Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load

Anthony Monday Ette; Joy Ulumma Chukwuchekwa; Williams Ifeanyichukwu Osuji; Atulegwu Chukwudi Osuji

doi:doi:10.11648/j.ajmcm.20200501.11

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Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load

Anthony Monday Ette, Joy Ulumma Chukwuchekwa, Williams Ifeanyichukwu Osuji, Atulegwu Chukwudi Osuji

Published in American Journal of Mathematical and Computer Modelling (Volume 5, Issue 1)

Received: 22 November 2019 Accepted: 13 December 2019 Published: 17 January 2020

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Abstract

This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.

Published in	American Journal of Mathematical and Computer Modelling (Volume 5, Issue 1)
DOI	10.11648/j.ajmcm.20200501.11
Page(s)	1-11
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Toroidal Shell, Airy Stress Function, Static and Dynamic Buckling, Asymptotic and Perturbation Technique

References

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[2]	Hutchinson, J. W. (1967), Initial post buckling behavior of toroidal shell segment, In. J. Solids Struct. 3, 97–115.
[3]	Oyesanya, M. O. (2002), Asymptotic analysis of imperfection sensitivity of toroidal shell segment modal imperfection, J. Nigerian Ass. Maths. Physics, 6, 197–2006.
[4]	Oyesanya, M. O. (2005), Influence of extra terms on asymptotic analysis of imperfection sensitivity of toroidal shell segment with random imperfection, Mechanics Research Communications, 32, 444–453.
[5]	Lockhart, D. and Amazigo, J. C. (1975), Dynamic buckling of externally pressurized imperfect cylindrical shells, J. of App. Mech. 42 (2), 316–320.
[6]	Ganaparthi, M., Gupta, S. S. and Patel, B. P. (2005), Nonlinear axisymmetric dynamic buckling of laminated angle–ply composite spherical caps, Composite Structures, 59, 89–97.
[7]	Kolakowski, Z. (2007), Semi–analytical method for the analysis of the interactive buckling of thin–walled elastic Structures in the second order approximation, Int. J. Solids Struct., 2790–3779.
[8]	Kolakowski, Z. and Mania, R. J. (2013), Semi–analytical method versus the FEM for analyzing the local post–buckling of thin–walled composite structures, Composite Structures, 97, 99–106.
[9]	Kubiak, T. (2013), Static and dynamic buckling of thin–walled plates structures, Springer–Verlag London, UK.
[10]	Magnucki, K., Paczos, P. and Kasprzak, J. (2010), Elastic buckling of cold–formed Thin–walled channel beams with drop flanges, J. of Structural Engrg. 136 (7), 886–896.
[11]	Tetter, A. and Kalakowski, Z. (2013), Coupled dynamic buckling of Thin–walled composite columns with open cross–section, Composite Structures, 95, 28–34.
[12]	Kriegesman, B., Mohle, M. and Rolfes, R. (2015), Sample size dependent probabilistic design of axially compressed cylindrical shells, Thin–Walled Structures, 96, 256–268.
[13]	Hu, N. and Burgueno, R. (2014), Elastic post buckling response of axially loaded cylindrical shells with seeded geometric imperfection design, Thin–Walled Structures, 74, 222–231.
[14]	Hutchinson, J. W. and Budiansky, B. (1966), Dynamic buckling estimates, AIAA J., 4, 525–530.
[15]	Amazigo, J. C. and Ette, A. M. (1987), On a two–small parameter nonlinear differential equation with application to dynamic buckling, J. of Nigerian math. Soc. 6, 91–102.
[16]	Amazigo, J. C. (1971), Buckling of stochastically imperfect columns on nonlinear elastic foundations, Quart. Appl. Math., 31 (1), 403–409.
[17]	Ette, A. M., Chukwuchekwa, J. U., Udo–Akpan, I. U. and Ozoigbo, G. E. (2019), On the normal response and buckling of a toroidal shell segment pressurized by a static compressive load, IJMTT, 65 (10), 15–26.

