application of the finite element method. Therefore, let us perform first an approximate technical calculation based on the estimate of torsional stiffness strips, like beams in a grid, without continuous dependencies. As w,xy is a continuous function in plates, we have w, xy = w, y x (Fig. 7c) and the comparative beams have the same relative twisting angle, and thus the ratio of moments (5.3.64) and (5.3.65) is M xy M yx

=

(GI k ) x (GI k ) y

(5.3.67)

Let us define a sum-relation between moment (5.3.64) and moments (5.3.65), (5.3.66) that is not in contrast with the isotropy. If we compare the formulas for energy (5.3.42) and (5.3.45) where there is a continuous mixed derivation of the function w, in which w, xy = w, yx , it follows: M xy + M yx = 2M xyF

(5.3.68)

This leads us to the following values: M xy =

(GI k ) x 2 M xyF (GI k ) x + (GI k ) y

M yx =

(GI k ) y (GI k ) x + (GI k ) y

2 M xyF

(5.3.69)

340


Fig. 7

Substituting (5.3.56) into the formula (5.3.69) we get w, xy = −

2 M xyF (GI k ) x + (GI k ) y

(5.3.70)

Comparing with the formula (5.3.64) w, xy = −

M xyF 2 D33

(5.3.71)

we obtain the formula that will be used to calculate the input value D33 , i.e. the torsional stiffness of the substitute physically orthotropic plate. D33 =

1 (GI k ) x + (GI k ) y  4

(5.3.72)

Quantities I k are related to a unit width of the section. Usually, they are calculated for another suitable width (beam-like section) b , so that I k = I kb b . If we calculate quantities I k from the accurate shear flow (5.3.65) and (5.3.66), the approach will be always acceptable. Exceptions, such as plates with encased I-beam etc, should be 341

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES consulted. Approximate calculations of common structures can be performed with D33 determined from the formula (5.3.63) that does not require identification of torsional stiffnesses. It can be proved that both the formula (5.3.63) and the future formula (5.3.88) for plates with lateral contraction are in the case of isotropic plates identical with the formula (5.3.72), see the reasoning in paragraph 6.2.1 following the formula (5.3.51). It generally follows from (5.3.68) that D33 =

1 ( D33 x + D33 y ) 2

(5.3.73)

and in our simplification we have D33 x =

1 (GI k ) x 2

D33 y =

1 (GI k ) y 2

(5.3.74)

Let us remind that the first subscript of τ and M always denotes the area (section x = constant) on which τ or M acts. The moment M xy thus shortens the plate strips parallel to the x -axis, and the moment M yx the strips parallel to the y -axis. See also Fig. 5 where the positive direction of these moments can be clearly seen. Two special situations for formulas (5.3.69): g) A plate with the same torsional stiffness in the x -direction and y -direction, i.e. I kx = I ky and M xy = M yx = M xyF is the printed value of the twisting moment. h) A plate with a predominant torsional stiffness in one direction, caused e.g. by thick ribs that are rigid in torsion in one direction – let us mark it x . It is characterised by a strong inequality I ky = I kx . In such a plate the formulas (5.3.68) and (5.3.72) give M xy ≅ 2M xyF

M yx ≅ 0

(5.3.75)

and thus the twisting moment in the x -direction is the double of the printed moment and it is zero in the y -direction. The real value is between the limits of type (a) and (b). If we deal with what is called design moments represented in outputs by values

( )( M

M xFdim = ( sign.M xF ) M xF + M xyF M xFdim = ( sign.M yF

F y

+ M xyF

) )

(5.3.76)

it must be considered that M x = M xF , M y = M yF (the superscript F still means the physically orthotropic plate for which the whole calculation is done), but M xyF is only a formal value. The application of values (5.3.76) is therefore useful for situations approaching the limit of type (a). Otherwise, it would be reasonable to use a correction in the meaning of the formulas (5.3.68) or (5.3.72).

Plates with lateral contraction 342


Coefficients of lateral contraction in plates with shape orthotropy can be, following from (5.3.23) to (5.3.25), assigned the following visual meaning (Fig. 8):

Fig. 8

Let us deform the plate element as in Fig. 8a into the shape corresponding to a cylindrical surface w( x) with a constant curvature w,xx . Then w, yy = 0 and, according to (5.3.23), the following moments are necessary to cause such deformation: M x = − D11w, xx

M y = − D21w, xx = ν 21M x

(5.3.77)

343

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES If we subIect the element only to moment M x , it would lead to a state shown in Fig. 8b, because the following condition follows from the second line of (5.3.23) with M y = 0 : D21w, xx + D22 w, yy = 0

(5.3.78)

and therefore, the element would be distorted also in the y -direction with the curvature w, yy = − D21w, xx D22 = −ν 12 w, xx . To produce the same curvature w,xx , smaller moment would be sufficient: M x = − D11 (1 −ν 21ν 12 ) w, xx

(5.3.79)

Similarly, the following moments are necessary to produce a cylindrical deflection w( y ) of the element in the y -direction (Fig. 8c): M y = − D22 w, yy

M x = − D12 w, yy = ν 12 M y

(5.3.80)

If the load is formed only by moments M y , i.e. if M x = 0 , it leads, according to the first line of (5.3.23), to a non-zero curvature w, xx = − D12 w, yy D11 = −ν 21w, yy . To produce the same w, yy , a smaller moment is necessary: M y = − D22 (1 −ν 12ν 21 ) w, yy

(5.3.81)

Therefore, for plates with shape orthotropy we can introduce the following definition of the coefficients of lateral contraction: The coefficient µ21 is numerically equal to the moment M y that must be applied to y = constant-edges of the element that is subIected to moment M x = 1 on edges where x = constant, in order to bend the element into a cylindrical surface w( x) . Similarly, the coefficient ν 12 is the value of M x on x = constant-edges of the element subIected to moment M y = 1 on edges where y = constant and bent to a cylindrical surface w( y ) . It can be easily verified that in the case of physically orthotropic plates this definition is equivalent to the original definition (5.3.8) and in the case of isotropic plates it results in the known relation ν 12 = ν 21 = ν . At the same time, with this definition it is clear that if the structure resembles a strong grid with a thin plate, it is practically true that ν 12 = ν 21 = 0 and formulas from paragraph 6.2.2.1 are applicable. The Maxwell-Betti theorem for plates with shape orthotropy: If a plate element is subIected only to moment M x = 1 , it leads to curvatures w, xx = −1 D11 (1 −ν 12ν 21 )

w, yy = +ν 12 D11 (1 −ν 12ν 21 )

(5.3.82)

If it is subIected only to moment M y = 1 , the curvatures are

344

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES w, yy = −1 D22 (1 −ν 12ν 21 )

(5.3.83)

w, xx = ν 21 D22 (1 −ν 12ν 21 )

If the deflection is small, the radii of curvature are Rx = 1 w, xx , Ry = 1 w, yy . If we relate the moment to a unit width, i.e. if we think about an element with sides bx = by = 1 , then the angles of relative rotation of the originally parallel vertical sides of the element are α = bx Rx = w, xx

β = by Ry = w, yy

(5.3.84)

The Maxwell-Betti theorem α = β results in the equality (5.3.82) = (5.3.83), i.e. ν 12 ν 21 = D11 D22

(5.3.85)

which is identical with the equality (5.3.25) for physically orthotropic plates. However, in that case it was a result of a general symmetry of physical constants (5.3.8) that follows directly from the requirement that the potential energy of internal forces of the body be a homogenous quadratic function of stress components or deformation components. Also in plates with shape orthotropy it must be ensured that the relation (5.3.85) is satisfied. If we use a technical reasoning or an experiment to determine e.g. coefficient ν 21 for the situation shown in Fig. 8a, also the other coefficient (for the situation in Fig. 8c) is determined. ν 12 =

D11 ν 21 D22

(5.3.86)

Satisfying this relation does not result in large values of µ for technical materials that behave like physically orthotropic materials (plywood, fibreglass, etc.). For example one specific type of pressed plywood has D11 305 = D22 46, 7

ν 21 = 0,02

ν 12 = 0,13

or another type of cross-glued plywood gives D11 120 = D22 60

ν 21 = 0, 0355

ν 12 = 0, 071

In practice, there may be a big ratio D11 D22 in shape-orthotropic plates, which may be in the range of 10 to 20. Usually however, it is found that, at the same time, the coefficient ν 21 is very small, and therefore ν 12 does not exceed 0.5 or 1.0. With regard to the definition of ν 12 , ν 21 by means of bending moments (Fig. 8), also values exceeding 0.5 or even 1.0 are not, in technical point of view, faulty. After all, they do not represent physical coefficients of contraction, which would 345

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES be limited by volume changes of the substance. In real situations and with proper thinking about transverse moments, such values are really rare. As the determination of the actual coefficient ν 12 or ν 21 can itself represent quite a difficult problem, simple approximate formulas are used in practice for what is called nondiagonal stiffness element D12 that does not include these coefficients, but only the coefficient ν of the isotropic material that the plate is made of, for example ν = 0.15 for concrete or 0.30 for steel. In ribbed plates and hollow core slabs, we can use the simplified formula (5.3.35): D12 = ν D11 D22

(5.3.87)

Similarly, instead of (5.3.63) the simplified formula (5.3.30) can be used: D33 =

1 −ν 2

D11 D22

(5.3.88)

and the increased modulus of elasticity (5.3.48) is substituted into the formulas (5.3.60) to (5.3.62), which represents a broadly small increase of 2.25 % in concrete and 9 % in steel. Conclusions from the previous paragraph 6.2.2.1 are applicable for the utilisation of values of M xy . Note concerning the plate mid-plane: It can be seen in the previous figures that, generally, the centre of gravity of sections x = constant is located in a different distance ex from the top fibre than the centre of gravity of sections y = constant ey . Therefore, we may ask a question about where the mid-plane of the plate is located. This question disappears if we consider that we calculate with a twodimensional plate continuum where ex , ey belong in fact among the “physical” properties. More serious error, however, occurs in plates with different ribs as a result of neglecting the Et planar stiffness D0 = of the top plate, or the area of thickness t . What proved useful (1 −ν 2 ) for such situations is the Giencke formula for mixed stiffness (5.3.27) D3 = C + ν ex e y D0 + ( ex + ey ) ⋅ 2

1 +ν D0 4

(5.3.89)

that is based on the total torsional stiffness of the plate C=

Et 3

(

12 1 −ν 2 )

+ (GI k ) x + (GI k ) y

(5.3.90)

And this can be used to determine the orthotropy constant (5.3.33).

5.3.6.2.3 Shear forces and reactions Shear forces result from the requirement of moment equilibrium of a plate element around the 346

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES y -axis and x -axis (Fig. 2): ∂M x ∂M xy + ∂x ∂y ∂M y ∂M xy Ty = + ∂y ∂x Tx =

(5.3.91) (5.3.92)

After substitution of (5.3.23), (5.3.65), (5.3.66) we have Tx = − ( D11w, xx + D12 w, yy ) + ( D33 y ⋅ 2 w, yx )  = − D11w, xxx − ( D12 + 2 D33 y ) w, xyy ,x ,y   Ty = − ( D21w, xx + D22 w, yy ) + ( D33 x ⋅ 2 w, xy )  = − D22 w, yyy − ( D21 + 2 D33 x ) w, xxy ,y ,x  

(5.3.93) (5.3.94)

If we compare this formula with the fourth and fifth line of matrix (5.3.23) for physically orthotropic plates, we can see that instead of the mixed stiffness D3 according to (5.3.27) the fourth line now contains the element D3 x = D12 + 2 D33 y

(5.3.95)

and the fifth line contains D3 y = D12 + 2 D33 x

(5.3.96)

In the basic plate equation (5.3.32) - which is the condition of vertical equilibrium of a plate element ∂Tx ∂Ty + + p=0 ∂x ∂y

(5.3.97)

and simultaneously the Euler's differential equation of variational plate problem - the application of (5.3.95) and (5.3.96) changes the expression 2D3 into the expression ( D3 x + D3 y ) , so that we have the following formula for the determination of the value of D3 D3 =

1 ( D3x + D3 y ) 2

(5.3.98)

Approximately (see 63a) we have D33 x =

1 (GI k ) x 4

D33 y =

1 (GI k ) y 4

(5.3.99)

However, programs for physically orthotropic plates calculate Tx and Ty according to matrix (5.3.23). Values applicable for plates with shape orthotropy can be derived from these values only by means of a rather complex calculation. If we already know twisting moments M yx and M xy , this calculation can be inspected directly by (5.3.91) and (5.3.92). In both situations, the derivatives are substituted by differences of values in adIacent finite element nodes of the analysed plate and, therefore, we cannot get the full conformity. 347

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES The first members (5.3.91) and (5.3.92) prevail over the second ones in the larger part of the plan-area of common plates and, in addition, the difference between (5.3.95), (5.3.96) and (5.3.27) is not too big. Therefore, values Tx and Ty can be used for an approximate shear design. The reactions of the plate are calculated from (5.3.36) or (5.3.37) also for the plates with shape orthotropy.

5.3.6.3 Plates with the effect of transverse shear taken into account This is an analogy to short, high, etc. beams in which it is not possible to neglect the effect of shear forces T on the deformation, as that effect is comparable to the effect of moments M . This influences the shape of deflection line w( x) and thus also the values of all statically determined quantities, e.g. hogging moments that are decisive for the design. Current programs have been developed for physically orthotropic plates following the paragraph 5.2 with the matrix of physical constants (5.3.40) and optional expansion into a full matrix of (5,5) type in the case of a general anisotropy. Therefore, it is first necessary to transform a shape-orthotropic plate into a physically orthotropic plate, i.e. to determine constants Dik in matrix (5.3.40). Instructions from paragraph 6.2.2 apply to constants D11 , D12 , D22 , D33 . What remains is to determine constants D44 and D55 in formulas Tx = D44γ xz

Ty = D55γ yz

(5.3.100)

that specify the relation between shear forces and transverse shear components of deformation γ , i.e. the change of right angles between the normal of the mid-plane of the plate after its transformation into the flexural surface w( x, y ) , see Fig. 9. These deformations (5.3.39) are zero only if w,x = −ϕ y , w, y = ϕ x (see the sign convention, Fig. 1 and 9), i.e. the Kirchhoff hypothesis (5.3.4) is valid. The most important formulas are obtained if the transverse shear stress τ xz , τ yz is assumed distributed uniformly across the sectional area Fx , Fy of sections with the width of b = 1 constructed though planes x = constant and y = constant. For a ribbed plate in Fig. 9 we have

Fx =

Fx1 bx

Fy =

Fy1 by

(5.3.101)

348


Fig. 9

In that case Tx = τ xz Fx , Ty = τ yz Fy and with identical shear modulus G in both directions we have D44 = GFx

D55 = GFy

(5.3.102)

If there are no ribs in one or both directions and if the plate thickness is h , then in that direction Fx = h

Fy = h

(5.3.103)

and thus in the solution of an isotropic thick plate D44 = D55 = Gh

(5.3.104)

These formulas are sufficient for the estimate of the magnitude of shear stress and for the estimate of required shear reinforcement in concrete. More detailed analysis however requires that the actual distribution of shear stress τ xz ( z ), τ yz ( z ) in interval − h1 ≤ z ≤ h2 be taken into account, where h1 + h2 = h is the plate thickness and h1 , h2 the distance of extreme fibre from the centroid of the section. In plates with shape orthotropy, it is possible to use the well known Grashof – Zhuravsky formula (with plate indexes) τ xz ( z ) =

Tx S ( z ) I x 2η ( z )

(5.3.105)

349

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES and similarly for τ yz where 2η ( z ) is the width of a section in the point specified by the coordinate z , and S ( z ) is the first moment of the section above this width related to the horizontal centroidal axis (Fig. 10). In rectangular section of constant width η this formula gives the distribution τ yz ( z ) that follows a parabola of the second order with its maximum 3 2

T h in the centroid. The same distribution can be obtained in isotropic plates in the Kirchhoff theory (5.3.4) from Cauchy equations of equilibrium, and therefore, we can assume that the application of (5.3.105) in plates with shape orthotropy will be fairly accurate. An uneven distribution of shear stress τ xz ( z ), τ yz ( z ) results in shear deformations γ xz ( z ), γ yz ( z ) , distributed non-uniformly across the plate thickness. If we introduce the assumption of a rigid normal (Fig. 10), the consequence is that we get constant γ xz , γ yz and thus also τ xz , τ yz and therefore the formulas (5.3.101) – (5.3.104) are justified. If we apply the more accurate distribution (5.3.105) and want to stick to the procedure of the finite element method, we have to find out the relation between Tx , Ty and values γ xz , γ yz , that are independent on z and represent angular changes in the sense of the equivalence of the potential energy of internal forces. For plates with shape orthotropy let us again proceed on the assumption of beam theorization: In a beam element of length l , in which a constant force Tx is acting, the accumulated potential energy is:

Fig. 10

1 π i = l ∫∫ (τ xz γ xz + τ xy γ xy ) dFx 2 Fx If we substitute into this formula from the Grashof hypothesis τ xz (5.3.105) and 350

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES τ xy ( x, z ) = τ xz ( z )

y tg ϕ ( z ) η ( z)

and from Hook’s law γ xz =

τ xz G

γ xy =

τ xy G

we get lTx2 ⌠⌠ S 2 ( z )  y 2 tg 2 ϕ ( z )  πi = 1 +  dFx 2 2GI x2  η 2 ( z)  ⌡⌡ 4η ( z )  Fx

Let us introduce the usual formula for the energy with a corrective coefficient β that expresses the variation of τ across the section 1 T2 πi = l x β 2 GFx β=

(5.3.106)

Fx ⌠ ⌠ S 2 ( z )  y 2 tg 2 ϕ ( z )  1 +  dydz  2  η 2 ( z)  4 I x2  ⌡ ⌡ η ( z) 

(5.3.107)

Fx

Let us write the energy (5.3.106) in the form of a half the product of the force Tx and path wT (Fig. 10): 1 π i = Tx wT 2 β Tx wT = l = γ xz l GFx Then it is clear that a constant angular change, equivalent in terms of energy to variables γ xz ( z ) , is γx = β Tx =

Tx GFx

1 GFxγ x β

Instead of formulas (5.3.102) that are valid for a constant τ xz across the whole section, we may use the following elements of the matrix of physical constants: D44 =

1 GFx βX

D55 =

1 GFy βY

(5.3.108)

351

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES where subscripts β indicate that β can be different for sections x = constant and y = constant.

5.3.6.4 Box-sections 5.3.6.4.1 Thick-walled box-section – non-solid slabs A non-solid plate with continuous hollow cores in one direction (let us denote it x ) can be calculated as a shape-orthotropic plate with lateral contraction and thus formulas from paragraph 6.2.2 apply to it. The assumption is that the webs of the box-sections are thick enough. Their thickness ti (it can be even variable) should roughly satisfy the inequality ti >

h 10

(5.3.109)

where h is the total plate thickness. If the vertical webs are thick enough (ratio bs b > 1 10 ), the influence of shear forces on the deformation of the plate can be neglected and the following formulas from paragraph 6.2.2 can be used to determine the matrix of physical constants: EI xb b (1 −ν 2 )

D22 =

D12 = ν D11 D22

D33 =

D11 =

EI y

(5.3.110)

1 −ν 2 1 −ν 2

D11 D22

Section Rectangle (plate without ribs) Solid circle and approximately also solid n -angle (n ≥ 6 ) Circle – thin-walled circular rings and approximately n -angle ( n ≥ 6 ) Steel I-beam Concrete profiles (I, square, T, etc.) with sectional area F and web area Fs I-sections, squares, etc. with the sum of vertical webs thickness t1 and radius of gyration to the neutral axis r :

(5.3.111)

β 65 32 27 (TP 3) 10 9 (ROARK) 2 2.8 to 2.1 F Fs t  4er 2 1 + k  2 − 1 22  t1  10r 352

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES t2

k=

e1

3 ( e22 − e12 ) e1 2e23

e2

Σ = t1

The value I x is calculated for section y = constant with a unit width that goes through the thinnest part of horizontal webs of the box-sections. Values I xb are calculated for an I-section where the flange width b is always the distance between the vertical axes of the boxes. In case of unequal boxes also the I-sections are asymmetrical. I xb is always related to the horizontal centroidal y -axis. The difference in the height of the centroids of sections x = constant and y = constant (see the note at the end of 6.2.2) is not significant. To determine the torsional element D33 , also formula (5.3.72) can be used: D33 =

1 (GI k ) x + (GI k ) y  4

Because in the inner webs no substantial shear flow can develop so as for the torsion stiffness one can easily neglect them and assume that multi-cell bridge is just one wide box beam.

5.3.6.4.2 Thin-walled box-sections They can be approximately analysed as plates with the influence of transverse shear taken account. In this analysis we use input data based on the comparison with modified formulas of V. Křístek: D11 = D22 =

E I xa 1 −ν 2 a

D12 = ν D11

D33 =

1 −ν D11 2

(5.3.112)

where I xa is the moment of inertia of the I-section of width a (Fig. 11) between box axes. It may however vary across elements, i.e. different boxes. The stiffness D22 is not too overestimated as the influence of the web I xa is small. Similarly, D33 is quite accurate especially in internal boxes, as the amount of shear flow that gets into the internal thin webs is I 1 very small. We may approximately calculate with a proximate value xa = th 2 , where t is a 2 the thickness of horizontal plates and h is their centre-to-centre distance. For unequal 353

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES thicknesses this value may be approximately determined from formula (5.3.116) presented later. The shear stiffness D55 in transverse direction y (see the fifth line of the matrix (5.3.40)) can be found through the comparison of the formula Ty = D55γ yz with the formula defining a relation between the transverse force Ty = Va + Vb in the highlighted I-shape frame of unit width and total skewing γ yz = γ 1 + γ 2 , where γ 1 , γ 2 are beam deflections of webs and flanges. Taking dimension as in Fig. 11 we get h a + th3 6ta3 α= h a + th3 6tb3

Vb = αVa

γ yz =

a h   3 + 3 ( 2 − α ) E (1 + α )  ta th  2aTy

(5.3.113)

(5.3.114)

Fig. 11

Therefore: D55 =

E (1 + α ) a h  2a  3 + 3 ( 2 − α )   ta th 

(5.3.115)

354


The shear modulus G23 (paragraph 5) was not needed for this calculation, but it could be determined for the substitute physical plate continuum e.g. on the assumption of (5.3.41) from the formula G23 = D55 h without introducing the conception of a sandwich plate with a soft core. Stiffness D55 is substantially increased in the location of transverse diaphragms. If they are very rigid and if they provide for non-deformability of the section proIection into the vertical plane, then γ yz can be neglected in elements located in the vicinity of these diaphragms (see the procedure for D44 ). This should happen with reasonably designed diaphragms, the web stiffness of which is higher by a factor of ten in comparison with the previously stated stiffness. The variation of D55 over elements can be easily taken into account in FEM. In case of densely located diaphragms, if one diaphragm relates to each finite strip, we can calculate with the average value of D55 , but the effect of shear γ yz will be small, otherwise the diaphragm would not meet one of its main purposes. The stiffness D44 (4th line of the matrix (5.3.40)) is according to (5.3.115) larger than D55 by a factor of ten and the shear changes γ xz can be neglected. In the input it can be expressed by the value D55 = 10−α D44

α = 2 or 3

It could be also possible to modify the program for plates with the effect of shear in one direction that nullifies γ xz beforehand. Processing of plate internal forces received from the FEM analysis: Mx

per one I-section according to Fig. 12 we have M = aM x , stress extremes

σx = Mz I , σx = ±M W .

Tx

similarly, per one I-section, we have T = aTx , stress τ xz according to (5.3.105), approximately for very thin webs τ xz = T hth only in the web.

My

In the top and bottom plate, axial forces in the transverse y -direction will become apparent N y = ± M y h and stress σ yb = − M y htb , σ ya = M y hta .

355

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES Ty

In the top and bottom plate, the transverse shear forces Vb and Va will

occur following from (5.3.113), i.e. Va = Ty (1 + α ) , Vb = αVa , with identical plate thickness. 1 Va = Vb = Ty 2 The maximum elastic transverse shear stress is approximately: 3 τ yzb = Vb tb 2

M xy

3 τ yza = Va ta 2

In the top and bottom plate, the horizontal shear forces Txy = ± M xy h occur and stress τ xyb = Txy tb , τ xya = Txy ta .

These values can be used in steel plates to calculate principal stresses that can be used for the assessment of their safety. It is similar in reinforced concrete structures (thin-walled), where the tension is carried by normal or prestressing reinforcement and the effect of M xy is reflected in the design moments, i.e. in the substitution of M x , M y by M x dim , M y dim .

5.3.6.5 Multi-cell slabs with linear hinges in longitudinal direction This means perpendicular or oblique plates assembled from prefabricated blocks, in which we assume that no reliable monolithic connection exists in the transverse direction (e.g. through transverse prestressing, which would cause that they would be treated as monolithic plates in accordance with the previous sections).

356


Fig. 12

Longitudinal joints transfer only shear forces Ty and no bending moments M y . There are many types of multi-cell slabs with linear hinges in longitudinal direction. In terms of input data Dik , all of them fall between two limit situations: lll) The vertical webs of the box-sections are so thin that practically no shear flows gets into them and the twisting moment M xy transfers only the shear in the top and bottom web. Then we may consider the equality M xy = M yx and I kx ≈ I ky ≈ I k , and therefore, following from (5.3.72), the torsional stiffness is 1 GI kb D33 = (5.3.116) 4 b if we calculate I kb for one prefabricated block of width b . Similarly to box-sections, the formula (5.3.110) applies to other stiffnesses. mmm) T he vertical webs of the sections are so thick that a continuous circulation of shear flow occurs in sections x = constant, which (considering the theorem of reciprocity τ xy = τ yx ) influences also the shear flow in sections y = constant. Then, we apply the formula (5.3.72), with varying torsional stiffnesses I kx , I ky and constant shear modulus G:

1 D33 = G ( I kx + I ky ) 8

(5.3.117)

nnn) A special situation arises when the longitudinal joints cannot transfer any torsional moment. Then we have I ky = 0

357

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES 1 D33 = GJ kx 8 Formulas (5.3.69) generally apply to the evaluation of printed M xy . If the webs in plates in Fig. 12a are so slim that transverse skewing occurs, the appropriate stiffness D55 from paragraph 6.3 can be found on the basis of the formula (5.3.115) from paragraph 6.42.

5.3.6.6 Other plate types Reinforced concrete plates with different reinforcement Fax , Fay in x - and y direction behave in the first phase practically like isotropic plates until cracks form in the tensile part of the concrete. In the second phase, we have to find moments of inertia I x , I y of the non-homogenous sections of unit width composed of (i) steel and (ii) concrete in compression. The ratio I x I y is approximately equal (plus or minus a few per cent) to the ratio Fax Fay . As the zone of concrete in compression, where some lateral contraction (or dilatation) still occurs, is small, the effective value µ is lower than 0.15. The value can become practically zero. The stiffnesses are calculated from the formulas stated earlier D11 = E ′I x

D22 = E ′I y

D12 = ν D11 D22

D33 =

D44 = D55 =

1 Gh β

1 −ν 2

(5.3.118)

D11 D22 E′ =

β ≈1

E 1 −ν 2

(5.3.119)

The effect of shear is taken into account only in plates whose thickness h > L 5 , see paragraph 5.2 and 6.3. Deviations that are of practical significance occur only if h > L 3 . Corrugated plates are calculated using the formula (5.3.118), and for sinusoidal shape of sections x = constant we take: Ix =

  1 0,81 hH 2 1 − 2 2  2  1 + 2,5 H 4l 

Iy =

l h3 s 12

(5.3.120)

where the shape of the corrugation is z = H sin π x l , the thickness of the corrugated plate is h , the amplitude of waves is H , and the length of the chord of one half-wave is l . This can be used approximately also for non-sinusoidal shape of the waves. Doubled corrugated plates (system BEHLEN, PUMS, etc.) require a special calculation of I x . In extra-thin corrugated

358

5.3 PHYSICAL AND SHAPE ORTHOTROPY OF PLATES plates, I x can be calculated on the curve z = f ( x) and then multiplied by the thickness h . The ratio I y I x is almost zero. Double-layer braced plates are beam systems used for roofing of extensive areas (stadium, etc.). For the needs of a preliminary calculation, they can be considered as orthotropic plates, with moments of inertia derived from sectional areas Fax , Fay of the beams in strips in x - and y - direction and lever arms rx , ry to the centroids of these areas, approximately rx ⋅ ry = 1 2 H . It is possible to establish the effect of transverse shear similarly to paragraph 6.3. The torsional stiffness is practically zero ( D33 = 0, χ = 0 ) in common right-angled systems of strips without diagonals in horizontal planes; in other systems it must be determined by means of comparative calculations. The values M x , Tx , Ty can be used for the calculation of axial forces in beams by means of the procedure that is common for lattice girders (intersection method). The procedure can be extended to generally anisotropic plates and is suitable for a preliminary design or for the determination of an optimal variant, etc. After the final design of the system, the accurate assessment can be performed following the calculation of axial forces in beams of the given system. The beams are, depending on the nature of Ioints, considered as a three-dimensional lattice girder or as a frame.

359

6.1 INTRODUCTION

6 Modelling Interaction

of

Structure-Soil

6.1 INTRODUCTION 6.1.1 Origin and Development of the Efficient Subsoil Model Rapid progress is now being made in many different aspects of structure models and also of soil and rock models. Today no difficulty arises in computing any common structure with the given boundary conditions which have an a priori known form, e.g. rigid or elastic bonds etc. Numerical methods of' geomechanics have lead to algorithms and programs for three-dimensional analysis of a subsoil which can be applied to very complicated problems of soil and rock mechanics in engineering practice. Unfortunately, this method of determining soil stress and strain state is very expensive and requires a thorough previous geological investigation in situ if the results are to be useful for further design. For this reason such methods are only applied when the knowledge of the stresses, strains and displacements in the whole three-dimensional domain of the subsoil is needed for the final engineering decision on a project, as in the case of dams, underground structures, tunnels, etc. The cost of the structure must be in proportion to the very high price of the geotechnical in situ and laboratory tests required for reliable input data and the very long computing time which is due to the large number of unknowns and great band width of the equation coefficient matrix pertaining to the three-dimensional elements of finite soil models. By the very well-known formula for the computing time t = aNB 2 , where a is a constant of the program and computer, N the number of unknowns and B the band width, we can state (and practice has proved it) that the computing time required for such an analysis is many orders greater (1,000 to 10,000) than in the two-dimensional case. When designing an ordinary structure the price of computing and the price of the appropriate prior investigation of the subsoil input data can outweigh price of the structure itself. In normal design practice the above-mentioned reality leads to the omission of any subsoil modelling at all, which is the opposite extreme. Perhaps for this reason the programs for structural analysis mostly do not contain any information about introducing the given subsoil properties into the calculation. But there is no structure without foundations (with the exception of some special cases, such as rockets flying in space) and any designer knows very well the difficulties connected with this. In many programs the designer must model the subsoil by some nodal springs arising as a nodal substitution of Winkler's foundation, or he can join that foundation continuously to some elements. Some programs use the pseudoelastic halfspace divided into three-dimensional elements, which results in an enormous computing time. Introducing the idea of a connecting matrix of the layered subsoil complicates matters further, requiring extra calculations and producing possibly unreliable results. 360

6.1 INTRODUCTION Theoretical investigation and research has been oriented towards the exacting cases mentioned above, where the soil mass modelling must be as precise as the modelling of the structure, expressing the displacement vector, stress and strain tensors, pore pressures etc. at any soil mass point. Today, many soil and interface elements are known and programmed, including such complicated effects as lateral earth pressure, stress-paths, size of load increments, disturbance around structures due to driving and installation, adhesion and cohesion, real constitutive laws, interface behaviour, strain-softening, construction sequences (excavation, dewatering) etc. Hundreds of references are included in recent books, journals and proceedings. The present book does not pertain to this geomechanically oriented literature. The designer of the structure has mostly no time for an accurate investigation of the subsoil behaviour, nor is he really very interested in it. Supposing the subsoil to be sufficiently stable, the designer has only to prove that the relative and absolute settlements and the stresses of the structure agree with technical conditions, standards and other requirements of safety and economy. As far as the subsoil is concerned, he only wants to know about its behaviour on the surface where it is connected with the structure e.g. through a foundation plate, grillage etc., which is included in the structure model. He is not interested in the exact values of stress and components in the subsoil mass under the surface, course of the displacement vector (u, v, w) in the space ( x, y, z ) , pore pressure etc., and has no time or money to investigate them by means of modern geomechanics, which are designed for other purposes and other problems of engineering. The great gap between common design practice with its „Winklerian“ ideas and the current state of geomechanics can be bridged only by an efficient subsoil model with the following characteristics: The model should be simple enough for straightforward cases, but it should also be capable of producing more accurate and truthful analysis if necessary. The simplest model form must be two-dimensional, condensing the subsoil properties into the surface where it is connected with the structure. All effective modern program systems are based on the finite element method in one or other of its well-tried forms, therefore the model must be based on proven mechanics theorems such as the principle of virtual work or Lagrange's variational principle etc., on which the modern finite element method (FEM) is based. The physical behaviour of the subsoil mass can be expressed only by a simple pseudoelastic continuum model with constants able to express all necessary facts which might influence the structure settlements and stresses, e.g. nonhomogeneity and anisotropy in a general layered subsoil of any thickness. The subsoil displacement, strain and stress state should be described by the same functions on its surface as those describing the state of the structure, because of the full compatibility demand. The number of model constants must not be too great and their technical meaning must be clear to any designer. In any case the model must express the real behaviour of the subsoil surface, not only under the foundation, but also in the surrounding area where some neighbouring structures may interact with the structure investigated or some surface loads may influence the settlements, etc. From the numerical point of view, the model must express the „infiniteness” of the subsoil area in the x, y, z directions by the modern idea of infinite elements leading to the possibility of analysing only the structure and its foundation domain without the expensive finite element division of a greater domain. In difficult cases this must be done in a reliable way by including a number of settlement functions under the subsoil surface in the number of 361

6.1 INTRODUCTION the unknown deformation functions of the problem. When analysing the problem by the finite element method the parameters pertaining to all the above-mentioned deformation functions are solved by only one linear equation set. Despite the three-dimensional nature of such a model the numerical algorithm must remain in the two dimensions of the subsoil surface, condensing all parameters in its division nodes. This concept is also very useful in the dynamic analysis of eigenfrequencies of the fundaments on an arbitrary layered foundation, because it involves the inertia forces of the subsoil elements too. The authors designed a basic two-dimensional model, initially for purely practical purposes, and incorporated it in the first programs of the NE-XX program package in 1975. They presented it at the 5th Danube Conference of Soil Mechanics and Foundation Engineering, 1977, published in EUROMECH Proceedings 97, May 1978, IBA-DAT '82 Proceedings, Berlin and in many publications later. After about ten years of practical applications and further development of their model the authors prepared the first general publication containing all necessary derivations and information about the theory of both model forms (surface model and its generalization), program algorithms, explicit matrix formulae, tables, examples and comparisons with the previous models in the form of numerical tests deep knowledge of [8].

6.1.2 The Main Ideas of the Efficient Subsoil Model A new efficient structure-subsoil model is defined and analysed in two forms: The first one is a simple two-dimensional (surface) or one-dimensional (line) model with a set of constants CS or CS * attached to the two- and one-dimensional elements, as will be explained later. The second one (designed for more exacting cases) is a three-dimensional model expressing an arbitrary layered soil medium with the pseudoelastic constants E , ν and G , in general unhomogeneous and anisotropic. The definition pertains to the general modelling of large or infinite domains in geomechanies. The properties of the model will be derived in Chapter 6.2. The effectiveness of the model is increased by the further domain restriction in the interface plane or surface, introducing special boundary conditions. The twice integral reduction of the original problem domain leads to the solution only in a small domain of the actual soil-structure interface what is derived in [8]. The first form of the efficient subsoil surface model (2D model, Section 6.2.1.2) will be derived in detail for the most frequent case of the horizontal soil-structure interface basing on some physical and geometrical assumptions which allow one to describe the subsoil nature by means of seven constants: C1Sz [MNm −3 , MPa ⋅ m −1 ] – foundation compression modulus of the Winkler type, expressing resistance to the vertical displacement of the subsoil surface. C2Sx , C2Sy , C2Sxy [MNm −1 ] – foundation shear moduli expressing resistance to the shear components in the x and y directions of the subsoil surface, generally different in positive and negative shears γ xz , γ yz (dilatancy and contractancy effects). 362

6.1 INTRODUCTION C1Sx , C1Sy , C1Sxy [MNm −3 , MPa ⋅ m −1 ] —foundation friction moduli expressing the resistance of the subsoil surface to its horizontal displacement components. When the interface between subsoil and foundation lies in a horizontal plane ( x, y ) and the axis z is vertical, then, according to the IASMFE rules, the constants can be named as follows: C1Sz – modulus of subgrade reaction, C2Sx , C2Sy , C2Sxy – moduli of subgrade shear reactions. C1Sx , C1Sy , C1Sxy – moduli of subgrade friction reactions. These seven constants are contained in the expression for the virtual work or potential energy of the subsoil internal forces pertaining to the above-mentioned seven surface deformation components. This virtual work or energy is added to the energy of the structure's internal forces and external loads. The total energy balance is governed by the virtual work principle. In special cases the common energy variational principles hold and no complication in numerical analysis arises. In this book, Lagrange's variational principle and the principle of total virtual work will be applied. The above principle is also valid in the case of a one-dimensional element (member). The property of its elastic medium is described by seven constants, depending not only on the soil properties but also on the cross section of the member: C1Sx* [MNm −1 ] – foundation-member friction modulus expressing the resistance to displacement in the axial direction x , C1Sy* , C1Sz* [MNm −1 ] – foundation-member compression or friction moduli expressing the resistance to both displacements v, w in the directions y, z normal to the member axis x , C2Sy* , C2Sz* [MN] – foundation-member shear moduli expressing the resistance to the rotations dv dx , dw dx of the model around the axis z , y normal to the member axis x, C1Sϕ*x [MNm] – foundation-member rotation modulus expressing the resistance to the member's rotation around its own axis x , i.e. ϕ x . The second form of the efficient subsoil model designed for more exacting analysis, is more sophisticated and introduces some further unknown functions, namely the settlements under the soil surface. These unknown functions, which lead to additional degrees of freedom in the numerical solution, enable one to minimize the extent of the designer's subjective assumptions. Only the upper part of the subsoil is usually modelled in this way. The lower part can be modelled to advantage in a manner similar to the first model form. It has only a small influence on the structure's internal forces, and therefore need not be expressed as precisely as the upper subsoil part.

363

6.1 INTRODUCTION

6.1.3 The Efficient Structure-Soil Interaction Model Assuming an Arbitrary Shape of Structure-Soil Interface The idea of expressing the surrounding soil mass by properties of structure-soil interface can be generalized. The structure-soil interface can be of any arbitrary shape (surface Ω1 , with the boundary Γ1 ; in Fig. la). At each point of Ω1 a local coordinate system x, y, z can be introduced, z being the surface normal direction and x, y lying in the tangent plane. The orientation of x, y axes can be defined by a suitable rule and the axes x, y, z form a positive coordinate trihedral (Fig. la). With a phenomenological approach to the problem, regarding the surrounding soil or rock mass as a black box and defining the relevant properties of the interface by in situ measuring or from experience, almost all explications of the 2D surface model in the present book hold also in the general case when the structure-soil interface is not plane but curved or generally shaped. The displacement vector u = [ u , v, w ]

T

(6.1.1)

and approximate shear components w′ = [ ∂w ∂x , ∂w ∂y ]

T

(6.1.2)

are defined in the coordinates x, y, z described above (Fig. la). In these coordinates the relevant seven physical constants of the 2D model are defined:  C1Sx  C1S = C1Sxy  0 

C1Sxy C1Sy 0

0   0  C1Sz 

 CS C2S =  S2 x C2 xy

C2Sxy   C2Sy 

(6.1.3)

364

6.1 INTRODUCTION

Fig. 1. a) Structure-soil interface Ω1 b) Three spring groups illustrating the interaction. c) One-dimensional case of structure-soil interaction.

A scheme of the mechanical behaviour of the interface Ω1 is shown in Fig. 1b, discretizing the domain for better illustration (the real model is, of course, continuous). The three spring groups in Fig. 1b illustrate the influence of the pressure constants C1Sz , (group 1), shear constants C2Sx , C2Sy , C2Sxy (group 2) and friction constants C1Sx , C1Sy , C1Sxy (group 3) respectively. The different lengths of springs of the same group express the fact that all constants can depend on coordinate of Ω1 points, i.e. they can be generally variable. The influence of the stress components σ x , σ y , τ xy , parallel to the tangent plane ( x, y ) , on the virtual work or potential energy of internal forces is ignored in defining the 2D model with constants of the C1S and C2S type, as will be explained in Section 6.2.1.2.3, formulae (6.2.41) to (6.2.48). The 1D model of structure-soil interaction with constants of the type CS * (Section 365

6.1 INTRODUCTION 6.2.2) can also be generalized to the curved line case (Fig. lc), introducing at each line point a local coordinate system x, y, z , x being the tangent of the curved line where the interaction between a curved beam and soil medium is concentrated. The trihedral x, y, z forms a positive rectangular system whose orientation varies with the position of the model point. A generalization of the 3D efficient structure-soil interaction model (Section 6.2.3) assuming a curved interface surface is also possible, defining additional surfaces with further unknown functions wi ( x, y, zi ) in each surface i in the soil mass besides the function w( x, y, 0) defined in the interface domain.

6.1.4 Some Remarks about Soil-Foundation-Structure Interaction The common meaning of the soil-structure interaction is very well known. There exist many references and much experience in the simple cases, where „structure” means only a raft, a thick or thin plate, a grid, a pile or a beam, which represents only the foundation of a real upper structure with its substructures, establishments etc. The interaction between the foundation and the building can be of the same order as the soil-foundation interaction. Therefore, the subsoil model is only one part of the global design modelling, and in this book, oriented as it is towards practical engineering, information about that effect must be included. The lack of a simple theory of interaction and appropriate programs for mini and personal computers with no expensive input data, i.e. without expensive geotechnical investigation, leads to faults in the routine design of the less important structures caused by the lack of time and money for better computing. The first example in Fig. 2a is based on the assumption of a linear course

366

6.1 INTRODUCTION

Fig. 2. a) Interaction between structure, foundation plate and subsoil. b) Elementary solution. c) More precise solution of reaction r .

of Winklerian reaction r ( x) (Fig. 2b) and a linear relation r = kw between the reaction r and the settlement w with a constant k („soil reaction modulus” or „modulus of subgrade reaction”, when the interface between the foundation and subsoil is horizontal). There are only two unknown boundary values r which can be calculated by two equilibrium conditions: the sum of all vertical forces and their moments must be equal to zero. The real course of the r ( x) and w( x) functions is mostly different (Fig. 2c) because of the foundation and structure elasticity and the real behaviour of the subsoil, whose surface not only settles under the foundation, etc. Errors in the w -values, the slopes dw dx and the curvatures 1 R ≈ d 2 w dx 2 have a direct influence on the design, showing up in tests and inequalities in standards, bending moments, reinforcement, etc.

367

6.1 INTRODUCTION

Fig. 2. d) Foundation plate loaded by a statically indeterminate structure. e) The first estimation of loads Pi assuming a rigid foundation. f)g) Influence of settlements on the loads Pi ′ .

Winklerian models are subject to a great discrepancy between the reality and the local relation r = kw , despite the generalization for unhomogeneous cases r ( x, y ) = k ( x, y ) ⋅ w( x, y ) and respecting the time-effect. The real r - and w -courses are not affine. The condition r = 0 does not implicate the state w = 0 in the same place. There is interaction between two neighbouring structures, but the Winklerian model cannot express it. From the security point of view the bending moments in the foundation plate or grid are most significant. When the loads are concentrated towards the foundation centre, the Winklerian bending moment values are too small, i.e. dangerous. But the Winklerian model has a great numerical advantage over the three-dimensional models, e.g. halfspace (see Fig. 10 in Section 6.3.2.6.1.). The advantage of the 2D-solution is also present in Pasternak's model (1936, 1954) with two constants C1S [MNm −3 ] , C2S [MNm −1 ] , which is a special case of the efficient surface model presented when introducing C2Sx = C2Sy = C2S , C3Sx = C3Sy = 0 , i.e. in the isotropic case without friction effect. But it requires the solution of the whole region, where the surface settlements cannot be omitted, which leads to a more expensive numerical solution. The advantage of the model presented here is based on the fact that it requires solution only in the foundation region, as simple as Winkler's introducing special boundary line bounds expressing the influence of the rest of the subsoil surface. 368

6.1 INTRODUCTION The first form of the efficient model with more constants described in the previous section is numerically as advantageous as the classical Winkler model, and at the same time its capability is greater than that of the Pasternak model. The idea common to all surface models is that they provide a cumulative expression of subsoil deformability. After the calculation of the C S -constants the subsoil itself is a „black box“ for the structure designer. It can be investigated separately in the second job, loaded by the resulting displacements on its surface, which is in structural design mostly unnecessary. The „interaction“ definition depends on the actual level of theory, experiments, computer hardware and software, and its significance will change in the future, when large, complex systems will be solved by one big equation set, without numerical instability in a reasonable time. In Clapeyron's time only continuous beams were solved, and it could be useful to define the „interaction” between the beam and columns when solving the frame in Fig. 2d. The „interaction forces and moments” are the common internal forces of the frame solution, also in the spatial case when all displacement and rotation components of the adjacent cross sections are respected. Similarly the real loads P1 to P5 and the real reactions r ( x) of a plate strip in Fig. 2e are internal forces of the whole system (structure + foundation + subsoil). Also, the simplest calculation of the isolated plate strip as depicted in Fig. 2e requires some speculation about the influence of settlements on the values of P1 to P5 respecting the real structure stiffness and an assumption concerning the r ( x) -course. Measuring the real loads and reactions in situ is very expensive. Comparison between the measured values r ′( x) , P1′ to P5′ (Fig. 2f, g) and the assumed values r ( x) , P1 to P5 is the last test of design reliability. But mostly only settlements are measured. Finally it must be stated that the behaviour of a structure also depends on the technology employed in its construction and on all installations in it. For instance, the rigid raft under a turbine (Fig. 3a) linearizes the foundation plate settlements in its region. The rigid walls change the stiffness of a steel frame or truss structure, which results in changing of settlements and bending moments of the foundation plate (Fig. 3b). The subsoil surface can be loaded directly, and its settlements can cause dangerous column slopes and horizontal displacements a1 , a2 (Fig. 3c). Also, a case which cannot be solved by any Winklerian model can be expressed by the efficient subsoil model in a simple way.

369

6.1 INTRODUCTION

Fig. 3. a) Elastic plate stiffened by a rigid raft. b) Steel frame structure stiffened by walls. c) Horizontal displacements a1 , a2 caused by stock loading of subsoil.

370

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL 6.2.1 Reduction of the Three-dimensional Model to the Two-dimensional Model

6.2.1.1 Three-dimensional Models in Geomechanics There exist more or less elaborated three-dimensional models of the soil medium including two- and three-phase models describing in detail time dependent processes such as consolidation, creep and relaxation, construction sequences, the influence of underground excavations, rock caverns etc. Some of them are described in the quoted references (Abel, Awojobi, Z. Bažant, Z. P. Bažant, Betles, Beer, Bowles, Brown, Carrier, Christian, Curnier, Davies, Desai, Feda, Fraser, Gatti, Gibson, Gioda, Gudehus, Hruban, Jori, Kolář, Maier, Medina, Meek, Meigh, Němec, Nova, Pircher, Poulos, Pruška, Rodriquez, Sacchi, Selvadurai, Simons, Smith, Šimek, Wardle, Zienkiewicz etc.). There is special journal devoted to these problems, the International Journal for Numerical and Analytical Methods in Geomechanics: it has been coming out since 1977 and its scope has been extended every year until it now involves hundreds of very significant papers covering a range of complicated cases, where time and money are no object because of the nature and significance of the cases. A new journal, Computers and Geotechnics, appeared in 1985. Despite this, three-dimensional models cannot be (and are not) used in general design practice. The reason can be demonstrated by the simplest case: a pseudoelastic soil medium (Fig. 4), which is mostly layered. The number n of different geological layers depends on the depth H n of undeformable rock surface or on the so-called „effective depth H n “ where no settlements occur (Altes, 1976). The subsoil surface in the plane z = 0 is loaded by the structure and its foundation in a region Ω1 with the boundary Γ1 (Fig. 4b). Generally the T

loading course p ( x, y ) represents an unknown vector function p =  px , p y , pz  of variables ( x, y ) . The vertical component pz . (mostly effect of gravity) as well as both horizontal components px , p y (friction effects) depend on the whole structure-soil interaction. Because of this even the most precise analysis of the subsoil displacements, strains and stresses without a knowledge of the p -function, i.e. with some practical design approximation p∗ , cannot lead to more precise results than would be found by any simplified analysis which took into account the structure-soil interaction and minimized the error p − p∗ in the sense of some norm in functional space. For instance, the total potential energy of the system (structure and soil) can be minimized. For this reason, no analysis which does not take into 371

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL account structure and its foundations can be used by the structure designer.

Fig. 4. a) Stresses in a layered subsoil. b) The foundation-soil interface

Ω1 .

Supposing the program and computer capability to be sufficient for the calculation of the structure, the foundations and their subsoil in one system, a further difficulty arises: subsoil input data, e.g. the real pseudoelastic constants of the type ( E , G,ν )i for any layer i = 1, 2, K , n (Fig. 4a) and in a two-phase medium the real pore pressures at any point ( x, y, z ) at the time t = 0 , cannot be obtained without very expensive geotechnical investigation, the results of which depend on many factors. Unfortunately, no time and money can be devoted to this investigation when designing most structures, but even if the 372

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL geotechnical data are presented, their reliability concerning the real behaviour of the subsoil surface contacting the foundations is mostly no better than that of a simple technical approximation and results in about the same reliability of structure design. The discrepancies between the laboratory tests or standard data and the actual in situ values of the type ( E , G,ν )i are cumulated in the interval 0 ≤ z ≤ H n . From the structural point of view mere surface settlement investigation would be more efficient, provided the size and shape of design loading could be retained. The real loads are very great. Therefore surface settlement measuring in situ, under and outside the real structures, is most significant far design in the same subsoil conditions. But such an investigation cannot be used directly to determine the physical constants of three-dimensional models. The results of this investigation replace the information included in these constants as far as the usual structure designs are concerned. The same holds in the case of more complicated three-dimensional soil models, e.g. unhomogeneous (R. E. Gibson 1967, 1974; R. E. Gibson, P. T. Brown and K. R. F. Adrews 1971; A. O. Awojobi 1972 to 1976; P. T. Brown and R. E. Gibson 1973; W. D. Carrier and J. T. Christian 1973 etc.). The value of these models lies in their application to the geomechanics of soil mass, where they can be appreciated, for example when comparing the results with other model results or when investigating the influence of some physical constant values on the settlement.

6.2.1.2 Two-dimensional Efficient Subsoil Model 6.2.1.2.1 Pseudoelastic Soil Medium The first form of the efficient subsoil model presented in this book represents a pure two-dimensional surface model situated in the plane ( x, y ) of the foundation bottom (Kolář, Němec, 1977, 1978). It can also represent only the lower part of the whole subsoil in the second form of the efficient subsoil model (Section 6.3.4). Then the plane ( x, y ) lies at the depth z = H n , where 0 ≤ z ≤ H n is the interval of the layered model which is essentially three-dimensional. In deriving the properties of the first model form we set out from Fig. 4a, where a generally layered subsoil of the depth H n is presented. The subsoil deformation caused by the surface loading in the plane z = 0 depends on its physical properties. Assuming a pseudoelastic soil medium without the filtration consolidation effect, i.e. omitting the pore pressure p( x, y, z , t ) investigation, all quantities can be deduced from the vector function u with three displacement components u , v, w in the direction of the axis x, y, z. u( x, y, z , t ) = [u ( x, y, z , t ), v( x, y, z , t ), w( x, y, z , t ) ]

T

(6.2.1)

The symmetrical strain tensor ε with six strain components can he derived as follows: ε = ∂T u

(6.2.2)

where ∂ represents a (6,3) -matrix of differential operators, generally nonlinear in the case of great deformation. 373

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL The symmetrical stress tensor σ with six stress components (Fig. 4a) is connected with the strain tensor by a more or less complicated physical (constitutive) relation L( σ , ε, t ) = 0

(6.2.3)

where L denotes some operator, including generally the time effect (creep, relaxation), mostly in the form of integrals along the time interval 0 ≤ t ≤ ta where ta , denotes the actual investigation time and t = 0 means the starting time of the process. The analytical solution of the complex problem with four variables x, y , z , t can be performed only in a few straightforward cases. The numerical solution is theoretically possible in any case where all desired input data required for relation (6.2.3) are known, but in practice this does not always give the desired reliability. The designers of most structures must therefore divide the problem into two independent tasks; the solution in spatial coordinates x, y, z and the analysis of the time effect. This can be carried out accurately only in one particular synchronous rheological case, where the so-called equivalent values u∗ , ε∗ , σ ∗ depending only on ( x, y, z ) coordinates, can be introduced (Sobotka, 1981). They can be solved by any static method. Then the course of u, ε, σ in time is analysed by means of integral equations with a sole variable t where the equivalent values u∗ , ε∗ , σ ∗ , present the known (given) functions. In the basic case that analysis can be replaced by the changing of the pseudoelastic constants in time, which only affects relation (6.2.10). Special contributions to this problem can be found in the references, e.g. G. Gatti and I. Jori 1981; G. Gioda 1980; G. Gioda and O. de Donato 1979; G. Gudehus 1977; G. Maier and G. Gioda 1981; R. Nova 1981; R. Nova and G. Sacchi 1982; H. G. Poulos and E. H. Davies 1973. We will not complicate the main idea that the first model form is based essentially on the reduction of the 3D-domain to the 2D-domain. Therefore we will firstly omit all time effects and investigate only the statical case of formula (6.2.1), i.e. the displacement vector u( x, y, z ) = [u ( x, y, z ), v(v( x, y, z ), w( x, y, z ) ]

T

(6.2.4)

For the same reason we will suppose only small strains and displacements, i.e. the geometrically linear case of small deformation. Than the operator ∂ in (6.2.2) involves only linear differential operation known from the standard textbooks (e.g. Kolář et al., 1979): 0 0 0 ∂ ∂z ∂ ∂y   ∂ ∂x  ∂= 0 ∂ ∂y 0 ∂ ∂z 0 ∂ ∂x   0 0 ∂ ∂z ∂ ∂y ∂ ∂x 0 

T

(6.2.5)

with the following sequence of strain tensor components written in the usual matrix vector form: ε = ε x , ε y , ε z , γ yz , γ zx , γ xy  = [ε1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 ] T

T

(6.2.6)

The normal strain components are denoted ε with the suffix index of the appropriate direction x, y, z and are positive when denoting extensions. The shear strain components are denoted γ with two suffixes denoting the directions which are originally perpendicular and the angle between them after deformation is (π 2 ) − γ . The positive sign of γ is also defined 374

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL by this fact. Because of the technical character of this book we will use only the simple common denotation. Relation with the more condensed form 1  ∂u ∂u  ε ik =  i + k  2  ∂xk ∂xi 

i, k = 1, 2, 3

(6.2.7)

where the indices 1, 2, 3 pertain to the axis x, y, z (the case i = k means the normal components ε and i ≠ k the shear components), shows only the well known difference in the definition of shear components ε ik = γ ik / 2 , i.e. only a half of the angle change γ ik represents a component of the strain tensor (6.2.7). Only the form (6.2.7) has the common tensor properties which must be taken into account, e.g., when transforming the coordinate system. Denoting the partial derivatives by the second index after a comma, we can write the relation (6.2.7) in the most condensed form: ε ik =

1 ( ui , k + u k , i ) 2

i, k = 1, 2,3

(6.2.8)

These remarks can be useful for the designer when studying the references and comparing the results. The stress components pertaining to the strain components (6.2.6) will also be denoted in a technical way and written in the matrix vector form: σ = σ x , σ y , σ z ,τ yz ,τ zx ,τ xy  = [σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 ] T

T

(6.2.9)

The normal stress components in the direction of the x, y, z -axis are denoted σ with the appropriate index, and are tensions when positive. The shear stress components τ have two indices: the first denotes the direction perpendicular to the surface on which the component acts, the second denotes the direction in which it acts. We shall use the classical Boltzmann's axiom of continuum mechanics where the tensor σ is symmetrical, i.e. τ yz = τ zy , τ zx = τ xz , τ xy = τ yx . The deformation of such a continuum is fully described by the three functions (6.2.1) or (6.2.4), because each point ( x, y, z ) has only three degrees of freedom: displacements u, v, w in the x, y, z -directions. Only when modelling the plate – or shell – structure or its beam stiffeners and pile members, the Mindlin's and Cosserat's continuum models with the rotational degrees of freedom at the point ( x, y, z ) , i.e. the rotational components ϕ x , ϕ y , ϕ z independent of u , v, w will be introduced. The physical relation (6.2.3) between the stress and strain tensors will be simplified to the common relation in the matrix form: σ = Cε

(6.2.10)

C represents a (6, 6) – symmetrical matrix of 21 independent pseudoelastic constants

C = cik

i, k = 1, 2, K , 6

cki = cik

(6.2.11) 375

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL This fully anisotropic case requires a very expensive triaxial investigation of the constant values cik . Mostly only a special orthotropic case is supposed. In the rectangular coordinates the orthotropy results in the dividing of the relation (6.2.11) into two isolated relations between σ − ε and τ − γ , the last of which is diagonal. Only nine constants c11 , c22 , c33 , c44 , c55 , c66 , c12 , c13 , c23 are introduced and investigated. Similarly in the polar coordinates simple cases of orthotropy can be defined. In the case of isotropy all constants cik depend only on two basic values. E (Young's modulus of elasticity) and µ (Poisson's ratio). Pseudoelastic soil models (Gibson, Awojobi etc.) often also introduce the shear modulus G, which depends in the isotropic case on the E , µ by the relation G = E (2 + 2 µ ) .

