It is seen in the previous chapter that the membrane theory cannot in all instances provide solutions compatible with the actual conditions of deformation. This theory also fails to predict the state of stress at the boundaries and in certain other areas of the shell. As is illustrated in this chapter, these shortcomings are avoided by application of bending theory by considering membrane forces, shear forces, and moments to act on the shell structure. Formulation of the governing differential equations for the midsurface displacements u, υ, and w, which define the geometry or kinematics of deformation of a shell, is discussed.
First, the basic relationship between the stress resultants and the deformations for shells of general shape is derived. Then, the relationships for the stresses and strain energy under an arbitrary loading are developed. The complete bending theory is mathematically intricate. Consideration here is limited to the most significant practical case involving rotationally symmetrical loading. Stress analysis of cylindrical shells under general loads is postponed until the final chapter.