结构力学英文课件-Chapter5

Chapter5PrincipleofVirtualWorkandDisplacementofStructure

Section1Introduction1、DisplacementofstructurePositionofastructurewillbechangedundertheactionofload,temperature,foundationsettlementetc.(1)、linedisplacementahorizontaldisplacement：

Hbverticaldisplacement：V(2)、angulardisplacement：

(3)、Alldisplacementsdescribedabovearecalled“generalizeddisplacement”,forcescorrespondingto“generalizeddisplacement”are“generalizedforce”.

(4)、Examplesofdisplacements2、AimsatstructuredisplacementLimitthedisplacementofstr.i.edeflectionofmainbeamislimitedwithin(1/1000)L.(2)Solutestaticallyindeterminateproblemslater.Inotherwords,staticallyindeterminateproblemsmustberequireddisplacementofthestructures.3、ASSUMPTIONS(1)、MaterialmustcomplywithHuke’slaw.(2)、Onlysmalldisplacementstakeplaceinthestructure.(3)、Thereisnofrictioninlinkage.

PPBASection2RealworkPThestaticloadPisgraduallyappliedtoastructure,itcannotgiverisetoVibration.P:0P:0Therealwork：1,RealworkdonebyexternalloadAbeamissubjectedtoanexternalconcentratedstaticloadP.Displacement△takeplaceunderthestaticload.

2,Realworkdonebyinternalforces

Asmallsegmentdsisisolatedforanalyzing.a)realworkdonebyNWhere:Eistheelasticitymodulus;Aisthecross-sectionarea.

2,Realworkdonebyinternalforces

Asmallsegmentdsisisolatedforanalyzing.b)realworkdonebyMWhere:Iisthecross-sectionmomentofinertia

2,Realworkdonebyinternalforces

Asmallsegmentdsisisolatedforanalyzing.c)RealworkdonebyshearingforceQWhere:kisacoefficientassociatedwiththeshapeofcross-section.Gismodulusinshear.Rectangularcross-sectionCircularcross-section

ThetotalrealworkdonebyN,Q,MinthesegmentdsisAsmallsegmentdsisisolatedforanalyzing.

ThetotalrealworkdonebyN,Q,MinthesegmentdsisAsmallsegmentdsisisolatedforanalyzing.Forthemember

Forthestructure(allmembers)

Bytheconservationlawofenergy,thetotalexternalrealworkmustbeequaltotheinternalrealwork(strainenergy).Section3.Virtualwork

Abeamiselongatedalengthunderactingof,later,anotherloadisappliedtothebeam.Displacementiscausedby.Obviously,theworkdonebyinthedisplacementTherearetwosubscripts,thefirstsubscriptstandfor,thesecondsubscriptmeansthedisplacementthatiscausedbyiscalledvirtualwork

ThevirtualworkisdefinedastheproductoftheforcePandthecorrespondingdisplacementthatisproducedbyanyreasonotherthantheforceP.ThemagnitudeofvirtualworkisequaltotheshadedrectangularareashowninFigVirtualworkisdifferentfromrealworkinfollowingaspects:(1)Thereiscoefficientinrealworkexpression.Andinvirtualworkformula,thereisno.(2)Valueofreadworkisalwayspositive,virtualworkispositionornegative.ItisdeterminedbydirectionsbetweenPand.ForconditionshowninfollowingFig.,iscausedby,afterP2isappliedtostructure.ItcausesatpointP1andatpointP2.SothetotalworkdonebyexternalloadsP1,P2is:ThetotalworkdonebyinternalforcesisBytheconservationlawofenergy,thetotalexternalworkmustbeequaltothetotalinternalworkBytheconservationlawofenergy,thetotalexternalworkmustbeequaltothetotalinternalworkBecauseofTheformulashowsthatexternalvirtualworkisequaltotheinternalvirtualwork.ItiscalledthePrincipleofVirtualworkTheprincipleofvirtualworkcanbeusednotonlytostaticallydeterminatestructure,butalsostaticallyindeterminatestructure.Becausethetwogrouploadsareindependent,sowecandividetwogrouploadsintotwoindependentbalancestates：Section4Displacementanalysis1,DisplacementequationInordertousetheprincipleofvirtualworktosolutethedisplacement,WesetFig.asanexample.

AbeamABisactedbyload.Howcanwedeterminethedisplacementatthemiddlepointofthebeam?

ThedisplacementcausedbyrealforcePatmiddlepointofthebeamis,theinternaldisplacementscausedbyrealforcePatsmallsegmentdsare:

TheexternalvirtualworkdonebyinthedisplacementcausedbyPis:

Tosolutethisproblem,wesetupanotherforcestate.Atthemiddleofthebeam,weapplyaforcetothebeamTheexternalvirtualworkdonebyinthedisplacementcausedbyPis:Theinternalvirtualworkdonebyinternalforces,,inthedisplacementdsduetoN,M,Qis

