 Open Access
 Total Downloads : 215
 Authors : D. T. Pham, P. H. Hoang, T. P. Nguyen
 Paper ID : IJERTV6IS030244
 Volume & Issue : Volume 06, Issue 03 (March 2017)
 DOI : http://dx.doi.org/10.17577/IJERTV6IS030244
 Published (First Online): 22032017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Dynamic Response of Beam on a New NonUniform Dynamic Foundation Subjected to a Moving Vehicle Using Finite Element Method
D. T. Pham#1
Faculty of Civil Engineering,
Quang Trung University, Dao Tan St., Nhon Phu Ward, Qui Nhon City, Vietnam
P. H. Hoang#2
Department of Construction of Bridge and Road, University of Science and Technology, The University of Da Nang, Da Nang City, Vietnam
T. P. Nguyen#3
Department of Civil Engineering – Architecture, Ho Chi Minh City Open University,
Ho Chi Minh City, Vietnam
Abstract– The purpose of paper analyzes influence of foundation mass on dynamic response of beam on nonuniform foundation subjected to a moving vehicle. This foundation model includes nonuniform linear elastic springs, shear layer, viscous damping and special consideration of the influence of a characteristic parameter of foundation mass. The moving vehicle is assumed to be consisted of two nodal masses that are connected by means of a springdamper components. The equation of motion for the beamvehiclefoundation interaction element is derived by means of dynamic balance principle. After by assembling the stiffness, damping and mass matrices, and vectors of nodal loads of all elements based on finite element method, the governing equations of motion for the integrated system are obtained and solved by stepbystep integration method procedure. The accuracy of the algorithm is verified by comparing the numerical results with the other numerical results in the literature. Also, the effects of characteristic parameter of foundation mass on dynamic analysis of beam vehicles interaction are investigated detail. The results show that the influence of foundation mass has effects significantly on dynamic response of beamvehicle interaction and more increasing dynamic response than others without influence of foundation mass.
Keywords– Foundation mass; moving vehicle; nonuniform foundation; dynamic analysis of beamvehicle interaction

INTRODUCTION
One of the most fundamental foundations suggested quite early is Winkler model in 1867. It has been commonly used in engineering application and attracted attention of many researchers in during many last decades with the uniform or nonuniform foundation stiffness considered as linear or nonlinear elastic springs [16].
But, one of the most important deficiencies of the Winkler model is appearance a displacement discontinuity between the loaded and the unloaded part of the foundation surface. Hence, several other foundation models had proposed by introducing some kind of interaction between the independent springs by visualising various types of interconnections to overcome the deficiency of Winkler model such as: Filonenko [7]; Hetenyi, [8]; Pasternak [9]; Reissener [10]; Kerr [11]; Vlasov [12].
It can be seen that the foundation always has foundation mass in reality, so that the foundation mass have to effect on dynamic response of structurefoundation interaction in during vibration of its. But, one of the most important deficiencies of the above foundation models overlooks the influence of the foundation mass. Hence, a new foundation model called dynamic foundation model including elastic spring, shear layer, viscous damping and special consideration of the influence of a characteristic parameter of foundation mass had proposed by Pham [13]. The dynamic foundation model applied to analyze response of beam and plate structures subjected to moving load [14, 15] and the results show that the influence of foundation mass has effects significantly on dynamic response of structures.
To continuously attention to effects of foundation mass on dynamic responses of structures, this study analyzes effects of the foundation mass on dynamic response of beam subjected to a moving vehicle. This foundation model includes the non uniform linear elastic springs, the uniform linear shear layer, viscous damping and special consideration of the influence of a characteristic parameter of foundation mass. The moving vehicle is assumed to be consisted of two nodal masses that are connected by means of a springdamper components. By means of dynamic balance principle and finite element method, the equation of motion for the beamvehicle foundation interaction element is derived. The governing equations of motion for the integrated system are obtained by assembling the stiffness, damping and mass matrices, and vectors of nodal loads of all elements, and solved by stepby step integration method procedure such as Newmarks method. The accuracy of the algorithm is verified by comparing the numerical results with the other numerical results in the literature. Also, the effects of characteristic parameter of foundation mass on dynamic analysis of beam vehicle interaction are discussed.

