Performance Verification for Railway Extradosed Bridges by

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Performance Verification for Railway Extradosed Bridges
by Dynamic Interaction Analysis
Masamichi SOGABE1, Tsutomu WATANABE1, Keiichi GOTO1, Munemasa TOKUNAGA1
Makoto KANAMORI2 and Shinichi TAMAI2
1
Railway Dynamics Div., Railway Technical Research Institute, 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo, Japan
2
Japan Railway Construction, Transport and Technology Agency, 6-50-1 Honcho, Naka-ku, Yokohama City, Kanagawa, Japan
email: [email protected]
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, bilinearnonlinear 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.


M V X V C V X V  K V X V  FLV  FV ( X V , X B )  FNV ( X V )
(1)
where affixing character V and B were the vehicle and the
bridge, respectively; X V was a displacement vector of the
vehicle; M V , C V and K V were the mass, damping and
stiffness matrices of the vehicle, respectively; F LV was load
vectors of the wind pressure; FV ( X B , X V ) was interaction
load vectors with the bridge; FNV ( X V ) was load vectors of
the nonlinear spring force of the vehicle model assumed
outside load.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
150m
1st Main tower
150m
2nd Main tower
3.8m
6.5m
P1
10m×4m
P2
22m 22m
40m
8.0m
10m×5m
P3
Φ14.1m
75m
3rd Main tower
10m×4m
P4
20.0m 19.5m 17.5m
75m
40m
P5
Side view
P2Cross-section
Figure 1. Outline of PC the extradosed bridge.
Table 1. Notations of vehicle dynamic model.
Not.
Half of lateral distance between contact
points of wheel and rail
Half of lateral distance between yaw
dampers
Half of lateral distance between axle
springs
Half of lateral distance between air
springs
Height of center of gravity of car body
from rail head
Height of center of gravity of truck from
rail head
Vertical distance between center of
gravities of wheelset and car body
Vertical distance between center of air
spring and center of gravity of car body
Vertical distance between center of
gravity of truck and center of air spring
Nominal radius of wheel
Half of length of car body
b
L
a
b0
b1
b2
Hb
HT
h1
h2
hs
r
Lc
Items
Half of mass of car body
m
Half of inertial moment of car body
around x axis
Half of inertial moment of car body
around y axis
Half of inertial moment of car body
around z axis
Mass of truck
Inertial moment of truck around x axis
Iy
Iz
MT
ITy
Inertial moment of truck around z axis
ITz
Mass of wheelset
Mw
Inertial moment of wheelset around x
axis
Inertial moment of wheelset around z
axis
Bridge dynamic model
Ix
ITx
Inertial moment of truck around y axis
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].
1204
Not.
Iwx
Iwz
Items
Longitudinal spring constant for air
spring (half side of one truck)
Longitudinal damping constant for
yaw damper (half side of one truck)
Lateral spring constant for air spring
(half side of one truck)
Damping constant for lateral damper
(half side of one truck)
Vertical spring constant for air spring
(half side of one truck)
Vertical damping constant for air
spring (half side of one truck)
Longitudinal spring constant for
wheelset (half side of one wheelset)
Lateral spring constant for wheelset
(half side of one wheelset)
Vertical spring constant for axle
spring (half side of one wheelset)
Vertical damping constant for axle
damper
Static wheel force
z
Body
φ
Truck
frame
Wheelset
Air spring
zT
ψT
φT
Axle spring
zW ψW
φW
Wheelset
Kwy
Kwx
Truck
frame
Body
Not.
K1
C1
K2
C2
K3
C3
Kwx
Kwy
Kwz
Cwz
Ps
z
φ
ψ
θ
y
zT
Coupler
K3, C3
θT
φT
zW
Kwz, Cwz
φW
K1 , C1
Force
2.2
Items
Half of longitudinal distance between
center pivots of fore and rear truck
Half of wheelbase
K2 , C2
θW
yT
yW
Displacement
Non-linear spring (stopper)
Non-linear spring
Damper
Figure 2. Vehicle dynamic model.
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
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Rail
Rigid beam to cable anchoring points
Rail pad
Diagonal steel cable: truss element
Girder
Angular
rotation
Tower: beam element
Main girder: beam element
Rail: beam element
Rail pad: spring (3DOF)
Connecting areas:
assumed an appropriate rigid zone
Rigid beam to rail support points
Simple support
P5
Train running
P4
Z
Y
P3
P2
Simple support
Main girder： beam element
Pier: beam element
X
P1
Main tower: beam element
Pier bottom: fix
Global coordinate
Figure 3. Bridge dynamic model.
B
B

