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CEL760 : FINITE ELEMENT METHOD IN GEOTECHNICAL ENGINEERING 3 Cr ed i t s (3-0-0)
Co o r d i n at o r : K .G. Sh ar m a
Course Content •Introduction. •Steps in FEM. •Stress-deformation analysis: One-, Two- and Three-dimensional formulations; • •Solution algorithms; •Discretization; •Use of FEM2D Program and Commercial packages. • Analysis of foundations, foundations, dams, underground underground structures and earth retaining structures.
Course Contents
Contd.
•Analysis of flow (seepage) through dams and oun a ons. •Linear and non-linear analysis. •Insitu stresses. •Sequence construction and excavation. •Joint/interface elements. • . . • Evaluation Evaluation of material material parameters parameters for linear linear and non-linear analysis. •Recent developments.
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References Desai, C.S. and Kundu T. T. (2001) Introductory Finite Element Method. CRC Press. Desai, C.S. and Abel, J.F. J.F. (1972) Introduction to Finite Element Method. Van Nostrand Reinhold, New York. Bathe, K.J. (1982) Finite Element Procedures in Engineering Analysis. Prentice-Hall, Inc. Zienkiewicz, O.C. and Taylor, R.L. (1989) The Finite Element Method. Vols. 1 & 2, 4th Edition, McGraw-Hill Book Company. Company. Desai, C.S. and Christian, J.T. J.T. (1977) Numerical Methods in Geotechnical Engineering. John Wiley & Sons.
References Naylor, Naylor, D.J. and Pande, G.N. (1981) Finite Elements in Geotechnical Engineering. Pineridge Press. Hinton, E. and Owen, D.R.J. (1977) Finite Element Programming. Academic Press. Evaluation Minor Test I : 20% Minor Test II : 20 Major Test : 40% Assignments : 20% Note: Students having l ess than 75% 75% attendance will be given one grade less than the grade scored by t hem.
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References Desai, C.S. and Kundu T. T. (2001) Introductory Finite Element Method. CRC Press. Desai, C.S. and Abel, J.F. J.F. (1972) Introduction to Finite Element Method. Van Nostrand Reinhold, New York. Bathe, K.J. (1982) Finite Element Procedures in Engineering Analysis. Prentice-Hall, Inc. Zienkiewicz, O.C. and Taylor, R.L. (1989) The Finite Element Method. Vols. 1 & 2, 4th Edition, McGraw-Hill Book Company. Company. Desai, C.S. and Christian, J.T. J.T. (1977) Numerical Methods in Geotechnical Engineering. John Wiley & Sons.
References Naylor, Naylor, D.J. and Pande, G.N. (1981) Finite Elements in Geotechnical Engineering. Pineridge Press. Hinton, E. and Owen, D.R.J. (1977) Finite Element Programming. Academic Press. Evaluation Minor Test I : 20% Minor Test II : 20 Major Test : 40% Assignments : 20% Note: Students having l ess than 75% 75% attendance will be given one grade less than the grade scored by t hem.
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Introduction to Finite Element Method Dr. K. G. Sharma Professor, Department of Civil Engineering, Indian Institute of Technology Delhi
Methods of Solution Closed
for m (Analytical) Solutio ns (CFS) (CFS)
•Many problems in engineering and applied science are over overne ned d b diff differ eren enti tial al or or inte inte ral ral e uati uation ons. s. • The solutions to these equations would would provide an exact, exact, closed-form solution to the particular problem being studied. •Gives the values of unknown quantity at any location in a body •. , simplified situations. •However, •However, complexities in the geometry, properties and in the boundary conditions that are seen in most real-world engineering problems usually means that an exact solution cannot be obtained or obtained in a reasonable amount of time.
