Equilibrium finite elements for plane problems of the elasticity theory

Structural mechanics

The work is devoted to the finite elements construction, based on the stresses approximation, for solving plane problems of the elasticity theory. Such elements are alternative to existing finite elements obtained using displacements approximation. Alternative solutions allow more accurate assessment of the structure stress-strain state. The proposed method for constructing finite elements is based on the principles of minimum additional energy and possible displacements. Various stress approximation variants are considered. All approximations variants satisfy the differential equilibrium equations for the case of no distributed load. A comparison is made of the solutions which are obtained by the proposed method with analytical solutions for the ring and the bent beam. The considered stress approximation variants show for test problems good accuracy and convergence, when we grind finite elements grid. It is shown that the best accuracy in calculating stresses and displacements is provided by the finite element with piecewise constant approximations of stresses. In addition, such finite element ensures the displacements convergence to exact values from above. Other finite element variants may be convenient for calculating branched and combined structures. The proposed equilibrium finite elements can be used to more accurately determine the stresses in the calculated structures. The proposed technique can be used to build volumetric finite elements.