Class Material 18.pdf

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr

457.646 Topics in Structural Reliability In-Class Material: Class 18

 FORM approximation (Hohenbichler & Rackwitz 1983)

① Series system

1

1

1

( ) ( )

( ( ) 0)

( 0)

m

sys i

i m

i i FORM m

i

P E P E

P g

P

=

=

=

=

= ≤

≅ ≤

x







Let

Z

_i

=

α

ˆ

_i

u

,

i = 1,



, m

[ ]

_i

E Z =

[ _i] 2

Var Z = = Therefore,

Z

_i

~ ( , )

T

[ , ]

[ ] [ ] [ ]

[ ] [ ] [ ]

i j

Z Z

T

E E E

E E E

ρ = = − ⋅

= ⋅ = = uu =

∴

Z ~ ( , )

^,

ρ

_{Z Zi j}

=

∴

1

( ) ( 0)

FORM m sys

i

P E P

=

≅



≤

1

1 ( )

m

i

P

=

= −



≤

1

_m

( , , ; )

= − Φ



R

Joint normal CDF of

Z ~ N ( ; ) 0 R

Where

Φ

_m

(

β

;R )

1 φ ( )

m

m d

β β

−∞ −∞

=

∫ ∫

^{ Z;R z}

# rv's

# comp's n

m

→

→

* * *

* *

( ) ( ) ( )( )

( )( ) 0

i i i i i i

i i i

G G G

G

+ ∇ −

= ∇ − ≤

u u u u u

u u u



0

⇔ ≤

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr

② Parallel system

1

1

1

1

1

( ) ( )

( ( ) 0)

( 0)

( )

( )

( , , ; )

m

sys i

i m

i i FORM m

i m

i sym m

i m

P E P E

P g P P

P

=

=

=

=

=

=

= ≤

≅ ≤

= ≥

= ≤

= Φ

x



R











→ may have huge errors due to curvatures

→ better linearization point?

“joint design point”

Hard to find or may not exist

Note: One could find such important domain using an adaptive sampling technique Kurtz, N., and J. Song (2013). Cross-entropy-based adaptive importance sampling using Gaussian mixture. Structural Safety. Vol. 42, 35-44.

③ General system?

⇒ No direct FORM approximation

Step 0

X₁

X2

-6 -4 -2 0 2 4 6

-6 -4 -2 0 2 4 6

Step 2

X₁

X2

-6 -4 -2 0 2 4 6

-6 -4 -2 0 2 4 6

Step 4

X₁

X2

-6 -4 -2 0 2 4 6

-6 -4 -2 0 2 4 6