FORMULATION
A. Model of beamvehiclefoundation interaction
Consider an EulerBernoulli beam resting on the dynamic foundation subjected to moving vehicles is shown in Fig.1.
The mass matrix of the beam element including the effects of the foundation mass of both the beam and foundation based on the kinetic energy of the beam element, can be expressed as
M
Mb MF
(5)
e
where Mb
e e e
is the mass matrix of the beam element and
e
MF
is the mass matrix for the influence of the foundation
Sub 1 Sub 2
Fig1. The beam subjected to moving vehicle on nonuniform foundation.
mass, is also written as
w we
MF N T mN dx
(6)
e
w w
The nonuniform foundation based on the dynamic foundation model [13], which fully describes dynamic characteristic parameters for behavior of foundation including the elastic stiffness idealized based on the linear elastic
The viscous damping property of the foundation is considered to be the dashpots system and based on the dissipated energy of these disputes the damping matrix can be expressed as
springs modulus
kW , the shear foundation modulus
kS ,
CF
N
T cN
dx
(7)
viscous damping c and the foundation mass F are respectively replaced by lumped mass m at the top of the elastic spring connected between elastic layer and shear layer. The pressuredeflection relationship at the time t due to a
pressure q(x, y,t) is determined based on dynamic balance principle, can be expressed mathematically as follows
q(x, y, t) k w(x, y, t) c w(x, y, t)
W t
where Ns and Nw are the matrix of interpolation functions for displacements and rotation, respectively, studied in many research related to finite element method.
C. The governing equation of motion
The moving vehicle model is regarded as a twonode with one is associated with each of two concentrated masses. The stiffness and damping coefficients of the oscillator are denoted
2 w(x, y, t)
m
k 2 w(x, y, t)
(1)
by kv
and
cv , respectively. The mass of the vehicle and the
t 2 s
where, the lumped mass m is given by
mass of the wheel is denoted by M v and mw, respectively. In addition, zv and zw denote the vertical displacements of two
m f
(2)
nodes measured from the static equilibrium position. At any
time t , the position of the moving vehicle is xm vt and the
in which is an experimental parameter characterized the influence of foundation mass.
left end of the beam element in global coordinate (node ith ) is to be x Int x / ll . Then, one ca find the element number
i m
B. Formulation of element matrices
The beam modeled as uniform EulerBernoulli beam is assumed that the beam material is isotropic; the vibration amplitudes of beam are sufficiently small and the bond between the beam and the foundation is perfect. Each beam element has two nodes, each node having two degree of freedom including vertical displacement and rotation displacement.
Based on the strain energy of the beam element, stiffness matrix of the beam element resting on nonuniform foundation
ith Int x / l1, nodes ith and i 1th , which the moving vehicle is applied to at any time t . Therefore, can be rewritten in terms of the global instead of the local
m
th
t xm i l (8)
By assuming the nojump condition for the moving vehicle, the contact force can be related to displacement of the contact force and its derivatives. Equations of motion of the moving vehicle can be written as follows
v
including the effects of both bending deformation of the beam
M 0 z
c c z
and nonuniform foundation is given by
v v v
v
K KB KW KS
(3)
0 mw zw cv cv zw
(9)
e e e e
kv kv zv 0
where KB
is the normal bending stiffness matrix; KW
and
k k
z
f

M

m g
e e
v v w
c v w
e
KS
are the nonuniform elastic stiffness matrix and the
where
fc is the contact force.
shear stiffness matrix, respectively, are given by
e w W w
KW N T k N dx
KS N T k N dx
(4)
Assuming that all information of the system at time t is known and t is a small time increment, the first row of Eq.
(9) can be expanded in an incremental form at time t t
e s S s
[16]Mv zv,t t cv zv,t t kv zv,t t qvc,t t
(10)
with
q c z
k z (11)
delta function and i denotes contact element between the beam and contact force.
By assembling the stiffness, damping and mass matrices,
vc,t t v w,t t v w,t t
Based on Newmarks method, average acceleration method ( 0.5 and 0.25 ), the displacement zw and its derivatives at time t t can be written as
and vectors of nodal loads of all elements corresponding degrees of freedom in the global coordinate, the governing equation of motion of the system beamvehiclesfoundation interaction in each time step is defined as follows
z b0 q
q
b z


