The mathematical model of the behavior of a flexible bar under eccentric tension

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Dudaev M. A., Aleskovskii S. L. Matematicheskaya model' povedeniya gibkogo brusa pri vnetsentrennom rastyazhenii [The mathematical model of the behavior of a flexible bar under eccentric tension]. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie [Modern Technologies. System Analysis. Modeling], 2020, No. 4 (68), pp. 10–18. – DOI: 10.26731/1813-9108.2020.4(68).10-18

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The article considers the influence of bar deflections on the values of normal stresses in the eccentric tension of a constant stiffness rod with a straight axis. A mathematical expression of the curve of the bending axis of the bar in the two main planes of bending (inertia of the section) is obtained using the methods of material resistance. This expression is based on the approximate differential equation of the elastic line and can be used in practical engineering calculations. The obtained expression of the curve of the bending axis of the bar is used to consider for the real eccentricity of the force application relative to the line of the centers of gravity of the cross sections, which changes with increasing or decreasing the applied load. Besides, the limits of applicability of the described mathematical model are defined and the restrictions of limiting physical quantities, by respecting the which the mathematical model retains its adequacy, are numerically indicated. The physical side of the problem is founded on the linear behavior of the material obeying Hooke's law so the article assumes that the maximum stresses in the material of the loaded bar do not exceed the limit of proportionality. Based on the finite element method, a numerical experiment is performed that takes into account the geometric nonlinearity of the problem. The values of discrepancies between the calculation results and the analytical solution are obtained and the values of these discrepancies are given during this experiment. All the calculation results are presented in the form of diagrams constructed in relative coordinates (which highlights the qualitative side of the problem), depicting the change in the studied values along the length of the bar at different flexibilities and different levels of average normal stress.

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