Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness

and


INTRODUCTION
The problems of nonlinear vibrations of plates and shallow shells are topical for both theory and application in many areas of modern industry.Especially, it concerns the space industry, where the plates and shells are used as members of many structural components.In practice, these elements can have a variable thickness, different form of the middle surface and boundary conditions, as well as different orientation of the anisotropy axes.The studies of linear vibrations of anisotropic shells have attracted the attention of many researchers for a long time [4,6,7,9,10].Great progress has been made over the past decades to develop numerical approximate methods as the most effective tools for studying nonlinear vibrations of the composite plates and shallow shells [1-3, 5, 7, 9-11].This is confirmed by a large number of papers and books.The finite elements method (FEM) is one of the most widely applied approach to non-linear vibration problems of continuous mechanical systems [10,11].However, it should be emphasised that even for linear vibrations of shells with variable thickness numerical results are not so widely presented.Furthermore, in the case of nonlinear vibrations of anisotropic shells of variable thickness the computational results are rather marginally discussed.This is due to the difficulties that arise while solving this class of problems.First of all, it is difficult to construct the system of eigenfunctions in an analytical form in the case of an arbitrary shape of a shallow shell.However, the latter approach is used mainly to solve nonlinear Latin American Journal of Solids and Structures 10(2013) 149 -162 problems.The second complex question refers to a transition from continuous to discrete models with respect to time.In this paper we propose a method to solve this class of problems using the Rfunctions theory and variational methods.In the literature devoted to the study of plates/shells statics and dynamics this approach is known as RFM which is an abbreviation for the R-function Method [3,7,9].It should be noted that the use of RFM allows researchers to take into account not only variable thickness of a shell, but also to design eigenfunctions in an analytical form that are then used to solve the problem of geometrically nonlinear vibrations of the shell.Further on in this paper we develop this approach to investigate non-linear free vibrations of orthotropic shallow shells with variable thickness.

MATHEMATICAL FORMULATION
The present formulation of the problem is based on Donell's nonlinear shallow shell theory, which adopts Kirchhoff's hypothesis.Transverse shear deformations and rotary inertia of a shell are neglected.According to this theory, the non-linear strain-displacement relations at the shell midsurface can be written as follows Here u,v,w are the displacements of the shell in directions Ox, Oy and Oz, respectively, whereas , xy RR-are radii of the shell curvature (Fig. 1).
The constitutive relations of the shell can be expressed as follows ,, ,, ,, ,, ,, The equation of equilibrium for free geometrically nonlinear vibration of a shallow shell may be written in the following form

METHOD OF SOLUTION
(i) Solving the linear vibration problem.The linear vibration problem for an orthotropic shallow shell with variable thickness is solved using the Ritz variational method.
The variational statement of the linear problem is reduced to finding the minimum of the following functional The R-functions theory is used to find a minimum of the functional including the basic functions satisfying the given boundary conditions.The main advantage of the R-functions method relies on the possibility of constructing these functions in an analytical form.For some kinds of boundary conditions such basic functions have been already presented in references [3,7,8,9,12].For example, the system of admissible functions corresponding to a clamped edge, and to in-plane immovable simply supported edges follows 2 ,, In the above 0 means the equation of the domain border, whereas , in are the elements of some complete systems 1 2 3 ,, Observe that the natural modes corresponding to linear vibrations of the shells serve as basic functions to represent the unknown functions., , , u x y v x y are solutions to a system similar to the Lamè system of the following form 11 11 12 11 1 Symbol in equations (27-28) is equal to 1 for shells and and it is equal to 0 for plates.The above mentioned problem is solved by the RFM.Note that the solution to this problem has been already described in references [7,9].Substituting expressions ( 27)-( 29) for ( , , ), u x y t ( , , ), v x y t ( , , ) w x y t into equations ( 12)-( 14) and ignoring inertia terms in equations ( 12)-( 13), one may see that equations ( 12)-( 13) are satisfied identically.Therefore, applying the Bubnov-Galerkin procedure to equation ( 14) we obtain the following equation where , , , , , which are defined as follows ,, In order to find a backbone curve, let us put cos N A and let us apply again the Bubnov-Galerkin procedure [9,13].Then, the approximate relation between maximum amplitude A and the ratio of the nonlinear vibration to linear one / NL is as follows:

NUMERICAL RESULTS
The so far developed approach is validated on some tested problems and will be applied to solve new problems regarding nonlinear vibrations of shallow shells with variable thickness.
Problem 1.The correctness, validity and reliability of the developed method have been studied by solving the linear vibration problem for an orthotropic clamped spherical shallow shell with square plane-form and variable thickness of the following form , for a clamped spherical panel versus results reported in reference [4] is given in Table 1.In what follows we study the influence of parameter variation .This problem has been solved in [4], using a spline -approximation to the assumed solution.One may see that the difference between our results and those given in [4] is less than 1.5%.It confirms the validation of the RFM method.Results reported in [4] are in bold.          ) are given in Table 6.The so far illustrated and discussed examples regarding nonlinear analysis for the given shells indicate the efficiency of our approach.Influence of curvatures and thickness parameter 0.5;0.5 on backbone curves is shown in Fig. 8 and Fig. 9.

CONCLUSIONS
Analysis of the geometrically nonlinear vibrations of the shallow shells with variable thickness and complex shape has been carried out using the R-functions theory and variational methods.A distinctive feature of the proposed approach is also the original construction of approximate solutions.In a single-mode approximation the solution, this approach allows to investigate the dynamical behavior of shallow shells with an arbitrary form of their plans and various types of boundary conditions.First, the validity and reliability of the proposed approach has been illustrated and discussed, and then a few examples of either linear or non-linear dynamics of the shells with variable thickness and complex shapes have been presented and discussed.

Fig. 1 .
Fig. 1.Geometry of a shallow shell are curvatures of the shell in directions Ox, O, respectively.The obtained system of equations is supplemented by boundary conditions defined by the way of shell fixation.Latin American Journal of Solids and Structures 10(2013) 149 -162

(
ii) Solving the non-linear vibration problem.Let us denote the natural frequency and the corresponding eigenfunctions by L and () cw , of Solids and Structures 10(2013) 149 -162 .

Table 1 .
[4]parison of non-dimensional frequencies for the clamped spherical panel with square plane-form using the RFM and spline approximation (see[4])

Table 2 .
Influence of the curvature thickness and turn angle of the orthotropic axes on non-dimensional frequencies of the clamped spherical panel with square plane-form xy kk ; 0 45 are presented in Table3.

Table 3 .
Influence of the clamped spherical panel curvatures on vibration modes

Table 4 .
Non-dimensional frequencies for simply supported spherical shells

Table 5 .
Dependence of the ratio of the nonlinear frequency to linear one on the vibration amplitude of spherical panels with variable thickness

Table 6 .
Influence of on frequencies