6.2.1.2.2 Basic Assumptions and Relations The reduction of the three-dimensional soil model defined in the previous section to the two-dimensional one can be done in a very general way, introducing a set of functions n

u ( x , y , z ) = ∑ ui ( x , y ) g i ( x , y , z ) i =1 n

v( x, y, z ) = ∑ vi ( x, y ) hi ( x, y, z )

(6.2.12)

i =1 n

w( x, y, z ) = ∑ wi ( x, y ) fi ( x, y, z ) i =1

where gi , hi , f i , i = 1, 2, K , n are selected functions and ui ( x, y ), vi ( x, y ), wi ( x, y ) are unknown functions of two variables x, y . The functions gi , hi , f i determine the course of displacement components along the variable z , i.e. under the subsoil surface, where some decrease in their values is mostly expected. In simple cases only one term (n = 1) can be taken into account, omitting the sole index i = 1: u ( x, y , z ) = u ( x , y ) g ( x, y , z ) v ( x, y , z ) = v ( x, y ) h ( x , y , z ) w( x, y, z ) = w( x, y ) f ( x, y, z )

(6.2.13)

The special case arises when the horizontal displacement components u, v of the soil mass points A (Fig. 5a) have practically no influence on the amount of energy of internal forces in the subsoil and we can calculate with the assumption about the settlements: w( x, y, z ) = w( x, y, 0) f ( z )

(6.2.14)

Concerning the chosen function f ( z ) only two conditions need to be fulfilled (Fig. 5): f (0) = 1 and f ( H n ) = 0 which also holds in the limit case H n → ∞ , modelling a half space. Mostly the depth H n will be finite. When it is small compared to the extent of the loaded

376

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL subsoil surface Ω1 (Fig. 4b), the linear course between the end values 1 and 0 can be assumed and the settlement w( z ) along a vertical line decreases linearly from the surface value w( x1 , y1 , 0) to the zero value at the bottom of the whole deformable subsoil layer: w( x1 , y1 , z ) = w( x1 , y1 )

Hn − z Hn

f ( z) =

Hn − z Hn

Fig. 5. a) 3D model, b) 2D model of a subsoil. c) Soil shear forces

(6.2.15)

t x , t y and reaction r ,

Relation (6.2.15) was first proposed by Pasternak (1936, l1954). A more general function f ( z ) was introduced by Vlasov—Leontjev (1960)

377

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL f ( z) =

sh γ ( H n − z ) sh γ H n

(6.2.16)

with a further constant γ in the argument of the hyperbolic function sh γ . The constant γ governs the velocity of w -decrease and can be considered as a geomechanical parameter of the subsoil. With the formula (6.2.16) the limit case H n → ∞ can also be calculated, which is impossible in the linear case (6.2.15). The settlement w decreases nonlinearly (the case w∗∗ ( z ) in Fig. 5a). Any other assumption about the „decrease function” f ( z ) can be made, e.g. α

H −z f ( z) =  n   Hn 

(6.2.17)

with some exponent α in the role of a geomechanical parameter etc. From the point of view of the investigation of model constants by surface measuring in situ (Section 6.2.2.1) the „decrease function” f ( z ) can be considered as a property of a „black box” and may not be analysed at all. In the following text we will assume generally any form of the function f ( z ) which fullfils the boundary conditions f (0) = 1, f ( H n ) = 0 and includes only known geomechanical parameters. Later on (second model form, Section 6.3.4), the more general case of a layered subsoil (denoted as w∗ ( z ) in Fig. 5a) with unknown settlement functions defined in all layer boundaries, will be presented. The virtual work of internal forces i.e. stresses σ on the virtual deformation defined by a strain tensor ε , is given by the common formula Π iv = ∫∫∫ δε v σdV = V

= ∫∫

Hn

∫ (σ δε x

Ω 0

x

(6.2.18)

+ σ yδε y + σ zδε z + τ yzδγ yz + τ zxδγ zx + τ xyδγ xy ) d Ωdz

The virtual work of external forces, i.e. volume (body) forces X = [ X , Y , Z ] acting on any T

T

point ( x, y, z ) of the whole volume V = ΩH n and surface forces p =  px , p y , pz  acting in the region Ω will be defined as follows: Π ev = − ∫∫∫ uTv XdV − ∫∫ uTv pd Ω V

(6.2.19)

Ω

with the displacement vector (6.2.4). Mostly the volume forces are reduced to the mere dead T weight X = [ 0, 0, γ ] where γ denotes soil density. This case can be investigated separately. Then the formula (6.2.19) includes only the second term due to the surface load p . In the above definitions the p – vector, u – vector, ε – tensor and σ – tensor are fully independent. 378

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL Let us define an equilibrium state of stresses and loads {σ,p} and a geometrically

compatible displacement vector and strain tensor {δ u, δε} as a virtual displacement, i.e. small

enough not to disturb the equilibrium state {σ, p} Then the virtual work principle holds and can be written in short form as follows: Π iv + Π ev = ∫∫∫ (δε ) σdV − ∫∫ (δ u ) pd Ω = 0 T

T

(6.2.20)

Ω

V

Generally the equilibrium state {σ, p} can arise in the course of some non-linear process and the virtual displacements can be regarded as a variation of the actual displacements and strains {u, ε} . Therefore the general virtual work principle (6.2.20) enables one to solve the nonlinear problems in most cases by the well-known incremental procedure. By defining the first surface model form without any insubstantial details, the simplest physically and geometrically linear case can be analysed. In this case the virtual work principle (6.2.20) with the relation (6.2.10) results in the well-known elementary form of Lagrange’s variational principle δΠ = δ (Π i + Π e ) = 0

(6.2.21)

where Π i denotes the potential energy of internal forces Πi =

1 εT σdV ∫∫∫ 2 V

(6.2.22)

and Π e the potential energy of external loads Π e = − ∫∫ uT pd Ω

(6.2.23)

Ω

Numerical analysis shows that in most loadings of a subsoil the terms σ xε x , σ yε y and τ xy γ xy in the formula (6.2.18) are very small compared with the other terms σ zε z , τ xzγ xz and τ yz γ yz and the formula (6.2.22) can be simplified to the form Πi =

1 (σ zε z + τ yzγ yz + τ xzγ xz ) dV 2 ∫∫∫ V

(6.2.24)

On the basis of the same analysis it can be shown that in the basic relation (6.2.2) with the operator matrix (6.2.5) the terms pertaining to the horizontal displacement components u, v can be neglected compared with the terms due to the vertical settlements w . Therefore we can write the following approximate relations: T

 ∂w ∂w ∂w  ε = ε z , γ xz , γ yz  =  , ,  = ∂T w z x ∂ ∂ ∂y  

(6.2.25)

∂ ∂ ∂ ∂= , ,   ∂z ∂x ∂y 

(6.2.26)

T

379

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL T

σ = σ z ,τ xz ,τ yz  = Cs ε

(6.2.27)

 Ez  Cs =  0 0 

0   Gxy , z  Gyz 

(6.2.28)

H  1 1  n T T Π i = ∫∫∫ ε σdV = ∫∫  ∫ ε Cs εdz  d Ω  2 V 2 Ω  0 

(6.2.29)

0 Gxz Gxy , z

The assumption (6.2.14) about the settlements w( x, y, z ) can be introduced in (6.2.25) and (6.2.29):   ∂f ( z ) ∂w( x, y ) ∂w( x, y ) ε =  w( x, y ) , f ( z ), f ( z) ∂z ∂x ∂y  

T

(6.2.30)

1 ⌠⌠   ∂w( x, y )  Π i =  C1S w2 ( x, y ) + C2Sx   +  2 ⌡⌡   ∂x  2

Ω

2

 ∂w( x, y )  S + C2Sy   + 2C2 xy ∂ y  

∂w( x, y ) ∂w( x, y )   dΩ ∂x ∂y 

(6.2.31)

The constants resulting from the integration in the interval 0 ≤ z ≤ H n are denoted as follows: Hn

⌠  df ( z )  C =  Ez   dz ⌡  dz  2

S 1

(6.2.32)

0

C

S 2x

=

Hn

∫G

xz

f 2 ( z )dz

yz

f 2 ( z )dz

0

C2Sy =

Hn

∫G

(6.2.33)

0

C

S 2 xy

=

Hn

∫G

xy , z

f 2 ( z )dz

0

These relations include the four physical constants of the subsoil mass: soil modulus Ez in the relation σ z = Ezε z = Ez ∂w ∂z and the shear soil moduli Gxz , Gxy , z , Gyz in the relations τ xz ≅ Gxz ∂w ∂x + Gxy , z ∂w ∂y and τ yz ≅ Gyz ∂w ∂y + Gxy , z ∂w ∂x , which are simplified according to the above mentioned suppositions. Generally the soil moduli can be variable with the depth z , e.g.

380

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL Hn

⌠  df ( z )  C =  Ez ( z )   dz  dz  ⌡ 2

S 1

(6.2.34)

0

They can vary with the position of the surface point ( x1 , y1 ) as well, and the „decrease function“ f ( z ) can also depend on it, which results in a more general formula with variable constants, e.g. Hn

⌠  df ( x1 , y1 , z )  C ( x1 , y1 ) =  Ez ( x1 , y1 , z )   dz dz   ⌡ 2

S 1

(6.2.35)

0

In practice this heterogeneous case will be analysed numerically as simply as the homogenous case, using the finite element technique. From the modelling point of view the only difficulty is in the proper definition of input data. The physical behaviour of a soil mass is essentially nonlinear. Therefore two basic methods of analysis can be used: the finite method, calculating the actual (finite) state, and the incremental method, starting from an initial state and itroducing sufficiently small loading increments dp . During one increment dp the physical relation between stress and strain increments d σ and d ε must be linearized and a tangent physical law similar to the law (6.2.27) d σ = CsT d ε

(6.2.36)

with the matrix of tangent soil moduli  EzT  CsT =  0  0 

0 GxzT Gxy , zT

0   Gxy , zT  G yzT 

(6.2.37)

must be used. Introducing these moduli into the formulae (6.2.32), (6.2.33) results in tangent constants C1ST , C2SxT , C2SyT , C2SxyT . Due to the change of the tensors σ and ε and their increments d σ and d ε with the depth z and mostly with the position ( x1 , y1 ) of the surface point as well, the form (6.2.34) or (6.2.35) should be applied, which of course needs some further investigation. Therefore, in the most practical structure designs, only approximate values are used, cumulating the z -effect in some reasonable way in the chosen values of C S -constants. The finite method without increments can be used only in so-called „conservative” cases, when the actual state does not depend on the previous states or loading history. The values of the moduli Ez , Gxz , Gyz , Gxy , z in the relations (6.2.32) (6.2.33) correspond to the actual stress-strain state, i.e. they mean secant moduli values. The real physical behaviour of the soil mass can be modelled approximately as conservative only in the case of small monotonic loading when the stresses fulfill some limit state inequality I (σ ) < I L (σ ) or the equality I (σ ) = aI L (σ ) with a small coefficient a , e.g. a = 0,3. ” I ” denotes some stress invariant or other limit state operator. In most building 381

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL designs this condition is fulfilled. In the more exacting cases including unloadings and the other nonmonotonic loading cases great soil stresses σ etc., the actual state of stress depends on the previous history or the so-called stress and strain paths in time, starting at some initial time t = 0 where all quantities are known. Then the incremental method must be used and at least an approximative estimation of the tangent soil moduli (6.2.37) must be made. For instance the unloading process is governed by an almost linear law with considerably greater soil moduli than the initial loading process.

6.2.1.2.3 Effects of Horizontal Displacements and Stresses In designing a structure the real behaviour of the interface between structure foundation and soil surface should be taken into account. Special models of the interface region with or without tensions and shears, physically linear or non-linear, viscoelastic or viscoplastic etc. have been elaborated and the appropriate interface finite elements can be found in references, e.g. G. Beer 1985; R. Frank et al. 1978; E. L. Wilson 1977; C. S. Desai 1981 to 1985. In usual structure design practice no precise information about the physical constants of foundation-soil contact is generally available. The designer decides between two limiting cases: contact without friction or contact with elastic resistance against horizontal displacement components u , v of the foundation bottom. The first case needs no further adaptation of the subsoil model, the second introduces some surface „friction stresses.” τ zx0 = C1Sx u + C1Sxy v τ zy0 = C1Sy v + C1Sxy u

(6.2.38)

depending linearly on the surface horizontal displacement componets u, v . The physical constants C1Sx , C1Sy [MPa ⋅ m −1 = MNm −3 ] are generally different in the x - and y - directions (orthotropic case) and only in the isotropic case does the relation C1Sx = C1Sy and C1Sxy = 0 hold, reducing the number of „friction constants” to one. No proper „friction” depending on the normal stress component σ z is described by the law (27). The nature of the law corresponds more to the so-called „surface friction”, well-known when analysing the behaviour of bored piles etc. The law (6.2.38) holds up to certain displacement limits u = uL , v = vL and for u > uL , v > vL the stresses τ zx0 , τ zy0 remain constant (full mobilization state). For u ≤ u L , v ≤ vL the virtual work of the stresses τ zx0 ,τ zy0 done on the virtual displacements uv , vv is given by the following formula: Πτ v = ∫∫ (τ zx0 uv + τ zy0 vv ) d Ω

(6.2.39)

Ω

The potential energy Πτ i of the stresses τ zx0 , τ zy0 , regarded as internal stresses of the system, can be evaluated as a half of the value (6.2.39), putting in uv = u, vv = v and using the 382

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL law (6.2.38): Πτ i =

1 C1Sx u 2 + C1Sy v 2 + 2C1Sxy uv ) d Ω ( ∫∫ 2Ω

(6.2.40)

In deriving the basic properties of the efficient subsoil model in the previous section, all terms depending on horizontal stress components σ x , σ y , τ xy were negligible compared with the other terms with the significant components σ z , τ xz , τ yz . Without this omission, the energy expressions (6.2.24) or (6.2.29) cannot be used and the full equation (6.2.18) holds. The difference is mostly small but in some special cases there can arise an essential energy due to the ignored components σ x , σ y , τ xy . Integration along the z -axis similar to formula (6.2.31) results in further constants which can be designated the C3S -constants with suffixes of appropriate stress and displacement components. An analysis reveals that these constants are identical to the physical constants C3S introduced in the bending of plates with the same course of σ x , σ y , τ xy – components along the plate thickness. For clarity’s sake we shall write the common expression of the potential energy Π i 3 of plate internal forces respecting only Kirchhoff’s terms arising due to the curvatures, assembled in the following vector: w′′ = ∂ 2 w ∂x 2 , ∂ 2 w ∂y 2 , 2∂ 2 w ∂x∂y 

T

(6.2.41)

The potential energy Π i 3 is a quadratic function of w ′′ : Πi3 =

1 w′′T C3S w′′d Ω ∫∫ 2Ω

(6.2.42)

where C3S denotes the symmetrical matrix (3,3) of physical constants C3,S ik , i, k = 1, 2, 3 : S S  C3,11 C3,12  S S C3S = C3,21 C3,22 S S C3,31 C3,32 

S  C3,13 S  C3,23  S  C3,33 

(6.2.43)

Similarly the virtual work Π v ,3 done by the actual internal forces C3S w′′ pertaining to the actual deflections w on the arbitrarily selected virtual displacement wv is defined by the product formula: Π v ,3 = ∫∫ w′′T C3S w′′d Ω

(6.2.44)

Ω

which must be used during the incremental procedure solving a nonlinear and time path dependent problem. Six independent constants of matrix (6.2.43) must be retained only in the most general case of full anisotropy. Mostly an orthotropic model with only four constants S S S S (C3,13 = C3,23 = 0) or an isotropic model with two independent constants ( C3,11 = C3,22 ,

383

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL S S S S C3,12 = ν C3,11 , C3,33 = 1 2 (1 −ν ) C3,11 ) will be sufficient to express the effect of horizontal stress

components σ x , σ y , τ xy in the subsoil mass. In the numerical analysis this effect can be modelled simply by defining a „plate stiffness” of the subsoil surface when calculating the whole system. In this way the three-dimensional soil medium (Fig. 5a) can be replaced by the twodimensional surface model (Fig. 5b), whilst retaining all properties which affect the practical structure design. The physical constants of the three-dimensional soil medium, assembled in a matrix C of the stress-strain relation, are replaced by the physical constants of the surface model assembled in a row matrix CS . In the unhomogeneous and anisotropic case the matrix C contains 21 independent functions Cik ( x, y, z ) of three variables, i, k = 1, 2, K , 6 and the S ( x, y ) of two variables: matrix CS 13 independent functions C1Sz ( x, y ) to C3,33 S S S S S S  CS = C1Sz ; C2Sx , C2Sy , C2Sxy ; C1Sx , C1Sy , C1Sxy ; C3,11 , C3,12 , C3,13 , C3,22 , C3,23 , C3,33

(6.2.45)

We order the terms CS according to their significance in practical design, which corresponds to their historical origin. C1Sz expresses the surface stiffness against vertical settlements w (a Winklerian constant), C2Sx , C2Sy , C2Sxy pertain to the shears wx , wy and C1Sx , C1Sy , C1Sxy to the S S horizontal displacements or „surface friction”. C3,11 to C3,33 express the stiffness against the

curvatures ∂ 2 w ∂x 2 , ∂ 2 w ∂y 2 , ∂ 2 w ∂x∂y i.e. the influence of the energy terms (6.2.42) or (6.2.44). In the case of orthotropy in the x and y directions the number of independent terms in the matrix C decreases to 9 and the matrix CS contains only 9 independent subsoil surface stiffnesses: S S S S  CS = C1Sz ; C2Sx , C2Sy ; C1Sx , C1Sy ; C3,11 , C3,12 , C3,22 , C3,33

(6.2.46)

Mostly the isotropic case will be sufficient to model the main subsoil behaviour concerning the common structure design. Then the terms of the matrix C depend only on two physical constants, e.g. E (pseudoelastic soil modulus) and µ (Poisson’s ratio of the soil mass). The number of terms in the matrix CS will be reduced to five, because C1Sx = C1Sy , C2Sx = C2Sy and the last two constants can be expressed through the 6th and 7th constant as follows: S 3,22

C

=C

S 3,11

,

S 3,33

C

S 1  C3,12  S =  1 − S  C3,11 2  C3,11 

(6.2.47)

Then the matrix CS can be written in the following form: S S  = C1S , C2S , C1Sf , C3S , C3′S  CS = C1Sz , C2Sx , C1Sx , C3,11 , C3,12

(6.2.48)

This is not the simplest form. In actual practice, when no information as to the destination of the „friction” and „bending” effect is available or cannot be taken into account due to the nature of the structure, lack of time or whatever, only the first two constants in (6.2.48) are usually retained. Then we can omit their second indices and write the form of the 384

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL matrix CS as follows: CS = C1S , C2S 

(6.2.49)

Frankly speaking the „two-constants-model” as a first improvement of the old Winkler model with only one constant C1S , fulfils the main demand of the structure designer by introducing the „shear effect”, which cannot be included in any form of Winkler’s model since its actual physical manifestation is as a liquid with the specific density C1S [MNm −3 ] The idea of the two-constants-model is old (1936) and was published in comprehensive form by Pasternak (1954), who first applied it to the case where the foundation of a structure (machine) is formed by a massive frame. In the reference quoted (1954), Pasternak presents some remarks about the C1S and C2S values in the case of a subsoil and recommends an experimental investigation by means of a circular plate but doesn’t give any further details. The influence of experimental scale, effective depth etc. was neglected, but this does not detract from the crucial theoretical significance of Pasternak’s idea in the case of homogeneous isotropic subsoil. Bearing it in mind the efficient subsoil surface model presented can be regarded as a full generalisation using the energy concept. But it is also much more efficient in the twoconstants-case because it reduces the analysis to computing in the foundation region only, as it will be explained latter. The ideas in this section can be summarized as follows: Any three-dimensional pseudoelastic soil medium (Fig. 5a) whose physical properties are defined by a C -matrix can be replaced by a two-dimensional surface model (Fig. 5b) for the purposes of designing a structure and its foundation. The physical properties of the surface model are described by at least two (and at most 13) constants C S which can be assembled in a row matrix CS , defining the physical behaviour of the co-called „efficient subsoil model”. Both constants C and CS can be attached to the final actual system state (secant moduli) or to the steps in an incremental procedure (tangent moduli). Generally, in an unhomogeneous case the constants C are functions of three variable coordinates ( x, y, z ) and the constants CS depend on only two variables ( x, y ) . In time-dependent problems the time variable must also be taken into account. A way of introducing the t -dimension into the calculation is to change the C and CS values with time; this applies to the special synchronous case of rheology.

6.2.1.2.4 Full Generalization of the Efficient Surface Model of Subsoil Medium The purpose of introducing a 2D subsoil model is to enable an efficient solution of structure-soil interaction. When reducing a 3D subsoil problem in a domain Ω3 to a 2D model in a domain Ω 2 the real behaviour of the 3D soil mass must be expressed by the properties of the subsoil surface. The main demand is the fulfillment of virtual work equivalence

385


∫∫∫ Π

* v3

Ω3

d Ω3 = ∫∫ Π *v 2 d Ω 2

(6.2.50)

Ω2

Π *v 3 and Π *v 2 are the densities of virtual work in the 3D domain Ω3 and 2D domain Ω 2 respectively. Both densities pertain to the same virtual displacement defined in the Ω 2 domain, where the contact with the foundation of the structure occurs. There arises the problem of defining the work density Π *v 2 in a 2D surface model, where unknown functions of only two coordinates x, y are introduced, and therefore only partial derivatives of these functions with respect to x and y can be formed and connected to the relevant physical properties of subsoil. For instance introducing only one unknown function w( x, y, 0) , according to section 6.2.1.2.2 and formula (6.2.14), and constants Cs relevant to the derivatives wlm =

∂ l +m w ∂xl ∂y m

(6.2.51)

the generalized forces can be defined as the sum of products of the following form. S Flm = ∑ Clmjk w jk

(6.2.52)

j ,k

doing on the generalized virtual displacements wv ,lm virtual work with the density S Π*v 2,lm = wv ,lm Flm = wv ,lm ∑ Clmjk w jk

(6.2.53)

j ,k

The total density of virtual work is the sum of work done by individual generalized forces: Π*v 2 = ∑ Π*v 2,lm

(6.2.54)

l ,m

In finite problems, where the potential energy Π i of internal forces (stresses) can be defined potential energy equivalence must be assumed:

∫∫∫ Π d Ω = ∫∫ Π d Ω * 3

Ω3

* 2

3

(6.2.55)

2

Ω2

with the density Π *2 expressed by the following sum: Π *2 =

1 1 S Flm wlm = ∑ wlm ∑ Clmjk w jk ∑ 2 l ,m 2 l ,m j ,k

(6.2.56)

The generalized displacements wlm include the settlements w = w00 , slopes wx = w10 , S wy = w01 etc. Each constant Clmjk can be defined as a generalized stiffness of subsoil surface. S The more terms wlm and constants Clmjk are taken into account the better the equivalence to the energy of the 3D model that can be achieved.

Let the column matrix W = [ w00 , w10 ,K , wnn ] include all defined derivatives wlm and T

386

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL S the square matrix CS = C(Slm ),( jk )  all constants Clmjk on the appropriate places. Then the densities of virtual work and potential energy can be written in the matrix form

Π *v 2 = WvT CS W

Π *2 =

1 T S W C W 2

(6.2.57)

The most general model introduces the vector U = [u00 , u10 ,K , w0 n ]

T

of all

displacement components u = [u , v, w] and all their derivatives by x and y up to the n -th T

order. The virtual work Π v of the generalized forces CS U on the virtual displacement U v is expressed by the formula Π v = ∫∫ UTv CS Ud Ω

(6.2.58)

Ω

Further on the simpler case will be presented, supposing the „diagonal” form of the CS matrix, i.e. S  CS = DIAG C1S , CS2 ,K , Cn+ 1

(6.2.59)

S (n) Then the virtual work done by the generalized internal forces C1S u, C2S u′, C3S u′′, K , Cn+ on 1u

the generalized virtual displacements u v , u′v , u′′v , K , u (vn ) reads as follows:

(

)

Π v ,i = ∫∫ uTv C1S u + u′vT C2S u′ + u′′v T C3S u′′ + K + u (vn ) CnS +1u ( n ) d Ω Ω

T

(6.2.60)

The conservative problems can be solved by a finite procedure based on the definition of the potential energy of generalized internal forces: Πi =

1 ( uT C1S u + u′T C2S u′ + u′′T C3S u′′ + K + u(n)CnS+1u(n) ) d Ω 2 ∫∫ Ω

(6.2.61)

In both cases the following matrix vectors assembling the geometrical quantities are defined: Vector of displacement components: u = [ u , v, w ]

T

(6.2.62)

Vector of the first derivatives of displacement components by the variables x, y (coordinate axis in the surface region Ω ): u′ = u x , u y , vx , v y , wx , wy 

T

(6.2.63)

Vectors of the 2nd, 3rd, …, n -th derivatives: u′′ = u xx , u xy , u yy , vxx , vxy , v yy , wxx , wxy , wyy 

M

u ( n ) = u xxKx , u xxK y ,K , u yyK y ,K , wyyK y 

T

T

(6.2.64) (6.2.65)

Matrices of the order (3,3), (6, 6), (9,9), K , (3n + 3,3n + 3) assembling the generalized 387

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL physical constants of the model: C1S = C1Sik  C2S = C2Sik  C3S = C3Sik  M CnS+1 = CnS+1,ik 

i, k = 1, 2, 3

(6.2.66)

i, k = 1, 2, K , 6

(6.2.67)

i, k = 1, 2, K , 9

(6.2.68)

i, k = 1, 2, K , 3n + 3

(6.2.69)

The further analysis can be done in a standard way starting with the principle of virtual work (general case) or Lagrange’s variational principle (conservative problems). Note. Not all terms of the matrices (6.2.66) to (6.2.69) need to be different from zero, e.g. the matrix (6.2.66) can be diagonal. Also the technical meaning and influence of various constants can be different. Many of them can be completely ignored and just one such case of omission results in the above defined effective surface model. The general case can also serve as an explanation of the chosen denotation concerning the indices of the C S -constants. The first index (n + 1) pertains to the constants attached to (n) -th derivatives. The zeroderivatives (n = 0, C1S ) mean the functions. The constants at the first derivatives ( n = 1, C2S )

are used to express the shear stiffness properties of the model, the constants at the second derivatives ( n = 2, C3S ) describe the „bending” or curvature stiffness properties. The general model form is mathematically connected with the idea of Taylor’s series and other ideas for reducing the dimensions of a problem from 3D to 2D (e.g. Reissner’s series). There is no space in the present book for further details.

6.2.1.2.5 Equilibrium Condition and Technical Remarks The equilibrium condition of the whole system (structure and subsoil) can be deduced from the virtual work principle (6.2.20) which represents the most general equilibrium principle and holds in all cases including the nonlinear ones. In [8] the equilibrium condition of the actual system state is derived as the Euler differential equation of the variational problem based on the common Lagrange variational principle of the minimum of total potential energy Π . Respecting only the terms due to the vertical displacement components (settlements) w and using the formulea (6.2.23) and (6.2.31) holding for the subsoil alone without structure, we obtain the Π - expression in the following form: Π = Πi + Πe = =

1 C1Sz w2 + C2Sx wx2 + C2Sy w2y + 2C2Sxy wx wy ) d Ω − ∫∫ pwd Ω ( ∫∫ 2Ω Ω

(6.2.70)

We denote briefly the partial derivatives ∂w ∂x and ∂w ∂y with wx , wy and omit the notation 388

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL of variables ( x, y ) in the region Ω of the investigated subsoil surface. The standard procedure of setting up the Euler differential equation of the variational problem Π = min . leads to the following equation: C1Sz w − ( C2Sx wxx + C2Sy wyy + 2C2Sxy wxy ) = p

(6.2.71)

where wxx = ∂ 2 w ∂x 2 , wyy = ∂ 2 w ∂y 2 represent in the case of small settlements w the curvatures of the sections of w( x, y ) -graph, arising in the plane y = const . and x = const . , wxy = ∂ 2 w ∂x∂y being the graph distortion. In the special case of C2Sx = C2Sy = C2Sxy = 0 (pure Winkler’s model) the equilibrium equation (6.2.71) takes the plain form: C1Sz w = p

(6.2.72)

which coincides with Winkler’s original assumption and gives the name „Winklerian constant“ to the constant C1Sz . But in the case that at least one of the constants C2Sx , C2Sy , C2Sxy is not zero, the constant C1Sz is not identical to the Winklerian constant. The physical dimension [MPa ⋅ m −1 = MNm −3 ] is the same but the value can be substantially different, as is shown in [8]. Due to the shear effects, the value C1Sz pertaining to the efficient subsoil model is smaller than the pure Winklerian value obtained with the same measurements but ignoring the settlements beside the loaded area. Some technical explanation would be useful here, using Fig. 5c, where a vertical soil element dxdy with its load pdxdy is presented. The Winklerian part of the subsoil reaction r = C1Sz w

(6.2.73)

maybe defined and its sign may be positive in the direction − z , i.e. opposite to the positive load p . the plain equilibrium condition of the pure Winkler model states only that p = r . The other terms in equation (6.2.71) may be regarded as the resulting vertical forces due to the shear forces t x , t y of the efficient subsoil model: t x = C2Sx wx + C2Sxy wy t y = C2Sy wy + C2Sxy wx

(6.2.74)

The shear forces t x , t x + d x t x [MNm −1 ] act in the element planes parallel to the ( y, z ) -plane. Their difference d xtx =

∂t x dx = C2Sx wxx dx + C2Sxy wyx dx ∂x

(6.2.75)

on the length dy results in the force (C2Sx wxx + C2Sxy wyx )dxdy in the direction + z , i.e. with negative sign compared to the + r direction. Similarly the difference d yt y =

∂t y ∂y

dy = C2Sy wyy dy + C2Sxy wxy dy

(6.2.76)

389

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL of the shear forces t y , t y + d y t y acting in the element planes parallel to the ( x, z ) -plane in the length dx results in the force (C2Sy wyy + C2Sxy wxy )dxdy, also oriented in the + z -direction. The vertical equilibrium condition can be written in the technical form: rdxdy = ( d x t x ) dy − ( d y t y ) dx = pdxdy

(6.2.77)

Introducing the above defined and calculated values: C1Sz wdxdy − ( C2Sx wxx + C2Sxy wyx ) dxdy − (C2Sy wyy + C2Sxy wxy )dxdy = pdxdy

(6.2.78)

and comparing the coefficients at the common term dxdy the equation (6.2.71) arises which proves its meaning as the equilibrium condition. At the same time it proves the possibility of the technical idea of Fig. 5c. The shear forces t x , t y [MNm −1 ] of the surface model replace in the energy sense the actual shear stress components τ xy , τ yz [MPa] arising in the subsoil mass: ∂w( x, y, 0) = ∂x

Hn

∂w( x, y, 0) = t y ( x, y ) ∂y

Hn

t x ( x, y )

∫τ

xz

( x, y, z )γ xz ( x, y, z )dz

0

∫τ

(6.2.79) yz

( x, y, z )γ yz ( x, y, z )dz

0

Using the formulae (6.2.73) and (6.2.74), the variational problem (6.2.70), i.e. Lagrange’s variational principle for the mere subsoil without structure can be written in technical form: Π = Πi + Πe =

1 ( rw + tx wx + t y wy ) d Ω − ∫∫ pwd Ω = min . 2 ∫∫ Ω Ω

(6.2.80)

In the pure Winkler model the terms with t x , t y are omitted. Supposing the most simple case of p( x, y ), i.e. p = const . and constant surface stiffness (6.2.46) in the whole area Ω , only the constant reaction r = C1Sz w and constant settlement w will arise. Then the integration will be reduced to a simple area measuring

∫∫

Ω

dΩ = A

(6.2.81)

and the most simple problem arises: 1  Π =  C1Sz w2 − pw  A = min 2  ∂Π = ( C1Sz w − p ) A = 0 ∂w p w= S C1z

(6.2.82) (6.2.83) (6.2.84)

390

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL The real energy balance is of course more complicated, as will be explained in Chapter 6.3.

6.2.2 One-dimensional Efficient Subsoil or Soil Medium Model

6.2.2.1 Introduction The basic models of structural analysis (simple or continuous beams, bars, trusses, frames, grids) are composed of one-dimensional elements, and it is very useful to model their interaction with the soil mass as a one-dimensional problem with one variable x . Even the oldest model of a beam on an elastic foundation was a one-dimensional one and introduced the well-konwn Winkler relation r1 ( x) = k ( x) w1 ( x) between the reaction r1 [kNm −1 ] and the settlement w1 (m) with a physical constant k ( x) [kNm −2 ] which holds in the interval 0 ≤ x ≤ L , L being the beam length. The subscript 1 denotes the dimension of the model. The constant k of the one-dimensional model is defined by the relation k = r1 w1 and its value can be investigated by measuring the reaction r1 pertaining to the settlement w1. The constant rate r1 w1 is a necessary assumption of the physical linearity. Subsoils are substantially nonlinear and the law r1 = kw1 holds only for very small settlements w . Generally only pseudoelastic behaviour and the incremental form dr1 ( x) = kT ( x, w)dw1 ( x) can be supposed with the „tangent stiffness” kT depending on the settlement level. In any case Winkler’s model can be represented by a system of isolated linear or nonlinear springs whose axial stiffness is equal to kdx [kNm −1 ] and the dimension dx tends to zero. The distinct difference between one-and two-dimensional subsoil models can be shown even in the case of Winkler’s model. Its two-dimensional form is included in the first form of the efficient subsoil surface model (Section 6.2.1.2) as a special case with only one non-zero constant C1Sz [kNm −3 ] and the relation r ( x, y ) = C1Sz ( x, y ) w( x, y ) between the reaction r [kNm −2 = kPa] and the settlement w(m). The physical dimension of the reaction r is the dimension o a stress component σ z , and there is a difference between the dimension of C1Sz [kNm −3 ] and k [kNm −2 ] . Note 1. There does not really exist a contact between soil and structure other than a two-dimensional one. The one-dimensional soil-structure interaction represents a useful abstraction directed towards the structure parts replaced by one-dimensional (beam) models, which is a frequent case in design practice e.g. foundation grids or piles. Note 2. The physical dimension of the model must not be confused with the

391

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL mathematical dimension of the problem. The two-dimensional model leads to onedimensional problem in two special cases: prismatic problems with only one variable, independent of the second one, axisymmetrical problems. The physical nature and quantity dimensions remain twodimensional. Note 3. The physical constants of both the one- and the two-dimensional model are essentially fully independent and can be investigated once they have been defined. Nevertheless, by introducing assumptions concerning the surface settlement course some simple relations between the above-mentioned constants can be derived, as is shown in [8].

6.2.2.2 An Example of the Relation Between the Constants of One- and Twodimensional Models The important question of the relation between the constants of 1D and 2D models is discussed later, in [8]. Here we will present only a simple idea illustrating the relation between Winkler’s constant k [kNm −2 ] and the similar constant C1Sz [kNm −3 ] , perhaps also the further constant C2Sy [kNm −1 ] of the two-dimensional model. We start with the plain case of a beam on an elastic foundation, modelled by Winkler’s medium. Let b( x) be the width of soil-structure interaction area, generally variable with the coordinate in the beam axis direction x , and r ( x, y ) [kPa] the Winklerian reaction (6.2.73) defined in the interval 0 ≤ x ≤ L , −b 2 ≤ y ≤ b 2 assuming symmetry conditions r ( x, y ) = r ( x, − y ) and rigid cross sections: w( x, y ) ≡ w1 ( x). Integrating the relation (6.2.73): b2

∫

b2

r ( x, y )dy =

−b 2

∫

C1Sz ( x, y ) w1 ( x)dy

(6.2.85)

−b 2

and comparing it with the equation r1 ( x) = k ( x) w1 ( x) , one gets the following formulae: b2

r1 ( x) =

∫

r ( x, y )dy

(6.2.86)

C1Sz ( x, y )dy

(6.2.87)

−b 2 b2

k ( x) =

∫

−b 2

connecting the one-dimensional quantities r1 , k with the previous two-dimensional r , C1Sz . In the case of the constant C1Sz ( x, y ) = C1Sz ( x) independent of y , the reaction r ( x, y ) will be constant in the interval −b 2 ≤ y ≤ b 2 and the formulae (6.2.86), (6.2.87) lead to the common relations:

392

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL r1 ( x) = b( x)r ( x)

(6.2.88)

k ( x) = b( x)C1Sz ( x)

(6.2.89)

Supposing a constant interaction width b and omitting the denotation of the x -variable, the following plain formulae can be written: r1 = br

(6.2.90)

k = bC1Sz

(6.2.91)

An improvement on the formulae (6.2.90), (6.2.91) can be based on rewriting equation (6.2.85) for the two-dimensional model with non-zero shear stiffness C2Sy . Along the beam edges y = b 2 and y = −b 2 the line reactions R [kNm −1 ] occur, expressing the influence of the subsoil in the regions y > b 2 and y < −b 2 . When the ratio L b is sufficiently great, allowing the transition to the limit case L → ∞ , then the uniform load p causes a settlement w( y ) independent on x . The course of w( y ) for y > b 2 and y < −b 2 is given by the formula (see [8]): w( y ) = w(0)e

− y −b 2 s

s=

C2Sy C1Sz

(6.2.92)

and for −b 2 ≤ y ≤ b 2 w( y ) = w(0) = w1

(6.2.93)

Then the one-dimensional Winkler constant k can be derived from the equilibrium (or equivalence) condition ∞ b 2 S  r1 = ∫ rdy = ∫ C w( y )dy = 2  ∫ C1z w1dy + ∫ C1Sz w1e− y −b / 2 / s dy  = kw1  0  −∞ −∞ b2

(6.2.94)

k = C1Sz b + 2 C1Sz C2Sy

(6.2.95)

∞

∞

S 1z

where

The same results can be obtained from the potential energy or virtual work equivalence. Thirty years ago, E.E. de Beer, H. Grasshoff and M. Kany investigated the value k at various conditions, A. D. Kerr designed a plain elastic or viscoelastic subsoil model and many other results were obtained which are useful till today. The above relations show the possibility of problem dimension reduction from 2D to 1D by replacing the y -dimension with an integration in the y -interval. The 2D model is based on a similar z -integration of the 3D relations leading to the formulae (6.2.32), (6.2.33). Thus double integration can reduce the 3D problem to the 1D problem and at the same time the relations between the appropriate physical constants are derived. All physical constants must be investigated by laboratory or in situ tests. The direct 393

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL measuring of their values, without any recalculation, seems, therefore, to be most useful for design practice. Unfortunately, it is very expensive in time and money and the use of approximate formulae connecting the constants of models of various dimensions is often inevitable. Nevertheless, the direct definition of all constants of one-dimensional soilstructure interaction forms the necessary basis for the theory, as well as investigation and design, and it will therefore be presented in the next section.

6.2.2.3 Basic One-dimensional Relations In order not to complicate the introduction we will start the explanation of the main ideas of a one-dimensional soil-structure interaction with a basic case: a straight line 0 ≤ x ≤ L may model the subsoil or soil/rock medium of a one-dimensional structural element, e.g. a part of a pile, frame, grid etc. Generally the x -axis is identical to the central axis of the element and forms with the y - and z -axis a positive rectangular coordinate system whose origin (0,0,0) lies at the centre of gravity of the end cross section x = 0. The y and z -axes represent the main cross-section axes. The element deformation is fully described by four functions u , v, w, ϕ x (classical theory) or six functions u , v, w, ϕ x , ϕ y , ϕ z (Cosserat’s one-dimensional continuum) denoting six independent degrees of freedom of a point, modelling a cross section (see Chapter 6 in [8]). In both cases the deformation of the straight line 0 ≤ x ≤ L modelling the soil medium need be described by only four functions u( x) = [u ( x), v( x), w( x), ϕ x ( x) ]

T

(6.2.96)

including their derivatives by x , when necessary. The first three functions mean the displacement components in the x, y and z direction, i.e. axial (u ) and transversal (v, w). The fourth function describes the line torsion, which could also be significant. According to the general concept of Section 6.2.1.2.4, equations (6.2.60) to (6.2.69), S* (n) the generalized soil medium reactions C1S *u, CS2 *u′, C3S *u′′, K , Cn+ can be defined, 1u denoting the derivatives by x briefly with prime, e.g. u′( x) = [u ′( x), v′( x), w′( x), ϕ x′ ( x) ]

T

 du ( x) dv( x) dw( x) dϕ x ( x)  = , , , dx dx dx   dx

T

(6.2.97)

The asterisk beside the C S * constants serves to differentiate between the 2D and 1D model constants. A basic case arises when all constant matrices are diagonal and the following set of reaction components can be defined:

394

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL T

r1

= C1S *u

= C1Sx*u , C1Sy*v, C1Sz* w, C1Sϕ*x ϕ x 

r2

= C2S *u′

= C2Sx*u ′, C2Sy*v′, C2Sz* w′, C2Sϕ*x ϕ x′ 

M

T

(6.2.98)

rn +1 = CnS +*1u ( n ) = CnS+*1, xu ( n ) , CnS+*1, y v ( n ) , CnS+*1, z w( n ) , CnS+*1,ϕ x ϕ x( n ) 

T

The denotation of the sole variable x is omitted. The virtual work Π v ,i done by the generalized reactions (6.2.98) on an arbitrary virtual displacement uv is equal to the following sum of products: L

Π v ,i = ∫ ( r1u v + r2u′v + K + rn +1u (vn ) ) dx 0

L

= ∫ ( uT C1S *u v + u′T C2S *u′v + K + u ( n )T CnS +*1u (vn ) ) dx =

(6.2.99)

0

L

(

= ∫ uC1*x uv + vC1*y vv + wC1*z wv + ϕ x C1*ϕ x ϕ xv + 0

)

+ u ′C2*x uv′ + v′C2* y vv′ + w′C2*z wv′ + K + ϕ x( n )Cn*+1,ϕ x ϕ xv( n ) dx Some problems with conservative external forces and the behaviour of a soil-structure system independent of stress and strain path can be solved using the idea of total potential energy Π . The part Π i , pertaining to the above defined soil reactions can be written in a form similar to equation (6.2.61): L

1 Π i = ∫ ( r1u + r2u′ + K + r2u′ + K + rn +1u ( n ) ) dx = 20 L

=

1 ( uT C1S*u + u′T CS2 *u′ + K + u(n )T CnS+*1u(n) ) dx = 2 ∫0 L

=

(6.2.100)

(

1 C1Sx*u 2 + C1Sy*v 2 + C1Sz* w2 + C1Sϕ*x ϕ 2 + ∫ 20

)

+ C2Sx*u ′2 + C2Sy*v′2 + C2Sz* w′2 + K + CnS+*1,ϕ x ϕ x( n ) dx 2

The influence of individual constants C S * on the soil-structure interaction is different and generally it decreases with the increasing order of derivatives. A technical analysis (see Chapter 6 in [8]) based on the great series of practical foundation calculations leads to the following result: The first seven constants C S * , explicitly written in the formulae (6.2.99) and (6.2.100), may be sufficient in everyday design practice to express all significant subsoil or soil medium properties which influence the structure behaviour. They can be written in a compact matrix form:

395

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL C1S * = C1Sx* , C1Sy* , C1Sz* , C1Sϕ*x  C2S * = C2Sx* , C2Sy* , C2Sz* 

(6.2.101)

CS * = C1S * , C2S *  = C1Sx* , C1Sy* , C1Sz* , C1Sϕ*x , C2Sx* , C2Sy* , C2Sz*  All constants can be given a distinct technical meaning by the following relations: r1x = C1Sx*u

r1 y = C1Sy*v

r1z = C1Sz* w

r1ϕ = C1Sϕ*x ϕ x

(6.2.102)

r2 x = C2Sx*u ′ r2 y = C2Sy*v′

r2 z = C2Sz*w′

r1x [kNm −1 ] denotes the reaction against axial displacement, i.e. a special form of friction, e.g. the resultant of the side shear stresses of the soil adjacent to the axially loaded pile, r2 x [kNm −1 ] denotes the reaction against axial strains ε x = du dx , which may be neglected in almost every case. Further on r1y and r1z [kNm −1 ] denote the Winklerian parts of transverse reactions against the deflections v and w (Section 6.2.2.1), r2 y , and r2 z [kNm −1 ] being Pasternak's parts of them, proportional to the derivatives v′ = dv dx and w′ = dw dx , i.e. slopes of the spatial deflection components. Finally r1ϕ [kNm m = kN] denotes the torsional reaction against rotation on the line axis x , which is significant, for example, for foundation grid elements with a substantial cross-section width. On the basis of this explanation the following practical denotation of C S * – constants can be used: Friction constants: C1Sx* [kNm −2 ], C2Sx* [kNm −1 ] Winklerian constants: C1Sy* , C1Sz* [kNm −2 ] Pasternakian constants: C2Sy* , C2Sz* [kNm −2 ] Torsional constant: C1Sϕ*x [kN]

6.2.3 Three-dimensional Efficient Subsoil Model as an Improvement on the Two-dimensional Model

6.2.3.1 Main Idea of the Improvement of the Two-dimensional Efficient Subsoil Model The modelling of large or infinite domains in geomechanics can be done with various 396

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL degrees of precision, corresponding to various numbers of functions describing the state of the soil or rock medium. In the numerical analysis, as a rule, the more precise models lead to a greater number of unknowns in equation sets. In the present section we will analyse the reduction of the solution domain only in the vertical z -direction see section 6.2.1.2.2, where basic assumptions and relations of the two-dimensional efficient subsoil surface model were introduced and the appropriate physical constants of the C S -type were defined. In section 5.3 in [8] the direct investigation of these constants based on some exact solutions will be presented. The dimensional analysis shows a great influence of the investigation size on the constant values even in homogeneous cases, and in the case of a generally layered subsoil the direct measuring in situ on the foundations of the same size is the most reliable (but very expensive) way of investigating C S -constants. Therefore it may be useful to know some of the various relations between the laboratory and the in situ tests. Nevertheless the uncertainty arising from the single „decrease function“ f ( z ) in assumption (6.2.14) may be unacceptable in some of the more demanding design cases. Therefore a more flexible model with more parameters of the decreasing function allowing its variability with ( x, y ) will be presented as a second form, improving on the first form defined in section 6.2.1.

6.2.3.2 Basic Geometrical and Physical Relations The second form of the model, i.e. the generalized efficient subsoil model, includes the first one as a special case because the last soil layer can be supported by an efficient twodimensional model expressing the influence of soil mass below that layer. The generalization relates to the assumption (6.2.14), where the course of the vertical displacement components, i.e. settlements, w , is simplified to: w( x, y, z ) = w( x, y ) f ( z )

(6.2.103)

This assumption is rather bold and needs some refinement if we wish to be more precise. On the other hand, neglect of the horizontal displacement components u , v in the total potential energy expression is physically justified in any case. In both model forms, formulae (6.2.24) to (6.2.29) will hold, defining the soil mass potential energy Π i , shear deformation components γ xz , γ yz , stress and strain components σ, ε and the matrix C p of the pseudoelastic constants.

397


Fig. 6. a) 3D model of layered subsoil. b), c) The real and approximate settlement decrease. d) Three displacement components in a soil mass point. c) So-called monoparametric model.

The settlement w( x, y, z ) is generally the function of all three variable coordinates ( x, y, z ) . In the following let us define the course w( x, y, z ) by (n + 1) functions of only two variables ( x, y ) denoted wi +1 ( x, y, H i ) , i = 0, 1, 2, K , n These functions describe the settlement of the horizontal planes z = H i , e.g. z = 0, z = H1 , z = H 2 and z = H 3 (Fig. 6a). Generally the depth H i can be a function of ( x, y ) and H i ( x, y ) denotes a set of n + 1 given surfaces (horizons) which form the inclined and curved boundaries of layers (see also Fig. 8). Any geological description of the subsoil can be included in the defined model geometry. Also a homogeneous subsoil can be divided into layers. A simple example of an in-plane problem depending only on ( x, z ) is presented in Fig. 7.

398


Fig. 7. a) Settlement decrease or increase with the depth

z . b) Soil shear stresses which are ignored in Winklerian models.

The settlements w( z ) vary with depth z as follows: On the vertical straight line we define the nodal points ( x1 , y1 , 0), ( x1 , y1 , H1 ), K , ( x1 , y1 , H n ) and their settlements w1 , w2 , K , wn +1 . The first denotes the surface settlement (Fig. 6b, n = 3 ). The value w1 may not be identified with w0 (Fig. 4). The surface z = 0 of the 2D model can be defined separately, e.g. in the depth z = H n as it will be explained later. The course of the function w( z ) will be interpolated by a linear formula in the local coordinates zi of any layer (Fig. 6c): 399

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL i = 1, 2, K , n

wi ( zi ) = wi fi ( zi ) + wi +1 f i +1 ( zi )

(6.2.104)

where the interpolating functions f i ( zi ) =

hi − zi hi

f i +1 ( zi ) =

zi hi

(6.2.105)

fulfill the general conditions f i (0) = 1

f i (hi ) = 0

f i +1 (0) = 0

f i +1 (hi ) = 1

(6.2.106)

Below the last layer, at the depth z = H n ( zn = hn ) the efficient surface model defined in Section 6.2.1 will be introduced. Its physical constants of the C S -type express the stiffness of the further subsoil. In the case of a rigid subsoil (rock) the last settlements wn +1 are equal to zero. It is advantageous to calculate only with (n − 1) layers and to express the influence of the n -th layer by the appropriate C S -constants of the efficient surface model. In general the subsoil mass points ( x, y, z ) are also displaced in a horizontal direction (Fig. 6d) but the influence of displacement components (u , v) in the ( x, y ) -direction on the total potential energy can be disregarded (Section 6.2.1 ). The same holds for the products σ xε x , σ yε y , τ xy γ xy in the energy expression. There are two limiting cases following the boundary conditions of the subsoil mass: a) high values of σ x , σ y , τ xy and low ones of ε x , ε y , γ xy , b) small stresses σ x , σ y , τ xy and large strains ε x , ε y , γ xy . In both cases the products σ xε x , σ yε y , τ xy γ xy are small compared with the values of σ z ε z , τ xzγ xz , τ yz γ yz , which can be interpolated in the other cases. Independently of the solution method, the settlement w( x, y, z ) of the second model form will be defined by (n + 1) functions wi ( x, y ) , i = 1, 2, K , n + 1 , wi ( x, y ) being the surface settlement ( z = 0) and wi the settlements of the horizons z = H i . The depth H i need not be uniform and can vary with ( x, y ) leading to „curved horizons“, (Fig. 8), as follows from the geological investigation. Between two adjacent horizons i − 1, i , i.e. from the depth H i −1 to the depth H i the physical properties of the i -th soil layer are defined by the four soil moduli of the matrix (6.2.28) pertaining to the middle horizon z = H i −1 + 1 2 hi and a constant k (−1 ≤ k ≤ 1) governing the moduli course along the depth z : Ei =  Ez 0 , Gxz 0 , G yz 0 , Gxy , z 0 , k  i

(6.2.107)

400


Fig. 8. a) Division of subsoil surface into finite elements which form the upper base of the 3D-elements. b) Plate finite element on the three layers. c) Unit cube. d) A brick element of a layer. e) Soil moduli can vary linearly with the depth in one element.

