Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refined plate theory

Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refined plate theory

Accepted Manuscript Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refin...

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Accepted Manuscript Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refined plate theory Hien - Ta Duy, Hyuk-Chun Noh PII: DOI: Reference:

S0307-904X(15)00065-7 http://dx.doi.org/10.1016/j.apm.2015.01.062 APM 10413

To appear in:

Appl. Math. Modelling

Received Date: Revised Date: Accepted Date:

17 June 2014 1 December 2014 22 January 2015

Please cite this article as: H.-.T. Duy, H-C. Noh, Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refined plate theory, Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm.2015.01.062

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Analytical solution for the dynamic response of functionally graded rectangular plates resting on elastic foundation using a refined plate theory Hien - Ta Duy 1, Hyuk-Chun Noh1* 1

Department of Civil and Environmental Engineering, Sejong University, Seoul, South Korea * Corresponding author ([email protected])

A new refined plate theory for functionally graded material (FGM) plate is developed. Also, an analytical solution for the dynamic response of functionally graded rectangular plates resting on the Pasternak foundation under the transverse loading is investigated. By extending classical plate theory, the displacement field is assumed as the in-plane and transverse displacements consist of bending and shear components and it therefore does not need to use the shear correction factor. The material properties are assumed to vary continuously in the thickness direction according to the power-law form. The equations of motion are derived by using Hamilton’s principle. An analytical solution of simply supported FGM rectangular plates is presented by using state-space methods. The results obtained using the proposed refined plate theory are extensively compared with those obtained by the classical plate theory and finite element method. The accuracy of the new refined plate theory is demonstrated by comparing the results of chosen examples with ones predicted by other higher-order plate theories in previous studies. The effect of the power-law exponent index and the stiffness of the foundation on the behavior of the FGM plate are discussed in detail. Keywords: Dynamic response, analytical solution, FGM plate, refined plate theory, Pasternak foundation. 1. Introduction The functionally graded material (FGM) is a material in which the volume fractions of two or more material components are created to vary continuously with their position along the thickness direction. The functionally graded materials continuously improve structural performance by tailoring the material architecture at microscopic scales to optimize certain functional properties of structures [1]. The most common functionally graded materials are metal/ceramic composites because the metallic part has superior fracture toughness and the ceramic part has good thermal resistance [1, 2]. In fact, the concept of functionally graded materials is not new due to the fact that it has always occurred in nature. Examples for natural FGMs have been included in the bones and bamboo trees which have functional grading. Although the concepts of FGMs were proposed by Japanese scientists in 1984 [3], recently many researchers have interest in the mechanical behavior of FGM [4-9]. As demonstrated in the literature, the

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study on the transient response of functionally graded plates can be applied to a variety of engineering structures, including automobiles, space vehicles and so on [1-3]. The dynamic transient response of the plate problem is of importance in the various engineering fields, and has been investigated by many researchers. The dynamic problems of plate can be solved by an analytical method or semi-analytical method, numerical methods such as the finite element method, mesh-free method, etc. The analytical solution can be obtained for simple cases such as rectangular or circular plates. However, to obtain any analytical solution is virtually not possible if the geometry, boundary conditions, or types of loads are not simple. As an alternative, numerical methods can be used for many complex dynamic plate problems. In the case of analytical and semi-analytical methods, for determining transient response, the Navier approach has been used for simple rectangular plate [10, 11], and the Ritz method [12], Green function [13], Fourier-Bessel series [14] have been used for circular plate. Moreover, numerical methods have been applied to the dynamics of plate problems, e.g., the finite element method is used to study dynamics of laminated composite plates [15-18]. Chen et al. [19] studied the dynamic stability of the orthotropic plates subjected to an arbitrary dynamic load using the Galerkin method. Huang and Thambiratnam [20] studied the dynamic response of plates on an elastic foundation subject to moving loads using the strip method. Jiann-Quo and Yung-Ming [21] used an asymptotic theory to study the dynamic response of anisotropic inhomogeneous and laminated plate. Zafer Kazanc [22] used the Galerkin method to deal with the analysis of the nonlinear dynamic response of a laminated composite plate subjected to blast loading. Rahbar-Ranji et al. [23] analyzed the flexural behavior of thin plates with variable thickness resting on one parameter elastic foundation using the element-free Galerkin method.

In recent years, numerous studies have been dedicated to the investigation of FGM plates for the static, free vibration or buckling problems based on the first-order shear deformation theory (FSDT) [4, 8, 24], the third-order shear deformation theory (TSDT) [4, 7, 25, 26] and three dimensional elasticity [27, 28], with or without foundation interaction effects. However, only a few studies have been performed for the dynamic response of FGM plates: the dynamic response of FGM plate using hybrid Fourier-Laplace transform method [4], Galerkin’s method [4], the finite element method [5], and the meshless local Petrov–Galerkin method [29]. On the other hand, transient waves in FGM plates excited by impact loads is investigated using a numerical method [5]. Sun and Luo [9] studied the dynamic response of the rectangular FGM plates with clamped supports under an impulse load.

