Random Dynamic Response Analysis of Bridge Subjected to Moving Vehicles

Random Dynamic Response Analysis of Bridge Subjected to Moving Vehicles

Xuan-Toan 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

Duy-Thao 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 vehicle-bridge interaction due to road unevenness based on the Finite element method and Monte-Carlo simulation method. The road unevenness are described by a zero-mean 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 two-span slab beam concrete, are simulated as bending beam elements. The finite element method is applied to established the overall model of vehicle-bridge interaction. Galerkin method and Green theory are used to discrete the motion equation of vehicle-bridge system in space domain. Solutions of the motion equations are solved by mean of Runge-Kutta-Mersion method (RKM) in time domain. The numerical results are in good agreement with full-scale 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 A-road to Grade E-road according to ISO 8608:1995 [1]Mechanical vibration – Road surface profiles Reporting of measured data.

Keywords– Monte-Carlo simulation method, finite element method, road unevenness, moving vehicles, vehicle-bridge interaction, slab beam concrete.

1. INTRODUCTION

   Vehicle-bridge 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 vehicle-bridge 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, Frýba [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 10-20 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 single-degree-of-freedom system was used for the vehicle and a Rayleigh-Ritz 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: 1-The effect of vehicle mass on the bridge response is more significant for high speeds; 2-The 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 side-by-side vehicle configurations. Values of the DAF were computed using prismatic beam models for the bridges and step-by-step integrations. It was found that: 1-the DAF decreases with an increase in vehicle weight; 2- the DAF for two side-by-side 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 long-term deflection on the dynamic impact factor of prestressed concrete girder and cable-stayed bridges due to moving vehicles. The results showed that the effects of random road unevenness and the long-term 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 vehicle-bridge interaction due to road unevenness by an original frequency domain method. The road unevenness was modeled by the random nonstationary

   where 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. Xuan-Toan 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 three-axle

   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 multi-span 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 full-scale 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 three-axle dumper truck vehicle and two-span prestressed slab beam bridge with link deck due to road unevenness. The road unevenness is simulated by a zero-mean stationary Gaussian random process. The bridge is modelled by finite element method. The dumper-truck has three axles. Each axle is idealised by two mass, in which a

   x3

   x

   x2

   1

   L

   Fig 2. Schematic of vehicle-bridge interaction

   Base on the model of dynamic vehicle-bridge 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 vehicle-bridge 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 Runge-Kutta-Mersion 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 full-scale field test results of two-span prestressed slab beam at Nguyen-Tri-Phuong 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

2. THE MODEL OF VEHICLE-BRIDGE

   1 if t

   t t T ; T L

   i

   INTERACTION

   Consider a two-span slab beam with link deck subjected to a three-axle 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 three-axle 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 (x-xi) 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 and

   stiffness 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 fix-datum wave number 0 is set as 1/2 cycle/m. The measurement shows that various values exist for exponential and the so-called roughness coefficient Sr(0), ranging from 1.5 to 3.0 for and from 2×106 m3/cycle to

   8192×106 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) [10-6 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

   1. 2i

      2i i

      2i i

      1. Properties of structural bridge and vehicle

         m2n .g k2n .rn d2n .rn n1

         Eq.10 can be solved by means of the direct step-by-step integration method based on Runge-Kutta-Mersion method to obtain responses of girder elements.

3. 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 Nguyen-Tri-Phuong Bridge, which is a two-span 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 three-axle 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 three-axle 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

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 Runge-Kutta- 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)

1. Numerical results

   Base on the survey on site, assume that the road surface condition of Nguyen-Tri-Phuong bridge is Grade C-road (ISO 8608:1995): roughness coefficient Sr(¬0)= 256×106 m3/cycle; exponential =2; M=1000; the range of spatial frequency (wave number) k = [0.011÷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) ¼ of 1st span; b) ½ 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.

