1Finit e Finite Element Analysis in Structures Zahit Mecitoğlu Istanbul January 2008 . Maslak.2 Finite Element Analysis in Structures Zahit Mecitoğlu İstanbul Technical University Faculty of Aeronautics and Astronautics. geomechanics. it is applied to other fields of continuum mechanics. fluid mechanics. This iterative process must be repeated until the design meets all design constraints. and the particular arrangement of elements is called a mesh.2. from start to finish. 1. and sporting goods. A field problem is formulated by differential equations or by an integral expression. this book is devoted solely to the topic of finite elements for the analysis of structures. Then. electromagnetics. such as polynomials. In the finite element analysis (FEA). such as heat transfer. electronics and appliances. biomechanics. automotive. bridges and buildings. FEM changes the governing differential equations or integral expressions into a set of linear algebraic equations to solve the nodal unknowns. heavy equipment and machinery. so FEA provides an approximate solution. micro electromechanical systems (MEMS).2 ANALYSIS STAGE IN A DESIGN PROCESS The structural design process. biomedical. The cautions which must be taken care about are denoted. One of the most important design constraints is that the structure must withstand the design . Either description may be used to formulate finite elements. is often outlined as in Fig. the variation of the field variable on the element is approximated by the simple functions. The finite element method is originally developed to study the stresses in complex aircraft structures. 1. Elements are connected at points called nodes. the solution can be improved by using more elements to represent the structure. acoustics. Otherwise it is necessary to use an approximate numerical solution such as FEM. such as aerospace. However. loading and boundary conditions of the problem are simple. FEA is used in industries.1 FINITE ELEMENT ANALYSIS Finite Element Analysis (FEA) is a method for numerical solution of field problems. The actual variation on the element is almost certainly more complicated. or calculation of the distribution of displacements and stresses on a helicopter rotor blade. The application steps of the method and software usage are discussed. The assemblage of elements is called a finite element structure. These pieces with simple geometry are called 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. 1. 1. The value of field variable and perhaps also its first derivatives are defined as unknowns at the nodes. Fig. However. In the FEA the structure is modeled by the assemblage of small pieces of structure. The advantages of the method over the other analysis methods are explained.1.Chapter 1 INTRODUCTION Finite element analysis is introduced in this chapter. The word “finite” distinguishes these pieces from infinitesimal elements used in calculus. A field problem may be determination of the temperature distribution in a turbine blade. 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 Presentation Figure 1.4 Figure 1.1 Finite element mesh.2 The structural design process. . A geometric model becomes a mathematical model when its behavior is described. or approximated. A model for analysis can be devised after the physical nature of the problem has been understood. Analyst makes some assumptions related to the geometry. deformations. loads. A diagram for the solution process of engineering problems is shown in Figure 1.2. In modeling the superfluous details are excluded but all essential features are included. Thus the resulting model is desired to be simple but to be capable of describing the actual problem with sufficient accuracy. In the structural design. input heat. An analysis is applied to a model problem rather than to an actual physical problem. etc. at first the analyses are used to improve the design. the structural designer has to predetermine the geometric shape and material makeup of the structure. In practice. test of some systems can be dangerous. The shortcomings of the both methods are the approximations during the modeling and solution/measurement phases. materials.3. Even laboratory experiments use models unless the actual physical structure is tested. 1. Furthermore. There are two ways to ensure design constraints: Analysis and experiment. Therefore it is desirable to develop a theory that will adequately predict failure analyze the particular design using this theory. stress field and so on. by differential equations or integral expressions.3 Steps of problem solving in engineering . The advantage of this method is that the engineer can predict failure of his design without having to actually construct and test it. analysis and experiment should both be viewed as dispensable in the design process. Thus the number of experiments is decreased and the stupidly accidents during the experiments are prevented. 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. ! aproximations ! mathematical model aproximations method of solution engineering problem results ! aproximations ! Experimental model aproximations measurements Figure 1.5 loads without failure.1 Mathematical Model Before the analysis step. and applied loads such as mechanical loads. Numerical Methods: They provide discrete form approximate solution to the mathematical model of engineering problems. Approximate Methods. They can be used only if the geometry. Integration methods and other analytical solution methods of differential equations are the examples of the analytical methods. Analytical Methods: They provide closed form exact solutions to the mathematical model of engineering problems. When a vertical distributed load P is applied.1 Consider a beam with length L as shown in Fig. and Numerical Methods.2 Solution Methods The solution methods can be classified in three categories: Analytical Methods. Collocation Methods. Finite Element Method.6 1. etc. etc. and the moment of inertia is I. loading and boundary conditions of the problem are simple. 