Teaching Fundamentals of Finite-Element Theory and Applications Using Sophomore-Level Strength-of-Materials Theory

Abstract

This paper presents a study module that is incorporated into a formal introductory undergraduate course on finite-element theory and applications. The module consists of an integrative project and homework exercises based upon sophomore-level education in strength-of-materials. The objective of the module is to support the teaching of the finite-element method and to emphasize assumptions and limitations in the application of the technique. The project centers on a simply supported beam with geometric discontinuities. This beam is investigated using commercial finite-element software in five different phases. Each phase uses a different solution model, consisting of a hand calculation, beam, two-dimensional area, and three-dimensional solid elements. The solution from each phase is compared to the solution from traditional strength-of-materials beam theory in terms of the following: weight and center of gravity, deflection, and stress. Static-failure theories, stress concentrations and a redesign are also considered. The approach of a comparative solution to a problem using different types of element has not been considered in any finite-element textbook to date and few books consider stress concentrations and failure theories. The homework exercises involve solving similar though smaller problems using commercial software and verifying the finite-element solutions against the strength-of-materials solutions. The five homework exercises considered here involve: a truss, an L-bracket with re-entrant corner, a plate with a notch, a thick-walled pressure vessel, and a pin joint connection. The project and some of the homework exercises reinforce the following: an understanding of finite-element theory, an understanding of strength-of-materials theory, knowledge of the physical behavior and usage of each type of element, the ability to select a suitable element for a given problem, and the ability to interpret and evaluate finite-element solutions. Emphasis is placed on the importance of verification. The project and several of the homework exercises also illustrate common major conceptual mistakes made by students and, often, by practitioners using commercial software.