Applied Mathematical Modelling 36 (2012) 3870–3882
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Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory Huu-Tai Thai, Seung-Eock Kim ⇑ Department of Civil and Environmental Engineering, Sejong University, 98 Gunja Dong, Gwangjin Gu, Seoul 143-747, Republic of Korea
a r t i c l e
i n f o
Article history: Received 10 January 2011 Received in revised form 18 October 2011 Accepted 2 November 2011 Available online 9 November 2011 Keywords: Levy-type solution Refined plate theory Free vibration Orthotropic plate
a b s t r a c t Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail. Ó 2011 Published by Elsevier Inc.
1. Introduction Orthotropic plates are widely used in civil infrastructure systems and other structural applications because of their advantageous features such as high ratio of stiffness and strength to weight. Thus, the knowledge of their free vibration characteristic is very important to the structural designers. Unlike any other isotropic plate, the vibration of orthotropic plate is more complicated due to the material anisotropy. Thus, an accurate and reliable vibration analysis of the orthotropic plates is required to develop accurate and reliable design. In company with the increase in the application of orthotropic plates in engineering structures, a number of plate theories have been developed based on considering the transverse shear deformation effects. The classical plate theory (CPT), which neglects the transverse shear deformation effect, provides reasonable results for thin plate. This theory was employed for free vibration analysis of orthotropic plate by Tretyak [1], Sakata and Hosokawa [2], Jayaraman et al. [3], Harik et al. [4], Biancolini et al. [5], Xing and Lui [6], among others. Since it overpredicts natural frequencies of moderately thick plate, many shear deformation plate theories which account for the transverse shear deformation effects have been developed to overcome the limitation of the CPT. The Reissner [7] and Mindlin [8] theories are known as the firstorder shear deformation plate theory (FSDT), and account for the transverse shear effects by the way of linear variation of in-plane displacements through the thickness. Many studies of free vibration analysis of orthotropic plates have been carried out using FSDT [9–12]. Since these models violate the equilibrium conditions at the top and bottom surfaces of the plate, shear correction factors are required to correct the unrealistic variation of transverse shear stresses and shear strains through the thickness. These shear correction factors are sensitive not only to the geometric parameters of plate, but also ⇑ Corresponding author. Tel.: +82 2 3408 3291; fax: +82 2 3408 3332. E-mail addresses:
[email protected] (H.-T. Thai),
[email protected] (S.-E. Kim). 0307-904X/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.apm.2011.11.003
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to the boundary conditions and loading conditions. To overcome the drawbacks of the FSDT, a number of higher-order shear deformation plate theories (HSDT), which involve the higher-order terms in power series of the coordinate normal to the middle plane, have been proposed. Although the HSDTs have been adopted for predicting the natural frequencies of orthotropic plates [13–16], they are not convenience to use due to the higher-order terms introduced into the theory [17]. Therefore, Shimpi [18] has developed a two variable refined plate theory which is simple to use. The most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. In addition, it has strong similarities with the CPT in some aspects such as governing equation, boundary conditions and moment expressions. Recently, this theory was successfully extended