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Methods of Solution
Numerical Methods
Problems involving complex material properties & boundary conditions: Numerical Methods Numerical Methods provide approximate but acceptable solutions.
o u on o a ne on y a a points in the body.
scre e num er o
Numerical Methods contd. Process of selecting only a certain number of discrete points In the body is termed as Discretization. Divide the body into an equivalent system of smaller bodies /units. The assemblage of these units then represents the original body. We do not solve the problem for the entire body in one . Instead, solutions are formulated for each unit and combined to obtain the solution for the original body. This approach is known as Going from Part to Whole.
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Discretization of Body
Numerical Methods Contd. Analysis procedure is considerably simplified. Amount of data to be handled depends upon the number of small bodies. Manual calculations for 1-D problems. Computer required for large data: 2-D, 3-D problems.
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Methods of Solution contd. Numerical Methods Finite Element Method (FEM) Boundary Element Method (BEM) Coupled Finite Element Boundary Element Method (FEBEM) Distinct Element Method (DEM) Infinite Elements
Best known numerical method is Finite Difference Method. The method has been adopted for use with computers. . FEM is essentially a product of electronic computer age. Many of the features common to the previous numerical methods. FEM possesses certain characteristics that take advantage of the special facilities offered by computers. FEM can be systematically programmed to accommodate Such complex and difficult problems as non-homogeneous materials, nonlinear stress-strain behaviour and complicated boundary conditions. It is difficult to accommodate these complexities in other methods.
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The approach is similar to the extension of the familiar concepts of analysis of framed structures as 1-D bodies to problems involving 2-D and 3-D structures. Physical or Intuitive approach to the learning and using the method. Or a rigorous mathematical interpretation of the method.
FAMILY TREE OF FINITE ELEMENT METHODS ENGINEERING
MATHEMATICS Trial Functions Weighted residuals
Variational
Gauss 1795 Galerkin 1915 Biezeno Koch 1923
Rayleigh 1870 Ritz 1909 Structural analogue substitution Hrenikoff 1941 McHenry 1943 Newmark 1949
Finite differences Richardson 1910 Liebman 1918 Southwell 1940
Piecewise continuous trial Functions Courant 1943 Prager Synge 1947 Direct continuum elements
Variational finite differences
Argyris 1955 Turner et al. 1956
Varga 1962
PRESENT – DAY FINITE ELEMENT METHOD
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Methods of Solution
Empirical Methods
Observational Methods
CONTINUUM
contd.
DISCONTINUUM
, Isotropic
Anisotropic
DISCRETIZATION Going from Part to Whole
Discreti zatio n Scheme
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Discreti zatio n Scheme
PROTYPE-TRACK ANALYSIS Discretization
Mesh Diagnostics Model
Total No. of Elements = 971 Total No. of Nodes = 5771 Prototype Track
Track Component
Nodes
Elements
Rail and Sleepers
644
62
Ballast
592
72
Sub-ballast
660
81
Subgrade
3875
756
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Finite Element Mesh of Dam-Foundation System
70 m
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Analysis Using UDEC
Discretisation
3-D layout of the underground storage caverns from the reference of Benardos and Kaliampakos, (2004). (ref 4 )
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PHASE 2 model of the oil storage cavern
Finite Element Method • In the FEM, a complex region defining a continuum is discretized into simple geometric shapes called elements.
• e proper es an e govern ng re a ons ps are assume over these elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes. • An assembly process is used to link the individual elements . conditions are considered, a set of linear or nonlinear algebraic equations is usually obtained. • Solution of these equations gives the approximate behavior of the continuum or system.
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Finite Element Method (cont.) • The continuum has an infinite number of degrees-of-freedom (DOF), while the discretized model has a finite number of DOF. This is the origin of the name, finite element method.
• The number of equations is usually rather large for most realworld applications of the FEM, and requires the computational power of the digital computer. The FEM has little practical value if the digital computer were not available. • vances n an rea y ava a y o compu ers an so ware has brought the FEM within reach of engineers working in small industries, and even students.
