Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips

European Journal of Mechanics A/Solids 30 (2011) 442e448

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Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips S. Nobakhti 1, M.M. Aghdam* Department of Mechanical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 1591634311, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 September 2010 Accepted 22 December 2010 Available online 30 December 2010

In this paper, the Generalized Differential Quadrature (GDQ) method is used to obtain bending solution of moderately thick rectangular plates. The plate is resting on two-parameter elastic (Pasternak) foundation or strips with a finite width. Various combinations of clamped, simply supported and free boundary conditions are considered. According to the first-order shear deformation theory, the governing equations of the problem consist of three second-order partial differential equations (PDEs) in terms of displacement and rotations of the plate. The governing equations and solution domain is discretized based on the GDQ method. It is demonstrated that the method converges rapidly while providing accurate results with relatively small number of grid points. Accuracy of the results is examined using available data in the literature for Pasternak foundation. Furthermore, due to lack of data for Pasternak strips, all predictions are verified by finite element analysis which can be used as benchmark in future studies. Ó 2010 Elsevier Masson SAS. All rights reserved.

Keywords: Pasternak foundation strips Generalized Differential Quadrature Reissner plate Rectangular plate Bending analysis

1. Introduction Plates and shells are considered as prominent structural elements in solid mechanics which are widely used in various engineering applications. Therefore, several theories have been presented to predict the behavior of structures under different loading conditions (Woinowsky-Krieger, 1931; Hencky, 1947; Reissner, 1954; Timoshenko and Woinowsky-Krieger, 1959). Among these theories, one may refer to the first-order shear deformation theory (FSDT) which is widely used to model moderately thick plates and shells by taking into account effects of both transverse shear deformation and normal stress. The theory was firstly proposed by Reissner (1944) and was further developed for dynamic problems by Mindlin (1951). One of the important engineering problems arises when a mechanical structure is rested on a non-ideal real foundation such as soil or foam. In this case, the governing equations should be modified to include the foundation effects on plate behavior. The most common way of modeling foundations is the Winkler elastic foundation model (Winkler, 1867). In this model, only effect of normal stresses of the foundation is considered using separate normal springs. Various techniques such as Fourier series, finite * Corresponding author. Tel.: þ98 21 64543429; fax: þ98 21 66419736. E-mail addresses: [email protected] (S. Nobakhti), [email protected] (M.M. Aghdam). 1 Present address: School of Engineering Sciences, University of Southampton, Highfield Campus, Southampton SO17 1BJ, UK. 0997-7538/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2010.12.016

element, finite difference and boundary element have been used to find solutions for plates resting on Winkler foundation (Svec, 1976; Henwood et al., 1981, 1982; Costa and Brebbia, 1985; Voyiadjis and Kattan, 1986; Kobayashi and Sonada, 1989; Feng and Owen, 1996; Liu, 2000). In order to increase accuracy of the model, various two-parameter elastic models such as Pasternak model (Pasternak, 1954) are developed. For instance, the effect of shear interactions between springs is also considered in the Pasternak model. A literature search reveals several studies related to free vibration and buckling (Gbadeyan, 1990; Takahashi and Sonoda, 1992; Xiang et al., 1994; Lam et al., 2000; Civalek, 2006, 2007) and bending and stress analysis (Wang et al., 1992; Shi et al., 1994; El-Zafrany and Fadhil, 1996; Han and Liew, 1997; Teo and Liew, 2002) of plates resting on the Pasternak foundation. Lam et al. (2000) presented the canonical exact solution for elastic bending, buckling and free vibration of levy-type plates on Pasternak foundation. Their solution and results includes plates with two opposite sides simply supported while the other two edges may be arbitrarily restrained. Wang et al. (1992) presented a fundamental solution for Reissner plate on two-parameter foundation and validated their solution for fully clamped and fully simply supported plates. Also, Shi et al. (1994) obtained the exact solutions for rectangular thick plates with four free edges on Pasternak foundation using the superposition technique. Differential quadrature (DQ), differential cubature (DC), harmonic differential quadrature (HDQ), discrete singular convolution (DSC)

