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.