弹性力学课件9

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将应力表表达式代代入边界界条件(2)考虑边界条件considerboundaryconditionsSubstitutingthestresscomponentsexpressionsintotheseequationsandnoticingthatE=F=G=0,wehave根据次要要边界（（左右两两面，占占很小部部分）条条件，确确定H、K。。若不能完完全满足足，可用用圣维南南原理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：最后求得得简支梁梁在均布布荷载作作用下的的应力分分量为：：（3）式式恒满足足应力分量量沿垂直直方向（（横截面面）的变变化规律律如图：：ThestressdistributiononatypicalcrosssectionisapproximatelyshowninFig.：x图y图xy图x不是直线线规律分分布；挤挤压应力力y发生在梁梁顶，材材料力学学中不考考虑将上述结结果与材材料力学学中的解解答比较较Comparethesolutionobtainedhereandthatgiveninmechanicsofmaterials,对板厚为为1的矩矩形截面面梁：所以，应应力分量量的表达达式为：：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Weseethatthebendingstressxgiveninmechanicsofmaterialsmustbesupplementedwithacorrectiontermwhiletheshearingstressxyneedsnocorrection.挤压应力力在材料料力学中中不考虑虑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际边边界上面面力的分分布情况况不清楚楚，只知知道其合合力为零零，和力力矩为零零：根据圣维维南原理理，不管管这些面面力是否否存在，，如何分分布，在在离边界界较远处处，应力力与上述述式子是是完全一一样的3.5TRIANGULARGRAVITYWALLxyggyExp.Consideradamoraretainingwallwithtriangularsectionsubjectedtotheactionofgravityandthepressureofimpoundedliquid.Letthedensityofthewallmaterialbeandthatoftheliquidbe.3—5楔楔形形体受重重力和液液体压力力xyggy图示楔形形体，左左面铅直直，右面面与铅直直成角，下端端无限长长，承受受重力和和液体压压力，楔楔形体密密度为，液体密密度为，计算算应力分分量Triangulargravitywall例.取图示坐坐标，应应力由两两部分引引起：xyggy（1）重重力：与与g成正比（2）液液压：与与g成正比即，应力力可能是是gx、gy、gx、gy的组和项项Atanypointinthewall,eachofthestresscomponentsmusyconsistoftwoparts：producedbygravity：isproportionaltogproducedbythepressure：isproportionaltogHence,ifthestresscomponentscanbeexpressionsintheformofpolynomials,theymustbecombinationsoftheexpressionsintheformsofAgx,Bgy,Cgx,Dgy.可假设应力力函数是x、y的纯三次式式，即X=0Y=gWemayassumethestressfunctionisapolynomialofthirddegreeTheseexpressionshavealreadysatisfiedthedifferentialequationsofequilibriumandthecompatibilityequation.Itremainstoinspectwhethertheboundaryconditionscanalsobesatisfiedbypropervaluesofthearbitraryconstantsa,b,c,d.xyggyBoundaryconditions：(1)theverticalsurfacex=0xyggy(2)ontheinclinedsurface:Withthesevaluesofa,b,candd,thestresscomponentsbecame：ThedistributionofstresscomponentsalongahorizontalsectionofthewallisshowninfollowingFig.：xyggy---xyxy应力分量x沿水平方向向无变化这这个结果在在材料力学学中得不出出来应力分量y沿水平按直直线规律变变化在左面：在斜