1-s2.0-S0020740315002519-main

International Journal of Mechanical Sciences 100 (2015) 322–327

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences
journal homepage: www.elsevier.com/locate/ijmecsci

A semi-analytical study on static behavior of thin skew plates
on Winkler and Pasternak foundations
Amin Joodaky n, Iman Joodaky
Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Islamic Republic of Iran

art ic l e i nf o

a b s t r a c t

Article history:
Received 19 April 2012
Received in revised form
9 March 2015
Accepted 28 June 2015
Available online 17 July 2015

This study presents a semi analytical closed-form solution for governing equations of thin skew plates
with various combination of clamp, free and simply supports subjected to uniform loading rested on the
elastic foundations of Winkler and Pasternak. The governing forth-order partial differential
equation (PDE) of two-variable function of deflection, w(X,Y), is defined in Oblique coordinates system.
Application of EKM together with the idea of weighted residual technique, converts the forth-order
governing equation to two ODEs in terms of X and Y in Oblique coordinates. Both resulted ODEs, are then
solved iteratively in a closed-form manner with a very fast convergence. Finally deflection function is
obtained. It is shown that some parameters such as angle of skew plate and stiffness of elastic foundation

have an important effect on the results. Also it is investigated that shear stresses exist considerably in
skew plates comparing to the corresponding rectangular plates. Comparisons of the deflection and
stresses at the various points of the plates show very good agreement with results of other analytical and
numerical analyses.
& 2015 Elsevier Ltd. All rights reserved.

Keywords:
Bending
Skew plate
PDE
Galerkin
Extended Kantorovich
Pasternak foundation

1. Introduction
Kerr [1] developed the idea of the well-known Kantorovich
method [2] to obtain highly accurate approximate closed-form solution for torsion of prismatic bars with rectangular cross-section. The
method employs the novel idea of Kantorovich to reduce the governing partial differential equation of a two-dimensional (2D) elasticity
problem to a double set of ordinary differential equations. Since then,
the Extended Kantorovich Method (EKM) extensively has been
applied for various 2D elasticity problems in Cartesian coordinates
system. Among these applications, one can refer to eigenvalue
problems [3], buckling [4] and free vibrations [5] of thin rectangular
plates, bending of thick rectangular isotropic [6,7] and laminated
composite [8] plates and free-edge strength analysis [9]. Most recent
EKM based articles include vibration of variable thickness [10] and
buckling of symmetrically laminated [11] rectangular plates. Accuracy
of the results and rapid convergence of the method together with
possibility of obtaining closed-form solutions for ODE systems have
been discussed in these articles and others [12]. Finally, a few research

consider polar coordinates e.g. using EKM for sector plates [13]. All
these applications of EKM, are devoted and restricted to the problems
in the Cartesian and polar coordinate systems. The authors of the

n

Corresponding author.
E-mail address: (A. Joodaky).

/>0020-7403/& 2015 Elsevier Ltd. All rights reserved.

present paper, for the first time applied EKM in Oblique coordinate
system for bending of skew plates under clamp boundary conditions
without considering foundations and stress analysis [14]. Based on the
other solution methods, several research have studied bending,
buckling, vibration and other analysis for skew plates in term of
Oblique coordinate system [15–20]. Winkler and Pasternak foundations are considered in the design of structures rested on elastic
mediums. Winkler model considers the foundation as a series of
springs which do not have any interaction with each other. More
advanced models like Pasternak simulate the coupling between these
springs too [21,22].
This study aims to examine the applicability of the EKM to obtain
highly accurate approximate closed-form solutions for 2D elasticity
problems in Oblique coordinate system. Applying Extended Kantorovich Method (EKM) with the aid of a weighted residual technique
(Galerkin method), the governing PDE, is converted to two
uncoupled ordinary differential equations (ODE) of f(X) and g(Y).
Then an initial guess function is considered for one of those functions
to obtain the constants of the ODE of the other function. After solving
the first ODE, constants of the second ODE are achieved. Then the
second ODE is solved for obtaining the first ODE's constants. These

iterations continues unless a good convegence is achieved. In every
iteration step, exact closed-form solutions are obtained for two ODE
systems. Deflection and stress analysis of thin isotropic skew plates
with a various combinations of clamp, free and simply supports
subjected to uniform loading and resting on the Winkler and


