 Open Access
 Total Downloads : 108
 Authors : XuanToan Nguyen, Kuriyama Yukihisa, DuyThao Nguyen
 Paper ID : IJERTV6IS060349
 Volume & Issue : Volume 06, Issue 06 (June 2017)
 Published (First Online): 30062017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Random Dynamic Response Analysis of Bridge Subjected to Moving Vehicles
XuanToan Nguyen#1
Faculty of Road and Bridge Engineering, University of Danang – University of Science and Technology, Danang City, Vietnam
Kuriyama Yukihisa#2 Research into Artifacts, Center for Engineering,
The University of Tokyo, Japan
DuyThao Nguyen#3
Faculty of Road and Bridge Engineering, University of Danang
University of Science and Technology, Danang City, Vietnam
Abstract– This article presents random vibration analysis of dynamic vehiclebridge interaction due to road unevenness based on the Finite element method and MonteCarlo simulation method. The road unevenness are described by a zeromean stationary Gaussian random process. The vehicle is a dumper truck with three axles. Each axle of vehicle is idealised as two mass dynamic system, in which a mass is supported by a spring and a dashpot. The structural bridge are twospan slab beam concrete, are simulated as bending beam elements. The finite element method is applied to established the overall model of vehiclebridge interaction. Galerkin method and Green theory are used to discrete the motion equation of vehiclebridge system in space domain. Solutions of the motion equations are solved by mean of RungeKuttaMersion method (RKM) in time domain. The numerical results are in good agreement with fullscale field testing results of the slab beam concrete at NguyenTriPhuong bridge in Danang city, Vietnam. Also, the effects of road surface condition on dynamic impact factor of bridge are investigated detail. The numerical results show that dynamic impact factor of bridge has increased significantly when road unevenness have varied from Grade Aroad to Grade Eroad according to ISO 8608:1995 [1]Mechanical vibration – Road surface profiles Reporting of measured data.
Keywords– MonteCarlo simulation method, finite element method, road unevenness, moving vehicles, vehiclebridge interaction, slab beam concrete.

