Axisymmetric bending and free vibration of symmetrically laminated circular and annular plates having elastic edge constraints

Ain Shams Engineering Journal 10 (2019) 343–352

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Mechanical Engineering

Axisymmetric bending and free vibration of symmetrically laminated circular and annular plates having elastic edge constraints Sumit Khare ⇑, Narain Das Mittal Department of Mechanical Engineering, Maulana Azad National Institute of Technology, Bhopal 462003, India

a r t i c l e

i n f o

Article history: Received 5 November 2017 Revised 13 October 2018 Accepted 28 October 2018 Available online 23 February 2019 Keywords: Bending Vibration Circular plates Axisymmetric Elastic edge DQM

a b s t r a c t To fill the gap noticed in the literature, this paper presents free vibration results of symmetrically laminated circular and annular plates having elastic edge constraints. Some static bending results have also been presented. Each layer is having cylindrical/polar orthotropy. A Differential Quadrature Method (DQM) based methodology is utilized for spatial discretization. The problem of shear-deformable laminated circular/annular plates having translational as well as rotational edge constraints is studied. By varying the edge stiffnesses, several combinations of free, simply supported, clamped and free edge conditions have been simulated. The results have been verified with the help of convergence study in terms of the number of discretization nodes and by comparison with the results of isotropic circular plates and of laminated circular/annular plates available in the literature. Some parametric studies in terms of material and geometrical variables are also reported. Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-ncnd/4.0/).

1. Introduction Due, primarily, to their relatively high strength to weight ratio, laminated plates are being used in several engineering applications like aeronautics, space, automobiles, bridges, etc. In comparison to laminated rectangular plates, the literature corresponding to laminated circular plates is relatively scant. Bending, buckling and vibration problems of isotropic, orthotropic and Functionally Graded Material (FGM) circular plates of different levels of computational complexity have been the subject of many research papers [1–13]. Timoshenko and Woinowsky-Krieger [1] had presented axisymmetric static deflection results of thin isotropic and rectangularly orthotropic circular plates. Most of the recent research is focused on FGM circular plates, on circular plates of variable thickness and on large deformation analyses [5–11]. Li et al. [5] presented elasticity solutions of transversely isotropic circular plates with elastic compliance coefficients being vari⇑ Corresponding author. E-mail addresses: [email protected] (S. Khare), ndmittal1956@gmail. com (N.D. Mittal). Peer review under responsibility of Ain Shams University.

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able in thickness direction. Hashemi et al. [6] presented FirstOrder Shear Deformation Theory based free vibration results of piezoelectric coupled FGM circular/annular plates under different combinations of boundary conditions. Wang [8] established the relationships between the Classical Laminated Plate Theory (CLPT) and First-Order Shear Deformation Theory (FSDT) based axisymmetric bending solutions of tapered isotropic circular plates. Hichem Bellifa et al. [9] presented the first-order shear deformation theory with four unknowns to study the bending and dynamic behavior of FG plates. Also, various refined and shear deformation theories have been proposed for bending and free vibration analysis of beam, plates and shells like structure made up of isotropic, laminated composite and functionally graded material [9–23]. But due to its high efficiency and simplicity, first-order shear deformation theory (FSDT) still propitious for analysis of laminated composite structures [9–16]. A closed form solution for linear axisymmetric bending analysis of circular plates having non-linearly varying thickness was given by Vivio and Vullo [24]. Dumir et al. [25] applied Chebyshev polynomials for studying the nonlinear large deflection transient response of isotropic circular plates. Huang and Li [26] studied the axisymmetric bending problem of isotropic circular plates the formulation being rendered non-linear due to a radial edge force. Oveisi and Shakeri [27] examined the forced vibration response of simply supported circular plates having piezoelectric layers. Boutaharband Benamar [28] presented the free vibration response of thin Functionally Graded Annular Plates with porosi-

https://doi.org/10.1016/j.asej.2018.10.006 2090-4479/Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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ties resting on elastic foundations. Recently, Wang et al. [29] have demonstrated a unified approach for free in-plane vibration of orthotropic circular, annular and sector plates. The formulation was limited in the sense that only in-plane displacements were considered and the plate was single layered. In the course of review of the literature, it is observed that lot of work has been done on the axisymmetric vibration analysis of polar orthotropic circular and annular plates of variable thickness using the different numerical techniques [30–39]. Among these works of great value, the literature pertaining to the relatively simpler analysis of axisymmetric static and free vibration response of laminated circular plates of uniform thickness is scarce. The practical significance of such results cannot be over-emphasized. Han and Liew [3] applied the DQM to report the axisymmetric linear static deflection results for isotropic circular plates. Hwang and Chen [40] applied Finite Element Method to find the frequencies of laminated circular plates with the laminates having layers comprising Aluminium, Steel, and GFRP. Lin and Tseng [41] employed finite element method to develop the free vibration results for laminated circular plates. Their formulation was asymmetric. However, the lamination scheme was mostly limited in the sense that the multilayered plates were having layers of different orthotropic materials. Thus, in general, the more practical case of laminates made up of cylindrically orthotropic materials differing only in fiber orientation is, somehow, found missing except for a few free vibration results given by Lin and Tseng [41]. During the above survey of the literature, authors found that the vibration analysis of moderately thick laminated circular and annular plates has not been proposed yet. Also, there is further need to study the vibration analysis of laminated circular and annular plates with elastically restrained edges. Among the methods used to study such types of problems, the DQM is increasingly being used to study the problems modelled by a set of differential equation(s) - linear or nonlinear, ordinary or partial [44–50]. As compared to the finite difference and finite element method the differential quadrature method can obtain very accurate numerical results using a considerably small number of grid points and hence required relatively little computational work [50]. Shu and Richards [44] used the generalized differential quadrature method to solve two-dimensional incompressible Navier-Stokes equations. Shu [50] has presented a good study of the differential quadrature technique and its various applications to different engineering problems. As already stated, Han and Liew [3] employed the DQM for determining the static response of isotropic circular plates. Recently, Sharma [42] employed the DQM for reporting the free vibration results of laminated sector plates having elastically supported edges. In this paper, an attempt is being made to archive the axisymmetric static and free vibration response of symmetrically laminated circular and annular plates of uniform thickness. Each layer of the circular plate will be made up of a fiber reinforced polar orthotropic material. The fiber orientation will be either radial (R) or circumferential (C). The FSDT formulation will be discretized using the DQM. The boundary conditions will be comprising the circumferential edge(s) being elastically supported - both in terms of linear as well as rotational displacement. Effect of important material and geometrical parameters will be studied.

