Vibration And Buckling Of Circular Cylindrical Panels By Higher Order Shear Deformation Theory

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Vibration And Buckling Of Circular Cylindrical Panels By Higher Order Shear Deformation Theory

Vibration And Buckling Of Circular Cylindrical Panels By Higher Order Shear Deformation Theory

Mr. Shrikant M. Harle, Dr. A. V. Asha

ABSTRACT:

The present study deals with the stability and vibration analysis of cross-ply cylindrical panels by a higher-order but simple shear deformation theory as suggested by Reddy and Liu. The theory is based on a displacement field in which the displacements of the middle surface are expanded as cubic functions of the thickness coordinate and the transverse displacement is assumed to be constant through the thickness. This displacement field leads to the parabolic distribution of the transverse shear stresses (and zero transverse normal strain) and therefore no shear correction factors are used. The analysis is also based on the assumption that the thickness to radius ratio of shell panel is small compared to unity and hence negligible. The eigenvalues, and hence, the frequency parameters and buckling parameter are calculated by using a standard computer program. To check the derivation and computer program, the frequencies for homogenous, isotropic circular cylindrical panels are calculated which compare very well with earlier available results.

Keywords: Stability, vibration, buckling, shell panels, higher order theory

  1. INTRODUCTION:

    An increasing number of structural designs, especially in the aerospace, automobiles and petrochemicals industries, are extensively utilizing fiber composite laminated plates and shell panels as structural elements. Structural components, including shell panels, are increasingly being fabricated as laminates, each lamina (or layer) consisting of parallel fibers (e.g. glass, boron, graphite) embedded in a matrix material (e.g. epoxy resin). The laminated orthotropic shell panel belongs to the composite shell panels category. It may be of an arbitrary number of bonded layers, each of which possesses different elastic properties, thickness, etc. The study of the static and dynamic behavior of such shell panels is very important in structures like pressure vessels, aircrafts, storage tanks and manned and unmanned spacecrafts and submersibles.

    A higher order shear deformation theory of elastic shell was developed for shell laminated of orthotropic layers by Reddy and Liu [23]. The theory was a modification of the Sanders theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the shell and tangential stress-free boundary conditions on the boundary surfaces of the shell. The Navier-type exact solutions for bending and natural vibration were presented for cylindrical and spherical shell under simply supported boundary conditions.

    A higher order shear deformation theory was used by Reddy and Phan [22] to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions for simply supported plates were obtained and the results were compared with the exact solutions of three dimensional elastic theories, the first order shear deformation theory and the classical plate theory.

    A mixed shear flexible finite element, with relaxed continuity, was developed for the geometrically linear and non-linear analysis of layered anisotropic plates by Putcha and Reddy [17]. The element formulation was based on the refined higher order theory which satisfied the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients. The mixed finite element developed consists of eleven degrees of freedom per node which include three displacements, two rotations and six moment resultants. The element was evaluated for its accuracy in the analysis of the stability and vibration of anisotropic rectangular plates with different lamination schemes and boundary conditions.

    The present study is carried out to determine the natural frequency of vibration and stability analysis of laminated cross-ply cylindrical panels that are simply supported. The formulation has been done using higher order shear deformation theory as proposed by Reddy and Liu. The transverse displacement is assumed to be constant through the thickness. The thickness coordinate multiplied by the curvature is assumed to be small in comparison to unity and hence negligible.

    The governing equations have been developed. These equations are then reduced to the equations of motion for cylindrical panel and the Navier solution has been obtained for cross-ply laminated composite cylindrical panels. The resulting equations are suitably nondimensionalised. The eigen value problem is to be solved to obtain the free vibration frequencies.

  2. THEORY AND FORMULATION:

    The shell panel under consideration is composed of a finite number of orthotropic layers of uniform thickness. Let n denote the number of layers in the shell panel and hk and hk+1 be the top and bottom -coordinates of the kth lamina. The following displacement field is assumed:

    , , , = 1 + 1 + 1 + 21 + 31 ,

    , , , = 1 + 2 + 2 + 2 2 + 3 2 , , , , = (1) Where t is the time, ( , , ) are the displacements along the (, , ) coordinates, (u, v, w) are the displacements of a point on the middle surface and 1 and 2 are the rotations at =0 of normal to the mid-surface with respect to and axes, respectively. The displacement fields in equations are so chosen that the transverse shear strains will be quadratic functions of the thickness coordinate, , and the transverse normal strain will be zero.