INTRODUCTION
Vehiclebridge interaction has been a subject of significant research for a long time. The aim of these studies is to investigate the structural behaviour of bridge under moving vehicles, as well as the ride comfort of vehicles travelling a bridge. Dynamic vehiclebridge interaction results in a increase or decrease of the bridge deformation, which is described by the dynamic impact factor (IM) or the dynamic amplification factor (DAF) or dynamic load allowance (DLA) that reflects how many times the constant load must be multiplied to cover additional dynamic effects, FrÃ½ba [2]. The dynamic IM plays a vital role in the practice of bridge design and condition assessment. Accurate evaluation of IMs will lead to safe and economical designs for new bridges and provide valuable information for condition assessment and management of existing bridges.
Honda et al. [3] derived the power spectral density (PSD) of road surface roughness on 56 highway bridges, measured using a surveyor's level. For each bridge, 84 lines at 1020 cm intervals and 0.5 and 2.0 m from the centerline of the road were measured. The authors observed that the PSD of roadway roughness can be approximated by an exponential function, and proposed different functions for certain bridge structural systems. Palamas et al. [4], Coussy et al. [5] presented a theoretical study of the effects of surface road unevenness on the dynamic response of bridges under suspended moving loads. A singledegreeoffreedom system was used for the vehicle and a RayleighRitz method was used for the dynamic analysis. This study showed that in some cases, the DAF could be two to three times that recommended by current international design codes, suggesting that road unevenness could no longer be neglected. Inbanathan and Wieland [6] presented an analytical investigation on the dynamic response of a simply supported box girder bridge due to a moving vehicle. In particular, they considered the profile of the roadway using a response spectrum and 10 artificially generated time history loads for speeds of 19 and 38 km/h. The study of the response of a bridge due to a generated dynamic force was justified in view of the random nature of the problem. Some of the findings reported were the following: 1The effect of vehicle mass on the bridge response is more significant for high speeds; 2The maximum response is not affected by damping; 3 The stresses developed by a heavy vehicle moving over a rough surface at high speeds exceed those recommended by current bridge design codes. Hwang and Nowak [7] presented a procedure to calculate statistical parameters for dynamic loading of bridges, to be used in design codes. These parameters, based on surveys and tests, included vehicle mass, suspension system and tires, and roadway roughness, which was simulated by stochastic processes. This procedure was applied to steel and prestressed concrete girder bridges, for single and sidebyside vehicle configurations. Values of the DAF were computed using prismatic beam models for the bridges and stepbystep integrations. It was found that: 1the DAF decreases with an increase in vehicle weight; 2 the DAF for two sidebyside vehicles is lower than that for a single vehicle; and 3 the dynamic load is generally uncorrelated with the static live load. But the vehicle model of Hwang and Nowak didnt consider the dashpot of suspension system and tires. Au et al.
[8] presented a numerical study of the effects of surface road unevenness and longterm deflection on the dynamic impact factor of prestressed concrete girder and cablestayed bridges due to moving vehicles. The results showed that the effects of random road unevenness and the longterm deflection of concrete deck on bridge vary a lot at the sections closed to the bridge tower, with significant effects on the short cables. Lombaert and Conte [9] proposed the random vibration analysis of dynamic vehiclebridge interaction due to road unevenness by an original frequency domain method. The road unevenness was modeled by the random nonstationarywhere wi(xi,t) is the vertical displacement of girder element at ith axle of vehicle; ri is road unevenness at ith axle of vehicle; z1i is the vertical displacement of chassis at ith axle of vehicle; z2i is the vertical displacement of ith axle of vehicle; y1i is the relative displacement between the chassis and ith axle; y2i is the relative displacement between ith axle and girder element; Gi.sini is the engine excitation force at ith axle, it is assumed as a harmonic function ; k1i and d1i are the spring and dashpot of suspension at ith axle respectively; k2i and d2i are the spring and dashpot of tire at ith axle respectively; xi is the coordinate of the ith axle of the vehicle at time t (i=1, 2, 3).
process. Due to the complexity of the problem, the authors
w G Sin
G Sin
G Sin
presented only the results of simple supported beam model subjected to a moving concentrated load. XuanToan Nguyen
(y)
(z)
3 3
m13
m13 .g z
2 2
m12
m12 .g z
1 1
m11
m11 .g z
d13
k13 .. 13
d12
k12 .. 12
d11
k11 .. 11
et al. [10],[11] and [12] analyzed the dynamic threeaxle
k .y
m13 .z13
.
k .y
m12 .z12
.
k .y
m11 .z11
.
vehicle – bridge interaction considering the change of vehicle
velocity through braking force by finite element method. The
13 13+d13 .y13
12 12+d12 .y12
11 11+d11 .y11
numerical results showed that the influenc of braking force
m23
m23 .g m22
m22 .g m21
m2i .g
has effects significantly on dynamic impact factor of bridge.
d23
F = k .y
k23 ..
m .z
23 23
.
z23
d22
F = k .y
k22 ..
m .z
22 22
.
z22
d21
F = k .y
k21 ..
m .z
21 21
.
z21
However, most of the previous research studied on dynamic
interaction between the vehicle and simply supported bridge, very few studies have focused on the multispan slab beam
3 23 23+d23 .y23
r3
w3
O
2 22 22+d22 .y22
r2
w2
1 21 21+d21 .y21
r1
w1 x
bridge with link deck considering the random road unevenness effects. Additionally, the fullscale field test is needed in order to obtain a clearer understanding of the relationship between dynamic interaction for bridge types and vehicle models.
This study develops the FEM to analyze the random dynamic interaction between threeaxle dumper truck vehicle and twospan prestressed slab beam bridge with link deck due to road unevenness. The road unevenness is simulated by a zeromean stationary Gaussian random process. The bridge is modelled by finite element method. The dumpertruck has three axles. Each axle is idealised by two mass, in which a
x3
x
x2
1
L
Fig 2. Schematic of vehiclebridge interaction
Base on the model of dynamic vehiclebridge interaction in Fig.2 and using dAlemberts principle, the dynamic equilibrium of each mass m1i, m2i on the vertical axis can be written as follows:
m1i .z1i d1i .z1i d1i .z2i k1i .z1i k1i .z2i Gi .sini m1i .g (1)
m .z d .z d d .z k .z k k .z
mass is supported by a spring and dashpot. The governing
2i 2i
1i 1i
1i 2i 2i
1i 1i
1i 2i 2i
(2)
equation of random dynamic vehiclebridge interaction is derived by means of dynamic balance principle. Galerkin method and Green theory are employed to discrete the
where ri
k2i wi ri d2i .wi ri m2i .g
is the first derivation of road unevenness at ith axle
governing equation in space domain. The solutions of equation are solved by RungeKuttaMersion method. Monte
of vehicle. Adding on the logic control function, Eq.(1) and
(2) can be rewritten as follows:
Carlo simulation is applied to generate the random road
t .m .z

