Numerical and physical modeling of bar deformation under axial loading in the channel. Part 1
In two parts of the work, numerical and physical modeling of the deformation of the bar in the channel under axial compression is carried out. The regularities of nonlinear bending of the bar in the plane are revealed. Bar shapes are determined by the load history and can differ at the same force value. The solution is to find the shape with the lowest potential energy. The first part of the work describes the numerical model of the bar and the results of its application. The shapes of the bar bending under gradual loading are obtained, the studies coinciding with V.I. Feodosev’s analytical solution. Further research shows that the solution to the problem has a more complex ramified structure with various additional shapes. Deformation of the bar under gradual loading occurs in the form of a sequential variant appearance of bending waves in the bar under forces determined by the degree of non-uniformity of the lengths of potentially unstable sections and forming a range of shape instability. In variant transitions from one initial shape with a loss of stability, it is possible to obtain various subsequent shapes that differ in the sequence of deformation of the sections with one number of half-waves, or the number of generated half-waves. When a straight bar is loaded in one step, an increase in the force leads to a sequential increase in the number of bending half-waves in the corresponding ranges of the existence of shapes. The results obtained can be applied to the analysis of the operation of such bar objects as drill, casing, tubing strings in the well and cased pipelines, pipelines in the well and tunnel.
 Volmir A.S. Ustoychivost deformiruemykh system [Stability of deformable systems]. Moscow, Nauka Publ., 1967, 987 p.
 Timoshenko S.P. Ustoychivost sterzhney, plastin i obolochek [Stability of bars, plates and shells]. Moscow, Nauka Publ., 1971, 808 p.
 Svetlitskiy V.A. Mekhanika truboprovodov i shlangov [Mechanics of pipelines and hoses.]. Moscow, Mashinostroenie Publ., 1982, 279 p.
 Svetlitskiy V.A. Mekhanika sterzhney. Ch. 1: Statika [Bar mechanics. Part 1: Statics]. Moscow, Vysshaya shkola Publ., 1987, 320 p.
 Feodosev V.I. Izbrannye zadachi i voprosy po soprotivleniyu materialov [Selected problems and questions on the strength of materials.]. 5th ed., revised and enlarged, Moscow, Nauka, Fizmatlit Publ., 1996, 368 p.
 Griguletskiy V.G. Neft, gaz i biznes (Oil, gas and business), 2016, no. 12, pp. 3–13.
 Silina I.G., Gilmiyarov E.A., Ivanov V.A. Nauka i tekhnologii truboprovodnogo transporta nefti i nefteproduktov — Science and technology of oil and oil products pipeline transport, 2019, vol. 9, no. 4, pp. 387–393.
 Trutaev S.Yu., Kuznetsov K.I. Territoriya Neftegaz — Oil and gas territory, 2019, no. 7-8, pp. 56–62.
 Morozov N.F., Tovstik P.E. Doklady Akademii nauk — Proceedings of the Russian Academy of Sciences, 2007, vol. 412, no. 2, pp. 196–200.
 Dorogov Yu.I. Vestnik Tomskogo gosudarstvennogo universiteta, Matematika i mekhanika — Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 4 (36), pp. 196–200.
 Liakou A., Detournay E. Constrained buckling of variable length elastics: solution by geometrical segmentation. International Journal of Non-Linear Mecha-nics, 2017, vol. 99, pр. 204–217.
 Holmes P., Domokos G., Shmitt J. Constrained Euler buckling: аn interlay of computation and analysis. Computer Methods in Applied Mechanics and Engineering, 1999, vol. 170, pp. 175‒207.
 Domokos G., Holmes P., Royce B. Constrained Euler buckling. Journal of Nonlinear Science, 1997, vol. 7, рp. 281–314.
 Chen, J.-S., Lu, C.-J., & Lee, C.-Y. On the use of energy method with element splitting to determine the stability of a constrained elastics. International Journal of Non-Linear Mechanics, 2015, vol. 76, 77–86. DOI: 10.1016/j.ijnonlinmec.2015.06.002