Keywords: hybrid element, anisotropic shells, vibrations, flowing fluid.

With the widespread use of shell structures in numerous fields of industry (e.g. aerospace, power generation, construction, transportation, shipbuilding), design and research engineers have been compelled over the years to understand and predict ever more accurately the behaviour of shell structures under static and dynamic loading conditions. Considerable attention has been focused on the design of composite shells that present low weight and high performance characteristics due to high specific strength (failure stress/ weight) and specific stiffness (stiffness/weight). For more than the three last decades, the aircraft and aerospace industry has been rapidly advancing our knowledge of composite materials. The new ASME section (section X) on boilers and pressure vessel codes includes the use of laminate theory for the design and analysis of fibre-reinforced plastic vessels to reduce weight, save materials and alleviate shipping and erection problems. In addition, a great deal of research has been concentrating on thorough flow-induced vibration analyses at the design stage of nuclear components. Flow-induced vibrations are recognised as one of the most significant problems in the design of steam generator components. Excessive flow-induced vibrations often limit the performance of the nuclear components, piping system and shell-tube heat exchanger. These vibrations may result in mechanical failure; i.e. fretting-wear and fatigue cracking. Flow-induced vibration of a tube can cause it to hit or rub against a support plate or adjacent tubes, resulting in fretting-wear of the tube. Thus, designers and researchers have been called upon to provide a shell structure exhibiting not only maximum reliability but also minimum weight, which can only be verified following careful static and/or dynamic analysis of the entire structure and a thorough knowledge of material behaviour. It is known that laminated composite shells exhibit greater thickness effects than corresponding structures using homogeneous isotropic materials. The application of conventional theories to layered composite shells could thus lead to large errors in deflection, stress and frequency values.

This paper proposes a semi-analytical approach to predict flow-induced vibrations of isotropic and anisotropic cylindrical shells filled with or subjected to an ideal fluid, that is, internal or external flows. Shear deformation and rotary inertia effects are taken into consideration. An efficient hybrid finite element method has been developed to permit analytical determination of the stiffness, mass and damping matrices. The non-viscous, irrotational and incompressible fluid motion is described by general velocity potential, which must satisfy the Laplace equation. The present model is an extension of the expanded model appearing in Ref. [1], which is itself based on an element and algorithm developed in Ref. [2]. It is a hybrid finite element method that combines the flexibility of the conventional finite element approach with the accuracy of exact displacement functions as determined by shearable shell theory. This method yields both high and low eigenvalues and eigenmodes with comparably high accuracy. The displacement functions are determined by exact solution of the equilibrium equations of the anisotropic cylindrical shells rather than the more commonly used and more arbitrary interpolating polynomial method. In this way, the accuracy of the formulation is less affected since the number of elements used is decreased (thus reducing computation time). This hybrid approach also presents a significant advantage over polynomial interpolation while the dynamic characteristics of the shell are required at higher axial and circumferential mode numbers. The structural and fluid mass, stiffness and damping matrices are then derived by precise analytical integration. All details of the lengthy derivation are given in the full-length article.

The formulation developed in this research provides the natural frequencies and mode shapes of shells (empty, partially and completely filled with or subjected to flowing fluid) defined by arbitrary boundary conditions, and without changing the displacement functions in each case. The axisymmetric (n=0), beam-like mode (n=1) and shell modes (n>=1) of the structural vibrations are also discussed. The proposed method is also effective in investigating vibration characteristics of non-uniform cylindrical shells.

Further calculations are carried out to verify convergence of the solution. The results show that the vibration characteristics can be obtained to a very good level of accuracy using 10 elements for different ranges of the length-to-radius and radius-to-thickness ratios and also for isotropic and anisotropic materials. Several numerical examples are given to demonstrate the effect of transverse shear deformations on the dynamic behaviour of cylindrical shells with different geometric (R/t, L/R and L/t) and material (isotropic, symmetric and anti-symmetric cross-ply and angle-ply lamination) parameters, as well as axial and circumferential wave numbers (m, n). Results throughout show good agreement with experimental and other theoretical results. It is clear that this approach is not only sound but has advantages over other methods.

1

Toorani M.H., Lakis A.A., "General Equations of Anisotropic Plates and Shells Including Transverse Shear Deformations, Rotary Inertia and Initial Curvature Effects", Journal of Sound and Vibration, Vol. 237 (4), pp. 561-615, 2000. doi:10.1006/jsvi.2000.3073

2

Lakis A.A., Païdoussis M.P., "Free Vibration of Cylindrical Shells Partially Filled with Liquid", Journal of Sound and Vibration, Vol. 19, pp 1-15, 1971. doi:10.1016/0022-460X(71)90417-2

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