工字梁有限元报告.docx

工字梁有限元报告.docx

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工字梁有限元报告

FiniteElementRaport

1.Analyticalsolution

Displacementcreatedbythebendingmoment:

Displacementcreatedbytheshearforce:

∙Simplifiedtheory:

∙Beamtheory:

So,totaldisplacementis

Thenormalstressis：

∙Simplifiedtheory:

∙Beamtheory:

Theshearstressis：

∙Simplifiedtheory:

∙Beamtheory:

2.FiniteElementsolution

1．Themodelingofthisbeamwitha3Dmodel:

The3Dmodelwithahalfofdimension:

Weaddthelimitconditionbyusingthesymmetry.WeobtainforthesymmetryplanXOYTz=0.Theleftextremityisfixed,sotheTx,Ty,Tz=0.

Wecangetthemodelwithlimitcondition:

Thuswecangettheresult.Thedisplacementisshownasfollow.

Wecanseethatthedisplacementofthebeamin3Dischangedlinearly.Attheendofthebeamthedisplacementis1.092503E+01mm.Itisclosetotheanalyticalsolution,andtheerroris2.6%.

ThenormalstressofcomponentX:

Fortheleftsection:

WecanseethatthestressofcomponentXofthebeamin3Dischangedlinearlyintheleftsection.Itreachesthemaximumattheterminalvertexes.Themax

is

1.435663E+02attheseterminalvertexes,andtheerroris9.1%.

TheshearstressofcomponentXY:

Wecanseethattheshearstressofthebeamin3DisnearlyconstantinthemiddlepartofthebeaminthedirectionofX.WechoseseveralnodesinthedirectionYinthemiddlepart.Theprecedentfigureshowsarangeasaquadraticfunctionofyalongtheweb,whichverifiesthebeamtheory.Therangeoftheshearstressisaround8-9Mpa,sowecanmakethesimplifiedmodelwhichhasconstantshearstressalongtheweb.Themaximumshearis9.570804E+00,theerrorwithbeamtheoryis1.8%.

Comparingwiththeanalyticalsolution,wemakeatabletocritiquetheresults.

Displacement

Max（mm）

Normalstressmax（Mpa）

Shearstress

（Mpa）

FiniteElementin3D

1.092503E+01

1.435663E+02

9.570804E+00（max）

Analyticalsolution

11.22

131

9.4（max）

Theresultsof3Dfiniteelementmodelareclosetotheanalyticalsolutionofbeamtheorynomatterinthedisplacementandthestress.Sothe3Dmodelmakesagoodapproximationtotheanalyticalresults.

Weaugmenttwicethesizeoftheelementwiththesame3Dmodel.Wecangettheresultofthenormalstress.

Themaxnormalstressis183Mpa,whichisfarmorethantheprecedentmodelcomparedwiththeanalyticalsolution,butthedelayofthecalculationismuchlessthantheprecedentmodel.

Soitisnotawisechoicetouse3Dmodel,becauseweshouldtrytofindabalanceoftheaccuracyandthedelay.

2．Themodelingofthisbeamwitha2Dmodel:

Weconstructthemodelwithmembraneandbeamelementsby5surfaces.Amongthesefivesurface,thewebsurfaceismodeledbyamembrane,andtheothersaremodeledbyshells.

Fromthemodelwecangettheresults

Wecanseethatthedisplacementofthebeamin3Dischangedlinearly.Attheendofthebeamthedisplacementis1.0702785E+01mm.Itisclosetotheanalyticalsolution,andtheerroris4.6%.

Thenormalstressdoesn’treachethemaximumattheterminalvertexes,butattheleftsection.Themax

is

7.5603534,andtheerrorcomparedwithbeamtheoryis32.6%

Wecanseethattheshearstressofthebeamin2DmodelisnearlyconstantinthemiddlepartofthebeaminthedirectionofX.WechoseseveralnodesinthedirectionYinthemiddlepart.Theprecedentfigureshowsarangeasaquadraticfunctionofyalongtheweb.Themaxis-9.624660E+00mm,andtheerroris2.3%.

