Static and vibration analysis of cross-ply laminated composite doubly curved shallow shell panels with stiffeners resting on Winkler–Pasternak elastic foundations

In this paper, the analytical solution for static and vibration analysis the cross-ply laminated composite doubly curved shell panels with stiffeners resting on Winkler–Pasternak elastic foundation is presented. Based on the first-order shear deformation theory, using the smeared stiffeners technique, the motion equations are derived by applying the Hamilton’s principle. The Navier’s solution for shell panel with the simply supported boundary condition at all edges is presented. The accuracy of the present results is compared with those in the existing literature and shows good achievement. The effects of the number of stiffeners, stiffener’s height-to-width ratio, and number of layers of cross-ply laminated composite shell panels on the fundamental frequencies and deflections of stiffened shell with and without the elastic foundation are investigated.


Introduction
The composite materials, with excellent mechanical properties such as high-strength, light-weight, and tailor ability, make it ideal for aircraft, aerospace, and marine application. Stiffened shell structures are extensively used for the construction of a variety of engineering structures such as commercial vehicles, road tankers, aircraft fuselages, wings, naval vessels, ship hulls, submarine, etc. These structures are very often subjected to both static and dynamic loads. Hence, it becomes necessary to carry out a static and dynamic analysis to know the actual deformation and vibration characteristic of these structures.
Laminated composite shallow shells can be formed as rectangular, triangular, trapezoidal, circular, or any other plan forms and various types of curvatures such as singlycurved (e.g., cylindrical), double-curved (e.g., spherical), or other complex shapes such as turbo machinery blades.
During the years, many researches have been devoted to the static and dynamic analysis of doubly-curved shell structures. To determine the natural frequencies of simply supported cross-ply laminates cylindrical and doubly curved shells, a set of layerwise three-dimensional equations of motion in terms of displacements has been presented by Bhimaraddi (1991) and Huang (1995). Based on three-dimensional elasticity, Wu et al. (1996) performed the bending and stretching problem of doubly curved laminated composite shells. However, these models require huge computational cost for multilayered structures.
To overcome these difficulties, typically, researchers make simplifying assumptions for particular applications, and reduce the 3D shell problems to various 2D representations with reasonable accuracy. Among the 2D theories for composite laminated shells have been developed, which can be classified into two different models, such as the equivalent single-layer model and the layerwise model. Review articles and monographs oriented to such contributions may be found in works of Carrera (2002Carrera ( , 2003, Toorani and Lakis (2000), Noor and Burton (1990) and Qatu (2004) and Reddy (2004). The equivalent single-layer theories can be classified into the three major theories, i.e., the classical shell theory (CST), the first-order shear deformation theory (FSDT), and the higher order shear deformation theory (HSDT).
The classical shell theory (CST) is based on the Kirchhoff-Love assumptions, in which transverse normal and shear deformations are neglected. Depending on different assumptions made during the derivation of the strain-displacement relations, stress-strain relations, and the equilibrium equations, various thin shell theories may be obtained within the Kirchhoff-Love framework. Among the most common of these are Donnell's, Love's, Reissner's, Novozhilov's, Vlasov's, Sanders', and Flügge's shell theories, for which a detailed description can be found in the monograph by Leissa (1993). For moderately thick shells, the effects of transverse shear deformations must be considered, and the first-order shear deformation theories (FSDTs) are developed. Although the FSDT describes more realistic behavior of thin-to-moderately thick plates, the parabolic distribution of transverse shear stress through the thickness of the plate is not properly reflected, thus the shear correction factor is introduced. To avoid using shear correction factor, higher order shear deformation theories (HSDTs) are proposed. Exact solutions of the equations and fundamental frequencies for simply supported, doubly curved, cross-ply laminated shells were presented by Reddy (1984). Khdeir et al. (1989) and  developed a shear deformable theory of cross-ply laminated composite shallow shells using state space concept in conjunction with the Levy method to analyze their static, vibration, and buckling response. Khdeir and Reddy (1997) presented a model for the dynamic behavior of a laminated composite shallow arch from shallow shell theory. Free vibration of the arch is explored and exact natural frequencies of the third-order, second-order, first-order, and classical arch theories are determined for various boundary conditions. The stiffeners are used to make shells with significantly increasing stiffness. Thus, to study these structures have been a remarkable trend of researchers in the recent years. The most powerful numerical tool for investigation of mechanical response of stiffened laminated composite shell structures with stiffeners is finite-element method: Bucalem and Bathe (1997), Scordelis and Lo (1964), Prusty (2003), Mukhopadhyay (1994, 1995). The literature on the analytical free vibration analysis of stiffened shell is limited to a few published articles. Mustafa and Ali (1989) presented the energy method to determine the natural frequency of orthogonally stiffened isotropic cylindrical shells. Lee and Kim (1998) studied the vibration of the rotating composite cylindrical shell with orthogonal stiffeners using energy method. Zhao et al. (2002) used the Love's shell theory and the energy approach, and carried out the vibration analysis of simply supported rotating cross-ply laminated composite cylindrical shells with stringer and ring stiffeners. Bich et al. ( , 2013, Bich and Nguyen (2012) used the smeared stiffeners technique, carried out the nonlinear analysis of eccentrically stiffened functionally graded cylindrical shell/panel, and eccentrically stiffened functionally graded shallow shell, (Bich and Van Tung 2011;. Nonlinear dynamic response, buckling and post-buckling analysis of imperfect eccentrically stiffened functionally graded doubly curved shallow shell resting on elastic foundation in thermal environment using smeared technique are presented in works of Duc and Cong (2014), Duc (2013), Duc and Quan (2012). Orthotropic circular cylindrical shells with closed ends stiffened by equally spaced stringers and rings subjected to combinations of uniform internal pressure, constant temperature change, and axial load are investigated by Wang and Hsu (1985). Wattanasakulpong and Chaikittiratana (2015) investigated the free vibration characteristic of stiffened doubly curved shallow shells made of functionally graded materials under thermal environment. The first-order shear deformation theory is employed to derive the governing equations used for determining natural frequencies of the stiffened shells. The governing equations can be solved analytically to obtain exact solutions for this problem.
To the best of the authors' knowledge, there is no published research in the literature conducted on the static and free vibration analysis of the stiffened laminated composite doubly curved shells by analytical approach. Thus, the purpose of the present paper is to develop an analytical solution for static and vibration analysis of cross-ply laminated composite stiffened doubly curved shallow shell panels resting on elastic foundation. In this study, the first-order shear deformation theory and smeared technique are used. Parametric studies are carried out and may be useful for the preliminary design of dynamically loaded, stiffened laminated composite shells resting on elastic foundation.

