This paper is concerned with the 2D elasticity solution for the buckling of a thick orthotropic ring under external hydrostatic pressure loading. The bifurcation buckling problem is first formulated using two methods, distinguished by the manner in which the external work done by the pressure loading during the buckling transition is treated. In doing so, the correct buckling equations and associated traction boundary conditions are derived. The resulting sets of equations and associated boundary conditions are then cast in a weak form, amenable to a numerical solution using the finite element method. The necessity of using the correct pairs of energetically conjugate stress and strain measures for the buckling problem is pointed out. Errors in using the incorrect traction boundary condition and terms that influence the buckling load and that have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. Results from the present two-dimensional analysis to predict the critical pressure are compared with previous theoretical results. The formulation and results presented here can be used as the correct benchmark solution to establish the accuracy in computing the buckling load of thick orthotropic composite structures, of contemporary interest, due to the increased use of thick-walled composite shell type structures in diverse engineering applications.