ABSTRACT: In recent years, PC extradosed bridges have been widely used for long road bridges featuring practical and economical structures. In this study, we discuss the applicability of 4-span continuous PC extradosed bridges (bridge length 450 m, span length 75 + 150 + 150 + 75 m) to high-speed railways through the analysis of dynamic interaction between vehicle and structures. As a result, we clarify that when the designed maximum speed is 360 km/h (1) a structure design impact factor of 0.35 guarantees safety with respect to resonance, (2) a car axle load decrease ratio of 9.0% shows no problem related to running safety and (3) a car acceleration of 0.44 m/s2 ensures high-level ride comfort

KEY WORDS: PC extradosed bridge; Dynamic interaction; Impact factor; Running safety; Ride comfort.

1 INTRODUCTION

In this study, we discussed the applicability of a four-span continuous PC extradosed bridge (bridge length 450 m, span length 75 + 150 + 150 + 75 m) to high-speed railways. The 150 m span length of the bridge is the longest among the Shinkansen concrete bridges in Japan [1].

Figure 1 shows the outline of the PC extradosed bridge. This bridge is used as a double-track railway bridge with 3 box section main girder which is stayed by diagonal cables in two planes, simply supported at the ends and rigidly connected with main towers.

In recent years PC extradosed bridges have been widely used for long road bridges featuring, practical and economical structures, and already built as railway bridges having a span length of about 100 m. However, their dynamic characteristics for high-speed train operation have yet to be clarified. Under these circumstances, therefore, we now shall address the following subjects specific to railways.

PC extradosed bridges are high-order indeterminate structures composed of members with different characteristics, such as main girders, main towers, bridge piers and diagonal cables. This means that we must clarify the dynamic loads (values of impact factor) caused by high-speed train operation on various members [2].

Furthermore, long extradosed bridges tend to cause large degrees of deformation and have low-frequency vibration modes close to the natural frequency of car bodies, in the vertical direction. Therefore, we must clarify the running quality (that is, running safety and ride comfort) of high-speed vehicles with respect to the dynamic deformation of the main girders [2].

In this study, we discuss the above problems using the Dynamic Interaction Analysis for Shinkansen Trains And Railway Structures (DIASTARS), a program for analyzing the dynamic interaction between vehicles and railway structures [3] as well as establish a system for visualizing the analytical results by applying existing computer graphics technologies and understand the dynamic behaviors from animations.

2 ANALYSIS METHOD

2.1 Vehicle dynamic model

Figure 2 and Table 1 show a vehicle dynamic model. The vehicle model was created by connecting each element of a vehicle body, two truck frames and four wheelsets which were modeled as rigid masses with springs and dampers. Then, a vehicle has 31 degrees of freedom. The actual vehicle has stoppers between each element part to control significant relative displacement. In order to consider this, bilinear-nonlinear springs were used for springs. In the analysis, we used 8 or 12 coached trains. Adequacy of these dynamic models has already been verified through running tests on the actual bridges and vibration experiment using a vehicle test plant and an actual vehicle model [4] ,[5] ,[6].

Vehicle specifications were assumed in reference to a recent high-speed Shinkansen train vehicle. The main input data were 25m of vehicle length, 40.0t of body mass, 3.3t of truck frame mass, 2.0t of wheelset mass, 300kN/m of vertical spring constant for the air-spring (half side of one truck), 30kN/s･m of damping constant for the air-spring (half side of one truck), 1300kN/m of spring constant for the axle spring (half side of one wheelset), and 40kN/s ･ m of damping constant for the axle spring (half side of one wheelset).

Equations of motion of the vehicle system in the vehicle coordinate system can be shown as equation (1) after transposing nonlinear spring terms between each element to the right-hand side.

)(),( VVN

BVVVL

VVVVVV XFXXFFXKXCXM

(1)

where affixing character V and B were the vehicle and the bridge, respectively; VX was a displacement vector of the vehicle; VV CM , and VK were the mass, damping and stiffness matrices of the vehicle, respectively; V

LF was load vectors of the wind pressure; ),( VBV XXF was interaction load vectors with the bridge; )( VV

N XF was load vectors of the nonlinear spring force of the vehicle model assumed outside load.

