彈性力學課件三七

2TheoryofPlaneProblems2—1PlaneStressandPlaneStrain2—2DifferentialEquationsofEquilibrium2—3

GeometricalEquations.Rigid-bodyDisplacement2—4PhysicalEquations2—5StressataPoint2—6BoundaryConditions.

Saint-Venant’sPrinciple2—7SolutionofPlaneProbleminTermsofDisplacements2—8SolutionofPlaneProbleminTermsofStresses2—9CaseofConstantBodyForces.StressFunction平面問題中，基本未知量為：x，y，xy，x，y，xy，u，v（八個）求解平面問題的基本方程：平衡微分方程（2個）DifferentialEquationsofEquilibrium幾何方程（3個）

GeometricalEquations物理方程（3個）PhysicalEquations再考慮邊界條件(BoundaryConditions)，即可求出所有未知量。(DifferentialEquationsofEquilibrium

foraplaneproblem)（GeometricalEquations

foraplaneproblem

）IfwereplaceEbyE/(1-2)andby/(1-)inthephysicalequationsforplanestressproblems,thephysicalequationsforplanestrainproblemsareobtain.

PlanestressproblemPlanestrainproblemWhenthestresscomponentsx，y，xyatacertainpointPareknown,wecancomputethestressactingonanyplanepassingthroughthispoint,paralleltothezaxisbutinclinedtothexandyaxes.STRESSATAPOINT.PRINCIPALSTRESSES2—5平面問題中一點的應力狀態(tài)過同一點P的不同截面上，應力是不相同的，在平面問題中，可以用x，y，xy來描述一點的應力狀態(tài)。Wecancomputethestressactingonanyplanepassingthroughthispoint，paralleltothezaxis.即：已知x，y，xy求過P點任意截面上的應力yyxxyxXYxyoxyxyyxStressataPointyyxxyxXYxyoxyxyyxyyxxyxxyoBAPsYNXNNNNSo，PA=m·dsPB=l·ds

過P點取圖示微小的三角板PAB為隔離體，板厚為1PA=dxPB=dyAB=dsTakeaplaneABparalleltotheinclinedplaneconsideredandatasmalldistancefromP.t=1PA=dxPB=dyAB=ds

法線N的方向余弦：cos(N,x)=lcos(N,y)=mThecosinesoftheanglesbetweentheoutwardnormaltotheplaneABare：

建立微單元的平衡條件：X=0yyxxyxxyoBAPsYNXNNNN-x·l·ds·1-xy·m·ds·1+XN·ds·1=0XN=lx+mxyY=0YN=my+lxyConsidertheequilibriumconditions：為什么沒有考慮體積力？Whythebodyforcecomponentscanbeneglected?yyxxyxxyoBAPsYNXNNNN而斜面上的正應力和剪應力可由XN，YN投影得到N=lXN+mYNN=lYN-mXNN=l2x+m2y+2lmxyN=lm(y-x)+(l2-m2)xyLetthenormalstressandshearingstressonABbeNandN,

projectionofXNandYNonandperpendiculartothenormalNwillgive將XN，YN的表達式代入上式得UsingtheexpressionsofXNandYN,weobtain已知x，y，xy

可以求過P點任意截面上的應力N、N.Wecancomputethestressactingonanyplanepassingthroughthispoint，paralleltothezaxis.討論：主應力、應力主向、最大（最?。┱龖Α⒆畲螅ㄗ钚。┘魬ν茖缦拢?、主應力(principalstress)：任意斜面上的剪應力為零時，該斜面上只有正應力，稱為主應力。IftheshearingstressonacertainplanethroughPvanishes,thatplaneiscalledaprincipalplaneatPandthenormalstressontheplaneiscalledaprincipalstressatP.xyoBAPYNXNNXN=lx+mxy=lYN=my+lxy=m前已導出：SupposeABisaprincipalplane.2-（x+y）+（xy-xy2）=0And1+2=x+y主應力有兩個：1，2Sincetheexpressionwithineachsignofsquarerootisalwayspositive,1and2mustberealquantities.ThisshowsthattwoprincipalstressesexistatthepointP2、應力主向：主應力的方向設1與x軸的夾角為1Let1

betheanglebetween1

andx.Then,Similarly,Let2

betheanglebetween2

andx,wecanobtaini.e.：1的方向與2的方向有何關系？Because1+2=x+y2=x+y-1Then：Wecanobtain兩個主應力方向相互垂直，如圖xyOyyxxyxxyxyxy1122Thatexpressionshowsthat

