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Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10310-10315 10310  
 

www.etasr.com Hien et al.: Finite Element Analysis of a Continuous Sandwich Beam resting on Elastic Support and … 

 

Finite Element Analysis of a Continuous 

Sandwich Beam resting on Elastic Support and 

Subjected to Two Degree of Freedom Sprung 

Vehicles 
 

Ta Duy Hien 

University of Transport and Communications, Vietnam 

tdhien@utc.edu.vn 

 

Nguyen Duy Hung 

Campus in Ho Chi Minh City, University of Transport and Communications, Vietnam 

hungnd_ph@utc.edu.vn 

 

Nguyen Trong Hiep   

University of Transport and Communications, Vietnam 

nguyentronghiep@utc.edu.vn 

 

Giap Van Tan 

Hanoi Architectural University, Vietnam 

tangv@hau.edu.vn 

 

Nguyen Van Thuan  

Nha Trang University, Vietnam 

thuannv@ntu.edu.vn 

(corresponding author)  
 

Received: 3 November 2022 | Revised: 4 December 2022 | Accepted: 12 December 2022 

 

ABSTRACT 

This paper has developed a Finite Element Method (FEM) to calculate the dynamic response of a 

continuous sandwich beam resting on elastic support subjected to moving vehicles. The equation of motion 

is derived using the classical beam theory and FEM. The vehicle model is a two Degree of Freedom (2DOF) 

system that moves with a constant velocity. The governing equation of motion is integrated by applying the 

Wilson-θ time integration method to obtain the dynamic response in each time step. Numerical examples 

investigate the displacement of the sandwich beam with various values of the structure and vehicle velocity. 

The effects of the stiffness of elastic support and the vehicle velocity on displacement are studied.  

Keywords-FEM; forced vibration; continuous beam; elastic support 

I. INTRODUCTION  

Structures in the construction industry can be multiform 
and diverse. The most commonly used structures in civil 
engineering are columns [1, 2], piles [3], beams [4-8], frames 
[9, 10], and plates [11-19]. In general, a beam is a flexural 
member and the normal stress has maximum values at the top 
and the bottom of the beam. So, sandwich beams will have 
optimal bearing capacity if the outer layer is a high-strength 
material. In practice, the fibers at the top and bottom of the 

beam can be made of steel and the core layer from concrete or 
wood. If the sandwich beam is made of a suitable material, it 
will have a better load capacity, reduced height, and a larger 
span. Thus, the sandwich beam has many advantages if it is 
made of appropriate materials. A sandwich beam usually has 
two layers, i.e. the outer layer and the core layer, as shown in 
Figure 1.  

Structures are subjected to various dynamic loads, e.g. from 
wind, moving vehicles, impulsive loading, etc. Structure 



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www.etasr.com Hien et al.: Finite Element Analysis of a Continuous Sandwich Beam resting on Elastic Support and … 

 

dynamics have many practical aspects, e.g. the dynamic 
behavior of beams [20-23] and the dynamics of plates [24, 25]. 
The dynamic response of an elastic plate in a viscoelastic 
medium, resting on a viscoelastic Winkler foundation is 
investigated in [26]. The forced vibration of functionally 
graded plates resting on a viscoelastic elastic foundation was 
solved analytically in [27]. The effect of randomness in the 
elastic modulus on the eigenvalue of free vibration of non-
uniform beams is studied using stochastic finite elements in 
[28]. The variability dynamic response of a beam subjected to a 
moving load with various uncertain parameters is investigated 
by Monte Carlo simulation in [29]. The dynamic response of a 
beam resting on a viscoelastic foundation was analyzed with a 
novel state-space formulation considering the interaction of the 
oscillator-beam-foundation system in [30]. The dynamic 
response of sandwich beams with a viscoelastic core subjected 
to moving loads was studied by FEM in [31]. An analytical 
solution was preformed to investigate the transient response of 
sandwich beams in [32]. A new semi-analytical method was 
applied in [33] to study the dynamics of beams. The transient 
response of double beams with viscoelastic boundary 
conditions under a moving load was investigated by the 
analytical method in [34]. The nonlinear dynamic response of 
the coupled vehicle-pavement system was computed using the 
Galerkin truncation method in [35], in which the pavement was 
modeled as a Timoshenko beam on the six-parameter 
foundation. The dynamic response of bridges to moving 
vehicle loads was investigated using a semi-analytical approach 
in [36]. The arch bridge subjected to the moving loads was 
modeled as a continuous beam resting on interconnected 
springs to compute the dynamic response using the analytical 
method [37]. Author in [38] presented an analytical and 
numerical method to study the dynamics of a beam–mass 
system. Experiments and finite element simulations studied the 
dynamic response of metallic sandwich beams under impact 
loading in [39].  

