Progressive limit state at critical levels of internal potential energy of deformation
Introduction. The work is devoted to one of the main issues of structural mechanics - the determination of the elements in which the limiting state occurs first. At first glance, the task has an infinite number of results, meaning an infinite number of options for loading the system. The problem becomes solvable if one examines the structure of a building (structure) for possible variations in displacements (forces) in the nodes of the structure. For this approach, it becomes possible to determine the main values and vectors of displacement of the system, which correspond to the maximum (minimum) values of deformations (forces) in the rods of the system. As close approaches to the formulation of the problem, one should indicate the theory of the limiting equilibrium of structures under the assumption of the work of the material under flow conditions, where the equality of the work of external forces and internal forces is considered (kinematic method), or possible static stress states of the system for maximum limiting loads (static method). The theory of protecting buildings and structures from progressive collapse seeks to solve similar problems, focusing on options for design solutions that prevent destruction from non-design loads. Materials and methods. To determine the options for the distribution of extreme values of internal forces (deformations) in the system, the problem is formulated in the form of an eigenvalue problem. The latter turns out to be the most convenient mathematical model of the problem, since, in addition to extreme values (as in the optimization problem), it allows one to take into account the values of the problem on the upper and lower bounds. The theoretical basis for the formulation of the problem is the criterion of the critical levels of the internal potential energy of the system, which makes it possible to find the self-stress states of the structure corresponding to the limiting states of the structural elements. Results. The methodology for solving the problem is illustrated by the example of a statically indeterminate five-rod truss, which was also considered by other authors. The matrix formulation of the problem and a detailed algorithm for its solution are given. It is shown that the values of the internal forces in the rods, obtained using the traditional method, are in the interval between the maximum and minimum main values of the self-stress state of the system. Solutions are given at each of the critical energy levels corresponding to the disconnection of bonds from work. Conclusions. The examples of the possible use of the technique in computational practice are given. Possible options for application in design practice, as well as further development of the theory are indicated.