component reliability analysis

7/23/2019 Component Reliability Analysis

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Chapter 3Chapter 3

Component Reliability Analysis

of Structures


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Chapter 3: Element Reliability Analysis of StructuresChapter 3: Element Reliability Analysis of Structures

3.2 AFOSM Advanced First Order Second Moment Method

3.3 CMethod Recommended by the CSSCommittee

3.! M"FOSM Mean "alue First Order Second Moment

Method

Contents

3.# MCS Monte Carlo Simulation Method


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3.! M"FOSM

Mean ValueFirst Order Second Moment Method

Chapter 3Chapter 3

Component Reliability AnalysisComponent Reliability Analysisof Structuresof Structures


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3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "13.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "1

M"FOSM Mean "alue

First Order Second Moment

First Order: The first-order terms in the Taylor series expansion

is used.

This method is also namedMean Value Methodor

Center Point Method.

Second Moment: Only means and variances of the asic variales

are needed.

Mean Value orCenter Point: The Taylor series expansion is

on the means values.


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%&ample 3.!

Please refer to the textoo*

+(eliaility of Structures,

y Professor $. S. o!a*.

Turn to Pa%e &/" loo* at the example 0.&carefully1

3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "$3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "$


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3.!.2 'onlinear $imit State Functions

!. Assumptions

!here"

1 2( ) ( , , , )nZ g X g X X X= = L

the terms are uncorrelated random variales"iX

2. Formula

2e can otain an approximate solution y lineari3in% the nonlinear

function usin% a Taylor series expansion. The result is

Consider a nonlinear limit state function of the form

and its mean and standard deviation are " respectively .iX

iX

* * *1 2

* * * *

1 2

1 ( , , , )

( , , , ) ( )

n

n

n i i

i i x x x

gZ g x x x X x

X=

+

L

L

3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "%3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "%


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One choice for this lineari3ation point is the point

correspondin% to the mean values of the random variales.

1 2

1 21 ( , , , )

*

1

( , , , ) ( )

( ) ( )

n i

X X Xn

n

X X X i X

i i

n

i i

i i M

gZ g XX

gg M X x

X

=

=

+

= +

L

L

!here" is the point aout !hich the expansion is performed.* * *1 2( , , , )nx x xL

From no! on "this point is represented y . Therefore" the aove formula

can e re!ritten riefly as follo!s:

*P

*

* *

1

( ) ( )n

i i

i i P

gZ g P X x

X=

+

M

1 2( , , , )

nX X XM = L

The point is also called mean value pointor central point.M

3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "&3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "&


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Moments of the performance function '

1 2( , , , )nZ X X Xg L

( )2

2

1 1i i

n n

Z X i X

i ii M

ga

X

= =

=

!here"i

i M

ga

X=

( )

1 2

2 2

11

( , , , ) ( )n

ii

X X XZ

nnZ

i XX i

i i M

g g M

g a

X

=

=

= = =

L

Formula of (eliaility )ndex

3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "'3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "'


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3.!.3 Comments on M"FOSM

!. Advanta(es

2. )isadvanta(es

)t is very easy to use.

)t does not re5uire *no!led%e of the distriutions of the randomvariales.

(esults are inaccurate if the tails of the distriution functionscannot e approximated y a normal distriution.

There is an invariance prolem: the value of the reliaility indexdepends on the specific form of the limit state function.

That is to say" for different forms of the limit state e5uation!hich have the same mechanical meanin%s" the values ofreliaility index calculated y MVFOSM may e different 1

3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! ")3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! ")


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%&ample 3.3


+(eliaility of Structures,y Professor $. S. o!a*.

Turn to Pa%e &6" loo* at the example 0.7

carefully1

The invariance prolem is est clarified y

3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "1*3.1 MVFOSM Mean Value First Or!er Secon! Moment Metho! "1*


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3.2 AFOSM

$dvanced First Order Second Moment

Method

Chapter 3Chapter 3



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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "13.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1

AFOSM Advanced First Order Second Moment

To overcome the invariant prolem" 8asofer and 9ind propose an

advanced FOSM method in &64 " !hich is called $FOSM .