    The functions and are determined using the condition that the transverse shear stresses, 4 and 5 vanish on the top and bottom surfaces of the shell panel:

    4 , , ± , = 0, 5 , , ± , = 0 (2)

    2 2

    For shell panels laminated of orthotropic layers, the conditions are equivalent to the requirement that the corresponding strains be zero on these surfaces. The transverse shear strains of a shell panel with two principal radii of curvature are given by

    4 = + 1 1 , 5 = + 1 2 (3)

    1

    1

    Substituting , , in the above equations, and neglecting the term multiplied by 1 and 2

    we have,

    4 = 1 + 21 + 321 + 1 , 5 = 2 + 22 + 322 + 1 , 4 , , ± , = 0,

    1 2 2

    5

    5

    , , ± , = 0 (4)

    2

    Setting 1 and 2 to zero,

    1 = 4 (1 + 1 ), 2 = 4 (2 + 1 ) (5)

    3 2

    1

    3 2

    2

    Substituting equations (5) into equations (4),

    = 1 + 1 + 1 + 3 4

    1 1 , = 1 + 2 + 2 + 3 4

    (2 1 )

    3 2

    1

    32

    2

    (6)

    Substituting above equations into the strain-displacement relations referred to an orthogonal curvilinear coordinate system, we get

    1 = 0 + 0 + 3 2, 2 = 0 + 0 + 32 , 4 = 0 + 2 1, 5 = 0 + 21,

    1 1 1

    2 2 2

    4 4 5 5

    6 = 0 + 0 + 3 2 (7)

    6 6 6

    Where,

    0 = 1 + 1

    1 + , 0 = 1

    + 1 2 +

    1 1

    2

    1 1

    2 2

    1

    2 2

    0 = 1

    + 1 + 1

    1

    + 1 + 1

    1

    1 1

    1

    2

    2

    2

    2

    0 = 1

    + 2 + 1 2 + 2 + 1

    2

    2 2

    2

    1

    1

    2

    1

    2 = 4 1 1 + 1

    1

    +

    1 + 1

    1

    2

    2

    1 32 1 2

    2

    1 2

    2 = 4 1 2 + 1

    2

    + 1 + 1 2

    1

    1

    2 32 2

    1

    1

    2

    2

    0 = 2 + 1 , 0 = 1 + 1 , 1 = 4 2 + 1 , 1 = 4 (1 + 1 )

    4 2 5

    1

    1. 2

      2

    2. 2

    1

    0 = 1

    + 1

    1

    1 1

    2

    6 1

    2

    12

    21

    0 = 1 2 + 2 1 1 1 + 2 1 1 1 + 1 1 + 1 1 2 2 +

    6 1

    2

    2

    2

    1

    1 1 2 2 (8)

    1

    For generalized plane stress condition, the elastic moduli are related to the engineering

    constants as follows:

    11 = 1 , 12 = 1 .21 = 2 .12

    , 22 = 2 , 44 = 23 , 55 = 13

    112 21

    66 = 12 , 1 = 12

    112 21

    112 21

    112 21

    (9)

    2 21

    Following are the expressions for stress resultants and stress couples:

    = 0 + 0 + 2, = 0 + 0 + 2 , = 0 + 0 + 2

    1 = 4 0 + 4 1, 2 = 5 0 + 5 1, 1 = 4 0 + 4 1, 2 = 5 0 + 5 1 (10)

    where , , etc. are the laminate stiffnesses

    , , , , , = +1 1, , 2 , 3, 4 , 6

    =1

    For I, j = 1, 2, 4, 5, 6, hk and hk+1 are the distances measured from the middle surface of the shell panel

    Following are the equations of equilibrium obtained:

