American Journal of Mechanics and Applications

| Peer-Reviewed |

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

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

Share This Article

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.

DOI 10.11648/j.ajma.20221001.11
Published in American Journal of Mechanics and Applications (Volume 10, Issue 1, March 2022)
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
[1] Kudiak, G. E. Asymptotic solution of nonlinear second order differential equations with variable coefficients, Pure Math, Manuscript, 1959; 23: 515–526.
[2] Kevorkian, J. Perturbation techniques for oscillatory systems with slowly varying coefficients, SIAMS Rev. 1987; 29: 391–461.
[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, 39th 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.
[19] Hutchinson, J. W and Budiansky, B. Dynamic buckling estimates, A. I. A. A. J. 1966; 4: 525–530.
[20] Amazigo, J. C. and Ette, A. M. On a two–small parameter nonlinear differential equation with application dynamic buckling. J. of Nig. Math. Soc. 1987; 6: 91–102.
Cite This Article
  • 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

    Copy | Download

    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

    Copy | Download

    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}
    }
    

    Copy | Download

  • 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  - 

    Copy | Download

Author Information
  • Department of Mathematics and Statistics, Alex Ekwueme Federal University, Ndufu-Alike, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

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

  • Sections