Structural Control for Seismic Protection of Structures 2010.7.27 APSS, Tokyo
Akira Igarashi Associate Prof., Graduate School of Engineering Kyoto University
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
Time
(a) Seismic Ground Motion Maximum acceleration (absolute value)
Seismic Ground Motion
(b) SDOF systems
Diagram representing “Natural period of structure” t t ” vs. “maximum “ i response”
Sa ( T )
Acceleration Response Spectrum (max. absolute acc. is used)
Response Sv ( T ) Velocity Spectrum (max.
Natural Period
relative velocity) Displacement Response Spectrum (max. relative displ.) where T =natural period
Sd ( T )
Response Spectra & Seismic Action •
From seismic design perspective…
：
：
Maximum load acting on the structure＝
a = x+z （ b l acc.）） （absolute Natural period ＝T
(d) Response Spectrum Time
(c) Response of each SDOF system
Demonstration of Seismic Action • Numerical Simulation
mass×maximum response acceleration
M F=Ma F=Ma
Maximum structural deformation＝maximum response displacement
(force acting on the structure）
Fmax = M ⋅ Sa (T ) xmax = Sd (T )
z
x (deformation)
where T = natural period of the structure
1
2010/8/6
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 Seismic Load Sa(T)
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 Seismic load Sa(T)
h=5%
T
Natural Period
h=5%
Low damping
＝ Greater seismic load
High damping ＝ Smaller seismic load
T
Natural period
Demonstration of the Effect of Damping on the Seismic Action • Numerical Simulation
2. Protective systems against seismic loads
Earthquake Protective Systems
Seismic Isolation System Also called as “Base Isolation” for the typical yp layout of isolators shown
Structure Energy Dissipation Device Isolator
2
2010/8/6
Seismic Isolation for Bridges (Example) Girder Isolator
Large Deformation Increased Damping Natural Period
Short period = Stiff
Long period = Soft
3
2010/8/6
Demonstration of Seismic Isolation Effect • Numerical Simulation
Energy Dissipation Devices • A type of passive structural control • Used combined /w isolators for seismic isolation • Adding damping capability of structures ddi d i bili f
Isolated bridge
Conventional
Oil Damper
Passive Control for Bridges (example) Girder Bearing Seismic damper Pier/abutment substructure
Laminated Rubber Dampers & Implementation for a Cable Stayed Bridge
Laminated Rubber Assembly
Girder Main Tower Damper cable
4
2010/8/6
Effect of Seismic Damper Seismic Load
Seismic load decreases
Summary for Seismic Load Protection • Seismic load decreases for – Longer natural periods – Higher damping (energy dissipation) capability
• Structural control technologies used in practice rely on (somehow) the above principle rely on (somehow) the above principle Natural Period
Short period = Stiff
Long period = Soft
Seismic Load Sa(T)
h=5%
・Also: effect of distributed lateral loads on plural piers for bridges
T
Active & Semi‐active controls 3. Active & semi‐active earthquake protection ideas q p
• Have been developed based on system control theory • Use of sensors, actuators and controllers Theoretical & Application development • Theoretical & Application development – Closed‐Loop Feedback Control – LQR, LQG, optimal control – H‐infinity control – Fuzzy control – …and more
Closed‐Loop Control
LQG Control dx (t ) = Ax ( t )dt + Bu( t ) dt + G c dW (t )
X (ω ) = H (ω ) F (ω )
X (ω ) =
H (ω ) F (ω ) 1 + H (ω )G (ω )
⎧ x1 (t ) ⎫ ⎪ M ⎪ ⎪ ⎪ ⎪ x (t ) ⎪ x(t ) = ⎨ n ⎬ v ( t ) ⎪ 1 ⎪ ⎪ M ⎪ ⎪ ⎪ ⎩⎪ vn (t ) ⎭⎪
⎡ 0 A=⎢ −1 ⎣ −M K
I ⎤ −M −1C⎥⎦
u(t ) = Fx (t )
J = E ⎡⎣ x T (t )R1x (t ) + uT ( t )R 2 u( t ) ⎤⎦
→
min
F = − R 2 −1BT P
PA + A T P − PBR 2 −1BT P + R1 = 0
• State‐feedback control
5
2010/8/6
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 • Control is designed based on the hysteretic li d i db d h h i response of the active/semi‐active devices • Avoiding complication of the control system
Motivations:
Motivation • What are the actual merits of introducing active/semi‐active systems over existing systems? – Flexibility against the change of dynamic Flexibility against the change of dynamic properties (natural periods etc.) of the structure to be controlled – Enhance control performance for limited control device loads
Active/Semi-active device Control…
maximum acceleration (=load) Limit the maximum damper of the structure forces maximum displ. response of the structure
- Effective for a wide range of excitation amplitude levels - Higher frequency components in damper loads
Base Isolation / Smart Base Isolation System • Control Device ‐ Passive
Background: Base isolation of liquid storage tanks • ‐ Greece Capacity: 65000m3 Diameter: 65.7m, Height: 22.5m. Each tank on 212 isolators Friction pendulum system (FPS)
(ex. Viscous damper, F i ti d Friction damper) )
Structure
Control Device Isolator
‐ Active ‐ Semi‐Active (ex. Variable Oil damper, MR damper)
• ‐ 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 )
6
2010/8/6
Pseudo Negative Stiffness Control for Semi‐ Active devices
4. Negative‐stiffness‐type Active & semi‐active control for earthquake protection
cns kns
Displ.
