Free vibration of antisymmetric angle-ply-laminated plates including transverse shear deformation: Spline method

ARTICLE IN PRESS International Journal of Mechanical Sciences 50 (2008) 1476–1485

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Free vibration of antisymmetric angle-ply-laminated plates including transverse shear deformation: Spline method K.K. Viswanathan , Kyung Su Kim Department of Naval Architecture and Ocean Engineering, School of Mechanical Engineering, Inha University, #253, Yonghyun Dong, Nam Ku, Inchon 402-751, South Korea

a r t i c l e in f o

a b s t r a c t

Article history: Received 13 April 2007 Received in revised form 26 June 2008 Accepted 22 August 2008 Available online 23 September 2008

A free vibration study of antisymmetric angle-ply composite plates including shear deformation and rotatory inertia using the point collocation method and applying spline function approximations is presented. The equations of motion for the plate are derived using the theory of Yang, Norris and Stavsky. The solution is assumed in a separable form to obtain a system of coupled differential equations in displacement and rotational functions and these functions were approximated by Bickley-type splines of order three. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibrations of two- and four-layered plates, made up of several types of layer materials and subjected to two types of boundary conditions are considered. Parametric studies were made of the variation of frequency parameters with respect to the aspect ratio, side-to-thickness ratio and ply angle. The numerical results are presented through diagrams and, in some cases, are compared with results obtained by FEM. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Free vibration Antisymmetric angle ply Shear deformation Laminated plates Spline method

1. Introduction Composite structural elements play a significant role in several fields including aerospace, automobile and ship building, because of their more desirable damping and shock absorbing characteristics than those of homogeneous ones. In composite plates, the influence of shear deformation becomes significant as the plate thickness increases and hence theories incorporating this aspect are highly desired, along with suitable numerical techniques. Several types of theories have been developed to treat the mechanical behavior of composite laminates. Classical laminate plate theory (CLPT) [1] due to Kirchoff and Love neglect shear deformation giving inaccurate results for moderately thick plates. The frequencies calculated by using CLPT are generally higher than those obtained by Mindlin plate theory[2]. Shear deformation theories have been proposed by many researchers among which the first theory for laminated isotropic plates was due to Stavsky [3]. This has been generalized to laminated anisotropic plates by Yang et al. [4] as the YNS theory. Sun and Whitney [5] and Whitney and Pagano [6] also discussed the YNS theory for laminated plates consisting of an arbitrary number of bonded anisotropic layers. Bert and Chen [7] presented a closed-form solution for the free vibration of simply supported rectangular

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plates of antisymmetric angle-ply laminates. Reddy [8] and Ghosh and Dey [9] presented a FEM solution for laminated plates using YNS and higher order theories, respectively. Recently, Ferreira [10] and Kabir et al. [11] used different methods for the analysis of laminated composite plates. Recently, Viswanathan and Sang-Kwon Lee [12] discussed the problem of vibration of cross-ply-laminated plates including shear deformation theory using spline function techniques. This paper deals with the free vibration of antisymmetric angle-ply-laminated composite plates including shear deformation using a method of collocation with splines. The same, as well as different types of materials are used in different layers. The problem is formulated using the YNS theory from which is obtained a system of coupled differential equations on a set of assumed displacement and rotational functions, which are functions of a space co-ordinate. A spline technique was used, which was chosen over a number of other methods available for such problems, like those of Galerkin, Runge–Kutta, Frobenius, Chebyshev collocation and FEM. The choice of this method is due to the possibility that a chain of lower order approximations, as used here, can yield greater accuracy than a global high-order approximation [13]. The spline strip method incidentally, was used by Mizusawa and Kito [14] to study the vibration of cross-ply-laminated cylindrical panels. Bickley [13] successfully tested the spline collocation method over a two-point boundary value problem with a cubic spline. Soni and Sankara Rao [15], Irie et al. [16], Irie and Yamada [17], Navaneethakrishnan and

ARTICLE IN PRESS K.K. Viswanathan, K.S. Kim / International Journal of Mechanical Sciences 50 (2008) 1476–1485

Chandrasekaran [18], Navaneethakrishnan [19], Navaneethakrishnan et al. [20] and Viswanathan and Navaneethakrishnan [21,22] have also successfully applied this method, but most of them used only a single spline function in their problems. In this work, three displacement functions and two rotational functions were approximated by splines, which are cubic, in a system of coupled equations. The convergence characteristics were revealed. These splines are simple and clear for analytical process and have significant computational advantage. Even on theoretical consideration spline functions are more elegant and convenient to conceive, construct and handle, as approximating, interpolating and curve fitting functions than many others. It is well known that polynomial approximations are always possible (Weierstrass’ Theorem) and of practical use over a given set of points. Spline is not only a polynomial approximation, but also a generalized polynomial in the sense that it is a piecewise polynomial that can be made as smooth as required, at the junction points. If the approximate solution is for a boundary (or, initial) value problem, comprised of a differential equation of order k, the order of the spline can be limited to k+1 and not n1, where n is the number of points over which the solution curve is approximated, with nbk. It is elegant since, to start with, the function is assumed in its final form, with only the coefficients to be determined; and only differentiations (as against integrations) are carried out to make it satisfy the boundary value problem and the associated boundary conditions (which may involve differential coefficients of order o ¼ k). Assuming the solution in a separable form, a system of coupled differential equations in displacement and rotational functions is obtained and these functions are approximated by Bickley-type splines of order three. Collocation with these splines yielded a set of field equations which, along with the equations of boundary conditions, reduce it to a system of homogeneous simultaneous algebraic equations on the assumed spline coefficients. Thus resulting generalized eigenvalue problem is solved for a frequency parameter, using eigensolution technique, to obtain as many frequencies as required, starting from the least. From the eigenvectors the spline coefficients are computed from which the mode shapes and shear rotations can be constructed. Parametric studies are made of the variation of frequency parameters with respect to the aspect ratio and side-to-thickness ratio for two and four layers and the ply angle for four layers. Three different layer materials were considered. Significant mode shapes were obtained. Numerical results are presented in terms of graphs and tables and discussed.

