Class 07 ( , , )n P y y = ( ) f y = ∑ ( ) Y g X X = = ( ) ( ) x h

457.646 Topics in Structural Reliability In-Class Material: Class 07

II-6. Functions of Random Variables (contd.)

 Derived Distribution of Functions (contd.)

②

m  n

, but NOT one-to-one mapping a) Discrete

( ,

1

,

_n

) P y

_Y

y 

( ,1 , ) roots of

P x_X xn

y = g(x) b) Continuous

all roots of Y

( )

f y  

y=g(x)

Example c)

( )

2

Y  g X  X

, X ~ N(0,1 )²

1 1

2 2

( )

( ) x h y

x h y

 

 

   



1 2

1 2

( ) ( ) ( )

Y X X

dx dx

f y f x f x

dy dy

 

2 2

1 2

1 1 1 1

exp exp

2 2

2 x 2 x

 

   

      

③

m  n

, one-to one mapping

1 1 1

1 1

( , , )

( , , )

n

m m n

m

Y g X X

Y g X X

Y

_

 

 

 

  

 

 



'

Y

Y Y' = g'(X)

Discrete

( ) ( )

P

_Y' y'

 P

_X x Then,

( ) ( )

P

^{Y y}

   P

^{X x} a) Continuous

1 1 1

( )

_m

( )

_{m m n}

f

_Y' y'

dy dy  f

_X x

dx dx dx

_

dx

( ) ( ) det 1

f_Y' y'  f_X x J_Y',X ^

1

( ) det x

f J m m^

 _X x _Y,X

1 1 1

1 2

', 1

m

m m

Y X m

y y y

x x x

y y

x x

J

  

 

   

 

 

  

 

  

 

 

 

 

 

 

 



1

1

( ) det 1

m n

m n

m m

x x

f f J dx dx





 



 

Y y X(x) Y,X

Example d)

1 2

Y   T T

←contd. From Example b)

( ) ?

f

Y

y

Y' _{1 1 2}

2

2

Y T T

Y T

  

 

  

( ) ( ) det 1

f_Y' y'  f_T t J_Y',T ^{ 1} detJ_Y,T1 1^_ 

1

 f_T( ) dett J_Y',T1 1^_



1 1 2

( ) _Y( )

f^{Y y}  f y 



dt

1( ) 2( ) 2

T T

f f dt





[exp( y) exp( y)], y 0

  



 

   



When

 

 , using I’Hopitals rule,

2

( )

lim ( ) lim exp( ), 0

Y ( )

f y y y y

   

  



 



    



④

m  n

, NOT one-to one mapping

Ⅲ. Structural Reliability (Component)

 Structural Reliability Analysis (contd.) e.g. Shear failure of RC beam w/o stirrups

Source: https://www.youtube.com/watch?v=DPQIpT1ZvXY

“Limit-state” function

( )

_{c d}

g

X

  V V

1 ^'

6 f b d^{c w}



V^d 0

   

where X { ,f b d_c^' _w, , ,



V_d, } random variables

Failure Probability ( ( ) ) Pf P g x



“Structural Reliability Analysis”

(Anatomical + Systematic)

Three important tasks for structural reliability analysis:

1) 2) 3)

 Joint Probability Distribution Models

① Joint Normal X~N(M_X,Σ_XX) a) Joint PDF

T 1

/ 2

1 1

( ) exp ( ) ( )

(2π)ⁿ det 2

f^X x   x M ^X Σ^XX x M ^X 

Σ

1 n 

1 1

2

( ) exp

2 2

X

f x x 

 

    

           

Uni-variate normal PDF (See supp.)

2 n 

1 2( ,1 2) ( )

fX X x x  f Bi-variate normal PDF (See supp.)

b) Properties

 Joint distribution completely defined by

 All lower order distribution are



 

 

 

 

  

 

 

 

 

X

 

 

 

 

  

 

 

 

 

MX

 

 

 

 

  

 

 

 

 



^XX

Given X₂ x₂, then X₁ ~N(M , Σ_{1|2 1,1|2}) Conditional mean and covariance

 

1

1|2 1 1,2 2,2 2 2

1

1,1|2 1,1 1,2 2,2 2,1





 



  



M = M + Σ Σ x M

Σ Σ Σ Σ Σ

e.g. n2, ^{i.e. 1}

2

X X

   

     

   

1 2

X X X

2 1~ ( 1 2, 1 2)

X N

 

if



0 (“ “)

2 2

1 1

1 2

2

x 

  



  

   

 



_{1 2}



2 2 2

1

1 2 (1 )













_{1 2}² 

 Uncorrelated ( ) s.i for jointly normal (in general,



0 s i. )

