英汉双语弹性力学2.ppt

1,Chapter2TheBasicTheoryofthePlaneProblem,2,第二章平面问题的基本理论,3,TheBasicTheoryofthePlaneProblem,Chapter2TheBasictheoryofthePlaneProblem,2-11Stressfunction.Inversesolutionmethodandsemi-inversemethod,2-1Planestressproblemandplanestrainproblem,2-2Differentialequationofequilibrium,2-3Thestressontheincline.Principalstress,2-4Geometricalequation.Thedisplacementoftherigidbody,2-5Physicalequation,2-6Boundaryconditions,2-7Saint-Venantsprinciple,2-8Solvingtheplaneproblemaccordingtothedisplacement,2-9Solvingtheplaneproblemaccordingtothestress.Compatibleequation,2-10Thesimplificationunderthecircumstancesofordinaryphysicalforce,ExerciseLesson,4,平面问题的基本理论,第二章平面问题的基本理论,2-11应力函数逆解法与半逆解法,2-1平面应力问题与平面应变问题,2-2平衡微分方程,2-3斜面上的应力主应力,2-4几何方程刚体位移,2-5物理方程,2-6边界条件,2-7圣维南原理,2-8按位移求解平面问题,2-9按应力求解平面问题。相容方程,2-10常体力情况下的简化,习题课,5,1.Planestressproblem,2-1Planestressproblemandplanestrainproblem,Inactualproblem,itisstrictlysayingthatanyelasticbodywhoseexternalforceforsufferingisaspacesystemofforcesisgenerallythespaceobject.However,whenboththeshapeandforcecircumstanceoftheelasticbodyforinvestigatinghavetheirowncertaincharacteristics.Aslongastheabstractionofthemechanicsishandledtogetherwithappropriatesimplification,itcanbeconcludedastheelasticityplaneproblem.Theplaneproblemisdividedintotheplanestressproblemandplanestrainproblem.,Equalthicknesslamellabearsthesurfaceforcethatparallelswithplatefaceanddontchangealongthethickness.Atthesametime,sodoesthevolumetricforce.z=0zx=0zy=0,Fig.21,TheBasicTheoryofthePlaneProblem,6,一、平面应力问题,2-1平面应力问题与平面应变问题,在实际问题中，任何一个弹性体严格地说都是空间物体，它所受的外力一般都是空间力系。但是，当所考察的弹性体的形状和受力情况具有一定特点时，只要经过适当的简化和力学的抽象处理，就可以归结为弹性力学平面问题。平面问题分为平面应力问题和平面应变问题。,等厚度薄板，板边承受平行于板面并且不沿厚度变化的面力，同时体力也平行于板面并且不沿厚度变化。z=0zx=0zy=0,图21,平面问题的基本理论,7,TheBasicTheoryofthePlaneProblem,Characteristics：,1)Thedimensionoflengthandbreadthisfarlargerthanthatofthickness.,2)Theforcealongtheplatefaceforsufferingisthefaceforceinparallelwithplateface,andalongthethicknesseven,thevolumetricforceisinparallelwithplateforceanddoesntchangealongthethickness,andhasnoexternalforcefunctiononthesurfacefrontandbackoftheflatpanel.,Attention:Planestressproblemz=0,but,thisiscontrarytoplanestrainproblem.,8,平面问题的基本理论,特点：,1)长、宽尺寸远大于厚度,2)沿板边受有平行板面的面力，且沿厚度均布，体力,平行于板面且不沿厚度变化，在平板的前后表面上,无外力作用。,9,2.Planestrainproblem,Verylongcolumnbearsthefaceforceinparallelwithplatefaceanddoesntchangealongthelengthonthecolumnface,atthesametime,sodoesthevolumetricforce.z=0zx=0zy=0,x,Fig.22,TheBasicTheoryofthePlaneProblem,Forexample:dam,circularcylinderpipingbytheinternalairpressureandlonglevellanewayetc.,Attention:Planestrainproblemz=0，but,thisiscontrarytoplanestressproblem.,10,二、平面应变问题,很长的柱体，在柱面上承受平行于横截面并且不沿长度变化的面力，同时体力也平行于横截面并且不沿长度变化。z=0zx=0zy=0,x,图22,平面问题的基本理论,如：水坝、受内压的圆柱管道和长水平巷道等。