In this chapter we encounter our first shells—shells that act like beams. The single greatest difference from the chapter on birods is that curvature effects now infest, at least implicitly, nearly every equation we set down. Consequent new features that we shall discuss at some length are: (1) necessary and sufficient conditions for a load potential P to exist if there are distributed as well as edge loads; (2) definitions of different extensional, shear, and bending strains;, (3) restrictions on the strain-energy density imposed by invariance requirements and implied by thinness; and (4) structural stability (buckling) including thermodynamic considerations.

Geometry of the Undeformed Shell and Planar Motion

An infinite, right cylindrical shell is a body whose initial shape is generated by translating a plane region normal to itself. See Fig. 4.1a. Mathematically, the plane region is the image of a rectangular region; the images of the top and bottom of the rectangular region generate the faces of the shell whereas the images of the ends of the rectangular region generate the edges of the shell. Although our intuitive picture is that of a thin-walled panel of, possibly, variable thickness, the notion of thinness shall not enter explicitly until we come to constitutive relations in Section N.

For a quantitative description, consider a fixed, right-handed Cartesian reference frame Oxyz. Let {ex, ey, ez} denote the associated set of orthonormal base vectors with ez perpendicular to the plane region that generates the shell.