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Reliability Aspects
Milan HolickýCTU in Prague
Leonardo da Vinci
Assessment of existing structures
Project number: CZ/08/LLP-LdV/TOI/134005
2
Uncertainties are always present•Uncertainties (aleatoric and epistemic)Decription
- randomness - natural variability
- statistical uncertainties - lack of data
- model uncertainties - simplified models
- vagueness - imprecision in definitions
- gross errors - human factors
- ignorance - lack of knowledge
•Tools- theory of probability and statistics, fuzzy logic
- reliability theory and risk engineering
Some uncertainties are difficult to quantify
Density Plot (Shifted Lognormal) - [A1_792]
210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 4200.000
0.005
0.010
0.015
0.020
Relative frequency
Yield strength [MPa]
Reliability - ability of a structure to fulfil all required
functions during a specified period of time under given conditions
Failure probability Pf
- most important measure of structural reliability
EurocodeEN 1990:
Limit State Approach
• Limit states - states beyond which the structure no longer fulfils the relevant design criteria
• Ultimate limit states– loss of equilibrium of a structure as a rigid body– rupture, collapse, failure – fatigue failure
• Serviceability limit states– functional ability of a structure or its part– users comfort– appearance
Reliability Methods
pf 10−1 10−2 10−3 10−−−−4 10−5 10−6 10−7
β 1,3 2,3 3,1 3,7 4,2 4,7 5,2
Reliability measures: failure probability pf and reliability index β
Historical methodsEmpirical methods
FORMLevel II
ProbabilisticLevel III
Level I
Partial Factor Method
Calibration
Calibration
Calibration
Probabilistic methodsDeterministic methods
Historical methodsEmpirical methods
FORMLevel II
ProbabilisticLevel III
Design point methodLevel I
Partial Factor MethodLevel I
Calibration
Calibration
Calibration
Probabilistic methodsDeterministic methods
Fundamental case for normal distribution E ≤ R
Prubab. Failure ìpod.
0
Reliability
βσG µG
( )ϕG x
µ µ µ σ σ σG R E G R E= − = +, 2 2 2
Reliability index : 2/1220)( ER
ER
G
Guσσ
µµσµβ
+−==−=
G = R - ETransformation of G to standardized variable U=(G – µG)/σG
For G = 0 the standardized variable u0 =(0 – µG)/σG
pf= Φ(−β)
Failure probability
Fundamental case E < R
probab.
e0
Reliability
βσG µG
Failure
( )G xφ
Reliability index for normal dist.: RR eu σµβ /)( 00 −=−=
Load E=e0 known, resistance R random: µR, σR, (αR)Limit state function:g(X) = G = R – E = 0
Transformation of R to standardized variable U=(R –µR)/σR
For R = e0 the standardized variable u0 =(e0 – µR)/σR
pf= Φ(−β)
pf = ΦR(e0)For normal distribution
Failure probability
E
R
An example of the fundamental case
Z = R - E
µZ = µR - µE = 100 – 50 = 50
β = µZ /σZ = 3.54
Pf = P(Z < 0) = ΦZ(0) = 0.0002
22E
2R
2Z 14σσσ =+=
β = µZ /σZ = 3.54 Pf = P(Z < 0) = ΦZ(0) = 0.0002,
β Is the distance of the mean of reliability margin fromthe origin
Relaibility marginand index β
Prubab. Failure ìpod.
0
Reliability
βσG µG
( )ϕG x
Z = R - E
∫∫<
ϕϕ=<=0)X(Z
ERf derd)e()r()0Z(PP
E
R
Probabilistic approach
Techniques:
Numerical integration (NI)
Monte Carlo (MC)
First order Second moment method (FOSM)
Third moment method (accounting forskewness)
First Order Reliability Methods (FORM)
E
R
distribution mean sd R resistance Lognormal 100 10 E load effect Gumbel 50 5
Probabilisticmodels
40 60 8 0 100 120 1 40 0 .00
0 .02
0 .04
0 .06
Probab ility density ϕ E(x), ϕ R(x)
R andom variab le X
Load effect E , G um bel d istribu tion , µ E = 50 , σ E = 5 R esistance R
log-no rm al d istr ibu tion , µ R = 100 , σ R = 10
12
Eurocodeconcepts of partial factors
Reliability index β
Action reliability indexβE = αE β
Resistance reliability indexβR = αR β
Main actionβE = - 0,7β
AccompanyingβE = - 0,28β
ResistanceβR = 0,8β
Partial factor
E
R
distribution mean sd R resistance Lognormal 100 10 E load effect Gumbel 50 5
Partial factor approach
40 60 8 0 100 120 1 40 0 .00
0 .02
0 .04
0 .06
Probab ility density ϕ E(x), ϕ R(x)
R andom variab le X
Load effect E , G um bel d istribu tion , µ E = 50 , σ E = 5 R esistance R
log-no rm al d istr ibu tion , µ R = 100 , σ R = 10
E k R k
Rk/γm > Ek γQ
Indicative target reliabilities in ISO 13822
Updating
distributions
fX(x), fX(x|I)
X
prior distribution fX(x)
updated distribution fX(x|I)
updated xdprior xd
P(x|I) = P(x) P(I | x) / P(I)
fX (x|I) = C fX(x) P( I | x)
updated prior likelihood
q)xL( q)fC)x|(qf QQ |ˆ|(ˆ ''' =
dq)x|(qfqxfxf QXUX ∫
∞
∞−
= ˆ)()( ''
Formal Updating formulas
Ask the expert !