2

In the present paper, a novel refined plate theory is proposed and used to find analytical solutions for the transient response of FGM rectangular plate resting on an elastic foundation. A new refined plate theory with four independent unknowns takes account of transverse shear effects and the trigonometric variation of the transverse shear strains through the thickness of plates, and without using any shear correction factors. The Navier solution is used to determine the analytical solutions of static and dynamic problems for simply supported FGM plates. The accuracy of the present formula is verified by comparing it with results in the literature. An analytical solution for the equations of motion of the proposed shear deformation theory as well as the classical laminate theory are obtained by using the statespace method. We investigate the behavior of the simply supported FGM rectangular plate, which is subjected to distributed dynamic loads having time functions of step, triangular and sine pulse. A comparison between the dynamic responses predicted by the new refined theory and finite element method is addressed in detail to validate the proposed scheme.

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2. Formulation for a new refined plate theory In this section, we provide basic equations for a new refined plate theory (RPT) introducing trigonometric functions for the transverse shear strain field. For the proposed RPT, employing Hamilton’s principle, the equations of motion for plates are also derived. The geometry of the FGM plate and coordinate system ( x, y, z) with the origin located at the corner of the middle plane is shown in Fig. 1: z b y

a

h

Grading direction

x Fig. 1. Geometry and coordinate system for a rectangular FGM plate

2.1 Basic equations for the new refined plate theory In order to extend and improve the performance of the classical plate theory (CPT), an additional component, which enables us to take into account of the shear deformation effect, is needed to be added to the displacement field. As a result, the transverse displacement W is assumed to include two components of bending wb and shear w s which are functions of coordinates x, y, and time t as well [3032]. W ( x , y , z , t ) = wb ( x , y , t ) + w s ( x , y , t )

(1)

The displacements U in x-direction and V in y-direction include extension, bending, and shear components. The bending components ub ,vb are assumed to be similar to the displacements in the classical plate theory: ∂wb ∂x ∂wb vb = − z ∂y

ub = − z

4

(2)

The shear components us and vs are used together with w s in establishing trigonometric variations of shear strains γ y z , γ xz . It guarantees shear strains γ y z , γ xz through the thickness of the plate in such a way that shear stresses τ yz , τ xz are zero at the top and bottom faces of the plate. The expression for us and vs are proposed as the following:  1 8  π z   ∂w us = z − + cos    s  h   ∂x  5 5π  1 8  π z   ∂w vs = z  − + cos    s  h   ∂y  5 5π

(3)

Consequently, the new displacement field is rearranged as follows: ∂wb  1 8  π z   ∂ws + z − + cos   ∂x  h   ∂x  5 5π ∂w  1 8  π z   ∂ws V ( x, y, z , t ) = v ( x, y , z , t ) − z b + z  − + cos   ∂y  h   ∂y  5 5π U ( x, y , z , t ) = u ( x, y, z , t ) − z

(4)

W ( x, y, z , t ) = wb ( x, y , t ) + ws ( x, y , t )

where U,V,W are the displacement components at an arbitrary point (x,y,z) in the plate and ( u, v, wb , ws ) are the displacement components on the mid-plane, respectively. The linear strains can be obtained by differentiating Eq. (4) as: ε  ε x0  κ xb   x   0   b  ε y  = ε y + z κ y  +    0  b γ xy  γ xy  κ xy 

κ xs    fˆ κ ys  ,  s κ xy 

s γ yz  γ yz  ˆ = g    s γ xz  γ xz 

(5)

where:  ∂ 2 ws  − 2   ∂ws κ xs   ∂x   s   ∂ 2 ws  γ yzs   ∂y ,  s = κ y  =  − 2   s   ∂y  γ xz   ∂ws κ xy   ∂ 2 w   ∂x s  −2   ∂x∂y  1 8 4 8  π z   π z  8z π z  fˆ ( z ) = z  − cos  cos    , gˆ ( z ) = +  − sin   5 5π  h   h  5h  h   5 5π

 ∂u    0 ε x   ∂x   0   ∂v  ε y  =  ,  0   ∂y  γ xy   ∂u ∂v   +   ∂y ∂x 

 ∂ 2 wb  − 2  κ xb   ∂x   b   ∂ 2 wb  , κ y  =  − 2   b   ∂y  κ xy   ∂ 2 w  b  −2   ∂x∂y 

5

     