b z
MU CU KU F
(19)
, t t
z b3 q
c , t t
,t 1 ,t

q b z
2 ,t
b z
(12)
where M, C, and K are the overall mass, damping and stiffness matrices of the system, respectively; U and F is
,t t
z z
c, t t
1 q
,t
4 ,t

q
5 ,t
the nodal displacement vector and the external force vector of the system, respectively. It is used for studying the dynamic
with
,t t
,t
c,t t v,t
response of the beamvehiclefoundation interaction and solved by means of the direct stepbystep integration method based on Newmarks algorithm.
v b0 Mv b3cv kv
qv,t Mv b1 zv,t b2 zv,t cv b4 zv,t b5 zv,t kv zv,t
and coefficients bi given by
(13)


NUMERICAL RESULTS

Verified examples
The first example analyzes free vibration of a simple
b 1 , b 1 , b 1 1, b ,
0 t 2 1 t 2 2 3 t
support beam resting on a uniform linear foundation without effect of foundation mass. The dimensionless parameters K1
(14)
and K2 representing the stiffness of linear elastic springs and
b 1, b
t 2
shear layer of the foundation and the dimensionless natural
4 5
2
frequency are defined as follows [17]
Substituting Eq. (12) into incremental form of the second
k L4
k L2 A
K W , K
S , L2
(20)
row of Eq. (9), the contact force fc in time t t is determined by
1 EI 2 2 EI EI
fc,t t mw zw,t t cc zw,t t kc zw,t t pc,t t qc,t (15) in which
in which is natural circular frequency of the beam. The
convergences of the lowest natural frequencies are compared with the results of Matsunaga [17], shown in Table 1.
Table 1. The dimensionless natural frequencies of the
v
c vw
vw
beam
c cv 1
v
, kc kv 1
(16)
K1 K2
Present Ref. [17]
p M m g, p
vw q q
0 13.9577 13.9577
with
c,t t v w c,t
v,t w,t
v
10 14.3115 14.3115
102 17.1703 17.1703
103 1 34.5661 34.5661
vw b3cv kv , qw,t cv b4 zv,t b5 zv,t kv zv,t
(17)
104 100.9694 100.9694
It assumes that there is no loss of contact between the tire and the upper surface of the beam, the displacement and acceleration of the wheel ( zw and zw , respectively) equals to the deflection and acceleration at the contact position of vehicle and beam. Therefore, the differential equation of motion of the beam element resting on the nonuniform
105 316.5356 316.5356
Table 2. The maximum vertical displacement at the midpoint
of the beam
k1 k2 Present SAP2000 (kN/m2) (kN/m2) (mm) (mm)
foundation subjected to the moving vehicle at time t can be
125 250
3.8586 3.8593
expressed as
Me ue +Ce ue +Ke ue = xi vt Nw, fc
(18)
250
500
750
500
1000
1500
2.3666 2.3674
1.3830 1.3834
1.0023 1.0026
where Nw,
is the value of interpolation function depended
1000
2000
0.7980 0.7982
on the coordinate corresponding with the position of wheel
on the beam element ith
at the time t ; xi vt is the Dirac
Continuously, the maximum vertical displacement of a simply supported beam resting on nonuniform linear elastic foundation subjected to a vertical static point load at the midpoint of the beam is investigated. The beam has length L 20 m , Youngs modulus E 210Gpa , moment of inertia
I 0.667 104 m4 , mass per unit volume 7800 kg / m3 and section area A 0.2 m2 . The point force acting on the beam has a value of P 10 kN and the nonuniform
foundation composed of two subdomains of equal length, the first (on the left side) with k1 and the second (on the right side) with k2 , see in Fig. 1. The maximum of vertical displacement of the midpoint of the beam are compared with result obtained by SAP2000 software, shown in Table 2.
In the next example is carried out to verify the present algorithm for a simply supported beam subjected to a moving vehicle. The following data are adopted the beam and the moving vehicle similar to Neves et al., 2012 [18]. The time history of vertical displacement of midpoint of the beam and body of the moving vehicle are plotted in Fig. 2.