1. The modulus of elasticity of beam is E. They can be used only if the geometry. d 4w dx 4 EI =P (1.1) with the boundary conditions for the clamped end dw w= = 0 at x = 0 dx and for the free end . Galerkin’s Method. loading and boundary conditions. Least Square Method. Mathematical model of the beam in differential equation form is P L EI Figure 1. Boundary Element Method. Finite Difference Method. Approximate Methods: They provide closed form approximate solutions to the mathematical model of engineering problems. Moment Method.4 Clamped beam under distributed load. loading and boundary conditions of the problem are simple.2. the beam deforms by w from the original horizontal line. Kantrovich’s Method. In particular finite elements can represent structures of arbitrarily complex geometry.4. Example 1. They can be used to solve the problems with relatively complex geometry. Ritz’s method. 1) in variational form as follows EI Π= 2 L ⎛ d 2w ⎞ dx − ⎜ ⎟ ∫ ⎜ dx 2 ⎟ ∫ Pwdx ⎠ 0⎝ 0 L 2 (1.’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. Solution: (i) Analytical Solution: Application of the integration method to Eq. (1.. (1.2) Here Π is the mechanical potential energy of the beam with deflection w under applied distributed force P. A trial function can be chosen as w( x) = x 2 (a1 + a2 x + a3 x 2 + L) If we take only two terms.C. and substitute the approximate solution into the potential energy expression Eq. d 3w dx 3 = = 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 d 2w The integration constants are obtained by applying the B.7 d 2w dx 2 d 3w dx3 = = 0 at x = L We can express the Eq.2) we obtain Π= EI 2 2 3 ∫ ( 2a1 + 6a2 x ) dx − ∫ P a1x + a2 x dx 2 0 0 L L ( ) .2) as a approximate solution method. (1. A solution of this problem statement can be obtained by minimizing the potential energy. (1.1) as an analytical solution method. ∂Π PL2 = 0 ⇒ 2a1 + 3La2 = ∂a1 6 EI ∂Π PL2 = 0 ⇒ a1 + 2 La2 = ∂a2 24 EI Solving the equations. We discretize the beam with two beam elements.5. w1 1 w2 2 P w3 θ3 θ1 1 L/2 θ2 2 L L/2 3 Figure 1.8 The potential energy is minimized by equating to zero its first derivatives with respect to unknown constants.5 Finite element model of the clamped beam. Fig. The unknown nodal parameters are the deflections and the slopes. After performing the integrations. 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. the nodal displacement are obtained . 1. 0 −12 3L ⎤ ⎧ w ⎫ ⎡ 24 ⎧ 12 ⎫ 2 ⎢ 2 2⎥⎪ 1 ⎪ ⎪0⎪ 2 L −3L 2 L ⎥ ⎪θ 2 ⎪ PL ⎪ 8 EI ⎢ 0 ⎪ ⎨ ⎬= ⎨ ⎬ ⎥ 3 ⎢ −12 −3L 12 −3L ⎥ ⎪ w3 ⎪ 24 ⎪ 6 ⎪ L ⎢ ⎪ ⎢ 3L 1 L2 −3L L2 ⎥ ⎪ θ ⎪ ⎩− L ⎪ ⎭ 2 ⎣ ⎦⎩ 3 ⎭ Then. the following equations are obtained. After the application of the FE procedure we reduce the problem to the following linear algebraic equation system. 125 1. 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. and so on.044271 -0.3 PROBLEM SOLVING BY FEA Solving a structural problem by FEA involves following steps [2].125 -0. creating more reliable and better-quality designs. The approximation is easily improved by grading the mesh (mesh refinement).125 L ⎪ ⎪ ⎪ ⎩ −1.041667 -0. Solution Techniques Analytical Approximate Numerical w2 (PL4/EI) -0. Computer simulation allows multiple “what-if” scenarios to be tested quickly and effectively.044271 w3 (PL4/EI) -0. 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. 1. Learning about the problem Preparing mathematical models Discretizing the model Having the computer do calculations . stress analysis.2. such as an artificial knee. Different elements (behavior and mathematical descriptions) can be combined in a single FE model. On the other hand.666667 ⎪ ⎭ ⎩ θ3 ⎪ ⎭ The numerical values at the middle span of the beam and the free beam are given at the Table 1. time savings. magnetic fields.9 ⎧ w2 ⎫ ⎧−0. There is no geometric restriction: It can be applied the body or region with any shape. the other reasons for preference of the FEM are cost savings.1 Table 1. The example for the second reason is surgical implants.14583 ⎪ ⎪ 2 ⎪ PL ⎪ ⎪ ⎨ ⎬= ⎨ ⎬ ⎪ w3 ⎪ EI ⎪ −0.125 -0. An FE structure closely resembles the actual body or region to be analyzed. such as heat transfer.1 The numerical values obtained from the different solution techniques.044271L ⎫ ⎪θ ⎪ 3 ⎪ −0.3 Advantages of FEA Advantages of FEA over most other numerical analysis methods: Versatility: FEA is applicable to any field problem. reducing time to market. Unnecessary detail can be omitted. This must enable that the analysis of the model is not unnecessarily complicated. The pressure produced by airflow on the panel deflects the panel and the deflection modifies the airflow and pressure. Devise a model problem for analysis. Therefore structural displacement and air motion fields cannot be considered separately.1 Learning About the Problem It is important to understand the physics or nature of the problem and classify it. how the problem is modeled. You must decide to perform a buckling analysis if the thin sections carry compressive load. 1. Some problems are interdisciplinary nature. Understanding the physical nature of the problem. and what method of solution is adopted. it is called indirect or sequential coupling.3. Because a model for analysis can be devised after the physical nature of the problem has been understood. 