d .z

d .z


k .z

k .z
unevenness input. The numerical results are in good i
1i 1i
1i 1i
1i 2i
1i 1i
1i 2i
(3)
agreement with fullscale field test results of twospan prestressed slab beam at NguyenTriPhuong bridge in Danang city, Vietnam. Also, the effects of the road surface
i t .Gi .sin i m1i .g
i t m2i .z2i d1i .z1i d1i d2i z2i k1i .z1i k1i k2i z2i
condition on dynamic impact factor of the prestressed slab
t k
w r d
.w r m
(4)
.g
beam bridge are discussed.
i 2i i i
2i i i
2i


THE MODEL OF VEHICLEBRIDGE
1 if t
t t T ; T L
i
INTERACTION
Consider a twospan slab beam with link deck subjected to a threeaxle dumper truck vehicle as in Fig. 1. Assume that the
t
0
i i i i v if t ti and t ti Ti
(5)
body weight of vehicle and goods on the vehicle distribute to
three axle m11, m12 and m13, respectively. The mass of three axles are m21, m22 and m23 respectively. The dynamic
From Fig.2, the contact force between the ith axle and girder element is described by:
interaction model between a threeaxle vehicle and a girder
Fi k2i . y2i d2i . y2i
(6)
element is described as in Fig. 2.
v
The combined Eq.(1),(2),(5) and (6), the contact force between the ith axle and girder element can be rewritten as follows:
pi x, z,t i t
L G .sin

m m
.g m .z

m .z
(7)
. x x
l l l
i i
1i 2i
1i 1i
2i 2i i
1 2 1
where (xxi) is the Dirac delta function.
Fig 1. Schematic of vehicle moving on bridge
According to Ray [13], the governing equation for the vibration of damped girder due to uniform loading p(x, z, t) can be written as follow:
m21
m22 0
…
4
5
2
M z 2 z 2 m
(15)
EJ. x4 x5 .t m . t2
n
p x, z,t i t
. t
p x, z,t
(8)
(9)
2i
0
…
m2n (nn)
i1
G .sin m m
.g m .z