Numerical investigation
In this section, the physical and geometric properties of a simply supported beam, vehicle and dynamic foundation analyzed the effects of foundation mass on dynamic response of beamvehiclefoundation interaction are listed in Table 3. The characteristic parameter of elastic stiffness of non uniform foundation is defined by ratio of elastic foundation stiffness of sub two (on the right) with lastic foundation stiffness of sub one (on the left), and the characteristic parameter of length of nonuniform foundation is also
defined by ratio of length of sub one (on the left) with length of sub two (on the right).
Table 3. Properties of railway track, vehicle and dynamic
Item
Notation
Unit
Value
Beam
Length
L
m
20
Youngs modulus
E
Gpa
24
Foundation mass
kg/m3
2500
foundation
Displacement (m)
(a)
(b)
0.001
0
0.001
0.002
0.003
0.001
0
Relative x /L
0 0.2 0.4 0.6 0.8 1
Ref. [18]
Present
Ralative x /L
0 0.2 0.4 0.6 0.8 1
Ref. [18]
Present
Cross sectional area A m2 0.3
Second moment of area I m4 2.25×103
Moving Vehicle
Mass of car body Mv kg 5×103
Spring stiffness kv N/m 1.5×106
Dashpot coefficient cv Ns/m 1.5 x104
Mass of axle mw kg 5×102
Nonuniform foundation
Linear stiffness kW N/m2 1.5×106
Shear parameter kS N 5×104
Viscous damping c Ns/m2 1.5×103 Density of foundation f kg/m3 1800
In the first investigation, the dynamic responses of the
Displacement of M v (m)
0.001
0.002
0.003
Fig 2. Time history of displacement: (a) Vertical displacement at the midpoint of beam; (b) Vertical displacement of body.
Through the above examples, the numerical results from the program based on the suggested formulation show quite good agreement with numerical results in the literature. Therefore, the program which will analyze the influence of many parameters on dynamic response of the beamvehicle foundation interaction is reliable.
beam on the foundation subjected to a moving vehicle with various values of the characteristic parameter of nonuniform foundation are studied. The effects of characteristic parameters of elastic stiffness of nonuniform foundation on time history of vertical displacement of the midpoint of the beam have been plotted in Fig. 3 and 4. The influence of foundation mass on vertical displacements of the midpoint of the beam is plotted in Fig. 5. It can be seen that the characteristic parameter of the nonuniform foundation and foundation mass of the foundation effect significantly on dynamic behavior of the beam. It increases the time history of vertical displacements of the midpoint of the beam with decrease values of the parameters and , shown in Fig 3
and Fig. 4. But, with an increase of values of the parameters of foundation mass also increase the time history of the vertical displacements of the midpoint of the beam, plotted in Fig. 5.
(a)
(b)
2.0
W(L/2,t) [mm]
0.0
2.0
4.0
6.0
8.0
2.0
W(L/2,t) [mm]
0.0
2.0
4.0
6.0
8.0
0 0.2 0.4 0.6 0.8 1
x/L
0 0.2 0.4 0.6 0.8 1
x/L
(c)
(d)
3.0
W(L/2,t) [mm]
0.0
3.0
6.0
9.0
12.0
4.0
W(L/2,t) [mm]
0.0
4.0
8.0
12.0
16.0
0 0.2 0.4 0.6 0.8 1
x/L
0 0.2 0.4 0.6 0.8 1
x/L
(c)
2.0
W(L/2,t) [mm]
0.0
Fig 4. Time history of displacement with =0.5 and =0.5: v=10 m/s, (b) v=25 m/s, (c) v=50 m/s, (d) v=75 m/s
2.0
(d)
2.0
4.0
6.0
8.0
W(L/2,t) [mm]
4.0
0.0
4.0
8.0
12.0
0 0.2 0.4 0.6 0.8 1
x/L
0 0.2 0.4 0.6 0.8 1
x/L
(a)
(b)
0.0
W(L/2,t) [mm]
2.0
4.0
6.0
8.0
2.0
W(L/2,t) [mm]
0.0
2.0
4.0
6.0
8.0
0 0.2 0.4 0.6 0.8 1
x/L
Fig 3. Time history of displacement with =0.5 and =0.5: (a) v=10 m/s, (b) v=25 m/s, (c) v=50 m/s, (d) v=75 m/s
2.5
W(L/2,t) [mm]
(a)
0.0
2.5
(c)
0 0.2 0.4 0.6 0.8 1
x/L
2.5
W(L/2,t) [mm]
0.0
2.5
(b)
5.0
7.5
10.0
2.5
W(L/2,t) [mm]
0.0
2.5
5.0
7.5
10.0
0 0.2 0.4 0.6 0.8 1
x/L
0 0.2 0.4 0.6 0.8 1
x/L
(d)
5.0
7.5
10.0
2.5
W(L/2,t) [mm]
0.0
2.5
5.0
7.5
10.0
0 0.2 0.4 0.6 0.8 1
x/L
0 0.2 0.4 0.6 0.8 1
x/L
Fig 5. Time history of displacement with =0.5 and =0.5: (a) v=10 m/s, (b) v=25 m/s, (c) v=50 m/s, (d) v=75 m/s
(a)
3.0
DMF
2.5
2.0
(c)
3.8
DMF
3.1
2.4
1.5 1.7
(b)
1.0
3.4
DMF
2.8
2.2
0 20 40 60 80 100
v [m/s]
(d)
1.0
3.0
DMF
2.5
2.0
0 20 40 60 80 100
v [m/s]
1.6 1.5
1.0
0 20 40 60 80 100
v [m/s]
1.0
0 20 40 60 80 100
v [m/s]
(c)
3.4
DMF
2.8
2.2
1.6
Fig 7. The effects of the parameter on the DMFs of the beam: =0.5, (b)
=0.75, (c) =1, (d) =2
DMFs
(a)
(d)
1.0
3.4
DMF
2.8
2.2
1.6
1.0
0 20 40 60 80 100
v [m/s]
0 20 40 60 80 100
v [m/s]
(b)
V [m/s]
DMFs
Fig 6. The effects of the parameter on the DMFs of the beam: (a) =0, (b)
=0.25, (c) =0.5, (d) =1
(a)
3.8
3.1
DMF
2.4
V [m/s]
(b)
1.7
1.0
3.8
DMF
3.1
2.4
1.7
0 20 40 60 80 100
v [m/s]
(a)
Fig 8. The effects of the parameter and on the DMFs of vertical displacement of the beam: (a) =0.3, (b) =0.6
DMFs
1.0
0 20 40 60 80 100
v [m/s]
V [m/s]
DMFs
(b)
V [m/s]
foundation and it also is useful for problems of practical design.
REFERENCES

