A scrutiny of the literature reveals that the free vibration characteristics of stiffened composite hypar shell with cutout are missing. So a generalized finite element formulation for the stiffened hyperbolic paraboloidal shells bounded by straight edges (commonly called as hypar shells) is attempted using an eight-noded curved quadratic isoparametric element for shell with a three-noded beam element for stiffener. Numerical problems of earlier investigators are solved as benchmark problems to validate the approach. A number of problems are further solved by varying the size of the cutouts and their positions with respect to the shell centre for different edge constraints. The results are presented in the form of figures and tables. The results are further analysed to suggest guidelines to select optimum size and position of the cutout with respect to shell centre considering the different practical constraints. 1. Introduction The advent of laminated composites in civil engineering applications has provided a new impetus to the researchers to explore the different aspects of composite structural elements including different forms of shells. A skewed hypar shell is aesthetically appealing and being doubly ruled is easy to cast. Moreover, this configuration can allow entry of north light and due to this advantage it finds use as roofing units in practical civil engineering situations. Shell structures that are normally thin-walled exhibit improved performances with stiffeners, particularly when the shell surface is provided with cutouts. Cutout is sometimes necessary in roof structure for the passage of light, to provide accessibility to other parts of the structure, for venting and also sometimes for alteration of resonant frequency. Basic knowledge of the free vibration characteristics of stiffened composite skewed hypar shell with cutout is essential for using these forms confidently. As early as in 1982, Reddy [1] carried out the finite element analysis of composite plate with cutout and presented the effects of parametric variations on linear and nonlinear frequencies. Later in 1989, Malhotra et al. [2] presented the effect of fibre orientation and size of cutout on natural frequency on orthotropic square plates with square cutouts for different boundary conditions using the Rayleigh-Ritz method. Sivasubramonian et al. [3] reported free vibration of curved panels with cutout. They analysed the effect of cutouts on the natural frequencies of plates with some classical boundary conditions. Later Sivakumar et al. [4], Rossi [5], Huang and Sakiyama
References
[1]
J. N. Reddy, “Large amplitude flexural vibration of layered composite plates with cutouts,” Journal of Sound and Vibration, vol. 83, no. 1, pp. 1–10, 1982.
[2]
S. K. Malhotra, N. Ganesan, and M. A. Veluswami, “Vibration of composite plates with cut-outs,” Journal of Aeronautical Society of India, vol. 41, pp. 61–64, 1989.
[3]
B. Sivasubramonian, A. M. Kulkarni, G. V. Rao, and A. Krishnan, “Free vibration of curved panels with cutouts,” Journal of Sound and Vibration, vol. 200, no. 2, pp. 227–234, 1997.
[4]
K. Sivakumar, N. G. R. Iyengar, and K. Deb, “Free vibration of laminated composite plates with cutout,” Journal of Sound and Vibration, vol. 221, no. 3, pp. 443–470, 1999.
[5]
R. E. Rossi, “Transverse vibrations of thin, orthotropic rectangular plates with rectangular cutouts with fixed boundaries,” Journal of Sound and Vibration, vol. 221, no. 4, pp. 733–736, 1999.
[6]
M. Huang and T. Sakiyama, “Free vibration analysis of rectangular plates with variously-shaped holes,” Journal of Sound and Vibration, vol. 226, no. 4, pp. 769–786, 1999.
[7]
S. S. Hota and P. Padhi, “Vibration of plates with arbitrary shapes of cutouts,” Journal of Sound and Vibration, vol. 302, no. 4-5, pp. 1030–1036, 2007.
[8]
D. Chakravorty, P. K. Sinha, and J. N. Bandyopadhyay, “Applications of FEM on free and forced vibration of laminated shells,” Journal of Engineering Mechanics, vol. 124, no. 1, pp. 1–8, 1998.
[9]
B. Sivasubramonian, G. V. Rao, and A. Krishnan, “Free vibration of longitudinally stiffened curved panels with cutout,” Journal of Sound and Vibration, vol. 226, no. 1, pp. 41–55, 1999.
[10]
S. S. Hota and D. Chakravorty, “Free vibration of stiffened conoidal shell roofs with cutouts,” Journal of Vibration and Control, vol. 13, no. 3, pp. 221–240, 2007.
[11]
N. Nanda and J. N. Bandyopadhyay, “Nonlinear free vibration analysis of laminated composite cylindrical shells with cutouts,” Journal of Reinforced Plastics and Composites, vol. 26, no. 14, pp. 1413–1427, 2007.
[12]
S. Sahoo, “Free vibration of laminated composite hypar shell roofs with cutouts,” Advances in Acoustics and Vibrations, vol. 2011, Article ID 403684, 13 pages, 2011.
[13]
S. Sahoo and D. Chakravorty, “Finite element vibration characteristics of composite hypar shallow shells with various edge supports,” Journal of Vibration and Control, vol. 11, no. 10, pp. 1291–1309, 2005.
[14]
A. Mukherjee and M. Mukhopadhyay, “Finite element free vibration of eccentrically stiffened plates,” Computers and Structures, vol. 30, no. 6, pp. 1303–1317, 1988.
[15]
A. N. Nayak and J. N. Bandyopadhyay, “On the free vibration of stiffened shallow shells,” Journal of Sound and Vibration, vol. 255, no. 2, pp. 357–382, 2003.
[16]
A. Dey, J. N. Bandyopadhyay, and P. K. Sinha, “Finite element analysis of laminated composite paraboloid of revolution shells,” Computers and Structures, vol. 44, no. 3, pp. 675–682, 1992.
