Plates and shells

ANALYSIS OF PLATES AND SHELLS, FRAME, COLUMN, AND PLATE INSTABILITY

2.1 Plates

2.1.1 Introduction

Plate bending analysis is similarly concerned with planarelements, but the loading is now normal to the plane of the element. It is presumed that the plate is free from in-planeloads, the effects of which, if present, may be analyzed separately and a resultant solution obtained by superposition.

Plate elements are usually of either metal or concrete construction. In the former case the elements are referred to simply as plates, while the latter are normally referred to as slabs. Both plates and slabs are used as flooring elements and as bridge decks.

2.1.2 Physical behaviour

2.1.2.1 Beam analogy

A plate subject to normal loading may be considered as an extension of a beam, in which the width of the beam has become of the same order as the length, while the depth (thickness) remains of a smaller order. The discrete supports applicable to beam analysis may also occur in plate problems but supports alonglines in the plane of the plate are commoner. With this relationship between beam and plate elements, some correspondence between their modes of structural behaviour can be expected. This is the case to the extent that a plate element resists normal loading bybending and shearing actions, as does a beam. However, in the case of a plate, the actions are clearly not restricted to the single direction – that along its length, which is available to abeam – but can occur in any direction in the plane of the plate. If rectangularCartesian coordinates are being used, a closer analogy to plate action is provided by a grid of rigidly interconnected beam elements set out along the coordinate directions. The behaviour of such a grid (Fig. 2.1) is therefore considered next.

Thin shells

2.2.1 Introduction

2.2.1.1 Generation, classification and application

A thin shell may be defined as comprising the materialcontained between two closely separated three-dimensional surfaces. As with plates, provided the two generating surfaces are sufficiently close, the deformation of the single surface formed by the mid-thickness points will be sufficient to describe thedeformation of the complete shell. Only the behaviour of a single,middle surface will therefore be considered in the followingtheory, and sufficient assumptions will be made to enable the deformation of any point in the shell to be determined from the deformation of such a middle surface.

Since general three-dimensional surfaces are of interest, the range of possible shell geometries is enormous. Some classification scheme is therefore essential so that advantage maybe taken of whatever similarities various shell forms may possess.Relevant strategies may then be adopted so that the potential analytical complexities are minimized using appropriate coordinatesystems and similar tactics. The most convenient primaryclassification is one based on shell geometry, and examples of the simplest geometric types are shown in Table 2.1. The forms included in Table 2.1 do cover many of the commonest types used in practice, but shells with other geometric regularity features arealso employed as, indeed, are shells of irregular geometry. As willbe seen from Table 2.1, shells find a wide range of application as storage and pressure vessels on land and on sea and in the air when employed as ship or aero plane hull structures. In civil engineering, the routine use of shell roofs has been curtailed by increased labour costs, and shell roofing (Cronowicz, 1968) tends to berestricted to structures which are of special architecturalsignificance, such as the Sydney Opera House in Australia. Majorapplications of shells remain, however, in the construction of cooling towers and water storage and retention structures (mainly circular tanks, cylindrical or conical towers, and arch dams). Shell theory may also be used for the analysis of box girders (see Fig.2.10) and of core-supported buildings (see Fig. 2.11), so that thefields of application are almost as diverse as the possible geometric forms.

Fig. 2.10 (a) Singly- connected box girder. (b) Multiply- connected box girder

Fig. 2.11 Concrete core construction plans

 

Table 2.1 Classification of shells

2.2.1.2 Structural behaviour

In general, shells resist loads by a combination of bending and in-plane actions. In the case of shells, in-plane action is characterized by the plane stress system of direct and shear stresses (see Fig. 2.12) and is normally referred to as membrane action.This terminology derives from the inability of membrane materials, such as fabrics, to resist any bending whatsoever, andtheir consequent total reliance on in-plane action. Examples of membrane shells are sails, tents, balloons, and inflated structures, each of which can only resist in-plane actions and must therefore adopt a shape which allows the imposed loading to be resisted inthis manner. Such structures do not therefore have a unique non-loaded geometry, as their rigid counterparts do, and deter-mining the form of such structures under their initial prestressing and/or self-weight effects becomes a problem in its own right (Firt, 1983).

Fig. 2.12 Plane stress components

Plates represent a special case of shell and may be considered to be the antithesis of membranes in the sense that, when normally loaded, no membrane stresses exist (see section 2.2) and resistance is provided by bending alone. Membrane resistance may be given to a thin plate by folding it, and the effect of the folding is todramatically increase the stiffness. Thus, if the flexible thin sheet of Fig. 2.13(a) is converted into the folded-plate type of shell shown in Fig. 2.13(b), then the sheet is able to sustain quite substantial loads, whereas it previously exhibited gross deformation under a much more modest load. Closed shells, in particular, exhibit high strength and stiffness, as evidenced by thefamiliar example of the ‘nut which is hard to crack’, or, in relationto its thickness, even an eggshell. The high stiffness is primarily due to membrane action, bending often being of secondary or localized significance.