Fig. 1. Geometry of the problem.

Based on first order shear deformation theory, the displacement components u*, v* and w* can be written as follows:

8
8 9 8 9 9 > > < u ðr; h; z; t Þ > < uðr; h; t Þ > < /r ðr; h; tÞ > = > = =
ðr; h; z; tÞ ¼ v ðr; h; tÞ þ z /h ðr; h; t Þ v > > > > > :
: : ; > ; ; w ðr; h; z; t Þ wðr; h; t Þ 0

where u, v and w are the middle surface displacements in the r, h and z directions, respectively, and /r , /h are the bending slopes in rz and hz planes, t is the time variable. The non-vanishing strain components at any point in the plate’s domain can thus be expressed as,

8 > > > > > > <

9

8

8

9

9

er > eðr0Þ > jðr0Þ > > > > > > > > > > > > > > > > ð0Þ > ð0Þ > > > > > > eh > > < eh > < jh > = > = = chz ¼ chz þ z 0 > > > > > > > > > > > >c > > > > > > > > 0 > > crz > > rz > > > > > > > > > > > : : ð0Þ > ; : ð0Þ ; ; crh jrh crh

ð2Þ

where

8 ð0Þ 9 er > > > > > > > ð0Þ > > > > < eh > =

8 > > > > > > <

8 ð0Þ 9 8 9 @/r jr > > > > @r > > > > > > >   > > > > ð0Þ > @/h 1 > > > > > > > > j / þ < h = < r r @h = ¼ 0 0 > > > > > > > > > > > 0 > > > > > 0 > > > > >  @/h > : ð0Þ > : 1 @/r ; > ; jrh  / þ h r @h @r

9 > > > > > > = /h þ 1r @w ; chz ¼ @h > > > > > > > > @w > > > > / þ > > crz > > >  r @r > > > > > : ð0 Þ > : 1 @u ; > ;  v þ @@rv crh r @h @u

 @r @ v  1 u þ @h r

ð3Þ The constitutive equations of a laminated composite structure which gives the in-plane resultant forces (Nr, Nh, Nrh) the resultant moments (Mr, Mh, Mrh) and the resultant shear forces (Qr, Qh) are as follows (Reddy [43])

8 9 8 N r > > A11 > > > > > > > > > > > A12 > Nh > > > > > > > > <0 = > rh ¼ > > Mr > > B11 > > > > > > > > > >B > > > M > h> > 12 > > > : : ; > Mrh 0

A12

0

B11

B12

A22

0

B12

B22

0

A66

0

0

B12

0

D11

D12

B22 0

0 B66

D12 0

D22 0

2. Mathematical formulation

 Fig. 1 shows the geometry of the circular plate. Each lamina is considered to be cylindrically orthotropic, with the fiber orientation being either in the radial or circumferential direction. The layers are assumed to be perfectly bonded.

ð1Þ

Qr Qh



 ¼ ks

A44

0

0 A55



crz chz

8 9 9 eð0Þ > > r > 0 >> > > > > > >> > > eðh0Þ > > > > 0 > > > > > > > > > ð0 Þ > < = B66 crh = 0 > > jðr0Þ > >> > > > > >> > ð0Þ > > > 0 > > jh > >> > > > > ;> > > > : ; ð 0 Þ D66

ð4Þ

jrh

 ð5Þ

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where Ks (¼ 5=6) is the shear correction coefficient. The extensional stiffness Aij , the bending- extensional coupling stiffness Bij , and the bending stiffness Dij of a laminate can be described by

Aij ¼

n X

Q kij ðzk  zk1 Þ; ði; j ¼ 1; 2; 6Þ;

ðpÞ

 C rik is the weighting coefficient of the value of u at the kth point for the pth order derivative with respect to r calculated at the ith point.  Wk is the value of u at the kth point in the 1-dimensional domain.

k¼1

Aij ¼ ks

n X

ðpÞ

The weighting coefficients C rik are evaluated using the recurrence relations [44],

