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Transcript of Structural Reliability
2013 Spring Structural Reliability Analysis 1 2013 Spring 2013 Spring
Structural Reliability Chapter I. Introduction
Jung J. Kim, Ph.D. Adjunct Professor Department of Civil & Environmental Engineering Sejong University
2013 Spring Structural Reliability Analysis 2 2013 Spring
Office: Chungmu-gwan 706
e-mail: [email protected]
Office hour: Mon/Tue 10:00~11:00 am
Mid-term (30%), Final (30%), Homework (40%)
Exams: Open book
Course information
2013 Spring Structural Reliability Analysis 3 2013 Spring
Schedule
week Date Contents Etc.
1 Mar 06 Introduction/ Probability applications to Engineering
2 Mar 13 Probability density functions, histogram
3 Mar 20 Cumulative probability functions, fitting Probability curve
4 Mar 27 Limit state function, Probability of failure and Reliability index
5 Apr 03 Integration at (G < 0), direct integration
6 Apr 10 Random Sampling
7 Apr 17 Monte Carlo method
8 Apr 24 Mid-term
9 May 01 FOSM
10 May 08 Calibration (1)
11 May 15 Calibration (2)
12 May 22 Models of Load and resistance
13 May 29 System reliability
14 June 05 Examples of application
15 June 12 Presentation
16 June 19 Final
2013 Spring Structural Reliability Analysis 4 2013 Spring
1. COURSE LAYOUT
• Design Philosophy
• Deterministic vs. Stochastic
• Reliability Analysis
• Partial Safety Factors
2013 Spring Structural Reliability Analysis 5 2013 Spring
Design Philosophy
R, S
R (resisting strength) > S (acting force)
R S
R, S f R g S
Engineer’s confidence to uncertainties,
reliability
Uncertainties in acting force and resisting strength
are considered in design by partial safety factors
(load factors and resisting factors)
f R > g S Wider gap, more confidence to designers
Narrower gap, less cost for contractors
2013 Spring Structural Reliability Analysis 6 2013 Spring
Deterministic vs. Stochastic
R, S R S
probability
Make acting force and resisting strength stochastic variables,
then calculate the probability that acting force is bigger than
resisting strength.
)( SRPp f
2013 Spring Structural Reliability Analysis 7 2013 Spring
Reliability analysis
A tool that can give theoretical background to consider
uncertainties of design variables in engineering problems.
)( SRPp f
General procedure
1. Determine the distributions of variables
2. Define Limit State Function
PDF, CDF
3. Calculate the probability of ‘violation of
the Limit State Function’
0 SRG
4. Check the probability with the target
probability of failure
Generally,
0.003%, (=4)
0.135%, (=3)
: Reliability index
2013 Spring Structural Reliability Analysis 8 2013 Spring
Partial Safety Factors
Although it is desirable to do reliability analysis for each design
case, it will be more practical to use partial safety factors for
most designers.
)( SRPp f gf
General procedure
1. Determine the distributions of variables
2. Define Limit State Function
PDF, CDF
3. Calculate the probability of ‘violation of
the Limit State Function’
0 SRG gf
4. Check the probability with the target
probability of failure (reliability index)
Generally,
0.003%, (=4)
0.135%, (=3)
5. Iteration to find f and g combination, which always give
over the target reliability index.
2013 Spring Structural Reliability Analysis 9 2013 Spring
2. PRERIMINARIES
• Events and Sets (Domain)
• De Morgan’s Rule
• Probability in Engineering
• Dependent Events and Sets
• Bayes’ Theorem
2013 Spring Structural Reliability Analysis 10 2013 Spring
Events and Set (Domain)
The formulation of a probabilistic problem starts with the
identification of the event of interests for a set.
Problems of “happen or not”
E11: Failure of link 1, E12: Failure of link 2
Parallel system Serial system
Event: failure.
Set: two set of link1 and link 2
Eij i: Event number
j: Set number
2013 Spring Structural Reliability Analysis 11 2013 Spring
De Morgan’s Rule
When does the following system fail?
E11: Failure of link 1, E12: Failure of link 2, E13: Failure of link 3, E14: Failure of link 4
Find when the system operates (gets connected).
Then, apply De Morgan’s Rule.
(Prob. 1-1) For the event “connected”, find the probability that system fails.
2013 Spring Structural Reliability Analysis 12 2013 Spring
Probability of events
P(E): probability that the event “E” occurs.
