This paper is concerned with an accurate local buckling analysis of the sheathing of a composite stabilising wall element in a new post-and-beam system for the market of non-residential multi-storey timber buildings, called Trä8. This element has a box-type cross-section with glulam framing members and sheathing on both sides. The wall element practically acts as a cantilever subjected to a shear load from wind. Consequently, the rectangular sheathing areas between the frames on both sides of the wall elements may buckle locally. These areas can be considered as thin-walled clamped rectangular orthotropic plates subjected to quasi-uniform distributed shear load around their edges. The differential quadrature method is employed to solve the governing equation of thin orthotropic plates and satisfy the boundary conditions. The eigenvalues for critical buckling loads are extracted and corresponding eigenfunctions are obtained for the mode shapes. An accurate finite element analysis is performed using the ANSYS software to ensure the correctness and accuracy of the results obtained. Comprehensive numerical results are presented to investigate the effect of different mechanical and geometric properties. Also, some interesting contour plots of mode-shapes are presented to get a better physical sense of the local buckling of the Trä8 wall element.