on the safety of high-speed trains running on bridges...

the 42nd Risk, Hazard & Uncertainty Workshop, 22-25 June 2016 Hydra Island, Greece

on the safety of high-speed trains

running on bridges during earthquakes

Qing ZENG

Elias G. DIMITRAKOPOULOS


motivation

proposed model

results – frequent earthquakes

results – rare earthquakes

conclusions

2

outline


motivation

proposed model



conclusions

3

outline


contemporary high-speed railways (HSR’s)

China’s HSR

• operational speed: 300-350 km/h

• total HSR length: 16, 000 km (to 28/12/2014) • longest bridge : 164 km (over 4000 bridges)

• bridge/line ratio :

Legend

high-speed railway lines

east-west lines

north-south lines

Beijing-Harbin

Shanghai-Chengdu

Xuzhou-Lanzhou

Qingdao-Taiyuan

Shanghai-Kunming

Shanghai-Shenzhen

Beijing-Hong Kong

Beijing-Shanghai

4

Beijing-Shanghai, China 80.5%

Taipei-Kaohsiung, Taiwan 73%

Beijing-Tianjin, China 86.6%

motivation


5

VBI: bridge resonance and cancellation

0 2 4 6 8 10-0.2

-0.1

0

0.1

0.2

0 2 4 6 8 10-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10-4

-3

-2

-1

0

1

0 2 4 6 8 10-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10-10

5

0

5

0 2 4 6 8 10-1

-0.5

0

0.5

1

1.5

● ten identical 3D vehicles ● single span curved simple bridge ● u: displacement; a: acceleration ● V: vehicle, B: bridge ● v: vertical, r: radial

aV

v (m

/s2)

uB

r (m

m)

dimensionless time (vt/L) dimensionless time (vt/L)

(c)

(a) (b)

aV

v (m

/s2)

(d)

v = 314 km/h

v = 458 km/h

vertical cancellation

radial cancellation

vertical resonance

radial resonance

5th car 9th car

1st car

vertical resonance

radial resonance

v = 314 km/h

v = 458 km/h

v = 391 km/h

v = 268 km/h

uB

r (m

m)

uB

v (m

m)

(e) (f)

R=5000 m

v

LB=32 m

uBv

uBr

aVvaVr

NV=10

uB

v (m

m)

5

motivation

vertical resonance radial resonance

vert. cancellation

radial cancellation


Chūetsu Earthquake, Japan (Mw = 6.8 23 Oct 2004) • Shinkansen railway, 200 km/h • broke the 48-year safety record

accidents: derailment of trains during earthquakes

Jiashian earthquake, southern Taiwan (Mw = 6.4 04 Mar 2010) • 300 km/h high-speed train

6

motivation

from VBI to SVBI: Seismic Vehicle-Bridge Interaction


motivation

proposed model



conclusions

7

outline


modelling approach

8

railway bridges

http://blog.sina.com.cn/s/blog_4d4c5f2b01014f60.html (2014)

vehicle

bridge

interaction

proposed SVBI model

multibody dynamics

finite element method

nonsmooth dynamics

http://blog.sina.com.cn/s/blog_4d4c5f2b01014f60.html

http://blog.sina.com.cn/s/blog_4d4c5f2b01014f60.html


brief history

current trend • beyond moving load analsis • vehicle-bridge interaction

analysis YB Yang (1995)

• interdisciplinary civil + mechanical engineering

vehicle modelling

9

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

moving loads

moving masses

moving sprung masses car body

wheelset

bogie


railway train vehicle

multibody assembly

• wheelsets, bogies, car-body → rigid bodies

• suspension → springs + dashpots

• e.g. 38 total DOF’s

Zti

Xti

XtiYti

car-body

wheelset

bogie

(a)

(c)

speed v

Yti

Zti

z

s

y

(rolling)

(lateral)

(vertical)

(pitching)

(yawing)ψ

φ

θ

(b)

(d)

(longitudinal)

(longitudinal)

(vertical)

(pitching)

(longitudinal)

(vertical)

(yawing)

(lateral)

(vertical)(rolling)

terminology

terminology

Zti

Xti

XtiYti

car-body

wheelset

bogie

(a)

(c)

speed v

Yti

Zti

z

s

y

(rolling)

(lateral)

(vertical)

