Challenge H: For an even safer and more secure railway

Risk Assessment Method for Train-Running Safety during Seismicity on Railway Line

1M. SOGABE, 1K. GOTO, 1M. TOKUNAGA and 1K. Asanuma

Railway Technical Research Institute, Tokyo, Japan1;sogabe@rtri.or.jp

Abstract: To evaluate train-running safety during seismicity on an entire railway line quantitatively, we should use risk analysis method. A lot of studies for seismic risk management were widely performed in various kinds of research fields. However, the study for fragility curve of train-running safety has not been conducted and its basic characteristics have not been obtained. Through this study, we clarified the PGA and PGV fragility curves on entire railway line based on a numerical analysis, as well as influence of train-running direction and phase difference of earthquake motions. We also investigated verification for vehicle guide device, which minimized damage when railway vehicle was derailed by a large-scale earthquake. 1. INTRODUCTION In order to examine seismic train-running safety and effects of seismic countermeasures of an entire railway line quantitatively, establishing a risk assessment system which is intended for an entire railway line and enables us to make a precise decision for investment is essential[1],[2]. For an earthquake hazard curve (which indicates year exceeding probability to earthquake ground motion index) among risk assessment components, a commercial software is on sale, and much progress is made in the other research field. On the other hand, discussions on an event tree (damage pattern classification), fragility curve of each damage pattern (probability of damage occurrence to earthquake ground motion index), and cost of each damage pattern, which are unique to railway, are not accurate enough and thought to be included in critical future agenda. Especially, fragility curve associated with seismic train-running safety of railway structures has not been studied at practical level yet, and we have not grasped a true figure of it. Against such a background, we authors have proceeded with the study on fragility curve computation. In this study, we set two goals described below. To clarify the fundamental properties of fragility curve on entire railway line in numerical experiments, specifically the effect of PGA, PGV, train-running direction and input phase difference of earthquake motion. To investigate verification for vehicle guide device on the fragility curves, that minimized damage when railway vehicle was derailed by a large-scale earthquake. 2. ANALYSIS METHOD A program called DIASTARSIII, which analyzes dynamic interaction between railway vehicles and railway structures, was used in the numerical analysis [3],[4]. 2.1 Dynamic Model of Vehicle Table 1 and Fig. 1 show a dynamic model of the vehicle. The vehicle model was created by connecting each element of a vehicle body, 2 truck frames and 4 wheelsets which were modeled at 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, bi-linear non-linear springs were used for springs. In analysis, a train is composed of several vehicles. Adequacy of these dynamic models has already been verified through vibration experiment using a vibration table and a full-scale vehicle model [5]. Tangible vehicle specifications were assumable in reference to a recent high-speed Shinkansen train vehicle. The main input data (for an empty vehicle) were 25m of a vehicle length, 32.0t of body mass, 3.0t of truck frame mass, 2.0t of wheelset mass, 200kN/m of vertical and horizontal spring constants for air-spring (half side of one truck), 25kN･s/m of damping constant for air-spring (half side of one truck), 20kN･s/m of damping constant for lateral damper (one damper of one truck), 1200kN/m of spring constant for axle spring (half side of one wheelset) and 40kN･s/m of damping constant for axle damper (half side of one wheelset). In addition, expansion gaps between each stopper were set to be 20-30mm. In this study, we used 8-car train. Equations of motion for vehicle system in vehicle coordinate system is shown as Eq. (1) after transposing

non-linear spring terms between each element to right- hand side.

)(+),(+=++ Γ

•••VV

NBVVV

LVVVVVV XFXXFFXKXCXM (1)

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

LF is load vectors of wind pressure; ),(Γ

BVV XXF is interaction load vectors with structure; )( VVN XF is load vectors of

non-linear spring force of vehicle model assumed external forces. 2.2 Dynamic Model between Wheels and Rails Figure 2 shows the vertical dynamic model between wheel and rail. The vertical relative displacement δz between the wheel and the rail can be shown as Eq. (2).

