Ike, C. (2024). An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory. , 42(1), 90-103. doi: 10.30684/etj.2023.142505.1538
Charles Chinwuba Ike. "An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory". , 42, 1, 2024, 90-103. doi: 10.30684/etj.2023.142505.1538
Ike, C. (2024). 'An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory', , 42(1), pp. 90-103. doi: 10.30684/etj.2023.142505.1538
Ike, C. An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory. , 2024; 42(1): 90-103. doi: 10.30684/etj.2023.142505.1538

An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory

Engineering and Technology Journal
Article 7, Volume 42, Issue 1, January 2024, Pages 90-103    PDF (618.7 K)
Document Type: Research Paper
DOI: 10.30684/etj.2023.142505.1538
Author
Charles Chinwuba Ike*
Department of Civil Enginnering, Enugu State, University of Science And Technology - Nigeria, Agbani, Enugu State, Nigeria.
Abstract
This paper presents analytical solutions for the buckling of thick beams. The Bernoulli-Euler beam theory (BEBT) overestimates their critical buckling load. This paper has derived a cubic polynomial shear deformation beam buckling theory (CPSDBBT) from first principles using the Euler-Lagrange differential equation (ELDE). It develops closed-form solutions to differential equations using the finite sine transform method. The formulation considers transverse shear deformation and satisfies the transverse shear stress-free boundary conditions. The governing equation is developed from the energy functional, Õ, by applying the ELDE. The domain equation is obtained as an ordinary differential equation (ODE). The finite sine transformation of the ODE transforms the thick beam, which is considered an algebraic eigenvalue problem. The solution gives the buckling load Nxx at any buckling mode n. The critical buckling load Nxx cr occurs at the first buckling mode and is presented in depth ratios to span (h/l). It is found that and agrees with previous solutions using shear deformable theories. For (a moderately thick beam), the Nxx cr is 2.50% lower than the value predicted using BEBT, confirming the overestimation by BEBT. The Nxx cr agrees with previous solutions, implying the shear deformation has been adequately accounted for, and the BEBT overestimates the Nxx cr. The value of Nxx cr found agrees with previous values in the literature. 

Highlights
  • Closed form solutions are derived for the buckling of thick beams under in-plane loading.
  • The buckling peoblem is derived using cubic polynomial shear deformation theory.
  • For thin beams ( ), Nxx cr has a negligible difference from the BEBT results.
  •  For  (thick beam), Nxx cr obtained is 13.44% lower than the value predicted by BEBT.
Keywords
cCubic polynomial shear deformation beam; Buckling theory; Finite sine transform; Potential energy functional; Euler-Lagrange equation; Algebraic eigenvalue problem
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