Chapter 1 - Introduction

Abstract

This chapter examines the relationships between the solutions of classical theories of beams and plates with those of the shear deformation theories. Shear deformation theories are those in which the effect of transverse shear strains is included. Relationships are developed for bending, buckling, and free vibration solutions. A plate is a structural element with plane form dimensions that are large compared to its thickness and is subjected to loads that cause bending deformation in addition to stretching. In most cases, the thickness is no greater than one-tenth of the smallest in-plane dimension. Because of the smallness of the thickness dimension, it is often not necessary to model the plate using three-dimensional elasticity equations. The chapter also discusses that beams are one-dimensional counterparts of plates. The governing equations of beams and plates can be derived using either vector mechanics or energy and variational principles. In vector mechanics, the forces and moments on a typical element of the plate are summed to obtain the equations of equilibrium or motion. In energy methods, the principles of virtual work or their derivatives, such as the principles of minimum potential energy or complementary energy, are used to obtain the equations. While both methods can give the same equations, the energy methods have the advantage of providing information on the form of the boundary conditions. Beam and plate theories are developed by assuming the form of the displacement or stress field as a linear combination of unknown functions and the thickness coordinate.