BENDING OF FUNCTIONALLY GRADED THICK PLATES RESTING ON WINKLER–PASTERNAK ELASTIC FOUNDATIONS

  

BENDING OF FUNCTIONALLY GRADED THICK PLATES RESTING

ON WINKLER–PASTERNAK ELASTIC FOUNDATIONS (1) (2) (1)

  BENYOUCEF Samir , BACHIR BOUIADJRA Rabbab ,ZIDI Mohamed , BOURADA

  (1) (1) (1)

  Mohamed , TOUNSI Abdelouahed ,ADDA BEDIA El Abbas (1) Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbes, BP 89 Cité Ben M’hidi 22000 Sidi Bel Abbes, Algérie.

  Samir.benyoucef@gmail.com (2) Université de Sciences et Technologie d’Oran (USTO), Algérie.

  

Abstract — The static response of simply supported functionally graded plates (FGP)

subjected to a transverse uniform load (UL) or sinusoidally distributed load (SL) and

resting on an elastic foundation is examined by a new hyperbolic displacement model in

this paper. Present theory exactly satisfies stress boundary conditions on the top and the

bottom of the plate. No transversal shear correction factors are needed because a correct

representation of the transversal shearing strain is given. Materials properties of the plate are assumed to be graded in the thickness direction according to a simple power-law

distribution in terms of the volume fractions of the constituents. In the analysis, the

foundation is modeled as two parameter Pasternak type foundation and Winkler type if the

second foundation parameter is zero. The equilibrium equations of a functionally graded

plate are given based on the present hyperbolic shear deformation plate theory. Effects of

stiffness of the foundation and gradient index on mechanical responses of the plates are

discussed. It is established that elastic foundations affects significantly the mechanical

behavior of functionally graded thick plates. Numerical results presented in the paper can

serve as benchmarks for future analyses of functionally graded thick plates on elastic

foundations.

  Keywords — FG plates; Winkler–Pasternak elastic foundation; Shear deformation

1. INTRODUCTION mechanical properties at the interface between

  In conventional laminated composite the layers. One way to overcome this problem structures, homogeneous elastic laminae are is to use functionally graded materials within bonded together to obtain enhanced which material properties vary continuously. mechanical and thermal properties. The main The concept of functionally graded material inconvenience of such an assembly is to create (FGM) was proposed in 1984 by the material stress concentrations along the interfaces, scientists in the Sendai area of Japan [1]. The more specifically when high temperatures are FGM is a composite material whose involved. This can lead to delaminations, composition varies according to the required matrix cracks, and other damage mechanisms performance. It can be produced with a which result from the abrupt change of the continuously graded variation of the volume fractions of the constituents. That leads to a continuity of the material properties of FGM: this is the main difference between such a material and an usual composite material. The FGM is suitable for various applications, such as thermal coatings of barrier for ceramic engines, gas turbines, nuclear fusions, optical thin layers, biomaterial electronics, etc. accurate solution of catamaran passenger vessel operated in the river. Several studies have been performed to analyze the behaviour of functionally graded plates and shells. Reddy [2] has analyzed the static behaviour of functionally graded rectangular plates based on his third-order shear deformation plate theory. Cheng and Batra [3] have related the deflections of a simply supported functionally graded polygonal plate given by the first-order shear deformation theory and a third-order shear deformation theory to that of an equivalent homogeneous Kirchhoff plate. Cheng and Batra [4] have also presented results for the buckling and steady state vibrations of a simply supported functionally graded polygonal plate based on Reddy’s plate theory. Loy et al. [5] have studied the vibration of functionally graded cylindrical shells using Love’s shell theory. By using the sinusoidal shear deformation theory (SSDT), Zenkour [6] presented Navier’s analytical solution of FG plates. However, bending analyses of FGMs are quite limited, especially of those on elastic foundations. To describe the interactions of the plate and foundation as more appropriate as possible, scientists have proposed various kinds of foundation models, as documented well in Ref. [7]. The simplest model for the elastic foundation is the Winkler model, which regards the foundation as a series of separated springs without coupling effects between each other, resulting in the disadvantage of discontinuous deflection on the interacted surface of the plate. Zhemochkin and Sinitsyn