The influence of bending effects on shell behaviour depends on the type of restraints and loading which are involved as well as the shape of the shell. In respect of shell shape, however, bending will always need to be considered in the cases of folded plates and open cylindrical sections (see Table 2.1). For axisymmetric shells, bending effects will tend to be localized, but the rate of decay of these effects will depend upon the nature of the principal radii of

Fig. 2.13 Flat and folded plate

 

 

 

Fig. 2.14 Gaussian curvature of surfaces: (a) positive; (b) zero; (c) negative

curvature which describe the shape of the shell. The product of the two principal curvatures of a surface is known as the Gaussian curvature of the surface, so that

where , are the principal curvatures, and r1, r2 are the principal radii of curvature.

If the two principal radii are of the same sign (Fig. 2.14(a)), as in a dome, then the surface has positive Gaussian curvature, andbending effects will tend to decay rather rapidly. If one of the principal radii is infinitely large, as in the case of a cylindrical shell (Fig. 2.14(b)), then the surface has zero Gaussian curvature and bending influence will persist over a greater region. Radii of curvature of opposing signs, as in the cooling tower of Table 2.1,produce negative Gaussian curvature (Fig. 2.14(c)), which is also susceptible to bending. Axisymmetric shells are therefore conveniently sub-divided into positive, zero, and negative Gaussian curvature types, it being anticipated that bendinginfluence will be greater for zero or negative Gaussian curvature forms than for shells of positive curvature.

As just described, membrane shells resist loads by in-plane forces alone, and membrane effects are also generally predominant in closed axisymmetric shells. The neglect of bending considerably simplifies shell analysis, and the initial treatment will therefore be based on such an assumption. In addition, for reasons of geometric simplicity, only axisymmetric shells will be considered. This restriction excludes the open cylindrical and folded-plate varietiesof Table 2.1, which has some consistency with the neglect of bending effects, since bending is of greater significance in these cases.

Further simplification of the analysis is produced if the loads are presumed to be axisymmetric, as well as the shell shape. This further assumption will therefore be made and will have the practical effect of restricting the discussion to dead, snow, over-pressure, and similar loads which may be presumed axisymmetric. Wind and other non-axisymmetric loads will have to be excluded.An analysis derived on the above basis is termed a membraneanalysis for thin elastic shells under axisymmetric loading.

A membrane analysis can be helpful even when significant bending occurs, since, as described, the bending effects will sometimes be localized, and a membrane solution will then represent the shell behaviour at regions distant from the areas of localized bending. Even when bending is significant throughout the shell, use can be made of a membrane analysis as a ‘primary’solution to which ‘corrections’ are made to allow for the bending effects (Baker et al., 1979; Billington, 1982; Zingoni, 1997). Shell analysis incorporating bending tends to be complex, whethertackled in the way just mentioned or by any other analytic closed-form approach. A numerical technique is therefore often preferred, and the finite element method generally offers the most flexible approach. As an example of the application of the finite elementmethod to shell analysis, the method is therefore applied to the analysis of circular cylindrical shells in the later sections of this chapter. Such an analysis is usually known as a linear thin shell bending analysis.

Summary:

  • Plates are structural elements analyzed for bending under normal loading, typically in construction as metal plates or concrete slabs. They resist loads through bending and shearing actions, with behaviors analogous to beams, but act in all directions.

  • Thin shells consist of material between two surfaces, described by middle surface deformation. Classified by geometry, they serve various applications such as pressure vessels and architectural structures. Membrane action is predominant in shells, allowing them to resist loads through in-plane forces.

Flashcards:

  1. Plate - A planar structural element that resists normal loading through bending.

  2. Shell - A three-dimensional structure that resists loads primarily through membrane action and bending.

  3. Gaussian Curvature - A measure of the curvature of a surface that influences resistance to bending in shells.

  4. Membrane Action - A load-resisting mechanism in which forces are handled through in-plane stresses only, typically seen in thin shell structures.

  5. Finite Element Method - A numerical technique often used for analyzing complex shell structures, providing flexibility in modeling.

Possible Questions:

  1. What are the key differences in behavior between plates and shells?

  2. How does Gaussian curvature affect the structural analysis of shells?

  3. What are common applications for thin shells in engineering?