δz= zR – zW + eZ + eZ0(y) (2)

Table 1 Notations of Dynamic Model of Vehicle

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 air spring (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 x axis 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 z axis Iwz Static wheel force Ps

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

Figure 1 Vehicle Dynamic Model

Figure 3 Horizontal Dynamic Model between Wheel and Rail

Figure 2 Vertical Dynamic Model between Wheel and Rail

zψ

φ

Wheelset

Truckframe

Body φ θz

y

T

zT

yTθT

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 Body

DisplacementNon-linear spring (stopper)

Forc

e

φ

φ

Rail displacement yR

Track irregularity ey

Rail

wheelWheel displacement yw

Flange force Qf

Creep force Qc

gap :u

Wheel

Rail Slip ratio S

Cre

ep fo

rce

Qc

Friction forceTread gradient γ

δy

Flan

ge fo

rce

Qf

Rail tiltingspring constant kp

gap :uInitial radius r

Flange

Relational displacement between the wheel and the rail flange δy

Relational displacement between the wheel and the rail δz

Rail displacement zR

Track irregularity ez

Rail

WheelWheel displacement zw Co

ntac

tfo

rce

H

Hertz contact spring

δzWheel jumping

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. 2; eZ0 is the amount of change of the wheel radius at the current contact point from initial wheel radius, which is 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 are 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 quadric surface respectively, the relation between the relative displacement δ of the wheel and the rail of the contact surface normal direction and the contact force H can be shown with the Hertz contact spring, as indicated in Eq. (3).

H = H(δ) = H(δz・cos a ) (3) The vertical component and the horizontal component of this contact force H were distributed to the wheel and the rail respectively to make the interaction force. Figure 3 shows the horizontal dynamic model between wheel and rail. The horizontal relative displacement δy between the wheel flange and the rail can be shown as Eq. (4).

δy = y– u (δz) = yw – yR – ey – u (δz) (4) where, y is the horizontal relative displacement between the wheel and the rail; yR and yW are horizontal displacements at the contact point of the rail and the wheel; ey is horizontal track irregularity existing on the rail shown in Fig. 3; u(δz) is 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 acted horizontally on the contact surface of the wheel and the rail. The creep force was the tangential force caused by creep of the wheel moving forward by rolling on the rail, which can be shown as Eq. (5). This creep force was saturated with the upper limit of friction force when the creepage became high.

vψvφryCSCQ wwwyc /)-+(•=•=••

(5)

where, C is the creep coefficient; Sy is the lateral creepage; v is the train speed; r is 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 was considered. The flange pressure Qf can be shown as Eq. (6) using the rail tilting spring constant kp.

Figure 4 Dynamic Model of Wheel and Vehicle Guide Device

Rail : Beam FE

Cross-section of rail : MBS

Cross-section of track with vehicle guide device : MBS

Structure：Beam, shell, solid FE

Track with vehicle guide device : Beam FE

Fastening system : Spring FE

Node

(a) Rigid Cross-Sectional MBS Model

Guide device

Frictional force

Impa

ct fo

rce

& d

ampi

ng

Impact force & damping

(b)Fall on track surface (c)Collision to guide

(d)Fall on guide (e)Collision to rail

Box approximation of rail in post-derailment

Impa

ct fo

rce

& d

ampi

ng Impact force& damping

※Impact force between wheels and track was computed using penalty function methodExternal

guide Internalguide

Rail

Wheel

Conical trapezoid approximation of wheel in post-derailment

δ•= ypf kQ (6)