Performance Verification for Railway Extradosed Bridges by Dynamic Interaction Analysis

Masamichi SOGABE1, Tsutomu WATANABE1, Keiichi GOTO1, Munemasa TOKUNAGA1

Makoto KANAMORI2 and Shinichi TAMAI2 1Railway Dynamics Div., Railway Technical Research Institute, 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo, Japan

2Japan Railway Construction, Transport and Technology Agency, 6-50-1 Honcho, Naka-ku, Yokohama City, Kanagawa, Japanemail: sogabe.masamichi.78@rtri.or.jp

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

1203

2.2 Bridge dynamic model

Figure 3 shows an analytical model of the bridge. DIASTARS can model structures of any type using beams, trusses, shells, solids, springs and other finite elements. We modeled all main girders and bridge piers with girder elements, while assuming that their rigidities are all linear (average weight of main girders: 600kN/m, cross-section area: 13 to 98m2, second moment of area: 29 to 544m4). We also modeled diagonal cables with truss elements and connected them to the main girder with a rigid beam installed to the diagonal steel cable anchoring points. In the connecting areas between main girders, main towers and piers, we assumed an appropriate rigid zone for each member. The main girders are simply supported on bridge piers P1 and P5. Each pier is fixed at the bottom end. The analytical model has 5,040 nodes and 7,085 elements in total. We applied a damping ratio of 0.7% in all modes by referring to the measurement made at PC cable-stayed railway bridges [6].

The axle load variation ratio of the vehicle is affected by the curvature of the wheel running surface (that is, the rail top curvature in the longitudinal direction). The deflection of girders causes angular rotations with an infinite curvature at their ends. In this study, however, we modeled the track

Figure 1. Outline of PC the extradosed bridge.

Table 1. Notations of vehicle dynamic model. Items Not. Items Not. Items Not.

Half of longitudinal distance between center pivots of fore and rear truck L Half of mass of car body m Longitudinal spring constant for air

spring (half side of one truck) K1

Half of wheelbase a Half of inertial moment of car body around x axis Ix

Longitudinal damping constant for yaw damper (half side of one truck) C1

Half of lateral distance between contact points of wheel and rail b Half of inertial moment of car body

around y axis Iy Lateral spring constant for air spring(half side of one truck) K2

Half of lateral distance between yaw dampers b0

Half of inertial moment of car body around z axis Iz

Damping constant for lateral damper(half side of one truck) C2

Half of lateral distance between axle springs b1

Mass of truck MT Vertical spring constant for air spring (half side of one truck) K3

Half of lateral distance between air springs b2

Inertial moment of truck around x axis ITx Vertical damping constant for airspring (half side of one truck) C3

Height of center of gravity of car body from rail head Hb

Inertial moment of truck around y axis ITy Longitudinal spring constant for wheelset (half side of one wheelset) Kwx

Height of center of gravity of truck from rail head HT Inertial moment of truck around z axis ITz

Lateral spring constant for wheelset(half side of one wheelset) Kwy

Vertical distance between center of gravities of wheelset and car body h1

Mass of wheelset Mw Vertical spring constant for axle spring (half side of one wheelset) Kwz

Vertical distance between center of air spring and center of gravity of car body h2

Inertial moment of wheelset around xaxis Iwx

Vertical damping constant for axle damper Cwz

Vertical distance between center of gravity of truck and center of air spring hs

Inertial moment of wheelset around zaxis Iwz

Static wheel force Ps

Nominal radius of wheel r Half of length of car body Lc

Figure 2. Vehicle dynamic model.

20.0

m 1

9.5m

17.

5m

75m 150m 150m 75m

10m×4m 10m×5m 10m×4m6.5m

3.8m

8.0m40m 22m 22m 40m

P2Cross-section

Φ14.1m

Side view

P1 P2 P3 P4 P5

1st Main tower 2nd Main tower 3rd Main tower

zψ

φ

Wheelset

Truckframe

Body φ θz

y

T

zT

yT

θT

θW

zW

Non-linear springDamper

Air spring

Axle springzT ψT

φT

zW ψW

φW

WyW

K1 , C1

K2, C2

K3, C3

Kwz, Cwz

Kwx

Kwy

Coupler

WheelsetTruckframe

BodyDisplacement

Non-linear spring (stopper)

Forc

e

φ

φ

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1204

structure by rails elastically supported with track pads so that the angular rotation can be eased, as shown in Fig. 3 [5].

