Lateral Free Vibration Analysis of Axially Functionally Graded Tapered Timoshenko Pipes with Variable Cross-Section Supported by a Variable Pasternak Foundation

Abstract

In this work, the Variational Iteration Method (VIM) is employed to examine the dynamic response of an axially inhomogeneous fluid-conveying pipe resting on a Pasternak elastic foundation. The pipe is modeled using Timoshenko beam theory, which accounts for shear deformation and rotary inertia, and is assumed to have a non-uniform cross-sectional configuration. Along the axial direction, the elastic foundation parameters, as well as the pipe’s mechanical and geometric properties including cross-sectional area, second moment of area, material density, and Young’s modulus are assumed to vary continuously. Three boundary conditions are considered in the analysis: Pinned–Pinned, Clamped–Pinned, and Clamped–Clamped. The governing coupled partial differential equations are solved using the variational iteration method, and the resulting natural frequencies of the system are evaluated and presented in tabular form. To validate the accuracy and reliability of the proposed approach, the numerical results obtained are compared with those in the literature, demonstrating excellent agreement. A comprehensive parametric study is further conducted to investigate the influence of the material gradient index, pipe non-uniformity parameter, foundation stiffness coefficients, rotary inertia, flow velocity, and mass ratio on the vibrational behavior of the system. The results reveal that, for both clamped–clamped and clamped–pinned boundary conditions and for a fixed value of the non-uniformity parameter, an increase in the material gradient index leads to a decrease in the fundamental (first-mode) natural frequency, while the higher-order natural frequencies increase.