Figure 4 shows a dynamic model of each wheels and vehicle guide device, which minimized damage when railway vehicle was derailed by a large-scale earthquake. Interaction force between wheels and rails after derailment was computed by modeling the track structure including wheels and vehicle guide devices with Multi Body System (MBS). The post-derailment wheels of the used model were approximated to a divisional straight line (conical trapezoid) to speed up the analysis. A “Rigid Cross-Sectional MBS Model” with internal and external vehicle guide devices was employed for the track structure. Collision of wheels and track structure members was expressed by a non-linear collision spring. Ladder-type sleepers with a 235mm vehicle guide device (external guard type) were used as an example for analysis in this study. Both of the two dynamic models explained above were used separately as running condition (derail or not) with each wheelset. 2.3 Dynamic Model of Structure Figure 5 shows the analysis system for the entire continuous structure group. This system picks up an analysis section automatically and creates an analysis model based on the structure database and analysis conditions, as shown in Fig. 5(a). After the analysis is done, the system integrates the analysis results and evaluates the entire railway line. Structure arrangement of model line shown in Fig. 5(b) was determined based on the statistical analysis of the actual structures. In evaluating the entire railway line, loading vehicles on the entire railway and analyzing them is ideal, however; it gives structures and vehicles an extensive degree of freedom. Thus, a method to divide an 8-car train into 1 sub case, as shown in Fig. 5(b) was suggested. In this method, an extracted section of each sub case was calculated based on dominance time (time when there is the strongest power to cause derailment) predetermined for each earthquake ground motion. For example, in sub case 11, necessary extraction section has distance from 1700 to 2400m. The extracted section was modeled by Finite Elements (FE) with automatic mesh. A pre-run section and post-run section were provided as semi-infinite running section, based on average vibration characteristics of structures in the vicinity of the extracted section. Thus, accuracy of the train-running safety evaluation unit is the interval of 25m, which is the length of one car, from the viewpoint of vehicles, and from the viewpoint of the wheel axis, it is the interval of wheel axis.

(c) FE model of extracted section (d) 2D analysis model for static non-linear model

Figure 5 Dynamic Model of Structure

010

20

0 1000 2000 3000 4000 5000 6000010

20

0.10.2 1.02.0200

1000

2000

L2Sp

c.IG

2

Acceleration (gal)

L2Sp

c.IIG

2

Tim

e (s

ec)

Distance (m)

500

-500

0

500

-500

0

1 11 21 31

Tim

e (s

ec)

Extraction section

8-car train

Structure hight

Sub case

Structure data baseTrain speed 　Vehicle numbers of train 　Earthquake motions

Analysis conditions

Elas

tic re

spon

se a

ccel

erat

ion

(gal

)

Equivalent period (sec)

L2Spc.IIG2

L2Spc.IG2

Distance 　Block length 　Block height 　Equivalent period 　Yield strength and displacement 　Maximum strength and displacement

Automatic division and extraction sectionAutomatic meshDivided analysis executionAutomatic composition of results

Analysis

Dom

inan

ce ti

me

Train position of each sub case

(a) Flowchart of entire system

(b) Relation between train-running position and earthquake motion

Rigid frame viaduct: rigid beam FE

Ground: (mass FE)Acceleration input

Column：Horizontal & yawing non-linear springs

Adjustment girder: rigid beam FE

M M

Over bridge: rigid beam FE

M

Rigid frame viaduct 3@8m Adjustment girder 8m

M M

Over bridge 30mRigid frame pier 3@8m

Columns:non-linear spring (M-θ)

Lateral beam :non-linear element (M-φ)

Vertical (pile bottom):non-linear spring

Horizontal spring：K =Σki(δ)Yawing spring：Kr =Σki (δ)・Li

2

ki

Li

DisplacementδLoad

P

Horizonral:non-linear spring

Vertical (pile surrounding):non-linear spring

Pile :non-linear element (M-φ)