Of course, the five constants assembled in a row matrix Ei can vary with x, y in the case of an unhomogeneous subsoil, where the notation Ei ( x, y ) may be more distinct. Thus the denomination „constant„ is really attached only to one mass point ( x, y, z ) and to one stress and strain path or also time in more complicated soil models. The meaning of the 5th constant k is demonstrated in Fig. 8. When k = 0 , the values of all four soil moduli do not change with depth in the i -th layer. When k ≠ 0 , then their course along the z -coordinate in the i -th layer is defined by the linear formulae:

401

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL Ez ( z )

= E0 (1 + kζ )

−1 ≤ k ≤ 1

Gxz ( z ) = Gxz 0 (1 + kζ ) G yz ( z ) = Gyz 0 (1 + kζ )

(6.2.108)

Gxy , z ( z ) = Gxy , z 0 (1 + kζ ) ζ =2

zi −1 hi

The limit case k = +1 leads to zero values of all soil moduli in the horizon zi = 0, ζ = −1 , which can be used in the first layer when modelling the so-called Gibson soil as a special case of a halfspace. All cases where k > 0 model the increasing soil stiffness in a layer interval; the cases where k < 0 are less interesting. Note. The division of the subsoil into n layers can follow the real geological layers according to their physical constants but any layer can be further divided into two or more thinner layers. Also, a homogeneous subsoil can and must be divided into layers when more precise results are desired. The division should be finer near the subsoil surface where large settlement gradients are expected. In design practice the number of layers is limited by the capacity of the program and computer. The solution by the finite element method introduces at each nodal point of the model (Fig. 6e) only one deformation parameter (settlement w ), which can be denoted by the finite element technique notation ∆ j with the global index j . The number N of all deformation parameters ∆ j assembled in the vector ∆ depends on the number m of the surface nodal points ( xi , yi ) and on the number n of layers: N = m(n + 1) In the numerical analysis only the surface division into elements will be performed. All deformation parameters pertaining to a vertical line in a surface node ( x1 , y1 ) will be attached to that node. In the problem of soil-structure interaction the further deformation parameters of the structure bottom will be added to the previous soil deformation parameters in the same surface node ( x1 , y1 ) . The parameters with the same physical meaning will be identified (in our case the displacement components). An example, a foundation plate of the Mindlin type, is presented in Fig. 8 and will be explained later, in section 6.3.2.6.

6.2.3.3 Some Special Cases The last n -th soil layer can be supported by the first (surface) model form as explained at the beginning of the previous section. Thus the case n = 0 (without any layer) is identical to the first model form which enables one to set up common programs for both model forms. In the actual design, the cases n = 3 with the first model form below the 3rd

402

6.2 ENERGY DEFINITION AND GENERAL THEORY OF THE EFFICIENT SUBSOIL MODEL layer were mostly applied. The layers, e.g. of the thickness h1 = 1 m , h2 = 2 m , h3 = 4 m , guarantee the proper modelling of the settlement decreasing or in some points ( x, y ) even increasing with z (Fig. 7) and give us a good expression of the settlement derivatives on which the structure forces are dependent. The 2D model form and its C S -constants must mainly express the global settlement of structure due to the deformation of the soil mass below the last layer. For this purpose the C1Sz constant is the most significant, i.e. the abovementioned influence on the absolute values can be obtained approximately by a pure Winklerian model below the last soil layer. When the depth H n of the layered model is small compared with the so-called effective or limit depth H (Section 5.4.1 in [8]), then the introduction of shear constants of the type C2S below the last layer can considerably improve the results and lead to smaller global settlement values which agree with the in situ measuring. The effectiveness of the subsoil model in its second form can be demonstrated in Fig. 9, where some special cases are presented: Fig. 9a: A homogeneous or unhomogeneous Winkler model with a sole physical constant or function C1Sz = C1S Fig. 9b: An isotropic Pasternak model with two physical constants (or functions) C1S [MPa/m] and C2S [MN/m] . It is the simplest model expressing the shear stiffness of the subsoil, according to formula (6.2.49). Fig. 9c: The unhomogeneous orthotropic efficient surface model of the Pasternak type with three physical constants or functions C1S , C2Sx and C2Sy expressing the different shear stiffness in the x - and y -directions (orthotropy axes). Fig. 9d, f: A pseudoelastic generally layered halfspace with orthotropic shear stiffness. In any layer i the physical constants (Young and shear moduli) can vary linearly with depth according to the formulae (6.2.108) and Fig. 8. The n -th layer can express the whole soil mass below the (n − 1) -th layer which is substantially deformed and influences the subsoil surface behaviour (Fig. 9d). The same effect can be expressed by the first model form below the last layer (Fig. 9f).

403


Fig. 9. Some special cases of the subsoil model. a) Winkler. b) Pasternak. c) Efficient 2D subsoil surface model. d) Layered halfspace. e) Layered subsoil on a rock base. f) Layered subsoil on a deformable sub-base. g) Winkler layer on a layered subsoil. h) Unhomogeneous and orthotropic soil mass. i) Simulation of a hole, cavern etc.

The classical Boussinesq halfsapce is included in the model as the special case with n = 1, E = 2(1 + µ )G, k = 0 . Gibson’s soil, which is well known in geomechanics, can be modelled bz lazers with k ≠ 0 beginning with k = 1 in the first layer. Fig. 9e: A pseudoelastic layered subsoil of finite depth resting on rock. The difference between the previous infinite halfspace and the finite layer is expressed by the values wn +1 = 0 , of the settlements, i.e. giving the appropriate input parameters values of zero. Fig. 9g: A combined subsoil, where the first layer models Winkler's behaviour

404

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL

(E

1

= C1Sz H1 , Gxz1 = Gyz1 = 0 ) of a cohesionless soil mass resting on a generally layered

orthotropic halfspace, with the orthotropy axes x, y . Fig. 9h: The boundaries of the layers (horizons) can be inclined or curved, which can be expressed in the geometrical input data. The physical constants can differ for each layer element. Fig. 9i: A small cavern can be modelled by the appropriate geometrical division of the subsoil and zero physical constants E = G = 0 in the cavern elements.

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL 6.3.1 Introductory Comment The efficient subsoil model derived in Chapter 6.2 is designed for the numerical solution of the soil-structure interaction from the point of view of the designer of an ordinary structure. The main purpose of the model is the structural analysis including the influence of any subsoil or soil medium without expensive three-dimensional modelling. The relevant programs and their input data must not be too complicated. These demands do not match the physical complexity of the soil medium and from the natural science point of view can never be fulfilled in view of the continuing rapid progress in geomechanics. The gap between the current state of knowledge and practical design, using in many cases only the old „Winklerian“ ideas, cannot be closed by further theoretical progress, even if it will prove most useful for the future. It seems necessary to include some engineering ideas which are not just based on theoretical analysis but also on professional judgement and instinct. The appropriate modelling must enable the engineer to make a decision in a reasonable time. The physical properties of the subsoil must be expressed in a condensed form, despite their complex nature. The aim of design is not research, but the completion of a structure with limited time and money. These comments are intended to explain the problem orientation of the model presented and forestan the questions of pure geomechanicians. It is only a part of the whole model (structure + foundation + subsoil) and Chapter 6 in [8] contains full information on this modelling in a general case, when the structure is composed of various two- and onedimensional elements. The present chapter deals only with plates on subsoil. It models plates simply lying on subsoil without structure or piles, for example, the concrete slabs of a motorway or airport runway and the foundation plates below ideal flexible structures, excluding their stiffness. These structures represent only a plate load and do not interact with the plate. This case can be generalized to a non-flexible structure, the stiffness of which can be expressed by the appropriate stiffening of some plate elements. This can be done very efficiently when using Mindlin's plate theory with shear stiffness independent of bending 405

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL stiffness. For example, the walls of a building may be represented by element strips whose bending stiffness is substantially greater than their shear stiffness, as follows from the real behaviour of the walls. Kirchhofl's plate theory does not allow such modelling because its bending and shear stiffnesses are interconnected by the geometrical restrictions of Kirchhoff's linked plate model (see Formula (6.3.12) in section 6.3.2.2) and no independent shear stiffness exists. This fact can be best demonstrated by the isotropic Kirchhofl's plate with only two input data: stiffness D and Poisson's ratio µ . Even the plate thickness h is only included in the value D = Eh3 [12(1 − µ 2 )] and the shear modulus G is assumed to be equal to E (2 + 2µ ) . Therefore Mindlin's plate theory will be used in subsequent plate analysis, alongside the classical Kirchhoff theory. Even the influence of piles can be approximately calculated by the plate model, introducing the elastic bonds into the supported plate points. In the plate examples, the properties of the subsoil model can be best explained and tested by comparing its results with exact or approximate results from other models and in situ investigation.

6.3.2 Variational Problem of the Plates on Efficient Subsoil Model

6.3.2.1 Total Virtual Work of the Structure-Soil System The whole system (structure with its foundation and subsoil) must at any instant be in an equilibrium state. The influence of static loading without dynamic effect can be analysed, ignoring the inertia forces, by the virtual work principle in its original static form: the total virtual work of any equilibrium force system performed on any virtual displacement must be equal to zero. The force system includes all internal and external forces, i.e. stresses and loads of the structure, foundation and subsoil. The virtual displacement can be an arbitrary system of displacement and strain components {uv , ε v } compatible with all geometric internal and external bonds of the whole system. We will use the following notation: Π v total virtual work, Π vi virtual work of internal forces (stresses), Π ve virtual work of external forces (loads), b, f , s indices pertaining to the structure (e.g. a building), foundation and subsoil, p index pertaining to the plate, modelling the structure with its foundation plate or just the plate itself, e.g. a pavement slab. The virtual work principle can be written in the following form: 406

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL Π v = Π vi + Π ve = ( Π vib + Π vif + Π vis ) + ( Π veb + Π vef + Π ves ) = 0

(6.3.1)

and in the case of a mere plate on subsoil and the load acting only on the plate: Π v = Π vip + Π vis + Π vep = 0

(6.3.2)

Between the plate and the subsoil surface a friction can arise, described by friction stress components τ zx0 , τ zy0 . It can be modelled by special interface models or elements (C.S. Desai et al. 1984, 1985; G. Beer 1985; E. L. Wilson 1977 etc.). Then the virtual work done by their internal forces must be added to the total virtual work. We shall simplify the following explanation and the term Π vis will contain friction effects represented by a special case governed by the physical laws (6.2.38)-(6.2.40). Thus, the friction stresses τ 0 will be incorporated into the subsoil stresses σ s and the virtual work Π vis will be the sum of the type (6.2.60) (Section 6.2.1.2.4): ⌠⌠  ∂w  ∂w ∂w  ∂wv  S ∂w ∂w  Π vis =   wv C1Sz w + v  C2Sx + C2Sxy + C2Sxy +  C2 y + ∂x  ∂x ∂y  ∂y  ∂y ∂x  ⌡⌡  Ω

(6.3.3)

+ uv ( C1Sx u + C1Sxy v ) + vv ( C1Sy v + C1Sxy u ) + w′′v T C3S w′′ d Ω

Denotation: u, v, w actual displacement components of the subsoil surface points ( x, y ) , uv , vv , wv virtual displacement components of the same points ( x, y ) , C1Sz , C2Sx , C2Sy , C2Sxy physical constants of subsoil defined in Section 6.2.1.2.2 by the formulae (6.2.32), (6.2.33), C1Sx , C1Sy , C1Sxy friction constants defined in Section 6.2.1.2.3 by the formulae (6.2.38), C3S matrix of constants (6.2.43), Section 6.2.1.2.3, expressing the influence of horizontal soil stress components on the solution, w′′v , w′′ virtual and actual curvatures assembled in a vector of the from (6.2.41). The general formula (6.3.3) pertaining to the full constant matrix CS (6.2.45) with 13 terms can of course be simplified in special cases. In the fundamental case (6.2.49) with only two constants C1S , C2S , i.e. an isotropic subsoil without friction and horizontal soil stress effect, the formula (6.3.3) reads as follows: ⌠⌠  ∂w ∂w ∂wv S ∂w  Π vis =   wvC1S w + v C2S + C2  dΩ ∂x ∂x ∂y ∂y  ⌡⌡ 

(6.3.4)

Ω

Concerning the integration region Ω , we return to the Fig. 4b (section 6.2.1.1) where Ω1 , and Γ1 denote the region and boundary of the contacting surface below the foundation 407

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL plate, and Ω and Γ denote the region and boundary of subsoil surface taken into account in the energetic sense, i.e. with a substantial virtual work Π vis . Outside the region Ω this work will be neglected. A more practical but bolder assumption can be used concerning the settlement w and its derivatives outside the region Ω ; they must be very small and can therefore be ignored. This assumption can be tested by in situ investigation using only a less expensive geodetic surface measuring before, during and after the construction and loading of the structure. The distance d n between the boundaries Γ1 and Γ depends on the accuracy required. For the decision on the d n -value the characteristic length sn of the subsoil surface model is significant. It will be derived in Chapter 3 in [8] and used in Chapter 4 in [8] in the form sn = C2Sn C1Sz [m]

(6.3.5)

The suffix n denotes the direction of the distance d n , i.e. the external normal to the boundary Γ1 , whose shape can be completely arbitrary. Numerical tests and in situ measuring with various shapes of Γ1 lead to a relation intended for practical design: d n = 4 sn to 5sn [m]

(6.3.6)

Pure Winklerian subsoils, represented by very loose and dry sands, can be analysed identifying the boundaries Γ ≡ Γ1 , i.e. with d n = 0 , which corresponds to the zero values of the C2Sn constant and sn length. The other subsoil below the foundation plates of the size 5 × 5 m to 15 × 30 m can be represented by an efficient model of the characteristic length interval 0 ≤ sn ≤ 1 m i.e. 0 ≤ d n ≤ 5 m. The values sn > 1 m, d n > 5 m can pertain only to larger plates on soils with a great internal friction and cohesion or high values of shear moduli Gnz – see Chapter 5 in [8]. In any case a greater d n can be supposed, leading only to a greater equation set in numerical solution the results of which tend to zero near the boundary Γ . This is helpful for an economical specification of the domain extent. The above-mentioned selection of the Ω size holds only in the case of a generally irregular shape of the foundation plate, when no assumptions about the real character of settlement decreasing (or even slightly increasing near the boundary Γ1 ) outside the domain Ω1 , can be made. It also holds when more foundations forming an irregular system of more domains Ω1 , have to be analysed regarding their mutual interaction or when the loads act directly on the subsoil surface beside the foundation plates. Generally it can be used in any case, but in some of the regular cases which are common in everyday design practice, it leads to superfluous calculation. Therefore a reduction of the Ω -domain to the mere Ω1 -domain introducing special boundary conditions which express the influence of the ( Ω − Ω1 ) domain on the solution will be presented in Chapter 4 in [8]. The reduction holds also in the limit case of an infinite Ω1 domain. In the present chapter the Ω -domain is general and independent of the Ω -domain. The other two terms Π vip and Π vep of the total virtual work (6.3.2) pertain to the plate and depend on the plate model (Kirchhoff, Mindlin); they can be found in textbooks and 408

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL references on plate theories and do not need further explanation here. They are fully independent of the above derived term Π vis pertaining to the soil mass energy and can simply be added in formula (6.3.2) following the energy addition principle. This fact is the main advantage of any method based on energy principles and leads to the simple algorithms and programming using the finite element technique (see Chapter 6 in [8]). For practical purposes the stiffness matrix of an element can be divided into two independent matrices: the element stiffness matrix and the subsoil stiffness matrix. In the present case, i.e. the virtual work principle (6.3.2), and in nonlinear analysis in general, this is a question of tangent stiffness matrices. The procedure for setting them up is in principle the same as in the linear case: a time and stress-strain path independent problem with monotonous load increase which can be solved using secant stiffness matrices and a procedure based on Lagrange's variational principle. The main ideas and algorithms will be presented in the next section.

6.3.2.2 Potential Energy of the Plate-Soil System We set out directly from the formula (6.3.2) which holds for the virtual work of a plate on an efficient subsoil surface model in its two-dimensional form (Section 6.2.1.2). Supposing the existence of the potential energy Π is of internal subsoil stresses, the value of Π is is a half of the virtual work value Π vis ; introducing the actual displacement components u , v, w instead of virtual uv , vv , wv into the formula (6.3.3) we get:  ∂w  1 ⌠⌠  ∂w ∂w  ∂w  Π is =  C1Sz w2 + C2Sx  + C2Sy  + 2C2Sxy +   2   ∂x  ∂y  ∂x ∂y   ⌡⌡ 2

2

Ω

(6.3.7)

+ C1Sx u 2 + C1Sy v 2 + 2C1Sxy uv + w′T C3 w′′ d Ω There are many special cases, the simplest of which corresponds to formula (6.3.4) pertaining to the isotropic subsoil without friction and horizontal soil stress influence: 2 2   ∂w    1 ⌠⌠  S 2 S  ∂w  Π is =  C1z w + C2    dΩ  + 2   ∂x   ∂y       ⌡⌡  

(6.3.8)

Ω

Concerning the last term in formula (6.3.7), which expresses the horizontal soil stress influence in the form of a product w′′T C3S w′′ , where w ′′ is the curvature vector (6.2.41) and C3S the modelling bending stiffness matrix, it can simply be added to the energy of plate internal forces. In the case of Kirchhoff’s bending theory a simplification is possible: the plate bending stiffness matrix Cs can be replaced by the sum Cs + C3S . This sum holds only when calculating the plate settlements and deformation. The plate internal forces are calculated using the proper stiffness matrix Cs . In Mindlin’s plate theory, where no curvature vector w ′′ 409

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL is introduced such a simplification is not possible. The friction terms with the horizontal displacement u, v in formula (6.3.7) are used directly in any case where these components are also defined in the structure, i.e. in a general analysis of space structures on an arbitrary subsoil or in a soil medium (Chapter 6 in [8]). In the special analysis of foundation plates loaded only in the vertical direction z , no horizontal components u, v of the plate middle plane z = 0 are defined and the settlement (bending, deflection) function w( x, y ) describes alone the whole deformation of Kirchhoff’s plate. In the case of a Mindlin plate, where two further independent functions ϕ x ( x, y ) and ϕ y ( x, y ) are defined, describing the material normal rotation components, three functions w, ϕ x and ϕ y are introduced but again no horizontal component u, v . Nevertheless, numerical tests and laboratory as well as in situ measurements show the friction effect also in the above mentioned „pure plate problems“, when the plate thickness h cannot be neglected compared to the plate plan size, i.e. when the plate is not thin. A plate can be of constant thickness h , as is the rule in common concrete plate foundations. But it can model a more complex case, e.g. a box structure or a ribbed plate. In any case two lengths rx , ry can be defined so that the bended plate positive surface (with the positive z -coordinate, where it contacts the subsoil) is horizontally displaced with components u = − rx

∂w ∂x

v = −ry

∂w ∂y

(6.3.9)

in Kirchhoff’s plate theory and u = rxϕ y

v = − ryϕ x

(6.3.10)

in Mindlin's plate theory, where ϕ x and ϕ y denote the components of material normal (points − h 2 ≤ z ≤ h 2 ) rotation, i.e. the indices x, y pertain to the rotation axes x, y . The rotation is positive in the clockwise sense regarded in the positive axis direction. The derivatives of the deflection w are positive in the common mathematical convention, i.e. when the function f ( x + dx, y ) > f ( x, y ) or f ( x, y + dy ) > f ( x, y )

(6.3.11)

This fact leads to a sign „–“ in the geometric relations ϕx =

∂w ∂y

ϕy =

∂w ∂x

(6.3.12)

holding in Kirchhofl's plate theory, where no independent rotations exist. It may be noted that in the simplest case of constant plate thickness h and symmetry to the plane z = 0 the lengths rx = ry = h 2 . Introducing the formulae (6.3.9) and (6.3.10) into (6.3.7) and omitting the above explained last term the potential energy of internal subsoil stresses can be written in the following form: Subsoil of Kirchhoff’s foundation plate:

410

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL 2 2  1 ⌠⌠  S 2 S  ∂w  S  ∂w  S ∂w ∂w Π is =  C1z w + C2 x  + C + C 2  dΩ +  2y  2 xy  ∂x  ∂y  ∂x ∂y  2      ⌡⌡ Ω

2 2 1 ⌠⌠  S 2  ∂w  ∂w ∂w  S 2  ∂w  S C r 2 C r r +  C1x rx  + +  d Ω1 1y y  1 xy x y   2  x y x y ∂ ∂ ∂ ∂       ⌡⌡ 

(6.3.13)

Ω1

Subsoil of Mindlin’s foundation plate: 2 2 ∂w ∂w  1 ⌠⌠  S 2 S  ∂w  S  ∂w  Π is =  C1z w + C2 x  + C2 y  + 2C2Sxy  dΩ +   x y x y 2  ∂ ∂ ∂ ∂       ⌡⌡  Ω

+

(6.3.14)

1 C1Sx rx2ϕ y2 + C1Sy ry2ϕ x2 − 2C1Sxy rx ryϕ xϕ y  d Ω1 2 ∫∫ Ω1

The potential energy Π ip of plate internal forces has been derived in many textbooks and references dealing with variational methods. Therefore, we present only the formulae which are necessary for the explanation which follows. Generally it holds that: Π ip =

1 εTp C p ε p d Ω1 ∫∫ 2 Ω1

(6.3.15)

where ε p is the „plate strain” vector and C p the plate stiffness matrix, defined by the relation σ p = C pε p

(6.3.16)

σ p being the plate internal forces virtually connected with the plate strains ε p . Kirchhoff’s plate theory, one unknown function w( x, y ) :  ∂2w ∂2w ∂2w  ε p = w′′ =  2 , 2 , 2 ∂x∂y   ∂x ∂y σ p = M p =  M x , M y , M xy   C11 C12 C p = C21 C22 C31 C32

T

(6.3.17)

T

(6.3.18)

C13  C23  C33 

(6.3.19)

Mindlin's plate theory, three unknown functions w( x, y ), ϕ x ( x, y ), ϕ y ( x, y ) :  ∂ϕ  ∂ϕ ∂ϕ ∂ϕ ∂w ∂w εp =  y , − x , y − x , + ϕy , − ϕx  ∂y ∂y ∂x ∂x ∂y  ∂x  σ p = M p =  M x , M y , M xy , Tx , Ty 

T

T

(6.3.20) (6.3.21)

411

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL  C11 C12 C  21 C22 C p = C31 C32  0  0  0 0

C13 C23

0 0

C33 0 0

0 C44 C54

0  0  0   C45  C55 

(6.3.22)

Internal forces: Bending moments and twisting moment: h2

∫σ

Mx =

h2 x

zdz

My =

−h 2

∫σ

y

zdz

(6.3.23)

−h 2

h2

M xy =

∫τ

xy

zdz

(6.3.24)

−h 2

Shear forces: h2

Tx =

∫

h2

τ xz dz

Ty =

−h 2

∫τ

yz

dz

(6.3.25)

−h 2

The formulae (6.3.25) can be directly used only in Mindlin's plate theory with uniform shear stress distribution, leading to the relations Tx = hτ xz

Ty = hτ yz

(6.3.26)

Kirchhoff’s plate theory cannot use Hooke's physical law τ xz = Gxz γ xz , τ yz = G yz γ yz because the shear strains γ xz , γ yz are zero according to Kirchhoff’s normal hypothesis: material normals remain straight and normal to the deflection surface w( x, y ) . Therefore the shear forces Tx , Ty must be computed from the moment equilibrium conditions in the form Tx =

∂M x ∂M xy + ∂x ∂y

Ty =

∂M y ∂y

+

∂M xy ∂x

(6.3.27)

The distribution of the shear stresses τ xz , τ yz can be found on the basis of the wellknown mutuality law τ xz = τ zx , τ yz = τ zy and the horizontal equilibrium condition of an elementary part Fx* ( z )dx or Fy* ( z )dy of the plate. Fx* ( z ) denotes the part of the plate cross section in the plane x = const . in the interval (h 2 ≥ z ∗ ≥ z ) , z ∗ being the variable in this interval; likewise the Fy* ( z ), for the cross section in the plane y = const . By making certain assumptions about the direction of shear stress resultant (Grashof, Zhurawski), the technical formula can be derived τ xz ( z ) =

Tx S x ( z ) bx ( z ) I y

τ yz ( z ) =

Ty S y ( z ) by ( z ) I x

(6.3.28)

412

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL S x ( z ), S y ( z ) are the static moments of the areas Fx* ( z ), Fy* ( z ) , bx ( z ), by ( z ) the section width generally variable with z and I x , I y moments of inertia of the whole sections with the vector indices x, y . In the simplest case of a full plate without ribs the shear stresses τ xz , τ yz are supposed to be distributed by the same law as on the rectangular beam cross section. It is a parabolic law with the maximum in the middle point z = 0 τ xz (0) =

3 Tx 2 h

τ yz (0) =

3 Ty 2 h

(6.3.29)

and zero values τ xz (± h 2), τ yz (± h 2) , which agrees with the mutuality law τ xz = τ zx , τ yz = τ zy because there are no stresses τ zx , τ zy on either of the plate surfaces. Mindlin's plate theory can fulfill this condition replacing the constants τ xz , τ yz in (6.3.25), (6.3.26) by the same parabolic law and formulae (6.3.29). Its results fit better in the case of thick foundation plates. Stiffness constants Cik = Cki i.e. the matrices C p are in any case symmetric. The physical properties of Kirchhoff’s plate are described by six independent constants and Mindlin's plate by 9 constants. Orthotropic cases with the orthotropy axes x, y are signified by the zero values C13 = C23 = 0, C45 = 0 and the number of constants decreases to four (Kirchhoff) or six (Mindlin). Isotropic cases are the most simple ones: the common foundation plate of the thickness h [m] , Young’s modulus of elasticity E [MPa] and Poisson’s ratio ν [1] , is physically defined by two constants C and ν or E and ν (Kirchhoff): C11 = C22 = C

C33 = (1 −ν ) C 2

C12 = ν C

C = Eh3 12 (1 −ν 2 ) 

(6.3.30) (6.3.31)

or by three constants E , ν , h or E , G, h , G = E [2(1 + ν )] (Mindlin), introducing besides the stiffnesses (6.3.29), (6.3.30) the shear stiffness C44 = C55 = Gh

(6.3.32)

The last term in the total potential energy expression represents the potential energy of external loads. According to the previous assumption the loads may act only on the plate, i.e. in the domain Ω1 , and consist of regular continuously distributed loading pz ( x, y ) and singular line loads qz ( s ) , concentrated forces Pzi and moments M jx , M jy (vector index)

acting at the points ( xi , yi ) , i = 1, 2, K , n p and ( x j , y j ) , j = 1, 2, K , nM . Their potential energy pertaining to the starting zero state without loads is given by the sum of products:

413

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL ⌠⌠  ∂w ∂w  Π ep = −  pz w + mx − my  d Ω1 − ∂y ∂x  ⌡⌡  Ω1

⌠ ∂w ∂w  −  qz w + qmx − qmy  d Λ1 − ∂y ∂x  ⌡

(6.3.33)

Λ1

nP nM  ∂w ∂w  −∑ Pw x y − − M jy , ∑  M jx  i ( i i) ∂y ∂x  i =1 j =1 

Λ1 denotes the one-dimensional line domain in which the line loads qz ( s ) act. Formula (6.3.33) holds in Kirchhoff’s plate theory with only one unknown deflection function w( x, y ) and allows for more complicated loading cases: continuously distributed moment loads mx , m y and moment line loads qmx , qmy . In Mindlin's plate theory with three independent functions w, ϕ x , ϕ y of variables ( x, y ) instead of formula (6.3.33) the following expression holds: Π ep = − ∫∫ ( pz w + mxϕ x + myϕ y ) d Ω1 − Ω1

− ∫ ( qz w + qmxϕ x + qmyϕ y ) d Λ1 −

(6.3.34)

Λ1

np

nM

i =1

j =1

−∑ Pw i i ( xi , yi ) − ∑ ( M jxϕ x + M jyϕ y ) Concerning the sign of the last term, which is different in Kirchhoff’s case (6.3.33), the relation (6.3.12) may be taken into account. Thus, the total potential energy of the plate-soil system is the sum of its three parts Π = Π is + Π ip + Π ep

(6.3.35)

defined by formulae (6.3.13), (6.3.15) to (6.3.19), and (6.3.33) in Kirchhoff’s plate theory and (6.3.14) to (6.3.16), (6.3.20) to (6.3.22), and (6.3.34) in Mindlin’s plate theory. Note. The formulae (6.3.33) and (6.3.34) also hold in the exceptional case of direct soil surface loading in the domain Ω − Ω1 , i.e. outside the foundation plate, when replacing the integration domain Ω1 by Ω . But some caution is necessary when using some finite elements in a numerical solution. The settlements w are always defined but the slopes ∂w ∂x , ∂w ∂y need not be directly included in the deformation parameters.

414


6.3.2.3 Potential Energy of the Improved Subsoil Model The basic geometrical and physical relations of the improved subsoil model, i.e. the second form of the efficient subsoil model, which is essentially three-dimensional, were presented in section 6.2.3.2, equations (6.2.104) to (6.2.108). The layered part of this model represents a pseudoelastic soil medium the physical properties of which were described generally in section 6.2.1.2.1, equations (6.2.1) to (6.2.11). When no further restrictions are applied, the potential energy of soil stress is defined by the well-known product formula H

Π is =

n 1 1 T ε σ dV = εTs Cs ε s d Ωdz s s ∫∫∫ ∫∫ ∫ 2 V 2Ω 0

(6.3.36)

The strain tensor ε s (2) is written in the matrix vector form (6.2.6) as well as the stress tensor σ s (8) in the form (6.2.9), with all six components. The symmetrical matrix Ds (9) generally includes 21 physical constants dik = d ki . The integration over the whole soil volume V can be divided into the integration over the soil surface Ω and over the interval 0 ≤ z ≤ H n , H n being the limit depth. In the domain z > H n no substantial energy value is accumulated or dissipated in the pseudoelastic problems (Section 5.4.1 in [8]). The effective improvement of the mere surface model (Section 6.2.3.2) neglects three terms due to the horizontal stress and strain components in the energy expression and uses only the formulae (6.2.24) to (6.2.29), i.e. vectors and matrix of the order 3. The quantities pertaining to the j -th layer in the interval H j −1 ≤ z ≤ H j or 0 ≤ z j ≤ h j in the local coordinate z j with the origin in the depth H j −1 will be denoted by indices s (subsoil) and j (number of a layer). The previous index i at the potential energy signifies internal forces, i.e. stresses. With this denotation the formula (6.3.36) can be rewritten in the form 1 ⌠⌠ n Π is =  ∑ 2  ⌡⌡ j =1 Ω

=

z j =h j

∫

εTsj Csj ε sj dz j d Ω =

z j =0

(6.3.37)

z j =h j

T 1 ∂ T wsj ) Csj ( ∂ T wsj ) dz j d Ω ( ∑ ∫∫ ∫ 2 j =1 Ω z j = 0 n

using the formulae (6.2.26), (6.2.28) and (6.2.104) to (6.2.108): ∂ = [ ∂ ∂z , ∂ ∂x , ∂ ∂y ] wsj ( x, y, z ) =  Ezj  Csj =  0 0 

w j ( x, y ) ( h j − z j )

0 Gxzj Gxy , zj

hj 0   Gxy , zj  G yzj 

(6.3.38) +

w j +1 ( x, y ) z j hj

(6.3.39)

(6.3.40)

415

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL Ezj ( z j ) = E0 zj (1 + k jζ j )

ζj =2

zj hj

−1

(6.3.41)

Gxzj ( z j ) = G0 xzj (1 + k jζ j ) G yzj ( z j ) = G0 yzj (1 + k jζ j )

(6.3.42)

Gxy , zj ( z j ) = G0 xy , zj (1 + k jζ j ) The other two terms Π ip , Π ep in formula (6.2.14) expressing the total potential energy of the whole system (plate and subsoil) pertain to the plate and remain unchanged, also in the present case with an improved subsoil model. Only the first term Π is is now replaced by the term following formula (6.3.37). For details see section 6.2.3.2 and for some special cases, Section 6.2.3.3.

6.3.2.4 Variational Principles of Structure-Soil Interaction The structure-soil interaction pertains to mechanical problems and its solution follows the principles of mechanics. Generally speaking, the basic principle must involve all the actual processes in a three-phase medium and can be written in a differential on integral (variational) form describing the motion in a time instant t or in a time interval t1 ≤ t ≤ t2 . In the time independent static problem the differential form describes the behaviour of a mass point and its neighbourhood. The integral form expresses the state of the whole body Ω , mostly using a variational principle. The last form has a numerical advantage based on the adding theorem of bounded integrals:

∫

Ω

N

f ( x)dV = ∑ ∫ f ( x)dV i =1 Ωi

N

Ω = ∑ Ωi

(6.3.43)

i =1

Any bounded integral of the continuous function f ( x) defined for all points x ∈ Ω in the domain Ω is equal to the sum of integrals in the subdomains Ωi , i = 1, 2 ,K , N . The continuity demand is very strong, the adding theorem holds under weaker conditions depending on integral definition. We will use the Lebesgue concept based on measure theory and the theorem (6.3.43) can be accepted in the following text. We will only use the integral form of mechanical principles. A survey of them may be found in references (J. Henrych 1985; V. Kolář et al. 1971, 1972, 1975; C. Lanczos 1970; K. Washizu 1975 etc.). We will omit the general principles and quote only some special cases dealing with the pseudoelastic concept (C. S. Desai and J. F. Abel 1972; C. S. Desai 1977; J. Feda 1978; G. Gatti and I. Jori 1981; G. Gioda 1980; R. Nova 1981, 1982; H. G. Poulos and E. H. Davies 1973; N. E. Simons and J. S. N. Rodriguez 1975; I. M. Smith 1981; L. J. Wardle and R. A. Fraser 1975; L. J. Wardle 1980 etc.). Concerning the dissipative part of energy, the unified approach to soil mechanics problems, including plasticity and viscoplasticity, leading to integral principles similar to the virtual work principle, will be assumed. The present book cannot include the generalization of the efficient subsoil model to these more complicated 416

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL problems. For details of such a generalization process, we refer the reader to the well-known ideas of Zienkiewicz (0. C. Zienkiewicz, C. Humpheson and R. W. Lewis 1977; O. C. Zienkiewicz 1979) but we will remain within the framework of Chapter 6.2, which is quite adequate for the model explanation. With regard to the models used in common design practice and incorporated into the well-known program packages mentioned in Section 6.3.2.6, three variational principles may be quoted: a) Lagrange’s variational principle of minimum total potential energy, introducing only one free vector field subject to variation, namely the displacement vector u , which generally represents three functions u , v, w of three variables x, y, z . It can be broadened to d’Alembert-Lagrange’s principle, which can be applied to problems of dynamics, replacing &&]T and the force vector P field with the (P − ma) field, a being the acceleration a = [u&&, v&&, w m the mass. We will also use another generalization, on Cosserat’s continuum with six independent components of displacement and rotation matrix vector u = [u, v, w, ϕ x , ϕ y , ϕ z ]T , where P denotes force and moment matrix vector P = [ Px , Py , Pz , M x , M y , M z ]T and &&, ϕ&&x , ϕ&&y , ϕ&&z ]T . The Euler equations pertaining to Lagrange's variational principle are a = [u&&, v&&, w equilibrium conditions; the solution can be denoted as the „deformation method” (the deformation parameters are unknown). b) Castigliano’s variational principle of maximum complementary energy, also introducing only one free field subject to variation. It is the stress tensor σ , written in the usual matrix vector form σ = [σ x , σ y , σ z ,τ yz ,τ zx ,τ xy ]T . In the 2D structures the resultants pertaining to their thickness h can be introduced (Chapter 6 in [8]) and denoted as internal forces S = [S m , S b ]T with plane stress forces S m = [ N x , N y , N xy ]T and bending the shear forces Sb = [ M x , M y , M xy , Tx , Ty ]T defined in planar coordinates ( x P , y P ) . The Euler’s equations pertaining to Castigliano’s variational principle are the continuity (compatibility) conditions; this solution is named the „force method” (the force parameters are unknown). c) Hellinger-Reissner’s variational principle introduces two free fields subject to variation, u and σ , and the solution is known as the so-called „mixed method“. No a priori physical connection between the fields u and σ is assumed. There exist many other variational principles which can be derived from the mechanical principles using Lagrangian multiplicators. The variational equations can also be set up in cases where no appropriate potential (functional, operator) exists by the weighted residual method (Galerkin). The structure-soil interaction can be solved at various levels of modelling and the soil medium behaviour can be described fairly accurately. In any case the main idea of the subsoil surface model, i.e. reducing the 3D model to the 2D one by integration along the z -coordinate and limiting the solution domain Ω to the foundation contact (interface) area Ω1 by integration in the Ω − Ω1 domain, can be used. The present book cannot include all these possible generalizations, which must be left to the reader.

417


6.3.2.5 Advantages of Lagrange's Variational Principle and Principle of Total Virtual Work Despite the great number of finite elements elaborated in the last thirty years, the most reliable are still those based on Lagrange's variational principle of total otential energy. It represents the most general equilibrium condition and leads to the well-known deformation method with unknown geometric quantities such as displacements and their derivatives, strains, curvatures, distorsions etc. The relevant elements can be referred to briefly as „deformation elements” or „pure Lagrangian elements”. They can be used only in conservative problems independent of the stress-strain path and time effect, when the given load implies a single solution. Only the finite (actual) state is analysed and all physical quantities must relate to the change between the primary unloaded state and the analysed state (secante values). More general cases can be analysed using the virtual work principle. It holds for the equilibrium force systems and virtual displacements which fulfil two basic conditions. During the displacement a) any geometric internal or external bond must be conserved, including compatibility conditions, and b) the force system must not be changed. Condition b) can generally be fulfilled only in the limit case when the virtual displacements tend to zero. The total virtual work (Section 6.3.2.1) of the structure-soil system forces done on any virtual displacement in any stage of a loading or unloading process must be equal to zero. Analogously to the above mentioned deformation method, we can reduce the general demands of the principle and use only a finite set of virtual displacements instead of all possible ones. Thus we obtain only a finite set of equations. But this procedure must be carried out at many points of the load-stress-strain path, starting with the original zero state and using at these points tangent values of physical moduli or the other constants. The main character of the deformation method is preserved and the algorithms need not be changed substantially. Stiffness matrices and load parameter vectors are generalized and involve the appropriate changes in similar form (geometric or initial stress matrices, residual vectors etc. ). The main advantage of the deformation method in its original or generalized form is its design reliability. Any numerical solution based on the finite element technique is only an approximate one. The results of the three above mentioned methods differ from the exact ones, and the designer must judge their quality accord ing to certain features of the output data. Assuming correctly defined elements and problem, the results of the deformation   force  method fulfil  mixed 

 compatibility   equilibrium  no 

   conditions  

418

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL and the quality of these results can be judged by the differences (residuals) in fulfilling the   compatibility  conditions . equilibrium and compatibility  equilibrium

The designer can very easily check the numerical stability of solution by comparing the printed, external nodal forces {f print } with the given values {f given } . The printed values

{f } are computed from the deformation parameters d obtained as the solution of the linear print

equation set Kd = f or a series of such sets in nonlinear cases. In the exact solution the equality with the {f given } is evident. The numerical solution leads to a difference (or residuum)

{df } = {f given − f print }

(6.3.44)

There are many sources for the difference, beginning with common errors in input data. For instance {f given } may not be equal to the {finput } due to numerical errors or changing the physical units. Some error sources are inevitable, e.g. arithmetical operations with a limited precision during the equation set solution. But the main error source in the finite element technique, independent of the other sources, is the nature of the domain Ω division into the elements (subdomains) Ωe and the properties of the base functions, i.e. the sort of elements. For a given program and computer, the results can only be improved when changing the division, e.g. refining the coarse division. The success can be measured by decreasing the difference (6.3.44). Any engineer can determine the value of the max df which does not devaluate the results from the point of view of further design. The vector (6.3.44) represents some unwanted nodal loads and moments which can be compared with the given ones in the internal as well as external nodes (static boundary conditions). In everyday design practice the usual condition may take the form max df < c max f given

(6.3.45)

with a constant c depending on the structure properties, numerical demands etc., mostly in the interval 0.001 < c < 0.020 . Checking the compatibility conditions, including geometrical boundary conditions, is not easy but does not lead to a general practical inequality or another test of the quality of the results. If a geometrical boundary condition is slightly disturbed then a great error in the estimation of the internal stresses or forces can follow. The oldest demonstration of this is the case of plate with clamped. simply supported and free edges. Solution by the force or mixed method cannot guarantee ideal clamping in advance. Therefore even fine division leads to a great error in the boundary bending moments despite relatively good deflection values. Some non-Lagrangian elements, e:g. Herrmann’s triangles, lead to asolution with the same convergence order in deflections as in moments, i.e. bending stresses. This is an advantage over most Lagrangian elements. The solution with them is signified by a rapid convergence in deflections and a smaller one in moments when refining the plate division. But the 419

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL geometrical boundary conditions are fulfilled in advance exactly and the test (6.3.45) is quite sufficient for the security demands. The same holds when using the principle of total virtual work in „Lagrangian form”, i.e. replacing all possible virtual displacements by a set of finite functions, fulfilling all geometric boundary conditions. On the boundaries with these conditions no virtual work is done by boundary forces. This original form of the principle seems to be most useful in engineering problems and underlies what follows.

6.3.3 Implementation of the Soil-Structure Interaction Model Using the Finite Element Technique

6.3.3.1 Dimension and Compatibility of Finite Elements At this stage of the explanation the theoretical description of the 3D → 2D reduction of the plate-soil interaction model is complete and all necessary relations have been prepared for deriving of the pertinent finite elements. The reduction of the 2D-domain to the soilstructure interface will be explained in Chapter 4 in [8]. Due to the complexity of practical demands concerning the plate shape, loads, subsoil properties etc. a general purpose program can be based only on a numerical method. Of course, the most efficient finite element (FE) technique is the best tool, meeting all the needs of the designer. Despite the great number of plate programs the subsoil influence is implemented only in some of them, either in 3D form or in 2D form, and the oldest programs enable the plate-soil interaction solution only by modelling it with concentrated springs or bonds. This idea will be completely omitted below. Regarding the 3D form of subsoil modelling we repeat the conclusions from the Introduction, section III. Three-dimensional modelling is inevitable in all cases when a detailed stress-strain analysis of the soil mass is essential for the design. Besides the analysis of soil body alone (dams, slopes, excavations etc.), structural problems may be encountered with a very exacting structure-soil interaction caused by a very complicated subsoil or soil medium geometry (holes, openings); physical properties (special nonlinear constitutive laws) and the other demands on the calculation arising from the technical nature of the structure and its environment must also be taken into account. These cases make up approximately 1 to 5 per cent of structures dealt with. Using 3D modelling when computing ordinary structures is not only very expensive, but impossible due to the lack of reliable information on the correct setting-up of 3D input data. Such an investigation could be more expensive than a whole design made in the usual way, and the chances of achieving the desired effect, i.e. the reliability of the design, may not be improved simply by 3D modelling (see Chapter 5 in [8]). The 2D modelling of soil-structure interaction implemented in most FE-programs remains Winklerian and introduces only one constant of the C1Sz type. It will be shown that on the basis of the efficient subsoil surface model presented here, these programs can be improved without any difficulty. The main advantage of this improvement (compared with the introduction of 3D models) is that it has the same number of equations N and band width B as the simplest Winklerian case, because only some non-zero terms of the global structure 420

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL stiffness matrix are influenced and no further non-zero term arises. The subsoil or soil medium is regarded as a physical property of the adjacent elements and no further elements are introduced (Fig. 10). Every programmer, engineer and designer knows the difficulties arising from the demands of compatibility between 3D- and 2D-elements, the deformation parameters of which do not coincide in many cases. For instance, the universally used 3D hexahedron bricks possess only displacement components u, v, w as deformation parameters and their deformation is mostly described by the same shape functions as their form (isoparametric elements). The only plate element compatible with them is a quadrilateral with the same number and position of nodes as are defined on one brick surface and the same course of u, v, w - components in this domain. Many existing program packages reffered to above and still in use, neglect this fact and recommend that all the elements involved, i.e. 3D, 2D and 1D ones, be connected in one system. In the case of 1D elements (beams, ribs, stiffeners, piles, frame or grid elements etc.) compatibility of their classical Bernoulli- Navier model with the modern Mindlin plate and shell theory is impossible (see Chapter 6 in [8]). No computer exists and no equations, however numerous, could be set up which would be capable of' correcting theoretical errors of this nature. Thus only a fully compatible system of finite elements should be defined in a universal program package, because users do not have the time or knowledge to find faults in the results, which may be to the detriment of any further design based on them.

421


Fig. 10. Various soil models and the relevant global stiffness matrix K . a) Winkler. b) Boussinesq c) Model with a connecting stiffness matrix of a halfspace, d) Efficient 3D or 2D subsoil model without domain restriction. e) Efficient 3D model using layered soil substructures. f) Efficient 2D model using special boundary bonds. It leads to the same number of unknowns N and band width BM as a), but its results are more precise, close to those of the most expensive cases b) and c).

422


6.3.3.2 Kirchhoff's Plate on the 2D-Efficient Model of the Subsoil The stiffness matrix K e and the load parameter vector fe of a finite element Ωe of a Kirchhoff plate can be derived from the potential energy of internal and external forces Π ip , Π ep eqs. (6.3.15), (6.3.33), where the formulae (6.3.17) to (6.3.19) are introduced. The subsoil can be regarded as a property of the supported element. Thus, the subsoil potential energy Π is (6.3.13) is added according to the general formula (6.3.35) and we obtain the resulting potential energy expression for one finite element e :  ∂2w ∂2w ∂2w  1 ⌠⌠   ∂ 2 w ∂ 2 w ∂ 2 w  D , 2 ,2 Π =    2 , 2 , 2 p   + 2 2    ∂x ∂y x y x y ∂x∂y  ∂ ∂ ∂ ∂   ⌡⌡  T

Ωe

2

 ∂w  S  ∂w  S ∂w ∂w + C w +C  +  + 2C2 xy  + C2 y  ∂x ∂y  ∂x   ∂y  2

S 1z

2

S 2x

(6.3.46)

2

∂w ∂w   ∂w  S 2  ∂w  S +C r   d Ωe −  + 2C1xy rx ry  + C1 y ry  ∂x ∂y   ∂x   ∂y  2

S 2 1x x

− ∫∫ pz wd Ω e Ωe

For the sake of brevity only the first term of (6.3.32) is written because the other terms can be regarded as particular cases of the first term (line and concentrated loads). Because of the general program algorithms (Chapter 6 in [8]), the integration area Ωe denotes in any case the two-dimensional finite element area both in Ω1 and (Ω − Ω1 ) . In the (Ω − Ω1 ) domain the plate stiffness matrix C p is zero and a pure subsoil finite element arises. On the other hand, when putting all C S -constants equal to zero in the Ω1 , domain a pure plate finite element is defined, i.e. without contact with the subsoil. Regarding loads, see the note at the end of Section 6.3.2.2. In principle the loads may act on the whole domain Ω , i.e. not only on the plate but also on the subsoil beside the plate. The element stiffness matrix K e is a square matrix of the type (n, n) and the load parameter vector fe is a column matrix of the type (n,1) . They depend on the number n of element deformation parameters d e = [ d1e , d 2 e ,K , d ne ]

T

(6.3.47)

which is equal to the number of polynomial coefficients ae = [ a1e , a2e ,K , ane ]

T

(6.3.48)

in the assumed course of the element deflection function we ( x, y ) = U( x, y ) a e

(6.3.49)

423

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL U( x, y ) denotes a set of the chosen mononomials of the form U i ( x , y ) = xα i y β i

(6.3.50)

assembled in a row matrix of the type (1, n) : U = [U1 ,U 2 ,K ,U n ]

T

(6.3.51)

All properties of a finite element expressed by its stiffness matrix K e and load parameter vector fe are predetermined by the chosen functions (6.3.51) and the appropriate choice of deformation parameters (6.3.47). The pure Lagrangian elements of the Kirchhoff plate introduce the values of deflection w and its partial derivatives wx , wy , wxy , wxx , wyy , K (index denotation) in element nodes as deformation parameters. Nodes (nodal points) with the coordinates ( x j , y j ), j = 1, 2, K , n

(6.3.52)

may lie in the element vertices or in the other element points, e.g. midpoints of its sides, centre of gravity etc. Two important conditions must be fulfilled: a) Compatibility condition (see Section 6.3.3.1) between two adjacent elements. Due to the relations (6.3.9) and (6.3.12) the continuity of the deflection w( x, y ) is not sufficient. Also the slopes ∂w ∂ne along the common side ( ne denotes its normal) must be continuous, i.e. of the same value in both adjacent elements. This condition can be fulfilled only when the course of ∂w ∂ne -function is uniquely destined by the deformation parameters defined in all side nodes. The equality of these parameters in both elements guarantees the equality of ∂w ∂ne -course. b) Numerical demands (small band width of the global stiffness matrix, see Chapter 6 in [8]) lead to the requirement of only vertex node parameters and, in cases where this is impossible, to the introduction of a minimum number of nonvertex node parameters on the element sides without any non-side parameters. In any case the deformation parameters (6.3.47) are connected with the deflection function (6.3.49) by relations of the type w j = U( x j , y j ) ae wx , j = U x ( x j , y j ) ae wy , j = U y ( x j , y j ) a e

(6.3.53)

wxy , j = U xy ( x j , y j ) ae These relations can be written in a common matrix form de = Seae

(6.3.54)

where S e denotes a square matrix of the type (n, n) . Its terms are constants depending on the nodal coordinates ( x j , y j ) , i.e. on the element shape, and on the vector of the chosen

424

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL mononomials U (6.3.51). S e can be called the „coordinate matrix“ and in any regular case there exists its inverse matrix S e−1 . Therefore, the multiplication of the relation (6.3.54) by S e−1 leads to the explicit expression for the coefficients: ae = S e−1d e

(6.3.55)

The assumed course of the element deflection function (6.3.49) can be expressed in the following form: we = Ua e = US e−1d e = Ved e

(6.3.56)

The parametrical influence functions are assembled in a square matrix of the type (n, n) . Ve = US e−1

(6.3.57)

The functions Ve can be constructed directly without choosing the polynomials (6.3.51), e.g. Ve can be defined by isoparametric shape functions in natural coordinates ξ , n . Then the coefficients ae and the relations (6.3.48) to (6.3.55) are omitted. The ξ , n -coordinate system permits the specification of a point within the element by the dimensionless values ξ , n which never exceed unity. Coordinates of the vertices are always equal to unities, positive or negative. The isoparametric concept will not be used when analysing the Kirchhoff plate because of the difficulties with the continuity of slopes between two adjacent elements on their common side. The general formula for the element stiffness matrix K e and load parameter vector fe can be derived from the energy expression (6.3.46) introducing the general parametric relation (6.3.56) in the form we = Ved e

(6.3.58)

instead of w -function. If the Ve -functions are not defined in x, y -coordinates, the appropriate transformation formulae must be used. A further one, the scalar equality we = weT = (Ved e )T = dTe VeT , can be used, leading to symmetric matrix expressions. The partial derivatives of the influence functions Ve by x or y may be denoted by the suffix x or y . The Kirchhoff plate bending operator in vector form  ∂2 ∂2 ∂2  ∂K =  2 , 2 ,2 ∂x∂y   ∂x ∂y

(6.3.59)

may be used for the short form: H e = G K Ve =  Vxx , Vyy , 2Vxy 

T

(6.3.60)

Thus the general formula (6.3.46) is transformed to the parametric form, where the element index e can, for the sake of brevity, be omitted (with the exception of the area Ωe ). Regarding the note on the page 304 we can write:

425

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL Π= +

1 dT HT C p Hdd Ω e + ∫∫ 2 Ωe 1 ( dT VT C1Sz Vd + dT VxT C2Sx Vxd + dT VyT C2Sy Vyd + 2 ∫∫ Ωe + 2dT VxT C2Sxy Vy d + dT rx2 VxT C1Sx Vx d +

(6.3.61)

+ dT ry2 VyT C1Sy Vy d + 2dT rx ry VxT C1Sxy Vy d ) d Ω e − − ∫∫ p z Vdd Ωe Ωe

Using the finite element notation and defining the plate element stiffness matrix K p = ∫∫ HT C p Hd Ω e

(6.3.62)

Ωe

the subsoil element stiffness matrix K s = ∫∫ ( VT C1Sz V + VxT C2Sx Vx + V Ty C2Sy Vy + VxT C2Sxy Vy + VyT C2Sxy Vx + Ωe

+ r V C Vx + r V C Vy + rx ry V C Vy + rx ry V C Vx ) d Ω e 2 x

T x

S 1x

2 y

T y

S 1y

T x

S 1 xy

T y

(6.3.63)

S 1 xy

and the load parameter vector of an element f = ∫∫ VT pz d Ω e

(6.3.64)

Ωe

the formula (6.3.61) can be written in short as follows: 1 Π = dT Kd − dT f 2

(6.3.65)

The element stiffness matrix K is the sum of the plate term K p and the subsoil influence Ks : K = K p + Ks

(6.3.66)

In the formula (6.3.63) the first term with C1Sz expresses a Winklerian reaction influence, the following three terms with C2Sx , C2Sy , C2Sxy express the subsoil shear stiffness and the last three terms with C1Sx , C1Sy , C1Sxy the linear friction between the plate and subsoil surface. If any of the effects is neglected, the appropriate terms are zeros. As regards the load parameter vector (6.3.64) it may be stated that the loads can act on the plate Ω1 , and/or on the subsoil in the domain Ω − Ω1 . In the latter case, the reduction to the domain Ω1 following Chapter 4 in [8] cannot be performed. The simple formula (6.3.64) is sufficient in the case of any regular load pz ( x, y ) which can be different in each element and 426

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL in the simplest case uniformly distributed in it, leading to formula fe = pze ∫∫ VT d Ω e

(6.3.67)

Ωe

Any singular load can be introduced following the generalization of formula (6.3.64) based on the general energy expression (6.3.33) in Section 6.3.2.2: np

fe = ∫∫ V pz d Ω e + ∫ V qz d Λ1 + ∑ ViT Pi + T

Ωe

T

Λ1

i =1

(6.3.68)

T  ∂V j T   ∂V j  + ∑   M jx −   M jy  j =1  ∂y    ∂x   nM

6.3.3.3 General Remarks on the Finite Element Technique The present book is not intended to offer detailed explanations of the finite element technique and the reader should follow up the references (O. C. Zienkiewicz 1979; V. Kolář et al. 1975 etc.). However some remarks should be presented at this stage, to help towards a full understanding of the efficiency of the derived subsoil model. These remarks are general and are valid not only in the present chapter dealing with plates, but also for other structures on subsoil or any arbitrary systems whose deformation is described by a set of parameters d of the type (6.3.47) and base functions of the type (6.3.51) irrespective of their number, complexity etc. The main aim of this section is to define the global deformation parameters d and the appropriate global stiffness matrix K and global load parameter vector f , and to describe their generation on the basis of the element quantities d e , K e , f e . The type of vectors and matrices may be d e (n,1), K e (n, n), fe (n,1) , n denoting the number of element parameters or so called degree of element deformation freedom. It need not be common for all elements but for the sake of simplicity no element index will be attached to the sign n . The type of global vectors and matrices may be d ( N ,1), K ( N , N ), f ( N ,1) where N denotes the total number of all the deformation parameters of the analysed structure or the degree of its global deformation freedom. N is equal to the number of equations in their set (6.3.85) which will be derived later in the form Kd = f and serves for the solution of values of deformation parameters d pertaining to the load parameter vector f . The potential energy of the internal and external forces of one element e can be written in any case in the form (6.3.69), assuming the pseudoelastic concept of physical behaviour of material: 1 Π e = dTe K e d e − dTe fe 2

