A justification of the Marguerre-von Kármán equations

The method of asymptotic expansions, with the thickness as the small parameter, is applied to the general three-dimensional equations for the equilibrium of nonlinearly elastic shells with specific geometries, subjected to suitable loadings and boundary conditions. Then it is shown that the leading term of the expansion is the solution of a system of equations equivalent to those of Marguerre-von Karman (the case of a clamped shell is also considered). In addition, without making any a priori assumption regarding the variation of the unknowns across the thickness of the shell, it is found that the displacement field is of Kirchhoff-Love type, and that the stresses have polynomial variations with respect to the thickness. This approach clearly delineates the types of three-dimensional loadings, boundary conditions, and "shallowness", for which a three-dimensional problem may be deemed asymptotically equivalent to a two-dimensional shallow shell model. © 1986 Springer-Verlag.