This paper deals with a numerical method for the free vibrational analysis of laminated deep shells. The strain-displacement relations are obtained for a general laminated shell geometry described by orthogonal curvilinear coordinates. Parabolic variation of transverse shear stresses along the thickness and the effects of rotary inertia are included in the formulation. The displacement fields are represented by Bezier patches. The shape and size of these patches are controlled by certain arbitrary points called control points. Owing to the special characteristics of these control points, the treatment of displacements, slopes, curvatures, etc., at a particular edge becomes very simple. Hence, the enforcement of boundary conditions along the edges is straightforward. Ritz-type solution procedure is used for the eigen-analysis of the shell structure. Numerical examples involving laminated spherical, conical, and cylindrical shells are investigated in detail. Such shell geometries usually have planes of symmetry; hence, only one-quarter of the shell is analyzed in this study. Good convergence of the natural frequencies is observed by using eighth-order Bezier functions. The results are compared with the existing sources in the literature. The influences of material strength and number of layers on the natural frequencies are also examined.

1.
Bert
C. W.
,
1979
, “
Recent Research in Composite Sandwich Plate Dynamics
,”
Shock and Vibration Digest
, Vol.
11
, pp.
13
23
.
2.
Bert
C. W.
, and
Francis
P. H.
,
1974
, “
Composite Material Mechanics: Structural Mechanics
,”
American Institute of Aeronautics and Astronautics Journal
, Vol.
12
, pp.
1173
1186
.
3.
Bezier, P., 1986, The Mathematical Basis of the UNISURF CAD System, Butterworths, London, U.K.
4.
Habip
L. M.
,
1965
, “
A Survey of Modern Developments in the Analysis of Sandwich Structures
,”
Applied Mechanics Reviews
, Vol.
18
, pp.
93
98
.
5.
Kapania
R. K.
,
1989
, “
A Review on the Analysis of Laminated Shells
,”
ASME JOURNAL OF PRESSURE VESSEL TECHNOLOGY
, Vol.
111
, pp.
88
96
.
6.
Kumar
V.
, and
Singh
A. V.
,
1993
, “
Vibration Analysis of Non-Circular Cylindrical Shells Using Bezier Functions
,”
Journal of Sound and Vibration
, Vol.
161
, No.
2
, pp.
333
354
.
7.
Kumar
V.
, and
Singh
A. V.
,
1995
, “
Vibrations of Composite Noncircular Cylindrical Shells
,”
ASME Journal of Vibration and Acoustics
, Vol.
117
, pp.
470
476
.
8.
Levinson
M.
,
1980
, “
An Accurate Simple Theory of the Statics and Dynamics of Elastic Plates
,”
Mechanics Research Communications
, Vol.
7
, pp.
343
343
.
9.
Logan
D. L.
, and
Widera
G. E. O.
,
1989
, “
Membrane Theory for Anisotropic Laminated Shells of Revolution
,”
ASME JOURNAL OF PRESSURE VESSEL TECHNOLOGY
, Vol.
111
, pp.
130
135
.
10.
Murthy, M. V. V., 1981, “An Improved Transverse Shear Deformation Theory for Anisotropic Plates,” NASA-TP-1903.
11.
Noor
A. K.
, and
Burton
W. S.
,
1990
, “
Assessment of Computational Models for Multilayered Composite Shells
,”
Applied Mechanics Reviews
, Vol.
43
, No.
4
, pp.
67
97
.
12.
Qatu
M. S.
, and
Leissa
A. W.
,
1991
, “
Natural Frequencies for Cantilevered Doubly-Curved Laminated Composite Shallow Shells
,”
Composite Structures
, Vol.
17
, No.
3
, pp.
227
256
.
13.
Reddy
J. N.
,
1984
a, “
A Simple Higher Order Theory for Laminated Composite Plates
,”
ASME Journal of Applied Mechanics
, Vol.
51
, pp.
745
745
.
14.
Reddy
J. N.
,
1984
b, “
Exact Solutions of Moderately Thick Laminated Shells
,”
Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers
, Vol.
110
, pp.
794
809
.
15.
Reddy
J. N.
, and
Liu
C. F.
,
1985
, “
A Higher Order Shear Deformation Theory of Laminated Elastic Shells
,”
International Journal of Engineering Science
, Vol.
23
, No.
3
, pp.
319
330
.
16.
Saada, A. S., 1974, Elasticity: Theory and Applications, Pergamon Press Inc.
17.
Singh
A. V.
,
1991
, “
On Vibrations of Shells of Revolution Using Bezier Polynomials
,”
ASME JOURNAL OF PRESSURE VESSEL TECHNOLOGY
, Vol.
113
, pp.
579
584
.
18.
Siu
C. C.
, and
Bert
C. W.
,
1970
, “
Free Vibrational Analysis of Sandwich Conical Shells with Free Edges
,”
Journal of the Acoustical Society of America
, Vol.
47
, pp.
943
945
.
19.
Soldatos
K. P.
,
1984
, “
A Flu¨gge-Type Theory for the Analysis of Anisotropic Laminated Non-Circular Cylindrical Shells
,”
International Journal of Solids and Structures
, Vol.
20
, No.
2
, pp.
107
120
.
20.
Soldatos, K. P., 1986, “Stability and Vibration of Thickness Shear Deformable Cross-ply Laminated Non-Circular Cylindrical Shells, Pressure Vessels and Piping, Vol. 115.
21.
Soldatos, K. P., 1987, “Buckling of Axially Compressed Antisymmetric Angle Ply Laminated Circular Cylindrical Panels According to a Refined Shear Deformable Shell Theory,” Pressure Vessels and Piping, Vol. 116.
22.
Soldatos
K. P.
,
1991
, “
A Refined Laminated Plate and Shell Theory with Applications
,”
Journal of Sound and Vibration
, Vol.
144
, pp.
109
129
.
23.
Tong
L.
,
1993
, “
Free Vibration of Composite Laminated Conical Shells
,”
International Journal of Mechanical Sciences
, Vol.
35
, No.
1
, pp.
47
61
.
24.
Vinson, J. R., and Sierakowski, R. L., 1986, The Behavior of Structures Composed of Composite Materials, Martinus Nijhoff Publishers.
25.
Weaver, W., and Johnston, P. R., 1984, Finite Elements for Structural Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ.
26.
Widera
G. E. O.
, and
Fan
H.
,
1988
, “
On the Derivation of a Refined Theory for Nonhomogenous Anisotropic Shells of Revolution
,”
ASME JOURNAL OF PRESSURE VESSEL TECHNOLOGY
, Vol.
110
, pp.
102
105
.
27.
Wilkins
D. J.
,
Bert
C. W.
, and
Egle
D. M.
,
1970
, “
Free Vibrations of Orthotropic Sandwich Conical Shells with Various Boundary Conditions
,”
Journal of Sound and Vibration
, Vol.
13
, pp.
211
228
.
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