(6.3.69)

The value of Π e can vary, i.e. need not be the same in each element Ωe . This fact is

expressed by the suffix e . In the whole body Ω = ∑ Ω e the total potential energy is equal to e

427

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL the sum: Π = ∑ Πe = e

1 (dTe K ed e − dTe fe ) ∑ 2 e

The summation by elements

∑

(6.3.70)

could be performed without further difficulty using

e

the known values de , i.e. after the solution of equation set Kd = f But before the solution, the parameters d e are unknown and the equations must be set up with respect to the equality of deformation parameters d e pertaining to the common nodes of neigbouring elements. Thus we must define the nodal parameters d of the whole system and their connection with the element parameters d e . It is only a topological problem, because the parameter values d e of the type (n,1) are not changed, but only rearranged in another vector d er , of the type ( N , 1). Each element parameter d je is written in a position (column) i of the global parameter d i identical to it, and all other terms of the vector d er are zeros: 1, 2, K ,

K , i, K ,

K, N

d er = 0, 0, K , ∆1e , K , 0, K , ∆ je , K , 0, K , ∆ ne , 0, K , 0  1, K , K, j, K, K, n

T

(6.3.71)

The same extension is performed on the load parameter vector: fer = 0, 0, K , f1e , K , 0, K , f je , K , 0, K , f ne , 0, K , 0 

T

(6.3.72)

and on the stiffness matrix: 1, 2, K 1, 0,  0,  M K er =    M M M  N 0, 1 2 M

2, K 0, K 0, K K k11e , k12 e K K K K kiie , ki ,i +1,e K K K K K knne K 0, K

K N

K N K 0  K 0  K M K 0  K 0 K 0  0, 0 

(6.3.73)

Each term k j1 j2e in the j1 -th row and j2 -th column of element stiffness matrix K e is written in the i1 -th row and i2 -th column of the widened matrix K er . When j1 < j2 it does not necessarily mean that i1 < i2 . The rearranging follows the global numbering of the system deformation parameters 1, 2, K , N , which should be reasonable in the sense of minimum band width B . The programming of operations with the widened vectors and matrices 428

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL d er , fer , K er uses the so called code numbers, i.e. sets of integer values (indices of global parameters) arranged by the natural numbering of element deformation parameters 1, 2, K , n . For the sake of general denotation and definition the following formulae may be applied: We define the localization matrix L e of the type ( N , n) where N and n represent the global and element number of deformation parameters respectively. In the j -th column of L e only one term, in row i , can be different from zero and equal to unity if i and j represent the global and element numbering of deformation parameters respectively: 1 2 K

j

0 0 K 0 0  M   0 M  0 0  0 0 K 0 1  0 0   M M  M  M N  0

1 2 M M Le = i M

K n

0 K 0     M  0 K 0  0   0  M   

(6.3.74)

The number of unities in each L e -matrix is equal to n and the other ( Nn − n) terms are zeros. It is possible to check the validity of the following equation: LTe L e = I

(6.3.75)

where I denotes the identity matrix of the type (n, n) , i.e. I = DIAG[1,1, K , 1] . Using the localization matrix L e , which is of course different for any element of the system, the previous large formulae can be rewritten shortly as follows: d er = Ld e

fer = Lfe

K er = LK e LT

(6.3.76)

Due to the orthogonality expressed by formula (6.3.75) the generalized inverse matrix L is equal to the transposed matrix LT −1

L−1 = LT

(6.3.77)

and the inverse relations to the equations (6.3.76) can be written in the following form: d e = LT d er

fe = LT fer

K e = LT K er L

(6.3.78)

The energy Π e of element - formula (6.3.69) - can be rewritten in broader terms putting in the relations (6.3.78): T T 1 T L d er ) ( LT K er L )( LT d er ) − ( LT d er ) ( LT fer ) = ( 2 1 1 = dTer LLT K er LLT d er − dTer LLT fer = dTer K er d er − dTer fer 2 2

Πe =

(6.3.79)

429

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL It can be shown that the widened deformation parameter vector d er can be replaced by the vector of all system parameters d = [ d1 , d 2 , d 3 ,K , di ,K , d N ]

T

(6.3.80)

without changing the value of Π . In any term the parameters d i lead to non-zero products only when they are contained in the element parameters. When they do not belong to them then they are multiplied only with zero terms of the K er and fer matrices. Therefore formula (6.3.69) can be further simplified: 1 Π e = dT K er d − dT fer 2

(6.3.81)

At this stage the summation of energy of all finite elements prescribed in formula (6.3.70) can be made without any difficulty because the vector d does not depend on the summation index e and the sum is performed only on the K er and fer terms: 1     1 Π = ∑ Π e = dT  ∑ K er  d − dT  ∑ fer  = dT Kd − dT f 2  e e   e  2

(6.3.82)

The finite element technique denotation is used to determine: Global stiffness matrix of the whole system, type ( N , N ) : K = ∑ K er

(6.3.83)

e

Global load parameter vector of the whole system, type ( N ,1) : f = ∑ fer

(6.3.84)

e

The conservative and time independent problems governed by Lagrange's variational principle Π = min . (Section 6.3.2.4) can be solved by the linear equation set, derived from formula (6.3.82) as follows: ∂Π = 0 ⇒ Kd − f = 0 ∂d

Kd = f

(6.3.85)

The left-hand side coefficients are assembled in the global stiffness matrix K and the right-hand side is formed by the global load parameter vector f . Physically nonlinear and geometrically linear problems can be solved by an incremental procedure based on the principle of total virtual work (Section 6.3.2.1) which leads directly to the equation set (6.3.85) in each step without defining an energy functional: K T dd = df

(6.3.86)

K T represents the tangent stiffness matrix, variable in each step according to the physical law which connects the stress and strain increments d σ and d ε ; dd and df are increments of deformation and load parameter vectors. The details of calculation need not concern us here but we will return to formula (6.3.86) in Chapter 5 in [8] where we deal with the values of C S constants. 430


6.3.3.4 Conclusions for the Solution of the Plate-Subsoil Interaction 1. Any finite element procedure and program algorithms which solve the plate problem can be applied to the plate-subsoil interaction solution when replacing the plate element stiffness matrix K p by the sum K = K p + K s , K s being the subsoil element stiffness matrix (6.3.63). The subsoil may be regarded as a property of finite element. The number N of unknown deformation parameters d and the band width B of the global stiffness matrix K are of the same order as when solving a plate without subsoil, because the subsoil surface elements alone should be attached only in a small domain Ω − Ω1 where substantial settlements are expected. Using the boundary conditions explained in (Chapter 4 in [8], the values of N and B in regular cases are the same as in the mere plate solution without subsoil and the influence of the domain Ω − Ω1 is included in the special boundary bonds. 2. The plate-subsoil interaction model presented here can be used in physically linear as well as nonlinear problems when sufficient physical constants or functions are known from the laboratory and/or in situ investigations. 3. In design practice and in common programs the most general formula (6.3.68) for computing the load parameter vector fe is never used. Mostly the simplest formula (6.3.67), holding for loading uniformly distributed in an element domain Ωe , is programmed and further only the nodal concentrated forces are respected, following the 3rd term of the general formula (6.3.68). This term also includes the concentrated loads P , which can act at any point. The simplest case arises when the point coincides with a node. Then all influence function vector terms are zeros with the exception of one which is attached directly to the nodal deflection w . This term is equal to unity and the load parameter vector contains only zero terms with the exception of one which is equal to the given load P . The value P influences directly the right-hand side of the appropriate equation, and when no other loading is prescribed, the right-hand side is equal to P . This case can be used as an approximation of complex loading and the value P is named the „lumped load”. 4. Only vertical loading in the z -direction and moments of horizontal vector direction can be prescribed. The x - and y -components of loads and z -components of moment vectors can be introduced only in more general models (Chapter 6 in [8]). The output settlements w must be tested by the condition w ≥ 0 . When w < 0 , then the resulting settlement wr including the influence of dead weight must be calculated and compared with zero. If in a domain Ω r the condition wr ≥ 0 is not fulfilled, then the contact with the subsoil is interrupted and the input data must be corrected, introducing in the domain Ω r only plate elements without subsoil, i.e. K e = K p and K s = 0 in formula (6.3.66). This correction can be repeated if the recalculation does not give the required precision in fulfilling the contact condition wr ≥ 0 . 5. The subsoil elements without plate part can be used either outside the complete plate-subsoil elements in the domain (Ω − Ω1 ) or individually when analysing the subsoil

431

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL loaded on its surface ( Ω1 = 0 ) . Their stiffness matrix K s 0 is defined by formula (6.3.63) omitting the last three terms pertaining to the friction described by the constants C1Sx , C1Sy , C1Sxy .

6.3.3.5 Mindlin's Plate on the 2D-Efficient Model of the Subsoil There exists an actual and formal difference between the Kirchhoff and the Mindlin plate theory expressed in the formulae (6.3.9) to (6.3.22) of Section 6.3.2.2. Firstly the description of the displacement components u, v, w of an arbitrary point ( x, y, z ) of the plate body by three independent functions w, ϕ x , ϕ y of two variables x, y – see formula (6.3.10) – leads to a generalization of formula (6.3.49): we ( x, y ) = U1 ( x, y ) a1e ϕ xe ( x, y ) = U 2 ( x, y ) a 2e ϕ ye ( x, y ) = U 3 ( x, y ) a3e

(6.3.87) (6.3.88) (6.3.89)

The common matrix form can be retained, defining the vector of generalized displacements of the type (3,1) : T

u e =  w, ϕ x , ϕ y  e

(6.3.90)

and the base function matrix of the type (3, n) : 0 0   U1 ( x, y )  U ( x, y ) =  0 U 2 ( x, y ) 0   0 0 U3 ( x, y ) 

(6.3.91)

and the common vector of coefficients of the type (n,1) : ae = a1Te , aT2 e , aT3e 

T

(6.3.92)

Then the formulae (6.3.87)-(6.3.89) can be written as follows: u e = U ( x, y ) a e

(6.3.93)

The vector (6.3.47) of element deformation parameters d e will contain only nodal values of deflection and/or rotations (6.3.87)-(6.3.89) without any derivatives. Therefore, the relation (6.3.54) can be obtained by introducing the nodal coordinates (6.3.52) into the formulae (6.3.87)-(6.3.89) which leads in the nodes j with the three parameters d ej = [ w, ϕ x , ϕ y ]Tj to the following equation: d ej = S ej ae

S ej = U( x j , y j )

(6.3.94)

In the nodes j , where only the deflection w will be defined as a deformation parameter, the

432

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL simple formula holds: d ej = {w j } = U1 ( x j , y j ) a1e

(6.3.95)

The relations (6.3.94) and (6.3.95) can be summarized for all m nodal points of the element in a common matrix form including all element parameters de = Seae

(6.3.96)

with the following notation:  S e1  S  2 Se =  e   M    S em 

d e = dTe1 , dTe 2 ,K , dTem 

T

(6.3.97)

S e denotes a square matrix of the type (n, n) the terms of which depend only on nodal coordinates ( x j , y j ) and are constant in one element. Using the inverse relation to (6.3.96) the course of deflection and rotations (6.3.93) can be written in the following form: u e = Ved e

Ve = US e−1

(6.3.98)

The difference from the simpler Kirchhoff case (6.3.56) consists in the fact that instead of one function we and row matrix Ve (1, n) the three functions u e = [ we , ϕ xe , ϕ ye ]T and a matrix Ve of the type (3, n) are defined. The energy of internal forces of the Mindlin plate defined in Section 6.3.2.2. formulae (6.3.15), (6.3.20), (6.3.22), introduces instead of the simple bending operator (6.3.59) the bending-shear operator ∂ TM of the type (5,3) , because there are five internal forces (6.3.21) and three generalized displacements (6.3.90):

∂ TM

0 ∂ ∂x   0  0 −∂ ∂y 0   = 0 −∂ ∂x ∂ ∂y    0 1  ∂ ∂x ∂ ∂y −1 0 

(6.3.99)

The ∂ TM -terms can easily be checked by comparing the matrix equation ε e = ∂ TM u e

(6.3.100)

with equation (6.3.20). Putting the relation (6.3.98) in formula (6.3.100) the Mindlin strains ε e of the element e can be expressed by its deformation parameters ε e = ∂ TM Ved e = H ed e

(6.3.101)

The strain influence function matrix of the type (5, n) is defined as follows:

433

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL H e = ∂ TM Ve

(6.3.102)

replacing the simpler matrix (6.3.60) of the Kirchhoff plate element which was of the type (3, n) . The element stiffness matrix K e can be divided into two parts (6.3.66) as in the Kirchhoff case. The first part is the (mere) plate element stiffness matrix K p following formula (6.3.62) where H and C p are introduced following formulae (6.3.102) and (6.3.22). The second part K s expresses the subsoil influence. The Kirchhoff case formula (6.3.63) cannot be used without a formal change: For the sake of clarity we omit in the next formulae (6.3.103) to (6.3.107) the element index e and define: The vector ε s of the subsoil surface deformation according all previous definitions (6.2.25) and (6.3.10): T

T  ∂w ∂w  , = ε s =  w, u , v, γ xy , γ yz  =  w, rxϕ y , − ryϕ x , ∂x ∂y   = ∂ Ts u = ∂ Ts Ua = ∂ Ts US −1d = ∂ Ts Vd = H s d

(6.3.103)

Beside the relations (6.3.93), (6.3.98) and a relation analogous to the (6.3.102) H s = ∂ Ts V = ∂ Ts US −1

(6.3.104)

an operator ∂ Ts is introduced: 1 0  0 T ∂s =  ∂   ∂x ∂   ∂y

0 0 −ry 0 0

0 rx  0  0   0 

(6.3.105)

The symmetrical matrix CSs of all above explained physical constants of the 2D subsoil model arranged as follows: C1Sz   0 S Cs =  0   0  0 

0 C1Sx C1Sxy 0 0

0 C1Sxy C1Sy 0 0

0 0 0 C2Sx C2Sxy

0   0  0   C2 xy  C2Sy 

(6.3.106)

434

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL Then the subsoil part K s of the element stiffness matrix K (6.3.66) can be written in a compact matrix form: K s = ∫∫ HTs CSs H s d Ω e

(6.3.107)

Ωe

Of course, every element e can possess various shape and physical constants, i.e. the matrices H s and CSs can be elementwise variable. This result corresponds to the subsoil energy Π is defined by formula (6.3.14) of Section 6.3.2.2. The common integration area Ωe can be introduced because a finite element belongs either to Ω1 (with plate part) or to Ω − Ω1 (without it). As regards the further finite element technique, all results of the preceding Section 6.3.3.3 and 6.3.3.4 can also be used in the Mindlin plate calculation. A generalization of loading is possible following the three degrees of freedom w, ϕ x , ϕ y of an arbitrary plate point. Formulae (6.3.87)-(6.3.89) can be rewritten in the same manner as Formula (6.3.63) to the form (6.3.108) using the potential energy of all external forces (6.3.34): np

fe = ∫∫ VeT pd Ω e + ∫ VeT qd Λ1 + ∑ VejT P j Ωe

p =  pz , mx , my 

Λ1

j =1

T

q =  qz , qmx , qmy 

(6.3.108) T

T

P j =  Pz , M x , M y  j

The subsoil elements without plate part need not be defined by the general formulae because only the settlements we are relevant to define their deformation. Their stiffness matrix K s is given by the first five terms of formula (6.3.107). If no full plate-subsoil element is attached at a node then no rotation ϕ x , ϕ y is defined in this node and no concentrated moments can act in it.

6.3.3.6 Mindlin's Plate on the 3D-Efficient Model of the Subsoil The 3D-efficient model of the subsoil as an improvement of the 2D-one was defined in Section 6.2.3 and 6.3.2.3. The main idea is based on the introduction of a layered subsoil mass in the interval 0 ≤ z ≤ H n where great settlement gradients can be expected and therefore no reliable assumption about the „effective depth“ can be made see Figs. 4 and 6 ÷ 9 of Chapter 6.2. A practical example of a thick foundation plate is presented in Fig. 12. The layer thickness hi (i = 1, 2, K , n) can be variable, as well as the soil moduli ( Ez , Gxz , G yz , Gxy , z )i and 435

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL their z -gradient ki , assembled in a common row matrix (6.2.107). Below the last layer, a 2D-efficient subsoil model can be defined with the C -constants (6.2.45) or its special cases (6.2.46) to (6.2.49). In the basic case only one Winklerian constant C1Sz can be prescribed, influencing mainly the total settlement of the foundation plate, the physical constants of which are assembled in a matrix C ; see Section 6.3.2.2, formula (6.3.22). S

As an example let us show a possible construction of a Mindlin´s plate element an a subsoil.

Fig. 12 Four types of finite elements: plate-subsoil elements in Ω1 ; the same elements also expressing the structure stiffness; subsoil elements only in

( Ω0 − Ω1 ) ; subsoil elements in ( Ω − Ω0 )

which can be omitted by

replacing the influence of subsoil outside the plate by bonds on the boundary

Γ0 .

436


6.3.3.6.1 Plate Element ISO 1 The natural coordinate system (ξ ,η ) will be introduced (Fig. 11h, i) by the transformation formulae x = N 2 (ξ ,η ) x = V2 (ξ ,η ) x

(6.3.109)

y = N 2 (ξ ,η ) y = V2 (ξ ,η ) y

V2 (ξ ,η ) = V2,1 , V2,2, , V2,3 , V2,4  V2,i (ξ ,η ) =

1 (1 + ξiξ )(1 + ηiη ) 4

i = 1, 2, 3, 4

(6.3.110)

x = [ x1 , x2 , x3 , x4 ]

T

y = [ y1 , y2 , y3 , y4 ]

T

The shape functions N 2i may be regarded as functions defining the influence of nodal coordinates ( xi , yi ), i = 1, 2, 3, 4 on the element shape. Their denotation „ N “ signifies the fact that they can be different from the base functions „ V “ defining the course of an unknown quantity in the element domain (sub- and superparametric elements). In the isoparametric elements the shape functions „ N “ are identical to the base functions and a common denotation Vi can be used. The first suffix 2 expresses the dimension of the element, i.e., 2D. The Mindlin plate parameters w, ϕ x , ϕ y , are defined in each vertex (nodal point) and assembled in a vector: d ep = dTep1 , dTep 2 , dTep 3 , dTep 4  d epi =  wi , ϕ xi , ϕ yi 

T

(6.3.111)

T

i = 1, 2, 3, 4

(6.3.112)

The suffix p pertains to the plate elements. The course of deflections w and rotations ϕ x , ϕ y in the element is defined only in natural coordinates (ξ ,η ) : w = V2 (ξ ,η ) [ w1 , w2 , w3 , w4 ]

T

ϕ x = V2 (ξ ,η ) [ϕ x1 , ϕ x 2 , ϕ x 3 , ϕ x 4 ]

T

ϕ y = V2 (ξ ,η ) ϕ y1 , ϕ y 2 , ϕ y 3 , ϕ y 4 

(6.3.113) T

Regarding the denotation (6.3.90), (6.3.111), (6.3.112) and local numbering of the twelve deformation parameters

437

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL d ep =  w1 , ϕ x1 , ϕ y1 , w2 , ϕ x 2 , ϕ y 2 , w3 , ϕ x 3 , ϕ y 3 , w4 , ϕ x 4 , ϕ y 4 

T

(6.3.114)

the formulae (6.3.113) can be written in a compact matrix form (6.3.98): u e = Vp d ep T

u e =  w, ϕ x , ϕ y  Vp =  Vp1 , Vp 2 , Vp 3 , Vp 4  Vpi = DIAG [V2i , V2 i , V2i ]

(6.3.115) i = 1, 2, 3, 4

The formulae (6.3.100) to (6.3.102) for the Mindlin plate strains can be used after the transformation of x, y derivatives to the ξ , η derivatives, because the operator ∂ TM (6.3.99) is defined in x, y and the generalized displacements (6.3.115) in ξ , η . The direct transformation will not be used because the derivatives ∂ξ ∂x , ∂ξ ∂y , ∂η ∂x , ∂η ∂y cannot be easily obtained. The functions ξ = ξ ( x, y ), η = η ( x, y ) are too complicated. In the isoparametric elements they are never constructed. Therefore, the transformation involving the derivatives ∂x ∂ξ , ∂x ∂η , ∂y ∂ξ , ∂y ∂η will be written, with an arbitrary function matrix V2 (ξ ,η ) of the type (1,4): ∂V2 (ξ ,η ) ∂V2 (ξ ,η ) ∂x ( ξ ,η ) ∂V2 (ξ ,η ) ∂y (ξ ,η ) = + ∂ξ ∂x ∂ξ ∂y ∂ξ ∂V2 (ξ ,η ) ∂V2 (ξ ,η ) ∂x ( ξ ,η ) ∂V2 (ξ ,η ) ∂y (ξ ,η ) = + ∂η ∂x ∂η ∂y ∂η

(6.3.116)

Omitting the denotation of variables (ξ ,η ) and introducing the common matrix form we can rewrite the formulae (6.3.116) as follows L 2ξη V2 = J 2L 2 xy V2

(6.3.117)

The Jacobian matrix J 2 of the type (2,2) can be written in the form J 2 = L 2ξη [ x, y ] = L 2ξη V2 (ξ ,η )[ x, y ]

(6.3.118)

representing the common expression  ∂x  ∂ξ J2 =   ∂x  ∂η 

∂y  ∂ξ   ∂y  ∂η 

(6.3.119)

and introducing the relations (6.3.109). The linear diferential operators are defined as follows: L 2ξη

∂ ∂  = ,   ∂ξ ∂η 

T

L 2 xy

∂ ∂ = ,   ∂x ∂y 

T

(6.3.120)

438

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL The relation inverse to the equation (6.3.117) reads: L 2 xy V2 = J 2−1L 2ξη V2

(6.3.121)

The principial difficulty of isoparametric finite elements is based on the fact that no inverse relation of the type ξ = ξ ( x, y ), η = η ( x, y ) to the equations (6.3.109) can be found. In other words: we can order a point ( x, y ) to a chosen point (ξ ,η ) but no such ordering is possible in the opposite direction. Therefore the direct calculation of the terms of the Jacobian matrix in the function form  ∂ξ  ∂x J −2 1 =   ∂ξ  ∂y

∂η  ∂x   ∂η  ∂y 

(6.3.122)

is impossible. But the inversion J −2 1 can be performed with constant values pertaining to an element point. This inversion is done only in the integration points (ξ k ,η k ) of numerical quadrature. Thus, the quantities in equation (6.3.121) may be denoted by the suffix k :

(L

2 xy

V2 ) = ( J 2−1 ) ( L 2ξη V2 )k k

k

(6.3.123)

The constant terms of the inverse matrix J −2k1 pertaining to a point ( xk , yk ) may be denoted in short as follows: j J −jk1 =  11  j21

j12  j22  k

(6.3.124)

Then the transformation (6.3.123) can be written in an explicit form for each derivative:  ∂V   ∂x   j11   =  ∂V   j21  ∂y  k

 ∂V  j12   ∂ξ    j22  k  ∂V   ∂η   k

(6.3.125)

The ∂ TM – operator (6.3.99) includes only the derivatives ∂ ∂x , ∂ ∂y zeros and units. At a point (ξ k ,η k ) (its xk , yk coordinates ned not be known) the following formulae hold: ∂   ∂   ∂     =  j11  +  j12   ∂x  k  ∂ξ  k  ∂η k  ∂   ∂   ∂    =  j21  +  j22   ∂y k  ∂ξ  k  ∂η  k

(6.3.126)

The operator can be rewritten from the ∂ TMxy – form (6.3.99) to the ∂ TMξη – form:

439


(∂ )

T M ξη k

 0    0    0 =  ∂ ∂   j11 ∂ξ + j12 ∂η  ∂ ∂   j21 ∂ξ + j22 ∂η 

0 ∂ ∂ − j22 ∂ξ ∂η ∂ ∂ − j11 − j12 ∂ξ ∂η

− j21

0 −1

∂ ∂  + j12 ∂ξ ∂η    0   ∂ ∂  j21 + j22 ∂ξ ∂η    1    0   j11

(6.3.127)

This form contains only constant values j11 , j12 , j21 , j22 of the terms of inverse matrix (6.3.120) calculated for one point (ξ k ,η k ) . The ξ , η derivatives of the functions (6.3.110) are as follows: ∂ 1 V2i (ξ ,η ) = ξi (1 + ηiη ) ∂ξ 4 ∂ 1 = V2 i (ξ ,η ) = ηi (1 + ξiξ ) ∂η 4

V2i ,ξ = V2i ,η

(6.3.128)

The derivatives of the functions (6.3.109) contained in the Jacobian matrix (6.3.121) which must be inverted can be written as follows:  ∂x   ∂ξ  = V2,ξ (ξ k ,ηk ) x k  k  ∂x   ∂η  = V2,η (ξ k ,ηk ) x k  k  ∂y    = V2,ξ (ξ k ,ηk ) y k  ∂ξ k

(6.3.129)

 ∂y    = V2,η (ξ k ,ηk ) y k  ∂η  k V2,ξ = V2,1,ξ ,V2,2,ξ ,V2,3,ξ ,V2,4,ξ  V2,η = V2,1,η ,V2,2,η ,V2,3,η ,V2,4,η 

(6.3.130)

At this stage all necessary formulae are prepared for the calculation of the following vectors and matrices: The Mindlin plate strain vector (6.3.20):

( ε )k = ( ∂ TM ξη )k u k = ( ∂ TM ξη )k Vpk d ep = H k dep

(6.3.131)

The „strain matrix” H k : 440

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL H k = ( ∂ TM ξη ) Vpk

(6.3.132)

k

The „stress matrix” Σ k in the relation

( σ )k = Σ k d ep

(6.3.133)

connecting the Mindlin plate internal forces (6.3.21) with the deformation parameters: Σk = C pH k

(6.3.134)

C p being the matrix of physical constants following the formula (6.3.22). The element stiffness matrix K p is calculated by the numerical integration over the unit element (Fig. 11h): K p = ∫∫ H C p Hd Ω e = T

Ωe

J2

k

1

∫

1

ni

T T ∫ H C p H J 2 dξ dη = ∑ ck H k C pk H k J 2 k

(6.3.135)

k =1

−1 −1

denotes the determinant of the Jacobian matrix (6.3.121) in the integration point

(ξ k ,ηk ) , k = 1, 2, K , ni :  ∂x ∂y ∂x ∂y  − J2 k =    ∂ξ ∂η ∂η ∂ξ k

(6.3.136)

It is calculated according to the formulae (6.3.128) to (6.3.130). The integration coefficients ck depend on the integration algorithm and the required precision of results, (see Section 6.3.3.1.). The consistent load parameter vector fe of the type (12,1) can be calculated from formula (6.3.108) after its transformation to the unit area of the element in Fig. 11h. Retaining only the regular load components p =  pz , mx , my 

T

(6.3.137)

we can write the following equation: fe = ∫∫ VeT pd Ω e = Ωe

1

1

∫ ∫V p J T e

2

dξ dη

(6.3.138)

−1 −1

In the most common case of a uniformly distributed load p = [ pze , 0, 0] in one element Ωe the value of the integral 1

1

∫ ∫V

T e

J 2 dξ dη

(6.3.139)

−1 −1

can be calculated numerically in advance and then multiplied by the pze values pertaining to the individual elements. The continuous moment loading can be introduced in the same 441

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL manner but its practical significance is small.

6.3.3.6.2 Layer Element B1 The finite element of soil mass will be represented by a simple and well-known brick element with trilinear shape and base functions, denoted B1 for short. Following the simplified model (Section 6.2.3 and 6.3.2.3), only vertical displacement components, i.e. settlements w , are taken into account. The brick B1 in Fig. 8d can be regarded as the transformation of the unit cube in Fig. 8c by the rule similar to the 2D-case (6.3.109): x = V3 (ξ ,η , ζ ) x y = V3 (ξ ,η , ζ ) y

(6.3.140)

z = V3 (ξ ,η , ζ ) z (ξ ,η , ζ ) being the natural coordinates and (x, y , z ) the Cartesian coordinates of the nodal points. The vertices i = 1, 2, K , 8 have the coordinates (ξ ,η , ζ ) = (±1, ± 1, ± 1) and ( xi , yi , zi ) , i = 1, 2, K , 8 according to Fig. 8c, d. For example, the natural coordinates (1, 1,1) and (−1, −1, −1) and the Cartesian ones ( x7 , y7 , z7 ) and ( x1 , y1 , z1 ) pertain to the nodes 7 and 1. The shape functions are a 3D extension of the functions (6.3.110): V3 (ξ ,η , ζ ) = V3,1 ,V3,2 ,K , V3,8  V3,i (ξ ,η , ζ ) =

1 (1 + ξiξ )(1 + ηiη )(1 + ζ iζ ) 8

x = [ x1 , x2 ,K , x8 ]

(6.3.141)

y = [ y1 , y2 ,K , y8 ]

T

T

The element is isoparametric, i.e., it has the same number of shape functions (denoted in references as N i ), as of base functions Vi , and it uses the same functions Vi to express the element shape and the course of unknown displacement component w in it. Therefore, the common notation Vi is possible: w = V3 w

(6.3.142)

w = [ w1 , w2 ,K , w8 ]

T

(6.3.143)

The element index e will be omitted because in this section only one element is analysed. There are eight nodal deformation parameters (6.3.143), i.e. eight unknown settlements in the element nodes. The strain and stress vectors ε, σ of the soil mass model are defined in Section 6.2.1.2.2 by the formulae (6.2.25) to (6.2.28). Introducing the relations (6.3.141) to (6.3.143), we obtain the direct isoparametric expressions: ε = ∂ T w = ∂ T V3 w = H s w

(6.3.144)

442

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL σ = Cs ε = Cs H s w = Σw

(6.3.145)

The „strain matrix” H s and „stress matrix” Σ are defined as follows: H s = ∂ T V3

(6.3.146)

∑ = Cs H s

(6.3.147)

Both are of the type (3,8) because the simplified strain vector ε = [ε z , γ xz , γ yz ]T and stress vector σ = [σ z ,τ xz ,τ yz ]T are composed of only three terms and the number of deformation parameters (6.3.143) is eight. The operator ∂ T is defined by formula (6.2.26) in Cartesian coordinates x, y, z : ∂ ∂ ∂ ∂T =  , ,   ∂z ∂x ∂y 

(6.3.148)

The derivatives in (6.3.148) must be expressed in terms of natural coordinate derivatives ∂ ∂ξ , ∂ ∂η , ∂ ∂ζ , because the base functions V3 (6.3.141) are written by means of these coordinates. For the same reason as in Section 6.3.3.6.1, we cannot find a general transformation formula with functions, but must be content with the transformation in the integration points of numerical quadrature, i.e. in a certain number of points (ξ ,η , ζ ) k . For this purpose we will widen the formulae (6.3.117) to (6.3.129) to the 3D-case, omitting all detailed component rewriting. V3 now represents the function vector V3 (ξ ,η , ζ ) ; of the type (1,8) : 113 L3ξηζ V3 = J 3L3 xyz V3 L3ξηζ

∂ ∂ ∂  = , ,   ∂ξ ∂η ∂ζ 

(6.3.149) T

L3 xyz

∂ ∂ ∂ = , ,   ∂x ∂y ∂z 

T

(6.3.150)

The Jacobian matrix of transformation:  ∂x  ∂ξ   ∂x J3 =   ∂η  ∂x  ∂ζ 

∂y ∂ξ ∂y ∂η ∂y ∂ζ

∂z  ∂ξ  ∂z  ∂η  ∂z  ∂ζ 

(6.3.151)

and its matrix-expression: J 3 = L3ξηζ ⋅ [ x, y, z ] = L3ξηζ V3 ⋅ [ x, y, z ]

(6.3.152)

The inverse relation written in the integration point of numerical quadrature reads as follows:

443

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL L3 xyz V3k = J 3−k1 L 3ξηζ V3 k

(6.3.153)

Denoting the individual terms of Jacobian matrix: J 3−k1

 j11 =  j21  j31

j 12 j22 j32

j13  j23  = L3ξηζ [ x, y, z ]k j33  k

(

)

−1

(

= L3ξηζ V3 ⋅ [ x, y, z ]k

)

−1

(6.3.154)

Widening the ∂ T operator:

∂ Tk = ( ∂ Tξηζ )

k

 ∂  j31 ∂ξ   ∂ =  j11  ∂ξ  ∂  j21 ∂ξ 

+ + +

∂ ∂η ∂ j12 ∂η ∂ j22 ∂η j32

+ + +

∂ ∂ζ ∂ j13 ∂ζ ∂ j23 ∂ζ j33

        k

(6.3.155)

The sole function to which the operator ∂ T will be applied is the settlement function w (6.3.142). Therefore only the following derivatives in the points (ξ ,η , ζ ) k are needed: ∂V3i 1 = ξi (1 + ηiη ) (1 + ζ iζ ) ∂ξ 8 ∂V 1 = 3i = ηi (1 + ξiξ ) (1 + ζ iζ ) ∂η 8 ∂V 1 = 3i = ζ i (1 + ξiξ ) (1 + ηiη ) ∂ζ 8

V3i ,ξ = V3i ,η V3i ,ζ

(6.3.156)

The ξ , η , ζ – derivatives of the functions (6.3.140) are terms of the Jacobian matrix J 3 (6.3.152), which must be inverted only with constant terms pertaining to the integration points (ξ ,η , ζ ) k of numerical quadrature. The x, y, z – derivatives of function V3 needed for evaluatin of the H -matrix (6.3.146) can be expressed by means of J 3−1 matrix in terms of ξ , η , ζ – derivatives by formula (6.3.153). The 3D extension of the formulea (6.3.129) can be written as follows:

444

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL  ∂x   ∂  = V3,ξ (ξ k ,η k , ζ k ) x k  ξ k  ∂x   ∂  = V3,η (ξ k ,ηk , ζ k ) x k  η k  ∂x    = V3,ζ (ξ k ,η k , ζ k ) x k  ∂ζ  k  ∂y   ∂  = V3,ξ (ξ k ,η k , ζ k ) y k  ξ k  ∂y   ∂  = V3,η (ξ k ,ηk , ζ k ) y k  η k  ∂y    = V3,ζ (ξ k ,η k , ζ k ) y k  ∂ζ  k  ∂z    = V3,ξ (ξ k ,η k , ζ k ) z k  ∂ξ k  ∂z   ∂  = V3,η (ξ k ,ηk , ζ k ) z k  η k

(6.3.157)

 ∂z    = V3,ζ (ξ k ,η k , ζ k ) z k  ∂ζ  k V3,ξ = V3,1,ξ , V3,2,ξ , K , V3,8,ξ  V3,η = V3,1,η , V3,1,η , K , V3,8,η 

(6.3.158)

V3,ζ = V3,1,ζ , V3,2,ζ , K , V3,8,ζ  The element stiffness matrix K s 3 is a square matrix of the type (8,8) and can be calculated using fomula (6.3.159), in a similar way as in the 2D-case, formula (6.3.135), using the numerical quadrature with integration points (ξ ,η , ζ ) k and their multiplying coefficient ck : K s 3 = ∫∫∫ H Cs H s d Ω e = T s

Ωe ni

1

1

1

∫ ∫ ∫H C H T s

−1 −1 −1

s

s

J 3 dξ dη dζ = (6.3.159)

= ∑ ck H Csk H sk J 3 k k =1

T sk

J 3 k denotes the value of the determinant of the Jacobian matrix (6.3.152) in the ni nodal points (ξ ,η , ζ ) k of numerical integration. It is composed of six products of three derivatives:

445

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL  ∂x ∂y ∂z ∂x ∂y ∂z  +K − J3 k =   ∂ζ ∂η ∂ξ  k  ∂ξ ∂η ∂ζ

(6.3.160)

following the known rule for calculating the value of the 3rd order determinant, and depends only on the values (6.3.157). The load parameter vector f could be defined analogously to the 2D-case by the appropriate extension of formulae (6.3.138)and (6.3.139). Because there is only one displacement component, i.e., the settlement w , the loading can consist only of vertical loads pz e.g. the dead weight of the soil mass or nodal vertical external forces Pz . The last case is the simplest one, because the values Pz are directly used as the terms of the load parameter vector. The first case leads to the following formula: f s = ∫∫∫ V p z d Ωe = T 3

Ωe

1

1

1

∫ ∫ ∫V

T 3

pz J 3 dξ dη d ζ

(6.3.161)

−1 −1 −1

and for the constant pz the numerical integration of the form 1

1

1

∫ ∫ ∫V

T 3

J 3 d ξ dη d ζ

(6.3.162)

−1 −1 −1

can be calculated in advance and then multiplied by the individual pz -values.

6.3.3.6.3 Element of the Subsoil Below the Last Layer Below the last layer introduced in the 3D soil mass model, various geological conditions are possible. In any case, we suggest defining a deformable subsoil under the last layer defined in the 3D soil mass model (Section 6.3.3.6.2). In the case of an undeformable rock at the depth H r , the designer may chose H n < H r and the soil mass below the depth H n in the interval H n ≤ z ≤ H r will be modelled by the 2D-efficient model of the subsoil which belongs to the surface defined at the bottom of the lowest layer at the depth H n . This case may be regarded as exceptional. In common practice, the geological investigation cannot be carried out at a great depth because of the expense. The input values E are known only in a reasonable interval 0 ≤ z ≤ H n (Chapter 5 in [8]) and below the depth H n only approximate deformation properties are predicted, which can best be taken into account by the efficient 2D model of the subsoil with cumulative constants of the type CS Mostly only two constants C1S , C2S can be deduced from the geological description. It is not necessary to follow the above-presented suggestion in the case of an undeformable rock at the depth H r , and H n can be chosen to equal H r , omitting the further subsoil (rock) mass. Following Fig. 8d, the deformation parameters w5 , w6 , w7 , w8 of the bottom of the lowest 3D-element must be made equal to zero. The disadvantage of this concept is based on the increasing number of 3D-elements and defined parameters N with no 446

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL corresponding influence on the precision and reliability of the results. Fortunately, the influence of the lowest layer on the internal forces of the structure and foundation plate is small because the curvatures of settlements w in the surface z = 0 are influenced mainly by the nearby layers where the gradients ∂w ∂z are great. To derive the element stiffness matrix K s 2 , the note after formula (6.3.108) in Section 6.3.3.5. dealing with Mindlin plates on the 2D-model of subsoil may be sufficient. The first five terms of the formula (6.3.107) should by rewritten in the sense of the isoparametric notation (6.3.113). Only the first row of (6.3.113) is used; the index 2 pertains to the 2D case and bilinear polynomials (6.3.110). The denotation (ξ ,η ) at the functions V2 (ξ ,η ) will be omitted: K s 2 = ∫∫ ( V2T C1Sz V2 + V2,T x C2Sx V2, x + V2,T y C2Sy V2 y + Ωe

(6.3.163)

+ V C V2, y + V C V2, x ) d Ω e T 2, x

S 2 xy

T 2, y

S 2 xy

Introducing a square matrix of physical constants  C2Sx C = S C2 xy S 2

C2Sxy   C2Sy 

(6.3.164)

and using the operator L 2 xy (6.3.120), the more condensed form of (6.3.163) can be written: K s 2 = ∫∫ ( V2T C1Sz V2 + HT2 CS2 H 2 ) d Ω e

(6.3.165)

H 2 = L 2 xy V2 = J −z 1L 2ξη V2

(6.3.166)

Ωe

where

The last formula must be introduced following the equation (6.3.123) because the bilinear functions V2 are defined only in natural coordinates ξ , η . Therefore the x, y derivatives of V2 must be expressed in terms of ξ ,η derivatives. The inverse matrix J −21 cannot be obtained in general function form but only in a set of constant points (ξ k ,η k ) . It will be denoted J −2k1 in short. The integration area Ωe will he transformed into the unit square −1 ≤ ξ ≤ 1, − 1 ≤ η ≤ 1 and the quadrature will be performed numerically: 1

K s2 =

1

∫ ∫ (V C T 2

S 1z

−1 −1

V2 + HT2 C2S H 2 ) J 2 d ξ dη = (6.3.167)

ni

= ∑ ( V C V2 + H C H 2 ) J 2 k ck k =1

T 2

S 1z

T 2

S 2

k

Index k is attached to the numerical integration points (ξ k ,η k ) with the multiplying coefficients ck . The inverse matrix J 2k−1 is calculated only at these points. J 2

k

signifies the

values of the determinant J 2 of the Jacobian matrix J 2 (6.3.121) in the same points. 447

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL Concerning the constants C1Sz and C2S (6.3.164) the same rules hold as in the case of the 2D-efficient subsoil model but the zero horizon is situated at the bottom of the lowest layer. Only in the special case when no layers are required or put in does the program solve the 2D-case and the zero horizon is identical to the subsoil surface.

6.3.4 Nonlinear Analysis of Structure-Soil Interaction using the 2D Efficient Subsoil Model

6.3.4.1 Introduction As it follows from the previous explanation, certain assumptions must have been introduced in the reduction of the 3D subsoil massif to the surface model: for example, the damping function f that determines the distribution of the settlement of a point on the subsoil surface over the depth. It can be easily shown that this distribution will differ significantly for different points on the foundation base and it will depend on the size and distribution of loads, even if the material of the subsoil is linear. However, this is not the case in the subsoil under structures. The parameters of the surface models are a function of (i) the position of the point on the subsoil surface, (ii) properties of subsoil material, and also (iii) the rigidity of the superstructure and (iv) loads. Therefore, the design and assessment of any structure that is in the contact with subsoil must deal with the interaction of the structure, foundation and subsoil. The load the foundation is subjected to is not transferred to the subsoil surface directly (if so, it would be possible to impose the corresponding reaction back to the foundation), but it depends on the distribution of the contact stress across the foundation base. This distribution, however, does not depend just on the load, but also on the relative rigidity of the foundation and superstructure in relation to the subsoil, on physical properties of the subsoil (heterogeneity, geological fractures), on adjacent constructions, etc. This chapter describes the procedure that makes it possible to apply the surface model of the structure-soil interaction and, at the same time, to take into account the abovementioned sources of nonlinearity. This procedure enables to take into consideration the national standards that define (i) the material properties of subsoil and (ii) the approach for the calculation of settlement of structures.

6.3.4.2 Stress in subsoil The initial problem in determining the “support conditions” of foundation structures is to find out the stress-state in the subsoil that, in general, covers the whole soil massif under the structure and in the vicinity that can influence the behaviour of the building on the foundation in question. The widespread approach (which is also implemented in numerous currently valid standards) today is that the stress-state in the subsoil can be obtained using the 448

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL model of a Boussinesq ideal homogenous half space. The previously used Winkler model did not make it possible to express (i) the decrease in the stress that occurs with the increasing depth and (ii) the formation of the subsidence basin, not mentioning (iii) the mutual impact of individual structures on each other. The application of the stress-state analysis using the half space inevitably raised questions concerning the conditions under which the solution is still satisfactory and under which a more detailed approach must be applied. In practice, the subsoil is almost always vertically non-homogenous (stratified). Quite often, the engineer has at hand geotechnical data that clearly prove also the horizontal heterogeneity. Therefore, various surveys have been performed to document tens of percents in the difference in stress σ z in case that soft and rigid strata alternate in the subsoil. Already in 1990 we carried out similar comparison [164] in which we analysed the impact of non-homogeneity of the half space on the stress tensor field due to varying thickness of inserted strata with two-, five- and ten-times greater deformation modulus E in comparison with the remaining strata. We made a set of calculations for a prismatic problem, which revealed that if, for example, the top stratum of the geological profile features larger E , the damping of the axial stress component σ z is faster and the difference between these values and the results for homogenous half space reaches up to 30%. On the other hand, if the stratum with the greater E is inserted in between other strata, the damping of σ z in higher-located strata is slower (the zone in question is “clamped” in between the loaded foundation base and the rigid stratum). It means that in the case of stratified geological profile with larger difference between the deformation modulus E in individual strata, it is more economical, and sometimes even safer, to analyse the stress-state of this non-homogenous half space. Naturally, it is clear that no such analyses will be performed in common practice. Therefore, the authors of technical standards included into the standards the article that the model of the ideal homogenous half space can be used also for non-homogenous and anisotropic subsoil (see e.g. art. 72, ČSN 73 1001 [169]). With regard to the fact that the error in the input geomechanical values required for the calculation of settlement (see chapter 6.3.4.6) is significantly greater than the error due to less accurate calculation of stress, the half space approach can be used for the calculation of σ z without scruples (being aware of possible deviations in the case of strata with significantly different deformation modulus E ).

6.3.4.3 Physical model of soil based on the formula stated in CSN 73 1001 The calculation model of subsoil is defined in art 116 – 122 of the above-mentioned standard [169]. There is no sense in rewriting the related paragraphs. We rather should try and comment on the provisions within the context of their application for the interaction of buildings with soil environment. The standard stipulates in page 33 the formula for the calculation of settlement s of the subsoil surface: n

σ z ,i − miσ or ,i

i =1

Eoed ,i

s=∑

hi

(6.3.1)

449

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL where σ z ,i is the vertical axial component of stress in an elastic homogenous infinite half space (or stratum), σ or ,i is the analogous component of the original geostatic stress, mi is the correction coefficient for surcharge loading (coefficient of structural strength), σ s ,i is the structural strength (i.e. miσ or ,i ) and n is the number of strata with thicknesses hi and constrained (oedometric) modulus Eoed ,i in which the effective stress is non-negative: σ zú = σ z ,i − σ s ,i = σ z ,i − miσ or ,i ≥ 0

(6.3.2)

Zero deformation is assigned to regions where the effective stress is negative. This represents mainly larger depths where the subsoil does not deform any more. The condition of zero effective stress then determines what is termed the depth of the deformed subsoil zone. The situation is, however, a bit more complex in regions outside of the foundation base. Even though the magnitude of the vertical axial component σ z is zero on the surface, it increases in deeper strata due to the effect of shear distribution - see fig. 13. The condition of zero effective stress thus determines the stratum of the deformed subsoil zone in which the upper level of the positive effect of the stress is not on the subsoil surface.

Fig. 13: Effective stress

The basic formula (6.3.1) for the calculation of settlement was derived in such a way that the standard replaced the integral over the vertical z H

s = ∫ ε z ( z )dz

(6.3.3)

0

by summation using the “rectangular rule” with the division of the interval 0 ≤ z ≤ H into n parts that may be generally of different size. The values of σ z , σ or , m, Eoed with the index i relate to the middle horizons of the strata. The accuracy of the summation can be increased 450

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL through finer mesh, i.e. greater n . Stress σ z is calculated from surcharge loading σ ol in the foundation base that must be in practice analysed mainly by means of interaction methods, i.e. through the solution of a contact problem: building + foundation + subsoil. The load of the foundation would be transferred directly to the subsoil surface only in case of a very flexible foundation. The result functions representing the distribution of the contact stress across the foundation base are, in fact, generally statically indeterminate quantities. As already stated, they depend not only on the load, but also on the relative rigidity of the foundation with the superstructure in relation to the subsoil, on physical properties of the subsoil, on the time factor (changes during construction and service life, consolidation), etc. The foundation base is subjected to the load directly only in special situations (e.g. load from embankments) and if this happens, the surcharge loading is known in advance. Stress σ or is determined by the weight of the soil above the given point that is in the middle of the thickness of the stratum. The calculation requires that the values of the effective unit weight (i.e. above the groundwater level) are known and it is assumed that its distribution is constant in every stratum. Considering formulas (6.3.1) and Chyba! Nenalezen zdroj odkazů., it is clear that our standard defines for each stratum a rigid-elastic stress-strain diagram with singularity in point σ z = σ s where the rigid model becomes elastic – see fig. 14.

Fig. 14: Stress-strain diagram of soil according to CSN

This strong physical nonlinearity results in the fact that the principles of linearity and superposition do not apply to the calculation of settlement and that it is not possible to use the same approach for the definition and assessment of the structure as in linear calculations. It would not be possible to evaluate separately individual load cases or use an automatic selection of the most effective combinations for each result quantity. Fortunately enough, the problem of the structure-soil interaction has certain specifics that allow for weakening of this 451

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL strict limitation. Short-term variable loads contained in combinations do not produce (through their effects) settlement of the subsoil and, therefore, they do not have to be included into the complete loading for which the support parameters will be evaluated. The principle of the solution is that the long-term loads are concentrated into one of just a few load cases and the supporting conditions are calculated for this linear combination. During the evaluation we have to bear in mind that the results correspond only to the given total load and not to individual load cases. Concerning the short-term loads, these are included only into the formulas for the calculation of the most effective combinations. They may have an impact on the deformations and internal forces of the superstructure, but not on the calculation of the interaction parameters. However, this approach is not recommended for problems in which the stress in the foundation base is influenced mainly by variable loads (e.g. light-weight steel halls) and the C S parameters must be calculated for each nonlinear combination separately.

6.3.4.4 Physical model of soil according to DIN 4019 Settlement is in the German standard calculated in the following way n

s = ∑ ds j , j =1

ds j = ε z ,efj dz j ,

ε z ,efj =

σ z ,efj Esj

(6.3.4)

where: ds j

compression of the j -th layer of the subsoil from the given surcharge loading

j

number of the layer with thickness dz j

n total number of layers into which the deformed zone of the subsoil from the loaded place up to the limit height H m is divided; geological strata represent a coarse division that is refined for mathematical reasons (numerical integration) similarly to formula (6.3.1) for CSN ε z ,efj

relative compression of the j -th layer from the given surcharge loading, it is the

vertical component of deformation ε z called in DIN 4019 “specific settlement” and marked s′z , which reminds derivational definition of ε z = ds dz . σ z ,efj vertical component of stress σ z in the centroidal level of the j -th layer due to the given surcharge loading. ef index used to emphasise that we deal with the effect of the given surcharge loading, because DIN 4019 introduces stress-strain diagrams σ z − ε z valid for the complete stress- and deformation-state of the stratum including the initial geostatic stress-state and corresponding deformation 452


this is termed “average modulus of deformation” of the j -th layer that corresponds to Esj the current stress-state and deformation of that layer. DIN 4019 defines it as a secant modulus of the graph σ z − ε z between two points: the original geostatic stress-state (σ z ,or , ε z ,or ) and the stress-state after applying the surcharge loading (σ z ,or , σ z ,ef , ε z ,or + ε z ,ef ) - see fig. 15.

Fig. 15 DIN – secant modulus Es

a) The standard assumes that the stress-strain diagram σ z − ε z (fig. 15) is provided for the calculation of settlement for every geological stratum starting from the initial state σ = 0, ε z = 0 and ending with the state σ z that is at least equal or greater than the expected state (σ z ,or + ε z ,or ) . It is apparently a uniaxial constrained deformation as in CSN.

b) According to DIN 4019, the j -th layer first gets into the state of the original geostatic stress-state (which corresponds to σ z ,or from CSN standard) and this happens along the same path as given in the diagram (fig. 15), i.e. by monotonous increase of σ z . This defines the first point of the graph (σ z ,or , ε z ,or ) . Then, the surface is subjected to the surcharge loading due to the structure and both stress and deformation increase. The second point of the graph (σ z ,or , σ z ,ef , ε z ,or + ε z ,ef ) is reached. What can be observed on the surface, i.e. settlement s , is just the effect of surcharge loading (σ z ,ef , ε z ,ef ) . That means that we are interested only in the secant modulus Es defined by the line connecting the two mentioned points of the graph. DIN draws stress σ z along the horizontal axis and deformation ε z (“specific variable settlement s′z ”) along the vertical axis downwards.

453

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL c) It is recommended to perform the summation (6.3.4) only within the limit depth H m defined by the requirement σ z ,ef = 0.2σ z ,or

( german iσ 1 = 0.2σ u&& )

(6.3.5)

which corresponds to the assumption that in depth H m the coefficient m always equals 0.2 ( m = 0.2 - see the coefficient of structural strength from CSN). d) Stress σ z due to the surcharge loading of the surface σ 1 is in DIN 4019 mentioned only in the form of the following formula σ z = iσ 1

(6.3.6)

where “ i ” is the coefficient found in some table of results for a sequence. About 30 collections of tables by different authors are recommended and they limit only to a rectangular or circular surcharge loading area. A warning is given concerning various difficulties related to table-defined parameters. Therefore, it is more convenient to use directly the exact solution of the stress tensor for the elastic homogenous half space, as implemented in the CSN standard. e) If the foundation is made in an excavation, the surcharge loading of the foundation base is reduced by the total original weight of the excavation σ z ,or ,v and, consequently, only the following load is taken into account p = σ ol − σ z ,or ,v

( german σ 1 = σ o − γ d )

(6.3.7)

This would correspond to CSN 73 1001 in a fictive situation with m = 1 for a stratum located at the foundation base in depth d . It means, that, most likely, only shallow excavations are considered. f) Graphs required in points a), b) assume the possibility to perform an experiment with a specimen of the soil from each stratum. At the beginning of the experiment the specimen must be in the ideal initial state of released stress and deformation and the subsequent process must follow exactly the same path as in reality (in situ). Such an experiment is impossible in terms of time, even though the geostatic stress-state itself could arise trough monotonous increase of load, e.g. in sediments. It is clear that the graphs required in DIN 4019 must include professional experience and possibly adaptation for a given locality with its geological history taken into account. Therefore, DIN 4019 requires that the graph be produced by an established soil-mechanics laboratory. Brief summary: The well known uncertainty following from the nature of geomechanical problems is in CSN 73 1001 implemented since 1988 mainly through the coefficient of

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6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL structural strength m ( 0.1 ≤ m ≤ 0.5 ) and modulus Eoed . In DIN 4019, this uncertainty is included into the graphs (stress-strain diagrams of soils as virgin materials). However, the German design practice often avoids using these graphs directly in favour of their cumulative consequences, for example in the form of “average modulus Es in the analysed problem”.

