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
10.26731/1813-9108.2020.4(68).10-18
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.
- Pisarenko G.S., Yakovlev A.P., Matveev V.V. Spravochnik po soprotivleniyu materialov [Reference book on the resistance of materials]. Kiev: Naukova dumka Publ., 1988. 736 p.
- Pisarenko G.S., Agaev V.A., Kvitka A.L. et al. Soprotivlenie materialov [Resistance of materials]. In Pisarenko G.S., Acad. Member of the AS USSR (ed.) Kiev: Vishcha Shk., Main publ., 1986. 775 p.
- Feodos’ev V.I. Soprotivlenie materialov: Ucheb. dlya vuzov. [Resistance of materials: a textbook for universities]. 10-th edition, revised and expanded. Moscow: N.E. Bauman MSTU Publ., 1999. 592 p.
- Birger I.A., Mavlutov R.R. Soprotivlenie materialov [Resistance of materials]. Moscow: Nauka Publ., 1986. 560 p.
- Darkov A.V., Shpiro G.S. Soprotivlenie materialov [Resistance of materials]. Moscow: Vysshaya shkola Publ., 1989. 622 p.
- Aleksandrov A.V., Potapov V.D., Derzhavin B.P. Soprotivlenie materialov [Resistance of materials]. Moscow: Vysshaya shkola Publ., 2000. 560 p.
- Mezhetskii G.D., Zagrebin G.G., Reshetnik N.N. et al. Soprotivlenie materialov: uchebnik [Resistance of materials: textbook for universities]. In Mezhetskii G.D., Zagrebin G.G. (gen. eds.) Moscow: Dashkov and KO Publ., 2008. 416 p.
- Styopin P.A. Soprotivlenie materialov [Resistance of materials]. Moscow: Vysshaya shkola Publ., 2012. 367 p.
- Vigodskii M.Y. Spravochnik po vysshei matematike [The handbook of higher mathematics]. Moscow: AST: Astrel Publ., 2006. 991 p.: il.
- Dudaev M.A. Matritsa zhestkosti balki Timoshenko v konechnoelementnom analize dinamicheskogo povedeniya rotornykh turbomashin [Timoshenko bar stiffness matrix in finite element analysis of dynamic behavior of rotary turbomachines]. [Proceedings of Irkutsk State Technical University], 2014. No.6. Pp. 59–65.
- Dudaev M.A. Vliyanie na dinamicheskie parametry izdeliya vneshnego neperiodicheskogo silovogo vozdeistviya [Influence of external non-periodic force impact on the dynamic parameters of the product]. [Transport infrastructure of the Siberian Region: the materials of the tenth International Scientific and Practical Conference, May 21–24 2019, Irkutsk: Vol. 1-2]. Irkutsk, 2019. Vol. 2. Pp. 826–830.
- Zienkiewicz O.S. Metod konechnykh elementov v tekhnike [The finite element method in engineering]. Moscow: Mir Publ., 1975. 542 p.
- Bathe K.J. Finite Element Procedures. Upper Saddle River, New Jersey: Prentice hall, 1996. 1038 p.
- Chen Z. Finite Element Methods and Their Applications. Berlin: Springer, 2005. 411 p.
- Cook R.D. Finite Element Modeling for Stress Analysis. New York: John Willey & Sons, Inc., 1995. 321 p.
- Crisfield M.A. Non-Linear Finite Element Analysis of Solids and Structures. Wiley: Essentials, 1996. Vol. 1-2.
- Zienkiewicz O.S., Taylor R.L. The Finite Element Method: The Basis. Oxford: Butterworth-Heinemann, 2000. Vol. 1-3.