A finite element model of vehicle - cable stayed bridge interaction considering braking and acceleration

*Toan Xuan Nguyen1), Duc Van Tran2)

1)Dept. of bridge and road, Danang University of Science and Technology, The

University of Danang city, Danang city, Vietnam 2) International school, Duytan University, Danang city, Vietnam

1) nguyenxuantoan2007@gmail.com, 2)duc98x3b@gmail.com

ABSTRACT

A finite element method (FEM) for the calculation of dynamic response and loading of cable-stayed bridge due to vehicle braking loads is described. The analysis takes account of vehicle acceleration or braking and eccentric placement of the vehicle on the bridge. The analysis was presented for a three-axle dumper truck vehicle and the cable-stayed bridge which was modeled as Bernoulli–Euler beam with uniform cross-section on linear spring supports and cables. The vehicle was modeled as a group of moving loads at a fixed spacing. An assumption of the averaged vehicle acceleration at current time is known. Subsequently, absolute vehicle displacements can be changed to relative vehicle displacements. Thus, the inertia force at the center of each vehicle mass can be generated. An experimental test was performed on a reality three spans cable-stayed bridge with double I steel-section beams subjected to vehicular braking load in order to test the performance of this investigation. Since braking in first span would create response in another spans, a more accurate definition of impact factor with vehicle braking based on a ratio of the maximum dynamic and static responses at the same station of span in which braking occurs. The main contribution of this paper is to validate the FEM results with the experimental results.

1. INTRODUCTION

The initial researchers had studied the response of bridges subjected to vehicles moving since the 50s of the 19th century. Recently, the study analyzed the complex problem of oscillations with the model interaction between vehicle and bridge closer to reality. Most studies focusing on the dynamic behavior of bridge subjected to vehicles moving with constant speed. Only a few research directions towards the dynamic behavior of bridge under the effect of vehicle with variable velocity (due to many reasons such as brakes, incident on the bridge ...). As Fryba (1974) introduced a fundamental study of beam due to mass roll on the train rails considering the braking force effect, and also studied the quasi-static distribution of braking. Then the study of Kishan and Trail-Nash (1977) concerned the dynamic response of highway bridges

1) Professor 2) Phd candidate

under the effect of vehicle braking force, and the resulting impact factors may larger than those adopted in the current design code. Gupta and Trail-Nash (1980) investigated the dynamic behavior of bridge model with single span uniform beam considering the road surface irregularities and vehicle braking force. Mulcahy carried out method for analysis of dynamic interaction between single span bridges and a three-axle tractor trailer vehicle considering vehicle acceleration or braking and roughness of the road surface. Krylov (1996) investigated the vibration of the ground under effect of vehicle with a constant acceleration from the rest to a constant speed. Toth and Ruge (2001) analyzed the longitudinal behavior of the railway bridge considering the braking force of the train. Yang and Wu (2001) applied and developed numerical method to investigate the dynamic behavior of a bridge when vehicle was decelerating. Hu and Han (2003) presented the non-linear behavior with four-axle vehicle model considering the braking force. Law and Zhu (2005) studied the dynamic behavior of continuous three spans under moving vehicle considering braking load and the roughness of the road surface. Ju and Lin (2007) used FEM to calculate the vertical vibration of beams caused moving vehicle due to the braking force. Toan and Duc (2007, 2011, 2012, 2014) studied the dynamic vibration of a beam subjected to moving vehicles considering the braking forces and the variable acceleration.

This study develops the FEM to analyze the interaction between three-axle dumper truck vehicle and three-span cable-stayed bridge considering the braking force and acceleration. In addition, this research evaluates the effects of the beam on vertical direction when the vehicle brakes. The moving vehicle modeled by four masses, including the mass m, m1, m2, m3, rotational inertia J of the chassis, the damping and stiffness of the suspension, spring and tires. Bridge structure is analysis as finite elements bending beams hanging on the cables subjected to direct effect of the vehicle loads. Numerical analysis results have been compared with the experimental testing results performed on PhoNam cable-stayed bridge in Danang city.

