Free vibrations of functionally graded porous hanging and standing cantilever beams

Abstract

The free oscillations of a functionally graded (FG) porous vertical cantilever beam in the frame work of Euler–Bernoulli beam theory is investigated. The beam is subjected to the gravity-load and the properties of the FG material such as the modulus of elasticity and the density are supposed to change through the thickness of the beam according to power-law relations. The equation of motion is derived using Newton’s second law. The Numerical Chebyshev collocation method is utilized to determine the transverse frequencies of the FG porous hanging and standing cantilever Euler–Bernoulli beams. A parametric study is conducted to determine the effects of various factors such as the transverse functionally graded index, the porosity factor, and the elastic and the mass density ratios on the natural frequencies and the mode shapes of the FG porous vertical hanging and standing cantilever thin beams under their self-weight. The accuracy of the proposed numerical method is evaluated through comparisons of the frequencies obtained from the present approach with those available in previous literature. In general, it was observed that the elastic ratio has a softening impact on the frequencies except for the fundamental frequency which remains constant as the elastic ratio increases. Moreover, the porosity parameter and the power-law index may have a softening or hardening impact on the frequencies, and the behavior of these frequencies depends on the values of the elastic and the mass density ratios.