Attenuation of Inelastic Spectra and Its Applications
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Transcript of Attenuation of Inelastic Spectra and Its Applications
Yousef Bozorgnia,Mahmoud Hachem,Kenneth Campbell
PEER GMSM Workshop,UC BerkeleyOctober 27, 2006
Attenuation of Inelastic Spectra and Its Applications
Background: Selected Ground Motions
3122 Horizontal records
64 Worldwide earthquakes
Magnitude: 4.3 - 7.9
Distance range: 0 – 200 km
Computation of inelastic spectra
Inelastic spectra were computed for 3122 horizontal records
For periods: 0.02 to 10 sec
1999 Duzce EQ (M W 7.1), Turkey, Duzce Station, H GM
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fy /
W
= 1
= 2
= 4
= 6
= 8
D eform ation
Fy
uy um ax
Fe
Elastic response
Fs
Inelastic response
Force
Fs
= F
e / R
Fy
= F
e / R
d
ue
1999 Duzce EQ (M W 7.1), Turkey, Duzce Station, H GM
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fy /
W
= 1
= 2
= 4
= 6
= 8
D eform ation
Fy
uy um ax
Fe
Elastic response
Fs
Inelastic response
Force
Fs
= F
e / R
Fy
= F
e / R
d
ue
Computed parameters: Strength ductility displacement energy,…
Inelastic spectra for: Constant ductility; Constant strength; Constant damage index
Overall: > 1,240,000 nonlinear runs
Computation of inelastic spectra
Attenuation of Inelastic Spectra
Having computed: Inelastic spectra: 3122 (records) * 5 (ductility ratios),
and Damage spectra: 3122 (records) * 5 (damage index
values)
For a given value of ductility or damage index
Nonlinear regression analyses were performed to correlate inelastic response to Magnitude Distance to fault Soil condition Style of faulting …
Attenuation of Inelastic Spectra -- Model
3 6 7( ) ( )RV F Nf F c F f H c F
1 2 3 4 5 6ln ( ) ( ) ( ) ( ) ( ) ( )Y f M f R f F f HW f S f D
0 1
1 0 1 2
0 1 2 2
5.5
( ) ( 5.5) 5.5 6.5
( 5.5) ' ( 6.5) 6.5
c c M M
f M c c M c M M
c c M c M c M M
30 309 2 30 1
1 1
5
309 2 30 1
1
ln ln ln
( )
( ) ln
n
s sr r s
ss
V Vc k PGA c PGA c V k
k kf S
Vc k n V k
k
2 22 3 4 5( ) ( ) ln rupf R c c M r c
4 8( ) ( ) ( ) ( )RV HW HW HWf HW c F f R f M f H
10
6
3 4
( 1) 1
( ) 0 1 3
exp 10 exp 3.333 exp 0.75 exp 0.25 3
c D D
f D D
k D k D D
1 0
( )0
jb
rup jbHWjb
rup
r
r rf Rr
r
0 6.0
6.0( ) 6.0 6.5
0.51 6.5
HW
M
Mf M M
M
0 20( ) 20
2020
HW
Hf H H
H
Attenuation of Inelastic Spectra:For Different Ductility Ratios
Strike-Slip Fault, M w=7.5, T =0.2, Vs30=760, D=1
0.1 1.0 10.0 100.0
Distance to Fault Rupture (km)
0.01
0.1
1
Fy /
W
= 1
= 2
= 4
= 6
= 8
Strike-Slip Fault, M w=7.5, T =1.0, Vs30=760, D=1
0.1 1.0 10.0 100.0
Distance to Fault Rupture (km)
0.01
0.1
1
Fy /
W
= 1
= 2
= 4
= 6
= 8
Attenuation of Inelastic Spectra:For Different Magnitude
Strike-Slip Fault, = 4, T =0.2, V s30=760, D=1
0.1 1.0 10.0 100.0
Distance to Fault Rupture (km)
0.01
0.1
1
Fy /
W
M W = 7.5
M W = 6.5
M W = 5.5
M W = 4.5
Strike-Slip Fault, = 4, T =1.0, V s30=760, D=1
0.1 1.0 10.0 100.0
Distance to Fault Rupture (km)
0.01
0.1
Fy /
W
M W = 7.5
M W = 6.5
M W = 5.5
M W = 4.5
Mag Saturation
Attenuation of Inelastic Spectra
Strike-Slip Fault, M w=7.5, rrup=0.1, Vs30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
Fy /
W
= 1
= 2