2. Formulation of the problem Consider a plate of length a, width b and constant thickness h composed of an even number of thin layers of equal thickness made up of anisotropic materials bonded together, with an orientation angle of y and y. The xy-plane (reference surface) is placed at mid-depth of the plate, while the z-axis is normal to it. Following Bert and Chen [7] and Reddy [8], the displacement components based on YNS theory are assumed to be u ¼ u0 ðx; y; tÞ þ zcx ðx; y; tÞ; v ¼ v0 ðx; y; tÞ þ zcy ðx; y; tÞ; w ¼ wðx; y; tÞ (1) where u, v and w are the displacement components in the x, y and z directions, respectively, u0 and v0 are the in-plane displacements of the middle plane and cx and cy are the shear rotations of any point on the middle surface of the plate.

1477

The displacement components u0, v0, w and shear rotations cx and cy are assumed in the form u0 ðx; y; tÞ ¼ uðx; yÞeiot v0 ðx; y; tÞ ¼ vðx; yÞeiot wðx; y; tÞ ¼ wðx; yÞeiot

cy ðx; y; tÞ ¼ cy ðx; yÞeiot cx ðx; y; tÞ ¼ cx ðx; yÞeiot

(2)

where o is the angular frequency of vibration and t is the time. Using Eq. (2) in the constitutive equations and the resulting expressions for the stress and moment resultants in the equation of motion [8], the governing equations of motion in u, v, w, cx and cy are obtained in the matrix form 2 32 3 L11 L12 L13 L14 L15 u 6L 7 6 7 6 21 L22 L23 L24 L25 76 v 7 6 76 7 6 L31 L32 L33 L34 L35 76 cx 7 ¼ f0g (3) 6 76 7 6L 76 7 4 41 L42 L43 L44 L45 54 cy 5 L51 L52 L53 L54 L55 w where Lij ¼ Lji are linear differential operators in x and y, given by

q2 q2 q2 þ A66 2 þ po2 ; L12 ¼ ðA12 þ A66 Þ qxqy qx2 qy q2 ¼ 2B16 qxqy q2 q2 q2 q2 ¼ B16 2 þ B26 2 ; L22 ¼ A66 2 þ A22 2 þ po2 qx qy qx qy q2 q2 ¼ B16 2 þ B26 2 qx qy q2 q2 q2 ; L ¼ D11 2 þ D66 2  KA55 þ Io2 ¼ 2B26 qxqy 33 qx qy q2 ¼ ðD11 þ D66 Þ qxqy q q2 q2 ¼ KA55 ; L44 ¼ D66 2 þ D22 2  KA44 þ Io2 qx qx qy q ¼ KA44 qy q2 q2 ¼ KA55 2  KA44 2  po2 and L15 ¼ L25 ¼ 0 qx qy

L11 ¼ A11 L13 L14 L23 L24 L34 L35 L45 L55

(4)

Here p and I are the normal and rotary inertia coefficients defined by Z ðp; IÞ ¼ rðkÞ ð1; z2 Þ dz (5) where r(k) is the material density of the k-th layer and K is the shear correction coefficient. The value of K for a general laminate depends on the lamina’s properties and lamination scheme, and may be calculated by various static and dynamic methods [4, 23–26]. Let the edges y ¼ 0 and b of the plate be simply supported. Then the displacements and rotational functions are assumed in the separable form as npy b npy vðx; yÞ ¼ VðxÞ sin b npy wðx; yÞ ¼ WðxÞ sin b npy cy ðx; yÞ ¼ Cy ðxÞ cos b npy cx ðx; yÞ ¼ Cx ðxÞ sin b uðx; yÞ ¼ UðxÞ cos

(6)

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respectively, by the splines

The following non-dimensional parameters are introduced: rffiffiffiffiffiffiffiffi p ; a frequency parameter l ¼ oa A11 a f ¼ ; the aspect ratio b x X ¼ ; a distance co-ordinate a a H ¼ ; side-to-thickness ratio h

U  ðXÞ ¼

L11 6L 6 21 6 6 L31 6 6L 4 41 L51

L12

L13

L14

L22

L23

L24

L32

L33

L34

L42

L43

L44

L52

L53

L54

L15

32

V  ðXÞ ¼

d

2

dX 2

(7)

3

U 6 7 L25 7 76 V 7 76 7 6 7 L35 7 76 aCx 7 ¼ f0g 7 6 L45 54 aCy 7 5 L55 W

2

L13 ¼ 2bS15 L21

2

 b S10 þ l ;