    : 2 1 + 1 6 2 2 + 6 1 + 1 2 1 + 6 1 2 2 + 1 6 +

    112 = 1 + 2 1 3

    1

    1 2

    1 2

    1

    = 2 6 + 1 2 1 1 + 6 2 + 2 2 6 1 1 + 2 1 + 2 6 +

    212 = 1 + 22 3 1 12

    2

    2 1 + 1 2 1 211 1 222 + 4

    1 2 1 +

    =

    1 1 2 +

    3p

    1

    2

    1 2 6 +

    1 1 6

    1 1

    2 2 +

    6 1 +

    6 2

    2

    1

    2

    1

    1

    2

    4 2 1 + 1 2 +

    +

    +

    =

    +

    2 +

    2

    1

    2

    1 2

    1 1 2

    3

    5

    2 1 + 3 1 + 5

    (1 2 ) 16 7

    2 +

    1

    9 4

    1

    2

    1 = 1 2 + 6 1 2 2 + 6 1 4 2 1 + 1 6 2 2 + 6 1

    3p

    2

    2

    112 + 4 112 = 2 + 4 1 5

    1

    1 2

    1 2

    1

    2 = 6 2 + 2 1 1 1 + 6 2 4 2 6 + 1 2 1 1 + 6 2

    3 2

    2

    2

    212 + 4 212 = 2 + 4 2 5

    1 (11)

    1 2

    1 2

    2

    1, 2, can be defined as the transverse loads

    The inertias 1

    (i=1, 2, 3, 4, 5) are defined by the equations,

    1

    1

    1

    = 1 + 212, 1 = 1 + 22 2, 2

    = 2 + 13 4

    4 41 5 , 2 + 2 3 4

    4 42 5

    3 2

    3 2

    3 2

    3 2

    3

    = 4 4 + 41 5,3 = 4

    4 + 41 5, 4

    = 3 8

    5 + 16 7, 5

    = 4 5 + 16 7 (12)

    3 2

    3 2

    3 2

    3 2

    3 2

    9 2

    3 2

    9 4

    =1

    =1

    Where 1, 2, 3 , 4, 5, 7 =

    +1 1, , 2, 3 , 4 , 6

    Where (k) is the density of the material of the kth layer.

    Considering the line integrals while integrating by parts the displacement gradients in the Hamilton principle, the boundary conditions at an edge =constant, are obtained as follows:

    [(N +k M ),u];[(N +k M ),v];[{Q + 4

    1 6 4 + 4

    1 2 1 + 1 6

    2 +

    1 1 1

    6 2 6

    1 3 2 2

    2 1

    3 2 1 2

    2

    6 1 3 + 5 1 16 7 1 }, w]; [(M1- 4

    1), 1]; [(M6- 4

    6), 2]; [P1, 1 ] (13)

    9 2 1

    3 2

    3 2

    1

    For the cylindrical shell panel configuration shown in the figure, the coordinates are given by = x/R, = , the Lames parameters 1 = 2 = and the principal curvatures

    1 2

    1 2

    ( ) = 0 ( ) = 1 , where R is the radius of the mid-surface of the cylindrical

    shell panel. Then the equations of motion in terms of the stress resultants and stress couples are obtained from equations.

    The strain-displacement relations of equations reduce to

    0 = 1 , 0 = 1 + , 0 = 1 1 , 0 = 1 1 + 2

    1

    2

    1

    2

    2 4 1

    1 1

    1 2

    2 4 1

    1 2

    1 2 0

    1

    1

    = 32

    + 2 , 2

    = 32

    + 2 , 4 = 2 + ,

    0 =

    + 1 , 1 = 4

    + 1 , 1 = 4

    + 1 , 0 = 1 +

    5 1

    4 2 2

    5 2 1

    6

    0 = 1 1 + 2 + 1 , 2 = 4 1 2 + 2 2

    + 1 (14)

    6 6 3 2

    For cylindrical shell panel configuration, the equations of equilibrium take the form