A. Joodaky, I. Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327

323

Pasternak foundations as Fig. 1, is considered. Comparisons of the
deflections and stresses at the various points of the skew plate show
very good agreement with the results of other valid literatures and
FEM analysis of ANSYS code. Discussions reveals the existence of
shear stresses in skew plates comparing to the corresponding
rectangular plates.
Fig. 1. Skew plate in Oblique coordinate (X,Y) resting on the elastic foundations
with the stiffness of k.

2. Governing equations
If no axial force exists, differential equation of motion is
expressed as [23]
∂2 M xy 2 M y
2 M x
ỵ
ỵ2
ỵ qx; yị  k0 wx; yị ỵk1 2 wx; yị ẳ 0
xy
x2

y2

1ị

and in terms of w as
4 wx; yị ẳ

qx; yị ỵ k0 wðx; yÞ  k1 ∇2 wðx; yÞ
D

multiplication of different single variable functions as
wij ðX; YÞ ffif i ðXÞ Ug j ðYÞ

where f i ðXÞ and g j ðYÞ are unknown functions to be determined and
subscripts i and j denote number of iterations. Using Eq. (6),
expanding of Eq. (7) is
4

ð2Þ

Eq. (2) is the governing equation for a thin plate, in which w(x,
y) is the deflection function, q is the applied distributed load, D is
flexural rigidity for isotropic plates, k0 and k1 are the stiffness of
Winkler and Pasternak foundation respectively. Having just
Winkler foundation it is enough to equal k1 to zero. Now,
consider a thin skew plate with dimensions of 2a  2b as Fig. 1.
For a clamp–support, deflection (w) and its first derivative with
respect to the normal direction of the boundary must be
vanished. For a simply-support, deflection and its second derivative with respect to the normal direction of the boundary must be
vanished. Considering Fig. 1, for example for SSSC boundary

conditions (S and C represent simply and clamp respectively)
we have



2
2

ð3Þ

Governing Eqs. (1) and (2) must be converted from Cartesian
coordinates system (x,y) to Oblique coordinates system (X,Y) as it
is shown in Fig. 1. The relations between Cartesian(x,y) and
Oblique(X,Y) are
X ¼ x  y tan φ and Y ¼ y= cos φ

Also, ∇4 becomes
 4

 4
1

4
ẳ
ỵ 2 1 ỵ 2 sin 2
4
4
cos X
X 2 Y 2

4


4 


4
ỵ
ỵ
 4 sin
X 3 ∂Y ∂X∂Y 3
∂Y 4

4

2

k0 cos 4 φ
k1 cos 3 φ
d f ðXÞ
ðf ðXÞ U g 0 ðYÞÞ 
g 0 ðYÞ
D
D
dX 2
!
2
dg Yị df Xị
d g 0 Yị
cos 4
ỵ f Xị
qX; Yị

ịẳ
 2 sin 0
2
dY
dX
D
dY
2

ỵ

10ị

For Eq. (7), according to the Galerkin weighted residual
method, we have [13]
Z 2a Z 2b
4
2
ðD∇ w  q ỵ k0 w  k1 wịwdXdY ¼ 0
ð11Þ
0

0

Now, for a prescribed function of g j ðYÞ, jẳ0 and referred to Eq.
(8), w becomes

w ẳ g0 Yị U δf i
ð6Þ

For the governing Eq. (2) in Oblique coordinates system we
have
D wX; Yị ỵ k0 wX; Yị  k1 wX; Yị ẳ qX; Yị

9ị

By considering assumption of Eq. (8), we have



4 
3
4
φ
φ ∂2 ðf ðXÞ:g 0 ðYÞÞ
f ðXÞ:g 0 ðYÞ  k1 cos
cos 4 φ U ∇ ðf Xị:g 0 Yịị ỵ k0 cos
D
D
X 2