(8)

d ¼ L31 ; dX

d ; ¼ bðS2 þ S10 Þ dX

L12 ¼ bðS2 þ S10 Þ 2

L14 ¼ S15 L22 ¼ S10

d

dX 2 2 d

dX 2

d dX

2

L15 ¼ L25 ¼ L51 ¼ L52

L55 ¼ KS14

d

2

2

 b S3 þ l

2

N 1 X

ei X i þ

i¼0

CY ðXÞ ¼

2 X

giXi þ

i¼0

W  ðXÞ ¼

2 X

f j ðX  X j Þ3 HðX  X j Þ

j¼0 N 1 X

pj ðX  X j Þ3 HðX  X j Þ

j¼0

li X i þ

i¼0

N1 X

qj ðX  X j Þ3 HðX  X j Þ

(11)

j¼0

Here H(XXj) is the Heaviside step function and N is the number of intervals into which the range [0, 1] of X is divided. The points X ¼ Xs ¼ (s/N), (s ¼ 0, 1, 2, y, N) are chosen as the knots of the splines, as well as the collocation points. Thus, the splines are assumed to satisfy the differential equations given by Eq. (8), at all Xs. The resulting expressions contain a (5N+5) homogeneous system of equations in the (5N+15) spline coefficients. The boundary conditions considered on the edges x ¼ 0 and a are

Each of these cases gives 10 more equations, thus making a total of (5N+15) equations, in the same number of unknowns. The resulting field and boundary condition equations may be written in the form 2

(12)

½Mfqg ¼ l ½Pfqg

where [M] and [P] are square matrices, {q} is a column matrix. This is treated as a generalized eigenvalue problem in the eigenparameter l and the eigenvector {q} whose elements are the spline coefficients.

2

 K b S13 þ l dX 2 ¼ 0 and b ¼ nf

(9)

Here n is the model wave number in y direction and we set n ¼ 1. The quantities Si (i ¼ 2, 3, y, 16) are defined by A12 A22 D11 D12 D22 A66 ; S3 ¼ ; S7 ¼ 2 ; S8 ¼ 2 ; S9 ¼ 2 ; S10 ¼ A11 A11 A11 a A11 a A11 a A11 D66 A44 A55 B16 B26 ¼ 2 ; S13 ¼ ; S14 ¼ ; S15 ¼ ; S16 ¼ A11 A11 aA11 aA11 a A11 (10)

S2 ¼

S12

2 X

dj ðX  X j Þ3 HðX  X j Þ

j¼0

2

d 2  b S16 ¼ L32 ; L24 ¼ 2bS16 ¼ L42 dX 2 2 d I 2 d 2 ¼ S7  b S12  KS14 þ 2 l ; L34 ¼ bðS8 þ S12 Þ ¼ L43 dX pa dX 2 2 d d I 2 2 ¼ L53 ; L44 ¼ S12 ¼ KS14  b S9  KS13 þ 2 l dX pa dX 2

L45 ¼ K bS13 ¼ L54 ;

N 1 X

ci X i þ

 b S16 ¼ L41

L23 ¼ S15

L35

CX ðXÞ ¼

bj ðX  X j Þ3 HðX  X j Þ

(1) (C–C): both the ends clamped, (2) (H–H): both the ends hinged.

2

L33

2 X

N 1 X j¼0

i¼0

where\cr

L11 ¼

ai X i þ

i¼0

Here a is the length of the plate along the x-direction, h is the total thickness and A11 is a standard extensional rigidity coefficient. Clearly 0pXp1. After substituting Eqs. (6) and (7) into Eq. (3), the modified matrix equation is in the form of\cr 2

2 X

3. Method of solution The differential equations in Eq. (8) contain derivatives of the second order in U(X), V(X), W(X), CX(X) and CY(Y). These functions can be approximated by using cubic spline functions, in the range of XAe[0, 1], since splines are relatively simple and elegant and use a series of lower order approximations rather than global higher order approximations, which afford fast convergence and high accuracy. The displacement functions U(X), V(X) and W(X) and the rotational functions aCX(X), aCY(Y) are approximated,

4. Numerical results, convergence, comparative studies and discussion Double precision arithmetic was used throughout for numerical computations, since the matrices of large orders are to be handled. The material properties were taken from Reddy [8] to analyze the convergence of the frequency parameter values with the number of subintervals N of the range of X, shown in Table 1. The program was run for several cases of parametric values and material values for the values of N ¼ 4 onwards. The computed Table 1 Convergence study for a four-layered simply supported plate of ply angle 451/451/451/451 N

l0 1

% Change

l0 2

% Change

l0 3

% Change

4 6 8 10 12 14 16 18

19.097004 18.644162 18.490203 18.419743 18.381682 18.358805 18.343986 18.333839

– 2.3713 0.8257 0.3812 0.2066 0.1244 0.0807 0.05531

38.402711 36.304809 35.485664 35.307242 35.195813 35.124824 35.070823 35.034735

– 5.4629 2.2563 0.5028 0.3156 0.2017 0.1532 0.1029

67.407101 60.354367 58.027225 56.982270 56.418146 56.166669 55.974129 55.797978

– 10.4629 3.8558 1.8008 0.9900 0.4457 0.3428 0.3147

a/b ¼ 1, a/h ¼ 10, (E1/E2) ¼ 40, (G12/E2) ¼ 0.6, (G13/E2) ¼ 0.5, n12 ¼ 0.25.

