Multivariate Fragility Curves for Performance-Based ...

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


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


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

0

1

ln1

p

p

p

+¥-¥

Binomial Logistic Regression

1 2 2 3 3ln ( ) ( )1

ph h

p

x x


0,1y

1 2 2 3 3ln ( ) ( )1

ph h

p

x x


( 1)p P y

1 2 2 3 3ln ( ) ( )1

ph h

p

x x


( 1)p P y

1 ( 0)p P y

1 2 2 3 3ln ( ) ( )1

ph h

p

x x


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


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


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


( )j jp P DS ds

1 2 1kp p p

2 2 3 3ln ( ) ( )1 j

j

ds

j

ph h

p

x x


( )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


jDS ds

1 1

( ) ij

n ky

ij

i j

L p

θ


1ijy jDS ds

1 1

( ) ij

n ky

ij

i j

L p

θ


1ijy ijL pjDS ds

1 1

( ) ij

n ky

ij

i j

L p

θ


1ijy ijL p

1L0ijy

jDS ds

jDS ds

1 1

( ) ij

n ky

ij

i j

L p

θ


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


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


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


f'cρlb

Step

Co

rrel

atio

n o

f

Var

iati

on

of

i


f'cρlb ρlw

Step

Co

rrel

atio

n o

f

Var

iati

on

of

i


f'cρlb ρlw ρhb

Step

Co

rrel

atio

n o

f

Var

iati

on

of

i


f'cρlb ρlw ρhb fylb

Step

Co

rrel

atio

n o

f

Var

iati

on

of

i



P/(Ag.f'c )

Step

Ch

ang

e in

dev

ian

ce

In e

ach

ste

p (

%)

Residual Deviance in Each Step


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

Random Number Generator

Yang et al. (2009)

Damage State


Random Number Generator

Yang et al. (2009)

Damage State

1 2 2ln ( ) ( )1

p p

ph h

p

x x

Damage State


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

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