Equations of motion of the bridge system can be shown as equation (2) after transposing nonlinear spring terms to the right-hand side.

)(),( BBN

BVBBL

BBBBBB XFXXFFXKXCXM

(2)

where BX was a displacement vector of the bridge; BB CM , and BK were the mass, damping and stiffness matrices of the bridge, respectively; B

LF was a load vector of earthquake or wind pressure of the bridge; ),( BVB XXF was an interaction load vector with vehicles; )( BB

N XF was a load vector of the nonlinear spring force of the bridge model assumed outside load.

2.3 Interaction model between the wheel and the rail

2.3.1 Vertical direction

Figure 4 shows the vertical dynamic interaction model between the wheel and the rail. The vertical relative displacement δz between the wheel and the rail can be shown as equation (3).

δz= zR – zW + eZ + eZ0(y) (3) where zR and zW were vertical displacements at the contact point of the rail and the wheel; eZ was vertical track irregularity existing on the rail shown in Fig. 4; eZ0 was the amount of change of the wheel radius between the current contact point and the initial contact point, which was shown as a function of horizontal relative displacement y between the wheel and the rail.

A contact point s and contact angle a for the relative displacement δz were calculated with the horizontal relative displacement y of the wheel and the rail and the contact function set in accordance with geometric shapes of the wheel and the rail. When the wheel and the rail consist of a quadratic surface respectively, the relation between the

relative displacement δ of the wheel and the rail of the normal direction of the contact surface and the contact force H can be shown with the Hertz contact spring, as indicated in equation (4).

H = H(δ) = H(δz・cos a ) (4) The vertical and horizontal components of this contact force H were distributed to the wheel and the rail respectively to make the interaction force.

Figure 4. Vertical interaction model between wheel and rail.

Figure 5. Horizontal interaction model between wheel and rail.

Figure 3. Bridge dynamic model.

Relational displacement between the wheel and the rail δz

Rail displacement zR

Track irregularity ez

Rail

Wheel

Wheel displacement zw Con

tact

forc

e H

Hertz contact spring

δzWheel jumping

Rail displacement yR

Track irregularity ey

Rail

wheel

Wheel displacement yw

Flange force Qf

Creep force Qc

gap :u

Wheel

Rail Slip ratio S

Cre

ep f

orce

Qc

Friction forceTread gradient γ

δy

Flan

ge f

orce

Qf

Rail tiltingspring constant kp

gap :uInitial radius r

Flange

Relational displacement between the wheel and the rail flange δy

Rail: beam element

Rail pad: spring (3DOF)

Main girder： beam element

Pier bottom: fix

Main girder: beam element

Rigid beam to rail support points

Pier: beam element

Rail

Angular rotation

Girder

Rail pad Diagonal steel cable: truss element

R igid beam to cable anchoring points

Main tower: beam element

Connecting areas:assumed an appropriate rigid zone

Tower: beam element

Simple support

XY

Z

Global coordinate

Train running

P1 P2

P3

P4 P5

Simple support

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1205

2.3.2 Horizontal direction

Figure 5 shows the horizontal dynamic model. The horizontal relative displacement δy between the wheel flange and the rail can be shown as equation (5).

δy = y– u (δz) = yw – yR – ey – u (δz) (5) where y was the horizontal relative displacement between the wheel and the rail; yR and yW were horizontal displacements at the contact point of the rail and the wheel; ey was horizontal track irregularity existing on the rail shown in Fig. 5; u(δz) was the gap between the wheel flange and the rail which was shown as a function of vertical relative displacement δz. A contact point s and contact angle a for the relative displacement δy were calculated with the vertical relative displacement δz of the wheel and the rail and the contact function set in accordance with geometric shapes of the wheel and the rail. When δy<0, it was considered that the wheel flange and the rail were not in contact. In this case, creep force Qc (slipping force) acted horizontally on the contact surface of the wheel and the rail. The creep force was the horizontal force caused by creep of the wheel moving forward by rolling on the rail, which can be shown as equation (6). This creep force reached the upper limit of friction force when the slip ratio became high.

vvryCSCQ wwwyc /)(

(6)

where C was the creep constant; Sy was the slipping ratio in the horizontal direction; v was the train speed; r was the nominal radius.

