Reliability Assessm ent of Structures Michael Havbro Faber Sw iss - - PowerPoint PPT Presentation
COST E5 5 W orkshop Graz University of Technology May, 1 4 -1 5 , 2 0 0 7 Reliability Assessm ent of Structures Michael Havbro Faber Sw iss Federal I nstitute of Technology ETH Zurich, Sw itzerland Sw iss Federal Institute of Technology
Sw iss Federal Institute of Technology
Michael Havbro Faber Sw iss Federal I nstitute of Technology ETH Zurich, Sw itzerland
COST E5 5 W orkshop Graz University of Technology May, 1 4 -1 5 , 2 0 0 7
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The Risk associated with a given activity RA may then be written as the Consequences of the event CEi The risk contribution REi from the event Ei is defined through the product between
Risk is a characteristic of an activity relating to all possible events nE which may follow as a result of the activity
= =
E i i E i
n i E E n i E A
1 1
the Event probability PEi and
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Uncertainties must be considered in the decision making throughout all phases of the life of an engineering facility
Idea & Concept Planning and feasibility study Investigations and tests Manufacturing Design Execution Operation & maintenance Decommissioning
Uncertainties Uncertainties Traffic volume Loads Resistances (material, soil,..) Degradation processes Service life Manufacturing costs Execution costs Decommissioning costs
Idea & Concept Planning and feasibility study Investigations and tests Manufacturing Design Execution Operation & maintenance Decommissioning
Idea & Concept Planning and feasibility study Investigations and tests Manufacturing Design Execution Operation & maintenance Decommissioning
Uncertainties Uncertainties Traffic volume Loads Resistances (material, soil,..) Degradation processes Service life Manufacturing costs Decommissioning costs
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Any activity carries a risk potential It is important that this potential is fully understood Only when the risk potential is fully understood can rational decisions be identified and implemented
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States of nature of which we have interest such as:
are in the following denoted „events“ we are generally interested in quantifying the probability that such events take place within a given „time frame“
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∞ → = for
exp exp
lim ) ( n n N A P
A
tot A
n n A P = ) (
will that belief
degree ) ( A A P =
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as there is we have
1
i i i i i n i i i
=
( ) ( ) ( ) ( ) ( )
i i i i
P A E P A E P E P E A P A = = I Likelihood Prior Posterior Bayes Rule
Reverend Thomas Bayes (1702-1764)
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Different types of uncertainties influence decision making
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frequency of floods
water levels and our knowledge about the input parameters is perfect then we can calculate the frequency of floods (per year) - a deterministic world !
have perfect information about it – so we might as well consider the world as random
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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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Observed annual extreme water levels Model for annual extremes Regression model to predict future extremes Predicted future extreme water level Aleatory Uncertainty Epistemic Uncertainty Observed annual extreme water levels Model for annual extremes Regression model to predict future extremes Predicted future extreme water level Aleatory Uncertainty Epistemic Uncertainty
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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 !
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Knowledge Time Future Past Present 100% Observation Prediction Knowledge Time Future Past Present 100% Observation Prediction
The uncertainty structure changes also as function of time – is thus time dependent !