m .z
. x x
M wz1 P.M z1z1 ; M wz2 P.M z2z2
(16)
i i
1i 2i
1i 1i
2i 2i i
where EJ is the bending stiffness of girder element; and are the coefficient of internal friction and external friction of girder element; m is the mass of girder per unit length; n is the number of axle (n=3).
The Galerkin method and Green theory are applied to Eq. (3),
P11
P21
P31
P41
P12
P22
P32
P42
…
…
…
…
P1i P2i P3i P4i
…
…
…
…
P1n P2n P3n
P4n (nn)
(17)
P
(4) and (8) transform into matrix form, and the differential
P (L 2x ).(L x )2
equations of girder element can be written in a matrix form as follow:
1i
P
i i
(t) L.x .(L x )2
P
2i i i i
(18)
[M ].{q} [C ].{q} [K ].{q} { f }(10)
i P
L3 x2.(3L 2x )
e e e e
3i
P4i
i
L.x2.(x
i
L)
where {q}, {q}, {q}, {fe} are the complex acceleration vector, complex velocity vector, complex displacement vector,
i i
d
complex forces vector, respectively. 11
w
w
w
F
d12 0
w
…
C
(19)
q z1 ;q z1 ;q z1 ;fe Fz1
(11)
z1z1 d
z
z
z
F
1i
2
wy1
2
2
z 2
0 …
d1n (nn)
{w} 1
(12)
d21
w
y 2
d22 0
2
in which w ,
are the vertical displacement and rotation
Cz2
…
d
(20)
y1 1
2i
angle of the left end of girder element, respectively; wy2,2 are the vertical displacement and rotation angle of the right end of element, respectively;
0 …
d2n (nn)
[Me], [Ce] and [Ke] are the mass matrix, damping matrix andstiffness matrix, respectively;
Cz1z2 = Cz2z1 = Cz1z1; Cz2z2 = Cz1z1 + Cz2; Cz2w = (Ni.Cz2)T(21)
M ww M wz1 M wz2
Cww 0 0
N11
N12
…
N1i
…
N1n
N21
N22
…
N2i
…
N2n
M e 0
M z1z1 0
Ce 0
Cz1z1
Cz1z 2
Ni
(22)
0
Kww
0 M z 2 z 2
0 0
C
z 2w
Cz 2 z1
Cz 2 z 2
N31
N41
N32
N42
…
…
N3i
N4i
…
…
<>N3n
N4n (nn)
Ke 0
K z1z1
K z1z 2
(13)
N 1 .L3 3Lx2 2×3
K z 2w K z 2 z1 K z 2 z 2
1i L3
i i
m11
N 1 .L2 x
2Lx2 x3
2i L2 i
i i
m12 0
…
N3i
1 .3Lx2 2×3
(23)
M
(14)
L3 i i
z1z1
m1i
0
…
N4i
1 .x3 Lx2
m1n (nn)
L2 i i
[Mww], [Cww] and [Kww] are mass, damping and stiffness matrices of the girder elements, respectively. They can be found in Zienkiewicz [14]in which frequency interval =(ml)/M and [l,m] is the range of frequency where Sr() has significant values. The amplitude Ak in Eq. (31) is represented by:
k11
Ak 2Sr k 2Sr k (33)
k12 0
in which S ( )=v.S () is the PSD roughness in terms of
r k r
K z1z1
…
0
k1i
…
k
(24)
wave number,, which represents spatial frequency; v is vehicle velocity. From ISO 8608:1995 [1], the PSD roughness in terms of wave number are described by:
0
1n (nn)
Sr Sr 0
(34)
K z 2
k21
k22 0
…
k
(25)
where the fixdatum wave number 0 is set as 1/2 cycle/m. The measurement shows that various values exist for exponential and the socalled roughness coefficient Sr(0), ranging from 1.5 to 3.0 for and from 2Ã—106 m3/cycle to
8192Ã—106 m3/cycle for S ( ). These different values reflect
2i r 0
0 …
k2n (nn)
the components of wavelength in elevation fluctuation and surface condition. Eq. (34) is used as PSD road unevenness later on to generate random road profile. The values of
Kz1z2 = Kz2z1 = Kz1z1; Kz2z2 = Kz1z1 + Kz2 (26)
.
roughness coefficient Sr(0) are classified by ISO 8608:1995 in Table 1.
Kz2w – (Ni .Kz2 )T – (Ni .Cz2 )T
N
Fw Gi .sin i m1i m2i g.Pi
(27)
(28)
Table 1: Road roughness values classified by ISO 8608:1995
roughness coefficient
i1
lower limit
geometric mean
upper limit
A (very good)
–
16
32
(29)
B (good)
32
64
128
C (average)
128
256
512
D (poor)
512
1024
2048
E (very poor)
2048
4096
8192
(30)
IV. ANALYSIS RANDOM VIBRATI NGUYENTRIPHUONG BRIDGE
ON OF
G1.sin 1 m11.g
Road class
Sr(o) [106 m2/(cycle/m)]
Fz1 Gi .sin i m1i .g ;
GN .sin N m1n .g
m21.g k21.r1 d21.r1
m .g k .r d .r
22 22 2 22 2
Fz 2
m
…
.g k .r d

2i
2i i
2i i

Properties of structural bridge and vehicle
m2n .g k2n .rn d2n .rn n1
Eq.10 can be solved by means of the direct stepbystep integration method based on RungeKuttaMersion method to obtain responses of girder elements.



