ABSTRACT

The finite element method (FEM) is a numerical method of solving differential equations, much like the classical variational methods. The major drawback of the traditional variational method is the unique derivation of approximation functions used to represent the dependent unknown. In the FEM, the domain is represented as a collection of non-overlapping subdomains, called finite elements, that cover the total domain. The Newmark family of time integration schemes is widely used in structural dynamics. The chapter presents the finite element models of the classical beam theory, Timoshenko beam theory, and Reddy beam theory beams. These are primarily displacement-based models but where possible, mixed models that include the bending moment as a nodal variable. The finite element solution from various models is the vector of nodal values. The nodal values can be used to determine the solution at any desired point of the beam using the finite element approximations of the variables.