Vol. 5 No. 02 (2024)
Articles

First Principles Variational Formulation and Analytical Solutions for Third Order Shear Deformable Beams with Simple Supports using Fourier Series Method

PDF
  • Prof Charles Chinwuba Ike,
  • Dr Benjamin Okwudili Mama,
  • Dr Onyedikachi Aloysius Oguaghamba
Prof Charles Chinwuba Ike
Enugu State University of Science and Technology, Agbani, Enugu State, Nigeria.
Dr Benjamin Okwudili Mama
Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria
Dr Onyedikachi Aloysius Oguaghamba
Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria
DOI: https://doi.org/10.38094/jocef50286

Published 2024-10-12

Keywords

  • Third order shear deformable beam theory, Fourier series method, warping function, deflection function

How to Cite

[1]
C. IKE, B. O. . Mama, and O. A. . Oguaghamba, “First Principles Variational Formulation and Analytical Solutions for Third Order Shear Deformable Beams with Simple Supports using Fourier Series Method”, JoCEF, vol. 5, no. 02, pp. 55-62, Oct. 2024.

Copyright (c) 2024

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

Abstract

The Fourier series method for solving bending problems of third-order shear deformable beams (TSBD) is presented in this paper. The theory accounts for transverse shear deformation and is suitable for moderately thick beams. Transverse shear stress-free conditions are valid at the beam surfaces for TSDB. The field equations are two coupled ordinary differential equations in terms of two unknown displacement functions – transverse deflection w(x) and warping function  For simply supported ends considered, the loading and unknown functions w(x) and are represented in Fourier series that satisfies the boundary conditions. The problem is reduced to a system of algebraic equations in terms of the Fourier coefficients wn and which is solved to obtain wn and  Axial and transverse displacements fields, axial bending stress  and transverse shear stress  are then determined for the cases of point load at midspan, uniformly and linearly distributed loads over the span. The results are identical with previous results obtained by other scholars who used Ritz variational methods and other mathematical tools. For moderately thick beams under uniform load with l/h = 4, the results obtained for is 0.516% greater than the exact result by Timoshenko and Goodier while for thick beams under uniform load with l/h = 2, the result for  is 1.96% greater than the exact result by Timoshenko and Goodier. Similar acceptable variation was obtained for  for beams under linearly distributed load with the variations being less than 2% for l/h = 2. However, unacceptable variations of 68.37% were found for in thick beams under point load, for l/h = 2.0. The variation for however reduced to 12.513% for moderately thick beams under point load for l/h =4.

PDF

Metrics

Metrics Loading ...

