桁架单元例子MATLAB 1Word文件下载.docx
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桁架单元例子MATLAB 1Word文件下载.docx
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Thedeflectionsandslopesatpointsinbetweennodescanbeinterpolatedusingtheshapefunctions.
Thedeflectionforthe1stelementofthebeamcanbewrittenintermsoftheshapefunctionsas,
Thedeflectionforthe2ndelementofthebeamcanbewrittenintermsoftheshapefunctionsas,
where,theshapefunctionsare,
and,theslopefor1stelementofabeamis,
Theslopefor2ndelementofabeamis,
Deflectionandslopeatx=0.5m:
Thepointx=0.5misinthe1stelement.
From(2.1),
From(3.1),
Deflectionandslopeatx=1m:
Thedeflectionandslopeforx=1miscalculatedaboveas,
Deflectionandslopeatx=1.5m:
Thepointx=1.5misinthe2ndelement.
From(2.2),
From(3.2),
(a)Deflectionandslopes
(b)Thebendingmomentandshearforcediagramsfortheentirebeam:
Thebendingmomentcanbefoundfrom,
Forthe1stelement:
++]
Forthe2ndelement:
Wehavetoexpressthesemomentsintermsofxtoplottheresults.Forthefirstelement,
Forthesecondelement,
From(6)and(8)weget,;
where,
From(7)and(9)weget,
where,
Theresultsareplottedinthefollowingfigure.
(b)BendingMomentdiagram
Theresultsareplottedinthefollowingfigure.
Theshearforcecanbefoundfrom:
Forelement1,
Forelement2:
(b)Shearforcediagram
(c)Thesupportreactions:
Thesupportreactionsarecalculatedbyputtingthevaluesofnodalparametersinthematrix
equation
(1)andwesolveforthefollowingmatrix,
(c)Supportreactions
(d)Usingequation
(2)andthenodalparameters
forelement1,weget,
Forelement2,weget,
Usingequation(8)and(9)intoequation(12)and(13)respectively,theshearforceexpressionsbecome,
Forelement1:
;
Forelement2
where
Thedeflectedshapewasplottedbelow.
(d)Deflectedshape
Problem2:
SolvethefollowingframestructureusingafiniteelementprogramintheAppendix.Theframeisunderauniformlydistributedloadofq=1000N/mandhasacircularcross-sectionwithradiusr=0.1m.Formaterialproperty,Young’smodulusE=207GPa.Plotthedeformedgeometrywithanappropriatemagnificationfactor,anddrawabendingmomentandshearforcediagrams.
Thefiniteelementprogramisgivenbelowtosolvethe5elementprogram.ThecodeiswrittenbyusingMATLABtoolbox.
%Definetheelements
Edof=[1123456;
2456789;
3789101112;
4101112131415;
5456101112];
K=zeros(15);
f=zeros(15,1);
%Definematerialproperties
E=207E9;
A=0.0314;
I=7.85E-5;
ep=[EAI];
ex=[00;
00;
01;
12;
01];
ey=[21;
10;
10];
eq=[0-1000];
ke1=beam2e(ex(1,:
),ey(1,:
),ep);
ke2=beam2e(ex(2,:
),ey(2,:
[ke3,fe3]=beam2e(ex(3,:
),ey(3,:
),ep,eq);
[ke4,fe4]=beam2e(ex(4,:
),ey(4,:
ke5=beam2e(ex(5,:
),ey(5,:
K=assem(Edof(1,:
),K,ke1);
K=assem(Edof(2,:
),K,ke2);
[Kf]=assem(Edof(3,:
),K,ke3,f,fe3);
[Kf]=assem(Edof(4,:
),K,ke4,f,fe4);
K=assem(Edof(5,:
),K,ke5);
bc=[10;
70;
80];
a=solveq(K,f,bc);
Ed=extract(Edof,a);
[es1edi1eci1]=beam2s(ex(1,:
),ep,Ed(1,:
),[00],20);
[es2edi2eci2]=beam2s(ex(2,:
),ep,Ed(2,:
[es3edi3eci3]=beam2s(ex(3,:
),ep,Ed(3,:
),eq,20);
[es4edi4eci4]=beam2s(ex(4,:
),ep,Ed(4,:
[es5edi5eci5]=beam2s(ex(5,:
),ep,Ed(5,:
sfac=scalfact2(ex(5,:
),es5(:
3),.2);
plotpar=[24];
figure
(1);
eldraw2(ex,ey,[110]);
figure
(2);
eldisp2(ex,ey,Ed,[110],3000);
figure(3);
eldia2(ex(1,:
),es1(:
1),plotpar);
eldia2(ex(2,:
),es2(:
eldia2(ex(3,:
),es3(:
eldia2(ex(4,:
),es4(:
eldia2(ex(5,:
axis([-0.52.5-0.52.5]);
figure(4);
3),plotpar,sfac);
figure(5);
2),plotpar);
Thedeformedgeometry,bendingmomentandshearforcediagramisplottedbelow:
Figure1:
Deformedgeometry
Figure2:
Bendingmomentdiagram
Figure3:
Shearforcediagram
Problem3:
Acantileveredframe(Element1)andauniaxialbar(Element2)arejoinedatNode2usingaboltedjointasshowninthefigure.Assumethereisnofrictionatthejoint.ThetemperatureofElement2israisedby2000Cabovethereferencetemperature.BothofelementshavethesamelengthL=1m,Young’smodulusE=1011Pa,cross-sectionalareaA=10-4m2.TheframehasmomentofinertiaI=10-9m4,whilethebarhasthecoefficientofthermalexpansion.Usingfiniteelementmethod,determines(a)displacementsandrotationatNode2;
(b)theaxialforceinbothelements;
(c)theshearforcesandbendingmomentsinElement1atNodes1and2;
and(d)drawthefree-body-diagramofNode2andshowtheforceequilibriumissatisfied.Hint:
TreatElement1asaplaneframeelement.
Theplaneframeisdirectlyorientedalongxaxis,i.e.,So,weneednottransformthelocalcoordinatetoglobalcoordinateandcanwritedirectlytheelementstiffnessmatrixequationforelement1as:
Puttingthesevaluesin
(1),weget,
Nowweapplyboundaryconditions,atnodeno.1,ie,andthematrixequationbecomes,
Strikingouttherowsandcolumnsofzeroelements,theabovematrixequationbecomes,
Theelementstiffnessmatrixinglobalcoordinates,
Here,
Theelementequationforelement2inglobalcoordinates,
Where,
Withthesevalues(3)becomes,
Nowweapplytheboundaryconditions,ieandtheabovematrixequationbecomes,
Strikingouttherowsandcolumnsofzeroelements,weget,
Assembling
(2)and(4),theglobalmatrixisformedas,
Puttingthevaluesoftheforcesandcouples,
Solvingthesematrixequations,
(a)Displacementandrotationatnode2:
(b)Axialforceinelement1:
Axialforceinelement2:
(c)Shearforceandbendingmomentforelement1atnode1and2:
Here,
Wepluginthevaluesandgetthefollowingmatrix,
(1.6962N
=
Wecanassumethatasitisverysmall,becausethereisnoexternallyappliedcoupleatnode2andthehingedjointofnode2cannotgiverisetoabendingmoment.
1.6962
(d)Freebodydiagramatnode2andforceequilibrium:
Fromforceequilibriumlaw,
Asandisverysmallsowecansetthemtozeroandconsiderpoint2isinequilibrium
(d)Forceequilibriumissatisfied
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