ABSTRACT
Torsion-less beams are designed as linear elements subjected to bending moments and shear forces. The values for freely supported beams and cantilevers are readily determined by the simple rules of static equilibrium, but the analysis of continuous beams and statically indeterminate frames is more complex. Historically, various analytical techniques have been developed and used as self-contained methods to solve particular problems. In time, it was realised that the methods could be divided into two basic categories: flexibility methods (otherwise known as action methods, compatibility methods or force methods) and displacement methods (otherwise known as stiffness methods or equilibrium methods). The behaviour of the structure is considered in terms of unknown forces in the first category, and unknown displacements in the second category. For each method, a particular solution, obtained by modifying the structure to make it statically determinate, is combined with a complementary solution, in which the effect of each modification is determined. Consider the case of a continuous beam. For the flexibility methods, the particular solution involves removing redundant actions (i.e. the continuity between the individual members) to leave a series of disconnected spans. For the displacement methods, the particular solution involves restricting the rotations and/or displacements that would otherwise occur at the joints.
