Free Vibration Analysis of Symmetrically Laminated Composite Plates on Elastic Foundation and Coupled with Stationary Fluid

Free vibration analysis of symmetrically laminated composite plates resting on Pasternak elastic support and coupled with an ideal, incompressible and inviscid fluid is the objective of the present work. The fluid domain is considered to be infinite in the length direction but bounded in the depth and width directions. In order to derive the eigenvalue equation, Rayleigh-Ritz method is applied for the fluid-plate-foundation system. The efficiency of the method is proved by comparison studies with those reported in the open literature. At the end, parametric studies are carried out to examine the impact of different parameters on the natural frequencies.


Introduction
Composite materials consist of two or more materials, which together produce desirable properties that cannot be achieved by any of the constituents alone (Reddy, 1997), received much attention in all types of engineering structures due to their outstanding properties such as high strength and stiffness to weight ratios and low cost. From the structural viewpoint, the vibration behavior of structural components like plates made of composite materials is important to design engineers. In addition, the plates on elastic foundations have considerable structural applications. The simple model of the elastic supports was introduced by Winkler (1867). The major deficiency of this model is having no interaction between the springs. Pasternak (1954) improved the Winkler model by attaching a shear layer to the springs. Many research works are conducted to the vibration analysis of laminated composite plates using the first-order shear deformation theory (FSDT) (Whitney and Pagano, 1970;Dawe and Craig, 1986;Xiang et al., 1996;Liew, 1996;Ferreira and Fasshauer, 2007;Cui et al., 2011;Yu et al., 2016), various higher-order and refined plate theories (Reddy, 1984(Reddy, , 1986Senthilnathan et al., 1987;Cho et al., 1991;Wu and Chen, 1994;Matsunaga, 2000;Kant and Swaminathan, 2001;Ray, 2003;Wu and Chen, 2006;Fiedler et al., 2010;Mantari et al., 2011;Hasani Baferani and Saidi, 2013;Vosoughi et al., 2013;Datta and Pay, 2016) and three-dimensional elasticity theories (Carrera, 2000(Carrera, , 2003a(Carrera, , 2003bCar-rera et al., 2011;Thai et al., 2016).
The fluid-structure interaction problem is a substantial criterion in design of many engineering systems. In recent years, the hydroelastic vibration analysis of plates has been the aim of many conspicuous research works. With the framework of a linear hydroelasticity theory, the investigation on the dynamic characteristics of a vertical and horizontal cantilever plate partially or totally submerged in fluid was carried out by Fu and Price (1987). By considering the classical plate theory, the analysis of rectangular plates coupled with fluid has been completed in the works done by Amabili (1997) and by Zhou and Cheung (2000). Liang et al (2001) studied the vibration of a cantilever plate totally immersed in fluid by applying the Ritz method based on an empirical added mass formula. By using a combination of the finite element method and Sanders' shell theory, Kerboua et al. (2008) studied the vibration of rectangular plates coupled with fluid. Uğurlu et al. (2008) examined the effects of elastic foundation and fluid on the dynamic characteristics of rectangular Kirchhoff plates using a mixed-type finite element formula. Uğurlu (2016) also studied free vibration of flexible base plates of rigid fluid storage tanks resting on elastic foundation using a higher-order boundary element procedure. Natural frequencies of laminated composite plates coupled with a bounded fluid considering the effect of free surface waves were investigated by Khorshid and Farhadi (2013) using the Ritz method. The same meth-od was applied by Hosseini-Hashemi et al. (2010) to examine the vibration characteristics of isotropic plates on elastic supports coupled with fluid. Hosseini-Hashemi et al. (2012) also presented an exact solution for free vibration of a horizontal rectangular plate either immersed in fluid or floating on its free surface. Vibration of multiple plates in contact with fluid which are applicable for fuel assemblies in a research reactor was studied by Jeong and Kang (2013). Shahbaztabar and Rahbar-Ranji (2016) examined the effects of uniform in-plane loads on natural frequencies of laminated composite plates resting on elastic supports and subjected to fluid. Cho et al. (2015) obtained natural frequencies of a rectangular bottom plate structure in contact with fluid. They examined thin and thick rectangular plates and stiffened panels with different framing types.
It is well noted that in many engineering problems the presence of fluid and foundation can affect simultaneously the natural frequencies of the plate. There are a few works related to free vibration analysis of rectangular laminated composite plates on elastic foundation and at the same time in contact with fluid and this is the motivation of this study. In the present study, the effects of the hydrostatic pressure and free surface waves are not considered. The accuracy of the results is confirmed by comparing them with other published papers. Numerical results are extracted in order to show the effects of various parameters on the frequencies such as the fluid levels and fiber orientations as well as the foundation stiffness in details.

Mathematical formulation
2.1 Geometrical configuration Fig. 1 depicts a thick laminated plate on the Pasternak support vertically coupled with an ideal fluid. a, b, and h denote the length, width and uniform thickness of the plate, respectively, and H refers to the depth of the fluid. The plate is made of N orthotropic layers.