6.3.4.5 Physical model of soil according to Eurocode 7 The EC 7 standard is rather tolerant and does not prescribe particular types of settlement calculation. It only recommends performing the calculation of stress using a homogenous half space and considering the value of the limit depth as defined by formula (6.3.5), the same way as stated in the DIN standard. As the recommended magnitude of the limit depth is defined by the value equal to 0.2 of the initial geostatic stress-state, we suggest that the calculation of the compression of soil strata according to EC be performed using the CSN variant with the coefficient of structural strength equal to the same value, i.e. m = 0.2 . The introduction of the limit depth means that the deformation modulus equal to infinity is assigned to deeper strata, which means zero compression. The strata just above should thus feature the minimum compression (in order to prevent the introduction of an unjustified singularity of the distribution of compression into the model), which is best suited just by the introduction of the term “structural strength”. This makes the formula for the determination of the limit depth logical and justified. Concerning the excavations, we tend to support the idea that it is not suitable to deduct the weight of the excavation in determining the surcharge loading (see chapter 6.3.4.6.5), i.e. we prefer to keep the value of the surcharge loading the same as if it acted on the original terrain.

6.3.4.6 Variability of subsoil input data Modelling of the structure-soil interaction is often accompanied with the uncertainty following from the determination of corresponding geomechanical properties of the subsoil. This is caused by qualitative differences between the data (that are available to the structural engineer) for the materials of the superstructure and the materials of subsoil strata – with the reliability of the latter being significantly lower. The situation is complicated not only by the physical nature of the soil, but also by the impossibility to know the corresponding geological profile under each point of the structure, as (at best) only the data from a few bores are available and the determination of the shape and thickness of geological strata is only a question of estimation. In practice it means that the problem of the structure-soil interaction should be solved repeatedly for different variants of geological input data. This is what is indirectly promoted in CSN 73 1001 (Subsoil under spread foundations) [169] that contains tables of indicative standard characteristics of soils with values of modulus of deformation Edef given as intervals. EC7 [170] in art. 2.4.3 and 2.4.6 directly says that the calculation must take into account incidental changes of soil properties and uncertainties related to the 455

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL geological data and that statistical methods can be used for these data. It means that the given issue of the structure-soil interaction is a typical example of a situation in which the variability of input data must be considered in the process of calculation. In what way can this probabilistic requirement be taken into account in the model? To be honest, we have to admit that today we are not capable of performing a correct probabilistic calculation in SCIA•ESA PT (see [166]), but, on the other hand, we are able to at least minimise the impact of this drawback by repetitive analyses of the problem (as different projects) with different variants of geomechanical input data. This approach allows us to find out partially the effect of the variability of input data on the values of resulting internal forces and deformation of the structure.

6.3.4.6.1 Modulus of deformation Practically the most important value for settlement of spread foundations is the modulus of deformation Edef . Depending on the soil class and other circumstances it reaches values of about 1 to 500 MPa. There are differences between the laboratory determined values and the real values measured in situ. The whole number of factors is to blame for these differences: ● Taking the samples – it can be said that this is the most significant factor. The sample is partially damaged already when taken and transported. ● Test methodology – the most often used test for the determination of the modulus of deformation is the oedometric test. In this case, the simplicity of the testing device negatively affects the produced stress-state. The stress-state in the oedometer corresponds to the quiescent state and not to the in-situ-pressure. ●

Humidity, temperature.

Probably the most serious problem is the variability of Edef depending on the stress level and stress path. In addition, it is necessary to realise that the soil belongs to materials that are particular in nature (multi-phase system). It brings the complexity into the deformation of soils that is, contrary to other materials, influenced by the history of loading and by the way in which the load increases. The structural engineer should cooperate with the contractor of the survey at least with regard to the prognosis of the most influential surcharge loadings. The deformation characteristics of the theory of elasticity apply to the determination of the corresponding increment in the settlement of spread foundations. For soils, however, the linear dependence is valid only for a certain small interval of surcharge loading. Consequently, the calculation introduces the modulus of deformation that expresses the linear substitution of the deformation curve for a certain interval of stress. The CSN standard uses the constrained modulus. Inevitably, the question arises concerning the way this value can be obtained: In general, we have two approaches: ● Tabulated values – in terms of our practice, this represents indicative standard characteristics: tables for individual soil classes define intervals of values that can be used for 456

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL the given soil. The values are average values of moduli of deformation for the axial stress within the range from zero to the value of calculated resistance. The application of these values is according to CSN conditioned by geotechnical category no. 2. What is also important, we must not forget that we do not deal with the constrained modulus ( Eoed ), but with the deformation modulus ( Edef ). Therefore, we have to perform recalculation using Poisson coefficient, which also influences the final value. ● Test values – these are obtained in laboratories and, therefore, are more accurate. This approach is required by geotechnical category no. 3. We obtain the stress-strain diagram that can be used to define corresponding secant constrained moduli of deformation for given intervals of load. To be exact, we should define the complete stress-strain diagram and the program itself should use, depending on the given axial stress in a certain depth, the appropriate modulus – as required in DIN [171]. The inevitable problem is the fact that the structural engineer usually does not have such a diagram. Even in Germany, in practice they work usually with one secant modulus Es (called “average modulus”) that corresponds to the constrained modulus for the given stress level – see fig. 15. So, what is the way out? The basis for the success is the awareness that the indicative value of modulus Edef is only approximate. Already this piece of knowledge can help us think about which modulus should be input in our specific situation. It is definitely possible to try and estimate the predominant value of axial stress on the basis of the position of the given geological stratum in the geological profile and corresponding contact stress, and then, using either the intervals of indicative values or more accurate survey, choose the appropriate value of Edef . This uncertainty in the determination of the “correct” value leads us to what was already mentioned earlier: to the fact that we must try to eliminate the variability of the input data through repetitive calculations with different variants of geomechanical input values.

6.3.4.6.2 Poisson coefficient of transverse contraction The uncertainty connected to the input of geomechanical soil properties is further increased by the estimate of the Poisson coefficient of transverse contraction ν in subsoil strata. This coefficient is difficult to determine in soils and, moreover, the question arises whether a particular (granular) material can be modelled by means of a continuum, i.e. continuous environment, at all. The standard-defined formula for settlement (6.3.1) contains constrained modulus Eoed that is related to Edef through formula (6.3.8) Eoed = Edef

(1 −ν ) (1 +ν )(1 − 2ν )

(6.3.8)

In particular in the area around the limit value of the Poisson coefficient 0.5 (noncompressible soil in terms of volume with infinitely large Eoed ) there is a significant sensitivity to a small change in coefficient ν . Therefore, it is necessary to be very cautious for

457

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL clays with high plasticity and, preferably, perform two calculations with different values of the Poisson coefficient.

6.3.4.6.3 Soil density Soil density has a significant impact on the determination of the structural strength in the given depth. If the analysed depth is above the groundwater level, the calculation is carried out with the density of dry soil. If permeable and impermeable soils are combined in the given geological profile, we have to take into account their mutual position. For example, if clay is below some permeable soil with groundwater, the stratum is subjected to the pressure of the water located above this stratum (that was subtracted from the uplift pressure), which is demonstrated by a horizontal step-change in the distribution of the geostatic stress σ or . In the model, this step-change can be achieved through a thin stratum (defined in the geological profile) with a significantly greater density.

6.3.4.6.4 Coefficient of structural strength The CSN standard [169] introduced the hypothesis of direct proportion between structural strength σ s and initial (original) geostatic stress σ or (see formulas in chapter 6.3.4.3). Coefficient m is for different foundation soils determined experimentally – see table 10 of the standard [169]. CSN specifies that the soil starts to deform if the stress due to the structure load reaches 10 to 50% of the geostatic stress in this depth (i.e. m = 0.1 to 0.5). Moreover, the structural strength influences the calculation of settlement, which means that, for example, two values m = 0.2 and m = 0.3 can give settlement results that differ in tens of percents. The soil is not a homogenous substance and its compression does not represent the deformation of the grains, but the deformation is caused by the change of the structure, positions of grains, their rotation, wedging in, etc. And it is just this coefficient m into which the Czech standard integrated the biggest uncertainty that we have to face when determining the settlement. Therefore, it is crucial to consider this table as a guide and to consult important cases with an engineer-geologist, or to perform a series of calculations with different values of coefficient m .

6.3.4.6.5 Excavations The depth of excavation (i.e. the depth of the location of the corresponding contact stress) influences coefficient κ1 and thus increases the speed of the decrease of the vertical stress component over the depth – see page 24 of the standard [169]. This manifests itself in the results by the fact that settlement is lower than if the surcharge loading would be applied to the original surface of the terrain. It is also due to the fact that the structural strength at the level of the foundation base in an excavation is considered to be zero, which significantly influences the values of the effective stress. A frequent question relating to foundations in excavations is the issue of surcharge loading. The DIN standard [171] strictly stipulates that the original weight of the excavation

458

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL is subtracted from the load. Both CSN [169] and EC [170] do not contain such a requirement and, therefore, we do not recommend that the original weight of the excavation be subtracted. If the excavation is done, the soil “breaths out” after some time, and if we applied load equal to the weight of the excavation, some settlement would occur, which must be taken into account in the model. Moreover, the calculation according to CSN (or EC) applies also the structural strength and, therefore, settlement (on condition that the weight of the excavations was subtracted) would occur only if the original load was greater than the weight of the excavation + the structural strength in the given depth ( γ h(1 + m) ). This is a strongly optimistic assumption and it would never happen in practice. Consequently, it is reasonable in calculations according to CSN and EC to leave the surcharge loading equal to the calculated contact stress. What is often also related to the problem of excavations is failure in the iteration of the calculation. Quite often, this happens due to such singularities in the model that are caused by zero settlement, which assigns large values of C S parameters in some isolated areas. However, zero settlement is often unrealistic and represents just an “error in the model” – for example the “breath-out” of soil strata in the excavation. Therefore, for deeper excavations (with a great structural strength) or small surcharge loadings, it is more convenient to “help” the model with a decreased value of the coefficient of the structural strength m (almost up to zero) in the stratum just below the foundation base.

6.3.4.7 Reduction of the dimension of the interactive problem In civil engineering practice, calculation models of structures are mostly created from planar and beam (finite) elements, i.e. 2D and 1D. The subsoil as soil environment is, however, a typical 3D medium and it should be analysed that way (i.e. 3D). In general, the system (structure + foundation + subsoil) is 3D in nature and if we wanted to know in detail the stress-state and deformation below the foundation, we would have to model the subsoil using 3D finite elements. This would, on the other hand, unproportionally increase the number of unknown parameters of deformation and – in practical models – we would exceed the time and capacity limits of contemporary computers. Moreover, if the application of 3D finite elements was driven by the attempt to perform a more detailed analysis, such a solution would make unproportionally big demands on the physical input data and, as a result, the geological survey would strongly increase the total costs of the whole project. Fortunately enough, the primary goal is the design of the structure and foundations and we are interested in the conditions in the subsoil only to be able to determine its effect on the response of the structure. In that situation we can swap to a solution in which the whole 3D subsoil is represented just by its 2D surface.

6.3.4.8 Surface model of subsoil The least credible is the Winkler model that considers the subsoil to be an infinitely dense system of springs or thick liquid. This model is not capable of expressing the creation of a subsidence basin or co-action of neighbouring buildings. The Winkler model was

459

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL improved by Pasternak who (in order to give a true picture of shear components that emerge in non-uniform settlement) established the second constant C2S . At around 1975, V. Kolar and I. Nemec introduced a new concept for this issue – see [22]. This is based on creation of such a 2D surface model the deformation of which produces the same virtual work as in 3D subsoil. With this, a whole hierarchy of parameters can be built (fig. 16). The most important are: ● C1S parameters of the interaction of the foundation with the surface 2D model of the subsoil in physical relations containing components of displacement u , v, w . Winkler formula for vertical components: σ z = r [kPa]

= C1Sz [MNm -3 ] ⋅ w [mm]

(6.3.9)

for horizontal shear components: τ zx = sx [kPa] = C1Sx [MNm −3 ] ⋅ u [mm]

(6.3.10)

τ zy = s y [kPa] = C1Sy [MNm −3 ] ⋅ v [mm]

(6.3.11)

● C2S parameters of the interaction of the foundation with the surface 2D model of the subsoil in physical relations containing the first derivative of settlement. Pasternak formula for shear forces: t x [kNm −1 ] = C2Sx [MNm −1 ] ⋅ ∂w ∂x [mm m]

(6.3.12)

t y [kNm −1 ] = C2Sy [MNm −1 ] ⋅ ∂w ∂y [mm m]

(6.3.13)

460


Fig. 16 Physical meaning of some C parameters

This new model of the subsoil is described in detail in [22] and, therefore, in this text we limit ourselves to a brief derivation for the purpose of the explanation that will follow: The formula for the potential energy of internal forces of the 3D model has the following form: i Π 3D =

1 T 1 T σ ε dV = ∫ ε Dε dV ∫ 2V 2V

(6.3.14)

Neglecting the effect of horizontal components of deformation, we get the following vectors: T

σ = σ z ,τ zx ,τ yz  = Dε  ∂w ∂w ∂w  ε = ε z , γ zx , γ yz  =  , ,   ∂z ∂x ∂y 

(6.3.15) T

T

(6.3.16)

This means the corresponding simplification of the matrix of physical constants D . 461

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL  Ez D =  0  0

0 G 0  0 G  0

(6.3.17)

In order to be able to reduce the problem from 3D to 2D, it is necessary to integrate formula (6.3.14) over the z -axis. For this reason, a certain “damping function” f ( z ) is introduced and it is defined by the ratio of the settlement in the given depth to the settlement of the surface w0 ( x, y ) f ( z) =

w( x, y, z ) w0 ( x, y )

(6.3.18)

Modifying (6.3.16) and (6.3.18) we get   ∂w ( x, y ) ∂f ( z ) ∂w0 ( x, y ) ε =  w0 ( x, y ) , f ( z ), 0 f ( z) ∂z ∂x ∂y  

T

(6.3.19)

Substituting (6.3.19) into the formula for the potential energy of body V = ΩH , where Ω is the extent of the 2D model and H is the depth of the deformed zone of the 3D model, we obtain the following formula: Π i2 D = Π i3D = =

1 σ zε z + τ zxγ zx + τ yzγ yz  dV = 2 V∫ 

1 ε z2 Ez + (γ zx2 + γ yz2 )G  dV = 2 V∫ 

(6.3.20)

2 ⌠ H  2 H 2  ∂w0  H 2 1   2 ⌠  ∂f   ∂w0  2  dΩ = w0  Ez   dz +  f Gdz + f Gdz    ∫ ∫   2  ∂x  0 ⌡  ∂z   ∂y  0  ⌡  0 Ω

Integrating over z , we get the formula for the potential energy of internal forces of the 2D model with two parameters: C1S , C2S  ∂w ( x, y )  1 ⌠⌠   ∂w ( x, y )  =  C1Sz w02 ( x, y ) + C2Sx  0 + C2Sy  0   2  ∂x ∂y     ⌡⌡  2

Π

i 2D

Ω

2

  dΩ 

(6.3.21)

Comparing (6.3.20) and (6.3.21), we can define the relation between the parameters of the general (3D) and surface (2D) model:

462

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL H

⌠  ∂f ( z )  C =  Ez   dz ⌡  ∂z  2

H

C = C = ∫ Gf 2 ( z )dz

S 1z

S 2x

S 2y

(6.3.22)

0

0

In this interpretation, the surface model has been implemented into the SCIA•ESA PT system [173] in such a way that the energy accumulated in the subsoil is added to the potential energy of the structure. What remains to be answered is how the appropriate C S parameters can be obtained with the best possible accuracy. This can be achieved using the SOILIN module (see [165], [172]) that – on the basis of the stress-state of the elastic homogenous half space and standard-defined model – determines in any location the distribution of settlement and subsequently the sought-after C S parameters. The calculation of parameter C1Sz is not carried out according to formula (6.3.22), but using directly the values of stress σ z and strain ε z from the stress-strain diagram, i.e. through the comparison of area density of the energy corresponding to compression ε z for 3D model and 2D model, which represents the first members in the expressions for the potential energy in formulas (6.3.20) and (6.3.21). That means H

1 1 σ zε z dz = C1Sz w02 ( x, y ) ∫ 20 2

(6.3.23)

Modifying this we get the following formula for the calculation of parameter C1Sz H

∫ σ ε dz z z

C = 2 1z

0

(6.3.24)

w02 ( x, y )

As the similar procedure (i.e. the determination of slopes γ zx and γ yz ) applied to the calculation of parameters C2S would cause numerical complications due to the necessity to determine other settlement values in ambiguously obtainable differences, it was decided to perform the calculation of parameters C2S by means of what is termed “isotropic form” using formulas (6.3.18) and (6.3.22). H

∫ Gw ( x, y, z )dz 2

H

C2Sx = C2Sy = ∫ Gf 2 ( z )dz = 0

0

w02 ( x, y )

(6.3.25)

As the C S parameters influence the contact stress and, at the same time, the distribution of the contact stress has an impact on the settlement of the foundation base and thus also on the C S parameters, it is necessary to perform the calculation of the structure-soil interaction in an iterative way – see fig. 17.

463


Fig. 17 Scheme of the iterative calculation

Therefore, the calculation of the superstructure and the determination of the C S parameters are performed in turns and, in majority of realistic problems, the relative concord can be obtained from an arbitrary initial assumption. Suitability of the initial assumption influences only the number of iteration steps. The results of the calculation are internal forces and deformations of the structure, settlement of the subsoil surface, contact stress in the foundation base in individual iteration steps and the final interaction parameters C S .

6.3.4.9 The effect of subsoil outside of the structure The subsidence basin of the subsoil does not end at the boundary of the foundation structure, but it extends further outside of the foundation. The reason is that the transfer of shear through components τ zx and τ yz produces vertical axial stress σ z even in the subsoil outside of the foundation base. It means that deeper subsoil strata are compressed in the location that is not subjected to any surface load. As a result, it is the value of the structural strength σ s that determines the boundary of the subsidence basin. This phenomenon is coped with in the SOILIN program through the approach in which the stress-state in the subsoil is solved according to the standard-defined half space model exactly using the analytical formulas in an arbitrary point ( x, y, z ) , i.e. even outside of the foundation base. The appropriate geological profile is taken into account in the given location and, therefore, the settlement of the surface and the interaction parameters C S can be determined. And just the C2S parameters that appear in the physical formulas containing the first derivative of settlement (it is in fact the elastic resistance of the surrounding against 464

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL skewing) allow in the model of the superstructure for the transfer of shear in the subsoil, which is successfully exploited in practice. The subsoil outside of the foundation itself can be taken into account, in general, in two ways: 1. Planar “subsoil” macroelements of an insignificant thickness are defined around the slabs modelling the foundation. The iteration process will calculate the settlement and the corresponding C S parameters even in these locations, which ensures correct supporting conditions of the edges of the foundation structure. In addition, this approach makes it possible to obtain an idea about the size of the subsidence basin. 2. Newer versions of the SOILIN module make the necessary steps to consider the effect of the surrounding subsoil automatically. The program detects the edge of a continuous region (or several regions) on the subsoil and vertical supports are assigned to the nodes at these edges. In every iteration, the stiffness of these springs is calculated from the just determined C S parameters of those elements that adjoin the given node. It is an approximate modelling, but the results are close to the results of variant 1. Moreover, the calculation is faster as it is not necessary to add “subsoil” elements. If the effect of the surrounding subsoil is not to be considered at a specific edge (e.g. in the vicinity of a sheet pile wall), it can be achieved through the input of a spring with a small stiffness. Such an input at a corresponding line overwrites the springs generated in SOILIN. The consideration of the effect of the subsoil outside of the foundation structure can have a significant impact on the behaviour of the whole structure. Simply said, the absolute value of the accompanying settlement decreases, but the relative settlement and internal forces increase, which is caused by the more definite support conditions at the edges of the foundation structure. The modelling of the interaction is more accurate, however, often at the cost of higher number of iterations. It can also happen that, due to singularities occurring at the boundary of the foundation, the required convergence does not happen even for increased maximum number of iteration. Then it is necessary to closely scrutinise the results of e.g. two consecutive iterations and make a “professional estimate”.

6.3.4.10 Implementation into SCIA•ESA PT system First of all, a linear combination of long-term load cases must be defined and this combination must be selected in the settings for the calculation of the structure-soil interaction. The geological profile is defined by means of geological bores directly in the location of drill holes. For that purpose, the geological profiles must be defined and several geological strata (see fig. 18) can be assigned to them. These are characterised by the stratum thickness, modulus of deformation Edef , Poisson coefficient of transverse contraction ν , unit weight of dry and wet (saturated) soil γ and coefficient of structural strength m (for the CSN standard). If the water table is found in a certain depth, this distance from the top of the bore can be specified and the unit weight of wet soil reduced by 10 kN/m3 is automatically 465

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL assigned to all the strata located below the water level. If the option “Non-compressible soil” is selected, the program automatically introduces the depth reduction coefficient according to CSN. Numerically it means that the damping of stress component σ z of the elastic half space is slowed down.

Fig. 18 Geological profile

The corresponding geological profile is positioned through the x, y, z global coordinates that denote the point on the surface of the terrain (see fig. 19). Therefore, the position of terrain must be harmonised with the already input superstructure, so that the global z -coordinates match each other. It is possible to define an arbitrary number of bores (borehole profiles) that must, however, contain the same number of geological strata. If some of the geological stratum is missing in a certain geological profile (it diminished), it still must be included in the corresponding geological profile. At least, some very small thickness must be input together with appropriate characteristics, so that the continuity of individual strata in the complete model of the subsoil is not interrupted.

466


Fig. 19 Location of bores

The program itself then interpolates both the surface of the terrain (“the digital model of the terrain” can be displayed – see fig. 20), the level of each geological stratum and all geomechanical characteristics.

Fig. 20 Display of the terrain

The interpolation is very robust and creates a smooth function surface from practically any input. It means that even a rough terrain can be easily modelled. In plan view, the surface is automatically defined by a rectangle whose edges are 10 m from the foundation structure. This makes it possible to avoid “sharp” transitions between geological zones that could – due to “step-changes” in stiffness – devaluate the determined supporting conditions in the given places and thus produce unreal internal forces in the superstructure. 467

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL As the user has defined the bore-holes and as the model of the superstructure has been already created, the program can in every place determine the corresponding depth of excavation, in which also the thickness of the foundation structure is taken into account. The same applies to the depth of excavation determined for the calculation of the C S parameters, as well as to the correct determination of the surcharge loading level that models the contact stress for the given load combination. The calculation of the settlement of the subsoil and subsequent determination of the C parameters is performed in a standard way using an iterative process. The result of this process is the state in which the contact stress or displacement u z in two subsequent iterations does not change significantly. For that reason, the following quadratic norms are evaluated in every j -th iteration: S

∑ (σ

− σ z ,i , j −1 ) Ai

n

εσ =

i =1

2

z ,i , j

(6.3.26)

n

∑σ i =1

∑ (u

z ,i , j

n

εu =

i =1

− u z ,i , j −1 ) Ai 2

z ,i , j

(6.3.27)

n

∑u i =1

⋅ σ z ,i , j −1 Ai

z ,i , j

⋅ u z ,i , j −1 Ai

where n

number of nodes

σ z ,i

contact stress in node i

Ai

area corresponding to node i

u z ,i

global displacement of node i in the z -direction

The iterative calculation is stopped if ε σ < 0.01 or ε u < 0.001 . Under these conditions, the settlement is “proclaimed to be tuned” and further we deal only with the results for the superstructure. It means that we are interested in the deformation, internal forces and stress in the building. If any problem occurs during the iterative calculation – e.g. the problem does not iterate, the number of iterations is too large, etc. – it is suitable to display the distribution of the contact stress in individual iterations, which may often help find the cause of the unfavourable behaviour. Moreover, we can display the C S parameters calculated by the SOILIN module (see fig. 21) and a table with the settlement of the subsoil. This table is useful e.g. to find out the settlement of points outside of the foundation structure.

468


Fig. 21 Calculated parameter C1z

6.3.4.11 Statistical analysis of the structure-soil interaction As already stated in chapter 6.3.4.6 (see also [179]), the variability of input data is very strong in soils, as the uncertainty (variability) of the data relating to soil properties is much bigger than in the case of structural materials (steel, concrete, timber). This is given in the first case by the physical nature of the soil. It is a strongly heterogeneous material with a huge influence of load history (e.g. compression, unloading) or with physical or chemical changes. Another significant factor is the variability in the geological structure of the subsoil. All this has an impact on the reliability of the structure (embankment, earth cut, retaining wall, foundation). The consequence of all this is that the probability and stochastic approach is currently being more and more applied in civil engineering, contrary to the deterministic approach used in the past – the fact that is reflected also in EUROCODE 7. This approach is most often applied in the calculation of stability of slopes and underground works. What also proves boom of the stochastic approach is the organisation of the symposium “Risk and Variability in Geotechnical Engineering” in 2003. As already mentioned above, the soil parameters (e.g. angle of internal friction - φ ′ , cohesion - c′ , compressibility – E Cc ; Cr and permeability - k ) are random variables and, therefore, it is necessary to determine the statistical parameters or models of distribution of the probability. This raises the question of what type of the distribution should be used for the monitored soil properties. There are two approaches possible. The first one is based on measured data which can be analysed and the best distribution can be determined. The second, more common, approach is based on the already verified previous experience – see [174], [175], [176]. This approach is characterised by the fact that the authors choose the type 469

6.3 THEORY OF PLATES ON THE EFFICIENT SUBSOIL MODEL of distribution without having performed the statistical analysis of the measured data (most commonly chosen distributions are normal, lognormal or Beta distribution). For the above-mentioned reasons, the crucial factor in the solution of the structure-soil interaction seems to be the focus on the examination of the variability and statistical characteristics of deformation properties of the soil under foundation. It means that, in the first place, it is necessary to deal with the values of the constrained (oedometric) modulus or Poisson coefficient (due to the recalculation to the deformation modulus). Also the unit weight of the soil and the variability of thickness in individual strata of the geological profile can have a significant influence – see [166]. These deformation characteristics in fine grain soils depend on the consistency or degree of saturation, while in coarse grain soils on their density. Moreover, the magnitude of the constrained modulus is also influenced by the size of stress or load and, therefore, with respect to the stated fact, the range of the stress for which the modulus was determined must always be stated – the higher the load, the greater the value of the modulus. The final solution thus features repeated deterministic calculations for different sets of input data in every run. These input data are obtained using the probabilistic approach. The object of the analysis is the deformation and internal forces that are crucial for the analysis of the structure. Some international studies deal with these approaches – see [174], [177], [178].

6.3.4.12 Conclusion It is clear from the previous chapters that correct modelling of the structure-soil interaction is not a simple issue. Even the selection of appropriate input data for the subsoil is not problem-free. The physical model of the soil is strongly nonlinear and, therefore, the complete solution of the structure-soil interaction is significantly influenced by this factor. In addition, it is necessary to realise that the interaction parameters that model the supporting conditions are not constants (of the subsoil), but that they depend on the whole range of factors and, consequently, they reach different value in different place. And last, but not least, the model must take into account also the effect of the surrounding subsoil and neighbouring buildings. SCIA•ESA PT with the integrated SOILIN module represents a versatile tool which can successfully handle all the above-mentioned pitfalls and produce reasonable results.

470

7.1 Introduction

7 Nonlinear Mechanics of Continua and Structures 7.1 Introduction 7.1.1 Selected Mathematical Concepts and Notations

7.1.1.1 Index, tensor and matrix notations Let us recall some mathematical notations that are used in publications on nonlinear mechanics. They are mainly of the following three types: index, matrix and tensor notations.

Index notation Under index notation rules, tensor or matrix components are specified explicitly. For instance, a vector, which is in fact a first order tensor, is written in index notation with one index, the value of which varies from one to the number of dimensions. So a position vector may be denoted as xi . If an index is repeated in an expression, the Einstein summarization rule applies, i.e. the expression represents a sum of products and the summarization is carried out with respect to the repeating index, e.g. c j = Aij bi .

Tensor notation In tensor notation, no subscripts or superscripts are given. The formulae are thus easier to remember. Tensors are denoted in bold. First order tensors are usually denoted by small letters, while for higher order tensors capital or Greek letters are used. In tensor notation, the sum of products with respect to the internal indices is denoted by a dot, to distinguish it from the matrix multiplication, which is denoted without a dot. E.g. A ⋅ B ≡ Aij B jk . The colon in tensor notation stands for the sum of products with respect to the pairs of internal indices repeated in the same order, e.g. A : B ≡ Aij Bij .

Matrix notation This notation can be often found in publications on the finite element method. It differs from the tensor notation in that no dot is placed in a matrix product. The matrices are 471

7.1 Introduction denoted by bold letters. One-dimensional matrices are often denoted by small letters and twodimensional matrices by capital letters. One-dimensional matrices are assumed to have one column, for two-dimensional ones the first index stands for the row and second for the column. Transposition is the interchange of the rows and columns. Notice that: A ⋅ A = AA

A ⋅ x = Ax

but

x ⋅ A = xT A

(7.1.1)

if A is a square matrix and x is a one-dimensional matrix. To compare the different notations, consider the following examples. a) A square of the size of a vector: r 2 = xi xi = x ⋅ x = xT x { { { index notation

tensor notation

(7.1.2)

matrix notation

b) Quadratic form: x Aij x j = x ⋅ A ⋅ x = xT Ax 3 1 424 3 1 4i 24 3 1424 index notation

tensor notation

(7.1.3)

matrix notation

c) Constitutive relations: σ ij = Cijkl ε kl

index notation

(7.1.4)

σ = C:ε

tensor notation

(7.1.5)

{σ} = [C]{ε}

matrix notation (Voigt)

(7.1.6)

It is necessary to keep in mind that while ε and σ are second order tensors and C is a fourth order tensor, {ε} and {σ} are one-dimensional matrices, derived from the corresponding

tensors by Voigt rule (see below) and [C] is a symmetric square matrix, Voigt notation of a fourth order tensor C . d) The density of the potential energy of internal forces: 1 1 1 T ε ij Cijkl ε kl = ε : C : ε = {ε} [C]{ε} 2 2 243 144 2 14243 14 2443 index notation

tensor notation

(7.1.7)

matrix (Voigt) notation

e) Cauchy equilibrium equation: Index notation: δσ ji δ xj

+ ρ bi = 0

(7.1.8)

472

7.1 Introduction Tensor notation: ∇ ⋅ σ + ρb = 0

(7.1.9)

ρb are body forces. The operator ∇ (pronounced del) is called “divergence operator”, sometimes also denoted div. Its application to a second order tensor follows from the comparison of index and tensor notation of the Cauchy equilibrium equations (equations (7.1.8) and (7.1.9)) – the result is a vector. Its application to a vector is defined by the index notation ∇ ⋅ v = div( v) =

∂v1 ∂v2 ∂v3 ∂vi + + = ∂x1 ∂x2 ∂x3 ∂xi

(7.1.10)

and the result is a scalar. Matrix notation (Voigt notation): ∂ {σ} + ρ b = 0

(7.1.11)

Operator ∂ in Voigt notation has the same meaning as divergence operator ∇ in tensor notation. Let us write it in the explicit form: ∂   ∂x  ∂= 0   0 

0

0

0

∂ ∂z

∂ ∂y

0

∂ ∂z

0

0

∂ ∂z

∂ ∂y

∂ ∂x

∂  ∂y  ∂  ∂x   0 

(7.1.12)

To remember it better, note the symmetry of the left and right half of the matrix and the scheme of successive derivatives, which resembles the Voigt rule:    sym.

0

0

0

0

0 sym.

   0 

The ∂ operator can be also used for geometrical equations in linear mechanics:

{ε} = ∂ T u

(7.1.13)

473

7.1 Introduction f) Gradient The same operator ∇ (speak del) is used as for the divergence, but without the dot behind the ∇ operator for a vector and a tensor. The gradient is also denoted as grad . The gradient applied to a scalar is defined as follows:  ∂a     ∂x1   ∂a  ∂a ∂a ∂a ∇a = grad ( a ) = i+ j+ k=  ∂x1 ∂x2 ∂x3  ∂x2   ∂a     ∂x3 

(7.1.14)

and the result is a vector. i, j, k are unit vectors in the direction of axes x1 , x2 , x3 . When applied to a scalar, the ∇ operator is also called the Hamilton operator. When applied to a vector field, it is called the “vector field gradient”. It indicates successive differentiation with respect to all coordinates. Let us show its application to vector v ∇v ≡ grad v =

∂v  vx , x = ∂x v y , x

vx , y  v y , y 

(7.1.15)

Let us recall that the differentiation is made with respect to the variable behind the comma in the subscript. The gradient of the vector is a second order tensor. It must be emphasised that ∇v = grad( v ) ≠ ∇ ⋅ v = div( v ) . The result of the divergence operation ∇ ⋅ v is a scalar which is the sum of the derivatives of all components of vector v with respect to the corresponding coordinates. Note: In some publications, ∇ denotes the left gradient, which, applied to a vector, is the transposition of the ∇ operator used here.

7.1.1.2 Voigt notation In the finite element method, second order tensors are often written as column matrices. This kind of notation is called Voigt notation. The procedure used to convert a symmetric second order tensor into a column matrix is called the Voigt rule. The Voigt rule depends on whether the tensor is a kinetic quantity, e.g. stress, or a kinematic quantity, for instance strain. For kinetic tensors such as stress tensor σ the following holds: For 2D

474

7.1 Introduction σ 11  σ 1   σ x  σ 11 σ 12        σ≡ → σ 22  = σ 2  =  σ y  = {σ}  σ 21 σ 22  σ  σ  σ   12   3   xy 

(7.1.16)

This rule may be symbolised by the following diagram:

For 3D

σ 11 σ 12 σ ≡ σ 21 σ 22 σ 31 σ 32

σ 11  σ 1  σ x  σ  σ  σ   22   2   y  σ 13  σ  σ  σ z  σ 23  →  33  =  3  =   = {σ} σ 23  σ 4  γ yz  σ 33  σ 31  σ 5  γ zx        σ 12  σ 6  γ xy 

(7.1.17)

This rule may be symbolised by the following diagram:

Note that while for the diagonal elements the components of the Voigt vector are arranged in the order of subscripts (1,2,3), for the non-diagonal elements the components are arranged according to the subscript which is missing in the relevant component. The assignment of subscripts can also be written into a table where i, j are the subscripts of the tensor and a is the Voigt notation index.

2D i 1 2 1

j 1 2 2

3D a 1 2 3

a 1 2 3 2 3 1

i 1 2 3 3 1 2

j 1 2 3 4 5 6

For kinematic tensors such as strain tensor ε the same principle applies as for the kinetic tensors, but the non-diagonal elements of the tensor must be multiplied by two in Voigt notation: 475

7.1 Introduction 2D :  ε11   ε1   ε x  ε11 ε12        ε≡ →  ε 22  = ε 2  =  ε y  = {ε}  ε 21 ε 22  2ε  ε  γ   12   3   xy 

(7.1.18)

3D :

ε11 ε12 ε ≡ ε 21 ε 22 ε 31 ε 32

 ε11   ε1   ε x   ε  ε   ε   22   2   y  ε13   ε  ε   ε z  ε 23  →  33  =  3  =   = {ε} 2ε 23  ε 4  γ yz  ε 33   2ε 31  ε 5  γ zx         2ε12  ε 6  γ xy 

(7.1.19)

A one-dimensional matrix is often called a vector, but it has to be borne in mind that it is not a vector in the physical sense of the word. The difference lies in the transformation. A physical vector is actually a first order tensor and in transformation it is multiplied by a rotation tensor (the matrix of directional cosines). For the transformation of tension and strain vectors in Voigt notation a more complex transformation matrix has to be used.

7.1.1.3 Voigt rule for higher order tensors The conversion of fourth order tensors into Voigt notation of a square matrix has practical applications. Assuming linear elasticity, the constitutive relations can be expressed as follows: σ ij = Cijkl ε kl in index notation or σ = C : ε in tensor notation,

where Cijkl or C is a forth order tensor. The Voigt matrix notation could be written as

{σ} = [C]{ε} or σ a = Cab ε b , where for a ← ij and b ← kl the same rule is used as for second order tensors. The Voigt notation of a constitutive matrix for a linearly elastic problem of plane stress can be written as follows: 476

7.1 Introduction  C11 C12 [C] = C21 C22 C31 C32

C13   C1111 C1122 C23  = C2211 C2222 C33   C1211 C1222

C1112  C2212  C1212 

(7.1.20)

The subscripts in the first matrix belong to Voigt notation and the subscripts in the second matrix correspond to index notation of a fourth order tensor. To verify the above conversion, let us show, for instance, element σ 12 in index notation: σ 12 = C1211 ε11 + C1212 ε12 + C1221 ε 21 + C1222 ε 22

(7.1.21)

The relation in Voigt notation can be written as follows: τ xy = σ 3 = C31 ε1 + C33 ε 3 + C32 ε 2

(7.1.22)

It can be shown that both relations are equivalent if we realize that γ xy = ε 3 = ε12 + ε 21 = 2ε12 and that tensor C is symmetric, i.e. C1212 = C1221 = C33

7.1.1.4 Tensors Since this term is often used in publications on nonlinear mechanics some information is worth mentioning.

Tensor transformation A tensor is a mathematical entity with certain properties. The most important property of a tensor is the way it transforms during the transition from one coordinate system to another.

Transformation of vectors The transformation relations between two coordinate systems are, in general (if the translation is neglected), defined by the following equation

xi = Fij X j =

∂xi Xj ∂X j

(7.1.23)

in tensor notation x = F⋅X

(7.1.24)

F is the Jacobi transformation matrix given by 477

7.1 Introduction Fij =

∂xi ∂X j

(7.1.25)

or in tensor notation F=

∂x ∂X

(7.1.26)

The inverse transformation is defined by the formula X = F −1 ⋅ x

(7.1.27)

where F −1 =

∂X ∂x

(7.1.28)

The same transformation equations apply also to other vectors. Therefore, for example, if vector v 0 is defined in X coordinates and v is the same vector defined in x coordinates, then we can write the following transformation equation vi (x) = Fij v0 j ( X) =

∂xi v0 j ( X) ∂X j

(7.1.29)

∂x ⋅ v 0 ( X) ∂X

(7.1.30)

or in tensor notation v ( x ) = F ⋅ v 0 ( X) =

The rotation of the solid as a rigid body plays an extremely important role. In that case, the derivatives in the Jacobi matrix F are what is termed “direction cosines”, i.e. the cosines of the angles between corresponding axes. We will call this special, but in technical terms very important, case of the Jacobi matrix a tensor, or rotation matrix and we will denote is R .

Fig. 1.1 Rotation of coordinate system

For 2D and configuration as in the figure, tensor R would be in the form: cos α R=  sin α

− sin α  cos α 

(7.1.31)

478

7.1 Introduction The movement of a solid as a rigid body consisting of translation and rotation around the origin of the coordinate system can be described by the following equation: x = R ⋅ X + xT = RX + xT

(7.1.32)

or, in index notation: xi = Rij X j + xTi

(7.1.33)

where xT is the vector of the displacement of origin and R is the rotation tensor, also known as the rotation matrix. Since the components of tensor R are the cosines of the angles formed by the corresponding axes of both systems of coordinates, tensor R is also called the matrix of directional cosines. The explicit expression of this matrix can be written as follows:  cxX R = c yX  czX

cxY c yY czY

cxZ  c yZ  czZ 

(7.1.34)

where ciJ is the cosine of the angle formed by axes i and J ( i = x, y, z , J = X , Y , Z ). The advantage of this notation lies in the fact that cosine is an even function and the relevant angles do not have to be defined as oriented angles. This matrix is an orthogonal matrix, which means that its inverse equals its transpose R −1 = R T . It can be easily shown from the assumption that the length (and consequently also its square) remains constant. x ⋅ x = X ⋅ RT ⋅ R ⋅ X tensor notation

(7.1.35)

xT x = XT RT RX

(7.1.36)

matrix notation

Equality R ⋅ X = X ⋅ RT was used in the tensor notation. It follows from the above-mentioned assumption that RT R = I and, therefore, R −1 = R T . Given the orthogonality of matrix R , the rotation is called an orthogonal transformation.

Transformation of second order tensors A physical vector (as distinguished from the term vector used for one- σ dimensional matrices) can be regarded as a first order tensor. Then the scalar is a zero order tensor. Unless indicated otherwise, the term tensor means a second order tensor which can be written as a square matrix. The mechanics of continuum uses, first of all, the tension or stress tensor (from which the very word for tensor derives) and the strain tensor. We can write the following transformation relations for the transformation of a second order 479

7.1 Introduction tensor: σ ij (x) =

1 ∂xi ∂x j σ PQ ( X) J ∂X P ∂X Q

(7.1.37)

∂X I ∂X J σ pq (x) ∂x p ∂xq

(7.1.38)

σ IJ ( X) = J

where J is the determinant of the Jacobi matrix  ∂x  J = det    ∂X 

(7.1.39)

σ is a tensor in the x coordinates and σ is a tensor in the X coordinates. In tensor notation, the transformation equation can be rewritten: 1 F ⋅ σ ( X) ⋅ FT J σ ( X) = JF −1 ⋅ σ (x) ⋅ F −T σ ( x) =

If we deal with the rotation of the solid as rigid body, we get J = 1 ,

(7.1.40) (7.1.41) ∂X I = cIp ∂x p

is cosine of the angle between axes X I and x p

Consequently, the following holds: σ (x) = R ⋅ σ ( X) ⋅ RT = Rσ ( X)RT σ ( X) = R T ⋅ σ ( x) ⋅ R = R T σ ( x) R

(7.1.42) (7.1.43)

R is the rotation tensor. The same transformation relations holds for any 2nd order tensors, e.g. tensors ε and ε in the x and X coordinates respectively. ε( x) = R ⋅ ε ( X) ⋅ R T = R ε ( X)R T ε ( X) = R T ⋅ ε ( x) ⋅ R = R T ε ( x ) R

(7.1.44) (7.1.45)

In the tensor notation tensors need not be distinguished as kinetic or kinematic tensors. When working in Voigt notation, we can introduce transformation matrix Tε which satisfies the relation

{ε(x)} = Tε { ε ( X)}

(7.1.46)

where ε is a kinematic tensor in the x coordinates and ε is the same tensor in the X coordinates. Because the Tε matrix is of considerable importance, let us introduce it in the explicit form:

480

7.1 Introduction 2  cxX  2  c yX 2  czX Tε =   2c yX czX  2c c  zX xX  2cxX c yX

2 cxY 2 c yY 2 czY 2c yY czY 2czY cxY 2cxY c yY

2 cxZ 2 c yZ 2 czZ 2c yZ czZ 2czZ cxZ 2cxZ c yZ

cxY cxZ c yY c yZ czY czZ c yY czZ + czY c yZ czY cxZ + cxY czZ cxY c yZ + c yY cxZ

cxZ cxX c yZ c yX czZ czX c yZ czX + czZ c yX czZ cxX + cxZ czX cxZ c yX + c yZ cxX

 cxX cxY  c yX c yY   czX czY  (7.1.47) c yX czY + czX c yY  czX cxY + cxX czY   cxX c yY + c yX cxY 

The matrix Tε is not orthogonal, and therefore:

{ ε ( X)} = Tε−1 {ε(x)}

(7.1.48)

Let us introduce the kinetic tensor σ (x) whose product with the conjugate kinematic tensor ε must be invariant, i.e. independent of coordinates system (e.g. virtual work). Let σ ( X) be the same tensor as σ (x) but in the X coordinates. Then we can write:

{δε} {σ} = {δε} {σ} = {δε} T

T

T

TεT {σ}

(7.1.49)

As a result we get the following relations for the transformation of stress:

{σ} = TεT {σ} = Tσ−1 {σ} {σ} = Tε−T {σ} = Tσ {σ}

(7.1.50) (7.1.51)

where Tσ = Tε−T

(7.1.52)

Let us introduce the Tσ matrix in the explicit form: 2  cxX  2  c yX  c2 Tσ =  zX cxX c yX c c  yX zX  czX cxX

2 cxY 2 c yY 2 czY cxY c yY c yY czY czY cxY

2 cxZ 2 c yZ 2 czZ cxZ c yZ c yZ czZ czZ cxZ

2cxX cxY 2c yX c yY 2czX czY cxX c yY + c yX cxY c yX czY + czX c yY czX cxY + cxX czY

2cxY cxZ 2c yY c yZ 2czY czZ cxY c yZ + c yY cxZ c yY czZ + czY c yZ czY cxZ + cxY czZ

 2cxZ cxX  2c yZ c yX   2czZ czX  (7.1.53) cxZ c yX + c yZ cxX  c yZ czX + czZ c yX   czZ cxX + cxZ czX 

Let us notice that the arrangement of the cosine functions in the matrices Tσ and Tε is identical and these matrices differ only by the fact that multipliers 2 from the left lower quadrant of the matrix Tε are transfered to the right upper quadrant of the matrix Tσ . Proof is simple: Let us denote the particular submatrices of the type (3,3) of the Tσ matrix as follows:

481

7.1 Introduction Tσ 12  T Τσ =  σ 11  Tσ 21 Tσ 22 

(7.1.54)

Let us introduce vectors

{ε ( x)} ∗

and

{ε ( X )} ∗

written from the ε(x) and ε ( X)

tensors by the Voigt rule for the kinetic tensors, i.e. without multiplication of the off-diagonal terms of the tensors by two. Let us divide these vectors into two parts, the first part containing the diagonal terms of the tensors ε and ε respectively and the second part containing the offdiagonal terms of these tensors. ε1∗  ∗ ε ( x ) = { } ε ∗   2 

 ε1∗  ∗ ε ( X ) = { } ε∗   2 

Then the following relations for the deformation vectors {ε} and  ε1∗  ε = { }  ∗ 2ε 2 

(7.1.55)

{ ε} hold :

 ε1∗  ε = { }  ∗ 2 ε2 

(7.1.56)

With respect to the fact that for notation of the vectors {ε ∗ (x)} and { ε ∗ ( X)} the Voigt rule for kinetic tensors was applied, the same transformation matrix Tσ which is applied for the

{σ}

and {σ} vectors, will be applied also for the vectors {ε ∗ (x)} and { ε ∗ ( X)} . Then we can

write:

{ε (x)} = T {ε (X)} ∗

σ

∗

(7.1.57)

For transformation of the vectors  T  ε   σ 11 {ε} =   =  2ε   2T  σ 21 ∗ 1 ∗ 2

{ε}

and

{ ε} the following relations must hold:

1  Tσ 12   ε1∗  2  ⋅  ∗  = Tε { ε} 1 2ε 2Tσ 22   2   2

(7.1.58)

Then we can write the following simple relation between the Tε and Tσ matrices:   Tσ 11 Tε =   2T  σ 21

1  Tσ 12   Tσ 11 2 =  1 2Tσ 22   2Tσ 21  2

1  Tσ 12  2  Tσ 22 

(7.1.59)

Transformation of fourth order tensors In the mechanics of continuum, we often come across a fourth order tensor. It is the tensor of material stiffness C which is defined by the following constitutive equation:

482

7.1 Introduction σ ij = Cijkl ε kl

(7.1.60)

or, in tensor notation: σ = C:ε

(7.1.61)

The following general relation holds for the transformation of this tensor between configurations X and x : Cijkl ( x ) =

ρ ( x ) ∂xi ∂x j ∂xk ∂xl 1 ∂xi ∂x j ∂xk ∂xl CPQRS ( X ) = CPQRS ( X ) ρ 0 ( X ) ∂X P ∂X Q ∂X R ∂X S J ∂X P ∂X Q ∂X R ∂X S (7.1.62)

CIJKL ( X ) =

ρ 0 ( X ) ∂X I ∂X J ∂X K ∂X L ∂X I ∂X J ∂X K ∂X L C pqrs ( x ) = J C pqrs ( x ) ρ ( x ) ∂x p ∂xq ∂xr ∂xs ∂x p ∂xq ∂xr ∂xs (7.1.63)

ρ 0 ( X ) and ρ ( x ) are the material densities in configurations X and x , respectively, C is the constitutive tensor in the X coordinates and C is the constitutive tensor in the x coordinates. J is the determinant of the Jacobi matrix. If only the rotation of a solid as a rigid body is concerned (there is no deformation), the transformation relation can be simplified. The density will be identical in both configurations ∂X i will be the cosines of angles formed by axes X i and x p , and the derivatives of the type ∂x p ∂X i i.e. = Cip ∂x p Therefore, we can write: CIJKL ( X ) = cIp cJq cKr cLs C pqrs ( x )

(7.1.64)

Cijkl ( x ) = ciP c jQ ckR clS CPQRS ( X )

(7.1.65)

or

In general, cIp and ciP are different values. In Voigt notation, we could write the transformation of the constitutive matrix [C] by using the transformation matrix Tε from the equation (7.1.46), on condition that we consider that the density of the work of deformation (or its double) cannot depend on the choice of the coordinates:

{ε} {σ} T

= {ε} [C]{ε} = ( Tε { ε} ) [C] Tε { ε} = T

T

= { ε} TεT [C] Tε { ε} = { ε} C { ε} T

T

(7.1.66)

therefore, 483

7.1 Introduction C  = TεT [C] Tε

(7.1.67)

Similarly, it holds: = { ε} C { ε} = ( Tε−1 {ε} ) C Tε−1 {ε} =

{ ε} {σ} T

T

T

= {ε} T

−T ε

T

C  T

−1 ε

{ε} = {ε} [C]{ε} T

(7.1.68)

therefore,

[C] = Tε−T C Tε−1

(7.1.69)

Tensor invariants Invariants are another important property of second order tensors. They are quantities independent on the choice of the system of coordinates.

There are three invariants for a general second order tensor A : I1 ( A ) = Aii I2 ( A ) =

1 2

{( A ) ii

2

− Aij A ji

}

I 3 ( A ) = det A

(the sum of the elements on the main diagonal)

(7.1.70)

(the sum of the main 2nd order subdeterminants)

(7.1.71)

(the determinant of the matrix)

(7.1.72)

If tensor A is asymmetric and λ1 , λ2 , λ3 are characteristic numbers of matrix A , then the invariants can be written as: I1 ( A ) = λ1 + λ2 + λ3

I 2 ( A ) = λ1λ2 + λ2λ3 + λ3λ1 I 3 ( A ) = λ1λ2 λ3

(7.1.73) (7.1.74) (7.1.75)

7.1.1.5 Transformation of finite elements matrices Let us consider that the following relations hold for the transformation of deformation parameters from one coordinate system to another (the matrix T need not be necessarily orthogonal): d = Td−1d and d = Td d

(7.1.76)

Let f and f be the nodal forces corresponding to deformation parameters d and d . Since the work of forces f over displacements d is independent of the choice of the system of 484

7.1 Introduction coordinates, the following relation must hold: dT f = ( Td d ) f = dT TdT f = dT f T

(7.1.77)

The expression for the transformation of nodal forces follows from the above: f = TdT f and f = Td−T f

(7.1.78)

It is necessary to emphasize that the column matrices d and f , which in the terminology of FEM are often called „vectors“, are not the physical vectors (i.e. the tensors of the first order), that are transformed as a tensors of the first order (see above). In order to derive the transformation relation for the element stiffness matrix, let us consider that the work of internal forces of the element is independent of the system of coordinates. Therefore, we can write: 1 1 1 W int = dT Kd = dT TdT K Td d = dT Kd 2 2 2

(7.1.79)

and consequently K = TdT K Td

(7.1.80)

Similarly, it holds: 1 1 1 W int = dT K d = dTd−T K Td−1d = dT K d 2 2 2

(7.1.81)

and consequently K = Td−T K Td−1

(7.1.82)

The same transformation relations as for the stiffness matrix apply to mass matrix M . An example where the transformatin matrix Td is not orthogonal is an eccentrical connection of the element nodes to the mesh nodes, assuming a rigid connection. Let us denote the global coordinate system with the origin in the mesh node x and the local coordinate system bounded with the given element (e.g. beam) with the origin in the beam node (e.g. end of the beam) x . For derivation we shall also need an auxiliary coordinate system xˆ , which is parallel to x , but having its origin in the mesh node.

485

7.1 Introduction

Fig. 1.2a Eccentrical connection of an element node to a mesh node, global and local coordinate system

Let us define the vectors of deformation parameters d , dˆ a d in the coordinate systems x , xˆ and x respectively. In the deformation parameters the components of the displacement vectors are introduced first and then the components of the rotation vectors. (It is necessary to notice that in fact rotation is not a vector. For its components the commutative law is not valid. In linearity, however, this law is assumed. It is implied by the principle of superposition. For rotation small enough, however, the rotation can be regarded as a vector with sufficient accuracy. When working with a larger rotation it is necessary to use an incremental approach when the increments are sufficiently small to satisfy the linearity assumption (including the commutative law) with satisfactory accuracy. The relation between the deformation parameters d defined in x and the deformation parameters dˆ defined in xˆ can be then written as follows: d = Td dˆ

(7.1.83)

where the transformation matrix Td reads I ∆  Td =   0 I 

(7.1.84)

where I is the unit diagonal matrix, 0 matrix of zeros and the matrix ∆ has the form  0  ∆ =  −ez  ey 

ez 0 −ex

−e y   ex  0 

(7.1.85)

The eccentricities (differences) of connection ex , ey , ez are defined in the coordinates xˆ . The relations between the deformation parameters d in the global coordinates and the parameters dˆ can be written as follows: ˆd dˆ = T d

(7.1.86) 486

7.1 Introduction where ˆ = R 0  T d  0 R  

(7.1.87)

R being the rotation matrix. The relation between the deformation parameters in the local (element) coordinates and those in the global coordinates can be then written by relation d = Td d

(7.1.88)

where the the transformation matrix Td can be written in the form ˆ =  R ∆R  Td = Td T d 0 R   

(7.1.89)

We can easily learn that the inverse of the Td matrix needed for the inverse transformation d = Td−1d has the following form:  R −1 T =  0 −1 d

− R −1∆RR −1   R −1 

(7.1.90)

It is seen that the matrix Td is not orthogonal ( Td−1 ≠ TdT ). Taking into account that the rotation matrix R is orthogonal ( R −1 = RT ) and the matrix ∆ fulfills the relation ∆ T = − ∆ ,

the formula for the Td−1 can be substantially simplified.  RT Td−1 =   0

RT ∆ T   RT = RT   0

( ∆R )

T

RT

  

(7.1.91)

It is seen that the inverse of the matrix Td can be then perfomed by the transposition of the particular submatrices.

487

7.1 Introduction

Fig. 1.2b Eccentrical connection of an element node to a mesh node, global and local coordinate system

Let us show the derivation of the transformation matrix Td in case that the eccentricities are defined in the global coordinates x (see Fig. 1.2b). The relation between the deformation parameters d defined in the local coordinates of ) ) the element x and the deformation parameters d defined in x can be then written as follows: ) ) d = Td d (7.1.92) where ) R 0  Td =    0 R

(7.1.93)

R being the rotation tensor and 0 being the matrix of zeros . The relations between the deformation parameters d in the global coordinates and the ) parameters d in the coordinates parallel to the global ones with the origin in the given node can be then written as follows: ) % d d=T (7.1.94) d % reads Where T d G % = I ∆  T   d 0 I 

(7.1.95)

∆ G consists of the global eccentricies and has the form 488

7.1 Introduction  0  G ∆ =  −ezG  eGy 

ezG 0 −exG

−eGy   exG  0 

(7.1.96)

The relations between the deformation parameters d in the global coordinates and the parameters d in the local coordinates of the element reads

d = Td d

(7.1.97)

)  R R∆ G  Td = Td T% d =   R  0

(7.1.98)

where

Matrix Td−1 which is needed for the inverse transformation d = Td−1d has the following form:  R −1 T =  0 −1 d

− R −1R∆ G R −1   R −1 

(7.1.99)

Because the rotation matrix R is orthogonal and the matrix ∆ G fulfills the relation

(∆ )

G T

= −∆ G , then Td−1 can be simplified to the following form:

 RT T =  0 −1 d

(∆ )

RT    RT

G T

(7.1.100)

Similarly as in the relation (7.1.91), the inverse matrix Td−1 can be created from the Td by transposition of its submatrices.

