"[W]e can consider the general equation for the deflection [w] of a shell as\mathcal{L}[w(x,y)] = f[p(x,y)]where \mathcal{L} is a and f[p] is some function of the given loading p. The general solution... will bew = w_h + w_pwhere w_p is the particular solution... that satisfies equilibrium and compatibility at all internal points... but not necessarily satisfying the boundary conditions. ...w_h is the solution of the homogeneous equation with p = 0 (...only edge loads can be present). ...[F]or solving practical problems, we can [find] w_p...by assuming moments and shears to be zero... the "membrane solution." ...similar to obtaining (fairly correctly) the forces in Truss members by assuming moments and shears in the members as zero (i.e., assuming the joints are perfect pins)... For obtaining w_h... we must... use the exact differential equation... the "bending theory" solutions... Fortunately, for most types of shells, they die out quickly as we move away from the boundaries. ...[O]ur general procedure ...obtain membrane forces under a given loading... then superimpose... the bending theory solutions for edge loads. ...[T]he membrane and edge-load solutions together satisfy the boundary conditions; i.e., the edge loads are obtained by solving equations of compatibility at the boundaries."