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Swiss Federal Institute of Technology
Risk and Safety
in
Civil, Surveying and Environmental
Engineering
Prof. Dr. Michael Havbro Faber Swiss Federal Institute of Technology
ETH Zurich, Switzerland
Swiss Federal Institute of Technology
Contents of Today's Lecture
• Probability theory• Uncertainties in engineering decision making• Probabilistic modelling• Engineering model building• Methods of structural reliability theory
- Linear normal distributed safety margins - Non-linear normal distributed safety margins- General case- SORM improvements- Monte-Carlo simulation
Swiss Federal Institute of Technology
Conditional Probability and Bayes‘s Ruleas there is
we have
1
( ) ( ) ( ) ( )( )
( ) ( ) ( )
i i i ii n
i ii
P A E P E P A E P EP E A
P A P A E P E=
= =∑
( ) ( ) ( ) ( ) ( )i i i iP A E P A E P E P E A P A= =∩
Likelihood Prior
PosteriorBayes Rule
Reverend Thomas Bayes(1702-1764)
Swiss Federal Institute of Technology
Overview of Uncertainty Modelling
• Why uncertainty modelling
Decision Making !
Risks
Consequences of eventsProbabilities of events
Probabilistic model
Data Model estimation
Decision Making !Decision Making !
RisksRisks
Consequences of eventsProbabilities of events Consequences of eventsConsequences of eventsProbabilities of eventsProbabilities of events
Probabilistic modelProbabilistic model
Data Model estimationData Model estimation
Uncertain phenomenonUncertain phenomenon
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Uncertainties in Engineering Problems
Different types of uncertainties influence decision making
• Inherent natural variability – aleatory uncertainty- result of throwing dices- variations in material properties- variations of wind loads- variations in rain fall
• Model uncertainty – epistemic uncertainty- lack of knowledge (future developments)- inadequate/imprecise models (simplistic physical modelling)
• Statistical uncertainties – epistemic uncertainty - sparse information/small number of data
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Uncertainties in Engineering Problems
• Consider as an example a dike structure
- the design (height) of the dike will be determining the frequency of floods
- if exact models are available for the prediction of future water levels and our knowledge about the input parameters is perfect then we can calculate the frequency of floods (per year) - a deterministic world !
- even if the world would be deterministic – we would not have perfect information about it – so we might as well consider the world as random
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Uncertainties in Engineering Problems
In principle the so-called
inherent physical uncertainty (aleatory – Type I)
is the uncertainty caused by the fact that the world is random, however, another pragmatic viewpoint is to define this type of uncertainty as
any uncertainty which cannot be reduced by means of collection of additional information
the uncertainty which can be reduced is then the
model and statistical uncertainties (epistemic – Type II)
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Uncertainties in Engineering Problems
Observed annual extreme water levels
Model for annual extremes
Regression model topredict future extremes
Predicted future extreme water level
Aleatory Uncertainty
Epistemic Uncertainty
Observed annual extreme water levels
Model for annual extremes
Regression model topredict future extremes
Predicted future extreme water level
Aleatory Uncertainty
Epistemic Uncertainty
Swiss Federal Institute of Technology
Uncertainties in Engineering Problems
The relative contribution of aleatory and epistemic uncertainty to the prediction of future water levels is thus influenced directly by the applied models
refining a model might reduce the epistemic uncertainty – but in general also changes the contribution of aleatory uncertainty
the uncertainty structure of a problem can thus be said to be scale dependent !
Swiss Federal Institute of Technology
Uncertainties in Engineering Problems
Knowledge
TimeFuture
Past
Present
100%
Observation
Prediction
Knowledge
TimeFuture
Past
Present
100%
Observation
Prediction
The uncertainty structure changes also as function of time – is thus time dependent !