7.1.2 Classification of Nonlinearity Two types of nonlinearities can be distinguished in structural mechanics: Geometrical nonlinearity – the source of nonlinearity is what is called geometrical equations, i.e. the relations between the displacement and strain.

489

7.1 Introduction Material or physical nonlinearity – the source of nonlinearity is nonlinear constitutive relations (physical equations), i.e. the relations between the stress and strain. This type of nonlinearity can logically include also nonlinearities caused by nonlinear behaviour of supports (e.g. exclusion of tension in supports or subsoil). Let us show the sources of nonlinearity on a simple example of an flexibly fixed cantilever:

Fig.1.3 Flexibly fixed cantilever

The moment in the support for the linear solution is defined by the expression: M = Fl

(7.1.101)

A geometrically nonlinear relation can be written, assuming the equilibrium at the deformed structure, as follows: M = Fl cos ϕ

(7.1.102)

If the stiffness of the support is linear, the relation between rotation ϕ and the moment can be expressed as: M = Kϕ ϕ

(7.1.103)

where Kϕ is the constant stiffness of the support, independent of rotation ϕ .

The linear solution of rotation ϕ yields: ϕ=

M Fl = Kϕ Kϕ

(7.1.104)

For a geometrically nonlinear solution, the relation between force F and rotation ϕ is: F=

Kϕϕ l cos ϕ

(7.1.105)

Let us show both relations in a graph:

490

7.1 Introduction

Fig.1.4 Linear and nonlinear relation between the force and rotation in a flexibly fixed cantilever

The figure demonstrates that a linear solution for larger rotations ϕ does not make sense. Physical nonlinearity could be introduced into the problem by a nonlinear relation between the moment and rotation: M = K s (ϕ ) ϕ

(7.1.106)

where the stiffness of the support K s is a function of rotation ϕ . In this case, K s (ϕ ) is called secant stiffness. If the relation is defined using a differential: dM = KT (ϕ ) dϕ

(7.1.107)

then it is called the tangent stiffness.

Fig.1.5 Nonlinear stress-strain diagram of flexible support

In the stress-strain diagram for the support, the secant stiffness KS is determined by the slope (gradient) of the secant s and the tangent stiffness K T by the slope (gradient) of the tangent t . 491

7.1 Introduction

7.1.3 Basic Equations, Eulerian and Lagrangean Elements The whole mechanics is based on five foundational systems of equations: 1. Law of conservation of mass 2. Law of conservation of momentum (linear as well as angular) 3. Law of conservation of energy 4. Constitutive relations (relations between stress and deformation) 5. Relations between displacement and deformation (also known as strain measure). In addition, the requirement of continuity or compatibility of deformations is raised. The equilibrium equations can be understood as a special case of the momentum conservation law with inertia forces neglected. These equations then transform into the Cauchy equilibrium equations. A force is equal to the rate of change of momentum, i.e. the derivative of momentum with respect to time, and a moment is equal to the rate of change of angular momentum, i.e. the derivative of angular momentum with respect to time. The last two systems of equations can be nonlinear – then we speak about physical or geometrical nonlinearity. However, the same numerical methods are used to solve all nonlinear problems, hence there is no principal difficulty if nonlinearity is solved in a comprehensive manner, i.e. if both physical and geometrical nonlinearities are treated simultaneously. In order to solve the geometrical nonlinearity, certain concepts need to be newly defined. Elements (meshes) in geometrical nonlinearity can be generally either geometrically constant or they can deform as the mass moves. From this aspect, we distinguish two types of elements (meshes): Eulerian elements which do not change their geometry and in which the mass passes from one element into another, and Lagrangean elements which deform together with the mass (as if they were drawn on it). While in gas or liquid mechanics Eulerian meshes often have to be used because any turbulence can occur there, Lagrangean meshes are popular in solid mechanics. In the following text we will deal exclusively with the latter. Eulerian mesh

492

7.1 Introduction Γ

Ω

Ω0

Γ0

Fig.1.6 Eulerian mesh of finite elements

Lagrangean mesh Γ

Ω

Γ0

Ω0

Fig.1.7 Lagrangean mesh of finite elements

493

7.2 GEOMETRICAL NONLINEARITY

7.2 GEOMETRICAL NONLINEARITY 7.2.1 Foundational Concepts

7.2.1.1 Systems of coordinates in nonlinear mechanics It is suitable to define two basic systems of coordinates which are used in geometrically nonlinear analysis. Spatial (also called Eulerian or global) coordinates shall be denoted by x . They determine the location of a point in space. Material (also called Lagrangean or local) coordinates, denoted by X , mark a point of a body. Each material point has one set of material coordinates, which are usually identical with the spatial coordinates in the initial configuration of the body. The displacement of a point in space is defined by the vector u ( X) = x − X

(7.2.1)

It also holds that x = X+u

(7.2.2)

Fig. 2.1 Non-deformed (initial) and deformed (current) body configuration

7.2.1.2 Deformation gradient Deformation gradient F is defined by the relation F=

∂x ∂u = ∇0 x = I + = I + ∇0 u ∂X ∂X

(7.2.3) 494

7.2 GEOMETRICAL NONLINEARITY In mathematics, the deformation gradient is also called the Jacobi transformation matrix. Note that the deformation gradient is identical to the material gradient of spatial coordinates x .

As the deformation gradient is important, let us write it in the expanded form.

F ≡ ∇ 0 x ≡ grad x

 ∂x1   ∂X 1  ∂x = 2  ∂X 1  ∂x3   ∂X 1

∂x1 ∂X 2 ∂x2 ∂X 2 ∂x3 ∂X 2

∂x1   ∂x  ∂X 3   ∂X  ∂x2   ∂y = ∂X 3   ∂X  ∂x3   ∂z  ∂X 3   ∂X

 ∂u1 ∂u1 1 + ∂X 2  ∂X 1  ∂u2 ∂u 1+ 2 = ∂X 2  ∂X 1  ∂u3 ∂u3  ∂X 2  ∂X 1

∂x ∂Y ∂y ∂Y ∂z ∂Y

∂x  ∂Z   ∂y  = ∂Z   ∂z  ∂Z 

∂u1   ∂X 3  ∂u2   ∂X 3  ∂u  1+ 3  ∂X 3 

(7.2.4)

Symbol ∇ 0 denotes the operator of the vector field gradient in the material coordinates (material gradient) and symbol ∇ the operator of the vector field gradient in the spatial coordinates (spatial gradient). The relation between the initial and the final configuration, which is defined by the deformation gradient F , can be divided to the rotation defined by rotation tensor R and the deformation defined either by the right stretch tensor U or by the left stretch tensor V . It depends on whether the imagined rotation in the infinitesimal part of the body comes first and is followed by the deformation, or vice versa. Consequently, the relationship between the deformation gradient F and tensors U and V can be described by the following equation: F = R⋅U = V⋅R

(7.2.5)

In other words, either deformation U is carried out first followed by rotation R , or rotation R comes first followed by deformation V . Tensors U and V are symmetric ( UT = U and VT = V ). It can be proved that: U = FT F = C

(7.2.6)

V = FFT = B

(7.2.7)

C and B are the right and left Cauchy-Green deformation tensor, respectively.

Let us remind that for involution and evolution of matrices the same rules apply as for involution and evolution of numbers. Thus if for a general square matrix A . A it stands A 2 = A ⋅ A = H (and not A 2 = AT ⋅ A ), then H = A .

495

7.2 GEOMETRICAL NONLINEARITY Let us prove the equation (7.2.6). Multiply the first part of equation (7.2.5) from the left by tensor RT . We get RT ⋅ F = U If we multiply both sides of this equation from the left by its transpose and consider that RRT = I , we get the equation (7.2.6). Similarly, we could prove the equation (7.2.7) as well. From the equation (7.2.5) we can write the following relation for the rotation tensor. R = F ⋅ U −1 = V −1 ⋅ F

(7.2.8)

It holds generally that each regular square matrix can be decomposed to the product of rotation matrix R and a symmetric matrix. This decomposition of the a regular square matrix is called polar decomposition. The polar decomposition theorem enables to extract from the deformation gradient F the rotation tensor R that represents the average rotation of the material point. If dX is an infinitesimal material line-segment in the initial configuration, i.e. in the material coordinates, then the corresponding line-segment in the current (deformed) configuration dx is defined by the relation dx = F ⋅ dX

(7.2.9)

The determinant of the deformation gradient F is denoted by J . In mathematics it is also called the Jacobian of the transformation. J = det ( F ) =

d Ω ρ0 = d Ω0 ρ

(7.2.10)

Where d Ω 0 = dxdydz , d Ω = dxdydz , ρ 0 and ρ are densities in material and space coordinates respectively. The Jacobian of the transformation is used in integrals in relations between different configurations of a body. For instance:

∫ f ( x ) d Ω = ∫ f ( X) J d Ω

Ω

0

(7.2.11)

Ω

Note that: F −1 =

∂X ∂x

(7.2.12)

This is because ∂x ∂X ⋅ =I ∂X ∂x

(7.2.13)

F ⋅ F −1 = I

(7.2.14)

and

In technical literature, the process of mapping from the initial configuration to the deformed configuration is called “push forward”. 496

7.2 GEOMETRICAL NONLINEARITY x = ϕ( X, t )

(7.2.15)

The reverse process, i.e. the mapping from the current (deformed) configuration back to the original one is termed ”pull back”. X = ϕ −1 (x, t )

(7.2.16)

7.2.1.3 Rate of deformation First, let us define the velocity gradient L L=

∂v = ∇v = grad v ∂x

where v is the velocity vector ( v =

(7.2.17) ∂u = u& , the dot denotes the derivative with respect to ∂t

time). Tensor L can be decomposed to a symmetric and skew-symmetric part. L=

1 1 L + LT ) + ( L − LT ) = D + W ( 2 2

(7.2.18)

The rate of deformation D is a tensor defined as the symmetric part of the velocity gradient L . D=

1 L + LT ) ( 2

(7.2.19)

Let us denote the second part of the tensor L as spin W . W=

1 L − LT ) ( 2

(7.2.20)

The rate-of-deformation tensor D can be expressed also by the Green deformation tensor (7.2.26). Let us consider that the derivative of the deformation gradient F with respect to time is the material velocity gradient: ∂  ∂x  ∂v F& =  = ∇0 v = ∂t  ∂X  ∂X

(7.2.21)

Using this relation, we can write: L=

∂v ∂v ∂X & −1 = = F⋅F ∂x ∂X ∂x

(7.2.22)

Let us introduce this expression into equation (7.2.19) D=

1 1 L + LT ) = ( F& ⋅ F −1 + F −T ⋅ F& T ) ( 2 2

(7.2.23)

Let us now differentiate the Green deformation tensor, defined by expression (7.2.38), with respect to time: & = 1 ∂ ( FT ⋅ F − I ) = 1 ( FT ⋅ F& + F& T ⋅ F ) E 2 ∂t 2

(7.2.24)

497

7.2 GEOMETRICAL NONLINEARITY Now let us multiply the equation (7.2.23) by FT from the left and by F from the right. FT ⋅ D ⋅ F =

1 T & &T ( F ⋅ F + F ⋅ F ) = E& 2

(7.2.25)

By multiplying this equation by F −T from the left and by F −1 from the right we obtain: & ⋅ F −1 D = F −T ⋅ E

(7.2.26)

7.2.2 Strain Measures Strain tensor ε used in linear mechanics is defined by the following formula, in Voigt notation:

{ε} = ∂ T u

(7.2.27)

Operator ∂ was defined in formula (7.1.12) Let us show that the relation is an approximate one, holding true with sufficient accuracy only for small rotations and small deformations. It should be pointed out that the rotation is of essential importance in nonlinear mechanics and that it is the largest source of problems. In linear mechanics the superposition principle can be applied. As a result, the commutative law is applicable to all quantities. Both the displacement and rotation can be considered as vectors. For rotation, however, this is true only approximately, in fact up to the rotation of about 0.1 rad. The superposition principle, or the commutative law, cannot be applied to larger rotation. Let us show this on a revealing example. Imagine e.g. a book in the x, y plane. Let us first rotate it around the x axis and then around the y axis, in both cases by angle π 2 . Then let us repeat the experiment in the reversed order of the rotations. The results will be very different. The approximate nature of the linear relation between the deformation and displacement can be shown on a fibre of initial length dS . Without any loss of generalization, let us introduce a system of coordinates x with the origin at the starting point of the fibre and with the x axis oriented in the original direction of the fibre. Let us denote by ds the length of the fibre in the deformed body.

Fig. 2.2 Elongation of fibre dS

Let us denote by u the vector of displacement of the starting point of the fibre. The end-point of the fibre will be displaced by vector u + du . 498

7.2 GEOMETRICAL NONLINEARITY Using the formula for the body-diagonal of a cuboid with dimensions dS + du , dv , dw , we can express the new length of the fibre using the following relation: ds =

( dS + du )

2

+ dv 2 + dw2

(7.2.28)

Let us introduce symbol λ for the new length of a unit fibre (dS = 1) and consider that du =

∂u ∂u dS = ∂x ∂x

then we can write the following relation for this quantity: ds  ∂u   ∂v   ∂w  = 1 +  +   +   = dS  ∂x   ∂x   ∂x  2

λ = 1+ εx =

2

∂u  ∂u   ∂v   ∂w  +  +  +  ∂x  ∂x   ∂x   ∂x  2

= 1+ 2

2

2

(7.2.29) 2

Let us consider the binomial theorem: 1+ A = 1+

A A2 A3 − + +K for A2 < 1 2 8 16

and let us take into account only the first two terms. Then we can write: 2 2 2 ∂u 1   ∂u   ∂v   ∂w   λ = 1+ +   +   +    ∂x 2   ∂x   ∂x   ∂x  

(7.2.30)

and for ε x 2 2 2 ∂u 1   ∂u   ∂v   ∂w   ε x = λ −1 = +   +   +    ∂x 2   ∂x   ∂x   ∂x  

(7.2.31)

If we want to be more accurate and take into account three terms of the binomial expansion, and if we neglect the third and higher powers of the derivatives of the displacement components, we get a more accurate expression for the elongation: λ = 1+

2 2 ∂u 1   ∂v   ∂w   +   +    ∂x 2   ∂x   ∂x  

(7.2.32)

and hence 2 2 ∂u 1   ∂v   ∂w   εx = +   +    ∂x 2   ∂x   ∂x  

(7.2.33)

For a 1D problem, therefore, this more accurate expression would be identical to the formula for ε x known from linear elasticity: εx =

∂u ∂x

(7.2.34) 499

7.2 GEOMETRICAL NONLINEARITY Having outlined the issues relating to the strain measures, let us now look at some of the measures used in practice and consider to what extent they satisfy our requirements which mainly require that a higher strain measure corresponds to larger deformation and that the perfectly rigid body has zero deformation.

7.2.2.1 Green – Lagrange strain tensor E The original non-deformed configuration forms the basis for this strain tensor. It means that differentiation is carried out with respect to the material coordinates. The first two terms from the binomial expansion of the square root are used, which means that quadratic terms are added to the linear expression known from the linear mechanics. Let us show several alternative notations for the definition of the Green – Lagrange strain tensor. Index notation: ∂u ∂u ∂uk 1  ∂u Eij =  i + j + k  2  ∂X j ∂X i ∂X i ∂X j

 1  = ( ui , J + u j , I + uk , I uk , J )  2

(7.2.35)

where ui , J =

∂ui ∂X j

(7.2.36)

Using the deformation gradient F , the expression for the Green – Lagrange strain tensor can be written as follows: 1 Eij = ( FikT Fkj − δ ij 2

)

(7.2.37)

where δ ij is the Kronecker delta (δ ij = 1, for j ≠ i δ ij = 0) .

Tensor, or matrix, notation: E=

1 T 1 1 1  ∂u ∂uT ∂uT ∂u  F ⋅ F − I ) = ( FT F − I ) = ( C − I ) =  + + ⋅ (  2 2 2 2  ∂X ∂X ∂X ∂X 

(7.2.38)

C = FT F is the right Cauchy – Green deformation tensor. If we use the material gradient of displacement vector u , we obtain the following formula: E=

(

1 T T ( ∇0u ) + ∇0u + ( ∇0u ) ⋅∇0u 2

)

(7.2.39)

∇ 0u is the gradient of the displacement vector field in the material coordinates. 500


7.2.2.2 Euler - Almansi strain tensor ( e ) This strain tensor (also known as Almansi – Hamel or Eulerian tensor) relates to the final, deformed configuration. Differentiation is carried out in spatial coordinates. Index notation: In index notation we can express the definition of the Euler – Almansi strain tensor as follows: 1  ∂u ∂u j ∂uk ∂uk  1 eij =  i + −  = ( ui , j + u j ,i − uk ,i uk , j ) 2  ∂x j ∂xi ∂xi ∂x j  2 where ui , j =

(7.2.40)

∂ui , ∂x j

or, using the deformation gradient F , eij =

1 (δ ij − Fik−T Fkj−1 ) 2

(7.2.41)

In tensor, or matrix, notation: e=

1 1 1 I − F −T ⋅ F −1 ) = ( I − F −T F −1 ) = ( I − B −1 ) = ( 2 2 2

1  ∂u ∂uT ∂uT ∂u  − ⋅   + ∂x ∂x  2  ∂x ∂x

(7.2.42)

B = F ⋅ FT is the left Cauchy – Green deformation tensor (Cauchy strain tensor, or Finger deformation tensor). If we use the spatial gradient of the displacement vector field, the formula can be written as follows: e=

(

1 T T ( ∇u ) + ∇u − ( ∇u ) ⋅∇u 2

)

(7.2.43)

The following hold for the relation between strain tensors E and e : e = F − T ⋅ E ⋅ F −1

(7.2.44)

E = FT ⋅ e ⋅ F

(7.2.45)

The first of these equations is called in publications “push forward” operation for the Green deformation, the other is termed “pull back” operation for the Almansi deformation. The validity of these equations can be easily verified if we multiply equation (7.2.38) by F −T from the left and by F −1 from the right, and equation (7.2.42) by FT from the left and by F from the right, and if we consider that F −T ⋅ FT = I and F ⋅ F −1 = I . Consequently: 501

7.2 GEOMETRICAL NONLINEARITY 1 −T T 1 F ⋅ F ⋅ F ⋅ F −1 − F −T ⋅ I ⋅ F −1 ) = ( I − F −T ⋅ F −1 ) = e ( 2 2 1 1 FT ⋅ e ⋅ F = ( FT ⋅ I ⋅ F − FT ⋅ F −T ⋅ F −1 ⋅ F ) = ( FT ⋅ F − I ) = E 2 2 F − T ⋅ E ⋅ F −1 =

(7.2.46) (7.2.47)

The validity of formulas (7.2.44) and (7.2.45) can be also proved when we mutually substitute one of the equations to the other one and we obtain equalities. e = F −T ⋅ E ⋅ F −1 = F −T ⋅ FT ⋅ e ⋅ F ⋅ F −1 = I ⋅ e ⋅ I = e E = FT ⋅ e ⋅ F = FT ⋅ F −T ⋅ E ⋅ F −1 ⋅ F = I ⋅ E ⋅ I = E

(7.2.48)

7.2.2.3 Logarithmic strain measure ( ε n ) This measure is defined on an incremental basis and in each increment it relates to the current configuration. Let us write its formula for 1D configuration. l

l  dl dl l l dε n = → εn = ⌠  = [ ln l ]l0 = ln − ln l0 = ln   = ln (1 + ε x ) ⌡ l l  l0 

(7.2.49)

l0

where ε x is the linear deformation, l0 is the initial length and l is the resulting length. For 2D and 3D we could define the logarithmic strain measure with help of the right stretch tensor U . ε n = ln U

(7.2.50)

U is the 2nd order tensor and the logarithmic strain tensor ε n can be determined through the spectral decomposition N

ε n = ∑ ln(λi )ei eTi

(7.2.51)

i =1

where λi and ei are the eigenvalues and eigenvectors of the matrix U respectively and N is the dimension of the space. The logarithmic strain measure is suitable for large deformations (ε x > 0, 05) . Structural materials do not reach such large deformations and, therefore, the application of this strain measure to structural building materials is not necessary.

7.2.2.4 Infinitesimal strain tensors ( ε ), ( eˆ ) If the deformation is so small that the second order terms in the Green strain tensor E can be neglected, then we get what is termed infinitesimal strain tensor, which is identical to the linear strain tensor ε . 502

7.2 GEOMETRICAL NONLINEARITY Therefore, in index notation we can write: ∂u 1  ∂u ε ij =  i + j 2  ∂X j ∂X i

 1  = ( ui , J + u j , I )  2

(7.2.52)

In the matrix, or tensor, notation the expression for ε can be written as follows:

(

)

1  ∂u ∂uT  1 1 T T ε=  +  = ∇ 0u + ( ∇ 0u ) = ( F + F ) − I 2  ∂X ∂X  2 2

(7.2.53)

Let us remind that ∇ 0 is the material gradient and hence ε is the symmetrical part of material displacement gradient ∇ 0u . The infinitesimal strain tensor in spatial coordinates eˆ must also be defined. The formulae are similar to the relations valid for ε but the differentiation is carried out in spatial coordinates x. In index notation we can write: 1  ∂u ∂u j  1 eˆij =  i +  = ( ui , j + u j ,i ) 2  ∂x j ∂xi  2

(7.2.54)

In tensor notation, the formula for eˆ can be written as: 1  ∂u ∂uT eˆ =  + 2  ∂x ∂x

(

)

 1 1 −1 T −T  = ∇u + ( ∇u ) = I − ( F + F ) 2  2

(7.2.55)

The relation of u = x − X was used to derive the last expression. Consequently ∂u = I − F −1 ∂x

(7.2.56)

Let us remind that ∇ is the spatial gradient, hence tensor eˆ is a symmetrical part of the spatial displacement gradient ∇u . The importance of the infinitesimal tensor eˆ is given by the fact that this strain tensor is energetically conjugate with Cauchy stress σ (which will be shown in Chapter 2.4. too).

7.2.2.5 Other strain measures Some other tensors can also be used as strain measures, e.g.: the deformation gradient F the left Cauchy – Green deformation tensor B = F ⋅ FT the right Cauchy – Green deformation tensor C = FT ⋅ F

503


7.2.2.6 Comparison of strain tensors The rigid body rotation test In order to assess different strain measures, it is important to find out how they behave if the solid moves as a rigid body, where only the translation and rotation are applied. It can be easily shown that all the strain measures mentioned above provide zero deformation in case of the translation of the body. All derivatives of the displacement components with respect to both material and spatial coordinates are zero. Let us now consider the rotation of the rigid body. Let us present a simple example of a rigid body rotation. Let us assume that the rigid body is rotated by 90 degrees.

It can be easily shown that under the assumption of perfect rigidity of the body, the following relations for the displacement components apply if the body is rotated by 90°: u = −X −Y v = X −Y

Using the well-known relation: x = X+u

the substitution yields the relations between the spatial and material coordinates: x = −Y y=X

The deformation gradient F then will have the following form:  ∂x  ∂X F=  ∂y  ∂X

∂x  ∂Y  0 −1 = ∂y  1 0  ∂Y 

Its inverse gives:  0 1 F −1 =    −1 0 By substituting into the formulae for the Green and Almansi strain tensors we find that both these tensors are zero.

504

7.2 GEOMETRICAL NONLINEARITY 1 T 1  1 0   F ⋅ F − I) =   −I = 0 ( 2 2  0 1   1 1  1 0  e = ( I − F −T ⋅ F −1 ) =  I −  =0 2 2  0 1   E=

Let us now prove generally that the two quadratic strain measures satisfy the requirement of zero deformation in case of rigid body rotation. The transformation equation for the translation and rotation of a solid as a rigid body is: x = R ⋅ X + xT

(7.2.57)

Therefore, it is evident that: F=

∂x =R ∂X

(7.2.58)

If we substitute the rotation tensor instead of the deformation gradient into the formula for the Green – Lagrange strain tensor, we get the following: E=

1 T 1 R ⋅ R − I) = (I − I) = 0 ( 2 2

(7.2.59)

Consequently, the Green – Lagrange strain tensor satisfies the important requirement for the strain measure – the requirement that the strain measure should be zero for rigid bodies. If we substitute the rotation tensor instead of the deformation gradient into the formula for the Euler – Almansi strain tensor, we get the following: e=

1 1 I − R − T ⋅ R −1 ) = ( I − R ⋅ R T ) = 0 ( 2 2

(7.2.60)

This is true because the following: if rotation matrix R is orthogonal, then so is matrix RT . Hence both the Green – Lagrange and the Euler – Almansi strain tensors satisfy the requirement of zero deformation if a solid rotates as a rigid body.

Maximum and minimum elongation test Let us consider a bar of initial length l0 and length after elongation l . The following table shows the comparison of strain measures:

Strain measure

Linear

Green– Lagrange

Expression for 1D configuration

l=∞

l=0

l − l0 l0

ε =∞

ε = −1

1 l 2 − l02 2 l02

E=∞

E=−

ε=

E=

1 2 505

7.2 GEOMETRICAL NONLINEARITY Euler–Almansi

logarithmic

e=

1 l 2 − l02 2 l2

e=

l  ε n = ln    l0 

1 2

e = −∞

εn = ∞

ε n = −∞

It follows from the table that only the logarithmic strain measure satisfies the requirement for infinite elongation and contraction. The linear and Green–Lagrange tensors do not satisfy the requirement in the case of an infinite contraction and the Euler – Almansi tensor in the case of an infinite elongation. Let us now investigate how the different strain measures satisfy the requirements in case of large deformations. It is natural that the largest possible elongation, namely the infinite one, should correspond to the elongation into infinite length and vice versa. The smallest possible (infinite) negative deformation should correspond to the contraction to zero length. Let us show that for 1D the following relations hold, which sometimes can be convenient to use. E=

1 dx 2 − dX 2 du 1 du 2 = + 2 dX 2 dX 2 dX 2

and

εE =

1 dx 2 − dX 2 du 1 du 2 = − 2 dx 2 dx 2 dx 2

Proof: 2 1 dx 2 − dX 2 ( dX + du ) − dX dX 2 + 2dXdu + du 2 − dX 2 E = = = = 2 dX 2 2dX 2 2dX 2 . 1 2dXdu + du 2 du 1 du 2 = = + 2 dX 2 dX 2 dX 2 2

2 2 2 1 dx 2 − dX 2 1 dx − ( dx − 2dxdu + du ) e = = = 2 dx 2 2 dx 2 1 2dxdu − du 2 du 1 du 2 = = − 2 dx 2 dx 2 dx 2

The relation dx = dX + du was used.

7.2.3 Stress Measures Due to the changes deformations cause both in the direction and size of the small area dA which serves to define stress, several stress tensors can be introduced, depending on what has been defined in the original configuration (i.e. in the material coordinates) and in the resulting configuration (i.e. in the spatial coordinates). 506

7.2 GEOMETRICAL NONLINEARITY Let us introduce some of the most important ones: 1. Cauchy stress σ 2. Nominal stress N and its transpose the first Piola – Kirchhoff stress P 3. Second Piola – Kirchhoff stress S 4. Corotation stress σˆ 5. Kirchhoff stress τ 6. Biot stress T First, let us define some concepts which will be used below.

Fig. 2.4 Original and resulting configuration of a body

dA0

infinitesimal small area of the body in the original configuration,

dA infinitesimal small area in the resulting (or current) configuration corresponding to the small area dA0 ,

df0 force acting on the non-deformed small area dA0 , transformed into the material coordinates, df

force acting on the small area dA (or dA0 ) in the spatial coordinates,

t0

stress vector acting on the small area dA0 ,

t

stress vector acting on the small area dA ,

n0

unit normal to the small area dA0 ,

n

unit normal to the small area dA ,

507

7.2 GEOMETRICAL NONLINEARITY dA 0

vector of oriented plane in the material coordinates (in the original configuration) and

dA

vector of oriented plane in the spatial coordinates.

The following holds for the relation between force df and stress vectors t 0 and t : df = t ⋅ dA = t 0 ⋅ dA0

(7.2.61)

Vectors t and t0 thus have the same direction in the spatial coordinates, but they are recalculated to different areas. Vector t0 is vector t recalculated to the original area. For the vectors of oriented planes the following holds: dA 0 = n 0 ⋅ dA0 a dA = n ⋅ dA

(7.2.62)

Let us discuss briefly each of the above stress tensors separately.

7.2.3.1 Cauchy stress ( σ ) Cauchy stress σ is defined by the Cauchy Theorem (1st Cauchy Theorem). Stress tensor σ is a linear mapping of stress vector t to normal vector n . dA ⋅ σ = n ⋅ σ ⋅ dA = df = t ⋅ dA ⇒ n ⋅ σ = t

(7.2.63)

Tensor σ is symmetrical ( σT = σ ). The Cauchy stress is fully defined in the resulting, or current, configuration of the body. Since it represents the real stress measured at a given moment on the deformed body, it is also called the “true stress”.

7.2.3.2 Nominal stress ( N ), First Piola – Kirchhoff stress ( P ) The nominal stress tensor is defined similarly to the Cauchy stress tensor but it relates to the non-deformed area dA0 . dA 0 ⋅ N = n 0 ⋅ NdA0 = dA 0 ⋅ N = df = t0 dA 0 ⇒ n 0 ⋅ N = t0

(7.2.64)

The first Piola-Kirchhoff stress is transpose of the nominal stress. P = NT

(7.2.65)

The stress tensors P and N are not symmetric. Note: Tensor P is in some publications called the nominal stress and its transpose is then called the first Piola- Kirchhoff stress.

508


7.2.3.3 Second Piola – Kirchhoff stress ( S ) Let us remind that dX = F −1 ⋅ dx By analogy, force df acting in the spatial coordinates on the deformed small area dA can be transformed to force df0 acting in the material coordinates on the non-deformed small area dA 0 df0 = F −1 ⋅ df = F −1 ⋅ ( dA 0 ⋅ N ) = dA 0 ⋅ N ⋅ F −T ≡ dA 0 ⋅ S

(7.2.66)

S = N ⋅ F − T = F −1 ⋅ P

(7.2.67)

Hence Formula (2.3.5) uses the equality A ⋅ a = a ⋅ AT . Consequently, by analogy to the Cauchy Theorem we can write the definition of tensor S in the original configuration (in the material coordinates) as follows: n 0 ⋅ S = F −1 ⋅ t0 = t 0

(7.2.68)

The right-hand side of this equation is the stress vector acting on the elementary small area dA 0 , transformed into the original configuration. Tensor S is symmetrical (ST = S) , analogously to σ .

7.2.3.4 Corotation stress ( σˆ ) This is in essence the Cauchy stress which is expressed in a system of coordinates that rotates together with the material. This concept is useful for the types of structures that work with internal forces, e.g. shell or rod. The corotation stress is simply obtained by the transformation of tensor σ into the rotated system. σˆ = RT ⋅ σ ⋅ R

(7.2.69)

Naturally, as tensor σ is symmetric, also σˆ is symmetric.

7.2.3.5 Kirchhoff stress ( τ ) This stress tensor is defined by the following equation: τ=Jσ

(7.2.70)

where J = det ( F ) . Since tensor σ is symmetric, so is tensor τ .

509


7.2.3.6 Biot stress ( T ) The Biot stress is useful because it is energy conjugate to the right stretch tensor U . The Biot stress is defined as the symmetric part of the tensor PT ⋅ R where R is the rotation tensor obtained from a polar decomposition of the deformation gradient. Therefore the Biot stress tensor is defined as T=

1 T ( R ⋅ P + PT ⋅ R ) 2

(7.2.71)

The Biot stress is also called the Jaumann stress.

7.2.3.7 Transformations between different types of stress The transformation relations among the above stress measures are summarised in the following table: σ σ N S σˆ τ

JF −1 ⋅ σ JF −1 ⋅ σ ⋅ F −T RT ⋅ σ ⋅ R Jσ

N −1 J F⋅N N ⋅ F −T J −1U ⋅ N ⋅ R F⋅N

S −1 J F ⋅ S ⋅ FT S ⋅ FT J −1U ⋅ S ⋅ U F ⋅ S ⋅ FT

σˆ R ⋅ σˆ ⋅ RT JU −1 ⋅ σˆ ⋅ R T JU −1 ⋅ σˆ ⋅ U −1

τ J −1τ F −1 ⋅ τ F −1 ⋅ τ ⋅ F − T J −1R T ⋅ τ ⋅ R

JR ⋅ σˆ ⋅ R T

U is the right stretch tensor.

U = RT ⋅ F

(7.2.72)

R is the rotation tensor.

7.2.3.8 Objective stress rate The concept of objectivity is based on the expectation that the stress and strain should not be affected by the movement of a solid as a rigid body. In other words, they should be independent of the point of view (or the system of coordinates). Tensors S and E defined in & E & ) satisfy this the material coordinates as well as their derivatives with respect to time (S, condition and, therefore, are called objective. Tensors σ and e defined in the spatial coordinates are not objective because they change with the rotation of the solid as a rigid body. Now we are going to show why objective stress rates are needed for constitutive relations. Imagine that there is a certain initial stress in a solid. Let us rotate the solid as a rigid body. We have already shown that E = e = 0 . It also holds that D = 0 . However, it does not generally hold that:

510

7.2 GEOMETRICAL NONLINEARITY Dσ =0 Dt D the material derivative with respect to time, i.e. the derivative with Dt respect to time for invariant material coordinates. Let us denote by

Consequently, the relation

Dσ = Cσ D : D cannot represent a valid constitutive equation. Dt

Dσ , therefore, the objective stress rate must be introduced. Let us denote it by Dt σ ∇ . Since there are more objective stress rates σ , let us affix another superscript to uniquely determine the specific objective stress rates. Instead of

Jaumann stress rate Jaumann stress rate of Cauchy stress σ is defined by the following equation: σ ∇J =

Dσ − W ⋅ σ − σ ⋅ WT Dt

(7.2.73)

W is the spin defined by (7.2.20)

Then the constitutive relation can be written as follows: σ ∇J = Cσ J : D

(7.2.74)

Truesdell stress rate Truesdell stress rate of the Cauchy stress is defined by the relation: σ ∇T =

Dσ + div( v )σ − L ⋅ σ − σ ⋅ LT Dt

(7.2.75)

where L is the velocity gradient defined by equation (7.2.17) and div( v ) = ∇v is the divergence of the velocity vector. Then the constitutive equation has the following form: σ ∇T = Cσ T : D

(7.2.76)

7.2.4 Energetically Conjugate Stress And Strain Measures

511

7.2 GEOMETRICAL NONLINEARITY Energy and work are scalar quantities independent of the system of coordinates. The principle of virtual work defined by: δ W = δ W int + δ W ext = 0

(7.2.77)

must be valid for any choice of strain measure. However, this holds only if an energetically conjugate stress measure is assigned to the corresponding strain measure. This is because the virtual work of internal forces is defined as the product of the stress tensor and strain increment. It can be shown that δ W int =

∫ S : δ E d Ω = ∫ σ : δ eˆ d Ω = ∫ T : δ U d Ω 0

Ω0

Ω

0

(7.2.78)

Ω0

Hence, the energetically conjugate stress and strain pair is represented in the first case by the Green – Lagrange strain tensor E and the second Piola – Kirchhoff stress S (both tensors are defined in the material coordinates, i.e. in the original configuration). In the second case, the energetically conjugate stress and strain pair is represented by the pair of tensors defined in the current configuration. It is the linear part of the Euler – Almansi strain tensor, or what is termed infinitesimal strain tensor defined in spatial coordinates eˆ , and the Cauchy stress tensor σ .  ∂u ∂uT  1  ∂u ∂X ∂XT ∂uT  ⋅ + ⋅ eˆ =  + =  = ∂x ∂X   ∂x ∂x  2  ∂X ∂x 1 = ( F − I ) ⋅ F −1 + F − T ⋅ ( F T − I ) = 2 1 1 = ( I − F −1 + I − F −T ) = I − ( F −1 + F −T ) 2 2

(

)

(7.2.79)

The following relations were used to derive the above: u = x−X ∂u ∂x = −I = F−I ∂X ∂X ∂uT ∂xT = − I = FT − I ∂X ∂X

(7.2.80)

The two above-mentioned pairs are of fundamental importance since they form the basis for two most important problem formulations in geometrical nonlinearity, namely (i) the “total Lagrangean” formulation, which is fully defined in the material coordinates and in which both stress and strain relate to the original, or non-deformed, configuration, and (ii) the “updated Lagrangean” formulation, which is defined in the spatial coordinates and in which both stress and strain relate to the last known configuration (i.e. the current configuration). In the “total Lagrangean” concept, therefore, the Green – Lagrange strain tensor E and the second Piola – Kirchhoff stress S are used, while in the “updated Lagrangean” concept, the Euler – Almansi strain tensor e , or its linear part eˆ , and the Cauchy stress tensor σ are used. Other energetically conjugate stress and strain pairs can also be defined but they are of lesser importance. For instance, for the first Piola–Kirchhoff stress P , the energetically conjugate strain measure is defined by the formula F − I ≡ ∇ 0u . For the corotation stress σˆ , the infinitesimal strain tensor eˆ is energetically conjugate (linear part of the Euler–Almansi strain tensor), but 512

7.2 GEOMETRICAL NONLINEARITY when transformed into the same coordinates as σˆ , i.e. RT ⋅ eˆ ⋅ R . The technical literature often states relations in which, instead of the deformation, the rate of deformation is used, i.e. the derivative of the deformation with respect to time. Then relationships for some conjugated pairs of stress tensors and rates of deformation can be written for the power of internal forces. W& int =

Deˆ & dΩ = ⌠ d Ω = ∫ σ : Dd Ω = ∫ P : F& d Ω0 = : : S E σ  0 ∫Ω ⌡ Dt Ω Ω0 0 Ω

ˆ dΩ = = ∫ σˆ : D Ω

∫ T : U& d Ω0

(7.2.81)

T

Ω0

The formula with tensor P takes into account that ∂I =0 ∂t

(7.2.82)

ˆ is the rate of deformation transformed into the same coordinates as corotation stress σˆ . D The following relation is used in the above equation: Deˆ =D Dt

(7.2.83)

which is easy to prove. The infinitesimal strain tensor in the current configuration is defined by the following formula: 1  ∂u ∂uT  eˆ =  +  2  ∂x ∂x 

(7.2.84)

Let us differentiate eˆ with respect to time: Deˆ 1  ∂u& ∂u& T =  + Dt 2  ∂x ∂x

 1  ∂v ∂vT =  +  2  ∂x ∂x

 1 T  = (L + L ) = D  2

(7.2.85)

Now we will show the proof of (7.2.83) for tensor eˆ expressed by means of the deformation gradient: eˆ = I −

1 −1 F + F −T ) ( 2

(7.2.86)

Let us differentiate it with respect to time: Deˆ 1 = − ( F& −1 + F& −T ) Dt 2

(7.2.87)

Comparing with (7.2.85) we get a new relation for the rate of deformation D . D=−

1 & −1 & −T (F + F ) 2

(7.2.88)

If we compare this relation with equation (7.2.23) we receive: 513

7.2 GEOMETRICAL NONLINEARITY F& ⋅ F −1 = −F& −1 F −T ⋅ F& T = −F& −T

(7.2.89) (7.2.90)

Let us prove that these equations hold. Let us start with the relations: x = X+u X = x−u

Then the following relations hold for the deformation gradient and its derivative with respect to time: ∂v ∂v ∂x F& = = ⋅ = L⋅F ∂X ∂x ∂X ∂v F& −1 = − = −L ∂x

(7.2.91) (7.2.92)

Let us substitute the formulas into (7.2.89) and (7.2.90). L ⋅ F ⋅ F −1 = L F −T ⋅ FT ⋅ LT = LT

(7.2.93) (7.2.94)

Therefore, both equations (7.2.89) and (7.2.90) are satisfied and relation (7.2.88) for the rate of deformation D holds true. Then also formula (7.2.83) holds, which shows that the rate of deformation is the derivative of the infinitesimal strain tensor with respect to time. We have defined energetically conjugate pairs of stress and strain measures on the basis of the fundamental requirement of independence of the energy and work of the system of coordinates and the choice of the stress measure.

7.2.5 Two Formulations of Geometrical Nonlinearity in FEM In solid mechanics, usually Lagrangean meshes are used. Two problem formulations are possible in the discretisation, depending on the configuration of the body which is used to describe the problem. If the problem is formulated in the current configuration of the body (i.e. in the spatial coordinates) it is the “updated Lagrangean” formulation, while if the problem is formulated in the reference (original) configuration (i.e. in the material coordinates) it is the “total Lagrangean” formulation. In the updated Lagrangean formulation the derivatives are carried out in the spatial (Eulerian) coordinates and integrals are carried out on the deformed body (on the “current” configuration). In the total Lagrangean formulation the derivatives are carried out in the material coordinates and integrals are carried out on the initial (reference) configuration (on the non-deformed body). In solid mechanics the following fundamental equations are used: • • • • •

Law of conservation of mass Law of conservation of momentum (linear as well as angular) Law of conservation of energy Constitutive equation, i.e. relations between stress and strain Geometrical equations, i.e. relations between displacement and strain 514

7.2 GEOMETRICAL NONLINEARITY The equations of the first group are conservation laws well known from physics, while the second group represents the properties of standard Boltzmann continuum. The whole mechanics of solids is based on these equations. Let us now describe in more detail both formulations of nonlinearity in FEM.

7.2.5.1 Formulation based on current configuration (updated Lagrangean) Let us first briefly show the formulation of the basic equations.

Law of conservation of mass This law determines the changes of density of a body in relation to deformation. ρ ( x) = where ρ 0 ρ

ρ 0 ( X) ρ 0 ( X ) = J det(F)

(7.2.95)

is the original density (in the reference configuration) is the current density (in the deformed body)

It holds that d Ω = Jd Ω 0 , dm = ρ d Ω and dm0 = ρ0 d Ω 0 . If we substitute these relations into the request for equality of masses dm = dm0 , the equation (7.2.95) will be obtained.

Law of conservation of momentum a) Law of conservation of linear momentum && ∇ ⋅ σ + ρ b = ρ v& = ρ u

(7.2.96)

&& is the vector of acceleration of the given point of the body. where v& = u

If inertial forces are neglected the equation is reduced to the Cauchy equilibrium equation: ∇ ⋅σ + ρ ⋅b = 0

(7.2.97)

where ∇ ⋅ σ is the spatial divergence of the Cauchy stress, ρb is the vector of body forces, b is usually the gravity acceleration vector

b) Law of conservation of angular momentum If inertial forces are neglected this law generates the momentum-related conditions of equilibrium. Another consequence is the symmetry of stress tensor σ σ = σT

(7.2.98)

This equation expresses the well-known theorem of reciprocity of tangential stresses ( σ ij = σ ji ). 515


Law of conservation of energy This law in solid mechanics means that the rate of change of the total energy of a body equals the sum of the rate of work of internal forces D : σ (which is equal to the rate of work of external forces i.e. load performance), the heat flux and the the rate of energy source. If the heat energy sources are neglected then the law expresses the fact that the rate of change of density of potential energy equals the difference between the load performance and the rate of dissipation (heat flux): Jw& int = D : σ − ∇ ⋅ q

(7.2.99)

∂w   w is the hyper-elastic potential in the original configuration  S =  ∂E   D is the rate-of-deformation tensor defined by D=

1 & ⋅ F −1 = Deˆ L + LT ) = F −T ⋅ E ( 2 Dt

(7.2.100)

where L is the velocity gradient L=

∂v = ∇v ∂x

(7.2.101)

q is the heat flux vector (note that the divergence ∇ ⋅ q is a scalar).

Constitutive equation This equation expresses the relation between the stress and strain in a current body configuration (in the deformed body) σ = σ( e, σ,K)

(7.2.102)

In the incremental form this relation can be linearised: δσ = Cσe (e, t ) : δ e

(7.2.103)

In Voigt notation we can write the incremental form of the constitutive equation as follows: δ {σ} = Cσe (e, t )  δ {e}

(7.2.104)

Cσ e is the tangential material module derived from the relation between the Cauchy stress tensor and Euler-Almansi strain tensor (it is a fourth order tensor). The most general form of the constitutive equation can be written in the infinitesimal (velocity) form: σ ∇ = Sσt D (D, σ,K)

(7.2.105) 516

7.2 GEOMETRICAL NONLINEARITY Sσt D is a function depending on the Cauchy stress, rate of deformation and possibly other variables. σ ∇ is one of objective stress rates. For a wide range of what is termed hypo-elastic materials the linear dependence between the rate of stress and deformation can be written as follows: σ ∇ = Cσ : D

(7.2.106)

where σ ∇ is one of the objective stress rates and Cσ is the tensor of elasticity modules which is defined for the given objective stress rate. In this way we can write for instance for the Jaumann stress rate σ ∇J = Cσ J : D

(7.2.107)

or for the Truesdell stress rate σ ∇T = Cσ T : D

(7.2.108)

Geometrical equations (strain measure) e=

1 I − F −T ⋅ F −1 ) ( 2

(7.2.109)

In the updated Lagrangean formulation the Euler–Almansi strain tensor e defined on the deformed body is used.

FEM discretisation for the formulation in the current configuration (updated Lagrangean) In this case the discretisation is defined on the deformed body Ω . If the relation between the virtual increment of the Euler – Almansi strain tensor in Voigt notation δ {e} and the virtual increment of the vector of strain parameters δ d in the current deformation is defined as follows: δ {e} = B ⋅ δ d

(7.2.110)

then the important vector of internal nodal forces can be calculated by the following formula: f int = ∫ BT {σ} d Ω

(7.2.111)

Ω

Let us briefly present how this formula can be derived. Let us start with the requirement of energetic equivalence of internal nodal forces and body stress. The virtual work performed by internal nodal forces f int on virtual deformation parameters δ d must be equal to the virtual work performed by stress {σ} on virtual deformation δ {e} .

Both formulae describe one and the same quantity, namely the virtual work of internal forces. 517

7.2 GEOMETRICAL NONLINEARITY W int = ∫ δ {e} {σ} d Ω = δ dT f int T

(7.2.112)

Ω

Let us substitute for {e} the expression from (7.2.110). Then we obtain

∫ δ d B {σ} d Ω = δ d f T

T

T int

(7.2.113)

Ω

Since the vector of virtual deformation parameters δ d is constant in relation to the integration, we can factor out δ dT and compare both sides of the equation. We receive formula (7.2.111).

Tangent stiffness matrix The tangent stiffness matrix serves to characterize the current stiffness at a given moment, i.e. one which respects the change of geometry, tangent stiffness of material as well as the effect of stress at the given moment. If the system of nonlinear equations for the deformation alternative of FEM is written in the form: K (d) ⋅ d = f

(7.2.114)

then K is a secant stiffness matrix. In the incremental form it could be written K T (d( i ) ) ⋅ δ d( i +1) = δ f ( i +1)

(7.2.115)

Where d( i+1) is obtained by the summation of the increments d( i +1) = d( i ) + δ d( i +1)

(7.2.116)

K T is a tangent stiffness matrix, that in the current configuration d ( i ) can be defined as: KT = K M + Kσ

(7.2.117)

Let us show the calculation algorithms for both components of the tangent stiffness matrix. It should be pointed out that the relevant integration is carried out in the current configuration Ω .

a) Material tangent stiffness matrix K M K M = ∫ BT Cσ (e, t )  Bd Ω

(7.2.118)

Ω

B is the matrix of spatial derivatives of base functions defined by formula (7.2.110), Cσ  is the tangent constitutive matrix for the formulation in the current configuration, or more precisely, the tangent material elasticity tensor defined for the given objective stress measure and written in Voigt notation.

518

7.2 GEOMETRICAL NONLINEARITY b) Geometrical tangent stiffness matrix K σ The concept of structural stiffness is well known to all structural engineers. They are acquainted with the deformation method for the analysis of frames and to a large degree also with the deformation variant of FEM, which is based on the Lagrange variational principle. But as structural engineers are usually not too familiar with nonlinear mechanics, they tend to misunderstand the term ‘structural stiffness’ and mostly think just about one of its components, which is the “material” or “initial” stiffness K 0 . This stiffness is the result of material properties and shape of the structure and does not include the effects of its stress. However, the stiffness component which is the result of the stress-state of the structure is intuitively understood by any musician who tunes their instrument by stretching a chord with a peg. The structural engineer knows that the tone frequency is directly proportional to the square root of the stiffness. But an unstressed chord has no stiffness and its stiffness in the instrument results exclusively from its stress-state, or its “geometrical” stiffness. The material used has no effect on this stiffness component, just the stress-state is important.

Let us show a simple way how to derive the geometrical stiffness for a planar lattice girder.

Let the vector of deformation parameters be d = [u1 , v1 , u2 , v2 ]

T

where u and v are the displacement components in the direction of the x - and y -axis, respectively. Then the material (or initial) stiffness matrix of the lattice girder is defined by the following relation:

KM

1  EA  0 = l  −1  0

0 −1 0 0 0 1 0 0

0 0 0  0

Note that the lattice girder has no lateral stiffness. Stiffness is the force to be exercised at the given point and direction in order to achieve a unit displacement at the given point and 519

7.2 GEOMETRICAL NONLINEARITY direction. This defines the diagonal terms of the stiffness matrix. Other terms in the relevant column or row are the resulting reaction components. Based on this definition, we can easily derive the geometrical stiffness matrix of the lattice girder subjected to tensile force N . The moment equilibrium condition for the girder in the configuration with the relevant lateral unit displacement in the given node is sufficient: N ⋅1 − Kσ ( 2, 2 ) ⋅ l = 0 ⇒ Kσ ( 2, 2 ) =

N l

From equilibrium equations and symmetry of the stiffness matrix it is easy to determine other coefficients of the geometrical stiffness matrix particularly Kσ ( 2, 4 ) , Kσ ( 4, 2 ) and Kσ ( 4, 4 ) . The remaining coefficients of the matrix are zeros. The geometrical stiffness matrix then has the following form: 0 0  N 0 1 Kσ = l 0 0   0 −1

0 0 0 −1 0 0  0 1

The resulting stiffness matrix K T is defined as the sum of the material and geometrical stiffness matrix: KT = K M + Kσ Before presenting the general calculation algorithm for the 3D geometrical stiffness matrix, let us first show the generalization of the algorithm for a beam subjected to tension, which serves to better understand the 3D algorithm.

520

7.2 GEOMETRICAL NONLINEARITY Let us think about the forces that act on an element of length dx . In addition to force N , which always acts in the same direction, there will be also a line moment of intensity m defined by the following equivalence:

m dx = N

∂v dx ∂x

i.e.

m= N

∂v ∂x

For the lattice girder in question the following holds: ∂v v2 − v1 = = Gd ∂x l  −1 1  where G =  ,   l l d = [ v1 , v2 ]

T

The contribution of the work of moment m on rotation

∂v can be written as: ∂x

∂v ∂v 1 1 ⌠  ∂v  1 T T = ⌠  m dl =    N dl = ∫ d G NGddl ∂x 2 ⌡ ∂x 2 ⌡  ∂x  2l T

W

int

l

l

(7.2.119)

1 1 1 = dT ∫ G T NGdl d = dT ∫ G T σ x Gd Ω d = dT K σ d 2 l 2 Ω 2 N and Ω = Al . If we A compare the resulting notation of the potential energy with the standard calculation algorithm for material stiffness matrix we can see that the stress-state of the beam produces additional (geometrical) structural stiffness which is defined by the geometrical stiffness matrix K σ . The following relations were used in obtaining the above: σ x =

K σ = ∫ G T σ x Gd Ω

(7.2.120)

Ω

Now let us show the general calculation algorithm for the geometrical stiffness matrix of an element. Let the following holds for each component u j of displacement vector u : n

u j = ∑ Ni u ji i =1

(7.2.121)

where u ji is the value of displacement u j in node i and n is the number of nodes.

Let us define matrix N as follows: N = N ⊗ I = [ N1I, N 2 I, K , N n I ]

(7.2.122)

where I is the unit diagonal matrix and the operator ⊗ means the Kronecker matrix product. Then for the displacement vector the following relation can be written: 521

7.2 GEOMETRICAL NONLINEARITY u = N ⋅d

(7.2.123)

where d is the vector of deformation parameters of the element containing all the components u ji in such arrangement that for each node i all components u j are listed.

Let us define matrix g i containing the first derivatives of base functions for node i with respect to spatial coordinates:  Ni, xI    gi =  Ni, y I   N i , z I 

(7.2.124)

and matrix G which is formed by sub-matrices g i G = [ g1 , g 2 , K , g i , K , g n ]

(7.2.125)

Further, let us define matrix Σ by multiplying each component of the Cauchy stress tensor σ by the unit diagonal matrix:  σ 11I σ 12 I Σ = σ ⊗ I =  σ 22 I sym.

σ 13I  σ 23I  σ 33I 

(7.2.126)

Then the following formula for the geometrical matrix of the element can be written: K σ = ∫ G T ΣG d Ω

(7.2.127)

Ω

The integration is carried out on the deformed body Ω (in the current configuration).

7.2.5.2 Formulation based on reference configuration (total Lagrangean) In this concept the differentiation is carried out in the material coordinates and integration is carried out on the non-deformed body (in the reference configuration). The fundamental equations for the reference configuration are formulated as follows:

Law of conservation of mass ρ J = ρ0

(7.2.128)

Law of conservation of momentum a) Law of conservation of linear momentum: && ∇ 0 ⋅ N + ρ 0 b = ρ 0u

(7.2.129) 522

7.2 GEOMETRICAL NONLINEARITY When inertial forces are neglected the equation is reduced to the statical equilibrium equation: ∇ 0 ⋅ N + ρ 0b = 0

(7.2.130)

&& is the acceleration vector. Let us remind that ∇ 0 is the material divergence and u

b) Law of conservation of angular momentum: F ⋅ N = NT ⋅ FT

(7.2.131)

S = ST

(7.2.132)

or also

The consequence of this law is the symmetry of tensor S .

