Introduction

In the last lecture, we derived two integral forms (Weighted Residual Integral and Variational Functional) corresponding to the model boundary value problem (i.e, the cantilever beam of Fig. 18.1). In this lecture, we shall develop the finite element formulation of the beam problem using the Weighted Residual Method. Further, we assume that the weight functions are the same as the shape functions of the approximation to the primary variable. Thus, we shall use the Galerkin finite element formulation. As shown in the last Lecture, for the beam problem, the approximation to the primary variable should possess -continuity. This means, there should be minimum two degrees of freedom (i.e, the primary variable and its first derivative) per node. We shall first develop the shape functions corresponding to this situation. These shape functions turn out to be Hermitian interpolation polynomials. Next, we shall derive the expressions for the element stiffness matrix and the element force vector. In the end, we shall discuss the connectivity matrix and the assembly procedure. Lecture 19A discusses the development of the shape functions and Lecture 19B describes the evaluation of element quantities and assembly procedure.