    1 + 6 + 1 = 1 + 2 1 3

    1

    6 + 2 + 1 6 + 1 2 + 2 = 1 + 22 3 1

    1 + 2 +

    +

    + 4

    1 2 1 + 2 2 + 2 2 6 4 1 + 2 +

    1

    2

    2 3 2

    2

    2

    2

    =

    +

    1 +

    +

    2 16 7 1 2 + 2 +

    3

    5

    3

    5

    9 4

    2

    2 1

    1 + 6 4

    1 + 6 1 + 4 1 = 2 + 4 1 5

    1

    3p

    2

    6 + 2 4

    6 + 2 2 + 4 2 = 2 + 4 2 5

    1 (15)

    3 2

    2

    Since the solution for the equations of motion is done by using the Navier solution, therefore such a solution exists only for a specially orthotropic shell panel for which the following laminate stiffnesses are zero;

    6 = 6 = 6 = 6 = 0, 45 = 45 = 0

    The equations of motion in terms of the displacement hence reduce to

    2 +

    2

    +

    + 1

    +

    2 +

    4 11 3 4

    +

    11 2

    66 2

    12 66

    12

    66

    12

    3 2 3

    3 2 12

    2

    3

    +

    4 11 2 1 +

    4 66 2 1 +

    +

    4

    +

    66 2

    11 3 2

    2

    66 3 2

    2

    12 66

    3 2 12

    2 2 = 2 + 3

    66

    1 2 1

    R

    1

    +

    + 1

    +

    2

    + 1

    + 266 + 66 2 + 1

    + 222 +

    12

    66 2 12

    66

    66

    2

    2

    22

    22 2 + 1

    + 22 4

    2

    +

    + 266 + 12 3

    4 +

    2

    2

    22

    3 2 2 66

    12

    2

    3 2 2 22

    22 3 + 1

    +

    + 66 + 12 4

    +

    + 66 + 12 2 1 + 1 +

    3

    66

    12

    3 2 66

    12

    66

    66 4

    + 66 2 2 + 1

    + 22 4

    + 22 2 2 = 2

    3 +

    22

    3 2 66

    2

    22

    3 2 22

    2

    1

    21 + 4

    3 + 2

    +

    3

    + 1

    + 22 + 4

    +

    3 2 2

    11 3

    66 12

    2

    22

    3 2 2 22

    22 3 + 2

    +

    + 12 + 266 3

    + 22 + 55 8 55 + 8

    12 +

    3

    66 12

    2

    2

    3 2 2

    16

    2 + 44 8 44 + 8

    22 + 16

    44 2 16 11 4 16 22 4

    4 55

    2

    2

    3 2 2

    4

    2

    9 4 3

    4

    9 4 3

    4

    32

    2

    +

    4

    +

    8

    12 + 412 + 16

    1 + 4

    12

    9 4 3 66

    12 2 2

    55 2 55

    3 2

    4 55

    3 2 2

    16 12 + 8 66 32 66 3 1

    + 8

    22 + 4 22 + 1644 2 + 4

    12

    9 4 2 3 2 2 9 4 2 2

    44 2 44

    3 2

    4

    3 2 2

    16 12 + 8 66 32 66 3 2 + 4

    22 16 22 3 2 = 2 +

    +

    9 4 2

    3 2 2

    9 4 2

    2

    3 2 2

    9 4 2

    3 1

    3

    3

    16 7 2 + 2 + 1 +

    2

    9 4 2 2 5

    5

    11 4 11 2 + 66 4 66 2 + 12 + 12 + 66 + 66 4

    12 + 12 + 66 +

    3 2 2 3 2

    2

    2

    2

    3 2 2

    66 2 + 12 4

    12

    + 8

    16

    + 4

    11 4

    11 3 +

    2

    3 2

    55 2 55

    4 55

    3 2

    2

    3 2 2

    3

    4

    12 + 266 4

    12 8

    66 3

    +

    + 8

    16

    +

    3 2

    2

    2

    3 2 2

    3 2 2

    2

    55 2 55 4

    55 1

    11 4

    211 4

    11 2 1 + 66 4

    266 4

    66 2 1 + 12 4

    212 + 266

    3 2

    3 2

    2

    3 2

    3 2

    2

    3 2

    4 12 4 66 + 66 2 2 = 5 +

    3 2

    3 2

    2 4 1

    66 + 12 4

    66 + 12 2

    + 66 + 66 4

    66 + 66 2 + 22 + 22

    3 2

    2

    3 2

    2

    2

    2

    4 22 + 22 2 + 22 4

    22

    + 8

    16

    + 4

    266 + 12

    3 2

    2

    2

    3 2

    44 2 44

    4 44

    3 2

    2

    2

    4 2

    +

    3

