Chapter 3
Component Reliability Analysis
of Structures
Chapter 3: Element Reliability Analysis of Structures
Contents
3.1 MVFOSM — Mean Value First Order Second Moment
Method
3.2 AFOSM — Advanced First Order Second Moment
Method
3.3 JC Method — Recommended by the JCSS Committee
3.4 MCS — Monte Carlo Simulation Method
Chapter 3
Component Reliability Analysis
of Structures
3.1 MVFOSM —
Mean Value First Order Second Moment Method
3.1 MVFOSM — Mean Value First Order Second Moment Method …1
MVFOSM — Mean Value
First Order Second Moment
–
Mean Value or Center Point: The Taylor series expansion is
on the means values.
–
First Order: The first-order terms in the Taylor series expansion
is used.
–
Second Moment: Only means and variances of the basic variables
are needed.
• This method is also named Mean Value Method or
Center Point Method.
3.1 MVFOSM — Mean Value First Order Second Moment Method …2
3.1.1 Linear Limit State Functions
1. Assumptions
Consider a linear limit state function of the form
Z  g ( X )  a0  a1 X 1  a2 X 2 
where, the ai terms (i  0,1, 2,
The
n
 an X n  a0   ai X i
i 1
, n) are constants;
X i terms are uncorrelated random variables.
2. Formula
According to the linear functions of uncorrelated random variables
introduced in Chapter 1, the mean and standard deviation of Z are:
n
Z  a0   ai  X
i 1
i
Z 
 a  
n
i 1
i
Xi
2
3.1 MVFOSM — Mean Value First Order Second Moment Method …3
According to the central limit theorem, as n increases, the random variable
Z will approach a normal probability distribution.
Formula of Reliability Index
n
Z


Z
a0   ai  X i
i 1
a  
n
i 1
i
2
Xi
Pf     
–
–
If the random variables are all normally distributed and uncorrelated,
then the above formula is exact.
Otherwise, it provides only an approximate estimate on the failure
probability.
3.1 MVFOSM — Mean Value First Order Second Moment Method …4
Example 3.1
Please refer to the textbook
“Reliability of Structures”
by Professor A. S. Nowak.
Turn to Page 102, look at the example 5.1
carefully!
3.1 MVFOSM — Mean Value First Order Second Moment Method …5
3.1.2 Nonlinear Limit State Functions
1. Assumptions
Consider a nonlinear limit state function of the form
Z  g ( X )  g ( X1 , X 2 ,
, Xn )
where, the X i terms are uncorrelated random variables,
and its mean and standard deviation are  X i ,  X i respectively .
2. Formula
We can obtain an approximate solution by linearizing the nonlinear
function using a Taylor series expansion. The result is
Z  g(x , x ,
*
1
*
2
g
, x )  (Xi  x )
X i
i 1
n
*
n
*
i
( x1* , x2* , , xn* )
3.1 MVFOSM — Mean Value First Order Second Moment Method …6
where, ( x1* , x2* , , xn* ) is the point about which the expansion is performed.
From now on ,this point is represented by P* . Therefore, the above formula
can be rewritten briefly as follows:
n
Z  g ( P* )   ( X i  xi* )
i 1
–
P*
One choice for this linearization point is the point M corresponding
to the mean values of the random variables.
M  (X1 , X 2 ,
–
g
X i
, X n )
The point M is also called mean value point or central point.
Z  g (  X1 ,  X 2 ,
n
g
,  Xn )   ( X i   Xi )
X i
i 1
 g ( M )   ( X i  xi* )
i 1
n
g
X i
M
(  X1 ,  X 2 , ,  X n )
3.1 MVFOSM — Mean Value First Order Second Moment Method …7
–
Moments of the performance function Z
Z  g (X , X , , X )
1
2
n
2
 g
 Z  

i 1  X i

 Xi  

M

g
where, ai 
X i M
n
a  
n
i
i 1
2
Xi
Formula of Reliability Index
Z g (  X ,  X ,


Z
n 
1
g


i 1  X i
2
M
, Xn )

