Nonlinear inelastic static analysis of plane frames with numerically generated tangent stiffness matrices

For the nonlinear analysis of structures using the well known Newton-Raphson Method, the tangent stiffness matrices of the elements must be constructed in each iteration. Due to the high expense required to find the exact tangent stiffness matrices, researchers have developed novel innovations into the Newton-Raphson method to reduce the cost and time required by the analysis. In this paper, a new approach is suggested to generate the tangent stiffness matrix numerically from internal forces for the materially nonlinear analysis of structures. The method is organized at the element level and, as is verified by numerical experiments, affords good stability and preserves the convergence rate near that of the original exact Newton-Raphson version. To implement the method, an appropriate configuration is first sought for the stiffness matrix of the finite element, which satisfies the element equilibrium requirement; then, the entries of this matrix template are generated from the generalized internal forces of the element by the numerical method of finite differences. The method is applied to construct the stiffness matrix of the plane frame element, which will be used in the analysis of some sample frame structures with materially nonlinear behavior, under monotonic static loading.