S. Nobakhti, M.M. Aghdam / European Journal of Mechanics A/Solids 30 (2011) 442e448

and GDQ methods have been used for bending, buckling and free vibration analysis of plates and shells with or without foundation effects (Han and Liew, 1997; Teo and Liew, 2002; Civalek, 2004, 2006, 2007; Tornabene and Viola, 2008; Xiang et al., 2010). Bending results of moderately thick plate resting on Pasternak foundation for various combinations of simply supported and clamped boundary conditions are also studied (Han and Liew, 1997; Teo and Liew, 2002). These methods are among reduced mesh techniques in which small number of grid points is enough to provide final results with desired accuracy. In particular, the GDQ method uses a simple procedure to obtain weighting coefficients of the first-order derivatives using a simple algebraic formulation and higher-order derivatives by a recurrence relationship. The other advantage of the GDQ in comparison with other techniques such as DQ and DC is ease of computation of weighting coefficients without any restriction on the choice of grid points. All previous studies in the literature are restricted to plates completely resting on Pasternak foundation. However, one of the interests in this study is plates partially rested on the Pasternak foundation, e.g. foundation strips along edges of the plate. This situation may occur in many engineering applications such as plates used to cover temporary holes in a street or cavities in various structures. Furthermore, apart from exceptional cases (Shi et al., 1994; El-Zafrany and Fadhil, 1996; Lam et al., 2000; Xiang et al., 2010), studies related to plates with at least one free edge is ignored in most previous studies. In this study, the GDQ method is used to find bending solution of moderately thick rectangular plates partially resting on Pasternak foundation. In particular, the study focuses on plates with at least one free edge. It is shown that reasonably accurate results can be obtained with relatively small number of grid points. Convergence and accuracy of the results are examined using available data in the literature. In the case of Pasternak strips, results are verified by commercial finite element code ANSYS to be used as reference values in future studies. 2. Governing equations A rectangular isotropic plate with length a, width b, thickness h subjected to a uniform load q is considered. The plate is partially rested on Pasternak type foundation along edges with width t as shown in Fig. 1. In the Pasternak model of elastic foundations, the end of elastic springs in Winkler model is connected to an incompressible shear layer which only deflects when shear stress applies. Therefore, the transverse deflections of the springs are correlated. kf and gf are normal and shear coefficients of the Pasternak foundation, respectively. The problem would be in the form of three governing equations with three deformation unknowns w, jx and jy which are

443

deflection of the plate, normal rotations about x and y axes, respectively. For simplicity, the dimensionless governing equations of a moderately thick plate on Pasternak foundation based on the first-order shear deformation theory are considered as (Han and Liew, 1997):

v2 JX ð1  nÞ 2 v2 JX ð1 þ nÞ v2 JY vW b aJ b þ  þ ¼ 0; þ ad X 2 2 vX vXvY vX 2 vY 2 (1) 2 ð1 þ nÞ v2 JX ð1  nÞ v2 JY vW 2 v JY b þ  aJY þ ag þb ¼ 0; 2 vXvY 2 2 vY vY vX 2 (2)


1þ

GF



d

a

v2 W v2 W þ gb 2 vX vY 2

! 


v JX v JY K d þQ  F W þb a vX vY

¼ 0:

ð3Þ

in which dimensionless parameters are defined as:

x y w a h h X ¼ ; Y ¼ ; W ¼ ; JX ¼ jx ; JY ¼ jy ; b ¼ ; g ¼ ; d ¼ ; Q a b h b b a 4k 2g a a q 6kð1 nÞ f f ; KF ¼ ¼ ;a¼ ; GF ¼ : kGh D D d2

(4)

where D is the plate flexural rigidity defined as D ¼ Eh3 =12ð1 n2 Þ, E is the Young’s modulus, G is the shear modulus, n is the Poisson’s ratio, k is the shear correction factor which is 5/6 for isotropic plates (Han and Liew, 1997). It should be noted that the plate is rested on strips with width t and therefore, the normalized form of the foundation parameters kf and gf in the governing Eq. (3) are zero except for part of the plate on the foundation strips. In addition, stress resultants in terms of dimensionless parameters can be considered as:


1 v JX n v JY ; MX ¼ D þ a vX b vY

(5)


n vJX 1 vJY ; MY ¼ D þ b vY a vX

(6)

MXY ¼ 


 1n 1 vJX 1 vJY D þ ; 2 b vY a vX  h vW  JX ; a vX

(8)

 h vW  JY : b vY

(9)


QX ¼ kGh
QY ¼ kGh

(7)

Finally, boundary conditions of the plate in terms of dimensionless parameters are: Clamped (C):

W ¼ 0; JX ¼ 0; JY ¼ 0:

(10)

Simply supported (S):

Fig. 1. A rectangular isotropic plate partially resting on Pasternak foundation.