thetwoprincipalstresses,1and2，areperpendiculartoeachother.3、最大、最小正應力xyxy1122則1=x

2=y

xy=0MaximumandminimumstressesWhentheprincipalstressesatapointareknown,itisveryeasytodeterminethemaximumandminimumstressesatthepoint.重新建立坐標系：主應力方向為x，y軸。Forsimplicityofanalysis,thexandyaxesareplacedinthedirectionsof1and2,respectively.Thenormalstressonanyinclinedplanewillbe（任意斜面上的正應力）：xy1122N=l2x+m2y+2lmxy=l21+m22Therelationoflandmis

l2+m2=1So,wecanobtainN=l21+（1-l2）2

=l2（1-2

）+2

Whenl2=1，N

ismaximum.Whenl2=0，N

isminimum.Whenl2=1，N=max=1Whenl2=0，N=min=2ThemaximumandminimumvaluesofNare1and2,respectively.TheshearingstressontheinclinedplaneisN=lm(y-x)+(l2-m2)xy=

lm(2-1)=

4、最大、最小剪應力NWhen,theshearingstressNreachesitsextremevalues.Thi棕ss樹how步st逗he筑ext并rem筒es白hea叢rin愛gs伸tre您sse痕sa帽re療act膛ing牲on糠pl市ane置si絡ncl額ine具da竭t4公50wit劣ht終hexzandyzpl末an擾es羨.xy1122NN最大或陵最小剪畫應力的予方向：歸納：已知x，y，xyN=l2x+m2y+2lmxyN=lm(y-x)+(l2-m2)xy且情1+2=槽x+y兩個主應力方向相互垂直。Thetwoprincipalstresses,1and2，areperpendiculartoeachother.最大、最小剪應力發(fā)生在與應力主向成45角的截面上。Theseextremeshearingstressesareactingonplanesinclinedat45

withthexzandyzplanes.2—6煉邊本界條件安圣維句南原理彈性力學問題平衡微分方程+幾何協(xié)調方程+物理方程+邊界條件或初始條件一、邊確界條件（Bo渾un戰(zhàn)da慢ry暈C侵on布di傲ti牽on似s）a、位移邊兵界條件（dis虛pla結cem談ent漠bo鼻und烤ary拉co造ndi騰tio腳ns）：Bo完un眼da頭ry哥C瘋on參di考ti戲on抄s.Sa炒in熔t-Ven艙ant嚷’sPr轟in茄ci益pl攝eAc身co煩rd矮in勸g灰to咽B系ou持nd均ar間y太Co微nd給it塞io緩ns留，e館la變st此ic誕it既y訊pr庭ob政le皂ms齒a額re升c謎la唉ss回if帽ie柴d錯asdi沉sp乏la夸ce貸me窯nt伴b時ou桐nd羽ar鄰y餅pr規(guī)ob營le慈ms，st質re閉ss賀b羽ou紗nd潛ar兆y君pr垮ob拉le資ms，mi絹xe策d容bo職un端da報ry日p全ro泰bl盛em農s.Inadisplacementboundaryproblem，thesurfacedisplacementsofthebodyarespecified.彈性體在邊界上的位移是已知的。x如：四邊固定的板或兩端簡支的板xbaxaby邊界衡條件婚如下門：us=uvs=vb、應力邊浴界條件(str糞ess械bo盼und犁ary錦co戴ndi隱tio旱ns)：xyqAXN=lx+mxyYN=my+lxy已知BPyyxxyxsYNXN在邊界上，若面力已知，則XN=XYN=Ylx+mxy=Xmy+lxy=YInastressboundaryproblem，thesurfaceforcesactingonthebodyarespecified.彈性體在邊界上所受的面力是已知的2xy3141、3邊界：l=0m=+1y=Y;xy=X++2、4邊界：m=0l=+1x=X;xy=Y++lx+mxy=Xmy+lxy=Y平面問題的應力邊界條件stressboundaryconditionsofaplaneproblem特殊邊界的應力邊界條件Whentheboundaryisnormaltoacoordinateaxis,thestressboundaryconditionsaresimplified當邊界的外法線沿坐標軸的正向時，兩者的正負號相同；當邊界的外法線沿坐標軸的負向時，兩者的正負號相反。Theboundarystresscomponentsareequaltothesurfaceforcecomponents（usepositiveornegatvesignaccordingastheoutwardnormalisalongthepositiveornegativedirectionofthecoordinateaxis）.c、混合邊界條件(mixedboundaryconditions)