!
2
2
f Xị:g 0 Yị f Xị:g 0 Yị
ỵ
 2 sin
XY

Y 2
(
4

d2 g Yị d2 f Xị
d f Xị
0
ỵ 2 1 ỵ 2 sin 2
ẳ g 0 Yị
4
dX
dY 2
dX 2
)
!
3
3
4
dg 0 ðYÞ d f ðXÞ d g 0 ðYÞ df ðXÞ
d g 0 Yị
ỵ
f
Xị
ỵ
 4 sin
dY
dX
dX 3
dY 3
dY 4

4ị

Consequently operator ∇2 in Cartesian coordinates could be
2
converted to Oblique coordinates as

2

1

2
2
2
ỵ 2
ẳ
 2 sin
5ị
2
cos X
XY Y

2

D w ỵ k0 wX; Yị  k1 wX; Yị ẳ qX; Yị
 4

 4 w
D
w
ỵ 2 1 ỵ 2 sin 2 φ

cos 4 φ ∂X 4
∂X 2 ∂Y 2



4
w
4 w 4 w
ỵ k0 w
ỵ
 4 sin φ
∂X 3 ∂Y ∂X∂Y 3 ∂Y 4

2

k1
∂ w
∂2 w 2 w
ỵ

2
sin

ẳ qX; Yị

XY Y 2
cos X 2

w ẳ d w=dx2 ¼ 0 for x ¼ 0; x ¼ 2a
w ¼ d w=dy2 ¼ 0 for y ¼ 0 and w ẳ dw=dy ẳ 0 for y ẳ 2b

8ị

12ị

Substitution of Eq. (8) into Eq. (11) in conjunction with Eq. (12)
leads to
#
Z 2a "Z 2b 

4
2
D∇ ðf i U g 0 ị  q ỵ k0 f i Ug 0 Þ  k1 ∇ ðf i Ug 0 Þ g 0 dY f i dX ẳ 0
0

0

7ị

13ị
Based on the existing rules in the Variational principle, Eq. (13)
is satisfied if the expression in the bracket is vanished

3. Iterative solution by EKM
According to the Extended Kantorovich Method (EKM) [1], the
two-variable-function of the plate deflection, w(X,Y) is assumed as

Z
0

2b




4
2
D∇ ðf i Ug 0 ị  q ỵ k0 f i Ug 0 ị  k1 ∇ ðf i Ug 0 Þ g 0 dY ẳ 0

14ị


324

A. Joodaky, I. Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327

Now referred to Eq. (10), Eq. (14) becomes


k1 cos 3 φ ∂2 f i ðXÞ:g 0 ðYÞ
k0 cos 4 φ 
f i ðXÞ U g 0 ðYÞ 
cos φ U ∇ f i ðXÞ U g 0 Yị ỵ
0
D
D
X 2
!#


2 f i Xị:g 0 Yị
2 f Xị:g Yịị

ỵ i Y 2 0
 2 sin
g 0 dY ¼
∂X∂Y
!
!
!
Z 2b
Z 2b

d2 g ðYÞ k cos 3 φ
4
2
1
2
f i Xị
d f i Xị
0
g
g 20 YịdY d dX
2 1 ỵ2 sin

Yị
g
Yị
dY
ỵ
4 ỵ
0
0

2
dX 2
D
dY
0
0
|{z}
|{z}


R 2b 

4



4



A4

A2

!
3

dg Yị
f i Xị
 4 sin g 0 Yị 0 dY d dX

3 ỵ
dY
0
|{z}
Z

2b



15ị

A3

!
!



Z 2b
4
k1 cos 3
dgYị
d g 0 Yị
df i Xị
g
2
sin

YịdY
ỵ

g
Yị
dY
f i Xị ỵ
0
0
dX
dY
D
dY 3
dY 4
0
0
|{z}
|{z}
Z

2b




 4 sin

dg 3 Yị

ỵ

A1

Z

2b

3

Z

2b

2

!