W ¼ 0; JY ¼ 0;

vJX v JY þ bn ¼ 0: vX vY

for X ¼ 0; 1

(11)

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W ¼ 0; JX ¼ 0; n

vJX vJY þb ¼ 0: vX vY

for Y ¼ 0; 1

(12)

Free (F):

ð1Þ

cij

ð1Þ

cij vJX vJY vJX vJY vW  JX ¼ 0: þ bn ¼ 0; b þ ¼ 0; d vX vX vY vY vX ¼ 0:

Mðxi Þ   ; ¼  xi  xj $M xj Pðyi Þ  ; ¼  yi  yj $P yj

for Y ð13Þ

 
a i1 p xi ¼ 1  cos 2 N1 yj ¼

i ¼ 1; 2; .; N; j ¼ 1; 2; .; M:

(15)

According to the GDQ method, the governing Eqs. (1)e(3) can be re-written in the discretized algebraic form as:

k¼1

M ð1  nÞ 2 X ð2Þ b cjl JX ði; lÞ  aJX ði; jÞ 2

ð2Þ

cik JX ðk; jÞ þ

¼

N X k¼1

ðmÞ

ðm1Þ

¼ m$@cii

cij

ðnÞ cii

¼ 

ðmÞ

¼ 

ð1Þ cik JX ðk; jÞ

a

xi  xj

1 A;

i; j ¼ 1; 2; .; N and isj:

ðm1Þ

ð1Þ

$cij 

cij

yi  yj

(23)

1 A;

i; j ¼ 1; 2; .; M and isj: (24)

N X

ðnÞ cij ;

j ¼ 1 jsi M X

i ¼ 1; 2; .; N:

ðmÞ

j ¼ 1 jsi

cij ;

(25)

i ¼ 1; 2; .; M:

(26)

2.3. Discretization of the boundary conditions

ð17Þ

l¼1

KF d



cij

j ¼ 1; 2; .; M: (27)

ð1Þ

þ Q ði; jÞ 

ðn1Þ

ðn1Þ ð1Þ n$@cii $cij

Wð1; jÞ ¼ 0; JX ð1; jÞ ¼ 0; JY ð1; jÞ ¼ 0;

cjl Wði; lÞ ¼ 0;

k¼1



0 ðnÞ cij

Clamped at X ¼ 0:

! 
N M X X G ð2Þ ð2Þ d cik Wðk; jÞ þ gb cjl Wði; lÞ 1þ F

a

(22)

For n,m > 1:

l¼1

k¼1

l¼1

yi  y j :

The discretized form of the boundary conditions based on the GDQ method can be considered as:

N ð1  nÞ X ð2Þ cik JY ðk; jÞ  aJY ði; jÞ þ 2 M X

(21)

j ¼ 1 jsi

l¼1

N X M M X ð1 þ nÞ X ð1Þ ð1Þ ð2Þ 2 b cik cjl JX ðk; lÞ þ b cjl JY ði; lÞ 2

þ ag

M  Y

Pðyi Þ ¼

cii

N X M N X ð1 þ nÞ X ð1Þ ð1Þ ð1Þ b þ cik cjl JY ðk; lÞ þ ad cik Wðk; jÞ ¼ 0; 2 k¼1 l¼1 k¼1 ð16Þ

k¼1 l¼1

(20)

(14)

2.2. Discretization of the governing equations

N X

i; j ¼ 1; 2; .; M and isj;

0


 b j1 p 1  cos 2 M1

(19)

N Y   xi  xj : j ¼ 1 jsi

Mðxi Þ ¼

2.1. Discretization of the domain The GDQ method is used to solve the governing differential equations of the plate on Pasternak foundation strips. The first step is dividing the plate into N  M grid points where N and M represent the number of nodes in the x and y directions, respectively. Although the simplest procedure for discretization of the domain is to select equally spaced points, it is shown (Shu et al., 2001) that one of the best options for obtaining grid points is zeros of the wellknown Chebyshev polynomials as:

i; j ¼ 1; 2; .; N and isj;