Finite Element Method (cont.) Two features of the finite element method are worth noting. • The iecewise a roximation of the h sical field (continuum) on finite elements provides good precision even with simple approximating functions. Simply increasing the number of elements can achieve increasing precision. • The locality of the approximation leads to sparse equation systems for a discretized problem. This helps to ease the solution of problems having very large numbers of nodal unknowns. It is not uncommon today to solve systems containing a million primary unknowns.
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Origins of the Finite Element Method • It is difficult to document the exact origin of the FEM,
because the basic concepts have evolved over a period of . • The term finite element was first coined by Clough in 1960. In the early 1960s, engineers used the method for approximate solution of problems in stress analysis, fluid flow, heat transfer, and other areas. • The first book on the FEM by Zienkiewicz and Chung was published in 1967. • In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering problems.
Origins of the Finite Element Method (cont.) • The 1970s marked advances in mathematical treatments,
including the development of new elements, and convergence studies. • Most commercial FEM software packages originated in the 1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s (FENRIS, LARSTRAN ‘80, SESAM ‘80.) • The FEM is one of the most important developments in computational methods to occur in the 20th century. In just a few decades, the method has evolved from one with applications in structural engineering to a widely utilized and richly varied computational approach for many scientific and technological areas.
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How can the FEM Help the Design Engineer? • The FEM offers many important advantages to the design
engineer : • , composed of several different materials and having complex boundary conditions. • Applicable to steady-state, time dependent and eigenvalue problems. • Applicable to linear and nonlinear problems. • One method can solve a wide variety of problems, including problems in solid mechanics, fluid mechanics, chemical reactions, electromagnetics, biomechanics, heat transfer and acoustics, to name a few.
How can the FEM Help the Design Engineer? (cont.) at reasonable cost, and can be readily executed on microcomputers, including workstations and PCs. •
• The FEM can be coupled to CAD programs to facilitate solid modeling and mesh generation. • Many FEM software packages feature GUI interfaces, auto-meshers, and sophisticated postprocessors and graphics to speed the analysis and make pre and postprocessing more user-friendly.
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How can the FEM Help the Design Organization? • Simulation
using the FEM also offers important business advantages to the design organization: • Reduced testing and redesign costs thereby shortening the product development time. • Identify issues in designs before tooling is committed. • Refine components before dependencies to other components prohibit changes. • Optimize performance before prototyping. • Discover design problems before litigation.
Theoretical Basis: Formulating Element Equations • Several approaches can be used to transform the physical formulation of a problem to its finite element discrete analogue.
• If the physical formulation of the problem is described as a differential equation, then the most popular solution method is the Method of Weighted Residuals. • e p ys ca pro em can e ormu a e as e m n m za on of a functional, then the Variational Formulation is usually used.
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Sources of Error in the FEM
The three main sources of error in a typical FEM solution are discretization errors, formulation errors and numerical errors.
Discretization error results from transforming the physical sys em con nuum n o a n e e emen mo e , an can e related to modeling the boundary shape, the boundary conditions, etc.
Discretization error due to poor geometry representation.
Discretization error effectively eliminated.
Sources of Error in the FEM (cont.)
Formulation error results from the use of elements that don't precisely describe the behavior of the physical problem.
Elements which are used to model physical problems for which they are not suited are sometimes referred to as ill-conditioned or mathematically unsuitable elements.
For example a particular finite element might be formulated on the assumption that displacements vary in a linear manner over the domain. Such an element will produce no formulation error when it is used to model a linearly varying physical problem (linear varying displacement field in this example), but would create a significant formulation error if it used to represent a quadratic or cubic varying displacement field.
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Sources of Error in the FEM (cont.)
Numerical error occurs as a result of numerical calculation procedures, and includes truncation errors an roun o errors.
Numerical error is therefore a problem mainly concerning the FEM vendors and developers.
The user can also contribute to the numerical accurac for exam le b s ecif in a h sical quantity, say Young’s modulus, E, to an inadequate number of decimal places.