to orthotropic plates [19,20], laminated composite plates [21], and functionally graded plates [22,23]. Although this theory was adopted for vibration analysis of orthotropic plates by Shimpi and Patel [19], their analytical solutions were limited to the rectangular plates with simply supported boundary conditions. Hence, it seems to be important to extend their analytical solutions to the rectangular plates with various boundary conditions. The purpose of this study is to derive the analytical solutions of refined plate theory for vibration problem of rectangular plates with various boundary conditions. Equations of motion are derived from the Hamilton’s principle. The Levy-type solution is employed for solving the governing equations of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. In addition, the closed-form solutions of orthotropic plate based on CPT are also generated for the verification purpose. The effects of boundary condition and variation of modulus ratio, aspect ratio, and thickness ratio on the natural frequencies of orthotropic plates are studied and discussed in detail. 2. Refined plate theory 2.1. Kinematics The two variable refined plate theory of Shimpi and Patel [19] accounts for quadratic variation of the transverse shear strains across the thickness of the plate, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate. The displacement field of this theory is as follows
@wb 1 5 z 2 @ws þz ; 4 3 h @x @x @wb 1 5 z 2 @ws þz ; Vðx; y; z; tÞ ¼ v ðx; y; tÞ z 4 3 h @y @y
Uðx; y; z; tÞ ¼ uðx; y; tÞ z
Wðx; y; z; tÞ ¼ wb ðx; y; tÞ þ ws ðx; y; tÞ;
ð1aÞ ð1bÞ ð1cÞ
where u and v are the mid-plane displacements of the plate in the x- and y-direction, respectively; wb and ws are the bending and shear components of transverse displacement, respectively; and h is the plate thickness. It should be noted that unlike the first-order shear deformation theory, this theory does not require shear correction factors. The kinematic relations can be obtained as follows:
ex ¼ e0x þ zjbx þ f jsx ; ey ¼ e0y þ zjby þ f jsy ; ez ¼ 0; cxy ¼ c0xy þ zjbxy þ f jsxy ; cyz ¼ g csyz ; cxz ¼ g csxz ;
ð2aÞ ð2bÞ ð2cÞ ð2dÞ ð2eÞ ð2fÞ
where
@u ; @x @v ¼ ; @y
@ 2 wb ; @x2 2 @ wb ¼ ; @y2
@ 2 ws ; @x2 2 @ ws ¼ ; @y2
e0x ¼
jbx ¼
jsx ¼
ð3aÞ
e0y
jby
jsy
ð3bÞ
@u @ v @ 2 wb @ 2 ws þ ; jbxy ¼ 2 ; jsxy ¼ 2 ; @y @x @x@y @x@y z 2 @w @w 1 5 z 2 5 csxz ¼ s ; csyz ¼ s ; f ¼ z þ z ; g ¼ 5 : 4 3 h 4 h @x @y
c0xy ¼
ð3cÞ ð3dÞ
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2.2. Constitutive equations For an orthotropic plate, the constitutive relations can be written as
8 > > > > > > <
9
38
2
9
rx > Q 12 0 0 0 > Q > ex > > > > 6 11 > > > 7> > > Q ry > Q 0 0 0 > ey > > > 7 6 12 22 = 6 = 7< 7 cxy ; 0 0 Q rxy ¼ 6 0 0 66 7> 6 > > > > 7> 6 > > > cyz > 0 0 Q 44 ryz > 0 5> > > > > 4 0 > > > > > > > > : : cxz ; 0 0 0 0 Q 55 rxz ;
ð4Þ
where Qij are the plane stress reduced elastic constants in the material axes of the plate, and are defined as
E1 ; 1 m12 m21
Q 11 ¼
Q 12 ¼
m12 E2 ; 1 m12 m21
Q 22 ¼
E2 ; 1 m12 m21
Q 66 ¼ G12 ;
Q 44 ¼ G23 ;
Q 55 ¼ G13
ð5Þ
in which E1, E2 are Young’s modulus, G12, G23, G13 are shear modulus, and m12, m21 are Poisson’s ratios. The subscripts 1, 2, 3 correspond to x, y, z directions of Cartesian coordinate system, respectively. 2.3. Equation of motion The strain energy of the plate is calculated by
P¼
Z
1 2
1 2
rij eij dV ¼
V
Z V
rx ex þ ry ey þ rxy cxy þ ryz cyz þ rxz cxz dV:
ð6Þ
Substituting Eqs. (2) into Eq. (6) and integrating through the thickness of the plate, the strain energy of the plate can be rewritten as
P¼
Z n
1 2
A
o Nx e0x þ Ny e0y þ Nxy c0xy þ M bx jbx þ Mby jby þ M bxy jbxy þ M sx jsx þ Msy jsy þ M sxy jsxy þ Q yz csyz þ Q xz csxz dx dy;
ð7Þ
where the stress resultants N, M and Q are defined by
ðNi ; M bi ; Msi Þ ¼ Qi ¼
Z
Z
h=2
