26 Shells

2.6.1 Stress Resultants in the Shell Element

A thin shell is defined as a shell with a thickness relatively small compared with its other dimensions. The primary difference between a shell and a plate is that the former has a curvature in the unstressed state, whereas the latter is assumed to be initially flat. The presence of initial curvature is of little consequence as far as flexural behavior is concerned. The membrane behavior, however, is affected significantly by the curvature. Membrane action in a surface is caused by in-plane forces. These forces may be primary forces caused by applied edge loads or edge deformations, or they may be secondary forces resulting from flexural deformations.

In the case of the flat plates, secondary in-plane forces do not give rise to appreciable membrane action unless the bending deformations are large. Membrane action due to secondary forces is, therefore, neglected in small deflection theory. In the case of a shell that has an initial curvature, membrane action caused by secondary in-plane forces will be significant regardless of the magnitude of the bending deformations.

A plate is likened to a two-dimensional beam and resists transverse loads by two-dimensional bending and shear. A membrane is likened to a two-dimensional equivalent of the cable and resists loads through tensile stresses. Imagine a membrane with large deflections (Figure 2.4 1 a), reverse the load and the membrane, and we have the structural shell (Figure 2.4 1 b), provided that the shell is stable for the type of load shown. The membrane resists the load through tensile stresses but the ideal thin shell must be capable of developing both tension and compression.

Consider an infinitely small shell element formed by two pairs of adjacent planes that are normal to the middle surface of the shell and contain its principal curvatures as shown in Figure 2.42a. The thickness of the shell is denoted as h. Coordinate axes x and y are taken as tangent at 0 to the lines of principal curvature and the axis z normal to the middle surface. rx and ry are the principal radii of curvature lying in the xz and yz planes, respectively. The resultant forces per unit length of the normal sections are given as


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