2. Field measurement results

   In order to validate the numerical results, field measurement of dynamic response of slab beam prestressed concrete was conducted at Nguyen-Tri-Phuong 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 ½ 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

   1. O v

      Displacement sensors (LVDT)

      0.2

      FEM results Experiment results

   2. 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)

   3. 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 Nguyen-Tri-Phuong 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

3. 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]×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, Monte-Carlo simulation method is applied to generate 100 road unevenness profiles. With each road profile input, the

1. 25 20

   Probability density function

   15

   10

   5

   (1+IM) Histogram (1+IM) Lognormal fit

   governing equation of vehicle-bridge 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

2. 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

3. 35

   Probability density function

   1. 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

   2. 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 A-road; b) grade B-road; c) grade C-road; d) grade D-road; e) grade E-road

Table 3. Statistical characteristics of dynamic impact factor at 1st midspan

Dynamic Impact Factor (1+IM)

Probability density function

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

x10-6

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 = 4.4e-005*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 A-road, grade B-road, grade C-road 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 D-road and grade E-road. 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 three-axle dumper truck vehicle and two-span slab beam prestressed concrete with link slab due to road unevenness is investigated by means of finite element method and Monte-Carlo simulation method. The road unevenness are described by a zero-mean stationary Gaussian random process. The bridge is modeled by finite element method. The dumper-truck 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 vehicle-bridge 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 Runge-Kutta-Mersion method in time domain. Monte-Carlo simulation is applied to generate the random road unevenness input. The numerical results are in good agreement with full-scale testing results at Nguyen-Tri-Phuong 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 E-road. 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.

REFERENCES

1. ISO 8608:1995 Mechanical vibration Road surface profiles Reporting of measured data issued by the International Organization for Standardiazation (ISO).

2. Frýba, L. (1996).Dynamics of railway bridges, Thomas Telford, London.

3. Honda, H.,Kajikawa, Y., and Kobori, T. (1982). Spectral of road surface roughness on Bridges. ASCE of Structural of the Division, 108(9): 1956-1966.

4. Palamas, J., Coussy, O., and Bamberger, Y. (1985). Effects of surface irregularities upon the dynamic response of bridges under suspended moving loads. Journal of Sound and Vibration, 99(2): 235-245.

5. Coussy, O., Said, M., and Van Hoore, J.P. (1989). The influence of random surface irregularities on the dynamic response of bridges under suspended moving loads. Journal of Sound and Vibration, 130(2): 3 13-320.

6. Inbanathan, M.J., and Wieland, M. (1987). Bridge vibrations due to vehicle moving over rough surface. ASCE Journal of Structural Engineering, 113(9): 1994-2008.

7. Hwang, E.S., Nowak, A.S. (1991). Simulation of dynamic load for bridges. ASCE Journal of Structural Engineering, 117(5): 1413-1434.

8. F.T.K Au, Y.S. Cheng, Y.K. Cheung. (2001). Effects of random road surface roughness and long-term deflection of prestressed concrete girder and cable-stayed bridges on impact due to moving vehicles, Computer and structures 79 (2001) 853-872.

9. Geert Lombaert and Joel P. Conte. (2012). Random vibration analysis of Dynamic Vehicle-Bridge interaction due to Road Unevenness. Journal of Engineering Mechanics.2012. 138:816-825.

10. Toan X.N, Duc V.T (2014). A finite element model of vehicle – cable stayed bridge interaction considering braking and acceleration. The 2014 World Congress on Advances in Civil, Environmental, and Materials Research. Busan, Korea. P.109(20p.)

11. Toan, X.N. (2014), Dynamic interaction between the two axle vehicle and continuous girder bridge with considering vehicle braking forces, Vietnam Journal of Mechanics, vol 36, p49-60.

12. Xuan-Toan Nguyen, Van-Duc Tran, Nhat-Duc Hoang. (2017) A Study on the Dynamic Interaction between Three-Axle Vehicle and Continuous Girder Bridge with Consideration of Braking Effects. Journal of Construction Engineering, Volume 2017 (2017), Article ID 9293239, 12 pages.

13. Ray, W. C., and Joseph, P. (1995). Dynamics of structures. 3rd Ed. Computers & Structures, Berkeley, California .

14. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals, Seventh Edition. Butterworth-Heinemann, 2013.

15. M. Shinozuka, Digital simulation of random processes and its applications, J. Sound Vib. 25 (1972) 111128.

16. L. Sun, Simulation of pavement roughness and IRI based on power spectral density, Mathematics and Computers in Simulation 61(2003) 7788.

17. AASHTO, LRFD Bridge Design Specifications, AASHTO, Washington, DC, USA, 2012.

18. Vietnammese Specification for Bridge Design, 2005. 22TCN 272-05.