FSDT assumptions
The displacement components, based on the FSDT, are given by: u(x, y, z, t) = u 0 (x, y, t) + zψ x (x, y, t); v(x, y, z, t) = v 0 (x, y, t) + zψ y (x, y, t); w (x, y, z, t) = w(x, y, t), where, , , , , and w are the rotations of a normal line about y and x axes, in-plane displacements, and transverse deflection of a point on the mid-plane, respectively. The strain-displacement relations can be written as follows: where, ω Ψ x Ψ y W where denotes the circular frequency. , and are the amplitude functions. For convenience, the following non-dimensional parameters are introduced: δ λ H where , and are the thickness to length ratio, aspect ratio and fluid depth ratio, respectively.
The strain energy of the plate can be calculated by the following equation: where, The extensional ( ), bending-extensional coupling ( ), bending ( ) and transverse shear ( ) stiffnesses are defined as follows: ij dz, i, j = 4, 5. (7) denotes the transverse shear correction factor and could be obtained from the following equations: where, and are the Young's moduli parallel and perpendicular to the fibers for a layer, respectively; , and are the shear moduli; and are the Poisson's ratios; and is the angle between the fiber orientation and rectangular coordinates x and y. It should be noted that the Poisson's ratios are governed by the equation . The maximum strain energy of the plate could be written for a symmetrically laminated plate ( ) as follows: where,

Energy of the foundation
The strain energy of the elastic foundation is calculated by: where and denote the Winkler and shear coefficients of foundation, respectively.

Kinetic energy of the plate
The maximum kinetic energy of the vibrating laminated plate is given by: where, refers to the plate mass density, and 2.4 Fluid formulation 2.4.1 Velocity potential of the fluid ϕ The velocity potential must satisfy the Laplace equation associated the boundary conditions in the fluid domain ( Fig. 1) as follows: ∂ϕ ∂y y=0 = 0, rigid walls; To allow the perfect contact between the plate surface and the tangential fluid layer, the impermeability condition is taken into account. To do so, ϕ One can take the solution of as: (21) By imposing the boundary conditions Eq. (16) through Eq. (19) and the method of separation of variables, the solution of the velocity potential of the fluid is derived as follows: By applying the boundary condition Eq. (20) into Eq. (22), the exact Fourier coefficients can be extracted as follows: where, where, 2.5.2 Kinetic energy of the fluid By assumption of the ideal fluid and no surface waves, the kinetic energy of the fluid can be obtained as follows: ρ f ∇ where refers to the fluid density and is the gradient operator. By applying the Green's theorem and using Eq. (15), a surface integral, surrounding the fluid domain boundary, can be obtained. By imposing the boundary conditions given in Eq. (16) through Eq. (19), the integral is simplified to a surface integral in the plate region as follows: Substituting Eqs. (20) and (24) into Eq. (27), one can obtain:

Admissible functions
The amplitude functions in Eq. (4) are assumed to be continuous and are approximated as follows: where , and are unknown coefficients; , , and are the Chebyshev polynomials multiplied by a boundary function to satisfy the essential boundary conditions. These functions related to different boundary conditions are defined in Table 1.

Rayleigh-Ritz procedure
The total energy function for the plate-fluid-foundation system can be given by: Truncating the number of the terms of the series in and directions up to and , respectively; and then minimizing the energy function in Eq. (30) with respect to the coefficients, i.e.

Results and discussion
Comparison and parametric studies are carried out in this section to demonstrate the exactness of the results and the effects of various parameters on the natural frequencies.
For simplicity, the boundary conditions are specified by capital letters; for example, CSCF indicates a plate which has clamped boundary condition (C) at the edges x=0 and 1,  simply supported (S) at y=0 and free (F) at y=1, respectively.

Comparison studies
As the first case, the comparison study between the present results with those reported by Hosseini Hashemi et al. (2010) is carried out for a simply supported isotropic square ( ) plate on the elastic foundation in contact with fluid. The results have been addressed in Table 2 for different fluid depth ratios and and 0.2. Table 2 demonstrates the fact that all results are in agreement with each other.
The second example is dedicated to non-dimensional fundamental natural frequency of simply supported symmetrically cross-ply laminated ( ) square ( ) plates. The numerical results are reported in Table 3, also including the results obtained by various plate theories, for graphite/epoxy composite plates with the material properties , , , υ 12 = ρ p = 1000 kg/m 3 0.25 and . It can be figured out that the results of the present method compared with the higher-order shear deformation and layer-wised theories are in good agreement.
For example, the relative error compared with the results based on the first-order shear deformation theory reported by Whitney and Pagano (1970) is 0.0% for all values of the thickness-to-width ( ) ratios. The maximum discrepancies for a moderately thick plate ( ) in comparison with the results based on the higher-order shear deformation plate theories are +0.233%, +0.506% and -0.153% reported by Reddy (1984), Cho et al. (1991) and Wu and Chen (2006), respectively. In addition, the maximum discrepancies for a moderately thick plate ( ) are +0.117%, +0.049% and -0.065% compared with the results obtained by Thai et al. (2016) using a isogeometric approach (IGA) based on the third-order, trigonometric and exponential-shear deformation plate theory, respectively.