 Xi 


2

g (M )
a  
n
i 1
i
Xi
2
3.1 MVFOSM — Mean Value First Order Second Moment Method …8
Example 3.2
Please refer to the textbook
“Reliability of Structures”
by Professor A. S. Nowak.
Turn to Page 104, look at the example 5.2
carefully!
3.1 MVFOSM — Mean Value First Order Second Moment Method …9
3.1.3 Comments on MVFOSM
1. Advantages
• It is very easy to use.
• It does not require knowledge of the distributions of the random
variables.
2. Disadvantages
• Results are inaccurate if the tails of the distribution functions
cannot be approximated by a normal distribution.
• There is an invariance problem: the value of the reliability index
depends on the specific form of the limit state function.
That is to say, for different forms of the limit state equation
which have the same mechanical meanings, the values of
reliability index calculated by MVFOSM may be different !
3.1 MVFOSM — Mean Value First Order Second Moment Method …10
The invariance problem is best clarified by
Example 3.3
Please refer to the textbook
“Reliability of Structures”
by Professor A. S. Nowak.
Turn to Page 107, look at the example 5.3
carefully!
Chapter 3
Component Reliability Analysis
of Structures
3.2 AFOSM —
Advanced First Order Second Moment
Method
3.2 AFOSM — Advanced First Order Second Moment Method …1
AFOSM — Advanced First Order Second Moment
–
To overcome the invariant problem, Hasofer and Lind propose an
advanced FOSM method in 1974 , which is called AFOSM .
–
The “correction” is to evaluate the limit state function at a point
known as the “design point” instead of the mean values.
Therefore, this method is also called “design point method” or
“checking point method”.
–
The “design point” is a point on the failure surface Z  0 .
–
Since the design point is generally not known a priori, an iteration
technique is generally used to solve for the reliability index.
3.2 AFOSM — Advanced First Order Second Moment Method …2
3.2.1 Principles of AFOSM
1. Assumptions
Consider a nonlinear limit state function of the form
Z  g ( X )  g ( X1 , X 2 ,
, Xn )
where, the X i terms are uncorrelated random variables,
and its mean value  X i and standard deviation  X i are known.
2. Transformation from X space into U space
–
The general random variable
form as follows:
Ui 
X i is transformed into its standard
X i   Xi
X
i
3.2 AFOSM — Advanced First Order Second Moment Method …3
–
The X space is then transformed into U space:
X  ( X1 , X 2 ,
–
U  (U1 ,U 2 ,
, Xn )
The limit equation in X space
Z  g ( X )  g ( X1 , X 2 ,
, Xn )
is transformed to U space as follows.
Z  G(U )  G(U1,U2 ,
–
The design point P* ( x1* , x2* ,
transformed
Pˆ * (u1* , u2* , to, un* )
,U n )
, xn* ) in X space is then
in U space.
,U n )
3.2 AFOSM — Advanced First Order Second Moment Method …4
3. Reliability Index in U Space
–
In U space, the tangent plane equation through the design point
on failure surface Z  G(U )  0 is
n
*
1
*
2
G (u , u ,
–
,u )  
*
n
i 1
G
U i
 (U i  ui* )  0
Pˆ *
Since the design point Pˆ * is a point on the failure surface Z  0 ,
then we have
G(u1* , u2* ,
–
, un* )  0
The hyper-plane equation can therefore be simplified as follows:
G

i 1 U i
n
 (U i  ui* )  0
Pˆ *
3.2 AFOSM — Advanced First Order Second Moment Method …5
–
The distance from the origin of U space to the tangent plane is
actually the reliability index

u2
Design point
Tangent
Failure surface
Pˆ *
u2*
U

2
U
G (u)  0
1
O
*
1
u
HL  arg min{u | G(u)  0}
HL  OPˆ *
u1
3.2 AFOSM — Advanced First Order Second Moment Method …6
–
From the geometric meaning of the reliability index, we know
n


i 1
G
U i
i 
G

U i
 G


i 1  U i
n
Pˆ *
 G


i 1  U i
n
Let
 ui*


Pˆ * 
2
i
is actually the direction cosine
of the distance OPˆ *
Pˆ *


Pˆ * 
2
i  cosU
i
ui*   cosUi  i 
3.2 AFOSM — Advanced First Order Second Moment Method …7
4. Reliability Index in X Space
–
The design point in X space
Since
we have
ui* 
xi*   X i
X
i
xi*  Xi  ui* Xi  Xi  i  Xi
g ( x1* , x2* ,
–
, xn* )  0
The direction cosine in X space
G
g X i
g