Law of conservation of energy W& int = F& T : N − ∇ 0 ⋅ q 0

(7.2.133)

q 0 = JF −1 ⋅ q

(7.2.134)

where q is the heat flux vector in the spatial coordinates q is the heat flux vector in the material coordinates ∇ 0 ⋅ q is the material divergence of the heat flux If energy sources are neglected, the law expresses the fact that the rate of change of potential energy equals the difference between the load performance (that is equal to the stress performace F& T : N ) and the rate of dissipation (i.e. the loss of energy in the form of dispersed heat).

Constitutive equation This equation expresses the relation between the second Piola – Kirchhoff stress S and the Green – Lagrange strain tensor E . S = S(E, K)

(7.2.135)

In the incremental linearised form the constitutive relation can be described as follows: δ S = CSE : δ E

(7.2.136)

or in Voigt notation δ {S} = CSE  δ {E}

(7.2.137)

CSE and CSE  is the tangent material stiffness in tensor and Voigt notation, respectively, expressed in relation to the reference configuration. 523

7.2 GEOMETRICAL NONLINEARITY For infinitesimal increments this relation acquires the velocity form, where the linearization is without objections. For hypo-elastic materials the following constitutive equation can be written: & S& = CSE : E

(7.2.138)

CSE is the tangent material stiffness tensor.

Geometrical equations (strain measure) The Green –Lagrange strain tensor E is used as the strain measure in the formulation for the reference configuration: E=

1 T (F ⋅ F − I) 2

(7.2.139)

Let us derive the formula for the variation of the Green–Lagrange strain tensor. We start with (7.2.139). I is the unit diagonal matrix and its variation is thus equal to zero. The variation of a tensor is the difference between the modified and initial tensor, i.e. in our case 1 1  ( FT + δ FT ) ⋅ ( F + δ F ) − I  −  FT ⋅ F − I     2 2 1 = FT ⋅ F + FT ⋅ δ F + δ FT ⋅ F + δ FT ⋅ δ F − I − FT ⋅ F + I  2 1 = (δ FT ⋅ F + FT ⋅ δ F ) 2

δE =

(7.2.140)

We neglected the variation of the second order with respect to the variation of the first order. As the following is true: ∂u ∂X

(7.2.141)

∂δ u ∂X

(7.2.142)

F=I+ and therefore δF =

the formula for the variation of the Green–Lagrange strain tensor can be written as: 1  ∂δ uT ∂δ u  δE =  ⋅ F + FT ⋅  2  ∂X ∂X 

(7.2.143)

Let us substitute F from (7.2.141). We obtain 524

7.2 GEOMETRICAL NONLINEARITY 1  ∂δ uT δE =  2  ∂X

T  δ u   δ u  ∂δ u  ⋅I +   + I +   ∂X   ∂X  ∂X 

1  ∂δ uT ∂δ u ∂δ uT ∂u ∂uT ∂δ u  =  + + ⋅ + ⋅  2  ∂X ∂X ∂X ∂X ∂X ∂X 

(7.2.144)

Or in index notation δ Eij =

1 (δ u j ,I + δ ui, J + δ u j ,I ui ,J + δ ui ,J u j ,I ) 2

(7.2.145)

The block letters behind the comma in the lower index means the derivative with respect to the material coordinates X . Let us rewrite the equation (7.2.143) in Voigt notation. Then we can, after substitution from (7.2.141), write u i ,1δ ui ,1 Fi1δ ui ,1      δ u1,1 δ E11         δ E  ui ,2δ ui ,2 Fi 2δ ui ,2      δ u 2,2  22        δ u3,3 ui ,3δ ui ,3 δ E33   Fi 3δ ui ,3 {δ E} =  +   (7.2.146) = =  2E 23  Fi 2δ u i ,3 + Fi 3δ ui ,2  δ u 2,3 + δ u3,2  ui ,2δ ui ,3 + ui ,3δ ui ,2   2E31   Fi 3δ u i ,1 + Fi1δ ui ,3   δ u3,1 + δ u1,3   u i ,3δ ui ,1 + ui ,1δ ui ,3           2E12   Fi1δ ui ,2 + Fi 2δ ui ,1   δ u1,2 + δ u 2,1   ui ,1δ u i ,2 + u i ,2δ ui ,1  Notice that the formula for the variation of the Green–Lagrange strain tensor has split into two parts. The first one is the variation of a linear vector and the second one depends on the achieved deformation.

FEM discretisation for the formulation based on the reference configuration (total Lagrangean) The discretisation in the total Lagrangean formulation is defined for the original body configuration, i.e. in the material coordinates. If the relation between the Green–Lagrange strain tensor written in Voigt notation {E} and the vector of deformation parameters d is: δ {E} = B(d)δ d

(7.2.147)

then the following formula can be written for the vector of internal nodal forces: f int =

∫B

T

(d) {S} d Ω0

(7.2.148)

Ω0

where {S} is the second Piola – Kirchhoff stress in Voigt notation. Matrix B is defined in the material coordinates for an increment (variation) of the GreenLagrange strain tensor and is constructed by the formula (7.2.146). 525

7.2 GEOMETRICAL NONLINEARITY Matrix B can be decomposed into the sub-matrices Bi for each node i . Hence B = B1 , B 2 , K , B n 

(7.2.149)

if n is the number of nodes. Each of these sub-matrices consists of a linear part, which is identical to a similar matrix from the linear solution of the problem, and of a nonlinear part, which depends on deformation: Bi = B 0i + B Li (d)

(7.2.150)

Therefore, also the resulting matrix B can be decomposed into a linear and a nonlinear part. B = B 0 + B L (d)

(7.2.151)

For a 3D problem the following explicit expression can be written for each sub-matrix B 0i and B Li :  N i ,1  0   0 B 0i =   0  N i ,3   N i ,2

0 N i ,2 0 N i ,3 0 N i ,1

0  0  N i ,3   N i ,2  N i ,1   0 

u1,1 N i ,1   u1,2 N i ,2   u1,3 N i ,3 B Li =  u1,2 N i ,3 + u1,3 N i ,2  u1,3 N i ,1 + u1,1 N i ,3   u1,1 N i ,2 + u1,2 N i ,1

(7.2.152)

u2,1 N i ,1 u2,2 N i ,2 u2,3 N i ,3 u2,2 N i ,3 + u2,3 N i ,2 u2,3 N i ,1 + u2,1 N i ,3 u2,1 N i ,2 + u2,2 N i ,1

u3,1 N i ,1   u3,2 N i ,2   u3,3 N i ,3  u3,2 N i ,3 + u3,3 N i ,2  u3,3 N i ,1 + u3,1 N i ,3   u3,1 N i ,2 + u3,2 N i ,1 

(7.2.153)

The number following the comma in the subscript denotes the derivative with respect to the corresponding material coordinate. For instance: u2,3 =

∂u2 ∂X 3

(7.2.154)

Tangent stiffness matrix The tangent stiffness matrix formulated in the reference configuration also consists of two components: the material tangent stiffness matrix K M and the geometrical tangent stiffness matrix K σ : KT = K M + Kσ

(7.2.155)

Material tangent stiffness matrix K M 526

7.3 MATERIAL NONLINEARITY

Since matrix B depends on deformation its component matrix B L (d) must be evaluated at the beginning of each iteration step on the basis of vector d calculated from the previous iteration. The matrix B 0 does not depend on deformation and can be evaluated only once.

Let us substitute B from (7.2.151) into the standard calculation formula for material stiffness matrix and let us substitute module CSE  for tangent material stiffness. Then the following relation for the material tangent stiffness matrix can be written: KM =

∫B

T

Ω0

=

∫B

CSE  B d Ω0 =

∫ (B

Ω0

+ B L ) CSE  ( B 0 + B L ) d Ω 0 = T

0

(

)

SE T SE T SE T SE C  B 0 d Ω0 + ∫ B 0 C  B L + B L C  B L + B L C  B 0 d Ω 0 = Ω0 0 14442444 3 Ω144444444424444444443 T 0

K0

(7.2.156)

KL

= K0 + KL

Geometrical tangent stiffness matrix The geometrical stiffness matrix in the total Lagrangean formulation is calculated similarly to the updated Lagrangean formulation, i.e. using formulae (7.2.121)–(7.2.127), with the exception that the differentiation is carried out in the material coordinates and the integration is carried out in the original (reference) body configuration. Therefore, an analogous formula for the geometrical tangent stiffness of an element can be written: Kσ =

∫G

T 0

Σ 0G 0 d Ω 0

(7.2.157)

Ω0

where matrices G 0 and Σ 0 are analogous to matrices G and Σ and matrix G 0 contains material derivatives and matrix Σ contains components of the second Piola–Kirchhoff stress S.

7.3 MATERIAL NONLINEARITY 7.3.1 Uniaxial Stress As an introduction into the problem let us first consider the case of uniaxial stress. Let us have a beam of initial length l0 and initial cross-sectional area A0 . If the beam is subjected to axial force F the nominal (engineering) stress, which is for 1D identical with first stress

527

7.3 MATERIAL NONLINEARITY Piola-Kirchhoff, is defined as N x = Px = εx =

F . The engineering (linear) strain is defined by A0

δl = λx − 1 l0

where δ l is the elongation of the beam and λx is the stretch of the beam λx =

(7.3.1) l . l0

Fig. 3.1 The relation between engineering strain and engineering stress

Alternatively, the response can be also expressed with respect to the true stress. The true (Cauchy) stress is defined by the following formula: σx =

F A

(7.3.2)

where A is the current area, i.e. it is variable during the stretching of the beam. The alternative stress measure is derived from the increment of strain as the change of the length per current length unit, i.e.: l

l  dl εn = ⌠  = ln   = ln λx ⌡ l  l0 

(7.3.3)

l0

Strain ε n is called the logarithmic, or true, strain.

528


Fig. 3.2 The relation between the true strain and true stress

Cross-sectional area A is changing during the stretching of the beam. It can be described as follows: A=

JA0l0 JA0 = l λx

(7.3.4)

J is the Jacobian of the transformation between the original and the current configuration. Then for the Cauchy stress we can write the following relations between the true and nominal stress:

σx =

F F = λx = λx J −1 N x A JA0

(7.3.5)

The relation between the strain and stress can depend on the rate of deformation, but let us limit ourselves to a material which is independent of the rate of deformation. So far, unloading has not been considered in the relation between the stress and strain. Let us show the influence of unloading on this relation for different types of material. For a perfectly elastic material the stress-strain curve is identical for both loading and unloading (Fig. 3.3(a)).

529


a) elastic

b) elastic with micro-cracks

c) elastic-plastic

530


d) general Fig. 3.3 Stress-strain curves for different materials during loading and unloading

For elastic-plastic material, the slope of the curve during unloading is typically the same as the linear (initial) part of the curve during loading (Fig.3.3(b)). A material with micro-cracks formed during loading can revert to its original shape during unloading if the cracks close (Fig.3.3(c)). A general material can be a combination of these ideal cases (Fig.3.3.(d)). Below we will limit ourselves just to the elastic material that is independent of the rate of deformation.

7.3.1.1 Uniaxial nonlinear elasticity The constitutive relation for a nonlinear elastic material in case of uniaxial stress-state can be written as follows: σ x = s (ε x )

(7.3.6)

where σ x is the Cauchy stress and ε x is the engineering (linear) strain. It is assumed that σs s (ε x ) is a monotonously increasing function. The case of < 0 would indicate material σε x instability. As already stated, the unloading curve on the stress-strain diagram for an elastic material is identical to the loading curve, which implies that no energy dissipation occurs during the deformation. All work used in the deformation of the body is conserved in the body in the form of potential energy of the elastic stress. It mans that there exists potential function w(ε x ) , for which: σ x = s (ε x ) =

dw(ε x ) dε x

(7.3.7)

where w(ε x ) is the density of potential energy of the elastic stress of the body. It follows from (7.3.7) that 531

7.3 MATERIAL NONLINEARITY dw(ε x ) = σ x d ε x

(7.3.8)

which after integration yields the relation for the density of the elastic stress energy. εx

w = ∫ σ xdε x

(7.3.9)

0

For the simplest case of the linear elasticity we can write the Hooke‘s law as follows σ = Eε

For the potential energy we can writet: ε

ε

ε

ε 2  1 w = ∫ σ d ε = ∫ Eε d ε =E   = Eε 2  2 0 2 0 0 The first derivative of the potential energy with respekt to strain is stress ∂w ∂ ( 2 Eε = ∂ε ∂ε 1

2

) = Eε = σ

Energy density w is usually a convex function of deformation, in other words: ∂2w ≥0 ∂ε x2

(7.3.10)

Fig. 3.4 Comparison of the w and s functions for a stable material a) Convex function w b) Stress-strain curve

532

7.3 MATERIAL NONLINEARITY If w is not a convex function then it is the case of deformation softening and instability of  ds  < 0 . material   dε x 

Fig. 3.5 Comparison of the w and s functions for a stable material a) A non-convex function w b) The corresponding stress-strain curve

The generalization of elasticity for large deformations is simple for uniaxial stress. What it takes is to choose a geometrically nonlinear strain measure and the corresponding energetically conjugate stress measure. If we take the Green–Lagrange strain measure and the second Piola–Kirchhoff stress, we can write: Sx =

∂w ∂Ex

(7.3.11)

7.3.2 General Stress 533

7.3 MATERIAL NONLINEARITY More general constitutive relations can be presented here. Since there are different strain and stress measures, the same constitutive relations can also be written in different ways.

7.3.2.1 Saint-Venant – Kirchhoff material Many engineering applications deal with small deformations but large rotations (e.g. a fishing rod). The response of such material can be modelled by means of a simple extension of the linear elasticity law by replacing the engineering strain with the Green – Lagrange strain tensor and the stress with the second Piola – Kirchhoff stress (PKZ). Hence we can write: Sij = Cijkl Ekl

(7.3.12)

or in tensor notation S = C:E

For the fourth order tensor C the following symmetry holds: Cijkl = C jikl = Cijlk

(7.3.13)

The Saint-Venant – Kirchhoff material is independent of the path and consequently it has an elastic energy potential. The formula for the density of energy can be written in the following form: 1 1 w = ∫ Sij dEij = ∫ Cijkl Ekl dEij = Cijkl Eij Ekl = E : C : E 2 2

(7.3.14)

The stress is defined by the following formula: Sij =

∂w ∂Eij

or in tensor notation S=

∂w ∂E

(7.3.15)

Energy density w is non-negative ( w ≥ 0 ) and equals zero for E = 0 . C is a positively definite fourth order tensor.

The smoothness of potential w (continuity of C1 ) implies symmetry of tensor C : Cijkl = Cklij

(7.3.16)

The matrix of elastic constants or the tangent material stiffness is usually written using Voigt notation:

{S} = [C]{E}

(7.3.17)

534

7.3 MATERIAL NONLINEARITY The symmetry (7.3.16) implies the symmetry of matrix [C] , that means that for 3D the following holds:  S11   C11 S    22    S33    =  S23    S13       S12  

C12 C22

C13 C23

C14 C24

C15 C25

C33

C34 C44

C35 C45 C55

sym.

C16   E11  C26   E22  C36   E33    C46  2 E23  C56   2 E13    C66   2 E12 

(7.3.18)

Consequently, matrix [C] contains 21 independent constants for a general anisotropic Kirchhoff material. For orthotropic material with the main orthotropic axes identical to axes X 1 , X 2 , X 3 the constitutive relation can be written in a simpler form:  S11   C11 S    22    S33    =  S23    S13       S12  

C12 C22

C13 C23

0 0

0 0

C33

0 C44

0 0 C55

sym.

0   E11  0   E22  0   E33    0  2 E23  0   2 E13    C66   2 E12 

(7.3.19)

7.3.2.2 Hyper-elastic materials An elastic material is a material for which the stress is uniquely determined by the strain. Elastic materials for which work is independent of the path are called hyper-elastic. For such materials the stress is obtained by differentiation of a strain energy function w with respect to strain. For them the following formula holds: S=

∂w(E) ∂E

(7.3.20)

Saint Venant – Kirchhoff material is an example of hyper-elastic materials. For linearized form of constitutive equation it is possible to define the tangent stiffness of the material by SE Cijkl =

∂Sij ∂Ekl

(7.3.21)

or

535

7.3 MATERIAL NONLINEARITY CSE =

∂S ∂ 2 w(E) = ∂E ∂E∂E

(7.3.22)

CSE means the material tangent stiffness for the incremental form of the relation between stress and strain: δ S = CSE : δ E

(7.3.23)

In terms of velocity the constitutive equation can be written as follows: & S& = CSE : E

(7.3.24)

The tensor of the material tangent stiffness CSE is also called the second elasticity tensor. In current configuration the formula (7.3.24) could be written as follows: τ ∇C = Cτ : D

(7.3.25)

τ ∇C is the convection Kirchhoff stress rate defined by τ ∇C = τ& − L ⋅ τ − τ ⋅ LT

L je gradient rychlosti

(7.3.26)

∂v ∂x

Cτ is called the fourth elasticity tensor and it is defined by: τ SE Cijkl = Fim F jn Fkp Flq Cmnpq

(7.3.27)

7.3.2.3 Hypo-elastic Materials Hypo-elastic materials are materials which can be defined by the constitutive relation: σ ∇ = f (σ, D)

(7.3.28)

σ ∇ is the objective stress measure. The rate of deformation D is objective as well. Function f must also be an objective function of stress and rate of deformation. The wide range of hypo-elastic constitutive relations can be expressed as a linear relation between the objective stress rate and the rate of deformation: σ ∇ = Cσ : D

(7.3.29)

Cσ is the material elasticity tensor defined for the given objective stress measure. In the material coordinates we can write the constitutive equation for the these materials as follows & S& = CSE : E

(7.3.30) 536

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS The FEM formulation of the solution of nonlinear differential equations leads to nonlinear algebraic equations which can be expressed in the following form: K (d) ⋅ d = f

(7.4.1)

where K is the structural stiffness matrix d is the vector of unknowns, usually nodal deformation parameters

f is the vector of right-hand sides, usually nodal forces. Matrix K is a function of d and, therefore, it cannot be solved without knowing vector d , i.e. the vector of the roots of the system. Since this nonlinear system cannot be solved directly, iterative procedures are used. These are based on gradual improvement of the solution accuracy. Each iteration step is linearised. If d ( i ) is the solution of the i -th step, then equation (4.1.1) can be reformulated as K (d (i ) ) ⋅ d (i+1) = f

(7.4.2)

d ( i +1) = K −1 (d ( i ) ) ⋅ f

(7.4.3)

i.e.

The procedure can be repeated until the desired precision is reached. It is defined by the difference of vectors d ( i ) and d (i+1) . We are going to show the three most often used methods: 1. Picard iteration method 2. Newton – Raphson iteration method 3. Riks method, also known as arc length. The methods will be demonstrated on one nonlinear equation. Let us consider nonlinear equation K (d ) ⋅ d = f

(7.4.4)

or 537

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS r (d ) = 0

(7.4.5)

where d is the unknown solution, K (d ) is the known function of d , f is the known righthand side (usually force) and r is the residuum (unbalanced load). r (d ) = K (d ) ⋅ d − f

(7.4.6)

The curve defined by equation r (d , f ) = 0 is the balanced path, also called the stress-strain curve. For any value of d ( i ) , curve K (d (i ) ) is the secant to the curve at the point d = d (i ) and  ∂r  KT =   (i ) is the tangent to the curve at d = d (i ) .  ∂d  d

7.4.1.1 Picard Iteration Method This method is also known as the “direct iteration method”. We start with an initial estimate of the unknown d , let us denote it by d (0) . The next approximation is calculated pursuant to equation (4.1.3), hence: d (1) = K −1 ( d (0) ) f

(7.4.7)

The following approximations of the unknown d proceed in the same way until the required elasticity is reached, measured as the difference between two immediately following approximations of the unknown d . The convergence test can be given in the following form: rj(i ) rj( i ) (i ) j

d d

(i ) j

<ε

rj(i ) = d (j i ) − d (ji −1)

(7.4.8)

For one variable the principle of the Picard method is depicted in Fig. 4.1

538

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS Fig. 4.1 The principle of the Picard method

Possible divergence is depicted in Fig.4.2

Fig. 4.2 Divergence in the Picard method

7.4.1.2 Newton – Raphson Iteration Method We seek a solution which makes the unbalanced forces r (d ) equal to zero. Let us transform series r (d ) around the known solution d (i −1) into the Taylor series. 1  ∂ 2r   ∂r  r (d ) = r (d ( i −1) ) +   ( i −1) δ d (i ) +  2  ( i −1) δ d ( i ) 2  ∂d  d  ∂d  d

(

)

2

+K = 0

(7.4.9)

δ d ( i ) is the increment δ d ( i ) = d (i ) − d (i −1)

(7.4.10)

If the terms of the second and higher order are neglected, the equation (7.4.9) can be rearranged as follows:  ∂r  r (d ( i −1) ) +   (i −1) δ d ( i ) = 0  ∂d  d

(7.4.11)

For an increment of the deformation parameter the following relation can be written

539

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS δ d ( i ) = − ( KT (d (i −1) ) ) r (d (i −1) ) = −1

= ( KT (d (i −1) ) )

−1

( f − K (d (i −1) )d (i −1) )

(7.4.12)

where  ∂r  KT =   ( i −1)  ∂d  d

(7.4.13)

is the slope (tangent) of curve r (d ) in point d (i −1) , K is the slope of the secant passing points r (d ) = 0 and r (d ) . In mechanics, when solving problems by the deformation variant of FEM, KT is called the tangent stiffness matrix and K is called the secant stiffness. The expression K (d (i −1) )d ( i −1) represents the transferred load in step (i − 1) . The residuum or unbalanced force r (d ) gradually decreases to zero on condition that the procedure converges. In each iteration, the increment of the unknown quantity d is calculated. The solution in the i -th iteration is obtained through gradual summation of increments δ d (i ) d (i ) = d ( i −1) + δ d (i )

(7.4.14)

For a system of nonlinear equations the Newton–Raphson procedure can be formulated as follows: δ d = −K T−1r

(7.4.15)

where KT is the tangent matrix K T(i ) =

∂r ∂d d (i−1)

(7.4.16)

r is the vector of unbalanced load r = f int − f ext

(7.4.17)

f ext is the load vector and f int is the vector of nodal internal forces (calculated as the energetic equivalent of internal forces). The principle of the Newton–Raphson method is graphically depicted in Fig. 4.3.

540


Fig. 4.3 The principle of the Newton–Raphson method

The Newton–Raphson method requires that the matrix of the left-hand sides of the equation system be assembled in each iteration step. Therefore, also the decomposition (factoring) of the matrices must be carried out repeatedly in each iteration step when the Gauss or Cholesky method is applied. Sometimes it is more convenient to leave the left-hand sides of the equation system unchanged and make changes only to the right-hand side. Such method is called the modified Newton–Raphson method. It generally requires far more iteration steps than the standard Newton–Raphson method but since the decomposition of the matrix of the equation system needs to be carried out only once, the iterations are much faster. The principle of the modified Newton–Raphson method is graphically depicted in Fig. 4.4.

541

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS Fig. 4.4 The principle of the modified Newton–Raphson method

Sometimes it is convenient to combine both methods. The goal is, on the one hand, to save time required for the solution of the problem, but, on the other hand, the combination of the methods can also allow for solution of problems for which the non-modified Newton– Raphson method would fail. Fig. 4.5 depicts one such possibility. At point 1 the solution “switches” to the modified Newton–Raphson method and at point 2 it “switches back” to the non-modified one.

Fig. 4.5 The combination of Newton–Raphson and modified Newton–Raphson method

7.4.1.3 Riks Method

The Newton–Raphson method or its modification is often used to solve nonlinear problems. However, if we want to follow the balanced path of the solution of a nonlinear problem, this method may fail. The difficulty often lies in overcoming what is termed limit points, i.e. points with a horizontal or vertical tangent on the stress-strain curve. While different modifications make it possible to overcome these points and find a stable solution for a higher load level, they do not make it possible to follow the solution also during unloading in the negative branch of the stress-strain curve.

542


Fig. 4.6 Stress-strain curve with limit points

The basic idea of the Riks method is to follow the solution path (the stress-strain curve) in equal intervals ∆s . This gives the method its alternative name of “arc length”. E. Riks proposed to define ∆s on the tangent at the given balanced point and to determine the next point as the intersection of the normal erected from such a point on the tangent and the stressstrain curve. M.A. Crisfield proposed to use circular arc instead of the normal. This condition determines the increment of the load. Within the increment (or decrement) defined in this way, the modified Newton-Raphson method is used to find the balanced solution. Both modifications of the Riks method are graphically depicted in Fig. 4.7. The first index represents the increment number and the second index the iteration step in the increment.

543


Fig. 4.7 Two modifications of the Riks method

Let us show the mathematical description of one of the possible alternatives of the Riks method.

Fig. 4.8 The principle of the Riks method

Instead of increments of load coefficient γ , increments of the length of the stress-strain curve ∆s are introduced. The increment of the load coefficient ∆γ is calculated from the parametric equation: p(d (i +1) , γ ( i +1) ) = 0

(7.4.18) 544

7.4 SOLUTION METHODS FOR NONLINEAR ALGEBRAIC EQUATIONS For the circular arc modification of the Riks method the equation has the following form: p(d, γ ) = ( d (i +1) − d (i ) ) ( d (i +1) − d (i ) ) + α 2 ∆γ 2f T f − ∆s 2 = 0

(7.4.19)

∆γ = γ (i +1) − γ (i )

(7.4.20)

T

where ∆s is the approximate arc length of the balanced curve r (d, f ) = 0 in space αf , d .

α is the scaling factor transforming the load into the same physical unit as d . If d is given in n [ m ] and f in [ N ] then α has the physical dimension mN -1  α = dof K

∑

ii

i

ndof is the total number of the degrees of freedom (the order of matrix K ). Hence α is the reciprocal value of the arithmetic mean of the diagonal terms of matrix K . The parametric equation for one unknown has the following form: ∆d 2 + α 2 ∆γ 2 f 2 − ∆s 2 = 0

(7.4.21)

∆d = d ( i +1) − d (i )

(7.4.22)

where

It follows from the equation that it is a circle in space d , α f . αf

∆s ∆γαf

∆d

d

Fig. 4.9 Determination of load increment by the Riks method

The parametric equation is added to the equilibrium equations and the resulting system of equations can be written in the following form:  r (d ( i +1) , γ ( i +1) )   f int − γ f ext  0  =  = ( i +1) ( i +1)  0  ( i +1) (i +1)     p (d ), γ   p (d ), γ

(7.4.23)

545


546

7.5 LINEAR AND NONLINEAR STABILITY, POST-CRITICAL ANALYSIS

7.5 LINEAR AND NONLINEAR STABILITY, POST-CRITICAL ANALYSIS 7.5.1 Introduction The definition of stability comes from Aleksandr Lyapunov and is based on the idea that a structure is stable if a small change in initial conditions is associated with a small change of the final state. Let us mark the solution for initial conditions d A(0) and d B(0) by symbols dA and dB respectively. Figure (5.1) shows the stable and unstable state.

Fig. 5.1 The solution trajectory for the stable state (top) and unstable state (bottom)

547


A

B

C

Fig. 5.2 Stable, unstable and indifferent state of a ball on a surface

As a practical example we may mention a little ball in equilibrium on a concave, convex and planar surface (Fig. 5.2). The concave surface represents the stable state. The convex surface corresponds to the unstable state and the sought after boundary between these two states (the critical, indifferent state) is represented by the plane. What happens in the critical state is what is termed bifurcation (fork) of equilibrium conditions. It means that for the same load there exist two or more solutions (Fig. 5.3).

Fig. 5.3 Graphical representation of a straight compressed beam without imperfections (b) and with imperfections (c).

7.5.2 Linear stability When we solve a problem of linear stability, we seek such stress-state of the structure where the deformation can arise without adding any load. In mathematical terms, we seek a non-trivial solution of a homogeneous system of linear equations. K ⋅d = 0

(7.5.1)

Stiffness matrix of the strucutre K is the sum of materiál and geometrical stiffnesses.

548

7.5 LINEAR AND NONLINEAR STABILITY, POST-CRITICAL ANALYSIS K = K 0 + Kσ

(7.5.2)

Geometrical stiffnesst K σ belong to the certain loading. We are looking for the multiplier of that loading λ , for which the following equation has a nontrivial solution.

( K 0 + λKσ ) ⋅ d = 0

(7.5.3)

It is well known that a homogeneous equation system, i.e. an equation system with zero right-hand side, has a non-trivial (non-zero) solution only if the determinant of the system equals zero. This means that we solve the determinant equations: det K 0 + λ K σ = 0

(7.5.4)

where K 0 is the material stiffness matrix, K σ is the geometrical stiffness matrix for the given load, λ is the unknown load coefficient for which the equation is satisfied, which is termed “eigenvalue”. The non-trivial solution for the given eigenvalue is the eigenvector, i.e. the mode of the loss of stability. Each vector of deformation parameters which is a multiple of the calculated eigenvector, i.e. also the mode resulting from multiplication of the calculated eigenvector, satisfies the homogeneous equation system. It is completely a matter of chance which of this infinite number of affine modes is found by the algorithm for finding the eigenvalues. The eigenmode is then usually appropriately normalized. But even after the normalization the sign remains undetermined. It is, therefore, not surprising if the found eigenmode has the opposite direction than what would correspond to the deformation arising from the applied load. The determinant equation (3.4.1) is an n-degree polygon where n is the order of matrices K 0 and Κ σ . In general, it can have n solutions. In solving the stability problem, however, usually only the lowest eigenvalue found makes sense. But there are cases when the corresponding solution is not technically relevant. In order to assess this, one has to look at the eigenvector and see which part of the structure lost its stability. For instance, if a compressed member of a wind brace buckles, then such solution is irrelevant and one has to check the next, higher, eigenvalue. If the program gives negative numbers, they are not considered at all since it means that the structure would buckle if the load signs changed, which is impossible, e.g. in the case of gravity. It has to be borne in mind that both matrices ( K 0 and Κ σ ) are calculated for the original geometry and material stiffness. Therefore, it is up to the user to decide whether a linear calculation of stability is sufficient for the given structure.

549


7.5.3 Nonlinear Stability In practice, structures deform when overloaded, compressed members buckle, material stiffness changes due to nonlinear constitutive relations and due to other changes in stiffness and stress-state of the structure. If such nonlinear effects are significant, the critical load coefficients obtained by solving the linear stability problem may not be of sufficient accuracy. For most types of civil engineering structures the geometrical conditions worsen with the deformation of the structure (e.g. an arc) and linear stability gives a solution which is on the dangerous side, i.e. the obtained critical load coefficients are higher than the actual ones. In many cases, therefore, the stability problem must be solved in a nonlinear way. It is in essence a nonlinear solution of the problem of incrementally increasing load until the limit load is reached, which is shown by the matrix of the left-hand sides of the system of equations that is no longer positively definite. A sufficiently accurate solution of this critical load can be obtained by two approaches. One assumes relatively small load increments. When such a load level is reached for which the left-hand side matrix is not positively definite, the last load still satisfying the condition of positive definiteness is proclaimed to be the critical load.

Fig. 5.4 Equilibrium path of the von Mises truss

550

7.5 LINEAR AND NONLINEAR STABILITY, POST-CRITICAL ANALYSIS A better method consists of a similar procedure except that in the last load level with a positively definite matrix the state of geometry, stiffness and stress is used to find the critical load more precisely by solving the eigenvalue problem in a way similar to the linear stability. But the difference is that the states of geometry, stiffness and stress are close to the state of collapse. It should be noted that sometimes in solving nonlinear stability the critical load is not reached even if the load is increased arbitrarily (i.e. no bifurcation of equilibrium conditions arrives). Then the stability problem changes into the resistance problem (there is no loss of stability but the first, or second, limit state is reached).

7.5.4 Post-critical Analysis For many types of structures the structure can still be used even after the critical load has been reached. For instance, buckling of one member or local bulging of a metal sheet does not necessarily cause the collapse of the structure. This possibility is taken into account for instance in the analysis of airplane wings. The most widely used method to solve the nonlinear problems is the Newton–Raphson method. However, it fails when the critical load is reached – the matrix of the equation system becomes negatively definite or indefinite. In order to follow the balanced curve even after the limit point has been reached, for instance the Riks method, also called “arc length”, can be used. However, when structures are solved, we usually do not need to follow the descending branch of the stress-strain curve, but we would like to know whether the structure stiffens up again. If not, the structure collapses after the critical load has been reached. For this purpose, a simpler method can be used, for instance the modified NewtonRaphson method or the controlled deformation method. Though for the Newton-Raphson method the matrix of the equation system remains positively definite, it often leads to too many iteration steps and does not necessarily converge to a more accurate solution. The following procedure, which represents a combination of several methods, provides a satisfactory solution of the problem. Let us start with the Newton–Raphson method and continue until the program finds that the matrix of the equation system is no longer positively definite (point 3 in Fig.5). Now we go one step back (point 1 in Fig. 5) and switch to the modified Newton–Raphson method. This procedure is graphically depicted in Fig. 5. We continue solving the structure until the rising branch of the stress-strain curve is reached (point 2 in Fig. 5). Then the solution switches back to the normal Newton–Raphson method which usually quickly converges to the precise solution. It must be ensured during the whole process that the rotation in one iteration step does not exceeds the allowed limit, which means that the superposition principle can be used also for the rotation angles, in other words that the rotation can still be handled as a vector. This limit lies approximately at 0.1 rad. There is no reason to artificially limit the size of the translation, unless it is required by another reason, for instance due to the robustness of the nonlinear solution. 551


552

8.1 Variational Formulation of the Inertial Problem

8 Linear and Nonlinear Dynamics of Structures 8.1 Variational Formulation of the Inertial Problem In the present chapter some basic relations and algorithms of the foundation plate dynamics will be deduced to an extent which may be sufficient to understand the incorporation of both 2D and 3D subsoil model forms in dynamic analysis and programs. The dynamics of elastic structures is based on Hamilton´s variational principle t2

δ ∫ Π dt = 0

(8.1.1)

t1

where Π denotes the Lagrangian functional Π = Π k − Πi − Π e

(8.1.2)

with the kinetic energy Π k and potential energy Π i and Π e of the internal and external forces respectively. The integration is performed in an arbitrary time interval t1 ≤ t ≤ t2 . The last two term is expressed by the following formula: Πk =

1 T u& I ρ u& I d Ω 2 Ω∫

(8.1.3)

u I represents the vector of generalized displacement components relevant to inertial forces. The dot denotes the partial derivative with respect to the time variable t , ρ the mass density matrix. Introducing the deformation parameters d ( t ) as functions of the time variable t in the usual formula uI = NI d

(8.1.4)

u& I = N I d&

(8.1.5)

the kinetic energy Π k becomes a function of deformation parameters d ( t ) ; the notation

(t )

will be omitted for the sake of brevity: 1 Π k = d& T 2

(∫

Ω

)

1 N TI ρ N I dΩ d& = d& T Md& 2

(8.1.6)

The matrix N I need not generally be the same as the matrix N used in the derivation of a 553

8.1 Variational Formulation of the Inertial Problem stiffness matrix. The matrix M can be named the “consistent mass matrix”: M = ∫ N TI ρ N I dΩ

(8.1.7)

Ω

Thus, the functional (8.1.2) can be written in the following form: 1 1 Π = d& T Md& = d T Kd + d T f 2 2

(8.1.8)

with the stiffness matrix K and load parameter vector f , fully described in Chapter 2 (static problem). The variation in Hamilton´s principle applied to a parametrized body is a sum of variations caused by the changes δ d and δ d& of individual parameters d and their tderivatives (velocities) d& which may be mutually independent: t2 t2  t2 ∂Π ∂Π  & & − δ dKd + δ df ) dt = 0 δ ∫ Π dt = ∫  δ d& +δd dt = ∫ (δ dMd  & t1 t1 t1 ∂ d ∂ d  

(8.1.9)

The first term can be integrated by parts:

∫

t2

t1

& & dt = δ dMd&  t2 - t2 δ dMd &&dt δ dMd   ∫ t1

(8.1.10)

t1

According to the geometrical restrictions the variations t = t2 must be equal to zero:

δ d at the time t = t1 and

δ d ( t1 ) = δ d ( t2 ) = 0 therefore the first term of the right-hand side of equation (8.1.10) vanishes. Only the second term appears in equation (8.1.9) and all its terms contain only the variation δ d :

∫

t2

t1

&& − Kd + f ) dt = 0 δ d ( −Md

(8.1.11)

Since the variation δ d of the parameters d is fully arbitrary and independent of any conditions, the equation (11) can be fulfilled generally only when the expression in parenthesis is equal to zero in the whole time interval t1 ≤ t ≤ t2 ; && − Kd + f = 0 −Md This expresses the dynamical equilibrium condition, i.e. the equilibrium condition including inertial forces according to d´Alembert´s principle: && = f Kd + Md

(8.1.12)

From the common statical equilibrium condition of the parametrized body Kd = f

&& . Therefore, the mass matrix the dynamical condition (8.1.7) differs in the term Md needs to be analyzed in the following sections.

M

554

8.2 Dynamics of Foundation Plates

8.2 Dynamics of Foundation Plates 8.2.1 Consistent Mass Matrix of the Plate on the 2D Subsoil Model

8.2.1.1 Consistent Mass Matrix of the Plate Let us assume the vector of generalized displacement components (including rotations) relevant to inertial for u I =  w, ϕ x , ϕ y 

T

(8.2.1)

and the appropriate relation of the type (8.1.4) to the deformation parameters discretized body: uI = NI d

d of the (8.2.2)

The general formula (8.1.11) for the consistent mass matrix M can rewritten in the case of a plate as follows: M = ∫ N TI μ p N I dΩ Ω

(8.2.3)

The matrix μ p is a square symmetric matrix of the form μ w  μp =  0 0 

0 μϕ x μϕ xy

0   μϕ xy  μϕ y 

(8.2.4)

where μ w denotes the planar mass density which can be obtained by the integration of plate mass density ρ p over the whole plate thickness h : μ w = ∫ ρ p dz

(8.2.5)

h

Further, µϕ x and µϕ y are the densities of rotational inertia pertaining to the rotation arround the x and y axis respectively: μϕ x = μϕ y = ∫ ρ p z 2 dz h

(8.2.6)

555

8.2 Dynamics of Foundation Plates In an everyday design practice the term μϕ xy

can be put equal to zero and the equality

μϕ x = μϕ y can be assumed. But in more sophisticated CASE, modelling a complicated structure by a plate with different horizontal displacement courses along the z axis the values μϕ x = μϕ y can be mutually different and a non-zero term μϕ xy can occur. In the case of a homogeneous plate, i.e. ρp ( z ) = ρ = const. , the following formulae hold: μ w = ρp h, μϕ x = μϕ y = μϕ =

1 ρp h3 , μϕ xy = 0 12

(8.2.7)

The above formulae (8.2.1) to (8.2.4) can be used directly when analyzing a Mindlin´s plate with the rotation components ϕ x ,ϕ y independent of the deflection w . The matrix N I is of the type (3,n) n denoting the number of deformation parameters, i.e. also the order of the square mass matrix M . Each row of the matrix N I is independent of the other rows. The vector u I (8.2.1) and matrix N I can be identical to vector u and matrix N defined in Chapter 2 for the statical analysis of Mindlin´s plate using its potential strain energy. Some rewriting is necessary when analyzing a Kirchhoff plate with only one independent function, i.e. the deflection w , because the rotations ϕ x = ϕ y depend on it by the relations (6.3.12) of Section 6.3.2.2.: ϕ x = ∂w / ∂y

ϕ y = −∂w / ∂x

Introducing only one relation w = Nd the matrix N I of the formula (8.2.2) can be written as follows:    N     ∂N  NI =    ∂y   ∂N  −   ∂x 

(8.2.8)

8.2.1.2 Consistent Mass Matrix of the Subsoil In the 2D efficient subsoil model the influence of all relevant subsoil mass must be expressed by the subsoil surface properties even in the dynamic problem i.e. the inertial properties of a subsoil mass. A planar mass density μ s attached to the subsoil surface must substitute he volume mass density ρ of the mass beneath the surface from the point of view of ints inertial properties. The kinetic energy of the infinitesimal area dxdy of the 2D model must be equal to the kinetic energy of the volume dxdyH of the 3D model. This equivalence is expressed by the following relation:

556

8.2 Dynamics of Foundation Plates 1 1 μ s w& 2 ( x, y, 0, t ) dxdy = 2 2

(∫

H

0

)

ρ w& 2 ( x, y, z , t ) dz dxdy

(8.2.9)

H denotes the effective depth of subsoil (Chapter 6) in a dynamic problem; it is usually smaller than its static value. Assuming the source of vibration to be situated on the plate and the decrease law (6.2.14) also in the dynamic problem: w( x, y, z , t ) = w ( x, y, 0, t ) f ( z )

(8.2.10)

with the function f ( z ) independent of the time variable t , i.e.: w& ( x, y, z , t ) = w& ( x, y, 0, t ) f ( z )

(8.2.11)

the relations (8.2.9) result in the formula: μ s = ∫ ρ f 2 ( z ) dz H

0

(8.2.12)

Assuming the linear decrease function f ( z ) and constant mass density ρ , the planar mass density μ s is defined by the following formula: μs = ρ H / 3

(8.2.13)

From the three components of the vector (8.2.1) only the first one is relevant to the defined subsoil inertia effect w = N Iwd

(8.2.14)

N Iw denoting the first row of the N I − matrix. Therefore, the formula for the consistent mass matrix of the subsoil alone is slightly different from the formula (8.2.3): M = ∫ N TIwμ s N IwdΩ Ω

(8.2.15)

8.2.1.3 Resulting Consistent Mass Matrix of the Plate on the 2D Subsoil Model To obtain the resulting consistent mass matrix of the plate on the 2D subsoil model the addition of formulae (8.2.3) and (8.2.15) is necessary. It can be easily shown that this addition can be performed in the formula (8.2.4) by replacing the planar mass density μ w by the sum μ w + μ s ,μ w and μ s being defined by formulae (8.2.5) and (8.2.12) respectively. Therefore, formula (8.2.3) can be rewritten to include the inertial properties of the subsoil as follows: M = ∫ N TI μN I dΩ Ω

(8.2.16)

557

8.2 Dynamics of Foundation Plates μ w + μ s  μ= 0  0 

0   μϕ xy  μϕ y 

0 μϕ x μϕ xy

(8.2.17)

In the basic case of a homogeneous plate and subsoil and linear decrease function f ( z ) the matrix (8.2.17) can be calculated according to the formulae (8.2.7) and (8.2.13): 1   ρ p h + 3 ρs H  μ= 0    0 

0

   0    1 3 ρp h  12 0

1 ρ p h3 12 0

(8.2.18)

In the term μ s the inertial properties of the subsoil mass beneath the plate are taken into account.

8.2.2 Consistent Mass Matrix of the Plate on the 3D Subsoil Model The 3D efficient subsoil model differs from the simpler 2D model in n layers i = 1, 2,..., n between the plate and 2D model; see Section 6.3.3.6, where all definitions and relations are presented. The consistent mass matrices of the plate and the 2D model are defined by the formulae (8.2.3) and (8.2.15) respectively. The integration is usually performed numerically. For example, in the NE-10 program (Chapter 6) the isoparametric quadrilateral Mindlin plate element with bilinear shape and base functions is implemented and the integration over the element region Ω e is replaced by the integration over the unit element: M = ∫ N TI μ p N I dΩ e = ∫

1

∫

1

−1 −1

Ω

N TI μ p N I J 2 dξ dη

(8.2.19)

The consistent mass matrix of a layer element can be calculated directly by the general formula (8.1.7), where ρ denotes the soil mass density and N the row matrix of base functions in the parametric expression of the settlement course in the element Ω e : w = Nd,

w& = Nd&

(8.2.20)

The kinematic energy of horizontal displacement components u , v is negligibly small and omitted in the same way as the potential strain energy of horizontal stresses when deriving the stiffness matrix of the layer subsoil element. For example, the isoparametric brick element defined in Section 6.3.3.6.2 and used in program NE-10 with Ahlin´s trilinear base and shape functions N3 (see Formula (6.3.141)) can be analysed by numerical integration similar to formula (6.3.159) holding for stiffness 558

8.2 Dynamics of Foundation Plates matrix. The consistent mass matrix is calculated by the following formula M = ∫ ∫ ∫ N 3T ρ N3dΩ e = ∫ Ωe

1

∫ ∫ 1

1

−1 −1 −1

N 3T ρ N 3 J 3 dξ dη dς

(8.2.21)

J 3 denotes the Jacobian matrix of transformation (6.3.152) of the analysed element Ω e to the unit cube, where the functions N are defined in natural coordinates ξ ,η , ζ see Section 6.3.3.6.2. It is not possible to add the consistent mass matrices in the 3D subsoil model in the same way as in the case of the plate on the 2D subsoil model (Section 8.2.1.3). The addition will be carried out in individual superelement nodes following the same addition theorem as for stiffness matrices and load parameter vectors, (Section 6.3.3.3) Note. The dynamic decrease function f ( z ) used in the 2D model is generally

different from the static decrease function f ( z ) defined in Chapter 1, which also influences the so-called dynamic limit depth H .

8.2.2.1 Damping Properties of the Plate-Soil System The dynamic equilibrium condition (8.1.12) can be extended by adding the viscous damping forces Cd& , proportional to the velocity, on the left-hand side: && + Cd& + Kd = f Md

(8.2.22)

A proper definition of the damping matrix C is not easy due to the lack of knowledge of damping properties. A simple assumption divides the matrix C into two parts: C = Cm + Cs

(8.2.23)

The parts Cm and Cs express the damping due to the velocity of mass points and strain changes respectively. The matrix Cm can be calculated in the same way as the mass matrix M by the formula (8.1.7), replacing the density matrix ρ by another matrix ϑm : Cm = ∫ N Tm ϑm N m dΩ 3 Ω3

(8.2.24)

The second matrix Cs can be calculated from the formula used for the stiffness matrix K (Chapter 6.3) replacing the matrix of physical constants D by another matrix ϑs ; Cs = ∫

Ω3

( GNs )

T

ϑs GNs dΩ 3

(8.2.25)

N , N m and Ns are generally four different matrices because the functions relevant to stiffness, inertia and damping properties need not always be identical. For instance a Kirchhoff plate with one unknown function w can be calculated whilst also respecting the rotary inertia of plate mass normals ( − h / 2 ≤ z ≤ h / 2 ) . This inertia is connected with the

559

8.2 Dynamics of Foundation Plates rotation components ϕ x , ϕ y in Mindlin Plate, while the damping is bonded with rotations wx and wy of the subsoil surface. Terms of matrices ϑm and ϑs can be obtained by experimental investigation. The 2D efficient subsoil model expresses all properties of subsoil mass by properties defined in the structure-soil interface; likewise in the case of damping. The derivation can proceed in the same way as for the stiffness and inertia properties. Let the unknown settlement function w ( x, y, z , t ) have the following form, assuming the vibration source on the plate: w ( x, y, z , t ) = w ( x, y, 0, t ) f ( z )

(8.2.26)

The same function may be expressed by nodal displacement parameters d ( t ) in a standard form: w ( x, y , z , t ) = N ( x , y , z ) d ( t )

(8.2.27)

The damping due to velocities in the x – and y – directions are not taken into account. Only one displacement component w is introduced in the calculation. Thus the matrices of damping properties ϑm and ϑs . The damping property Θ of the surface model must represent the damping of the whole subsoil mass. To determine it the equivalence of the rate of dissipation can be applied. The equality between the rates of dissipation of the 2D and 3D models can be written in the following form: Θ w& 2 ( x, y, 0, t )dxdy =

( ∫ ϑw& ( x, y, z, t ) dz ) dxdy H

0

2

(8.2.28)

Substituting the hypothesis (8.2.26) into equation (8.2.28) the following formula for the surface damping property Θ can be written: Θ = ∫ ϑ f 2 ( z ) dz H

0

(8.2.29)

In this derivation Θ and ϑ represent general damping properties due to velocity, hence the derivation holds for both kinds of the damping mentioned above and we can write for them the following formulae: Θ m = ∫ ϑm f 2 ( z ) dz

(8.2.30)

Θs = ∫ ϑs f 2 ( z ) dz

(8.2.31)

H

0

H

0

Introducing this reduction of the problem the expressions (8.2.24) and (8.2.25) for damping matrices can be rewritten as follows: Cm = ∫∫ N TmΘ m N m dΘ 2 Ω2

(8.2.32)

560

8.2 Dynamics of Foundation Plates Cs = ∫∫

Ω2

( GNs ) Θs ( GNs ) dΘ 2 T

(8.2.33)

561

8.3 Linear solution of structures subjected to vibration

8.3 Linear solution of structures subjected to vibration The linear solution of models of structures that are subjected to dynamic load is usually performed using one of the two following approaches: (i) the decomposition into eigenmodes (mode superposition method) or (ii) numerical methods of direct integration.

8.3.1 The decomposition into eigenmodes method A system of motion equations of a discrete model of a structure subjected to a dynamic load can be written &&(t ) + Cu& (t ) + Ku(t ) = F(t ) Mu

(8.2.34)

In general, the matrices in (8.2.34) are variable over time, and, therefore, system (8.2.34) can only be solved using the direct numerical integration methods. On condition that the mass matrix and stiffness matrix are constant and the damping matrix satisfies certain assumptions, system (8.2.34) can be solved using the decomposition into eigenmodes. The principle is that we seek the solution of equation (8.2.34) in the form of a linear combination of eigenmodes, i.e. in the form n

u(t ) = φ(1) q1 (t ) + φ(2) q 2 (t ) + K + φ( n ) q n (t ) = ∑ φ( j ) q j (t ) = Φq(t )

(8.2.35)

j =1

where q j (t ) are the coefficients of the linear combination, q(t ) is the vector composed of these coefficients and Φ is the matrix created from the eigenmode vectors. If we express (8.2.34) using formula (8.2.35), we get &&(t ) + CΦq& (t ) + KΦq(t ) = F(t ) MΦq

(8.2.36)

Now, let us multiply (8.2.36) from the left by matrix Φ T &&(t ) + Φ T CΦq& (t ) + Φ T KΦq(t ) = Φ T F(t ) Φ T MΦq

(8.2.37)

As the eigenmodes are orthogonal, we can transform (8.2.37) into the form &&(t ) + Φ T CΦq& (t ) + Ω 2q(t ) = Φ T F(t ) q

(8.2.38)

Damping matrix C is in practice often expressed as a linear combination of mass matrix M and stiffness matrix K using what is termed Rayleigh damping matrix C = α RM + β RK ,

(8.2.39)

562

8.3 Linear solution of structures subjected to vibration that can – through coefficients α R , β R - assign proportionally different (damping) weights to the velocity and to the speed of deformation change. If the damping matrix is expressed using (8.2.39), then we get Φ T CΦ = Φ T (α R M + β R K ) Φ = α R + β R Ω2

(8.2.40)

In that case, the simultaneous system of n differential equations breaks into n independent equations in the following form q&& j (t ) + 2ξ( j )ω( j ) q& j (t ) + ω(2j ) q j (t ) = Qj (t ) where 2ξ( j )ω ( j ) = (α R + β Rω (2j ) )

(8.2.41) (8.2.42)

m

Q j = φ(Tj ) F (t ) = ∑ Fi (t )φi ,( j )

(8.2.43)

i =1

After we solve n equations (8.2.41) and create the linear combination according to (8.2.35), we can obtain the sought after solution of system (8.2.34).

8.3.1.1 Calculation of seismic effects from response spectrum If a structure is subjected to seismic excitation, then the load vector in (8.2.34) has the following form: && g (t ) , F(t ) = −Mu

(8.2.44)

&& g (t ) is the distribution of the seismic excitation. The solution of the response to such where u a load is numerically demanding and, therefore, the calculation is often done using what is termed response spectra. A response spectrum is a diagram of response maximums (displacement, stress, acceleration, etc.) of a single DOF system to a given excitation with respect to a certain parameter (usually with respect to non-damped natural frequency). The response spectrum for displacement and acceleration can be written as S d (ω( j ) , ξ ) = max u (t )

(8.2.45)

S a (ω( j ) , ξ ) = max u&&(t )

(8.2.46)

As civil engineering structures assume small damping ( ξ ≤ 5% ), the relation between the two presented spectra can be written using the following formula Sa (ω( j ) , ξ ) ≅ ω(2j ) ⋅ S d (ω( j ) , ξ )

(8.2.47)

Moreover, for majority of structures we can suppose that the structure is subjected to only one seismic excitation at a given time instant and that this excitation can be decomposed into three && g (t ) can be written directions x, y, z . The distribution of acceleration u 563

8.3 Linear solution of structures subjected to vibration && g (t ) = I x u&&g , x (t ) + I y u&&g , y (t ) + I z u&&g , z (t ) u

(8.2.48)

where vectors I x , I y and I z are unit vectors containing ones (1) only in the positions corresponding to the x, y and z coordinate. Substituting (8.2.48) to (8.2.44) and then to (8.2.43), we get – after modification Q j = −φ(Tj ) M ( I x u&&g , x (t ) + I y u&&g , y (t ) + I z u&&g , z (t ) ) = =

∑ (φ

T k ,( j )

k =x, y,z

MI k u&&g ,k (t ) ) =

∑ (Γ

k = x, y , z

( j ), k

⋅ u&&g ,k (t ) )

(8.2.49)

where Γ ( j ),k is what is termed the participation factor. Using (8.2.49) in equation (8.2.41) we get q&&j (t ) + 2ξ jω j q& j (t ) + ω 2j q j (t ) = −

∑ (Γ

k =x, y,z

( j ), k

⋅ u&&g ,k (t ) )

(8.2.50)

Applying the response spectrum for displacement according to (8.2.45), we can write

∑ (Γ

max q j (t ) =

k = x, y , z

( j ), k

S d ,k (ω( j ) , ξ ( j ) ) )

(8.2.51)

If we have the response spectrum for acceleration, we can use (8.2.47) and modify formula (8.2.51) and obtain the eigenmode coefficient q( j ) = max q( j ) (t ) =

∑ (Γ

k = x, y , z

( j ), k

Sa ,k (ω( j ) , ξ ( j ) ) ) (8.2.52)

ω(2j )

Using q( j ) we can calculate the maximum of an arbitrary static quantity corresponding to the j-th eigenmode according to S( j ),max = q( j ) S( j )

(8.2.53)

The maximums in individual eigenmodes, however, do not happen simultaneously. To sum absolute values of all S( j ),max would lead to a very conservative estimate. Therefore, a formula known as SRSS method is used to estimate the maximums. S max =

n

∑S j =1

2 ( j ),max

(8.2.54)

Alternatively, a formula known as CQC method can be used.