þb

M X l¼1

Simply supported at X ¼ 1: N X

WðN; jÞ ¼ 0; JY ðN; jÞ ¼ 0;

k¼1

¼ 0;

! ð1Þ cjl JY ði; lÞ

ð1Þ

cNk JX ðk; jÞ þ bn

j ¼ 1; 2; .; M:

M X l¼1

ð1Þ

cjl JY ðN; lÞ ð28Þ

Free at Y ¼ 0:

Wði; jÞ ¼ 0:

ð18Þ

In these equations, (i,j) is an arbitrary grid point inside the ðnÞ

n

k¼1

ðmÞ

domain where i ¼ 2; .; N  1 and j ¼ 2; .; M  1. cpq and cpq are the weighting coefficients of the nth and mth order partial derivatives of W; JX and JY with respect to X and Y, respectively which could be calculated using the GDQ formulation (Shu and Richards, 1990) as: For n,m ¼ 1:

N P

b

M P l¼1

g

M P l¼1

M P

ð1Þ

cik JX ðk; 1Þ þ b ð1Þ

c1l JX ði; lÞ þ ð1Þ

l¼1

N P k¼1

ð1Þ

c1l JY ði; lÞ ¼ 0;

ð1Þ

cik JY ðk; 1Þ ¼ 0;

c1l Wði; lÞ  JY ði; 1Þ ¼ 0;

i ¼ 1; 2; .; N

(29)

S. Nobakhti, M.M. Aghdam / European Journal of Mechanics A/Solids 30 (2011) 442e448

445

Application of the GDQ method to the governing equations leads to a set of 3(N  2)  3(M  2) equations with the same number of unknowns for all nodes inside the domain. In addition, boundary condition equations should also be added to the system of equations for all boundary nodes. Therefore, the size of the final system of algebraic equations is different for various plates based on the type of boundary conditions. Furthermore, this may results in an overdetermined system of equations in which numbers of equations are more than unknowns. The least square technique is then used to obtain best possible solution for the resulted system of equations. Finally, it should be noted that the foundation parameters kf and gf exist for all nodes on the foundation strips and vanish for all other nodes. Solution to the resulted system of algebraic equations provides deflection and rotations of the plate at each grid point which can then be used to obtain stress resultants using Eqs. (5)e(9). 3. Results and discussions The presented GDQ method is used to obtain solution to the governing equations of rectangular plates with different combination of clamped (C), simply supported (S), and free (F) boundary conditions. In all examples, plate is subjected to uniformly distributed load and partially rest on Pasternak foundation. Dimensionless plate parameters are a/b ¼ 1 and h/a ¼ 0.1 and Poisson’s ratio is 0.3. The edges of the plate are numbered from 1 to Table 1 Convergence of normal deflection and stress resultants of SFSF square plate under uniform loading and resting on Pasternak foundation with various foundation parameters. KF

1

GF

1

3^4

5^4

3^4

1

3^4

5^4

5^4

1

3^4

5^4

Grid points

w*

M*x

M*y

Q*x

X ¼ 0.5

X ¼ 0.5

X ¼ 0.5

X ¼ 0.0

Y ¼ 0.5

Y ¼ 0.5

Y ¼ 0.5

Y ¼ 0.5

77 99 11  11 Lam et al. (2000) 77 99 11  11 Lam et al. (2000) 77 99 11  11 Lam et al. (2000)

12.210 12.200 12.200 11.890 1.587 1.591 1.591 1.571 0.229 0.230 0.231 0.228

11.100 11.060 11.080 11.100 1.313 1.325 1.325 1.334 0.176 0.183 0.182 0.185

2.377 2.263 2.314 2.443 0.224 0.215 0.219 0.239 0.025 0.025 0.025 0.028

0.436 0.426 0.430 0.421 0.103 0.099 0.100 0.101 0.027 0.024 0.024 0.041

77 99 11  11 Lam et al. (2000) 77 99 11  11 Lam et al. (2000) 77 99 11  11 Lam et al. (2000)

