General Bending of Beams

Abstract

The theory of homogeneous bending of non-symmetric elastic beams with constant cross-section forms a central part of the mechanics of structures. The theory combines the possibility of general cross-section properties with the simultaneous bending about two axes, and thus constitutes a natural extension of the simple plane bending treated in Chapters 3–4 and developed into simple finite elements for analysis of plane frames in Chapter 7. The general theory of beam bending has wide application, e.g. to beams in buildings, bridge decks in concrete, steel or composites, or in a very general form to wind turbine blades with changing aerodynamic closed cross-section. Homogeneous bending of a beam leads to a linear distribution of the axial strain over the cross-section and thereby to a simple explicit relation for the distribution of the axial stress component. The bending properties of the beam are characterized by the elastic center and the principal axes of bending. The axial stress constitutes an important design parameter. The axial stress distribution also serves as a step in the determination of the shear stress distribution associated with non-homogeneous bending as discussed in Chapter 11.