RITZ VARIATIONAL METHOD FOR THE FREE HARMONIC VIBRATION SOLUTIONS OF SLENDER BEAMS ON TWO-PARAMETER ELASTIC FOUNDATIONS
DOI:
https://doi.org/10.4314/njt.v44i2.3Keywords:
Ritz variational method, Two-parameter elastic foundation, Fundamental frequency, Slender beam, Shape functionAbstract
This study presents Ritz variational method for the free transverse harmonic vibration solutions of slender beams on two-parameter elastic foundations (SBo2PEFs). The studied problem is a soil-structure interaction problem of dynamics that is important in the dynamic design of foundations and buried pipelines. The domain equation is derived using variational calculus, and the total energy functional was found for harmonic vibrations in terms of the modal displacement W(x) and the derivatives Minimization criteria with respect to the generalized parameter of the displacement is used to find the characteristic frequency equation. The obtained Ritz equation is an eigenvalue problem. It was found that for simply supported SBo2PEF, the exact sinusoidal shape function used gave the exact eigenfrequency for any mode of vibration. For clamped-clamped SBo2PEF, a one-parameter shape function gave accurate fundamental frequency. For cantilever SBo2PEF, a one-parameter shape function gave accurate fundamental frequency solutions.
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