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Random Variables
• Probability distribution and density functions
A random variable is denoted with capital letters : X
A realization of a random variable is denoted with small letters : x
We distinguish between
- continuous random variables : can take any value in a given range
- discrete random variables : can take only discrete values
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Random Variables
• Probability distribution and density functions
The probability that the outcome of a discrete random variable X is smaller than x is denoted the probability distribution function
The probability density function for a discreterandom variable is defined by
( ) ( )i
X X ix x
P x p x<
=∑
)xX(P)x(p iX ==
Swiss Federal Institute of Technology
Random Variables
• Probability distribution and density functions
The probability that the outcome of a continuous random variable X is smaller than x is denoted the probability distribution function
The probability density function for a continuous random variable is defined by
( ) ( )XF x P X x= <
( ) ( )XX
F xf xx
∂=∂
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Random Variables
• Moments of random variables and the expectation operator
Probability distribution and density function can be described in terms of their parameters or their moments
Often we write
The parameters can be related to the moments and visa versa
),( pxFX ),( pxf X
Parameters
p
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Random Variables
• Moments of random variables and the expectation operator
The i‘th moment mi for a continuous random variable X is defined through
The expected value E[X] of a continuous random variable X is defined accordingly as the first moment
∫∞
∞−
⋅= dx)x(fxm Xi
i
[ ] ( )dxxfxXE XX ∫∞
∞−
⋅==μ
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Random Variables
• Moments of random variables and the expectation operator
The i‘th moment mi for a discrete random variable X is defined through
The expected value E[X] of a discrete random variable X is defined accordingly as the first moment
1( )
ni
i j X jj
m x p x=
= ⋅∑
[ ]1
( )n
X j X jj
E X x p xμ=
= = ⋅∑
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Random Variables
• Moments of random variables and the expectation operator
The standard deviation of a continuous random variable is defined as the second central moment i.e. for a continuous random variable X we have
for a discrete random variable we have correspondingly
[ ] [ ] ( ) ( )dxxfxXE XXXX ∫∞
∞−
⋅−=−== 222 )(XVar μμσ
Xσ
Variance Mean value
[ ]2 2
1
( ) ( )n
X j X X jj
Var X x p xσ μ=
= = − ⋅∑
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Random Variables
• Moments of random variables and the expectation operator
The ratio between the standard deviation and the expected value of a random variable is called the Coefficient of Variation CoV and is defined as
a useful characteristic to indicate the variability of the random variable around its expected value
[ ] X
X
CoV X σμ
=
Dimensionless
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Random Variables
• Typical probability distributionfunctions in engineering
Normal : sum of random effects
Log-Normal: product of randomeffects
Exponential: waiting times
Gamma: Sum of waiting times
Beta: Flexible modeling function
Distribution type Parameters MomentsRectangular
a x b≤ ≤
ab)x(f X −
= 1
abaxxFX −
−=)(
a b 2
ba +=μ
12ab −=σ
Normal
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−=
2
21
21
σμ
πσxexp)x(f X
dxxexp)x(Fx
X ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−= ∫
∞−
2
21
21
σμ
πσ
μ σ > 0
μ σ
Shifted Lognormal x > ε
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−−
=2
)ln(21exp
2)(1)(
ζλε
πζεx
xxf X
2
Φ ⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=ζ
λε )xln()x(FX
λ ζ > 0 ε
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
2exp
2ζλεμ
1)exp(2
exp 22
−⎟⎟⎠
⎞⎜⎜⎝
⎛+= ζζλσ
Shifted Exponential x ≥ ε
))(exp()( ελλ −−= xxf X ( )ex
X e)x(F −−−= λ1
ε λ > 0
λεμ 1+=
λσ 1=
Gamma x ≥ 0
1)exp()(
)( −−Γ
= pp
X xbxp
bxf
( ))(
,)(p
pbxxFX ΓΓ=
p > 0 b > 0 b
p=μ
bp
=σ
Beta a x b r t≤ ≤ ≥, , 1
( )( ) ( ) 1
11
−+
−−
−−−
Γ⋅Γ+Γ= tr
tr
X )ab()xb()ax(
trtr)x(f
( )( ) ( ) du
)ab()ub()au(
trtr)x(F tr
tru
aX 1
11
−+
−−
−−−⋅
Γ⋅Γ+Γ= ∫
a b r > 1 t > 1
1+−+=
rra)(baμ
1+++−=
trrt
trabσ
Swiss Federal Institute of Technology
Random Variables
• The Normal distribution
The analytical form of the Normal distribution may be derived byrepeated use of the result regarding the probability density function for the sum of two random variables
The normal distribution is very frequently applied in engineering modelling when a random quantity can be assumed to be composed as a sum of a number of individual contributions.
A linear combination S of n Normal distributed random variables is thus also a Normal distributed random variable , 1,2,..,iX i n=
01
n
i ii
S a a X=
= +∑
Swiss Federal Institute of Technology
Random Variables
• The Normal distribution:
In the case where the mean value is equal to zero and the standard deviation is equal to 1 the random variable is said to be standardized.