ARTICLE IN PRESS K.K. Viswanathan, K.S. Kim / International Journal of Mechanical Sciences 50 (2008) 1476–1485

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Table 2 0

Comparison of the effects of plate aspect ratio (a/b) and length-to-thickness ratio (a/h) on the non-dimensional frequency l ¼ oa2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r=E2 h2 of a simply supported

rectangular plate made of material I (451/451/451/451) Values compareda

a/h

Aspect ratio (a/b) 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

10

P G R B

8.767 4.93 8.724 8.664

10.416 10.31 10.535 10.42

12.810 12.65 12.965 12.82

15.488 15.29 15.712 15.54

18.359 18.06 18.609 18.46

21.355 21.02 21.567 21.51

24.450 24.05 24.602 24.67

27.639 27.18 27.736 27.95

30.923 30.45 30.981 –

34.305 31.28 34.247 34.87

20

P G R B

9.749 9.52 9.475 9.300

11.624 11.70 11.767 11.46

14.713 14.72 14.896 14.45

18.146 18.26 18.557 17.97

21.978 22.19 22.584 21.87

26.149 26.45 26.857 26.12

30.636 31.02 31.401 30.68

35.426 35.89 36.249 35.56

40.512 41.07 41.372 –

45.887 46.54 46.789 46.26

30

P G R B

10.411 9.72 9.667 9.64

12.083 12.02 12.074 11.70

15.459 15.22 15.385 14.84

19.045 19.01 19.304 18.56

23.121 23.28 23.676 22.74

27.636 27.97 28.381 27.35

32.573 33.07 33.457 32.38

37.928 38.59 38.940 37.82

43.700 44.53 43.832 –

49.886 50.89 51.132 49.98

40

P G R B

11.150 9.80 9.759 9.49

12.467 12.14 12.205 11.78

16.075 15.41 15.853 14.98

19.660 19.30 19.604 18.78

23.785 23.70 24.118 23.08

28.404 28.57 29.003 27.83

33.501 33.89 34.397 33.05

39.076 39.69 40.071 38.72

45.132 45.98 46.305 –

51.672 52.74 53.012 51.52

50

P G R B

12.001 9.84 9.816 9.51

12.877 12.20 12.280 11.82

16.727 15.50 15.689 15.04

20.248 19.44 19.759 18.89

24.348 23.91 24.343 23.24

28.979 28.86 29.321 28.06

34.125 34.30 34.742 33.37

39.787 40.24 40.653 39.17

45.970 46.70 47.067 –

52.677 53.68 53.989 52.59

a

P: represents the authors’ work; G: Ghose and Dey [9], R: Reddy [8] and B: Bert and Chen [7].

Table 3 0

Comparison of the effects of plate aspect ratio (a/b) and length-to-thickness ratio (a/h) on the non-dimensional frequency l ¼ oa2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r=E2 h2 of a simply supported

rectangular plate made of material II (451/451/451/451) Values compareda

a/h

Aspect ratio (a/b) 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

10

P R

6.750 6.670

7.922 7.890

9.532 9.533

11.368 11.378

13.318 13.303

15.332 15.259

17.394 17.256

19.504 19.310

21.668 21.437

23.891 23.642

20

P R

7.650 7.517

9.230 9.232

11.437 11.574

14.036 14.298

16.914 17.266

20.019 20.401

23.329 23.707

26.831 27.193

30.518 30.860

34.383 34.704

30

P R

8.083 7.733

9.735 9.584

12.080 12.137

14.894 15.151

18.073 18.499

21.573 22.100

25.377 25.967

29.480 30.117

33.874 34.551

38.556 39.266

40

P R

8.502 7.828

10.139 9.730

12.498 12.365

15.367 15.501

18.648 19.012

22.301 22.819

26.313 26.944

30.683 31.410

35.409 36.222

40.490 41.381

50

P R

8.969 7.883

10.559 9.808

12.883 12.484

15.746 15.679

19.051 19.273

22.762 23.188

26.866 27.449

31.363 32.083

36.256 37.101

41.546 42.504

60

P R

9.492 7.922

11.020 9.859

13.286 12.556

16.113 15.784

19.406 19.427

23.129 23.405

27.269 27.746

31.827 32.481

36.807 37.621

42.212 –

a

P: represents the authors’ work and R: Reddy [8].

values of l improved with an increase of N, but the improvement decreased steadily. It can be seen that the choice of N ¼ 14 is adequate, since for the next value of N the percent changes in values of l were very low, the maximum being 0.37%.