A0

!
k0 cos φ
k1 cos φ
d g 0 ðYÞ
cos φ
g 0 2 ðYÞdY 
g
Yị
dY
f
Xị
ẳ
0
i

D
D
D
dY 2
0
0
0
|{z}
|{z}
4

Z

4

2b

A5

A6

Using an arbitrary prescribed function for g 0 Yị, as initial guess,
the constants of Ai (i¼ 0–6), could be calculated and Eq. (15) turns
into a forth order ODE as
4

A4

d f i Xị
dX 4

3

ỵ A3

d f i Xị
dX 3

2

ỵ A2

d f i Xị
dX 2

ỵ A1

dX

4

3

ỵ

2

A3 d f i Xị A2 d f i Xị A1 df i Xị A0 ỵ A5 ị
A6
ỵ

ỵ
ỵ
f i Xị ẳ
A4
A4 dX 3
A4 dX 2
A4 dX
A4

17ị

The corresponding characteristic equation related to Eq. (17), is
m4 ỵ

A3 3 A2 2 A1
A0 ỵ A5 ị
m ỵ m ỵ mỵ
ẳ0
A4
A4
A4
A4

18ị

Eq. (18) may has four complex roots as: mr ¼ 7 a1 7b1 i that,
r ¼1–4 and first description of fi (X) as f1(X) is expressed as
f 1 Xị ẳ ea1 X C 1 cos b1 Xị ỵ C 2 sin b1 Xịị
ỵ e  a1 X C 3 cos b1 Xị ỵC 4 sin  b1 Xịị ỵ C 5

ð19aÞ

or Eq. (18) has four real roots as: m1;2 ¼ 7 a1 ; m3;4 ¼ 7 b1 , then
f1(X) is considered as
f 1 Xị ẳ C 1 ea1 X þ C 2 e  a1 X þC 3 eb1 X ỵ C 4 e  b1 X ỵ C 5

19bị

where C 5 ẳ A6 =A0 ỵ A5 ị.
It should be noted that after substituting the double-term
function of w(X,Y) with the multiplied single-term functions of
f(X) and g(Y), the new forms of boundary conditions must be
considered in terms of single-term functions. For example new
forms of boundary conditions of SSSC in Eq. (3), are
2

f Xị ẳ d f Xị=dX 2 ẳ 0 for X ẳ 0 and X ẳ 2a
2

gYị ẳ d gYị=dY 2 ẳ 0 for Y ẳ 0; and
gYị ẳ dgYị=dY ẳ 0 for Y ẳ 2b

20ị

Solving Eqs. (19) in conjunction with the new boundary data
leads to the first estimate of the function f 1 ðXÞ. Similarly, it is
possible to continue the procedure by introducing the obtained
function f 1 ðXÞ to Eq. (8). The new form of δw is

δw ¼ f 1 ðXÞ U δgj

Similarly Galerkin equation is obtained as
"Z
#

2a 
4
2
D f 1 U g j ị ỵ k0 f 1 U g j Þ  k1 ∇ ðf 1 U g j Þ  q f 1 dX δg j dY ẳ 0

2b

0

22ị

Dividing both sides by A4, yields
d f i Xị

Z
0

df i Xị
ỵ A0 ỵ A5 ịf Xị ẳ A6
dX
ð16Þ

4



q Ug 0 ðYÞ dY

ð21Þ

Again, in order to satisfy Eq. (22), the bracket must be vanished.
Using the already obtained f 1 ðXÞ for the expression in the bracket
and integration with respect to X leads to the second forth-order
ODE in terms of gYị for obtaining g1(Y) in
4

