Stress-calculation in frameworks by the method of systematic relaxation of constraints - III

Abstract An earlier paper bearing this title dealt in detail (Part I) with the application of relaxation methods to frameworks having frictionless joints, and in general terms (Part II) with their relation to the Moment Distribution Method of Professor Hardy Cross. Given any framework and a (self-equilibrating) system of applied forces, we may imagine constraints to be provided at every joint whereby the three components of displacement and the three components of rotation can be severally controlled. Then initially, If all constraints are fixed before the forces are applied, extension of members is prevented, and hence the forces (of known magnitude) are taken wholly by the constraints. Subsequently, by permitting a suitable relaxation, we can transfer any component of force or moment from a constraint to the framework, and in so doing we shall store strain-energy in the latter. If the successive relaxations are systematically chosen, we can bring about a continuous approximation to the required conditions, in which every constraint is relieved of load and all forces have been transferred to the framework. When all the joints are frictionless (as in the problems of Part I) the constraints at any one joint have only to control its three component displacements, because no stresses are entailed by joint rotations; but If any joint is rigid it can by rotating transmit couples to adjacent joints through the agency of shearing forces and the consequent bending moments. When the members have uniform flexural rigidity, it can be shown that any one which shares in the rotation of a joint, and which in consequence transfers a couple M away from that joint, transmits it necessary to the strict validity of a method which depends (as the relaxation method evidently does) upon the principle of superposition, although it appears that a solution for "P. y effects" could be obtained without difficulty by a slight modification of the standard procedure. some remarks on this aspect are given in Appendix C.