弹性力学课件三七

2TheoryofPlaneProblems2—1PlaneStressandPlaneStrain2—2DifferentialEquationsofEquilibrium2—3

GeometricalEquations.Rigid-bodyDisplacement2—4PhysicalEquations2—5StressataPoint2—6BoundaryConditions.

Saint-Venant’sPrinciple2—7SolutionofPlaneProbleminTermsofDisplacements2—8SolutionofPlaneProbleminTermsofStresses2—9CaseofConstantBodyForces.StressFunction平面问题中，基本未知量为：x，y，xy，x，y，xy，u，v（八个）求解平面问题的基本方程：平衡微分方程（2个）DifferentialEquationsofEquilibrium几何方程（3个）

GeometricalEquations物理方程（3个）PhysicalEquations再考虑边界条件(BoundaryConditions)，即可求出所有未知量。(DifferentialEquationsofEquilibrium

foraplaneproblem)（GeometricalEquations

foraplaneproblem

）IfwereplaceEbyE/(1-2)andby/(1-)inthephysicalequationsforplanestressproblems,thephysicalequationsforplanestrainproblemsareobtain.

PlanestressproblemPlanestrainproblemWhenthestresscomponentsx，y，xyatacertainpointPareknown,wecancomputethestressactingonanyplanepassingthroughthispoint,paralleltothezaxisbutinclinedtothexandyaxes.STRESSATAPOINT.PRINCIPALSTRESSES2—5平面问题中一点的应力状态过同一点P的不同截面上，应力是不相同的，在平面问题中，可以用x，y，xy来描述一点的应力状态。Wecancomputethestressactingonanyplanepassingthroughthispoint，paralleltothezaxis.即：已知x，y，xy求过P点任意截面上的应力yyxxyxXYxyoxyxyyxStressataPointyyxxyxXYxyoxyxyyxyyxxyxxyoBAPsYNXNNNNSo，PA=m·dsPB=l·ds

过P点取图示微小的三角板PAB为隔离体，板厚为1PA=dxPB=dyAB=dsTakeaplaneABparalleltotheinclinedplaneconsideredandatasmalldistancefromP.t=1PA=dxPB=dyAB=ds

法线N的方向余弦：cos(N,x)=lcos(N,y)=mThecosinesoftheanglesbetweentheoutwardnormaltotheplaneABare：

建立微单元的平衡条件：X=0yyxxyxxyoBAPsYNXNNNN-x·l·ds·1-xy·m·ds·1+XN·ds·1=0XN=lx+mxyY=0YN=my+lxyConsidertheequilibriumconditions：为什么没有考虑体积力？Whythebodyforcecomponentscanbeneglected?yyxxyxxyoBAPsYNXNNNN而斜面上的正应力和剪应力可由XN，YN投影得到N=lXN+mYNN=lYN-mXNN=l2x+m2y+2lmxyN=lm(y-x)+(l2-m2)xyLetthenormalstressandshearingstressonABbeNandN,

projectionofXNandYNonandperpendiculartothenormalNwillgive将XN，YN的表达式代入上式得UsingtheexpressionsofXNandYN,weobtain已知x，y，xy

可以求过P点任意截面上的应力N、N.Wecancomputethestressactingonanyplanepassingthroughthispoint，paralleltothezaxis.讨论：主应力、应力主向、最大（最小）正应力、最大（最小）剪应力推导如下：1、主应力(principalstress)：任意斜面上的剪应力为零时，该斜面上只有正应力，称为主应力。IftheshearingstressonacertainplanethroughPvanishes,thatplaneiscalledaprincipalplaneatPandthenormalstressontheplaneiscalledaprincipalstressatP.xyoBAPYNXNNXN=lx+mxy=lYN=my+lxy=m前已导出：SupposeABisaprincipalplane.2-（x+y）+（xy-xy2）=0And1+2=x+y主应力有两个：1，2Sincetheexpressionwithineachsignofsquarerootisalwayspositive,1and2mustberealquantities.ThisshowsthattwoprincipalstressesexistatthepointP2、应力主向：主应力的方向设1与x轴的夹角为1Let1

betheanglebetween1

andx.Then,Similarly,Let2

betheanglebetween2

andx,wecanobtaini.e.：1的方向与2的方向有何关系？Because1+2=x+y2=x+y-1Then：Wecanobtain两个主应力方向相互垂直，如图xyOyyxxyxxyxyxy1122Thatexpressionshowsthat

thetwoprincipalstresses,1and2，areperpendiculartoeachother.3、最大、最小正应力xyxy1122则1=x

2=y

xy=0MaximumandminimumstressesWhentheprincipalstressesatapointareknown,itisveryeasytodeterminethemaximumandminimumstressesatthepoint.重新建立坐标系：主应力方向为x，y轴。Forsimplicityofanalysis,thexandyaxesareplacedinthedirectionsof1and2,respectively.Thenormalstressonanyinclinedplanewillbe（任意斜面上的正应力）：xy1122N=l2x+m2y+2lmxy=l21+m22Therelationoflandmis

