# Guidelines of Designing Lead Rubber Bearing for a Cable-Stayed ...

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위한

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제안

Guidelines of Designing Lead Rubber Bearing for

a Cable
-
Stayed Bridge to Control Seismic Response

성진

:
한국과학기술원

건설

환경공학과

석사과정

규식

:
한국과학기술원

건설

환경공학과

박사과정

춘호

:
중부대학교

토목공학과

교수

인원

:
한국과학기술원

건설

환경공학과

교수

2003
년도

가을

학술발표회

Oct. 11. 2003

Structural Dynamics & Vibration Control Lab., KAIST

2

Contents

䥮瑲潤畣瑩潮

Design Procedure of LRB

Numerical Examples

Conclusions

Structural Dynamics & Vibration Control Lab., KAIST

3

Backgrounds

Design procedure of base isolation system for building and

short
-
span bridges.

-

Design natural period of structure or effective period of

base isolator

-

Then, the design parameters of isolator are determined.

Introduction

Structural Dynamics & Vibration Control Lab., KAIST

4

Long span bridge such as cable
-
stayed bridges

-

Flexible : long period modes and natural seismic isolation

-

Small structural damping

†††††

-

Structural Dynamics & Vibration Control Lab., KAIST

5

Objective

Suggest the design procedure and guidelines of LRB

for cable
-
stayed bridge.

Structural Dynamics & Vibration Control Lab., KAIST

6

Design Parameters

of LRB

p
K
e
K
y
Q
y
X
d
X
eff
K
y
F
u
F
e
K
p
K
y
Q
eff
K
y
F
u
F
y
X
d
X

, : elastic & plastic stiffness

: effective stiffness

: characteristic shear strength

, : yield and ultimate strength

, : yield and ultimate displacement

Fig. 1 Behavior and design parameters of LRB

䙬數F扩汩b礠潦⁲畢u敲›⁰敲楯搠獨楦t

†

Determine the , , to minimize the earthquake

forces and displacements.

e
K
p
K
y
Q
Design Procedure of LRB

Structural Dynamics & Vibration Control Lab., KAIST

7

(
DI
) is minimized or unchanged (less than 0.05) for variation

of design parameters.

Proposed Design Procedure

5
1
max
i
i
i
J
J
DI

i
= 1 ~ 5

)
1
(

-

Five important responses of cable
-
stayed bridge are

considered.

: base shear and moment at towers

: shear and moment at deck level at towers

: deck displacement (longitudinal direction)

Structural Dynamics & Vibration Control Lab., KAIST

8

Design procedure

-

Step 1

: design earthquake (history or artificial earthquake, etc.)

-

Step 2

: appropriate is selected for variation of .

: and are assumed.

-

Step 3

: appropriate is selected for variation of .

: use selected and assume .

-

Step 4

: appropriate is selected for variation of .

-

Step 5

: iterate step 2 ~ 4 until parameters remain unchanged.

p
K
p
K
y
Q
e
K
y
Q
e
K
y
Q
p
K
p
e
K
K
/
p
e
K
K
/
Structural Dynamics & Vibration Control Lab., KAIST

9

Bridge Model

Fig. 2 Bill Emerson Memorial Bridge
(Benchmark cable
-
stayed bridge model)

Benchmark cable
-
stayed bridges (Dyke et al. 2003)

142.7 m

350.6 m

142.7 m

g
x
Numerical Examples

Structural Dynamics & Vibration Control Lab., KAIST

10

Finite element evaluation model

-

Modeling : 162 beam elements, 420 rigid links

128 cable elements, 579 nodes

-

Stiffness matrix : nonlinear static analysis corresponding

-

Damping matrix : 3 % of critical damping to each mode

-

Control devices : longitudinal direction between the deck

and piers

-

Ground motion : longitudinal direction not considering

multi
-
excitation

Structural Dynamics & Vibration Control Lab., KAIST

11

Design Earthquakes

Scaled El Centro earthquake (1940)

-

The PGA of El Centro earthquake

: scaled to the design PGA of cable
-
stayed bridges (0.36 g’s.)