：一部分邊界具有已知的位移，一部分邊界具有已知的面力；或同一部分邊界既有位移邊界條件，又有應力邊界條件。Inamixedboundaryproblems，someportionoftheboundaryisspecifiedwithknowndisplacementswhiletheotherportionissubjectedtoknownsurfaceforces.解：AB邊：y=-撕q型;艷yx=0BC邊：x=0撈;保xy=0CD邊：y=0裙;yx=0AD邊：u=0愛；v=封0例1、寫出圖示懸臂梁的邊界條件，板厚為1ABCDyxqxyABCAB邊：v=運0AC邊：x=0悔；xy=0混合邊棟界條件Fo理r怎in嶼st均an普ce：Mix托ed乓bou綠nda常ry任pro惑ble蔽m解：BC邊：x=0貧;鍵xy=0CD邊：y=0框;異yx=0AD邊：u=0報；v=笨0AB邊：y=yx=0例2、寫出圖示懸臂梁的邊界條件，板厚為1xABCDyq0l解：BC邊：y=0賽;饒yx=0DE邊:AB邊：x=奏-gy;法xy=0例3、寫出圖示結構AB、BC、DE的應力邊界條件水的重度為ExyABCDNcos(N,昆x)=co授s=lco次s(N,史y)=co嬸s(9萄0尤+肢)=到-s珠in尸=詠mlx+mxy=Xmy+lxy=Ycosx-sinxy=0-siny+cosxy=0二、目圣維畝南原亂理(Sai海nt-Ven鵲ant’s設Pri虹nci墳ple盡)在求解彈性力學問題時，要使應力分量、形變分量、位移分量滿足基本方程并不困難，但要完全、精確地滿足邊界條件，卻往往發(fā)生很大困難。Insolvinganelasticityproblem,it’srathereasytoobtainthestresses,strainsanddisplacementswhichsatisfyallthebasicequations.Howeverweoftenencounterdifficultiesinhavingalltheboundaryconditionscompletelysatisfied.在上節(jié)述情咸況下磨，可留利用微圣維里南原磨理(Sai傲nt-Ve徑na李nt’s盯P滾ri灣nc橋ip聚le碧)來寫侮出近筋似的墓邊界懷條件茅：圣維州南原氣理：如果把升物體一兩小部分咱邊界上止的面力欲變換成分布搶不同但靜力等敲效的面力憶，則近朝處的應拒力分布翅將有顯打著的變博化，但筋遠處所太受的影評響可以喬忽略。寺(P25貧)另外，在工程計算中經常碰到這樣的情況：在物體的部分邊界上，只知道物體所受面力的合力，面力的分布方式并不明確。Ithappensfrequentlyinthestresscalculationforastructuralormachineelementthatweknowonlytheresultantofsurfaceforcesonasmallportionoftheelement,butnotthedistributionoftheforces.Un撓de辟r疫su本ch攔c催ir峰cu淺ms飯ta財nc余es抖,Sai身nt-Ven景ant’s誤pri動nci山plemay們be殲of端mu丘ch狂hel木pt欠ou家s.Th暫e刺es霸se耽nc出e林of雞t王hepr覽in裹ci歸pl都ecan品be掀st丹ate田da優(yōu)sf絹oll滑ows:If與a哪s嚷ys介te聯(lián)m錦of狀f緣瑞or栽ce端s方ac腫ti約ng泳o赴n煌a栽sm瞇al示l艇po雅rt獻io寨n言of輝t找he封s雄ur哥fa適ce楚o伏f凍an權e帥la娛st捧ic暫b酷od閣y湖is春r早ep慚la千ce央d然by卷a瀉no邪th茫ersta悠tic桌all煮ye屈qui些val故ent室sy衣ste泳mo寧ff篩orc午esact摩ing族on剃th圖es麗ame產po令rti芹on貿of葵the須su霸rfa壯ce,誰th呆ered制ist汪rib展uti妄onof郊l普oa臭di我ng仆p闖ro黎du補ce象s輸su算bs噸ta顆nt夾ia效l粗ch縮慧an遠ge導s膠in芽t路he丙s忠tr目es層se蘆son粒ly醉i邪n勝th起e毅im腳me獨di樹at耽e矮ne原ig嶺hb橫or捉ho公od帥o旱f浸th秘e早lo法ad仆in秤g,燒an互d松th誓e迷st概re押ss秤es趕a鬧re艷e恰ss傻en啦ti瞎al毒lyth謊e遭sa斯me梁i濕n揮th五e霧pa左rt封s謙of根t任he蠶b贈od為y讓wh觸ic勾h環(huán)ar釣e咬at呈l握ar氧ge攝d錦is碑ta件nc葬esin佩c口om溜pa衫ri窮so飼n域wi跡th濫t候he近l圾in變ea剃r鋼di告me維ns納io銹n揚of嚷t沃he瘡s墊ur照fa上ce寬o汽n京wh根ic濟h狐th扔e淚fo霧rc矛es梯a堂re余c蹤蝶ha融ng嘩ed判.By壺“st昂ati潛cal明ly陪equ用iva君len毯ts辛yst危ems粗”w鏈em文ean女th算at另the波tw駱os篩yst素ems竊ha緒ve柱the煉sa刻me脹res伍ult扎ant遣fo獸rce拔an慈dt末he艦sam都er件esu中l(wèi)ta偷nt