206 CHAPTER 6 Matrix Methods
Fig.6.14
Quadrilateral element subjected to nodal in-plane forces and displacements.
Asinthecaseofthetriangularelement,weselectadisplacementfunctionwhichsatisfiesthetotalof
eightdegreesoffreedomofthenodesoftheelement;again,thisdisplacementfunctionwillbeinthe
formofapolynomialwithamaximumofeightcoefficients.Thus,
u(x,y)=α 1 +α 2 x+α 3 y+α 4 xy
v(x,y)=α 5 +α 6 x+α 7 y+α 8 xy}
(6.96)
Theconstantterms,α 1 andα 5 ,arerequired,asbefore,torepresentthein-planerigidbodymotionofthe
element,whilethetwopairsoflineartermsenablestatesofconstantstraintoberepresentedthroughout
theelement.Further,theinclusionofthexytermsresultsinbothu(x,y)andv(x,y)displacementshaving
thesamealgebraicformsothattheelementbehavesinexactlythesamewayinthexdirectionasitdoes
intheydirection.
WritingEqs.(6.96)inmatrixformgives
{
u(x,y)
v(x,y)}
=
[
1 xyxy 0000
0000 1xyxy]
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α 1
α 2
α 3
α 4
α 5
α 6
α 7
α 8