When δy≧0, it was considered that the wheel flange and the rail were in contact. For the flange contact, only the flange pressure Qf which was equivalent to the horizontal component of contact force H was considered. The flange pressure Qf

can be shown as equation (7) using the rail tilting spring constant kp.

Qf = kp・δy (7)

2.4 Numerical analysis method

Equations of motions of the train and the bridge shown as equation (1) and (2) were solved in the modal coordinates for each time increment t by the Newmark time difference scheme. Since the equations were nonlinear, iterative calculations were necessary during each time increment until the unbalanced force between the train and the railway structures became small enough to be within the specified tolerance [3].

3 ANALYSIS RESULTS

3.1 Impact factors

Figure 6 shows the natural frequency modes obtained through eigenvalue analysis. A symmetric primary vibration mode at 1.04 Hz and an anti-symmetric secondary vibration mode at 1.23 Hz exist close to the vertically anti-symmetric primary vibration mode at 0.95 Hz. Figure 7 shows the time history response waveforms of the bridge under single-track loading by an 8-car train running at 360 km/h. In Fig. 7(a), the waveforms present an approximately static behavior generated by the moving load or a quasi static behavior slightly affected by the dynamic effect of the natural vibration. The main girders are of the

Mode 1 0.95Hz Mode 2 1.04Hz

Mode 3 1.11Hz Mode 4 1.23Hz Figure 6. Natural frequency modes.

Figure 7. Time history response waveforms of the bridge (single-track loading by 8-car train running at 360 km/h).

0.0 2.0 4.0 6.0 8.0 10.0-30-20-10

010

0.0 2.0 4.0 6.0 8.0 10.0-6-4-2

0246

0.0 2.0 4.0 6.0 8.0 10.020000

10000

0

-10000

0.0 2.0 4.0 6.0 8.0 10.0-4000-2000

020004000

0.0 2.0 4.0 6.0 8.0 10.0-15000-10000-5000

05000

1000015000

0.0 2.0 4.0 6.0 8.0 10.0

0

50

1001st span2nd span

3rd span4th span

Ver

tica

l

Time(sec)

Rai

l di

rect

ion

Time(sec)

1st tower2nd tower3rd tower

(a) Main gerder center displacement

(d) Tower top displacement

Ben

ding

mom

ent(

kN-m

)

Time(sec)(b) Main girder center bending moment

Time(sec)

1st tower2nd tower3rd tower

Ben

ding

mom

ent(

kN-m

)

(e) Tower base bending momentTime(sec)

P2 pierP3 pierP4 pier

Ben

ding

mom

ent(

kN-m

)

(f) P ier bottom bending moment

Time(sec)

Nor

mal

tens

ion

forc

e(kN

)

(c) 2nd tower cable normal force

1st span2nd span

3rd span4th span

1st en trance side 11th exit side11th entrance side 1st exit side

defl

ecti

on(m

m)

disp

lace

men

t(m

m)

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1206

double-plane suspension type, so the maximum rotational angle due to the torsion of the girders under one-line loading is as small as 10-5 rad, with the main towers and diagonal cables on the left and right sides presenting almost symmetric behaviors. Figure 8 shows a relation between the train speed and the impact factor of a 12-car train case. The impact factor i is the ratio of the increment of the dynamic deflection or the section force caused by the running train to their static values as expressed by equation (8) [2].

i = (fd – fs) / fs (8) where fd were dynamic deflection or section force; fs were static deflection or section force.

The impact factor, which depends on the kind of member and section force, tends to increase as a whole as train speed increases without significantly high resonance peaks. Although not shown in the figures, a 12-car train gives a slightly larger impact factor value than by an 8-car train. To calculate the value of the design impact factor to be applied to the bridge, we first summarized the analytical results on the values of impact factor effected by 8- and 12-car trains, without including the components by the bending moment of the main towers (having sectional dimensions

determined by verifying the earthquake resisting performance) and then added the component effected by track irregularities to the resultant value [2]. As a result, we obtain 0.35 as the value of the design impact factor to be applied to the bridge.