Some special cases of the general problem considered for which results are already available in literature were taken up for comparison in order to gain a conviction on the validity of the current result. The fundamental natural frequency l converted

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Table 4 0

Comparison of the effects of plate aspect ratio (a/b) and length-to-thickness ratio (a/h) on the non-dimensional frequency l ¼ oa2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r=E2 h2 of a simply supported

rectangular plate made of material II (301/301/301/301) Values compareda

a/h

Aspect ratio (a/b) 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

10

P R

8.435 8.198

9.144 8.937

10.199 10.019

11.481 11.361

12.903 12.739

14.411 14.237

15.981 15.785

17.603 17.382

19.278 19.038

21.011 20.771

20

P R

9.946 9.788

10.900 10.819

12.338 12.351

14.126 14.233

16.169 16.363

18.404 18.674

20.802 21.126

23.350 23.697

26.040 26.383

28.866 29.291

30

P R

10.493 10.214

11.502 11.333

13.038 13.009

14.971 15.088

17.207 17.471

19.688 20.094

22.386 22.913

25.291 25.906

28.400 29.064

31.705 32.397

40

P R

10.899 10.392

11.914 11.545

13.468 13.277

15.439 15.439

17.737 17.929

20.307 20.688

23.122 23.673

26.176 26.859

29.467 30.242

32.988 33.831

50

P R

11.308 10.490

12.308 11.659

13.848 13.418

15.817 15.619

18.127 18.163

20.725 20.992

23.585 24.062

26.704 27.351

30.079 30.855

33.706 34.582

60

P R

11.752 10.555

12.727 11.731

14.239 13.505

16.185 15.738

18.482 18.302

21.079 21.770

23.951 24.291

27.094 27.640

30.507 31.216

34.187 35.036

P: represents the authors’ work; and R: Reddy [8].

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 into the suitable parameter l ¼ oa2 r=E2 h and the material properties (typical of graphite epoxy) were taken as given by Reddy [8]: Material I :

Material II :

E1 G12 G13 G23 ¼ 40; ¼ 0:6; ¼ ¼ 0:5; n12 ¼ 0:25 E2 E2 E2 E2 E1 G12 G13 G23 ¼ 25; ¼ 0:5; ¼ ¼ 0:2; n12 ¼ 0:25 E2 E2 E2 E2

The value of the shear correction coefficient K ¼ 5/6 assumed by Bert and Chen [7], Reddy [8] and Perngjin [26] were used for comparison and for obtaining new results. In Table 2 the values of the fundamental frequency l0 obtained for material I of different aspect ratios (a/b) and side-to-thickness ratios (a/h) for four-layered antisymmetric angle-ply plates with the angle 451 are compared with those of Bert and Chen [7], Ghose and Dey [9] and Reddy [8]. It shows the close agreement of the present results with those obtained by FEM [8], a higher order theory [9] and closed-form solution by Bert and Chen [7]. The agreement is striking in some cases. For example, the current value of l0 for a/b ¼ 0.4 and a/h ¼ 10 agrees very well with Bert and Chen [7]. Similarly, for a/b ¼ 1.8 and a/h ¼ 10, the present value of l0 is almost equal to the value of Reddy [8]. The slight higher differences observed in some cases must be due to the differences in the formulations, methods of solution and numerical accuracy involved. Tables 3 and 4 show the comparison of the effects of the plate aspect ratio and side-to-thickness ratio on the fundamental frequency l0 for material II with the lamination angle of 451 and 301 between the current values and those of Reddy [8]. The agreement is very good. Fig. 1 provides a graphical comparison of current results with the results of references [7–9]. The manner of variation of fundamental frequencies with respect to the side-to-thickness ratio for three typical values of aspect ratio was studied. The results agreed very well, providing credibility to the method of analysis and the results.

30.00 a/b=1.2

Dimensionless frequency

a

25.00

a/b=1.0

20.00

a/b=0.8

15.00

10.00

Present Reference [7] Reference [8] Reference [9]

10

15

20

25 30 35 Side-to-thickness ratio

40

45

50

Fig. 1. Comparison of fundamental frequencies for a simply supported four-layered antisymmetric angly-ply plate with angle 451, with those of Refs. [7–9].

Some graphical results are shown for two-layered and fourlayered composite plates. Combinations of Kevler-49 epoxy (KGE), E-glass epoxy (EGE) and AS4/1350-6 Graphite epoxy (AGE) (see e.g. Bhimaraddi [27]) were considered. Fig. 2 describes the manner of variation of the values of the frequency parameters lm (m ¼ 1, 2, 3) with respect to the side-tothickness ratio (a/h) of two-ply plates of the same material (AGE), of ply angles 451/451, of three different aspect ratios (a/b), with all the four edges simply supported (S–S–S–S). Since the edged y ¼ 0 and b are simply supported for all cases, only the support conditions on the other two edges are indicated in this and all the figures that follow. From all three diagrams of Fig. 2, it is seen that with the increase of (a/h), lm values steadily decrease. The rate of decrease is higher for higher modes (m); for any m, it is higher for (a/h) o20 nearly, than for greater values of (a/h). Thus, the thinner the plate (i.e., h smaller), the smaller are the values of lm; and smaller are the changes in the values of lm with further reduction in thickness. The maximum relative differences in values of lm

ARTICLE IN PRESS K.K. Viswanathan, K.S. Kim / International Journal of Mechanical Sciences 50 (2008) 1476–1485

2

2

S-S

AGE

1.6

2.5

1.6

1.2

a/b = 2.0

(45°/−45°)

λm

1.5

λm

λm

(45°/−45°)

2

1.2 3

3

3

0.8

0.8

1 2

2 2

0.4 0

0.4

30 a/h

10

0.5

m=1

m=1

m=1

0

50

S-S

AGE

S-S

AGE a/b = 1.4

a/b = 1.0 (45°/−45°)

1481

10

20

30 a/h

40

0

50

10

20

30 a/h

40

50

Fig. 2. Variation of frequency parameter with respect to side-to-thickness ratio of a two-layered ply for three different aspect ratios under S–S–S–S condition.