Q kij ðzk  zk1 Þ; ði; j ¼ 4; 5Þ

k¼1 n n     1X 1X Q kij z2k  z2k1 ; Dij ¼ Q k z3  z3k1 ði; j ¼ 1; 2; 6Þ Bij ¼ 2 k¼1 3 k¼1 ij k

ðpÞ

ðp1Þ

C rik ¼ p C rii

C 1rik 

ð6Þ Q kij

where are the stiffness constants for a lamina. The stiffness constants Qij ‘s are defined for a layer as,

Q 11 ¼

Er ; ½1t2rh ðEh =Er Þ

Q 12 ¼ ½1tt2rhðEEh=Er Þ ; rh

h

Q 22 ¼

Eh ½1t2rh ðEh =Er Þ

Q 66 ¼ Grh ; Q 44 ¼ Ghz ; Q 55 ¼ Grz ; Q 45 ¼ 0

ðp1Þ

C rik ri  rk

! ð13Þ

for i, k = 1, 2,. . ., M with i – k, and, p = 1, 2,. . ..M – 1. ðpÞ

C rik ¼ 

M X

ðpÞ

ð14Þ

C rik

k¼1;k–i

ð7Þ

for i = 1, 2,. . ., M with p = 1, 2,. . .. M – 1



QM

j¼1;j–i r i  r j  Q rk Þ M j¼1;j–i r k



Here, notations follow from Reddy [46]. Hence, for example, (Er, Eh) are the Young’s moduli in the respective direction, trh is the Poisson’s ratio, etc. For the cases of symmetrically laminated plates made up of polar or cylindrically orthotropic layers having fiber orientations either in radial or circumferential directions,

for i, k = 1, 2,. . ., M with i – k. The grid point coordinates are generated using the cosine law as [47],

D16 ¼ D26 ¼ 0;

ri ¼ b þ

A45 ¼ 0

ð8Þ

The current problem being that of axisymmetric analysis of symmetrically laminated circular/annular plates with each layer being cylindrically orthotropic, the notable simplifications, will be:  All derivatives with respect to the circumferential (h) coordinate are zero.  The bending behavior will be decoupled from the in-plane deformations.  The only two displacement variables entering the formulation will be - (i) w, the transverse displacement of the mid-plane, and, (ii) /r , the angular displacement of the normal to the mid-plane about the h-direction, in the rz-plane (Fig. 1). The equations of motion in terms of the stress resultants (Mr, Mh and Qr) (Sharma [42]),

ðrMr Þ;r  Mh  rQ r ¼ ad rI2 /r;tt ðrQ r Þ;r ¼ ð1  ad Þrq þ ad rI0 w;tt

ð9Þ ð10Þ

In Eqs. (9) and (10), by putting the parameter ad = 1 or 0, the problem being analyzed becomes that of free vibration or of static bending, respectively. The inertias are defined as follows,

ðI 0 ; I 2 Þ ¼

n Z X k¼1

zk

zk1





qk 1; z2 dz

ð11Þ

with the index k representing one of the n number of layers. 3. Method of solution In this work, DQM is employed for spatial discretization of the governing equations. The pth spatial derivatives of a general function u (r) at the (i)th point in a 1-dimensional domain (having M divisions in that r-direction) can be approximated using the DQM as [44]:

 M X @ p ur  ðpÞ ¼ C rik Wk  p @r i

k¼1

ð12Þ

C 1rik

¼

ðr i 

 rj



   ða  bÞ ði  1Þp 1  cos 2 M1

ð15Þ

ð16Þ

for i = 1, 2,. . ., M. Where, as shown in Fig. 1, a and b are the outer and inner radii of the circular and annular plate. The elastic edge conditions applied in this work are as follows. For the annular plate, along the inner circumferential edge r = b:

Q r þ kw;b w ¼ 0;

M r þ k/r ;b /r ¼ 0

ð17Þ

Along the outer circumferential edge r = a:

Q r þ kw;a w ¼ 0;

M r þ k/r ;a /r ¼ 0

ð18Þ

For the solid circular plate, the continuity condition at the center r = 0 [41]

Q r ¼ 0;

/r ¼ 0

ð19Þ

Here, kij (i = w, /r and j = b, a) are stiffnesses associated with the corresponding degrees of freedom at the two circumferential edges. In the linear free vibration problem of a laminated circular/ annular plate, each of the two degrees of freedom (w, /r) at each of nodes can be assumed to be varying sinusoidally with respect to time, excluding the constrained degrees of freedom at the nodes located on boundary. Thus, for example, the displacement component w at the ith point is visualized as,

wij ¼ W ij sinxt

ð20Þ

Finally, using Eqs. (12) and (20), a linear algebraic equation is obtained corresponding to each of the two degrees of freedom at each node from the equations of motion 9–10, or from the boundary conditions – 17 and 18, depending on whether the node is on the boundary or not. The resulting set of algebraic equations define an eigenvalue problem as follows (for the free vibration case):

x2 ½M



W

Ur



  W þ ½K  ¼0

Ur

ð21Þ

Here, for example, the vector W contains the amplitudes for the displacement component w as follows,

346

8 9 W1 > > > > > > > > > W2 > > > > > > > > > > > ::: > > < = ::: W¼ > > > > > ::: > > > > > > > > > > > > W M1 > > > > > : ; WM

S. Khare, N.D. Mittal / Ain Shams Engineering Journal 10 (2019) 343–352

4. Results and discussions

ð22Þ

The system of simultaneous linear algebraic equations generated in the static bending problem, or, the eigenvalue problem given in Eq. (21) is then solved using the software –MATLAB [51]. A schematic diagram of solution procedure is given in Fig. 2.

