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Finit e Finite Element Analysis in Structures
Zahit Mecitoğlu
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Finite Element Analysis in Structures Zahit Mecitoğlu İstanbul Technical University Faculty of Aeronautics and Astronautics, Maslak, Istanbul
January 2008
Chapter 1
INTRODUCTION Finite element analysis is introduced in this chapter. The advantages of the method over the other analysis methods are explained. The application steps of the method and software usage are discussed. The cautions which must be taken care about are denoted.
1.1 FINITE ELEMENT ANALYSIS Finite Element Analysis (FEA) is a method for numerical solution of field problems. A field problem may be determination of the temperature distribution in a turbine blade, or calculation of the distribution of displacements and stresses on a helicopter rotor blade. A field problem is formulated by differential equations or by an integral expression. Either description may be used to formulate finite elements. Why the Finite Element Method (FEM) is necessary to solve the engineering problems? Analytical solutions to the engineering problems are possible only if the geometry, loading and boundary conditions of the problem are simple. Otherwise it is necessary to use an approximate numerical solution such as FEM. The finite element method is originally developed to study the stresses in complex aircraft structures. Then, it is applied to other fields of continuum mechanics, such as heat transfer, fluid mechanics, acoustics, electromagnetics, geomechanics, biomechanics. However, this book is devoted solely to the topic of finite elements for the analysis of structures. FEA is used in industries, such as aerospace, automotive, biomedical, bridges and buildings, electronics and appliances, heavy equipment and machinery, micro electromechanical systems (MEMS), and sporting goods. In the FEA the structure is modeled by the assemblage of small pieces of structure, Fig. 1.1. These pieces with simple geometry are called finite elements. The word “finite” distinguishes these pieces from infinitesimal elements used in calculus. In the finite element analysis (FEA), the variation of the field variable on the element is approximated by the simple functions, such as polynomials. The actual variation on the element is almost certainly more complicated, so FEA provides an approximate solution. However, the solution can be improved by using more elements to represent the structure. Elements are connected at points called nodes. The value of field variable and perhaps also its first derivatives are defined as unknowns at the nodes. The assemblage of elements is called a finite element structure, and the particular arrangement of elements is called a mesh. FEM changes the governing differential equations or integral expressions into a set of linear algebraic equations to solve the nodal unknowns.
1.2 ANALYSIS STAGE IN A DESIGN PROCESS The structural design process, from start to finish, is often outlined as in Fig. 1.2. This iterative process must be repeated until the design meets all design constraints. One of the most important design constraints is that the structure must withstand the design
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Figure 1.1 Finite element mesh.
Recognation of need
Definition of problem Determination of design constraints Design modifications
Structural design
Analysis
Experiment
Evaluation
No
Does design satisfy all design constraints
Yes
Figure 1.2 The structural design process.
Presentation
5 loads without failure. There are two ways to ensure design constraints: Analysis and experiment. Experimental way is based on the trial-and-error approach and for the large structures with expensive components the cost for a trial-and-error experiment approach is severe. Furthermore, test of some systems can be dangerous. Therefore it is desirable to develop a theory that will adequately predict failure analyze the particular design using this theory. The advantage of this method is that the engineer can predict failure of his design without having to actually construct and test it. A diagram for the solution process of engineering problems is shown in Figure 1.3. An analysis is applied to a model problem rather than to an actual physical problem. Even laboratory experiments use models unless the actual physical structure is tested. The shortcomings of the both methods are the approximations during the modeling and solution/measurement phases. In the structural design, analysis and experiment should both be viewed as dispensable in the design process. In practice, at first the analyses are used to improve the design. Thus the number of experiments is decreased and the stupidly accidents during the experiments are prevented.
1.2.1 Mathematical Model Before the analysis step, the structural designer has to predetermine the geometric shape and material makeup of the structure, and applied loads such as mechanical loads, input heat, etc. A model for analysis can be devised after the physical nature of the problem has been understood. In modeling the superfluous details are excluded but all essential features are included. Analyst makes some assumptions related to the geometry, loads, materials, deformations, stress field and so on. Thus the resulting model is desired to be simple but to be capable of describing the actual problem with sufficient accuracy. A geometric model becomes a mathematical model when its behavior is described, or approximated, by differential equations or integral expressions.