 Xi
U i X i U i X i
i 
g

X i
 g


i 1  X i
n
 Xi
P*

 Xi 

*
P

2
3.2 AFOSM — Advanced First Order Second Moment Method …8
–
The reliability index in X space
n


i 1
g
X i
 g


i 1  X i
n
ui* 
xi*   X i
X
i
  X i ui*
P*

 Xi 

P*

2

n

*
a


x
 i Xi i
i 1
a  
n
i 1
ai 
i
g
X i
Xi
P*
2

3.2 AFOSM — Advanced First Order Second Moment Method …9
Comparison of Formulas in X Space
n
MVFOSM: linear case
a0   ai  X i
Z


Z
i 1
a  
n
i
i 1
MVFOSM: nonlinear case
–
center point
Xi
Z g (  X ,  X , ,  X )
g


ai 
n
Z
2
X i
  ai X 
1
2

n
–
design point

*
a


X
 i Xi i
i 1

n
i 1
ai X i

n
i
i 1
AFOSM: nonlinear case
2
2

g
ai 
X i
P*
M
3.2 AFOSM — Advanced First Order Second Moment Method …10
3.2.2 Computation Formulas of AFOSM

i 
g
X i
 g


i 1  X i
n
 Xi
P*

 Xi 

P*

xi*  Xi  i  Xi
g ( x1* , x2* ,
, xn* )  0
p f  p f 1  1 ( )
2
(i  1, 2,
(i  1, 2,
, n)
, n)
… … … … …(1)
… … … … … …(2)
… … … … … … … … … …(3)
… … … … … … … … … …(4)
3.2 AFOSM — Advanced First Order Second Moment Method …11
3.2.3 Iteration Algorithm of AFOSM
1.
Formulate the limit state equation
g ( X1 , X 2 ,
, Xn )  0
Give the distribution types and appropriate parameters of all random
variables.
2.
Assume the initial values of design point X i* and reliability index 
In general, the initial value of design point is taken as mean value  X i .
Then the initial value of  is 0.
3.
Using Eq.(1) to calculate the n values of direction cosine i .
4.
Using Eq.(2) to calculate the n values of design point xi* .
5.
Using Eq.(3) to calculate the reliability index  .
6.
Using Eq.(2) to calculate the new design point .
3.2 AFOSM — Advanced First Order Second Moment Method …12
7.
Go back to Step 3 and repeat. Iterate until the  values converge.
Begin
Flowchart
Assume xi   Xi
*
Calculate
i
*
Calculate xi  X  i  X
i
Calculate
No