If the term ws in Eq. (4) is neglected, Eq. (4) corresponds to the classical plate theory. Consider a FGM plate made of two different materials, e.g., ceramic and metal, for which the mechanical properties of FGM such as Young’s modulus E and mass density ρ are expressed as p

 z 1 E ( z ) = ( Ec − Em )  +  + Em  h 2 p

 z 1 ρ ( z ) = ( ρc − ρm )  +  + ρm h 2

(6)

where the subscripts m and c represent the metallic and ceramic constituents, respectively; and p is the power law index of the volume fraction. The Poisson’s ratio ν is assumed to be constant. The linear constitutive relations of a FGM plate can be written as: σ x   Q11    σ y   Q12   σ yz  =  0    0 σ xz   σ xy   0  

Q12 Q22

0 0

0 0

0

Q44

0

0 0

0 0

Q55 0

0  ε x    0  ε y    0  γ yz    0  γ xz   Q66  γ xy   

(7)

where E (z) vE ( z ) , Q22 = Q11 , Q12 = 1 − v2 1 − v2 Q44 = G23 , Q55 = G13 , Q66 = G12 Q11 =

2.2 Equations of motion The kinetic energy of the mass system can be written as

T =

2 2   ∂w b ˆ ∂w s   ∂w b ˆ ∂w s  1 2   z u z f v z f ρ − − + − − + ( w b + w s )  dV ( )      ∫ 2 V ∂x ∂x   ∂y ∂y   

(8)

where ρ(z) is mass density. The strain energy of the foundation can be written as 1  2 Π F = ∫ K w ( wb + ws ) + K s A 2  

 ∂ ( w + w ) 2  ∂ ( w + w ) 2   b s b s   +    dxdy ∂x ∂ y      

6

(9)

where Kw and Ks are the transverse and shear stiffness coefficients of the Pasternak foundation, respectively. The potential energy of the applied load is P = − ∫ q ( wb + ws ) dxdy A

(10)

where q is the applied transverse load. The strain energy of the FGM plates can be written as

Π =

1 σ ij ε ij dV 2 ∫V

(11)

Substituting Eqs. (5) and (7) into Eq. (11) and integrating through the thickness of the plate, the strain energy of the plate can be rewritten as

Π =

1  M xbκ xb + M ybκ yb + M xyb κ xyb + M xsκ xs + M ysκ ys + M xys κ xys ∫ A 2 Nxε x0 + N yε y0 + Nxyγ xy0 + Qx γ xzs + Qyγ yzs  dxdy

(12)

where the force and moment resultants of the plate are defined by:  Nx  σ x  h /2    [ N ] =  N y  = ∫− h / 2  σ y dz N  σ   xy   xy 

σ x   M xb  h/2    b b  M  =  M y  = ∫ z  σ y dz − h/ 2  b σ   xy   M xy   M xs  h/2    M  =  M ys  = ∫ − h/ 2  s   M xy  s

σ x    fˆ  σ y  dz σ   xy 

h/2 Qx  σ xz    = ∫− h / 2 gˆ   dz Q  y  σ yz 

By substituting the stress-strain relations into Eq. (13) the following equations are obtained:

7

(13)

 [ N ]   [ A]     b    M   =  [ B ]    s s    M     B    Q y   A44s  =  Qx   0

[B] [D]  D s 

 B s    ε 0        D s    κ b      H s    κ s    

(14)

s 0  γ yz    A55s  γ xzs 

where: ε x0  κ xb  κ xs            ε 0  =, ε y0  κ b  = κ yb  , κ s  = κ ys   0  b  s γ xy  κ xy  κ xy 

 X11 [ X ] =  X12  0

0  0  , X 66 

X12 X 22 0

( X = A, B, B , D, D , H ) s

s

s

where Aij , Bij , etc., are the plate stiffness, defined by h /2

( A , B , D ) = ∫ (1, z, z )Q dz, ( i = 1, 2, 6) ij

ij

ij

Bijs = ∫

h/ 2

Dijs = ∫

h /2

fˆQij dz ,

−h / 2

− h /2

Hijs = ∫

h/ 2

−h / 2

Aijs = ∫

h/ 2

−h / 2

2

ij

− h /2

( i = 1, 2, 6)

ˆ dz , z fQ ij

( i = 1, 2, 6)

fˆ 2Qij dz,

( i = 1, 2,6)

ˆ ij dz , gQ

(i = 4,5)

The strains are computed from Eq. (5) by using the displacement fields in Eq. (4). Employing Hamilton’s principle the equations of motion can be derived. Collecting the coefficients for δ u , δ v , δ wb , and δ w s , respectively, the equations of motion of plate are obtained as:

8

b  ∂N x ∂N xy ∂w ∂w + = I 0 u − I1 − L1 s ∂x ∂y ∂x ∂x ∂N xy ∂x

+

∂N y ∂y

= I 0 v − I1

b  ∂w ∂w − L1 s ∂y ∂y

∂ 2 M xyb ∂ 2 M yb ∂ M + 2 + − K w ( wb + ws ) + K s ∇ 2 ( wb + ws ) + q ∂x 2 ∂x∂y ∂y 2 2

b x

(15)

