Finite element models in stresses for plane elasticity problems

Abstract

The solution of the plane problems of elasticity theory on the basis of stress approximation is considered. To construct the solution, the additional energy functional is used. With the help of the principle of possible displacements, algebraic equations of equilibrium of the nodes of the grid of finite elements are constructed. Equilibrium equations are included in the functional of additional energy by means of Lagrange multipliers. The necessary relations for rectangular and triangular finite elements are obtained. Variants with constant and piecewise-constant approximations of stresses in the region of the finite element are considered. The ribbon width of system of the solving linear equations is estimated. Calculations have been made for the bended beam and for stretched plate with the hole, for the different grids of finite element. It is made comparison of the solutions obtained in stresses with the solutions obtained by finite element method in displacements and with exact solutions. It is shown, that for plane problems in the theory of elasticity, solutions based on stress approximations make it possible to obtain convergence of displacements to exact values from above. For coarse grids, solutions based on piecewise constant stresses much more accurate results, but require large computational costs, since the width of the ribbon of non-zero elements of the resolving system of linear algebraic equations is approximately twice as large as in the other considered variants. Finite elements models in stresses allow constructing solutions, which are alternative to solutions obtained by finite element method in displacements.