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Vol.1, No.2, 120-123 (2009) Natural Science
http://dx.doi.org/10.4236/ns.2009.12014
Copyright © 2009 SciRes. OPEN ACCESS
An Improved Model for Bending of Thin Viscoelastic
Plate on Elastic Foundation
Zhi-Da Li1,2, Ting-Qing Yang1, Wen-Bo Luo3
1Department of Mechanics, Huazhong University of Science and Technology, Wuhan, China; [email protected]
2School of transportation, Wuhan University of Technology, Wuhan, China
3College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, China
Received 10 June 2009; revised 16 July 2009; accepted 20 July 2009.
ABSTRACT
An improved model for bending of thin viscoe-
lastic plate resting on Winkler foundation is
presented. The thin plate is linear viscoelastic
and subjected to normal distributed loading, the
effect of normal stress along the plate thickness
on the deflection and internal forces is taken
into account. The basic equations for internal
forces and stress distribution are derived based
on the general viscoelastic theory under small
deformation condition. The reduced equations
for elastic case are given as well. It is shown
that the proposed model reveals a larger flex-
ural rigidity compared to that in classic models,
in which the normal stress along the plate
thickness is neglected.
Keywords: Thin Viscoelastic Plate; Deformable
Foundation; Flexural Rigidity; Winkler Foundation
1. INTRODUCTION
The analysis of soil-structure interaction has a wide
range of applications in structural and geotechnical en-
gineering, for instance, in highway asphalt pavement
engineering, the pavement is usually treated as thin elas-
tic/viscoelastic plate structure resting on elastic/viscoe-
lastic foundation. Due to the complexity of the actual
behavior of foundations, many idealized foundation mo-
dels have appeared in the literature [1]. The simplest of
those models, which was proposed in 1867 by Winkler,
assumes that the soil medium consists of a system of
mutually independent spring elements. There are many
papers dealing with the elastic beam or plates resting on
the Winkler foundation in the literature [2,3]. As com-
putational power has developed, more realistic modeling
of soil-structure interaction has become possible. Be-
cause of the importance of viscoelastic nature of the ma-
terials used for structures, e.g. asphalt layer of pavement
structure, many works have been done to deal with the
bending behaviour of thin viscoelastic plate on elas-
tic/viscoelastic foundation. Most of such works utilized
the models similar to those for bending of elastic plate.
Mase [4] directly offered the fundamental equations for
bending of viscoelastic plate by replacing the flexural
rigidity of elastic plate with the rigidity of viscoelastic
plate. Radovskii [5] discussed the problem of treating
highway and airport pavement as thin viscoelastic plate.
Pister [6], Robertson [7] and Hewitt and Mazumdar [8]
applied the elasticity-viscoelasticity correspondence pri-
nciple to get the solution of the bending problem of vis-
coelastic plate. In contrast to dealing with the viscoelas-
tic plate on elastic foundation, some attempts have also
been made to solve the bending problem of elastic plate
resting on viscoelastic foundation. Sonoda et al [9,10]
studied the circular and rectangular plate on linear vis-
coelastic foundation. Lin [11] and Yang et al. [12] ana-
lyzed the dynamic response of circular plate resting on
viscoelastic half space. All of the above studies followed
the classic model and traditional flexural rigidity for thin
plate bending, in which the Kirchhoff hypothesis was
used and
z
z
was neglected [13]. However, in the case
of large lateral load subjecting to thin plate resting on a
deformable foundation with relatively large rigidity, the
bearing stresses along the plate thickness,
z
z
, may not
be ignored. Furthermore, it is the bearing stress of the
plate that transfer the active lateral load to the founda-
tion, the boundary condition on the main surfaces of the
plate should be satisfied. Therefore it is necessary to
develop a method to consider the effect of lateral normal
stress. In this paper, we first seek to develop a modified
Kirchhoff theory for thin viscoelastic plate resting on
Winkler foundation, in which the effect of the lateral
normal stress is considered. Then we reduce the obtained
results to the problem of thin elastic plate resting on
elastic foundation, and a different elastic flexural rigidity
is obtained.
Z. D. Li et al. / Natural Science 1 (2009) 120-123
Copyright © 2009 SciRes. OPEN ACCESS
121
Figure 1. Thin viscoelastic plate resting on a Winkler foundation.
2. VISCOELASTIC PLATE ON WINKLER
FOUNDATION: AN IMPROVED
MODEL
Figure 1 illustrates a thin viscoelastic plate of a thick-
ness h, its middle-plane coincides with the x-y plane of
the reference coordinate system. Let the upper surface
(2zh ) be subjected to a normal distributed loading
with intensity of (, )qxy , while the lower surface
(2zh) rests on a Winkler type foundation. In the
Winkler foundation model, the foundation for the plate is
assumed to act like a set of springs. Thus the foundation
reaction force can be written aspkw , where k de-
notes the elastic stiffness of the foundation and w is the
lateral deflection of the plate.
For a linear viscoelastic plate, the equilibrium equa-
tions, the strain-displacement relations and the constitu-
tive equations are given, respectively, by
;0
ij j
(1)

;;
1
2
ijijj i
uu

(2)
1
2
ij ij
s
Gde
, 1
3
kk jj
Kd
 (3)
where the body forces are neglected, 1()Gtand 1()Kt
are the shear modulus function and volume modulus
function, respectively, the * denotes convolution product.
The deviatoric components of stress and strain tensors
are
1
3
ijijij kk
s

 and kkijijij
e

3
1
 (4)
For simplicity, the displacement components along x,
y and z axis are denoted by u, v and w respectively. As-
suming 0
zz
,0
zx
, 0
zy
, from Eq.2 we have

