The basis equations of statics, geometry and physics are given. They concern a conical shell with a variable thickness along the meridian. The shell, however, undergoes arbitrary loading. The investigation is based on the general moment shell theory and the basic system of partial differential equations has been derived in displacements. Furthermore, differential relations, expressing loads by means of displacements, have been obtained. The problem is considered as physically and geometrically linear. The effect of the elastic foundation, following Winkler’s model, has been taken into account in meridional, parallel and normal direction. Effects of uniform and nonuniform temperature field are revealed as well. Equations, describing the special cases of cylindrical shells and circular and ring-shaped plates, have been simplified. The external load (including temperature effects) and the unknown displacement functions, forces and moments are expanded in Fourier series, but in parallel direction only. This reduces the problem to the one-dimensional case. The basic system of equations for the determination of the Fourier displacement coefficients, as well as formulas for the force and moment Fourier coefficients, have been derived.