Determination of the derivative of the tangent stiffness matrix with respect to the load parameter

Abstract

In order to solve the so‐called consistently linearized eigenproblem in the frame of the Finite Element Method (FEM), the derivative of the tangent stiffness matrix equation image with respect to the load parameter λ needs to be calculated. In this work, three schemes for calculation of equation image are presented. The first scheme is based on an analytical expression for the first derivative of e the element tangent stiffness matrix equation image with respect to λ for the special case of a co‐rotational beam element. The second one is a finite difference approach for computation of equation image. The third one is also a finite difference approach. However, it is based on a directional derivative of equation image. An elastic beam, subjected to a compressive axial force and a small transverse uniform load, is chosen as a numerical example. The effectiveness and the accuracy of the three schemes are compared. The third scheme is found to be not only very practical but also more effective than the two competing schemes.