When two events E11 and E21 are mutually exclusive event in a set S,
the probability of “E11 or E21”, P(E11 U E21) is P(E11)+P(E21) and
the probability of “E11 and E21”, P(E11 ∩ E21) is 0 (impossible).
e.g) En1: occurrence of number n for a dice (n=1,2,…,6)
P(E11 U E21) = 1/6+1/6 =1/3
P(E11 ∩ E21) =0, mutually exclusive event.
However, for different two sets S1, S2,
P(E11 U E22) = 1/6+1/6 =1/3
P(E11 ∩ E22) =1/6(1/6)=1/36, mutually exclusive set.
2013 Spring Structural Reliability Analysis 13 2013 Spring
Probability of events in Engineering
The probabilities of events in engineering problem are not such
random as previous examples, those are already engineered.
Sometimes, the events and sets are not mutually exclusive.
e.g.) Event 1: fracture, Set: link 1 and link 2 P(E11) = 0.02, P(E12) = 0.03
1. Engineered: the event is no more about “happen or not, 1/possible
cases problem”. The probability of the event was already engineered
or designed for preventing or holding the event.
2. The selection of events in a set is not necessary to be mutually exclusive.
e.g.) Event 1: fracture, Set: link 1 and link 2 from same manufacturer.
2111 EE
2212 EE
e.g.) Event 1: fracture, Event 2: Damage, Set: link 1
3. The selection of sets is not necessary to be mutually exclusive.
dependency
2013 Spring Structural Reliability Analysis 14 2013 Spring
Conditional Probability
Conditional probability: For some occasions when the probability of an
event depends on the occurrence of another event.
A For the case that event A and B
are in one set,
)(
)()|(
B
BABA
EP
EEPEEP
B
For the case that event A and B
are in different set, A B dependency
pEEP BA )|(
For this case, will be given such as “When rain falls (B), the probability to sell umbrella (A) is 70%.”
)()|( ABA EPEEP
If the probability that rain falls, what is the probability to sell umbrella? 56.0)8.0(7.0)()|( BBA EPEEP
If there is no dependency between A an B in different sets,
)()()( BABA EPEPEEP
)|( BA EEP
2013 Spring Structural Reliability Analysis 15 2013 Spring
Dependent events in a set
Intersect part needs to be omitted.
When the events in a set are not mutually exclusive,
2111 EE
e.g.) Event 1: fracture, Event 2: Damage, Set: link 1
)()()( 21112111 EPEPEEP
)()()()( 211121112111 EEPEPEPEEP
)()( 112111 EPEEP )()( 212111 EPEEP
Fracture
Damage
No damage
Link 1
2013 Spring Structural Reliability Analysis 16 2013 Spring
Dependent event in different sets
Intersect part needs to be omitted.
When the selected sets are not mutually exclusive,
e.g.) Event 1: fracture, Event 2: Damage, Set: link 1
)()()( 12111211 EPEPEEP
)|()()()(
)()()()(
1211121211
121112111211
EEPEPEPEP
EEPEPEPEEP
Fracture
Link 1
Link 2
Fracture
If there is no dependency,
)()()()()( 111212111211 EPEPEPEPEEP
If there is total dependency (so, if link 1 fractures then link 2
also fractures.), 1)|( 1211 EEP
)()()()( 1212111211 EPEPEPEEP
(Prob. 1-2) Which case will be better for engineers? Why? (Put numbers and
check.)
2013 Spring Structural Reliability Analysis 17 2013 Spring
Total Probability
For this case, will be given such as “When rain falls (B), the probability to sell umbrella (A) is 70%.”
If the probability that rain falls, what is the probability to sell umbrella? 56.0)8.0(7.0)()|( BBA EPEEP
)|( BA EEP
Remind
Is 56% the probability to sell umbrella?
The total probability to sell umbrella should include when rain does not fall.
Therefore, if 3.0)|( BA EEP
62.0)2.0(3.0)8.0(7.0)()|()()|( BBABBA EPEEPEPEEP
Generally, )()|()(
1
i
n
i
i EPEAPAP
2013 Spring Structural Reliability Analysis 18 2013 Spring
Bayes’ Theorem
The Bayes’ theorem (Bayesian update) provides a valuable and useful tool for revising
or updating a calculated probability as additional data or information becomes
available.
)()( ii EAPAEP )()|()()|( APAEPEPEAP iii
)()|()(1
i
n
i
i EPEAPAP
)(
)()|()|(
AP
EPEAPAEP ii
i
)()|(
)()|()|(
1
j
n
j
j
iii
EPEAP
EPEAPAEP
Previous slide
Updating
)|()|(
)|(
)|()|(
)(
)|(
1
p
j
n
jp
j
j
u
p
ip
i
i
u
u
i
AEPAEP
EAP
AEPAEP
EAP
AEP
For the previous probability of event Ei for sample size Ap,
the updated probability of event Ei for the new sample size
Au can be calculated as
Previously find )()|( i
p
i EPAEP ?)|( u
i AEP
2013 Spring Structural Reliability Analysis 19 2013 Spring
Bayes’ Theorem Example 1
The fraction defective of a product has been 5% after observing 100 samples.
