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A Brief HistoryA Brief History

Finite Element Analysis (FEA) was first developed in 1943 by R.Finite Element Analysis (FEA) was first developed in 1943 by R.Courant, who utilized the Ritz method of numerical analysis andCourant, who utilized the Ritz method of numerical analysis andminimization of variational calculus to obtain approximate solutionsminimization of variational calculus to obtain approximate solutionsto vibration systems. Shortly thereafter, a paper published in 1956to vibration systems. Shortly thereafter, a paper published in 1956

by M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Toppby M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Toppestablished a broader definition of numerical analysis. The paper established a broader definition of numerical analysis. The paper centered on the "stiffness and deflection of complex structures".centered on the "stiffness and deflection of complex structures".

By the early 70's, FEA was limited to expensive mainframeBy the early 70's, FEA was limited to expensive mainframecomputers generally owned by the aeronautics, automotive,computers generally owned by the aeronautics, automotive,defense, and nuclear industries. Since the rapid decline in the costdefense, and nuclear industries. Since the rapid decline in the cost

of computers and the phenomenal increase in computing power,of computers and the phenomenal increase in computing power,FEA has been developed to an incredible precision. Present dayFEA has been developed to an incredible precision. Present daysupercomputers are now able to produce accurate results for allsupercomputers are now able to produce accurate results for allkinds of parameters.kinds of parameters.



What is Finite Element Analysis?What is Finite Element Analysis?

FEAFEA consistsconsists of of aa computer computer modelmodel of of aa materialmaterialor or designdesign thatthat isis stressedstressed andand analyzedanalyzed for for specificspecific resultsresults.. ItIt isis usedused inin newnew productproduct design,design,andand existingexisting productproduct refinementrefinement.. A A companycompany isisableable toto verifyverify aa proposedproposed designdesign willwill bebe ableable totoperformperform toto thethe client'sclient's specificationsspecifications prior prior totomanufacturingmanufacturing or or constructionconstruction.. ModifyingModifying ananexistingexisting productproduct or or structurestructure isis utilizedutilized toto qualifyqualifythethe productproduct or or structurestructure for for aa newnew serviceserviceconditioncondition.. InIn casecase of of structuralstructural failure,failure, FEAFEA maymaybebe usedused toto helphelp determinedetermine thethe designdesignmodificationsmodifications toto meetmeet thethe newnew conditioncondition..



contdcontd

There are generally two types of analysis that are used in industry:There are generally two types of analysis that are used in industry: 22--D modeling, andD modeling, and

33--D modeling.D modeling.

While 2While 2--D modeling conserves simplicity and allows the analysis toD modeling conserves simplicity and allows the analysis tobe run on a relatively normal computer, it tends to yield lessbe run on a relatively normal computer, it tends to yield less

accurate results. 3accurate results. 3--D modeling, however, produces more accurateD modeling, however, produces more accurateresults while sacrificing the ability to run on all but the fastestresults while sacrificing the ability to run on all but the fastestcomputers effectively. Within each of these modeling schemes, thecomputers effectively. Within each of these modeling schemes, theprogrammer can insert numerous algorithms (functions) which mayprogrammer can insert numerous algorithms (functions) which maymake the system behave linearly or nonmake the system behave linearly or non--linearly. Linear systems arelinearly. Linear systems arefar less complex and generally do not take into account plasticfar less complex and generally do not take into account plasticdeformation. Nondeformation. Non--linear systems do account for plastic deformation,linear systems do account for plastic deformation,

and many also are capable of testing a material all the way toand many also are capable of testing a material all the way tofracture.fracture.



How Does Finite ElementHow Does Finite Element

Analysis

Work?An

alysis

Work? FEA uses a complex system of points calledFEA uses a complex system of points called nodesnodes which make awhich make a

grid called agrid called a meshmesh (Figure 2). This mesh is programmed to contain(Figure 2). This mesh is programmed to containthe material and structural properties which define how the structurethe material and structural properties which define how the structurewill react to certain loading conditions. Nodes are assigned at awill react to certain loading conditions. Nodes are assigned at acertain density throughout the material depending on the anticipatedcertain density throughout the material depending on the anticipated

stress levels of a particular area. Regions which will receive largestress levels of a particular area. Regions which will receive largeamounts of stress usually have a higher node density than thoseamounts of stress usually have a higher node density than thosewhich experience little or no stress. Points of interest may consist of:which experience little or no stress. Points of interest may consist of:fracture point of previously tested material, fillets, corners, complexfracture point of previously tested material, fillets, corners, complexdetail, and high stress areas. The mesh acts like a spider web in thatdetail, and high stress areas. The mesh acts like a spider web in thatfrom each node, there extends a mesh element to each of thefrom each node, there extends a mesh element to each of theadjacent nodes. This web of vectors is what carries the materialadjacent nodes. This web of vectors is what carries the material

properties to the object, creating many elements.properties to the object, creating many elements. (Theory)(Theory)