Theoretical formulations
Consider a cross-ply laminated composite doubly curved shallow shell panel with stiffeners in coordinate (x, y, z), as shown in Fig. 1; symbolize a, b, and h are lengths of the shell in the x-direction, y-direction, and thickness of the shell, respectively. R 1 and R 2 are middle surface radii of curvature in the x-and y-directions, respectively.
The displacement components at any point in the shell based on the first-order shear deformation theory (FSDT) are assumed as follows (Reddy 2004;Wattanasakulpong and Chaikittiratana 2015): where The constitutive equation of the kth layer of laminated composite shell can be written as Reddy (2004) r x r y r xy r yz r xz 8 > > > > < > > > > : where r x , r y , r xy , r yz , r xz and e x , e y , c xy , c yz , c xz are the stress and strain components in the global coordinate system-laminate coordinates system (x, y, z) of laminated composite shell; Q 0 ij s are the transformed elastic constants with respect to the global coordinate system (x, y, z), see more detail in Reddy (2004).
The geometry of the cross-ply laminated composite shallow shell panel with internal stiffeners in the x-z is plane, as illustrated in Fig. 2 l x. and E y , q y , A y , b y , h y , e y , l y are Young's modulus, mass density of stiffener material, cross-sectional area, the width, the height of the cross section, the eccentricity, and the space between stiffeners along the x-and y-directions, respectively. Stiffeners are assumed to be in uniaxial state of stress. Based on assumption that the stiffeners and the shell are perfectly bonded, the stress-strain relations of the stiffeners in the x-and ydirection can be written as The effects of the stiffness of the stiffeners are assumed to be smeared over the shell and the twist effect is ignored. Using the smeared stiffeners' technique, we have the internal moment and force resultants of the stiffened shell as follows Wattanasakulpong and Chaikittiratana 2015): Àh=2 zr xx zr yy zs xy 8 > < > : with ðijÞ ¼ 11; 12; 21; 22; 66 The parameter k s is the shear correction factor (k s = 5/ 6). Appling Hamilton's principle, the governing equations of stiffened shell resting on the elastic foundation using the FSDT can be expressed as follows (Kiani et al. 2012;Reddy 2004): where with q is the material mass density of the laminated composite shell and K w is the Winkler's elastic foundation coefficient and K p is a constant showing the effect of the shear interaction of vertical elements. The above internal moment and force resultants are expressed in displacement terms using Eqs. (1-3) and (7). In addition, then, substituting the obtained results to Eq. (11), we get the equilibrium equations with respect to displacement components.