B4

d g 1 Yị
dY

4

3

ỵ B3

d g1 Yị
dY

3

2

ỵ B2

d g 1 Yị
dY 2

ỵ B1

dg 1 Yị
ỵ B0 ỵ B5 ịg 1 Yị ẳ B6
dY

23ị

The corresponding characteristic equation related to Eq. (23) is
n4 ỵ

B3 3 B2 2 B1
B0 ỵ B5 ị
n ỵ n ỵ nỵ
ẳ0
B4
B4
B4
B4

24ị

Again Eq. (24) either has four complex roots as nk ¼ 7 a2 7 b2 i
and k¼ 1–4, so g1(Y) is shown as
g 1 Yị ẳ ea2 Y D1 cos b2 Yị ỵ D2 sin b2 Yịị ỵ ea2 Y D3 cos b2 Yị
ỵ D4 sin b2 Yịị ỵ D5

25aị

or Eq. (24) has four real roots as: n1;2 ¼ 7 a2 ; n3;4 ¼ 7 b2 , so g(Y)
could be expressed as
g 1 Yị ẳ D1 ea2 X ỵ D2 e  a2 X ỵ D3 eb2 X ỵ D4 e  b2 X ỵ D5

25bị

where D5 ẳ B6 =B0 ỵ B5 ị. The process is continued by solving Eqs.
(25) together with the new boundary data and obtaining the new
prediction for g 1 ðYÞ. This finishes the first iteration for determination of deflection function of w(X,Y). Eqs. (19) and (25) are then
solved iteratively and new updated estimates for functions f i ðXÞ
and g j ðYÞ are determined. Iteration procedure continues in the
Table 1
Sample mechanical properties for the skew plate on Winkler and Pasternak
foundations.
Properties
2a
1m

2b
1.5 m

h
0.01 m

ν
0.3

φ
151

k0
1e6 Pa/m

k1
1e5 N/m

q
1e4 Pa

E
70 GPa


Table 2
Convergence of the deflection, in the center of the skew plate (a,b) on the Winkler–
Pasternak foundation for different combination of boundary conditions.
Boundary conditions

Central deflections (m) during iterations (i¼ 0 to 4)

CCCC
SSSS
SCFC
SCSC
CFCF
SCCC

0

1

2

3

 0.202e–2
 0.334e–2
 0.466e–2
 0.334e–2
 0.202e–2
 0.261e–2

0.1882e–2
0.3719e–2
0.4647e–2
0.3181e–2
0.2131e–2
0.2377e–2

0.1881e–2
0.3734e–2
0.4637e–2
0.3179e–2
0.2172e–2
0.2375e–2

0.1881e–2

0.3734e–2
0.4637e–2
0.3179e–2
0.2172e–2
0.2375e–2

Deflection of the skew Plate,
z Axis (meter)

A. Joodaky, I. Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327

3.50E-03

k0=0, k1=0

3.00E-03

k0=1e6 , k1=1e5

2.50E-03

k0=1e5 , k1=1e6

2.00E-03

k0=1e7 , k1=1e7

1.50E-03
1.00E-03
5.00E-04

0.00E+00
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x Axis (meter) along y=0
Fig. 4. Deflection of the CCCF skew plate, for various foundation stiffness of k0 and
k1 along the Y ¼b and X axes.

4.00E-03
of the skew Plate (Pa)

2.00E+07

3.50E-03
3.00E-03
2.50E-03
2.00E-03
1.50E-03

CCCC

1.00E-03

SSSS
CCSS
CCSF

5.00E-04
0.00E+00
0.0

0.1

0.2

0.3

0.4

0.5

0.6

σ

Deflection of the skew
plate, Z Axis (meter)

325

1.00E+07
0.00E+00
-1.00E+07

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2.00E+07

phi=15
phi=30

-5.00E+07

0.8

0.9

1.0

phi=0

-3.00E+07
-4.00E+07

0.7

0.9

x Axis (meter) along y=b

1.0

Fig. 5. Stress of σ X along the Y ¼ b and X axes for the CCSS plate in Table 1 with
different skew angles.

X Axis (meter) along Y=b

2.5E-03

phi=0

2.0E-03

phi=15

1.5E-03

phi=30
phi=45

1.0E-03

σ of the skew Plate (Pa)

Deflection of the skew plate, Z
Axis (meter)

Fig. 2. Deflection of the skew plate, for various boundary conditions.

