Vol. 5 No. 02 (2024)
Articles

Deriving Ritz formulation for the Static Flexural Solutions of Sinusoidal Shear Deformable Beams

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  • Benjamin Okwudili Mama,
  • Onyedikachi Aloysius Oguaghamba,
  • Charles IKE
Benjamin Okwudili Mama
Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria
Onyedikachi Aloysius Oguaghamba
Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria
Charles IKE
Enugu State University of Science and Technology, Agbani, Enugu State, Nigeria.
DOI: https://doi.org/10.38094/jocef50284

Published 2024-09-18

Keywords

  • Trigonometric shear deformation beam theory, Ritz variational method, total potential energy functional, stress field, displacement field, thick beam

How to Cite

[1]
B. O. Mama, O. A. . Oguaghamba, and C. IKE, “Deriving Ritz formulation for the Static Flexural Solutions of Sinusoidal Shear Deformable Beams”, JoCEF, vol. 5, no. 02, pp. 49-54, Sep. 2024.

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Abstract

This paper presents the Ritz variational method for the static bending analysis of sinusoidal shear deformable beams. The theory accounts for transverse shear deformation and satisfies transverse shear stress-free conditions at the top and bottom surfaces of the beam. The total potential energy functional for the thick beam bending problem is formulated and minimized using a Ritz procedure. The problem considered simply supported boundary conditions and two cases of loading – uniformly distributed load over the span and point load at the center. The function is a function of two unknown displacement functions constructed in terms of unknown generalized displacement parameters and shape functions that satisfy the boundary conditions. Ritz minimization of the functional is used to find the generalized displacement parameters and then the displacements w(x) and  The displacements and stresses are found for the loading distributions considered. It is found that the results obtained agree remarkably well with the exact results obtained using the theory of elasticity. The differences between the present results and the exact solutions are less than 0.3% for maximum transverse displacement for both cases of loading considered. For uniform load over the beam, the result for l/t = 10 is 0.112% greater than the exact solution. For the point load at the center, the result for l/t = 10 is 4.468% greater than the exact solution. The increased difference in the point load case is due to the singular nature of the point load problem.

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