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A random variable is denoted with capital letters : X A realization of a random variable is denoted with small letters : x We distinguish between
can take any value in a given range
: can take only discrete values
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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 discrete random variable is defined by
i
X X i x x
<
i X
x x
A) B) 1
PX (x)
x1 x2 x3 x4 x1 x2 x3 x4
pX (x)
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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
X
X X
x x
A) B) 1
FX (x)
f X (x)
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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
p
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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
∞ ∞ −
X i i
X X
∞ ∞ −
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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
n i i j X j j
=
1
n X j X j j
=
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The standard deviation
second central moment i.e. for a continuous random variable X we have for a discrete random variable we have correspondingly
X X X X
∞ ∞ −
2 2 2
X
σ
2 2 1
n X j X X j j
=
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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
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functions in engineering Normal : sum of random effects Log-Normal: product of random effects Exponential: waiting times Gamma: Sum of waiting times Beta: Flexible modeling function
Distribution type Parameters Moments Rectangular
a x b ≤ ≤
a b ) x ( f X − = 1 a b a x x FX − − = ) (
a b
2 b a + = μ 12 a b − = σ
Normal
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − =
2
2 1 2 1 σ μ π σ x exp ) x ( f X dx x exp ) x ( F
x X
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − =
∫
∞ − 2
2 1 2 1 σ μ π σ
μ σ > 0 μ σ Shifted Lognormal
x > ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − − =
2
) ln( 2 1 exp 2 ) ( 1 ) ( ζ λ ε π ζ ε x x x f X
2
Φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = ζ λ ε ) x ln( ) x ( FX
λ ζ > 0 ε
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = 2 exp
2
ζ λ ε μ 1 ) exp( 2 exp
2 2
− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ζ ζ λ σ
Shifted Exponential
x ≥ ε )) ( exp( ) ( ε λ λ − − = x x f X
( )
e x X
e ) x ( F
− −
− =
λ
1
ε λ > 0
λ ε μ 1 + = λ σ 1 =
Gamma
x ≥ 0
1
) exp( ) ( ) (
−
− Γ =
p p X
x bx p b x f
( )
) ( , ) ( p p bx x FX Γ Γ =
p > 0 b > 0
b p = μ b p = σ
Beta
a x b r t ≤ ≤ ≥ , , 1
( ) ( ) ( )
1 1 1 − + − −
− − − Γ ⋅ Γ + Γ =
t r t r X
) a b ( ) x b ( ) a x ( t r t r ) x ( f
( ) ( ) ( )
du ) a b ( ) u b ( ) a u ( t r t r ) x ( F
t r t r u a X 1 1 1 − + − −
− − − ⋅ Γ ⋅ Γ + Γ =
∫
a b r > 1 t > 1
1 + − + = r r a) (b a μ 1 + + + − = t r rt t r a b σ
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values at different times or at new trials.
random quantity is called a random sequence failure events, traffic congestions,…
the random quantity is called a random process or stochastic process wind velocity, wave heights,…
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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.
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 1.50 2.00 2.50 3.00 3.50
Water level [m] Time [days]
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Extremes of a random process:
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Extreme Value Distributions In engineering we are often interested in extreme values i.e. the smallest or the largest value of a certain quantity within a certain time 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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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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I f the extrem es w ithin the period T of an ergodic random process X( t) are independent and follow the distribution: Then the extrem es of the sam e process w ithin the period: w ill follow the distribution:
max ,
T X
max max , ,
n X nT X T
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Extrem e Type I – Gum bel Max W hen the upper tail of the probability density function falls off exponentially ( exponential, Norm al and Gam m a distribution) then the m axim um in the tim e interval T is said to be Type I extrem e distributed
max ,
T X
max ,
T X
max max
0.577216 6
T T
X X
u u γ μ α α π σ α = + = + =
max max max
T T nT
X X X
For increasing tim e intervals the variance is constant but the m ean value increases as:
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Extrem e Type I I – Frechet Max W hen a probability density function is dow nw ards lim ited at zero and upw ards falls off in the form then the m axim um in the tim e interval T is said to be Type I I extrem e distributed
k X
x x F ) 1 ( 1 ) ( β − = ) exp( ) (
max , k T X
x u x F ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − =
) exp( ) (
1 max , k k T X
x u x u u k x f ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =
+
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − Γ − − Γ = − Γ = ) 1 1 ( ) 2 1 ( ) 1 1 (
2 2 2
max max
k k u k u
T T
X X
σ μ
Mean value and standard deviation
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Extrem e Type I I I – W eibull Min W hen a probability density function is dow nw ards lim ited at ε and the low er tail falls off tow ards ε in the form then the m inim um in the tim e interval T is said to be Type I I I extrem e distributed
k
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − − =
k T X
u x x F ε ε exp 1 ) (
min ,
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =
− k k T X
u x u x u k x f ε ε ε ε ε exp ) (
1 min ,
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + Γ − + Γ − = + Γ − + = ) 1 1 ( ) 2 1 ( ) ( ) 1 1 ( ) (
2 2 2
min min
k k u k u
T T
X X
ε σ ε ε μ
Mean value and standard deviation
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Return period for extreme events: The return period for extreme events TR may be defined as If the probability of exceeding x during a reference period of 1 year is 0.01 then the return period for exceedances is
max ,
T X R
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Combination of loads We are interested in the maximum
loads
t Wind t Earth-quake t t Snow Transient load Imposed load Time Week Days Seconds Minutes Hours Permanent load t
{ }
) t ( X ... ) t ( X ) t ( X max ) T ( X
n T max
+ + + =
2 1
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2.00 GENERAL PRINCIPLES 2.01 SELF WEIGHT 2.02 LIVE LOAD 2.06 LOADS IN CAR PARKS 2.12 SNOW LOAD 2.13 WIND LOAD 2.15 WAVE LOAD 2.17 EARTHQUAKE 2.18 IMPACT LOAD 2.20 FIRE
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3.00 GENERAL PRINCIPLES 3.01 CONCRETE 3.02 STRUCTURAL STEEL 3.0* REINFORCING STEEL 3.04 PRESTRESSING STEEL 3.05
TI MBER
3.07 SOIL PROPERTIES 3.09 MODELUNCERTAINTIES 3.10 DIMENSIONS 3.11 EXCENTRICITIES
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Different types of information is used when developing engineering models
Frequentistic
Subjektive
Distribution family Distribution parameters Probabilistic model
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Model building may be seen to consist of five steps 1) Assessment and statistical quantification of the available data 2) Selection of distribution function 3) Estimation of distribution parameters 4) Model verification 5) Model updating
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Reliability of structures cannot be assessed through failure rates because
nature
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:
) ( ≤ − = S R P Pf
r s R S
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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 / ( ) ( ) ( ≤ = = ≤ = s R P s F s R P P
R f
∞ ∞ −
= ≤ − = ≤ = dx x f x F S R P S R P P
S R f
) ( ) ( ) 1 ( ) (
) ( ), ( s f r f
S R
s r,
Load S Resistance R ) (x f
F
P
s r,
B A
x
) ( ), ( s f r f
S R
s r,
Load S Resistance R ) ( ), ( s f r f
S R
s r,
Load S Resistance R ) (x f
F
P
s r,
B A
x
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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
) ( ) ( ≤ = ≤ − = M P S R P P
F S R M
2 2 S R M
σ σ σ + =
) ( ) ( β σ μ − Φ = − Φ =
M M F
P
M M σ
μ β / =
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The normal distributed safety margin M
) (m f M
M
μ
Safe Failure
M
σ
M
σ
) (m f M
M
μ
Safe Failure
M
σ
M
σ
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In the general case the resistance and the load may be defined in terms
where X are basic random variables and the safety margin as where is called the limit state function failure occurs when
) ( ) (
2 1
X X f S f R = = ) ( ) ( ) (
2 1
X X X g f f S R M = − = − =
) ( ≤ x g
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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
i
1 + i
Failure domain Safe domain
) ,.., , (
2 1
>
n
x x x g
s
f
Ω
2 1
n
Failure event
( ) 0
( )
f g
P f d
≤
= ∫
X x
x x
( ) g = x
) ( ≤ = x F g
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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
) (
x X
g f
f
) (x
X
f
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When the limit state function is linear the saftey margin M is defined through with mean value and variance
=
⋅ + =
n i i i x
a a g
1
) (x
=
⋅ + =
n i i i X
a a M
1
=
+ =
n i X i M
i
a a
1
μ μ
≠ = = =
+ =
n i j j j i j i ij n i n i X i M
a a a
i
, 1 1 1 2 2 2
σ σ ρ σ σ
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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
) ( ) ) ( ( ≤ = ≤ = M P g P P
F
X
M M
σ μ β =
) ( β − Φ =
F
P
m
) (m fM
Basler and Cornell
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The reliability index β has the geometrical interpretation of being the shortest distance between the failure surface and the origin in standard normal distributed space u in which case the components of U have zero means and variances equal to 1
2 4 6 8 10 12
2 4 6 8 10 12 S R
x2 x1
) ( = x g
2 4 6 8 10 12
2 4 6 8 10 12 S R
x2 x1
2 4 6 8 10 12
2 4 6 8 10 12 S R
x2 x1
) ( = x g ) ( = x g ) ( = x g
u2 u1
) ( = u g
u2 u1
u2 u1
) ( = u g ) ( = u g ) ( = u g
i i
X X i i
X U σ μ − = Design point
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Example: Consider a steel rod with resistance r subjected 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 = =
R R
σ μ 40 , 200 = =
S S
σ μ
S R g − = ) (X ) ( ≤ − S R P
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Example: Consider a steel rod with resistance r subjected 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 = =
R R
σ μ 40 , 200 = =
S S
σ μ
S R g − = ) (X ) ( ≤ − S R P
S R M − =
150 200 350 = − =
M
μ
15 . 53 40 35
2 2
= + =
M
σ
84 . 2 15 . 53 150 = = β
3
10 4 . 2 ) 84 . 2 (
−
⋅ = − Φ =
F
P
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Usually the limit state function is non-linear
the so-called invariance problem Hasofer & Lind suggested to linearize the limit state function in the design point
problem The reliability index may then be determined by the following
Can however easily be linearized !