from g ()  0
 ( k 1)   ( k ) ≤ 
i
Output  and xi*
Yes
3.2 AFOSM — Advanced First Order Second Moment Method …13
Example 3.4
Assume that a steel beam carry a deterministic bending moment M  210kN  m ,
The plastic section modulus W and the yield strength Fy of the beam are
statistically independent, normal random variables. It is known that
W  0.02
W  692cm3
F  390Mpa  F  0.07
y
The limit state equation is
y
Z  g (Fy ,W )  FyW  M  0
Calculate the reliability index  of the beam as well as the checking
points of W and Fy by AFOSM method.
3.2 AFOSM — Advanced First Order Second Moment Method …14
Solution:
Z  FyW  M  FyW  210 103  0 ( N  m)
 F  F  F  27.3MPa
 W  W W  13.84cm3
g
Fy
g
W
y
y
y
  Fy  27.3W *
P*
F 
y
W 
  W  13.84 Fy*
P*
27.3W *
 27.3W   13.84F 
* 2
* 2
y
(a)
13.84 F
*
y
 27.3W   13.84F 
* 2
* 2
y
3.2 AFOSM — Advanced First Order Second Moment Method …15
Fy*  Fy   Fy  Fy  390  27.3 Fy
W *  W  W  W  692  13.84W
Fy*W *  210000  0
(b)
(c)
 2 F W   (50 F  14.29W )  158.4  0
y
(d)
y
Iteration cycle 1
(1) Let Fy*  F  390
y
W *  W  692
(2) Solve  Fy and W from formula (a)
 Fy  0.9615 W  0.2747
(3) Solve  from formula (d)
0.2642 2  51.97  158.4  0
Checking
 F2  W2  1
y
 (1)  3.095
3.2 AFOSM — Advanced First Order Second Moment Method …16
Iteration cycle 2
(1) Solve Fy* and W * from formula (b)
Fy*  390  (0.9615)  3.095  27.3  309
W *  692  (0.2747)  3.095 13.84  680
(2) Solve  Fy and W from formula (a)
 Fy  0.9745 W  0.2245
Checking
 F2  W2  1
y
(3) Solve  from formula (d)
2188 2  51.9  158.4  0
(2)  (1)  0.003  0.001
(2)  3.092
3.2 AFOSM — Advanced First Order Second Moment Method …17
Iteration cycle 3
(1) Solve Fy* and W * from formula (b)
Fy*  308 W *  682
(2) Solve  Fy and W from formula (a)
 Fy  0.9748 W  0.2232
(3) Solve  from formula (d)
(3)  3.092
 (3)  (2)  0.000
The final results:
  3.092 Fy*  308 W *  682
Pf  1  (  )  1  (3.092)  1  0.9993  0.0007
Chapter 3
Component Reliability Analysis
of Structures
3.3 JC Method —
Recommended by the JCSS Committee
3.3 JC Method — Recommended by the JCSS Committee …1
JC Method — Recommended by the JCSS Committee
–
The AFOSM method can only treat with the limit state equation
with normal random variables. To overcome this problem,
Rackwitz and Fiessler propose a procedure which can deal with
the general random variables in 1978. This method is then
recommended by the Joint Committee of Structural Safety,
Therefore it is also named JC Method.
–
The reliability index calculated by JC method is also called
Rackwitz—Fiessler reliability index.
–
The basic idea of JC method is to convert each non-normal
random variable into an equivalent normal random variable by
using the Principle of Equivalent Normalization.
3.3 JC Method — Recommended by the JCSS Committee …2
3.3.1 Basic Idea of JC Method
–
Convert each non-normal random variable into an equivalent normal
random variable by using the Principle of Equivalent Normalization.
–
After this transformation, the problem can then be solved by AFOSM
method.
3.3.2 Principle of Equivalent Normalization
1. Transformation Conditions of Equivalent Normalization
(1) At the design checking point P* , the CDF value of the equivalent
normal random variable is equal to that of the original non-normal
random variable.
(2) At the design checking point P* , the PDF value of the equivalent
normal random variable is equal to that of the original non-normal
random variable.
3.3 JC Method — Recommended by the JCSS Committee …3
f X i ( xi )
PDF of non-normal RV X i
X
f Xi ( x )  f X e ( x )
*
i
i
*
i
Xi
i
X
f X i ( xi )
FX i ( xi* )  FX e ( xi* )
i
xi*  X ie  X i
e
PDF of equivalent normal RV X i
xi
e
Xi
X
e
i
X
f X e ( xi )
i
e
i
i
3.3 JC Method — Recommended by the JCSS Committee …4
2. Formulas of Equivalent Normalization
 xi*   X e
i
FX i ( xi* )   
  Xe
i





*

x
i  X e
1
*
i
f X i ( xi ) 

 Xe   Xe
i
i

 X  xi*   1  FX ( xi* )   X
e
i
X
e
i
i
*

x
i  X e
1
i



f X i ( xi* )   X e
i

e
i




… … … … …(1)

1
1
*


[

(
F
(
x
Xi
i ))]
*
 f X i ( xi )

… … … … …(2)
3.3 JC Method — Recommended by the JCSS Committee …5
3. Formulas of Equivalent Normalization for lognormal RV
X
e
i



X
i

 xi* 1  ln xi*  ln
2

 … … … … …(3)
ln(1

V
)
X
i



 xi* 1  ln xi*  ln X i
 X  xi* ln(1  VX2 )
e
i
i
 xi* ln X i

… … … … …(4)
Please refer to the textbook “Reliability
of Structures” by Professor A. S. Nowak.
Turn to Page 122, look at the example 5.8 carefully!
3.3 JC Method — Recommended by the JCSS Committee …6
3.3.3 Procedure of JC Method
1.
Formulate the limit state equation
g ( X1 , X 2 ,
, Xn )  0
Determine the distribution types and appropriate parameters of all
random variables.
2.
Assume the initial values of design point X i* and reliability index 
In general, the initial value of design point is taken as mean value  X i .
Then the initial value of  is 0.
3.
For non-normal RV X i , the mean  X e and standard deviation  X e
i
i
should be calculated, and then, they replace the mean  X i and
standard deviation  X i of the non-normal RV.
X  X
i
e
i
X X
i
e
i
3.3 JC Method — Recommended by the JCSS Committee …7
4.
Calculate the direction cosine

