chapter 3 component reliability analysis of structures

Chapter 3Chapter 3

Component Reliability Analysis

of Structures

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

3.2 AFOSM — Advanced First Order Second Moment Method

3.3 JC Method — Recommended by the JCSS Committee

3.1 MVFOSM — Mean Value First Order Second Moment

Method

Contents

3.4 MCS — Monte Carlo Simulation Method

3.1 MVFOSM —

Mean Value First Order Second Moment Method

Chapter 3Chapter 3 Component Reliability Analysis Component Reliability Analysis of Structuresof Structures

3.1 MVFOSM — Mean Value First Order Second Moment Method …13.1 MVFOSM — Mean Value First Order Second Moment Method …1

MVFOSM — Mean Value

First Order Second Moment

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

is used.

• This method is also named Mean Value Method or Center Point Method.

– Second Moment: Only means and variances of the basic variables

are needed.

– Mean Value or Center Point: The Taylor series expansion is

on the means values.

3.1.1 Linear Limit State Functions

1. Assumptions

where, the termsia are constants;

0 1 1 2 2 01

( )n

n n i ii

Z g X a a X a X a X a a X

( 0,1,2, , )i n

The terms are uncorrelated random variables.iX

2. FormulaAccording to the linear functions of uncorrelated random variables introduced in Chapter 1, the mean and standard deviation of Z are:

01

i

n

Z i Xi

a a

2

1i

n

Z i Xi

a

Consider a linear limit state function of the form


According to the central limit theorem, as n increases, the random variable Z will approach a normal probability distribution.

01

2

1

i

i

n

i XiZ

nZ

i Xi

a a

a

Formula of Reliability Index

fP

– 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.


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!


3.1.2 Nonlinear Limit State Functions

1. Assumptions

where,

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

the terms are uncorrelated random variables,iX

2. FormulaWe can obtain an approximate solution by linearizing the nonlinear function using a Taylor series expansion. The result is

Consider a nonlinear limit state function of the form

and its mean and standard deviation are , respectively .iXiX

* * *1 2

* * * *1 2

1 ( , , , )

( , , , ) ( )n

n

n i ii i x x x

gZ g x x x X x

X


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

1 2

1 21 ( , , , )

*

1

( , , , ) ( )

( ) ( )

n i

X X Xn

n

X X X i Xi i

n

i ii i M

gZ g X

X

gg M X x

X

where, is the point about which the expansion is performed.* * *1 2( , , , )nx x x

From now on ,this point is represented by . Therefore, the above formula can be rewritten briefly as follows:

*P

*

* *

1

( ) ( )n

i ii i P

gZ g P X x

X

M

1 2( , , , )

nX X XM – The point is also called mean value point or central point.M


– Moments of the performance function Z

1 2( , , , )

nZ X X Xg

2

2

1 1i i

n n

Z X i Xi ii M

ga

X

where, ii M

ga

X

1 2

2 2

11

( , , , ) ( )n

ii

X X XZ

nnZ

i XX i

i i M

g g M

g aX

Formula of Reliability Index


Example 3.2




3.1.3 Comments on MVFOSM

1. Advantages

2. Disadvantages

• It is very easy to use.• It does not require knowledge of the distributions of the random

variables.

• 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 !


Example 3.3



The invariance problem is best clarified by


3.2 AFOSM —

Advanced First Order Second Moment

Method


3.2 AFOSM — Advanced First Order Second Moment Method …13.2 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 .0Z

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

3.2.1 Principles of AFOSM

1. Assumptions

2. Transformation from X space into U space

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

i

i

i Xi

X

XU


where,

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

the terms are uncorrelated random variables,iX

and its mean value and standard deviation are known.iX

iX

Consider a nonlinear limit state function of the form

iX

– The X space is then transformed into U space:


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

– The design point in X space is then

transformed to in U space.

* * * *1 2( , , , )nP x x x

* * * *1 2

ˆ ( , , , )nP u u u

– The limit equation in X space

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

is transformed to U space as follows.