0
5
10
15
F
r
e
q
u
e
n
c
y
(
H
z
)
0
2
4
6
8
10
P
o
w
e
r

S
p
e
c
t
r
a
l

D
e
n
s
i
t
y
P
o
w
e
r

S
p
e
c
t
r
a
l

D
e
n
s
i
t
y
0
50
100
150
200
t
i
m
e

(
s
e
c
)
-4
-2
0
2
4
A
c
c
e
l
a
t
i
o
n
(
m
/
s
^
2
)
T
i
m
e
-
A
c
c
e
l
e
r
a
t
i
o
n

G
r
a
p
h
Fig. 3 Design Earthquake (Scaled El Centro)

Structural Dynamics & Vibration Control Lab., KAIST

12

Kanai
-
Tajimi artificial earthquake

-

Stationary Kanai
-
Tajimi filter

-

Power spectral density

0
2
2
2
2
2
)
(
4
]
)
(
1
[
]
)
(
4
1
[
(
S
S
g
g
g

2
2
0
)
1
4
(
03
.
0
g
S
g
g
g



and : site dominant damping coefficient and frequency.

: constant power spectral intensity.

g

g

0
S
-

= 37.3 rad/s, = 0.3 (Spencer et al.)

g

g

)
2
(
)
3
(
Structural Dynamics & Vibration Control Lab., KAIST

13

Properties of LRB

DI
**

LRB I (Scaled El Centro)

1.4W
*

(tf/m)

0.13W
(tf)

11

3.334

LRB II (Kanai

Tajimi)

1.5W
(tf/m)

0.12W
(tf)

12

4.175

p
K
y
Q
p
e
K
K
/
Table 1. Properties of LRB

* : Pier 1,4
-

1557.18 (
tf
), Pier 2,3
-

5383 (
tf
) ** : Max. of DI =5

Need the stiffer rubber and bigger lead core size than

general buildings and short
-
span bridges.

reduce the seismic response for cable
-
stayed bridge.

Structural Dynamics & Vibration Control Lab., KAIST

14

El Centro : 1940, Imperial Valley, 0.348
g’s

Mexico City : 1985, Galeta de Campos, 0.143
g’s

䝥扺攠†††††e‱㤹㤬T畲u敹e䝥扺攬‰⸲㘵e
g’s

0
10
20
30
40
50
-3
-2
-1
0
1
2
3
4
A
c
c
e
l
e
r
a
t
i
o
n

(
m
/
s
2
)
E
l

C
e
n
t
r
o
0
10
20
30
40
50
T
i
m
e

(
s
e
c
)
-2
-1
0
1
2
M
e
x
i
c
o

C
i
t
y
0
10
20
30
40
50
-2
-1
0
1
2
3
G
e
b
z
e
Fig. 4 Time
-
history of input earthquakes

Performance of Designed LRB

Structural Dynamics & Vibration Control Lab., KAIST

15

Evaluation criteria under El Centro earthquake

J1

: Max. base shear

J2

: Max. shear at deck level

J3

: Max. base mom.

J4

: Max. mom. at deck level

J5

: Max. cable deviation

J6

: Max. deck displacement

J7

: Norm base shear

J8

: Norm shear at deck level

J9

: Norm base mom.

J10

: Norm base mom. at deck

level

J11

: Norm cable deviation

*
: Scaled El Centro

**
: Kanai
-
Tajimi Artificial Earthquake

***
: Naeim
-
Kelly Method (
T
eff

= 1.5 sec )

****
: Naeim
-
Kelly Method (
T
eff

= 2.0 sec )

0
0.5
1
1.5
2
J1
J2
J3
J4
J5
J6
J7
J8
J9
J10
J11
Evaluation Criteria
Control/Uncontrol
LRB I*
LRB II**
N-K I***
N-K II****
Structural Dynamics & Vibration Control Lab., KAIST

16

Evaluation criteria under Mexico City earthquake

J1

: Max. base shear

J2

: Max. shear at deck level

J3

: Max. base mom.

J4

: Max. mom. at deck level

J5

: Max. cable deviation

J6

: Max. deck displacement

J7

: Norm base shear

J8

: Norm shear at deck level

J9

: Norm base mom.