The present study gives free vibration results, and also some static bending results, of moderately thick symmetrically laminated circular plates made up of cylindrically orthotropic layers. The boundary conditions considered here are various combinations of elastic edge conditions given in Eqs. (17)–(19). The effects of boundary stiffnesses, moduli ratio, lamination scheme, and thickness ratio are studied. With each layer being cylindrically orthotropic, the rotation of material axes by 90° about the global z-axis gives the material constants for a layer having fibers oriented in circumferential

Fig. 2. A schematic diagram of solution procedure.

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direction, starting with a layer having radially oriented fibers, and, vice versa. Thus, a 10 layer (n = 10) cross-ply symmetric plate denoted in rectangular domain by (0°/90°/0°/90°/0°)2 is represented here by (R/C/R/C/R)2 with R representing radial and C representing circumferential orientation of the fiber. With fibers oriented in circumferential direction, the required material properties (Material I) are as follows unless otherwise specified [41]. Material I

Eh =Er ¼ 50;

trh ¼ 0:0052; Grh ¼ 0:6613Er ; Ghz ¼ Grz ¼ 0:5511Er ð23Þ

With fibers oriented in radial direction, Material II will be obtained. Another cylindrically orthotropic material (fibers in circumferential direction) is, Material III [41]: Material III

Eh =Er ¼ 5;

trh ¼ 0:06; Grh ¼ 0:35Er ; Ghz ¼ Grz ¼ 0:292Er ð24Þ

The other default parameters of the laminated plates are as follows.

h=r0 ¼ 0:2;

auv ¼ 1

ðC=R=C Þ2 ;

n ¼ 6;

ð25Þ

The total thickness (h) is equally distributed among all the layers.   The two pairs of edge stiffnesses kw;j ; k/r ;j given in Eqs. (17) and (18) are prescribed with the help of non-dimensionalized stiffness parameters ðkw;j ; k/r ;j Þ defined below.

kw;j ¼ kw;j a3 =D22

ð26Þ

k/r ;j ¼ k/r ;j a=D22

ð27Þ

The boundary condition corresponding to each of the two degrees of freedom at each of the two circumferential edges can be continuously varied from the classical natural boundary condition to the classical essential boundary condition by varying the appropriate stiffness parameters ðkw;j ; k/r ;j Þ from zero to a very large positive value (=1e12). For having a convenient printed representation of the (2  2) matrix ki;j , a format in software environments dealing with matrices will be used. Thus, considering an annular plate, for the clamped case ki;j ¼ ½1e12; 1e12; 1e12; 1e12 and for the commonly used simply supported case ki;j ¼ ½1e12; 1e12; 0; 0.

Table 1 Convergence study with respect to the number of nodes, and, a comparison with the results of Lee and Schultz [4] for a clamped isotropic circular plate ki;j ¼ ½0; 1e12; 1e12; 0. M



xi a 2

pffiffiffiffiffiffiffiffiffiffiffiffi   qh=D; D ¼ Eh3 =12 1  v 2

i=1

2

3

4

5

6

7

8

13 15 17 19 21 23 [4]

h/a = 0.2 9.25 9.25 9.25 9.25 9.25 9.25 9.24

30.283 30.283 30.283 30.283 30.283 30.283 30.211

56.87 56.87 56.87 56.87 56.87 56.87 56.682

85.907 85.907 85.907 85.907 85.907 85.907 85.571

116.05 116.06 116.06 116.06 116.06 116.06 115.56

146.79 146.64 146.63 146.63 146.63 146.63 145.94

175.67 175.83 175.89 175.89 175.89 175.89 174.97

179.82 179.71 179.74 179.73 179.73 179.73 178.76

13 15 17 19 21 23 [4]

h/a = 0.005 10.215 10.215 10.215 10.215 10.215 10.215 10.215

39.762 39.762 39.762 39.762 39.762 39.762 39.762

89.067 89.06 89.06 89.06 89.06 89.06 89.06

158.09 158.05 158.05 158.05 158.05 158.05 158.05

245.85 246.84 246.68 246.69 246.69 246.69 246.69

354.06 355.33 354.88 354.93 354.93 354.93 354.92

525.04 481.73 483.78 482.55 482.72 482.7 482.69

725.2 632.96 632.18 629.61 629.99 629.93 629.91

Table 2 Convergence study with respect to the number of nodes, and, a comparison with the results of Lee and Schultz [4] for a simply supported isotropic circular plate ki;j ¼ ½0; 1e12; 1e12; 0. M



xi a2

pffiffiffiffiffiffiffiffiffiffiffiffi   qh=D; D ¼ Eh3 =12 1  v 2

i=1

2

3

4

5

6

7

8

13 15 17 19 21 23 [4]

h/a = 0.2 4.7787 4.7787 4.7787 4.7787 4.7787 4.7787 4.7773

25.031 25.031 25.031 25.031 25.031 25.031 24.994

52.65 52.649 52.649 52.649 52.649 52.649 52.514

83.054 83.051 83.051 83.051 83.051 83.051 82.766

114.31 114.34 114.34 114.34 114.34 114.34 113.87

145.82 145.82 145.79 145.79 145.79 145.79 145.13

167.4 167.33 167.34 167.34 167.34 167.34 166.29

177.28 177.01 177.17 177.16 177.16 177.16 176.28

13 15 17 19 21 23 [4]