The +correction, is to evaluate the limit state function at a point

*no!n as the +desi%n point, instead of the mean values.Therefore" this method is also called +desi%n point method, or

+chec*in% point method,.

The +desi%n point, is a point on the failure surface .0Z=

Since the desi%n point is %enerally not *no!n a priori" an

iteration techni5ue is %enerally used to solve for the reliaility

index.


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3.2.! *rinciples of AFOSM

!. Assumptions

2. +ransformation from , space into - space

The %eneral random variale is transformed into its

standard form as follo!s:

i

i

i X

i

X

XU

=

3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "#3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "#

!here"

1 2( ) ( , , , )nZ g X g X X X= = L

the terms are uncorrelated random variales"iXand its mean value and standard deviation are *no!n.

iX

iX

Consider a nonlinear limit state function of the form

iX


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The ; space is then transformed into < space:


1 2( , , , )nX X X X= L 1 2( , , , )nU U U U = L

The desi%n point in ; space is then

transformed to in < space.

* * * *

1 2( , , , )nP x x xL

* * * *

1 2 ( , , , )

nP u u uL

The limit e5uation in ; space

1 2( ) ( , , , )nZ g X g X X X= = L

is transformed to < space as follo!s.

1 2( ) ( , , , )nZ G U G U U U= = L


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "$3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "$

)n < space" the tan%ent plane e5uation throu%h the desi%n pointon failure surface is( ) 0Z G U= =

*

* * * *

1 2

1

( , , , ) ( ) 0n

n i i

i i P

GG u u u U u

U=

+ =

L

Since the desi%n point is a point on the failure

surface " then !e have

0Z=*P

* * *

1 2( , , , ) 0nG u u u =L The hyper-plane e5uation can therefore e simplified as follo!s:

*

*

1

( ) 0n

i i

i i P

GU u

U=

=

3. Reliability nde& in - Space


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The distance from the ori%in of < space to the tan%ent plane is

actually the reliaility index

*

1u

*2u

1u

2u=esi%n pointTan%ent

Failure surface

1U

2U

*P

O

arg min{ | ( ) 0}HL G = =u u

3.# AFOSM3.# AFOSM A!+ance! First Or!er Secon! Moment Metho!A!+ance! First Or!er Secon! Moment Metho! %%

*HL O P

=

( ) 0G =u


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "&3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "&

From the %eometric meanin% of the reliaility index" !e *no!

*

*

*

1

2

1

n

i

i i P

n

i i P

G uU

G

U

=

=

=

9et

*

*

2

1

i Pi

n

i i P

G

U

G

U

=

=

is actually the direction cosine

of the distancei

*O P

cosii U =

* cosii U i

u = =


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "'3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "'

Since

*

* i

i

i X

i

X

xu

=

The desi%n point in ; space

!e have* *

i i i ii X i X X i X x u = + = +

The direction cosine in ; space

i

iX

i i i i

XG g g

U X U X

= =

*

*

2

1

i

i

X

i Pi

n

X

i i P

gX

g

X

=

=

* * *

1 2( , , , ) 0ng x x x =L

#. Reliability nde& in; Space


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "(3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "(

The reliaility index in ; space

*

*

*

1

2

1

i

i

n

X i

i i P

n

X

i i P

g uX

g

X

=

=

=

( )( )

*

1

2

1

i

i

n

i X ii

n

i X

i

a x

a

=

=

=

*

* i

i

i X

iX

x

u

=

*

i

i P

ga

X

=


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1*3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1*

3.2.2 Computation Formulas of AFOSM

* * *

1 2( , , , ) 0

ng x x x =L

*

*

2

1

i

i

X

i Pi

n

X

i i P

g

X

g

X

=

=

( 1, 2, , )i n= L

*

i ii X i X x = + ( 1, 2, , )i n= L / / / / / /021

/ / / / /0!1

/ / / / / / / / / /031

1

1 ( )f fp p = / / / / / / / / / /0#1


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3.2.3 teration Al(orithm of AFOSM

&. Formulate the limit state e5uation

1 2( , , , ) 0ng X X X =L

>ive the distriution types and appropriate parameters of all randomvariales.

