The governing equations for bending of truncated conical shells with multi-layer anisotropic construction are developed by a variational method. The shell is considered to consist of an arbitrary number of alternating soft and hard layers. It is assumed further that the n hard membrane layers are isotropic and may possess different elastic properties, while the (n-1) soft core layers are orthotropic in general and may take transverse shear only. The variations of stresses across the membranes are neglected, as are the surface-parallel stresses in the cores. These assumptions are consistent with those usually employed in single-core sandwich shells. The energy functional is formulated with the stresses considered as independent variables. The stresses are also dependent variables defined in the set of two curvilinear co-ordinates defining the surface of the shell. The stress resultants are introduced as constraint conditions utilising Lagrange multipliers. Successful definition of an elastic neutral surface ensures the uniqueness of shell constants and the equations obtained may be in a form comparable to that of classical shell theory. The system of equations is reduced to a form where Galerkin’s method can be applied directly.