Static and vibration analysis of cross-ply laminated composite doubly curved shallow shell panels with stiffeners resting on Winkler–Pasternak elastic foundations
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Abstract
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.
Keywords
Static analysis Vibration analysis Cross-ply laminated composite Doubly curved shallow shell Stiffened shell Elastic foundation First-order shear deformation theoryIntroduction
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 singly-curved (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 (2002, 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 Librescu et al. (1989) 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), Goswami and 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. (2012, 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; Bich et al. 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
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
Static analysis
In this paper, the uniformly distributed transverse load over the surface of shell panel is considered: q_{mn} = 16q_{0}/(mnπ^{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 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, ω_{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: \(\frac{{a_{{}} }}{{R_{1} }} = \frac{{b_{{}} }}{{R_{2} }} = 0 \,\) for a flat plate; \(\frac{{a_{{}} }}{{R_{1} }} = 0\) for a cylindrical shell panel; \(\frac{{a_{{}} }}{{R_{1} }} = \frac{{b_{{}} }}{{R_{2} }}\) for a spherical shell panel.
Non-dimensional fundamental frequencies and center deflections for the simply supported cross-ply laminated composite spherical shell panel (a/b = 1, a/h = 100, k_{s} = 5/6)
R/a | 0°/90° | 0°/90°/0° | ||||||
---|---|---|---|---|---|---|---|---|
\(\bar{\omega }\) | \(\bar{w}\) | \(\bar{\omega }\) | \(\bar{w}\) | |||||
Reddy (1984) | Present | Reddy (1984) | Present | Reddy (1984) | Present | Reddy (1984) | Present | |
3 | 46.002 | 46.002 | 0.6441 | 0.6441 | 47.265 | 47.265 | 0.6224 | 0.6224 |
4 | 35.228 | 35.228 | 1.1412 | 1.1412 | 36.971 | 36.971 | 1.0443 | 1.0443 |
5 | 28.825 | 28.825 | 1.7535 | 1.7535 | 30.993 | 30.993 | 1.5118 | 1.5118 |
10 | 16.706 | 16.706 | 5.5428 | 5.5428 | 20.347 | 20.347 | 3.6445 | 3.6445 |
Plate | 9.687 | 9.687 | 16.980 | 16.980 | 15.183 | 15.183 | 6.6970 | 6.6970 |
Non-dimensional fundamental frequencies of the simply supported cross-ply laminated composite square plate resting on elastic foundation
Configuration | K_{0} | J_{0} | Method | a/h | |||
---|---|---|---|---|---|---|---|
5 | 10 | 20 | 50 | ||||
[0°/90°/0°] | 0 | 0 | Akavci (2007) | 10.265 | 14.700 | 17.481 | 18.640 |
Present | 10.290 | 14.766 | 17.516 | 18.648 | |||
Discrepancy (%) | 0.24 | 0.45 | 0.20 | 0.04 | |||
100 | 0 | Akavci (2007) | 14.246 | 17.751 | 20.131 | 21.152 | |
Present | 14.263 | 17.806 | 20.161 | 21.158 | |||
Discrepancy (%) | 0.12 | 0.31 | 0.15 | 0.03 | |||
100 | 10 | Akavci (2007) | 19.880 | 22.595 | 24.535 | 25.390 | |
Present | 19.891 | 22.637 | 24.560 | 25.396 | |||
Discrepancy (%) | 0.05 | 0.19 | 0.10 | 0.02 |
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; ν_{12} = 0.24;and ρ = 1600 kg/m^{3}. The material properties of internal stiffeners (only considered internal stiffeners) are E = 3E_{2}; ρ = 1600 kg/m^{3}; and ν = 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