21 1( ) ( ) exp22Zf z z zϕ
π⎛ ⎞= = −⎜ ⎟⎝ ⎠
21 1( ) ( ) exp22
z
ZF z z x dxπ −∞
⎛ ⎞= Φ = −⎜ ⎟⎝ ⎠∫
X
X
XZ μσ−=
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-3 -2 -1 0 1 2 3
x
fX(x)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-3 -2 -1 0 1 2 3
Standardized random variable
Standard normal
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Stochastic Processes and Extremes
• Random quantities may be “time variant” in the sense that they take new values at different times or at new trials.
- If the new realizations occur at discrete times and have discrete values the random quantity is called a random sequence
failure events, traffic congestions,…
- If the new realizations occur continuously in time and take continuous values the random quantity is called a random process or stochastic process
wind velocity, wave heights,…
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Stochastic Processes and Extremes
• Random sequences
The Poisson counting process is one of the most commonly applied families of probability distributions applied in reliability theory
The process N(t) denoting the number of events in a (time) interval (t, t+Dt[is called a Poisson process if the following conditions are fulfilled:
1) the probability of one event in the interval (t, t+Dt[ is asymptotically proportional to Dt.
2) the probability of more than one event in the interval (t, t+Dt[ is a function of higher order of Dt for Dt→0.
3) events in disjoint intervals are mutually independent.
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Stochastic Processes and Extremes
• Random sequences
The probability distribution function of the (waiting) time till the first event T1 is now easily derived recognizing that the probability of T1 >tis equal to P0(t) we get:
1T 1 0 1
t
0
F (t )=1-P (t )
=1-exp(- ν(τ)dτ )∫
1T 1F (t )=1-exp(-νt)
Homogeneous case !
Exponential probability distributionExponential probability density
1T 1f (t )=ν exp(-νt)⋅
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Stochastic Processes and Extremes• Continuous random processes
A continuous random process is a random process which has realizations continuously over time and for which the realizations belong to a continuous sample space.
Variations of: water levelswind speedrain fall
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1001.50
2.00
2.50
3.00
3.50
Wat
er le
vel [
m]
Time [days]
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Stochastic Processes and Extremes• Continuous random processes
The mean value of the possible realizations of a random process is given as:
The correlation between realizations at any two points in time is given as:
[ ]( ) ( ) ( , )X Xt E X t x f x t dxμ∞
−∞
= = ⋅∫
Function of time !
[ ] 212121212121 dxdx)t,t;x,x(fxx)t(X)t(XE)t,t(R XXXX ⋅⋅== ∫ ∫∞
∞−
∞
∞−
Auto-correlation function – refers to a scalar valued random process
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Stochastic Processes and Extremes
Extreme Value Distributions
In engineering we are often interested in extreme values i.e. thesmallest or the largest value of a certain quantity within a certaintime interval e.g.:
The largest earthquake in 1 year
The highest wave in a winter season
The largest rainfall in 100 years
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Stochastic Processes and Extremes
Extreme Value Distributions
We could also be interested in the smallest or the largest value of a certain quantity within a certain volume or area unit e.g.:
The largest concentration of pesticides in a volume of soil
The weakest link in a chain
The smallest thickness of concrete cover
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Stochastic Processes and Extremes
Extremes of a random process:
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Overview of Estimation and Model Building
Different types of information is used whendeveloping engineering models
- subjective information- frequentististic information
Frequentistic- Data
Subjective- Physical understanding- Experience- Judgement
Distribution family
Distribution parameters
Probabilistic model
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Structural Reliability Analysis
Reliability of structures cannot be assessed through failure rates because
- Structures are unique in nature
- Structural failures normally take place due to extreme loads exceeding the residual strength
Therefore in structural reliability, models are established for resistances R and loads S individually and the structural reliability is assessed through:
)0( ≤−= SRPPf
rs
R
S
Swiss Federal Institute of Technology
Structural Reliability Analysis
If only the resistance is uncertain the failure probability may be assessed by
If also the load is uncertain we have
where it is assumed that the load and the resistance are independent
