An analytical solution for nonlinear vibration and post-buckling of functionally graded pipes conveying fluid considering the rotary inertia and shear deformation effects

Abstract

In this paper, a nonlinear Timoshenko beam model is considered to analytically investigate post-buckling and nonlinear vibration of flexible fluid-conveying pipes made of functionally graded materials. Closed-form expressions are presented to study the effects of shear deformation and rotary inertia on the post-buckling behavior and nonlinear vibration of fluid-conveying functionally graded pipes for the first time. It is assumed that the material property of the functionally graded material pipe is changed continuously along the direction of the pipe wall thickness based on a simple power law. Considering the Von-Karman nonlinear strain components and using the Hamilton's principle, the nonlinear governing equations for vibration and buckling of the functionally graded pipe conveying fluid are derived. These nonlinear governing equations are discretized by applying the Galerkin's procedure. The homotopy analysis method is employed to determine the closed-form expressions for the nonlinear frequency and the time history of vibration of the functionally graded pipe. The influences of rotary inertia and shear deformation on the buckling behavior, critical fluid velocity, nonlinear frequency and non-dimensional amplitude of the system are investigated and discussed in detail. The effects of physical parameters of the pipe conveying fluid on the divergence fluid velocity are studied by considering polypropylene random and functionally graded polymer pipes. The numerical results attest to the importance of considering rotary inertia and shear deformation in analyzing of short functionally graded material pipes especially for stiffer materials with higher initial amplitudes.