 ∂u ∂ v b + w s ) + I1  b − L2∇ 2 w s = I0 ( w +  − I2∇2 w ∂ x ∂ y   ∂ 2 M xys ∂ 2 M ys ∂Qxz ∂Qyz ∂ 2 M xs + + + − K w ( wb + ws ) + K s ∇ 2 ( wb + ws ) + q + 2 ∂x 2 ∂x∂y ∂y 2 ∂x ∂y  ∂u ∂ v b + w s ) + L1  b − Lˆ1∇ 2 w s = I0 ( w +  − L2 ∇ 2 w  ∂x ∂y 

(

)

(

)

b, s where q is the transverse load, Ni , Qi , Mi are the stress resultants and the inertias I i , Li , Lˆ1 are

defined by Ii = ∫

h/ 2

−h / 2

Li = ∫

ρ ( z )zi dz, ( i = 0,1,2)

h/ 2

−h / 2

ˆ , ( i = 1, 2) ρ ( z )zi −1 fdz

h/ 2 Lˆ1 = ∫ ρ ( z ) fˆ 2 dz −h / 2

9

3. Solution procedures The state-space method [10] which is popular in control theory can be used to determine the response of the system under consideration. The state-space representations of the dynamic systems in Eq. (15) will be used to analyze the transient response of the simply supported FGM rectangular plates. The simple support boundary conditions are: for the RPT: ∂wb ∂ws = = N x = M xb = M xs = 0 at x = 0, a ∂y ∂y ∂w ∂w u = wb = ws = b = s = N y = M yb = M ys = 0 at y = 0, b ∂x ∂x

(16)

∂wb = N x = M xb = 0 at x = 0, a ∂y ∂wb u = wb = = N y = M yb = 0 at y = 0, b ∂x

(17)

v = wb = ws =

for the CPT: v = wb =

The Navier approach is used to derive the analytical solutions of the equations of motion in Eq.(15). The sinusoidal function is chosen to satisfy all the boundary conditions as follows: ∞



u ( x , y , t ) = ∑ ∑ U mn ( t ) cos α x sin β y n =1 m =1 ∞



v ( x, y , t ) = ∑ ∑ Vmn ( t ) sin α x cos β y n =1 m =1 ∞

(18)



wb ( x, y , t ) = ∑ ∑ Wbmn ( t ) sin α x sin β y n =1 m =1 ∞



ws ( x, y , t ) = ∑ ∑ Wsmn ( t ) sin α x sin β y n =1 m =1

where: α =

mπ nπ ,β = a b

Substitution of Eq. (18) into Eq. (15) and using the orthogonal property of sinusoidal function in Eq. (18), Eq. (15) can be written as follows:

10

 s11   s12  s13   s14

s12 s22

s13 s23

s23 s24

s33 s34

s14   U mn   m11 s24   Vmn   0  + s34  Wbmn   m13   s44  Wsmn  m14

m14   Umn   0    m24   Vmn   0     =   m34  W bmn   Fmn        m44  Wsmn   Fmn 

0 m22

m13 m23

m23 m24

m33 m43

0 m 22 m23

m13   Umn   0      m23   Vmn  =  0  m33  Wbmn   Fmn 

(19)

For classical plate theory, the equations of motion are:  s11 s  12  s13

s12 s22 s 23

s13   U mn   m11   s 23   Vmn  +  0 s33  Wbmn   m13

(20)

where: m11 = I 0 , m22 = I 0 , m13 = −α I1 , m14 = −α L1 , m23 = − β I1 , m24 = −α L1 m = I + I (α 2 + β 2 ) , m = I + L (α 2 + β 2 ) , m = I + Lˆ (α 2 + β 2 ) 33

0

2

34

0

2

44

0

1

s11 = A11α 2 + A66 β 2 , s12 = ( A12 + A66 ) αβ s13 = −α  B11α 2 + ( B12 + 2 B66 ) β 2  , s14 = B11α 2 + B66 β 2 s22 = A66α 2 + A22 β 2 , s23 = − β ( B12 + 2 B66 ) α 2 + B22 β 2  s24 = − β ( B12s + 2 B66s ) α 2 + B22s β 2  s33 = D11α 4 + 2 ( D12 + 2D66 ) α 2 β 2 + D22 β 4 + K w + K s (α 2 + β 2 ) s34 = D11s α 4 + 2 ( D12s + 2 D66s ) α 2 β 2 + D22s β 4 + K w + K s (α 2 + β 2 ) s44 = H11s α 4 + 2 ( H12s + 2 H 66s ) α 2 β 2 + H 22s β 4 + A55s α 4 + A44s β 4 + K w + K s (α 2 + β 2 )

We use the following notations for respective matrices: M = {mij } , S = {sij } T