3.2 Train running quality

Figure 9 shows the time history response waveforms of the train under single-track loading by an 8-car train running at 360 km/h. It was clarified that car body acceleration presents a sinusoidal wave at a frequency equal to the ratio of train speed to span length, the axle load variation ratio is analogous to car body acceleration, so axle load variation are mostly caused by car body acceleration, and the axle load decreases are caused by the angular rotations at the girder ends of the bridge entering point [5], [6], [7]. The main girders are not twisted, so the lateral force was no more than 0.5 kN. Figure 10 shows a relation between the train speed and train running quality. We evaluated running safety, an item in the train running quality, in terms of the axle load reduction ratio (the axle load variation ratio on the negative side). We verified running safety under the condition of simultaneous double-track loading in two directions at the maximum seat-load factor (with passengers 3.5 times as many as the passenger capacity on board) according to the railway structure design standard and commentaries (limit of

Figure 9. Time history response waveforms of the train (single-track loading by 8-car train running at 360 km/h).

Figure 8. Relation between train speed and impact factor of the 12-car train case.

0.0 2.0 4.0 6.0 8.0 10.0-6-4-2

0246

0.0 2.0 4.0 6.0 8.0 10.0-6-4-2

0246

0.0 2.0 4.0 6.0 8.0 10.0-6-4-2

0246

0.0 2.0 4.0 6.0 8.0 10.0-0.6-0.4-0.2

00.20.40.6

0.0 2.0 4.0 6.0 8.0 10.0-0.6-0.4-0.2

00.20.40.6

0.0 2.0 4.0 6.0 8.0 10.0-0.6-0.4-0.2

00.20.40.6

Whe

el lo

ad v

aria

tion(

%)

Time(sec) Time(sec)

Reduction due to change of angle

Time(sec)

Car

bod

y ac

cele

ratio

n(m

/s2 )

Time(sec) Time(sec) Time(sec)

(a) 1st car wheel load variation (b) 4th car wheel load variation (c) 8th car wheel load variation

(d) 1st car body acceleration (e) 4th car body acceleration (f) 8th car body acceleration

Whe

el lo

ad v

aria

tion(

%)

Car

bod

y ac

cele

ratio

n(m

/s2 )

Whe

el lo

ad v

aria

tion(

%)

Car

bod

y ac

cele

ratio

n(m

/s2 )

1st wheelset3rd wheelset

At front truckAt rear truck

1st wheelset3rd wheelset

At front truckAt rear truck

1st wheelset3rd wheelset

At front truckAt rear truck

100 200 300 4000.0

0.1

0.2

0.3

0.4

0.5

100 200 300 4000.00.10.20.30.40.50.60.70.8

100 200 300 4000.00.10.20.30.40.50.6

100 200 300 4000.0

0.1

0.2

0.3

0.4

0.51st entrance side/exit side4th entrance side/exit side8th entrance side/exit side11th entrance side/exit side

100 200 300 4000.0

0.1

0.2

0.3

0.4

0.5

100 200 300 4000.00.10.20.30.40.50.6

Train speed(km/h)

Impa

ct fa

cto

r

Train speed(km/h)

Impa

ct fa

cto

r

Train speed(km/h)

1st span2nd span3rd span4th span

Train speed(km/h) Train speed(km/h)

Impa

ct fa

cto

r

1st tower entrance/exit side sway ing2nd tower entrance/exit side sway ing3rd tower entrance/exit side sway ing

Impa

ct fa

cto

r

P2 pier entrance/exit side sway ingP3 pier entrance/exit side sway ingP4 pier entrance/exit side sway ing