Solution procedures
In this study, analytical solution for the static and vibration analysis of the simply supported cross-ply laminated composite doubly curved shallow shell panels resting on elastic foundation is developed using Navier's solution. For symmetric or anti-symmetric cross-ply laminates, we have: The mathematical expressions for movable simply supported condition are given by Reddy (1984) and Kiani et al. (2012): The displacement expressions to satisfy boundary conditions (Eq. 13) and applied load are assumed as qðx; y; tÞ ¼ where u mn (t), v mn (t), w mn (t), / xmn (t), / ymn (t) are the coef- qðx; y; tÞ sin ax ð Þsin by ð Þdxdy: Substituting Eqs. (14a, 14b) into the equilibrium equations with respect to displacement components of Eq. (11), it can be obtained as follows: where The coefficients K ij and M ij are determined using symbolic toolbox in the MATLAB software, see more detail in ''Appendix''.

Static analysis
For static analysis, the solution can be obtained by solving the equations resulting from Eq. (15) by setting the time derivative terms to zero: In this paper, the uniformly distributed transverse load over the surface of shell panel is considered: q mn = 16q 0 / (mnp 2 ), q 0 is the intensity of the uniformly distributed load.
Solving Eq. (17), we can get the displacement components of Eq. (14a) and obtained the deflection of shell panel.

Vibration analysis
For vibration analysis, all applied loads are set to zero, {F(t)} = {0}, and we assume that the periodic solutions in Eq. (14a) are of the form: u mn ðtÞ ¼ u mn e ixt ; v mn ðtÞ ¼ v mn e ixt ; w mn ðtÞ ¼ w mn e ixt ; where i is the imaginary unit, i 2 = -1; x is natural frequency.
In addition, then, substituting Eq. (14a) into Eq. (15), it can be obtained the equation: For a non-trivial solution, the determinant of the coefficient matrix of Eq. (20) should be zero. Solving the resulted determinant, we get the natural frequency, x mn , corresponding to mode (m, n). The smallest of the frequencies is called the fundamental frequency.