2. COMPUTATIONAL MODELS AND ASSUMPTIONS

The cable-stayed bridge used in this study is the PhoNam Bridge located in Danang, Vietnam which is three span continuous cable-stayed bridge with main span is 80m; side span is 35.7m; and bridge length is 151.4m. The three-axle vehicle moving on bridge is shown on Fig. 1

80m35.7m 35.7m

O

xelf L

21.6

8m

Fig. 1 The three-axle vehicle move on the the PhoNam Bridge

The structures of cable-stayed bridge were modeled by following elements: Stayed cables were modeled as cable elements with considering tensional force

and deflection of the cables. The non-linear dynamic vibration of cable element in the cable-stayed bridges subjected to moving loads can be found in investigation of Raid K. (1999), Toan (2007).

The tower structures of the cable-stayed bridge were modeled as frame

elements which can be found in Zienkiewicz (2000). The girder beams of the cable-stayed bridge were modeled as beam elements

which interact directly to the three-axle vehicle. The dynamic interaction between a beam element and three-axle vehicle modeled by the braking forces and acceleration are shown in Fig. 2.

1G s

-m.s..

Pu

m.g

23-J.

m3

dt3 kt3u3

-m .s..

ds3 ks3

Fs3= ks3.ys3 ds3.ys3+.

3

Tt3

w3

h +

u3

h+um2

dt2 kt2

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s2 s2

.

2

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w2

h +

u2

u2

m1

dt1 k t1

-m .s..

1

Tt1

w1

h +

u

u1

x

1

d k s1 s1d k13

Fs2= ks2.ys2 ds2.ys2+.

Fs1= ks1.ys1 ds1.ys1+

Ft3= kt3 .yt3 dt3 .yt3+. .

Ft2= kt2 .yt2 dt2 .yt2+.

Ft1= kt1 .yt1 dt1 .yt1+

w

Lx1

xo

x2

x3O

-m.u

m .g1 m .g2 m .g1

m .u3..

3 m .u2..

2 m .u1..

1

..

u2

- - -

..

23 1

u u

Fig. 2 The dynamic interaction model between a three-axle vehicle and beam element

Where:

eibielfbii

biiibii

biielfii

itttxttv

ttattv

tttxttv

xif ;).(

2

).().(

if ;).(

; Lxi 0 (1)

L – the length of the beam element

xo – the coordinate of vehicle mass m at the time t xi – the coordinate of the ith axle at the time t (i= 1, 2, 3) xelf – the distance from the left end of bridge to the left end of beam element. vi – the velocity of the ith axle before the vehicle brakes ai – the acceleration of the ith axle at brake is used (ai<0) t – the time at current location ti – the time when the ith axle begin enter the bridge tbi – the time when vehicle brakes on the ith axle tei – the time when the ith axle stops

).sin(. tGP - the stimulating force cause by the eccentric mass of engine m – the mass vehicle and goods, not including the mass of the axles m1, m2, m3 – the mass of the 1st, 2nd, 3rd axle, respectively ks1, ks2, ks3 , ds1, ds2, ds3 – the stiffness and the damping of the three axles kt1, kt2, kt3, dt1, dt2, dt3 – the stiffness and the damping of the three tires s – the acceleration of vehicle – the rotation angle of chassis u – the vertical displacement of centre of the chassis ū1, ū2, ū3 – the vertical displacement of chassis at the three axles u1, u2, u3 – the vertical displacement of the three axles ys1, ys2, ys3 – the relative displacement between the chassis and the three axles yt1, yt2, yt3 – the relative displacement between the beam element and the three axles h1, h2, h3 – the height from the centre of beam element to centre of mass m1, m2, m3 Tt1, Tt2, Tt3 – the friction forces between tire and bridge surface Inertial forces, damping forces, elastic forces, stimulating forces and braking forces of system are shown in Fig. 2.