S max =

n

n

∑∑ S i =1 j =1

( i ),max

⋅ ρij ⋅ S( j ),max

(8.2.55)

where 8 ξiξ j ⋅ (ξi + rξ j ) ⋅ r 2 3

ρi , j =

(1 − r

)

2 2

+ 4ξiξ j r (1 + r 2 ) + 4 (ξi2 + ξ 2j ) r 2

, r=

ω( i ) ω( j )

(8.2.56)

564

8.3 Linear solution of structures subjected to vibration

8.3.2 Numerical methods of direct integration Numerical methods of direct integration solve system (8.2.34) in a finite number of time instants t0 , t1 , K , tm . The distance between individual time instants ∆ti = ti − ti −1 is called the length of the integration step. The lengths of integration steps ∆ti influence the accuracy, stability and speed of the solution. Defined initial conditions are an integral part of system (8.2.34). The time t = 0 is considered to be the starting point at which u(t0 ) = u 0 , u& (t0 ) = u& 0 . System (8.2.34) can be thus written as &&i + Cu& i + Kui = Fi Mu

(8.2.57)

Usually, we divide the numerical methods to: • explicit methods, • implicit methods and • predictor-corrector methods. The first two methods are considered the basic ones and the predictor-corrector method is in fact a simulation of the implicit method. Whether an integration method is explicit or implicit depends on the time instant in which the method uses system (8.2.34).

8.3.3 Explicit methods In explicit methods we make use of the assumption about the distribution of motion && in interval 〈ti , ti +1 〉 and the knowledge of these characteristics at time characteristics u, u& , u &&i +1 from (8.2.57). Neither triangulation nor instant ti , and we calculate vectors ui +1 , u& i +1 , u modification of the stiffness matrix is performed in explicit methods.

8.3.4 Method of central differences The numerical integration of differential equations uses the substitution of the derivative of the independent variable with respect to time. If we replace the derivatives in (8.2.57) by u& i =

1 ( ui +1 − ui −1 ) 2∆ti

(8.2.58)

&&i = u

1 ( ui +1 − 2ui + ui −1 ) ∆ti 2

(8.2.59)

565

8.3 Linear solution of structures subjected to vibration we get a recurrent formula for ui  1  1 C  ui +1 =  2 M+ 2∆ti   ∆ti    1  2 1 = Fi −  K − 2 M  ui −  2 M − C  ui −1 ∆ti 2∆ti     ∆ti

(8.2.60)

The method has all the advantages of explicit methods as long as [C] = [ 0] or [C] = α [ M ] . Its application is most effective for diagonal mass matrix. However, the method is only conditionally stable. The length of the integration step must meet the condition ∆ti ≤

Tn π

(8.2.61)

where Tn is the smallest vibration period.

8.3.5 Implicit methods Implicit methods are based on system (8.2.57) at time instant ti . The numerical integration of the system is carried out step by step using the following formula &&i = f ( ti , Fi −1 , u i −1 , u& i −1 , u &&i −1 ) u

(8.2.62)

with the necessity – in order to be able to start with the solution – to evaluate the acceleration at the beginning of the motion at time t0 directly from system (8.2.57) &&0 + Cu& 0 + Ku 0 = F0 Mu

(8.2.63)

The most common implicit methods include: • Newmark method and • Wilson method.

8.3.5.1 Newmark method The basic formula of the Newmark method that specifies the relations between displacement, velocity and acceleration vectors have the following form 1  &&i −1 + β ∆ti 2u &&i ui = u i −1 + ∆ti u& i −1 +  − β  ∆ti 2u 2 

(8.2.64)

566

8.3 Linear solution of structures subjected to vibration &&i −1 + γ∆tiu &&i u& i = u& i −1 + (1 − γ )∆ti u

(8.2.65)

where β and γ are what is termed Newmark’s parameters. After substituting the stated relations into (8.2.63) and modifying the obtained formula we get the relation for the &&}i at time ti calculation of the unknown acceleration vector {u

( M + γ∆t C + β∆t K ) u&& 2

i

i

i

&&i −1 ) − = Fi −C ( u& i −1 + (1 − γ )∆tiu   1  &&i −1  −K  u i −1 + ∆ti u& i −1 +  − β  ∆ti 2u 2   

(8.2.66)

The selection of Newmark’s parameters influences the accuracy and stability of the solution. For γ = 12 and β = 14 we obtain the constant acceleration method, for γ = 12 and β = 16 we get the linear acceleration method. The method is stable if β ≥ 14 . For 0 ≤ β ≤ 14 , the method is conditionally stable with the step length ∆ti being the main condition ω ∆ti < 2 1 − 4β

(8.2.67)

Let us introduce into (8.2.66) the following substitution G = M + γ∆tiC + β ∆ti 2 K

(8.2.68)

  1  &&i −1 Fi = Fi − Kui −1 − ( C + ∆ti K ) u& i −1 −  (1 − γ )∆ti C +  − β  ∆ti 2K  u 2   

(8.2.69)

As a result we obtain &&i = Fi Gu

(8.2.70)

It is suitable to select a constant numerical integration step ∆ti in the solution of linear problems where matrices K , M and C are also constant. In that case, also matrix G will be constant throughout the whole integration process. Consequently, for every step i, systems (8.2.70) represent systems of n equations for n unknowns that differ only in the right hand side Fi . The solution of such systems is relatively fast as the triangulation of matrix G is performed only in the first step. Subsequent steps are then used to calculate the sought after &&i from the changing right hand sides Fi . vectors u In nonlinear mechanics, matrices K , M and C are not constant, and, therefore, the above-mentioned advantages cannot be applied to the solution of system (8.2.70). In general, matrix G changes in every step and, as a result, the whole mentioned system must be calculated in every step. The calculation takes longer time but it makes it possible to take into account geometrical or material nonlinearities. The relations for displacement, velocity and acceleration vectors can be now written using the vectors of corresponding increments ui = u i −1 + ∆ui

(8.2.71)

u& i = u& i −1 + ∆u& i

(8.2.72)

&&i = u &&i −1 + ∆u &&i u

(8.2.73) 567

8.3 Linear solution of structures subjected to vibration Now we can prove that (8.2.70) can be modified to the following formula &&i = ∆Fi G∆u

(8.2.74)

  1  &&i −1 ∆Fi = ∆Fi − K∆u i −1 − ( C + ∆ti K ) ∆u& i −1 −  (1 − γ )∆ti C +  − β  ∆ti 2K  ∆u 2   

(8.2.75)

with

8.3.5.2 Wilson method This method is based on the assumption of a linear acceleration within the interval 〈t , t + ϑ ∆t 〉 . The basic relations then have the following form &&( t +τ ) = u &&(t ) + u

τ ( u&&(t +ϑ h ) − u&&(t ) ) ϑh

u& ( t +ϑ h ) = u& (t ) +

ϑ ∆ti ( u&&(t +ϑ h) + u&&(t ) ) 2

u ( t +ϑ h ) = u (t ) + ϑ ∆ti u& ( t ) +

ϑ 2 ∆ti 2 ( u&&(t +ϑ h ) + 2u&&(t ) ) 6

(8.2.76) (8.2.77) (8.2.78)

Relations (8.2.77) and (8.2.78) are analogous to formulas (8.2.64) and (8.2.65) in the Newmark method. The subsequent procedure is thus identical. Substituting into (8.2.57) we &&(t +ϑ h ) can be determined. Performing a backward obtain a formula from which vector u substitution into (8.2.77) and (8.2.78) and applying equality τ = ∆ti we get the sought after vectors ui+1 and u& i+1 at time ti +1 . The stability and accuracy of the method depends on the selection of coefficient ϑ . To have the method stable, the following condition must be met: ϑ ≥ 1,37 . Every numerical method has its advantages and disadvantages that must be taken into account when the numerical method for the solution of the given problem is being selected. It is necessary to consider the kind of technical problem, type of excitation, requirements on the results, capabilities of used computers, etc.

568

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load A nonlinear solution of structure models subjected to dynamic load can exploit procedures based on a connection of the Newmark method of direct integration of motion equations and the Newton-Raphson method for the solution of linear problems. To obtain a clearer description of such connection, it is suitable to modify the above-mentioned formulas.

8.4.1 Modification of relations in motion equations In a nonlinear solution of dynamic problems, the tangent stiffness matrix is used in motion equation (8.2.57) instead of the stiffness matrix Mai + Cv i + K T u i = Fi

(8.3.1)

&& and v = u& are used in equation In order to simplify the following formulas, substitutions a = u (8.3.1). The first two members of the left hand side of equation (8.3.1) represent inertia and damping forces of a vibrating system, the third member is what is termed restoring force. The inertia and damping forces are non-zero if the system is subjected to a dynamic load or if such a load caused vibration of the system. If we assume that the structure is subjected only to a static load that produces no vibration, the inertia and damping forces are zero. The first two members and the third member of the equation (8.3.1) can be denoted

FD,i = Mai + Cv i

(8.3.2)

FS,i = K T ui

(8.3.3)

where subscript D means the forces due to the dynamic load acting on the structure and subscript S denotes the forces originating from the static load. If we rewrite (8.3.1) in an incremental form M∆ai + C∆v i + K T ∆u i = ∆Fi

(8.3.4)

lze ve členy na levé straně obdobně jako vztahy (8.3.2) a (8.3.3) ∆FD,i = M∆ai + C∆v i

(8.3.5)

∆FS,i = K T ∆ui

(8.3.6)

Using (8.3.4) to (8.3.6) we can write ∆FS,i = ∆Fi − ∆FD,i

(8.3.7)

In a nonlinear solution, formula (8.3.6) becomes K T,i ∆ui = ∆FS,i ,

(8.3.8) 569

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load which is used in the Newton-Raphson method to determine the vector of displacement increments.

8.4.2 Modification of relations between displacement, velocity and acceleration vectors As formulas ui = u i −1 + ∆ui

(8.3.9)

v i = v i −1 + ∆vi

(8.3.10)

ai = ai −1 + ∆ai

(8.3.11)

hold for the vectors of displacement, velocity and acceleration between two time instants, the displacement vector can be modified according to (8.2.64) and its components can be marked 1 ui = u i −1 + hv i −1 + h 2ai −1 + β h 2 ∆ai 2

(8.3.12)

where h = ∆ti . Therefore, the following relation can be written for the vector of displacement and vector of displacement increments at time instant ti : = uˆ i + β h2 ∆ai

(8.3.13)

∆ui = ∆uˆ i + β h 2 ∆ai

(8.3.14)

ui

where uˆ i

= ui−1 + hv i−1 + 12 h2ai−1 = ui−1 + ∆uˆ i

∆uˆ i = hv i−1 + 12 h 2ai−1

(8.3.15) (8.3.16)

Similarly, formula (8.2.65) can be modified for the velocity vector v i = v i −1 + (1 − γ ) hai −1 + γ hai −1 + γ h∆ai The velocity vector and velocity increment vector at time instant ti can be, consequently, written as vi

= vˆ i + γ h∆ai

∆v i = ∆vˆ i + γ h∆ai where vˆ i

= v i −1 + hai −1 = v i −1 + ∆vˆ i

∆vˆ i = hai−1

(8.3.17) (8.3.18) (8.3.19) (8.3.20)

The above-mentioned relations are used to calculate the vectors of displacement and velocity (or their increments) at time instant ti from the values in the previous known time instant and, also, from the vector of acceleration (or its increment) in the analysed time instant 570

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load ti . If the procedures based on the connection of the Newmark and Newton-Raphson method are used, it is necessary to determine the vectors of velocity and acceleration (or their increments) from, among others, the values of the displacement vector (or its increment) at the analysed time instant ti . Modifying (8.3.13) and (8.3.14) in the following way ∆ai =

1 ( ui − uˆ i ) β h2

(8.3.21)

∆ai =

1 ( ∆ui − ∆uˆ i ) β h2

(8.3.22)

and substituting (8.3.16) into (8.3.22), we can get  1  1 1 ∆ai =  − v i −1 − ai −1  + 2 ∆ui 2β  βh  βh

(8.3.23)

As a result, we can write the following formula for the vector of acceleration increments: ∆ai = ∆ ai +

1 ∆ui β h2

 1  1 vi −1 − ai −1  where ∆ ai =  − 2β  βh 

(8.3.24)

(8.3.25)

Similarly, using (8.3.16), (8.3.22) and (8.3.20) in (8.3.18) we get  γ ∆v i = 1 −  2β

 γ γ  ha i −1 − β v i −1 + β h ∆ui 

(8.3.26)

We can write the following formula for the vector of velocity increments: ∆v i = ∆v i +

γ ∆u i βh

 γ where ∆v i = 1 −  2β

 γ  hai −1 − β v i −1 

(8.3.27)

(8.3.28)

The relations for the vectors of displacement increments, velocity vectors and acceleration vectors at the currently analysed time instant ti are thus always expressed using two members with the first member depending only on the values of individual vectors from the previous, already known, time instant ti −1 .

571

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load Using the above-mentioned formulas in (8.2.74), in which we consider the notation according to (8.3.1), we can obtain G∆ai = ∆Fi − C∆vˆ i − K T ∆uˆ i

(8.3.29)

8.4.3 Variants of connection of methods There are several variants of the connection of the Newton-Raphson and Newmark method. These variants differ in how and when the Newton-Raphson method is applied, or in its direct modification. As the Newmark method calculates the increments of individual quantities between two time instants and the Newton-Raphson method uses the forces increments from which the displacement increments are determined, it is convenient – in the connection of the two methods - to introduce the quantity called partial increment. The partial increment is the increment of any quantity at the given time instant corresponding to the current load increment in the Newton-Raphson method. We introduce symbol ∆∆S for the partial increment. We will use the following formulas for the k-th iteration of the vector of total increments in the i-th time step: k

∆S i = ∑ m∆∆S i

k

(8.3.30)

m=1

∆S i =

k

k −1

∆Si + k ∆∆Si

(8.3.31)

It is convenient, for the purpose of further explanation, realise that ∆Si = 1∆∆S i

(8.3.32)

∆Si = 0

(8.3.33)

1

0

The total increments of the displacement, velocity, acceleration and force vectors can be written as ∆ui = ∆v i = ∆ai = ∆Fi =

nITER

∑

∆∆ui

(8.3.34)

k

∆∆v i

(8.3.35)

k

∆∆ai

(8.3.36)

k

∆∆Fi

(8.3.37)

k

k =1

nITER

∑ k =1

nITER

∑ k =1

nITER

∑ k =1

where nITER is the total number of iteration steps at the given time instant i. Substituting (8.3.34) into (8.3.27) and into (8.3.24), we get

572

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load ∆v i = ∆v i +

γ nITER k ∑ ∆∆ui β h k =1

∆ai = ∆ ai +

1 β h2

nITER

∑ k =1

k

(8.3.38)

∆∆ui

(8.3.39)

Comparing (8.3.38) with (8.3.35) and comparing (8.3.39) with (8.3.36) we obtain   ∆v i + k ∆∆v i =   

γ k ∆∆ui ; βh γ k ∆∆ui ; βh

1 k  ∆ ai + β h 2 ∆∆ui ; k ∆∆ai =  1 k  ∆∆ui ;  β h2

pro k = 1 (8.3.40) pro k > 1 pro k = 1 (8.3.41) pro k > 1

Considering (8.3.5), we can write the relation for partial increments of forces due to dynamic load k

∆∆FD,i = C k ∆∆v i + M k ∆∆ai .

(8.3.42)

8.4.3.1 Algorithm of linear solution Before proceeding to the description of individual variants, let us present (for comparison) the algorithm of a linear solution of problems using the Newmark method. 1. Adjustment of initial conditions and calculation of values that do not vary over time i = 0, t = t0 u 0 = u(t0 ) , v 0 = v (t0 ) , a 0 = a(t0 ) G = M + γ hC + β h2 K 2. Determination of the load values at time step (i) i = i + 1, t = t + ∆t = t + h Fi ∆Fi = Fi − Fi−1 3. Calculation of the values resulting only from the time instant i 573

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load ∆uˆ i = hv i −1 + (α + β )h2ai −1 ∆vˆ i = hai −1 4. Calculation of acceleration increments from equation G∆ai = ∆Fi − C∆vˆ i − K∆uˆ i 5. Calculation of velocity and displacement increments ∆v i = ∆vˆ i + γ h∆ai ∆ui = ∆uˆ i + β h2 ∆ai 6. Calculation of current values of deformation, velocity and acceleration ui = u i −1 + ∆ui v i = v i −1 + ∆vi ai = ai −1 + ∆ai 7. Verification if the calculation has been performed throughout the whole interval t < tn

⇒

continue starting from point 2.

8. End of calculation

8.4.3.2 Variant I The first variant of the connection of the stated methods has no impact on the Newton Raphson method that is used to solve the nonlinear system (8.3.8). It is, however, necessary to resolve relations (8.3.5) and (8.3.7) prior to this solution. It can be done by means of what is termed linear estimates of vectors of acceleration increments and velocity increments. These will be marked in the text by subscript L, e.g. k S L,i . These linear estimates can be obtained the same way as if a linear problem was being analysed. Substituting in (8.3.29) the vectors by notation (8.3.31), we get G(

∆ai + k ∆∆a i ) = (

k −1

∆Fi + k ∆∆Fi ) − C∆vˆ i − K T ∆uˆ i

k −1

(8.3.43)

After a modification of (8.3.43) we obtain G k −1∆ai + G k ∆∆ai =

k −1

∆Fi − C∆vˆ i − K T ∆uˆ i + k ∆∆Fi

(8.3.44)

Considering relations (8.3.32) and (8.3.33), it is possible to use (8.3.44) to write the following formula that can be used for a linear estimate of the acceleration. Therefore, the relevant quantities in this formula will be marked by subscript L

574

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load k

∆∆a L,i

 G k ∆∆a L ,i = k ∆∆Fi − C∆vˆ i − K T ∆uˆ i ; :  k k  G ∆∆a L ,i = ∆∆Fi ;

pro k = 1 pro k > 1

(8.3.45)

Substituting (8.3.36) into (8.3.18) and comparing with (8.3.35) we get nITER

∑ k =1

k

nITER

∆∆vi = ∆vˆ i + γ h ∑ k ∆∆ai .

(8.3.46)

k =1

Modifying (8.3.46) we obtain the relation for a linear estimate of velocities k

∆∆v L ,i

k  ∆vˆ i + γ h ∆∆a L,i ; = γ h k ∆∆a L,i ; 

pro k = 1 pro k > 1

(8.3.47)

It is clear that these linear estimates of partial increments of velocities and accelerations can be in each iteration step k used to determine linear estimates of partial increments of inertia and damping forces k

∆∆FD, L ,i = C k ∆∆v L,i + M k ∆∆a L ,i

(8.3.48)

and the corresponding partial increments due to restoring forces k

∆∆FS, L ,i = k ∆R i − k ∆∆FD, L ,i

(8.3.49)

where k ∆R i is the vector of residual (unbalanced) forces that is specified later in the text. These linear estimates can be used – using the Newton-Raphson method – to determine the partial increments of displacement k ∆∆ui and the corresponding partial restoring forces k

∆∆FS,i . Substituting the calculated vector k ∆∆ui into (8.3.40) and (8.3.41) we obtain the

corresponding vectors

k

∆∆v i and

k

∆∆ai and from them it is possible to determine –

according to (8.3.42) – the corresponding vector of inertia and damping forces k ∆∆FD,i . Using (8.3.30) or (8.3.31) we can calculate the appropriate values of vectors of total increments of all required quantities in the k-th iteration. It is probable that the sum of increments of all forces in the k-th iteration will not be in equilibrium with the increment of the total force in the i-th time step. The imbalance can be expressed using the vector of residual forces ∆R i = ∆Fi − ( k ∆FS,i + k ∆FD,i ) .

k +1

(8.3.50)

If this vector is not negligible (with a satisfactory accuracy) we continue with next iteration step and the stated vector of residual forces is used for the calculation of new linear estimates of partial increments of accelerations and velocities. The advantage of this method is the possibility to use different modifications of the Newton-Raphson algorithm that is (in this case) independent on the Newmark integration. The whole procedure of variant I can be clearly seen in Fig 8.1 and in the following algorithm. 1. Adjustment of initial conditions 575

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load i = 0, t = t0 u 0 = u(t0 ) , v 0 = v (t0 ) , a 0 = a(t0 ) 2. Determination of the load values and matrices at time step (i) i = i + 1, t = t + ∆t = t + h G i = M + γ hC + β h 2 K T,i Fi , ∆Fi = Fi − Fi−1 3. Calculation of the values resulting only from the time instant i ∆uˆ i = hv i −1 + 12 h 2ai −1 ∆vˆ i = hai −1  1  1 ∆ ai =  − vi −1 − ai −1  2β  βh   γ ∆v i = 1 −  2β

 γ  hai −1 − β v i −1 

4. Adjustment of values prior to integration k =0

ui = ui−1

Fi = Fi−1

0

0

v i = v i−1

0

ai = ai−1

1

0

FS,i = FS,i−1 ∆R i = ∆Fi

0

5. Beginning of the iteration cycle, linear estimates of partial increments k = k +1 k

k

k

∆∆a L,i

∆∆v L ,i

 G k ∆∆a L ,i = k ∆R i − C∆vˆ i − K T,i ∆uˆ i ; :  k k  G ∆∆a L ,i = ∆R i ; k  ∆vˆ i + γ h ∆∆a L,i ; = γ h k ∆∆a L,i ; 

pro k = 1 pro k > 1

pro k = 1 pro k > 1

∆∆FD, L ,i = C ⋅ k ∆∆v L ,i + M k ∆∆a L ,i

6. Calculation of the linear estimate of partial increments of static forces and linear estimate of static forces k

∆∆FS, L ,i = k ∆R i − k ∆∆FD, L ,i 576

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load k −1

FS, L,i =

k

FS,i + k ∆∆FS, L ,i

7. Nonlinear calculation of partial increments of deformation using the Newton-Raphson method k

∆∆u i = f ( K T,i ,

k

∆∆FS,i = kFS,i −

ui −1 , k ∆∆FS,i , kFS,i )

k −1

k −1

FS,i

8. Calculation of partial increments of acceleration and velocity   ∆v i + k ∆∆v i =   



pro k = 1 pro k > 1 pro k = 1 pro k > 1

9. Calculation of current partial increments of dynamic and total forces k

∆∆FD,i = C k ∆∆v i + M k ∆∆ai

k

∆∆Fi = k ∆∆FS,i + k ∆∆FD,i

10. Calculation of current values of displacement, velocity, acceleration and forces after the kth iteration k

ui =

k −1

k

vi =

k −1

k

ai =

k −1

k

FD,i =

k

Fi =

ui + k ∆∆ui v i + k ∆∆v i

ai + k ∆∆ai

k −1

FD,i + k ∆∆FD,i

k −1

Fi + k ∆∆Fi

11. Calculation of residual forces k +1

∆R i = kFi − Fi

12. Test criterion

(

∆R i )

k +1

( Fi )

2

2

> ε

⇒

opakovat od bodu č. 5.

13. Adjustment of values of the current time step u i = ku i

Fi = kFi 577

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load v i = k vi

FS,i = k FS,i

a i = ka i 14. Verification if the calculation has been performed throughout the whole interval t < tn

⇒



Fig. 8.1a – Principle of variant I – first iteration

578


Fig. 8.1b – Principle of variant I – subsequent iterations

8.4.3.3 Variant II Variant II is in fact a modification of variant I. The difference is just in the calculation of partial increments of displacement k ∆∆ui and in the subsequent determination of partial increments of velocities k ∆∆v i and accelerations k ∆∆ai . In this variant, the partial increments of velocities and accelerations are calculated after each iteration step of the Newton-Raphson method, which updates partial increments of inertia and damping forces k ∆∆FD,i as well as static forces k ∆∆FS,i . As a consequence of these changes, the partial increments of displacement in the next iteration of the Newton-Raphson method are calculated for a different load level. This variant is applicable only if implemented directly into the NewtonRaphson method.

579


1K

F(u)

2K

1

1 3K

1

Fi 1F D,i

∆Fi

2F

2∆R 1∆R

Fi-1

1F S,i 1∆∆u

ui-1

2∆∆u

i

1u

i

i

3∆∆u

i

D,i

3∆R

i›

0

i

FD,i-1

FS,i-1

3F

D,i

2F

3F

S,i

S,i

i

2u 3 i ui

u

Fig. 8.2 – Principle of variant II

8.4.3.4 Variant III Variants I and II are less suitable for the solution of nonlinear problems with large displacements due to weaken convergence. This problem is usually related to the linear estimates that are used during the solution. Therefore, it is better to use the following variant III for the solution of such problems. In this variant, relations (8.3.24) and (8.3.27) are substituted into (8.3.4), and the obtained relation is modified so that it could be used in several iteration steps k     1 γ M  ∆ ai + 2 ∆u i  + C  ∆v i + ∆u i  + K T,i ∆u i = ∆Fi βh βh    

(8.3.51)

 1  k γ k k  β h 2 M + β h C + K T,i  ⋅ ∆u i = ∆Fi − M∆ ai − C∆v i  

(8.3.52)

The bracket on the left hand side of (8.3.52) represents what is termed modified stiffness matrix, which can be denoted ˆ =  1 M + γ C+ kK  K i T,i   β h2 βh  

(8.3.53)

Using the substitution according to (8.3.53) and applying the formula (8.3.31) to vectors k ∆ui and k ∆Fi in (8.3.52), we get

580

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load ˆ ⋅ ( k −1 ∆u + k ∆∆u ) = ( k −1 ∆F + k ∆∆F ) − M∆ a − C∆v K i i i i i i i

(8.3.54)

The presented formula can be written in a similar form as (8.3.45): k

ˆ k ∆∆u = k ∆∆F − M∆ a − C∆v ; K i i i i i ∆∆ui :  k k ˆ ∆∆u = ∆∆F ; K i i i

pro k = 1 pro k > 1

(8.3.55)

Using this relation (8.3.55), it is possible to calculate partial increments of displacement that are not burdened with any linear estimate Vectors of partial increments k ∆∆v i and k ∆∆ai can be determined from the calculated vector k ∆∆ui and from relations (8.3.40) and (8.3.41).Substituting into (8.3.42) we obtain vector k ∆∆FD,i . Using the vectors of partial increments we can - according to (8.3.30) or (8.3.31) – express the current values of the vectors of total increments of all necessary quantities in the k-th iteration. Vectors of total increments of inertia and damping forces and restoring forces will be used in (8.3.50) to calculate residual forces k +1∆R i . If this vector is not negligible (with a satisfactory accuracy) we continue with next iteration step and the stated vector of residual forces is used for the calculation of subsequent partial increments of displacement. This method does not make use of any linear estimates, which positively contributes to the convergence of this variant, but the Newton-Raphson algorithm is totally affected. The algorithm below describes the presented variant III.

1. Adjustment of initial conditions i = 0, t = t0 u 0 = u(t0 ) , v 0 = v (t0 ) , a 0 = a(t0 ) 2. Determination of the load values and matrices at time step (i) i = i + 1, t = t + ∆t = t + h Fi , ∆Fi = Fi − Fi−1 3. Calculation of the values resulting only from the time instant i  1  1 ∆ ai =  − vi −1 − ai −1  2β  βh 

581

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load  γ ∆v i = 1 −  2β

 γ  hai −1 − v i −1 β 

4. Adjustment of values prior to integration k =0

ui = ui−1

Fi = Fi−1

0

0

v i = v i−1

0

ai = ai−1

1

0

FS,i = FS,i−1 ∆R i = ∆Fi

0

5. Beginning of the iteration cycle, modified stiffness matrix k = k +1

ˆ =  1 M + γ C+ kK  K i T,i   β h2 βh   6. Calculation of partial increments of deformation k

ˆ k ∆∆u = k ∆∆F − M∆ a − C∆v ; K i i i i i ∆∆ui :  k k ˆ ∆∆u = ∆∆F ; K i i i

(

k −1

pro k = 1 pro k > 1

ui , k ∆∆u i , k K T,i )

k

FS,i = f

k

∆∆FS,i = kFS,i −

k −1

FS,i

7. Calculation of partial increments of acceleration and velocity   ∆v i + k ∆∆v i =   



pro k = 1 pro k > 1 pro k = 1 pro k > 1

8. Calculation of current partial increments of inertia and damping and total forces k

∆∆FD,i = C k ∆∆v i + M k ∆∆ai

k

∆∆Fi = k ∆∆FS,i + k ∆∆FD,i

582

8.4 Numerical methods for nonlinear solution of structure models subjected to dynamic load 9. Calculation of current values of displacement, velocity, acceleration and forces after the kth iteration k

ui =

k −1

k

vi =

k −1

k

ai =

k −1

k

FD,i =

k

Fi =

ui + k ∆∆ui v i + k ∆∆v i

ai + k ∆∆ai

k −1

FD,i + k ∆∆FD,i

k −1

Fi + k ∆∆Fi

10. Calculation of residual forces k +1

∆R i = kFi − Fi

11. Test criterion

(

∆R i )

k +1

( Fi )

2

2

> ε

⇒

opakovat od bodu č. 5.

12. Adjustment of values of the current time step u i = ku i

Fi = kFi

v i = k vi

FS,i = k FS,i

a i = ka i 13. Verification if the calculation has been performed throughout the whole interval t < tn

⇒



583


584

9.1 Bending with the shear deformation

9 Benchmarks examples

and

illustrative

9.1 Bending with the shear deformation 9.1.1 General remarks In the SCIA Engineer program the bending of beams, plates and shells is calculated including the shear deformation. The Mindlin´s bending theory is used introducing besides the deflection function also independent rotation functions. The difference between the first derivative of the deflection function and the rotation of the mass normal is the shear deformation. Generally the shear effect can be neglected for thin plates and shells. Therefore for these cases also the Kirchhoff theory can be used. Beams are generally thicker than plates so the Navier beam solution neglecting the shear effect is not offered in to the users of the SCIA Engineer and can be only modelled by entering a big shear cross section area. In the following examples it is shown that the shear deformation is substantial for beams and especially for steel structures. The use of the Navier approach can lead to error in order of tens procents, in many cases even more than 50%. So the Navier´s solution that was by some users regarded as „exact“ is only approximate for concrete structures, but quite unacceptable for steel structures.

B1 example A concrete clamped beam of the span l = 5m can represent an internal span of a continuous beam. The beam is loaded in the middle by the concentrated force P = 100KN. The cross section is 0,2 x 0,5m, Young modulus E = 26 x 103 MPa, Poisson ratio υ = 0,2. For this special case the contributions of the both components of deflection (bending and shear) can be calculated separately. The bending deflection (Navier solution): 1 Pl 3 wB = = 1, 20192 mm 192 EI the shear deflection ws =

1 Pl = 0,13846 mm 4 GAs

G is the shear modulus and As is the shear cross section area. The total deflection in the middle is then

585

9.2 Geometric nonlinearity w = wB + ws = 1,3404 mm .

The SCIA Engineer program gives the exact results for linear calculation of beams without necessity of division of the members into finite elements. (Division of beams into elements should be done for interaction with subsoil, dynamic calculations and the Newton-Raphson method of geometric nonlinearity.)

B2 example The similar beam as in the B1 example, only the concrete profile is replaced by steel I profile (HD400/187). E = 2,1 ⋅1011 ν = 0,3 As = 0, 0046658 I = 0, 000602 G = 8, 0769 ⋅1010 1 Pl 3 wB = = 0,51498 mm 192 EI 1 Pl = 0,33170 mm ws = 4 GAs w = wB + ws = 0,84668 mm . The SCIA Engineer gives the exact solution. Navier solution would give an error 64,4%!

9.2 Geometric nonlinearity 9.2.1 General remarks The SCIA Engineer introduces two methods for geometrically non-linear solution of structures. The first one we have denoted the "Timoshenko” method, because this method is based on the exact Timoshenko´s solution of beams with known normal force. It is 2nd order theory with equilibrium on the deformed structure, but it assumes small displacements and rotations, and small strains. The advantage of this method is that in case of beams which are not in contact with subsoil and which are not ribs of shells no division into beam elements is needed. When the normal force is lower than the critical 586

9.2 Geometric nonlinearity force this solution is robust. The method needs only two steps, which causes its big efficiency. The first step serves only for solution of normal force and the second step uses these normal forces for Timoshenko´s exact solution. Original Timsohenko´s solution was in the SCIA Engineer generalised and the shear deformations are taken into account. This method serves very well for solution of such problems where the normal or membrane forces are not substantially changed by deformation of the structure. The second method, denoted as the "Newton-Raphson" method is more general. From the mathematical point of view this approach is based on the Newton-Rapson method of solution of nonlinear equations. This method is very general and can be used for large displacements and large rotations. But limitation for small strains remains, so only non-rubber materials can be solved. The precision of a solution can be increased by refinement of division (even beams must be divided) and by increasing of the number of increments. In one increment the rotation should not increase 5° (i.e. 0,087 rad). In some cases the number of increments should be increased also if a singularity problem occurs during the iterations, which can happen especially when calculating post critical states. If there are not the above mentioned reasons only 1 increment may be used.

N1 example The cantilever beam loaded by a moment at the end. This example has the exact solution. The rotation ϕ at the end is given by the formula: ϕ=

M. l EI

When ϕ = 2π, this means that the beam creates the whole circle. The moment that would create it is M 2π = 2π

EI l

Let us have the steel cantilever beam of the cross section 5/20 mm and the length l = 1m. Then M 2π = 274,89 Nm . Because the rotation in one increment should not be larger than 5 deg, we should accept at least 80 increments. For such large displacement we should have a fine mesh. When having 80 increments and 40 elements we obtain such deformations at the end: u = -1001,7 mm v = 1,1 mm

(axial displacement) (transversal displacement)

587

9.2 Geometric nonlinearity ϕ = 6,29 rad The error is approximately 0,1 %. When you display the deformation with scale = 1 you can see that the deformed beam has created the circle.

N1-shell example A cantilever beam modelled as a shell loaded in the similar way as in the previous example (N1) by a moment at the end that should bend the shell into a circle. The picture shows that error is negligible.

588

9.2 Geometric nonlinearity

9.2.2 Axially and transversally loaded cantilever beam For more complex examples than the N1 example, the exact solution is not known. But very precise results were obtained by K. Mathiasson in 1979 in the Gőtteborg university (internal report 79:10 Dept. Struct. Mech.) by precise numerical solution of the elliptic integrals for axially and transversally loaded cantilever beams. The pressure and shear deformations are not taken into account in the solution which suits to slim beams well. Let us introduce some of his results in the tables for different cases of loading at the cantilever end. The tables contain displacements and rotations at the cantilever end for different ratios Pl2/EI.

Table 1 Transversal load at the cantilever end - global direction. PL2 EI 0,2 0,4 0,6 0,8 1,0

v/L

u/L

ϕ

M/PL

0,066 0,131 0,192 0,249 0,301

0,003 0,010 0,022 0,038 0,056

0,100 0,197 0,291 0,379 0,461

0,997 0,990 0,978 0,962 0,944

Table 2 Transversal load at the member end - local direction (the direction of the force remains normal to the beam). PL2 EI 0,2 0,4 0,6 0,8 1,0

v/L

u/L

ϕ

M/PL

0,066 0,132 0,197 0,260 0,321

0,003 0,011 0,024 0,042 0,064

0,100 0,200 0,299 0,398 0,496

0,997 0,996 0,991 0,984 0,975

Table 3 Cantilever beam loaded with the axial force P and the transversal load n . P at the end. n = 1/1000 2

PL EI 0,2 0,4 0,6 0,8 1,0 1,2

v/L

u/l

ϕ

0,00007 0,00016 0,00026 0,00039 0,00056 0,00077

0,00000 0,00000 0,00000 0,00000 0,00000 0,00000

0,00011 0,00024 0,00040 0,00060 0,00085 0,00119 589

9.2 Geometric nonlinearity 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0

0,00107 0,00150 0,00220 0,00348 0,00668 0,02859 0,39820 0,57077 0,66440 0,80245 0,79529 0,76100 0,72265 0,63618 0,65305 0,62337

0,00000 0,00000 0,00000 0,00001 0,00003 0,00050 0,10465 0,23564 0,34790 0,72593 0,94008 1,07737 1,17302 1,24376 1,29846 1,34227

0,00165 0,00232 0,00340 0,00541 0,01043 0,04486 0,65418 0,99616 1,22653 1,86270 2,19030 2,39844 2,54338 2,64996 2,73124 2,79491

Table 4 n = 1/10 2

PL EI 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0

v/L

u/L

ϕ

0,00725 0,01588 0,02632 0,03922 0,05553 0,07675 0,10531 0,14524 0,20290 0,28563 0,39208 0,50195 0,59424 0,66371 0,71371 0,80646 0,80184 0,77414 0,74202 0,71086 0,68221 0,65631

0,00003 0,00015 0,00042 0,00093 0,00187 0,00359 0,00678 0,01297 0,02557 0,05162 0,10057 0,17337 0,25844 0,34444 0,42574 0,73531 0,92773 1,05617 1,14787 1,21686 1,27091 1,31462

0,01091 0,02398 0,03989 0,05966 0,08481 0,11773 0,16234 0,22528 0,31746 0,45324 0,63698 0,84385 1,04083 1,21413 1,36342 1,86699 2,15694 2,34695 2,48110 2,58045 2,65656 2,71634

Table 5 n=1 590

9.2 Geometric nonlinearity PL2 EI 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0

w/L

u/L

ϕ

0,07207 0,15463 0,24533 0,33908 0,42922 0,51006 0,57869 0,63481 0,67971 0,71523 0,74318 0,76513 0,78236 0,79590 0,80653 0,83326 0,83915 0,83772 0,83368 0,82884 0,82392 0,81922

0,00313 0,01452 0,03712 0,07259 0,12000 0,17597 0,23617 0,29688 0,35563 0,41106 0,46259 0,51012 0,55380 0,59389 0,63071 0,77604 0,87728 0,95197 1,00968 1,05588 1,09392 1,12593

0,10859 0,23470 0,37667 0,52918 0,68412 0,83342 0,97148 1,09586 1,20634 1,30390 1,39000 1,46612 1,53366 1,59384 1,64767 1,84790 1,97559 2,06251 2,12446 2,17015 2,20473 2,23145

N2 example Cantilever beam L = 10m. Cross section properties: Ax = 1m2, Ay = 10m2, I2 = 0, 001m4, E = 1010 N/m2 . The above quantities were defined to fit with the Mattiasson´s solution, where both pressure and shear deformations are neglected. Load case 1 - transversal nodal force at the cantilever end. P = 105 N, 5 increments, 10 elements Engineer (N-R) Mattiasson

v/L 0,301 0,301

u/L 0,056 0,056

ϕ 0,462 0,461

M/PL 0,944 0,944

Load case 2 - member end transversal force. 591

9.2 Geometric nonlinearity P = 105 N, 5 increments, 10 elements Engineer (N-R) Mattiasson

v/L 0,321 0,321

u/L 0,064 0,064

ϕ 0,497 0,496

M/PL 0,976 0,975

Load case 3 - axial nodal force P and transversal force n . P, n = 0,001 . P = 105 N, 5 increments, 10 elements

linear Engineer (Timosh.) Engineer (N-R) Mattiasson

v/L 0,00033

u/L 0,00001

ϕ 0,00050

0,00056

0,00001

0,00085

0,00056 0,00056

0,00001 0,00000

0,00085 0,00085

P = 106 N, 200 increments, 20 elements

Engineer (N-R) Mattiasson

v/L 0,62285 0,62337

u/L 1,34325 1,34227

ϕ 2,80124 2,79491

Notice that the force P = 106 N overcomes 4 times the critical force that is approximately 2,5. 105 N (see "yielding" in the table 3 around the ratio 2,5). This postcritical state could be calculated only when increasing the number of increments.

592

9.2 Geometric nonlinearity Load case 4 - axial nodal force P and transversal force n . P, n = 0,1. P = 105 N, 100 increments, 20 elements linear Engineer (Timosh.) Engineer (N-R) Mattiasson

v/L 0,03323

u/L 0,00001

ϕ 0,050000

0,05574

0,00001

0,08508

0,05551 0,05553

0,00187 0,00187

0,08477 0,08481

P = 106 N, 100 increments, 20 elements Engineer (N-R) Mattiasson

v/L 1,31582 1,31462

u/L 0,65699 0,65631

ϕ 2,72354 2,71634

Load case 5 - axial nodal force P and transversal force n . P, n = 1. P = 105 N, 5 increments, 10 elements. Engineer (N-R) Mattiasson

v/L 0,42925 0,42922

u/L 0,11970 0,12000

ϕ 0,68540 0,68412

P = 106 N, 80 increments, 20 elements Engineer (N-R) Mattiasson

v/L 0,81898 0,81922

u/L 1,12756 1,12593

ϕ 2,23890 2,23145

Remark: The Timoshenko´s solution could be performed only in cases where rotations are small. In the remaining cases only the Newton - Raphson solution could be done. But this solution does not need a division of members into smaller finite elements. Moreover it needs only 2 iterational steps. So for small rotations (less than cca 0,1 rad) which is valid for great majority of civil engineering structures and cases where normal (or membrane) forces do not vary during deformation substantially, the Timoshenko´s method is extremely efficient. Large rotations (eve 2 or 3 radios) are not typical for real structures, but results of the Newton-Raphson method that differ from precise results by less than 1% prove correctness of the method.

593

9.3 Subsoil

N2-shell example A cantilever modelled as a shell loaded by a compression force that is 4 times greater than the critical force and by a transversal force being 1/1000 of the compression force. When divided into 40 elements and with 100 increments the error is smaller than 0.5%.

9.3 Subsoil 9.3.1 General remarks The SCIA Engineer introduces the Pasternak´s subsoil model with the settlement stiffness C1 and the shear stiffness C2. The old Winkler´s model with only settlement stiffness k works exactly as a liquid and the contact stress is nothing else than the hydrostatic pressure of a liquid with the volume weight k. Such model can never express the real behaviour of the subsoil and cannot provide a good prediction of the settlements and the contact stresses. E.g. this model gives the zero settlement of the subsoil surface 1 mm outside the structure - soil interface, which is nonsense at the first sight. The Pasternak´s model is able to substitute the real 3D continuum of the subsoil with the 2D surface properties but one problem still remains. It is the determination of the C1 and C2 parameters. The formulae for C1 and C2 are introduced in the book 594

9.3 Subsoil "Modelling of Soil - Structure Interaction" (V. Kolar, I. Nemec - Elsevier 1989). These formulae work with limit depth of the deformable zone that should be estimated by the designer. Much better way of determination of the C1 and C2 parameters is using the SOILIN (SOIL INteraction) program. The users do not need any speculations. They just have to define the subsoil layers properties (Young modulus, Poisson ratio, structural strength coefficient) and the cut - off depth. The contact stress that is necessary for the surface properties calculation is automatically passed from the structural analysis part of the SCIA Engineer program. The calculation itself is an iterative process. In the first step a typical C1, C2 values are entered and the structure analysis is performed to evaluate the contact stresses. These contact stresses and the subsoil layers properties are entered into the SOILIN program. This program calculates the stress distribution in the whole 3D subsoil mass using the Boussinesq half space solution and the distribution of the structural strength (the structural strength coefficient multiplied by the original σ2 (before excavations)). The zone of subsoil deformation is automatically determined by the condition that the σZ stress component is bigger that the structural strength. These results altogether with the layers properties enable the program to calculate the subsoil surface settlement and the C1, C2 parameters that would yield the same settlement. The new subsoil surface model parameters C1, C2 lead to another contact stress so iterations have to repeat until the precision required is achieved. The use of the SOILIN program gives the excellent accordance of the analysis results and the real building settlements if good soil mechanical properties are entered. Because of the use of the Boussinesq elastic half space, this solution satisfies also the national code.

Soilin 1 example This example shows that the SOILIN gives the exact solution for a case where such exact solution can be determined by manual calculation. The size of the plate (100 x 100 m) is large comparing to the subsoil thickness (10 m) so we can assume that the area around the centre of the plate can be regarded as one-dimensional problem with constant σz. The loading is uniform q = 40 kN/m2. The subsoil properties are: Modulus of deformationEdef = 40 Mpa Poisson ratio ν = 0.2 Structural strength coefficientm = 0 Then the oedometric modulus Eoed = (1-ν) Edef/((1+ν)(1-2ν)) = 44.444 MPa Then k = C1Z =Eoed/h = 4.4444 MN/m3 and the settlement w = q/k = 9 mm ESA gives: 595

9.3 Subsoil

C1z = 4.4445 MN/m3 w = 9.006 mm which is practically equal to the "manual" values.

Subsoil - C1z, Macros 2D

596

9.3 Subsoil

Deformations - Uz, Macros 2D, Load case 1

Soilin 2 example This illustrative example shows differences of the solution of the square plate with uniform loading. The Soilin_2b uses the Winkler´s subsoil, Soilin_2c uses the Pasternak´s subsoil model with the determination of the C1, C2 parameters by formulae and the most precise Soilin_2a example uses iteration process with the SOILIN program. In the results you can see the big differences of the settlements and internal forces. Notice that the Winkler´s model gives zero internal forces. Soilin_2b: Geology – Winkler – individual C1z

597

9.3 Subsoil

Uniform load


Soilin_2c: Geology - C1z = 5.5 MN/m3, C2x = C2y = 3.0 MN/m2 + line elastic bond kz 598

9.3 Subsoil = 8.0 MN/m2


Internal forces - mx, Macros 2D, Load case 1

599

9.3 Subsoil

Soilin_2a: Geology SOLIN


600

9.3 Subsoil Internal forces - mx, Macros 2D, Load case 1

Soilin 3 example This example shows ambiguous influence of plates which are independent but situated near one another. Notice that even if each particular plate with its loading is central symmetric the results are not symmetric and the edges of the plates close to one another have much bigger settlements than the remaining edges.

Uniform load

601

9.3 Subsoil


Soilin4 example This example shows the possibility of calculation of a declination of existing old structures after a construction of a new building in the near neighbourhood.

602

9.3 Subsoil

2D Macro

603

9.3 Subsoil

Uniform load

604

9.4 Cables


9.4 Cables Generally the Newton-Raphson nonlinear solution should be applied for a cable solution: then there is no limitation. More increments than 1 can be demanded for bigger displacements and rotation but for most cases this is not necessary. Solution is stable and very precise. For cases where the cables have only small traversal loading or if they are not too long also the Timoshenko´s method can be used. But the increase of the normal force due to the cable suspension should not be great comparing to the initial force. When checking the results all the equations, i.e. statical (equilibrium), geometrical (strain) and physical (Hook's law) must be satisfied.

Cable 1 example length cross seetion area Young modulus prestressing division

l = 20 m A = 0,001 m2 E = 2.1 . 1011 N/m2 Ninit = 1000 KN 0.5 m (40 elements)

605

9.4 Cables 2 increments

Load case 1 - transversal uniform load 1 KN/m. i) solution with prestressing wmax 49.8 mm N = 1003.5 KN checking: Moment in the middle: l 1 M   = pl 2 − N .wmax = 50 − 50 = 0 2 8 For calculation of the strain for the case of uniform load the formula supposing circular shape of displacement can be used 2

16  wmax  − 1 = 1.62 ⋅10−5   3 l  N d = ε ⋅ E ⋅ A = 3.4 kN

ε = 1+

The condition N init + N d = N is fulfilled almost exactly (1003.4 = 1003.5). Such case is possible to solve also using the Timoshenko's solution with sufficient precision. We obtain: wmax = 50 mm N = 1000 KN The results are very close to the Newton-Raphton solution.

j) solution without prestressing wmax = 329.7 mm N = 152.1 kN

606

9.4 Cables checking: l M   = 50 − 50.1 = 0 2 2

ε = 1+

16  0.3297  −4   − 1 = 7.2442 ⋅10 3  20 

N d = E ⋅ A ⋅ ε = 152.1 Nd = N The results have showed that cable without prestressing is also almost exact. Load case 2 - concentrated forces. P1 = -10 KN at 1/4 l P2 = -10 KN at 1/2 l P3 = +10 KN at 3/4 l This loading case was chosen to show the stability of calculation for very irregular loading. a) Solution with prestressing: l w   = 49.8 mm 2 N = 1005 checking: l M   = 50 − 50 = 0 2 For the calculation of the strain we use the fact that the deformed shape is straight by parts.

(

)

ε = 10 + 2 0.0482 + 52 − 20 / 20 = 2.48 ⋅10−5 N d = 52 N d + N init = 5.2 + 1000 = 1005.2 The slight difference between the N and the Nd + Ninit is caused by the strain calculations by hand. Solution by Timoshenko gives almost the same results

607

9.4 Cables l w   = 50  2 N = 1000 .

b) Solution without prestressing l w   = 290.4 mm 2 N = 172.6 checking: l M   = 50 − 50.1 2 ε = 8.403 ⋅10−3 N d = 173.9 This solution has showed the robustness of the solution even in the sensitive case of the irregular load on the cable without prestressings.

Cable 2 example Let us have a cable supported as a simple beam (with one moveable support). Conservative distributed load When loaded by conservative perpendicular distributed load it will hang down from the fixed support (see picture).

608

9.4 Cables

Following distributed load When this cable is loaded by normal following load it will create the exact half circle.

609

9.5 Membranes

9.5 Membranes The same as in the case of cables also for membranes analysis the NewtonRaphson method can be used in any case. Only for sufficiently high prestressing the Timoshenko's method can be used.

Membrane 1 example This example corresponds to the Cable 1 example. Cross section area the well as loading is 10times bigger, so the deformations of both examples can be compared. Lenght l = 20 m width b = 1 m thickness h = 0.01 m Yong modulus E = 2.1 . 1011 N/m2 prestressing x ninit = 10000 KN division 0.5 m 2 increments 610

9.5 Membranes

Load case 1 - transversal load 10 KN/m2. c) Solution with prestressing: Newton-Raphson: wmax = 49.7 mm nx = 10100 KN/m Timoshenko: wmax = 49.6 mm nx = 10000 KN/m Both solutions are very close to the cable solution (i.e. also to the exact solution). d) Solution without prestressing: wmax = 329.5 mm nx = 1525 KN The results are very precise (see the Cable 1 example).

Load case 2 - line loading p1 = - 100 KN/m p2 = - 100 KN/m p3 = 100 KN/m e) Solution with prestressing: Newton-Raphson: wmax = 49.5 mm nx = 10000 KN/m

611

9.6 Mechanisms

Timoshenko: wmax = 49.7 mm nx = 10000 KN/m f) Solution without prestressing: wmax = 290,2 mm nx = 1750 KN/m All the results of the load case 2 are very precise (see the Cable 1 example).

9.6 Mechanisms SCIA Engineer enables also solution of mechanisms. Any possible shape can be used as a starting geometry. The geometrical nonlinearity and the N - R method must be applied. The solver should be switched to the iterative mode (which is recommended for

612

9.6 Mechanisms any use of N - R method). Correctness of the solution can be checked using equilibrium equations. In cases where large rotations are expected many increments should be prescribed. In many cases the solution can be more efficient if additional, very weak springs, which cannot influence the results, are defined. The following examples demonstrate the possibilities.

Mechanism1 example The structure consists of two normal members. At one end is a hinge support and at the second one is a nodal force. After deformation it can be checked whether the force beam goes through the hinge support.

Mechanism1-shell example A similar structure as in the previous case (Mechanism1) but modelled by the shell elements. In the same way as in the previous case the beam of the force goes through the hinge on the deformed shape.

613

9.6 Mechanisms

Mechanism2 example Three members are connected by hinges. Both ends are supported by hinges. Internal nodes are loaded by uneven forces. Moments from forces and reactions to any hinge must be zero.

614

9.6 Mechanisms

Mechanism2-shell example A similar structure as in the previous case (Mechanism2), but modelled by the shell elements. In the picture you can see that in the unloaded hinge the structure becomes straight. 615

9.6 Mechanisms

616

9.7 Stability (Buckling)

Mechanism3 example A pendulum is supported by a hinge at the top and loaded at the bottom by vertical nodal force and the horizontal member end force. This example shows that also nonconservative loading can be applied for large rotation.

9.7 Stability (Buckling) Stability 1 example Simply supported steel beam of cross section 10/100 mm, with compression force 10 KN. Analytical solution for i-th buckling mode is:

PK ,i

i 2π 2 EI = l2

617

9.8 Dynamics Comparison of SCIA Engineer solution and analytical solution (beam is divided into 10 elements). MODE NO. 1 2 3 4

ANALYTICAL 1.727 6.909 15.446 27.635

ENGINEER 1.727 6.901 15.515 27.586

The precision of course decreases with complexity of mode.

Stability 1-shell example The same example as stability 1 but modelled by shell elements. The precision is slightly lower than in case of the beam model. The SCIA Engineer results for division into 160 elements are in the following table: MODE NO. 1 2 3 4

ANALYTICAL 1.727 6.909 15.446 27.635

ENGINEER 1.730 6.947 15.733 28.204

9.8 Dynamics Dynamics 1 example Free vibration of the same beam as the Stability1 example. Analytical solution for. Navier theory is as follows: i 2π 2 l2 w fi = i 2π

wi =

EI ∫A

Comparison of analytical calculation frequencies: MODE NO. 1 2 3

ANALYTICAL 23.453 93.811 211.075

ENGINEER 23.450 93.751 210.672 618

9.8 Dynamics

Slightly fewer frequencies in SCIA Engineer are caused by influence of shear that is not calculated in the analytical solution. In case of the compression forces the analytical solution will be as follows: wi =

i 2π 2 a Sl 2 1 − l2 i 2 EI π 2

where EI ∫A w fi = i 2π

a=

For compression force 10 KN and i=1: Analytical: 15.212 Hz Engineer: 15.213 Hz

Dynamics 1-shell example Free vibration of the same shell as in the Stability2 example. Analytical solution is the same as in the case Dynamics1. Comparison of analytical and SCIA Engineer calculated frequencies: MODE NO. 1 2 3

ANALYTICAL 23.453 93.811 211.075

ENGINEER 23.464 93.975 211.847

For compression force 100 KN the results are as follows: i= 1 Analytical: 15.212 Hz Engineer: 15.239 Hz

619

9.8 Dynamics

620

Literature

Literature [1]

Zienkiewicz O.C., Cheung Y.K.: The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967.

[2]

Zienkiewicz O.C.: The Finite Element Method in Engineering Science, McGraw-Hill, London, 1971.

[3]

Kolář V., Kratochvíl J., Zlámal M., Ženíšek A.: Technical, Physical and Mathematical Principles of the Finite Element Method, Rozpravy ČSAV, Vol. 81, No.2, 1971.

[4]

Chapelle D., Bathe K.J.: The finite element analysis of shells - fundamentals, Springer-Verlag, Berlin, 2003.

[5]

Kolář V., Kratochvíl J., Leitner F., Ženíšek A.: Berechnung von Flächen und Raumtragwerken nach der Methode der finiten Elemente, SPRINGER VERLAG, Wien-New York, 1975.

[6]

Bathe C.J.: Finite Elements Procedures, Prentice Hall, New Jersey, 1996.

[7]

Cook R.D., Malkus D.S., Plesha M.E., Witt R.J.: Concepts and applications of FE analysis, John Wiley & Sons., University of Wisconsin, Madison, 2002.

[8]

Kolář V., Němec I.: Modelling of Soil-Structure Interaction, ACADEMIA Praha ELSEVIER Amsterdam-New York, 1990.

[9]

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