Results and discussions Validation
In this section, the three examples for the verification of the present study are presented including the cross-ply laminated composite doubly curved shell without stiffeners, the cross-ply laminated composite plate without stiffeners resting on elastic foundation, and the stiffened isotropic plate without elastic foundation. It is noted that the doubly curved shell panel can changed to the various structural types by setting quantities as follows: a R 1 ¼ b R 2 ¼ 0 for a flat plate; a R 1 ¼ 0 for a cylindrical shell panel; a R 1 ¼ b R 2 for a spherical shell panel.
The non-dimensional fundamental frequency, non-dimensional deflection, and non-dimensional elastic foundation parameters form is used: First, the cross-ply laminated composite doubly curved shell without stiffeners is considered. The geometric parameters of the shell panel: a/b = 1; a/ h = 100; R 1 = R 2 = R (for spherical shell); and the material properties of the shell: E 1 = 25E 2 ; G 13 = G 12 = 0.5E 2 ; G 23 = 0.2E 2 ; and m 12 = 0.25. The non-dimensional fundamental frequencies x ¼ xa 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q=E 2 =h 2 p and non-dimensional center deflections w ¼ wh 3 E 2 =q 0 a 4 ð ÞÂ10 3 of the cross-ply laminated composite doubly curved shell without stiffeners are calculated and listed in Table 1. The results are compared with those based on exact solution of Reddy (1984) using first-order shear deformation theory, too. From Table 1, it can be observed that the present results are identical with those of Reddy (1984).
Next, the cross-ply laminated composite plate without stiffeners resting on elastic foundation is considered. The geometric parameters of shell panel: a = b = 1; a/R 1 = b/ R 2 = 0 (for plate); lamination scheme [0°/90°/0°]; and the material properties of the shell panel: E 1 = 40E 2 ; G 13 = G 12 = 0.6E 2 ; G 23 = 0.5E 2 ; and m 12 = m 13 = 0.25; q = const. The fundamental frequencies of the cross-ply laminated composite plate resting and not resting on elastic foundation are calculated with various side-to-thickness ratios. The results are compared with those of Akavci (2007) and presented in Table 2. The non-dimensional fundamental frequencies as: x ¼ xa 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q=E 2 =h 2 p . The comparison shows good agreement (maximum discrepancy is 0.45%).
The third example, the five natural frequencies of simply supported stiffened isotropic plate without elastic foundation are presented in Table 3. The comparison between the present results with those reported by Szilard (1974) and Troitsky (1976) is made. The dimensions of the plate (a/ R 1 = b/R 2 = 0) are: a = 0.6 m; b = 0.41 m; and h = 0.00633 m; the plate is stiffened with three stiffeners in the x-direction (n x = 3) and five stiffeners in the ydirection (n y = 5). The dimensions of stiffeners are b x = b y = 0.0127 m and h x = h y = 0.02222 m, and the isotropic material properties of the shell and stiffeners are: E = 211 GPa; q = 7830 kg/m 3 ; and m = 0.3. Results from Table 3 show good agreement, and the maximum discrepancy is 5.07% (Mode 3). Note that the results obtained by Szilard (1974) and Troitsky (1976) using the finite-element method. From the above verifications, it can be concluded that the present numerical results are reliable.

Parametric study
In the next investigations, the following geometric parameters and material properties of the shell panels are used: a = b = 1 m; b/h = 50; and a/R 1 = b/R 2 = 0.5 (for spherical shell); a/R 1 = 0 and b/R 2 = 0.5 (for cylindrical shell); and E 1 = 132.5 GPa; E 2 = 10.8 GPa; G 13 = G 12 = 5.7 GPa; G 23 = 3.4 GPa; m 12 = 0.24;and q = 1600 kg/m 3 . The material properties of internal stiffeners (only considered internal stiffeners) are E = 3E 2 ;q = 1600 kg/m 3 ; and m = 0.24. It is also noted that the stiffeners in the x-direction can be called stringers, and the stiffeners in the y-direction can be called rings for cylindrical shells.