ð1; z; f Þri dz; ði ¼ x; y; xyÞ;
ð8aÞ
ði ¼ xz; yzÞ:
ð8bÞ
h=2
h=2
g ri dz;
h=2
By substituting Eqs. (2) into Eq. (4) and the subsequent results into Eq. (8), the stress resultants are obtained as
8 9 9 2 38 @u=@x A11 A12 0 > > > < < Nx > = = 6 7 Ny @ v =@y ; ¼ 4 A12 A22 0 5 > > > > : : ; ; Nxy 0 0 A66 @u=@y þ @ v =@x 8 b9 2 9 38 2 2 > M > 0 > D11 D12 > < @ wb =@x > = < x > = 6 7 M by 0 5 ; ¼ 4 D12 D22 @ 2 wb =@y2 > > > > : ; > 2 : b > ; 0 0 D 2@ wb =@x@y 66 Mxy 8 s 9 9 2 38 2 2 D11 D12 0 > > < @ ws =@x > < Mx > = = 1 6 s 7 My 0 5 ; ¼ 4 D12 D22 @ 2 ws =@y2 > > > : : s > ; ; 84 0 0 D66 Mxy 2@ 2 ws =@x@y # " s @ws =@y Q yz A44 0 ¼ ; @ws =@x Q xz 0 As55
ð9aÞ
ð9bÞ
ð9cÞ
ð9dÞ
where Aij and Dij are called extensional and bending stiffness, respectively, and are defined in terms of the stiffness Qij as
ðAij ; Dij Þ ¼ Asii ¼
Z
Z
h=2
Q ij ð1; z2 Þdz ði; j ¼ 1; 2; 6Þ;
ð10aÞ
h=2 h=2
Q ii g 2 dz ¼
h=2
ðAii ; Dii ; F ii Þ ¼
Z
25 25 25 Aii 2 Dii þ 4 F ii 16 2h h
ði ¼ 4; 5Þ;
ð10bÞ
h=2
h=2
Q ii ð1; z2 ; z4 Þdz ði ¼ 4; 5Þ:
ð10cÞ
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The kinetic energy of the plate is can be written as
T¼
Z
1 2
" # _b _s 2 _b _s 2 @w @w @w @w _bþw _ s Þ2 dx dy dz; u_ z þ v_ z þ ðw f f @x @x @y @y
q
V
ð11Þ
where dot-superscript convention indicates the differentiation with respect to the time variable t; q is the mass density. Hamilton’s principle is used herein to derive the equations of motion. The principle can be stated in analytical form as
0¼
Z
t
dðP TÞdt;
ð12Þ
0
where d indicates a variation with respect to x and y. Substituting Eqs. (7) and (11) into Eq. (12) and integrating the equation by parts, collecting the coefficients of du, dv, dwb, and dws, the equations of motion for free vibration analysis of the orthotropic plates are obtained as follows:
@Nx @Nxy €; þ ¼ I0 u @x @y @Nxy @Ny dv : þ ¼ I0 v€ ; @x @y du :
ð13aÞ ð13bÞ
dwb :
@ 2 Mbxy @ 2 M by @ 2 M bx €b þ w € s Þ I2 r2 w € b; þ2 ¼ I 0 ðw þ 2 @x @x@y @y2
ð13cÞ
dws :
@ 2 Msxy @ 2 M sy @Q xz @Q yz @ 2 M sx I € s; €b þ w € s Þ 2 r2 w þ 2 þ þ þ ¼ I 0 ðw @x2 @x@y @y2 @x @y 84
ð13dÞ
where the inertias are given by
ðI0 ; I2 Þ ¼
Z
h=2
qð1; z2 Þdz:
ð14Þ
h=2
Since this paper deals with the out-of-plane vibration of orthotropic plates, the initial in-plane displacements u, and v have to equal to zero in Eqs. (13). The equations of motion for free vibration of orthotropic plate can be expressed in terms of wb and ws as
D11
@ 4 wb @ 4 wb @ 4 wb €b þ w € s Þ; € b I0 ðw þ 2ðD12 þ 2D66 Þ 2 2 þ D22 ¼ I 2 r2 w @x4 @x @y @y4
" # 1 @ 4 ws @ 4 ws @ 4 ws @ 2 ws @ 2 ws I2 2 € s Þ: € s I 0 ðw €b þ w A44 D11 þ 2ðD þ 2D Þ þ D A55 ¼ r w 12 66 22 4 2 2 4 2 84 @x @x @y @y @y @x2 84
ð15aÞ
ð15bÞ
Clearly, when the effect of transverse shear deformation is neglected (ws = 0), the governing equations (15) yield the equations of motion of orthotropic plate based on the classical plate theory. 3. Free vibration analysis 3.1. Boundary condition Consider a rectangular plate with length a and width b as shown in Fig. 1. It is assumed that the two opposite edges parallel to the x-axis are simply supported and the other two edges can have any arbitrary conditions such as free, simply supported or clamped conditions as shown in Fig. 2. The simply supported boundary conditions on two opposite edges parallel to the x-axis (y = 0 and y = b) are given as follows:
wb ¼ ws ¼ Mby ¼ M sy ¼ 0;
ð16Þ
while the boundary conditions for the remaining two edges (x = 0 and x = a) are: Clamped (C)
wb ¼ ws ¼
@wb @ws ¼ ¼ 0: @x @x
ð17Þ
Simply supported (S)
wb ¼ ws ¼ Mbx ¼ M sx ¼ 0:
ð18Þ
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y
b
x a Fig. 1. Levy-type plate with coordinate convention.
y
y
y
x
x SC plate
CC plate
SS plate
y
y
y
x FC plate
x
x FS plat e
x F F p la t e
Fig. 2. Boundary conditions of Levy-type plate.