(pitching)

(yawing)ψ

φ

θ

(b)

(d)

(longitudinal)

(longitudinal)

(vertical)

(pitching)

(longitudinal)

(vertical)

(yawing)

(lateral)

(vertical)(rolling)

terminology

vehicle modelling

10


equation of motion

Yti

Xtiwheel contact point i

ith beam element

Li

i-1 th beam element

i+1 th beam element

si=vtξ

speed v

• expressed in the inertia global system • direction matrices WB

N and WBT

e.g. for a beam element • linear shape functions for the axial

(longitudinal) and torsional DOF’s • cubic (Hermitian) shape functions for the

flexural DOF’s

• as vehicle wheel moves, different bridge element contact with wheel

• WBN and WB

T : time-dependent

• the only non-zero entries in WBN and WB

T →elements in contact with the wheel

B B B B B B B B B

N N H T M u C u K u W λ W λ F

bridge subsystem

11


railway train vehicle

multibody assembly

• wheelsets, bogies, car-body → rigid bodies

• suspension → springs + dashpots

vehicle modelling

12

(a) first lateral-rolling mode (b) second lateral-rolling mode (c) vertical mode

(d) yawing mode

(e) pitching mode

fV1 = 0.54 Hz fV

2 = 0.78 Hz fV3 = 0.90 Hz

fV4 = 1.43 Hz

fV5 = 1.60 Hz


Y

X

Z Yti

Xti

Zti

O

Oti

Yir

Xir

Oir

yir

zir

ψir

φir

θir

si=vt

space-fixed system I

trajectory system TI

body-fixed system IR

Rti

Ri

Rir

Zir

vehicle-dynamics

3 systems of reference (Shabana 2010) • inertial (space-fixed) system I • body-fixed system IR • moving trajectory system TI

moving trajectory system TI (for curved paths) • moves along the curve • longitudinal direction OtiXti : tangent to the curve • origin and orientation : defined by the arc length, si

vehicle modelling

13


equation of motion on a curved path

vehicle subsystem

14

05

10

15

20

25

30

-4-3-2-10123

Yti

Xti

Zti

Oti

Zir

Xir

Yir

Oir

yir

zir

ψir

φir

θir

si=vtmoving trajectory

system TI

body-fixed system IR

Z

X

Y

O

space-fixed(inertial) system I

ground-fixed system IG

ZG

XG

YG

OG

ground acceleration

irI OOr

I O OG tir

GI OOr

I O Oti irr

I O OG irrGI OO

r

T T

T T

T

G

i i i i i i i

I I IR IRIR

i i i i i i i i i i

v I R IR IR IR IR IR IRI

i i i

G I I OO

t t m t t t

m t t

m t

M L L H I H

F L γ H I γ ω I ω

F L r


( )V V V V V V V V V

N N T Tt M u C u K u W λ W λ F

• expressed in the moving trajectory system • MV(t) → time varying for curved path

• FV = gravity, inertia (centrifugal and coriolis forces) due to the curved path seismic loads if considered

equation of motion

FG

Fcϕ+ϕh

Fccosϕ

FGcosϕ

FGsinϕ

Y

Z

he1

e2

Δh

ϕ

φh

(a)

(b)yH

λN2 λN1

λT1

λT1

λN1λN2

wheelset

rail

bridge deck

• λN and λT = normal and tangential contact force vector • WV

N and WVT = direction matrices of λN and λT

vehicle subsystem

15


equation of motion

coupled vehicle-bridge system

16

Mu Cu Ku Wλ F

, ,

, , = ,

, ,

,

V V

B B

V V V

B B B

N

N H

T

V V

T N

T NB B

T N

t

t t

M 0 C 0M C

0 M 0 C

K 0 u FK u F

0 K u F

λW W W λ

λ

W WW W

W W

• global matrices • contact forces λ

FG

Fcϕ+ϕh

Fccosϕ

FGcosϕ

FGsinϕ

Y

Z

he1

e2

Δh

ϕ

φh

(a)

(b)yH

λN2 λN1

λT1

λT1

λN1λN2

wheelset

rail

bridge deck

W contains the coupling is time-dependent

contact forces (coupling)


wheel-rail contact interaction

point and direction of contact • continuous contact the wheel is in contact with the rail → nonlinear wheel, rail profile