!
! aproximations
method of solution
mathematical model aproximations
results
engineering problem
! aproximations
! Experimental model
measurements aproximations
Figure 1.3 Steps of problem solving in engineering
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1.2.2 Solution Methods The solution methods can be classified in three categories: Analytical Methods, Approximate Methods, and Numerical Methods. Analytical Methods: They provide closed form exact solutions to the mathematical model of engineering problems. They can be used only if the geometry, loading and boundary conditions of the problem are simple. Integration methods and other analytical solution methods of differential equations are the examples of the analytical methods. Approximate Methods: They provide closed form approximate solutions to the mathematical model of engineering problems. They can be used only if the geometry, loading and boundary conditions of the problem are simple. Ritz’s method, Galerkin’s Method, Collocation Methods, Least Square Method, Moment Method, Kantrovich’s Method, etc. Numerical Methods: They provide discrete form approximate solution to the mathematical model of engineering problems. They can be used to solve the problems with relatively complex geometry, loading and boundary conditions. In particular finite elements can represent structures of arbitrarily complex geometry. Finite Difference Method, Finite Element Method, Boundary Element Method, etc.
Example 1.1 Consider a beam with length L as shown in Fig. 1.4. The modulus of elasticity of beam is E, and the moment of inertia is I. When a vertical distributed load P is applied, the beam deforms by w from the original horizontal line. Mathematical model of the beam in differential equation form is P
L
EI
Figure 1.4 Clamped beam under distributed load.
EI
d 4w dx 4
=P
with the boundary conditions for the clamped end dw w= = 0 at x = 0 dx and for the free end
(1.1)
7 d 2w dx 2
=
d 3w dx3
= 0 at x = L
We can express the Eq. (1.1) in variational form as follows EI Π= 2
2
L ⎛ d 2w ⎞ dx − ⎜ ⎟ ∫ ⎜ dx 2 ⎟ ∫ Pwdx ⎠ 0⎝ 0 L
(1.2)
Here Π is the mechanical potential energy of the beam with deflection w under applied distributed force P. A solution of this problem statement can be obtained by minimizing the potential energy. Solution: (i) Analytical Solution: Application of the integration method to Eq. (1.1) as an analytical solution method.. d 3w dx
3
d 2w
=
1 Px + C1 EI
1 Px 2 + C1 x + C2 2 EI dx 1 dw = Px3 + C1 x 2 + C2 x + C3 dx 6 EI 1 w= Px 4 + C1x3 + C2 x 2 + C3 x + C4 24 EI 2
=
The integration constants are obtained by applying the B.C.’s and the exact solution is found as follows
Px 2 2 w=− x − 4 Lx + 6 L2 24 EI
(
)
(ii) Approximate Solution: Application of the Ritz method to Eq. (1.2) as a approximate solution method. A trial function can be chosen as
w( x) = x 2 (a1 + a2 x + a3 x 2 + L) If we take only two terms, and substitute the approximate solution into the potential energy expression Eq. (1.2) we obtain Π=
EI 2
L
L
(
)
2 3 ∫ ( 2a1 + 6a2 x ) dx − ∫ P a1x + a2 x dx 0
2
0
8 The potential energy is minimized by equating to zero its first derivatives with respect to unknown constants. After performing the integrations, the following equations are obtained. ∂Π PL2 = 0 ⇒ 2a1 + 3La2 = ∂a1 6 EI ∂Π PL2 = 0 ⇒ a1 + 2 La2 = ∂a2 24 EI Solving the equations, the constants are determined as a1 =
5PL2 24 EI
a2 = −
PL2 12 EIL
and the approximate solution is obtained as w( x) = −
(
PL 5 x 2 L − 2 x3 24 EI
)
(iii) Numerical Solution: Application of the FEA as a numerical solution method. We discretize the beam with two beam elements, Fig. 1.5. The unknown nodal parameters are the deflections and the slopes. After the application of the FE procedure we reduce the problem to the following linear algebraic equation system.
