Stability of a compressed-tensioned rod of variable cross-section with combined loading

Objective. The purpose of the study is to determine the stability of a straight rod of variable cross-section under combined axial loading. Method. The longitudinal bending of the rod is described by the classical theory using Bernoulli’s hypothesis, and the critical forces are determined from the Euler problem with appropriate assumptions. Result. An algorithm for a numerical method for solving the problem of determining the eigenvalues of the differential equation for longitudinal bending of a rod is proposed. External loads are considered “dead”. The functions of changing the variable cross-sectional area, variable stiffness and distributed load are considered given. The curved axis of the rod after bifurcation is described using a linear ordinary differential equation. Conclusion. The implementation of the numerical method was carried out by the finite difference method using numerical methods and modern computer software.

1. Karamansky T.D. Numerical methods of structural mechanics. M.: Stroyizdat, 1981;436. (In Russ)

2. Samarsky A.A., Gulin A.V. Numerical methods. M.: Nauka, 1989; 432. (In Russ)

3. Ilyin V.P., Karpov V.V., Maslennikov A.M. Numerical methods for solving problems of structural mechanics. – M.: Publishing house ASV; St. Petersburg: SPbGASU, 2005; 425. (In Russ)

4. Baragunova L.A. Determination of the critical force of a compressed rod by the finite difference method. Nalchik: Bulletin of Kabardino-Balkarian State University. 2000; 13. (In Russ)

5. Kulterbaev K., Lafisheva M., Baragunova L. (2022). Solving the Euler Problem for a Flexible Support Rod Base on the Finite Difference Method. In: Tchernykh, A., Alikhanov, A., Babenko, M., Samoylenko, I. (eds) Math-ematics and its Applications in New Computer Systems. MANCS 2021. Lecture Notes in Networks and Systems, vol 424. Springer, Cham.

6. Kulterbaev Kh.P., Baragunova L.A., Lafisheva M.M. Calculation of a beam on elastic base analytical and numerical methods. Structural mechanics and structures. 2022;4(35):16-23. (In Russ) DOI 10.36622/VSTU.2022.35.4.002

7. Kulterbaev, K.P., Lafisheva, M.M. (2022). Non-classical Methods of Determination Critical Forces of Compressed Rods. In: Tchernykh, A., Alikhanov, A., Babenko, M., Samoylenko, I. (eds) Mathematics and its Applications in New Computer Systems. MANCS Lecture Notes in Networks and Systems, vol 424. Springer, Cham. https://doi.org/10.1007/978-3-030-97020-8_9. 2021; 91–1021.

8. Kulterbaev, K., Lafisheva, M., Baragunova, L. (2022). Solving the Euler Problem for a Flexible Support Rod Base on the Finite Difference Method. In: Tchernykh, A., Alikhanov, A., Babenko, M., Samoylenko, I. (eds) Mathematics and its Applications in New Computer Systems. MANCS Lecture Notes in Networks and Systems, vol 424. Springer, Cham. https://doi.org/10.1007/978-3-030-97020-8_13. 2021;143–151.

9. Kulterbaev Kh.P., Abdul Salam I.M., Paizulaev M.M. Free longitudinal vibrations of a vertical rod with discrete masses in the presence of damping forces. Herald of the Daghestan State Technical University. Technical science. 2018; 45(3): 8-17. DOI:10.21822/2073-6185-2018-45-3-8-17 (In Russ)

10. Kagan-Rosenzweig L.M. On the influence of shear deformations on the stability of a rod of variable cross-section. Bulletin of Civil Engineers. 2019; 6 (77): 137-142. (In Russ)

11. Lampsey B.B., Khazov P.A., Kirillova N.A. Stability of a centrally compressed rod of variable cross-section. Bulletin of the Volga Regional Branch of the Russian Academy of Architecture and Construction Sciences. 2018; 21:139-142. (In Russ)

12. Bogdanovich A.U., Abdyushev A.A. Stability of a rod of variable elliptical cross-section under longitudinal compression. News of the Kazan State University of Architecture and Civil Engineering. 2006; 2 (6): 38-41.

13. Radin V.P., Chirkov V.P., Poznyak E.V., Novikova O.V. Stability of a rod with an elastic hinge under loading with a distributed non-conservative load. News of higher educational institutions. Mechanical engineering. 2023; 5 (758): 3-13. (In Russ)

14. Ilgamov M.A. Bending and stability of a cantilever rod under the influence of pressure on its surface and longitudinal force. Izvestia of the Russian Academy of Sciences. Mechanics of solids.2021;4:77-88.(In Russ)

15. Tarasov V.N. The influence of transverse load on the stability of a rod compressed by longitudinal force. News of the Komi Scientific Center of the Ural Branch of the Russian Academy of Sciences. 2023; 4 (62): 18-22. (In Russ)