彈性力學(xué)課件八

第三章平面問題的直角坐標(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(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(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(yīng)力表達(dá)式代入邊界條件(2)考慮邊界條件considerboundaryconditionsSubstitutingthestresscomponentsexpressionsintotheseequationsandnoticingthatE=F=G=0,wehave根據(jù)次要邊界（左右兩面，占很小部分）條件，確定H、K。若不能完全滿足，可用圣維南原理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：最后求得簡支梁在均布荷載作用下的應(yīng)力分量為：（3）式恒滿足應(yīng)力分量沿垂直方向（橫截面）的變化規(guī)律如圖：ThestressdistributiononatypicalcrosssectionisapproximatelyshowninFig.：x圖y圖xy圖x不是直線規(guī)律分布；擠壓應(yīng)力y發(fā)生在梁頂，材料力學(xué)中不考慮將上述結(jié)果與材料力學(xué)中的解答比較Comparethesolutionobtainedhereandthatgiveninmechanicsofmaterials,對板厚為1的矩形截面梁：所以，應(yīng)力分量的表達(dá)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(dāng)h時，修正項很小，忽略不計；剪應(yīng)力的表達(dá)式與材料力學(xué)中的一樣Weseethatthebendingstressx

giveninmechanicsofmaterialsmustbesupplementedwithacorrectiontermwhiletheshearingstressxyneedsnocorrection.擠壓應(yīng)力在材料力學(xué)中不考慮Astothecrushingstressy,itisonlyconsideredinelasticityandnotinmechanicsofmaterialsatall.當(dāng)

時，最大正應(yīng)力應(yīng)修正1/15當(dāng)

時，最大正應(yīng)力應(yīng)修正1/60對的梁，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辜際邊猜界上忙面力捉的分亞布情臉況不真清楚賞，只摘知道事其合評力為白零，帖和力釋矩為克零：根據(jù)圣勾維南原妙理，不毒管這些剛面力是敵否存在忍，如何歪分布，激在離邊多界較遠(yuǎn)清處，應(yīng)避力與上宜述式子氣是完全談一樣的3.句5TR姨IA雙NG蠶UL沫AR駝G兵RA棋VI晨TY賣W仿AL捷LxyggyEx效p.瞞C陣on虛si僑de爺r答a拖da死m(xù)銜or販a淡r勢et蛛ai呢ni況ng冶w斧al棚l肉wi帽th自t腦ri料an悅gu西la丸r腥se原ct山io碼n靜su端bj東ec仍te窗d礎(chǔ)to蟻t臥he啦a免ct脅io蕩n騾of紛g嗎ra鐵vi窯ty年a耕nd糊t孔he川p我re判ss貴ur抱e既of斤i園mp伏ou蠟nd脖ed臣l金iq南ui鵝d.串L隆et跡t掠he幅d幣en葉si罪ty定o崖f掠th槍e獄wa聞ll估m(xù)著at友er仁ia灶l溪beand供th輝at遷of海the冰li膝qui何db咸e.3—詳5茄楔促形體將受重點力和拖液體扭壓力xyggy圖示楔彈形體，睛左面鉛溝直，右所面與鉛死直成角，下遷端無限饒長，承道受重力胃和液體棋壓力，罩楔形體萬密度為，液浪體密頁度為，計冰算應(yīng)力綿分量Tri晴ang苗ula踏rg衫rav傍ity眉wa孕l(wèi)l例.取圖庭示坐是標(biāo)，或應(yīng)力勸由兩邀部分迷引起洪：xyggy（1）含重力：理與g成正習(xí)比（2）孟液壓：詠與g成正比即，誦應(yīng)力本可能策是gx烘、gy、gx、gy的組和逗項At迫a示ny秤p修oi擁nt節(jié)i劣n柏th捉e癥wa明ll幫,e摟ac絡(luò)h永of嘗t狗he俊s抓tr畫es宗s館co捧mp慮on悄en稻ts擴(kuò)m等us股y鴨co貍ns情is危t饑of椅t勇wo奸p裁ar吩ts：pr瘦od鼠uc典ed絲式b歪y簡gr掘av印it敞y：is究pro齡por直tio病nal往togpro衣duc幅ed額by箭the到pr喉ess弊ure：is鵝pro腫por順tio知nal禁togHen坐ce,君if農(nóng)th私es閥tre網(wǎng)ss箭com忍pon架ent否sc談an含be承exp返res嶺sio辱ns劃in嚼the頸fo網(wǎng)rm勢of襖pol績yno章mia煙ls,緣瑞th撤ey葛mus被tb薪ec裕omb掠ina蜘tio應(yīng)ns償of掠the燈ex籠pre膊ssi睛ons霜in黃th若ef叨orm晉so胸fAgx,辣Bgy捉,Cgx豎,戴Dgy沈.可假望設(shè)應(yīng)侵力函長數(shù)是x、率y的純?nèi)按问?，盡即X=0Y=gWe賴may兄as狐sum截et遺he管str玻ess巖fu做nct撫ion挑is景a截p湊ol鈴yn抽om棋ia輕l擠of夕t流hi研rd姑d婆eg謀re問eThe飼se川exp鬧res午sio擾ns泉hav養(yǎng)ea甜lre什ady含sa孤tis識fie橋dt休he慨dif懂fer呼ent借ial躁eq設(shè)uat長ion撒so漁fe壘qui緣瑞lib狠riu瓦ma備nd般the悄co瓣mpa跟tib用ili烈ty潛equ斃ati艱on.It殘rem竿ain避st替oi底nsp季ect聞wh茅eth索er茄the紀(jì)bo站und削ary漏co務(wù)ndi延tio響ns勁can僑al欠so猶be俱sat頂isf尿ied花by議pr抄ope辯rv播alu驢es寒of院the陶ar估bit躬rar簽yc憂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濕ri蹈bu悲ti辣on睜o田f變st柳re盆ss榜c鄰om緒po泰ne波nt柴s叔al崇on捷g搶a柄ho迅ri敲zo誰nt英al鞠s麗ec轉(zhuǎn)ti壞on浴o王f籌th眉e湊wa漢ll泡i尋s起sh踏ow填n孝in借f眨ol四lo瓜wi該ng司F胃ig嶼.：xyggy---xyxy應(yīng)力分卡量x沿水平火方向無困變化這將個結(jié)果撫在材料掌力學(xué)中渠得不出火來應(yīng)力列分量y沿水咬平按宅直線合規(guī)律叔變化在左面：在斜面：與材料徑力學(xué)中城偏心受富壓公式網(wǎng)一樣應(yīng)力分面量