Mu Cu Ku Wλ F

wheelsetrail

λN7

λN8

λT8 λT7

dL

rL rR

δL δR

-850 -800 -750 -700 -650-100

-50

0

lateral [mm]

vert

ical

[m

m]

tangential direction

wheel/rail

normal direction

-40 -20 0 20 40-60

-40

-20

0-100 -50 0 50 100-40

-20

0

rail head

flange

thread



point and direction of contact

• wheel-rail separation/ detachment (uplifting)

• single/double separation


Mu Cu Ku Wλ F

660 680 700 720 740 760 780 800 820-50

0

50

gN is the minimum distance



normal direction • continuous contact vs detachment (separation)

• impact and recontact


Mu Cu Ku Wλ F

contact-detachment transition is formulated as a Linear Complimentary Problem (LCP) nonsmooth approach • contact: zero contact acceleration • detachment: zero normal contact force

• Newton’s law • normal contact displacement 𝑔𝑁 = 0 and

(𝑔 𝑁 < 0) • velocity jump on the velocity level

0 0 continuous contact

0 0 detachment

N N

N N

g

g

continuous

0

detachement

contact

λN

· · gN

- kinematic constraint

(c) normal contact model

nonsmooth dynamics



tangential direction • creep lateral / longitudinal forces • creep spin moment • rolling contact


Mu Cu Ku Wλ F

zir(vertical)(yawing)ψir

(rol

ling)

si

φ

ir

Oir

λTx2

λTy2

λMz2

λTx1λTy1

λMz1

y ir

θ ir(lateral)(pitching)

Yti XtiZti

Oti



tangential direction • creep lateral / longitudinal forces • creep spin moment


Mu Cu Ku Wλ F

zir(vertical)(yawing)ψir

(rol

ling)

si

φ

ir

Oir

λTx2

λTy2

λMz2

λTx1λTy1

λMz1

y ir

θ ir(lateral)(pitching)

Yti XtiZti

Oti

wheel-rail friction model

nonlinear relationship between creep forces

and creepage


rail irregularities creep forces normal contact forces

* * * *t t t t t t t M u C u K u F

final equations of motion

• M* → time-dependent (for curved path) • K* and C* → time-dependent and coupled • F* → time-dependent

• E → identity matrix; • G(t)-1 = (WTM-1W) -1 → the effective mass during contact • ()' and ()'' → the derivatives with respect to the arc length


22

* 1 T 1 1 T

* 1 T 1 2 1 T

* 1 T 1 2 1

2

N NN N N NN N

N NN N N NN N

N NN N NN cN

v

v

v

C E W G W M C C W G W

K E W G W M K W G W

F E W G W M F F WG r

effect of:


final equations of motion

23


sources of excitation • external excitation: earthquake ground motion, wind loading etc.

• Coriolis and centrifugal forces: curved bridges and/or curved rails

• self-excitation : wheel-hunting, creep forces, rail irregularities

2la

rw

AH

2sinH H H

H

vty A

L

φh

yH

wheelset

rail

v

1

2 cos( )cN

i

c n n n

n

r A s

(a)

(b) (c)

car body

wheelset

bogie

λN7 λN5 λN3 λN1

λN8 λN7λT4

λT1λT2λT4 λT3

5

6

7

8

1

2

3

4

λN7 λN5 λN3 λN1

Yti

Xti

Xti

Zti

Zti

Yti

(d)

y

x

z(rolling)

(lateral)

(vertical)

(pitching)

(yawing)ψ

ϕθ

speed v

Node 1si = vt

Li

Node 2

Yti

Xti

Zti

elevationirregularities rcN

ith element

speed v

contact point i

* * * *t t t t t t t M u C u K u F


finite element software

multibody software

in-house algorithm

time-history solutions

24

proposed analysis platform

pre-processing ANSYS → bridge FEM model (MB, CB and KB ) Workbench → vehicle multibody model (MV(t), CV and KV )

post-processing deformed shapes, member forces diagrams ANSYS → bridge Workbench → vehicle

processing system global matrices

(mass/stiffness/damping)

M*, C* and K*

perform VBI analysis in MATLAB →

time-history solutions


motivation

proposed model



conclusions

25

outline


straight continuous highway bridge interacting with trucks during earthq. (2D)

curved continuous railway bridge interacting with train during earthq. (3D)

straight steel arch truss railway bridge interacting with train no earthq. (3D)