F * = {0 0

Fmn

Fmn } for the RPT

F * = {0 0

Fmn }

T

for the CPT

where, b a

4 Fmn = ∫∫ q sin α x sin β ydxdy ab 0 0 We consider two kinds of loads as follows:

11

(21)

 q0 F ( t ) : for uniformly distributioned pressure  q= πx π y  F ( t ) : for sinusoidally distributioned pressure q0 sin sin  a b

(22)

Substitution Eq. (22) into Eq. (21) we obtain:  16q0  F ( t ) : for uniformly distributioned pressure  Fmn =  mnπ 2  q F ( t ) : for sinusoidally distributioned pressure  0



Three types function of times F ( t ) of dynamic loadings are considered as follow: For sine loading  sin ( π t t1 ) 0 ≤ t ≤ t1 F (t ) =  t ≥ t1 0

For step loading  1 0 ≤ t ≤ t1 F (t ) =  0 t ≥ t1

For triangular loading  (1 − t t1 ) 0 ≤ t ≤ t1 F (t ) =  0 t ≥ t1

To determine the natural frequency, the displacement fields are represented in harmonic functions. Thus, the natural frequency ω can be obtained by solving the following equation: S − ω2 M = 0

(23)

For solving Eqs. (19), (20) by using the state space methods, they can be rewritten as: Z = AZ + b

where, for the RPT:

12

(24)

Z = {U mn

{

b= 0

Vmn

0 0

Wbmn 0 bˆ1

Wsmn bˆ2

U mn

bˆ3

bˆ4

Vmn

Wbmn

T W smn }

T

}

and for the CPT: Z = {U mn

Vmn

{

0

b= 0

0

Wbmn 0 bˆ1

U mn bˆ2

bˆ3

Vmn

T W bmn }

T

}

and bi are the coefficients of the column matrix:

{bˆ} = M

−1

F*

The block matrix A is:  0 A= -1 -M S

I 0 

where I is a unit matrix. Solving Eq. (24), we obtain the following solution:

Z (t ) = e

A( t − t 0 )

t

Z ( t0 ) + ∫ e A(t −τ ) b (τ ) dτ

(25)

t0

where t0 is the initial time and Z ( t0 ) initial response while e A is exponential matrix which is constructed from eigenvalues

e A(t −τ )

λi and the eigenmatrix L

of A, as follow:

e λ1 (t −τ ) 0    −1 = [ L]    [ L]  λ8 ( t −τ )  e  0 

We assume that at the time t=0, the system is stationary at equilibrium position.

13

4. Numerical examples In order to investigate the adequacy of the proposed scheme, an Al/Al2O3 plate composed of aluminum (as metal) and alumina (as ceramic) is considered. The elastic moduli are chosen to be the same as [33] : Em = 7 0 GPa , ρ m =27 07 kg / m 3 , Ec = 38 0GPa , ρ c =38 00 kg/m 3 . The Poisson’s ratio of the plate is assumed

to be constant through the thickness and is equal to 0.3. The dynamic loading is uniformly distributed on the plate with the magnitude of q 0 .

z b y

a

q(x,y,t)

h x

Fig. 2 Example FGM plate z

Ec

Ec

Ec

h/2 0 -h/2

Ec (p=0)

Em (p=1)

Em (p=2)

Fig. 3 Elastic modulus variation of the FGM plate along the thickness direction according to Eq.(6) We can obtain the concept of the variation of the elastic modulus and the mass density of the FGM plate along the thickness direction by referring to Fig 3, which is drawn complying with Eq. (6). For convenience, the non-dimensional natural frequency normalized stress

σx ,σy

are defined as :

14

ϖ

, the equivalent stiffness of foundation

k1 ,k2

,

ϖ = ωh

ρm Em 4

k1 =

Kwa K a2 , k2 = s Dm Dm

10h3 Ec  a b  w ,  a 4 q0 2 2 a b h σ x = σ x  , , ±  q0 2 2 2 a b h σ y = σ y  , , ±  q0 2 2 2 w=

where Dm =

Em h 3

12 (1 − v 2 )

is the flexural rigidity of a full-metal plate.

The displacement and stress at the center of the plate are considered for every case. 4.1 Comparison with precedent research in statics and free vibration To validate the present shear deformation theory, comparisons are carried out with the available solutions for statics and free vibration cases. As the first example for the statics case, the square FGM plate

πx  π y   sin   is considered using both CPT and RPT.  a   b 

subjected to a sinusoidal load defined as q0 sin  The non-dimensional displacement

w and axial stresses σ x of FGM plate with a/h = 10 and different

values of p are compared in Table 1 with those given by Benyoucef et al.[34] using the hyperbolic shear deformation theory (HSDT). The foundation stiffness is not considered in this example.

