Swiss Federal Institute of Technologyarchiv.ibk.ethz.ch/emeritus/fa/education/Seminare/... · 2015....

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


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


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)


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


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



• 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



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)



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



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 !



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 !


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


Random Variables


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 ==


Random Variables


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

∂=∂


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


Random Variables


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


Random Variables


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=

= = ⋅∑


Random Variables


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=

= = − ⋅∑


Random Variables


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


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


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=

= +∑


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


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,…



• 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.



• 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)⋅


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]


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



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



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



Extremes of a random process:


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


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



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



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 /=



The normal distributed safety margin M

)(mfM

mMμ

SafeFailure

Mσ Mσ

)(mfM

mMμ

SafeFailure

Mσ Mσ



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



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


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



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 !



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



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



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


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


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



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


- 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



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



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



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



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



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



The procedure can be extended to consider

Correlated random variables UYX →→

Correlatedrandomvariables

Un-correlatedrandomvariables

Translation and scaling Orthogonal

transformation(rotation)

Standardizedrandom variables



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




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




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



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*



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…



Transformation

g(Z): linear g(U): non linear

μZ1, μZ2 R μU1= μU2= 0

σZ1, σZ2 R σU1= σU2= 1

∈∈∈∈



joint probability density function

“Limit state function”

g(U) = R-S



Start point X1



Linearization of Limit state function in starting point



Calculation of new design point X2



Linearisation of Limit statefunction in X2



Calculation of new design point X3



Linearization of Limit statefunction in X3



β1=3.556

β2=3.607

β3=3.608

β4=3.608

Convergency Criteria: εβββ ≤−=Δ + nn 1



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



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



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 =


0

2

4

6

8

10

12

14

16

18

20

-2 0 2 4 6 8 10 12Load

Res

ista

nce


• 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

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