    + 4

    22 4

    22 3 + 66 + 12 4

    266 + 212

    3 2 2 66

    12 2

    3 2

    2

    3 2 2

    3

    3 2

    4 66 + 12 2 1 +

    + 8

    16

    +

    3 2

    44 2 44

    4 44 2

    66 4

    266 4

    66 2 2 + 22 4

    22 4

    222 4

    22 2 2 =

    5

    +

    3 2

    3 2

    2

    3 2

    3 2

    3 2

    2

    2

    4 2 (16)

    The boundary conditions for a simply supported cylindrical shell panel are given by N1= 0, v = 0, w = 0, M1= 0, 2= 0 At = 0 and = L/R

    Following the Navier solution procedure, the following form which satisfies the boundary condition in equations (2.9.1) is assumed for the stability analysis

    = cos sin , = sin cos , = sin cos

    1 = 1 cos sin , 2 = 2 sin cos (17)

    Following the Navier solution procedure, the following form which satisfies the boundary condition in above equations is assumed for the free vibration analysis

    = cos sin , = sin cos , 2 = 2 sin cos

    = sin cos , 1 = 1 cos sin (18)

    For convenience, the elements of the above matrices are suitably non-dimensionalised as follows:

    = , = , = , = , = , = , 1 = 1, 2 = 2 , = 2,

    1

    1

    = 22 , = 23 , = 24 , = 25 , = 27, 1 = (1)

    2 = (1)2, 3 = (1)3 , 4 = (1)4 , 5 = (1)5 , 7 = (1)7

    2 3 4 5 7

    ( , , .are the non-dimensionalised quantities)

    1 2 3

    1

    = 1(1), = + 2 1 = 1(1), 2 = 4 1 2 = 2(1)2

    1 1 2 2 3 4

    = [ + 4 4 ](1)2 = 2 (1)2, 3

    = 4 1 2 = 3(1)2

    2 2 3

    3 4 3 5 3 4

    = 4 + 4 (1)2 = 3 (1)2 , 4

    = [ 8 + 16 ](1)3 = 4(1)3

    3 3 4

    3 5

    3 3 5 9 7

    5

    = [4 16 ](1)3 = 5(1)3 (19)

    3 5 9 7

  3. NUMERICAL RESULTS

    1

    1

    In the case of vibration analysis, 2 is the eigenvalue for free vibration and n

    for the buckling load and {X} is the eigenvector. For convenience the above equation is non- dimensionalised. Then if the equation is premultiplied by [ ]1, one obtains the following standard eigenvalue problem,

    = 2{ } , = 1

    { } , 2 = 22 1

    22

    , 1

    = 1

    22 2

    , = [ ]1[ ]

    In table 1, a two-layer cross-ply cylindrical shell panel has been analysed. The frequency parameter is 2 = 20 Ls 2/A11 where Ls = 2R. The results given by Dong and Tsos theory and present theory are in very close agreement for all cases.

    Table 1: Comparison of frequency parameter of a two layer cross-ply cylindrical shell panel (E1/E2 = 40; G12/E2 = 0.5; h/R = 0.01; G23/E2 = 0.6; 12 = 0.25; G13 = G12; =1)

    L/R

    n

    Dong & Tso

    Present Theory

    1.0

    1

    2.106

    2.106

    2

    1.344

    1.344

    3

    0.9587

    0.9587

    4

    0.7493

    0.7493

    5

    0.6419

    0.6420

    6

    0.6130

    0.6130

    2.0

    1

    1.073

    1.073

    2

    0.6710

    0.6710

    3

    0.4710

    0.4710

    4

    0.3773

    0.3773

    5

    0.3629

    0.3630

    6

    0.4160

    0.4162

    5.0

    1

    0.4212

    0.4212

    2

    0.2470

    0.2470

    3

    0.1733

    0.1734

    4

    0.1818

    0.1819

    5

    0.2516

    0.2517

    6

    0.3567

    0.3569

    The dimensionless natural frequencies = 2 0

    2 3

    for various values of R/h ratios for two-

    layered cylindrical panels having aspect ratio S/L =1 are given in table 2. In all cases, the results given by Dong and Tsos theory and present theory are quite agreeable.

