Free vibration of cylindrical functionally graded tubes on bi-parameter foundation based on a higher-order beam theory

Based on the theory of high-order shear deformation beam, the governing equations for the free vibration behavior of Winkler-Pasternak elastic foundation radial function gradient hollow cylindrical pipe were derived. This method does not require the introduction of shear correction coefficient and automatically satisfies the free boundary condition of shear stress on the inner and outer surfaces of the hollow cylindrical pipe. By introducing auxiliary functions, the coupled equations for deflection and angle were transformed into a single high-order differential equation. The frequency and mode shapes of the function gradient hollow cylindrical pipe under typical boundary conditions were given. The calculation results were compared with the results in the existing literature to verify the accuracy of the proposed theory. It can provide higher precision one-dimensional elastic theoretical solutions for the common Winkler-Pasternak elastic beam structures in engineering. Research results show that the gradient parameter and elastic foundation stiffness coefficient of the function gradient material have a significant impact on the natural frequency value. Compared with the high-order natural frequency, the stiffness coefficient has a more significant impact on the low-order natural frequency.