OVING load problems are commonly encountered in a
wide range of engineering applications (a helpful review
can be found in [1]). Many of such applications deal with
materials that behave viscoelastically under applied loads, that
is, they exhibit a combination of both viscous and elastic
characteristics in their response. Examples include asphalt
roads, concrete airport runways, floating ice sheets, and
constrained layer viscoelastic laminated dampers, of which all
can be modeled as viscoelastic beams or plates under an
external moving load.
One of the earliest contributions to the foregoing problem
was made by Kelly [2] who examined the response of a
viscoelastic beam acted upon by a moving force. Several
studies have been conducted since then, focusing on the
analysis of viscoelastic beams, including those by Flügge [3],
Lv et al. [4], and Louhghalam et al. [5]. In all of these studies,
the inertial effects of the moving object due to convective
acceleration terms are neglected in order to further simplify
the analysis. However, this assumption may cause a
significant error, usually in cases involving a large high-speed
moving mass. In addition, the solutions proposed in those
studies are restricted to the special case of a Kelvin model for
the viscoelastic material, so that they cannot be generalized to
cover various types of viscoelastic behavior. Therefore, in the
current study, we aim at introducing a new analytical-
numerical solution that can be used to evaluate the dynamic
response of a rectangular plate made of a general viscoelastic
material and subjected to a moving inertial load. In this study,
we take advantage of the Laplace transform to derive the
governing equation of motion. This equation is then
transformed into a system of linear differential equations in
the time domain, of which the solution leads to the dynamic
response of the plate.
Consider a rectangular plate made of a viscoelastic material
and subjected to a moving inertial load m traveling along a
rectilinear trajectory at a constant speed v0 on line Y0 = b/2.
The schematic of the plate is shown in Fig. 1. In this figure, a,
b, and h represent the length, width and thickness of the plate,
respectively, and k and c are the stiffness and damping factors
of the supporting foundation. Any type of boundary condition
that can guarantee the stability conditions may be assumed for
the plate. Considering the equilibrium equation of an element
of the plate leads to
22
2
22
2 ( , , ) (1)
xy y
xMM
Mf x y t
xy
xy

+ + = −


where Mx, My and Mxy are the internal moments, and
M
A semi-analytical solution for the dynamic analysis of a rectangular
viscoelastic plate subjected to a moving inertial load
M. MOFID, M. A. FOYOUZAT
Department of Civil Engineering, Sharif University of Technology, Azadi Ave., Tehran, P.O. Box: 11155-
9313,. IRAN
Abstract: A semi-analytical method is developed to determine the response of a thin rectangular plate made of
a general viscoelastic material to the excitation of a moving inertial load. The governing equation of the general
problem is derived in the Laplace domain, which, for any particular viscoelastic model, is transformable into a
system of differential equations in the time domain. Any standard procedure can then be readily employed to
solve this system of equations. Using this method, sample response spectra are presented, through which the
effect of viscosity, mass and velocity is scrutinized. The results show that, when the viscosity is not large
enough, inertial terms cannot be ignored, especially when a heavy load is travelling at a high velocity.
Keywords: Moving inertial load, Plate, Viscoelastic material, Laplace transform.
Received: September 21, 2021. Revised: June 18, 2022. Accepted: July 14, 2022. Published: September 13, 2022.
1. Introduction
2. Method of Solution
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DOI: 10.37394/232021.2022.2.22
M. Mofid, M. A. Foyouzat
E-ISSN: 2732-9976
141
Volume 2, 2022
2
2
0
22
0
( , / 2, )
( , , )
( , , )
( , , )
( ) ( / 2) ( , , ) (2)
d z v t b t
z x y t
f x y t h m g
t dt
z x y t
x v t y b k z x y t c t

= − + −
− − − −



where ρ, g, δ and z(x, y, t) are the mass density of the plate, the
acceleration of gravity, the Dirac delta function and the
displacement field of the plate, respectively, and
is the inertial term of the mass. One can express the stress-
strain relation of a viscoelastic material as [3]
0( ) d (4)
'
tkl
ij ijkl
d
tdt
= −
where σij, εkl, and ijkl are respectively the stress, strain, and
relaxation components. The relaxation tensor in the above
equation can be represented in terms of relaxation functions
0(t) and 1(t) as
10
( )( ) ( )
ijkl ik jl il jk ij kl
tt
+ =+
.
By imposing a Laplace transform on Eq. (1) with respect to
time, taking advantage of Eq. (4) along with compatibility
equations resulted from the well-known Kirchhoff’s
hypothesis, one will get
3 4 4 4
01
4 2 2 4
3
4
01
ˆ ˆ ˆ
ˆˆˆ
( 2 ) ( 2 )
12
ˆˆˆ
( 2 ) (5)
12
h z z z
fs x x y y
hsz
= + + +
= +
where carets denote the Laplace transform. Assuming a
solution of the form for the
response, where ψn’s are the eigenfunctions of the plate, and
exploiting the orthogonality property, one will reach
( )
20
0
23
42
01 30 (6)
( , , )
12 2
( , ) ( ) ( ))
2d
12
ˆ ˆ ˆ
( 2 ) ) ( )
n
A
nn
b
dz A
dt
v t t
mb
x y g x v t y
h
s hs cs k F s
h

