引用本文: 吴昊. 多结点力矩分配法精确计算研究. 力学与实践, 2023, 45(6): 1414-1419.
Wu Hao. Exact solution of moment distribution. Mechanics in Engineering, 2023, 45(6): 1414-1419.
 Citation: Wu Hao. Exact solution of moment distribution. Mechanics in Engineering, 2023, 45(6): 1414-1419.

## EXACT SOLUTION OF MOMENT DISTRIBUTION

• 摘要: 针对多结点力矩分配法轮流放松结点只能获得渐近解的不足，提出了一种将分配结点分层并按顺序放松、相邻结点不同时放松、放松后不重新固定的放松策略，即从远端只与基础相连的结点开始将其划分为若干层，按照同层相邻分配结点逐个或间隔放松的原则由外层向内层逐层移除结点约束，引入弹性支座模拟相邻杆件的转动约束作用，通过推导远端转动弹性支承等直杆弯矩形常数，确保结点放松时转动刚度和传递系数已知或可计算，结果表明，按顺序逐层移除约束的放松约束方式一方面各分配弯矩能够直接或通过已放松结点向支座传递，其中，向放松结点传递的弯矩通过结点协调变形依次向远端传递直至支座，而不产生新的不平衡力矩，从而快速消除结点不平衡力矩，结果精确；另一方面各分配结点只经历一轮完整力矩分配与连续传递过程即恢复至最终变形位置，计算量小。

Abstract: Owing to the repetitive unclamping and re-clamping of joints, moment distribution gives only successive approximation solution for multiple-joint structures. An alternative relaxation method for unclamping joints is presented that divides the joints into several layers and removes the artificial clamp at joint from layer to layer without re-clamping again in its new position. To ensure that the rotational stiffness and carry-over factors are known or computable when unclamping joints, each member can be idealized as a member with elastic rotational support. A derivation of the end moments corresponding to a unit rotation of straight member of constant cross-section with the far end restrained by elastic rotational support was carried out which can be expressed in terms of the end-connection spring stiffness factor. The results show that the distributed moments cause a spread of moment, not only to adjacent clamped joints in its path, but also back to released joints in its wake until all the released joints are eased down into equilibrium. Since the released joints are free to rotate, no unbalanced moments develop there whenever the joints remain unclamped. As the distribution process proceeds, the number of clamps decreases rapidly and the corresponding unbalanced moments in the clamps can be eliminated consecutively. The carry-over moment is transferred from far end of member to other members framing into the joint and additional moments can be received from the future release of clamps by satisfying compatibility of deformations and static equilibrium. Only one full cycle of the balancing procedure is required before each free joint rotates to its position of final equilibrium, thereby saving the computational effort for analysis. And the terminal member-end moments in each member can be obtained exactly according to the proposed method.

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