Development and evaluation of solution procedures for nonlinear structural analysis

Abstract

Solution procedures are developed and evaluated for the analysis of nonlinear structural problems. The principle of virtual work is first developed in undeformed, Lagrangian coordinated for the case of large strains. The virtual work equations are then specialized for the case of small strains and the matrix displacement approach (finite element method) is used to obtain the governing equilibrium equations for the general nonlinear structural problem involving geometric and material nonlinearities. It is shown that the solution of the nonlinear problem reduces to the solution of a system of nonlinear algebraic or differential equations. Various solution procedures are developed for the solution of the non-linear equations. These include the Newton-Raphson method, incremental methods, iterational methods, and an initial-value formulation. The initial-value approach is a new and unique approach on that it yields a system of nonlinear first-order differential equations of the initial-value type. Several integration schemes are developed to integrate these equations. Each solution technique is applied first to simple geometrically non-linear truss problems and second to the analysis of geometrically nonlinear shells of revolution subjected to symmetric and asymmetric loadings. A comparison of the numerical results provides an index of the applicability of the solution technique to highly non-linear problems. The general conclusion is reached that a modified form of the basic Newton-Raphson procedure is the "best" solution technique from the standpoint of computational efficiency, accuracy, reliability, and ease of usage..