Table 1 Comparison of non-dimensional displacement

p 1 2 3 4 5 6

w and axial stress σ x with Benyoucef et al.[34] σx

w CPT 0.5623 0.7206 0.7917 0.8281 0.8521 0.8713

Benyoucef 0.5889 0.7572 0.8372 0.8810 0.9108 0.9345

Present 0.5892 0.7567 0.8359 0.8791 0.9084 0.9318

CPT 30.537 35.657 38.217 40.101 41.848 43.572

15

Benyoucef 30.870 36.094 38.742 40.693 42.488 44.244

Present 30.828 36.022 38.64 40.566 42.343 44.086

7 8 9 10

0.8885 0.9048 0.9204 0.9355

0.9552 0.9741 0.9917 1.0083

0.9523 0.9711 0.9887 1.0054

45.280 46.957 48.591 50.173

45.971 47.661 49.303 50.890

45.806 47.491 49.130 50.716

As can be seen in Table 1, the results of the new refined plate theory and hyperbolic shear deformation theory are in good agreement. The displacements predicted by the present RPT and HSDT are clearly larger than those predicted by the CPT because the plate modeled by CPT is relatively stiffer. However, the deviation of the axial stress σ x between CPT and higher order plate theories is small. It is clear that the effect of shear deformation is more significant in predicting displacement than in predicting stresses.

The accuracy of the new refined plate theory is also verified with free vibration analysis. The nondimensional fundamental frequency of the square FGM plate with various values of power law index p, thickness-to-length ratios, and the stiffness of foundation are compared with those of Hasani et al. [33] in Table 2. From Table 2, it can be noticed that the non-dimensional fundamental frequency predicted by the present RPT is in good agreement with the results in [33] based on Reddy’s shear deformation plate theory. Furthermore, since the fundamental frequencies are smaller than Hasani, we note that the present results correspond to the more flexible cases, which means that the results are closer to the exact one than the precedent research in the literature. This assertion is reasonable since the formulation is based on displacement fields for which the convergence of analysis results comes from a stiff state to a flexible state.

Table 2 Comparison of non-dimensional fundamental frequency

p=0 k1, k2

h/a

(0,0)

(0,100)

p=1

ϖ

with Hasani et al. [33]

p=2

Present

Hasani Present

Hasani Present

Hasani

0.05

0.0291

0.0291

0.0222

0.0227

0.0202

0.0209

0.1

0.1134

0.1134

0.0869

0.0891

0.0788

0.0819

0.15

0.2453

0.2454

0.1885

0.1939

0.1708

0.1778

0.2

0.4152

0.4154

0.3205

0.3299

0.2897

0.3016

0.05

0.0406

0.0406

0.0378

0.0382

0.0374

0.038

0.1

0.1599

0.1599

0.1495

0.1517

0.1479

0.1508

0.15

0.3514

0.3515

0.3305

0.3365

0.3271

0.3351

16

0.2

0.6079

0.6080

0.5755

0.5876

0.5698

0.5861

0.05

0.0298

0.0298

0.0233

0.0238

0.0214

0.0221

0.1

0.1162

0.1162

0.0911

0.0933

0.0837

0.0867

0.15

0.2518

0.2519

0.1982

0.2036

0.1820

0.1889

0.2

0.4270

0.4273

0.3381

0.3476

0.3101

0.3219

(100,100) 0.05

0.0411

0.0411

0.0384

0.0388

0.0381

0.0386

0.1

0.1619

0.1619

0.1520

0.1542

0.1505

0.1535

0.15

0.3560

0.3560

0.3361

0.3422

0.3330

0.3412

0.2

0.6160

0.6162

0.5855

0.5978

0.5804

0.5970

(100,0)

4.2 Verification of RPT for dynamic responses In order to verify the proposed RPT, a simply supported square FGM plate subjected to uniformly distributed dynamic loading is taken as an example and the results are compared with finite element solution obtained by using ABAQUS commercial software. The parameters of the example are: dimension of plate a = b = 0.4m, h =

a , power law index p = 2 , magnitude of load q 0 = 2 × 10 6 N / m 2 , 10

duration of load application time t1 = 0.002s , the equivalent stiffness of foundation k1 = 200, k 2 = 0 . In the finite element analysis, the plate is divided into sixteen layers in the thickness direction with average properties within each layer and modeled as composite layup. We use 30x30 uniform mesh and

0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5

Forced Vibration

Free Vibration

Displacement (mm)

Displacement (mm)

time step ∆t = 2 ×10−6 s .