 

 

Fig. 1.  Cross section of a sandwich beam. 

In this study, FEM is used to study the forced vibration of a 
continuous sandwich beam resting on an elastic support.  

II. FINITE ELEMENT FORMULATION  

Let us consider a continuous sandwich beam resting on 
elastic supports, subjected to moving vehicles, with the 
coordinate system placed at the mid-plane of the beam as 
shown in Figure 2.  

The deflection w of the beam can be written in a matrix 
form as [40]: 

          

 
 
 
 

     

1

1

1 2 3 4

2

2

,

e

w t

t
w x t N x N x N x N x

w t

t

N x q t





 
 
 

  
 
 
 



 (1) 

where   
e

q t denotes the displacement vector of the finite 

element and   N x are Hermite interpolation functions, 
defined as follows: 

   

   

2 3 2

1 22 3 2

2 3 2

3 42 3 2

1 3 2 ;   1 2

3 2 ;    

e e e e

e e e e

x x x x
N x N x x

L L L L

x x x x
N x N x x

L L L L

  
       

  


 
     

 

 (2) 

 
Fig. 2.  Continuous sandwich beam resting on elastic supports and 
subjected to a moving vehicle. 

The strain energy of the beam element is:  

2
2

0

2

20
2

0

2

1

2

e

Face
A

L Face

e

Core
A

Core

w
E y dA

x

U dx
w

E y dA
x

       
     

 
  

    
   

    
  






�����������

�����������

 (3) 

The potential energy of the elastic support is defined as: 

 
2

0

1

2
s sU k w x        (4) 

where ks is the stiffness of the elastic support.  

The kinetic energy of the beam element is given as: 

2 2

0 0

0

1

2

eL

e Face Core
A A

Face Core

w w
dx

t t
 

 
            

         
           

  

  
��������� ���������

 (5) 

The governing equation of the sandwich beam is derived as: 

Core

Face

Face



Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10310-10315 10312  
 

www.etasr.com Hien et al.: Finite Element Analysis of a Continuous Sandwich Beam resting on Elastic Support and … 

 

      Tb b s wM w K K w N f            ɺɺ  (6) 

where , ,
s b b

K K M           are the stiffness matrix of the elastic 

support and the stiffness and mass matrices of the sandwich 
beam, respectively. 

The equation of motion of the moving mass m1 and the 
contact force fw are determined as follows: 

   1 0m m m mm y c y w k y w    ɺɺ ɺ ɺ   (7) 

 1 2 2 1w mf m m g m w m y   ɺɺ ɺɺ    (8) 

where wm is the vertical dynamic deflection of the beam at 
mass m2. 

Combining (6) to (8), the governing equation is obtained: 

     
 

 

   
 

   

1

1

1 2

* 0*

0

* 0 ( )
+

0

b T

m m

b s T

m m mx

cw wM m N m

c N cy ym

K K k w N m m g

yc x N k N k

      
            

           
          

ɺɺ ɺ

ɺɺ ɺ

ɺɺ

 (9) 

where: 

   
   

       

2

2

2

2 2

*

* 2 ( )

* ( ) ( )

T

T

x

T T

xx x

m m N N

c m x t N N

k m x t N N m x t N N





 

ɺ

ɺ ɺɺ

 (10) 

III. SOLVING THE EQUATION OF MOTION WITH 
THE WILSON- METHOD 

The equation of motion of the vehicle-bridge interaction 
can be written in the following general form:  

                 M u t C u t K u t F t  ɺɺ ɺ  (11) 

where         , , ,M C K F t  are the mass, damping, and 
stiffness matrices, and the force vector, respectively. In order to 
use the Wilson scheme, we introduce the effective stiffness 
matrix as form: 

     0 1K K a M a C     ɶ    (12) 

where: 
 

0 12

1
;  a a

tt


 

 


              (13) 

The vector of effective forces at time t+θ can be calculated 
as follows: 

            
           
           

1 4 5

0 1 2

                

                

F t F t F t P t

C a u t a u t a u t

M a u t a u t a u t

       

  

  

ɶ

ɺ ɺ ɺɺ

ɺ ɺ ɺɺ

 (14) 

where 
2 4 5

1
;  1;  2

2

t
a a a

t

  
  

  
     

  
. The values of 

the parameters are selected as 1 / 2,  =1 / 6  , and 1.37  . 

The following linear system of equations computes the 
displacements at time after initialization of displacement, 
velocity, and acceleration vectors: 

     K̂ u t F t       ɶ    (15) 
The vector of acceleration at the time t+θ is calculated by: 

         
     

0

2 3
                   

u t a u t u t

a u t a u t

    

 

ɺɺ

ɺ ɺɺ
  (16) 

where 
3

1
1

2
a


  . 