Additional observation by an excellent expert (assume he/she would not miss detecting
any significant flaw) detected a defective product at the 3rd product. Show the
sequence of the fraction defective using Bayesian update.
1st observation: No defective
Put Event 1: Defective, Event 2: No defective
0495.0
95.095.0
101/9605.0
05.0
101/5
05.005.0
101/5
)|()|(
)|(
)|()|(
)(
)|(
1
p
j
n
jp
j
j
u
p
ip
i
i
u
u
i
AEPAEP
EAP
AEPAEP
EAP
AEP
2nd observation: No defective
049.0
9505.09505.0
102/970495.0
0495.0
102/5
0495.00495.0
102/5
)|(
u
i AEP
3rd observation: defective
0583.0
951.0951.0
103/970495.0
0495.0
103/6
049.0049.0
103/6
)|(
u
i AEP
2013 Spring Structural Reliability Analysis 20 2013 Spring
Bayes’ Theorem Example 2
For the previous example, if the reliability of expert is only 90%, what is the updated
fraction defective when the first observation detect a defective product.
321.0)95.0)(1.0()05.0)(9.0(
)05.0)(9.0(
)()|(
)()|()|(
2
1
111
j
j
j EPEDP
EPEDPDEP
(Prob. 1-3) What is the updated fraction defective when the first observation is not a defective product.
006.0)95.0)(9.0()05.0)(1.0(
)05.0)(1.0(
)()|(
)()|()|(
2
1
111
j
j
j EPENP
EPENPNEP
(Prob. 1-4) What happens when the reliability of expert is only 50%. Discuss the results.
2013 Spring Structural Reliability Analysis 21 2013 Spring 2013 Spring
Structural Reliability Chapter II. Stochastic Variables
Jung J. Kim, Ph.D. Adjunct Professor Department of Civil & Environmental Engineering Sejong University
2013 Spring Structural Reliability Analysis 22 2013 Spring
1. UNCERTAIN VARIABLES
• Stochastic variables and others
• Probability Density Function (PDF)
• Generation of PDFs
2013 Spring Structural Reliability Analysis 23 2013 Spring
Evidence theory
belief, plausibility
Probability theory
probability
Possibility theory
necessity, possibility
Modeling of Uncertain Variables
There exist many methodology to model uncertain variables according to the types of
uncertainties such as likelihood, non-specificity, ambiguity and so on.
Fuzzy logic
membership
Fuzzy set Crisp set
2013 Spring Structural Reliability Analysis 24 2013 Spring
Distributions of variables
What is the probability that the event x1 occurs?
For },...,,,{ 10321 xxxxX
How about that of x3?
10321 ... xxxx
Uniform distribution,
if there are no intentions, no
constraints and so on.
p
Engineering properties are
designed to have an
intended value, such as
fy = 400 MPa, fck = 25 MPa.
0
2
4
6
8
15 20 25 30 35 Over
Fre
qu
ency
Strength (MPa)
2013 Spring Structural Reliability Analysis 25 2013 Spring
Discrete vs. Continuous
Discrete set Continuous function },...,,,{ 321 nxxxx
},...,,,{ 321 npppp
2
2
1
2
1)(
cx
d expe.g.)
Easy to understand the concepts of probability.
For practical use, it is necessary for the probability concepts to be extended to continuous domain
0
0.02
0.04
0.06
0.08
0.1
15 20 25 30 35 Over
Fre
qu
ency
den
sity
Strength (MPa)
Histogram
0
0.02
0.04
0.06
0.08
0.1
0 10 20 30 40 50
Pro
ba
bil
ity
den
sity
Strength (MPa)
Normal distribution function
2013 Spring Structural Reliability Analysis 26 2013 Spring
• Probability Density Function (PDF)
2013 Spring Structural Reliability Analysis 27 2013 Spring
Probability Density Function (PDF)
Normalized distribution of a continuous variable x.
The types of PDFs Uniform, Normal, Lognormal, Gamma, Gumbel (Extreme type I),
Frechet (Extreme type II), Weibull(Extreme type III), Poison distributions
… discussed later!
Central Limit Theorem
If the number of samples for a PDF are infinity, the PDF approaches
a normal distribution.