  [8] introduced an elastic combined foundation which is a classical foundation covered by a layer of a Winkler foundation. Filonenko- Borodich [9] developed a model that improved upon the Winkler model by connecting the top ends of the springs with an elastic membrane stretched to a constant tension. Hetényi [10, 11] created an interaction among the springs in the foundation by imbedding an additional plate with flexural rigidity in the Winkler foundation. Vlasov [12] proposed also a more refined model, mainly the two-parameter model. Gorbunov-Posadov [13] presented also the problems of flexure of plates and beams lying on a linearly deformable foundation. Pasternak [14] improved upon the Winkler model by connecting the ends of the springs to a plate, or "shear layer," consisting of incompressible, vertical elements, which can deform only by lateral shear. From then on, the Pasternak model was widely used to describe the mechanical behavior of structure– foundation interactions [15–20]. Cheng and Kitipornchai [21] proposed a membrane analogy to derive an exact explicit eigenvalues for compression buckling, hydrothermal buckling, and vibration of FGM plates on a Winkler–Pasternak foundation based on the first-order shear deformation theory. The same membrane analogy was later applied to the analyses of FGM plates and shells based on a third-order plate theory [22, 23]. This paper presents a new hyperbolic shear deformable plate theory for analyzing the static response of functionally graded plates resting on a Winkler–Pasternak foundation. The present theory and the well-known Reddy’s higher-order plate theory (HPT) [2] contain the same number of dependent variables as in the first-order plate theory (FPT), but results in more accurate prediction

  E = E − E ,

  of deflections and stresses, and satisfy the zero where _{CM C M} tangential traction boundary conditions on the surfaces of the plate. However, the present

  − h ≤ z ≤ h p

  where /

  2 / 2 and is the power law

  theory and HPT do not require the use of shear index which takes values greater than or equal correction factors. In conclusion, the present

  M C

  to zero. Subscripts and refer to the metal theory gives accurate results, especially and ceramic constituents which denote the transverse shear stresses, than other theories material property of the bottom and top including HPT. surface of the plate, respectively. The FGM plate is made of an isotropic material with material properties varying in the

  2.1. Constitutive equations

  thickness direction only. The governing partial For elastic and isotropic FGMs, the differential equations are reduced to a set of constitutive relations can be written as: coupled ordinary differential equations in the thickness direction, which are then solved by using the Navier solutions for simply

      Q Q

  σ   ε _{x 11 12 x}

  supported rectangular plates. Numerical results    

    = Q Q σ ε and  y  ^{12 22  y }  

  are presented for an aluminum/alumina    

    Q

  τ γ _{xy 66 xy}      

  functionally graded plate. To make the study

   τ   Q   γ  _{yz 44 yz}

  reasonably, effects of the foundation stiffness (3)

  =       τ Q γ _{zx zx 55}      

  and gradient index of the Young’s modulus on the mechanical behavior of FGM plates are where ( σ , σ , τ , τ , τ ) and ( ε , ε , _{x y xy yz yx x y} investigated.

  γ , γ , γ ) are the stress and strain _{xy yz yx}

2. THEORETICAL FORMULATIONS

  components, respectively. Using the material Consider a rectangular FG plate having the properties defined in Eqs. (1), stiffness thickness h , length a , and width b , as

  Q

  coefficients, ij , can be expressed as depicted in Fig. 1. It is assumed to be rested on (4a)

  E ( z )

  a Winkler–Pasternak type elastic foundation

  Q = Q = , _{11 22 2} 1 − ν k

  with the Winkler stiffness of and shear

  k

  stiffness of . The FGM plate is subjected to ₁ E ( z )

  ν (4b) Q = , ₁₂ q ( x , y ) ²

  a transverse load . The material

  1 − ν

  properties of the plate are assumed to vary continuously through the thickness of the

  E ( z ) Q = Q = Q = , _{44 55} ^{66 (4c)} +

  plate. Also, it is assumed that the Poisson’s

  2 1 ν ( )

  ν

  ratio is constant. Based on the power law distribution, the relationship between the Based on the thick plate theory, the assumed

  E z

  Young’s modulus and for ceramic and displacement field can be defined in unified metal FGM plate is assumed as [6]: form as follows:

  (5a)

• u = u ( x , y ) − z w f ( z ) _{p , x x}

  θ

  (1)

  1 z 

• E ( z ) = E E  + _{M CM}

    ,

  (5b)

   _{, y y} θ

• 2 h  v = v ( x , y ) − z w f ( z )

  (2) (5c)

  w = w x y ( , )

  The stress and moment resultants of the FGM where, u , v , w are displacements in the x , plate can be obtained by integrating Eq. (3)

  y z u v w

  , directions, , and are midplane over the thickness, and are written as: displacements, θ and θ are rotations of _{x y}

  normals to the midplane about -axis and _a

• y x

     N  A B B  ε   

f (z )   a  

  axis, respectively, and represents shape a (9)