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


– In U space, the tangent plane equation through the design point

on failure surface is ( ) 0Z G U

*

* * * *1 2

1 ˆ

( , , , ) ( ) 0n

n i ii i P

GG u u u U u

U

– Since the design point is a point on the failure

surface , then we have

0Z *P̂

* * *1 2( , , , ) 0nG u u u

– The hyper-plane equation can therefore be simplified as follows:

*

*

1 ˆ

( ) 0n

i ii i P

GU u

U

3. Reliability Index in U Space

– The distance from the origin of U space to the tangent plane is

actually the reliability index

*1u

*2u

1u

2uDesign pointTangent

Failure surface

1U2U

*P̂

O

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


*ˆHL O P

( ) 0G u


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

*

*

*

1 ˆ

2

1 ˆ

n

ii i P

n

i i P

Gu

U

GU

Let

*

*

ˆ

2

1 ˆ

i Pi

n

i i P

GU

GU

is actually the direction cosine

of the distance i

*ˆO P

cosii U

* cosii U iu


Since

** i

i

i Xi

X

xu

– The design point in X space

we have* *

i i i ii X i X X i Xx u

– The direction cosine in X space

i

iX

i i i i

XG g g

U X U X

*

*

2

1

i

i

Xi P

in

Xi i P

gX

gX

* * *1 2( , , , ) 0ng x x x

4. Reliability Index in X Space


– The reliability index in X space

*

*

*

1

2

1

i

i

n

X ii i P

n

Xi i P

gu

X

gX

*

1

2

1

i

i

n

i X ii

n

i Xi

a x

a

*

* i

i

i Xi

X

xu

*

ii P

ga

X


Comparison of Formulas in X Space

*

1

2

1

i

i

n

i X ii

n

i Xi

a X

a

*

ii P

ga

X

01

2

1

i

i

n

i XiZ

nZ

i Xi

a a

a

1 2

2

1

( , , , )n

i

X X XZ

nZ

i Xi

g

a

i

i M

ga

X

MVFOSM: linear case

MVFOSM: nonlinear case

AFOSM: nonlinear case

– center point

– design point


3.2.2 Computation Formulas of AFOSM

* * *1 2( , , , ) 0ng x x x

*

*

2

1

i

i

Xi P

in

Xi i P

gX

gX

( 1,2, , )i n

*

i ii X i Xx ( 1,2, , )i n … … … … … …(2)

… … … … …(1)

… … … … … … … … … …(3)

11 ( )f fp p … … … … … … … … … …(4)


3.2.3 Iteration Algorithm of AFOSM

1. Formulate the limit state equation

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

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

2. Assume the initial values of design point and reliability index*iX

In general, the initial value of design point is taken as mean value .iX

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 .*ix

5. Using Eq.(3) to calculate the reliability index .6. Using Eq.(2) to calculate the new design point .


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

Begin

Assume*

ii Xx

Calculate i

Calculate*

i ii X i Xx

Calculate from ( ) 0g

( 1) ( )k k ≤

Output and *ix

No Yes

Flowchart

Example 3.4


210M kN m Assume that a steel beam carry a deterministic bending moment ,

The limit state equation is

The plastic section modulus and the yield strength of the beam are

statistically independent, normal random variables. It is known that

W yF

3692W cm 0.02W 390

yF Mpa 0.07yF

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

yF

Calculate the reliability index of the beam as well as the checking

points of and by AFOSM method.

W


Solution:

3210 10 0 ( )y yZ F W M F W N m

27.3y y yF F F MPa 313.84W W W cm

*

*27.3yF

y P

gW

F

*

*13.84W yP

gF

W

*

2 2* *

27.3

27.3 13.84yF

y

W

W F

*

2 2* *

13.84

27.3 13.84

yW

y

F

W F

(a)


Iteration cycle 1

(1)

* 390 27.3y y y yy F F F FF

(b)* 692 13.84W W W WW

* * 210000 0yF W (c)

2 (50 14.29 ) 158.4 0y yF W F W (d)

Let * 390yy FF * 692WW

(2) Solve and from formula (a)yF W

0.9615yF 0.2747W

(3) Solve from formula (d)20.2642 51.97 158.4 0 (1) 3.095

2 2 1yF W

Checking


Iteration cycle 2

(1) Solve and from formula (b)


0.9745yF 0.2245W

(3) Solve from formula (d)22188 51.9 158.4 0 (2) 3.092

*yF *W

* 390 ( 0.9615) 3.095 27.3 309yF

* 692 ( 0.2747) 3.095 13.84 680W

2 2 1yF W

Checking

(2) (1) 0.003 0.001


Iteration cycle 3

(1) Solve and from formula (b)


0.9748yF 0.2232W

(3) Solve from formula (d)

(3) 3.092

*yF *W

* 308yF * 682W

(3) (2) 0.000

The final results: 3.092 * 308yF * 682W

1 ( ) 1 (3.092) 1 0.9993 0.0007fP

3.3 JC Method —

Recommended by the JCSS Committee


3.3 JC Method — Recommended by the JCSS Committee …13.3 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.


3.3.1 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.