A1 P1 P2 A2

straight continuous slab highway bridge interacting with trucks no earthq. (3D)

straight cable-stayed bridge interacting with train no earthq. (3D)

26

case studies


27

horizontally curved continuous bridge

A0 A5P1 P2 P3 P4(a)36 m 45 m 45 m 45 m 36 m

10.38 m

24.99 m 25.97 m 13.80 m

Y

XO

(b)

R = 750 m

ZX

O

railway double track

27

SVBI frequent earthquakes


deformation animation of the whole bridge

top view

elevation view

• 10 passing vehicles, without earthquake


28


animation/curved bridge VBI.avi


deformation animation of the whole bridge

• 10 passing vehicles, with earthquake


29


top view

elevation view

animation/curved bridge earthquake.avi


deformation animation of the vehicle

front view

• 10 passing vehicles, with earthquake


elevation view

30


3D view

animation/curved bridge vehicle earthquake.avi


0 2 4 6 8 10 12 14-0.6

-0.4

-0.2

0

0.2

0 2 4 6 8 10 12 14

0

1

2

2.5

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

0.2

0.3

uB

v (

mm

)

uB

r (m

m)

time (s)

(a) (b)

● VBI model ● no earthquake excitation ● u: displacement ● a: acceleration

aV

r (m

/s2)

aV

v (

m/s

2)

(c) (d)

time (s)

first vehicle

first vehicle

effective

uncracked

effective stiffness of piers


uBvuBr

aVvaVr

L=207 m

R=750 m

v = 120 km/h

10 vehicles

• vertical and radial disp. of bridge midpoint • vertical and radial accel. of vehicle car-body

0 2 4 6 8 10 12 14-0.6

-0.4

-0.2

0

0.2

0 2 4 6 8 10 12 14

0

1

2

2.5

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

0.2

0.3

uB

v (

mm

)

uB

r (m

m)

time (s)

(a) (b)


aV

r (m

/s2)

aV

v (

m/s

2)

(c) (d)

time (s)

first vehicle

first vehicle

effective

uncracked



uBvuBr

aVvaVr

L=207 m

R=750 m

v = 120 km/h

10 vehicles

0 2 4 6 8 10 12 14-0.6

-0.4

-0.2

0

0.2

0 2 4 6 8 10 12 14

0

1

2

2.5

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

0.2

0.3

uB

v (

mm

)

uB

r (m

m)

time (s)

(a) (b)


aV

r (m

/s2)

aV

v (

m/s

2)

(c) (d)

time (s)

first vehicle

first vehicle

effective

uncracked



uBvuBr

aVvaVr

L=207 m

R=750 m

v = 120 km/h

10 vehicles


31


riding comfort of the passengers → acceleration of the car-body • vertical ≤ 2.0 m/s2 • radial (lateral) ≤ 1.5 m/s2

.


• 0.3Qk,1: the additional mass due to traffic

0 2 3 5 10 15 20-2

0

2

2 4 6 8 100

2

4x 10

-5

2 4 6 8 100

0.5

1

1.5x 10

-3

0 5 10 15 20-10

0

10

uB

r (m

m)

(c) (d)

FS

1.57 Hz

without 0.3Qk,1

without 0.3Qk,1

with 0.3Qk,1

uB

v (

mm

)

(a)

● no passing vehicles ● earthquake excitation in 3 directions ● u: displacement ● FS: Fourier Spectrum

FS

(b)

5.87 Hz

without 0.3Qk,1

time (s)

frequency (Hz)

umax: 1.33 mm

umin: -1.53 mm

umax: 9.54 mm

umin: -9.03 mm

uBv

uBr L=207 m

R=750 m

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

ground acc. XY

Z

05

1015

2025

30-4-3-2-10123

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

2 2.5 3

-2

-1

0

0 2 3 5 10 15 20-2

0

2

2 4 6 8 100

2

4x 10

-5

2 4 6 8 100

0.5

1

1.5x 10

-3

0 5 10 15 20-10

0

10

uB

r (m

m)

(c) (d)

FS

1.57 Hz

without 0.3Qk,1

without 0.3Qk,1

with 0.3Qk,1

uB

v (

mm

)

