prof.igarashi july27 apss2010

8/20/2019 Prof.igarashi July27 APSS2010

1/11

2010/8

Structural Control

for

Seismic

Protection of Structures

2010.7.27

APSS, Tokyo

Akira Igarashi Associate Prof., Graduate School of Engineering

Kyoto University

1. Earthquake

ground

motion

and seismic loads

Particular Feature of Seismic Loads

• Level of force can be greatly high

• Highly nonstationary

Seismic Load

Representation

• “Response Spectrum” Response of SDOF systems to seismic ground motion

Diagram representing

“Natural period of

” “

(a) Seismic Ground Motion

Time

(b) SDOF systemsSeismic Ground MotionMaximum acceleration

(absolute value)

s rucure vs. maxmum

response”

S T a ( )

where T =natural period

Acceleration Response

Spectrum (max.

absolute acc. is used)

S T v ( )Velocity Response

Spectrum (max.

relative velocity)

S T d

( ) DisplacementResponse Spectrum(max. relative displ.)

(c) Response of each SDOF system

Time

(d) Response Spectrum

Natural Period

Response Spectra & Seismic Action

• From seismic design perspective…

a = x + z

：：

Maximum load acting on

the structure＝

mass×maximum response

accelerationM

z

F = M aa soute acc.

F = M a

(force acting on the structure）

x (deformation)

F M S T

x S T

a

d

max

max

( )

( )

= ⋅

=where

T = natural period of thestructure

deformation＝maximumresponse displacement

Natural period

＝T

Demonstration of Seismic Action

• Numerical Simulation


2/11

2010/8

Property of Seismic Loads

(Relationship with Natural Period)• Amplitude of seismic load acting on the structure changes depending on

the natural period of the structure, even under the identical ground motion

• (Overall tendency)

Seismic

load decreases as

the

natural

period

increases

than a certain value, even though the load increases in a short natural period range

T

S a(T ) h=5%

Natural

Period

Seismic

Load

Property of Seismic Loads

(Relationship with Damping)

• Amplitude of seismic load acting on the structure changes depending on the damping of the structure,

even under

the

identical

ground

motion

• Seismic load decreases as the damping increases

T

S a(T ) h=5%＝

Greaterseismic

load

Low

damping

＝

Smallerseismic

load

High

damping

Natural

period

Seismic

load

Demonstration of the Effect of

Damping on the Seismic Action


2. Protective systems against

seismic loads

Earthquake Protective Systems Seismic Isolation System

Structure

Also called as

“Base Isolation”

for the typical

Isolator

Energy

Dissipation

Device

layout of

isolators shown


3/11

2010/8

Seismic Isolation for Bridges (Example)

Isolator

Girder

SteelPlates

Rubber

• Isolator：Laminated Rubber (elastomeric) bearing, friction bearing

substructure

High Damping Rubber Bearing

Rubber Bearing with Lead plugs (LRB)

Lead Plug

Laminated Layers

Steel Plate

Rubber

Steel Plate

Laminated Rubber Bearing

Seismic Isolation Principle (1)

ForceSeismic force decreases

Natural Period

Short period

= Stiff

Long period

= Soft

Seismic Isolation Principle (2)

Large

DeformationDeformation

Damping

Natural Period

Short period

= Stiff

Long period

= Soft

Load‐Displacement Performance of

Isolators


4/11

2010/8

Demonstration of Seismic Isolation

Effect


Isolated

bridge

Conventional

Energy Dissipation Devices

• A type of passive structural control

• Used combined

/w

isolators

for

seismic

isolation

• A ng amp ng capa ty o structures

Oil Damper Passive Control for Bridges (example)

Bearing

Girder

Seismic

damper

Pier/abutment

substructure

Laminated Rubber Dampers &

Implementation for a Cable Stayed Bridge

Laminated

RubberAssembly

Damper cable

Girder

Main

Tower


5/11

2010/8

Effect of Seismic Damper

Seismic

Load Seismic load decreases

Natural Period

Short period

= Stiff

Long period

= Soft

・ Also: effect of distributed lateral loads on plural piers for

bridges

Summary for Seismic Load Protection

• Seismic load decreases for

– Longer natural periods

– Higher damping

(energy

dissipation)

capability

• Structural control technologies used in practice

T

S a(T ) h=5%

Seismic

Load

3. Active & semi‐active

earth uake rotection ideas

Active & Semi‐active controls

• Have been developed based on system control theory

• Use of sensors, actuators and controllers

•

– Closed‐Loop Feedback Control

– LQR, LQG, optimal control

– H‐infinity control

– Fuzzy control

– …and more

Closed‐Loop Control

( ) ( ) ( ) X H F ω ω ω =

X H

H GF ( )