;;
,, , ,
xy
wwxytuzwv zw
(5)
and
2
;;;
, , ,
xx xxyyyyxy xykk
zwzwzw εzw

   (6)
Combining Eq.4 and Eq.3, along with substitution of
Eq.6 into it, the stress components can be written as

2
1;1 1
2
23
xx xx
zGdwz KGdw

 

 (7)

2
1;1 1
2
23
yy yy
zGdwz KGdw



 (8)
1;
2
xy xy
zG dw
 (9)
The other three components of the stress tensor can be
obtained by using the equilibrium equations, Eq.1. Inte-
grating the first and second representations of Eq.1 over
z and considering the boundary condition
x
0
zy
for 2zh
(i.e. there is no shear stress on the plate
surface and no friction between plate and foundation),
we get

2
22
1
1;
2
423
z
xx
K
h
zGdw







(10)

2
22
1
1;
2
423
z
yy
K
h
zGdw







(11)
The third representation of Eq. 1 is
z
y
zx
zz
zxy
 

Substituting Eqs.10 and 11 into the above equation
and integrating it over z yields

23
34
1
1
12
423 823
zz
K
hh h
zz Gdwkw




 


 
 


(12)
It should be noted here that the stress boundary condi-
tion zz kw
at the lower surface (2zh) of the plate
is used to determine the integral constant. Moreover,
zz q
at the upper surface (2zh ) of the plate, thus

3
4
11
4
12 3
hKGdwqkw



 (13)
This is a differential-integral equation interrelating the
materials’ properties, the applied normal loads and the
corresponding lateral deflection.
We denote by ij
M
the bending moments and the
twisting moments, and
j
Q the shear forces
22
22
,
hh
ij ij jzj
hh
M
zdz Qdz




Z. D. Li et al. / Natural Science 1 (2009) 120-123
Copyright © 2009 SciRes. OPEN ACCESS
122
Table 1. Variation of D1/D with Poisson ratio.
0 0.2 0.25 0.30 0.35 0.40 0.45 0.49 0.5
D1/D 1 1.067 1.125 1.225 1.408 1.80 3.025 13.005
∞
Substituting Eqs.7-11 into the above expressions, we
get

3
2
1;1 1
2
2
12 3
xx xx
h
M
GdwKG dw


 




(14)

3
2
1;11
2
2
12 3
yy yy
h
M
GdwKG dw


 




(15)
3
1;
6
xy xy
h
M
Gdw (16)

3
2
11 ;
4
123
xx
h
QKGdw

 

 (17)

3
2
11 ;
4
12 3
yy
h
QKGdw

 

 (18)
Furthermore, the stress components can be expressed
in the form of ij
M
and
j
Q as follows:
3
12 , ( ,,)
ij ij
zMijx y
h

(19)
2
32
1, (,)
2
zj j
zQ jxy
hh


 





(20)

2
1
21
2
zz
zz
qkw kw
hh

 


(21)
The forms of Eqs.19-21 are as same as those for elas-
tic bending plate [13,14], however it should be noted
that ij
M
and
j
Q involved are the moments and the shear
forces in viscoelastic cases.
3. ELASTIC PLATE ON WINKLER
FOUNDATION
In the case of thin elastic plate with material constants of
G andK, resting on a Winkler foundation, the Eq.13
reduces to the following equation
4
1
D
wqkw (22)
in which


2
33
12
1
4
1231 2
12 1
hEh
DKG

 


(23)
and
are the elastic modulus and Poisson ratio of
the elastic plate, respectively.
Again, it can be seen that the form of the control equa-
tion for elastic plate bending is similar to that derived
from classic theory. However the flexural rigidity 1
D is
different from that in classic theory, which gives
32
12 1DEh

. Obviously,
2
1112DD
 . It
can be seen that 1
D
D increases with the Poisson's ratio.
Several results are listed in Table 1. For most engineer-
ing structural materials,
is about 0.3 [13], thus the
flexural rigidity given in the present model is about 22%
larger than that in the classic model.
In elastic cases, Eqs.14-18 reduces to the followings
in terms of1
D:
1; ;
1
xxxx yy
MDww
 
(24)
1; ;
1
yyyy xx
MDw w
 
(25)
1;
12
1
xy xy
M
Dw

 

(26)
2
1;
xx
QDw  (27)
2
1;yy
QDw (28)
Moreover, the stress distribution in the elastic plate
can be read in the same forms as Eqs.19-21 if the mo-
ments and shear forces are given by Eqs.24-28.
4. CONCLUDING REMARKS
We have derived in this paper the basic equations for
bending of viscoelastic thin plate on Winkler foundation.
In classic treatment, the normal stress
z
z
is assumed to
be zero, which is the main characteristic of the plane
stress state in classic elastic theory, and thus the corre-
sponding classic bending model of thin elastic plate can
be considered as a plane stress model. In contrast with
the classic treatment, in the proposed model in this paper,
z
z
is taken into account and the general viscoelastic
constitutive equation is used to represent the thin plate
behavior, and the plate thickness is further set to be con-
stant, i.e. 0
zz
, which is the main characteristic of the
plane strain state in classic elastic theory. Thus we may
consider this improved model as a quasi-plane strain
model.
5. ACKNOWLEDGMENTS
The study was supported by the National Natural Sci-
Z. D. Li et al. / Natural Science 1 (2009) 120-123
Copyright © 2009 SciRes. OPEN ACCESS
123
ence Foundation of China (No.10372074).
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