Load
F ≡ − k ns x + c ns x& ⎧ F (F ⋅ x& > 0 ) Fd = ⎨ ⎩0 (F ⋅ x& ≤ 0 )
• Pseudo Negative Stiffness (PNS) Control Fd = −k ns x + cns x&
Shift of natural periods →seismic load decreases more efficiently
• The value of the negative stiffness kns is recommended to be identical to the structural stiffness – Relationship between the required damper load l i hi b h i dd l d and the d h 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
Natural Period
Short period = Stiff
Load
PNS Control
Idea of optimization
Effect of Negative Stiffness Damper Seismic Load
Displ.
Long period = Soft
Optimal PNS Control Concept
Effects of parameter values on PNS control Hysteretic Response
Base shear vs. Displacement Hysteretic response
Str. Stiffness
x(t) m
+ St t l restoring Structural t i force f
40
+ Structural stiffness
kns, cns k0 Increased Acc Acc.
D Damper lload d
=
Increased damping
Base Shear Possible Maximum Base Shear
Str. Stiffness
+ Structural stiffness
Affecting the parameter determination
41
42
7
2010/8/6
Optimality criterion for PNS control Str. stiffness
Base Shear Indicator
Base shear
Increase of damping
Bmax = −m0 × &x&max
Ba = k0 × xmax Base shear at Point a
DIspl
Maximum base shear
Bmax = max(Ba , Bb ) Base shear at Point b
ΔB ≡ Bmax − Ba
Maximum energy dissipation w/o increasing base shear
( B a < Bb ) ( B a ≥ Bb )
⎧ Bb − Ba =⎨ ⎩0
ΔB Area I
Area I ： Ba < Bb Area II : Ba ≥ Bb
Ba = Bb Area II
Optimality condition 43
Example SDOF model mass m0 stiffness k0 damping coef c0 damping ratio h0 Natural freq f0 N t l period Natural i d T0
• Different combinations of PNS control parameters kns and cns 45
Theoretical representation of optimal PNS control parameters Bb = xmax
(cnsω )2 + (k 0 − k ns )2
f = 0.5[Hz] (f / f0 = 1)
46
f = 1.0[Hz] (f / f0 = 2)
Derivation of optimal PNS control parameter representation Ba = k 0 x max
Bb = xmax
(cnsω )2 + (k 0 − k ns )2
Ba = k 0 x max PNS control
Fd = −k ns x + cns v
k0：structural stiffness xmax：maximum displ displ.