( )

( ) ( )( )ω

ω

ω ω ω =

+1

LQG Control

1

1

( )

( )( )

( )

n

x t

x t t

v t

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪

= ⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪

x

M

M1 1− −

⎡ ⎤= ⎢ ⎥− −

⎣ ⎦

0 IA

M K M C

( ) ( ) ( ) ( )d t t dt t dt c dW t = + +x Ax Bu G

• State‐feedback control

nv t ⎩ ⎭

( ) ( )t t =u Fx

1 2( ) ( ) ( ) ( ) minT T J E t t t t ⎡ ⎤= + →⎣ ⎦x R x u R u

1

2 10T T −+ − + =PA A P PBR B P R

1

2

T −= −F R B P


6/11

2010/8

Application to Earthquake Protection

• Relationship between the “seismic load”

principle &

control

concepts?

Hysteretic‐response‐based active &

semi‐active control

• Use of the seismic loading principle

• Taking advantage

of

active

and

semi

‐active

control technologies

• Contro s es gne ase on t e ysteret c

response of the active/semi‐active devices

• Avoiding complication of the control system

Motivation

• What are the actual merits of introducing active/semi‐active systems over existing

systems?

– –

properties (natural periods etc.) of the structure

to be controlled

– Enhance control performance for limited control

device loads

Motivations:

Act ive/Semi-act ive

device

Act ive/Semi-act ive

device

Control…

maximum acceleration (=load)of the structure

maximum acceleration (=load)of the structure

maximum displ. response ofthe structure

maximum displ. response ofthe structure

- Effective for a wide range of excitation amplitude levels

- Higher frequency components in damper loads

Limit the maximum damperforces

Limit the maximum damperforces

Base Isolation / Smart Base Isolation

System

• Control Device

‐ Passive

(ex. Viscous damper, Structure r ct on amper

‐ Active

‐ Semi‐Active

(ex. Variable Oil damper,

MR damper)

Isolator

Control Device

• ‐ Greece

Capacity: 65000m3

Diameter: 65.7m, Height: 22.5m.

Each tank on 212 isolators

Friction pendulum system (FPS)

• ‐ Greece

Capacity: 65000m3

Diameter: 65.7m, Height: 22.5m.

Each tank on 212 isolators

Friction pendulum system (FPS)

Background: Base isolation of liquid storage tanks

• ‐ Korea

• Three LNG storage tanks

• Capacity of 100000 m3

• Diameter 68 m, Height 30

• steel laminated rubber bearings

• The design isolation period ~ 3 Sec.

• ‐ Korea

• Three LNG storage tanks

• Capacity of 100000 m3

• Diameter 68 m, Height 30

• steel laminated rubber bearings

• The design isolation period ~ 3 Sec.

(Tajirian, 1998 )


7/11

2010/8

4.

Negative‐

stiffness‐

type

Active

&

semi‐active control for earthquake

pro ec on

Pseudo Negative Stiffness Control for Semi‐

Active devices

k nscns

PNS ControlDispl.Displ.

LoadLoad

xc xk F nsns &+−≡

⎩⎨⎧

=0

F F d

( )( )0

0

≤⋅

>⋅

xF

xF

&

&

Effect of Negative Stiffness Damper

Seismic

Load Shift of natural periods

→seismic load decreases

more efficiently

Natural Period

Short period

= Stiff

Long period

= Soft

Idea of optimization

• Pseudo Negative Stiffness (PNS) Control

• The value of the negative stiffness k ns is recommended to be identical to the structural stiffness

xc xk F nsnsd

&+−=

40

Relationship between the required damper load and the negative stiffness value

– Principle to determine the damping parameter for compromised values of negative stiffness

• Parameter determination procedure considering the balance between the limited damper load and the structural displacement reduction should be established

Optimal PNS Control Concept

Base shear vs. Displacement Hysteretic

response

+

m

k 0

x(t)

k ns , c ns

4141

=

Possible Maximum Base Shear

Affecting the parameter determination

Base Shear

ruc ura res or ng orce amper oa

Effects of parameter values on PNS control

Hysteretic ResponseStr. Stiffness

+ Structural stiffness

42

Str. Stiffness

Increased damping

+ Structural stiffness

.


