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1、结构力学第十一章第十一章 结构结构的弹性稳定的弹性稳定11-1 概述11-2 有限自由度体系的稳定11-3 用静力法确定弹性压杆的临界力11-4 用能量法确定弹性压杆的临界力11-5 组合压杆的的稳定11-6 刚架的稳定11-7 拱和窄梁的稳定11 Stability of Elastic Structures311-1 概述概述不同的平衡状态稳定不稳定随遇平衡临界状态11-1 Introduction哪个是稳定的？重心高度不同稳定不稳定稳定临界状态稳定不稳定力的方向不同支点的相对位置不同4pFO第一类失稳PcrF分支点分支点失稳pFlAB非线性分析pFpF直线状态（未失稳）平衡；弯曲状态（失
2、稳）也平衡。两个状态之间有明确分界点直线和弯曲状态都能平衡5第二类失稳极值点pFO极值点失稳PcrFqpFpFpFpFpFlAB进入失稳状态没有明确分界点荷载偏心荷载偏心侧向荷载偏心距不同轴线不直侧向荷载6跳跃失稳pFOPcrFpFABC第三类失稳电灯按键式开关关开发卡的两个平衡状态7分支点pFOPcrFy11-2 有限自由度体系的稳定有限自由度体系的稳定11-2-1 静力法pcr30EIF llpcr23EIFlpcr0F ykylpcrFklpFllEI1EI ABC0AM0AMpFl1EI ABpcrFkyypcrF3EIl11-2 Stability of Finite Degrees
3、 of Freedom Systemsk在变形状态8例11-11y2ylllpFEI EI EI kkABCD230yByCyDFlFlFl yBFyCFyDF0AM12(2)/3yDFkyky整体CD杆0CMp20yDFyFl 1yBFky2yCFkyp121p2(32)0(32)0Fkl yklyklyFkl yBD杆0BMp120yCyDFyFlFl p1p232032FklklyklFklypp32032FklklklFkl22pp34()0FklFkl解：p1212p2122(2)/30(2)/30F yklyl kykyF yl kyky稳定方程（特征方程）一般地，n个稳定自由度的
4、结构的稳定方程为n阶行列式，有n个解答，有n个失稳位移形态BC9例11-1p121p2(32)0(32)0Fkl yklyklyFkl ypcr2Fklpcr1/3Fklp2132Fklyykl/3kl2y1ykl2y1y解：211yy 22pp34()0FklFkl211yy1y2ylllpFEI EI EI kkABCDyBFyCFyDF1011-2-2 能量法122221()xByllylll 2(1)(1)12!nn nxnxx 22ylppxBUF 212UkyxByllpFAB12n 2()1yxl 22(1)2ylll ypFl1EI AB22xBylkcosxBll 21(1)
5、2!ll 22lyl22xBl2411cos12!4!1111-2-2 能量法ppxBUF 212Uky22p122ykyFl2pppxB12EUUkyF2p1()2klF ylp0E势能驻值原理体系取得平衡的充分和必要条件是：任意可能的位移和变形均使势能取得驻值。pp1()0EklF yylpcrFkl2xB2yl2xB2l2pp21()0EklFlypFklxByllpFABypFl1EI ABk121y2ylllpFEI EI EI kkABCDp1121p1()0EyyykyFyll212xDyl2221122ppDp()222xyyyyUFFlll 22121122Ukyky例11-
6、1另解：222221122pp12p()1122222yyyyEUUkykyFlllp1p2(2)0klF yF yp1222p2()0EyyykyFyllp1p2(2)0F yklF ypp1pp2202klFFyFklFy22pp34()0FklFklpcr21pcr211klFyyFpcr2Fklpcr1/3Fkl212()2yyl222yl1311-3 用静力法确定弹性压杆的临界力用静力法确定弹性压杆的临界力无限自由度体系的稳定问题pFlABxy0M pR()0F yF lxM0M xy0y MEIy pR()EIyF yF lx pR()FFyylxEIEI 22Rp()FyylxF
7、 p2FEIRp()cossin()Fy xAxBxlxFRp(0)0FyAlF通解pFQFpFxyABxMRFRp(0)0FyBF()cossin0y lAlBlRp10010cossin0/lABllFF边界条件tan0ll4.493l2pFEI2pcr2220.19(0.7)EIEIFlly10010cossin0lll11-3 Critical Load of Elastic Columns-Static Method14各种压杆的临界力2pcr2()EIFlpFpFpFpFlpFEI0.50.71122cr2EIli2IiA15pFpFlhEIEI1EIpFpFEIEI1EIpFpF
8、EIEI1EIpFEI2kpFEI1k2lEIpF2lEIpF16例11-32Pcr2EIFlpFpFlEIEI1EI 1EI ACEBDFEI(1)AB杆单独失稳(2)结构侧移失稳NpBDFFlR33EIFklpRN()()()DBMFyF lxFlx EIyM pp33()()()EIFylxF lxllppp313()()FEIyyFlxF lxEIEIllNBDFpFABRFEFpFpFABCEFpFpFABCDFEypFRFNFBDxxMCDy1723()cossin2()lxyAxBxlxll23sincos1()yAxBxll pFRFNBDFyxxyM23(0)20()lyAl