w1
θ1
1
w2
2
P
w3 θ3
1
L/2
θ2
2
L/2
3
L Figure 1.5 Finite element model of the clamped beam. 0 −12 3L ⎤ ⎧ w ⎫ ⎡ 24 ⎧ 12 ⎫ 2 ⎢ 2 2⎥⎪ 1 ⎪ 2 L −3L 2 L ⎥ ⎪θ 2 ⎪ PL ⎪⎪ 0 ⎪⎪ 8 EI ⎢ 0 ⎨ ⎬ ⎢ ⎥⎨ ⎬= L3 ⎢ −12 −3L 12 −3L ⎥ ⎪ w3 ⎪ 24 ⎪ 6 ⎪ ⎪⎩− L ⎪⎭ ⎢ 3L 1 L2 −3L L2 ⎥ ⎪⎩ θ3 ⎪⎭ 2 ⎣ ⎦ Then, the nodal displacement are obtained
9 ⎧ w2 ⎫ ⎧−0.044271L ⎫ ⎪θ ⎪ 3 ⎪ −0.14583 ⎪ ⎪ 2 ⎪ PL ⎪ ⎪ ⎨ ⎬= ⎨ ⎬ ⎪ w3 ⎪ EI ⎪ −0.125 L ⎪ ⎪⎩ θ3 ⎪⎭ ⎪⎩ −1.666667 ⎪⎭
The numerical values at the middle span of the beam and the free beam are given at the Table 1.1 Table 1.1 The numerical values obtained from the different solution techniques. Solution Techniques Analytical Approximate Numerical
w2 (PL4/EI) -0.044271 -0.041667 -0.044271
w3 (PL4/EI) -0.125 -0.125 -0.125
1.2.3 Advantages of FEA Advantages of FEA over most other numerical analysis methods:
Versatility: FEA is applicable to any field problem, such as heat transfer, stress analysis, magnetic fields, and so on. There is no geometric restriction: It can be applied the body or region with any shape. Boundary conditions and loading are not restricted (boundary conditions and loads may be applied to any portion of the body) Material properties may be change from one element to another (even within an element) and the material anisotropy is allowed. Different elements (behavior and mathematical descriptions) can be combined in a single FE model. An FE structure closely resembles the actual body or region to be analyzed. The approximation is easily improved by grading the mesh (mesh refinement).
In industry FEA is mostly used in the analysis and optimization phase to reduce the amount of prototype testing and to simulate designs that are not suitable for prototype testing. Computer simulation allows multiple “what-if” scenarios to be tested quickly and effectively. The example for the second reason is surgical implants, such as an artificial knee. On the other hand, the other reasons for preference of the FEM are cost savings, time savings, reducing time to market, creating more reliable and better-quality designs.
1.3 PROBLEM SOLVING BY FEA Solving a structural problem by FEA involves following steps [2].
Learning about the problem Preparing mathematical models Discretizing the model Having the computer do calculations
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Checking results
Generally an iteration is required over these steps.
1.3.1 Learning About the Problem It is important to understand the physics or nature of the problem and classify it. The first step in solving a problem is to identify it. Therefore an engineer has to identify the problem asking the following questions. What are the more important physical phenomena involved? Is the problem time-independent or time dependent? (static or dynamic?) Is nonlinearity involved? (Is iterative solution necessary or not?) What results are sought from analysis? What accuracy is required? From answers it is decided that the necessary information to carry out an analysis, how the problem is modeled, and what method of solution is adopted. Some problems are interdisciplinary nature. There are some couplings between the fields. If the fields interacts each other, it is called direct or mutual coupling. If one field influences the other, it is called indirect or sequential coupling. An example of direct coupling is flutter of an aircraft panel. The pressure produced by airflow on the panel deflects the panel and the deflection modifies the airflow and pressure. Therefore structural displacement and air motion fields cannot be considered separately. Cautions: Without this step a proper model cannot be devised. At present, software does not automatically decide what solution procedure must apply to the problem.
