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transcript
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
Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
Outline
Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
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
ATC-58 Project
ln
( )i
i
i
D
F D
Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
Multivariate Model
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
Multivariate Model
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
Multivariate Model
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
0
1
ln1
p
p
p
+¥-¥
Binomial Logistic Regression
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
Binomial Logistic Regression
0,1y
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
Binomial Logistic Regression
( 1)p P y
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
Binomial Logistic Regression
( 1)p P y
1 ( 0)p P y
1 2 2 3 3ln ( ) ( )1
ph h
p
x x
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
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
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
Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
Multinomial Logistic Regression
( )j jp P DS ds
2 2 3 3ln ( ) ( )1 j
j
ds
j
ph h
p
x x
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
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
Maximum Likelihood Method
1 1
( ) ij
n ky
ij
i j
L p
θ
Maximum Likelihood Method
jDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
Maximum Likelihood Method
1ijy jDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
Maximum Likelihood Method
1ijy ijL pjDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
Maximum Likelihood Method
1ijy ijL p
1L0ijy
jDS ds
jDS ds
1 1
( ) ij
n ky
ij
i j
L p
θ
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
Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
RC Shear Wall
u
wh
wl
DS for RC Shear Wall
d
V
cuy
1ds 2ds3ds 4ds
Observations for Model Development
d
V
cuy
2 observations
of drift in ds2
Observations for Model Development
d
V
cuy
2 observations
of drift in ds2
d
V
cuy
5 observations
of drift in ds2
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
Model Development1
0
1
0
ij
y
0.07 0.012
0 4.56
X
Model Development1
0
1
0
ij
y
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
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
p
Comparison
Logistic Regression
Multivariate Model
Selection of
Influential Variables
Application:
RC Shear Walls
Fragility Function
Univariate Model
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
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
ρlb
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw ρhb
Step
Co
rrel
atio
n o
f
Var
iati
on
of
i
Stepwise Deletion of hi(x)
f'cρlb ρlw ρhb fylb
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 )
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 )
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
Implementation in PBEE
Implementation in PBEE
Random Number Generator
Yang et al. (2009)
Damage State
Implementation in PBEE
Random Number Generator
Yang et al. (2009)
Damage State
Implementation in PBEE
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
Conclusion
1P( | ) lnj
j j
edpDS ds EDP edp
Conclusion
1P( | ) lnj
j j
edpDS ds EDP edp
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)
Thank you for your Attention