9、ll23(0)10()yBll()cossiny lAlBl223102()0301()/cossin0llAlBllll22331tan20()()llll1.645l22.706EIl22(1.91)EIl2223()2()lxyylxllp2FEIpp23(2)()FFyylxlxEIEIll2pFEI1811-4 用能量法确定弹性压杆的临界力用能量法确定弹性压杆的临界力EIyM 201()d2lEI yxppEUU201d2lMUxEIppUF ddxdyABdxxyxdx201()d2lyx2(d)d2dyx21()d2yx2p01()d2lFyx 22p0011()d()d22ll
10、EI yxFyxpFlAB()sinxy xal()ax lx2012()()x lx aa xa x()()iiiy xax()ix满足边界条件22xByl11-4 Critical Load of Elastic Columns-Energy Method19()()jjy xxa22pp0011()d()d22llEEI yxFyx1()()niiiy xax0ijiic a 111211212222120nnnnnnncccacccaccca1,2,jnp00()()d()()dllijijijcEIxxxFxxx1,2,jnp001()()d()()d 0nlliijijiaEIxx
11、xFxxx pp00dd0lljjjEyyEIyxFyxaaa1()()niiiy xax1()()niiiy xax()()jjy xxap0011()()d()()d0nnlliijiijiiEIaxxxFaxxx1122()()()nnaxaxax()()jjy xxa0ijc1112121222120nnnnnnccccccccc1,2,jn20例11-5 能量法求临界力pFl(1)1()(1 cos)2xy xal1()sin22axy xll212()cos24axy xll201()d2lUEI yx421364aEIl2pp01()d2lUFyx 221p16aFl ppEUU
12、422211p31664aaEIFll42p11p3101664EaaEIFall2p2(2)EIFl(0)(0)0yyxy与精确解一致，因为采用的变形曲线就是真实的变形曲线。21(2)321()()3xy xa xl(0)(0)0yy21()(2)xy xaxl1()2(1)xy xal201()d2lUEI yx2pp01()d2lUFyx ppEUU3pp111420315EF l aEIlaa2123EIla3p21415F la 3p221142315F lEIlaaPcr22.5EIFl比精确解高1.32%pFlxy22(3)2312()y xa xa x(0)(0)0yy212(
13、)23y xa xa x12()26y xaa x201()d2lUEI yx2pp01()d2lUFyx 4pp2121214pp212122()2(23)(4045)030()2(36)(4554)030UUF lEIlaa laa laUUF lEIlaa laa la22 211222(33)EIl aa a la l3p22 21122(204527)30F laa a la l 2pF lEI1212(248)(369)0(205)(406)0alaala248(369)0205(406)ll2.486p22.486EIFl比精确解高0.75%。也比前面的（1.32%）精确高，因为
14、前面的函数只是这里任意3次函数的子集。pFlxy23(4)21()y xa x(0)(0)0yy1()2y xa x1()2y xa201()d2lUEI yx2pp01()d2lUFyx 4pp2111()4403UUF lEIl aaa212EIla32123pF la p23EIFl比精确解高21.59%pFlxy因为()y x=常数，意味着弯矩为常数，与事实不符。误差过大。从数学角度理解：求导降低了计算精度！临界力的所有近似解答都比精确解大，因为位移近似假设相当于增加了约束，提高了结构刚度，所以提高了临界力。所以：在假设位移的方法中，较小的值更精确。24例11-32Pcr2EIFlpF
15、pFlEIEI1EI 1EI ACEBDFEI(1)AB杆单独失稳(2)结构侧移失稳321()()3xy xa xl21()(2)xy xaxl2p2p01()d2lUFyx 222p101(2)d2lxF axxl 32p1415F l a 323112()()33ly la ll al211()2Uky l32131 32()23EIa ll2123EIla211423UUEIa l2p1p1()2UFy ll 32p129F l a pFpFABCEFx25例11-3p3p11d422()0d345EEIlF laapcr22302.72711EIEIFll比精确解高0.78%pp1p2
16、EUUU2143UEIa l32p1p129UF l a 32p2p1415UF l a 22 31p1422345EIa lF a lpFpFlEIEI1EI 1EI ACEBDFEI26例11-722()()y xax lx(0)()0yy l22()(3)y xa lx()6y xax 201()d2lUEI yx2201(6)d2lEIaxx326EIl a2p21d()()d2Uq lx yx 2p201()()d2lUqlx yx 222201()(3)d2lqalx lxx 62320ql a 223p101()d2lUqlyx 22222301(3)d2lqlalxx 6274
17、5ql a ql23lqlyxABpp1p2EUUU332376()2045EIql l apd0dEacr3637()2045EIql319.64EIlxyxdx2711-6 刚架的稳定刚架的稳定刚架在竖向荷载作用下的失稳通常属于丧失第二类稳定经常简化为第一类稳定hl4lqEIEI2EI2ql34ql11-6 Stability of Frames23il128轴向压力的增加会降低弯曲刚度pF13ipF考虑轴力对弯曲刚度影响（二阶效应）的结构分析称为二阶分析。（几何非线性分析）当轴力较大或挠度较大时需要二阶分析。如超高层建筑2911-6-1 刚架稳定性分析的位移法Qp()ABEIyMF xF
