On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load

Gerald Ozoigbo; Anthony Ette; Joy Chukwuchekwa; Williams Osuji; Itoro Udo-Akpan

doi:doi:10.11648/j.ajma.20221001.11

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On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load

Gerald Ozoigbo, Anthony Ette, Joy Chukwuchekwa, Williams Osuji, Itoro Udo-Akpan

Published in American Journal of Mechanics and Applications (Volume 10, Issue 1)

Received: 6 February 2022 Accepted: 26 March 2022 Published: 31 May 2022

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Abstract

This investigation is concerned with the determination of the dynamic buckling load of a Pre – Statically loaded imperfect elastic cubic model structure that is later struck by a dynamically slowly varying explicitly time - dependent load which is infinitely differentiable and has right hand derivatives of all orders at the initial time. Our initial pre–occupation is the determination of a uniformly valid asymptotic expression of the maximum displacement by means of multi–timing regular perturbation procedures. This is finally followed by a determination of the dynamic buckling load of the structure. The result shows, among other things, that the dynamic buckling load depends on the first derivative of the load function evaluated at the initial time. Besides, the dynamic buckling load is related to the static buckling load and this relationship is independent of the imperfection parameter. The result is, in the final analysis, particularized to cases of a step load with or without a pre–load. All results are asymptotic in nature and so, are valid as the small parameters approach zero.

Published in	American Journal of Mechanics and Applications (Volume 10, Issue 1)
DOI	10.11648/j.ajma.20221001.11
Page(s)	1-15
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), 2024. Published by Science Publishing Group

Keywords

Nonlinear, Slowly Varying, Infinitely Differentiable, Explicitly Time Dependent, Pre – Statically Loaded

References

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[3]	Kevarian, J. and Li, Y. P. Explicit approximations for strictly nonlinear oscillations with slowly varying parameters with application to free–electron lasers, Stud. Appli. Math. 1988; 78 (2): 111–165.
[4]	Kroll, N. M, Morton, P. L and Rosenbhuth, M. N. Free–electron lasers with variable parameter, wigglers, IEEE J. Quantum Electron. 1981; QE–17: 1436–1468.
[5]	Boslly, D. L. Transient and sustained reverence in very slowly varying oscillatory Hamiltonian systems with application to free–electron lasers, Ph.D. dissertation, Univ. of Washington, Seattle, Dec. 1989.
[6]	Luke, J. C. A perturbation method for nonlinear dispersive wave problems, Proc. Roy. Soc., London, Ser. A. 1966; 403–412.
[7]	Kubiak, T. Dynamic bucking of thin–walled composite plate with varying widthwise material properties, Int. J. of solids and Struct. 2005; 42: 5555–5567.
[8]	Kolakowski, Z. Static and dynamic iteration buckling regarding axial extention mode of thin–walled channels, J. theoretical and Appl. Mechanics. 2010; 48 (3): 703–714.
[9]	Simitses, G. J. Effect of static preloading on the dynamic stability of structures. A. I. A. A. J. 1983; 12 (8): 1174–1180.
[10]	Simitises, G. J. Instability of dynamically loaded structures, Appl. Mech. Rev. 1987; 40 (10): 1403–1408.
[11]	Tanov, R., Tabiei, A. and Simitses, G. J. Effect of static pre - loading on the dynamic buckling of laminated cylinder under sudden pressure. Mech. Compos. Mater. Struct. 1999; 6: 195–206.
[12]	Tabiei, A., Tanov, R. and Simitses, G. J. Numerical simulation of cylindrical laminated shells under impulse lateral pressure," in a collection of Technical Papers, 39^th A. I. A. A./ASME/ASCE/ AHS/ASC structures. Structural Dynamics and Material Conference A. I. A. A. J. 1998; 37: 509–514.
[13]	Ozoigbo, G. E and Ette, A. M. Perturbation Approach to Dynamic Buckling of a Statically Pre–Loaded, but Viscously Damped Elastic Structure. J. App. Mech. Tech. Phys. 2020; 61: 1001–1015.
[14]	Kolakowski, Z. and Teter, A. Coupled static and dynamic buckling modeling of thin–walled structure in elastic reage, Rev., of selected problem, Acta Mech., of Automatica. 2016; 10 (2): 141–149.
[15]	Russel, B. P., Vikran, S. P. and Haydan, N. G. W. Quasi–static deformation and failure modes of composite square Honeycombs, J. of Mech. Mat. Structures. 2005; 3 (7): 1315–1342.
[16]	Ozoigbo, G. E., Nwaeze, E. and Okpala, M. Dynamic buckling load of an imperfect viscously damped spherical cap stressed by a step load, Physical Science International Journal. 2015; 7 (3): 192–213.
[17]	Belyaev, A. K., Qlen, D. N. and Morovo, N. F. Stability of transverse vibration of rod under longitudinal step–wise loading, J. of Physics Conference Series. 2013; 45 (1): 1–6.
[18]	Budiansky, B. Dynamic buckling of elastic structures, criteria and estimation, in dynamic stabilities of structures Ed. by Hermann Pergamon Press, Oxford. 1966; 83–106.
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APA Style

