Deformation of Beams and Frames

Abstract

The internal forces in beams and frames lead to deformations, and methods for analysis of these deformations are the subject of the present chapter. Typically the satisfactory performance of a structure will impose some limitation on the displacements. However, the deformation of beams also plays another important role, namely in the distribution of the internal forces in statically indeterminate structures. A statically indeterminate structure permits several ways of distributing the internal forces to carry a given load, and the actual distribution depends on the relative stiffness of the individual parts of the structure. The linear theory for deformation of elastic beams is presented in the form of differential equations. The displacements of elastic beam and frame structures can be determined by solving the corresponding differential equations. However, for frames this would involve special transition conditions at joints, and even for beams the integration of the differential equations may be quite laborious for non-trivial load distributions and non-homogeneous beams. Therefore the principle of virtual work, derived and illustrated in this chapter, plays an indispensable role in the analysis of displacements of structures.