Class Material 09.pdf

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Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering [email protected]

457.646 Topics in Structural Reliability In-Class Material: Class 09

III. Structural Reliability (Component) - continued

 Joint Probability Distribution Models

⑤ Joint distribution models with marginal & corr. coeff (contd.)

a) Morgenstern: FX_i( ), xi i=1,..., & n αij but ρ_ij <0.30 b) Nataf model (Nataf, 1962)

★ Joint PDF by Nataf model

( ) ( ) det

φ_n

( )

f = f ⋅ J

=

X x Z z Z,X

z;R'

J

 

 

=  

 

 

Z,X

1

i

( )

n

X i

i

f x

=

 

=   ⋅

 ∏ 

Note:

( ) ( )

( ) φ( )

i

X i i

X i i i i

F x z

f x dx z dz

= Φ

=

★ ρ_ij

′

(corr. coeff. b/w Z_i and Z_j)?

ρ

( , )

i j

ij

f

X X

x x dx dx

i j i j

∞ ∞

−∞ −∞

  

=   

  

∫ ∫

2

ρ φ

( , ;

ρ

)

ij

   z z

i j ij

′ dz dz

i j

∴ =   

  

∫∫

In general, ρ′ij ρij

ρ_ij A 1

∴ ≤ < may not cover the whole range of

ρ

_ij

ρ

′ ≅

_ij F ⋅ρij Liu & ADK (Table 4~6) for pairs of selected distribution types

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Table 9: Range of ρij ~ wider (than Morgenstern)

Later used for transformation of dependent RVs intoU ~N(0, I)

X Z U

 Elementary Structural Reliability Problem

Describe the failure event in terms of _________ & __________

① Failure : g( )x =g( , )= ≤0

② Failure probability : P_f =P( ≤0)

,

( , )

( ) ( )

( )

f R S

R S s

s R S

s

P f r s drds f r s f s drds f r s dr f s ds

f s ds

=

= ⋅

=

∫∫

∫

If R&S are s.i P_f =

∫

ds

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OR

[ ]

( ) ( )

( ) if s.i

f S R R

r s S R R

R

P f s r f r dsdr f s r dsf r dr

f r dr dr

≤

=

∫∫

∫

③ Reliability Index by “Safety Margin,” βSM

: Safety Margin

Failure : {R− ≤S 0} ⇔{ ≤0}

{U_M }

⇔ ≤

※ Standardization

M

U = M

[ ]

M M

E U Var U

 =

 =



For n RVs, U = L D (X M)⁻¹ ⁻¹ −

( ) ( )

( )

M

f M U

U

P P U F

F

∴ = ≤ =

=

M :

FU depends on distribution of R and S e.g. special case ~ R and S are jointly normal Then U_M ~

Therefore ( β )

f UM SM

P =F − = M =

2 2 2

β : reliability index by safety margin

=

1 , μ μ

δ δ 2 δ δ ρ

SM

R S

R s R S RS

r r

=

= − =

+ −

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※ A. Cornell (1968. ACI codes)

Assumed R&S are jointly normal & used βSM to compute P_f

④ Reliability Index by “Safety Factor”

ln ln ln

F = = −

Failure :{ ≤0} (※ used for LRFD

φ R

_n

≥ ∑ γ

_k

Q

_k⁾

2

{ 0}

{ }

μ

β

σ

( )

F

F SF

F

f u

u

P F

⇔ ≤

∴ = =

=

= −

⇒ special case: R & S are jointly lognormal

F ~ U

( )

2 2

( )

2 2

( ) μ

σ

1 δ

ln 1 δ μ

, μ

ln(1 δ ) 2 ln(1 ρ δ δ ) ln(1 δ )

f LN F

LN F

S

LN R R

SF

R RS R S S s

P

r

β

r

∴ = Φ

=

 ⋅ + 

 

 + 

 

= =

+ − + + +

Safety factor-based reliability-index when R & S are jointly lognormal