2 4 6 8 10 12
2 4 6 8 10 12 S R
u2 u1
) ( = ′ u g ) ( = u g
2 4 6 8 10 12
2 4 6 8 10 12 S R
u2 u1
2 4 6 8 10 12
2 4 6 8 10 12 S R
u2 u1
) ( = ′ u g ) ( = u g
{ } ∑
= = ∈
=
n i i g
u
1 2 ) (
u u
β
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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
) (
x X
g f
f
Random number 1
) (
i X
x F
i
j
z
j
x
i
x
f
f f =
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2 4 6 8 10 12 14 16 18 20
2 4 6 8 10 12 Load Resistance
Monte Carlo Simulation
2 4 6 8 10 12 14 16 18 20
2 4 6 8 10 12 Load Resistance 2 4 6 8 10 12 14 16 18 20
2 4 6 8 10 12 Load Resistance 2 4 6 8 10 12 14 16 18 20
2 4 6 8 10 12 Load Resistance
m n p
f f =
are generated and the number of
are recorded and summed
is then
2 4 6 8 10 12 14 16 18 20
2 4 6 8 10 12 Load Resistance
Safe Failure
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following components Design situations Ultimate, serviceability, accidental Design equations Design variables Characteristic values Partial safety factors Design values
C Q c G m c
a
C
G
C
Q
m
G
Q
d c m
x x = γ
c d Q
x x = γ
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C
S
C
R
) ( ), ( s f r f
S R
d c m
c d Q
d d
z x R S ,
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to maximize the life-cycle benefit obtained from the structures by „calibrating“ (adjusting) the partial safety factors
m i t s P C C C B w W
u i i l i L j Fj Fj Rj Ij j j
,..., 1 , . . ) ( ) ( ) ( ) ( max
1
= ≤ ≤ ∑ − − − =
=
γ γ γ γ γ γ γ
γ
( )
j
min ( ) . . , , , z , 1,...,
Ij c j l u i i i
C s t G z z i N
γ
γ ≥ ≤ ≤ = z x p z
) , ( ≥ γ z , p , x
j c j
G
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Relative cost of safety measure Minor consequences
Moderate consequences
Large consequences
High β=3.1 (
F
P ≈10-3) β=3.3 (
F
P ≈5 10-4) β=3.7 (
F
P ≈10-4) Normal β=3.7 (
F
P ≈10-4) β=4.2 (
F
P ≈10-5) β=4.4 (
F
P ≈5 10-5) Low β=4.2 (
F
P ≈10-5) β=4.4 (
F
P ≈10-5) β=4.7 (
F
P ≈10-6)
Relative cost of safety measure Target index (irreversible SLS) High β=1.3 (
F
P ≈10-1) Normal β=1.7 (
F
P ≈5 10-2) Low β=2.3 (
F
P ≈10-2)
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equations
∑ − =
= L j t j j
w W
1 2
) ( ) ( min β γ β γ
γ
∑ − =
= L j t F Fj j
P P w W
1 2
) ( ) ( ' min γ γ
γ
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Feasible decisions Optimal decision
Utility Decision alternative
Acceptable decisions
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Facility Facility boundary
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………. Exposure events Constituent failure events and direct consequences Follow-up consequences
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1 1 1
( ) ( , ) ( , ) ( , ) ( , ) ( , ( ), ) ( , )
EXP EXP STA
j n n n ij k j D ij j k j l k j ID l D j k j k k l
E U a p C EX a c C a p EX a p S EX a c S c a p EX a
= = =
⎡ ⎤ = ⎣ ⎦ +
C
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Direct risks: Indirect risks: Robustness Index:
1
( ) ( ) ( )
EXP
n D ij k D ij k k
R p C EX c C p EX
=
= ∑
1 1
( ) ( , ( )) ( )
STA EXP n
n ID l k ID l D k k l
R p S EX c S c p EX
= =
= ∑∑ C
Exposure Vulnerability Robustness Exposure Vulnerability Robustness
( )
k
p EX ( ) ( )
ij k D ij
p C EX c C
( ) ( , ( ))
l k ID l D
p S EX c S c C
D R D ID
R I R R = +
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:
Flood Ship impact Explosion/Fire Earthquake Vehicle impact Wind loads Traffic loads Deicing salt Water Carbon dioxide Yielding Rupture Cracking Fatigue Wear Spalling Erosion Corrosion Loss of functionality partial collapse full collapse Use/functionality Location Environment Design life Societal importance Design codes Design target reliability Age Materials Quality of workmanship Condition Protective measures Ductility Joint characteristics Redundancy Segmentation Condition control/monitoring Emergency preparedness Direct consequences Repair costs Temporary loss or reduced functionality Small number of injuries/fatalities Minor socio-economic losses Minor damages to environment Indirect consequences Repair costs Temporary loss or reduced functionality Mid to large number of injuries/fatalities Moderate to major socio-economic losses Moderate to major damages to environment