You must decide to do a nonlinear analysis if stresses are high enough to produce yielding. You must decide to perform a buckling analysis if the thin sections carry compressive load. 1.3.2 Preparing Mathematical Models
FEA is applied to the mathematical model. FEA is simulation, not reality. Even very accurate FEA may not match with physical reality if the mathematical model is inappropriate or inadequate. Devise a model problem for analysis, Understanding the physical nature of the problem. Because a model for analysis can be devised after the physical nature of the problem has been understood. Excluding superfluous detail but including all essential features. Unnecessary detail can be omitted. This must enable that the analysis of the model is not unnecessarily complicated. Decide what features are important to the purpose at hand. This provides us to obtain the results with sufficient accuracy. A geometric model becomes a mathematical model when its behavior is described, or approximated, by selected differential equations and boundary conditions.
11 Thus, we may ignore geometric irregularities, regard some loads as concentrated, say that some supports are fixed and idealize material as homogeneous, isotropic, and linearly elastic. What theory or mathematical formulation describes behavior? Depending on the dimensions, loading, and boundary conditions of this idealization we may decide that behavior is described by beam theory, plate-bending theory, equations of plane elasticity, or some other analysis theory Modeling decisions are influenced by what information is sought, what accuracy is required, the anticipated expense of FEA, and its capabilities and limitations. Initial modeling decisions are provisional. It is likely that results of the first FEA will suggest refinements, in geometry, in applicable theory, and so on. 1.3.3 Preliminary Analysis
Before going from a mathematical model to FEA, at least one preliminary solution should be obtained. We may use whatever means are conveniently available – simple analytical calculations, handbook formulas, trusted previous solutions, or experiment. Evaluation of the preliminary analysis results may require a better mathematical model.
1.3.4 Discretization A mathematical model is discretized by dividing it into a mesh of finite elements. Thus a fully continuous field is represented by a piecewise continuous field. A continuum problem is one with an infinite number of unknowns. The FE discretization procedures reduce the problem to one of finite number of unknowns, Figs. 1.6.
Figure 1.6 Finite element model of a stair (from ANSYS presentation). Discretization introduces another approximation. Relative to reality, two sources of error have now been introduced: modeling error and discretization error. Modeling error can be reduced by improving the model; discretization error can be reduced by using more elements. Numerical error is due to finite precision to represent data and the results manipulation.
12 The FEA is an approximation based on piecewise interpolation of field quantity. By means of this FE method [3,4], Solution region is divided into a finite number of subregions (elements) of simple geometry (triangles, rectangles …) Key points are selected on the elements to serve as nodes. The nodes share values of the field quantity and may also share its one or more derivatives. The nodes are also locations where loads are applied and boundary conditions are imposed. The nodes usually lie on the element boundaries, but some elements have a few interior nodes. The unknown field variable is expressed in terms of interpolation functions within each element. The interpolation functions approximate (represent) the field variable in terms of the d.o.f. over a finite element. Polynomials are usually chosen as interpolation functions because differentiation and integration is easy with polynomials. The degree of polynomial depends on the number of unknowns at each node and certain compatibility and continuity requirements. Often functions are chosen so that the field variable and its derivatives are continuous across adjoining element boundaries. Degrees of freedom (d.o.f.) are independent quantities that govern the spatial variation of a field. In this way, the problem is stated in terms of these nodal values as new unknowns. Now, we can formulate the solution for individual elements. There are four different approaches to formulate the properties of individual elements: Direct approach, variational approach, weighted residuals approach, and energy balance approach. Stiffness and equivalent nodal loads for a typical element are determined using the mentioned above. The element properties are assembled to obtain the system equations. The equations are modified to account for the boundary conditions of the problem. The nodal displacements are obtained solving this simultaneous linear algebraic equation system. Once the nodal values (unknowns) are found, the interpolation functions define the field variable through the assemblage of elements. The nature of solution and the degree of approximation depend on the size and number of elements, and interpolation functions. Support reactions are determined at restrained nodes.
