弹性力学课件八

第三章平面问题的直角坐标解答3—1逆解法与半逆解法3—2矩形梁的纯弯曲3—3位移分量的求出3—4简支梁受均布荷载3—5楔形体受重力和液体压力SolutionofPlaneProblemsinRectangularCoordinatesInverseMethodandSemi-InverseMethodPureBendingofaRectangularBeamDeterminationofDisplacmentsBendingofaSimpleBeamUnderUniformLoadsTriangularGravityWall3—4简支梁受均布荷载作用例2.图示矩形截面简支梁，梁高为h，长为2，受均布荷载作用，取板厚度为1，两端的支反力为q，体力不记，求应力函数及应力分量（P41）。xh1yh/2h/2oqqqSimplebeamunderuniformloadConsiderasimplebeam,withlength2landdepthh,subjectedtoauniformlydistributedloadofintensityq.forconvenience,onlyaunitwidthofthebeamisconsidered,sothereactionateachendwillbeql.1、求应力分量（用半逆解法）由材料力学已知：弯曲应力x主要由弯矩引起，剪应力xy由剪力引起，挤压应力y由荷载q引起。由于q不随x而变化，所以y不随x而变，可以假设：y=f（y）而Justasthebendingstressxandtheshearingstressxyaremainlyproducedbythemomentandtheshearingforcerespectively,thecrushingstressyismainlyproducedbythedirectloadonthebeam.Sincethedirectloadqdoesnotvarywithx,wemayassumethaty

doesnotvarywithxeitherandconsequentlyitisonlyafunctionofy：f1(y)、f2（y）是待定函数wheref1(y)andf2(y)arearbitraryfunctions.考察是否满足相容方程？Todeterminethefunctionf(y),f1(y)andf2(y),wesubstitutetheexpressionforintocompatibilityequation,obtaining代入相容方程得（1）（2）这是关于x的二次方程，要使此方程在x为任何值时恒满足，则此方程中x前的系数和自由项必须为零，即Thisisaquadraticequationofx,butitmustbesatisfiedforallvaluesofxbetween–landl,astheconditionofcompatibilityrequires.Thisispossibleonlywhenthecoefficientsofx2andx,aswellasthetermindependentofx,arezero:（3）由（2）得：略去常数项由（1）得：Integrationof(1)and(2)yields：由（3）得：略去一次式和常数项Substitutingfinto(3)andintegrating,wehave：Theconstanttermandthetermlinearinyareneglect,becausetheywillnotaffectthestress.Thestresscomponentswillbe:Theseexpressionssatisfythedifferentialequationsofequilibriumandthecompatibilityequation.Hence,ifthearbitraryconstantsA,B,,Kcanbechosentosatisfyalltheboundaryconditions,theseexpressionswillbetherightsolutionoftheproblem.上述应力分量满足平衡微分方程和相容方程，根据边界条件确定积分常述数，即可得正确的解答x、y关于yz面正对称（是x的偶函数）xy关于yz面反对称，是x的奇函数xyh/2h/2oqqq(1)考虑对称性considertheconditionsofsymmetrySincetheyzplaneisaplaneofsymmetryofthebeamandtheloading,thestressdistributionmustbesymmetricwithrespecttotheplane.Thus,theexpressionsforx、ymustbeevenfunctionofx,whilethatforxymustbeoddfunctionofx.Thisrequires于是得到：E=0F=0G=0xyh/2h/2oqqq将应力表达式代入边界条件(2)考虑边界条件considerboundaryconditionsSubstitutingthestresscomponentsexpressionsintotheseequationsandnoticingthatE=F=G=0,wehave根据次要边界（左右两面，占很小部分）条件，确定H、K。若不能完全满足，可用圣维南原理NowinordertodetermineHandK,wecanconsidertheboundaryconditionsattheendsofthebeam(theleftandrightendsofthebeamareonlysmallportionsoftheboundary).Iftheboundaryconditionstherecannotbesatisfiedexactly,wemayapplytheSaint-Venant’sprincipletohavetheconditionsapproximatelysatisfied.xyh/2h/2oqqq边界条件：（1）（2）（3）Boundaryconditionsatends：From(1),wehave：K=0From(2),wehave：(3)issatisfiedNowtheexpressionsforthestresscomponentsare：最后求得简支梁在均布荷载作用下的应力分量为：（3）式恒满足应力分量沿垂直方向（横截面）的变化规律如图：ThestressdistributiononatypicalcrosssectionisapproximatelyshowninFig.：x图y图xy图x不是直线规律分布；挤压应力y发生在梁顶，材料力学中不考虑将上述结果与材料力学中的解答比较Comparethesolutionobtainedhereandthatgiveninmechanicsofmaterials,对板厚为1的矩形截面梁：所以，应力分量的表达式为：Forthebeamofunitwidth,wehave：So,thestresscomponentscaberewrittenas：x中，第一项为主要项，与材力中的解答相同，第二项为修正项，当h时，修正项很小，忽略不计；剪应力的表达式与材料力学中的一样Weseethatthebendingstressx

giveninmechanicsofmaterialsmustbesupplementedwithacorrectiontermwhiletheshearingstressxyneedsnocorrection.挤压应力在材料力学中不考虑Astothecrushingstressy,itisonlyconsideredinelasticityandnotinmechanicsofmaterialsatall.当