B
B
B
B
L
B

V
B
B
N
B
(2)
where X B was a displacement vector of the bridge; M B , C B
and K B were the mass, damping and stiffness matrices of the
bridge, respectively; FLB was a load vector of earthquake or
wind pressure of the bridge; FB(XV , XB) was an interaction
load vector with vehicles; FNB ( X B ) was a load vector of the
nonlinear spring force of the bridge model assumed outside
load.
2.3
2.3.1
Rail
Rail displacement zR
Relational displacement between the
wheel and the rail δz
wheel
Track
irregularity ey
Tread gradient γ
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
Wheel
jumping
Figure 4. Vertical interaction model between wheel and rail.
Interaction model between the wheel and the rail
Vertical direction
Wheel displacement zw
Hertz contact
spring
Contact
force H
Track
irregularity ez
Initial radius r
Rail displacement yR
Rail
Relational displacement between the
wheel and the rail flange δy
Flange
Rail tilting
spring constant kp
Wheel
Friction force
Creep force Qc
Flange force Qf
Wheel displacement yw
Flange force Qf

M X C X  K X  F  F ( X , X )  F ( X )
B
Wheel
Creep force Qc
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.
Rail
gap :u
Slip ratio S
gap :u
δy
Figure 5. Horizontal interaction model between wheel and rail.
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.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Mode 1 0.95Hz
Mode 2 1.04Hz
Mode 3 1.11Hz
Mode 4 1.23Hz
4.0
6.0
8.0 10.0
Time(sec)
(a) Main gerder center displacement
6
4
2
0
-2
-4
-6
0.0
2.0
1st tower
2nd tower
3rd tower
8.0
10.0
2.0
4.0
6.0
Time(sec)
(d) Tower top displacement
-10000
1st span
2nd span
3rd span
4th span
0
10000
20000
0.0
2.0
4.0
6.0
8.0 10.0
Time(sec)
(b) Main girder center bending moment
4000
2000
0
1st tower
-2000
2nd tower
-4000
3rd tower
0.0
2.0
4.0
6.0
8.0
10.0
Time(sec)
(e) Tower base bending moment
Normal tension force(kN)
3rd span
4th span
Bending moment(kN-m)
1st span
2nd span
Bending moment(kN-m)
10
0
-10
-20
-30
0.0
Bending moment(kN-m)
Rail direction
displacement(mm)
Vertical
deflection(mm)
Figure 6. Natural frequency modes.
1st en trance side
11 th entrance sid e
100
11th exit side
1st exit side
50
0
0.0
4.0
6.0
8.0 10.0
Time(sec)
(c) 2nd tower cable normal force
15000
10000
5000
0
-5000
-10000
-15000
0.0
2.0
2.0
P 2 pier
P 3 pier
8.0
10.0
P 4 pier
4.0
6.0
Time(sec)
(f) P ier bottom bending moment
Figure 7. Time history response waveforms of the bridge (single-track loading by 8-car train running at 360 km/h).
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.


