Progressive limit state at critical levels of internal potential energy of deformation

Abstract

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.

Methodology for determining progressing ultimate states based on the displacement method

“…The criterion of critical energy levels is based on the separation of the energy of external influences and the energy of internal deformations [22]. This criterion can be formulated in the form of equations describing the requirement of a minimum variation of the deformation energy of a structure, including the condition of orthonormality of the design parameters of the structure, and boundary conditions for the range of permissible design parameters:…”

“…To resolve the mentioned difficulties, it is proposed to use the criterion of critical levels of internal energy of deformation [22]. It can be used to find the maximum possible deformation energy of a structure and based on it evaluate the remaining load-bearing capacity of the structure [23].…”

Experimental Determination of the Bearing Capacity of a Reinforced Concrete Floor Slab

“…Это обусловлено теоретическими предпосылками расчетов, основанных на гипотезах о наступлении предельного состояния, невозможности учета всего многообразия строительных конструкций и возможных воздействий на них [1][2][3][4][5][6][7][8][9][10]. Одной из причин является бурное развитие композитных конструкций, для которых пока не существует достаточно адекватных расчетных моделей [11][12][13][14][15][16][17][18][19], но требуются экспериментальные результаты, позволяющие их построить.…”