,11,2-2DifferentialEquationofEquilibrium,Whetherplanestressproblemorplanestrainproblem,istheresearchprobleminplanexy,allthephysicsquantityhasnothingtodowithz.,Discussbelowthecorrelationbetweenanypointstressandvolumetricforcewhentheobjectisplacedinthestateofequilibrium,andleadanequilibriumdifferentialequationfromhere.FromthelamellashowninFig.2-1,wetakeoutasmallandpositiveparallelepipedPABC,andtakeforanunitlengthinthedirectionaldimensioninz.,Establishingthefunctionofthepositivestressforceinanunitontheleftsideis,thecoordinateontherightsidexgetstheincrement,thepositivestressonthefaceis,spreadingtheformulaabovewillbeTaylorsseries:,TheBasicTheoryofthePlaneProblem,12,2-2平衡微分方程,平面问题的基本理论,13,Afteromittingsmallquantityofthetworankandabovethetworank,canget,atthesametime,aregetthestateofstressfromthedrawingshow.,Whileconsideringthevolumetricforcetotheplanestressstate,stillprovemutualandequaltheoryofshearingstrength.RegardthecenterDandstraightlineinparallelwiththeshaftofzasthemomentshaft,listtheequilibriumequationofthemomentshaft:,TheBasicTheoryofthePlaneProblem,14,略去二阶及二阶以上的微量后便得同样、都一样处理，得到图示应力状态。,对平面应力状态考虑体力时，仍可证明剪应力互等定理。以通过中心D并平行于z轴的直线为矩轴，列出力矩的平衡方程：,将上式的两边除以得到：,平面问题的基本理论,15,Deducetheequilibriumdifferentialequationoftheplanestressproblembelow,listtheequilibriumequationtotheunit:,TheBasicTheoryofthePlaneProblem,16,下面推导平面应力问题的平衡微分方程，对单元体列平衡方程：,平面问题的基本理论,17,Sortingthemgets:,TheBasicTheoryofthePlaneProblem,18,整理得:,平面问题的基本理论,19,2-3ThestressontheInclinedPlane.Principalstress,1.ThestressontheinclinedplaneHavingknownthestressweightofanypointPinsidetheelasticbody,wetrytogetthestresswhichpassthepointPonthearbitrarilyinclinedcrosssection.FromneighborhoodofpointPtakingaplaneAB,whichisinparallelwiththeinclinedplaneabove,anddrawsasmallsetsquareorthreecolumnPABontwoplaneswhichpasspointPandhaveperpendicularityintheshaftofxandy.WhentheplaneABapproachespointPinfinitely,themeanstressontheplaneABwillbecomethestressontheinclinedplaneabove.,EstablishthelengthofthefaceABintheplanexyisdS,Nistheexteriornormaldirection,anditsdirectioncosineis:,TheBasicTheoryofthePlaneProblem,20,2-3斜面上的应力、主应力,一、斜面上的应力已知弹性体内任一点P处的应力分量，求经过该点任意斜截面上的应力。为此在P点附近取一个平面AB，它平行于上述斜面，并与经过P点而垂直于x轴和y轴的两个平面划出一个微小的三角板或三棱柱PAB。当平面AB与P点无限接近时，平面AB上的应力就成为上述斜面上的应力。,设AB面在xy平面内的长度为dS，厚度为一个单位长度，N为该面的外法线方向，其方向余弦为：,平面问题的基本理论,21,TheprojectionofthewholestressontheinclinedplaneABisXNandYNrespectivelyalongwiththeshaftofxandy.FromthePABequilibriumtermcanget:,Divideandget:,Samefromandget:,ThepositivestressontheinclinedplaneAB,fromtheprojectioncanget:,TheshearingstrengthontheinclinedplaneAB,fromtheprojectioncanget:,TheBasicTheoryofthePlaneProblem,22,斜面AB上全应力沿x轴及y轴的投影分别为XN和YN。由PAB的平衡条件可得：,除以即得：,同样由得出：,斜面AB上的正应力，由投影可得：,斜面AB上的剪应力，由投影可得：,平面问题的基本理论,23,3.Principalstress,IftheshearingstressofsomeinclinedplanethroughpointPisequaltozero,thenthepositivestressofthatinclinedplanecallsaprincipalstressofpointP,butthatinclinedplanecallsthemainplaneofthestressatpointP,andthenormaldirectionofthatinclinedplanecallsthemaindirectionofthestressatpointP.,1.Thesizeoftheprincipalstress,2.Thedirectionoftheprincipalstressisintheperpendicularitywithforeachother.,TheBasicTheoryofthePlaneProblem,24,二、主应力,如果经过P点的某一斜面上的切应力等于零，则该斜面上的正应力称为P点的一个主应力，而该斜面称为P点的一个应力主面，该斜面的法线方向称为P点的一个应力主向。