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APA Style

Anthony Monday Ette, Joy Ulumma Chukwuchekwa, Williams Ifeanyichukwu Osuji, Atulegwu Chukwudi Osuji. (2020). Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load. American Journal of Mathematical and Computer Modelling, 5(1), 1-11. https://doi.org/10.11648/j.ajmcm.20200501.11

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Anthony Monday Ette; Joy Ulumma Chukwuchekwa; Williams Ifeanyichukwu Osuji; Atulegwu Chukwudi Osuji. Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load. Am. J. Math. Comput. Model. 2020, 5(1), 1-11. doi: 10.11648/j.ajmcm.20200501.11

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AMA Style

Anthony Monday Ette, Joy Ulumma Chukwuchekwa, Williams Ifeanyichukwu Osuji, Atulegwu Chukwudi Osuji. Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load. Am J Math Comput Model. 2020;5(1):1-11. doi: 10.11648/j.ajmcm.20200501.11

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@article{10.11648/j.ajmcm.20200501.11,
  author = {Anthony Monday Ette and Joy Ulumma Chukwuchekwa and Williams Ifeanyichukwu Osuji and Atulegwu Chukwudi Osuji},
  title = {Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {5},
  number = {1},
  pages = {1-11},
  doi = {10.11648/j.ajmcm.20200501.11},
  url = {https://doi.org/10.11648/j.ajmcm.20200501.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20200501.11},
  abstract = {This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.},
 year = {2020}
}

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TY  - JOUR
T1  - Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load
AU  - Anthony Monday Ette
AU  - Joy Ulumma Chukwuchekwa
AU  - Williams Ifeanyichukwu Osuji
AU  - Atulegwu Chukwudi Osuji
Y1  - 2020/01/17
PY  - 2020
N1  - https://doi.org/10.11648/j.ajmcm.20200501.11
DO  - 10.11648/j.ajmcm.20200501.11
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 1
EP  - 11
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20200501.11
AB  - This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.
VL  - 5
IS  - 1
ER  -

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Author Information

Anthony Monday Ette

Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, Imo State, Nigeria
Joy Ulumma Chukwuchekwa

Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, Imo State, Nigeria
Williams Ifeanyichukwu Osuji

Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, Imo State, Nigeria
Atulegwu Chukwudi Osuji

Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, Imo State, Nigeria

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APA Style

Anthony Monday Ette, Joy Ulumma Chukwuchekwa, Williams Ifeanyichukwu Osuji, Atulegwu Chukwudi Osuji. (2020). Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load. American Journal of Mathematical and Computer Modelling, 5(1), 1-11. https://doi.org/10.11648/j.ajmcm.20200501.11

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ACS Style

Anthony Monday Ette; Joy Ulumma Chukwuchekwa; Williams Ifeanyichukwu Osuji; Atulegwu Chukwudi Osuji. Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load. Am. J. Math. Comput. Model. 2020, 5(1), 1-11. doi: 10.11648/j.ajmcm.20200501.11

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AMA Style

Anthony Monday Ette, Joy Ulumma Chukwuchekwa, Williams Ifeanyichukwu Osuji, Atulegwu Chukwudi Osuji. Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load. Am J Math Comput Model. 2020;5(1):1-11. doi: 10.11648/j.ajmcm.20200501.11

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@article{10.11648/j.ajmcm.20200501.11,
  author = {Anthony Monday Ette and Joy Ulumma Chukwuchekwa and Williams Ifeanyichukwu Osuji and Atulegwu Chukwudi Osuji},
  title = {Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {5},
  number = {1},
  pages = {1-11},
  doi = {10.11648/j.ajmcm.20200501.11},
  url = {https://doi.org/10.11648/j.ajmcm.20200501.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20200501.11},
  abstract = {This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.},
 year = {2020}
}

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TY  - JOUR
T1  - Perturbation Procedures in the Dynamic Analysis of a Toroidal Shell Segment Pressurized by a Step Load
AU  - Anthony Monday Ette
AU  - Joy Ulumma Chukwuchekwa
AU  - Williams Ifeanyichukwu Osuji
AU  - Atulegwu Chukwudi Osuji
Y1  - 2020/01/17
PY  - 2020
N1  - https://doi.org/10.11648/j.ajmcm.20200501.11
DO  - 10.11648/j.ajmcm.20200501.11
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 1
EP  - 11
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20200501.11
AB  - This paper uses perturbation techniques in asymptotic procedures to determine the normal displacement, the associated Airy stress function and the dynamic buckling load of an imperfect, finite toroidal shell segment pressurized by a step load. The adoption of asymptotic and perturbation procedure is made possible by the presence of small non-dimensional parameter on which asymptotic expansions are made possible. It is assumed here that the imperfection can be regarded as the first term in the Fourier Sine series expansion. The buckling modes are also assumed to be strictly in the shape of the imperfection which is in turn given in the shape of the classical buckling mode. In the final analysis, a simple but implicit formula for determining the dynamic buckling load was obtained. The dynamic buckling load was related to the corresponding static buckling load and that relationship is independent of the imperfection parameter. It is observed, that this procedure, perhaps more than other ones, can be used to analyze relatively more complicated problems particularly where more demands and restrictions are placed on the imperfection parameter. The results are strictly and are valid as far as the imperfection parameter is relatively small compared to unity.
VL  - 5
IS  - 1
ER  -

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