The following assumptions are adopted:

The mass of the vehicle, excluding the mass of the axles is transferred to the mass center of the system. It is equivalent to the mass m and rotational inertia J. The mass of the 1st axle is m1, which is considered as a mass point at the center of the corresponding axle. Similarly, the mass of the 2nd and 3rd axles are m2 and m3. The chassis is assumed to be absolutely rigid. The materials of beam are linear elastic stage. The bridge surface has the homogeneous friction coefficient over the entire bridge surface. Braking forces of the axles of vehicle are assumed to occur simultaneously. The forces direction between bridge surface and tires are assumed to be in the opposite direction of the vehicle moving as shown in Fig. 2. The friction forces between bridge surface and tires are Tt1, Tt2, Tt3, will decelerate gradually the vehicle and produce inertia forces sm .1 , sm .2 , sm .3 , sm . .

When the vehicle is sudden braked, the forces Tt1, Tt2, Tt3 are assumed to be directly proportional the weight of vehicle.

.).( 321321 gmmmmTTT ttt (2)

Where: - the coefficient of friction between the tire and the bridge surface g - the acceleration of gravity

3. BENDING VIBRATION OF BEAM ELEMENTS DUE TO BRAKING APPLIED ON THREE-AXLE VEHICLE Base on the calculation model and assumptions in Section 2, the system of masses m, m1, m2, m3, inertia forces, damping forces, elastic forces, stimulating force and braking forces are taken into account. In this case, braking forces are converted to friction forces Tt1, Tt2, Tt3 as shown in Fig. 2. Using d’Alembert’s Principle and considering the equilibrium of each mass m, m1, m2, m3 on the vertical axis and that of whole system on the longitudinal axis, we can be obtained:

0.)(

0.

0

11

1

smmT

gmumFF

mgFumP

n

ii

n

iti

iiitisi

n

isi

(3)

Where i is the number of axles (i= 1, 2, 3). Considering the equilibrium equation of the system with the O point, we can be obtained:

0)..().(.)...(.).(....111

itiiti

n

i

n

iiii

n

iiiiio xFwTuhsmxgmumJuhsmxgmumP (4)

Where:

iiti

ioiiisi

oioii

tititititi

sisisisisi

wuy

uxxuuuy

xxutgxxuu

ydykF

ydykF

)(

)()(

(5)

Substituting Eq. (5) in to Eq. (3) and (4), the combined Eq. (2), (3) and (4), we can be obtained a set of equations:

.

0...).(

.).().(.).(.

0....)(...)(.

0)...(.]..)([]..)([

.)().(.)(.)(.

111111

1111

1

2

111

2

gs

gmwkwdukk

ukxxkuddudxxdum

gmPukukxxkududxxdum

shmhmwTusmxxkusmxxk

xxkuxxduxxdxxdJ

iitiitiitisi

sioisiitisisioisiii

n

iisi

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isi

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ioisi

n

iii

n

iitii

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iioisi

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ioisi

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ioisii

n

ioisi

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ioisi

n

ioisi

(6)

According to Ray, W.C (2003), the differential equation of motion for the flexure vibration of damped beam due to uniform loading ),,( tzxp can be written as follow:

n

iidd tzxp

t

w

t

wF

tx

w

x

wEJ

12

2

4

5

4

4

),,(...

.. (7)

The differential equation of longitudinal motion for a beam element due to uniform loading ),,( tzxq can be written as follow:

n

ii

xxd

xd tzxptzxq

t

u

t

uF

x

uEF

12

2

4

2

),,(.),,(... (8)

Where:

dF - the cross-sectional area

dJ - the second moment of area

E - the Young's modulus

dEJ - the flexural rigidity of the beam element

dEF - the axial rigidity of the beam element

dF - the mass of beam per unit length

and - the coefficient of internal friction and external friction

xu - the axial displacement of the beam element

)(.]...).(

..).(.[).()(.).(),,(

iiisisioisi

isisioisiiiiitiii

xxgmukukxxk

ududxxdumxxxFxtzxp

(9)

Where: )( ixx is the Dirac delta function

Lxxif

Lxifx

ii

ii and00

01)( is the logic control signal function

The Galerkin method and Green theory are applied to Eq. (6), (7), (8) and (9) transform into matrix form, and the differential equations of beam element can be written as follow:

eeee fqKqCqM ... (10)

efqqq ,,, – the complex acceleration vector, complex velocity vector, complex

displacement vector, complex forces vector, respectively

z

w

e

eee

f

ff

z

wq

z

wq

z

wq ;;;