Seismic force Yeild Max. bear capacity

In this study, the target of our analysis was a 6.3km-long double track viaduct. We set a three-span rigid-frame viaduct of adjustment girder type (block length of 24m) as a fundamental unit of the structures. Figure 5(c) shows a specific FE model of the extracted section. Each block of the rigid frame and adjustment girder is modeled as a rigid beam element. The mass and the non-linear property of each structure are determined based on the design calculation for the actual structure (an item to input in the database). Using a 2D analysis model, a static non-linear push-over analysis is conducted on the cross-section of each column, and the relation between the load and displacement is researched [6]. Horizontal and yawing springs on the center of gravity are computed from these obtained skeleton lines as shown in Fig. 5(d). A hysteresis model of the horizontal non-linear spring is a standard tri-linear. The damping ratio ξ of the structures were fixed to 5%. Equations of motion for structure system is shown as Eq. (7)

)(+),(+=++ Γ

•••BB

NBVBB

LBBBBBB XFXXFFXKXCXM (7)

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

LF is a load vector of earthquake or wind pressure of bridge; ),(ΓBVB XXF is an interaction

load vector with vehicles; )( BBN XF is a load vector of non-linear spring force of bridge model assumed outside

load. As stipulated in Design Standards for Railway Structures and Commentary (for Seismic Design), L2 spectrum I of ocean-trench type and L2 spectrum II of inland active faults type (level 2 means large scale earthquake) were used for input earthquake ground motion [6]. The simulated surface ground was G2 (diluvium: 0.25 and shorter ground natural period Tg). Since the vehicle response had strong non-linearity, seismic peak ground acceleration (PGA) was gradually and linearly increased in the study. Figure 6 shows the setting method of seismic phase difference. The phase difference was simply computed by difference of shear wave transmission pathway as shown in Eq. (8).

Δti = xi / Vs = Lbi ･ sinφ / Vs (8) where, Δti is arrival time difference between structure i and i -1; xi is pathway difference of shear wave transmission between structure i and i -1; Vs is shear wave velocity in the bedrock surface in seismic design; φ is incident angle. In the setting, Vs=400m/s and φ=30° were assumed. The phase difference between each block was 0.04 (s). 2.4 Numerical Method Equations of motion of the train and stracture shown as Eq. (1) and (7) were solved in the modal coordinates for each time increment Δt by the Newmark time difference scheme. Since the equations were non-linear, iterative calculations were necessary during each time increment until the unbalanced force between the train and viaduct became sufficiently small within the specified tolerance. 2.5 Evaluation Criterion Figure 7 shows the derailment mode and the criterion of train-running safety during seismicity. The criterion used the wheel horizontal displacement in accordance with the Design Standards and Interpretation of Railway Structures (for Displacement Limits). In addition, the limit value was set to be 70mm [7]. As the wheel horizontal displacement rapidly increase when the wheel rise exceeds the flange height (30mm), the wheel rise was also as shown in Fig. 7. The vehicle derailment modes during an earthquake were classified broadly into upper and lower center rolling which were distinguished at less than 0.7 Hz and more than 1.3 Hz respectively. However, this was determined by phases of horizontal and rolling movements of the vehicle shown in Fig.7 (refer to Fig. 1 for the coordinate system) .

Figure 6 Time Difference of Shear Wave Figure 7 Derailment Mode and Criterion of Train-running Safety

Incidentangleφ

Lbi

xi

Surface ground

Bedrock surfacein seismic design

⊿ti=xi/Vs=ΣLbi・sinφ/Vs

Wheel horizontaldisplacement

Wheel

Rail

Whe

el ri

se

Roll angle φ

Horizontaldisplacement y

Wheel horizontal displacement

Rail

TruckframeWheelset

Body

3. ANALYSIS RESULT 3.1 Time series waveforms Figure 8 shows an example of time series waveforms when a derailment occurs. A response of the L2 spectrum I of ocean-trench type with PGA=240gal is shown. The rigid frame viaduct and pier turned out to have 50mm relative displacement at 7 seconds. It was discovered that the structural horizontal displacement directly under the wheel axis vibrated at 2Hz and total amplitude 80mm in around 7 seconds. Wheels started rising on the adjustable girders between the rigid frame viaduct and pier, and derailment occurred when wheel horizontal displacement exceeded 70mm. The derailed wheels collided with the vehicle guide device in 9.7 seconds. The