Advantages of the Finite Element Method
Can readily handle complex geometry: •
Can handle complex analysis types: •
ra on
•
Transients
•
Nonlinear
•
Heat transfer
•
Fluids
Can handle complex loading: •
The heart and power of the FEM.
-
.
•
Element-based loading (pressure, thermal, inertial forces).
•
Time or frequency dependent loading.
Can handle complex restraints: •
Indeterminate structures can be analyzed.
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Advantages of the Finite Element Method (cont.)
Can handle bodies comprised of nonhomogeneous materials: •
Every element in the model could be assigned a different set of material properties.
Can handle bodies comprised of nonisotropic materials: •
Orthotropic
•
Anisotropic
Special material effects are handled: •
Temperature dependent properties.
•
Plasticity
•
reep
•
Swelling
Special geometric effects can be modeled: •
Large displacements.
•
Large rotations.
•
Contact (gap) condition.
Disadvantages of the Finite Element Method
A specific numerical result is obtained for a specific problem. A general closed-form solution, which would permit one to examine system response to changes in various parameters, is .
The FEM is applied to an approximation of the mathematical model of a system (the source of so-called inherited errors.)
Experience and judgment are needed in order to construct a good finite element model.
A powerful computer and reliable FEM software are essential. Input and output data may be large and tedious to prepare and interpret.
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Disadvantages of the Finite Element Method (cont.)
Numerical problems: •
Computers only carry a finite number of significant digits.
•
Round off and error accumulation.
•
Can help the situation by not attaching stiff (small) elements to flexible (large) elements.
Susceptible to user-introduced modeling errors: •
Poor choice of element types.
•
Distorted elements.
•
Geometry not adequately modeled.
•
Buckling
•
Large deflections and rotations.
•
Material nonlinearities .
•
Other nonlinearities.
Advantages of Finite Element Method Nonhomogeneity Material & Geometric Nonlinearities
as c,
on near e as c,
as o – p as c, creep, v scop as c
Irregular Geometry Any Boundary Conditions Generality 1-D 2-D 3-D Applicable to Wide Range of Problems
Structural En ineerin Geotechnical Engineering Water Resources Engineering Mechanical Engineering Nuclear Engineering Heat Transfer
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Advantages of Finite Element Method
-Contd…
Applicable to Wide Range of Problems
Biomedical Engineering Electro-magnetism
Y
Z
X
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Discretization Examples
One-Dimensional Frame Elements
Two-Dimensional Triangular Elements
Three-Dimensional Brick Elements
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Field near a Magnet
Anatomy of Hip Joint
Largest bearing joint
weight
Composed of rounded head of the femur joining the acetabulum of pelvis in a ball and socket arrangement 46
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Composite Model
Basic Composite Model With Elements
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Maximum Shear Stress Region n arge ew o e e orme em an or ca one Showi ng th e Maximum Shear Stress Region (Path Aa)
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Natur al Knee Joint
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Idealization of Natural Femur
Fixation of Endoprost hesis
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Idealization with Endoprosthesis
Problems & Loads
Problem Types
One – Dimensional Problems Two – Dimensional Problems • • •
Plane Stress Plane Strain Axisymmetric
Three – Dimensional Problems
Loads
Point loads Pressure loading o y orces Due to insitu stresses Due to Temperature
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BODY FORCE Volume element dV
Xc dV
Body force: distributed force per unit volume (e.g., weight, inertia, etc)
Xb dV a
w
Volume (V)
⎧X a ⎫ ⎪ ⎪ X = ⎨X b ⎬ ⎪X ⎪ ⎩ c⎭
v
u
Surface (S)
z x
NOTE: If the body is accelerating, then the ⎧ ρ &u& ⎫ ⎪ ⎪ inertia force ρ &u&
y x
may be considered as part of X
X
=
~
X
=
ρ &v&
⎪ ρ w ⎪ ⎩ && ⎭
− ρ &u&
SURFACE TRACTION Traction: Distributed force per unit surface area Volume element dV
pz
Xc dV Xb dV w
Xa dV
px
Volume (V) v
u
ST
py
⎧ p x ⎫ ⎪ ⎪ T S = ⎨ p y ⎬ ⎪ p ⎪ ⎩ z⎭
z
y x
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Analysis Types
Analysis: Stress – Deformation Linear elastic
Elasto – Plastic Elasto – Viscoplastic Simulation of Sequential Construction Excavation Joint / Interfaces
Confined flow Unconfined flow Finite elements Infinite elements Joint / Interface elements Line / Bar elements
The width of the concrete face is 0.3 m at the top and 1 m at the bottom of the dam. Upstream slope of the dam is of 1V : 1.4H. Downstream slope is of 1V : 1.5H.
