3. Generalization to Variable Bending Moment

In this third Chapter, the theoretical model proposed in Chap. 1 for the bending of fully nonlinear beams is generalized to the case of variable bending moment. Such a generalization focuses on the local determination of curvature and bending moment along the deformed beam axis. Once the moment-curvature relationship has been derived, the equilibrium problem for nonlinear beams subjected to variable bending moment has been formulated. The governing equations assume the form of a coupled system of three equations in integral form. To solve this highly nonlinear system, an iterative numerical procedure has been proposed. Definitively, the analysis developed in this Chapter allows considering a very wide class of equilibrium problems for nonlinear beams. By way of example, the Euler beam and a cantilever beam loaded by a concentrated force of the dead or live (follower) type, applied in its free end, has been studied, showing the shape assumed by the deformed beam axis as the load multiplier increases.