Figure 8 Example of Time Series Waveforms (L2 spectrum I of ocean-trench type with PGA=240gal)

Figure 9 Specifications of Entire Model Line

Figure 10 Maximum Response of Structure and Train (L2 spectrum I of ocean-trench type with PGA=240gal)

Figure 11 Limits of Peak Ground Acceleration PGAL for Train-running Safety

4 5 6 7 8 9 10 11 12-100

0

100

-500

0

500

4 5 6 7 8 9 10 11 12-200

0

200

-100

0

100

4 5 6 7 8 9 10 11 12-200-100

0100200

4 5 6 7 8 9 10 11 12-200-100

0100200

2nd wheelset

Wheel rise

Whe

el h

oriz

onta

l di

spla

cem

ent (

mm

)

Struck to vehicle guide device

wheel riseDerailment

11.0

11.0

11.5

12.0

12.0

12.0

10.5

16.0

12.5

12.0

12.0

11.5

11.5

11.0

11.0

11.0

Numbers：Hight (m)

Adjustment girder

Whe

el ri

se (m

m)

Wheel horizontal displacement

Body horizontal displacement

Rol

ling

angl

e (m

rad)

H

oriz

onta

l di

spla

cem

ent (

mm

)

Body rolling angle

Time (sec)

Upper center rolling

Rigid frame viaduct ( h=10.5m)Rigid frame pier ( h=16.0m)

Derailment

Dis

plac

emen

t dire

ctly

un

der w

heel

set (

mm

)D

ispl

acem

ent (

mm

)

Time (sec)

Rigid frame pier ( h=16.0m)Rigid frame viaduct(h=10.5m)

5

10

0 1000 2000 3000 4000 5000 6000

0.5

1.0

0.5

1.0

1st r

iver

brid

ge

2nd

river

brid

ge

1st o

ver b

ridge

2nd

over

brid

ge

3rd

over

brid

ge

4th

over

brid

ge

3rd

river

brid

ge

Dist ance (m)

Stru

ctur

e hi

ght h

(m)

Yei

ld se

ismic

co

effic

ient

kh

Equi

vale

nt p

erio

d

T eq(

sec)

Yeild seismic coefficient khEquivalent period Teq

4th

river

brid

ge0

100

200

300

0

10

20

30

0 1000 2000 3000 4000 5000 60000

100

200

0

100

200

Distance (m)

H

oriz

onta

l di

spla

cem

ent (

mm

)

A

ngul

ar

rota

tion

(mra

d)

Whe

el ri

se (m

m)

Whe

el h

oriz

onta

l di

spla

cem

ent (

mm

)

Horizontal displacementAngular rotat ion

Wheel riseWheel horizontal displacement

Rise

Displacement

0 1000 2000 3000 4000 5000 60000

200

400

600

Distance (m)

PGA L

(gal

)

L2 spect rum I of ocean−t rench typeL2 spect rum II of inland act ive fault s type

vehicle horizontal displacement and vehicle roll angle were in the same phase at the time of derailment occurrence, and it was assessed that upper center rolling resulted in derailment. 3.2 Basic Characteristics of Train-Running Safety Figure 9 and Figure 10 show specifications of the entire model line and maximum response of analysis results respectively. Structure arrangement of model line was determined based on the statistical analysis of the actual structures as described previously. A response of the L2 spectrum I of ocean-trench type with PGA=240gal is shown. The figure shows the maximum value of the vehicle response in each one vehicle at the running position of 8 seconds for the ocean trench type, and 6 seconds for the inland active faults type. Horizontal movement of wheels which exceeds 200mm is expressed as 200mm. Equivalent natural period of a high pier river bridge is longer than that of rigid frame viaduct, and the river bridge tends to have larger displacement. On the other hand,