(b) Main girder center bending moment

(e) Tower base bending moment (f) Pier bottom bending moment

Impa

ct fa

cto

r

Impa

ct fa

cto

r

Train speed(km/h)(c) 2nd tower cable tension force

1st span2nd span3rd span4th span

1st tower2nd tower3rd tower

(a) Main girder center vertical displacement

(d) Tower top rail direction displacement

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1207

displacement). We set the limit value of the axle load reduction ratio at 37% [8]. This limit value of structure displacement is set to guarantee that cars don’t reach the criteria for running safety, even when track irregularity exists on a bridge. Figures 10(a) and 10(b) show that the axle load reduction ratio tends to increase as train speed increases; for example, 9.0% at a train speed of 360 km/h. This value has an ample margin with respect to the limit value of 37%. We evaluated ride comfort, the other item pertaining to train running quality, in terms of the maximum car body acceleration directory above the trucks. We verified ride comfort in running under the condition of single-track loading at the rated seat-load factor with the limit value of car body acceleration set at 2.0 m/s2, according to railway structure design standards and commentaries (limit of displacement) [8]. Figures 10(c) and 10(d) show that the maximum car body acceleration tends to increase as train speed increases; for example, 2.0 m/s2 at a train speed of 360 km/h. This value has an ample margin with respect to the threshold value of 2.0 m/s2.

4 ANALYSIS VISUALIZATION

Various time history data for infinitesimal increments calculated by the direct integral method are normally too large in volume, so it is extremely difficult to grasp all analytical results and appropriately understand their dynamic behaviors. Therefore, we established a system for DIASTARS to visualize their dynamic characteristics, by applying the existing computer graphics technologies. Figure 11 shows an outline of the visualization system. In the recent film industry, development in CG (Computer Graphics) technology has advanced significantly, and the technology is becoming available with ease. Therefore, we tried to make the most of the existing CG technology to establish the visualization system. We created new modules to make a motion capture of analytical results, perform conversion to the coordinates in the visualizing space and adjust the enlargement ratio of responses, while, combining several existing modules, we created objects of vehicle and structures, arranged these objects in a visualizing space, set cameras, specified light sources, rendered pictures and compressed images.

Figure 12 shows a vehicle rigid object model. As mentioned earlier, in DIASTARS, component elements such as a vehicle body, a truck and a wheelset were considered rigid masses, which were connected with springs and dampers. Therefore, the analysis results can be presented as a six-degree-of-freedom response of a rigid object (running at a constant speed in the rail direction).

Normally, a method of presenting the element mesh as an ordinal wire frame is common in dynamic analysis visualization with the commercial based finite element method. In this study, we decided to separately create geometry data based on the actual vehicle and express each rigid masses with shading display (removal of black lines, shade and shadow, region fill) due to each vehicle’s component element being rigid masses.

The vehicle configuration was created by statically combining polygon (square surface elements). In addition,

Figure 10. Relation between train speed and train running quality.

Figure 11. Outline of the visualization system.

(a)Polygon model

(b)Shading model

Figure 12. Vehicle dynamic model

100 200 300 4000.0

5.0

10.0

15.0

20.0

100 200 300 4000.0

0.5

1.0

100 200 300 4000.0

5.0

10.0

15.0

20.0

100 200 300 4000.0

0.5

1.0

Whe

el lo

ad r

edu

ctio

n ra

te(%

)

Train speed(km/h)

Car

bo

dy a

ccel

erat

ion(

m/s2 )

Train speed(km/h)

1st car3rd car5th car7th car

Limit 37%

(max. load)

Limit 2.0m/s2

(a) Wheel load reduction rate

(c) Car body acceleration

Double truckloading

Single truckloading

1st car3rd car5th car7th car (capacity load)

(8 car train)

(8 car train)

Whe

el lo

ad r

edu

ctio

n ra

te(%

)

Train speed(km/h)

Car

bo

dy a

ccel

erat

ion(

m/s2 )

Train speed(km/h)

1st car3rd car5th car7th car

Limit 2.0m/s2

9th car12th car

(b) Wheel load reduction rate

(d) Car body acceleration

Double truck loading(max. load)

Single truck loading

(capacity load)

Limit 37%

(12 car train)

(12 car train)

1st car3rd car5th car7th car9th car12th car

Rendering

Dynamic analysis

Bridge time series data

Vehicle timeseries data

To capture result motions of analysis resultsTo convert to visualized space coordinates

Motion capture

To adjust response enlargement ratios

Create object

・polygons・Shading

DIASTARSⅡ

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1208

the color specified texture (image data) was attached on the surface of each polygon to create the object. The vehicle object model in the figure consists of each rigid object such as a vehicle body, a truck and a wheelset with a total of 3521 pieces of polygons for each vehicle.

