### Calculation of the stress state of a noncircular helicoidal tube based on the tensor theory of shells

**Published:**20.07.2021

**Authors:** Sorokina A.G., Kirsanova V.A., Fomicheva V.F.

**Published in issue: **#7(115)/2021

**DOI: **10.18698/2308-6033-2021-7-2092

**Category:** Mechanics | **Chapter:** Dynamics, Strength of Machines, Instruments, and Equipment

The purpose of the study was to develop an algorithm for calculating helical-symmetric shells with a closed contour in oblique Gaussian coordinates. The twist and length of the shell were taken unchanged. The method is based on the representation of the generating contour of the helicoidal surface by a discrete set of points with the replacement of differentiation along the angular coordinate by finite differences. The unknown were the displacement vectors at the indicated points of the contour. Due to the helicoidal symmetry, the differentiation of vector quantities with respect to the helical coordinate was replaced by vector multiplication. The tensor of deformations and the tensor of the parameters of the change in curvature were calculated using the nabla operator, represented in oblique Gaussian coordinates. Integration over the contour coordinate was replaced by summation over discrete points. The tensors found, which characterize the deformed state, were used to calculate the strain energy of one period of the helicoidal shell, and then the total potential of the mechanical system was compiled. The unknown displacements were determined by minimizing the total potential, taking into account the constraints that prohibit the displacement of the shell as a rigid whole. The study gives a numerical example of the application of the developed approach.

**References**

**[1]**Sorokina A.G. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie — BMSTU Journal of Mechanical Engineering, 2017, no. 10 (691), pp. 10–17.

**[2]**Vasin V.A. Izvestiya MGTU MAMI, 2008, no. 1 (5), pp. 212–229.

**[3]**Shevelev L.P., Korikhin N.V., Golovin A.I. Stroitelstvo unikalnykh zdaniy i sooruzheniy — Construction of Unique Buildings and Structures, 2015, no. 2 (17), pp. 25–38.

**[4]**Sorokina A.G. Izvestiya vysshikh uchebnykh zavedeniy. Mashinostroenie — BMSTU Journal of Mechanical Engineering, 2012, no. 7, pp. 22–26.

**[5]**Rekach V.G., Krivoshapko S.N. Raschet obolochek slozhno geometrii [Calculation of shells of complex geometry]. Monograph. Moscow, UDN Publ., 1988, 176 p.

**[6]**Rekach V.G., Krivoshapko S.N. K voprosu rascheta uprugikh tonkikh obolochek v neortogonalnykh krivolineynykh koordinatakh [On the problem of calculating elastic thin shells in non-orthogonal curvilinear coordinates]. Collection of works. Moscow, UDN Publ., 1977, pp. 3–14.

**[7]**Khalabi S.M. Stroitelnaya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2001, no. 10, pp. 61–67.

**[8]**Rynkovskaya M.I. Stroitelnaya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2006, no. 2, pp. 63–66.

**[9]**Rynkovskaya M.I. Vestnik RUDN. Seriia: Inzhenernye issledovaniya — RUDN Journal of Engineering Researches, 2012, no. 4, pp. 84–90.

**[10]**Rynkovskaya M.I. Stroitelnaya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2015, no. 6, pp. 13–15.

**[11]**Tupikova E.M Stroitelnaya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2015, no. 3, pp. 23–27.

**[12]**Tupikova E.M. Stroitelnaya mekhanika i raschet sooruzheniy — Structural Mechanics and Analysis of Constructions, 2015, no. 4, pp. 24–28.

**[13]**Tupikova E.M. Stroitelnaya mekhanika i raschet sooruzheniy — Structural Mechanics and Analysis of Constructions, 2016, no. 1, pp. 14–20.

**[14]**Tupikova E.M. Stroitelnaya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2016, no. 3, pp. 3–8.

**[15]**Biderman V.L. Mekhanika tonkostennykh konstruktsiy. Statika [Mechanics of thin-walled structures. Statics]. Moscow, URSS, Lenand Publ., 2017, 485 p.

**[16]**[Dyakonov V.P. Mathematica 5.1/5.2/6. Programmirovanie i matematicheskie vychisleniya [Mathematica 5.1/5.2/6. Programming and mathematical calculations]. Moscow, DMK-Press Publ., 2008, p. 574.