TABLE OF CONTENTS Curved beams can be subdivided into two major categories beams loaded out of plane and beams loaded in their planes. In the first category, discussed in Sections 8-3-1 and 8-3-2, the application of the load does not change the natural curvature of the beam (within the assumption of small deformation). In the second category a change occurs affecting the moment-deflection relation (see also Chapter 5) and thus the differential equation for deflection. On the other hand, while torsion always occurs in a beam bent out of plane, even though no external torque is applied, it is never present in a beam bent in its own plane unless a torque is applied.
A thin curved beam loaded out of its plane is assumed to behave similarly to a straight beam to the extent that any plane cross section remains plane after the deformation. In order to determine the deflection of such a beam, we could derive a differential equation for the deflection by analyzing the equilibrium of an infinitesimal element, or we might apply Castigliano's theorem. Using the second approach, which is easier, we disregard the effect of shear on the strain energy and assume that there are no axial forces. Thus we take into account only the strain energy due to the bending and to the torsion since both will obviously be generated by any load. Considering for simplicity a curved cantilever with a circular axis (Fig. 8-17), let us determine the...