,1.主应力的大小,2.主应力的方向与互相垂直。,平面问题的基本理论,25,2-4GeometricalEquation.TheDisplacementoftheRigidBody,Inplaneproblem,everypointinsidetheelasticbodycanproducethearbitrarilydirectionaldisplacement.TakeanunitPABthroughanypointPinsidetheelasticbody,suchasFig.2-5show.Aftertheelasticbodysuffersforce,thepointP,A,BmovetothepointP、A、Brespectively.,Fig.25,一、ThepositivestrainatpointP,Herebecauseofsmalldeformation,PAforcausingstretchandshrinkfromtheydirectiondisplacementvisthesmallquantityofahighrankandthissmallquantitymaybeomitted.,TheBasicTheoryofthePlaneProblem,26,2-4几何方程、刚体位移,在平面问题中，弹性体中各点都可能产生任意方向的位移。通过弹性体内的任一点P，取一单元体PAB，如图2-5所示。弹性体受力以后P、A、B三点分别移动到P、A、B。,图25,一、P点的正应变,在这里由于小变形，由y方向位移v所引起的PA的伸缩是高一阶的微量，略去不计。,平面问题的基本理论,27,Thesamecanget:,2.ShearingstrainatpointP,ThecornerofthelinesegmentPA:,ThesamecangetthecornerofthelinesegmentPB:,Thus,TheBasicTheoryofthePlaneProblem,28,同理可求得：,二、P点的切应变,线段PA的转角：,同理可得线段PB的转角：,所以,平面问题的基本理论,29,Thereforegetthegeometricalequationoftheplaneproblem,Fromthegeometricalequationabove,whenthedisplacementweightoftheobjectiscompletelycertain,thedeformationweightiscompletelycertain,uniqueweightcannotbemadesurethoroughly.,TheBasicTheoryofthePlaneProblem,30,因此得到平面问题的几何方程：,由几何方程可见，当物体的位移分量完全确定时，形变分量即可完全确定。反之，当形变分量完全确定时，位移分量却不能完全确定。,平面问题的基本理论,31,2-5ThePhysicalEquation,Intheisotropyofthecompleteelasticity,therelationbetweenthedeformationweightandthestressweightisestablishedaccordingtotheHookeslawasfollows:,TheBasicTheoryofthePlaneProblem,32,2-5物理方程,在完全弹性的各向同性体内，形变分量与应力分量之间的关系根据虎克定律建立如下：,平面问题的基本理论,33,Insidetheformula,theEisamodulusofelasticity;theGisastiffnessmodulus;theuisapoissonratio.Therelationofthreeonesabove:,1.Thephysicsequationoftheplanestressproblem,Andhave:,theBasicTheoryofthePlaneProblem,34,式中，E为弹性模量；G为刚度模量；为泊松比。三者的关系：,一、平面应力问题的物理方程,且有：,平面问题的基本理论,35,2.Thephysicsequationoftheplanestrainproblem,3.Thetransformationrelationoftherelationtypebetweenthestressstrainandtheplanestrain.,Therelationtypeoftheplanestress:,TheBasicTheoryofthePlaneProblem,36,二、平面应变问题的物理方程,三、平面应力的应力应变关系式与平面应变的关系式之间的变换关系,将平面应力中的关系式：,平面问题的基本理论,37,Forchange,Cangettherelationtypeintheplanestrain:,Becauseofthesimilarityofthiskind,whilesolvingplanestrainproblem,thecorrespondingequationoftheplaneproblemandtheelasticconstantintheanswercanbeexchangedasabove,cangetthesolutionofthehomologousplanestrainproblem.,TheBasicTheoryofthePlaneProblem,38,作代换,就可得到平面应变中的关系式：,由于这种相似性，在解平面应变问题时，可把对应的平面应力问题的方程和解答中的弹性常数进行上述代换，就可得到相应的平面应变问题的解。,平面问题的基本理论,39,2-6BoundaryConditions,Whentheobjectisplacedinthestateofequilibrium,itsinternalstateofstressatallpointshouldsatisfytheequilibriumdifferentialequationandalsosatisfytheboundarytermontheboundary.Accordingtothedifferenceoftheboundarycondition,theelasticityproblemisdividedintothedisplacementboundaryproblem,stressboundaryproblemandmixedboundaryproblem.,1.DisplacementBoundaryTerm,Whenthedisplacementhasbeenknownontheboundary,thedisplacementofthepointontheobjectboundaryandtheequaltermofthefixeddisplacementshouldbeestablished.Forexample,ifmakingtheboundaryofthefixeddisplacementis,andhave(onthe):,Amongthem,andmeansthedisplacementweightontheboundary,however,andisthecoordinatefunctionwehaveknowtheboundary.,TheBasicTheoryofthePlaneProblem,40,2-6边界条件,当物体处于平衡状态时,其内部各点的应力状态应满足平衡微分方程；在边界上应满足边界条件。