;

ny

x

y

x

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u

uu

z

wu

w

u

w

1

2

2

2

1

1

1

;

(11)

Where:

111 ,, yx wu – the axial displacement, flexural deflection and rotation angle of the left

end of element, respectively

222 ,, yx wu – the axial displacement, flexural deflection and rotation angle of the

right end of element, respectively Me, Ce and Ke – the mass matrix, damping matrix and stiffness matrix, respectively

;

zzzw

wzww

e MM

MMM ;

zzzw

wzww

e CC

CCC

zzzw

wzww

e KK

KKK (12)

wwww CM , and wwK are mass, damping and stiffness matrices of the beam elements

Where:

nnii

nnii

nnii

nnii

nnii

nnii

wz

PmPmPm

PmPmPm

PmPmPm

PmPmPm

PmPmPm

PmPmPm

M

66611

551511

44411

33311

22211

11111

......00

......00

......00

......00

......00

......00

; (13)

)......(

)......(

)......()......(

)......(

)......(

66611

55511

44411

33311

22211

11111

nnii

nnii

nnii

nnii

nnii

nnii

w

PmPmPmg

PmPmPmg

PmPmPmgPmPmPmg

PmPmPmg

PmPmPmg

f ; (14)

nsnisis

n

iisi

n

iioisi

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PdPdPdPdPxxd

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C

661611

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11

1

......)(

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; (15)

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wz

PkPkPkPkPxxk

PkPkPkPkPxxk

PkPkPkPkPxxk

PkPkPkPkPxxk

PkPkPkPkPxxk

PkPkPkPkPxxk

K

666111

61

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555111

51

5

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41

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31

3

222111

21

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111111

11

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......)(

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; (16)

n

i

zz

m

m

m

m

J

M

...0...000

.....................

0......000

.....................

0...0...00

0...0...00

0...0...00

1; (17)

gm

gm

gmmgP

shmmh

f

n

i

n

iii

z

...

...

).(

1

1

; (18)

tnsnsnonsn

tisisioisi

tssos

snsis

n

isi

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ioisi

onsnoisios

n

ioisi

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zz

dddxxd

dddxxd

dddxxd

ddddxxd

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...0...0)(

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0...0...)(

......)(

)(...)(...)()()(

11111

111

1111

2

;(19)

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tisisioisi

tssos

snsis

n

isi

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ioisi

onsnoisios

n

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zz

kkkxxk

kkkxxk

kkkxxk

kkkkxxk

xxkxxkxxkxxkxxk

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...0...0)(

.....................

0......0)(

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0...0...)(

......)(

)(...)(...)()()(

11111

111

1111

2

;(20)

0zwM ;

ntnntnntnntn

itiitititi

tttt

zw

PdPdPdPd

PdPdPdPd

PdPdPdPd

C

6532

651312

611511311211

00

..................

00

..................

00

000000

000000

; (21)

ntnntnntnntnntnntnntnntn

ttitiitiitiitiitiiti

tttttttt

n

iiti

n

iiti

n

iiti

n

iiti

zw

PkPdPkPdPkPdPkPd

PkPdPkPdPkPdPkPd

PkPdPkPdPkPdPkPd

PTPTPTPT

K

66553311

611611553322

611611511511311311211211

16

15

13

12

....0....0

..................

....0....0

..................

....0....0

000000

..0..0

; (22)

In which:

)(.)(

).2..()(

)23(.)(

..)(

)2.3.()(

).(.)(

2

26322

23

2

35

4

323

32

1

LxxL

xPxxLxL

L

xP

xLxL

xP

xL

xP

xxLLL

xP

xLL

xP

iii

iii

i

iii

i

ii

i

ii

i

ii

i

ii

i

ix is computed by Eq. (1).