18、 y AxylABMABBBAMQFQFpFpFyxABMQFpFMQFpFpQABFFMyyxEIEIEI pFEIQ2()ABMF xuyylEI QcossinABpMF xuxuxyABllF0y 当 x=0时Ay y当 x=l时By p0ABMAFQpAFuBlFQpcossinABMF lAuBuFQpsincosBFuuAuBullFpQp01010001/cossin11sincos01ABABABMulFluuF luuuuF pQpABFMF xFEI Q2p()ABMF xuFl ulxy302121()()()ABABMiuuuul pFulEIEIil2p()uFEI
19、l2pF liupQp01010001/cossin11sincos01ABABABMulFluuF luuuuF 1pQp01010001/cossin11sincos01ABABABMulFluuF luuuuF p112Qp0*/*()()()*ABABABMlFluuuF lF 1214()2()6()ABABMiuiuiul Qp()BAABMMF lF 2112()4()6()ABiuiuiul Q11226612()()()ABiiiFuuulll AxylABMABBBAMQFQFpFpFyx31轴力作用下等截面超静定杆的杆端弯矩和剪力1214()2()6()ABABMiuiu
20、iul QQ11226612()()()ABBAABiiiFFuuulll 11tan()tan2412uuuuu21sin()tan2412uuuuu1121()2()()3uuu14()iu22()iuQFQ16()iFulpFullEI1pF2112()4()6()BAABMiuiuiul 32位移法分析第一类稳定性1p2p3p0RRR1112132122233132330rrrrrrrrr如果平衡，上式成立；1111221331p2112222332p3113223333p000r Zr Zr ZRr Zr Zr ZRr Zr Zr ZRlllpFpFI2III1Z2Z3Z111122
21、133211222233311322333000r Zr Zr Zr Zr Zr Zr Zr Zr Z显然Mp图是零，0iZ 如果发生变形并且平衡，上式有非零解上式有非零解的条件是：基本结构如果发生变形33例11-7 求临界力12uu111114()riiu22184()riiu12214rri11114()40484()iiuiiiiu211()4.75()4.50uu1()1.307u 3.4435.46u 2Pcr2229.81EIEIFulllllpFpFI2IIIpFpFI2III1Z8i4i14i11Z 2ZpFulEI4i8i14()iu22()iu11r21r12r3i22i1
22、1tan()tan2412uuuuupFpFI2III22r21Z 111221220rrrr3411-6-2 刚架稳定性分析的有限单元法21d()d2evxd()(d)deev v xIpdd()WFep dF v v xd()ed()vdvdxdeIp0dlWF vv xTp220()()dlFxNN 23223()1 3()2()(1)3()2()(1)iijjxxxxxxxv xvxvxlllllllT2 N T232232()1 3()2()(1)3()2()(1)xxxxxxxxxxlllllllNT()iijjxvvTTp220dlFxNNTTTdeeeeVEV FBBTdeVE
23、VkB BTTp220dlFxNNT22deplFxkNNT()ee kk()eeeFkk35几何刚度矩阵Tp22delFxkNNT232232()1 3()2()(1)3()2()(1)xxxxxxxxxxlllllllN2p2236336334336336330334llFlllllllllll2p220000000363036303403000000300363036303034ellFlllllllllllk考虑轴向变形36临界力方程()eeeFkk当Fp压为正时无弯曲荷载时有非零解时p0FKK单元刚度方程p()FFKK结构刚度方程p()FFKKp()0FKK临界力方程的一般推导方法3
24、7例11-8 求临界力pFl/2l/2l1234yx213233v32322232322212612664621261266264eeEIEIEIEIllllEIEIEIEIllllEIEIEIEIllllEIEIEIEIllllk0ivijjvEAexylyiFyjFjMiMijivjvjiivjvij0010001012301232300230022322122442419204016lllEIllllK1 结构标识2 结点位移向量3 单元刚度矩阵4 结构刚度矩阵38例11-8 求临界力0ivjvjiivjvij0010001012301230022p2246628806008lllFll
25、llK22236336334336336330334pellFlllllllllllk1 结构标识2 结点位移向量3 单元几何刚度矩阵4 结构几何刚度矩阵232300pFl/2l/2l1234yx213233v390KK22322122442419204016lllEIllllK222246628806008plllFllllK222232222122444624192062880060401608pllllllFEIllllllllpcr228.97EIFl22224(3)6(4)(4)6(4)96(23)00(4)08(2)llllll2pF lEI28.97222222221224446241920628800401608llllllllllll5 结构稳定方程临界力40总结总结22122yll201()d2lyxMEIy 201()d2lUEI yx2pp01()d2lUFyx p()0iUUa静力法能量法刚杆弹性杆ppUF 应变能刚杆 建立平衡方程弹性杆pFlABxypFQFpFxyABxMRFy结束