Exposure Vulnerability Robustness Exposure Vulnerability Robustness Exposure Vulnerability Robustness Exposure Vulnerability Robustness
Physical characteristics Scenario representation Indicators Potential consequences
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Exposure Vulnerability Robustness Risk reduction m easures Exposure Vulnerability Robustness Risk reduction m easures Exposure Vulnerability Robustness Risk reduction m easures Exposure Vulnerability Robustness Risk reduction m easures
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Define Context and Criteria Define System Identify Hazard Scenarios
Analysis of Consequences Analysis of Probability Identify Risk Scenarios Analyse Sensitivities Assess Risks Risk Treatment Monitor and Review
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Life Safety – and the Performance of Society
sectors and through very different activities
Efficiency is markedly different from sector to sector and from activity to activity ! It is a societal responsibility to spend public resources efficiently ! If this is not done – life is taken away from some individuals in society
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Life Safety – and the Performance of Society
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Life Safety – and the Performance of Society
Development Index (HDI)
World map indicating Human Development Index (2004).
██ 0.950 and over ██ 0.900-0.949 ██ 0.850-0.899 ██ 0.800-0.849 ██ 0.750-0.799 ██ 0.700-0.749 ██ 0.650-0.699 ██ 0.600-0.649 ██ 0.550-0.599 ██ 0.500-0.549 ██ 0.450-0.499 ██ 0.400-0.449 ██ 0.350-0.399 ██ 0.300-0.349 ██ under 0.300 ██ n/a
1 1 1 3 3 3 HDI GDP Index EI LEI = + +
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Life Safety – and the Performance of Society
between the individuals of the nations (Gini – Index)
1 1 1 3 3 3 HDI GDP Index EI LEI = + +
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Modelling Socio-Economical Acceptable Risks
is time – Nathwani, Pandey and Lind developed the Life Quality Index – a preference model – which at a societal level acts as a revealed preference on how we weight money against life time and time for private activities
( , ) : is the part of the GDP available for investment into life safety : is the life expectancy at birth : is the part of life spent for work 1 1 : is a factor which takes
q
L g g g w w q w β β = = − l l l into account that only a part of the GDP is based on humal labour
Sw iss Federal Institute of Technology
Modelling Socio-Economical Acceptable Risks
should lead to an increase in life-expectancy results in a risk acceptance criterion: which leads to the important Societal Willingness To Pay (SWTP) criterion:
= = − l l g d SWTP dg q
GDP 59451 SFr
l
80.4 years
w
0.112
β
0.722
g
35931 SFr
q
0.175
Sw iss Federal Institute of Technology
Modelling Socio-Economical Acceptable Risks
acceptable structural failure probabilities
x x x
Sw iss Federal Institute of Technology
Modelling Socio-Economical Acceptable Risks
y x PE y PE
structural failure probabilities
Sw iss Federal Institute of Technology
Modelling Socio-Economical Acceptable Risks
( ) m p p
dp
maximum acceptable failure rate acceptable decisions
( )
y x PE
q dC p C N k g
( )
y x PE
C p q C N k g
( ) ( )
y x PE
g dC p C N kdm p q ≥ −
Sw iss Federal Institute of Technology
Modelling Socio-Economical Acceptable Risks
assessed – Societal Value of a Statistical Life (SVSL). For Switzerland this amounts to about 6 million SFr
= g SVSL E q
Sw iss Federal Institute of Technology
Modelling Socio-Economical Acceptable Risks
Acceptable decisions are limited by the SWTP criterion Costs of failure include compensation – through the SVSL
Feasible decisions Optimal decision
Utility Decision alternative
Acceptable decisions
Sw iss Federal Institute of Technology
COST E5 5 W orkshop Graz University of Technology May, 1 4 -1 5 , 2 0 0 7