2.50

2.50

S-S

KGE a/b = 0.4

1.50

(45°/−45°/45°/−45°) 3

1.50

λm

3

2

1.00

1.00

1.00

a/b =1.6

2.00

(45°/−45°/45°/−45°)

1.50

3

λm

a/b =1.0

2.00

(45°/−45°/45°/−45°)

S-S

KGE

λm

2.00

2.50

S-S

KGE

2

2

0.50

0.50

0.50

m=1

0.00

m=1

m=1

10 20

30 40 a/h

50

60

0.00

10

20

30 40 a/h

50

60

0.00

10

20

30 40 a/h

50

60

Fig. 3. Effect of side-to-thickness ratio and an aspect ratio on frequencies.

S-S

KGE-AGE-AGE-KGE

2.00

S-S

AGE-KGE-KGE-

2.00

a/b = 1.0

a/b = 1.0

(45°/−45°/45°/−45°)

(45°/−45°/45°/−45°)

1.50

λm

λm

1.50

3

1.00

2

2

0.50

0.50

m=1

m=1

0.00

0.00 10

20

3

1.00

30

40 a/h

50

60

10

20

30

40

50

60

a/h

Fig. 4. Variation of frequency parameter of four-layered 451 ply plates with respect to side-to-thickness ratio under S–S–S–S condition.

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3

3 S-S

EEGE-EGA-EGA-EG a/h = 10 (45°/−45°/45°/−45°)

2.5

2.5 3

3

2

λm

2

λm

S-S

AEGA-EGE-EGE-EG a/h = 10 (45°/−45°/45°/−45°)

1.5

1.5

2

2

1

1

m=1

0.5

0.5

m=1

0

0 0.2

0.5

0.8

1.1 a/b

1.4

1.7

2

0.2

0.5

0.8

1.1 a/b

1.4

1.7

2

Fig. 5. Variation of frequency parameter of four-layered 451 ply plates with aspect ratio and the effect of the sequence of stacking of layers.

2 KGE a/b = 1.0 (30°/−30°/30°/−30°)

1.6

2

S-S

1.6

1.2

2

S-S KGE a/b = 1.4 (30°/−30°/30°/−30°)

1.6

λm

1.2

λm

λm

1.2 3

0.8

S-S KGE a/b = 2.0 (30°/−30°/30°/−30°)

3

0.8

0.8 3

2

0.4 0

0.4

20

0.4

2

m=1

m=1

10

2

30 40 a/h

50

60

0

10

20

30

40 a/h

50

60

0

m=1

10

20

30 40 a/h

50

60

Fig. 6. Variation of frequency parameter with side-to-thickness ratio for four-layered 301 plies under S–C–S–C boundary condition.

(m ¼ 1, 2, 3), over the range of values of (a/h) considered, were 0.766, 0.665 and 0.585 for the square plate, and 0.787, 0.719 and 0.634 for the rectangular plate with (a/b) ¼ 2. Fig. 3 describes results of a similar study, but with the difference that the plates are four-layered antisymmetric angle plies, with a different material. Qualitatively the vibrational behavior of these plates is similar to that of the cases of Fig. 2. In Fig. 4 similar studies are made but with plates of four layers of two different materials, arranged in two different orders. The ordering of the plates seemed to have no effect on fundamental frequencies, but some significant and increasing effect for higher modes was displayed. From Figs. 2 to 4 it is also seen that with an increasing aspect ratio, for any fixed (a/h), the value of lm increases. This phenomenon is further clarified in Fig. 5 which describes the variation of lm with the aspect ratio for two cases of four-ply-laminated plates described in the figure. The values of

lm increase monotonically with (a/b). The maximum relative increase for the first three modes, for the range 0.2o(a/b)o2.0 considered, were 3.395, 0.998 and 0.434, respectively. A change in the order of stacking seemed to have no appreciable effect on the frequencies. In Fig. 6 four-layered angle-ply plates of same material KGE, with a ply angle of 301 under S–C–S–C boundary conditions and three different aspect ratios are considered. The variation of lm with respect to (a/h) is studied. Clearly, the vibrational pattern is similar to that under S–S–S–S condition. The maximum relative changes in the values of lm (m ¼ 1, 2, 3) over the range 10o(a/h)o60 considered were 0.662, 0.628 and 0.551, respectively, for the square plate and 0.721, 0.643 and 0.564 for the rectangular plate of aspect ratio 2. Figs. 7–9 describe the variation of the frequency parameter with the aspect ratio for the plates of features indicated, under S–C–S–C boundary conditions. One significant aspect is that all

ARTICLE IN PRESS K.K. Viswanathan, K.S. Kim / International Journal of Mechanical Sciences 50 (2008) 1476–1485

1483

2

2.5 C-C

KGE

C-C

KGE a/h = 10

a/h = 10

2

3

3

1.5 (30°/−30°)

(45°/−45°)

1.5 1

λm

λm

2

2

1 0.5

m=1

m=1

0.5

0

0.2

0.5

0.8

1.1 a/b

1.4

1.7

0

2

0.2

0.5

0.8

1.1 a/b

1.4

1.7

2

Fig. 7. Variation of frequency parameter with two two-layered plies with angles 451 and 301 with respect to aspect ratio under S–C–S–C boundary condition.