RANDOM ROAD UNEVENNESS
Assume that the PSD (Power spectral density) roughness represented by the angular frequency of a pavement section is known as Sr(). According to Shinozuka [15], Honda [3] and Sun [16] the temporal random excitation formed by a road unevenness can be expressed by means of :
M
NguyenTriPhuong bridge located in Danang city, Vietnam. The approach bridge of NguyenTriPhuong Bridge, which is a twospan slab beam prestressed concrete. The deck of slab beam is connected in the flexible joint between two span, shown in Fig.1. The cross section of the prestressed concrete slab beam and position of vehicle is shown in Fig.3. The threeaxle vehicle used in the numerical simulation and the field test is FOTON dumper truck as shown in Fig. 4.
r t Ak cos k t k
k 1
(31)
where M is a positive integer and k is an independent random variable with uniform distribution at range [0,2). Also, the discrete frequency k is given by:
k 1
(32)
k 1 2
Fig 3. Cross section of slab beam
Elevation, r(m)
0.05
0
0.05
0.05
Road unevenness profile 1
0 10 20 30 40 50 60
Distance (m)
Road unevenness profile 2
Fig 4. The FOTON dumper truck
The properties of slab beam are collected from design documents of the bridge management unit; the properties of threeaxle dumper truck FOTON are given by the manufactory company and checked on site. The parameters of slab beam and dumper truck are listed in Table 2
Table 2. Properties of slab beam and dumper truck FOTON
Item Notation Unit Value
0
Elevation, r(m)
0.05
Elevation, r(m)
0.05
0
0.05
0 10 20 30 40 50 60
Distance (m)
Road unevenness profile 3
0 10 20 30 40 50 60
Distance (m)
Fig 5. Typical random road unevenness profiles
Slab beam concrete 

Lenght 
L 
m 
22.35 

Youngs modulus 
E 
Gpa 
36 

Density 
kg/m3 
2500 

Coefficient of internal friction 
– 
0.027 

Coefficient of external friction 
– 
0.01 

Cross sectional Ã¡rea 
A m2 0.723 

Second moment of area 
I m4 0.097 

Link slab (deck) 

Height 
h 
m 
0.15 

Cross sectional area 
A m2 0.15 

Second moment of area 
I m4 0.28×103 

Dumper truck vehicle 

Mass m11 
m11 
kg 
5200 

Mass m21 
m21 
kg 
260 

Mass m12,m13 
m12,m13 
kg 
8900 

Mass m22,m23 
m22,m23 
kg 
870 

Suspensions spring k11;k12;k13 
k1i 
N/m 
2.6×106 

Tires spring k21;k22;k23 
k2i 
N/m 
3.8×106 

Suspensions dashpot d11;d12;d13 
d1i 
Ns/m 
4000 

Tires dashpot d21;d22;d23 
d2i 
Ns/m 
8000 
Using the finite element method, the bridge structure was
discrete as Fig.6. Th deck of slab beam prestressed concrete are connected in the flexible joint with 1.4m of length. Setting vehicle velocity moving on the bridge is 10 m/s. For each road unevenness input, Eq. 10 is solved by the RungeKutta Mersion method to obtain the static and dynamic displacements of slab beam, shown in Fig.7.
1 2 3 4 5 6 7 8 9 10
22.35m
1.4m 22.35m
Fig 6. Schematic of discrete bridge structure
a)
b)

Numerical results
Base on the survey on site, assume that the road surface condition of NguyenTriPhuong bridge is Grade Croad (ISO 8608:1995): roughness coefficient Sr(Â¬0)= 256Ã—106 m3/cycle; exponential =2; M=1000; the range of spatial frequency (wave number) k = [0.011Ã·2.83] cycle/m. Monte Carlo simulation method is applied to generate road unevenness profiles. Some of random road unevenness profiles are described as follows:
Fig 7. Static and dynamic displacement of 1st span: a) Â¼ of 1st span; b) Â½ of 1st span
From the time history of 1st span displacements in Fig.7. it can be seen that displacements of 1st span decreased quickly when the dumper truck went over the 1st span. The cause for that
issue is that bending stiffness of concrete link slab in the flexible joint is very smaller than bending stiffness of slab beam prestressed concrete.