6.747 6.743 6.745 6.663 1.429 1.432 1.433 1.417 0.225 0.227 0.227 0.225

6.004 5.987 5.994 6.094 1.169 1.181 1.181 1.190 0.173 0.180 0.180 0.182

1.330 1.271 1.296 1.380 0.204 0.197 0.200 0.217 0.025 0.025 0.025 0.027

0.286 0.279 0.282 0.276 0.099 0.095 0.096 0.097 0.027 0.024 0.024 0.028

77 99 11  11 Lam et al. (2000) 77 99 11  11 Lam et al. (2000) 77 99 11  11 Lam et al. (2000)

1.656 1.656 1.656 1.660 0.852 0.854 0.854 0.851 0.203 0.204 0.204 0.203

1.271 1.270 1.271 1.310 0.645 0.658 0.658 0.664 0.153 0.160 0.159 0.162

0.323 0.315 0.317 0.337 0.125 0.124 0.125 0.134 0.023 0.023 0.023 0.026

0.141 0.138 0.139 0.132 0.083 0.080 0.080 0.082 0.026 0.024 0.023 0.044

Fig. 2. Convergence of dimensionless central deflection of SCSF square plate under normal pressure that rest on Pasternak foundation (kf ¼ 50, gf ¼ 25).

4 which represents the X ¼ 0, Y ¼ 1, X ¼ 1 and Y ¼ 0, respectively as shown in Fig. 1. For example, the symbol SCSF identifies a plate with edges 1 and 3 simply supported, edge 2 clamped and edge 4 free. To facilitate reporting and comparison of the results, the following set of non-dimensional quantities are also used (Han and Liew, 1997):

w* ¼

Qj WD Mi  103 ; Mi* ¼  102 ; Qj* ¼ ; qa qa2 qa4



i ¼ x; y; xy j ¼ x; y (30)

3.1. Convergence study To study the efficiency and accuracy of the GDQ method, results for deflection and stress resultants of SFSF plate rested entirely on Pasternak foundation are presented in Table 1 for several number of grid points and compared with Lam et al. (2000). It could be concluded from Table 1 that results of GDQ have good agreement with Lam et al. (2000). The best results would be obtained with relatively small number of grid points of 9  9. It is further to be noted that by increasing the foundation parameters, the error decreases. In the case that foundation parameters are relatively small, a maximum error of 2% exists. In Fig. 2, convergence of the GDQ method for central deflection of SCSF plate resting on Pasternak foundation is studied. It can be

Fig. 3. Deflection of FFFF square plate on Pasternak foundation subjected to unit pressure (E ¼ 207 GPa, kf ¼ 100, gf ¼ 30).

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S. Nobakhti, M.M. Aghdam / European Journal of Mechanics A/Solids 30 (2011) 442e448

Fig. 4. Dimensionless deflection of CCCC square plate under normal pressure along Y ¼ 0.5 that rest on Pasternak foundation strips (t/a ¼ 0.1, kf ¼ 50, gf ¼ 25).

concluded from results of Table 1 and Fig. 2 that the GDQ method converges rapidly to reasonably accurate results with relatively small number of grid points. 3.2. Plate resting on Pasternak foundation In order to examine the accuracy of the GDQ predictions, another interesting case study is a full free square plate on Pasternak foundation subjected to unit pressure. The geometric and material properties of the plate are b/a ¼ 1; h/a ¼ 0.1; n ¼ 0.3 and E ¼ 207 GPa and foundation parameters are kf ¼ 100, gf ¼ 30. Fig. 3 shows deflection of FFFF plate on Pasternak foundation using 13  13 grid point mesh. As expected, the plate should deform uniformly and therefore, a constant deflection should be obtained for all of the domain and boundary nodes. It should be noted that due to lack of shear effects in this particular problem, the constant deflection is totally related to the effects of kf and can simply be determined using 1/kf. 3.3. Plate resting on Pasternak strips Convergence and accuracy of the presented method are studied by providing comparison of the results with published data for plates resting on Pasternak foundation. In this section, results are

Fig. 5. Dimensionless deflection of SSSS square plate under normal pressure along Y ¼ 0.5 that rest on Pasternak foundation strips (kf ¼ 50, gf ¼ 25).

