Энергетические свойства симметричных деформируемых систем

Energy methods for calculating structures, which have become popular for a century, are based on the Lagrange principle and have the meaning of equality of work of external forces and internal forces. Having proved their effectiveness in the overwhelming majority of problems of structural mechanics, they became the dominant approach in formulating the problems of studying solid deformable systems and gave rise to the main methodology for solving problems. As a result, a situation has arisen that the internal potential energy of a deformed body remains insufficiently studied. The paper develops an approach to the study of the symmetric structure at critical levels of strain energy. The criterion of critical levels of strain energy, based on the concepts of "self-stress" ("self-balance") of a deformable body. Limiting values of the structure strain energy may get by varying the reactions and deflections in the nodal points. The extreme values of forces and displacements of the rods are calculated in matrix form from the values of nodal reactions (displacements). Methodology for studying the energy properties of a system   is shown on the examples of the study of symmetric rod systems without involving the concept of external forces. The technique is based on matrix methods of structural mechanics and the mathematical apparatus of eigenvalue problems. The comparison of structural design and structural analysis solution of structural mechanics tasks by traditional methods and with the proposed methodology is carried out.