  M = B D D k and Q = A θ       _{a a a}    

   

  function determining the distribution of the

  P B D F k θ

       

  transverse shear strains and stresses along the thickness. ( ) and ( ) are partial derivative _{, x , y} in which:

  x y

  with respect to and , respectively. In this

      u w _{, x , xx}

     

  study a new hyperbolic shear deformation

  = v k = − w ε , ,  , y   , xx     

• plate theory is obtained by setting u v w _{, y , x , xy}

  2       θ _{x , x} θ

   π      x

  (10a)

  h / sinh z

π   k = =

  ( ) θ , θ ,  y , y   

  θ cosh / 2 h

  π   θ ( ) ^y

     

• f ( z ) = z −

  (6) θ θ _{x , y y , x}

    −

  [ cosh π / 2 − 1 ] [ cosh π /

  2 1 ] ( ) ( ) _{t t}

  N = N N N M = M M M

  This function helps to satisfy zero transverse , , , , , ,

  { x y xy } { x y xy }

  shear stresses at top and bottom surfaces of the t (10b)

  P = P , P , P

  ,

  { x y xy }

  plate. The parabolic distributions of transverse shear stresses through the plate thickness are

  A A B B     _{11 12 11 12}

  taken into account for the analysis, by means

      A = A A B = B B _{12 22} , , _{12 22}

     

  of the hyperbolic function of the assumed

      A B _{66 66}

      displacement field.

  D D   _{11 12} By substituting the displacement relations (5)  

  D = D D (10c) _{12 22} ,  

  into the strain-displacement equations of the

    D ₆₆

   

  elasticity, the normal and shear strain components are obtained as: (7a)

      = u − zw f ( z ) B B D D ^{a a a a} ε θ _{x , x , xx x , x 11 12 11 12}

• _{a a a a a a}

      = = B B B , D D D , _{12 22 12 22}

      _{a a}

  (7b)    

  = v − zw f ( z ) ^{66 66} ε θ

_{y , x , yy y , y    }

_{a a}   F F _{11 12 a a a}  

• B D

  (7c)

  γ = + 2 z ( ) θ θ + + u v − w f z F = F F , (10d) _{xy , y , x , xy x , y y , x  } ( ) ^{12 22} _a  

  F ₆₆  

  (7d)

  γ = f ' ( z ) θ _{yz y a}   _{t a} A ₄₄

  (10e)

  Q = Q , Q A =

  , ,

  { xz yz }   _a

  (7e)

  A γ = f ' ( z ) θ _{xz x}

    ₅₅   N M where and are the basic components of stress resultants and stress couples, P are

  where

  additional stress couples associated with the df ( z )

  (8)

  f ' ( z ) = Q transverse shear effects, is transverse shear dz stress resultant. Note that the superscript t denotes = = = = = = u w θ N M P at y = , b _{x y y y} the transpose of the given vector. The stiffness

  To solve this problem, Navier presented the

  A B coefficients and ,… etc., are defined as _{ij ij}

  external force in the form of a double _{a a a  } trigonometric series:

    ∞ ∞ A B D B D F _{11 11 11 11 11 11 h / 2}

  1   _{a a a 2 2   (11a)} q ( x , y ) = q sin( x ) sin( y )

  A B D B D F = Q ,1 z , z , f ( z ), z f ( z ), f ( z ) ν dz λ µ , _mn  ^{12 12 12 12 12 12  11 ( )  }

  ∑∑

  (11a)

  ∫ _{a a a − ν}

  1     _{h / 2 m 1 n 1} − = =

  A B D B D F _{66 66 66 66 66 66}    

  2   and

  = m a n / b m n _{a a a a a a} where λ π / and µ = π , and and A , B , D , B , D , F = A , B , D , B , D , F

  ( ^{22 22 22 22 22 22 ) ( 11 11 11 11 11}

^{11 )}

  are modes numbers. The coefficients q for _mn

  E ( z ) , Q = _{11 (11b) 2}

  the case of uniformly distributed load are

  1 − ν _{h / 2} defined as follows: _{a a} E ( z ) ₂ A = A = f z dz (11c) _{44 55} [ ' ( ) ] 16 q 

  ∫

  1 ν

• 2

  ( ) _{h / 2} for m , n odd,

  − ₂  mn

  π q = _{mn } (15)

   

  Since the bottom surface of the plate is for m , n even,

  

  assumed subjected to Winkler–Pasternak elastic foundation (see Fig. 1), the reaction–

  q

  where represents the intensity of the load at deflection relation at the bottom surface of the the plate center. For the case of sinusoidally model is expressed by ₂ distributed load, f = k w − k ∇ w . _{e 1} (12)

   π x   π y  q ( x , y ) = q sin sin   (16)

    , a b

      f

  where is the density of reaction force of _e

  q = q

  we have m = n = 1 , and . ₁₁ foundation. If the foundation is modeled as the

  Following the Navier solution procedure, the linear Winkler foundation, the coefficient k ₁ static bending solution can be obtained as in Eq. (12) is zero.

