彈性力學課件9

第三章平面問題的直角坐標解答3—1逆解法與半逆解法3—2矩形梁的純彎曲3—3位移分量的求出3—4簡支梁受均布荷載3—5楔形體受重力和液體壓力SolutionofPlaneProblemsinRectangularCoordinatesInverseMethodandSemi-InverseMethodPureBendingofaRectangularBeamDeterminationofDisplacmentsBendingofaSimpleBeamUnderUniformLoadsTriangularGravityWall3—4簡支梁受均布荷載作用例2.圖示矩形截面簡支梁，梁高為h，長為2，受均布荷載作用，取板厚度為1，兩端的支反力為q，體力不記，求應力函數(shù)及應力分量（P41）。xh1yh/2h/2oqqqSimplebeamunderuniformloadConsiderasimplebeam,withlength2landdepthh,subjectedtoauniformlydistributedloadofintensityq.forconvenience,onlyaunitwidthofthebeamisconsidered,sothereactionateachendwillbeql.1、求應力分量（用半逆解法）由材料力學已知：彎曲應力x主要由彎矩引起，剪應力xy由剪力引起，擠壓應力y由荷載q引起。由于q不隨x而變化，所以y不隨x而變，可以假設(shè)：y=f（y）而Justasthebendingstressxandtheshearingstressxyaremainlyproducedbythemomentandtheshearingforcerespectively,thecrushingstressyismainlyproducedbythedirectloadonthebeam.Sincethedirectloadqdoesnotvarywithx,wemayassumethaty

doesnotvarywithxeitherandconsequentlyitisonlyafunctionofy：f1(y)、f2（y）是待定函數(shù)wheref1(y)andf2(y)arearbitraryfunctions.考察是否滿足相容方程？Todeterminethefunctionf(y),f1(y)andf2(y),wesubstitutetheexpressionforintocompatibilityequation,obtaining代入相容方程得（1）（2）這是關(guān)于x的二次方程，要使此方程在x為任何值時恒滿足，則此方程中x前的系數(shù)和自由項必須為零，即Thisisaquadraticequationofx,butitmustbesatisfiedforallvaluesofxbetween–landl,astheconditionofcompatibilityrequires.Thisispossibleonlywhenthecoefficientsofx2andx,aswellasthetermindependentofx,arezero:（3）由（2）得：略去常數(shù)項由（1）得：Integrationof(1)and(2)yields：由（3）得：略去一次式和常數(shù)項Substitutingfinto(3)andintegrating,wehave：Theconstanttermandthetermlinearinyareneglect,becausetheywillnotaffectthestress.Thestresscomponentswillbe:Theseexpressionssatisfythedifferentialequationsofequilibriumandthecompatibilityequation.Hence,ifthearbitraryconstantsA,B,,Kcanbechosentosatisfyalltheboundaryconditions,theseexpressionswillbetherightsolutionoftheproblem.上述應力分量滿足平衡微分方程和相容方程，根據(jù)邊界條件確定積分常述數(shù)，即可得正確的解答x、y關(guān)于yz面正對稱（是x的偶函數(shù)）xy關(guān)于yz面反對稱，是x的奇函數(shù)xyh/2h/2oqqq(1)考慮對稱性considertheconditionsofsymmetrySincetheyzplaneisaplaneofsymmetryofthebeamandtheloading,thestressdistributionmustbesymmetricwithrespecttotheplane.Thus,theexpressionsforx、ymustbeevenfunctionofx,whilethatforxymustbeoddfunctionofx.Thisrequires于是得到：E=0F=0G=0xyh/2h/2oqqq將應力表表達式代代入邊界界條件(2)考慮邊界條件considerboundaryconditionsSubstitutingthestresscomponentsexpressionsintotheseequationsandnoticingthatE=F=G=0,wehave根據(jù)次要要邊界（（左右兩兩面，占占很小部部分）條條件，確確定H、K。。若不能完完全滿足足，可用用圣維南南原理NowinordertodetermineHandK,wecanconsidertheboundaryconditionsattheendsofthebeam(theleftandrightendsofthebeamareonlysmallportionsoftheboundary).Iftheboundaryconditionstherecannotbesatisfiedexactly,wemayapplytheSaint-Venant’’sprincipletohavetheconditionsapproximatelysatisfied.xyh/2h/2oqqq邊界條件：（1）（2）（3）Boundaryconditionsatends：From(1),wehave：K=0From(2),wehave：(3)issatisfiedNowtheexpressionsforthestresscomponentsare：最后求得得簡支梁梁在均布布荷載作作用下的的應力分分量為：：（3）式式恒滿足足應力分量量沿垂直直方向（（橫截面面）的變變化規(guī)律律如圖：：ThestressdistributiononatypicalcrosssectionisapproximatelyshowninFig.：x圖y圖xy圖x不是直線線規(guī)律分分布；擠擠壓應力力y發(fā)生在梁梁頂，材材料力學學中不考考慮將上述結(jié)結(jié)果與材材料力學學中的解解答比較較Comparethesolutionobtainedhereandthatgiveninmechanicsofmaterials,對板厚為為1的矩矩形截面面梁：所以，應應力分量量的表達達式為：：Forthebeamofunitwidth,wehave：So,thestresscomponentscaberewrittenas：x中，第一項為主要項，與材力中的解答相同，第二項為修正項，當h時，修正項很小，忽略不計；剪應力的表達式與材料力學中的一樣Weseethatthebendingstressxgiveninmechanicsofmaterialsmustbesupplementedwithacorrectiontermwhiletheshearingstressxyneedsnocorrection.擠壓應力力在材料料力學中中不考慮慮Astothecrushingstressy,itisonlyconsideredinelasticityandnotinmechanicsofmaterialsatall.當