(a)


FS

(b)

5.87 Hz

without 0.3Qk,1

time (s)

frequency (Hz)

umax: 1.33 mm

umin: -1.53 mm

umax: 9.54 mm

umin: -9.03 mm

uBv

uBr L=207 m

R=750 m

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

ground acc. XY

Z

05

1015

2025

30-4-3-2-10123

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

2 2.5 3

-2

-1

0

0 2 3 5 10 15 20-2

0

2

2 4 6 8 100

2

4x 10

-5

2 4 6 8 100

0.5

1

1.5x 10

-3

0 5 10 15 20-10

0

10

uB

r (m

m)

(c) (d)

FS

1.57 Hz

without 0.3Qk,1

without 0.3Qk,1

with 0.3Qk,1

uB

v (

mm

)

(a)


FS

(b)

5.87 Hz

without 0.3Qk,1

time (s)

frequency (Hz)

umax: 1.33 mm

umin: -1.53 mm

umax: 9.54 mm

umin: -9.03 mm

uBv

uBr L=207 m

R=750 m

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

ground acc. XY

Z

05

1015

2025

30-4-3-2-10123

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

2 2.5 3

-2

-1

0

• vertical and radial disp. of bridge midpoint


32



0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

aV

v (

m/s

2)

V-1 passing

over the bridge

13.575 s

aV

r (m

/s2)

time (s)

(a)

(c)

V-1 - vertical

strong shaking

(b)

aV

v (

m/s

2)

aV

r (m

/s2)

(d)

V-1 - radial

6.75 s

time (s)

6.82 s

V-1 running

on the bridge

13.575 s

5 s

0 s

0 s

strong shaking

6.82 s

5 s

0 s

0 s

5 s

0 s

V-10 running on the bridge

13.575 s

V-10 running on the ground

13.575 s

strong shaking

V-10 - vertical

6.75 s

5 s

0 s

strong shaking


13.575 s


13.575 s

V-10 - radial

V-1 running

on the bridge

13.575 s

● VBI model ● earthquake excitation in 3 directions ● a: acceleration ● V-1 and V-10: the first and tenth vehicle

0 s

0 s

13.58 s

13.58 s

V-1 passing

over the bridge

13.575 s

aVvaVr

v = 120 km/hL=207 m

R=750 m

10 vehicles

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

ground acc. XY

Z

05

1015

2025

30-4-3-2-10123

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

radial

vertical

10th vehicle 1st vehicle

• accel. of the vehicle on bridge > accel. of the vehicle on ground

• vertical and radial accel. of car-body


0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

aV

v (

m/s

2)

V-1 passing

over the bridge

13.575 s

aV

r (m

/s2)

time (s)

(a)

(c)

V-1 - vertical

strong shaking

(b)

aV

v (

m/s

2)

aV

r (m

/s2)

(d)

V-1 - radial

6.75 s

time (s)

6.82 s

V-1 running

on the bridge

13.575 s

5 s

0 s

0 s

strong shaking

6.82 s

5 s

0 s

0 s

5 s

0 s


13.575 s


13.575 s

strong shaking

V-10 - vertical

6.75 s

5 s

0 s

strong shaking


13.575 s


13.575 s

V-10 - radial

V-1 running

on the bridge

13.575 s


0 s

0 s

13.58 s

13.58 s

V-1 passing

over the bridge

13.575 s

aVvaVr

v = 120 km/hL=207 m

R=750 m

10 vehicles

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

ground acc. XY

Z

05

1015

2025

30-4-3-2-10123

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

0 2 4 6 8 10 12 13.58-2

-1

0

1

2

aV

v (

m/s

2)

V-1 passing

over the bridge

13.575 s

aV

r (m

/s2)

time (s)

(a)

(c)

V-1 - vertical

strong shaking

(b)

aV

v (

m/s

2)

aV

r (m

/s2)

(d)

V-1 - radial

6.75 s

time (s)