  (17) The governing equations of equilibrium can be K ∆ = F

  

[ ]

{ } { }

  derived by using the principle of virtual displacements in the same way as described by

  ∆ F

  where and denotes the columns

  { } { } Akavci [24]. _T ∆ = A , B , C , X , Y ,

  and

  { } { mn mn mn mn mn }

  2.2. EXACT SOLUTIONS FOR FGM _T

  (18) F = , , q , , .

  { } { mn } PLATes

  and Rectangular plates are generally classified in

  a a a a a   _{11 12 13 14 15}

  accordance with the type support used. We are

    a a a a a _{12 22 23 24 25}

  here concerned with the exact solutions for

      K = a a a a a

  [ ] , (19) _{13 23 33 34 35}

  simply supported FGM plate. The following

    a a a a a _{14 24 34 44 45}

   

  boundary conditions are imposed at the side

    a a a a a _{15 25 35 45 55}

    edges.

  A B C

  X Y

  , , , , and are arbitrary (13) mn mn mn mn mn

  v = w = θ = N = M = P = x = , a _{y x x x} at parameters to be determined.

3. NUMERICAL RESULTS

  _{τ yz τ , } h _yz  

  2 h b a aq

  _{σ x σ ,} h _x    

    =

  3 , 2 ,

  2 h b a aq h _{y y}

  σ σ ,   

     − =

  3 , , h aq h _{xy xy}

  τ τ ,    

    =

  6 , ,

  2 h a aq

     =

       =

  ,

  2 , b aq

  _{τ xz τ ,} h _xz / h z z = .

  It may be observed from Table 1 that results of the present theory are compared with those obtained using sinusoidal shear deformation plate theory (SSDPT). As the plate becomes more and more metallic, the difference increases for deflection

  w

  and in-plane longitudinal stress ^x

  σ while it decreases for

  in-plane normal stress ^y

  σ . It is important to

  observe that the stresses for a fully ceramic plate are the same as that for a fully metal plate. This is because the plate for these two cases is fully homogeneous and the stresses do not depend on the modulus of elasticity. In order to validate the present method in the case of plates resting on elastic foundation, the results for the dimensionless deflections of isotropic thick plate are compared with the previously published results. Table 2 presents the deflections of a uniformly loaded homogeneous square simply supported plate on Winkler foundation. The results are compared with those exact values for Mindlin plates obtained by Kobayashi and Sonoda [26] who used the Levy-series method and by Lam et al. [27] using Green’s functions. It can be seen that the results are in close agreement. Table 3 presents the central deflections of a uniformly loaded homogeneous square simply supported plate on Winkler–Pasternak foundations. The results are compared with those obtained by Lam et al. [27] using Green’s functions. It can be seen that the results are in close agreement. Figs. 2–7 depict the through-the-thickness distributions of the shear stresses ^{τ xz} ; the in- plane longitudinal and normal stresses ^{σ x} , and the longitudinal tangential stress ^{τ xy} in the square FGM plate under sinusoidal distributed load. The volume fraction exponent of the FGM plate is taken as

  2 = p in these figures.

  Distinction between the curves in Figs. 2 and 3 is obvious. It is observed that transverse shear

  2 , 2 ,

  In this study, bending analysis of FG plates on elastic foundation by a new hyperbolic shear deformation theory is suggested for investigation. For simplicity, Poisson’s ratio of plate is assumed to be constant in this study for that the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus [25]. The Poisson’s ratio is fixed at

  = 3 . ν . Comparisons are made with

  kg/m

  available solutions in literature. In order to verify the accuracy of the present analysis, some numerical examples are solved. The material properties used in the present study are: Metal (Aluminium, Al): ⁹

  10 = 70 × _M E

  N/m

  2 = 3 . ν ; ; 2702 = _M ρ kg/m

  3 .