References

  1. C.C. Ike and E.U. Ikwueze, “Ritz method for the analysis of statically indeterminate Euler-Bernoulli beams,” Saudi Journal of Engineering and Technology, Vol. 3, Issue 3, pp. 133 – 140, 2018.
  2. C.C. Ike and E.U. Ikwueze, “Fifth degree Hermittian polynomial shape functions for the finite element analysis of clamped simply supported Euler-Bernoulli beam,” American Journal of Engineering Research, Vol. 7, Issue 4, pp. 97 – 105, 2018.
  3. C.C. Ike, “Fourier sine transform method for the free vibration of Euler-Bernoulli beam resting on Winkler foundation,” International Journal of Darshan Institute on Engineering Research and Emerging Technologies (IJDI-ERET), Vol. 7, No. 1, pp. 1 – 6, 2018.
  4. C.C. Ike, “Point collocation method for the analysis of Euler-Bernoulli beam on Winkler foundation,” International Journal of Darshan Institute on Engineering Research and Emerging Technologies (IJDI-ERET), Vol. 7, No. 2, pp 1 – 7, 2018.
  5. S.P. Timoshenko, “On the correction for shear of the differential equation for transverse vibration of prismatic bars,” Philosophical Magazine, Vol. 41 No. 6, pp. 742 – 746, 1921.
  6. M. Levinson, “A new rectangular beam theory,” Journal of Sound and Vibration, Vol. 74 No. 1, pp. 81 – 87, 1981.
  7. R.P. Shimpi, P.J. Guruprasad, and K.S. Pakhare, “Simple two variable refined theory for shear deformable isotropic rectangular beams,” Journal of Applied and Computational Mechanics, Vol. 6 No. 3, pp. 394 – 415, 2020.
  8. Y.M. Ghugal and R. Sharma, “A refined shear deformation theory for flexure of thick beams,” Latin American Journal of Solids and Structures, Vol. 8, pp. 183 – 195, 2011.
  9. S.A. Ambartsumyan, “On the theory of bending of plates,” Izv Otd Tech Nauk ANSSSR, Vol. 5, pp. 67 – 77, 1958.
  10. E.T. Kruszewski, Effect of transverse shear and rotary inertia on the natural frequency of a uniform beam. National Advisory Committee for Aeronautics Technical Note 1909 (NACA-TN-1909), pp. 1 – 16, 1949.
  11. S.S. Akavci, “Buckling and free vibration analysis of symmetric and antisymmetric laminated composite plates on an elastic foundation,” Journal of Reinforced Plastics and Composites, Vol. 26 No 18, pp. 1907 – 1919, 2007.
  12. J.N. Reddy, “A general non-linear third order theory of plates with moderate thickness,” International Journal of Non-linear Mechanics, Vol. 25 No. 6, pp. 677 – 686, 1990.
  13. S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3rd International Edition. Singapore: McGraw Hill, 1970.
  14. C.C. Ike, “Ritz variational method for the flexural analysis of third order shear deformable beams,” Conference paper presented at Conference on Engineering Research Technology Innovation and Practice (CERTIP). Faculty of Engineering, University of Nigeria, Nsukka, 3rd – 6th November, 2020.
  15. C.C. Ike and O.A. Oguaghamba, “Trigonometric shear deformation theory for the bending analysis of thick beams: Fourier series method,” Conference paper presented at Conference on Engineering Research Technology Innovation and Practice (CERTIP). Faculty of Engineering, University of Nigeria, Nsukka, 3rd – 6th November, 2020.
  16. C.C. Ike, Timoshenko beam theory for the flexural analysis of moderately thick beams – variational formulation and closed form solutions. Tecnica Italiana – Italian Journal of Engineering Science, Vol. 63 No. 1, pp. 34 – 45, 2019.
  17. H.N. Onah, C.U. Nwoji, M.E. Onyia, B.O. Mama, and C.C. Ike, “Exact solutions for the elastic buckling problem of moderately thick beams,” Revue des Composites et des Materiaux Avances, Vol. 30, No. 2, pp. 83 – 93, 2020.
  18. C.C. Ike, C.U. Nwoji, B.O. Mama, H.N. Onah, and M.E. Onyia, “Laplace transform method for the elastic buckling analysis of moderately thick beams,” International Journal of Engineering Research and Technology, Vol. 12, No. 10, pp. 1626 – 1638, 2019.
  19. Y. Ghugal, A two-dimensional exact elasticity solution of thick beams. Departmental Report – 1. Department of Applied Mechanics, Government Engineering College, Aurangabad, India, pp 1 – 96, 2006.
  20. Y.M. Ghugal and R.P. Shimpi, “A review of refined shear deformation theories for isotropic and anisotropic laminated beams,” Journal of Reinforced Plastics and Composites, Vol. 20 No. 3, pp. 255 – 272, 2001.
  21. P.R. Heyliger and J.N. Reddy, “A higher order beam finite element for bending and vibration problems,” Journal of Sound and Vibration, Vol. 126 No. 2, pp. 309 – 326, 1988.
  22. Y.M. Ghugal and A.G. Dahake, “Flexural analysis of deep beam subjected to parabolic loads using refined shear deformation theory,” Applied Computational Mechanics, Vol 6 No. 2, pp 163 – 172, 2012.
  23. A.S. Sayyad, “Comparison of various shear deformation theories for the free vibration of thick isotropic beams,” Latin American Journal of Solids and Structures, Vol. 2 No 1, pp. 85 – 97, 2011.
  24. A.S. Sayyad, “Comparison of various refined beam theories for the bending and free vibration analysis of thick beams,” Applied and Computational Mechanics, Vol. 5, pp. 217 – 230, 2011.