Introduction to the Variational Formulation in Mechanics

7Bending of Beams

7.1 Introduction

This chapter is devoted to the presentation of the simplest beam model (studied by Bernoulli, Euler, and others), which is characterized by a very particular geometry, as illustrated in Figure 7.1. First, a beam is a structural component which has one of its characteristic dimensions, say and called the beam length, much larger than the other characteristic dimensions, which are the height, , and the width, . That is and . Second, a beam has a longitudinal plane of symmetry, implying that transversal sections have symmetry with respect to this plane. Thirdly, in a beam, when it is deflected as in Figure 7.1, the superior longitudinal fibers are shortened, while inferior longitudinal fibers are elongated. In the middle of the beam, there are fibers that maintain their length. When such a condition is satisfied, fibers are said to be in a neutral state. The set of these neutral fibers forms the beam neutral surface. It will be considered that the intersection ...