Direct Method Lecture34

FiniteElementMethodFEM

Di t M th dLecture3

DirectMethod

Dr.MohammedAlHazmi

Dr.MohammedAlHazmiFEM

DIRECTFORMULATIONThe following problem illustrates the steps and the procedure involved inThefollowingproblemillustratesthestepsandtheprocedureinvolvedindirectformulation.

EXAMPLEEXAMPLE

Consider a bar with a variable cross sectionsupporting a load P, as shown in Figure. The barpp g , gis fixed at one end and carries the load P a t theother end. Let us designate the width of the barat the top by w1 at the bottom by w2 itsat the top by w1 at the bottom by w2 itsthickness by t, and its length by L. The bar'smodulus of elasticity will be denoted by E. Weare interested in approximating how much theare interested in approximating how much thebar will deflect at various points along its lengthwhen it is subjected to the load P. We will neglecth h f h b h f ll lthe weight of the bar in the following analysis,assuming that the applied load is considerablylarger than the weight of the bar:


PreprocessingPhaseDiscrete the solution domain into finite elements.Discretethesolutiondomainintofiniteelements.

Webeginbysubdividingtheproblemintofivenodesandfourelements,witheachsegmenthavingauniformcrosssection,asshowninFigure.


2.Assumeasolutionthatapproximatesthebehaviorofanelement.

The average stress a in the member is given byTheaveragestressainthememberisgivenby

Th l t i l f th b i d fi d th h i l thTheaveragenormalstrain lofthememberisdefinedasthechangeinlengthAtperunitoriginallengtheofthemember:


Overtheelasticregion,thestressstrainarerelatedbyHookesLaw,g , y ,accordingtotheequation:

WhereE isthemodulusofelasticityofthematerial.Andbycombiningtheequationsandsimplifying,wehave:

Wherekeq istheequivalentstiffnessfortheuniformcrosssectionismodeledasaspring


ByturningtotheExample

The bar is represented by model consisting of four elastic springs (elements) inThe bar is represented by model consisting of four elastic springs (elements) inseries, and the elastic behavior of an element with nodes i and i+1 is modeledby an equivalent linear spring according to the equation

where


Byconsideringtheforcesactingoneachnode.

The free body diagram of nodes which showsy gthe forces acting on nodes 1 through 5 of thismodel, is depicted in the figure.

Static equilibrium requires that the sum of theforces acting on each node be zero. So that, thisrequirement creates the following five equation


B i th ilib i tiByrearrangingtheequilibriumequations:

Bypresentingtheequilibriumequationinthematrixform


The matrix equation can be written in the form as:Thematrixequationcanbewrittenintheformas:

Sothat,itcanbewritteninthegeneralformas:g


ModelboundaryconditionsTh b i fi d d h h di l d 1 iThebarisfixedattopend,sothatthedisplacementatnode1iszero(u1 =0 )

Hence,therearefourunknownnodaldisplacementsu2 ,u3 ,u4 ,u5andthereactionforceatnodeoneisalsounknownR1

Sothattheapplicationoftheboundaryconditionleadsthefollowingmatrixequation:


SolutionPhase:Themodelmaterialisaluminum:Modulus of Elasticity E = 10 4 x 106 Ib/in2 w = 2 in w = 1 in t = 0 125ModulusofElasticityE=10.4x106 Ib/in2 ,w1 =2in,w2 =1in,t=0.125in,L=10in,andP=1000Ib.

Theelementpropertiesare:p p


Assemblingtheelementmatricesleadstothegenerationoftheg gglobalstiffnessmatrix:

Applyingtheboundaryconditionu1 =0andloadP=1000Ib,wegetpp y g y 1 , g


Direct Method Lecture34

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