Solution of the External Pochhammer-Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuu

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The possibility of partial splitting of the dynamic equations of the linear theory of elasticity for bulk expansion and the components of the rotation vector of medium particles in cylindrical coordinates in the general case of a nonstationary problem is proved. This result is a generalization of a similar fact established by A. Love with a simple harmonic dependence of the named functions on time and two spatial coordinates. An exact analytical solution of the problem on bending vibrations of an elastic space with a circular cylindrical cavity (external Pochhammer-Chree problem) is given. It is shown that the study published by K. Toki and Sh. Takada on this problem does not provide a solution to the posed problem. On the basis of the solution obtained for the external Pochhammer-Chree problem, bending vibrations of an underground pipeline caused by the action of a seismic wave are studied. The results obtained in this case provide, apparently, the first theoretical substantiation of the engineering theory of rigid jamming of the pipeline in the soil for the case of bending vibrations, widely used in regulatory documents on seismic construction.

About the authors

M. Sh. Israilov

Kh. Ibragimov Complex Institute of the Russian Academy of Sciences

Author for correspondence.
Email: israiler@hotmail.com
Grozny, Chechen Republic, 364051 Russia

References

  1. Окамото Ш. Сейсмостойкость инженерных сооружений. М.: Стройиздат, 1980. 272 с.
  2. Нормы проектирования атомных станций: НП 031–01. Приложение 6. Основные положения расчета линейно-протяженных конструкций. Москва, 2001. С. 23–25.
  3. O'Rourke M.J., Liu X. Response of buried pipelines subject to earthquake effects. Buffalo: Univ. of Buffalo, 1999. 250 p.
  4. Рашидов T. Динамическая теория сейсмостойкости сложных систем подземных сооружений. Ташкент: Изд-во “ФАН”, 1973. 180 с.
  5. Kouretzis G.P., Bouckovalas G.D., Gantes C.J. 3-D shell analysis of cylindrical underground structures under seismic shear (S) wave action // Soil Dyn. Earthquake Eng. 2006. V. 26. P. 909–921. https://doi.org/10.1016/j.soildyn.2006.02.002
  6. Kouretzis G.P., Bouckovalas G.D., Karamitros D.K. Seismic verification of long cylindrical underground structures considering Rayleigh wave effects // Tunneling Underground Space Technol. 2011. V. 26. P. 789–794. https://doi.org/10.1016/j.tust.2011.05.001
  7. Toki T., Takada S. Earthquake response of underground tubular structure // Bull. Disaster Preventation Res. Inst. Kyoto Univ. 1974. V. 24. Pt. 2. № 221. P. 107–125.
  8. Исраилов М.Ш. Динамическая теория упругости и дифракция волн. М.: Изд-во Моск. ун-та. 1992, 206 с.
  9. Ляв А. Математическая теория упругости. М.: ОНТИ НКТП СССР, 1935. 674 с.
  10. Pochhammer L. Ueber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegrenzten isotropen Kreiscylinder // J. Reine Angew. Math. 1876. V. 81. P. 324–336.
  11. Chree C. Longitudinal vibrations of a circular bar // Quart. J. Pure Appl. Math. 1886. V. XXI. P. 287–298.
  12. Исраилов М.Ш. Действие наклонной сейсмической волны на подземный трубопровод // Известия РАН. МТТ. 2022. № 5. С. 58–69.
  13. Трехмерные задачи математической теории упругости и термоупругости / Под ред. В.Д. Купрадзе. М.: Наука, 1976. 664 с.
  14. Bell W.W. Special functions for scientists and engineers. London: Van Nostrand, 1968. 248 p.
  15. Поручиков В.Б. Методы динамической теории упругости. М.: Наука, 1986. 328 с.
  16. Eringen A.C. Mecanics of continua. N. Y.: Wiley, 1967. 592 p.
  17. Magnus W., Oberhettinger F., Soni R.P. Formulas and theorems of the special functions of mathematical physics. 3-d ed. N. Y.: Springer, 1966. 508 p.
  18. Miklowitz J. The theory of elastic waves and waveguides. Amsterdam: North-Holland Publ., 1978. 618 p.
  19. Исраилов М.Ш. Связанные сейсмические колебания трубопровода в бесконечной упругой среде // Изв. РАН. МТТ. 2016. № 1. С. 57–66.
  20. Лебедев Н.Н. Специальные функции и их приложения. М.: ГИТТЛ, 1963. 380 с.
  21. Берч Ф., Шерер Дж., Спайсер Г. Справочник для геологов по физическим константам. М.: Изд-во иностр. лит., 1949. 304 с.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2.

Download (19KB)
Indexing metadata

Copyright (c) 2022 М.Ш. Исраилов