Fundamental frequencies (Hz) of cross-ply laminated composite cylindrical shell panels resting on elastic foundation
K_{0} | J_{0} | Stiffener type | Number of stiffeners | |||||
---|---|---|---|---|---|---|---|---|
0 | 1 | 3 | 5 | 12 | 18 | |||
0 | 0 | Stringers (x-direction) | 162.559 | 161.658 | 159.912 | 158.237 | 152.859 | 148.754 |
Rings (y-direction) | 162.559 | 162.028 | 160.970 | 159.921 | 156.340 | 153.399 | ||
Orthogonal stiffeners | 162.559 | 161.152 | 158.539 | 156.153 | 149.127 | 144.246 | ||
100 | 10 | Stringers (x-direction) | 215.091 | 213.500 | 210.443 | 207.537 | 198.394 | 191.595 |
Rings (y-direction) | 215.091 | 213.788 | 211.270 | 208.860 | 201.162 | 195.324 | ||
Orthogonal stiffeners | 215.091 | 212.245 | 207.012 | 202.301 | 188.841 | 179.896 |
Fundamental frequencies [Hz] of cross-ply laminated composite spherical shell panels resting on elastic foundation
K_{0} | J_{0} | Stiffener type | Number of stiffeners | |||||
---|---|---|---|---|---|---|---|---|
0 | 1 | 3 | 5 | 12 | 18 | |||
0 | 0 | Stringers (x-direction) | 287.146 | 284.565 | 279.620 | 274.942 | 260.361 | 249.664 |
Rings (y-direction) | 287.146 | 284.743 | 280.131 | 275.762 | 262.092 | 252.012 | ||
Orthogonal stiffeners | 287.146 | 282.237 | 273.225 | 265.135 | 242.202 | 227.178 | ||
100 | 10 | Stringers (x-direction) | 319.408 | 316.486 | 310.888 | 305.595 | 289.101 | 277.011 |
Rings (y-direction) | 319.408 | 316.647 | 311.352 | 306.338 | 290.673 | 279.142 | ||
Orthogonal stiffeners | 319.408 | 313.810 | 303.533 | 294.307 | 268.156 | 251.030 |
Center deflection (m) of cross-ply laminated composite cylindrical shell panels resting on elastic foundation
K_{0} | J_{0} | Stiffener type | Number of stiffeners | |||||
---|---|---|---|---|---|---|---|---|
0 | 1 | 3 | 5 | 12 | 18 | |||
0 | 0 | Stringers (x-direction) | 0.0450 | 0.0447 | 0.0440 | 0.0433 | 0.0412 | 0.0397 |
Rings (y-direction) | 0.0450 | 0.0446 | 0.0437 | 0.0428 | 0.0399 | 0.0378 | ||
Orthogonal stiffeners | 0.0450 | 0.0442 | 0.0427 | 0.0413 | 0.0370 | 0.0341 | ||
100 | 10 | Stringers (x-direction) | 0.0252 | 0.0251 | 0.0249 | 0.0247 | 0.0241 | 0.0235 |
Rings (y-direction) | 0.0252 | 0.0251 | 0.0249 | 0.0246 | 0.0237 | 0.0230 | ||
Orthogonal stiffeners | 0.0252 | 0.0250 | 0.0246 | 0.0242 | 0.0227 | 0.0217 |
Center deflection (m) of cross-ply laminated composite spherical shell panels resting on elastic foundation
K_{0} | J_{0} | Stiffener type | Number of stiffeners | |||||
---|---|---|---|---|---|---|---|---|
0 | 1 | 3 | 5 | 12 | 18 | |||
0 | 0 | Stringers (x-direction) | 0.01325 | 0.01327 | 0.01330 | 0.01331 | 0.01329 | 0.01325 |
Rings (y-direction) | 0.01325 | 0.01327 | 0.01329 | 0.01329 | 0.01318 | 0.01310 | ||
Orthogonal stiffeners | 0.01325 | 0.01330 | 0.01335 | 0.01336 | 0.01326 | 0.01316 | ||
100 | 10 | Stringers (x-direction) | 0.01065 | 0.01067 | 0.01069 | 0.01070 | 0.01071 | 0.01069 |
Rings (y-direction) | 0.01065 | 0.01067 | 0.01069 | 0.01070 | 0.01065 | 0.01060 | ||
Orthogonal stiffeners | 0.01065 | 0.01069 | 0.01074 | 0.01076 | 0.01073 | 0.01068 |
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 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
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
For the vibration analysis, when the stiffener’s height-to-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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