To express texture of the object surface, characteristics such as environment light, glazing, reflection, transparency and shadow were specified to match the material set of each polygon. The bump (bumpy) method with texture was used for the details of the window and the door instead of using polygons in an attempt to decrease burdens on drawings.

As explained above, in DIASTARS, the bridge model was created with finite elements such as beams, tresses, shells and solids, whose response analysis results are provided momentary by node displacement. As for visualization, since all finite element node behaviors are not always required, the entire structural behavior was expressed by connecting created rigid objects with selected representative nodes. In this study, the main girder was divided into rigid objects of 5m. A model of the main girder was created with beam elements in numeric analysis. In contrast, a rigid body model was created with polygon in the same way as the case of the vehicle. This object behavior is presented by displacement time history data of the centric position node in a section of 5m.

With the established motion capture modules, the displacement response history obtained by DIASTARS was converted into rigid object motion data in the visualization space coordinate. However, real behaviors of the vehicle and the structure are minute compared with the entire structure size. For this, the response needs to be enlarged by a constant fraction in order to understand dynamic behavior. Therefore, arbitrary enlargement factors were made to be specified for the displacement and the rotating angle during the process of converting than into visualization space coordinate.

After allocating each object in the visualization space and specifying motion data for each, camera setting, light source specification, and background image were determined in reference to the simple rendering. The rendering was performed in 30fps (frame per second), and final movie files were created with image compression in the frame and the time direction.

(a)Vibration behavior of bridge (b)Vibration behavior of vehicle

Figure 13. Example of visualization of the analysis results (single-track loading by 8-car train running at 360 km/h).

Figure 14. Video-type displacement sensor.

(a) Time history response waveforms of main girder deflection

(b)Relation between vertical deflection and train speed

Figure 15. Validation result of the actual train running test.

非線形ばね(ストッパ)

Video camera

Target

0.0 10.0 20.0 30.0-10

-5

0

5

0.0 10.0 20.0 30.0-10

-5

0

5

1 2 310

-3

10-2

10-1

100

1 2 310

-3

10-2

10-1

100

Time(sec)

Time(sec)

1st test f=1.31Hz2nd test f=1.34Hz3rd test f=1.34Hz

(a) 2nd span

Ver

tica

l def

lect

ion(

mm

)

(b) 3rd span

Ver

tica

l def

lect

ion(

mm

) 4th test f=1.43Hz5th test f=1.33Hz6th test f=1.39Hz

Frequency(Hz)

Pow

err

spec

trum

(m

m2 /H

z)

Frequency(Hz)

Pow

err

spec

trum

(m

m2 /H

z)

100 200 300 4000.0

-5.0

-10.0

-15.0

-20.0

-25.0

Ver

tica

l def

lect

ion

(mm

)

Train speed (km/h)

1st span2nd span3rd span4th span

Analysis Running test

2nd span3rd span

Static Axis load =110kN

(8-car train) (12-car train)

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1209

Figure 13 shows an example of visualization of analysis results of the single-track loading by an 8-car train running at 360 km/h. Figure 13(a) shows a picture of a case where the vibration behavior of the whole bridge was analyzed with a camera fixed in a three-dimensional visualizing space. Then Figure 13(b) shows another case where the vibration behavior of the vehicle was analyzed with a camera run in parallel with the vehicle. In this manner, we are able to precisely and visually grasp the behavior of the structures coupled with the vehicle as one object.

5 ANALYSIS VALIDATION

In order to verify the numerical analysis results shown in Section 3, actual 10-car train running tests were conducted on actual structures. Figure 14 shows a video-type displacement sensor. Since the bridge is a long and high structure, the video-type non-contact displacement sensor consisted of a high-vision (1920 x 1080 dots) video camera and a target 150 by 150 mm in size was used to measure vertical deflection. Sampling frequency was set at 30 Hz (30 fps). In this way, we analyzed a key shape printed on the target which was set at the main girder center in each image frame of the movie, and estimated the bridge vertical deflection from the target transfer amount. The natural frequency of the bridge was calculated by using the Autoregressive moving average model (AR model) [9].