Free (F)
Mbx ¼ Msx ¼ 0; @M bx
ð19aÞ
@Mbxy
€b @w þ I2 ¼ 0; @y @x s s @Mxy €s @M x I2 @ w þ2 þ Q xz þ ¼ 0: @x @y 84 @x @x
þ2
ð19bÞ ð19cÞ
3.2. Levy-type solution In order to solve the governing equation with the above-prescribed boundary conditions, a generalized Levy-type approach [9,15,24] is employed to obtain the closed-form solutions. The following solutions for the transverse displacement wb and ws are chosen to automatically satisfy the boundary conditions in Eq. (16)
wb ðx; y; tÞ ¼ ws ðx; y; tÞ ¼
1 X n¼1 1 X n¼1
W b ðxÞeixn t sin by;
ð20aÞ
W s ðxÞeixn t sin by;
ð20bÞ
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pffiffiffiffiffiffiffi where b ¼ np=b; i ¼ 1, and Wb(x), Ws(x) are unknown functions to be determined, xn denotes the natural frequency of the nth mode. Substituting Eqs. (20) into Eq. (15), a system of ordinary differential equations along the x-axis is obtained as follows: 0000
W b ¼ C 1 W 00b þ C 2 W b þ C 3 W s ; 0000
Ws ¼
C 4 W 00s
ð21aÞ
þ C5 W s þ C6 W b;
ð21bÞ
where ( )0 d/dx and the coefficients Ci are given as
2b2 ðD12 þ 2D66 Þ x2n I2 x2 I0 þ x2n I2 b2 D22 b4 ; C2 ¼ n ; D11 D11 2 2 2 x I0 84A55 þ 2b ðD12 þ 2D66 Þ xn I2 ; C3 ¼ n ; C4 ¼ D11 D11 2 4 2 2 2 84xn I0 þ xn I2 b D22 b 84A44 b 84x2n I0 ; C6 ¼ : C5 ¼ D11 D11
C1 ¼
ð22aÞ ð22bÞ ð22cÞ
By applying the concept of state space, the coupled system of Eqs. (21) can be converted into matrix form as
Z 0 ðxÞ ¼ TZðxÞ;
ð23Þ
0 00 000 T ZðxÞ ¼ W b W 0b W 00b W 000 ; b Ws Ws Ws Ws
ð24Þ
where
2
3
0
1
0
0
0
0
0
0
6 0 6 6 6 0 6 6C 6 2 T¼6 6 0 6 6 0 6 6 4 0
0
1
0
0
0
0
0
0
1
07 7 7 07 7 07 7 7: 07 7 07 7 7 15
C6
0
0
0
0 C1
0 C3
0
0
0 0
0 0
0 0
0 0
1 0
0 1
0
0
0
0
0
0
0
0
0 C5
0 C4
ð25Þ
0
For the case of classical plate theory, vector Z(x) and matrix T can be simplified by setting the shear component of transverse displacement to zero (ws = 0) as
T ZðxÞ ¼ W b W 0b W 00b W 000 ; b 2
0 6 0 6 T¼6 4 0 C2
3
1
0
0
0 0
1 0
07 7 7: 15
0 C1
ð26Þ
ð27Þ
0
A formal solution of Eq. (23) is given by
ZðxÞ ¼ eTx K;
ð28Þ
where K is a constant column vector determined from the boundary conditions of the two edges parallel to the y-axis; and eTx is the general matrix solution of Eq. (23) which can be expressed by
2
ek1 x
6 eTx ¼ ½E4
0 ..
0
3 7 1 5½E ;
.
ð29Þ
ek8 x
where ki ði ¼ 1; 8Þ and [E] are distinct eigenvalues and corresponding matrix of eigenvectors, respectively, associated with the matrix T. By substituting Eqs. (20) into Eqs. (17–19), the boundary conditions of the two edges along the y-axis (x = 0 and x = a) can be expressed in terms of unknown functions Wb and Ws as Clamped (C)
W b ¼ W s ¼ W 0b ¼ W 0s ¼ 0:
ð30Þ
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H.-T. Thai, S.-E. Kim / Applied Mathematical Modelling 36 (2012) 3870–3882
Simply supported (S)
W b ¼ W s ¼ D11 W 00b þ D12 b2 W b ¼
1 D11 W 00s D12 b2 W s ¼ 0: 84
ð31Þ
Free (F)
D11 W 00b þ D12 b2 W b ¼ 0; 1 D11 W 00s D12 b2 W s ¼ 0; 84
0 2 2 D11 W 000 b þ ðD12 þ 4D66 Þb I2 xn W b ¼ 0;
0 2 2 D11 W 000 s þ ðD12 þ 4D66 Þb I2 xn þ 84A55 W s ¼ 0:
ð32aÞ ð32bÞ ð32cÞ ð32dÞ
Substituting Eq. (28) into the appropriate eight boundary conditions of the two edges along the y-axis (x = 0 and x = a), a homogeneous system of equations is obtained as follow:
Gij K j ¼ 0;
ði; j ¼ 1; 8Þ;
ð33Þ
where
2
ek1 x
6 ½GðxÞ ¼ ½E4
0 ..
