![Page 1: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/1.jpg)
Multivariate Fragility Curves for Performance-Based
Earthquake Engineering
by:
Abbas Javaherian Yazdi
Dr. Terje Haukaas
Dr. Tony Yang
2013 UBC-Tongji-CSRN Symposium
August 19-22, 2013
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Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
Outline
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Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
![Page 4: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/4.jpg)
Fragility Functions
P(D
S ≥
ds j
)
1 2 3 4 5 6 700.0
0.2
0.6
0.4
0.8
1.0
edp – Interstory drift ratio (%)
ds2
ds3
ds4
1( ) ( ) ( )i i iP DS ds P DS ds P DS ds
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ATC-58 Project
ln
( )i
i
i
D
F D
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Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
![Page 7: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/7.jpg)
Multivariate Model
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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Multivariate Model
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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Multivariate Model
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
0
1
ln1
p
p
p
+¥-¥
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Binomial Logistic Regression
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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Binomial Logistic Regression
0,1y
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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Binomial Logistic Regression
( 1)p P y
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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Binomial Logistic Regression
( 1)p P y
1 ( 0)p P y
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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Binomial Logistic Regression
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
( 1)p P y
1 ( 0)p P y
1
1
( ) (1 ) i i
ny y
i i
i
L p p
θ
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Binomial Logistic Regression
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
( 1)p P y
1 ( 0)p P y
1
1
( ) (1 ) i i
ny y
i i
i
L p p
θ 1y L p
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Binomial Logistic Regression
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
( 1)p P y
1 ( 0)p P y
1
1
( ) (1 ) i i
ny y
i i
i
L p p
θ 0 1y L p
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Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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Multinomial Logistic Regression
( )j jp P DS ds
1 2 1kp p p
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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Multinomial Logistic Regression
( )j jp P DS ds
1 2 1kds ds ds
1 2 1kp p p
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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Maximum Likelihood Method
1 1
( ) ij
n ky
ij
i j
L p
θ
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Maximum Likelihood Method
jDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
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Maximum Likelihood Method
1ijy jDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
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Maximum Likelihood Method
1ijy ijL pjDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
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Maximum Likelihood Method
1ijy ijL p
1L0ijy
jDS ds
jDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
![Page 28: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/28.jpg)
Maximum Likelihood Method
ln ( )0
i
L
θ
1,2, ,i n
1 1
( ) ij
n ky
ij
i j
L p
θ
1ijy ijL p
1L0ijy
jDS ds
jDS ds
1 2 1kds ds ds
![Page 29: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/29.jpg)
Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
![Page 30: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/30.jpg)
RC Shear Wall
u
wh
wl
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DS for RC Shear Wall
d
V
cuy
1ds 2ds3ds 4ds
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Observations for Model Development
d
V
cuy
2 observations
of drift in ds2
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Observations for Model Development
d
V
cuy
2 observations
of drift in ds2
d
V
cuy
5 observations
of drift in ds2
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Arrangement of Test Data
DS P/(Ag.f'c ) hw/lw f'c fyl fylb ρlw ρlb ρhw ρhb
1 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 0.012
1 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 0.60
2 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 0.63
2 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 2.16
3 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 2.18
3 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 2.86
4 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 2.88
4 0.073 2.00 47.1 399.91 471.62 0.27 3.17 0.26 1.55 3.55
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Model Development1
0
1
0
ij
y
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0.07 0.012
0 4.56
X
Model Development1
0
1
0
ij
y
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0.07 0.012
0 4.56
X
1 1
( ) ij
n ky
ij
i j
L p
θ
Model Development1
0
1
0
ij
y
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0.07 0.012
0 4.56
X
Parameter 2 observations 5 observations
ds1-6.28 -7.81
ds2 -3.54 -4.44
ds3 -1.57 -2.26
2 -2.85 -3.82
3 0.88 1.18
4 0.0021 0.003
5 0.0013 0.0016
6 0.0011 0.0015
7 -0.29 -0.32
8 0.0078 0.0039
9 1.58 2.09
10 -0.16 -0.21
11 -2.72 -3.52
1 1
( ) ij
n ky
ij
i j
L p
θ
Model Development1
0
1
0
ij
y
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p
Comparison
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Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
![Page 41: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/41.jpg)
Select a comprehensive
set of hi(x)
Compute COV of i
and D1
Identify j with largest
COV
Remove hj(x),
compute D2
Is
(D2- D1)/D1
negligible?
Accept elimination of
hj(x)
Yes
NoThe model is accepted
with hj(x)
Model Selection
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Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
ρlb
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Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb
![Page 44: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/44.jpg)
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw
![Page 45: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/45.jpg)
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw ρhb
![Page 46: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/46.jpg)
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw ρhb fylb
![Page 47: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/47.jpg)
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw ρhb fylb
P/(Ag.f'c )
![Page 48: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/48.jpg)
Step
Ch
ang
e in
dev
ian
ce
In e
ach
ste
p (
%)
Residual Deviance in Each Step
f'cρlb ρlw ρhb fylb
P/(Ag.f'c )
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Model parameterExplanatory
functionMean of i COV of i in %
ds1- -7.78 3.1
ds2- -4.47 4.1
ds3- -2.32 7.2
2P/Agf ’c -3.41 12
3hw/lw 1.19 4.7
5fyl 0.0028 7.9
9rhw 1.65 7.1
11ln( -3.44 2.6
Model Statistics
![Page 50: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/50.jpg)
Implementation in PBEE
![Page 51: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/51.jpg)
Implementation in PBEE
![Page 52: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/52.jpg)
Random Number Generator
Yang et al. (2009)
Damage State
Implementation in PBEE
![Page 53: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/53.jpg)
Random Number Generator
Yang et al. (2009)
Damage State
Implementation in PBEE
![Page 54: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/54.jpg)
Random Number Generator
Yang et al. (2009)
Damage State
1 2 2ln ( ) ( )1
p p
ph h
p
x x
Damage State
Implementation in PBEE
![Page 55: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/55.jpg)
Conclusion
1P( | ) lnj
j j
edpDS ds EDP edp
![Page 56: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/56.jpg)
Conclusion
1P( | ) lnj
j j
edpDS ds EDP edp
![Page 57: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/57.jpg)
Conclusion
1P( | ) lnj
j j
edpDS ds EDP edp
2 3'
4 5 6
ln1
ln( )
j
wd
j
j
hw
s
g c w
yl
p hP
p A f l
f
r
hw/lw
p
P(DS≥ds1)
P(DS≥ds3)
P(DS≥ds2)
![Page 58: Multivariate Fragility Curves for Performance-Based](https://reader030.vdocuments.site/reader030/viewer/2022012510/61887d18b8c3a6210832585a/html5/thumbnails/58.jpg)
Thank you for your Attention