Mining Journal of Kryvyi Rih National University

Abstract

Purpose. To review the methodology for solving problems of structural mechanics based on the criterion of critical levels of energy. Based on the solution of the classical problem of material resistance, to determine the effectiveness of the proposed solution method and to compare the solution methodology based on the criterion of critical levels of energy with traditional methods. Research methods. A complex of research methods was used, which includes analysis and generalization of literary sources in accordance with the purpose of the work, formulation of concepts and their verification by performing practical calculations. Scientific novelty. The possibility of studying deformation processes of structures based on the division of fields of external influences and the field of deformations of a solid deformed body, which exist according to their own laws, has been shown. The concept of critical levels of energy has been introduced, and the periodic nature of problems of critical energy levels and their connection with states of self-stress in structures has been demonstrated. Using the example of calculating a rod under tension, a methodology for calculating structures at critical energy levels has been shown, and explanations for existing discrepancies between "classical" calculation theory and experimental data have been provided. Practical significance. The hypothesis of the existence of critical levels of potential deformation of solid deformed bodies has found its confirmation both in theory and in the implementation of the methodology for solving practical problems. The variation criterion of critical levels of energy has demonstrated its effectiveness in solving practical problems. The criterion of critical energy levels provides a theoretical-mathematical basis for formulating problems in structural mechanics from a unified perspective in the form of an eigenvalue problem. Results. The criterion of critical energy levels allowed obtaining new results that refine the data of linear theory of elongation of rod structures, as well as demonstrated its effectiveness compared to iterative procedures of geometrically nonlinear problems. The internal potential energy has an extreme value at critical levels and is limited to the range of permissible parameter values, which allows to formulate and solve boundary value problems for the construction of any formulation of the boundary state in terms of eigenvalues. The theory of critical levels of internal potential energy of deformed bodies allows more accurate and efficient solution of problems of optimal design of structures

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Retrieved from Vol. 57, No. 1, 2023

Pages 54–60