6.82 s

V-1 running

on the bridge

13.575 s

5 s

0 s

0 s

strong shaking

6.82 s

5 s

0 s

0 s

5 s

0 s


13.575 s


13.575 s

strong shaking

V-10 - vertical

6.75 s

5 s

0 s

strong shaking


13.575 s


13.575 s

V-10 - radial

V-1 running

on the bridge

13.575 s


0 s

0 s

13.58 s

13.58 s

V-1 passing

over the bridge

13.575 s

aVvaVr

v = 120 km/hL=207 m

R=750 m

10 vehicles

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

ground acc. XY

Z

05

1015

2025

30-4-3-2-10123

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

33




34



• vertical and radial disp. of bridge midpoint • the conventional seismic analysis overestimates the response of the bridge • the multibody vehicle acts as a additional damping to the bridge and reduce its

response

vertical

radial


35



• 20 different earthquakes • comfort: vertical and radial accel. of car-body • safety: derailment and offload factor


comfort:

safety

36


Derail factor

Ti

Ni

Offloading factor

Nli Nri

Nli Nri


motivation

proposed model



conclusions

37

outline



38

SVBI rare earthquakes

0 10 20 300

0.2

0.4

0.6

0.8

0 10 20 300

0.2

0.4

15.4 15.42 15.44 15.460

100

200

0 10 20 30

-20

0

20

0 10 20 30-100

0

100

0 10 20 30

-5

0

5

0 10 20 30

-5

0

5

● a

Vr and a

Vv: radial and vertical accel. of car body

● uVr

and uVФ

: vertical and rolling disp. of car body ● λN1: normal contact force of wheel 1 ● P and Q: dynamic vertical and horizontal force

uVФ

(m

m r

ad)

uV

r (

mm

)

zoom

in

λN

1 (

kN

)

λ N1 (

kN

)

aV

v (

m/s

2)

(b)

offlo

ad f

acto

r

(f)

(h)

der

ailm

ent

fact

or

(e)

(a)

(g)

aV

r (

m/s

2)

time (s)

P

Q λN

λTy

contact angle δ

max: 0.41 max: 0.64

t = 15.455 s detachment

t = 15.430 s recontact


max: 6.73 m/s

2

max: 2.05 m/s

2

ground acc.

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

X

YZ

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

05

1015

2025

30-4-3-2-10123

derailment factor = Q / P

offload factor = |P - Pst| / Pst Pst: static vertical force

time (s)

(d) (c)

max: 111.1 mm

min: 57.6 mm min: -28.2 mm rad

max: 17.6 mm rad


0 10 20 300

100

200

• contact: nonzero normal contact force • detachment: zero normal contact force

• three components of the ground motion recorded on April 25, 1992 in Cape Mendocino USA, at Petrolia station (0.59g, 0.66g, 0.19g)

0 10 20 300

0.2

0.4

0.6

0.8

0 10 20 300

0.2

0.4

15.4 15.42 15.44 15.460

100

200

0 10 20 30

-20

0

20

0 10 20 30-100

0

100

0 10 20 30

-5

0

5

0 10 20 30

-5

0

5

● a

Vr and a


● uVr

and uVФ

: vertical and rolling disp. of car body ● λN1: normal contact force of wheel 1 ● P and Q: dynamic vertical and horizontal force

uVФ

(m

m r

ad)

uV

r (

mm

)

zoom

in

λN

1 (

kN

)

λ N1 (

kN

)

aV

v (

m/s

2)

(b)

offlo

ad f

acto

r

(f)

(h)

der

ailm

ent

fact

or

(e)

(a)

(g)

aV

r (

m/s

2)

time (s)

P

Q λN

λTy

contact angle δ

max: 0.41 max: 0.64




max: 6.73 m/s

2

max: 2.05 m/s

2

ground acc.

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

X

YZ

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

05

1015

2025

30-4-3-2-10123



time (s)

(d) (c)

max: 111.1 mm


max: 17.6 mm rad


0 10 20 300

100

200


separation sequence

39


-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

list of state change of wheel 1 ● t = 3.188 s 1

st detachment

● t = 3.228 s 1st recontact

● t = 3.232 s 2nd

detachment ● t = 3.263 s 2

nd recontact

(b)

(a)

(c)

ground acc.

