第13章-结构弹性稳定

第十三章结构弹性稳定§13-1概述一.第一类稳定问题(分支点失稳)lEIP22lEIPcr---临界荷载crPP稳定平衡crPP随遇平衡crPP不稳定平衡qPP不稳定平衡状态在任意微小外界扰动下失去稳定性称为失稳(屈曲).两种平衡状态:轴心受压和弯曲、压缩。---第一类稳定问题完善体系二.第二类稳定问题(极值点失稳)偏心受压三.分析方法大挠度理论。第二类稳定问题PP有初曲率小挠度理论。静力法能量法四.稳定自由度在稳定计算中,一个体系产生弹性变形时,确定其变形状态所需的独立几何参数的数目,称为稳定自由度。非完善体系PEI1个自由度PPEI2个自由度无限自由度§13-2.用静力法确定临界荷载一.一个自由度体系0AM0sinPlk小挠度、小位移情况下：kPEIlk1抗转弹簧Asink0)(Plk00Plk----稳定方程（特征方程）lkPcr/---临界荷载二.N自由度体系0BM0)(121yyPlky（以2自由度体系为例）0)2()(plkPPklkl----稳定方程02klPklPPkl---临界荷载klAPEIlk1y2y1ky2kyB0AM02112Pylkylky0)(21PyyPkl0)2(21klyyPlk03222lkklPPklklklP382.0618.2253klPcr382.0618.112yy---失稳形式P11.618三.无限自由度体系)()(xMxyEI00sincos1001nlnlnl)(xlQpyMEIPn2PEIlxyxy挠曲线近似微分方程为QPMQ)()(xlQPyxyEI或)()(xlEIQyEIPxy令)()(22xlPQnynxy通解为)(sincos)(xlPQnxBnxAxy由边界条件0)(,0)0(,0)0(lyyy得0lPQA0PQBn0sincosnlBnlA稳定方程0sincosnlnlnlnlnltan00sincos1001nlnlnlPEIlxyxyQPMQ得0lPQA0PQBn0sincosnlBnlA稳定方程0sincosnlnlnlnlnltannly22325nlnly)(nlnlytan)(经试算493.4nl485.4tannlEInPcr222/19.20)493.4(lEIEIl§13-3.具有弹性支座压杆的稳定lEIk3PEIlEIkPk1练习:简化成具有弹簧支座的压杆PEIlEIlEIPEIlEIEAkPlEIk6PEIk33lEIkEIkPlAyyxkQPMQ)()(xMxyEI)(xlQpyM挠曲线近似微分方程为)()(xlQPyxyEI0AMkQlEIPn2令)()(2xllEIkynxy通解为)(sincos)(xlPlknxBnxAxy边界条件0)(,)0(,0)0(lyyy0PkA0)1(PlkBn0sincosnlBnlA00sincos)1/(0/01nlnlPlknPk稳定方程2)(1tannllkEInlnl解方程可得nl的最小正根EInPcr2EIkPlAyyxkQPMQ00sincos)1/(0/01nlnlPlknPk稳定方程2)(1tannllkEInlnl解方程可得nl的最小正根EInPcr2lEIP22lEIPcrnl0k若0tannl0sinnlk若nlnltan2/19.20lEIPcrPEIllEIP22lEIPcrnl0k若0tannl0sinnlk若nlnltan2/19.20lEIPcrPEIlEIkPlEIlknlnltanPEI33)(tanklnlEInlnl例:求图示刚的临界荷载.PlPII21IIlPPPP正对称失稳反对称失稳正对称失稳时PPkk1lEIlEIk/42/22)(1tannllkEInlnl4/)(12nlnl83.3nl22/67.14lEIEInPcr例:求图示刚的临界荷载.PlPII21IIlPPPP正对称失稳反对称失稳反对称失稳时PklEIlEIk/122/2312tanEIlknlnl45.1nl22/67.14lEIEInPcrP0k122/10.2lEIEInPcr原结构的临界荷载为:2/10.2lEIPcr§13-4用能量法确定临界荷载一.势能原理2.外力势能1.应变能弯曲应变能P2/PVeldxM021拉压应变能2/PVeldxN021PP剪切应变能2/PVeldxQ0211231P2P3P外力从变形状态退回到无位移的原始状态中所作的功.iiePV*y(x)q(x)ledxxyxqV0*)()(3.结构势能*PePVVEEAlPPPViie2111*结构势能例:求图示桁架在平衡状态下的结构势能.EA=常数.45P1llA45解:杆件轴力2/211PN杆件伸长量EAlP112EAlPEAlN1122A点竖向位移外力势能应变能EAlPNVe2221211*PePVVEEAlPEAlPEAlP22212121EAlPPPViie2111*结构势能45P1llA45杆件轴力2/211PN杆件伸长量EAlP112EAlPEAlN1122A点竖向位移外力势能应变能EAlPNVe2221211*PePVVEEAlPEAlPEAlP222121214.