Component Reliability Analysis

1
Chapter 3

• Component Reliability Analysis
• of Structures

2
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
3
Chapter 3 Component Reliability Analysis of
Structures

• 3.1 MVFOSM
• Mean Value First Order Second Moment Method

4
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.

5
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
where, the terms
are constants
2. Formula
According to the linear functions of uncorrelated
random variables introduced in Chapter 1, the
mean and standard deviation of Z are
6
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

• 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.

7
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!
8
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
where,
the terms are uncorrelated random variables,
and its mean and standard deviation are ,
respectively .
2. Formula
We can obtain an approximate solution by
linearizing the nonlinear function using a Taylor
series expansion. The result is
9
3.1 MVFOSM Mean Value First Order Second
Moment Method 6
where,
is the point about which the expansion is
performed.
From now on ,this point is represented by .
Therefore, the above formula can be rewritten
briefly as follows

• One choice for this linearization point is the
  point corresponding to the mean values of
  the random variables.

• The point is also called mean value point
  or central point.

10
3.1 MVFOSM Mean Value First Order Second
Moment Method 7

• Moments of the performance function Z

where,
Formula of Reliability Index
11
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!
12
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 !

13
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!
14
Chapter 3 Component Reliability Analysis of
Structures

• 3.2 AFOSM
• Advanced First Order Second Moment
• Method

15
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.

• Since the design point is generally not known a
  priori, an iteration technique is generally used
  to solve for the reliability index.

16
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
where,
the terms are uncorrelated random variables,
and its mean value and standard deviation
are known.
2. Transformation from X space into U space

• The general random variable is transformed
  into its standard form as follows

17
3.2 AFOSM Advanced First Order Second Moment
Method 3

• The X space is then transformed into U space

• The limit equation in X space

is transformed to U space as follows.

• The design point in
  X space is then transformed to
  in U space.

18
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
  is

• Since the design point is a point on the
  failure surface , then we have

• The hyper-plane equation can therefore be
  simplified as follows

19
3.2 AFOSM Advanced First Order Second Moment
Method 5
20
3.2 AFOSM Advanced First Order Second Moment
Method 6

• From the geometric meaning of the reliability
  index, we know

Let
21
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

• The direction cosine in X space

22
3.2 AFOSM Advanced First Order Second Moment
Method 8

• The reliability index in X space

23
3.2 AFOSM Advanced First Order Second Moment
Method 9
Comparison of Formulas in X Space
MVFOSM linear case
MVFOSM nonlinear case

• center point

AFOSM nonlinear case

• design point

24
3.2 AFOSM Advanced First Order Second Moment
Method 10
3.2.2 Computation Formulas of AFOSM
(1)
(2)
(3)
(4)
25
3.2 AFOSM Advanced First Order Second Moment
Method 11
3.2.3 Iteration Algorithm of AFOSM

1. Formulate the limit state equation

Give the distribution types and appropriate
parameters of all random variables.

1. Using Eq.(2) to calculate the n values of design
   point .

1. Using Eq.(2) to calculate the new design point .

26
3.2 AFOSM Advanced First Order Second Moment
Method 12

1. Go back to Step 3 and repeat. Iterate until the
   values converge.

Begin
Flowchart
Assume
Calculate
Output and
No
Yes
27
3.2 AFOSM Advanced First Order Second Moment
Method 13
Example 3.4
Assume that a steel beam carry a deterministic
bending moment ,
The limit state equation is
28
3.2 AFOSM Advanced First Order Second Moment
Method 14
Solution
(a)
29
3.2 AFOSM Advanced First Order Second Moment
Method 15
(b)
(c)
(d)
Iteration cycle 1
(1)
Let
(2)
Solve and from formula (a)
Checking
(3)
Solve from formula (d)
30
3.2 AFOSM Advanced First Order Second Moment
Method 16
Iteration cycle 2
(1)
Solve and from formula (b)
(2)
Solve and from formula (a)
Checking
(3)
Solve from formula (d)
31
3.2 AFOSM Advanced First Order Second Moment
Method 17
Iteration cycle 3
(1)
Solve and from formula (b)
(2)
Solve and from formula (a)
(3)
Solve from formula (d)
The final results
32
Chapter 3 Component Reliability Analysis of
Structures

• 3.3 JC Method
• Recommended by the JCSS Committee

33
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 RackwitzFiessler 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.

34
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 , 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 , the
PDF value of the equivalent normal random
variable is equal to that of the original
non-normal random variable.
35
3.3 JC Method Recommended by the JCSS
Committee 3
PDF of non-normal RV
36
3.3 JC Method Recommended by the JCSS
Committee 4
2. Formulas of Equivalent Normalization
(1)
(2)
37
3.3 JC Method Recommended by the JCSS
Committee 5
3. Formulas of Equivalent Normalization for
lognormal RV
(3)
(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!
38
3.3 JC Method Recommended by the JCSS
Committee 6
3.3.3 Procedure of JC Method

1. Formulate the limit state equation

Determine the distribution types and appropriate
parameters of all random variables.

• For non-normal RV , the mean and
  standard deviation
• should be calculated, and then, they replace
  the mean and standard deviation
  of the non-normal RV.

39
3.3 JC Method Recommended by the JCSS
Committee 7

1. Calculate the direction cosine using

1. Calculate the design point using

1. Calculate the reliability index using

1. Calculate the new design point using

1. Repeat Steps 3-7 until and the design
   points converge.

40
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
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
Normal 50 2.5 0.05
Extreme ? 85 17 0.2
Lognormal 250 25 0.1
41
Chapter 3 Component Reliability Analysis of
Structures

• 3.4 MCS
• Monte Carlo Simulation

42
3.4 MCS Monte Carlo Simulation 1
3.4.1 Procedure of MCS
1. Formulate the limit state equation
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
of the basic variables in the limit state
equation.
5. Calculate a simulated value z of Z of the
limit state function for each set of random
number values of the basic variables.
6. Calculate the times of the simulated are
less than zero. Assume that it is denoted as
.
7. Calculate the estimated probability of failure
according to the following formula
43
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 be the theoretical correct
probability that we are trying to estimate by
calculating . The probability estimate
accuracy is
44
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
45
3.4 MCS Monte Carlo Simulation 4
Solution
Lognormal
Normal
Extreme ?
46
3.4 MCS Monte Carlo Simulation 5
Simulated values of RVs in MATLAB
Lognormal
Normal
Extreme ?
47
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
48
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, re-calculate 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
49
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.
50

• End of
• Chapter 3