1.3.5 Results Checking and Model Revising The analysts are responsible for interpreting the results and taking whatever action is proper. The analysts must estimate the validity of the result first. This is very important because the tendency is to accept the result without question. First we examine results qualitatively. They look right, that is, are there obvious errors? Have we solved the problem we intended to solve, or some other problem? Are boundary conditions applied on the model correctly? Does the deformed geometry reflect the boundary conditions? Are the expected symmetries seen on the deformation and stress results? If the answers to such questions are satisfactory, FEA results are compared with solutions from preliminary analysis and with any other useful information that may be available. Rarely is the first FE analysis satisfactory. Obvious blunders must be corrected. Either physical understanding or the FE model, or both, may be at fault. Disagreements
13 must be satisfactorily resolved by repair of the mathematical model and/or the FE model. After another analysis cycle, the discretization may be judged inadequate, perhaps being too coarse in some places. Then mesh revision is required, followed by another analysis. Do not forget: Software has limitations and almost contains errors. Yet the engineer, not to software provider, is legally responsible for results obtained. 1.4 USAGE OF A FEA SOFTWARE
There are three stages which describe the use of any existing finite element program: Preprocessing, solution and postprocessing. Before entering the program’s preprocessor, the user should have planned the model and gathered necessary data [5].
1.4.1 Steps of Analysis Preprocessing: Input data describes geometry, material properties, loads, and boundary conditions, Fig. 1.7. Software can automatically prepare much of the FE mesh, but must be given direction as to the type of element and mesh density desired, Fig.1.8. Review the data for correctness before proceeding. The completion of the preprocessing stage results in creation of an input data file for the analysis processor.
Figure 1.7 Solid model of a rail vehicle body.
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Figure 1.8 FE model of a rail vehicle body. Solution: This processor reads from the input data file each element definition. Software automatically generates matrices that describe the behavior of each element, and combines these matrices into a large matrix equation that represents the FE structure, applies enough displacement boundary conditions to prevent rigid body motion, solves this equation to determine values of field quantities at nodes and performs additional calculations for nonlinear or time-dependent behavior. Element and node values of strains and stresses are computed for each solution. The processor then produces an output listing file with data files for postprocessing. Postprocessing: This processor takes the results files and allows the user to create graphic displays of the structural deformation and stress components. The node displacements are usually very small for most engineering structures so they are exaggerated to provide visible deformed shapes of structures, Fig. 1.9. Sometimes the animation of structural behavior will be useful to acquire a good understanding.
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Figure 1.9 Deformations of a rail vehicle body.
1.4.2 Expertise on FEA Why study the theory of FEA? It is possible to use FEA programs while having little knowledge of the analysis method or the problem to which it is applied, inviting consequences that may range from embarrassing to disastrous. Even an inept user can obtain a result using a software. However, reliable results are obtained only when the analyst understand the problem, how to model it, behavior of finite elements, assumptions and limitations built in the software, input data formats and when the analyst checks for errors at all stages. It is not realistic to demand that analyst understand details of all elements and procedures, but misuse of FEA can be avoided only by those who understand fundamentals. Students and young engineers can begin to learn the FEA with the examples which has already reliable numerical or analytical solutions. They solved the problem using FEA and see their mistakes during the modeling, data input, and software options. They carry on analysis repeatedly until results are in a good agreement. These exercises will improve analytical skills as well as FE skills. The failures in the problem solving may be discouraging but they should be conscious of that the failures are more instructive than successes. A study on the misuse of computers in engineering [6] shows the most of the faults due to the user error. In the study 52 cases caused damage is cited. The damages are in the form of expensive delay, a need to redesign, poor performance, or collapse. The distribution of errors is given in Table 2. Table 2. Error distributions. Error Type
Case number
16 Hardware error Software error User error Other causes
7 13 30 2
User error was usually associated with poor modeling, and with poor understanding of software limitations and input data formats. References:
[1] R., Szilard, Theory and Analysis of Plates, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1974. [2] R.D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt, “ Concepts and Applications of Finite Element Analysis,” John Wiley and sons, Inc., USA, 2002. [3] W. Weaver, Jr. and P. R. Johnston, “Finite Elements for Structural Analysis,” Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984. [4] T.J.R. Hughes, “The Finite Element Method – Linear Static and Dynamic Finite Element Analysis,” Prentice-Hall, Inc., NJ, 1987. [5] C.E. Knight, Jr., “The Finite Element Method in Mechanical Design,” PSW-KENT Publishing Co., Boston,1993. [6] “Computer Misuse – Are We Dealing with a Time Bomb? Who is to Blame and What are We Doing About It? A Panel Discussion,” in Forensic Engineering, Proceedings of the First Congress, K.L. Rens (ed.), American Society of Civil Engineers, Reston, VA, 1997, pp. 285-336.