Gerald Ozoigbo, Anthony Ette, Joy Chukwuchekwa, Williams Osuji, Itoro Udo-Akpan. (2022). On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load. American Journal of Mechanics and Applications, 10(1), 1-15. https://doi.org/10.11648/j.ajma.20221001.11

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ACS Style

Gerald Ozoigbo; Anthony Ette; Joy Chukwuchekwa; Williams Osuji; Itoro Udo-Akpan. On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load. Am. J. Mech. Appl. 2022, 10(1), 1-15. doi: 10.11648/j.ajma.20221001.11

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AMA Style

Gerald Ozoigbo, Anthony Ette, Joy Chukwuchekwa, Williams Osuji, Itoro Udo-Akpan. On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load. Am J Mech Appl. 2022;10(1):1-15. doi: 10.11648/j.ajma.20221001.11

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@article{10.11648/j.ajma.20221001.11,
  author = {Gerald Ozoigbo and Anthony Ette and Joy Chukwuchekwa and Williams Osuji and Itoro Udo-Akpan},
  title = {On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load},
  journal = {American Journal of Mechanics and Applications},
  volume = {10},
  number = {1},
  pages = {1-15},
  doi = {10.11648/j.ajma.20221001.11},
  url = {https://doi.org/10.11648/j.ajma.20221001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20221001.11},
  abstract = {This investigation is concerned with the determination of the dynamic buckling load of a Pre – Statically loaded imperfect elastic cubic model structure that is later struck by a dynamically slowly varying explicitly time - dependent load which is infinitely differentiable and has right hand derivatives of all orders at the initial time. Our initial pre–occupation is the determination of a uniformly valid asymptotic expression of the maximum displacement by means of multi–timing regular perturbation procedures. This is finally followed by a determination of the dynamic buckling load of the structure. The result shows, among other things, that the dynamic buckling load depends on the first derivative of the load function evaluated at the initial time. Besides, the dynamic buckling load is related to the static buckling load and this relationship is independent of the imperfection parameter. The result is, in the final analysis, particularized to cases of a step load with or without a pre–load. All results are asymptotic in nature and so, are valid as the small parameters approach zero.},
 year = {2022}
}

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TY  - JOUR
T1  - On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load
AU  - Gerald Ozoigbo
AU  - Anthony Ette
AU  - Joy Chukwuchekwa
AU  - Williams Osuji
AU  - Itoro Udo-Akpan
Y1  - 2022/05/31
PY  - 2022
N1  - https://doi.org/10.11648/j.ajma.20221001.11
DO  - 10.11648/j.ajma.20221001.11
T2  - American Journal of Mechanics and Applications
JF  - American Journal of Mechanics and Applications
JO  - American Journal of Mechanics and Applications
SP  - 1
EP  - 15
PB  - Science Publishing Group
SN  - 2376-6131
UR  - https://doi.org/10.11648/j.ajma.20221001.11
AB  - This investigation is concerned with the determination of the dynamic buckling load of a Pre – Statically loaded imperfect elastic cubic model structure that is later struck by a dynamically slowly varying explicitly time - dependent load which is infinitely differentiable and has right hand derivatives of all orders at the initial time. Our initial pre–occupation is the determination of a uniformly valid asymptotic expression of the maximum displacement by means of multi–timing regular perturbation procedures. This is finally followed by a determination of the dynamic buckling load of the structure. The result shows, among other things, that the dynamic buckling load depends on the first derivative of the load function evaluated at the initial time. Besides, the dynamic buckling load is related to the static buckling load and this relationship is independent of the imperfection parameter. The result is, in the final analysis, particularized to cases of a step load with or without a pre–load. All results are asymptotic in nature and so, are valid as the small parameters approach zero.
VL  - 10
IS  - 1
ER  -

