The study object is a three-dimensional triangular prism finite element based on piecewise constant approximations of stresses. The use of such a finite element makes it possible to obtain more accurate stress values, especially at the boundaries of the region and in the stress concentration zones. The solution of the volume theory elasticity problem was obtained on the basis of the additional energy functional and the possible displacements principle. With the help of the possible displacement principle, algebraic equilibrium equations of finite element grid nodes are formed. The resulting equilibrium equations sum up with the additional energy of the system using the Lagrange multiplier method. In this case, the stresses are determined directly at the nodal points, and not at the finite element centers. The stress fields are continuous along finite element boundaries and discontinuous inside them. The paper shows that the displacements obtained by the proposed method, when refining the finite elements mesh, tend to exact values from above. As a test, the article provides calculations for bending plates and beams. As the test problems solutions showed, the proposed finite elements allow obtaining more accurate stress values compared to traditional finite elements based on stress approximation. Comparison of the stresses obtained by the proposed method with analytical solutions shows the high accuracy of the proposed method.