# Dynamic characteristics of laminated angle-ply
noncircular cylindrical shells

Hitendra G Patel, Badri Prasad Patel, Manickam Ganapathi, Maurice

Touratier

► To cite this version:

Hitendra G Patel, Badri Prasad Patel, Manickam Ganapathi, Maurice Touratier. Dynamic characteristics of laminated angle-ply noncircular cylindrical shells. 6th International Conference on Computational Structures Technology 2002, 2002, Prague, Czech Republic. 10.4203/ccp.75.89. hal-00019482

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Submitted on 11 May 2023

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# Dynamic Characteristics of Laminated Angle-Ply Noncircular Cylindrical Shells

H.G. Patel†, B.P. Patel†, M. Ganapathi† and M. Touratier‡

**Keywords:** laminated shell, angle-ply, free vibration, non-circular, modes, finite element, elliptical cross section.

The study of response of shells of revolution under static and dynamic loading situations has received considerable attention in the literature compared to the analysis that deals with shells of noncircular cross section. This is possibly due to the difficulty introduced in the governing equations because of varying nature of the radius of curvature of the noncircular cylinders with the circumferential co-ordinate. Studies of such shells made of advanced composite materials that are preferred in the design of lightweight and efficient shell structures are further limited because of the increased complexity due to the inherent directional properties of these materials. Although many structural components of noncircular cross-section can be adequately treated as equivalent shells of revolution, this approach may be unacceptable for shells with significantly noncircular curvature.

The extensive studies available in the literature on dynamic analysis of circular cylindrical shells/panels have been reviewed in the work of Leissa [1], Noor [2] and Soldatos [3]. Recently, the limited progress made in understanding the behavior of the non-circular cylinders has been reviewed by Soldatos [4]. It may be concluded that few contributions are available concerning with the free vibration analysis of anisotropic laminated non-circular cylindrical shells compared to those of isotropic case and they are all concerned with cross-ply case. Some of the available works on dynamics of cross-ply noncircular shells are of Soldatos and Tzivanidis [5], and Soldatos [6]. All these investigations have been carried out by the various analytical methods. However, the analysis of angle-ply noncircular shells appears to be scarce in the literature because of increased complexity due to various couplings arising from anisotropicity.

The present paper deals with the study of the dynamic behavior of anisotropic laminated angle-ply noncircular cylindrical shells based on finite element method. The element employed here is a $C^0$ continuous, eight-noded serendipity quadrilateral shear flexible shell element with five nodal degrees of freedom ($u_0, v_0, w_0, \theta_x, \theta_y$) developed based on field consistency approach. A detailed study is carried out to highlight the effects of lap-up and ply-angle on the natural frequencies pertaining to different types of modes of vibrations of layered elliptical cylinders. The radius of curvature of the middle surface of the elliptical cross-section is described as $R = (b^2/R_o)(1 + \mu_o \cos 2\theta)^{-3/2}$ where $R_o = [(a^2 + b^2)/2]^{1/2}$, $\mu_o = (a^2 - b^2)/(a^2 + b^2)$; and $\theta$ denotes an angle between the

<footnote>†Institute of Armament Technology, Girinagar, Pune, India</footnote>

<footnote>‡LMSP-UMR CNRS-ENSAM-ESEM, Paris, France</footnote>
tangent at the origin of y (circumferential co-ordinate) and the one at any point on the center-line. The parameters a and b are the lengths of the semi-major and -minor axes, respectively.

Results obtained for the first few natural frequencies of two- and eight-layered angle-ply $(30°/30° \& (30°/30°)_4)$ laminated thin elliptical shells ($L/R_o = 0.5$; $R_o/h = 100$) for simply supported boundary conditions are presented in the full paper. It is observed from the results that the 15° ply-angle case results in lowest frequency values pertaining to all asymmetric modes of vibration and the next higher one is mostly yielded by 30° case. The maximum values occur either for the ply-angle 45° or for 60° case, depending on the frequency order and type of asymmetric vibration mode. The detailed studies show that the increase in eccentricity value affects the frequencies quantitatively but the nature of variation of the frequency parameter does not change.

## References

[1] Leissa, A.W., "Vibration of shells," NASA SP-288, 1973.

[2] Noor, A.K., "Bibliography of monographs and surveys on shells," Applied Mechanics Review, 43, 223-234, 1990.

[3] Soldatos, K.P., "Review of three-dimensional dynamic analysis of circular cylinders and cylindrical shells," Applied Mechanics Review, 47, 501-516, 1994.

[4] Soldatos, K.P., "Mechanics of cylindrical shells with non-circular cross section: A survey," Applied Mechanics Review, 52, 237-274, 1999.

[5] Soldatos, K.P., Tzivanidis, G.J., "Buckling and vibration of cross-ply laminated non-circular cylindrical shells," Journal of Sound and Vibration, 82, 425-434, 1982.

[6] Soldatos, K.P., "A Flugge-type theory for the analysis of anisotropic laminated non-circular cylindrical shells," Int. J. Solids Structures, 20, 107-120, 1984.