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1、第六章弹塑性地震反应分析GeneralThe analysis process, which leads to the evaluation of seismic actions and deformations, invokes knowledge from several sub-disciplines in engineering. To perform reliable seismic analyses of structural systems, a conceptual framework is required.DefinitionsStiffnessThe ability of
2、 a component or assembly of components to resist deformations when subjected to actionsStrengthThe capacity of a component or an assembly of components for load resistance at a given response state.DuctilityThe ability of a component or an assembly of components to deform beyond the elastic limit.De
3、finitionsDemandThe action or deformation imposed on a component or assembly of components when subjected to earthquake ground motion.It continuously varies as the structural characteristics vary during inelastic response.SupplyThe action or deformation capacity of a component or assembly of componen
4、ts when subjected to earthquake ground motion.It may continuously vary as the structural characteristics change during inelastic response.General GeneralGeneralTypical methods of analyses and relative earthquake input representations目录6.1 弹塑性动力分析的一般过程6.2 结构模型6.3 多维地震波作用下的平扭耦联系统6.4 非线性地震反应分析的Ritz向量叠加
5、法6.1 弹塑性动力分析的一般过程6.1.1 动力方程6.1.2 刚度修正技术6.1.3 一般分析过程6.1.1 动力方程MU+CU+KU=-MUg1 jjjFFKU结构非线性增量方程MDUj+CDUj +KDUj=-MDUgjMU+CU+F=-MUgfj+1= fj+kjDUj全量 vs 增量?负刚度条件下数值积分法的稳定性刚度条件正刚度负刚度中心差分法条件稳定无条件稳定Z变换法无条件稳定无条件稳定Wilson 法1.37无条件稳定条件稳定1.37条件稳定条件稳定二步Adams显式方法条件稳定无条件稳定Newmark 法 = 1 /4、 = 1 /2常用精度较低6.1.2 刚度修正技术截面M
6、-关系截面抵抗矩单元刚度矩阵总刚度矩阵层间力变形关系刚度系数恢复力模型的运行 与刚度修正广义加载卸载两类拐点广义加载点卸载点拐点的处理近似方法二分法：计算工作量很大，且在卸载点 处，由于相应结构单元的应变速度为零， 或与相应结构构件有关的运动质点间的 相对速度为零等缘故不易恰当地确定 应给出的允许误差。插值法：采用线性插值，精度较低；泰勒展开法：会导致错误的时间点，这 是由于只利用了t时刻的结果，没有利用 t+t时刻的信息。拐点的处理精确方法对于Newmark-法和Wilson-法，由于它们不 仅在积分点上给出数值计算结果，同时方法本身已 经确定了整个积分时间步长上的加速度、速度和位 移分布，
7、可以精确地确定拐点：平均常加速度法 拐点的确定mut+Dt+cut+Dt+kt+Dtut+Dt+qt+Dt=gt+Dtut+t=ut+t 2(ut+ut+Dt)ut+t=ut+tut+t24(ut+ut+Dt)(0) t gt+Dt=gt+Dt-qi+Dt-c(ut+Dt 2ut)-kt+Dt(ut+Dtut+Dt24ut)(D t )A,C=2( ut-(ut+ut+Dt)(ut-Dyg) -ut/(ut+ut+Dt)(D t )B=-2ut/(ut+ut-Dt)(D t )B=Dtut/(ut-ut+Dt)mt+Dtut+Dt=gt+Dt224ttttttmmck或或拐点处理存在的问题拐点
8、“死循环”：由于拐点处理后可能出现很小的补 差步长，而精确法使得卸载点处的速度值非常准确地 位于零附近，造成舍入误差引起的构件受力状态的反 向前进，形成两类拐点之间的交替循环。对于层间模型分析问题，采用拐点处理尚有价值，而 对于自由度较多的杆系模型，则意义不大，有时细分 不当甚至会导致计算的发散。由于动力问题荷载反复 作用的特点，当时间步长取得较为合适时，动力非线 性分析不会如静力非线性分析那样出现严重的位移反 应失真现象，对于杆件较多的大型结构，尤其如此。拐点处理的缓和曲线法非平衡力的处理成因 1.非线性恢复力在t时间内的割线斜率 以切线斜率代替 2. 假定t时间内，刚度矩阵保持不变处理方法
9、 积分一步后，计算出不平衡力，加到下 一步的外力增量中去6.1.3 一般 分析过程输入：结构参数，地震波计算恢复力基本参数形成初始刚度矩阵及质量 矩阵与阻尼矩阵形成T=0时的初值T=T+T计算位移增量、内力增量迭加给出T时刻的地震反应值将位移速度反应值转化为单元变形、 变形速度值判断刚度是否发生变化修正单元刚度重新合成总刚度矩阵是否迭代计算重新计算T时刻地震反应输出控制点反应值是是计算是否结束 是否否否结构总刚计算流程图t=0输入各单元初始内力和初 始变形状态由各单元内力及内力增量判别单元变形状 态（弹性或非弹性）由各单元变形状态、单元内力及滞回模 型确定单元刚度由各单元刚度形成结构 总刚K(
10、t)计算结构变形增量由结构变形增量确定单元 变形增量由单元变形增量和单元刚度确定 单元内力增量将单元内力与单元内力增量相加赋值 为新的单元内力t=t+tt预定时 间结束是否6.2 Structural Modelling6.2.1 General6.2.2 Lumped mass stick model6.2.3 Frame model6.2.4 Fiber model and multiple spring model6.2.1 GeneralThree main levels of modelling are generally used for earthquake structural
11、 response analysis (combinations and intermediate levels exist): Substitute models; Stick models; Detailed models.6.2.1 GeneralThree main levels of modelling are generally used for earthquake structural response analysis (combinations and intermediate levels exist): Substitute models; Stick models;
12、Detailed models.6.2.1 GeneralSubstitute (or SDOF) modelsThe structure is idealised as an equivalent SDOF system or substitute system. Four parameters are needed to define the substitute system: effective mass Meff, effective height Heff, effective stiffness keffand effective damping eff.For inelasti
13、c systems, the effective stiffness keffmay be assumed as the secant stiffness at some given displacement, while eff, which is utilised to quantify the energy dissipation, is assumed as the equivalent viscous damping.Substitute models are inadequate to assess local response of structures, although th
14、ey are very efficient for global analyses.6.2.1 GeneralSubstitute (or SDOF) modelsThe structure is idealised as an equivalent SDOF system or substitute system. Four parameters are needed to define the substitute system: effective mass Meff, effective height Heff, effective stiffness keffand effectiv
15、e damping eff.For inelastic systems, the effective stiffness keffmay be assumed as the secant stiffness at some given displacement, while eff, which is utilised to quantify the energy dissipation, is assumed as the equivalent viscous damping.Substitute models are inadequate to assess local response of structures, although they are very efficient for global analyses. 1and are the natural frequency of the system and the frequency of the applied load6.2.1 GeneralStick models These consist of MDOF systems in which each element repesents a number of members of the p
