ABSTRACT
The boundary-value problems of linear elastostatics formulated in the last chapter generally do not admit an exact solution in terms of elementary or special functions. An accurate numerical solution for any fully three-dimensional problem is usually not feasible or practical with the computing power available to many engineers and designers. In the presence of layer phenomena, even two-dimensional problems (resulting from some kind of symmetry inherent in the geometry and loading) require an unacceptaly high level of computing. Historically, this situation gave considerable impetus to the search for adequate approximate theories for special classes of problems. One approach takes advantage of the special geometrical features of the elastic body to be analyzed. Intuition and experience gained from exact solutions for specific problems suggest simplyfying approximations for broad classes of problems involving thin or slender bodies. Applications of these approximations lead to simpler boundary value problems in lower dimensions, such as the one-dimensional theory of beams and two-dimensional theories of plates and shells. In this chapter, we discuss three approximate theories for the three-dimensional elastostatics of thin flat bodies bounded by two parallel planes and one or more cylindrical edge surfaces (see Fig. 17.1). We show how the variational formulations of the three-dimensional elasticity theory can be used in conjunction with the semidirect method to derive these approximate theories. In the process, we also resolve a long-standing difficulty concerning the appropriate stress boundary conditions for thin plate theory. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/fig17_1_OB.tif"/>