Figure 12 Fragility Curve of Seismic Train-Running Safety of Modeled Railway Line

Figure 13 Effects of Vehicle Running Direction on Fragility Curve

Figure 14 Effects of Earthquake Motion Phase Difference on Fragility Curve

0 500 10000

0.5

10 50 100 150

0.0

1.0

2.0

3.0

0 500 10000

0.5

10 50 100 150 200 250

0.0

1.0

2.0

3.0

Expe

cted

dam

age

num

ber o

f car

Peak Ground Acceleration PGA (gal)Dam

age

occu

rren

ce p

roba

bilit

y144 of 16-car train a day : 1 train each 360 sTrain interval : 27000m 　Model line length : 6300mTrain speed : 270km/h(75m/s)

Vehicle existing probability in 25m evaluation unit : 0.0148Expected value of number of running car on model line : 3.674

(a) L2 spectrum I of ocean-trench type

Peak Ground Velocity PGV (kine)

Peak Ground Acceleration PGA (gal)(b) L2 spectrum II of inland active faults type

Safety 　　Derailment

Peak Ground Velocity PGV (kine)

Deviation

Computation assumption of fagility curve Wheel horizontal displacement = 70mmOverriding device (Height = 135mm)

Wheel horizontal disp. Vehicle guidedevice overriding

Original peak = 400gal

Original Peak = 870 gal

Overriding device (Height = 235mm)

Evaluation unit : 25m of vehicle length Occupation time : 0.333 sDeviceHight

PGA=325ζ =0.312R2=0.99

Safety 　　Derailment　　Deviation

Eq.(9)PGA=416ζ =0.265R2=1.00

Eq.(9)

Dam

age

occu

rren

ce p

roba

bilit

y

Expe

cted

dam

age

num

ber o

f car

0 500 10000.0

0.5

1.00 50 100 150

0.0

1.0

0 500 10000.0

0.5

1.00 50 100 150 200 250

0.0

1.0

Expe

cted

dam

age

num

ber o

f car

Peak Ground Acceleration PGA (gal)Dam

age

occu

rren

ce p

roba

bilit

y

(a) L2 spectrum I of ocean-trench type

Peak Ground Velocity PGV (kine)

Expe

cted

dam

age

num

ber o

f car

Peak Ground Acceleration PGA (gal)(b) L2 spectrum II of inland active faults type

Safety 　　Derailment

Peak Ground Velocity PGV (kine)

Deviation

Wheel horizontal displacement = 70mmD

amag

e oc

curr

ence

pro

babi

lity

Original Peak = 400gal

Original Peak = 870gal

Overriding Device (Height = 235mm)

Safety 　　Derailment　　　　Deviation

PGA=416ζ =0.250R2=1.00

Eq.(9)PGA=332ζ =0.275R2=0.99

Eq.(9)

Down Up

0 500 10000.0

0.5

1.00 50 100 150

0.0

1.0

2.0

3.0

0 500 10000.0

0.5

1.00 50 100 150 200 250

0.0

1.0

2.0

3.0

Expe

cted

dam

age

num

ber o

f car

Peak Ground Acceleration PGA (gal)Dam

age

occu

rren

ce p

roba

bilit

y

(a) L2 spectrum I of ocean-trench type

Peak Ground Velocity PGV (kine)

Peak Ground Acceleration PGA (gal)(b) L2 spectrum II of inland active faults type

Safety 　Derailment

Peak Ground Velocity PGV (kine)

Deviation

Wheel horizontal displacement = 70mm

Original peak = 400gal

Original peak = 870gal

Overriding device (Height = 235mm)

Safety 　　Derailment 　Deviation

PGA=477ζ =0.312R2=0.99

Eq.(9)PGA=333ζ =0.312R2=0.99

Eq.(9)PGA=222ζ =0.259R2=0.99

Eq.(9)