1.20E+07
1.00E+07
8.00E+06
6.00E+06
4.00E+06
2.00E+06

0.00E+00
-2.00E+06 0.0
-4.00E+06
-6.00E+06

phi=0
phi=15
phi=30

0.1

0.2

0.3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.5

0.6

0.7

0.8

0.9

1.0

x Axis (meter) along y=b

5.0E-04
0.0E+00

0.4

Fig. 6. Stress of σ XY along the Y ¼ b and X axes for the CCSS plate in Table 1 with
different skew angles.

X Axis (meter) along Y=b

same way until a reasonable level of convergence achieves. After
obtaining the deflection function, one can determine all other
mechanical parameters in terms of deflection (w), i.e. stresses and
moments using well-known expressions presented elsewhere, see
for instance, [23]. One can consider φ equals to zero and develop
all above equations and results to rectangular plates.

σ of the skew Plate (Pa)

Fig. 3. Deflection of the CSCS skew plate, for various skew angles of φ.

1.50E+07
1.00E+07
5.00E+06
0.00E+00
-5.00E+06

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1.00E+07

0.9

1.0

phi=15

-1.50E+07
-2.00E+07

0.8
phi=0
phi=30

x Axis (meter) along y=b

Fig. 7. Stress of σ Y along the Y ¼b and X axes for the CCSS plate in Table 1 with
different skew angles.

4. Results and discussion
Consider a skew plate resting on the Winkler–Pasternak
foundation as Fig. 1. The plate is subjected to a uniform loading
and different combinations of clamp, free and simply supports.
2
The initial guess of g 0 Yị ẳ Y 2  b ị2 that does not satisfy all

boundary conditions necessarily, is considered. As it was mentioned, this g0(Y) is applied to obtain f1(X)'s constants in Eqs. (19)
and w0(X,Y)¼f1(X)g0(Y) is the result for the iteration #0. Then the
obtained f1(X) is used to calculate g1(Y) and w1(X,Y) ¼ f1(X)g1(Y)
completes the iteration #1. This procedure continues up to less
than four iterations when a high convergence is achieved. In
Table 1, the sample's mechanical properties and geometry are
presented although in some cases a parameter may be considered

as a variable for studying its effects on the results. Considering
Fig. 1 and downward pressure loading of q, it is clear that plate also
deflects downward although, for brevity, the signs of minus (  )
for deflection results are neglected and all diagrams are shown
inversely along z direction. When the skew angle is equaled to
zero, the results are developed to rectangular plates.
Table 2 shows deflection convergence in the center of the skew
plate for different boundary conditions. It reveals that the convergence of the method is very fast as there are no major changes after
the second iteration. It also shows that different boundary conditions
lead to obtain different results for the center deflection of the skew
plate. It is clear that more clamp supports decreases deflection.


326

A. Joodaky, I. Joodaky / International Journal of Mechanical Sciences 100 (2015) 322–327

Table 3
Comparison of the maximum deflection and stresses in the isotropic 151 skew plate
(a,b) on Winkler foundation in Table 1 but E¼ 380 GPa, q¼ 1e6 Pa, k0 ¼ 1e8 Pa/m
with the results of ANSYS code.
Deflections (m) and stresses (Pa)

w
σX
σY
σ XY

Boundary conditions

Present
ANSYS
Present
ANSYS
Present
ANSYS
Present
ANSYS

CCCC

CSCS

0.548e–4
0.555e–4
 0.415e10
 0.404e10
 0.151e10
0.146e10
0.762e9
0.718e09

0.609e–4
0.623e–4
 0.450e10
 0.438e10
 0.164e10
 0.158e10
0.827e9
0.779e09

Table 4
Comparison of deflection, w, in the center of the SSSS isotropic rectangular plate on
Winkler foundation, which is obtained from a plate by considering φ¼ 0, q ¼1000
and k0 ¼ 1e6 with a similar plate from [23] for different plate dimensions.
Source

Present
[23]

of a simply supported isotropic rectangular plate on Winkler
foundation with the properties in Table 1 but φ ¼0, q¼ 1000 and
k0 ¼1e6 are compared with the bending results of a similar SSSS
plate in the reference [23].

