The functional of additional energy for stability analysis of spatial rod systems

Structural mechanics

The problem solutions of stability of spatial rod systems by finite elements method in stresses were considered. The proposed method is based on a combination of functional additional energy and the principle of virtual displacements, used for the construction of the equilibrium equations. After discrediting of the subject field, solution of the problem is reduced to the search of the minimum of additional strain energy functional with constraints in the form of the system of linear algebraic equilibrium equations of the nodes. The equilibrium equations are included in the functional with the help of Lagrange multipliers, which are displacements of the nodes. Equations are derived for the static analysis based on approximations of internal forces (stresses) for the spatial rod systems. To solve the stability problems, in the functional of additional energy there are added additional energy the longitudinal deformations, arising due to the bending of rods. Form of the rod buckling is approximated by a linear function on finite element field. Two variants of the internal forces approximations on the finite element field: linear and piecewise constant were considered. Calculations of critical forces (loads) have been performed by the proposed method for the straight rods with different variants of the ends support and the spatial frameworks. The calculation results were compared with the analytical solutions and the solutions obtained by the method of finite elements in displacements. Analysis of the results shows that the use of piecewise constant approximations of internal forces leads to convergence to the exact values of the critical forces (loads) is strictly from below and provides solution with the reserve of stability.