A general stiffness method for the solution of nonlinear cable networks with arbitrary loading

A general matrix stiffness method for the solution of nonlinear elastic cable networks with distributed loading is presented. The procedure is applicable to interconnected networks with irregular boundaries with member, joint, and temperature loading. Accurate numerical evaluation of the stiffness coefficients was found to be critical to the convergence and stability of the solution. An algorithm is presented which meets these requirements. The governing equations are the equations of equilibrium for the joints and the equations of equilibrium for the members. The equations of equilibrium for each member are satisfied at each iteration by solving for the required member end forces assuming that all joints are fixed. The linearized equations describing the joint force-displacement relationships are then formulated. Each joint may have up to three degrees of freedom. These equations are solved for corrections to the joint coordinates. The member forces are then recomputed using the loading corresponding to the new network configuration. Each joint is checked for equilibrium and the process is repeated until equilibrium is achieved. Relative errors of 10 −3 have been shown to be readily achievable for all joints. The method may be used to compute the prestressed configuration of the network if either the length or desired tension of the members is given. The linearization occurs in the numerical evaluation of the stiffness coefficients. The numerical procedure for evaluating the stiffness coefficients is critical to the convergence of the entire process. The stiffness coefficients must possess uniform accuracy to prevent divergence in nonlinear networks. A stiffness coefficient accuracy parameter was identified. The procedure must be applicable to a wide range of cable force-displacement relationships which are not known a priori . An algorithm has been devised and implemented that yields very good convergence. Each member is characterized by nine stiffness coefficients which are functions of the member loading as well as the location of the end points. The stiffness coefficients Kxx, Kxy, Kxz, Kyz, Kyy, Kyz, Kzx, Kzy and Kzz are evaluated numerically by considering small displacements of the joint in each of the x , y and z directions for each cable. The stiffness coefficients are defined in the usual manner, i.e. Kxy is the force component in the x direction which results from a displacement in the y direction. Each cable is treated as a catenary with uniform loading over its length for both the preload and loaded conditions. For the preload ease only the cable weight and any concentrated joint loads are considered. For the case where wind and ice are considered, the cable shape between joints is still considered to be a catenary whose plane and curvature have been changed. Comparisons of special cases with results of previous investigations are given.