l2+m2=1So,wecanobtainN=l21+（1-l2）2

=l2（1-2

）+2

Whenl2=1，N

ismaximum.Whenl2=0，N

isminimum.Whe内nl2=1，N=max=缝1Whe睡nl2=0，N=mi乐n=谣2Th策e姜ma蛋xi勒mu滩m婚an哀d猜mi堤ni坚mu买m愿va玩lu技es鬼o绑fNar龄e1and2,盯res追pec珍tiv泳ely猜.The慎sh区ear病ing历st竟res台so引nt拐he点inc立lin背ed泄pla糠ne逼isN=lm(y-x)+(l2-m2)xy=

lm(2-1)=

4、垫最大台、最寨小剪翁应力NWhen,theshearingstressNreachesitsextremevalues.Th矿is盯s雕ho从ws修t宿he猴e承xt傲re慢me揭s苏he幸ar着in料g环st纠re柜ss蚁es辟a蚁re志a工ct营in钳g兰on稳p垒la捏ne支s票in则cl废in凤ed密a叔t单450wit酬ht首hexzan词dyzpl胶an捉es极.xy1122NN最大或工最小剪悼应力的巩方向：归纳：已知x，y，xyN=l2x+m2y+2lmxyN=lm(y-x)+(l2-m2)xy且1+2=比x+y两个主应力方向相互垂直。Thetwoprincipalstresses,1and2，areperpendiculartoeachother.最大、最小剪应力发生在与应力主向成45角的截面上。Theseextremeshearingstressesareactingonplanesinclinedat45

withthexzandyzplanes.2—劈燕6矛边哗界条容件肃圣知维南守原理弹性力学问题平衡微分方程+几何协调方程+物理方程+边界条件或初始条件一、盒边界牧条件（Bo宋un职da牢ry凯C句on巷di安ti痕on酒s）a、位移畜边界城条件（dis稿pla稻cem震ent暑bo疯und挤ary雷co沿ndi辈tio剖ns）：Bou针nda夜ry局Con稼dit的ion屑s.Sa踪蝶in夹t-Ven撞ant朋’sPr羽in哭ci嫌pl登eAcc质ord行ing河to愧Bo毙und肺ary借Co叮ndi胳tio祥ns，殿ela隆sti细cit秤yp挠rob砖lem殃sa镇re测cla尽ssi起fie影da厉sdi矿sp俯la资ce卧me匀nt糕b亡ou攻nd匆ar回y气pr趴ob虽le揭ms，st杨re鸣ss像b转ou阅nd种ar省y贴pr替ob存le退ms，mix赛ed追bou遣nda陪ry耀pro谎ble洪ms.Inadisplacementboundaryproblem，thesurfacedisplacementsofthebodyarespecified.弹性体在边界上的位移是已知的。x如：四边固定的板或两端简支的板xbaxaby边界条闯件如下档：us=uvs=vb、应力边研界条件(st槽re等ss危b邻ou照nd谨ar挽y月co悬nd当it梢io耕ns图)xyqAXN=lx+mxyYN=my+lxy已知BPyyxxyxsYNXN在边界上，若面力已知，则XN=XYN=Ylx+mxy=Xmy+lxy=YInastressboundaryproblem，thesurfaceforcesactingonthebodyarespecified.弹性体在边界上所受的面力是已知的2xy3141、3边界：l=0m=+1y=Y;xy=X++2、4边界：m=0l=+1x=X;xy=Y++lx+mxy=Xmy+lxy=Y平面问题的应力边界条件stressboundaryconditionsofaplaneproblem特殊边界的应力边界条件Whentheboundaryisnormaltoacoordinateaxis,thestressboundaryconditionsaresimplified当边界的外法线沿坐标轴的正向时，两者的正负号相同；当边界的外法线沿坐标轴的负向时，两者的正负号相反。Theboundarystresscomponentsareequaltothesurfaceforcecomponents（usepositiveornegatvesignaccordingastheoutwardnormalisalongthepositiveornegativedirectionofthecoordinateaxis）.c、混合边界条件(mixedboundaryconditions)

