We approximate the displacement field in a shell by a fifth-order Taylor–Young expansion in thickness. The model is derived from the truncation of the potential energy at fifth order. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field. This leads to an analytical expression for the two-dimensional potential energy of a shell in terms of the zero-order displacement field and its derivatives that include non-standard transverse shearing and normal stress energy. Then we derive the equation of equilibrium and the boundary conditions.