J10

: Norm base mom. at deck

level

J11

: Norm cable deviation

0
0.5
1
1.5
2
2.5
J1
J2
J3
J4
J5
J6
J7
J8
J9
J10
J11
Evaluation Criteria
Control/Uncontrol
LRB I*
LRB II**
N-K I***
N-K II****
*
: Scaled El Centro

**
: Kanai
-
Tajimi Artificial Earthquake

***
: Naeim
-
Kelly Method (
T
eff

= 1.5 sec )

****
: Naeim
-
Kelly Method (
T
eff

= 2.0 sec )

Structural Dynamics & Vibration Control Lab., KAIST

17

Evaluation criteria under Gebze earthquake

J1

: Max. base shear

J2

: Max. shear at deck level

J3

: Max. base mom.

J4

: Max. mom. at deck level

J5

: Max. cable deviation

J6

: Max. deck displacement

J7

: Norm base shear

J8

: Norm shear at deck level

J9

: Norm base mom.

J10

: Norm base mom. at deck

level

J11

: Norm cable deviation

0
0.5
1
1.5
2
2.5
3
3.5
J1
J2
J3
J4
J5
J6
J7
J8
J9
J10
J11
Evaluation Criteria
Control/Uncontrol
LRB I*
LRB II**
N-K I***
N-K II****
*
: Scaled El Centro

**
: Kanai
-
Tajimi Artificial Earthquake

***
: Naeim
-
Kelly Method (
T
eff

= 1.5 sec )

****
: Naeim
-
Kelly Method (
T
eff

= 2.0 sec )

Structural Dynamics & Vibration Control Lab., KAIST

18

Table 2. Maximum evaluation criteria for three historical earthquake

Evaluation Criteria

LRB I

LRB II

N
-
K I

N
-
K II

: Max base shear

0.7410

0.7389

0.6118

0.6100

: Max shear at deck level

1.0938

1.1134

1.1027

1.4220

: Max base moment

0.7317

0.7183

0.5852

0.7028

: Max moment at deck level

0.6145

0.6718

0.6484

1.0271

: Max cable deviation

0.1526

0.1550

0.1693

0.1973

: max deck displacement

1.3811

1.3042

1.5733

3.3021

: Norm base shear

0.5547

0.5500

0.5139

0.4841

: Norm shear at deck level

0.8423

0.8610

0.9673

1.4240

: Norm base moment

0.5732

0.5637

0.5279

0.5070

: Norm moment at deck level

0.5262

0.5409

0.6389

1.2043

: Norm cable deviation

0.0167

0.0163

0.0141

0.0209

1
J
2
J
3
J
4
J
5
J
6
J
7
J
8
J
9
J
10
J
11
J

The performance of designed LRB is good for several

historical earthquakes.

Structural Dynamics & Vibration Control Lab., KAIST

19

Design Properties of LRB for Earthquake Frequency

The behavior of structure is affected by not only PGA but also

the dominant frequency of earthquake.

The PGA of earthquakes : 0.36g’s

0
5
10
15
F
r
e
q
u
e
n
c
y
(
H
z
)
0
2
4
6
8
10
P
o
w
e
r

S
p
e
c
t
r
a
l

D
e
n
s
i
t
y
S
c
a
l
e
d

E
l

C
e
n
t
r
o

(
1
.
5

H
z
)
0
5
10
15
F
r
e
q
u
e
n
c
y
(
H
z
)
0
10
20
30
40
P
o
w
e
r

S
p
e
c
t
r
a
l

D
e
n
s
i
t
y
S
c
a
l
e
d

M
e
x
i
c
o

C
i
t
y

(
0
.
5
H
z
)
0
5
10
15
F
r
e
q
u
e
n
c
y
(
H
z
)
0
4
8
12
16
P
o
w
e
r

S
p
e
c
t
r
a
l

D
e
n
s
i
t
y
S
c
a
l
e
d

G
e
b
z
e

(
2
.
0
H
z
)
Fig. 8 Power Spectral Density of input earthquakes

Structural Dynamics & Vibration Control Lab., KAIST

20

Properties of LRB

Frequency

Scaled Mexico City

0.5 Hz

0.9W
(tf/m)

0.15W
(tf)

10

Scaled El Centro

1.5 Hz

1.4W
(tf/m)

0.13W
(tf)

11

Scaled Gebze

2.0 Hz

1.5W
(tf/m)

0.16W
(tf)

9

p
K
y
Q
p
e
K
K
/
Table 3. Properties of LRB for earthquake frequency

-

and of LRB

: affected by dominant frequency of earthquake.