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Author Information

Gerald Ozoigbo

Department of Mathematics and Statistics, Alex Ekwueme Federal University, Ndufu-Alike, Nigeria
Anthony Ette

Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Joy Chukwuchekwa

Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Williams Osuji

Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Itoro Udo-Akpan

Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria

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APA Style

Gerald Ozoigbo, Anthony Ette, Joy Chukwuchekwa, Williams Osuji, Itoro Udo-Akpan. (2022). On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load. American Journal of Mechanics and Applications, 10(1), 1-15. https://doi.org/10.11648/j.ajma.20221001.11

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ACS Style

Gerald Ozoigbo; Anthony Ette; Joy Chukwuchekwa; Williams Osuji; Itoro Udo-Akpan. On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load. Am. J. Mech. Appl. 2022, 10(1), 1-15. doi: 10.11648/j.ajma.20221001.11

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AMA Style

Gerald Ozoigbo, Anthony Ette, Joy Chukwuchekwa, Williams Osuji, Itoro Udo-Akpan. On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load. Am J Mech Appl. 2022;10(1):1-15. doi: 10.11648/j.ajma.20221001.11

Copy | Download

@article{10.11648/j.ajma.20221001.11,
  author = {Gerald Ozoigbo and Anthony Ette and Joy Chukwuchekwa and Williams Osuji and Itoro Udo-Akpan},
  title = {On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load},
  journal = {American Journal of Mechanics and Applications},
  volume = {10},
  number = {1},
  pages = {1-15},
  doi = {10.11648/j.ajma.20221001.11},
  url = {https://doi.org/10.11648/j.ajma.20221001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20221001.11},
  abstract = {This investigation is concerned with the determination of the dynamic buckling load of a Pre – Statically loaded imperfect elastic cubic model structure that is later struck by a dynamically slowly varying explicitly time - dependent load which is infinitely differentiable and has right hand derivatives of all orders at the initial time. Our initial pre–occupation is the determination of a uniformly valid asymptotic expression of the maximum displacement by means of multi–timing regular perturbation procedures. This is finally followed by a determination of the dynamic buckling load of the structure. The result shows, among other things, that the dynamic buckling load depends on the first derivative of the load function evaluated at the initial time. Besides, the dynamic buckling load is related to the static buckling load and this relationship is independent of the imperfection parameter. The result is, in the final analysis, particularized to cases of a step load with or without a pre–load. All results are asymptotic in nature and so, are valid as the small parameters approach zero.},
 year = {2022}
}

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TY  - JOUR
T1  - On the Analysis of a Pre – Statically Loaded Nonlinear Cubic Structure Pressurized by an Explicitly Time Dependent Slowly Varying Load
AU  - Gerald Ozoigbo
AU  - Anthony Ette
AU  - Joy Chukwuchekwa
AU  - Williams Osuji
AU  - Itoro Udo-Akpan
Y1  - 2022/05/31
PY  - 2022
N1  - https://doi.org/10.11648/j.ajma.20221001.11
DO  - 10.11648/j.ajma.20221001.11
T2  - American Journal of Mechanics and Applications
JF  - American Journal of Mechanics and Applications
JO  - American Journal of Mechanics and Applications
SP  - 1
EP  - 15
PB  - Science Publishing Group
SN  - 2376-6131
UR  - https://doi.org/10.11648/j.ajma.20221001.11
AB  - This investigation is concerned with the determination of the dynamic buckling load of a Pre – Statically loaded imperfect elastic cubic model structure that is later struck by a dynamically slowly varying explicitly time - dependent load which is infinitely differentiable and has right hand derivatives of all orders at the initial time. Our initial pre–occupation is the determination of a uniformly valid asymptotic expression of the maximum displacement by means of multi–timing regular perturbation procedures. This is finally followed by a determination of the dynamic buckling load of the structure. The result shows, among other things, that the dynamic buckling load depends on the first derivative of the load function evaluated at the initial time. Besides, the dynamic buckling load is related to the static buckling load and this relationship is independent of the imperfection parameter. The result is, in the final analysis, particularized to cases of a step load with or without a pre–load. All results are asymptotic in nature and so, are valid as the small parameters approach zero.
VL  - 10
IS  - 1
ER  -

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