i 
g
X i
 g


i 1  X i
 Xi
P*
n
5.
i using
P*

 Xi 


2
(i  1, 2,
, n)
Calculate the design point xi* using
xi*  Xi  i  Xi
6.
Calculate the reliability index  using
g ( x1* , x2* ,
7.
, xn* )  0
Calculate the new design point using
xi*  Xi  i  Xi
8.
Repeat Steps 3-7 until  and the design points {xi*} converge.
3.3 JC Method — Recommended by the JCSS Committee …8
Example 3.5
Assume that a reinforced concrete short column that carry a dead load and a
live load. The limit state equation is
Z  g ( R, G, Q)  R  G  Q  0
The random variables are dead load effect G, live loaf effect Q, and section
resistance . The parameters of these RV are listed in the following table:
Random
Variables
Types of
Distribution
Mean (kN)
Standard
deviation (kN)
C.o.V
G
Normal
50
2.5
0.05
Q
Extreme Ⅰ
85
17
0.2
R
Lognormal
250
25
0.1
Calculate the reliability index  of the column by JC method .
Chapter 3
Component Reliability Analysis
of Structures
3.4 MCS —
Monte Carlo Simulation
3.4 MCS — Monte Carlo Simulation …1
3.4.1 Procedure of MCS
1. Formulate the limit state equation: Z  g ( X1 , X 2 ,
, XM )  0
2. Determine the necessary distribution information.
3. Determine the number N of simulated values of the limit state
equation to be generated according to the following formula:
4. Generate the random number values xij (i  1, , M ; j  1,
of the basic variables in the limit state equation.
N
, N)
100
Pˆf
5. Calculate a simulated value z of Z of the limit state function for each
set of random number values xij of the basic variables.
6. Calculate the times of the simulated
that it is denoted as N f .
zi are less than zero. Assume
7. Calculate the estimated probability of failure according to the
following formula:
Nf
ˆ
Pf 
N
3.4 MCS — Monte Carlo Simulation …2
3.4.2 Application Area of MCS
1. It is used to solve complex problems for which closed-form solutions
are either not possible or extremely difficult.
2. It is used to solve complex problems that can be solved in closed form
if many simplifying assumptions are made.
3. It is used to check the results of other solution techniques.
3.4.3 Accuracy of Probability Estimate of MCS
Let Ntrue be the theoretical correct probability that we are trying to
estimate by calculating Pˆ . The probability estimate accuracy is:
f
E ( Pˆf )  Ptrue
 Pˆ 
f
Ptrue (1  Ptrue )
N
VPˆ 
f
(1  Ptrue )
Ptrue N
3.4 MCS — Monte Carlo Simulation …3
Example 3.6
Please refer to the textbook
“Reliability of Structures”
by Professor A. S. Nowak.
Turn to Page 138, look at the example 5.16
carefully! We will demonstrate this example
in MATLAB immediately……
3.4 MCS — Monte Carlo Simulation …4
Solution:
R
Lognormal
R  2300  R  RVR  299
ln R  ln
R
1V
2
R
 7.732
 ln R  ln(1  VR2 )  0.1295
 D  DVD  90
D
Normal
D  900
L
Extreme Ⅰ
L  675  L  LVL  168.75
  1.282 /  L  0.0076
u  L  0.45 L  599.06
3.4 MCS — Monte Carlo Simulation …5
Simulated values of RVs in MATLAB
R
Lognormal
R  lognrnd(7.732,0.1295,1000,1)
D
Normal
D  normrnd(900,90,1000,1)
L
Extreme Ⅰ
p  rand(1000,1)
Lu
log( log( p ))

Chapter3: Homework 3
Homework 3.1
3.1 Programming the AFOSM in MATLAB environment
according to the flow chart proposed by this course.
(1) By using your own handwork, re-calculate the example
5.4 in text book on P.112
(2) By using your own subroutine, calculate the problem 5.3
in text book on P.142
Chapter3: Homework 3
Homework 3.2
3.2 Programming the JC Method in MATLAB environment
according to the procedure proposed by this course.
(1) By using your own handwork, re-calculate the example
3.5 by assuming the initial iteration value at the means.
(2) By using the procedure proposed by this course, recalculate the example 5.9 on Page 123 and the example
5.10 in the textbook on Page 125.
(3) By using your own subroutine, calculate the example 5.11
on P.127 and the problem 5.4 in text book on P.142
Chapter3: Homework 3
Homework 3.3
3.3 Programming the MCS Method in MATLAB environment
according to the procedure proposed by this course.
By using your own subroutine, re-calculate the example
5.11 in P.127 and the problem 5.4 in text book on P.142 by
Monte Carlo Simulation.
End of
Chapter 3