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

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응답

제어를

위한


납고무

받침의

설계

기준

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




Lead Rubber Bearing (LRB)



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




楴 楳⁤i晦楣畬琠瑯⁡灰汹⁴桩猠灲潣敤畲攠慮搠杵楤敬楮敳i

†††††
潦⁩獯污s楯渠獹獴敭 摩d散e汹l瑯⁣慢汥
-
獴慹敤a扲楤来b.

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




偬慳P楣i扥桡癩潲映敡搠㨠敮敲杹⁤楳獩灡s楯i
†



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



The design parameters of LRB are chosen that design index


(
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


to the deformed stated of bridge with dead loads


-

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.



周攠灬慳瑩挠扥桡癩潲潦敡搠捯c攠潦⁌剂楳⁩浰潲m慮琠瑯


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


晬數楢汥e䱒䈮

-

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.




周攠捡扬c
-
獴慹敤e扲楤来b楳 湥敤敤⁳瑩t晥f⁲畢扥 慮搠


扩杧敲 汥慤 捯c攠獩攠瑨慮 来湥牡氠獴牵捴畲敳.




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,

††
晬f硩扬攠䱒䈠楳 湥敤敤n

Structural Dynamics & Vibration Control Lab., KAIST


23

Thank you for your attention!!

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.



周攠捨慲慣e敲e獴楣i獴r敮杴栠潦䱒䈠楳⁲散潭浥湤敤n瑯⁵獥


five percent of weight carried by LRB

to obtain additional


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


: Yielding displacement of lead


: 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

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