0
3 7 1 5½E :
. e
ð34Þ
k8 x
By setting the determinant of Gij equal to zero, one can determine the natural frequency x2n associated with the nth mode. It should be noted that this solution procedure cannot provide natural frequency directly because the undetermined natural frequency load x2n is included in matrix T. Hence, a trial and error procedure needs to be used to obtain the natural frequency results. The following iteration procedure has been used in the present study to calculate the natural frequency x2n : Step 1: Step 2: Step 3. Step 4.
Assign a small initial value of x2n . Form matrix T and compute the eigenvalues ki and eigenvectors [E] of T Form matrix [G] according to appropriate boundary conditions in Eqs. (30–32) Check if the determinant of matrix [G] changes sign. (a) If no, increase the natural frequency and go back to Step 2. (b) If yes, decrease the natural frequency by a small amount and go to next step.
Step 5. Check if the relative error between two successive iterations is within a given tolerance, stop the iteration. Otherwise, return to Step 2.
4. Results and discussion In this section, an orthotropic rectangular plate with two opposite edges simply supported and the other two edges having arbitrary boundary conditions is considered. For convenience, a two-letter notation is used to describe the boundary conditions of the remaining edges as shown in Fig. 2. For instance, FC indicates that one edge is free (F) and the other is clamped (C). For verification purpose, the results obtained by present theory are compared with those predicted by other theories. The effects of boundary condition and variation of modulus ratio, aspect ratio, and thickness ratio on the natural frequencies of orthotropic plates are studied and discussed in detail. The following material properties are used: Material 1 [16]
E1 ¼ 20; 830 ksi; G13 ¼ 3710 ksi;
E2 ¼ 10; 940 ksi; G23 ¼ 6190 ksi;
G12 ¼ 6100 ksi;
m12 ¼ 0:44; m21 ¼ 0:23:
ð35Þ
Material 2 [24]
E1 =E2 varied; G12 =E2 ¼ G13 =E2 ¼ 0:5;
G23 =E2 ¼ 0:2;
m12 ¼ 0:25:
ð36Þ
For convenience, the following nondimensional natural frequency load is used in presenting the numerical results in graphical and tabular forms
¼x x
a2 pffiffiffiffiffiffiffiffiffiffiffi q=E2 : h
ð37Þ
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H.-T. Thai, S.-E. Kim / Applied Mathematical Modelling 36 (2012) 3870–3882 Table 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ xn h q=Q 11 of simply-supported square plate (a = 10h). Comparison of nondimensional natural frequencies x Method
Mode
Exact [25] CPT [16] FSDT [16] HSDT [16] Present
n=1
Err. (%)
n=2
Err. (%)
n=3
Err. (%)
n=4
Err. (%)
0.0474 0.0497 0.0474 0.0474 0.0477
– 4.63 0.00 0.00 0.63
0.1033 0.1120 0.1032 0.1033 0.1040
– 7.77 0.10 0.00 0.67
0.1888 0.2154 0.1884 0.1888 0.1898
– 12.35 0.21 0.00 0.53
0.2969 0.3599 0.2959 0.2969 0.2980
– 17.50 0.34 0.00 0.37
Table 2 Comparison of nondimensional fundamental frequencies of plate with different boundary conditions (a/h = 100). a/b
E1/E2
Method
0.5
10
CPT Present
25
CPT Present
40
CPT Present
1.0
10
CPT Present
25
CPT Present
40
CPT Present
2.0
10
CPT Present
25
CPT Present
40
CPT Present
a
Boundary conditions CC
SC
SS
20.6530 (20.6543)a 20.5591 (20.5603) 32.4370 (32.4390) 32.0777 (32.0795) 40.9608 (40.9633) 40.2455 (40.2478)
14.3442 (14.3450) 14.3128 (14.3137) 22.4246 (22.4259) 22.3056 (22.3069) 28.2839 (28.2855) 28.0464 (28.0480)