/. $ssume the initial values of desi%n point and reliaility index*iX

)n %eneral" the initial value of desi%n point is ta*en as mean value .iX

Then the initial value of is .

7.


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1#3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1#

6. >o ac* to Step 7 and repeat. )terate until the values conver%e.

Begin

Assume *

ii Xx =

Calculate i

Calculate *

i ii X i X x = +

Calculate from ( ) 0g =

( 1) ( )k k +

Outut an! *ix

o es

Flo!chart


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%&ample 3.#


210M kN m= $ssume that a steel eam carry a deterministic endin% moment "

The limit state e5uation is

The plastic section modulus and the yield stren%th of the eam are

statistically independent" normal random variales. )t is *no!n that

W yF

"#$2

W

cm = 0%02W

="$0

yF Mpa = 0%0&

yF =

( , ) 0y yZ g F W F W M= = =

yF

Calculate the reliaility index of the eam as !ell as the chec*in%

points of and y $FOSM method.

W


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1$3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1$

Solution:

"210 10 0 ( )y yZ F W M F W N m= = =

2&%"y y yF F F

MPa = = "1"%'W W W cm = =

*

*2&%"yF

y P

g WF

=*

*1"%'W y

P

g FW

=

( ) ( )

*

2 2* *

2&%"

2&%" 1"%'

yF

y

W

W F

=

+

( ) ( )

*

2 2* *

1"%'

2&%" 1"%'

y

W

y

F

W F

=+

0a1


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1%3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1%

)teration cycle &

0!1

* "$0 2&%"y y y yy F F F F

F = + = +

0b1* #$2 1"%'W W W W W = + = +

* * 210000 0yF W = 0c1

2 (0 1%2$ ) 1'% 0y yF W F W

+ + + = 0d1

9et* "$0

yy FF = =

* #$2WW = =

021 Solve and from formula @aAy

F W

0%$#1yF

= 0%2&&W =

031 Solve from formula @dA

20%2#2 1%$& 1'% 0 + = (1) "%0$ =

2 2 1yF W

+ =

Chec*in%


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1&3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1&

)teration cycle /

0!1 Solve and from formula @A

021 Solve and from formula @aAyF W0%$&

yF = 0%22W =


2

21'' 1%$ 1'% 0 + + = (2) "%0$2 =

*yF *W

* "$0 ( 0%$#1) "%0$ 2&%" "0$yF = + =

*#$2 ( 0%2&&) "%0$ 1"%' #'0W = + =

2 2 1yF W

+ =

Chec*in%

(2) (1) 0%00" 0%001 = >


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3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1'3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "1'

)teration cycle 7

0!1 Solve and from formula @A

021 Solve and from formula @aAyF W

0%$&'yF

= 0%22"2W

=


(") "%0$2 =

*yF *W

* "0'yF = * #'2W =

(") (2) 0%000 =

The final results: "%0$2 = * "0'yF =

* #'2W =

1 ( ) 1 ("%0$2) 1 0%$$$" 0%000&fP = = = =


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3.3 C Method

(ecommended y the DCSS Committee

Chapter 3Chapter 3



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3.3 ,C Metho! Recommen!e! by the ,CSS Committee "13.3 ,C Metho! Recommen!e! by the ,CSS Committee "1

C Method Recommended by the CSS Committee

The $FOSM method can only treat !ith the limit state e5uation

!ith normal random variales. To overcome thisprolem"

(ac*!it3 and Fiesslerpropose a procedure !hich can deal !ith

the %eneral random variales in &6E. This method is then

recommended y theDoint Committee ofStructuralSafety"Therefore it is also namedDCMethod.

The reliaility index calculated y DC method is also called

(ac*!it3Fiessler reliaility index.