Figure 15 shows a validation result of the actual train running test. The design maximum speed of this bridge is 260km/h, however the actual train operation speed is set at about 130km/h at present because there is a terminal station near the bridge. From this figure, we can estimate that the actual bridge rigidity is 1.6 times larger than the design one and the actual natural frequency of 1.35 Hz is 1.4 times than the design one of 0.95 Hz. This increasing tendency of rigidity is the same as that observed in previous measurement [4], [5], [6] and the reasons for this are considered to be the influence of non-structural members, such as concrete used for water discharge gradients and track structure, and also the influence of the increases in the actual concrete strength and the Young’s modulus.

6 CONCLUSIONS

In this study, we discussed applicability of the 4-span continuous PC extradosed bridge to the high-speed railway bridge by applying a technique for analyzing the dynamic interaction between structures and vehicles. The knowledge obtained through this study is as follows.

(1) At the maximum speed of 360 km/h, the design impact factor is 0.35, which guarantees that all members and sectional force are safe with respect to resonance despite the complicated construction of the bridge. (2) At the maximum speed of 360 km/h, the maximum axle load reduction ratio is 9.0%, a value that is not problematic at all to guarantee running safety, in comparison to the limit value of 37% and the maximum car body acceleration is 0.44 m/s2, a value that is not problematic at all either, from the viewpoint of ride comfort, in comparison with the limit value of 2.0 m/s2.

(3) From the actual vehicle running tests, the actual bridge rigidity is 1.6 times larger than the design one and the actual natural frequency of 1.35 Hz is 1.4 times than the design one of 0.95 Hz.

REFERENCES [1] S. Tamai and K. Shimizu, The long spanned bridge for deflection-

restricted high speed rail –SANNAI-MARUYAMA Bridge-, World Congress of Railway Research, E2, USB, Lille, France, 2011.

[2] Railway Technical Research Institute, Design Standard for Railway Structures (for seismic design), Maruzen, Tokyo, Japan, 1999.

[3] H. Wakui, N. Matsumoto and M. Tanabe, A Study on Dynamic Interaction Analysis for Railway Vehicles and Structures - Mechanical model and practical analysis method -, Quarterly Report of RTRI, Vol. 35, No. 2, pp. 96-104, RTRI, Tokyo, Japan, 1994.

[4] M. Sogabe, N. Matsumoto, M. Kanamori, T. Sato and H. Wakui, Impact Factors of Concrete Girders Coping with Train Speed-up, Quarterly Report of RTRI, Vol.46, No.1, RTRI, Tokyo, Japan, 2005.

[5] M. Sogabe, A. Furukawa, T. Shimomura, T. IIDA, N. Matsumoto and H. Wakui, Deflection Limits of Structures for Train Speed-up, Quarterly Report of RTRI, Vol.46, No.2, RTRI, Tokyo, Japan, 2005.

[6] M. Sogabe, N. Matsumoto, H. Wakui, M. Kanamori and T. Shiimoto, Study on Dynamic Response and Ride Comfort of a PC Multicable-stayed Bridge (Daini-Chikumagawa Bridge on Hokuriku Shinkansen Line) (in Japanese), Proceedings of Structural Engineering, Vol.44A, pp.1333-1340, JSCE, Japan, 1998.

[7] S. Miyazaki, M. Kanamori, H. Wakui, N. Matsumoto and M. Sogabe, Analytical Study on Response and Riding Comfort of PC Multicable-Stayed Railway Bridge, International Conference of Speed-up Technology for Railway and Maglev Vehicles, pp.424-429, Japan, 1993.

[8] Railway Technical Research Institute, Design Standard for Railway Structures (for displacement limits), Maruzen, Tokyo, Japan, 2006.

[9] K. Matsuoka and K. Kaito: Variable Estimation of Bridge Frequency under Train Loads, The 31th International Modal Analysis Conference, IMAC-XXXI, CD-ROM, No.429, California, U.S., 2013.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

1210