h/a = 0.005 4.935 4.935 4.935 4.935 4.935 4.935 4.9349

29.716 29.716 29.716 29.716 29.716 29.716 29.716

74.134 74.131 74.131 74.131 74.131 74.131 74.131

138.27 138.23 138.23 138.23 138.23 138.23 138.23

221.26 222.08 221.98 221.99 221.99 221.99 221.99

323.77 325.88 325.3 325.37 325.36 325.36 325.37

475.07 445.91 449.06 448.19 448.31 448.29 448.29

694.5 590.86 593.64 590.3 590.79 590.72 590.71

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Also, as validated later in this work, even free circumferential edge condtions, stated normally in literature (Lin and Tseng [41]) as,

Q r ¼ 0;

Mr ¼ 0

ð28Þ

can be implemented by taking the two stiffnesses at that edge as very small ( 0). The reference boundary conditions for further studies are developed using the simply supported conditions given by Lee and

Table 3 Convergence study with respect to the number of nodes, and, a comparison with the results of Lin and Tseng [41] for clamped symmetrically laminated circular plates ki;j ¼ ½0; 1e12; 1e12; 1e12. M

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi a2 Þ qh=D11 i=1

2

3

4

5

6

7

8

13 15 17 19 21 23 [41]

h/a = 0.1 24.113 24.113 24.113 24.113 24.113 24.113 23.422

70.348 70.348 70.348 70.348 70.348 70.348

119.94 119.94 119.94 119.94 119.94 119.94

173.34 173.31 173.31 173.31 173.31 173.31

229.82 229.75 229.76 229.76 229.76 229.76

289.97 288.79 288.61 288.62 288.62 288.62

349.01 349.27 349.32 349.34 349.34 349.34

455.63 418.46 412.38 411.5 411.5 411.5

13 15 17 19 21 23 [41]

h/a = 0.01 27.636 27.636 27.636 27.636 27.636 27.636 27.487

89.658 89.659 89.659 89.659 89.659 89.659

167.62 167.7 167.69 167.69 167.69 167.69

264.9 264.2 264.27 264.26 264.26 264.26

384.06 379.14 379.71 379.61 379.62 379.62

530.55 517.82 513.3 513.74 513.66 513.67

713.63 684.8 665.4 666.7 666.11 666.19

1437.5 920.99 855.58 836.65 837.27 836.85

Table 4 Convergence study with respect to the number of nodes, and, a comparison with the results of Lin and Tseng [41] for simply supported symmetrically laminated circular plates ki;j ¼ ½0; 1e12; 1e12; 0. M

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi a2 Þ qh=D11 i=1

2

3

4

5

6

7

8

13 15 17 19 21 23 [41]

h/a = 0.1 19.935 19.935 19.935 19.935 19.935 19.935 19.758

62.186 62.186 62.186 62.186 62.186 62.186

110.82 110.82 110.82 110.82 110.82 110.82

164.31 164.32 164.31 164.31 164.31 164.31

221.25 221.42 221.4 221.4 221.4 221.4

282.39 281.22 281.15 281.15 281.15 281.15

344.66 342.17 342.99 342.86 342.88 342.88

452.28 412.34 406.78 406.06 406.06 406.06

13 15 17 19 21 23 [41]

h/a = 0.01 21.783 21.783 21.783 21.783 21.783 21.783 22.337

75.564 75.564 75.564 75.564 75.564 75.564

147.8 147.87 147.87 147.87 147.87 147.87

239.24 239.28 239.27 239.27 239.27 239.27

354.42 349.18 349.81 349.72 349.72 349.72

490.52 479.03 479.12 479.03 479.04 479.04

679.39 646.91 625.9 627.49 626.91 626.99

1028.5 847.98 796 794.02 793.21 793.28

Table 5 Convergence study with respect to the number of nodes, and, a comparison with the results of Lin and Tseng [41] for symmetrically laminated annular plates. M

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi a2 Þ qh=D11 i=1

4

5

6

7

8

13 15 17 19 [41]

h=a ¼ 0:1; b=a ¼ 0:1; ki;j ¼ ½1e12; 1e12; 1e12; 1e12 41.197 90.062 145.1 41.198 90.064 145.1 41.199 90.065 145.1 41.199 90.065 145.11 39.74

2

3

205.19 205.19 205.19 205.19

269.1 269.05 269.05 269.05

339.2 335.85 335.72 335.71

408.18 404.57 404.47 404.45

551.64 487.79 475.87 474.79

13 15 17 19 [41]

h=a ¼ 0:1; b=a ¼ 0:5; ki;j ¼ ½0; 1e12; 0; 1e12 35.904 114.67 212.91 35.904 114.67 212.9 35.904 114.67 212.9 35.904 114.67 212.9 35.01

325.9 325.89 325.89 325.89

447.68 447.81 447.8 447.8

572.4 573.8 573.65 573.66

704.96 699.46 699.58 699.53

796.72 794.67 795.78 795.66

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Table 6 Comparison with the static deflection results (at r = 0) of Han and Liew [4] for isotropic circular plates. M = 9; ki;j ¼ ½0; 1e12; 1e12; 1e12 for clamped (CL), ki;j ¼ ½0; 1e12; 1e12; 0 for simply supported (SS). 