Acceleration, velocity, and displacement at time t t  are 
calculated from: 

            
           
           

  

6 7

8

9

1

                     

u t t u t u t u t

u t t u t a u t a u t t

u t t u t t u t a u t

a u t t





    


     

    

  

ɺɺ ɺɺ ɺɺ ɺɺ

ɺ ɺ ɺɺ ɺɺ

ɺ ɺɺ

ɺɺ

 (17) 

where:   2 26 7 8 9
1

1 ;  ;  ;  
2

a t a t a t a t   
 

          
 

. 

IV. NUMERICAL EXAMPLES   

A. Example 1 

Considering the continuous sandwich beam shown in 
Figure 3, subjected to a moving vehicle with 2DOF. The 
rectangular sandwich beam has 3 uniform spans of 4m, the 
dimensions of cross-section are h=60cm, b=25cm, and the 
thickness of the face layer is 1cm. The parameters for the 
vehicle in Figure 3 are: m1=1680kg and m2=840kg. The 
structural properties of the sandwich beam are given in Table I. 

TABLE I.  MATERIAL PROPERTIES OF THE SANDWICH 
BEAM  

Fiber 

Material properties 

Material 
Young's modulus 

(GPa) 

Mass density 

(kg/m
3
) 

Face layer Steel 200 7800 

Core Wood 10 700 

 

Figure 4 shows the displacement at the center of the beam 
for 2 cases of stiffness of elastic support. The speed of the 
vehicle is set to 5, 10, and 15m/s. The Figure clearly shows that 
the displacements at the center of the beam are highly similar. 
So, the effect of speed of the vehicle on the displacement of the 
beam is small.    

 



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Fig. 3.  Three-span sandwich beam subjected to a moving vehicle. 

(a) 

 

(b) 

 

Fig. 4.  Displacement at the center of the beam for Kgoi equal to (a) 10
7
, (b) 

10
8 
N/m. 

(a) 

 

(b) 

 

Fig. 5.  Displacement at the contact point with the m2 mass of the vehicle 
for Kgoi equal to (a) 10

7
, (b) 10

8 
N/m. 

The displacement of the mass of the vehicle m2 is 
investigated with vehicle speed values of 5, 10, and 15m/s and 
stiffness of the elastic support Kgoi equal to 10

7
 and 10

8
N/m. 

When the speed is slow, the displacement is close to the static 
displacement. At high vehicle speed, the displacement is 
fluctuated around the displacement line at low speed.   

Figure 6 illustrates the displacement at the center of the 
beam and at the contact point m2 of the moving vehicle with 
speed of 10m/s for 3 stiffening cases of elastic support. As the 
stiffness of the support increases, the stiffness of the structure 
increases, leading to a decrease in displacement. When the 
stiffness of the support is quite large, the displacement at the 
center of the beam has an almost sinusoidal shape. It is clear 
that the stiffness of elastic support affects the displacement of 
the beam. 

 

(a) 

 

(b) 

 

Fig. 6.  Displacements for 3 stiffening cases of elastic support: (a) at the 
center of the beam, (b) at the contact point. 

 

Fig. 7.  Displacements of sandwich beam resting on elastic support. 

B. Example 2 

In this example, the dynamics of a sandwich beam is 
investigated by replacing two rigid supports of the example 1 

L

k c

m
1

m
2

y

wm 

v

m m

w

x

Kgoi

L/3 L/3L/3

Kgoi



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with elastic supports as described in Figure 7. The 
displacements at the center of the beam and at the contact point 
are shown in Figure 8 for 3 stiffening cases of elastic support. 
Figure 8 clearly shows that the displacements are greatly 
increased. It also shows the strong influence of the stiffness of 
the elastic support on the displacements. Increased stiffness 
causes the displacement to decrease drastically. 

 

(a) 

 

(b) 

 

Fig. 8.  Displacements for 3 stiffening cases of elastic support. (a) at the 
center of the beam, (b) at contact point. 

V. CONCLUSIONS 

The forced vibration of continuous sandwich beams 
traversed by a moving vehicle has been studied in this paper. 
Finite element method based on the classical beam theory was 
used to derive the governing equations of motion of the 
sandwich beams. The governing equations were solved with 
the Wilson method. Numerical examples were employed to 
investigate the effects of vehicle velocity and the stiffness of 
elastic support on the displacement response of the beam. The 
result of the numerical examples showed that changes in the 
stiffness of the elastic support have a significant effect on the 
displacement response of the beam.    

ACKNOWLEDGMENT 

This research is funded by University of Transport and 
Communications (UTC) under grant number T2021-CT-
007TĐ. 

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