Applications of Linear Elastic Behavior

Abstract

The theory of linear elasticity is a complete mathematical description of the behavior of solids under the action of mechanical and thermal loads. However, because of the large number of equations to be solved, it is not useful for simple problems. On the other hand, simple theories of beam bending and twisted bars have a proven record of applications. The conventional beam bending theory for example, can be applied to many practical problems. Although historically the beam theory precedes the elasticity theory, we can visualize it as a logical subset of the latter. Table 2.1 presents the theory of beams, the theory of torsion, and the theory of plates and shells as subsets of elasticity theory. The assumptions underlying the formulation of these theories are also briefly mentioned in the table. In this chapter, we discuss the applications of these theories to certain practical situations. In doing so, the topics have been arranged as one and two dimensional problems. Sections 2.2 through 2.9 deal with the one dimensional problems, and the last section deals with two dimensional problems. No effort has been made in this chapter to restate the well known beam theory or plate theory. Instead, this chapter is devoted to examples of applications that are germane to the plastics industry. In-depth development of the theory of beams with examples may be found in References 1, 2, 3, and 4. For plate theory, References 3, 4, 5, 6, and 7 are a few of the well-known texts. Reference 8 is the “cookbook” of stress analysis, and is a large collection of pre-engineered solutions for beams, plates, and shells of various shapes.