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Discretization
The mesh for the analysis of the dam consists of 7500 elements
Sequential loading
The loading is done in 23 layers with each layer being 5 m high.
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Excavation sequence for the power house cavern
Categories of problem in Geotechnical Engineering FEM is applicable to a wide range of boundary . Boundary Value Problem: A solution is sought in the region of the body, while on the boundaries of the region, the values of unknowns are prescribed. Initial Value Problems: Initial values of the unknowns are also prescribed.
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Steady
or Equilibrium Static stress-deformation analyses for foundations, , , , Steady state fluid flow
Eigen
value Natural frequencies of foundations and structures
Categories of problem in Geotechnical Engineering
Transient or Dynamic
ress e orma on e av our o oun a ons, s opes, banks, tunnels, and other structures, under time dependent forces
Viscoelastic analysis
Consolidation
Transient fluid flow
Wave propagation
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Governing Equations
Static and Steady State
nergy
r nc p es,
eren a
qua ons
Transient or Dynamic
Energy Principles, Differential Equations
Initial Value Problems
Time Marchin Schemes
•
Forward difference method (Euler): Explicit
•
Backward difference: Implicit
•
Central difference: Implicit
Time Step, Δt
Modules
Pre – Processor: Data, Mesh Analysis FEM , , , MIDAS, UDEC, 3DEC Post – Processor Mesh Deformed mesh / shape Deformation vector / contour plots Stress vector / contour plots
,
,
2,
Flow vectors Analysis Module Solid Elements: Finite Infinite Joint Elements: Zero thickness Bar Elements
Thin
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PROTYPE-TRACK ANALYSIS
Deformed Shape Displacement Contours
Intact D-F Case in CC
Dam-Foundation in CC
Displacement Vectors for D-F
Major Principal Stress Contours
Minor Principal Stress Contours
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Flow vectors at 4th Stage with grout zone in the presence of water table
Flow net for cavern with oil with grout zone in the presence of water table
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Approaches
Displacement Method
Equilibrium Method
Mixed Method
Displacement Formulation Primary & Secondary Unknowns Problem
Primary
Secondary
Stress – Deformation Static Dis lacements Dynamic foundations, dams, embankments, Slopes, pavements
Strains Stresses Accelerations, velocities
Seepage, flow
Fluid potentials
Velocities, Discharge, Quantity of flow
Coupled consolidation, Liquefaction
Displacements, Pore pressure
Strains, Stresses, Quantity of flow
Non-homogeneity Complex Boundaries Material
Non-linearity Geometric Non- Linearity
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Commercial Software
ABAQUS ax s
,
ax s
Phase2, Midas GTS
SOFiSTiK, CESAR-LCPC
Examine2D, Examine3D
FLAC, FLAC3D
UDEC, 3DEC
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Books Desai & Kundu: Introductory Finite Element Method Desai & Abel: Introduction to Finite Element Method Bathe: Finite Element Procedures in Engineering Analysis Zienkiewicz: Finite Element Method
Rao SS: Finite Element Method in Engineering Krishnamoorthy: Finite Element Analysis
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