7. Kirchhoff Plate Theory in Cartesian Coordinates

This chapter is dedicated to the so-called Kirchhoff plate theory in Cartesian coordinates. Starting from the assumptions of this elementary important plate theory, both the displacement field and the strain field of the Kirchhoff plate are derived and discussed in detail, from which finally the stress field of the plate can be determined. The internal forces and moments of the Kirchhoff plate then follow directly from the stresses. The effective properties of plates for various special cases are then discussed before the basic equations of plate bending are derived. An important part is the discussion of the so-called equivalent shear forces and the plate boundary conditions. After elementary solutions of the plate equation, boundary value problems in the form of the bending of plate strips as well as Navier-type solutions and Lévy-type solutions are treated. This is followed by the energetic treatment of plate problems, and both the principle of virtual displacements and the principle of the stationary value of the total elastic potential are used to derive the plate equation and boundary conditions. Furthermore, plates with arbitrary boundaries are considered as well. The discussion of two interesting special cases, namely the plate on an elastic foundation and the membrane, completes the present chapter.