Intermediate Mechanics of Materials
This book covers the essential topics for a second-level course in strength of materials or mechanics of materials, with an emphasis on techniques that are useful for mechanical design. Design typically involves an initial conceptual stage during which ma
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A shell is any structure or part of a structure in which one local dimension is much smaller than the other two. Shells abound in all branches of engineering and in the natural world. Examples include gas tanks, thin-walled pipes and air ducts, grain hoppers, car bodies, aircraft wings and fuselages, window panes, eggshells, wineglasses, concrete roofs and the human skull. A shell which is plane in the undeformed state is known as a plate. Plates are generally much less efficient than curved shells as structural elements, being more flexible and less capable of transmitting transverse loads (i.e. loads normal to the plane) without failure. You can verify this by pushing against a plate glass window or the plane side of a metal filing cabinet. You will find that it is quite easy to produce an observable deflection. The same material formed into a curved shell (e.g. a car body panel or a drinking glass) will sustain an equal transverse load without observable deformation. The basic idea behind the analysis of shells is that the stresses can’t vary much across the thin dimension and hence we can approximate the stress distribution across the thickness by a Taylor series. When the shell is very thin, only the first term of this series will be important — this is the ‘membrane theory’, in which the stresses are assumed to be uniform through the thickness. In many problems — particularly those involving shells with discontinuities or built-in boundary conditions — it is necessary to include both the first and second terms of the series. This constitutes the ‘bending theory of shells’. It might be helpful to draw an analogy with beams, which of course are structures in which two dimensions are much smaller than the third. The ‘membrane theory of beams’ assumes that the axial stress σzz is uniform across the section and hence corresponds only to the transmission of an axial force passing through the centroid. The addition of one extra term in the Taylor series permits the stress to vary linearly across the section and defines the bending theory of beams. In this chapter, we shall restrict attention to the membrane theory of shells, which therefore implies that bending moments are everwhere negligible. It follows that derivative of the bending moment must also be everywhere negligible and hence
J.R. Barber, Intermediate Mechanics of Materials, Solid Mechanics and Its Applications 175, 2nd ed., DOI 10.1007/978-94-007-0295-0_8, © Springer Science+Business Media B.V. 2011
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8 Membrane Stresses in Axisymmetric Shells
there can be no shear force or transverse shear stress, by extension of equation (6.1). In other words, the only non-zero stresses are in the local (tangential) plane of the shell and are uniform through the thickness. Despite these restrictions, we shall find that the curvature of a membrane shell permits it to transmit a wider range of loads than either a straight beam or a plate under similar restrictions, making the shell a useful and versatile lightweight structural member. We shall illustrate this here
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