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

X

YZ

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

05

1015

2025

30-4-3-2-10123

(d)

t = 3.188 s 1

st detachment

of wheel 1

t = 3.263 s 2

nd recontact

of wheel 1

wheel 1

wheel 2

wheel 2

wheel 2

wheel 2

wheel 1

wheel 1

wheel 1

t = 0 s initial condition

flange contact of wheel 2

t = 3.254 s uplifting of wheel 1 after 2

nd detachment

contact of wheel 2

contact of wheel 2

left wheel/rail coordinate

right wheel/rail coordinate

tangential

vector

normal

vector

wheel profile

rail profile

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

-850 -800 -750 -700 -650

-40

-20

0

20

650 700 750 800 850

-40

-20

0

20

list of state change of wheel 1 ● t = 3.188 s 1

st detachment

● t = 3.228 s 1st recontact

● t = 3.232 s 2nd

detachment ● t = 3.263 s 2

nd recontact

(b)

(a)

(c)

ground acc.

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

X

YZ

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

05

1015

2025

30-4-3-2-10123

(d)

t = 3.188 s 1

st detachment

of wheel 1

t = 3.263 s 2

nd recontact

of wheel 1

wheel 1

wheel 2

wheel 2

wheel 2

wheel 2

wheel 1

wheel 1

wheel 1

t = 0 s initial condition

flange contact of wheel 2

t = 3.254 s uplifting of wheel 1 after 2

nd detachment

contact of wheel 2

contact of wheel 2

left wheel/rail coordinate

right wheel/rail coordinate

tangential

vector

normal

vector

wheel profile

rail profile

separation • no need for high

lateral accelerations


separation sequence

40


derailment

600 650 700 750 800-50

0

50

-900 -850 -800 -750 -700-50

0

50

0 2 4 6 8 10 11.230

100

200

300

0 2 4 6 8 10 11.23-10

0

10

0 2 4 6 8 10 11.23-10

0

10

0 2 4 6 8 10 11.230

0.5

1

1.5

0 2 4 6 8 10 11.230

0.5

1

0 2 4 6 8 10 11.23-200

0

200

400

600

0 2 4 6 8 10 11.23-100

0

100

200

300

0 2 4 6 8 10 11.23-40

-20

0

● a

Vr and a


● uVr

and uVФ

: vertical and rolling disp. of car body

● uVwr

: radial disp. of the wheelset ● λN1: normal contact force of wheel 1 ● P and Q: dynamic vertical and horizontal force

uVФ

(m

m r

ad)

uV

r (

mm

)

λ N1 (

kN

)

uV

wr (

mm

)

aV

v (

m/s

2)

(b)

offlo

ad f

acto

r

(f)

(h)

der

ailm

ent

fact

or

(e)

(a)

(g)

aV

r (

m/s

2)

time (s)

P

Q λN

λTy

contact angle δ

max: 1.19 max: 0.89

max: 9.38 m/s

2

max: 7.88 m/s

2

ground acc.

05

1015

2025

30

-4

-3

-2

-1

0

1

2

3

X

YZ

0

5

10

15

20

25

30

-4

-3

-2

-1

0

1

2

3

05

1015

2025

30-4-3-2-10123



time (s)

(d) (c)

max: 211.0 mm


max: 14.1 mm rad

derailment max: 50 mm

max: 217.8 kN

wheelset 4 (wheels 7 and 8)

wheel 8

wheel 7

wheel 8

left wheel/rail coordinate (mm)

right wheel/rail coordinate (mm)

rail 7

rail 8

(i) (j)


motivation

proposed model



conclusions

41

outline


• scheme for seismic response of interacting vehicle and horizontally curved bridges

• the mass / stiffness / damping matrix of the coupled vehicle-bridge system become time-dependent

• the scheme accommodates easily different number of vehicles, and with various DOFs

• the seismic response of the bridge affects (adversely) the safety of the vehicle

• Vehicles-bridge-earthquake timing problem

• lack of performance (comfort and safety) criteria for vehicles → probabilistic

• complicated dynamics, multi-parametric problem

42

conclusions


list of relevant journal publications

1. Zeng Q., Dimitrakopoulos Ε.G. (2016) “Seismic Response Analysis of an Interacting Curved Bridge-Train System Under Frequent Earthquakes” Earthquake Engineering and Structural Dynamics, published online, 13 Jan 2016 DOI 10.1002/eqe.2699

2. Zeng Q., Yang Y.B., Dimitrakopoulos Ε.G. (2016) “Dynamic response of high speed vehicles and sustaining curved bridges under conditions of resonance” Engineering Structures, published 1 May 2016, vol. 114 pp 61-74, DOI: 10.1016/j.engstruct.2016.02.006

3. Dimitrakopoulos E.G. & Zeng Q. (2015) "A Three-dimensional Dynamic Analysis Scheme for the Interaction between Trains and Curved Railway Bridges" Computers & Structures, vol. 149 43-60.

