Strength and stiffness are considerations of basic importance to the engineer. The stress level is frequently used as a measure of strength. Stress in members under various loads was discussed in Chapter 3. We now turn to deflection, the analysis of which is as important as that of stress. Moreover, deflections must be considered in the design of statically indeterminate systems, although we are interested only in the forces or stresses.

Stiffness relates to the ability of a part to resist deflection or deformation. Elastic deflection or stiffness, rather than stress, is frequently the controlling factor in the design of a member. The deflection, for example, may have to be kept within limits so that certain clearances between components are maintained. Structures such as machine frames must be extremely rigid to maintain manufacturing accuracy. Most components may require great stiffness to eliminate vibration problems. We begin by developing basic expressions relative to deflection and stiffness of variously loaded members using the equilibrium approaches. Then the integration, superposition and moment-area methods are discussed. Following this, the impact or shock loading and bending of plates are treated. The theorems are based upon work-energy concepts, classic methods, and finite element analysis (FEA) for determining the displacement on members or structures are considered in the chapters to follow.

Comparison of various deflection methods shows when one approach is preferred over another and the advantages of each technique. The governing differential equations for beams on integration give the solution for deflection in a problem. However, it is best to limit their application to prismatic beams; otherwise, considerable complexities arise. In practice, the deflection of members subjected to several or complicated loading conditions are often synthesized from simpler loads, using the principle of superposition.

The dual concepts of strain energy and complementary energy provide the basis for some extremely powerful methods of analysis, such as Castigliano’s theorem and its various forms. These approaches may be employed very effectively for finding deflection due to applied forces and are not limited at all to linearly elastic structures. Similar problems are treated by the principles of virtual work and minimum potential energy for obtaining deflections or forces caused by any kind of deformation. They are of great importance in the matrix analysis of structures and in finite elements. The moment-area method, a specialized procedure, is particularly convenient if deflection of only a few points on a beam or frame is desired. It can be used to advantage in the solution of statically indeterminate problems as a check. An excellent insight into the kinematics is obtained by applying this technique. The FEA is perfectly general and can be used for the analysis of statically indeterminate as well as determinate, both linear and nonlinear, problems.