Abstract Flexural response of isotropic, orthotropic, and cross-ply laminated shear deformable plates has been presented in this paper using a new first-order shear deformation theory. The theory involves only two primary unknown functions, as compared to widely used Mindlin’s first-order shear deformation theory, which involves three primary unknown functions. This reduction in the number of primary variables make this theory simple and efficient to use. The principle of minimum total potential energy is utilised to obtain governing differential equations and variationally consistent boundary conditions. The Levy-type plates accompanied by the generalised segmentation technique is used to obtain the closed form analytical solution of isotropic, orthotropic and cross-ply laminated shear deformable plates with two opposite edges simply-supported and the other two edges having different combination of simply-supported, clamped and free boundary conditions. Comparative studies are performed to show the efficacy of the present theory. Effects of plate length-to-thickness ratio, aspect ratio and modulus ratio on transverse displacement and in-plane normal stresses are also presented and discussed. New benchmark solutions in the form of parametric study are presented for the future reference.