EXACT SOLUTION OF MOMENT DISTRIBUTION

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.