1. FEA is simulation. There are some couplings between the fields. Even very accurate FEA may not match with physical reality if the mathematical model is inappropriate or inadequate. Therefore an engineer has to identify the problem asking the following questions.10 Checking results Generally an iteration is required over these steps. . At present. Cautions: Without this step a proper model cannot be devised. 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. not reality. or approximated. If the fields interacts each other. An example of direct coupling is flutter of an aircraft panel. This provides us to obtain the results with sufficient accuracy.2 Preparing Mathematical Models FEA is applied to the mathematical model. A geometric model becomes a mathematical model when its behavior is described. The first step in solving a problem is to identify it. by selected differential equations and boundary conditions. Excluding superfluous detail but including all essential features. it is called direct or mutual coupling. If one field influences the other. 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. Decide what features are important to the purpose at hand.3. handbook formulas. Thus a fully continuous field is represented by a piecewise continuous field. Relative to reality. The FE discretization procedures reduce the problem to one of finite number of unknowns. and its capabilities and limitations. isotropic. we may ignore geometric irregularities. two sources of error have now been introduced: modeling error and discretization error. Figs.3 Preliminary Analysis Before going from a mathematical model to FEA. trusted previous solutions. or experiment. 1. and linearly elastic. in geometry. Discretization introduces another approximation. . Numerical error is due to finite precision to represent data and the results manipulation.11 Thus. 1.3. We may use whatever means are conveniently available – simple analytical calculations. Figure 1. and so on. or some other analysis theory Modeling decisions are influenced by what information is sought. at least one preliminary solution should be obtained. in applicable theory.4 Discretization A mathematical model is discretized by dividing it into a mesh of finite elements.3. and boundary conditions of this idealization we may decide that behavior is described by beam theory.6. the anticipated expense of FEA. discretization error can be reduced by using more elements.6 Finite element model of a stair (from ANSYS presentation). It is likely that results of the first FEA will suggest refinements. regard some loads as concentrated. A continuum problem is one with an infinite number of unknowns. equations of plane elasticity. Evaluation of the preliminary analysis results may require a better mathematical model. say that some supports are fixed and idealize material as homogeneous. plate-bending theory. Modeling error can be reduced by improving the model. What theory or mathematical formulation describes behavior? Depending on the dimensions. loading. Initial modeling decisions are provisional. what accuracy is required. 1. Either physical understanding or the FE model. the interpolation functions define the field variable through the assemblage of elements. 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. Once the nodal values (unknowns) are found. Rarely is the first FE analysis satisfactory.) are independent quantities that govern the spatial variation of a field. that is. 1. Obvious blunders must be corrected. Solution region is divided into a finite number of subregions (elements) of simple geometry (triangles. This is very important because the tendency is to accept the result without question. The nodes usually lie on the element boundaries. The element properties are assembled to obtain the system equations. the problem is stated in terms of these nodal values as new unknowns.5 Results Checking and Model Revising The analysts are responsible for interpreting the results and taking whatever action is proper. The equations are modified to account for the boundary conditions of the problem. The nodes share values of the field quantity and may also share its one or more derivatives. The degree of polynomial depends on the number of unknowns at each node and certain compatibility and continuity requirements. variational approach. but some elements have a few interior nodes. The nodes are also locations where loads are applied and boundary conditions are imposed. By means of this FE method [3. and interpolation functions. Disagreements . or both. over a finite element. Polynomials are usually chosen as interpolation functions because differentiation and integration is easy with polynomials. are there obvious errors? Have we solved the problem we intended to solve. and energy balance approach. may be at fault.4]. The nodal displacements are obtained solving this simultaneous linear algebraic equation system. Often functions are chosen so that the field variable and its derivatives are continuous across adjoining element boundaries. The interpolation functions approximate (represent) the field variable in terms of the d.f. weighted residuals approach. Support reactions are determined at restrained nodes. There are four different approaches to formulate the properties of individual elements: Direct approach. First we examine results qualitatively.3. rectangles …) Key points are selected on the elements to serve as nodes. The nature of