彈性力學(xué)課件9

第三章平面問(wèn)題的直角坐標(biāo)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(jiǎn)支梁受均布荷載3—5楔形體受重力和液體壓力SolutionofPlaneProblemsinRectangularCoordinatesInverseMethodandSemi-InverseMethodPureBendingofaRectangularBeamDeterminationofDisplacmentsBendingofaSimpleBeamUnderUniformLoadsTriangularGravityWall3—4簡(jiǎn)支梁受均布荷載作用例2.圖示矩形截面簡(jiǎn)支梁，梁高為h，長(zhǎng)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(yīng)力函數(shù)及應(yīng)力分量（P41）。xh1yh/2h/2oqqqSimplebeamunderuniformloadConsiderasimplebeam,withlength2landdepthh,subjectedtoauniformlydistributedloadofintensityq.forconvenience,onlyaunitwidthofthebeamisconsidered,sothereactionateachendwillbeql.1、求應(yīng)力分量（用半逆解法）由材料力學(xué)已知：彎曲應(yīng)力x主要由彎矩引起，剪應(yīng)力xy由剪力引起，擠壓應(yīng)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(shí)恒滿足，則此方程中x前的系數(shù)和自由項(xiàng)必須為零，即Thisisaquadraticequationofx,butitmustbesatisfiedforallvaluesofxbetween–landl,astheconditionofcompatibilityrequires.Thisispossibleonlywhenthecoefficientsofx2andx,aswellasthetermindependentofx,arezero:（3）由（2）得：略去常數(shù)項(xiàng)由（1）得：Integrationof(1)and(2)yields：由（3）得：略去一次式和常數(shù)項(xiàng)Substitutingfinto(3)andintegrating,wehave：Theconstanttermandthetermlinearinyareneglect,becausetheywillnotaffectthestress.Thestresscomponentswillbe:Theseexpressionssatisfythedifferentialequationsofequilibriumandthecompatibilityequation.Hence,ifthearbitraryconstantsA,B,,Kcanbechosentosatisfyalltheboundaryconditions,theseexpressionswillbetherightsolutionoftheproblem.上述應(yīng)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(duì)稱（是x的偶函數(shù)）xy關(guān)于yz面反對(duì)稱，是x的奇函數(shù)xyh/2h/2oqqq(1)考慮對(duì)稱性considertheconditionsofsymmetrySincetheyzplaneisaplaneofsymmetryofthebeamandtheloading,thestressdistributionmustbesymmetricwithrespecttotheplane.Thus,theexpressionsforx、ymustbeevenfunctionofx,whilethatforxymustbeoddfunctionofx.Thisrequires于是得到：E=0F=0G=0xyh/2h/2oqqq將應(yīng)力表表達(dá)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(jiǎn)支梁梁在均布布荷載作作用下的的應(yīng)力分分量為：：（3）式式恒滿足足應(yīng)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(yīng)力力y發(fā)生在梁梁頂，材材料力學(xué)學(xué)中不考考慮將上述結(jié)結(jié)果與材材料力學(xué)學(xué)中的解解答比較較Comparethesolutionobtainedhereandthatgiveninmechanicsofmaterials,對(duì)板厚為為1的矩矩形截面面梁：所以，應(yīng)應(yīng)力分量量的表達(dá)達(dá)式為：：Forthebeamofunitwidth,wehave：So,thestresscomponentscaberewrittenas：x中，第一項(xiàng)為主要項(xiàng)，與材力中的解答相同，第二項(xiàng)為修正項(xiàng)，當(dāng)h時(shí)，修正項(xiàng)很小，忽略不計(jì)；剪應(yīng)力的表達(dá)式與材料力學(xué)中的一樣Weseethatthebendingstressxgiveninmechanicsofmaterialsmustbesupplementedwithacorrectiontermwhiletheshearingstressxyneedsnocorrection.擠壓應(yīng)力力在材料料力學(xué)中中不考慮慮Astothecrushingstressy,itisonlyconsideredinelasticityandnotinmechanicsofmaterialsatall.當(dāng)

時(shí)，最大正應(yīng)力應(yīng)修正1/15當(dāng)

時(shí)，最大正應(yīng)力應(yīng)修正1/60對(duì)的梁，材料力學(xué)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(shí)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(yuǎn)處處，應(yīng)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(wú)限長(zhǎng)長(zhǎng)，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(jì)算算應(yīng)力分分量Triangulargravitywall例.取圖示坐坐標(biāo)，應(yīng)應(yīng)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(yīng)力力可能是是gx、gy、gx、gy的組和項(xiàng)項(xiàng)Atanypointinthewall,eachofthestresscomponentsmusyconsistoftwoparts：producedbygravity：isproportionaltogproducedbythepressure：isproportionaltogHence,ifthestresscomponentscanbeexpressionsintheformofpolynomials,theymustbecombinationsoftheexpressionsintheformsofAgx,Bgy,Cgx,Dgy.可假設(shè)應(yīng)力力函數(shù)是x、y的純?nèi)问绞剑碭=0Y=gWemayassumethestressfunctionisapolynomialofthirddegreeTheseexpressionshavealreadysatisfiedthedifferentialequationsofequilibriumandthecompatibilityequation.Itremainstoinspectwhethertheboundaryconditionscanalsobesatisfiedbypropervaluesofthearbitraryconstantsa,b,c,d.xyggyBoundaryconditions：(1)theverticalsurfacex=0xyggy(2)ontheinclinedsurface:Withthesevaluesofa,b,candd,thestresscomponentsbecame：ThedistributionofstresscomponentsalongahorizontalsectionofthewallisshowninfollowingFig.：xyggy---xyxy應(yīng)力分量x沿水平方向向無(wú)變化這這個(gè)結(jié)果在在材料力學(xué)學(xué)中得不出出來(lái)應(yīng)力分量y沿水平按直直線規(guī)律變變化在左面：在斜