Normalized means that the area under the distribution is 1.0.
1)pdf(
dxx
This can be the basis of normal approximation of
distributions.
11
N
i
ipDiscrete set
2013 Spring Structural Reliability Analysis 28 2013 Spring
Cumulative Density Function (CDF)
Definition of CDF of a continuous variable x.
dxx
)pdf()cdf(
Is the inverse of CDF available? How about PDF?
i
k
ki pcp1
Discrete set
)(X)cdf( xprobx where X is a continuous set.
2013 Spring Structural Reliability Analysis 29 2013 Spring
Moments of PDF
Moments
The first moment
The r-th moment r is defined as dxxpxr
r
)(
Expected value
The r-th central moment r is defined as dxxpx r
r
)()( 1
dxxxp
)(1
The second central moment
dxxpx
)()( 2
12 Variance
e.g.)
With the uniformly distributed assumption
},...,,{ 21 NxxxX For the discrete set of
Nppp N
1...21
cxN
xpxN
i
i
N
i
ii 11
1
1)(
2
1
2
2 )(1
N
i
cxN
average square of the standard deviation
Those are the first two moments, which are most useful.
2013 Spring Structural Reliability Analysis 30 2013 Spring
Characteristics of PDF
Coefficient of Variation (COV)
c
1
2/1
2COV
Coefficient of Skewness (COS)
2/3
2
3COS
Coefficient of Kurtosis (COK)
2
2
4COK
Figure
(+) COS
Right
(-) COS
Left
COK <3.0
flatter COK >3.0
narrower
2013 Spring Structural Reliability Analysis 31 2013 Spring 2013 Spring
Types of PDF (uniform)
Uniform PDF
CDF
The uniform distribution is the most uncertain distribution.
(Mathematically, the minimum information and the maximum
entropy principle)
otherwise0
1
)(bxa
abxpU
2
ba
12
)( 22 ab
(Prob. 2-1) Derive the equation for 2 (The second central moment)
Figure
2013 Spring Structural Reliability Analysis 32 2013 Spring 2013 Spring
Normal PDF
Types of PDF (Normal or Gauss)
CDF
2
2
1
2
1)(
x
N exp
The most important distribution in structural reliability theory.
(Central limit theorem)
dexPx
N
2
2
1
2
1)(
Figure
No closed-form solution
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
10 20 30 40 50
Pro
ba
bil
ity d
en
sity
Strength (MPa)
PDF (Normal distribution)
2013 Spring Structural Reliability Analysis 33 2013 Spring 2013 Spring
Standard Normal PDF Normal PDF for = 1, = 0
Types of PDF (Standard Normal)
CDF
)(2
1)(
2
2
1
zezpz
SN f
)()( zzPSN
)(1)( zz
2013 Spring Structural Reliability Analysis 34 2013 Spring 2013 Spring
PDFs are standardized to the Standard Normal PDF, not only for the normal PDFs, but also for the other type of PDFs based on Second Moment (SM) approximation, which is that most distributions can be described by the first two moments, and .
Standard form
XZ ZX
xxZprobxZprobxP )()(
then Put
X
2
2
Z
-2 -1 0 1 2
Moving and scaling
f
xx
dx
dxP
dx
dxp
1)()(
X
2
2
Z
-2 -1 0 1 2
2013 Spring Structural Reliability Analysis 35 2013 Spring 2013 Spring
Only defined in the positive region, which gives reasons to be used
for most physical quantities.
Types of PDF (Lognormal)
2ln
2
1exp
2
1)(
xxpLN
Lognormal PDF
Normal PDFs for ln(X)
2)ln(
2
X
2
XX
2 /1ln
2exp
2X CDF 1)exp( 22 XX
For COVX <20%
Xln
XCOV
2013 Spring Structural Reliability Analysis 36 2013 Spring 2013 Spring
Normal vs. Lognormal
Normal and lognormal Distributions for (, ) =(30, 3), (30,9) MPa.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
10 20 30 40 50
Pro
ba
bil
ity
den
sity
Strength (MPa)
PDF s(Normal and Lognormal distributions)
(Prob. 2-2) Draw figures
2013 Spring Structural Reliability Analysis 37 2013 Spring 2013 Spring
Types of PDF (Binomial)
The probability that no bar fails under 210 MPa
The probability that a steel bar fails under 210 MPa by tension is
0.05. If we do tensile strength tests of 5 steel bars, what is the
probability that at least one bar fails under 210 MPa.
