<p>Asymptotic averaging of elastic homogeneous, continuously inhomogeneous in transverse direction and layered plates up to the fifth order of approximation in the case of small strains and deflection was carried out. The asymptotic decomposition reduces 3D equilibrium equations of linear elasticity to a series of 2D forth-order plate equations and 1D boundary-value problems in the transverse direction. Boundary conditions on the plate top and bottom surfaces were derived. It is shown that averaged equations of equilibrium are equivalent to the boundary conditions on the top surface if the bottom surface of plate is free. On the contrary, the boundary conditions on the lateral surface, as well as the edge effect that occurs near the lateral surface, were not covered by asymptotic representation. The zone of edge effect is investigated using numerical example for cylindrical bending. A novel feature of the work is the consideration of the fourth and fifth approximations. Bending and in-plane tension/shear coupling occurs in fourth-order asymptotic term even in the case of a homogeneous plate and small deflections. While this effect does exist, it is small quantitatively. The forth-order asymptotic term gives a refinement to the transverse shear stresses for a strongly orthotropic plate regardless the deflection is calculated by the asymptotic method or the finite element method.</p>