按照边界条件的不同，弹性力学问题分为位移边界问题、应力边界问题和混合边界问题。,一、位移边界条件,平面问题的基本理论,41,2.Stressboundaryterm,Whentheboundaryoftheobjectisgiventosurfaceforce,thenthestressoftheobjectontheboundaryshouldsatisfytheequilibriumtermofforceswiththeequilibriumofthesurfaceforce.,Amongthem,andarethesurfaceforceweightsand,arethestressweightsontheboundary.,Whentheboundaryfaceisinperpendicularityinshaftx,stressboundarytermcanbechangedbrieflyinto:,Whentheboundaryfaceisinperpendicularityinshafty,stressboundarytermcanbechangedbrieflyinto:,TheBasicTheoryofthePlaneProblem,42,二、应力边界条件,当物体的边界上给定面力时，则物体边界上的应力应满足与面力相平衡的力的平衡条件。,其中和为面力分量，、为边界上的应力分量。,当边界面垂直于轴时，应力边界条件简化为：,当边界面垂直于轴时，应力边界条件简化为：,平面问题的基本理论,43,3.Mixedboundarycondition,1.Thedisplacementhasbeenknownonapartofboundariesoftheobject,theresultofwhichhavethedisplacementboundaryterm,theboundariesofotherpartshavethesurfaceforcewehaveknow.Andthenthereshouldbestressboundarytermanddisplacementboundarytermrespectivelyontwopartsoftheboundaries.Theleftsurfaceofthecantilevercontainsdisplacementboundaryterm,suchasshowninFig.2-6.,Topandbottomsurfacecontainsstressboundaryterm:,Therightsurfacecontainsstressboundaryterm:,Fig.2-6,TheBasicTheoryofthePlaneProblem,44,三、混合边界条件,1.物体的一部分边界上具有已知位移，因而具有位移边界条件，另一部分边界上则具有已知面力。则两部分边界上分别有应力边界条件和位移边界条件。如图2-6，悬臂梁左端面有位移边界条件：,上下面有应力边界条件：,右端面有应力边界条件：,图2-6,平面问题的基本理论,45,2.Onthesameboundary,therearenotonlystressboundarytermbutdisplacementboundaryterm.Couplersustainstheboundaryterm,suchasshowninFig.2-7.,ThealveolusboundarytermshowninFig.2-8.,Fig.2-7,TheBasicTheoryofthePlaneProblem,46,2.在同一边界上，既有应力边界条件又有位移边界条件。如图2-7连杆支撑边界条件：,如图2-8齿槽边界条件：,图2-7,平面问题的基本理论,47,2-7Saint-VenantPrinciple,1.Saint-VenantsPrinciple,Iftransformingasmallpartofthesurfaceforceontheboundaryintothesurfaceforcethathasequaleffectbutdifferentdistribution(Themainvectorisequal,soisthemainquadraturetothesamepointaswell),andthenthedistributionofthestressforcenearbywillhaveprominentchanges,buttheinfluencefromthedistantplacecannotbeaccounted.,2.GiveExamples,Establishingthecomponentofthecolumnforms,thecentroidofareaincrosssectionsofbothendssuffersthetensibleforcewhichisequalinsizebutcontraryindirection,suchasshowninFig.2-9a.Iftransforminganorbothendsoftensileforceintotheforceatthesameeffectasthestaticforce,suchasshowninFig.2-9borFig.2-9c,thedistributionofstressforcedrawnonlybybrokenlinehasprominentchanges,whereas,theinfluenceoftherestpartscannotbeaccounted.Ifchangingbothendsoftensileforceintothatofuniformdistributionagain,thegatheringdegreeisequaltoP/AandamongthemAisthecross-sectionareaofthecomponent,suchasshowninFig.2-9d,thereisstillthestressclosetobothendsunderthenoticeableinfluence.,TheBasicTheoryofthePlaneProblem,48,2-7圣维南原理,一、圣维南原理,如果把物体的一小部分边界上的面力，变换为分布不同但静力等效的面力（主矢量相同，对于同一点的主矩也相同），那么，近处的应力分布将有显著的改变，但是远处所受的影响可以不计。