End Effects

Abstract

The solution of §5.2.2 must be deemed approximate insofar as the boundary conditions on the ends x = ±a of the rectangular beam are satisfied only in the weak sense of force resultants, through equations (5.45–5.47). In general, if a rectangular beam is loaded by tractions of finite polynomial form, a finite polynomial solution can be obtained which satisfies the boundary conditions in the strong (i.e. pointwise) sense on two edges and in the weak sense on the other two edges.

The error involved in such an approximation corresponds to the solution of a corrective problem in which the beam is loaded by the difference between the actual tractions applied and those implied by the approximation. These tractions will of course be confined to the edges on which the weak boundary conditions were applied and will be self-equilibrated, since the weak conditions imply that the tractions in the approximate solution have the same force resultants as the actual tractions.

For the particular problem of §5.2.2, we note that the stress field of equations (5.78–5.80) satisfies the boundary conditions on the edges y =±b, but that there is a self-equilibrated normal traction

$$\sigma _{xx} = \frac{p}{{10b^3 }}(3b^2 y - 5y^3 )$$

((6.1))

on the ends x = ±a, which must be removed by superposing a corrective solution if we wish to satisfy the boundary conditions of Figure 5.3 in the strong sense.