Forthebeam

ABAccordingtothePrincipleofvirtualwork,wehave:WhatvaluedoesPkhasforconveniencetocalculatethedisplacement?So,thedisplacementofthebeamcanbecalculatedbyusingformula:Formembersofastructure,wehave:Stepsindetailtosolutethedisplacement:(1)listequationsofN,M,Qcausedbyrealexternalloads.(2)Setupafictitiousstate.Unitloadorisappliedtothepointwherethedisplacementwillbedetermined,listequationsofallmembers’,,causedbyunitload.(3)SubstituteallinternalforcestoequationNote:①payattentiontosignsofN,Q,M.iftheordinatesofandMarethesameside,thesignofandMispositive,ifarenotthesameside,isnegative.②ifsignofispositive,itshowsthatthedirectionofdisplacementisidenticaltothedirectionof.2、Howtoapplyunitload（linedisp.、relativelinedisp.）

Thedirectionofunitloadisidenticaltothedirectioninwhichthedisp.isdesired2、Howtoapplyunitload（angulardisp.、relativeangulardisp

）

Section5Simplificationoftheequation1、Beamandframe（N,Qareomitted）：2、Truss（onlyNinmembers）：3、Forcompositestructure：4、EX.

EX.1Findverticaldisplacement

AVatpointA.EIofallmembersisconstant.Solution：（1）MequationsunderrealloadSegmentAB：SegmentBC：

（2）Setupfictitiousstate,unitloadisappliedtoA.(3)MequationsundertheunitloadinthefictitiousstateSegmentAB：SegmentBC：

（4）SubstituteMequationstodisplacementexpression

EX.2FindverticaldisplacementatjointC

CVintruss.EAisconstantSolution：（1）Findaxial(normal)forcesofalllinksinthetruss,theirvaluesaremarkedinthetruss.

（2）Setupfictitiousstate,unitloadisappliedtojointC,findallforces

underthisstate,marktheminFig.

（3）Substitutetheforces,toformula

EX.3FindverticaldisplacementatpointBinaquarterofring.RadialisR,EIisconstant.Solution：

（1）Isolateanarbitrarysmallsegmentds

fromthering,MequationinthearbitrarycrosssectionIswrittenas:

（2）Setupafictitiousstate,theMcanbeexpressedas:

（3）DisplacementFinddeflectionatmiddlepointCofthebeam.

Solution:(1)Reactions:

（2）PointAisassumedtobeoriginalpoint.Anarbitrarycross-sectionfarfrompointA,xisselected.Thebendingmomentinthecross-sectionis(3)setupafictitiousstateofexternalload.ShowninFig.P=1isexertedonpointCofthebeam.

Thereactions:

（4）Thebendingmomentinarbitrarycross-sectionis:Forthisbeam:Q,Nareomitted.

Theispositive.Thedirectionofdisplacementisdownward.

Findverticaldisp.AtpointCandangulardisp.AthingeAinthefollowingbeam.EIisconstant.PCBAl/2l/2CBAl/2l/2M(1)(2)Problems:P179.9—5(b),9--14Section6GraphMultiplicationThegraphmultiplicationcanbeappliedtodisplacementcalculationprovidedthat:

(1)

Theaxisofmemberisastraightline;(2)TherigidityEI(orGA,EA)isaconstant;(3)Oneofthetwointernalforcefoundationsisalinearfunction.Iftherearemanymembersinastructure,andintegralwillbecalculatedmanytimes.Graphmultiplicationcanbeusedeasily.1、CertificationConclusion:undertheconditionsdescribedabove，displacement

istheproductbetweenareaofadiagramandcentroidordinateoftheotherdiagram,thendividetheEI。3、Notice:

(1)、yomustbeobtainedfromalinerdiagramωjyoMMω1y1(2)、Iflinearfunctiondiagramconsistsofzigzagline,thecalculationmustbedonebydividingthelineintosegmentsω2y2MMω1y1(3)、Ifcrosssectionisnotuniform,calculationmustconductbysegmentsω2y2(4)、If2diagramsareonthesamesideofmember,theproductispositive,or,isnegativeMMωPyoωPyoMMω1y1ω2y2(5)、If2diagramsarebothliner,theareaandordinatecanbeobtainedfromeitherofthediagramsMM(6)、Ifthediagramiscomplex，thediagramcanbedividedinto2ormoreeasierdiagrams,thenusesuperposition.y1y2ω1y3y4ω2abcdlMM(6)、Ifthediagramiscomplex，thediagramcanbedividedinto2ormoreeasierdiagrams,thenusesuperposition.acdlalSUMMARY：(1)yomustbeobtainedfromalinerdiagram(2)Iflinearfunction

diagramconsistsofzigzagline,thecalculationmustbedonebydividingthelineintosegments.(3)Ifcrosssectionisnotuniform,calculationmustconductbysegments.(4)If2diagramsareonthesamesideofmember,theproductispositive,or,isnegative.(5)If2diagramsarebothliner,theareaandordinatecanbeobtainedfromeitherofthediagrams.(6)Ifthediagramiscomplex，thediagramcanbedividedinto2ormoreeasierdiagrams,thenusesuperposition.

Themagnitudeofareasandthelocationoftheircentroidforsomecommondiagramsaregivenasfollowing.

Solution1：（1）DrawM（2）Unitloadisappliedonpointc,drawM（M）（M）

Findverticalandangulardisplacement

CVand

CSolution2：

AnunitmomentisappliedtopointC（M）（M）

P170;Ex.9-6P171;Ex.9-7

Problem:9--19Section7Displacementscausedbyotherfactors1、DisplacementcausedbymanufacturingerrorsThedisplacementcausedbymanufacturingerrorsintrussisWhere:istheerror.

ExInthistruss,membe