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

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.

Keywords

• ccubic polynomial shear deformation beam
• buckling theory
• finite sine transform
• potential energy functional
• euler-lagrange equation
• algebraic eigenvalue problem