PGA=307ζ =0.178R2=0.98

Eq.(9)

None　Positive　Negative　 Phase difference

Dam

age

occu

rren

ce p

roba

bilit

y

Expe

cted

dam

age

num

ber o

f car

there is a tendency an over bridge has a shorter equivalent natural period and smaller displacement than the rigid frame viaduct do. Since angular rotations are likely to occur on the border of such structures, wheel rise and horizontal displacements were inclined to increase. Figure 11 shows the limit of peak ground acceleration PGAL, which contains horizontal displacement of wheel exceeding 70mm. The L2 spectrum I of ocean-trench type, which has more frequent repeats of equivalent waves, has more severe PGAL than L2 spectrum II of inland active faults type. It indicates that PGAL is apt to decrease at the point of vibration property change in the railway line. 3.3 Fragility Curves of Train-Running Safety Figure 12 shows a fragility curve of seismic train-running safety of the modeled railway line. Based on the assumption in the figure, numerical analysis results shown in Fig. 11 were organized. In Fig.11, we performed running safety evaluation aimed at 248 of vehicle running position (evaluation unit was 25m length of vehicle). In Fig. 12, we firstly computed vehicle existing probability in 25m evaluation unit (about 1.5%). Probabilities of damage occurrence were computed by counting derailment or deviation vehicles, increasing seismic peak ground acceleration (PGA) gradually and linearly. Each plot in Fig.12 becomes the fragility curves which separate safety, derailment and deviation zone respectively. In addition, we show the seismic peak ground velocity (PGV) as 2nd horizontal axis in Fig.12. In the risk assessment, the fragility curves are generally expressed by the lognormal cumulative distribution function as shown in Eq. (9).

ζPGAPGA

PGAPdr)ln(

=)(-

Φ (9)

where, Pdr(PGA) is probability of damage occurrence; Φ is standard normal cumulative distribution function; PGA is average PGA of damage occurrence; ζ is the standard deviation of lognormal distribution function. The standard deviation of lognormal distribution function ζ became 0.32 for the L2 spectrum I of ocean-trench type, and 0.26 for L2 spectrum II of inland active faults type.

Figure 15 Effects of track surface shape difference on the fragility curve

Figure 16 Example of time series waveforms when wheelset derail, fall and run on track surface.

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The ladder type sleepers with vehicle guide device were found out to have about 10% of deviation damage occurrence rate to original wave of each earthquake ground motion and to be very effective to large scale earthquake motions. The figure also implies that lowering the height of device (height 135mm) reduces the effect of deviation prevention performance. Figure 13 indicates effects of a vehicle running direction on the fragility curve. The analysis target model did not show any effects of a vehicle running direction on the fragility curve. Figure 14 indicates effects of earthquake motion phase difference on the fragility curve. The earthquake motion transmitted in the same direction as vehicles running is defined as positive, while that in the opposite direction is defined as negative. As seen in the figure, the train-running safety degrades when the earthquake motion is transmitted to the opposite direction of the vehicle running. This is because the time axis of the earthquake motion is compressed apparently due to train running. The degradation is consistent with the tendency of safety limit curve by sinusoidal wave vibration, that is, higher vibration frequency with the same amplitude increases risk of derailment. Figure 15 indicates effects of track surface shape difference on the fragility curve. When wheelsets fall and run on track surface in Fig. 4(b), impact and damping force between wheel and track surface are affected by the track surface shape such as ballasted track with cross sleepers. In order to consider this, we defined vertical shape as function of rail direction position on track surface in Fig. 4(b). The figure implies that track surface shape of ballasted track using type 4H PC cross sleepers changes deviation prevention performance slightly. Figure 16 shows an example of time series waveforms when wheelset derail, fall and run on track surface. The case of ballasted track using cross sleepers tends to have larger wheel jumping rise on track surface than the case of concrete flat shape surface such as slab or ladder track. The maximum impact force between a wheel and a cross sleeper surface is 1500 kN. The type 4H PC cross sleepers reaches flexural failure limit when impact force is 900kN in impact experiment using a wheel shape drop weight and actual type 4H PC cross sleepers. Thus, the sleeper which wheel contact with first will be broken by the impact force. On the other hand, the other sleepers have enough bearing capacity for flexural failure. We used 150kN/mm of contact spring between wheel and track surface as standard case based on above-mentioned impact experiment using a wheel shape drop weight and actual type 4H PC cross sleepers. As seen in the figure, the case 400kN/mm of contact spring increases wheel jumping height. In the case of ballasted track using sleepers, contact spring value has a great influence on jumping height. Thus, we will conduct more detailed static and impact experiment for this problem. 4. CONCLUSIONS (1)We clarified the basic characteristics of the fragility curves of train-running safety during seismicity on railway