時，最大正應力應修正1/15當

時，最大正應力應修正1/60對的梁，材料力學中的結(jié)果足夠精確由于所以梁在左右兩邊的水平面力為而實際邊邊界上面面力的分分布情況況不清楚楚，只知知道其合合力為零零，和力力矩為零零：根據(jù)圣維維南原理理，不管管這些面面力是否否存在，，如何分分布，在在離邊界界較遠處處，應力力與上述述式子是是完全一一樣的3.5TRIANGULARGRAVITYWALLxyggyExp.Consideradamoraretainingwallwithtriangularsectionsubjectedtotheactionofgravityandthepressureofimpoundedliquid.Letthedensityofthewallmaterialbeandthatoftheliquidbe.3—5楔楔形形體受重重力和液液體壓力力xyggy圖示楔形形體，左左面鉛直直，右面面與鉛直直成角，下端端無限長長，承受受重力和和液體壓壓力，楔楔形體密密度為，液體密密度為，計算算應力分分量Triangulargravitywall例.取圖示坐坐標，應應力由兩兩部分引引起：xyggy（1）重重力：與與g成正比（2）液液壓：與與g成正比即，應力力可能是是gx、gy、gx、gy的組和項項Atanypointinthewall,eachofthestresscomponentsmusyconsistoftwoparts：producedbygravity：isproportionaltogproducedbythepressure：isproportionaltogHence,ifthestresscomponentscanbeexpressionsintheformofpolynomials,theymustbecombinationsoftheexpressionsintheformsofAgx,Bgy,Cgx,Dgy.可假設(shè)應力力函數(shù)是x、y的純?nèi)问绞?，即X=0Y=gWemayassumethestressfunctionisapolynomialofthirddegreeTheseexpressionshavealreadysatisfiedthedifferentialequationsofequilibriumandthecompatibilityequation.Itremainstoinspectwhethertheboundaryconditionscanalsobesatisfiedbypropervaluesofthearbitraryconstantsa,b,c,d.xyggyBoundaryconditions：(1)theverticalsurfacex=0xyggy(2)ontheinclinedsurface:Withthesevaluesofa,b,candd,thestresscomponentsbecame：ThedistributionofstresscomponentsalongahorizontalsectionofthewallisshowninfollowingFig.：xyggy---xyxy應力分量x沿水平方向向無變化這這個結(jié)果在在材料力學學中得不出出來應力分量y沿水平按直直線規(guī)律變變化在左面：在斜