9.3416 (9.3421) 9.3326 (9.3331) 14.4571 (14.4578) 14.4238 (14.4245) 18.1866 (18.1876) 18.1205 (18.1215)
FC 3.5613 (3.5614) 3.5599 (3.5600) 5.3050 (5.3051) 5.3002 (5.3003) 6.6028 (6.6030) 6.5935 (6.5937)
FS 1.3190 (1.3190) 1.3189 (1.3190) 1.3193 (1.3193) 1.3192 (1.3192) 1.3193 (1.3194) 1.3192 (1.3193)
FF 0.7124 (0.7124) 0.7123 (0.7123) 0.7123 (0.7123) 0.7122 (0.7122) 0.7123 (0.7123) 0.7122 (0.7122)
21.2870 (21.2889) 21.2059 (21.2078) 32.8434 (32.8464) 32.5486 (32.5515) 41.2828 (41.2866) 40.7027 (40.7062) 25.5129 (25.5184) 25.4372 (25.4427) 35.7226 (35.7303) 35.5162 (35.5237) 43.6060 (43.6154) 43.2325 (43.2415)
15.2028 (15.2042) 15.1734 (15.1747) 22.9826 (22.9847) 22.8815 (22.8835) 28.7279 (28.7305) 28.5312 (28.5337) 20.5897 (20.5941) 20.5500 (20.5543) 26.8480 (26.8537) 26.7603 (26.7659) 31.9032 (31.9099) 31.7564 (31.7630)
10.4954 (10.4963) 10.4854 (10.4863) 15.2265 (15.2278) 15.1960 (15.1972) 18.8036 (18.8052) 18.7462 (18.7477) 17.1329 (17.1364) 17.1094 (17.1129) 20.3640 (20.3682) 20.3247 (20.3288) 23.1574 (23.1622) 23.0996 (23.1043)
5.0583 (5.0586) 5.0561 (5.0564) 6.4142 (6.4146) 6.4096 (6.4100) 7.5248 (7.5253) 7.5173 (7.5178) 12.9353 (12.9377) 12.9215 (12.9238) 13.5537 (13.5562) 13.5380 (13.5404) 14.1245 (14.1271) 14.1067 (14.1093)
3.6112 (3.6114) 3.6104 (3.6105) 3.6116 (3.6118) 3.6108 (3.6110) 3.6119 (3.6121) 3.6111 (3.6112) 12.2357 (12.2379) 12.2238 (12.2259) 12.2284 (12.2305) 12.2164 (12.2186) 12.2280 (12.2301) 12.2160 (12.2182)
2.8502 (2.8503) 2.8495 (2.8496) 2.8492 (2.8493) 2.8485 (2.8486) 2.8491 (2.8492) 2.8484 (2.8485) 11.4075 (11.4094) 11.3963 (11.3981) 11.3974 (11.3993) 11.3862 (11.3880) 11.3958 (11.3977) 11.3846 (11.3864)
Ignored rotary inertia.
4.1. Comparison studies The first comparison is carried out for simply supported square plate using Material 1. The side-to-thickness ratio is assumed to be 10. Table 1 shows the comparisons of the natural frequencies predicted by the present theory with exact solution given by Srinivas and Rao [25] based on three-dimensional elasticity theory, and those reported by Reddy and Phan [16] based on CPT, FSDT, and HSDT. It can be seen that the present theory gives very good accuracy with the maximum error is only 0.67%. Both FSDT and HSDT give slightly better accuracy than the present theory, whereas CPT overpredicts the natural frequencies. The error between the results of CPT and shear deformation theories is more considerable for higher modes. It should be noted that the present theory involves only two unknown functions and two differential equations compared to three unknown functions and three differential equations in case of FSDT and HSDT. Moreover, both two differential equations in present theory are only inertial coupling and there is no elastic coupling, hence, these equations can be solved easily. Whereas, all three differential equations in FSDT and HSDT are inertial as well as elastic coupling, hence, these
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50 CPT
40
Present
30 ω
With rotary inertia Without rotary inertia
20
10 0
0
20
40
60
80
100
a/h Fig. 3. The effect of thickness ratio on nondimensional fundamental frequency of CC square plate (E1/E2 = 40).
50
40
CPT 30
With rotary inertia Without rotary inertia
ω 20
Present 10
0
0
10
20
30
40
50
E1/E2 Fig. 4. The effect of modulus ratio on nondimensional fundamental frequency of CC square plate (a = 5h).