viaduct group. The standard deviation of lognormal distribution function ζ became 0.32 for the L2 spectrum I of ocean-trench type, and 0.26 for L2 spectrum II of inland active faults type.

(2)The analysis target model did not show any effects of a vehicle running direction on the fragility curve. (3)The train-running safety degrades when the earthquake motion is transmitted to the opposite direction of the

vehicle running. (4)We clarified that vehicle guide device were very effective to large scale earthquake motions, and higher

vehicle guide devices increased the effect of deviation prevention performance. References: [1] Sogabe, M., Harada, K. and Watanabe, T.: “Vehicle Running Quality during Earthquake on Railway Viaducts,” Proceedings

of the International Symposium on Speed-up, Safety and Service Technology for Railway and Maglev System 2009 (STECH’09), CD-ROM No.360713, (2009).

[2] Miyamoto, T., Sogabe, M., and Murono, Y.: “Dynamic Simulation of Shinkansen Cars in Seismic Motion during the Niigataken Chuetsu Earthquake,” Proceedings of the International Symposium on Speed-up, Safety and Service Technology for Railway and Maglev System 2009 (STECH’09), CD-ROM No.362681, (2009).

[3] Sogabe, M., Matsumoto, N., Kanamori, M., Sato, T. and Wakui, H.: “Impact Factors of Concrete Girders Coping with Train Speed-up,” Quarterly Report of RTRI, Vol. 46, No. 1, 46-52, (2005).

[4] Sogabe, M., Furukawa, A., Shimomura, T., Iida, T., Matsumoto, N. and Wakui, H.: “Deflection Limits of Structures for Train Speed-up,” Quarterly Report of RTRI, Vol. 46, No. 2, 130-136, (2005).

[5] Miyamoto, M., Matsumoto, N., Sogabe, M., Shimomura, T., Nishiyama, Y. and Matsuo, M.: “Railway Vehicle Dynamic Behavior against Large-Amplitude Track Vibration – A Full-scale Experiment and Numerical Simulation,” Quarterly Report

of RTRI, Vol. 45, No. 3, 111-115, (2004). [6] Railway Technical Research Institute: “Design Standard for Railway Structures (for Seismic Design),” Maruzen, (1999) , (in

Japanese). [7] Railway Technical Research Institute : “Design Standard for Railway Structures (for Displacement Limits),” Maruzen,

(2006) , (in Japanese). [8] Goto, K., Sogabe, M., Asanuma, K. and Watanabe, T.: “Study on Evaluation of Contact Force between a Train Wheel and

Prestressed Concrete Sleeper,” Proceedings of the Japan Concrete Institute, Vol. 31, No. 2, pp.769-774, (2010), (in Japanese).

[9] Brabie, D.: “Wheel-Sleeper Impact Model in Rail Vehicles Analysis,” Journal of System Design and Dynamics, Vol.1, No.3. pp.468-480, (2007).