This is called the
„Fundamental Case“
)1/()()( ≤==≤= sRPsFsRPP Rf
∫∞
∞−
=≤−=≤= dxxfxFSRPSRPP SRf )()()1()(
)(),( sfrf SR
sr ,
Load SResistance R
)(xfFP
sr ,
B
A
x
)(),( sfrf SR
sr ,
Load SResistance R
)(),( sfrf SR
sr ,
Load SResistance R
)(xfFP
sr ,
B
A
x
Swiss Federal Institute of Technology
Structural Reliability Analysis
In the case where R and S are normal distributed the safety margin M is also normal distributed
Then the failure probability is
with the mean value of M
and standard deviation of M
The failure probability is then
where the reliability index is
SRM −=
)0()0( ≤=≤−= MPSRPPF
SRM μμμ −=
22SRM σσσ +=
)()0( βσ
μ −Φ=−Φ=M
MFP
MM σμβ /=
Swiss Federal Institute of Technology
Structural Reliability Analysis
The normal distributed safety margin M
)(mfM
mMμ
SafeFailure
Mσ Mσ
)(mfM
mMμ
SafeFailure
Mσ Mσ
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Structural Reliability Analysis
In the general case the resistance and the load may be defined in terms of functionswhere X are basic random variables
and the safety margin as
where is called the
limit state function
failure occurs when
)()(
2
1
XX
fSfR
==
)()()( 21 XXX gffSRM =−=−=
0)( ≤xg
0)( ≤xg
Swiss Federal Institute of Technology
Structural Reliability Analysis
Setting defines a (n-1) dimensional surface in the space spanned by the n basic variables X
This is the failure surface separating the sample space of X into a safe domain and a failure domain
The failure probability may in general terms be written as
0)( =xg
ix
1+ix
Failure domain
Safe domain
0),..,,( 21 >nxxxg
sΩ
fΩ
0),..,,( 21 ≤nxxxg
Failure event
( ) 0
( )fg
P f d≤
= ∫ Xx
x x
( ) 0g =x
{ }0)( ≤= xF g
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The probability of failure can be assessed by
where is the joint probability density function for the basic random variables X
For the 2-dimensional case the failure probability simply corresponds to the integral under the joint probability density function in the area of failure
{ }∫
≤=Ω
=0)(
)(x
X xxg
f
f
dfP
)(xXf
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The probability of failure can be calculated using- numerical integration
(Simpson, Gauss, Tchebyschev, etc.)
but for problems involving dimensions higher than say 6 the numerical integration becomes cumbersome
{ }∫
≤=Ω
=0)(
)(x
X xxg
f
f
dfP
Other methods are necessary !
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
When the limit state function islinear
the saftey margin M is definedthrough
with
mean value
and
variance
∑=
⋅+=n
iii xaag
10)(x
∑=
⋅+=n
iii XaaM
10
∑=
+=n
iXiM i
aa1
0 μμ
∑∑∑≠===
+=n
ijjjijiij
n
i
n
iXiM aaa
i,111
222 σσρσσ
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The failure probability can then be written as
The reliability index is defined as
Provided that the safety margin is normal distributed the failure probability is determined as
)0()0)(( ≤=≤= MPgPPF X
M
M
σμβ =
)( β−Φ=FP
m
)(mfM
Basler and Cornell
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The reliability index β has the geometrical interpretation of being the shortest distance between the failure surface and the origin instandard normal distributed spaceu
in which case the components of U have zero means and variances equal to 1
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
x2
x1
0)( =xg
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
x2
x1-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
x2
x1
0)( =xg 0)( =xg 0)( =xg
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1β
0)( =ug
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1ββ
0)( =ug 0)( =ug 0)( =ug
i
i
X
Xii
XU
σμ−
=
Design point
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Basics of Structural Reliability MethodsExample:
Consider a steel rod with resistance rsubjected to a tension force s
r and s are modeled by the random variables R and S
The probability of failure is wanted
35,350 == RR σμ40,200 == SS σμ
SRg −=)(X
)0( ≤− SRP
Swiss Federal Institute of Technology
Basics of Structural Reliability MethodsExample:
Consider a steel rod with resistance rsubjected to a tension force s
r and s are modeled by the random variables R and S
The probability of failure is wanted
The safety margin is given as
The reliability index is then
and the probability of failure
35,350 == RR σμ40,200 == SS σμ
SRg −=)(X
)0( ≤− SRP
SRM −=150200350 =−=Mμ
15.534035 22 =+=Mσ
84.215.53
150 ==β
3104.2)84.2( −⋅=−Φ=FP
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Usually the limit state function is non-linear- this small phenomenon caused
the so-called invariance problem
Hasofer & Lind suggested to linearizethe limit state function in the design point- this solved the invariance
problem
The reliability index may then be determined by the following optimization problem
Can however easily be linearized !