Effects of number of the stiffeners on the fundamental frequency and central deflection
In this investigation, the cross-ply laminated composite doubly curved shallow shell panels with lamination  x-direction only or y-direction only or orthogonal stiffeners. From Fig. 3, it can be seen that the fundamental frequencies of both stiffened spherical and cylindrical shell panels decrease with increased number of stiffeners for both cases: resting and not resting on elastic foundation. This phenomenon can be explained by the fact that mass effect by stiffeners activates larger than stiffness effect.
In particular case, for stiffened spherical shell panel, at the beginning, the deflection of shell increases up and then   Fig. 4 Effects of number of stiffeners on the center deflection (m) of cross-ply shell panels resting and not resting on elastic foundation decreases down. Besides, the fundamental frequencies of stiffened shell panel with elastic foundation are higher than those without elastic foundation, but the deflections are smaller than those. Figure 3 also shows that the fundamental frequency of both cylindrical shell panel and spherical shell panel with ring stiffeners is the highest (only for this dimension of stiffener and their material properties). Figure 4a depicts the decrease of central deflection with increased number of stiffener for cylindrical panel and shows that the deflection of cylindrical shell panel with orthogonal stiffeners is the smallest (i.e., the stiffest).

Effects of stiffener's height-to-width ratio on the fundamental frequency and central deflection
The effects of stiffener's height-to-width ratio (h s /b s ) on the fundamental frequencies and central deflections of the laminated cross-ply spherical shell and cylindrical shell panels are plotted in Figs. 5 and 6. The cross-ply shell panels with lamination scheme [0°/90°/0°/90°], the number of the stiffeners in the x-direction (stringers), the number of stiffeners in the y-direction (rings), and orthogonal stiffeners (n x = n y ) are nine stiffeners. The dimension of the Ratio hs/bs (spherical shell) The center deflection of shell stringers(x-direction) rings(y-direction) orthogonal stiffeners Fig. 6 Effects of stiffener's height-to-width ratio of stiffener on the center deflection (m) of the cross-ply shell panels resting on elastic foundation Int J Adv Struct Eng (2017) 9:153-164 161 stiffeners used is b x = b y = h and h x = h y = (0.5-5) h; h is the thickness of the shell. As shown in Fig. 5, we see that the fundamental frequency of both stiffened cylindrical shell panel and spherical shell panel has a same trend of decrease to the minimum and then increase up, while the deflection of stiffened shell panels decreases, as shown in Fig. 6. Besides, the fundamental frequencies of stiffened shell panels with elastic foundation are higher than those without elastic foundation, while the deflections are smaller than those.
In Fig. 5a, b, it can be seen that the highest fundamental frequency of shell changes from shell panel with ring stiffeners to shell panel with orthogonal stiffeners.
However, the deflection of both cylindrical shell panel and spherical shell panel with orthogonal stiffeners is the smallest (the stiffest), see Fig. 6.
Effects of the number of layers of cross-ply laminated composite shell on the fundamental frequency and central deflection Both the figures indicate clearly that with the increase of number of shell layers, the fundamental frequencies are increased and the central deflections are decreased. This may be explained by the fact that the number of shell layers is increased, the laminate becomes stiffer.
In addition, from Figs. 3, 4, 5, 6, 7, and 8, it can be found that the fundamental natural frequency is increasing and the deflection is decreasing in the addition of the foundations.

Conclusions
In this work, the analytical solution for static and vibration analysis of stiffened cross-ply laminated composite doubly curved shallow shell panels resting on the elastic foundation with the simply supported boundary condition is presented. From the previous investigations, it can be noted that: • For the vibration analysis, when the stiffener's heightto-width ratio increases, the highest fundamental frequency of shell changes from shell panel with ring stiffeners to shell panel with orthogonal stiffeners. • For static analysis, when the stiffener's height-to-width ratio increases, the deflection of shell panel with orthogonal stiffeners (both cylindrical shell panel and spherical shell panel) is the smallest (stiffest). • The elastic foundations make the fundamental natural frequency of stiffened cross-ply laminated composite doubly curved shallow shell increased and the deflection decreased. • The number of shell layers is increased; the laminate doubly curved shell becomes stiffer.
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