457.646 Topics in Structural Reliability In-Class Material

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

x

n

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

π π

   

=   −   +   −   =

③

m < n

, one-to one mapping

1 1

1 1

( , , )

( , , )

m 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}

det J

1 1⁻

×

=

Y,T

1

f ( ) det J

1 1⁻

=

_T

t

_Y',T ×

=

1 1 2

( )

_Y

( )

f

^Y

y = f y = ∫ dt

1

( )

2

( )

2

T T

f f dt

= ∫

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

αβ β α

= α β − − − >

−

When

α β =

, using I’Hopitals rule,

2

( )

lim ( ) lim exp( ), 0

Y

( )

f y y y y

β α β α

β α α

β

→ →

∂

= ∂ = − >

∂

∂

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}

ε V

^d

0

= + − ≤

where

X = { , f b d

_c^' _w

, , , ε V

_d

,



}

random variables

Failure Probability

( ( ) ) P

f

= 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 µ

πσ σ

  −  

=    −       

Uni-variate normal PDF (See supp.)

2 n =

1 2

( ,

1 2

) ( )

f

X X

x x = f

Bi-variate normal PDF (See supp.) b) Properties

• Joint distribution completely defined by

• All lower order distribution are

•

 

 

 

 

=  

 

 

 

 

X

 

 

 

 

=  

 

 

 

 

M

X

 

 

 

 

=  

 

 

 

 

∑

^XX

Given

X

₂

= x

₂

, then X

₁

~ N ( M ,

_1|2 Σ_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 µ σ

if

ρ = 0

(“ “)

2 2

1 1

1 2

2

x µ µ µ ρσ

σ

 − 

= +  

 

µ

_{1 2}

=

2 2 2

1 2 1

(1 )

σ = σ − ρ

σ

_{1 2}²

=

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

ρ = 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^−xx

x M −

^x

  

In summary,

X ~ N (M ,

_X Σ_XX

)

⇒

Y ~ N (M ,

_Y Σ_YY

)

Y

= M

YY

=

Σ c) Standard Normal

For univariate, ‘standard normal’ means,

µ = , σ =

∴ For jointly normal,

M =

X

ΣXX

=

T / 2

1 1

~ ( , ) ( , ) exp (2 ) det 2

n n

N ϕ

π

 

= ⋅   −

^XX

 

Z 0 z z R z

/ 2

1 1

~ ( , ) ( , ) exp (2 ) det 2

n n

N ϕ

π

 

= ⋅   −  

U 0 u

1 n

i=

= ∏

U used for FORM/SORM For normal,

1 1

− −

 

  −



x = DLu + M

u = L D (x M)