Stupishin investigation of the stability of systems with lumped parameters based on critical energy level criterion

The article analyses the stability of systems with lumped parameters from the standpoint of the criterion of critical energy levels. Until now, there is no consensus both on the reasons for the manifestation of the buckling of the deformation form in building structures, and on the formulation of criteria that determine the critical state. Most often in structural mechanics and the theory of stability of structures, energy criteria are used in the form of Timoshenko and Brian. Despite the simplicity of the formulation of the first criterion and the generality of the second, it is difficult to assert that they can cover the entire spectrum of stability problems that arise in technology. The criterion of critical energy levels allows one to set and solve stability problems without restrictions on the smallness of displacements, the type of impact on the system, and is intended for formulating boundary states. To understand the essence of the mentioned criteria and to illustrate the differences, simple problems are proposed in the form of systems with lumped parameters with several degrees of freedom. The design research technique is illustrated by the example of a system model with lumped parameters in the form of elastic hinges. Energy relations are given that describe the state of the system at critical energy levels. The results obtained for critical internal forces coincide with the values of external critical loads known from the literature, which confirms the well-known fact of the theory of eigenvalue problems that the branch points of the solution coincide in linear and nonlinear formulations. A technique for determining the values of the deviation angles of the system elements in elastic hinges is shown. A comparison is made of the rotation angles of the system elements in the formulation of the problem in a linear (infinitely small deviation angles) and non-linear formulation.