*P

*P


iX

ix

( )iX if x

iXeiX

*ix

* *( ) ( )ei i

X i iXf x f x

* *( ) ( )ei i

X i iXF x F x

PDF of non-normal RV

iXiX

( )iX if x

iX

eiXPDF of equivalent normal RV

eiX

eiX

( )ei

iXf x

e

iX


2. Formulas of Equivalent Normalization*

*( )ei

iei

i XX i

X

xF x

*

* 1( )

ei

ie ei i

i XX i

X X

xf x

* 1 *( )e eii i

i X iX Xx F x

*

1 ** *

1 1[ ( ( ))]

( ) ( )

ei

eii

ei ii

i XX iX

X i X iX

xF x

f x f x

… … … … …(1)

… … … … …(2)


3. Formulas of Equivalent Normalization for lognormal RV

* *

2

* *ln

1 ln lnln(1 )

1 ln

iei

i

i

Xi iX

X

i i X

x xV

x x

* 2

*ln

ln(1 )eii

i

i XX

i X

x V

x

… … … … …(3)

… … … … …(4)




3.3.3 Procedure of JC Method

1. Formulate the limit state equation

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

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

2. Assume the initial values of design point and reliability index*iX

In general, the initial value of design point is taken as mean value .iX

Then the initial value of is 0.3. 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.

iX

iXeiX

eiX

iXe

i iX X

ei i

X X


4. Calculate the direction cosine using i

5. Calculate the design point using*ix

6. Calculate the reliability index using

7. Calculate the new design point using

*

*

2

1

i

i

Xi P

in

Xi i P

gX

gX

( 1,2, , )i n

*

i ii X i Xx

* * *1 2( , , , ) 0ng x x x

*

i ii X i Xx

8. Repeat Steps 3-7 until and the design points converge. *{ }ix

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:

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


G

Random Variables

Types of Distribution

Mean (kN)Standard

deviation (kN) C.o.V

Normal 50 2.5 0.05

Extreme Ⅰ 85 17 0.2

Lognormal 250 25 0.1

Q

R

Calculate the reliability index of the column by JC method .

3.4 MCS —

Monte Carlo Simulation


3.4 MCS — Monte Carlo Simulation …13.4 MCS — Monte Carlo Simulation …1

3.4.1 Procedure of MCS

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:

100ˆf

NP

4. Generate the random number values of the basic variables in the limit state equation.

( 1, , ; 1, , )ijx i M j N

1. Formulate the limit state equation: 1 2( , , , ) 0MZ g X X X

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 .

izfN

7. Calculate the estimated probability of failure according to the following formula:

ˆ ff

NP

N

ijx


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:ˆ

fPtrueN

ˆ( )f trueE P P

ˆ

(1 )f

true trueP

P P

N

ˆ

(1 )f

trueP

true

PV

P N

Example 3.6


Turn to Page 138, look at the example 5.16 carefully! We will demonstrate this example in MATLAB immediately……


R Lognormal


Solution:

2300R 299R R RV

ln 2ln 7.732

1R

R

RV

2ln ln(1 ) 0.1295R RV

D Normal 900D 90D D DV

L Extreme Ⅰ 675L 168.75L L LV

1.282 / 0.0076L

0.45 599.06L Lu

R Lognormal


Simulated values of RVs in MATLAB

lognrnd(7.732,0.1295,1000,1)R

D Normal

L Extreme Ⅰ

normrnd(900,90,1000,1)D

log( log( ))pL u

rand(1000,1)p

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 3Chapter3: 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.

(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

(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.


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 3Chapter 3

chapter 3 component reliability analysis of structures

Documents

mean value method

mean value point

mean values

order terms

center point method

momentfirst order

jc method

moment methodcontents3