  Ceramic (Alumina, Al

  2 O

  3 ⁹ 10 380 × = _C E

  ): ; N/m

  2 = 3 . ν

  ; ;

  _C 3800 = ρ

  3 D a k K / ⁴ = .

  E h w ^c

  In all examples, the foundation parameters are presented in the non-dimensional form of and D a k K / ² _{1 1}

  =

  , where

  ) 1 ( 12 / ^{2 3} ν − = Eh D is a reference bending rigidity of the plate.

  As a first example; the deflections and the dimensionless stresses of the square FG plate (

  = 10 / h a ) for different values of the power

  law p are compared with those of sinusoidal shear deformation plate theory (SSDPT) of Zenkour [6] in Table 1. The various non-dimensional parameters used are

     

    =

  2 ,

  2

  10 ₄ ³ b a w q a

  ,  stress ( xz τ ) increases gradually with decreasing

  K

  . This means that the deflections are computed for plates with different ceramic–metal mixtures. It is clear that the deflections decrease smoothly as the ratio of metal-to-ceramic moduli increases. Further, the deflections decrease gradually as either K or ₁ K increases.

  Num. Meth. Eng., 47 , 663–684.

  2. Reddy J.N. (2000), “Analysis of functionally graded plates”, Int. J.

  1. Koizumi M. (1997), “FGM Activities in Japan”, Composites, 28, 1–4.

  REFERENCES

  In the present study, closed form solutions for bending analysis of functionally graded plates resting on a Winkler–Pasternak elastic foundation are developed on the assumption that transverse shear displacements vary as a hyperbolic function through the thickness of plate. The stress and displacement response of the plates have been analyzed under sinusoidal loading. The gradation of properties through the thickness is assumed to be of the power law type and comparisons have been made with homogeneous isotropic plates. Non- dimensional stresses and displacements are computed for plates with ceramic–metal mixture. It is seen that the basic response of the plates that correspond to properties intermediate to that of the metal and ceramic, is necessarily lie in between that of ceramic and metal. This behaviour is found to be true irrespective of boundary conditions. Thus, the gradients in material properties play an important role in determining the response of the FGM plates. It can be seen from the results that deflections and the stresses decrease gradually as either K or ₁ K increases. Numerical results given in the present paper render a benchmark for the analyses of FGM thick plates on elastic foundations in the future

  4. CONCLUSION

  E E /

  or ₁ K . It is indicated that large moduli of elastic foundation can enhance bending rigidity of the plate. The through-the- thickness distributions of the shear stress ^{τ xz} are not parabolic as in the plate made of pure material. It is to be noted that the maximum value occurs at

  Contrary to the in-plane longitudinal and normal stress, the tensile and compressive values of the longitudinal tangential stress, xy τ (Figs. 6 and 7), is maximum at a point on the bottom and top surfaces of the FGM plate, respectively. It is clear that the minimum value of zero for all in- plane stresses x σ and xy τ occurs at 153 . = z . Finally, the exact maximum deflections of simply supported FGM square plate are compared in Figs. 8 and 9 for various ratios of moduli, _{c m}

  or ₁ K .

  K

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  z

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  6. Zenkour AM. (2006), “Generalized shear deformation theory for bending analysis of functionally graded materials”, Appl. Math. Modell., 30, 67–84.

  Sic., 41, 309–324.

  (1999), “Vibration of functionally graded cylindrical shells”, Int. J. Mech.

  5. Loy C.T., Lam K.Y., Reddy J.N.

  21. Cheng ZQ, Kitipornchai S. (1999), “Membrane analogy of buckling and vibration of inhomogeneous plates”,

  ASCE. J. Eng. Mech., 125(11), 1293– 1297.

  22. Cheng ZQ, Batra BC. (2000), “Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plate”, J.

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  23. Reddy JN, Cheng ZQ. (2002), “Frequency correspondence between membranes and functionally graded spherical shallow shells of polygonal

  Figure1: FGM plate resting on elastic plan form”, Int J Mech Sci., 44(5), foundation 967–985.

  24. Akavci S. S. (2005), “Analysis of _0,5 thick laminated composite plates on an _0,4 elastic foundation with the use of _0,3 various plate theories” Mechanics of _0,2 Composite Materials, 44(5), 445 – 460. _0,1

  25. Delale F, Erdogan F. (1983). The _0,0 crack problem for a nonhomogeneous

• _{-0,1 K =0} plane, Journal of Applied Mechanics, -0,2 K =20 _{-0,3 K =40} 50, 609–614.
• _{-0,4 K =100 K =80} ^{K =60}

  26. Kobayashi H, Sonoda K. (1989), _-0,5 “Rectangular Mindlin plates on elastic _{0,00 0,02 0,04 0,06 0,08 0,10 0,12} foundations”, Int. J. Mech. Sci., 31, 679–692.