TSDT CPT Abaqus

0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5

Forced Vibration

Free Vibration

TSDT CPT Abaqus

0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

Fig. 4a. Comparison of the transient displacement Fig. 5a. Comparison of the transient displacement due to step load

due to triangular load

17

Free Vibration

Forced Vibration

Normalized Stress

Normalized Stress

125 100 75 50 25 0 -25 -50 -75 -100

TSDT CPT Abaqus 0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

100 80 60 40 20 0 -20 -40 -60 -80

3.0

TSDT CPT Abaqus 0.0

Fig. 4b. Comparison of the transient normalized

Free Vibration

Forced Vibration

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

Fig. 5b. Comparison of the transient normalized

stress σ x at the top center of the plate (z=h/2) due stress σ x at the top center of the plate (z=h/2) due to step load 40

Forced Vibration

TSDT CPT Abaqus

30 20

Free Vibration

30

TSDT CPT Abaqus

20 Normalized Stress

Normalized Stress

to triangular load

10 0 -10 -20 -30

Forced Vibration

Free Vibration

10 0 -10 -20 -30

-40 0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

Fig. 4c. Comparison of the transient normalized Fig. 5c. Comparison of the transient normalized stress σ x at the bottom center of the plate (z=-h/2) stress σ x at the bottom center of the plate (z=-h/2) due to step load

due to triangular load

The time history of the deflection and normalized stress at the center point of the plate are given in Figs. 4a-c, 5a-c due to the step load and triangular load, respectively. It can be seen that the oscillation of the plate deflection for FGM plate using RPT almost perfectly coincides with the finite element solution, showing that the RPT proposed is reasonable. The CPT gives a different displacement history to the RPT, since CPT does not consider the shear effect and accordingly the stiffness is different from each other. 4.3 Investigation into the effect of power law index p and foundation stiffness In this example, we consider a simply supported rectangular FGM plate subjected to time varying uniformly distributed load in the form of sine or step. The parameters of the example are: dimension of plate a = 0.3m, b=0.5m, h =

a , magnitude of load q 0 = 2 × 10 6 N / m 2 , duration of load application time 10

t1 = 0.002s , the equivalent stiffness of foundation k1 = 100, k 2 = 10 . Using solution (25), the displacement

18

history at the center of the plate based on the refined plate theory is obtained, and shown in Figs. 6a-c and 7a-c. 0.8 Forced Vibration

Free Vibration

0.2

0.1 p=1 p=3

0.0

Forced Vibration

0.6 Displacement (mm)

Displacement (mm)

0.3

Free Vibration

0.4 0.2 0.0 p=1 p=3

-0.2 -0.4

-0.1

-0.6 0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

0.0

80 70 60 50 40 30 20 10 0 -10 -20 -30

Free Vibration

Forced Vibration

0.0

σx, p=1;

σx, p=3

σy, p=1;

σy, p=3

0.5

1.0

1.5 Time (ms)

2.0

2.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

Fig. 7a. The transient displacement due to step load

Normalized Stress

Normalized Stress

Fig. 6a. The transient displacement due to sine load

0.5

175 150 125 100 75 50 25 0 -25 -50 -75 -100 -125 -150

3.0

Forced Vibration

0.0

σ x, p=1;

σx, p=3

σ y, p=1;

σy, p=3

0.5

1.0

1.5 Time (ms)

Free Vibration

2.0

2.5

3.0

Fig. 6b. The transient normalized stress σ x , σ y at Fig. 7b. The transient normalized stress σ x , σ y at the top center of the plate (a/2,b/2,h/2) due to sine the top center of the plate (a/2,b/2,h/2) due to step load

load

10 5

σx, p=1;

σx, p=3

σy, p=1;

σy, p=3

40 20 Normalized Stress

Normalized Stress

0 -5 -10 -15 -20 -25

Forced Vibration

σx, p=1;

σx, p=3

σy, p=1;

σy, p=3

0 -20 -40

Free Vibration

-30

Forced Vibration

-60 0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

0.0

0.5

1.0

1.5 Time (ms)

2.0

Free Vibration 2.5

3.0

Fig. 6c. The transient normalized stress σ x , σ y at Fig. 7c. The transient normalized stress σ x , σ y at the bottom center of the plate (a/2,b/2,-h/2) due to the bottom center of the plate (a/2,b/2,-h/2) due to sine load

step load

19

The response of deflection and normalized stress at the center of the plate for sine and step loads are shown in Figs. 6a-c and Figs. 7a-c, respectively. Two cases with different power law index p are calculated, i.e., p=1 and p=3. It can be seen in Figs. 6a and 7a that the transient deflection predicted for FGM plate with p= 3 is moderately lager than the plate with p= 1 since the flexural rigidity is higher for p=1 than p=3. Also, from Figs. 6b-c and 7b-c, we can observe the effect of power law index p on the normalized stress σ x and σ y . 1.25

1.25 Forced Vibration

1.00

Free Vibration

Free Vibration

0.75 Displacement (mm)

0.75

Displacement (mm)

Forced Vibration

1.00

0.50 0.25 0.00 -0.25 (k1,k2) = (0,0)

-0.50

(k1,k2) = (100,0)

-0.75

0.50 0.25 0.00 -0.25 (k1,k2) = (0,0)

-0.50

(k1,k2) = (0,10)

-0.75

(k1,k2) = (1000,0)

-1.00

(k1,k2) = (0,100)