4. Paraskeva T.S., Dimitrakopoulos Ε.G., Zeng Q., (2016) “Dynamic Vehicle-Bridge Interaction Under Simultaneous Vertical Earthquake Excitation” Bulletin of Earthquake Engineering, accepted 28/5/2016

43

references


thank you for your kind attention!

44


45

low-cost high impact bamboo bridges


steel arch truss bridge

http://campus.liepin.com/tsy/enterprisefeatures (2015)

Ting Sihe Bridge of High-speed railway line in China

46

case studies and results

http://campus.liepin.com/tsy/enterprisefeatures

http://campus.liepin.com/tsy/enterprisefeatures


47


• deformed shape of the whole bridge • 1 vehicle: speed of train = 300 km/h

47


animation/arch bridge one vehicle.avi


48


deformation animation of the vehicle

3D view front view

• 1 vehicle: speed of train = 300 km/h • vibration due to the track irregularities

elevation view

48


animation/arch vehicle.avi



49

• deformed shape of the whole bridge • 10 vehicle: speed of train = 300 km/h

49


animation/arch bridge ten vehicles.avi


cable-stayed bridge with the passage of MTR

http://en.wikipedia.org/wiki/Kap_Shui_Mun_Bridge

Kap Shui Mun Bridge (KSMB) in Hong Kong

50


http://en.wikipedia.org/wiki/Kap_Shui_Mun_Bridge


uB

v u

Bh (

mm

)

aV

v (

m/s

2)

uB

t (ra

d)

(a)

(b)

(c)

dimensionless time (vt/L)

aV

h (

m/s

2)

(d)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-20

-10

0

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4-2

-1

0

1x 10

-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

u

Bv

u

Bh

● ten 3D vehicles ● L = 750 m ● v = 144 km/h

10 vehiclesuBv

uBh uBt

aVv

aVh

uB

v u

Bh (

mm

)

aV

v (

m/s

2)

uB

t (ra

d)

(a)

(b)

(c)


aV

h (

m/s

2)

(d)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-20

-10

0

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4-2

-1

0

1x 10

-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

u

Bv

u

Bh


10 vehiclesuBv

uBh uBt

aVv

aVh

uB

v u

Bh (

mm

)

aV

v (

m/s

2)

uB

t (ra

d)

(a)

(b)

(c)


aV

h (

m/s

2)

(d)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-20

-10

0

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4-2

-1

0

1x 10

-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

u

Bv

u

Bh


10 vehiclesuBv

uBh uBt

aVv

aVh

• (a) and (b) bridge midpoint vertical horizontal and torsional displacements

• (c) and (d) vehicle vertical and horizontal accelerations

horizontal

vertical

horizontal

vertical

torsional

bridge displacement vehicle acceleration

uB

v u

Bh (

mm

)

aV

v (

m/s

2)

uB

t (ra

d)

(a)

(b)

(c)


aV

h (

m/s

2)

(d)

0 0.2 0.4 0.6 0.8 1 1.2 1.4-20

-10

0

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4-2

-1

0

1x 10

-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

u

Bv

u

Bh


10 vehiclesuBv

uBh uBt

aVv

aVh

(a)

(c)

10 vehicles

(b) vehicle position

vehicle position


vehicle position

kPa

kPa

kPa


51




52


animation/KSM MATLAB animation.avi


straight highway bridge

P1 P2

A1

A2

63.80m 118.60m 63.80m

speed v speed v

uB

36

.00

m

46

.00

m

ground accel.

x

z

O

L=246.20m

53



5-span highway continuous bridge with the passage of trucks

• deformed shape of the whole bridge and the vehicle

• 1 vehicle: speed of truck = 80 km/h

54


animation/slab bridge and vehicle.avi


5-span highway continuous bridge with the passage of trucks

• deformed shape of the whole bridge

• 5 vehicle: speed of truck = 80 km/h

55


animation/slab bridge 5 vehicles.avi

on the safety of high-speed trains running on bridges...

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