Post on 16-Feb-2018
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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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%&le 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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%&le 3.3
Please refer to the textoo*
+(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
Component Reliability AnalysisComponent Reliability Analysisof Structuresof Structures
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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:
3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "33.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "3
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.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "113.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "11
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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%&le 3.#
3.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "133.# AFOSM A!+ance! First Or!er Secon! Moment Metho! "13
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%$&
yF = 0%22W =
031 Solve from formula @dA
2
21'' 1%$ 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
=
031 Solve from formula @dA
(") "%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
Component Reliability AnalysisComponent Reliability Analysisof Structuresof Structures
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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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%&le 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
Component Reliability AnalysisComponent Reliability Analysisof Structuresof Structures
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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%&le 3.:
Please refer to the textoo*
+(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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