时，最大正应力应修正1/15当

时，最大正应力应修正1/60对的梁，材料力学中的结果足够精确由于所以梁在左右两边的水平面力为而实辜际边猜界上忙面力捉的分亚布情脸况不真清楚赏，只摘知道事其合评力为白零，帖和力释矩为克零：根据圣勾维南原妙理，不毒管这些刚面力是敌否存在忍，如何歪分布，激在离边多界较远清处，应避力与上宜述式子气是完全谈一样的3.句5TR姨IA双NG蚕UL沫AR驼G兵RA棋VI晨TY卖W仿AL捷LxyggyEx效p.瞒C阵on虚si侨de爷r答a拖da死m衔or贩a淡r势et蛛ai呢ni况ng冶w斧al棚l肉wi帽th自t脑ri料an悦gu西la丸r腥se原ct山io码n静su端bj东ec仍te窗d础to蚁t卧he啦a免ct胁io荡n骡of纷g吗ra铁vi窑ty年a耕nd糊t孔he川p我re判ss贵ur抱e既of斤i园mp伏ou蜡nd脖ed臣l金iq南ui鹅d.串L隆et迹t掠he幅d币en叶si罪ty定o崖f掠th枪e狱wa闻ll估m着at友er仁ia灶l溪beand供th辉at迁of海the冰li膝qui何db咸e.3—详5茄楔促形体将受重点力和拖液体扭压力xyggy图示楔弹形体，睛左面铅沟直，右所面与铅死直成角，下迁端无限饶长，承道受重力胃和液体棋压力，罩楔形体万密度为，液浪体密页度为，计冰算应力绵分量Tri晴ang苗ula踏rg衫rav傍ity眉wa孕ll例.取图庭示坐是标，或应力劝由两邀部分迷引起洪：xyggy（1）含重力：理与g成正习比（2）孟液压：咏与g成正比即，诵应力本可能策是gx烘、gy、gx、gy的组和逗项At迫a示ny秤p修oi拥nt节i劣n柏th捉e症wa明ll帮,e搂ac络h永of尝t狗he俊s抓tr画es宗s馆co捧mp虑on悄en稻ts扩m等us股y鸭co狸ns情is危t饥of椅t勇wo奸p裁ar吩ts：pr瘦od鼠uc典ed丝式b歪y简gr掘av印it敞y：is究pro龄por直tio病nal往togpro衣duc幅ed额by箭the到pr喉ess弊ure：is鹅pro肿por顺tio知nal禁togHen坐ce,君if农th私es阀tre网ss箭com忍pon架ent否sc谈an含be承exp返res岭sio辱ns划in嚼the颈fo网rm势of袄pol绩yno章mia烟ls,缘瑞th撤ey葛mus被tb薪ec裕omb掠ina蜘tio应ns偿of掠the灯ex笼pre膊ssi睛ons霜in黄th若ef叨orm晋so胸fAgx,辣Bgy捉,Cgx竖,戴Dgy沈.可假望设应侵力函长数是x、率y的纯三野次式，尽即X=0Y=gWe赖may兄as狐sum截et遗he管str玻ess岩fu做nct抚ion挑is景a截p凑ol铃yn抽om棋ia轻l挤of夕t流hi研rd姑d婆eg谋re问eThe饲se川exp闹res午sio扰ns泉hav养ea甜lre什ady含sa孤tis识fie桥dt休he慨dif懂fer呼ent借ial躁eq设uat长ion撒so渔fe垒qui缘瑞lib狠riu瓦ma备nd般the悄co瓣mpa跟tib用ili烈ty潜equ毙ati艰on.It残rem竿ain避st替oi底nsp季ect闻wh茅eth索er茄the纪bo站und削ary漏co务ndi延tio响ns劲can侨al欠so犹be俱sat顶isf尿ied花by议pr抄ope辩rv播alu驴es寒of院the陶ar估bit躬rar签yc忧ons落tan黑tsa,b,c,d.xyggyBoundaryconditions：(1)theverticalsurfacex=0xyggy(2)ontheinclinedsurface:Withthesevaluesofa,b,candd,thestresscomponentsbecame：Th戴e泛di伸st湿ri蹈bu悲ti辣on睁o田f变st柳re盆ss榜c邻om绪po泰ne波nt柴s叔al崇on捷g抢a柄ho迅ri敲zo谁nt英al鞠s丽ec转ti坏on浴o王f筹th眉e凑wa汉ll泡i寻s起sh踏ow填n孝in借f眨ol四lo瓜wi该ng司F胃ig屿.：xyggy---xyxy应力分卡量x沿水平火方向无困变化这将个结果抚在材料掌力学中渠得不出火来应力列分量y沿水咬平按宅直线合规律叔变化在左面：在斜面：与材料径力学中城偏心受富压公式网一样应力分面量