    Table 2 Comparison of frequency parameter of a two-layered circular cylindrical panel

    E1/E2=40.0, 12=0.25, G12/E2=1, G23/E2=0.6, G13=G12

    (radians)

    R/h

    Dong & Tso

    Present Theory

    0.16

    312.2

    14.04

    14.04

    0.32

    156.25

    19.14

    19.14

    0.16

    62.5

    11.16

    11.24

    0.32

    31.25

    11.32

    11.39

    The variation of non-dimensional frequency with various parameters is now studied. The material properties of the first layer are given in Table 3.

    Table 3: Material Properties (first layer) of laminated cylindrical panels

    E1

    (x 106psi)

    E2

    (x 106psi)

    12

    G12

    (x 106psi)

    G23

    (x 106psi)

    G13

    (x 106psi)

    25.0

    1.0

    0.25

    0.5

    0.2

    0.5

    1.0

    Fig. 1 Section Properties for different layer of panels

    In Table 4, the values of non-dimensional frequencies for the various cross-sections for m =1 and various included angles are shown.

    Table 4: Non-dimensional frequency parameters of cross-ply circular cylindrical panels

    (radian)

    Number of Layers

    n =1

    n = 2

    n =3

    n = 4

    0.5235

    2

    0.47

    1.098

    2.031

    3.082

    3

    0.6203

    1.08

    1.75

    2.47

    5

    0.6209

    1.33

    2.269

    3.249

    6

    0.6011

    1.1

    1.822

    2.591

    10

    0.6802

    1.72

    3.53

    5.89

    1.048

    2

    0.205

    0.313

    0.623

    1.026

    3

    0.247

    0.237

    0.581

    0.905

    5

    0.23

    0.416

    0.803

    1.25

    6

    0.226

    0.336

    0.621

    0.971

    10

    0.26

    0.44

    0.925

    1.6

    1.57

    2

    0.172

    0.157

    0.291

    0.494

    3

    0.185

    0.166

    0.28

    0.454

    5

    0.178

    0.203

    0.395

    0.651

    6

    0.169

    0.167

    0.301

    0.493

    10

    0.212

    0.207

    0.41

    0.72

    2.094

    2

    0.157

    0.109

    0.168

    0.284

    3

    0.163

    0.115

    0.164

    0.267

    5

    0.16

    0.13

    0.23

    0.389

    6

    0.149

    0.112

    0.175

    0.291

    10

    0.195

    0.136

    0.229

    0.401

    2.618

    2

    0.145

    0.092

    0.11

    0.18

    3

    0.149

    0.0956

    0.111

    0.174

    5

    0.146

    0.101

    0.15

    0.254

    6

    0.136

    0.091

    0.117

    0.19

    10

    0.18

    0.116

    0.148

    0.253

    Table 6: Frequency parameters of cross-ply circular cylindrical panels for different material property

    Material property 1: E1/E2 = 25.0, 12 = 0.25, G12/E2 =1, G23/E2 = 0.2, G13 = G12 Material property 2: E1/E2 = 40.0, 12 = 0.25, G12/E2 =1, G23/E2 = 0.6, G13 = G12

    (radian)

    Number of Layers

    Material property 1

    Material property 2

    0.5235

    2

    0.473

    0.431

    3

    0.657

    0.619

    5

    0.604

    0.582

    6

    0.6029

    0.584

    10

    0.731

    0.7045

    Table 6: Frequency parameters of cross-ply circular cylindrical panels for different ratio of L/h

    (radian)

