Introduction

So far, we have dealt with only the problem of axial extension of bar in our discussion on finite element formulation. Now, we shall consider another one-dimensional problem, namely the bending of beam. As far as the finite element formulation is concerned, the beam problem differs from the bar problem in certain respects. The integral formulation of the bar problem contains only the first derivative of the primary variable and the weight function. To make the integral finite, the approximation to the primary variable needs to satisfy the continuity of the primary variable only and not of its first derivative. In the beam problem, its integral formulation contains the second derivative of the primary variable and the weight function. Therefore, the continuity requirement extends to the first derivative of the primary variable also. Approximations in which both the primary variable and its derivative are continuous are called as the -continuity approximations. Thus, in the beam problem, we need to use -continuity shape functions. This makes it necessary to have minimum two degrees of freedom per node, at least at the end nodes. Increase in the degrees of freedom per node requires modifications in the assembly procedure as well.

In the present lecture, we shall discuss the integral formulations of the beam problem. In the next lecture, we shall develop the finite element formulation.