On the dynamics of axially functionally graded CNT strengthened deformable beams

In this study, by functionally grading carbon nanotube (CNT) fibres through the axial direction of deformable beams, a new model for strengthening such structures is formulated. The strengthened deformable beam is additionally supported by a varying elastic foundation which is modelled via the Winkler–Pasternak elastic foundation. CNT distribution in the longitudinal direction is modelled using a power-law function presenting general forms of variation from linear to parabolic models. Equations of motion are determined using Hamilton’s principle and are solved by employing the generalised differential quadrature approach method. A comprehensive parametric investigation is presented in order to indicate the influence of having CNT fibres distributed functionally through the length. It is shown that grading CNT fibres axially have a significant effect in varying the natural frequency parameter. Moreover, the influence of having Winkler–Pasternak elastic bed on the free vibration of such structures is discussed considering different types of foundation stiffness variation through the length. A comprehensive comparison study is presented to verify the current methodology and formulation by using previous literature for foundation modelling and FE modelling for linearly varying CNT’s longitudinal distribution, which for all the cases, a good agreement between the results is observed.