4. APPLY THE FEM TO ANALYSE THE DYNAMIC VIBRATION OF THE PHONAM BRIDGE IN DANANG CITY

The cable-stayed bridge used in this study is the PhoNam Bridge located in Danang, Vietnam which is three span continuous cable-stayed bridge with main span is 80m; side span is 35.7m; and bridge length is151.4m. The three-axle vehicle move on bridge is shown on Fig. 1

Apply the FEM and the algorithm of the FEM can be found in Zienkiewicz (2000), we have the dynamic vibration differential equation for the system as in Eq. (23).

FQKQCQM eee ... (23)

In which:

CKM ,, – the mass matrix, damping matrix and stiffness matrix of system.

FQQQ ,,, – the acceleration vector, the velocity vector, the deflection vector and the force vector of the system.

After imposing boundary and initial conditions on Eq. (23), we can solve this

equation by the Runge-Kutta-Merson method. The numerical values of the vehicle and bridge parameters were used in the computer simulation and the field test as follows:

The parameters of the PhoNam Cable-stayed bridge:

1 2 3 4 5

10 9 8 7 6

20 19 18 17

16

11 12

13 14 15

T1

T2

T1

T2

Fig. 3 The cables plane of PhoNam bridge

- The Cable properties:

Cross sectional area of Cable No. 4, 5, 6, 7, 14, 15, 16, 17 : A=980 mm2 Cross sectional area of Cable No. 2, 3, 8, 9, 12, 13, 18,19 : A=1260 mm2 Cross sectional area of Cable No. 1, 10, 11, 20 : A=1680 mm2

- The Rigid Beam properties:

E= 2.1x108 T/m2, Jd= 0.001702m4, Fd = 0.02568m2, qy=Fd = 2.035T/m, =0.027; =0.01, g = 9,81m/s2, = 0.25.

- The Tower properties:

The part T1: E= 2.1x108 T/m2, Jd=0.00598036m4, Fd = 0.04706m2, qx=0.406T/m, =0.027; =0.01.

The part T2: E= 2.1x108 T/m2, Jd=0.00439952 m4, Fd = 0.0346m2, qx=0.406T/m, =0.027; =0.01.

The three-axle vehicle used in the computer simulation and the field test is KAMAZ-55111(Russia) dumper truck as shown in Fig. 4.

1.32m 2.48m

x

b1=2.09m

O

b3=2.07m

b2=0.39m

0.51

m

w

Fig. 4. The three-axle vehicle dimensional parameters

- The three-axle vehicle parameters: m= 8.56 T, m1= 0.06 T, m2=0.11 T, m3=0.11 T, P=0, b1=x1-x0 = 2.09 m, b2= x0-x2 =0.39 m, b3= x0-x3 =2.07 m, h= 0.95m, h1=h2=h3= 0.51m, k1s=120 T/m, k1t=220 T/m, k2s=k3s=160 T/m, k2t=k3t=320 T/m, d1s=0.734 Ts/m, d1t=0.367 Ts/m,d2s=d3s=0.4 Ts/m, d2t=d3t=0.8 Ts/m.

The weights of vehicle: Gross weight, kg...........8560 Front axle load, kg........3650 Rear bogie load, kg.......4910

The brakes of vehicle: Drive.............................................pneumaticSize: drum diameter, mm............400 Brake linings width, mm...............14

The FEM results are compared with those of corresponding experimental

investigation for cable-stayed bridge response with three-axle vehicle loading. The rigid beam deflections of PhoNam bridge caused by three-axle vehicle moving on the side span and used sudden braking at the velocities of 20km/h and 25km/h. The FEM results are shown in Figs. 5-12. The experimental investigation results are shown in Figs. 13-20.

Fig. 5 The Node 1st deflection for braking at the quarter side span (v=20km/h).

Fig. 6 The Node 2nd deflection for braking at the quarter side span position (v=20km/h).

Fig. 7 The Node 1st deflection for braking at the half side span position (v= 20km/h).

Fig. 8 The Node 2nd deflection for braking at the half side span position (v= 20km/h).

Fig. 9 The Node 1st deflection for braking at the quarter side span position (v=25km/h).

Fig. 10 The Node 2nt deflection for braking at the quarter side span position(v=25km/h).

Fig. 11 The Node 1st deflection for braking at the half side span position (v= 25km/h).