    Number of Layers

    L/h = 20

    L/h =100

    0.5235

    2

    0.268

    0.157

    3

    0.390

    0.169

    5

    0.357

    0.165

    6

    0.363

    0.155

    10

    0.418

    0.186

    Table 7: Lowest frequency parameters of various cross-ply circular cylindrical panels

    (m =1, n =1)

    0.212

    (radian)

    Number of Layers

    Frequency parameter

    0.5235

    2

    0.473

    3

    0.657

    5

    0.604

    6

    0.6029

    10

    0.6802

    1.048

    2

    0.205

    3

    0.247

    5

    0.230

    6

    0.226

    10

    0.2609

    1.57

    2

    0.172

    3

    0.185

    5

    0.178

    6

    0.169

    10

    2.094

    2

    0.157

    3

    0.163

    5

    0.16

    6

    0.149

    10

    0.195

    2.618

    2

    0.145

    3

    0.149

    5

    0.146

    6

    0.136

    10

    0.18

    The frequency envelopes of a two-layer, three-layer, four layer, five layer and ten-layer cross-ply circular cylindrical panel are plotted for m =1 and for various shallowness angles for a thickness to radius ratio of 0.05 and are shown in figure3.

    0.5

    0.4

    0.3

    0.2

    0.1

    0

    n=1

    n=2

    Frequency

    0.5

    0.4

    0.3

    0.2

    0.1

    0

    n=1

    n=2

    Frequency

    0 0.5 1 1.5 2 2.5 3 3.5

    phi (radian)

    0 0.5 1 1.5 2 2.5 3 3.5

    phi (radian)

    omega

    omega

    Fig. 2 Frequency envelope for two layer cross-ply circular cylindrical panel

    0.7

    0.6

    omega

    omega

    0.5

    0.4

    0.3

    0.2

    0.1

    0

    n=1

    n=2

    0 0.5 1 1.5 2 2.5 3 3.5

    phi (radians)

    Frequency

    Fig. 3 Frequency envelope for three layer cross-ply circular cylindrical panel

    0.7

    0.6

    0.5

    0.4

    n=1

    0.7

    0.6

    0.5

    0.4

    n=1

    0.2

    n=2

    0.2

    n=2

    0.1

    0

    0.1

    0

    0

    1

    2

    3

    4

    0

    1

    2

    3

    4

    phi (radians)

    phi (radians)

    0.3

    0.3

    Frequency

    Frequency

    omega

    omega

    omega

    omega

    Fig. 4 Frequency envelope for four layer cross-ply circular cylindrical panel

    0.7

    0.6

    0.5

    0.7

    0.6

    0.5

    n=1

    n=1

    0.4

    0.3

    Frequency

    0.4

    0.3

    Frequency

    0.2

    n=2

    0.2

    n=2

    0.1

    0

    0.1

    0

    0

    1

    2

    3

    4

    0

    1

    2

    3

    4

    phi (radians)

    phi (radians)

    Fig. 5 Frequency envelope for five layer cross-ply circular cylindrical panel

    0.8

    0.7

    0.6

    n=1

    0.8

    0.7

    0.6

    n=1

    0.5

    0.4

    0.3

    Frequency

    0.5

    0.4

    0.3

    Frequency

    0.2

    n=2

    0.2

    n=2

    0.1

    0

    0.1

    0

    0

    1

    2

    3

    4

    0

    1

    2

    3

    4

    phi (radians)

    phi (radians)

    omega

    omega

    omega

    omega

    Fig. 6 Frequency envelope for ten layer cross-ply circular cylindrical panel

    0.8

    0.7

    0.6

    0.5

    0.4

    0.3

    0.2

    0.1

    0

    n=1

    n=2

    two layer

    three layer four layer five layer ten layer

    0.8

    0.7

    0.6

    0.5

    0.4

    0.3

    0.2

    0.1

    0

    n=1

    n=2

    two layer

    three layer four layer five layer ten layer

    0 0.5 1 1.5 2 2.5 3 3.5

    phi (radians)

    0 0.5 1 1.5 2 2.5 3 3.5

    phi (radians)

