Andreas Öchsner - Structural Mechanics with a Pen: A Guide to Solve Finite Difference Problems

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The classical approximation methods known in continuum mechanics are the finite difference method, the finite element method, the finite volume method, and the boundary element method. Each method has its advantages and disadvantages and a different spread in different areas of applied mechanics, e.g. solid or fluid mechanics. The oldest approximation method to solve partial differential equations is the finite difference method since the mathematical set of tools is basically limited to the series expansion of derivatives. More precisely, truncated Taylors series is used for the local expansions of the variables. This simple derivation of the method makes it quite attractive to introduce approximation methods in tertiary engineering education, e.g. in mechanical or civil engineering. This allows acquiring a general understanding of numerical approximation methods and the involved common steps such as discretization, assembly of a global system of equations, consideration of boundary and load conditions as well as the solution of a linear or nonlinear system of equations.

This book is focused on the introduction of the finite difference method based on the classical one-dimensional structural members, i.e. rods/bars and beams. It is the goal to provide a first introduction to the manifold aspects of the finite difference method and to enable the reader to get a methodical understanding of important subject areas in structural mechanics. The reader learns to understand the assumptions and derivations of different structural members. Furthermore, she/he learns to critically evaluate the possibilities and limitations of the finite difference method. Additional comprehensive mathematical descriptions, which solely result from advanced illustrations for two- or three-dimensional problems, are omitted. Hence, the mathematical description largely remains simple and clear.

Chapter introduces a simple treatment of elasto-plastic bending problems under the consideration of the thin beam formulation. The so-called layered approach is applied to linear-elastic/ideal-plastic material behavior and is restricted to monotonic loading.

All derivations in chapters two to four follow a common approach: based on the three basic equations of continuum mechanics, i.e. the kinematics relationship, the constitutive law, and the equilibrium equation, the partial differential equations, which describe the physical problem, are presented. The finite difference method is then used to derive approximate equations for the corresponding structural member.

In order to deepen the understanding of the derived equations and theories, each technical chapter collects at its end supplementary calculation problems. A short solution for each problem is included at the end of this book. It should be noted that these short solutions contain major steps for the solution of the problem and not only, for example, a numerical value for the final result. This should ensure that students are able to successfully master these problems. I hope that students find this book a useful complement to many classical textbooks. I look forward to receiving their comments and suggestions.

It is important to highlight the contribution of the students which helped to develop the concept and content of this book. Their questions, comments, and struggles during difficult lectures, assignments, and final exams helped me to develop the idea for this approach. In addition, it is important to highlight the contribution of many undergraduate and postgraduate students. Furthermore, I would like to express my sincere appreciation to the Springer publisher, especially to Dr. Christoph Baumann, for giving me the opportunity to realize this book.