=



−−




+

−




+ + + +
L
where denotes the Laplace transform,
42
nn
hD
=
, ωn
are the natural frequencies, and D is the flexural rigidity.
Substituting the relaxation functions of the viscoelastic
material to Eq. (6) and imposing an inverse Laplace transform
on either side of it, the equation is transformed into the time
domain. For example, for the Kelvin model with
1 0 1
( ) ( ), 3 ( ) 2 ( ) 3t G t t t K

= + + =
(see [3]), where G, K
and μ are the shear modulus, bulk modulus, and viscosity
coefficient, respectively, application of this method leads to
KV KV KV KV KV KV KV
( ) ( ) ( ) ( ) (7)t F t F t F t+ + =M C K f
where
( )
( )
KV 3
KV 0 ,
33
12 ( );
4 12 24 ;
3
ij i j
ii ij i j x
ij
ij
M h m
h
cm
Cv
hh
+
+ +
=
=
( )
KV ,
33
T 4 4 4
KV 1 2 1 2
2
0
4 12 12 ;
3
{ } { ( ), ( ), ... , ( )}; diag ( , , , ) (8)
() ij i j xx
N N N N
ii
ij
km
K K G v
hh
F F t F t F t
+ + +
==
=
Ω
and N is the number of considered modes in the solution. The
system of Eqs. (7) can be readily solved, employing any
standard procedure.
In Fig. 2, the effect of higher modes in the response of the
system is examined. It is understood from this figure that the
heavier the moving inertial load becomes, the more the higher
modes affect the response. The same conclusion also holds
true as the viscosity level decreases, or as the speed of the
moving object further increases.
(a)
Fig. 1. Viscoelastic plate subjected to a moving inertial load
3. Results
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DOI: 10.37394/232021.2022.2.22
M. Mofid, M. A. Foyouzat
E-ISSN: 2732-9976
142
Volume 2, 2022
(b)
(c)
(d)
Fig. 2. Effect of higher modes in the response: (a) γ = 0.1, H = 0; (b) γ = 0.1,
H = 10-2; (c) γ = 0.7, H = 0; (d) γ = 0.7, H = 10-2.
In the current study, a semi-analytical method was put forward
to the dynamic response of a viscoelastic plate to a moving
inertial load. The Laplace transform employed in the solution
made possible the treatment of any type of viscoelastic
material. To verify the solution, the numerical results were
compared with those coming from the Moving Least Square
Method (MLSM), where an excellent agreement was obtained.
A numerical example was also solved to examine the effect of
viscosity and inertial terms on the response. It was shown that
the inertial terms of the moving object cannot be ignored in
low levels of viscosity, especially when a heavy inertial load is
travelling at high velocities.
The effect of considering a higher number of modes in the
solution was also investigated in the current study. It was
shown that as the inertial load or the speed of the moving
object rises, the effect of higher modes becomes more crucial.
Conversely, at a high viscosity level, the effect of higher
modes is proved to be less strong.
[1] H. Ouyang, Moving-load dynamic problems. Mech. Sys.
Signal Process. 25 (2011) 2039–2060.
[2] J.M. Kelly, Moving Load Problems in the Theory of
Viscoelasticity. Ph. D. Diss., Stanford 1962.
[3] W. Flügge, Viscoelasticity, Springer, 1975.
[4] P. Lv et al., Dynamic response solution to transient state
of viscoelastic road under moving load and its application,
J. Eng. Mech., ASCE 136 (2) (2010) 168-173.
[5] A. Louhghalam, et al., Flügge’s conjecture: dissipation
versus deflection-induced pavement-vehicle interactions, J.
Eng. Mech., ASCE 140 (8) (2013) 04014053.
[6] K.M. Liew et al., Moving least squares differential
quadrature method and its application to the analysis of shear
deformable plates, Int. J. Numer. Meth. Engng 56 (2003) 2331–
2351.
Contribution of individual authors to the
creation of a scientific article (ghostwriting
policy)
Massood Mofid conceived and designed the analysis and
supervised the findings of this work as well.
Mohammad Ali Foyouzat performed the analytic
calculations and the numerical simulations, and also
wrote the manuscript.
4. Conclusion
References
EQUATIONS
DOI: 10.37394/232021.2022.2.22
M. Mofid, M. A. Foyouzat
E-ISSN: 2732-9976
143
Volume 2, 2022