Deflections (m) in (a, b)
2a¼ 1, 2b¼ 1

2a¼ 1, 2b¼ 1.5

2a ¼1, 2b¼ 2

0.44780e–3
0.44842e–3

0.66567e–3
0.66679e–3

0.74845e–3
0.74969e–3

Fig. 2 shows the deflection diagrams of the skew plate with
various boundary conditions, which are not symmetric in some
cases. Note that maximum deflection occurs in the center, if the
plate is under symmetric boundary conditions e.g. SSSS.
By increasing the angle of φ edges of the skew plate become
closer to each other, so deflections decrease. Fig. 3 shows the effect
of φ on the deflection results of CSCS skew plate.
Fig. 4 for CCCF boundary conditions shows that foundation
stiffness of ki changes the deflection diagram considerably. In
addition, k1 changes the results much more than k0. In other word,
Pasternak's foundation limits deflection more than Winkler does
with the same stiffness. For this sample, these changes are not
considerable when the stiffness factor of ki is lesser than 1e4. Also,
if the foundation is rigid enough, for example for this case larger
than 1e7, the plate does not deflect at all.
The functions of stresses and moments are obtained in terms of
lateral deflection of w [23]. Therefore, one can achieve stress
components for every point of the skew plate. Figs. 5–7 show
amounts of the stress components of σ X ; σ Y ; σ XY of the CCSS skew
plate along the Y ¼b and X axes respectively for different skew
angle of φ. The effect of the skew angle is more tangible in σ XY .

Since the first and second boundaries are clamp in this case, the
beginning points of the skew plate have primary stresses. For the
ending edge of the plate, stresses are zero since boundaries are
simply and do not have resistance to bending. Increasing the angle
of the skew plate causes a considerable change in the stress
components particularly for σ XY . For the rectangular plates
(φ ¼0), this stress is zero in the edges and shows limited variations
along the plate. The shear stresses that are a consequence of the
skew angle, need to be considered during the applied designs
calculations of skew structures.
In Table 3, bending results of an isotropic skew plate on
Winkler foundation with the plate's properties in Table 1 but for
q ¼1e6 Pa, k0 ¼1e8 Pa/m, E ¼380 GPa, for two boundaries of CCCC
and CSCS are compared with the similar modeled plate in ANSYS
code. It shows that there is a good agreement between the results.
Considering φ ¼0 the skew plate is converted to a rectangular
plate. In Table 4, for different plate dimensions, the bending results

5. Concluding remarks
Application of EKM based on Galerkin method successfully
obtains a highly accurate approximate closed-form solution for
deflection and stress analysis of the skew plates resting on
Winkler–Pasternak foundation and subjected to uniform loadings
with various boundary conditions. EKM extracts two sets of
decoupled ordinary differential equations in terms of X and Y in
Oblique coordinates from the forth-order partial differential governing equation of the main problem. The solution procedure then
completes by presenting an exact approximate closed-form solution for two sets of ODE systems in an iterative scheme. The
method provides very fast convergence and highly accurate predictions. Angle of skew plates has an important role on the
deflection function and consequently on stresses diagrams, that
are completely different from corresponding rectangular plate. In

other words, even a small angle of φ that makes a deviation from
rectangular shape, causes a change in the stress distributions that
need to be considered in the designing calculations of these
structures. Pasternak foundation imposes a larger deflection limitation on the plate comparing to Winkler foundation. Comparing
to the results of the other valid literatures and FEM analysis of
ANSYS code, there are very good agreements with the results of
the present study in every case. Finally, by equaling inclination
angle of the skew plate to zero, the present study could be
developed and compared to the studies for rectangular plates.
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