势能驻值原理设A点发生任意竖向位移是的函数.PE,杆件伸长量2/2lEAN/杆件轴力lEA2/2应变能lEANVe22212外力势能1*PVe结构势能122PlEAEP])[(22121lEAPE10)(1lEAddEP1EAlPPlEAEP22)(2111211EAlP2214.势能驻值原理设A点发生任意竖向位移是的函数.PE,杆件伸长量2/2lEAN/杆件轴力lEA2/2应变能lEANVe22212外力势能1*PVe结构势能122PlEAEP])[(22121lEA0)(1lEAddEP1EAlPPlEAEP22)(2111211PE1EAlP221在弹性结构的一切可能位移中，真实位移使结构势能取驻值。满足结构位移边界条件的位移对于稳定平衡状态,真实位移使结构势能取极小值.二.能量法确定临界荷载例一:求图示结构的临界荷载.PEIlkyP解:应变能ykyVe21PPViie*外力势能2sin2cos2llllylyll2)(21)2(2222lPy22结构势能*PePVVE22ylPlk0ylPlkdydEPlkPcr由势能驻值原理得临界荷载例二:求图示结构的临界荷载.解:应变能22212121kykyVe]2)(2[21222*lyylyPPViie外力势能结构势能*PePVVE]2)(2[2121212222221lyylyPkykyklPEIlk1yP2y])2(2)[(21222121yPklyPyyPkll02211yyEyyEEPPP01yEP02yEP0])[(1211PyyPkllyEP0])2([1212yPklPylyEP02PklPPPkl03222lkklPPklklklP382.0618.2253klPcr382.0三.瑞利里兹法)(xyEIMPEIlPEIxyx)(xydsdxdsdxdy应变能ledxEIxMV02)(21ledxxyEIV02)]([21dxydxdxds2)(1]1))(1[(2/12ydx]1)(211[2ydxdxy2)(21dxydxdsll200)(21)(ledxyPPV02*)(2外力势能结构势能*PePVVElldxyPdxyEI0202)(2)(21设)()()()(2211xaxaxaxynn)(1xaiini将无限自由度化为有限自由度.结构势能则为的多元函数,求其极值即可求出临界荷载.naaa,,21lEIP22lEIPcrlxaxysin)(例:求图示体系的临界荷载.xyx)(xy解:1.设234024)]([21alEIdxxyEIVle2202*4)(2PaldxyPVle2234)44(aPllEIEP0)22(234aPllEIdadEP022234PllEI精确解:22lEIPcr212lEIPcr例:求图示体系的临界荷载.lEIPxyx)(xy解:)(4)(22xlxlaxy2.设精确解:22lEIPcr误差:+21.6%3.设杆中作用集中荷载所引起的位移作为失稳时的位移.l/2l/2Q)(xy)20()1216()(32lxxxlEIQxy令EIQla348)43()(33lxlxaxy210lEIPcr误差:+1.3%EIGAlPxyx)(1xy)(2xy)(1xy设弯矩和剪力影响所产生的挠度分别为和)(2xy22221222)()(dxydxdxyddxxydEIMy1同时考虑弯矩和剪力对变形的影响时的挠曲微分方程的建立:二者共同影响产生的挠度为)()()(21xyxyxy近似的曲率为弯矩引起的曲率为dxxdy)(2dxdMGAGAQdx2dyQQ截面形状系数矩形截面为1.2圆形截面为1.1122222)(dxMdGAdxxyd挠曲微分方程为2222)(dxMdGAEIMdxxyd§13-6剪力对临界荷载的影响EIGAlPxyx)(1xy)(2xydx2dyQQ22222)(dxMdGAdxxyd挠曲微分方程为2222)(dxMdGAEIMdxxyd对于图示两端铰支的等截面杆,有yPMPyM,2222)(dxydGAPEIPydxxyd0)1(yEIPGAPy令)1(2GAPEIPm0)()(2xymxy方程的通解mxBmxAxysincos)(边界条件0)(0)0(lyyEIGAlPxyx)(1xy)(2xydx2dyQQ对于图示两端铰支的等截面杆,有yPMPyM,2222)(dxydGAPEIPydxxyd0)1(yEIPGAPy令)1(2GAPEIPm0)()(2xymxy方程的通解mxBmxAxysincos)(边界条件0)(0)0(lyy0sinmlB0sinml稳定方程lmml/,)1(22GAPEIlPEIlGAEIlPcr22221kPEIGAlPxyx)(1xy)(2xydx2dyQQ0sinmlB0sinml稳定方程lmml/,)1(22GAPEIlPEIlGAEIlPcr22221kPEIlPk22不计剪变的欧拉临界力E