x  64D= qa4 ; D ¼ Eh3 =12 1  v 2 ; v ¼ 0:3 h/a = 0.001 CL SS

Present 1.000 4.0774

h/a = 0.1 [3] 1.000 4.0769

Schultz [4]. In that work, the simply supported circumferential edge condition was implemented as,

w ¼ Mr ¼ 0

ð29Þ

and, clamped circumferential edge as,

w ¼ /r ¼ 0

ð30Þ

The developed formulation is validated by doing the convergence study with respect to the number of nodes (M), in parallel with carrying out comparisons with available results for isotropic and laminated plates. For the free vibration case of isotropic circular plates, Tables 1 and 2 give the convergence study and comparison of results with the results of Lee and Schultz [4], for clamped and simply supported cases, respectively. In these two tables, convergence to 5 significant digits (and also, an agreement with the results given by Lee and Schultz [4]) can be seen, even up to the fourth frequency, starting at (M = 13). Tables 3 and 4 demonstrate the convergence results along with comparison with the results of Lin and Tseng [41] for symmetrically laminated circular plates. In this study, the plate is 4 layered (I/III/III/I). Again, convergence of results can be seen even at (M = 13). As the reference Lin and Tseng [41] only has the fundamental frequency given, the comparison is done accordingly, and, agreement up to 1% is observed even at (M = 13). For the annular plates, the formulation is validated in Table 5. Here, the convergence study and the comparison are done for the cases of clamped-clamped and free-clamped cases of annular plates as presented by Lin and Tseng [41]. As mentioned in Table 5 itself, the edge stiffnesses are specified as ki;j ¼ ½1e12; 1e12; 1e12; 1e12 for the clamped-clamped case, and as ki;j ¼ ½0; 1e12; 0; 1e12 for the free-clamped case. Here also, convergence to five significant digits and comparison to about 2% is seen starting at (M = 13). As a result of the above-mentioned validation results, all the further results in this work are generated with (M = 15). Regarding the static deflection results, to the best of authors’ knowledge, the only available results are the ones given by Han and Liew [3] for isotropic circular plates. The excellent agreement obtained with the results of this work is given in Table 6. After having thus validated the present formulation, results of certain parametric studies are now presented. The effect of variation of the stiffness parameter ðki;j Þ on the lowest three nondimensional natural frequencies (wi, i = 1, 2, 3) of laminated circular and annular plates are given in Figs. 3 and 4. Each of these figures ratios (h/a = 0.1, 0.01). The applicable non-dimensional edge stiffnesses associated with the transverse displacement kw;j are kept constant at 1e12 (=1012) while the ones associated with the rotational degree of freedom ðk/r ;j Þ are varied from 105 to 104. The three frequencies increases, only slightly for (h/a = 0.1), and  = 1 to 10). For a little more for (h/a = 0.01) at around a value of (k the thinner plates, particularly for the annular ones, there is a significant jump for the two higher modes. Also, for both increasing as

Present 1.0457 4.1226

[3] 1.0457 4.1226

well as decreasing values of ðk/r ;j Þ, there is a consistent plateauing of the values of all the frequencies. The lamination scheme used in Fig. 3 is the same as used in Lin and Tseng [41]. Consequently, the fundamental frequencies in this figure, at the two extremes of the k-axis, match with the ones given in Tables 3 and 4. Variation of fundamental frequency with the annularity ratio is depicted in Fig. 5. Expectedly, as the plates become stiffer with the increasing annularity ratio, their fundamental frequency also increases.

Fig. 3. Variation of first three natural frequencies of the 4-layered (I/III/III/I) circular  plate, as used by Lin and Tseng [16], with the edge stiffness, ki;j ¼ [0, 1e12; 1e12, k].

Fig. 4. Variation of first three natural frequencies of the 6-layered (C/R/C)2 annular  k].  plate (b/a = 0.5) with the edge stiffness, ki;j ¼ [1e12, 1e12; k,

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Most of the results given in this work are for plates made of the Material I (Eq. (23)), in continuity with the results given by Lin and Tseng [41]. Another material used in this work is Material III (Eq. (24)) which differs significantly in terms of the moduli ratio (Eh/ Er). Hence, to obtain the effect of moduli ratio, Figs. 6 and 7 show the variation of the first three frequencies for 6-layered plates made of Materials I and III. The plates here are having annularity ratio b/a as 0.4. It can be seen that material-wise difference is more prominent for the two higher modes. All the figures up to this point have had the stiffness kw;j ¼ 0; and, the stiffness k/r ;j is varied, along with the k-axis, to get the edge condition varying from a simply supported edge to a clamped edge - as mentioned in Eqs. (29) and (30). As a significant variation, Fig. 8 demonstrates the effect of variation of edge conditions at the two edges of annular plates from free-simply supported to  in place of clamped-clamped. This is done by taking (kw;b ¼ k) (kw;b ¼ 1e12), as done for previous plots. The frequency variations are, again, more prominent for thinner plates. Fig. 7. Variation of first three natural frequencies of the 6-layered (C/R/C)2 annular plates (h/a = 0.05, b/a = 0.4) with the edge stiffness for materials I and III, ki;j ¼  k].  [1e12, 1e12; k,

