Theory of Shells

This chapter begins with the shell geometry using the surface theory. Defining the Lamè parameters, derivations continue to discuss the continuity of surface upon satisfying the Codazzi and Gauss conditions. These continuity conditions are derived in detail. The geometry of the shells of revolution is discussed and the Euclidean metric tensor, associated metric tensor, and the Christoffel symbols are derived in the cylindrical and spherical coordinates. Derivation of general strain–displacement relations, using the general expression for strain tensor in curvilinear coordinates, is presented in detail. Different classical second-order shell theories are described and their associated expressions for the strains and curvatures are derived. Assuming the first-order shear deformation theory including the effect of normal stress and strain, the general strain–displacement relations are derived. These equations are reduced to those for the spherical, conical, and cylindrical shells. Considering Hooke’s law and the strain–displacement relations, the stresses are derived in terms of the displacement components. Considering the second-order shell theory, Hamilton’s principle is employed to derive the most general form of the equations of motion for shells of revolution. These equations are reduced for different shell geometries under the axisymmetric loading conditions.