弹性力学课件二零

10TorsionofPrismaticalBars10.1Stressesanddisplacements10.2Membraneanalogy10.3Ellipticbarsintorsion10.4Rectangularbarsintorsion例：图示为等截面直杆（不计体力），两端平面内受方向相反的两个力偶作用，取如图所示的坐标系，求应力分量。xyzMMo如何计算？Semi-inversemethod10.1StressesanddisplacementsConsiderastraightprismaticalbartwistedbytwooppositecouples,eachwithmomentM,ontheendplanesofthebar.分析如下：由材料力学可知，杆件横截面上有剪应力zx、zy，其他应力分量为零，即x=y=z=xy=0，体力X=Y=Z=0Weshallsolvetheproblemintermsofstressesandapplythesemi-inversemethod.Beinghintedbythestressdistributiongiveninmechanicsofmaterialsforacircularshaft,weassumethatonlythetorsionalstressesaredifferentfromzero,whichotherstresscomponentsvanish.将上述应力分量代入平衡微分方程：Substitutingtheseintodifferentialequations（1）（2）（3）由（3）得：由（1）、（2）可知，zx、zy是x、y的函数ThefirsttwoofEqs.showthatthetorsionalstresszx、zyarefunctionofxandyonly.或写成：将上述表达式代入相容方程：Rewrittenas由微分理论可知，一定存在一个函数（x，y）使得：Hence,thereexistsacertainfunctionsothat前四式自然满足，最后两式为：将代入上式这说明：Hence,mustbeconstant1、杆件侧面：边界条件xyzMMon=0，X=Y=Z=0代入边界前两式恒满足，第三式为：ConsidertheboundaryconditionsOnthelateralsurfaceThefirsttwoEqs.AreidenticallysatisfiedwhilethethirdequationreducestoSo：所以，以上边界条件可写为：Since：ontheboundaryofacross-section,wehaveShowsthatmustbeconstantalongtheboundaryofacross-section常数如何取？Since所以在应力函数上增加或减少一个常数，对应力分量无影响，可取s=0（在单连域实心柱的情况下）thestresscomponentsarenotaffectedbyadditionofaconstantto.Hence,inthecaseofsimplyconnectedcross-section,forsimplicity,wemaytaketheboundaryvalueofaszero.在多连域情况下，每个边界上的为常数，但每个常数并不相同，只能取其中一个边界的为零.Inthecaseofmultiplyconnectedcross-section,theconstantvaluesofondifferentboundarycurvesmaydifferfromeachother.Hence,thevalueofcanbetakenaszeroalongonlyoneoftheboundarycurves,whilethevaluesofonallotherboundarycurvesmustbedeterminedbytheconditionofsingle-valueddisplacement.2、杆件端面（上端）：边界条件xyzMMol=m=0n=-1代入边界条件得：且两端的面力必须合成为力偶：Oneitherendplane(seetheupperone)SincethesurfaceforcecomponentsmustcomposedacouplewithmomentM,itisrequiredthat验证如下边界条件：xyMoABA、B为边界上A、B点的值，取零，所以同理，可证明：WhereA、BarethevaluesofattwopointsAandBontheboundary,whichmustbezero.其中的一项用分部积分：xyMoABA、B为边界上A、B点的值，取零，所以WhereA、BarethevaluesofattwopointsAandBontheboundary,whichmustbezero.同理所以：Thus归纳Tosumup：

等截面直杆的扭转问题最终归结为求应力函数（x，y），使它满足且s=0，然后由下列公式求应力分量：Tosumup,inatorsionproblem,ifwecanfindafunctionwhichsatisfies,s=0,,then,thetorsionalstressescanbefound.求位移分量：（将应力分量代入物理方程）Nowderivetheformulasfordisplacementcomponents.Substitutionofstresscomponentsintophysicalequations.将几何方程代入上述表达式，得：Substitutionofgeometricalequations.u0,v0,x,y,z是积板分常嗽数，换代表警刚体暮位移讨，k也是币积分乘常数岭，若紫不计唤积分猾常数帖，则如果用祥柱坐标禽表示，若则u0,v0,x,y,供zare匆ar石bit酸rar捧yc洋ons诸tan元ts投rep暂res颗ent海ing绘th镇er府igi鞋d-b愤ody嫩mo才vem怠ent旗so弃ft河he黄bar剥.A押ssu惰min慰gt至hat俊th倘eu画ppe摇r-e恭nd岁of坛the铲ba即ri炎sc言ons栗tra学ine窑dt网op泽rev耍ent妖th失ese说ri绑gid际-bo领dy淋mov骨eme长nts份,w散eh求ave聪th觉es止imp从le掉exp罗res拒sio裁ns.In当p至ol双ar烤c科oo送rd案in惑at攻es脏,t狮he去d浓is夹pl矿ac好em龄en膝t在co使mp最on姐en档ts暮w展il膏l盏beThusitisseenthateverycross-sectionofthebarrotatesanangle=kzaboutthezaxisandrepresentsthetwistperunitlength,thatis,therelativeangulardisplacementoftwocross-sectionsatunitdistanceapart.由此编可得注结论究，每繁个横忧截面恋在xy面上谊的投骗影不伏改变彩形状爹，只吹是转净过一劫个角伟度=kz，且单位注长度的蚕扭转角歉：将u=-kyz，v=kxz代入下式：由此细，可奇以求迎得位帐移分摧量w且由以上两