Describe the probability of occurrence of x among n trials for the
occurrence probability of p.
nxppx
nxp xnx
BN ,...,2,1,0for)1()(
x
k
knk
EX ppk
nxP
0
)1()(
CDF
774.0)05.01()05.0)(1()1(0
5)0( 50050
pppBN
Therefore, the probability that at least one bar fails under 210 Mpa is then
226.0774.01)0(1 BNp
(Prob. 2-3) The probability that no more than two bars fails under 210 Mpa (CDF)
Bernoulli Sequence
2013 Spring Structural Reliability Analysis 38 2013 Spring 2013 Spring
Types of PDF (Poisson)
(Prob. 2-4) The probability that such large earthquakes occurs at least two times in the next 5 years.
Describe the probability of random occurrence at any given time or
space interval. e.g.) earthquake, tsunami…
,...2,1,0forexp
!)( xvt
x
vtxXp
x
tPSPDF
x is the average number of occurrences in a time [0, t].
v is the mean occurrence rate (time/number of event).
In the last 50 years, there were three large earthquakes (with M > 6)
in Southern California. So, the mean occurrence rate is 3/50=0.06
per year. The probability that such large earthquakes in the next 10
years is then
Poisson Sequence
329.0)10(06.0exp
!0
)10(06.0)0(
0
10 XpPS
671.0329.01)0(1)1( 1010 XpXp PSPS
2013 Spring Structural Reliability Analysis 39 2013 Spring 2013 Spring
Types of PDF (Exponential)
No occurrence of event during time t
vtvt
vtXp tPS expexp
!0)0(
0
The probability that an event occurs in the next t years (CDF).
Poisson Sequence
)exp(1)0(1)1( vtXpXp tPStPS
The probability that an event occurs at t year is then (PDF).
)exp()0(1)1( vtvXpXp tPStPS
For the previous example, the probability that such large
earthquakes in the next 20 years is then
699.0)]20(06.0exp[1)20( EXp
)exp(1)( vttpEX
)exp()( vtvtpEX
(Prob. 2-5) What are the differences between Poisson and Exponential distributions?
2013 Spring Structural Reliability Analysis 40 2013 Spring 2013 Spring
Types of PDF (Geometric)
Describe the probability of occurrence at n-th trial for the occurrence
probability of p.
integerfor)1()( 1 nppnp n
GMPDF
(Prob. 2-6) The probability that the first occurrence of 50-yr wind velocity within 5 years (sum).
Mean Recurrence Time (Return period)
In design code, design wind is defined as such a way that wind
velocity with return period 50yr (return period).c
yearper02.050
11
Tp
For a reference time (completion of building), the probability that the
first occurrence of 50-yr wind velocity on the second year is
196.0)98.0(02.0)1()2( 12 pppGM
2013 Spring Structural Reliability Analysis 41 2013 Spring 2013 Spring
Gamma PDF
X
2
X 2
k
k
Types of PDF (Gamma)
Usually used for modeling of sustained live load.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 1 2 3 4 5 6 7 8 9 10
Pro
ba
bil
ity
den
sity
Sustained live load (kN/m2)
Gamma PDFs
1,1 k
1,2 k
1,3 k1,1 k 0forexp)( xxxpG
1,2 k 0forexp)( xxxxpG
1,3 k 0for
2
exp)(
2
xxx
xpG
0for
)(
exp)()(
1
xk
xxxp
k
G
0integerfor)!1()(0
1
kkduuek ku
(Prob. 2-7) Draw figures for various parameters (2)
2013 Spring Structural Reliability Analysis 42 2013 Spring 2013 Spring
(x )
X (x) exp (x ) for xuf u e
Extreme Type I PDF
X
22
X 2
0.577
6
u
Types of PDF (Gumbel)
Extreme values, wind loads
(Prob. 2-8) Draw figures for various parameters (2)
2013 Spring Structural Reliability Analysis 43 2013 Spring 2013 Spring
1
X (x) exp for 0 xx x
k kk u u
fu
Extreme Type II PDF
X
2 2 2
X
11 for 1
2 11 1 for 2
where ( ) ( 1)!
u kk
u kk k
y y
Types of PDF (Frechet)
Extreme values, the maximum seismic load
Prob. 12: Draw figures for
various parameters (2)
(Prob. 2-9) Draw figures for various parameters (2)
2013 Spring Structural Reliability Analysis 44 2013 Spring 2013 Spring
Extreme Type III PDF
Types of PDF (Weibull)
Important distribution for most research works.
Used to simulate a lifetime event of a structure
Bathtub curve: Summation of three phase curves
(Prob. 2-10) Draw figures for various parameters (3)