equations are difficult to solve. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of orthotropic plates. The next comparison is performed for thin plates with different boundary conditions using Material 2. Table 2 shows a comparison of fundamental frequencies of thin plate (a/h = 100) obtained by present analysis based on present theory and CPT. It can be seen that the results are in good agreement, although the CPT slightly overestimates the fundamental frequency. The maximum difference between present theory and CPT appears in the case of CC plate. This difference increases with the decrease of thickness ratio a/h as shown in Fig. 3 and the increase of modulus ratio E1/E2 as shown in Fig. 4. This difference becomes considerable for thick plate with high modulus ratio. This is due to the fact that increasing the plate thickness and modulus ratio increases the transverse shear deformation effects. It is observed from Figs. 3,4 that the effect of rotary inertia is to decrease the frequencies. This effect is negligible in present theory, whereas it is considerable in CPT only for very thick plates. It should be noted that when the rotary inertia is ignored, the nondimensional frequencies predicted by CPT are independent of the side-to-thickness ratio a/h. 4.2. Parameter studies Parameter studies are carried out to investigate the effects of boundary condition and variations of thickness ratio, modulus ratio, and aspect ratio on the nondimensional fundamental frequencies of Levy-type plates. Material 2 is used. Based on
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Table 3 Nondimensional fundamental frequencies CC plate. a/b
a/h
E1/E2 3
10
20
30
40
50
0.5
5 10 20 50
7.9535 10.2041 11.1943 11.5352
9.9278 15.0373 18.6411 20.2829
10.8258 17.6572 24.1768 28.0499
11.3284 18.9857 27.5890 33.7397
11.7027 19.8333 29.9782 38.3200
12.0159 20.4445 31.7716 42.1776
1.0
5 10 20 50
8.7867 11.2124 12.2683 12.6301
10.8428 16.0752 19.4909 20.9637
11.8374 19.0023 25.2595 28.6650
12.3765 20.5455 28.9746 34.4146
12.7642 21.5377 31.6502 39.1030
13.0803 22.2500 33.6985 43.0937
2.0
5 10 20 50
12.3302 16.2738 18.1623 18.8424
13.9744 20.0853 23.7190 25.1992
15.1006 23.2497 29.2243 32.0377
15.7484 25.2219 33.2481 37.5067
16.2003 26.5937 36.3948 42.1473
16.5517 27.6162 38.9547 46.2147
10
20
30
40
50
Table 4 Nondimensional fundamental frequencies SC plate. a/b
a/h
E1/E2 3
0.5
5 10 20 50
6.4917 7.6335 8.0410 8.1690
8.7219 11.9527 13.6063 14.2179
9.7709 14.7852 18.2286 19.7562
10.2901 16.3600 21.3634 23.9265
10.6323 17.3918 23.7191 27.3668
10.8921 18.1334 25.5851 30.3311
1.0
5 10 20 50
7.4686 8.8947 9.4219 9.5898
9.5462 12.8632 14.4921 15.0824
10.6878 15.8014 19.0510 20.4197
11.2725 17.5431 22.2738 24.5288
11.6549 18.7235 24.7632 27.9597
11.9399 19.5882 26.7775 30.9416
2.0
5 10 20 50
11.6089 14.8681 16.3187 16.8218
12.7991 17.3070 19.5797 20.4196
13.8229 19.7820 23.2516 24.6437
14.4727 21.5656 26.1972 28.1975
14.9333 22.9280 28.6627 31.3099
15.2847 24.0101 30.7812 34.1026
10
Table 5 Nondimensional fundamental frequencies SS plate. a/b
a/h
E1/E2 20
30
40
50
0.5
5 10 20 50
4.8399 5.3202 5.4685 5.5126
6.9857 8.5241 9.1141 9.3044
8.2401 11.0551 12.4009 12.8804
8.8813 12.6703 14.7974 15.6246
9.2832 13.8239 16.7105 17.9239
9.5660 14.7001 18.3073 19.9333
1.0
5 10 20 50
6.1425 6.9515 7.2194 7.3012
7.8304 9.5628 10.2349 10.4530
9.0458 11.9334 13.2676 13.7360
9.7339 13.5598 15.5845 16.3474
10.1864 14.7744 17.4839 18.5726
10.5121 15.7267 19.1002 20.5377
2.0
5 10 20 50
10.9975 13.7909 14.9772 15.3796
11.6394 14.9934 16.5030 17.0294
12.3588 16.4739 18.4742 19.1992
12.9019 17.7038 20.2036 21.1436
13.3277 18.7467 21.7468 22.9151
13.6720 19.6461 23.1427 24.5504
3
the present closed-form solutions, comprehensive results are given in Tables 3–8 for orthotropic plates with different boundary conditions. For each case of boundary conditions, the aspect ratios are taken to be 0.5, 1 and 2, while four different thickness ratios of 5, 10 (corresponding to thick plates), 20, and 50 (corresponding to thin plates) are examined. In addition, the variations of nondimensional fundamental frequency with respect to thickness ratio a/h and modulus ratio E1/E2 are also illustrated in Figs. 5 and 6, respectively, for all boundary conditions. The following remarks can be made from Tables 3–8 and Figs. 5 and 6:
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Table 6 Nondimensional fundamental frequencies FC plate. a/b
a/h
E1/E2
0.5
5 10 20 50
2.1667 2.2876 2.3219 2.3319
3.0779 3.4166 3.5232 3.5552
3.8027 4.4688 4.7058 4.7803
4.2572 5.2333 5.6189 5.7450
4.5780 5.8366 6.3805 6.5655
4.8206 6.3336 7.0411 7.2906
1.0
5 10 20 50
3.7396 4.1039 4.2162 4.2497
4.2840 4.8210 4.9954 5.0483
4.8450 5.6262 5.8973 5.9816
5.2661 6.2861 6.6621 6.7816
5.5990 6.8489 7.3356 7.4938
5.8710 7.3401 7.9419 8.1417
2.0
5 10 20 50
9.1579 11.3590 12.2644 12.5667
9.3062 11.6008 12.5539 12.8735
9.4897 11.9172 12.9416 13.2876
9.6517 12.2071 13.3024 13.6751
9.8003 12.4801 13.6466 14.0462
9.9384 12.7400 13.9779 14.4050
10
20
30
40
50
3
10
20
30
40
50
Table 7 Nondimensional fundamental frequencies FS plate. a/b
a/h
E1/E2
0.5
5 10 20 50
1.2779 1.3081 1.3160 1.3183
1.2781 1.3083 1.3163 1.3186
1.2783 1.3085 1.3165 1.3188
1.2785 1.3086 1.3166 1.3189
1.2785 1.3087 1.3166 1.3189
1.2786 1.3087 1.3167 1.3189
1.0
5 10 20 50
3.2611 3.5147 3.5900 3.6121
3.2577 3.5102 3.5851 3.6072
3.2582 3.5105 3.5854 3.6074
3.2586 3.5108 3.5857 3.6077
3.2589 3.5110 3.5858 3.6079
3.2591 3.5111 3.5859 3.6080
2.0
5 10 20 50
8.9954 11.0973 11.9528 12.2370
8.9727 11.0557 11.9012 12.1817
8.9715 11.0515 11.8952 12.1752
8.9722 11.0512 11.8945 12.1743
8.9730 11.0515 11.8944 12.1742
8.9737 11.0517 11.8946 12.1742
10
20
30
40
50
3
Table 8 Nondimensional fundamental frequencies FF plate. a/b
a/h
E1/E2
0.5
5 10 20 50
2.2305 2.3049 2.3252 2.3310
2.2309 2.3053 2.3256 2.3314
2.2313 2.3055 2.3258 2.3316
2.2315 2.3056 2.3258 2.3316
2.2318 2.3056 2.3258 2.3316
2.2320 2.3056 2.3259 2.3317
1.0
5 10 20 50
2.5821 2.7801 2.8384 2.8556
2.5756 2.7721 2.8301 2.8470
2.5749 2.7713 2.8292 2.8462
2.5748 2.7712 2.8290 2.8460
2.5747 2.7711 2.8289 2.8459
2.5747 2.7711 2.8289 2.8459
2.0
5 10 20 50
8.3908 10.3538 11.1510 11.4156
8.3646 10.3087 11.0961 11.3572
8.3607 10.3019 11.0879 11.3485
8.3597 10.3003 11.0860 11.3464
8.3593 10.2996 11.0852 11.3456
8.3591 10.2993 11.0848 11.3452
3
(1) Regardless of boundary conditions, aspect ratios and modulus ratios, the nondimensional fundamental frequencies are increased by increasing the side-to-thickness ratio a/h (Fig. 5). The effect of thickness ratio becomes significant for thicker plate with high modulus ratio. This is due to the effect of shear deformations. (2) Regardless of aspect ratios and modulus ratios, the nondimensional fundamental frequencies are increased by increasing the modulus ratio E1/E2 (Fig. 6) except for FS and FF plates where the value of nondimensional fundamental frequency is almost constant with respect to the change of modulus ratio.
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CC SC SS FC FS FF
35 30 25
ω
20 15 10 5 0 0
20
40
60
80
100
a/h Fig. 5. The effect of thickness ratio on nondimensional fundamental frequency of square plate (E1/E2 = 25).
25
CC SC SS FC FS FF
20
15
ω 10
5
0 0
10
20
30
40
50
E1/E2 Fig. 6. The effect of modulus ratio on nondimensional fundamental frequency of square plate (a/h = 10).
(3) Nondimensional fundamental frequency increases when higher restraining boundary condition is used at the other two edges of Levy-type plates. In other words, the lowest and highest values of nondimensional fundamental frequency correspond to the FF and CC plates, respectively. Such behavior is due to the fact that higher constraints at the edges increase the flexural rigidity of the plate, leading to a higher frequency response.
5. Conclusions The closed-form solution for free vibration of orthotropic plate with two opposite edges simply supported and the other two edges having arbitrary boundary conditions has been developed based on the two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The governing equations of plate are solved by applying the concept of state space to the Levy-type solution. The accuracy and efficiency of the present theory have been demonstrated for free vibration analysis of orthotropic plates. It can be concluded that the present theory is not only accurate but also simple in predicting natural frequency of orthotropic plates compared to other shear deformation plate theories such as FSDT and HSDT. Due to the interesting features of the present theory, the present findings will be a useful benchmark results for researchers to validate their analytical and numerical methods in the future.
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