finite element analysis in structures

7/28/2019 Finite Element Analysis in structures.

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Zahit Mecitolu

stanbul Technical UniversityFaculty of Aeronautics and Astronautics,

Maslak, Istanbul

January 2008


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Chapter 1

INTRODUCTIONFinite 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 rotorblade. 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 ofthe most important design constraints is that the structure must withstand the design


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Figure 1.1 Finite element mesh.

YesNo

Recognation of need

Definition of problem

Structural design

Analysis

Evaluation

Presentation

Experiment

Design modifications

Does design

satisfy all design

constraints

Determination of

desi n constraints

Figure 1.2 The structural design process.


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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 designusing 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.

engineering

problem

mathematical

model

method of

solution

results

Experimental

modelmeasurements

aproximations

!

!

aproximations

aproximations

!

!

aproximations

Figure 1.3 Steps of problem solving in engineering


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1.2.2 Solution MethodsThe 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. Ritzs

method, Galerkins Method, Collocation Methods, Least Square Method,

Moment Method, Kantrovichs 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

Figure 1.4 Clamped beam under distributed load.

P

EIL

4

4

d wEI P

dx

= (1.1)

with the boundary conditions for the clamped end

0 at x = 0dw

wdx

= =

and for the free end


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2 3

2 30 at x = L

d w d w

dx dx

= =

We can express the Eq. (1.1) in variational form as follows

22

20 0

2

L LEI d w

dx Pwdxdx

=

(1.2)

Here is the mechanical potential energy of the beam with deflection w under applieddistributed 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..

3

13

22

1 22

1

1

2

d wPx C

EIdx

d wPx C x C

EIdx

= +

= + +

3 21 2 3

4 3 21 2 3

1

6

124

dwPx C x C x C

dx EI

w Px C x C x C x C EI

= + + +

= + + + 4+

The integration constants are obtained by applying the B.C.s and the exact solution is

found as follows

( )2

2 24 624

Pxw x Lx

EI= + L

(ii) Approximate Solution: Application of the Ritz method to Eq. (1.2) as a

approximate solution method. A trial function can be chosen as

2 21 2 3( ) ( )w x x a a x a x= + + +L

If we take only two terms, and substitute the approximate solution into the potential

energy expression Eq. (1.2) we obtain

( ) ( )2 2 31 2 1 20 0

2 62

L LEI

a a x dx P a x a x d = + + x


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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.

2

1 21

2

1 22

0 2 3 6

0 224

PL

a Laa E

PLa La

a E

= + =

= + =

I

I

Solving the equations, the constants are determined as

2 2

1 2

5

24 12

PL PLa a

EI EIL= =

and the approximate solution is obtained as

( )2 3( ) 5 224

PLw x x L x

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.

2

w3

21

w1 w2P

L/2 L/2

L

1 2

3

13

Figure 1.5 Finite element model of the clamped beam.

22 21

2 2

33

2 21 32

24 0 12 3 12

0 2 3 08

12 3 12 3 624

3 3

Lw

L L LEI PL

wL LL

LL L L L

=

Then, the nodal displacement are obtained


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2

32

3

3

0.044271

0.14583

0.125

1.666667

w L

PL

w LEI

=

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 w2 (PL4/EI) w3 (PL

4/EI)

Analytical -0.044271 -0.125

Approximate -0.041667 -0.125

Numerical -0.044271 -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, stressanalysis, magnetic fields, and so on.

There is no geometric restriction: It can be applied the body or region with anyshape.

Boundary conditions and loading are not restricted (boundary conditions andloads may be applied to any portion of the body)

Material properties may be change from one element to another (even within anelement) 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 costsavings, 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 resultsGenerally an iteration is required over these steps.

1.3.1 Learning About the ProblemIt 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 thefields. 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 analysiscan be devised after the physical nature of the problem has been understood.

Excluding superfluous detail but including all essential features. Unnecessarydetail 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 isdescribed, or approximated, by selected differential equations and boundary

conditions.


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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 planeelasticity, 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 theresults manipulation.


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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) ofsimple geometry (triangles, rectangles )

Key points are selected on the elements to serve as nodes. The nodes sharevalues of the field quantity and may also share its one or more derivatives. Thenodes 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 functionswithin 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 fourdifferent 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 usingthe 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 algebraicequation 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 isproper. 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


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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 programs 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 areexaggerated 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 Expert ise on FEA

Why study the theory of FEA? It is possible to use FEA programs while having littleknowledge 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 softwareoptions. 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


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Hardware error 7

Software error 13

User error 30

Other causes 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.

finite element analysis in structures

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