Numerical results
In this section the influence of different parameters on the natural frequencies such as the plate geometrical dimensions and fluid characteristics as well as the elastic foundation parameters, boundary conditions and fiber orientations are investigated. Throughout the study the material properties of graphite/epoxy are taken, i.e.
, G 12 = G 13 = 0.6E 2 , , and , and . Fig. 2 depicts the variation of the frequency parameter for the SCSC plate with 50% fluid depth ratio as a function of the elastic foundation parameters when and . It can be observed that the natural frequency of the plate resting on the Pasternak elastic foundation is more sensitive to the shearing coefficient rather than the Winkler parameter. As can be seen, the frequency parameters for increase by 14.6% and 168% when the Winkler and shearing layer coefficients vary from zero to 100, respectively.
The change in the wet natural frequency parameters versus the fluid levels for the SSSS boundary condition are illustrated in Fig. 3 when and . It can be inferred from this figure that the frequency parameters decrease as the level of the fluid increases. Table 4 provides the information about the variation of the natural frequencies of a four-ply ( ) lamin-δ = 0.1 ated plate ( ) with respect to various fluid levels and side ratios. As could be observed from this table, the aspect ratio has direct effects on the frequencies and Fig. 4 clearly demonstrates this trend.
In Tables 5 and 6, the first three wet natural frequency parameters for three-ply ( ) laminated composite square ( ) plates with and without considering the elastic foundation are listed for two combinations of boundary conditions namely, SCSC and SFSF. In these tables three thickness to length ratios such as, , and δ = 0.2 have been considered and the levels of the fluid vary from zero to one. The effects of the thickness to width ratio,   elastic coefficients of foundation and fluid depth ratios are discussed above thoroughly in graphical forms. Fig. 5 gives information about the variation of wet frequency parameters of four-ply ( ) laminated square ( ) plates as a function of the fluid-plate density ratio for 50% waterline level and . The horizontal axis is attributed to the fluid-plate density ratios and provides a wide range of 0 to 0.4. It could be derived that the values of the fundamental wet natural frequency parameter diminish as the fluid-plate density ratio increases.
6-11 illustrate changes in the values of wet frequency parameters of laminated composite square plates on Pasternak foundation ( ) versus the fiber orientations when the fluid depth ratio is 0.5. Three different stacking sequences including , , and are considered while the fiber orientation angle varies from 0° to 180°. It can be observed that, for all combinations of the boundary conditions, the largest value of the wet frequency parameters belongs to the stacking sequence ( ). An interesting point attracting the authors' attention is that the behavior of this variation is different for various combinations of the boundary conditions. For example, for the SSSS boundary condition the fundamental wet frequency parameter increases and reaches a peak as the lamination angle changes from to and afterward decreases when the angle increases up to the angle of . The symmetric manner could be observed across the mirror line at the angle of . In other words, the highest and the lowest wet frequency parameters occur at For CCCC and SCSC, a slightly different manner compared with the SSSS boundary condition can be seen. For these boundary conditions the wet frequency parameters increase to a peak at the angle and then decrease as the lamination angle varies from to . For the CFFF, SFSF and CFCF boundary conditions a different behavior can be seen. In these cases, the wet frequency parameters decrease as the orientation angle changes . To be exact, for these combinations of the boundary conditions the highest and lowest wet frequency parameters occur at the lamination angles and , respectively. contact with fluid when the fluid depth ratio is 0.6. Three thickness to length ratios including , and have been considered which could be referred as thin to moderately thick plates. The fiber orientation is assumed to be . It could be seen that the fluid can affect the mode shapes of the plate and sever distortion occurs for .

Conclusions
In this study, the simultaneous effects of the fluid and foundation on the vibratory characteristics of the laminated composite plates were studied. The Rayleigh-Ritz method was employed to derive the eigenvalue equation. To verify the accuracy of the present method different comparison studies were carried out. It was concluded that the obtained results were in excellent agreement with those acquired by other methods based on various higher order shear deformation theories. Numerical results were presented in order to study the influence of different variables which support the following conclusions.
(1) The wet natural frequencies decrease by taking large values of the ratio of the fluid depth to fluid-plate density.
(3) The elastic foundation parameters influence the natural frequencies directly, however, the impact of the shearing layer is more tangible.
(4) The variation of the wet natural frequencies with respect to the fiber orientations is different for various combinations of the boundary conditions. For the SSSS boundary condition, the highest and lowest wet frequency parameters occur at the lamination angles and 180°, respectively. For CCCC and SCSC, the maximum wet frequency parameter occurs at the lamination angle , while for the CFFF, SFSF and CFCF boundary conditions, the minimum wet frequency parameter occurs at the lamination angle .