Vibration of prestressed thin cylindrical shells conveying fluid
Introduction
The beam- and shell-mode vibration of cylindrical shells conveying fluid has been extensively investigated, both theoretically and experimentally, for a variety of different industrial scenarios, including steel conduits for power plants [1] and subsea/ground oil pipelines [2]. Reviews of this research include those by Païdoussis [3] and Chen [4]. All these investigations were primarily motivated by the need to achieve a fundamental understanding of the dynamic behaviour of pipes conveying fluid flow, establishing: (1) the critical flow velocity at which cylindrical shells conveying fluid, either cantilevered or clamped at both ends, lose stability by the Païdoussis flutter or divergence [5] (the stability of a fluid-conveying shell, simply supported at both ends, is the subject of a benchmark paper by Weaver and Unny [6]); (2) the dynamic response of systems without initial axial tensions at a sub-critical flow velocity [7]; and (3) the influence of initial axial tension, flow velocity and fluid pressure on natural frequency [8], [9]. In most of these studies, the dynamics of stiff cylindrical shells with low hydrostatic pressures and cantilevered cylindrical shells were analysed, so the effect of hydrostatic pressures was neglected and initial stresses cancelled out in the models.
The presence of an internal pressure in a thin cylindrical shell, however, could appreciably affect the natural frequency. Such an effect has been observed experimentally [10]. Similarly, Miserentino and Vosteen [11] tested various thin shells over a wide range of internal pressures and observed that natural frequencies increased with increasing initial pressures. Penzes and Kraus [12] investigated the free vibration of prestressed cylindrical shells with arbitrary homogeneous boundary conditions. Initial tensions within the shell affect the lateral vibration of a prestressed flexible system causing stiffening of the tube and an increase in natural frequencies [13]. Additionally, the published literature on finite element modelling of the vibration of prestressed cylindrical shells conveying fluid, examining the influence of initial tension and fluid pressure [9], is very sparse. Lakis et al. [14] presented an approach which combined the finite element method (FEM) and Sanders’ shell theory, derived from Love’s shell equation, for application to the dynamic problem of anisotropic fluid-filled conical shells. In this hybrid FEM approach, an exact displacement function, derived from Sanders’ shell theory, was used. Later, this study was extended by presenting an improved FEM approach incorporating the influence of flowing fluid on the vibration of an open cylindrical shell in the absence of fluid pressures and initial tensions. The effect of the presence of internal or/and external fluid on the free vibration was then demonstrated [15].
In this study, we use the theory of thin shells and classical potential flow theory to establish a new FEM model for the dynamic problem of initially tensioned cylindrical shells conveying steady fluid flow. The fields of the flowing fluid and the moving shell are fully coupled. The present paper describes the formulation, implementation, and experimental verification of the model. The effect of initial tensions, internal pressures, flow velocities and geometric properties on the vibratory behaviour of prestressed cylindrical shells conveying steady fluid flow is investigated.
Section snippets
The governing equations
In this paper, the dynamic problem of a prestressed circular cylindrical shell containing steady fluid flow is formulated in a cylindrical polar coordinate system (x, θ, r). The coordinate axis x is chosen to coincide with the cylindrical shell centreline, while the coordinate axes r and θ are taken along the radial and circumferential directions, respectively.
The method of solution
, , governing the motion of the shell and fluid subject to boundary conditions , , are solved by a FEM in which the shell is discretized into cylindrical frustums and the shell displacement, u, may be expressed in the form:where n denotes the circumferential wave number; ; Nt denotes the shape function matrix, i.e. ; represents the nodal displacement vector, i.e.
Numerical examples
We have developed a finite element model for the dynamic problem of prestressed thin circular cylindrical shells conveying fluid. We choose a cylindrical frustrum finite element for the numerical implementation. The model is validated by comparison with published experimental results for a cylindrical shell (both without and containing fluid), prestressed cylindrical shells, pressurized thin cylindrical shells, and a cylindrical shell conveying fluid. A series of examples are given and compared
Conclusions
In this paper, a finite element model for the vibration of prestressed thin cylindrical shells conveying fluid has been presented. The method is based on Sanders’ nonlinear theory of thin shells and classical potential flow theory. A nonlinear strain–displacement relationship was employed to derive the geometric stiffness matrix due to the initial stresses and hydrostatic pressures. Hydrodynamic pressure acting on the shell is developed through dynamic interfacial coupling conditions using the
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- 1
Present address: Faculty of Engineering, Tsinghua University, P.R. China.
- 2
Present address: Dept. of Mechanical Engineering, University of Strathclyde, James Weir Building, Montrose Street, Glasgow, G1 1XJ, UK.