## 276 Methods of Analysis 2761 Response Behavior

Thin shells may resist external loading through forces acting parallel to the shell surface, forces acting perpendicular to the shell surface, and moments. While the analysis of such shells may be formulated within the three-dimensional theory of elasticity, there are reduced theories that are two-dimensional and are expressed in terms of force and moment intensities. These intensities are traditionally based on a reference surface, generally the middle surface, and are forces and moments per unit length of the middle surface element upon which they act: they are called stress resultants and stress couples, respectively, and are associated with the three directions — circumferential, 01; meridional, y2; and normal, 03. In Figure 27.15, the extensional stress resultants, n11 and n22, the in-plane shearing stress resultants, n12 = n21, and the transverse shear stress resultants, q12 = q21, are shown in the left diagram along with the components of the applied loading in the circumferential, meridional, and normal directions, pl, p2, and p3, respectively. The bending stress couples, m11 and m22, and the twisting stress couples, m12 = m21, are shown in the right diagram along with the displacements v1, v2, and v3 in the respective directions.

Historically, doubly curved thin shells have been designed to resist applied loading primarily through the extensional and shearing forces in the ''plane'' of the shell surface, as opposed to the transverse shears and bending and twisting moments, which predominate in flat plates loaded normally to their surface. This is known as membrane action, as opposed to bending action, and is consistent with an accompanying theory and calculation methodology having the advantage of being statically determinate. This methodology was well suited for the precomputer age and enabled many large thin shells, including cooling towers, to be rationally designed and economically constructed [5]. Because the conditions that must be provided at the shell boundaries in order to insure membrane action are not always achievable, shell bending should be taken into account, even for shells designed by membrane theory. Remarkably, the accompanying bending often is confined to narrow regions in the vicinity of the boundaries and other discontinuities and may have only a minor effect on the shell design, such as local thickening and additional reinforcement. Many clever and insightful techniques have been developed over the years to approximate the effects of local bending in shells designed by the membrane theory.

As we have passed into and advanced in the computer age, it is no longer appropriate to use the membrane theory to analyze such extraordinary thin and massive shells, except perhaps for preliminary design purposes. The finite element method is widely accepted as the standard contemporary analysis technique and the attention shifts to the level of sophistication required. Usually, the more sophisticated the analysis, more data are required. Consequently, a model may evolve through several stages, starting with a relatively simple version that enables the structure to be sized, to the most complex version that may depict such phenomena as the sequence of progressive collapse of the as-built shell under various static and dynamic loading scenarios, the incremental effects of the progressive stages of construction, the influence of the operating environment, aging and deterioration on the structure, etc. The techniques described in the following paragraphs form a hierarchical progression from the relatively simple to the very complex, depending on the objective of a particular analysis.

In modeling cooling tower shells using the finite element method, there are a number of options. For the shell wall, a variety of ring elements, triangular elements, and quadrilateral elements have been used. Early on, flat elements adapted from two-dimensional elasticity and plate formulations were used to n22 02 1 n2i .

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