    Fig. 7 Frequency envelope for cross-ply circular cylindrical panel

    The non-dimensionalised critical buckling loads for cross-ply laminated plates are presented in the form of table. The following table shows the comparison of results of present

    analysis and Reddy [22] for the non-dimensionalised buckling parameter of two layer, three layer and four layer of simply supported plates. The geometrical and material properties used are

    E1/E2 = 40; G12/E2 = 0.5; h/R = 0.01; G23/E2 = 0.6; 12 = 0.25; G13 = G12; =1

    Table 8: Comparison of buckling parameter of cross-ply laminated plates:

    a/h

    Two Layers 0°/90°

    Three Layers 0°/90°/0°

    Four Layers 0°/90°/90°/0°

    Reddy

    Present

    Error in

    %

    Reddy

    Present

    Error in

    %

    Reddy

    Present t

    Error in

    %

    50

    12.569

    12.494

    -0.6

    34.936

    35.873

    2.68

    35.100

    35.277

    0.5

    100

    12.614

    12.531

    -0.66

    35.602

    35.959

    1.0

    35.645

    35.369

    -0.77

    Error in Percentage = 100 * (Present Theory Reddy Theory)/ Reddy theory

    From the results presented in the above table, it is clear that results of present theory in good agreement with those obtained by Reddy theory.

    The non-dimensional buckling loads for different R/h values for cylindrical panels of L/h

    = 500 are tabulated in Table 9. It is seen that the values of the buckling load increase as the r/h ratio decreases.

    Table 9: Non-dimensional Buckling load of cross-ply circular cylindrical panels for l/h = 100

    R/h

    Two Layer 0°/90°

    Three Layer 0°/90°/0°

    Four Layer 0°/90°/90°/0°

    1000

    21.508

    44.13

    50.55

    500

    25.53

    59.15

    61.09

    400

    31.89

    73.94

    76.34

    200

    34.81

    147.84

    152.49

    In Table 10, the non-dimensional buckling loads for cylindrical panels with R/h = 500 and different material properties are shown. It is seen that as the E1/E2 ratio is increased, the buckling load increases.

    Table 10: Non-dimensional buckling load of cylindrical panels for different material properties

    Material property 1: E1/E2 = 25.0, 12 = 0.25, G12/E2 = 1, G23/E2 = 0.2, G13 = G12 Material property 2: E1/E2 = 40.0, 12 = 0.25, G12/E2 = 1, G23/E2 = 0.6, G13 = G12

    R/h

    No. of Layers

    Material Property1

    Material Property2

    500

    2 [0°/90°]

    17.70

    25.53

    3 [0°/90°/0°]

    38.016

    59.15

    4 [0°/90°/90°/0°]

    37.411

    61.09

    Table 11: Non-dimensional buckling load of circular cylindrical panels for different ratio of l/h

    l/h

    Two layers [0°/90°]

    Three layers [0°/90°/0°]

    Four Layers [0°/90°/90°/0°]

    100

    25.53

    59.15

    61.09

    200

    44.35

    150.59

    176.57

    In Table 11, the non-dimensional buckling loads have been calculated for different l/h ratios (varying thickness). It is seen that as the l/h ratio decreases, the buckling load decreases. Table 12: Non-dimensional buckling load of circular cylindrical panel for different cross-

    section (l/h=100, R/h=500)

    l/h=100, R/h=500

    Two layers

    Three layers

    Four layers

    Five Layers

    Ten Layers

    25.53

    59.15

    61.09

    56.66

    66.45

    CONCLUSIONS

As the number of layers increases, the non-dimensional frequency increases. As increases, the value of the non-dimensional frequency parameter decreases in all cases. As the degree of orthotropy increases, the value of the frequency parameter decreases. As the thickness decreases, the frequency parameter decreases owing to a decrease in the stiffness matrix. As the L/h ratio increases, the frequency parameter decreases. As E1/E2 ratio increases the frequency parameter decreases. As the thickness decreases buckling parameter increases. As the number of layers increases the buckling parameter increases. As the R/h ratio decreases buckling parameter increases. As E1/E2 ratio increases the buckling parameter

increases. As the l/h ratio increases the buckling parameter increases. The above conclusions establish that the present theory, though a simple one including the shear deformation effects, gives comparable results.

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