Nonprismatic Shear Beams

A solution of the problem of bending of nonprismatic beams, including the effect of shear deformations, is presented. Such beams—called here shear beams—occur with some frequency as elements of 'stocky' rigid frames which are used in the mining and power industries and, most importantly, as both the horizontal diaphragms and the verical shear elements which are parts of structural systems resisting lateral loads due to wind or seismic excitation. The governing differential equations are derived, and their general solution is obtained in terms of a displacement function. This formulation underscores the near analogy between the problem of a shear beam and a standard beam and, in a number of cases, permits obtaining solutions directly from the known solutions of the corresponding problems of standard beams. Finally, a detailed solution is obtained for the problem where the depth of the beam varies linearly with the span. A numerical example illustrates the procedure. The results show that, in nonprismatic shear elements, the distribution of internal forces may be markedly different from that encountered in nonprismatic standard beams.