Fig. 5. Variation of fundamental frequency of the 6-layered (C/R/C)2 circular/ annular plate (h/a = 0.2) with the edge stiffness for different annularities (b/a), ki;j ¼  k].  [1e12, 1e12; k,

Fig. 6. Variation of first three natural frequencies of the 6-layered (C/R/C)2 annular plates (h/a = 0.2, b/a = 0.4) with the edge stiffness for materials I and III, ki;j ¼ [1e12,  k].  1e12; k,

Fig. 8. Variation of first three natural frequencies of the 6-layered (C/R/C)2 annular  1e12; k,  k].  plates (b/a = 0.1) with the edge stiffness, ki;j ¼ [k,

Fig. 9. Variation of static deflection (at r = 0) of the 6-layered (C/R/C)2 circular plates  k].  with the edge stiffness, ki;j ¼ [0, 1e12; k,

S. Khare, N.D. Mittal / Ain Shams Engineering Journal 10 (2019) 343–352

Fig. 10. Variation of static deflection (at r = b) of the 6-layered (C/R/C)2 annular  1e12; k,  k].  plates with (b/a = 0.4) with the edge stiffness, ki;j ¼ [k,

Finally, variation of the static deflection with the edge stiffnesses is presented. In Fig. 9, static deflection of a circular plate at its center (r = 0) is shown, and, in Fig. 10, static deflection at the inner edge (r = b) of an annular plate is shown. Notably, the variation of edge condition in Fig. 10 is same as in Fig. 8. In Fig. 9, there is continuous decrease in deflection with increasing edge stiffnesses. However, the Fig. 10 shows that the deflection at the inner edge increases slightly and then again decreases with the edge stiffnesses. 5. Conclusions Using a DQM based procedure, the axisymmetric free vibration and bending problems of shear deformable symmetrically laminated circular and annular plates having elastically supported circumferential edges have been solved. Both translational, as well as rotational edge restraints, have been considered. The method facilitates a simple and computationally efficient solution in the sense that a one-dimensional grid of 15 was found satisfactory for all the cases dealt herein. The methodology developed proved robust and effective for all combinations of edge conditions varying continuously from the classical clamped to the classical simply supported edge conditions. Even free edge condition results were successfully generated. Detailed parametric analyses with respect to the effects on natural frequencies and static deflection have been carried out and the results have been graphically presented for future reference. References [1] Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. New York: McGraw-hill Book Company; 1959. [2] Lakshmi Kantham C. Bending and vibration of elastically restrained circular plates. J Franklin Inst 1958;265(6):483–91. [3] Han JB, Liew KM. Analysis of moderately thick circular plates using differential quadrature method. J Eng Mech Div-ASCE 1997;123(12):1247–52. [4] Lee J, Schultz WW. Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method. J Sound Vib 2004;269(3–5):609–21. [5] Li XY, Ding HJ, Chen WQ. Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk. Int J Solids Struct 2008;45(1):191–210. [6] Hashemi SH, Khorshidi K, Eshaghi M, Fadaee M, Karimi M. On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates. Appl Math Model 2012;36 (3):1132–47. [7] Khan K, Patel BP, Nath Y. Dynamic characteristics of bimodular laminated panels using an effcient layerwise theory. Compos Struct 2015;132:759–71.

