Recovery of stresses in a beam from those in a cone

We consider a trapezoidal thin plate, as shown in Figure 1, which has one end loaded by a concentrated transverse unit force and the other clamped at zero displacement. The stress is determined according to a method proposed by Galerkin which involves the superposition of two plane elastic stress distributions. To access the accuracy of the method, the oblique sides of the trapezium are allowed to become parallel so that, in the limit, the trapezium becomes a rectangle of the same area. The limiting values of the maximum stress components are exactly those obtained by the Saint-Venant method for a bar. However, over the clamped end, the displacement vanishes only at three arbitrarily selected points. A similar procedure is applied to the transversely loaded truncated cone, but now the limiting values of the stress do not entirely match those obtained by the Saint-Venant theory for a circular cylinder. We also briefly discuss, for completeness, the same bodies under axial loadings, and arrive at similar conclusions which therefore hold for general loads.