结构力学(第五版)第十章-矩阵位移法

第十章矩阵位移法§10-1概述§10-2矩阵位移法解连续梁§10-3矩阵位移法解平面刚架§10-4矩阵位移法解平面桁架矩阵位移法是以结构位移为基本未知量，借助矩阵进行分析，并用计算机解决各种杆系结构受力、变形等计算的方法。理论基础：位移法分析工具：矩阵计算手段：计算机§10—1概述基本思想:•化整为零------结构离散化将结构拆成杆件,杆件称作单元.单元的连接点称作结点.•单元分析对单元和结点编码.356412123456e单元杆端力•集零为整------整体分析单元杆端力结点外力单元杆端位移结点外力单元杆端位移(杆端位移=结点位移)结点外力结点位移基本未知量:结点位移§10—2矩阵位移法解连续梁一.离散化----整体编码1P2P3Pii1ii2ll1ll212123(1)(2)(3)12----单元编码1,2,3----结点编码(1),(2),(3)----结点位移编码结点位移逆时针为正,结点弯矩逆时针为正二.单元分析建立单元杆端力和单元杆端位移的关系.1P2P3Pii1ii2ll1ll212123(1)(2)(3)----单元杆端力1,2----局部编码单元分析的目的:e1eeeFFF21eieeF1eF2e212----单元杆端位移eee21单元杆端弯矩和单元杆端转角逆时针为正.e1eieeF1eF2e212ei4e1eF1eF2e2ei21e1ei4ei2e21eeeeeiiF21124eeeeeiiF21242eeeeeeiiiiFF21214224简记为eeekF---单元刚度方程其中称作单元刚度矩阵(简称作单刚)ek单元刚度矩阵中元素的物理意义e1eieeF1eF2e212e1ei4eF1eF2e21e1ei2ei4e2ei21eeeeeeeeeiiiikkkkk422422211211eijk---发生位移时在i端所需施加的杆端力.0,1eiej单元刚度矩阵性质:对称矩阵三.整体分析整体分析的目的:建立结点力与结点位移的关系.3312211111kkkP1P2P3Pii1ii2121231231=111k31k=1212k=1322k32k13k23k33k21k3322221212kkkP3332321313kkkP321333231232221131211321kkkkkkkkkpPP简记为kP---结构刚度方程--结构刚度矩阵(总刚)k11111kk1111k121k12121kk031k11212kk21112222kkk22132kk013k21223kk22233kk1112k122k1212k222k211k221k333231232221131211kkkkkkkkkk简记为kP1P2P3Pii1ii212123123---结构刚度方程--结构刚度矩阵(总刚)k11111kk12121kk031k11212kk21112222kkk22132kk013k21223kk22233kk单元刚度矩阵中元素的物理意义ijk---发生其它结点位移为零位移时在i结点所需加的结点力.,1j结构刚度矩阵性质:对称矩阵总刚的形成方法---“对号入座”k1221211121111kkkkk21212121321111k112k121k122k3212222212122112kkkkk21213232211k221k212k222k00kP四.计算杆端力计算结点位移eeekF计算杆端力kN.m611i22ikN.m3kN.m3例:计算图示梁,作弯矩图解:1.离散化求单元刚度矩阵121232.计算总体刚度矩阵、总荷载8404122024k42241k2121212184482k21213232336P4.求杆端力2/766/112/1742241F32/124/116/184482F67/21/233.解方程,求位移kP24/116/112/173218404122024336eeekFM-图(kN·m)3321368404122024P03002/33602/342241F000084482F063五.(零位移)边界条件处理方法:先处理法后处理法kN.m611i22ikN.m33P12123(1)(2)(3)后处理法:置0置1法乘大数法(1)置0置1法000100361000122024321M-图(kN·m)11i22ikN.m3作弯矩图练习:123(1)(2)(3)123132138404122024PP031030100012000132104/1032112/14/1042241F1204/184482F1/2121M-图(kN·m)3321368404122024P03(2)乘大数法若,则将总刚主对角元素乘以大数N.0iiik3321368404122024PN第三个方程变为:3321840PN)8/()40(2133NP03kN.m611i22ikN.m33P12123六.非结点荷载(1).等效结点荷载1EP2EP3EP321EEEEPPPP---结构等效结点荷载“等效”是指等效结点荷载引起的结点位移与非结点荷载引起的结点位移相同(2).等效结点荷载的计算P8/)12/8/(12/22321PlqlPlqlPPPPEEEEqPl2/l2/l12123q12/2ql8/Pl12/2ql12/2ql8/Pl12/2ql1282qlPl8/Pl8/Pl12/2ql1282qlPl8/Pl012/012/224321qlqlPPPPPEEEEElqll123412312/2ql12/2qllqll练习:求图示结构的等效结点荷载.12/2ql12/2ql1012/2ql12/2ql212/2ql12/2ql304(2).等效结点荷载的计算qPl2/l2/l12123“对号入座”形成结构的等效结点荷载1qF单元固端力:荷载引起的固端弯矩.逆时针为正.记作eqFq112/2ql12/2ql12/2ql12/2ql8/Pl2P8/Pl2qF8/Pl8/Pl单元固端力改变符号称为单元等效结点荷载,记作eEF1EF12/2ql12/2ql2EF8/Pl8/Pl由单元等效结点荷载“对号入座”可形成结构等效结点荷载2132EP8/Pl8/Pl32112/2ql12/2ql2121q112/2ql12/2ql1qF12/2ql12/2qlEP0432112/2ql12/2ql练习:求图示结构的等效结点荷载.lqll12341232q12/2ql12/2ql1EF12/2ql12/2ql21212qF12/2ql12/2ql32212EF12/2ql12/2ql12/2ql12/2ql12/2ql012/2ql0(3).结构综合结点荷载qPl2/l2/l12123Pl00PlPD---直接结点荷载8/)12/8/(12/22PlqlPlqlPE---等效结点荷载8/8/12/12/22PlPlqlPlqlPPPED---结构综合结点荷载(总荷)(4).最终杆端力计算qPl2/l2/l12123Pl8/8/12/12/22PlPlqlPlqlPkP---计算结点位移eeekF---计算杆端力12/2ql8/912/2Plql8/PlPq8/Pl12/2qleqF七.例题矩阵位移法解图示梁,作M图.m4kN/m4kN1061EI242EI61EI2m1m8m4解:1.离散化12341232.求总刚35.15.18/641k84412/2442k35.15.133k21212121213232212143214335.1005.1114004115.1005.13km4kN/m4kN1061EI242EI61EI2m1m8m412341233.求总荷0DPkN/m424812/2ql48108/Pl1kN101010101qF48482qF2132212110101EF48482EF0483810EP0483810Pm4kN/m4kN1061EI242EI61EI2m1m8m4123412335.1005.1114004115.1005.13k总荷：0483810P4.边界条件处理048381035.1005.1114004115.1005.1343210483801000011400411000014321总刚：m4kN/m4kN1061EI242EI61EI2m1m8m4123412304838010000114004110000143215.解方程0476.681.5043216.求杆端力