 Linear functions of X~N(M Σ, ) ^→ follow _____________



₀ Y = AX A

( ) ( )

f_Y y  f_X x  J_Y,X 

 det 

T 1

( ) exp 1( ) ( )

f^Y y  2 x M ^x Σ x M^xx  ^x  In summary, X ~N(M , Σ_{X XX})

⇒Y~ N(M , Σ_{Y YY})

Y  M

YY



Σ c) Standard Normal

For univariate, ‘standard normal’ means,



 ,





 For jointly normal,

M =X

ΣXX =

T / 2

1 1

~ ( , ) ( , ) exp

(2 ) det 2

n n

N 



 

    

^XX

 

Z 0 z z R z

/ 2

1 1

~ ( , ) ( , ) exp

(2 ) det 2

n n

N 



 

      

U 0 u

1 n i





U used for FORM/SORM For normal,

1 1

 

 

  



x = DLu + M

u = L D (x M)

457.646 Topics in Structural Reliability In-Class Material: Class 08

III. Structural Reliability (Component)

 Joint Probability Distribution Models

② Joint Lognormal

1

, ,

_n

X X

are jointly lognormal if

ln X

₁

, , ln X

_n are jointly _________

a) Parameters

 

²

λ

_i

 E  ln μ

_i

 0.5ln(1 δ ) 

_i

 

2 2 2

ξ

_i

 Var  ln(1 δ ) ( δ for δ 1) 

_i



_i

ln ,ln

ρ 1 ln(1 ρ δ δ )

ξ ξ

i j

X X ij i j

i j

 

b) Properties

 Completely defined in terms of ( ) & ( )

 All lower order distribution are jointly

 Conditional distribution are jointly

 Uncorrelated ⇄ S.I.

 Product / Quotient of jointly lognormal r.v.’s follows

 _,ln

1

ρ δ ρ

ξ

i j

X X j ij

i



③ General Joint Distribution Forms e.g. Johnson & Kotz (1976)

 on multivariate prob. distribution models

④ Joint Distribution by conditioning (e.g. Bayesian Networks)

1 1 1

( , ,

_n

) (

_n

, ,

_n

) f x x  f x x x

_



⑤ Joint Distribution model with : Prescribed marginals: ,i1, ,n and correlation coefficient matrix :

 Read CRC Ch.14

 See Liu & Der Kiureghian (1986) a) Morgenstern b) Nataf

※ “Copula”: a class of functions satisfying certain conditions that can be used to construct joint probability functions using marginal probability functions

𝐹(𝑥₁, … , 𝑥_𝑛) =𝐶(𝐹₁(𝑥₁), … , 𝐹_𝑛(𝑥_𝑛))

Nataf transformation/model employs “normal copula” to describe the dependence structure

See Lebrun and Dutfoy (2009a,b,c) for details about copulas a) Morgenstern distribution

1

( ) ( ) 1 α [1 ( )][1 ( )]

i i j

n

X i ij X i X j

i j i

F F x F x F x





 

      

  



X x

Q) Can we derive ( )

Xi i

F x from

F

_X

( )

x ?

i.e.

x x

₂

,

₃

, , x

_n



then

F

_X

( )

x

 ?

Q) Can we describe dependence using α ?_ij

( , )

i j

X X i j

F x x 

( , )

i j

X X i j

f x x



 

( ) ( ) 1 α [1 2 ( )][1 2 ( )]

i j i j

X i X j ij X i X j

f x f x F x F x

     



_ij

  

α 0

α 0

ij ij

 

  

Therefore, α_ij is a parameter that represents (corr coeff.) But α_ij ρ_ij

Liu & Der Kiureghian (1986) showed

μ μ

ρ ( , )

σ σ ^{i j}

j j

i i

ij X X i j i j

i j

x x

f x x dx dx

 

 

  

  



 

   4α_ijQ Q_{i j}

  ρ_ij 0.30

Where μ

( ) ( ) 0.28

σ ^{i i}

i i

i X i X i i

i

Q x F x f x dx





   

    

 

 



Table 1: Distributions considered Table 2:

Q

_i

Table 3 : maximum



_ij

⇒ In summary, using Morgenstern’s model, you cannot describe X X_i, _j whose ρ_ij 0.30

b) Nataf model (Nataf, 1962) (“Gaussian Copula”)

~ ( ), 1, , [ρ ]

Xi i ij

F x i  n



X

R

(1) (2) Nataf

 



'

~ ( )

[ρ ]

_ij

N



Z 0, R'

R'

Transformation to Z

Z

i



Why?

( ) ( )

( ) ( )

i i

i i

i

Z i X i

i

Z i X i

f z f x dx dz

f z f x

 

  