  Figure2: Variation of transversal shear stress 27. Lam K.Y., Wang C.M., He X.Q.

  τ xz

  ( ) through-the-thickness of a square FGM (2000), “Canonical exact solutions for

  K a / h =

  10

  plate for different values of . ( , Levy-plates on two-parameter

  p = 2 and K = ₁ 10 ).

  foundation using Green’s functions”, Eng. Struct., 22, 364–378.

• 0,5 ^{-0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5} _{0,00 0,05 0,10 0,15 0,20 0,25 0,30} ^K
^{K 1 =0 1 =2 K 1 =4 K K 1 =6 K 1 =8 1} =10<
• 0,5 ^{-0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5} _{-1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4} ^{K =0 K =20 K =40 K =60 K =80 K =100}

• 0,5 ^{-0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5} _{-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6} ^{K =0 K =20 K =40 K =60 K =80 K =100}

  = 10 / h a , 2 = p and

  (

  τ ) through-the-thickness of a square FGM plate for different values of K .

  Figure.6: Variation of longitudinal tangential stress ( ^xy

  = 10 / h a , 2 = p and K 10 = ).

  (

  FGM plate for different values of ₁ K .

  σ ) through-the-thickness of a square

  Figure5: Variation of in-plane longitudinal stress ( ^x

  10 ₁ = K ).

• ^{-0,5 -0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 K}
^{K 1 =0 K 1 =2 K 1 =4 K 1 =6 K 1 =8 1 =10}

  and

  2 = p

  ,

  = 10 / h a

  (

  σ ) through-the-thickness of a square FGM plate for different values of K .

  Figure4: Variation of in-plane longitudinal stress ( ^x

  10 = K ).

  2 = p and

  ,

  10 / = h a

  . (

  Figure.3: Variation of transversal shear stress ( xz τ ) through-the-thickness of a square FGM plate for different values of ₁ K

  10 ₁ = K ).

• 0,5 ^{-0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5} _{-2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0} ^K
^{K 1 =0 K 1 =2 K 1 =4 K 1 =6 K 1 =8 1 =10}

  Figure7: Variation of longitudinal tangential stress ( ^{τ xy} ) through-the-thickness of a square FGM plate for different values of ₁ K . (

  = 10 / h a , 2 = p and K 10 = ).

  0,0 0,1 0,2 0,3 0,4 0,5 ^{0,20 0,25 0,30 0,35 0,40 0,45 K =0 K =20 K =40 K =60 K =80 K =100}

  Figure.8: The effect of material anisotropy on the dimensionless maximum deflection ( w ) of a square FGM plate for different values of K .

  ( = 10 / h a , 2 = p and

  10 ₁ = K ). ^{0,0 0,1 0,2 0,3 0,4 0,5 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 K K 1 =0 K 1 =2 K 1 =4 K 1 =6 K 1 =8 1 =10}

  Figure.9: The effect of material anisotropy on the dimensionless maximum deflection ( w ) of a square FGM plate for different values of ₁ K . (

  = 10 / h a

  ,

  2 = p

  and

  K 10 = ). Table. 1: Comparison of the dimensionless Table. 3: Comparisons of central deflections _{4 3} deflections and stresses in square FG-plates Dw ( .

  5 a , . 5 a ) / qa

  10

  ( ) of a uniformly subjected to sinusoidal distributed load. loadedhomogeneous square simply supported plate ( a / h = 100 ) on Winkler–Pasternak foundations.

  K K Results ₁ Lam et al. Present [27] method 1 3.853 3.8550 ₄

  1 3 0.763 0.7630 ₄ 5 0.115 0.1153 ₄ 1 3.210 3.2108 ₄

  3 3 0.732 0.7317 ₄ 5 0.115 0.1145 ₄ 1 1.476 1.4765 ₄

  5 3 0.570 0.5704 ₄ 5 0.109 0.1095

  Table2: Comparison of the deflection _{4 3} ( Dw ( .

  5 a , . 5 a ) / qa 10 ) of a uniformly loaded

  homogeneous square simply supported plate on Winkler foundation.

  Results K

  Kobayashi and Lam et Present sonoda [26] al. [27] method 1 4.052 4.053 4.053 ₄ 3 3.347 3.349 3.348 ₄ 5 1.506 1.507 1.506