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BarElement&LinearStaticAnalysisLinearStaticAnalysis

,thefollowingassumptions:1.Smalldeformations(loadingpatternisnotchangedduetothedeformedshape)2.Elasticmaterials(noplasticityorfailures)3.Staticloads(theloadisappliedtothestructureinasloworsteadyfashion)

Linearanalysiscanprovidemostoftheinformationaboutthebehaviorofastructure,

andcanbeagoodapproximationformanyanalyses.Itisalsothebasesofnonlinear

analysisinmostofthecases.

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BarElement

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StiffnessMatrix DirectMethod

ssum ng a e sp acemen u svary ng near ya ong eax so e ar, .e.

Forcein ar

Thebarisactinglikeaspring:

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DegreeofFreedom(DOF)Number

of

components

of

the

displacement

vector

at

a

node.For1Dbarelement:onedofateachnode.

StiffnessMatrix AFormalApproach

Define

two

linear

shape

functions

as

follows

where

Thedisplacementcanbewrittenas

or

Straincanbeexpressesas

whereB istheelementstraindisplacementmatrix

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Stresscanbewrittenas

Thestrainenergystoredinthebarisgivenby

Thework

done

by

the

two

nodal

forces

is

It is stated that: U = W

istheelementstiffnessmatrix

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Theelementstiffnessmatrixofabarelement

Thestrainenergyintheelementcanbewrittenas

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Example1

whichisloadedwithforceP,andconstrainedatthetwoends.Pleaseusetwo1Dbar

elements.

Element1: Element2:

Imagineafrictionlesspinatnode2,whichconnectsthetwoelements.Theglobal

FEequationcanbeconstructedasfollows,

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Loadandboundaryconditions(BC)are

FEequationbecomes:

Stressinelement1is Stressinelement2is

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Notes:

In this case, the calculated stresses in elements 1

structures.It will not help if we further divide element 1 or 2 into

smaller finite elements.

For tapered bars, averaged values of the cross

sectional areas should be used for the elements.

We need to find the displacements first in order to

find the stresses, since we are using the displacement

.

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Example2

Wefirstchecktoseeifornotthecontactofthebarwiththewallontherightwill

occur. o o s,we mag ne ewa on er g sremove an ca cu a e e

displacementattherightend

,contact

occurs.

TheglobalFEequationisfoundtobe

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Theloadandboundaryconditionsare,

The2ndequationgives

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Supportreactions:

The1stequationgives