力矩分配法两种平衡方法的一致性证明  

Testification of Consistency for Two Joint Balancing Processes of Moment Distribution Method

作　　者：张年文 薛志成 ZHANG Nianwen;XUE Zhicheng(Architecture and Civil Engineering Institute,Guangdong University of Petrochemical Technology,Maoming 525000,China)

机构地区：[1]广东石油化工学院建筑工程学院,广东茂名525000

出　　处：《广东石油化工学院学报》2024年第1期91-95,101,共6页Journal of Guangdong University of Petrochemical Technology

基　　金：广东石油化工学院博士启动项目(2019bs009)。

摘　　要：同时节点平衡法和先后节点平衡法是力矩分配法的两种节点弯矩平衡方法,基于矩阵位移法的节点弯矩平衡方程组构建了这两种平衡方法的加权转角迭代式,结合两种平衡方法的求解过程,证明了:(1)两种平衡方法求解的方程组与节点弯矩平衡方程组相同;(2)同时节点平衡法和先后节点平衡法分别采用增量雅可比迭代法和增量高斯-赛德尔迭代法求解节点弯矩平衡方程组,增量雅可比迭代法、增量高斯-赛德尔迭代法分别与雅可比迭代法、高斯-赛德尔迭代法是一致的;(3)两种平衡方法计算的杆端弯矩与转角位移方程计算的杆端弯矩相同。分析结果表明,力矩分配法求解的变量是加权节点转角,权值为杆件在该节点的转动刚度之和,符号相反;加权节点转角大小等于迭代的节点不平衡弯矩之和。Simultaneous joint balancing(SJB)processes or consecutive joint balancing(CJB)processes are balancing processes for the moment distribution method.Based on the equilibrium equations of moments at joints by the matrix displacement method,the iteration formulations were formed to solve the weighted rotations for both processes.According to both solution procedures,the following were proven:(1)the system of equations solved by both processes were identical with the system of the equilibrium equations for the moments of the joints;(2)SJB process and CJB process adopted the incremental Jacobi method and the incremental Gauss-Seidel method respectively to solve the system of equations,and the incremental Jacobi method and the incremental Gauss-Seidel method were consistent with the Jacobi method and the Gauss-Seidel method respectively;(3)the moments at the ends of all the members solved by both processes were identical with the moments figured out by the slope-deflection equation.They demonstrated that the unknowns solved by the moment distribution method were the weighted rotations whose weights were the total stiffness factors at the joints with opposite signs.A weighted rotation equaled the sum of the unbalanced moments of the joint in the iterations of all the cycles.

关 键 词：力矩分配法 雅可比法 高斯-赛德尔法 矩阵位移法 

分 类 号：TU311.4[建筑科学—结构工程] O342[理学—固体力学] O241.6[理学—力学] O151.21