Field measurement results
In order to validate the numerical results, field measurement of dynamic response of slab beam prestressed concrete was conducted at NguyenTriPhuong bridge in Danang city, Vietnam. This section presents a measurement system and results obtained so far.
mentioned above are quite reliable. This numerical model are used continuously to investigate the influence of the road surface condition on dynamic impact factor of slab beam prestressed concrete in the next section.
FEM results
0.2
Experiment results
a)
Displacement (mm)
0
0.2
Since large vibration of slab beam prestressed concrete had been observed, displacement sensors (LVDT) were placed on Â½ of 1st span as Fig.8.
Dumper Truck
0.4
0.6
0.8
0 0.5 1 1.5 2
Vehicle position (mm)
4
x 10

O v
Displacement sensors (LVDT)
0.2
FEM results Experiment results

0
Dynamic strainmeter
0.2
Displacement (mm)
0.4
Computer
b)
0.6
0.8
0 0.5 1 1.5 2
Vehicle position (mm)
FEM results Experiment results
0.2
Displacement (mm)

0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2
Vehicle position (mm)
4
x 10
4
x 10
Fig 8. Measurement system in NguyenTriPhuong bridge a)Diagram of installing system; b) Measurement system on site
Slab beam vibration was measured after the slab beam was excited by a dumper truck with various velocity. Properties of slab beam prestressed concrete and dumper truck FOTON are listed in Table 2. Since traffic velocity have been limited by the bridge management company, the testing velocity of dumper truck was suggested 10, 20,30 and 40 km/h. For each velocity level of testing vehicle, dynamic displacement of slab beam prestressed concrete was recorded and compared with the numerical results, shown in Fig.9
FEM results Experiment results
From the time history of 1st midspan displacement in Fig.9, it can be seen that the numerical results (FEM results)
0.2
d)
Displacement (mm)
0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2
Vehicle position (mm)
4
x 10
show quite good agreement with the experiment results at the field. The difference of maximum dynamic displacement between them are 3.83%; 4.77%, 5.24% and 6.12%, respectively with the moving vehicle velocity 10, 20, 30 and
40 km/h. Therefore, the algorithm and numerical model
Fig 9. Time history of displacement at 1st midspan: a) v=10 km/h; b) v=20 km/h; c) v=30 km/h; d) v=40 km/h


Numerical investigation
Base on the validated numerical model with experiment results above, it carried out to investigate dynamic vehicle bridge interaction for this numerical model with various road surface condition. Assume that the roughness coefficient changed in the range, Sr(0) = [0, 32, 128, 512, 2048, 8192]Ã—106 m3/cycle, corresponding to the road surface condition: ideal smooth, Grade A, B, C, D and E (ISO 8608:1995). The velocity of dumper truck was 10 m/s. The other parameters of structural bridge and dumper truck vehicle was given as Table.2. For each the road surface condition, MonteCarlo simulation method is applied to generate 100 road unevenness profiles. With each road profile input, the

25 20
Probability density function
15
10
5
(1+IM) Histogram (1+IM) Lognormal fit
governing equation of vehiclebridge interaction is solved to obtain static and dynamic displacements output of slab beam concrete. From static and dynamic displacement, it can determine dynamic impact factor of slab beam prestressed concrete as shown in Eq.35:
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
1+IM

30
(1+IM) Histogram
Probability density function
25 (1+IM) Lognormal fit
x
D
s
(1 IM ) j
R j x
R j x
(35) 20
D
s
where Rj (x) is the dynamic displacement of slab beam prestressed concrete at position x due to dumper truck moving on jth road unevenness profile; Rj (x) is the static displacement of slab beam prestressed concrete at position x due to dumper truck moving on jth road unevenness profile.
After analyzing with a series of road profiles input, it can obtain a series of dynamic impact factor output which are also random process as shown in Fig.10. The statistical characteristics of the dynamic impact factor (IM) at 1st
midspan are described in Table 3.
15
10
5
0
0.8 1 1.2 1.4 1.6 1.8
1+IM

