Free vibration of functionally graded beam embedded in Winkler-Pasternak elastic foundation with geometrical uncertainties using symmetric Gaussian fuzzy number

This article seeks to investigate the effect of geometrical uncertainties on the free vibration of Euler-Bernoulli Functionally Graded (FG) beams resting on Winkler-Pasternak elastic foundation. In this scenario, the uncertainties are linked to length and thickness of FG beam using Symmetric Gaussian Fuzzy Number (SGFN). The governing equations of motion regarding free vibration of the uncertain model are derived by combining the Symmetric Gaussian Fuzzy Number with Hamilton's principle and double parametric form of fuzzy numbers. The natural frequencies of the uncertain models are computed using the double parametric form-based Navier's approach for Hinged-Hinged (H-H) boundary condition. The double parametric form-based Hermite-Ritz approach was also used to calculate the natural frequencies of Hinged-Hinged (H-H), Clamped-Hinged (C-H), and Clamped-Clamped (C-C) boundary conditions. Natural frequencies obtained using Navier's method and Hermite-Ritz method are used to validate the results of the uncertain model which exhibit strong agreement. A comprehensive parametric analysis is also conducted with respect to various graphical and tabular results to investigate the fuzziness or spreads of natural frequencies in relation to various uncertain parameters.