351

[8] Wang CM. Relationships between Mindlin and Kirchhoff bending solutions for tapered circular and annular plates. Eng Struct 1997;19(3):255–8. [9] Bellifa H, Benrahou KH, Hadji L, Houari MS, Tounsi A. Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position. J Braz Soc Mech Sci Eng 2016;38(1):265–75. [10] Thai HT, Choi DH. A simple first-order shear deformation theory for laminated composite plates. Compos Struct 2013;106:754–63. [11] Kahya V, Turan M. Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Compos Part B-Eng 2017;109:108–15. [12] Jouneghani FZ, Dashtaki PM, Dimitri R, Bacciocchi M, Tornabene F. First-order shear deformation theory for orthotropic doubly-curved shells based on a modified couple stress elasticity. Aerosp Sci Technol 2018;73:129–47. [13] Izyan MN, Aziz ZA, Viswanathan KK. Free vibration of anti-symmetric angleply layered circular cylindrical shells filled with quiescent fluid under first order shear deformation theory. Compos Struct 2018;193:189–97. [14] Amabili M. Nonlinear vibrations and stability of laminated shells using a modified first-order shear deformation theory. Eur J Mech A Solids 2018;68:75–87. [15] Liu Q, Paavola J. General analytical sensitivity analysis of composite laminated plates and shells for classical and first-order shear deformation theories. Compos Struct 2018;183:21–34. [16] Sofiyev AH. Application of the first order shear deformation theory to the solution of free vibration problem for laminated conical shells. Compos Struct 2018;15(188):340–6. [17] Mahi A, Tounsi A. A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Appl Math Model 2015;39(9):2489–508. [18] Tounsi A, Houari MS, Benyoucef S. A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerosp Sci Technol 2013;24(1):209–20. [19] Meziane MA, Abdelaziz HH, Tounsi A. An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J Sandw Struct Mater 2014;16(3):293–318. [20] Belabed Z, Houari MS, Tounsi A, Mahmoud SR, Bég OA. An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates. Compos Part B-Eng 2014;60:274–83. [21] Bessaim A, Houari MS, Tounsi A, Mahmoud SR, Bedia EA. A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J Sandw Struct Mater 2013;15(6):671–703. [22] Hebali H, Tounsi A, Houari MS, Bessaim A, Bedia EA. New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. J Eng Mech Div-ASCE 2014;140(2):374–83. [23] Bennoun M, Houari MS, Tounsi A. A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mech Adv Mater Struct 2016;23(4):423–31. [24] Vivio F, Vullo V. Closed form solutions of axisymmetric bending of circular plates having non-linear variable thickness. Int J Mech Sci 2010;52:1234–52. [25] Dumir PC, Nath Y, Gandhi ML. Nonlinear transient response of isotropic circular plates. Comput Struct 1984;18(6):1009–18. [26] Huang Y, Li XF. Effect of radial reaction force on the bending of circular plates resting on a ring support. Int J Mech Sci 2016;119:197–207. [27] Oveisi A, Shakeri R. Robust reliable control in vibration suppression of sandwich circular plates. Eng Struct 2016;116(1):1–11. [28] Boutahar L, Benamar R. A homogenization procedure for geometrically nonlinear free vibration analysis of functionally graded annular plates with porosities, resting on elastic foundations. Ain Shams Eng J 2016;7(1):313–33. [29] Wang Q, Shi D, Liang Q, e Ahad F. A unified solution for free in-plane vibration of orthotropic circular, annular and sector plates with general boundary conditions. Appl Math Model 2016;40(2122):9228–53. [30] Gupta US, Lal R, Jain SK. Effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness. J Sound Vib 1990;139 (3):503–13. [31] Gupta US, Lal R, Jain SK. Buckling and vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation. J Sound Vib 1991;147(3):423–34. [32] Gupta US, Lal R, Sagar R. Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic Mindlin circular plates. Indian J of Pure Ap Mat 1994;25:1317-. [33] Gupta US, Ansari AH. Free vibration of polar orthotropic circular plates of variable thickness with elastically restrained edge. J Sound Vib 1998;213 (3):429–45. [34] Gupta AP, Bhardwaj N. Free vibration of polar orthotropic circular plates of quadratically varying thickness resting on elastic foundation. Appl Math Model 2005;29(2):137–57. [35] Bhardwaj N, Gupta AP. Axisymmetric vibration of polar orthotropic circular plates of quadratically varying thickness resting on elastic foundation. Int J Struct Stab Dy 2005;5(03):387–408. [36] Sharma S, Gupta US, Lal R. Effect of pasternak foundation on axisymmetric vibration of polar orthotropic annular plates of varying thickness. J Vib Acoust 2010;132(4):041001. [37] Viswanathan KK, Javed S, Aziz ZA, Prabakar K. Free vibration of symmetric angle-ply laminated annular circular plate of variable thickness under shear deformation theory. Meccanica 2015;50(12):3013–27.

352

S. Khare, N.D. Mittal / Ain Shams Engineering Journal 10 (2019) 343–352

[38] Lal R, Rani R. On radially symmetric vibrations of circular sandwich plates of non-uniform thickness. Int J Mech Sci 2015;99:29–39. [39] Yuan J, Chen W. Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness. Appl Math Mech-Engl Ed 2017;38(4):505–26. [40] Hwang JR, Chen LW. Vibrations of thick composite circular plates. Comput Struct 1992;43(1):129–35. [41] Lin CC, Tseng CS. Free vibration of polar orthotropic laminated circular and annular plates. J Sound Vib 1998;209(5):797–810. [42] Sharma A. Free vibration of moderately thick antisymmetric laminated annular sector plates with elastic edge constraints. Int J Mech Sci 2014;83:124–32. [43] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. New York: CRC Press; 2004. [44] Shu C, Richards BE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. Int J Numer Methods Fluids 1992;15:791–8. [45] Liew KM, Han JB, Xiao ZM. Differential quadrature method for thick symmetric cross-ply laminates with first-order shear exibility. Int J Solids Struct 1996;33 (18):2647–58. [46] Shu C, Wang CM. Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates. Eng Struct 1999;21:125–34. [47] Karami G, Malekzadeh P. Static and stability analysis of arbitrary straightsided quadrilateral thin plates by DQM. Int J Solids Struct 2002;39:4927–47. [48] Taha M, Essam M. Stability behavior and free vibration of tapered columns with elastic end restraints using the DQM method. Ain Shams Eng J 2013;4 (3):515–21. [49] Eftekhari SA. A simple and accurate mixed Ritz-DQM formulation for free vibration of rectangular plates involving free corners. Ain Shams Eng J 2016;7 (2):777–90. [50] Shu C. Differential quadrature and its applications in engineering. London: Springer-Verlag Limited; 2012. [51] https://www.mathworks.com/products/matlab/.

Sumit Khare is research student in the department of Mechanical Engineering, Maulana Azad National Institute of Technology, Bhopal. His research interests include vibration of continuous systems, solid mechanics, mechanics of composite structures.

Prof. N.D. Mittal is a faculty member in the department of Mechanical Engineering, Maulana Azad National Institute of Technology, Bhopal. He has published over 100 research papers in international journal of foreign repute and guided more than 8 Ph.D. students. His fields of interest include solid mechanics, structure dynamics, and numerical methods.