: Low frequency

-

and of LRB

: not related to dominant frequency of earthquake.

p
K
e
K
y
Q
p
e
K
K
/
Structural Dynamics & Vibration Control Lab., KAIST

21

The guidelines and procedure of designing LRB for

seismically excited cable
-
stayed bridge are investigated.

-

The plastic behavior of lead core of LRB is important

to reduce the seismic response of cable
-
stayed bridge.

Conclusions

Structural Dynamics & Vibration Control Lab., KAIST

22

The performance of designed LRB is good for several

historical earthquakes.

䅳A瑨攠摯浩湡湴n晲敱略湣礠潦⁥慲瑨煵ok攠楳 汯lⰠ瑨,

††

Structural Dynamics & Vibration Control Lab., KAIST

23

This research is supported by the
National Research Lab.

Grant (No.: 2000
-
N
-
NL
-
01
-
C
-
251) in Korea.

Acknowledgments

Structural Dynamics & Vibration Control Lab., KAIST

24

Structural Dynamics & Vibration Control Lab., KAIST

25

Previous Application of LRB for cable
-
stayed bridge

Ali and Abdel
-
Ghaffar

-

Efficiency of LRB for cable
-
stayed bridge

Wesolowsky and Wilson

-

Design the LRB for cable
-
stayed bridge using N
-
K method.

-

Effective period of LRB

Structural Dynamics & Vibration Control Lab., KAIST

26

Design Procedure of LRB for General Structures

The natural period of general building and continuous

bridge is 0.3 sec ~ 0.6 sec.

The main design aim for these structures is shifting the

natural period of these structures.

The stiffness of LRB is designed that the natural period of

structure or effective period of isolator is
1.4 sec ~ 2.0 sec.

five percent of weight carried by LRB

damping effect.

(Ghobarah, A. and Ali, H. M., 1988)

Structural Dynamics & Vibration Control Lab., KAIST

27

Design Procedure

(N
-
K Method)

1. Maximum allowable displacement( ) and shear
-
force( )

of isolator is established.

2. Effective stiffness and period of isolator is calculated.

where, M is the structural mass assigned to the isolator

3. The effective damping ( ).

4. Energy dissipation of isolator per one cycle.

d
X
u
F
d
u
eff
X
F
K

eff
eff
K
M
T

2

)
1
(
eff

eff
d
eff
D
X
K
E

2
2

)
2
(
Structural Dynamics & Vibration Control Lab., KAIST

28

5. Shear strength ( )

-

In the first step, is neglecting.

6. Post
-
yield stiffness ( )

7. Yield displacement ( )

8. Repeat 6~8 until converges.

y
Q

y
d
D
y
X
X
E
Q
(
4
)
3
(
y
X
p
K
d
eff
p
X
Q
K
K

)
4
(
2
1
K
K
Q
X
y

)
5
(
y
X
y
X
Structural Dynamics & Vibration Control Lab., KAIST

29

LRB Model (Bouc
-
Wen Model)

)
)(
/
1
(
1
(
1
n
r
n
r
r
y
y
e
r
e
Z
X
Z
Z
X
X
A
D
Z
Z
D
K
X
K
F

: Post to pre
-
yielding stiffness ratio of LRB

: Linear stiffness of LRB

: Relative displacement & velocity

: Dimensionless parameter to represent shape of

hysteretic curve

e
Κ
y
D
r
r
X
X

,

,
,
A
n
where

)
2
(

††
㴠ㄬ†††㴠ㄬ†††‽1〮㔬†††‽‰⸵

n
A

)
3
(
Structural Dynamics & Vibration Control Lab., KAIST

30

Evaluation Criteria

Five important responses of cable
-
stayed bridge are considered.

