Skip to main content
main-content

Über dieses Buch

The study ofthree-dimensional continua has been a traditional part of graduate education in solid mechanics for some time. With rational simplifications to the three-dimensional theory of elasticity, the engineering theories of medium-thin plates and of thin shells may be derived and applied to a large class of engi­ neering structures distinguished by a characteristically small dimension in one direction. Often, these theories are developed somewhat independently due to their distinctive geometrical and load-resistance characteristics. On the other hand, the two systems share a common basis and might be unified under the classification of Surface Structures after the German term Fliichentragwerke. This common basis is fully exploited in this book. A substantial portion of many traditional approaches to this subject has been devoted to constructing classical and approximate solutions to the governing equations of the system in order to proceed with applications. Within the context of analytical, as opposed to numerical, approaches, the limited general­ ity of many such solutions has been a formidable obstacle to applications involving complex geometry, material properties, and/or loading. It is now relatively routine to obtain computer-based solutions to quite complicated situations. However, the choice of the proper problem to solve through the selection of the mathematical model remains a human rather than a machine task and requires a basis in the theory of the subject.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
The theory of elasticity is the basis for several engineering theories which, in turn, are applied to mechanical and structural design. The basic components of the elasticity problem are often designated as equilibrium, compatibility, and a constitutive law. The equilibrium equations represent a statement of Newton’s laws, which are restricted here to the static case. The compatibility conditions express the kinematic relationships between strains and displacements, and the constitutive law embodies the stress-strain behavior of the material which is presumed here to be linear elastic. In general, this set of basic components may be collected as a set of differential equations or as an energy principle. The simultaneous satisfaction of each component of the elasticity problem often is foreboding from the mathematical standpoint; therefore, engineers have naturally looked to simplifications and approximations.
Phillip L. Gould

Chapter 2. Geometry

Abstract
Consider a portion of the middle surface of a shell as shown in figure 2-1.
Phillip L. Gould

Chapter 3. Equilibrium

Abstract
Consider the element of the shell shown in figure 3-1(a), bounded by the normal sections α, α + dα, β, and β + dβ. The geometry of the middle surface of such an element was considered in chapter 2 (see figures 2-2 and 2-4). Here, we show the entire thickness h with the coordinate ζ defined in the direction of t n and depict a differential volume element dα dβ dζ with thickness dζ, parallel to and displaced from the middle surface a distance ζ.
Phillip L. Gould

Chapter 4. Membrane Theory

Abstract
We examine the individual terms of the force equilibrium equations, (3.17a-c), and the moment equilibrium equations, (3.22a-c). We see that the equations are coupled only through the transverse shear stress resultants, Q α and Q β If we suppose that for a certain class of shells, the stress couples are an order of magnitude smaller than the extensional and in-place shear stress resultants, we may deduce from equations (3.22a-c) that the transverse shear stress resultants are similarly small and thus may be neglected in the force equilibrium equations, (3.17). This implies that the shell may achieve force equilibrium through the action of in-plane forces alone. From a physical viewpoint, this possibility is evident for the first two equilibrium equations which reflect in-plane resistance to in-plane loading, a natural and obvious mechanism. On the other hand, the third equilibrium equation refers to the normal direction, and the possibility of resisting transverse loading with in-plane forces alone is not as apparent. It is evident from equation (3.17c) that this mode of resistance is possible only if at least one radius of curvature is finite; i.e., R α and/or R β ≠∞. Thus, flat plates are excluded from resisting transverse loading in this manner, within the limitations of small deformation theory (assumption [2], table 1-1.
Phillip L. Gould

Chapter 5. Deformations

Abstract
In the earlier chapters, we developed a geometric description of the middle surface of a shell which proved to be adequate to derive the equations of equilibrium. In turn, a simplified subset of the equilibrium equations formed the basis of the membrane theory of shells, for which many important practical applications were illustrated. Although membrane action in a shell is desirable from the dual standpoints of mathematical simplification and material efficiency, the requisite conditions for corresponding behavior cannot always be simulated in an actual structure. Consequently, to expand our base of understanding of shell behavior, we must develop relationships between the forces and the deformations of the shell. The first step is a description of the displacements, where we follow the vector approach suggested by Novozhilov.1
Phillip L. Gould

Chapter 6. Constitutive Laws, Boundary Conditions, and Displacements

Abstract
The constitutive law or stress-strain relationship is the third basic component of the elasticity problem. This subject is quite broad, but the detailed examination of various alternatives that are dependent on the characteristics of particular engineering materials is not properly within the scope of this book. Rather, it remains within the purview of the theory of elasticity, since we may accommodate a variety of material laws within our formulation of the shell or plate problem. Initially, we use the basic Hooke’s law for isotropic materials, and then we illustrate how some extended material laws can be accommodated.
Phillip L. Gould

Chapter 7. Energy and Approximate Methods

Abstract
In solid mechanics, an energy formulation is often viewed as an alternative to the differential equation statement. There exists a direct connection between the energy and the differential equation approaches through the principle of virtual displacements and through various extremum principles. However, we will not pursue this connection, since it is beyond our immediate scope.
Phillip L. Gould

Chapter 8. Bending of Plates

Abstract
The equilibrium equations for initially flat plates are stated as equations (3.25a-e); the strain-displacement relations are given by equations (5.54) or, with transverse shearing strains suppressed, as equations (5.55). Taken together with the stress resultant-strain relationships in the form of equation (6.10), the requisite boundary conditions discussed in section 6.2, and specifically the Kirchhoff conditions equations (6.25) and (6.27), the elements of a quite general plate theory are available and substantiated.
Phillip L. Gould

Chapter 9. Shell Bending and Instability

Abstract
In earlier chapters, we derived the equilibrium, strain-displacement, and constitutive equations and stated the required boundary conditions for the bending theory of shells, referred to a system of orthogonal curvilinear coordinates. Also, we developed strain energy and potential energy expressions that can be incorporated into an energy formulation of the shell theory. In this chapter, these equations are specialized for various classes of shells, as we have done for the membrane theory equations in chapter 4.
Phillip L. Gould

Chapter 10. Conclusion

Abstract
The preceding chapters have treated the analysis of thin elastic plates and shells from a unified viewpoint, insofar as possible. Specializations to specific forms— e.g., membrane shells, flat plates—were introduced in a logical sequence after the general aspects of the theory were set forth. The emphasis was on explaining the mechanics of surface structures, with ample examples to illustrate the most important aspects. In general, relatively simple and idealized cases, which could readily be solved analytically, were used to illustrate the salient points. Fairly complete compilations of known analytical solutions for both plates1 and thin shells2 are available elsewhere to supplement the examples given in the text. The solution of more complex structures is relegated to the domain of numerical analysis, and several citations to the literature in this area were provided. Although the calculations for those situations may be more involved, such structures generally reflect the same basic behavioral characteristics as the relatively simple illustrations contained in this book.
Phillip L. Gould

Backmatter

Weitere Informationen