presented for plates partially rested on Pasternak type strips. Due to lack of published data in the open literature for this case, another study was also carried out using commercial finite element code ANSYS mainly to validate predictions of the presented method. The next example is a fully clamped (CCCC) plate resting on Pasternak type strips with t/a ¼ 0.1. Foundation parameters are kf ¼ 50, gf ¼ 25. Dimensionless deflection along Y ¼ 0.5 is presented using presented GDQ method and FEM. It can be concluded from Fig. 4 that using a mesh of 11 11 grid points is enough for the GDQ method to provide reasonably accurate results in comparison with the FEM predictions. The other example includes a fully simply supported (SSSS) plate rested on Pasternak strips with foundation parameters kf ¼ 50, gf ¼ 25. Fig. 5 presents predictions of dimensionless deflection of the plate along centerline (Y ¼ 0.5) obtained by both the GDQ method and ANSYS for several “t/a” ratios. A 21  21 mesh is used to produce the GDQ results for t/a ratios of 0.1, 0.2 while 23  23 is used for t/a ¼ 0.3. It can be observed that by increasing the foundation strips width and placing a larger section of the plate on the foundation, the results from GDQ have the least deviance from ANSYS predictions. However, even in the case of thin strips of t/ a ¼ 0.1, maximum deviation is about 6%. The next example is dimensionless deflection of SCSC plate along diagonal path, i.e. from (X ¼ 0, Y ¼ 1) to (X ¼ 1, Y ¼ 0) which is presented for three different “t/a” ratios in Fig. 6. A 9  9, 25  25 and 31  31 mesh is used to produce the GDQ results for t/a ratios of 0.1, 0.2 and 0.3, respectively. The ‘X’ axis represents dimensionless diagonal path of the square plate and foundation parameters are kf ¼ 50, gf ¼ 25. Again, decreasing the width of the strips leads to greater maximum discrepancies. A maximum error of 9% occurs when t/a ¼ 0.1 while in other cases the maximum error is less than 3%. In Table 2, numerical results for dimensionless deflection and stress resultants of moderately thick plate resting on Pasternak foundation strips with various foundation parameters are presented for several boundary conditions. The strips width is considered as t/a ¼ 0.2 with 15  15 mesh to produce the results. Of interest in Table 2 are plates with at least one free edge. It could be seen that by increasing the foundation normal parameter kf, when foundation shear parameter gf is constant, the magnitude of deflection, bending and twisting moments at the plate center point decrease. Although it is not the same when kf is constant and gf increases, but reverse case happens. In Figs. 7 and 8, the effects of foundation parameters on deflection and bending moment of SCCC plate are studied using GDQ predictions with a mesh of 15  15. It can be inferred from

Fig. 6. Dimensionless deflection of SCSC square plate under normal pressure along its diagonal resting on Pasternak foundation strips (kf ¼ 50, gf ¼ 25).

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Table 2 Numerical results for a square plate under normal loading with various boundary conditions resting on Pasternak foundation strips for several foundation parameters (t/a ¼ 0.2). Boundary condition

