Keywords: non-linear, vibrations, anisotropic shells, shear deformations.

A semi-analytical method is developed in conjunction with shearable shell theory and modal expansion approach to predict the influence of geometrical non- linearities, incorporating large displacements and large deformations, on free vibrations of anisotropic laminated cylindrical shells. The shear deformation and rotary inertia effects are taken into account in the equations of motion. The hybrid method developed in this theory is a combination of classical finite element approach, shearable shell theory (as described in reference [1,2]) and modal coefficient procedure [3]. The displacement functions are obtained by the exact solution of the equilibrium equations of anisotropic cylindrical shells instead of the usually used and more arbitrarily interpolating polynomials. The mass and linear stiffness matrices are then derived by exact analytical integration. Green's exact strain-displacement relations are used to obtain the modal coefficients for these displacement functions. The second- and third-order non-linear stiffness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. The linear and non-linear natural frequency variations are determined as a function of shell parameters for different cases.

A good description of geometrical non-linear effects on the vibration behaviour of shells could be presented when the coupling between different modes is taken into account by considering the cross-product terms of non-linear stiffness matrices. The non-linear strain terms arising from products of in-plane strain terms may be important in buckling problems. Also, it is well known from experimental observations that the classical theory of plates and shells, which neglects the transverse shear strains, leads to underestimation of deflections and over-predictions of natural frequencies and buckling loads. These errors are even higher in the case of plates and shells made up of advanced anisotropic composite materials.

In the present method the equations of anisotropic cylindrical shells are used in full to obtain the pertinent displacement functions, instead of using the more common arbitrarily polynomial forms. The finite element method is a cylindrical panel segment, as developed in reference [2], rather than the more commonly used triangular or rectangular elements. The displacement functions, mass, linear and non-linear stiffness matrices over an element are derived by exact analytical integration [4]. In doing so, the accuracy of the formulation will be less affected as the number of elements used is decreased (thus reducing computation time) and as the dynamic characteristics of the shell are required at higher beam-mode () or higher shell-mode (), a significant advantage over polynomial interpolation. Therefore, this method is more accurate than the more usual finite element methods.

The available solution based on Sanders' theory can also be obtained from the present theory in limiting case of infinite stiffness in transverse shear. The present method offers many advantages, some of which are; a) simple inclusion of thickness discontinuities, material property variations and difference in material comprising the shell; b) arbitrarily boundary conditions without changing the displacement functions in each case; c) high and low frequencies are obtained with high accuracy; d) this approach has also been applied, with satisfactory results, to the dynamic analysis of shells containing flowing fluid or partially filled with liquid. Stability problems in several cylindrical geometry, different multi-layered layouts (cross-ply, angle-ply symmetrically and asymmetrically laminated) and circumferentially non-uniform and various boundary conditions can be analysed by the present formulations. A full implementation of the non-linear dynamic equations can be conducted to show the reliability and effectiveness of the present formulation along with the linear and non-linear effects of flowing fluid to study the non-linear flow-induced vibrations of anisotropic cylindrical shell. This later is under way and the results will be presented in future publications.

1

M.H. Toorani, A.A. Lakis, "General Equations of Anisotropic Plates and Shells Including Transverse Shear Deformations, Rotary Inertia and Initial Curvature Effects", Journal of Sound and Vibration, 237(4), pp.561-565, 2000. doi:10.1006/jsvi.2000.3073

2

M.H. Toorani, A.A. Lakis, "Shear Deformation Theory in Dynamic Analysis of Anisotropic Laminated Open Cylindrical Shells Filled With or Subjected to a Flowing Fluid", Computer Methods in Applied Mechanics and Engineering, 190, 4929-4966, 2001. doi:10.1016/S0045-7825(00)00357-1

3

H. Radwan, J. Genin, "Non-Linear Modal Equations for Thin Elastic Shells", International Journal of Non-Linear Mechanics, 10, 15-29,1975. doi:10.1016/0020-7462(75)90026-8

4

M.H. Toorani, A.A. Lakis, "Geometrically Non-Linear Dynamics of Anisotropic Open Cylindrical Shells with a Refined Shell Theory", Technical Report, EPM-RT-01-07, Polytechnique Montreal, Canada, 2002.

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