KGE a/h = 10

2.5

AGE-KGE-KGE-

3 (45°/−45°/45°/−45°)

2

(45°/−45°/45°/−45°)

1.5

2

2

m=1

0.6

1 1.4 a/b

0 0.2

C-C

3 (45°/−45°/45°/−45°)

1.5 1

m=1

0.5

1.8

KGE-AGE-AGEKGE a/h = 10

2

1

1

0 0.2

2.5

3

2

1.5

0.5

3

C-C

a/h = 10

2.5

λm

λm

2

3 C-C

λm

3

0.6

1 1.4 a/b

m=1

0.5

1.8

0

0.2

0.6

1 1.4 a/b

1.8

Fig. 8. Variation of frequency parameter with aspect ratio under S–C–S–C boundary condition and the effect of stacking of layers.

the lm(a/b) curves in each figure were almost identical, suffering only parallel shifts. The same phenomenon prevailed for plates under S–S–S–S conditions also. For the same material, the vibrational frequencies were affected by the ply angles (Fig. 7(a) and (b)) and the number of layers (Figs. 7(a), 8(a), 7(b) and 9(a)), and the constituent layer materials (Fig. 9(a) and (b)) and the order of stacking of layers (Fig. 8(a) and (c)). Order of stacking of layers does affect frequencies significantly now, under S–C–S–C boundary conditions, while not so under the S–S–S–S conditions. Other conditions being the same, the values of the frequency parameter were found to be higher under S–C–S–C boundary condition than under S–S–S–S condition (figure not presented). Fig. 10 describes the variation of the frequency parameter with respect to ply angle of a four-layered plate of the same material KGE under different boundary conditions. Fig. 10(a) shows the variation of frequency parameter with angle under S–C–S–C condition and Fig. 10(b) shows the variation of frequency parameter with angle under S–S–S–S condition. The nature of curves lmy were almost identical in these two figures and it is

also seen that the effect of increasing lamination angle, up to a value of 451, increases the frequency parameter lm (m ¼ 1, 2, 3). The first three mode shapes of vibration of an antisymmetric square two-layer ply plate of KGE material, with the ply angle 451 is presented in Fig. 11. Normalization is done with respect to the maximum transverse displacement W. As expected, the transverse displacements are most predominant.

5. Conclusion The equations of free vibration of antisymmetric angle-ply composite rectangular plates of several layers were derived, accounting for rotatory inertia. A pair of opposite edges were simply supported. Assuming the solution in a suitable separable form, a system of five differentiable equations in a single unknown was obtained. The solution was approximated in terms of cubic splines and point collocation was used to yield a generalized eigenvalue problem. Thus a solution for the first

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2.5

2.5 C-C

KGE

a/h = 10

(30°/−30°/30°/−30°)

2

C-C

AGEC

a/h = 10

(30°/−30°/30°/−30°)

2

3 3

1.5

λm

λm

1.5

1

1

2

0.5

2

0.5

m=1

m=1

0 0.2 0.4 0.6 0.8

0 1

1.2 1.4 1.6 1.8 a/b

2

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8 a/b

2

Fig. 9. Variation of frequency parameter with aspect ratio and material under S–C–S–C boundary condition.

2.5

2.5

C-C

KGE a/b = 1.0 a/h = 10

2

S-S

KGE a/b = 1.0 a/h = 10

2

3

1.5

3

λm

λm

1.5

1

1

2

0.5

2

0.5

m=1

m=1

0

0 0

5

10

15

20

25

30 35

40

θ

45

0

5

10

15

20

25

30

35

40

45

θ

Fig. 10. Variation of frequency parameter with respect to angle for four-layered plate with aspect ratio, side-to-thickness ratio and boundary conditions.

three eigenvalues giving the corresponding frequency parameters, and the related eigenvectors from which the mode shapes can be constructed. The manner of variation of the eigenfrequencies with respect to side-to-thickness ratio, aspect ratio, ply angle and the types of boundary conditions on the other edges were studied. Three types of layer materials, two types of boundary conditions, two and four layers of the plate and their effects on the free vibrational behavior were studied and presented. The frequency parameter values tend to decrease, in general, with an increase of the side-to-thickness ratio of the plate, the

change becoming smaller around layer values of this ratio. However, the frequencies increase with the aspect ratio. The ply angles and the number of layers also affect the frequency parameter values. The influence of the order of stacking was larger for a S–C–S–C plate than for a S–S–S–S plate. Further the frequency parameter values were higher when the support conditions were S–C–S–C, than when the conditions were S–S–S–S. The study has also brought out the elegance and usefulness of the spline function collocation method of solution for boundary value problems of the type discussed.

