The paper proposes the method for calculating of thin plates stability by the finite element method based on piecewise constant approximations of the moments’ fields. Using this approach may allow us to obtain the lower limit of critical stresses. We build the solution based on the extended functional of additional energy. The functional, using the method of Lagrange multipliers, includes algebraic equations of nodes equilibrium of the finite elements mesh. Using the possible displacements principle, we get equilibrium equations. The plate vertical displacements function after stability loss, is combination of linear basis functions. For rectangular and triangular finite elements there are the necessary expressions for the stresses work, acting in the plate median plane, from bending deformations. There are critical stress calculations for rectangular plates with different supporting conditions. The options for the action of compressive and shear stresses are considered. It is shown, that when the finite element mesh is refining up, the critical stress value in all the considered examples tends to the exact value from below. We perform comparison of the obtained solutions with the analytical solutions and the solutions by the program based on the finite element method in displacements. Comparison of solutions showed good accuracy in determining critical stresses by the proposed method.