ABSTRACT
The primary objective of this book is to study theories and analytical as well as numerical solutions of plate and shell structures, i.e., thin structural elements undergoing stretching and bending. The plate and shell theories are developed using certain assumed kinematics of deformation that facilitate writing the displacement field explicitly in terms of the thickness coordinate. Then the principle of virtual displacements and integration through the thickness are used to obtain the governing equations. The theories considered in this book are valid for thin and moderately thick plates and shells. The governing equations of solid and structural mechanics can be derived by
either vector mechanics or energy principles. In vector mechanics, better known as Newton’s second law, the vector sum of forces and moments on a typical element of a structure is set to zero to obtain the equations of equilibrium or motion. In energy principles, such as the principle of virtual displacement or its derivative, the principle of minimum total potential energy, is used to obtain the governing equations. While both methods can give the same equations, the energy principles have the advantage of providing information on the form of the boundary conditions suitable for the problem. Energy principles also enable the development of refined theories of structural members that are difficult to formulate using vector mechanics. Finally, energy principles provide a natural means of determining numerical solutions of the governing equations. Hence, the energy approach is adopted in the present study to derive the governing equations of plates and shells. In order to study theories of plates and shells, a good understanding of the basic
equations of elasticity and the concepts of work done and energy stored is required. A study of these topics in turn requires familiarity with the notions of vectors, tensors, transformations of vector and tensor components, and matrices. Therefore, a brief review of vectors and tensors is presented first, followed by a review of the equations of elasticity. Readers familiar with these topics may skip the remaining portion of this chapter and go directly to Chapter 2, where the principles of virtual work and classical variational methods are discussed.