Qc  C  S y  C  ( yw  r  w  v w ) / v
(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.
1206
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
2.4
(7)
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
3.1
ANALYSIS RESULTS
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
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
200
300
400
Train speed(km/h)
(a) Main girder center vertical displacement
1st tower
2nd tower
3rd tower
100
200
300
400
Train speed(km/h)
(d) Tower top rail direction displacement
0.5
0.4
0.3
0.2
0.1
0.0
Impact factor
1st span
2nd span
3rd span
4th span
200
300
400
Train speed(km/h)
(b) Main girder center bending moment
Impact factor
Impact factor
0.6
0.5
0.4
0.3
0.2
0.1
0.0
100
0.5
0.4
0.3
0.2
0.1
0.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
100
1st tower entrance/exit side sway ing
2nd tower entrance/exit side sway ing
3rd tower entrance/exit side sway ing
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Impact factor
1st span
2nd span
3rd span
4th span
Impact factor
Impact factor
0.5
0.4
0.3
0.2
0.1
0.0
100
200
300
400
Train speed(km/h)
(e) Tower base bending moment
1st entrance side/exit side
4th entrance side/exit side
8th entrance side/exit side
11th entrance side/exit side
100
200
300
400
Train speed(km/h)
(c) 2nd tower cable tension force
P2 pier entrance/exit side sway ing
P3 pier entrance/exit side sway ing
P4 pier entrance/exit side sway ing
100
200
300
400
Train speed(km/h)
(f) Pier bottom bending moment
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0.0
2.0
4.0
6.0
8.0 10.0
Time(sec)
(d) 1st car body acceleration
4.0
6.0
8.0 10.0
Time(sec)
(b) 4th car wheel load variation
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0.0
At front truck
At rear truck
2.0
4.0
6.0
8.0 10.0
Time(sec)
(e) 4th car body acceleration
Wheel load variation(%)
Reduction due to change of ang le
2.0
6
4
2
0
-2
-4
-6
0.0
1st wheelset
3rd wheelset
2.0
4.0
6.0
8.0 10.0
Time(sec)
(c) 8th car wheel load variation
2
At front truck
At rear truck
1st wheelset
3rd wheelset
Car body acceleration(m/s)
4.0
6.0
8.0 10.0
Time(sec)
(a) 1st car wheel load variation
Wheel load variation(%)
2.0
6
4
2
0
-2
-4
-6
0.0
2
1st wheelset
3rd wheelset
Car body acceleration(m/s)
6
4
2
0
-2
-4
-6
0.0
2
Car body acceleration(m/s)
Wheel load variation(%)
Figure 8. Relation between train speed and impact factor of the 12-car train case.
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0.0
At front truck
At rear truck
2.0
4.0
6.0
8.0 10.0
Time(sec)
(f) 8th car body acceleration
Figure 9. Time history response waveforms of the train (single-track loading by 8-car train running at 360 km/h).
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 seatload 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
1207
20.0
15.0
(max. load )
10.0
5.0
Lim it 37%
0.0
2
100 200 300 400
Train speed(km/h)
(a) Wheel load reduction rate
(8 car train)
1st car
1.0
3rd car
truck
5th car Single
loading
7th car (capacity load)
Wheel load reduction rate(%)
1st car
3rd car
5th car Double truck
7th car loading
Car body acceleration(m/s)
2
Car body acceleration(m/s)
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.
Wheel load reduction rate(%)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
2
0.5 Limit 2.0m /s
0.0
100 200 300 400
Train speed(km/h)
(c) Car body acceleration
(8 car train)
20.0
15.0
10.0
1st car
3rd car
5th car
7th car Double truck
9th car loading
(max. load )
12th car
5.0
Limit 37%
0.0
100 200 300 400
Train speed(km/h)
(b) Wheel load reduction rate
(12 car train)
1st car
1.0
3rd car
5th car Single truck
loading
7th car (capacity
load )
9th car
0.5
12th car
2
Lim it 2.0m/s
0.0
100 200 300 400
Train speed(km/h)
(d) Car body acceleration
(12 car train)
Figure 10. Relation between train speed and train running quality.
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 sixdegree-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,
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Dynamic analysis
Create object
DIASTARSⅡ
Bridge time
series data
・polygons
・Shading
Vehicle time
series data
Motion capture
Rendering
To capture result motions of
analysis results
To convert to visualized space
coordinates
To adjust response enlargement
ratios
Figure 11. Outline of the visualization system.
(a)Polygon model
(b)Shading model
Figure 12. Vehicle dynamic model
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
(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).
Target
非線形ばね
(ストッパ)
Video camera
10.0
5
20.0
30.0
Time(sec)
(a) 2nd span
4th test f=1.43Hz
5th test f=1.33Hz
6th test f=1.39Hz
0
-5
-10
0.0
2
10
0
10
-1
10
-2
10
-3
1
2
3
Frequency(Hz)
10
0
10
-1
10
-2
10
-3
20.0
30.0
1
2
3
Frequency(Hz)
Time(sec)
(b) 3rd span
(a) Time history response waveforms of main girder deflection
-25.0
-20.0
-15.0
-10.0
-5.0
0.0
10.0
Powerr spectrum (mm /Hz)
-5
2
0
-10
0.0
Vertical deflection(mm)
1st test f=1.31Hz
2nd test f=1.34Hz
3rd test f=1.34Hz
5
Powerr spectrum (mm /Hz)
Vertical deflection(mm)
Figure 14. Video-type displacement sensor.
Vertical deflection (mm)
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.
Analysis
(8-car train)
1st span
2nd span
3rd span
4th span
Running test
(12-car train)
2nd span
3rd span
Static Axis load =110kN
100
200
300
400
Train speed (km/h)
(b)Relation between vertical deflection and train speed
Figure 15. Validation result of the actual train running test.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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 noncontact 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.
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(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.
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