## A SOlVING EQUATION FOR MOMENT DISTRIBUTION METHOD

ZHU Xiaojiang, ZHANG Linnan,1), QIN Taiyan

China Agricultural University, Beijing 100083, China

Abstract

This paper transformed the gradual calculation process in the traditional moment distribution method into the solution of equations. The method can improve the calculation efficiency and accuracy. The equations obtained by this method are similar to the equations obtained by the matrix displacement method, but the unknowns are completely different. The derivation process of the equation is more concise and easier to understand. It can inspire students to think about the analysis of the matrix displacement method in teaching.

Keywords： moment distribution method; computational efficiency; precision

ZHU Xiaojiang, ZHANG Linnan, QIN Taiyan. A SOlVING EQUATION FOR MOMENT DISTRIBUTION METHOD. Mechanics in Engineering, 2022, 44(2): 415-418 DOI:10.6052/1000-0879-21-271

## 1 力矩分配法的方程解法原理

### 图1

$\left. \begin{array}{l} M_{B}+C_{CB}\mu_{CB}M_{C}=-M_{B}^{f} \\ {C_{BC}\mu_{BC}M_{B}+M}_{C}+C_{DC}\mu_{DC}M_{D}=-M_{C}^{f} \\ C_{CD}\mu_{CD}M_{C}+M_{D}=-M_{D}^{f} \\ \end{array} \right\}$

$M_{ij}=M_{ij}^{f}+\mu_{ij}M_{i}+C_{ji}\mu_{ji}M_{j}(i,j=B,C,D)$

## 2 力矩分配法的方程解法算例

$\left.\begin{array}{l}6.4 i \theta_{B}+2 i \theta_{C}=3.31 \\2 i \theta_{B}+8 i \theta_{C}+2 i \theta_{D}=1.38 \\2 i \theta_{C}+6.4 i \theta_{D}=-7.62\end{array}\right\}$

## 3 力矩分配法的方程解法一般方程及其解

$\begin{array}{l}M_{1}+C_{21} \mu_{21} M_{2}+0+\cdots+0=\\-M_{1}^{f}\\C_{12} \mu_{12} M_{1}+M_{2}+C_{32} \mu_{32} M_{3}+\\0+\cdots+0=-M_{2}^{f}\\0+C_{23} \mu_{23} M_{2}+M_{3}+C_{43} \mu_{43} M_{4}+\\0+\cdots+0=-M_{3}^{f}\\0+\cdots+0+C_{(n-2)(n-1)} \mu_{(n-2)(n-1)} \text {. }\\M_{(n-2)}+M_{(n-1)}+C_{n(n-1)} \mu_{n(n-1)} \text {. }\\M_{n}=-M_{(n-1)}^{f}\\0+\cdots+0+C_{(n-1) n} \mu_{(n-1) n} M_{(n-1)}+\\M_{n}=-M_{n}^{f}\end{array}$

$M_{ij}=M_{ij}^{f}+\mu_{ij}M_{i}+C_{ji}\mu_{ji}M_{j}$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Liu Tianyi, Chen Suwen.

Moment distribution formula method to seek exact solution of structures with three distribution points

Mechanics in Engineering, 2014, 36(2):207-209 (in Chinese)

Liu Maosui, Cheng Weimin.

Moment distribution method of one-off distribution

Mechanics in Engineering, 2007, 29(4):73-75 (in Chinese)

Zhang Yu.

A multi-joint moment distribution method for continuous beams

Mechanics in Engineering, 2003, 25(5):77-80 (in Chinese)

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