Scaled El Centro

Kanai
-

Tajimi

J
1

Max. base shear at tower

RMS

J
1

RMS base shear at tower

J
2

Max. base moment at tower

RMS J
2

RMS base moment at tower

J
3

Max. shear at deck level

RMS J
3

RMS shear at deck level

J
4

Max. moment at deck level

RMS J
4

RMS moment at deck level

J
6

Max. deck displacement

RMS J
6

RMS deck displacement

Table 1 Evaluation criteria (Control/Uncontrolled)

5
1
max
i
i
i
J
J
DI
Structural Dynamics & Vibration Control Lab., KAIST

31

Performance of Designed LRB

Design Earthquake

LRB I
*

LRB II
**

J
1
or RMS J
1

0.3322

0.5293

J
2
or RMS J
2

0.9368

0.7998

J
3
or RMS J
3

0.2996

0.4349

J
4
or RMS J
4

0.3801

0.6385

J
6
or RMS J
6

0.9821

1.2539

Table 3 Performance of designed LRB for design earthquake

* : Scaled El Centro earthquake ** : Kanai

Tajimi artificial earthquake

Structural Dynamics & Vibration Control Lab., KAIST

32

Evaluation Criteria

LRB I
*

LRB II
**

N
-
K
***

: Max base shear

0.3210

0.3229

0.3103

: Max shear at deck level

0.8764

0.8716

0.9505

: Max base moment

0.3010

0.3057

0.2936

: Max moment at deck level

0.3593

0.3532

0.4796

: Max cable deviation

0.1526

0.1550

0.1693

: max deck displacement

0.8743

0.9350

1.1192

: Norm base shear

0.2557

0.2482

0.2429

: Norm shear at deck level

0.7439

0.7428

0.8378

: Norm base moment

0.2728

0.2630

0.2574

: Norm moment at deck level

0.4038

0.4069

0.4732

: Norm cable deviation

0.0167

0.0163

0.0141

Table 4. Performance of Designed LRB (
El Centro)

1
J
2
J
3
J
4
J
5
J
6
J
7
J
8
J
9
J
10
J
11
J
* : Scaled El Centro ** : Kanai

Tajimi Artificial Earthquake

*** : Naeim and Kelly

Structural Dynamics & Vibration Control Lab., KAIST

33

Evaluation Criteria

LRB I
*

LRB II
**

N
-
K
***

: Max base shear

0.7410

0.7389

0.6118

: Max shear at deck level

1.0938

1.1134

1.1027

: Max base moment

0.7317

0.7183

0.5852

: Max moment at deck level

0.3961

0.3991

0.3659

: Max cable deviation

0.0763

0.0783

0.0593

: max deck displacement

1.3811

1.3042

1.5733

: Norm base shear

0.5547

0.5500

0.5139

: Norm shear at deck level

0.8197

0.8610

0.7770

: Norm base moment

0.5732

0.5637

0.5279

: Norm moment at deck level

0.5088

0.5145

0.4875

: Norm cable deviation

0.0101

0.0098

0.0073

Table 5. Performance of Designed LRB (
Mexico City)

1
J
2
J
3
J
4
J
5
J
6
J
7
J
8
J
9
J
10
J
11
J
* : Scaled El Centro ** : Kanai

Tajimi Artificial Earthquake

*** : Naeim and Kelly

Structural Dynamics & Vibration Control Lab., KAIST

34

Table 6. Performance of Designed LRB (
Gebze)

Evaluation Criteria

LRB I
*

LRB II
**

N
-
K
***

: Max base shear

0.3453

0.3565

0.4038

: Max shear at deck level

1.0251

1.1190

1.0543

: Max base moment

0.3586

0.3795

0.4174

: Max moment at deck level

0.6145

0.6718

0.6484

: Max cable deviation

0.0813

0.0913

0.1022

: max deck displacement

1.0458

1.1626

1.4685

: Norm base shear

0.3467

0.3462

0.3360

: Norm shear at deck level

0.8423

0.8583

0.9673

: Norm base moment

0.3852

0.3863

0.3788

: Norm moment at deck level

0.5262

0.5409

0.6389

: Norm cable deviation

0.0093

0.0093

0.0078

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* : Scaled El Centro ** : Kanai

Tajimi Artificial Earthquake

*** : Naeim and Kelly