,二、举例,设有柱形构件，在两端截面的形心受到大小相等而方向相反的拉力，如图2-9a。如果把一端或两端的拉力变换为静力等效的力，如图2-9b或2-9c，只有虚线划出的部分的应力分布有显著的改变，而其余部分所受的影响是可以不计的。如果再将两端的拉力变换为均匀分布的拉力，集度等于，其中为构件的横截面面积，如图2-9d，仍然只有靠近两端部分的应力受到显著的影响。,平面问题的基本理论,49,Underthefourkindsofcircumstancesabove,partsofdistributionofstressforcedistantfrombothendshavenomarkeddifference.,Attention:,TheapplicationoftheSaint-VenantsprincipleisbynomeansseparatedfromthetermofEqualEffectofStaticForce.,TheBasicTheoryofthePlaneProblem,50,在上述四种情况下，离开两端较远的部分的应力分布，并没有显著的差别。,注意：,应用圣维南原理，绝不能离开“静力等效”的条件。,平面问题的基本理论,51,2-8SolvingthePlaneProblemaccordingtothedisplacement,Therearethreekindsofbasicmethodstosolvetheprobleminelasticity:thesolutiontotheproblemaccordingtodisplacement,stressforceandadmixture.,Whilesolvingproblemsusingdisplacementmethod,weregarddisplacementweightasthebasicfunctionunknown.Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofthedisplacementweight,thengetthedeformationweightusinggeometricalequation,therefore,getthestressweightwiththephysicsequation.,1.PlaneStressProblem,Inplanestressproblem,thephysicsequationis:,TheBasicTheoryofthePlaneProblem,52,2-8按位移求解平面问题,在弹性力学里求解问题，有三种基本方法：按位移求解、按应力求解和混合求解。,按位移求解时，以位移分量为基本未知函数，由一些只包含位移分量的微分方程和边界条件求出位移分量以后，再用几何方程求出形变分量，从而用物理方程求出应力分量。,一、平面应力问题,在平面应力问题中，物理方程为：,平面问题的基本理论,53,Fromthreeformulasabovementionedtosolvethestressweight,canget:withthesubstitutionofgeometricalequation,wecangettheelasticityequation:,Againequilibriumdifferentialequationwithsubstitutioninformula(a),simplificationhereafter,canget:,（a),Thisistheequilibriumdifferentialequationtomeanwiththedisplacement,ie,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weadoptabasicdifferentialequationforneeds.,（1）,TheBasicTheoryofthePlaneProblem,54,由上列三式求解应力分量，得：,将几何方程代入，得弹性方程：,再将式（a）代入平衡微分方程，简化以后，即得：,（a),这是用位移表示的平衡微分方程，也就是按位移求解平面应力问题时所需用的基本微分方程。,（1）,平面问题的基本理论,55,Thestressboundarytermwithsubstitutioninformula(a),simplificationhereafter,canget:,Thisisthestressforceboundarytomeanwiththedisplacement,ie,weadopttheboundarytermofthestressforcewhensolvingtheplanestressproblemaccordingtodisplacementmethod.,（2）,Sumup,whensolvingtheplanestressproblemaccordingtodisplacementmethod,weshouldmakethedisplacementweightsatisfydifferentialequation(1)andcombinetosatisfydisplacementboundarytermorstressboundarytermorstressboundaryterm(2)ontheboundary.Aftergettingdisplacementweight,wecangetthedeformationweightwithgeometricalequationandthengetthestressforceweightwiththephysicsequation.,2.Planestrainproblem,Makethesubstitutionbetweenandineachequationoftheplanestrainproblem:,TheBasicTheoryofthePlaneProblem,56,将（a）式代入应力边界条件，简化以后，得：,这是用位移表示的应力边界条件，也就是按位移求解平面应力问题时所用的应力边界条件。,（2）,总结起来，按位移求解平面应力问题时，要使得位移分量满足微分方程（1），并在边界上满足位移边界条件或应力边界条件（2）。求出位移分量以后，用几何方程求出形变分量，再用物理方程求出应力分量。