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1β
0)( =′ ug
0)( =ug-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1ββ
0)( =′ ug
0)( =ug
{ }∑
==∈=
n
ii
gu
1
2
0)(min
uuβ
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- 6
- 4
- 2
0
2
4
6
8
10
12
- 2 0 2 4 6 8 10 12
R
u2
u1
- 6
- 4
- 2
0
2
4
6
8
10
12
- 2 0 2 4 6 8 10 12
R
u2
u1ββ
α0)( =ug
0)( =′ ug
*u
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The optimization problem can be formulated as an iteration problem
1 ) the design point is determined as
2) the normal vector to the limit state function is determined as
3) the safety index is determined as
4) a new design point is determined as
5) continue the above steps until convergence in
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1β
0)( =′ ug
0)( =ug-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1
-6
-4
-2
0
2
4
6
8
10
12
-2 0 2 4 6 8 10 12
S
R
u2
u1ββ
0)( =′ ug
0)( =ug
α
ni
dug
dug
n
j i
ii ,..2,1 ,
)(
)(
2/1
1
2
=
⎥⎦
⎤⎢⎣
⎡⋅∂
⋅∂−=
∑=
α
α
β
βα
0),...,( 21 =⋅⋅⋅ ng αβαβαβ
( )Tnu αβαβαβ ⋅⋅⋅=∗ ,..., 21
* β= ⋅u α
β
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Basics of Structural Reliability Methods
Example :
Consider the steel rod with cross-sectional area a and yield stress r
The rod is loaded with the tension force s
The limit state function can then be written as
r, a and s are uncertain and modeled by normal distributed random variables
we would like to calculate the probability of failure
rah ⋅=
sarg −⋅=)(x
35,350 == RR σμ1,10 == AA σμ
300,1500 == RS σμ
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The first step is to transform the basic random variables into standardized normal distributed space
Then we write the limit state function in terms of the realizations of the standardized normal distributed random variables
R
RR
RUσ
μ−=
A
AA
AUσ
μ−=
S
SS
SUσ
μ−=
200035300350350u )1500300()10)(35035(
)())(()(
R ++−+=+−++=
+−++=
ARSA
SAR
SSSAAARRR
uuuuuuu
uuuug μσμσμσ
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The reliability index iscalculated as
the components of theα-vector are then calculate as
where222SARk ααα ++=
ARSAR αβααααβ
353003503502000
+−+−=
)35350(1AR k
βαα +−=
)35350(1RA k
βαα +−=
kS300=α
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
following the iteration schemewe get the following iterationhistory
Iteration Start 1 2 3 4 5β 3.0000 3.6719 3.7399 3.7444 3.7448 3.7448αR -0.5800 -0.5701 -0.5612 -0.5611 -0.5610 -0.5610αΑ -0.5800 -0.5701 -0.5612 -0.5611 -0.5610 -0.5610αS 0.5800 0.5916 0.6084 0.6086 0.6087 0.6087
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The procedure can be extended to consider
Correlated random variables UYX →→
Correlatedrandomvariables
Un-correlatedrandomvariables
Translation and scaling Orthogonal
transformation(rotation)
Standardizedrandom variables
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Correlated random variables
The covariance matrix for the randomvariables is given as
and the correlation coefficient matrix is
The first step is the standardization
[ ] [ ] [ ]
[ ] [ ] ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
nn
n
XVarXXCov
XXCovXXCovXVar
1
1211
,
,...,
XC
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
11
1
1
1
n
n
ρ
ρ
Xρ
niX
Yi
i
X
Xii ,..2,1 , =
−=
σμ
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Correlated random variables
The transformation of the correlated random variables into non-correlated random variables can be written as
where is a lower triangular matrix
then we can write
with T standing for transpose matrix
Y = TU
T T T T T TE E E⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = =⎣ ⎦ ⎣ ⎦ ⎣ ⎦Y XC Y Y T U U T T U U T T× T ρ
T
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Correlated random variables
In the case of 3 random variables wehave
As is a lower triangular matrix wehave
11 12 13
22 23
33.
T
sym
ρ ρ ρρ ρ
ρ
⎡ ⎤⎢ ⎥⋅ = = ⎢ ⎥⎢ ⎥⎣ ⎦
XT T ρ
232
23133
22
21312332
22122
1331
1221
11
1
1
1
TTT
TTTT
TT
TTT
−−=
⋅−=
−=
===
ρ
ρρ
11 11 12 13 11 12 13
21 22 22 23 22 23
31 32 33 33 33
0 00 0
0 0 .