：一部分边界具有已知的位移，一部分边界具有已知的面力；或同一部分边界既有位移边界条件，又有应力边界条件。Inamixedboundaryproblems，someportionoftheboundaryisspecifiedwithknowndisplacementswhiletheotherportionissubjectedtoknownsurfaceforces.解：AB边：y=-汉q镇;愧yx=0BC边：x=0稼;xy=0CD边：yAD边：u=崭0；墨v=巡寿0例1、写出图示悬臂梁的边界条件，板厚为1ABCDyxqxyABCAB边：v=0AC边：x=0近；xy=0混合外边界猛条件For湿in略sta蔽nce：Mix肆ed荐bou夺nda邀ry务pro俘ble疏m解：BC边：x=0逆;xy=0CD边：y=0捞;辣yx=0AD边：u=台0；辟v=教0AB边：y=yx=0例2、写出图示悬臂梁的边界条件，板厚为1xABCDyq0l解：BC边：y=0腔;简yx=0DE边:AB边：x=落-gy;xy=0例3、写出图示结构AB、BC、DE的应力边界条件水的重度为ExyABCDNcos(N它,x条)=co全s=lcos(N,峰y)=co盲s(90活+雷)=-冈sin脂=mlx+mxy=Xmy+lxy=Ycosx-sinxy=0-siny+cosxy=0二、雁圣维浙南原阵理(Sai加nt-Ven肠ant’s葛P哪ri怨nc漠ip贴le只)在求解弹性力学问题时，要使应力分量、形变分量、位移分量满足基本方程并不困难，但要完全、精确地满足边界条件，却往往发生很大困难。Insolvinganelasticityproblem,it’srathereasytoobtainthestresses,strainsanddisplacementswhichsatisfyallthebasicequations.Howeverweoftenencounterdifficultiesinhavingalltheboundaryconditionscompletelysatisfied.在上征述情余况下该，可顽利用淹圣维违南原昌理(Sa哲in蕉t-Ven慨ant’s挣Pri睬nci幼ple填)来写础出近呈似的影边界肿条件箱：圣维雪南原休理：如果把每物体一脑小部分婶边界上扁的面力饥变换成分布钻不同但静力等旷效的面锁力，慨则近橡处的估应力滑分布队将有侄显著寒的变桶化，丢但远惧处所太受的判影响为可以北忽略兔。(P2走5)另外，在工程计算中经常碰到这样的情况：在物体的部分边界上，只知道物体所受面力的合力，面力的分布方式并不明确。Ithappensfrequentlyinthestresscalculationforastructuralormachineelementthatweknowonlytheresultantofsurfaceforcesonasmallportionoftheelement,butnotthedistributionoftheforces.Un作de戒r距su障ch摩c初ir千cu故ms投ta饱nc由es光,Sai油nt-Ven咏ant’s英p礼ri长nc平ip牺lema坑y开be妈o倡f纱mu晋ch约h博el俭p耽to巷u议s.The劲es营sen瓜ce怪of琴thepr原in奔ci及pl牺eca仁n涨be骡s雀ta纪te咸d迫as宁f遍ol斥lo弦ws:If舍a笔s充ys真te逗m代of疯f罗or用ce泰s挣ac培ti丢ng阁o移n何a脆sm呜al伶l芹po便rt饿io桑n倒of烈t心he扰s黎ur挤fa锤ce径o厘f界an县e珠la世st趁ic捞b矛od欣y脱is签r历ep春la驶ce象d戴by雅a筐no爱th佛ersta项tic饿all幅ye挤qui雅val坟ent艳sy扯ste断mo邀ff灯orc狂esac出ti接ng析o刊n竭th余e袍sa桌me园p宿or之ti会on坏o连f拆th误e该su途rf粘ac跨e,刑t卷here奥di扒st愚ri馆bu惭ti邻onof妇l如oa贤di达ng配p渠ro跪du寺ce百s验su赌bs谱ta诵nt稠ia粒l证ch总an立ge灾s雅in著t匙he选s替tr竿es谈se姓sonl骄yi役nt坏he缴imm漆edi咽ate弹ne丢igh邀bor英hoo右do常ft招he镰loa种din融g,变an沈d详th苦e冬st瘦re哄ss恩es触a仗re屿e讯ss废en摇ti劈燕al飞lyth代e后sa涨me估i暗n炎th有e湿pa减rt税s帆of扒t最he姓b链od惠y柄wh伴ic们h辰ar稍e骄at誉l拐ar稍ge辛d掠is投ta膏nc盐esin流com海par兴iso鲜nw设ith拴th宁el犬ine旧ar俘dim伞ens油ion张of虾th戏es浅urf堡ace日on港wh抢ich支th炎ef肾orc债es居are海ch防ang摄ed.By谎“野st色at派ic腾al馅ly涨e鸡qu旷iv每al没en蓄t暑sy愉st确em督s”掏w卡e散me裕an愁t亮ha侦t尝th仿e愁tw暴o瞒sy辩st堆em卧s崇ha狡ve痒t挺he桑s叛am伙e顶re瓶su寻lt雾an帖t肯fo尚rc或e闲an统d着th堡e遮sa排me睬r杂es族ul逐ta街nt投m恭om嘉en月t.Fo槐r侵in戴st蚕an泳ce学