,二、平面应变问题,只须将平面应力问题的各个方程中和作代换：,平面问题的基本理论,57,2-9SolvingthePlaneProblemAccordingtotheStressForce.CompatibleEquantion,Whilesolvingtheplaneproblemaccordingtothedisplacement,wemustcombinetwopartialdifferentialequationofthesecondrankstosolvetheproblem,thisisverydifficultonthemathematics.Butwhilesolvingtheplaneproblemaccordingtothestressforce,wecanavoidthisdifficultyandsowhatweadoptmoreistogetthesolutionaccordingtothestressforce.,Whilegettingthesolutionaccordingtothestressforce,weregardstressweightasthebasicfunctionunknown.Aftergettingdisplacementweightfromonlyincludingthedifferentialequationandboundarytermofdisplacementweight,thengetthedeformationweightusingphysicsequation,therefore,getthedisplacementweightwithgeometricalequation.,CompatibleEquation,Fromgeometricalequationoftheplaneproblem:,TheBasicTheoryofthePlaneProblem,58,2-9按应力求解平面问题。相容方程,按位移求解平面问题时，必须求解联立的两个二阶偏微分方程，这在数学上是相当困难的。而按应力求解弹性力学平面问题，则避免了这个困难，故更多采用的是按应力求解。,按应力求解时，以应力分量为基本未知函数，由一些只包含应力分量的微分方程和边界条件求出应力分量以后，再用物理方程求出形变分量，从而用几何方程求出位移分量。,相容方程,由平面问题的几何方程：,平面问题的基本理论,59,Canget:,ie,Thisrelationtypecallsthedeformationmoderatesequationorcompatibleequation.,1.Compatibleequationinplanestressforce,2.Compatibleequationinplanestrainforce,TheBasicTheoryofthePlaneProblem,60,可得：,即：,这个关系式称为形变协调方程或相容方程。,（一）平面应力问题的相容方程,（二）平面应变问题的相容方程,平面问题的基本理论,61,Whilesolvingtheplaneproblemaccordingtothestressforce,thestressweightshouldnotonlysatisfyboththeequilibriumdifferentialequationandcompatibleequation,butsatisfythestressboundarytermontheboundarywhetherisaplanestressproblemorplanestrainproblem.,TheBasicTheoryofthePlaneProblem,62,按应力求解平面问题时，无论是平面应力问题还是平面应变问题，应力分量除了满足平衡微分方程和相容方程外，在边界上还应当满足应力边界条件。,平面问题的基本理论,63,2-10TheSimplificationUndertheCircumstancesofOrdinaryPhysicalForce,Underthecircumstancesofordinaryphysicalforce,thecompatibleequationoftwokindsofplaneproblemsissimplifiedas:,Therefore,underthecircumstancesofordinaryphysicalforce,shouldsatisfyLaplacedifferentialequation(inharmonywithequation),shouldbeharmonicfunctions.Representwiththemark,theformulaabovecanbesimplifiedas:,Conclusion,Inthestressboundaryproblemofsingleconnectioniftwoelasticbodieshavethesameboundaryshapeandsuffertheexternalforceofthesamedistribution,andthenstressforcedistribution,shouldbethesamewhetherthematerialsoftwoelasticbodiesaresameornotandwhethertheyareundertheplanestresscircumstancesorundertheplanestraincircumstances(Twokindsofthestressforceweightintheplaneproblem,thedeformationandthedisplacementareuncertainlythesame).,TheBasicTheoryofthePlaneProblem,64,2-10常体力情况下的简化,常体力下，两种平面问题的相容方程都简化为：,可见，在常体力的情况下，应当满足拉普拉斯微分方程（调和方程），应当是调和函数。用记号代表，上式简写为：,结论,在单连体的应力边界问题中，如果两个弹性体具有相同的边界形状，并受到同样分布的外力，那么，不管这两个弹性体的材料是否相同，也不管它们是在平面应力情况下或是在平面应变情况下，应力分量、的分布是相同的（两种平面问题中的应力分量，以及形变和位移，却不一定相同）。,平面问题的基本理论,65,Inference2,Whenmeasuringtheabovestressweightofthestructureorcomponentwiththemethodofexperiment,wecanmakethemodelusingthematerialoftheconvenientmeasurementinordertoreplaceoriginalstructureorcomponentmaterialsoftheinconvenientmeasurement;wealsocanadoptstructureorcomponentoflongcolumns