Theory of Plates and Shells

In this chapter all necessary basics of linear elasticity theory of three-dimensional anisotropic bodies are discussed, which are necessary for the understanding of the contents of this book. As a special case, at the end of this chapter we also discuss the simplifications as they are assumed for plane problems. In all explanations of this chapter we restrict ourselves to the absolutely necessary contents to make this book self-contained.

This chapter is devoted to the presentation of all the necessary basics of energy-based methods which are indispensable for the understanding of the contents of this book. At the beginning, the concepts of work and energy are introduced. This is followed by an introduction to the principle of virtual displacements and its application to rods, beams and continua, before the so-called principle of the stationary value of the total elastic potential is introduced and discussed as a particularly important mechanical principle.

This chapter is devoted to the consideration of isotropic disk structures in Cartesian coordinates. After a short definition of what constitutes a disk, the two basic analytical approaches, namely the displacement method and the force method, are motivated and, for the force method, all basic equations necessary for the description of a disk are compiled. This is followed by an energetic consideration of the disk problem, before the solutions of the disk equation and elementary disk problems are discussed in detail.

This  chapter deals with isotropic disks in polar coordinates. After a short presentation of all necessary basic equations, as already in the case of Cartesian coordinates, the force method is discussed in detail, where besides the derivation of the disk equation and solution approaches, also an energetic consideration is presented. In addition to elementary basic cases, selected technically relevant cases are discussed, namely rotationally symmetric disks as well as non-rotationally symmetric circular arc disks and wedge-shaped disks, before this chapter concludes with the consideration of disks containing circular holes.

In this chapter we will discuss approximation methods of structural mechanics for application to isotropic disk structures. First of all, we will discuss the classical Ritz method, where we will consider two versions, namely a displacement-based formulation on the one hand, and a force-based formulation on the other hand. Furthermore, we will discuss the finite element method (FEM).

In this chapter we want to extend the considerations made so far with respect to isotropic disks to anisotropic disk structures, whereby we will mainly restrict ourselves to orthotropic disks due to their high practical relevance. All considerations will focus on the plane stress state, the transfer of the results to the plane strain state is then very easy to accomplish by replacing the elastic constants (see also Chap. 1). Let the assumed anisotropic materials be such that they have at least one symmetry plane coincident with the disk middle plane.

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.

This chapter is devoted to the application of classical approximation methods to plate bending problems. Specifically, we discuss both the Ritz method, which is a very universal method of approximation, and the Galerkin method.

In this chapter we consider plate problems in the framework of Kirchhoff’s plate theory which can be advantageously described by polar coordinates. Among these are naturally circular plates and circular ring plates. At the beginning of this chapter all necessary basic equations of the Kirchhoff plate theory are provided in polar coordinates. Hereafter, the case of bending of circular and annular plates is discussed where both the structural situation and the loading are rotationally symmetric, and some exemplary solutions are presented. The chapter concludes with a discussion of the case of asymmetric bending. The interested reader can find further elaborations on this subject in Altenbach et al. (2016), Girkmann (1974), Hake and Meskouris (2007), Reddy (2006), Szilard (2004), Timoshenko and Woinowsky-Krieger (1964), Ugural (1981), among others.

The  Kirchhoff plate theory discussed in the previous chapters has proven itself for many technical applications and is widely used in many technical fields of application. It does have some inconsistencies, but these are mostly negligible when considering sufficiently thin plate structures.

Plates are  thin-walled structures. In this respect, attention must be paid to their buckling behavior. Accordingly, this chapter is devoted to the analytical treatment of the buckling behavior of plates within the framework of the Kirchhoff plate theory. First, all necessary basic equations describing the buckling of orthotropic plates are provided. Then, the Navier solution is discussed, i.e. the exact analytical determination of the buckling load of rectangular simply supported orthotropic plates under uniaxial and biaxial loading. The chapter concludes with the energetic treatment of plate buckling problems and describes as a simple method the so-called Rayleigh quotient and as a more universally applicable method the Ritz method.

Plates  are thin-walled structures, so in addition to the already discussed plate bending in the context of a geometrically linear analysis, the geometrically nonlinear behavior is also of quite fundamental importance.

Laminates are thin-walled structures in the form of disks, plates or shells consisting of any number of individual layers, each of which can have different properties (thickness, material properties, principal material directions, etc.). A very common application in lightweight construction is laminates whose layers consist of fiber-reinforced plastics. Such a fibrous composite material is a composite of at least two different components (constituents) that are combined by means of a specific manufacturing process to form a new material, i.e. the composite material.

This chapter introduces the basics of the analysis of shells. It provides a very basic introduction to the consideration of shells as load-bearing structures and clarifies some basic concepts. In addition to some geometric relationships, the basic assumptions of Classical Shell Theory are explained and the stress, displacement, and strain quantities that occur are introduced along with relevant load types.

In this chapter, the so-called membrane theory  is described as a very useful analytical approach to the analysis of shell structures. After describing the limits of applicability of this theory, the equilibrium conditions for shells of revolution are derived and then specialized for the case of rotationally symmetric loading, from which the membrane force flows of the shells can be determined. Basic examples describe the application of the membrane theory to circular cylindrical, spherical and conical shells.

This  chapter is devoted to the bending theory of shells of revolution. After deriving the equilibrium conditions for arbitrary shells of revolution, the corresponding kinematic equations and the constitutive law in the presence of bending action are discussed. This is followed by the discussion of the so-called container theory of the circular cylindrical shell, i.e. a shell of revolution under rotationally symmetric loading.