Document Type : Research Paper


Civil Engineering Dept., Enugu State University of Science and Technology, Agbani, Enugu State, Nigeria.


The determination of free vibration frequencies of thin beams on Winkler foundations is important in their design to avoid resonant failures when the excitation frequency equals the least natural frequency. This article presents the determination of natural transverse vibration frequencies of Euler-Bernoulli beams on Winkler foundations using the Generalized Integral Transform Method (GITM). The problem is governed by a fourth-order partial differential equation (PDE) and boundary conditions dependent on the end restraints. The governing PDE is transformed into an algebraic eigenvalue problem for harmonic vibrations and harmonic response. Analytical solutions of the governing equations are difficult and complex. Hence, other solution methods are needed. The main merit of the GITM is the apriori selection of kernel functions as orthogonal functions of vibrating thin beams with equivalent boundary conditions. This prior selection of the kernel functions and their orthogonality properties simplify the resulting integral equation formulation to an algebraic problem in the GITM space. Eigenfunctions of freely vibrating thin beams with identical restraints are used in the GITM to construct the displacement function as infinite series in terms of unknown parameters. The solution of the algebraic equation yielded exact solutions to the candidate problem. Exact solutions for frequency parameters are obtained for the four cases of boundary conditions for the various values of foundation parameters considered. The effectiveness of the GITM in obtaining a simplification of the governing PDE and reducing the problem to an algebraic problem is illustrated.

Graphical Abstract


  • Eigenfunctions of thin beam vibrations used as kernel functions
  • Generalized integral transform method converts governing equations to integral and then algebraic equations
  • Exact solutions obtained for an infinite spectrum of natural frequencies of beams on Winkler foundations
  • Method provides accurate solutions compared to previous approaches


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