Numerically Generated Tangent Stiffness Matrices for Geometrically Non-Linear Structures

Abstract

The aim of this thesis is to develop a general numerical solution method for geometrically non-linear structures. Most common work involves tedious derivations of analytic tangent stiffness matrices. The major objective of the current work is to develop a numerically generated tangent stiffness matrix that allows for a general and easily implementable solution method. The thesis begins with the definition of the tangent stiffness matrix and a discussion of the Newton-Raphson incremental-iterative method typically used to solve geometrically non-linear problems. This is followed by a detailed description of how the tangent stiffness matrix is numerically generated using complex variable differentiation to approximate sensitivities. The thesis proceeds with details of the solution method applied to three different structural elements: 3D truss, membrane, and 3D beam. These discussions include numeric examples for each type of structure, the results of which are compared with the literature and ANSYS solutions. The results from the present work show that solutions obtained using the general numerically generated tangent stiffness matrix are accurate. While computational effort is increased, the method is especially attractive in the context of research involving small finite element models.