35
Probability density function

30 25
20
30
Probability density function
(1+IM) Histogram 25
(1+IM) Lognormal fit
20
15
(1+IM) Histogram (1+IM) Lognormal fit
15 10
10 5
5
0
1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17
1+IM

40

(1+IM) Histogram
0
0.5 1 1.5 2 2.5
1+IM
Fig 10. Dynamic impact factor at 1st midspan due to dumper truck: a) grade Aroad; b) grade Broad; c) grade Croad; d) grade Droad; e) grade Eroad
Table 3. Statistical characteristics of dynamic impact factor at 1st midspan
Dynamic Impact Factor (1+IM)
Probability density function
m3/cycle 
deviation 

0 
1.12 
– 
– 
– 
32 
1.126 
1.159 
1.093 
0.019 
128 
1.134 
1.199 
1.068 
0.038 
512 
1.175 
1.34 
1.011 
0.096 
2048 
1.249 
1.579 
0.918 
0.194 
8192 
1.488 
2.18 
0.96 
0.405 
35 (1+IM) Lognormal fit Sr(o)
30
25
20
15
10
5
0
1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22
1+IM
x106
Mean Max Min
Standard
Base on statistical characteristics of IM in Table.3, the relationship between the mean value of IM and the road unevenness condition can be established in Fig 11. In the Fig.11, the correlation equation between the mean value of IM and road surface condition are also found out, in which x is the roughness coefficient.
y = 4.4e005*x + 1.1
Maximum (1+IM) recommended by [17],[18]
1.6
1.55
1.5
Mean (1+IM)
1.45
1.4
1.35
1.3
1.25
Mean (1+IM) linear
1.2
1.15
1.1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Roughness coefficient, Sr(o)
Fig 11. Mean value of IM versus the roughness coefficient
From the investigation results in Fig.11, it can be seen that when the road surface condition changes to be grade Aroad, grade Broad, grade Croad the mean value of IM increases 0.54%, 1.25% and 4.91%, respectively. Specially, the mean value of IM reaches 11.52% and 32.85%, respectively, while the road surface condition changes to be grade Droad and grade Eroad. This increase in dynamic impact factors are quite large and exceed those recommended by current bridge design codes as AASHTO [17] and Vietnamese Specification for Bridge Design [18]. Therefore, it is necessary to consider the influence of road surface condition on analyzing dynamic response of structural bridge, especially bridges have passed long time in operation and the pavement have been damaged as well as seriously downgraded.
V. CONCLUSIONS
In this paper, the analysis of random dynamic interaction between threeaxle dumper truck vehicle and twospan slab beam prestressed concrete with link slab due to road unevenness is investigated by means of finite element method and MonteCarlo simulation method. The road unevenness are described by a zeromean stationary Gaussian random process. The bridge is modeled by finite element method. The dumpertruck has three axles. Each axle is idealised by two mass, in which a mass is supported by a spring and dashpot. The governing equation of random dynamic vehiclebridge interaction is derived by means of dynamic balance principle. Galerkin method and Green theory are employed to discrete the governing equation in space domain. The solutions of governing equation are solved by RungeKuttaMersion method in time domain. MonteCarlo simulation is applied to generate the random road unevenness input. The numerical results are in good agreement with fullscale testing results at NguyenTriPhuong bridge in Danang city, Vietnam. In addition, this research evaluates the effects of the road surface condition on dynamic impact factor of slab beam prestressed concrete. The numerical results showed that the road surface
condition has significantly effects on dynamic impact factor of slab beam prestressed concrete. Specially, the mean value of IM reaches 32.85%, respectively, while the road surface condition changes to be grade Eroad. This value of dynamic impact factors are quite large and exceed those recommended by current bridge design codes. Therefore, it is necessary to consider the influence of road surface condition on analyzing dynamic response of structural bridge subjected to moving vehicles, especially bridges have passed long time in operation and the pavement have been damaged as well as seriously downgraded.
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