= 4
= 6
= 8
C&B NGA Elastic
Strike-Slip Fault, M w=7.5, rrup=10.0, V s30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
Cy
= F
y /
W
= 1
= 2
= 4
= 6
= 8
C&B NGA Elastic
Attenuation of Elastic Spectrum Becomes a Special Case
Damage Spectra (Bozorgnia & Bertero, 2003)
Normalized Ductility Normalized Hysteretic Energy
DI1= (- e)/(mon-1) (1-1) + 1 EH/EHmon
DI=0, if Elastic response DI=1, if it reaches deformation capacity under monotonic lateral deformationOther performance states fall between DI=0,1
Damage Spectra: An example
D eform ation
Fy
uy um ax
Fe
Elastic response
Fs
Inelastic response
Force
Fs
= F
e / R
Fy
= F
e / R
d
ue
1978 Tabas EQ (M W 7.35), Iran, Tabas Station, HGM
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Fy /
W
DI = 0.0
DI = 0.2
DI = 0.4
DI = 0.6
DI = 0.8
DI=0, Elastic spectrum
Attenuation of Damage Spectra
Strike-Slip Fault, M w=7.5, rrup=0.1, Vs30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
Fy /
W
DI= 0.0
DI= 0.2
DI= 0.4
DI= 0.6
DI= 0.8
Strike-Slip Fault, M w=7.5, rrup=10.0, V s30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
Fy /
W
DI= 0.0
DI= 0.2
DI= 0.4
DI= 0.6
DI= 0.8
DI=0, Elastic spectrum
Displacement Inelastic Spectra
DI= 0.4, =0.3,Strike-Slip Fault, M W=6.5, V s30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
5.0
10.0
15.0
20.0
25.0
um
ax
( c
m )
rrup= 0.1 km
rrup= 10 km
rrup= 40 km
rrup= 200 km
DI = 0.4, =0.3, Strike-Slip Fault, rrup=10.0, Vs30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
5.0
10.0
15.0
20.0
25.0
um
ax
( c
m )
M w= 7.5
M w= 6.5
M w= 5.5
M w= 4.5
Use of Inelastic Spectra for GMSM
1. Simple scaling using attenuation of inelastic spectra
2. Inelastic spectrum matching
Scaling of Ground Motion
Scaling Law of Inelastic Spectra:Bertero, Mahin, Herrera (August 1976)
For given ductility and period, If you want to scale “Yield strength” Cy
by a factor of λ You will have to scale time history by
scale factor λ
Selection and Scaling of THs: Step1
Select: Target Performance Level Select ductility; or Damage Index, or
…
Select: period
Select: site of the structure Mag, Style of Faulting, Rrup, Vs30, …
Step2: Initial Selection of Time Series
Various options … M,R,… bin M,R,ductility,… bin M,R,DI,… bin …
Example
Attenuation of Inelastic Spectra
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4
Period (sec)
Cy
= F
y /
W mu = 1
mu = 2
mu = 4
mu = 6
Inelastic Spectrum Matching
Given Inelastic Design Spectrum (e.g., attenuation of inelastic spectra)
Strike-Slip Fault, M w=7.5, rrup=10.0, V s30=760, D=1
0.0 1.0 2.0 3.0 4.0
Period (sec)
0.0
0.2
0.4
0.6
0.8
1.0
Cy
= F
y /
W
= 1
= 2
= 4
= 6
= 8
C&B NGA Elastic
And a Record
Inelastic Spectrum, Manjil EQ, Abbar Station, Ductility =4
00.10.20.30.40.50.60.70.8
0 1 2 3 4Period (sec)
Cy
(=F
y /
W )
Abbar_0
Is it possible to modify the record to match the Inelastic Spectrum?
Inelastic Spectra, Ductility =4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4Period (sec)
Cy
(=F
y /
W )
Target
Abbar_0
Answer: Yes
Inelastic Spectra; Ductility=4
Ground Motions
0 10 20 30 40 50-0.5
0
0.5
0 10 20 30 40 50
-0.2
0
0.2
Initial
Final
It is relatively fast …
Thank You!