We propose a new numerical method for the determination of stress and deformation in laminated and inhomogeneous, anisotropic, elastic and thermoelastic plates and shells. The method determines numerical solutions of the three–dimensional elasticity equations, without recourse to any thin plate or shell approximations. It is based on a transfer matrix formulation of the elasticity equations and proceeds by using finite–difference approximations to derivatives in the in–surface coordinate directions, while retaining the derivatives in the through–thickness direction. This leads to a system of linear first–order differential equations, for the through–thickness values at the grid–points, of the relevant stress and displacement variables. These equations are solved numerically by standard methods. In this paper we consider the specific problem of an anisotropic laminated circular thermoelastic cylinder in plane strain, but various other problems may be formulated in a similar way. We present numerical solutions to four test problems for a laminated circular cylinder under: (i) sinusoidally varying internal pressure; (ii) mixed boundary conditions with a through–thickness temperature variation; (iii) a spatially discontinuous internal pressure; and (iv) both angular and radial temperature variation. For problems (i), (ii) and (iv) analytical solutions are also derived, and excellent agreement between the numerical and analytical solutions is obtained.