Jeroen H. Hoefakker

Introduction to Shells

This book is concerned with thin elastic shells. A thin shell has a very small thickness-to-minimal-radius ratio, often smaller than 1/50. As with plates, an applied load that acts out-of-plane leads to larger displacements than those generated by a load acting in-plane with the same intensity. Due to its initial curvature, a shell is able to transfer an applied load by in-plane as well as out-of-plane actions. A thin shell subjected to an applied load therefore produces mainly in-plane actions, which are called membrane forces. These membrane forces are actually resultants of normal stresses and in-plane shear stresses that are uniformly distributed across the thickness.

Membrane Theory for Thin Shells of Arbitrary Curvatures

The three sets of equations of Chap. 2 derived for the membrane behaviour of thin shells relate to a co-ordinate system placed according to the principal curvatures. In practice it may be useful to choose the co-ordinate system in such a way that a co-ordinate axis is placed along an edge of the shell, which does not necessarily coincide with a principal curvature. This is the subject of this chapter. Different from the approach of Chap. 2, we now choose a reference system of axes in the tangent plane of a point O at the middle surface of the shell. The x-axis and y-axis are in the tangent plane, and the z-axis is normal to the plane. Instead of the principal curvatures k 1 and k 2, we now work with curvatures k x and k y . On top of that, it will appear to be convenient to define a twist k xy . The expressions for the curvatures k x and k y are in fact the same as in Chap. 2, but we will derive them again in an alternate way, such that it is easy to extrapolate to the derivation of the twist k xy .

Introduction to Buckling

The investigation of stability of shell structures is a specialty which in its own right deserves a complete separate book. Here just some main aspects are explained, and the correspondence and difference with beam-column buckling is touched. We will successively discuss buckling of uni-axially loaded plates as a limit case of shells, arched beams, arched circular roofs, axially-pressed shells of rotation and domes.

Membrane Theory for Shells with Principal Curvatures

The basic assumption of membrane theory is that a thin shell produces a pure membrane stress field, and that no bending stresses occur. This assumption is applicable if certain boundary and loading conditions, exemplified in Chap. 1, are met. In this pure membrane stress field, only normal and in-plane shear stresses are produced. They are due to stretching and shearing of the middle plane of the shell. Bending, torsion and transverse shear stresses are not accounted for.

Morley Bending Theory for Circular Cylindrical Shells

The Donnell theory of shallow shells does not accurately apply to fully-closed circular cylindrical shells like chimneys and storage tanks. The main reason is the simplifying assumption that we can use the formulas of the flat plate theory for the change of curvatures in the shell. In full circular cylindrical shells, this is not sufficiently accurate, because then rigid body motions would lead to non-zero changes of curvatures. Especially for structures like long industrial chimneys, storage tanks and pipelines, the imperfections are undesirable and result in substantial errors.

Edge Disturbance in Shell of Revolution Due to Axisymmetric Loading

Tens of years ago, theories were proposed for the edge disturbance problem in shells of revolution. Because a rigorous bending theory is complicated, attempts were made for reliable approximations. A well-known one was published by Geckeler [1, 2], who obtained his approximation by simplifying mathematical considerations to the exact equations. We adopt in the present chapter the approximation of Geckeler, but will arrive at it in an alternative, simpler way, which is inspired by the stave-ring model for edge disturbances in circular cylindrical shells in Chap. 5.

Circular Cylindrical Roof

In this chapter, we study the bending behaviour of a circular cylindrical roof shell under asymmetric loading. The shell is part of a full circular cylinder, and is supported at the curved edges by a diaphragm (tympan) which is infinitely rigid in its plane and perfectly flexible out-of-plane. The shell is pin-connected to the diaphragm. The roof structure can be considered as a shallow shell, so the Donnell theory [1] of Chap. 6 is applicable. The theory presented here was worked out by Bouma et al. Bouma was the leader of a project team in The Netherlands, in which also Loof and Van Koten contributed, making the demanding theory accessible to practitioners [2, 3]. Von Karman et al. extended Donnell’s theory to the study of buckling [4] and Jenkins [5] was the first to apply it to circular cylindrical roofs (applying matrix analysis). Because of the respective contributions of Donnell, Karman and Jenkins, the Dutch team used to refer to the theory as the DKJ-method.

Donnell Bending Theory for Shallow Shells

In this chapter we will extend the membrane theory for shells of arbitrary curvatures, as presented in Chap. 3, to a theory in which we account for both membrane and bending action. This theory, developed by Donnell [1], is applicable to shallow shells like roof shells. The Donnell theory is not sufficiently accurate for circular cylindrical shells like chimneys and storage tanks. These structures are deep shells instead of shallow ones. Hereafter, in Chap. 9, we will present a more rigorous theory for this type of shell.

Hyperbolic- and Elliptic-Paraboloid Roofs

In the previous chapters, we have discussed cylindrical shells, which are special cases of doubly curved shells. We now proceed to more general cases of such shells, the elliptic paraboloid shell and the hyperbolic paraboloid shell. Structural engineers refer to the first category as elpar and to the second one as hyppar. If built, elpars have a rectangular plan with curved edges (left shell in Fig. 8.1). Hyppars also have rectangular plans, but may have either curved edges (middle shell in Fig. 8.1) or straight edges (right shell in Fig. 8.1). Because hyppars on straight edges are applied most, we will pay most attention to this type, and start with them.

Semi-Membrane Concept Theory for Circular Cylindrical Shells

In the previous chapter, we learned about short and long influence lengths for full circular cylindrical shells. For axisymmetric loading and beam-type loading, only the short influence length plays a role, and the edge disturbance is governed by a fourth-order differential equation. For the self-balancing modes, both the short and long influence lengths play a role, and the more complex eighth-order differential equation of the Morley theory is needed. It is possible for this loading type to reduce also to a fourth-order differential equation, an approximation in which only the long influence length will play a role. This so-called semi-membrane concept (SMC), introduced by Pircher et al. [1], is the subject of the present chapter.