A wide range of objective functions A wide range of objective functions

(variables within the system) are available(variables within the system) are available

for minimization or maximization:for minimization or maximization: Mass, volume, temperatureMass, volume, temperature

Strain energy, stress strainStrain energy, stress strain

Force, displacement, velocity, accelerationForce, displacement, velocity, acceleration

Synthetic (User defined)Synthetic (User defined)



There are multiple loading conditionsThere are multiple loading conditionswhich may be applied to a system. Next towhich may be applied to a system. Next toFigure 3, some examples are shown:Figure 3, some examples are shown:

Point, pressure (Figure 3), thermal, gravity,Point, pressure (Figure 3), thermal, gravity,and centrifugal static loadsand centrifugal static loads

Thermal loads from solution of heat transfer Thermal loads from solution of heat transfer analysisanalysis

Enforced displacementsEnforced displacements

Heat flux and convectionHeat flux and convection

Point, pressure and gravity dynamic loadsPoint, pressure and gravity dynamic loads



Each FEA program may come with an elementEach FEA program may come with an elementlibrary, or one is constructed over time. Somelibrary, or one is constructed over time. Somesample elements are:sample elements are: Rod elementsRod elements

Beam elementsBeam elements

Plate/Shell/Composite elementsPlate/Shell/Composite elements

Shear panelShear panel

Solid elementsSolid elements

Spring elementsSpring elements Mass elementsMass elements

Rigid elementsRigid elements

Viscous damping elementsViscous damping elements



Many FEA programs also are equippedMany FEA programs also are equipped

with the capability to use multiple materialswith the capability to use multiple materials

within the structure such as:within the structure such as:

Isotropic, identical throughoutIsotropic, identical throughout

Orthotropic, identical at 90 degreesOrthotropic, identical at 90 degrees

General anisotropic, different throughoutGeneral anisotropic, different throughout



Types of Engineering AnalysisTypes of Engineering Analysis

Structur alStructur al analysis consists of linear and nonanalysis consists of linear and non--linear linear models. Linear models use simple parameters andmodels. Linear models use simple parameters andassume that the material is not plastically deformed.assume that the material is not plastically deformed.

NonNon--linear models consist of stressing the material pastlinear models consist of stressing the material pastits elastic capabilities. The stresses in the material thenits elastic capabilities. The stresses in the material thenvary with the amount of deformation as in Figure 4.vary with the amount of deformation as in Figure 4.

Vibr ationalVibr ational analysis is used to test a material againstanalysis is used to test a material againstrandom vibrations, shock, and impact. Each of theserandom vibrations, shock, and impact. Each of theseincidences may act on the natural vibrational frequencyincidences may act on the natural vibrational frequencyof the material which, in turn, may cause resonance andof the material which, in turn, may cause resonance andsubsequent failure.subsequent failure.



FatigueFatigue analysis helps designers to predict the life of aanalysis helps designers to predict the life of amaterial or structure by showing the effects of cyclicmaterial or structure by showing the effects of cyclicloading on the specimen. Such analysis can show theloading on the specimen. Such analysis can show theareas where crack propagation is most likely to occur.areas where crack propagation is most likely to occur.

Failure due to fatigue may also show the damageFailure due to fatigue may also show the damagetolerance of the material (Figure 5).tolerance of the material (Figure 5).

Heat Tr ansfer Heat Tr ansfer analysis models the conductivity or analysis models the conductivity or thermal fluid dynamics of the material or structurethermal fluid dynamics of the material or structure(Figure 1). This may consist of a steady(Figure 1). This may consist of a steady--state or state or

transient transfer. Steadytransient transfer. Steady--state transfer refers tostate transfer refers toconstant thermoproperties in the material that yield linear constant thermoproperties in the material that yield linear heat diffusion.heat diffusion.



Results of Finite ElementResults of Finite Element

AnalysisAnalysis

FEA has become a solution to the task of FEA has become a solution to the task of

predicting failure due to unknown stressespredicting failure due to unknown stresses

by showing problem areas in a materialby showing problem areas in a material

and allowing designers to see all of theand allowing designers to see all of the

theoretical stresses within. This method of theoretical stresses within. This method of

product design and testing is far superior product design and testing is far superior

to the manufacturing costs which wouldto the manufacturing costs which wouldaccrue if each sample was actually builtaccrue if each sample was actually built

and tested.and tested.



1.1. IntroductionIntroduction

2.2. PrePre- -processing processing

3.3. SolutionSolution

4.4. Post Post- -processing processing

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