T
T T T TT T T TT T T T sym
ρ ρ ρρ ρ
ρ
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
T T
T
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
The normal-tail approximation
)()(*
*
i
i
iiX
XiiX
xxF
σμ
′′−
Φ= )(1)(*
*
i
i
i
iiX
Xi
XiX
xxf
σμ
ϕσ ′
′−=
)()))(((
*
*1
iX
iXX xf
xF
i
i
i
−Φ=′
ϕσ
iii XiXiX xFx σμ ′Φ−=′ − ))(( *1*
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Non-normal distributed random variables
Rosenblatt Transformation
)(),,(),,()( 12211121 11xFxxxxFxxxxFxF XnnXnnXX nn
……… −−− −⋅=
),,()(
)()(
)()(
121
122
11
2
1
−=Φ
=Φ
=Φ
nnXn
X
X
xxxxFu
xxFu
xFu
n…
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Transformation
g(Z): linear g(U): non linear
μZ1, μZ2 R μU1= μU2= 0
σZ1, σZ2 R σU1= σU2= 1
∈∈∈∈
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Basics of Structural Reliability Methods
joint probability density function
“Limit state function”
g(U) = R-S
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Start point X1
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Linearization of Limit state function in starting point
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Basics of Structural Reliability Methods
Calculation of new design point X2
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Linearisation of Limit statefunction in X2
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Basics of Structural Reliability Methods
Calculation of new design point X3
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Basics of Structural Reliability Methods
Linearization of Limit statefunction in X3
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
β1=3.556
β2=3.607
β3=3.608
β4=3.608
Convergency Criteria: εβββ ≤−=Δ + nn 1
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
SORM Improvements
{ }∫
≤=Ω
=0)(
)(x
X xxg
f
f
dfP
- 6
- 4
- 2
0
2
4
6
8
10
12
- 2 0 2 4 6 8 10 12
R
u2
u1
- 6
- 4
- 2
0
2
4
6
8
10
12
- 2 0 2 4 6 8 10 12
R
u2
u1ββ
α0)( =ug
0)( =′ ug
*u0)( =′′ ug
FORM
SORM
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
SORM Improvements
Asymptotic Laplace integral solutions
( )
∏∫ −
=
+
+−−
≤ −≈=
1
1
)1(
)1(2/)1(
0)(
)(
)1(
2n
ii
n
nn
g
h deIκλ
λπλ
x
x x
∫≤
=0)(
)(
x
x xg
h deI λ
( ) ∏∏∏∫ −
=
−
=
−
=
−
≤
−
−
−Φ≅−
−=−
≈∑
==
1
1
1
1
1
1
2
0)(
21
)1(
)(
)1(
)(
)1(22
2
1
2
n
ii
n
ii
n
ii
gn
x
fedeP
n
ii
κ
β
κβ
βϕ
κβππ
β
x
x
Main curvatures
n
n-1
0)( ≤xg
Swiss Federal Institute of Technology
Basics of Structural Reliability Methods
Simulation methods may also be used to solve the integration problem
1) m realizations of the vector X are generated
2) for each realization the value of the limit state function is evaluated
3) the realizations where the limit state function is zero or negative are counted
4) The failure probability is estimated as
{ }∫
≤=Ω
=0)(
)(x
X xxg
f
f
dfP
Ran
dom
num
ber
1
)( iX xFi
jz
jx ix
fn
mn
p ff =
Swiss Federal Institute of Technology
0
2
4
6
8
10
12
14
16
18
20
-2 0 2 4 6 8 10 12Load
Res
ista
nce
Basics of Structural Reliability Methods
• Estimation of failure probabilities usingMonte Carlo Simulation
0
2
4
6
8
10
12
14
16
18
20
-2 0 2 4 6 8 10 12Load
Res
ista
nce
0
2
4
6
8
10
12
14
16
18
20
-2 0 2 4 6 8 10 12Load
Res
ista
nce
0
2
4
6
8
10
12
14
16
18
20
-2 0 2 4 6 8 10 12Load
Res
ista
nce
mn
p ff =
• m random outcomes of R und S are generated and the number of outcomes nf in the failure domainare recorded and summed
• The failure probability pfis then
0
2
4
6
8
10
12
14
16
18
20
-2 0 2 4 6 8 10 12Load
Res
ista
nce
SafeFailure