solution and the degree of approximation depend on the size and number of elements. Now.f.o. The analysts must estimate the validity of the result first. The unknown field variable is expressed in terms of interpolation functions within each element.o. we can formulate the solution for individual elements. FEA results are compared with solutions from preliminary analysis and with any other useful information that may be available.12 The FEA is an approximation based on piecewise interpolation of field quantity. Degrees of freedom (d. In this way. Stiffness and equivalent nodal loads for a typical element are determined using the mentioned above. They look right. Figure 1. material properties. loads.7 Solid model of a rail vehicle body.8. After another analysis cycle. Before entering the program’s preprocessor. followed by another analysis. Then mesh revision is required. 1. 1. the discretization may be judged inadequate. Do not forget: Software has limitations and almost contains errors.13 must be satisfactorily resolved by repair of the mathematical model and/or the FE model. Review the data for correctness before proceeding. but must be given direction as to the type of element and mesh density desired.1. the user should have planned the model and gathered necessary data [5]. not to software provider. .7. Fig. Software can automatically prepare much of the FE mesh. is legally responsible for results obtained. and boundary conditions.4. Fig.4 USAGE OF A FEA SOFTWARE There are three stages which describe the use of any existing finite element program: Preprocessing. The completion of the preprocessing stage results in creation of an input data file for the analysis processor. perhaps being too coarse in some places. solution and postprocessing.1 Steps of Analysis Preprocessing: Input data describes geometry. 1. Yet the engineer. Element and node values of strains and stresses are computed for each solution. Software automatically generates matrices that describe the behavior of each element.8 FE model of a rail vehicle body. applies enough displacement boundary conditions to prevent rigid body motion. Postprocessing: This processor takes the results files and allows the user to create graphic displays of the structural deformation and stress components. Fig. solves this equation to determine values of field quantities at nodes and performs additional calculations for nonlinear or time-dependent behavior. Solution: This processor reads from the input data file each element definition. 1. The processor then produces an output listing file with data files for postprocessing. .14 Figure 1. Sometimes the animation of structural behavior will be useful to acquire a good understanding.9. and combines these matrices into a large matrix equation that represents the FE structure. The node displacements are usually very small for most engineering structures so they are exaggerated to provide visible deformed shapes of structures. but misuse of FEA can be avoided only by those who understand fundamentals. reliable results are obtained only when the analyst understand the problem. They solved the problem using FEA and see their mistakes during the modeling.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. However. Error distributions. assumptions and limitations built in the software. a need to redesign. or collapse. data input. input data formats and when the analyst checks for errors at all stages.9 Deformations of a rail vehicle body. These exercises will improve analytical skills as well as FE skills.4. behavior of finite elements.15 Figure 1. and software options. They carry on analysis repeatedly until results are in a good agreement. Table 2. In the study 52 cases caused damage is cited. poor performance. The failures in the problem solving may be discouraging but they should be conscious of that the failures are more instructive than successes. Students and young engineers can begin to learn the FEA with the examples which has already reliable numerical or analytical solutions. inviting consequences that may range from embarrassing to disastrous. how to model it. It is not realistic to demand that analyst understand details of all elements and procedures. Error Type Case number . The damages are in the form of expensive delay. The distribution of errors is given in Table 2. A study on the misuse of computers in engineering [6] shows the most of the faults due to the user error. Even an inept user can obtain a result using a software. 1. Szilard. Weaver.E. pp. Inc.L.. Proceedings of the First Congress. Plesha and R.” Prentice-Hall. NJ. Cook. [2] R. NJ. “Finite Elements for Structural Analysis. 2002. Boston. [6] “Computer Misuse – Are We Dealing with a Time Bomb? Who is to Blame and What are We Doing About It? A Panel Discussion. D. USA. Englewood Cliffs. . [4] T. Witt. Theory and Analysis of Plates.D. [5] C.1993..” Prentice-Hall. 285-336. References: [1] R. R. Rens (ed. and P. “The Finite Element Method – Linear Static and Dynamic Finite Element Analysis.” PSW-KENT Publishing Co.. Jr. Jr. Inc.). VA. 1987. 1984. 1974.. Hughes. “ Concepts and Applications of Finite Element Analysis. Reston. Johnston.E. Malkus.J.. Prentice-Hall.. Inc. Knight.. 1997.” in Forensic Engineering. Englewood Cliffs. and with poor understanding of software limitations and input data formats. K. Inc. [3] W.” John Wiley and sons.16 Hardware error Software error User error Other causes 7 13 30 2 User error was usually associated with poor modeling.R. “The Finite Element Method in Mechanical Design. American Society of Civil Engineers. M. NJ.J.S.