3.Themodelingofthisbeamwithmembraneandbeamelements:

Weconstructthemodelwithmembraneandbeamelements

Display:

1Dline

Display:

3DFullSpan

Fromthemodelwecangettheresults

Wecanseethatthedisplacementofthebeaminmodelofmembraneandbeamischangedlinearly.Attheendofthebeamthedisplacementis9.814795E+00mm.Theerrorwiththeanalyticalsolutionreaches12.5%.

ThenormalstressofcomponentX:

WecanseethatthenormalstressofcomponentXofthebeaminthismodelisalsochangedlinearlyintheleftsection.Itreachesthemaximumattheterminalvertexes.Themax

is

1.093545E+02attheseterminalvertexes.Theerrorreaches16.5%.

WechoseseveralnodesinthedirectionYinthemiddlepartofthemodel.Theprecedentfigureshowsarangeasaquadraticfunctionofyalongtheweb.Themaxis-6.271123E+00.Theerrorreaches33.3%.

4.Themodelingofthisbeamwitha1Dmodel:

Weconstructthemodelin1Dmodel

Thesectionofbeam:

Display:

3DFullScan

Fromthemodelwecangettheresults

Wecanseethatthedisplacementofthebeamin1Dischangedlinearly.Attheendofthebeamthedisplacementis1.1136911E+01mm.Theerrorwiththeanalyticalsolutionreaches0.7%.

Thenormalstress:

reachesthemaximumattheterminalvertexes.Themax

is

1.31282E+02attheseterminalvertexes,C/D/E/F.Theerrorreaches0.2%.

5.Themodelingofthisbeamwithmembraneandrodelements:

Weconstructthemodelwithmembraneandrodelements.Thesectionofthetworodsis960mm2.Themodelisshownasfollow.

Wecanseethatthedisplacementofthebeamin1Disalsochangedlinearly.Attheendofthebeamthedisplacementis1.0735427E+01mm.Theerrorwiththeanalyticalsolutionreaches4.3%.

WecanseethatthenormalstressofcomponentXofthebeaminthismodelisalsochangedlinearlyintheleftsection.Itreachesthemaximumattheterminalvertexes.Themax

is

1.190676E+02mmattheseterminalvertexes.Theerrorreaches6.1%.

WecanseethattheshearstressofthebeamwithmembraneandrodsisnearlyconstantinthemiddlepartofthebeaminthedirectionofX.WechoseseveralnodesinthedirectionYinthemiddlepart.Theprecedentfigureshowsarangeasaquadraticfunctionofyalongtheweb.Themaxis-9.084330E+00mm,theerroris3.4%.

3.Conclusion:

Inthefollowingtable,welistthecomparisonofdifferentmodelswiththeanalyticalsolutionofbeamtheory.

Differencewith

AnalyticalsolutionofBeamtheory

Errorofthedisplacement

（%）

Errorofthenormalstress

（%）

Errorofthe

shearstress

（%）

Calculation

Time

Simplified

theory

0.4

12.9

-

-

3D

2.6/√

9.1/√

1.8/√

×

2D

4.6/√

32/×

2.3/√

√

1D

0.7/√

0.2/√

-

√

Membraneandbeams

12.5/√

16.5/×

33.3/×

√

Membraneandrods

4.3/√

6.1/√

3.4/√

√

Fromthetablewecanseethe3Dmodelisgoodenoughfortheaccuracy,butthecalculationtimeofthemodelislong.Themodelwithmembraneandbeamhasbigerrorinthenormalstress.The2Dmodelislessaccurateinthenormalstress,butaccurateindisplacementandshearstress.The1Dmodelisthemostaccurate,butthemodelisnotsuitableforeverysituation.Themodelwithmembraneandrodswhichisusuallyusedforanaeronauticalstructureisaccurateforalltheparameter.