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Fig 9. The effects of the parameter and on the DMFs of vertical displacement of the beam: (a) =0.5, (b) =2
To show more clear the influence of the characteristic arameters of the nonuniform foundation on dynamic analysis of the interaction between the beam and foundation subjected to a moving vehicle, the effects of the above parameters on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving vehicle, shown in Fig. 6 and Fig. 7. At the same time, the effects of the between parameters or and the parameters
of foundation mass for various values of the velocity of the moving vehicle on DMFs of the beam are also studied, shown in Fig. 8 and Fig. 9.
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CONCLUSIONS
In this paper, the dynamic analysis of the beam on the non uniform foundation subjected to a moving vehicle is investigated by means of finite element method. The non uniform foundation includes nonuniform linear elastic spring, shear layer, viscous damping and special consideration influence of foundation mass of it. The beam, vehicle and nonuniform foundation are regarded as an integrated system and the governing equation of motion of the system is derived based on dynamic balance principle and solved by stepby step integration method procedure. The accuracy of the numerical results is verified by comparing its numerical solutions with those of other available numerical results. The results show that the characteristic parameters of nonuniform and foundation mass effect significantly on dynamic response of the beam. A comparison shows that the foundation mass in the dynamic foundation model is more increasing dynamic behavior of the beam than others without the influence of foundation mass. The presented results can be employed to perform the parametric studies about various dynamic and structural properties of the structuralvehiclefoundation interaction model such as tracktrainfoundation, roadvehicle
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