Determination of the Residual Strain Energy of a Structure and the Method of Progressive Limiting State

Abstract

The technique of progressive limiting state of the structure, which is implemented in the form of the displacement method, is considered. The existing formulations of the problems in structural mechanics are based on the principle of the minimum of the total energy of deformation of structures (Lagrange approach), according to which the resulting project corresponds to a predetermined load. But using this approach, it was not possible to obtain information about its residual load-bearing capacity in a deterministic form. To overcome these difficulties, it is proposed to use the criterion of critical levels of deformation energy. The equations of state of the structure at critical energy levels are derived from the variational principle of the minimum of the potential energy of deformation at the critical energy level. The problem is formulated as an eigenvalue problem and the values of the maximum values of the design parameters are found for the stiffness matrix. It becomes possible to calculate the displacements for the critical energy level, and the maximum possible value of the deformation energy of the structure. Subtracting the magnitude of the work of external forces, we obtain the value of the residual potential energy, which determines the residual load-bearing capacity of the structure. When analyzing internal forces in a structure in a critical condition, we determine their greatest values, which cause the limiting state. We call these elements the ‘weak link’ because they will be the first to stop working on an external load. Sequentially removing these elements from the design scheme, we obtain new states of self-tension of the structure until it becomes an unstable system. Thus, we obtain the technique of progressive limiting state of the structure.