8/11

2010/8

Optimality criterion for PNS control

Base shear

DIspl

Str. stiffness

Increase of damping

4343

Maximum energy dissipation

w/o increasing base shear

Optimality condition

ba B B =

Base Shear Indicator

max0 xk Ba ×= max0max xm B &&×−=

( )ba

B B B ,maxmax =

−≡Δ a B B B max

Base shear at Point a Maximum base shear

Base shear at Point b

4444

( )

( )⎩⎨⎧

≥


9/11

2010/8

Comparison between numerical and

theoretical solutions

2

0 22 ⎟

⎟ ⎞

⎜⎜⎛

−⋅=k

k

k

k h nsnsns

ω

ω

4949

f = 0.125[Hz] ( f / f 0 = 0.25)

f = 0.5[Hz] ( f / f 0 = 1)

f = 0.25[Hz] ( f / f 0 = 0.5)

f = 1.0[Hz] ( f / f 0 = 2)

Good agreement

Optimal PNS control formula

Application to seismic ground motion

• Optimal PNS for sinusoidal input2

00

0 22 ⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ −⋅=k

k

k

k h nsnsns

ω

ω ba B B =

50

– Frequency dependency

• Optimal PNS for earthquake input?

Optimality condition ba B B =

Earthquake ground motion for input

Name Descriptioin Peak Acc.

KobeHyogoken Nanbu Eq. Kobe JMA

(NS)817.8gal

ElCentro El Centro (NS) 341.7gal

Kawaguchi Niigataken Chuetu EQ. Kawaguchi

(NS)972.8gal

Ja anese Desi n S ecification for

51

T221

Highway Bridge Type2-II-1686.8ga

51 Acc. Response spectra

Direct numerical computation for

optimal PNS (base shear indicator)

Boundary lines for

Ba = Bb

5252

Kobe

Kawaguchi

El Centro

T221

Theoretical

solution？

Idea for theoretical solution of optimal PNS

control for seismic input

10

=k

k ns

DaS k B

0=

V nsb

S C B =

For the extreme case

5353

V ns D S cS k =0

⎪⎩

⎪⎨

⎧

=

=

V

Dns

ns

S

S k c

k k

0

0

S D：Max. displacement

response to EQ input

S V ：Max. velocity response

to EQ input

Fundamental solution

Idea for theoretical solution of optimal PNS

control for seismic input (2)

• The fundamental solution in terms of damping

ratio parameter is0

0 02 2

ns Dns

V

c S h

m S

ω

ω

= =

• Su st tut on nto t e opt ma PNS so ut on or sinusoidal input yields

• Proposed approximate solution:

20 0 02 1 12 2 2

D

V

S

S

ω ω ω

ω ω = ⋅ − =

D

V

S

S =ω

2

00

0 22 ⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ −⋅=

k

k

k

k h nsnsns

ω

ω

D

V

S

S =ω


10/11

2010/8

Estimation of SD and SV for a given

earthquake ground motion

• Assumption of k ns=k 0 implies that the system

is

responding

with

ideal

PNS

control• The SDOF system to be considered in

calculatin the maximum res onse SD and SV is such that

– stiffness k = 0

– damping ratio h = 0.534.

Numerical Example

• Original structural system model

with a semi

‐active

damper

x(t)

• SDOF system to calculate SD and

SV

x(t)c=2013 kN/m s

m=600 t

k 0=6100kN/mk ns , c ns

inputground motion( ) z t &&

k 0=0 kN/m

m=600 t

inputground motion( ) z t &&

Result

-Good agreement

between the

approximate

solution for the

optimal control

parameters and

hose determined

5757

Kobe

Kawaguchi

ElCentro

T221

by direct numerical

computation

Implication in the advantage in terms

of control device loads

Optimal PNS control

58

• Achieves mimimum acceleration

• Suggests the highest performance under damper

load constraint

Damper load

Fb

=Fs+F

d

u

Other ideas for Semi-Active Negative-

stiffness-type Control Algorithm

Fy

Fs

u

Fd

u

+Semi-Active

Damper ForceIsolator

- Proper energy dissipation for various amplitudes

- Limit the Absolute Base Acceleration

- Limited hysteretic l oop shape for large amplitude response

- Proper energy dissipation for various amplitudes

- Limit the Absolute Base Acceleration

- Limited hysteretic l oop shape for large amplitude response

=Base Shear

Hysteretic Loop

Schematic of the damping load components

uu

F3

u

F2F1

Fy

u

Fd =F1+F2+F3F b

u

Semi-Active

Damper ForceBase shear


11/11

2010/8

⎩⎨⎧

×++=

00

0321

uF if

uF if F F F F

d

d d

&

&

Z F F y=1 u A Z u Z Z u Z nn

&&&& +−−= − β γ 1

Key Expressions for the Control Algorithm

⎪⎪

⎩

⎪⎪

⎨

⎧

>++−

>×

=

prof.igarashi july27 apss2010

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