Vibration of Continuous Systems, 2nd Edition

18Numerical Methods: Finite Element Method

18.1 INTRODUCTION

All structures and structural elements have distributed mass, stiffness and damping properties and hence can be considered to have an infinite number of degrees of freedom. The main objective of selecting a mathematical model is to reduce the infinite number of degrees of freedom to a finite number which can still exhibit the essential physical behavior of the system. The finite element method represents a type of mathematical model in which a continuous structure is approximated by a finite number of degrees of freedom [1, 2]. In this method, the actual structure is replaced by several pieces or elements, each of which is assumed to behave as a continuous structural member called a finite element. The elements are assumed to be interconnected at certain points known as joints or nodes. Since it is very difficult to find the exact solution (such as the displacements) of the whole structure under the specified loads, a simple approximate solution is assumed in each finite element. The idea is that if the solutions of the various elements are selected properly, they can be made to converge to the exact solution of the total structure as the element size is reduced. During the solution process, the equilibrium of forces at the joints and the compatibility of displacements between the elements are satisfied so that the entire structure (assemblage of elements) is made to behave as a single entity. A simple finite element ...