The asic idea of DC method is to convert each non-normal

random variale into an e5uivalent normal random variale y

usin% the Principle of ?5uivalent ormali3ation.


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3.3 ,C Metho! Recommen!e! by the ,CSS Committee "#3.3 ,C Metho! Recommen!e! by the ,CSS Committee "#

3.3.! asic dea of C Method

Convert each non-normal random variale into an e5uivalent

normal random variale y usin% the Principle of ?5uivalent

ormali3ation. $fter this transformation" the prolem can then e solved y $FOSM

method.

3.3.2 *rinciple of %uivalent 'ormali4ation

!. +ransformation Conditions of %uivalent 'ormali4ation

0!1 $t the desi%n chec*in% point " the C=F value of the e5uivalent

normal random variale is e5ual to that of the ori%inal non-normalrandom variale.

021 $t the desi%n chec*in% point " the P=F value of the e5uivalentnormal random variale is e5ual to that of the ori%inal non-normalrandom variale.

*P

*P


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3.3 ,C Metho! Recommen!e! by the ,CSS Committee "33.3 ,C Metho! Recommen!e! by the ,CSS Committee "3

iX

ix

( )iX i

f x

iXe

iX*

ix

* *( ) ( )ei i

X i iXf x f x=

* *( ) ( )ei i

X i iXF x F x=

P=F of non-normal (V

iX

iX

( )iX i

f xi

X

e

iXP=F of e5uivalent normal (V

eiX

eiX

( )ei

iXf x

e

iX


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3.3 ,C Metho! Recommen!e! by the ,CSS Committee "$3.3 ,C Metho! Recommen!e! by the ,CSS Committee "$

2. Formulas of %uivalent 'ormali4ation

*

*( )ei

i

ei

i X

X i

X

xF x

=

*

* 1

( )

ei

ie ei i

i X

X iX X

x

f x

=

* 1 *( )e eii i

i X iX Xx F x =

*

1 *

* *

1 1 ( ( ))+

( ) ( )

ei

eii

ei ii

i X

X iXX i X iX

xF x

f x f x

= =

/ / / / /0!1

/ / / / /021

C CSS C


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3.3 ,C Metho! Recommen!e! by the ,CSS Committee "%3.3 ,C Metho! Recommen!e! by the ,CSS Committee "%

3. Formulas of %uivalent 'ormali4ation for lo(normal R"

( )

* *

2

* *

ln

1 ln lnln(1 )

1 ln

i

ei

i

i

X

i iX

X

i i X

x xV

x x

= + +

= +

* 2

*

ln

ln(1 )eii

i

i XX

i X

x V

x

= +

=

/ / / / /031

/ / / / /0#1

Please refer to the textoo* +(eliaility

of Structures, y Professor $. S.

o!a*.

Turn to Pa%e &//" loo* at the example 0.Ecarefully1


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3 3 ,C M th ! R ! ! b th ,CSS C itt3 3 ,C M th ! R ! ! b th ,CSS C itt '


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3.3 ,C Metho! Recommen!e! by the ,CSS Committee "'3.3 ,C Metho! Recommen!e! by the ,CSS Committee "'

4. Calculate the direction cosine usin%i

0.

Calculate the desi%n point usin%

*

ix

B. Calculate the reliaility index usin%

6. Calculate the ne! desi%n point usin%

*

*

2

1

i

i

X

i P

i

n

X

i i P

g

X

g

X

=

=

( 1, 2, , )i n= L

*

i ii X i X x = +

* * *

1 2( , , , ) 0ng x x x =L

*

i ii X i X x = +

E. (epeat Steps 7-6 until and the desi%n points conver%e. *

{ }ix

3 3 ,C M th ! R ! ! b th ,CSS C itt (3 3 ,C M th ! R ! ! b th ,CSS C itt (


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%&ample 3.5

$ssume that a reinforced concrete short column that carry a dead load and a

live load. The limit state e5uation is

The random variales are dead load effect >" live loaf effect G" and section

resistance . The parameters of these (V are listed in the follo!in% tale:

( , , ) 0Z g R G Q R G Q= = =

3.3 ,C Metho! Recommen!e! by the ,CSS Committee "(3.3 ,C Metho! Recommen!e! by the ,CSS Committee "(

G

Random"ariables

+ypes of)istribution

Mean 06'1 Standarddeviation 06'1

C.o."