ARTICLE IN PRESS K.K. Viswanathan, K.S. Kim / International Journal of Mechanical Sciences 50 (2008) 1476–1485

1.2

1485

0.08

0.08 S-S

S-S

S-S

0.8 0.04

0.04

0

0.2

0.4

-0.4

0.6

0.8

1

KGE

a /b = 1 a / h = 10

-0.8 -1.2 1.2

0.2

0.4

0.6

0.8

0

1

-0.04

-0.08 0.06

-0.08

U 0

0.2

-0.4 -0.8 -1.2

0.4

0.6

0.8 KGE a /b = 1 a / h = 10

0.4

0.6

0.8

1

C-C

0.1

0

1

0.2

0.2

0.02

0

0

0.3 C-C

0.04

0.4 W

0

-0.04

C-C

0.8

0

V

0

U

W

V

0.4

0

0.2

0.4

0.6

0.8

1

0

-0.02

-0.1

-0.04

-0.2

-0.06

-0.3

0.2

0.4

0.6

0.8

1

Fig. 11. Mode shapes of vibration of an antisymmetric angle-ply plate:—, Mode 1; - - - - - -, Mode 2; -  -  -  -, Mode 3.

Acknowledgement This work was supported by INHA UNIVERSITY Research Grant, South Korea. References [1] Reissner E, Stavsky Y. Bending and stretching of certain types of heterogeneous aelotropic elastic plates. Journal of Applied Mechanics 1961;28: 402–12. [2] Mindlin RD. Influence of rotary inertia and shear on flexural motion of isotropic elastic plates. Journal of Applied Mechanics 1951;18:31–8. [3] Stavsky Y. On the theory of symmetrically heterogeneous plates having the same thickness variation of the elastic moduli. In: Abir D, Ollendorff F, Reiner M, editors. Topics in applied mechanics. New York: American Elsevier; 1965. p. 105. [4] Yang PC, Nooris CH, Stavsky Y. Elastic wave propagation in heterogeneous plates. International Journal of Solids and Structures 1966;2:665–84. [5] Sun CT, Whitney JM. Theory for the dynamic response of laminated plates. AIAA Journal 1973;11:178. [6] Whitney JM, Pagano NJ. Shear deformation in heterogeneous plates. Journal of Applied Mechanics 1970;37:1031–6. [7] Bert CW, Chen TLC. Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates. International Journal of Solids and Structures 1978;14:465–73. [8] Reddy JN. Free vibration of antisymmetric angle-ply laminated plates including transverse shear deformation by the finite element method. Journal of Sound and Vibration 1978;66(4):565–76. [9] Ghosh AK, Dey SS. Free vibration of laminated composite plates—a simple finite element based on higher order theory. Computers and Structures 1994; 52(3):397–404. [10] Ferreira AJM. A formulation of the multiquadric radial function method for the analysis of laminated composite plates. Composite Structures 2003;59:385–92. [11] Kabir HRH, Al-Khaleefi AM, Chaudhuri Reaz A. Free vibration analysis of thin arbitrarily laminated anisotropic plates using boundary-continuous displacement Fourier approach. Journal of Sound and Vibration 2001;53:469–76. [12] Viswanathan KK, Lee Sang-Kwon. Free vibration of laminated cross-ply plates including shear deformation by spline method. International Journal of Mechanical Sciences 2007;49:352–63.

[13] Bickley WG. Piecewise cubic interpolation and two point boundary problems. Computer Journal 1968;11:206–8. [14] Mizusawa T, Kito H. Vibration of cross-ply laminated cylindrical panels by the spline strip method. Computers and Structures 1995;57(2):253–65. [15] Soni SR, Sankara Rao K. Vibration of non-uniform rectangular plates: a spline technique method of solution. Journal of Sound and Vibration 1974;35: 35–45. [16] Irie T, Yamada G, Kanda R. Free vibration of rotating non-uniform discs: spline interpolation technique calculation. Journal of Sound and Vibration 1979;66: 13–23. [17] Irie T, Yamada G. Analysis of free vibration of annular plate of variable thickness by use of a spline technique method. JSME Bulletin 1980;23: 286–92. [18] Navaneethakrishnan PV, Chandrasekaran K. A spline function analysis of axisymmetric free vibrations of layered conical shells and plates of variable thickness. ASME Structural Vibration and Acoustics 1989;18:153–9. [19] Navaneethakrishnan PV. Vibration of layered shells and plates: a unified formulation and spline function study. In: Proceedings of the international noice and vibration control conference NOICE-93, vol. 7. St. Petersburg, 1993. p. 71–6. [20] Navaneethakrishnan PV, Chandrasekaran K, Ravi Srinivas N. Axisymmetric vibration of layered tapered plates. ASME Journal of Applied Mechanics 1992; 59:1041–3. [21] Viswanathan KK, Navaneethakrishnan PV. Buckling of non-uniform plates on elastic foundation: spline method. Journal of Aeronautical Society of India 2002;54:366–73. [22] Viswanathan KK, Navaneethakrishnan PV. Free vibration study of layered cylindrical shells by collocation with splines. Journal of Sound and Vibration 2003;260:807–27. [23] Whitney JM, Sun CT. A higher order theory for extensional motion of laminated composites. Journal of Sound and Vibration 1973;30:85. [24] Whitney JM. Shear correction factors for orthotropic laminates under static loads. Journal of Applied Mechanics 1973;40:302. [25] Bert CW. Simplified analysis of static shear factors for beams of nonhomogeneous cross section. Journal of Composite Materials 1973;7:525. [26] Perngjin F Pai. A new look at shear correction factor d and warping functions of anisotropic laminates. International Journal of Solids Structures 1995;32: 2295–313. [27] Bhimaraddi A. Large amplitude vibrations of imperfect antisymmetric angleply laminated plates. Journal of Sound and Vibration 1993;162(3):457–70.