• Do the optimal combinations of PNS control parameters allow mathematical expressions? • Assumption : structural response is approximated by a sinusoidal function 47
Relationship between hns and ω ：frequency ratio ω0
2
k ns k0 48
8
2010/8/6
Comparison between numerical and theoretical solutions
Application to seismic ground motion • Optimal PNS for sinusoidal input Ba = Bb
hns = f = 0.125[Hz] (f / f0 = 0.25)
f = 0.25[Hz] (f / f0 = 0.5)
ω0
2ω
2⋅
k ns ⎛ k ns ⎞ ⎟ −⎜ k 0 ⎝⎜ k0 ⎟⎠
2
Optimal PNS control formula
Good agreement
hns =
ω0
2⋅
2ω
k ns ⎛ k ns ⎞ −⎜ ⎟ k 0 ⎜⎝ k0 ⎟⎠
2
– Frequency dependency
• Optimal PNS for earthquake input? Optimality condition
Ba = Bb
49
f = 0.5[Hz] (f / f0 = 1)
50
f = 1.0[Hz] (f / f0 = 2)
Direct numerical computation for optimal PNS (base shear indicator)
Earthquake ground motion for input Name
Descriptioin Hyogoken Nanbu Eq. Kobe JMA (NS) ElCentro El Centro (NS) Niigataken Chuetu EQ. Kawaguchi Kawaguchi (NS) Japanese Design Specification for T221 Highway Bridge Type2-II-1 Kobe
Peak Acc. 817.8gal
Boundary lines for Ba = Bb
341.7gal 972.8gal 686 8 l 686.8gal Kobe
Acc. Response spectra
51
52 Kawaguchi
Idea for theoretical solution of optimal PNS control for seismic input For the extreme case k ns =1 k0
Ba = k0 S D SD：Max. displacement response to EQ input SV：Max. velocity response to EQ input
Theoretical solution？
El Centro
T221
Idea for theoretical solution of optimal PNS control for seismic input (2) • The fundamental solution in terms of damping ratio parameter is hns =
Bb = Cns SV
cns ωS = 0 D 2m0ω0 2 SV
• SSubstitution into the optimal PNS solution for b i i i h i l S l i f sinusoidal input yields
k0 S D = cns SV
ω0 S D
⎧k ns = k0 ⎪ k0 S D ⎨ ⎪cns = S V ⎩
2 SV
=
ω0
2ω
2 ⋅1 − 12 =
ω0
ω=
2ω
• Proposed approximate solution:
Fundamental solution
hns = 53
ω0
2ω
2⋅
k ns ⎛ k ns ⎞ −⎜ ⎟ k0 ⎜⎝ k0 ⎟⎠
2
ω=
SV SD
SV SD
9
2010/8/6
Estimation of SD and SV for a given earthquake ground motion
Numerical Example
• Assumption of kns=k0 implies that the system is responding with ideal PNS control • The SDOF system to be considered in g p calculating the maximum response SD and SV is such that
• Original structural system model • SDOF system to calculate SD and with a semi‐active damper SV x(t)
x(t)
– stiffness k= 0 – damping ratio h = 0.534.
m=600 t
kns, cns k0=6100 kN/m
k0=0 kN/m
&& z (t )
&& z (t )
input ground motion
Result
Kobe
c=2013 kN/m s
m=600 t
-Good agreement between the approximate solution for the optimal control parameters and those determined by direct numerical computation
ElCentro
input ground motion
Implication in the advantage in terms of control device loads Optimal PNS control
Damper load
• Achieves mimimum acceleration • Suggests the highest performance under damper load constraint Kawaguchi
57
T221
58
Other ideas for Semi-Active Negativestiffness-type Control Algorithm Fs
Schematic of the damping load components Fb=Fs+Fd
Fd
F1
Fy
+ Isolator
Semi-Active Damper Force
=
u
u
u
u
u
F3
F2
Fy u
Computed components of damper force commands Base Shear Hysteretic Loop
Fb
Fd=F1+F2+F3
u
u
- Proper energy dissipation for various amplitudes - Limit the Absolute Base Acceleration - Limited hysteretic loop shape for large amplitude response
Semi-Active Damper Force
Base shear
10
2010/8/6
Key Expressions for the Control Algorithm ⎧ F + F2 + F3 Fd = ⎨ 1 ⎩0 F1 = Fy Z
Response of a Controlled SDOF System to Harmonic Loads: Example
if Fd × u& > 0 if Fd × u& < 0
Z& = −γ u& Z Z
n −1
m Fd
− β u& Z + Au&
Fs
Base shear
Damper force
n
Ag
T=2 Sec.
⎧ k d 1 .Z .u y + C d 1 .u& ⎪ ⎪0 F2 = ⎨ ⎪0 ⎪ F y − F1 ⎩
If Z < 1.0
Amplitude= 1m/s2
Z = 1.0 if F2 × u& > 0 if ( F1 + k d1 Z .u y + C d 1 .u& ) > F y If
