Class Material 25.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 25

VIII-3. Random fields

~ Random quantity distributed over _______________ field (space or time)

Ex1) Spatial Distribution of Random Ground Motion Intensity)

Ex2) Spatial distribution of material property (Young’s Modulus)

Ex3) Ground acceleration time history x t_g( )

e.g.

random process, stochastic process

⇒ ( ) # of random variables

⇒ ( ) representation is required 𝒙𝒙̈𝒈𝒈 (𝒕𝒕)

(Song & Ok, 2010)

𝒕𝒕

( , )

E x y

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

 Discretization of Random field → Random vector

→ v={ ,v₁ ,v_n}^T Random n-vector

[ ] { }

[( )( ) ]

covariance matrix

? where diag[ ]

[ ]

( ) joint PDF of

i

i

i j

v

T

v v

v v

vv v v

E E

D R f

µ

σ ρ

= =

 

= − −

  =

  =

  =

  →



v

vv

v vv v

v

M v

Σ

v M v M

D R D

v v

 Theoretical Representation of R.F

( ),

v x x ∈Ω

random field in domain Ω Partial descriptors:

µ ( ) x

: mean function

E v x [ ( )]

2

( )

σ x

: variance function

E v [ ( )]

²

x − µ

²

( ) x

( , )

'

ρ x x

: correlation coefficient function

ρ

_{v( ) ( ')xvx}

For Gaussian R.F. the above gives a complete specification For Nataf R.F., also specify

F v x

_v

( ; )

For general RF’s, specify joint PDF of ( ) and ( ) for,

x x , ' ∈Ω

,

f

_vv

( ( ), ( ')) v x v x

e.g. _____________ Random field

~ _____________ does not change over the domain Ω

( )

v x

, x∈Ω

µ ( ) x

=

2

( ) σ x

=

( , )

'

ρ x x

⁼

( ; )

F v x

=

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr Note; This doesn’t mean

v ( ) x = v

(not constant over the domain)

Scenario 1

Scenario 2

¹

1.5 1.5

²

( ) 1.5

x x

µ µ

µ µ

= =

= =

x

v ( ) x = v

Q: Correlation Function

ρ ( , ) x x

^' ←meaning?

How to capture this from

ρ ( , ) x x

^' ?

 Correlation length

0

( x dx ) θ =

^∞

∫ ρ ∆

~ measure of the distance over which significant loss of correlation occurs

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

Examples

• ( ) exp x

x a

ρ

∆ = ^−^{∆ }

0

0

exp exp

x d x a

a x a

a θ

^∞

∞

 ∆ 

=   −   ∆

 ∆ 

= −   −   =

∫

•

2

( ) exp x

2

x a

ρ ∆ =   − ∆  

 

2 2 0

2 2

exp

1 exp

2

1 2

x d x a

x d x a

a a

θ

π θ

∞

∞

−∞

 ∆ 

=  −  ∆

 

 ∆ 

=  −  ∆

 

= ∝

∫

∫