-1.00 0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

3.0

(a) Effect of the foundation stiffness k1 ,k2 on the displacement at the center of the plate 400

400

300

200

200

100 0 -100 (k1,k2) = (0,0)

-200 -300

100 0 -100 (k1,k2) = (0,0)

-200

(k1,k2) = (100,0)

-300

(k1,k2) = (1000,0)

-400

-400

(k1,k2) = (0,10) (k1,k2) = (0,100) 0.0

0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

Free Vibration

Forced Vibration

Free Vibration

Normalized Stress

Normalized Stress

Forced Vibration 300

0.5

1.0

3.0

1.5 Time (ms)

2.0

2.5

3.0

(b) Effect of the foundation stiffness k1 ,k2 on the transient normalized stress σ x at (a/2,b/2,h/2) 125

125

100

(k1,k2) = (0,0)

75

(k1,k2) = (100,0)

75

(k1,k2) = (0,10)

50

(k1,k2) = (1000,0)

50

(k1,k2) = (0,100)

(k1,k2) = (0,0)

Forced Vibration

Free Vibration Normalized Stress

Normalized Stress

100

25 0 -25 -50 -75

Forced Vibration

Free Vibration

25 0 -25 -50 -75 -100

-100

-125

-125 0.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

0.0

3.0

0.5

1.0

1.5 Time (ms)

2.0

2.5

(c) Effect of the foundation stiffness k1 ,k2 on the transient normalized stress σ x at (a/2,b/2,-h/2) 20

3.0

Fig. 8 Effect of the foundation stiffness on dynamic behaviors of the FGM square plate under triangular load

To investigate the effect of the foundation stiffness, a simply supported square FGM plate is considered with parameters as: the dimension of plate a = 0.3m, b=0.3m, h =

q0 = 2×106 N / m2 , the duration of load application time

a , magnitude of load 20

t1 = 0 .002 s , power law index p = 2 . Figs. 8a-c

illustrate the effect of the foundation stiffness k1 ,k2 on dynamic behaviors of the FGM plate, respectively. It can be seen that the foundation stiffness has a significant effect on the dynamic behavior of the FGM plate. Also, it is apparent that deflection and normalized stress at the center point of the plate decrease if the foundation stiffness k1 ,k2 increases.

21

5. Conclusions A new refined plate theory for FGM plates on elastic foundation is successfully developed. The theory accounts for a trigonometric variation of the transverse shear strains through the plate thickness without using shear correction factors. Analytical solutions are used to solve the bending, free vibration of FGM plates using both the classical theory and refined plate theory. The validation of the proposed theory is ascertained by comparing deflection, axial stress, natural frequency predicted by present theory with previous studies in the literature. The results of present theory are comparable with those created by other shear deformation plate theories in literature. The dynamic responses are obtained by using the statespace method. In addition, the comparison between the dynamic responses obtained using the refined theory with those obtained by the classical plate theory and finite element solution show that the new refined plate theory gives more reasonable results. It can be concluded we can be confident in using the proposed refined theory for solving the static and dynamic behavior of plates. Moreover, numerical results show that the parameters of structure such as the foundation stiffness, and the power law have significant effects on the behavior of FGM plates. In addition, we expect that the analytical solutions given in this study can guide numerical analysts to validate their numerical approaches. Acknowledgements This work was supported by the Human Resources Development program (No. 20124030200050) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy. References [1] CPM SA, Varghese B, Baby A. A review on functionally graded materials. The International Journal Of Engineering And Science 2014;3(6):90-101. [2] Jha DK, Kant T, Singh RK. A critical review of recent research on functionally graded plates. Composite Structures. 2013;96(0):833-49. [3] Koizumi M. FGM activities in Japan. Composites Part B: Engineering. 1997;28(1–2):1-4. [4] Akbarzadeh AH, Abbasi M, Hosseini zad SK, Eslami MR. Dynamic analysis of functionally graded plates using the hybrid Fourier-Laplace transform under thermomechanical loading. Meccanica. 2011;46(6):1373-92. [5] Han X, Liu GR, Lam KY. Transient waves in plates of functionally graded materials. International Journal for Numerical Methods in Engineering. 2001;52(8):851-65. [6] Hao YX, Zhang W, Yang J, Li SY. Nonlinear dynamic response of a simply supported rectangular functionally graded material plate under the time-dependent thermalmechanical loads. J Mech Sci Technol. 2011;25(7):1637-46. [7] Reddy JN. Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering. 2000;47(1-3):663-84. [8] Singha MK, Prakash T, Ganapathi M. Finite element analysis of functionally graded plates under transverse load. Finite Elements in Analysis and Design. 2011;47(4):453-60. [9] Sun D, Luo S-N. The wave propagation and dynamic response of rectangular functionally graded material plates with completed clamped supports under impulse load. European Journal of Mechanics - A/Solids. 2011;30(3):396408. 22

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23