CFCF

SSFF

CCCF

SFSF

SCSF

gf

w*

M*x

M*y

M*xy

Q*x

Q*y

X ¼ 0.5

X ¼ 0.5

X ¼ 0.5

X ¼ 0.0

X ¼ 0.0

X ¼ 0.5

Y ¼ 0.5

Y ¼ 0.5

Y ¼ 0.5

Y ¼ 0.0

Y ¼ 0.5

Y ¼ 0.0

1.5225 1.5744

1.8099 1.8337

0.3293 0.3510

0.0000 0.0000

0.1781 0.2582

1.3868 1.4429

1.5748 1.6228

0.1832 0.2007

0.0000 0.0000

0.1147 0.1643

2.1032 2.1401

1.3917 1.4623

0.7254 0.6966

0.0213 0.0203

0.0694 0.0563

1.8382 1.8985

1.1457 1.2312

0.4423 0.4564

0.0081 0.0054

0.0322 0.0270

1.7937 1.8250

1.4324 1.4708

0.0000 0.0000

0.2326 0.3346

0.3100 0.4131

1.6838 1.7257

1.1938 1.2584

0.0000 0.0000

0.1475 0.2312

0.2473 0.3334

2.0695 2.0951

1.4543 1.5271

0.0005 0.0005

0.0000 0.0000

0.0689 0.0561

1.8416 1.8958

1.1706 1.2587

0.0005 0.0005

0.0000 0.0000

0.0320 0.0270

2.0788 2.1002

1.3961 1.4687

0.3832 0.4027

0.2440 0.3733

0.0727 0.0585

1.8444 1.8943

1.1491 1.2335

0.2605 0.2839

0.1469 0.2450

0.0344 0.0288

kf ¼ 50 25 0.9672 100 1.0056 kf ¼ 100 25 0.7501 100 0.8021 kf ¼ 50 25 1.1584 100 1.2055 kf ¼ 100 25 0.8444 100 0.9138 kf ¼ 50 25 0.9173 100 0.9547 kf ¼ 100 25 0.7430 100 0.7950 kf ¼ 50 25 1.1436 100 1.1855 kf ¼ 100 25 0.8446 100 0.9105 kf ¼ 50 25 1.0783 100 1.1212 kf ¼ 100 25 0.8222 100 0.8850

Fig. 7 that the effect of foundation parameter kf on the plate deflection is more than gf. It is noticeable that when kf is small, by increasing the gf parameter, deflection of the plate increases. However, in the cases where kf is relatively large, the reverse happens. Deflection of a SCCC plate with same characteristics and with no foundation parameters are presented and compared with ANSYS. As expected, it can be seen that when foundation parameters shift to zero, the deflection of the plate moves toward

Fig. 7. Dimensionless deflection of SCCC square plate under normal pressure along Y ¼ 0.5 resting on Pasternak foundation strips (t/a ¼ 0.2).

Fig. 8. Dimensionless bending moment of SCCC square plate under normal pressure along Y ¼ 0.5 resting on Pasternak foundation strips (t/a ¼ 0.2).

deflection of a plate with no foundation. It can also be seen that the predictions of the GDQ method are in good agreement with those obtained from ANSYS. Finally, Fig. 8 shows that for small values of foundation parameters, the plate global behavior does not affected by foundation strips. However, increasing foundation parameters leads to different slopes of bending moment at the edge of the foundation strips, i.e. X ¼ 0.2 and 0.8. 4. Conclusion The Generalized Differential Quadrature method is used to find bending solution of moderately thick rectangular plates partially rested on two-parameter elastic foundation. Of interest are plates with at least one free edge. It is shown that reasonably accurate results can be obtained with relatively small number of grid points. Convergence and accuracy of the results are examined using available data in the literature for full free plate. In the case of no published result for plate rested on Pasternak strips, GDQ predictions are verified by finite element results and can be used as reference values in future studies. References Civalek, Ö., 2004. Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng. Struct. 26 (2), 171e186. Civalek, Ö., 2006. Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation. J. Sound Vib. 294, 966e980. Civalek, Ö., 2007. Nonlinear analysis of thin rectangular plates on WinklerePasternak elastic foundations by DSC-HDQ methods. Appl. Math. Model. 31, 606e624. Costa, J.A., Brebbia, C.A., 1985. The boundary element method applied to plates on elastic foundation. Eng. Anal. 2, 174e183. El-Zafrany, A., Fadhil, S., 1996. A modified Kirchhoff theory for boundary element analysis of thin plates resting on two-parameter foundation. J. Eng. Struct. 18 (2), 102e114. Feng, Y.T., Owen, D.R.J., 1996. Iterative solution of coupled FE/BE discretizations for plate-foundation interaction problems. Int. J. Numer. Methods Eng. 39, 1889e1901. Gbadeyan, J.A., 1990. Lateral vibration analysis of an initially stressed rectangular plate on Pasternak elastic foundation. Model. Simul. Control B: Mech. Therm. Eng., Mater. Resour. Chem. 27 (3), 9e35. Han, J.B., Liew, K.M., 1997. Numerical differential quadrature method for Reissner/ Mindlin plates on two parameter foundations. Int. J. Mech. Sci. 39 (9), 977e989. Hencky, H., 1947. Uber die Brucksichtigung der Schubver Zerruhgen in ebehen platten. Ing. Arch. 16, 72. Henwood, D.J., Whiteman, J.R., Yettram, A.L., 1981. Finite difference solution of a system of first order partial differential equations. Int. J. Numer. Methods Eng. 17 (9), 1385e1395.

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