Objective functions for nonlinear structural analysis

An objective function that is suitable for both stable and unstable equilibrium states since it is guaranteed to assume a relative minimum at any equilibrium state is presented. The objective function, which is the inner product of the equations of equilibrium (algorithm 2), is compared with another objective function which, for a static problem, is the total potential energy (algorithm 3). The method of formulating the mathematical model for algorithm 2 is presented in detail.

Algorithm 2 is found to be less efficient than algorithm 3. However, it is demonstrated that algorithm 2 is able to solve for equilibrium states on either the fundamental path or a bifurcation path. Hence algorithm 2 is a powerful tool for nonlinear structural analysis since it is able to predict the existence of limit and bifurcation points and to determine post-buckled equilibrium states.

Also addressed are the methods of formulating mathematical models for nonlinear structural analysis. A comparison of the methods of integrating the equations of motion for nonlinear dynamics problems and the selection of an appropriate time step for the time integration schemes are presented.