Fig. 12. The Node 2nd deflection for braking at the half side span position (v= 25km/h).

Fig. 13 The Node 1st deflection for braking at the quarter side span (v= 20km/h)

Fig. 14 The Node 2nd deflection for braking at the quarter side span position (v=20km/h)

Fig. 15 The Node 1st deflection for braking at the half side span position (v= 20km/h)

Fig. 16 The Node 2nd deflection for braking at the half side span position (v= 20km/h)

Fig. 17 The Node 1st deflection for braking at the quarter side span position (v= 25km/h)

Fig. 18 The Node 2nd deflection for braking at the quarter side span position (v=25km/h)

Fig. 19 The Node 1st deflection for braking at the half side span position (v= 25km/h)

Fig. 20 The Node 2nd deflection for braking at the half side span position (v= 25km/h)

The results of maximum dynamic deflections and the dynamic factors of the PhoNam bridge caused by a three-axle vehicle due to braking force at speeds of 20km/h and 25km/h on the side span are shown in Tab. 1.

Tab.1. The dynamic deflections of rigid beam on side span due to braking position

Deflection positions of rigid beam

Braking positions on side span

Velocity at

sudden braking (km/h)

The FEM results

The experimental

results

The difference between FEM analysis and experimental

results Ud

(mm) Kd

Ud

(mm) Kd

∆Ud (%)

∆Kd (%)

Node 1st a quarter 20 4.918 1.227 4.424 1.164 11.2 5.4 Node 2nd a quarter 20 5.293 1.154 4.529 1.105 16.9 4.5 Node 1st a half 20 4.711 1.175 4.048 1.079 16.4 8.9 Node 2nd a half 20 6.048 1.225 5.393 1.135 12.1 7.9 Node 1st a quarter 25 5.292 1.320 4.770 1.255 10.9 5.2 Node 2nd a quarter 25 5.831 1.206 5.395 1.148 8.1 5.1 Node 1st a half 25 5.092 1.260 4.508 1.156 13.0 9.0 Node 2nd a half 25 5.780 1.163 5.279 1.111 9.5 4.6

In which: s

dd U

UK

Kd – the dynamic factor of deflection, Ud – the maximum dynamic deflection of rigid beam, Us – the maximum static deflection of rigid beam,

∆Ud – the difference of Ud between FEM and experimental results, ∆Kd – the difference of Kd between FEM and experimental results. The FEM results are agree well when compared with the experimental results.

The difference of dynamic deflection between the FEM results and the experimental results vary from 8.1% to 16.9% while as for the dynamic factor of deflection vary from 4.5% to 9.0%. Base on the FEM and experimental results, the maximum dynamic factor of deflection are appreciable. We can see the errors between the FEM and experimental results are significant which is caused by the following reasons:

- When the vehicle braked, the braking forces did not happen simultaneously at all wheels.

- The chassis stiffness for the FEM is assumed absolutely rigid, while the actual stiffness of chassis is comparative.

- The input parameters of the three-axle vehicle and the cable-stayed bridge for the FEM analysis are not measured exactly as desired.

5. CONCLUSIONS

This paper introduces the results of research on dynamic interaction model between a three-axle vehicle and a cable-stayed bridge considering braking forces. The FEM has been applied to vibration analysis of the PhoNam bridge considering braking force and acceleration. The finite element analysis results were verified by the experiments. The FEM results are agree well when compared with the experimental results. The results of dynamic impact factors are appreciable. Therefore, the authors recommend that in bridge design, engineers should take into account the dynamic interaction caused by the vehicle moving on bridge and the sudden braking.

REFERENCES

[1] Chan, T.H.T. and O’Conor (1990), “Wheel Loads from Highway Bridge Strains: Field Studies”. J. Structural Engineering, ASCE, 116(7), 1751-1771.

[2] Curtis, F.G., Patrick, O.W. (2003), “Applied Numerical Analysis”. Addison-Wesley, New York.

[3] Fafard, M., Bennur, M. (1997), “A general multi-axle vehicle model to study the bridge vehicle interaction”, J. Engineering Computations, 14(5), 491-508.

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