'ormal 57 2.5 7.75

%&treme 85 !9 7.2

$o(normal 257 25 7.!

Q

R

Calculate the reliaility index of the column y DC method .


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3.# MCS

Monte Carlo Simulation

Chapter 3Chapter 3


3 $ MCS M t C l Si l ti 13 $ MCS M t C l Si l ti 1


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3.$ MCS Monte Carlo Simulation "13.$ MCS Monte Carlo Simulation "1

3.#.! *rocedure of MCS

/. =etermine the necessary distriution information.

7. =etermine the numer of simulated values of the limit statee5uation to e %enerated accordin% to the follo!in% formula:

100

f

N

P

4. >enerate the random numer values

of the asic variales in the limit state e5uation.

( 1, , 1, , )ijx i M j N= =L L

&. Formulate the limit state e5uation: 1 2( , , , ) 0MZ g X X X= =L

0. Calculate a simulated value 3 of ' of the limit state function for eachset of random numer values of the asic variales.

B. Calculate the times of the simulated are less than 3ero. $ssumethat it is denoted as .

ifN

6. Calculate the estimated proaility of failure accordin% to thefollo!in% formula:

ff

NP

N

ijx

3 $ MCS M t C l Si l ti #3 $ MCS Monte Carlo Sim lation #


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3.$ MCS Monte Carlo Simulation "#3.$ MCS Monte Carlo Simulation "#

3.#.2 Application Area of MCS

&. )t is used to solve complex prolems for !hich closed-form solutionsare either not possile or extremely difficult.

/. )t is used to solve complex prolems that can e solved in closed formif many simplifyin% assumptions are made.

7. )t is used to chec* the results of other solution techni5ues.

3.#.3 Accuracy of *robability %stimate of MCS

9et e the theoretical correct proaility that !e are tryin% toestimate y calculatin% . The proaility estimate accuracy is:

fP!"ue

N

( )f !"ue# P P=

(1 )

f

!"ue !"ue

P

P P

N

=

(1 )

f

!"ue

P!"ue

PV

P N

=

3 $ MCS M t C l Si l ti 33 $ MCS Monte Carlo Simulation 3


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%&ample 3.:


+(eliaility of Structures,

y Professor $. S. o!a*.

Turn to Pa%e &7E" loo* at the example 0.&Bcarefully1 2e !ill demonstrate this example

in M$T9$H immediatelyII

3.$ MCS Monte Carlo Simulation "33.$ MCS Monte Carlo Simulation "3

3 $ MCS Monte Carlo Simulation $3 $ MCS Monte Carlo Simulation $


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R 9o%normal

3.$ MCS Monte Carlo Simulation "$3.$ MCS Monte Carlo Simulation "$

Solution:

2"00R = 2$$R R RV = =

ln2

ln &%&"21

RR

RV

= =

+

2

ln ln(1 ) 0%12$R RV = + =

$ ormal $00$ = $0$ $ $V = =

L ?xtreme #&L = 1#'%&L L LV = =

1%2'2 - 0%00L = =

0% $$%0#L Lu = =

3 $ MCS Monte Carlo Simulation %3 $ MCS Monte Carlo Simulation %


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R 9o%normal

3.$ MCS Monte Carlo Simulation "%3.$ MCS Monte Carlo Simulation "%

Simulated values of (Vs in M$T9$H

lognrn!(&%&"2,0%12$,1000,1)R=

$ ormal

L ?xtreme

normrn!($00,$0,1000,1)$=

log( log( ))pL u

=

ran!(1000,1)p=

Chapter3: -omeor/ 3Chapter3: -omeor/ 3


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;ome


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;ome


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component reliability analysis

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