Concluding remarks on reliability aspects
� Uncertainties are always present
� Probability Theory may be helpful
� Reliability targets depends on consequences of failure
� Reliability targets depend on costs of improving
� Existing structures may have a lower target reliability
� Reliability may be updated using inspection results
� There is a relation partial factor – reliability index
Fatigue steel structures
no cracks found, but? measured 1 mm, but?
Find
P(a(t+∆∆∆∆t) > d | a(t) = .. of a(t)<..)
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25Scheurafmeting [mm]
PO
D
alpha = 1 ; beta = 3 alpha = 3 ; beta = 10
d crack a
Example: Resistance with unknown mean mR and known stand. Dev. sR =17,5
Assume we have 3 observations with mean mm = 350
Then mR has sm = 17,5/√3 = 10.
If the load is to 304 then:
mZ= 350-304=46
sZ =√(17,52 +102) = 20,2
β=2,27
Pf=0,0116
Now we have one extra observation equal to 350.
In that case the estimate of the mean mm does not change.
The standard deviation of the mean changes to 17,5/√(4) = 8,8
mZ= 350-304=46,
sZ =√(17,52 +8,82) = 19,6,
β=2,35
Pf=0,0095
Reliabilty level Beta (one year periods)
given a crack found at t=10 a
0
2
4
6
8
0 5 10 15 20 25
reli
abil
ity
ind
ex
time [year]
without inspection
after inspection
fully correlated
Existing Structures (NEN 8700)
Reliability index in case of assessment
Minimum β < βnew – 1.0
Human safety: β > 3.6 – 0.8 log T
limit
Condition
⇑⇑⇑⇑
Time ⇒⇒⇒⇒
Inspection en monitoring
E
R
EXAMPLE
Resistance: R = ππππ d2 fy / 4Load effect: E = Vρρρρ
Failure if E>R or: V ρρρρ > ππππ d2 fy / 4Limit state: Vρρρρ = ππππ d2 fy / 4
Limit state function: Z = R-E = ππππ d2 fy / 4 - Vρρρρ
Reliability index
Probability of Failure = Φ(-β) ≈≈≈≈ 10 - ββββ
ββββ 1.3 2.3 3.1 3.7 4.2 4.7
P(F)=ΦΦΦΦ(-ββββ) 10-1 10-2 10-3 10-4 10-5 10-6
Relation Partial factors and beta-level:
α = 0.7-0.8
β = 3.3 - 3.8 - 4,3 (life time, Annex B)
k = 1.64 (resistance)
k = 0.0 (loads)
V = coefficient of variation
γ = exp{α β V – kV} ≈≈≈≈ 1 + α β V
Extensions
• load fluctuations
• systems
• degradation
• inspection
• risk analysis
• target reliabilities
JCSS TARGET RELIABILITIES ββββfor a one year reference period
Consequences of failure ⇒⇒⇒⇒
Minor Moderate Large
Large ββββ=3.1 (pF≈≈≈≈10-3) ββββ=3.3 (pF ≈≈≈≈ 5 10-4) ββββ=3.7 (pF ≈≈≈≈ 10-4)
Normal ββββ=3.7 (pF≈≈≈≈10-4) ββββ=4.2 (pF ≈≈≈≈ 10-5) ββββ=4.4 (pF ≈≈≈≈ 5 10-6)
Small ββββ=4.2 (pF≈≈≈≈10-5) ββββ=4.4 (pF ≈≈≈≈ 5 10-5) ββββ=4.7 (pF ≈≈≈≈ 10-6)
Cost to
increase
safety ⇓⇓⇓⇓
Human life safety
• Include value for human life in D
• Still reasons for IR and SR
• Example: p < 10-4 / year
optimal annual failure probability
0
0,05
0,1
0,15
0,2
0,25
0 10 20 30 40 50 60
design working time [year]
tim
es 0
,001
Consequence class
Minimum reference period
for existing building
β-NEW β-EXISTING
wn wd wn wd
0 1 year 3.3 2,3 1.8 0.8
1 15 years 3.3 2,3 1.8a 1.1a
2 15 years 3.8 2.8 2.5a 2.5a
3 15 years 4.3 3.3 3.3a 3.3a
Class 0: As class 1, but no human safety involved wn = wind not dominant wd = wind dominant (a) = in this case is the minimum limit for personal safety normative
Example NEN 8700 (Netherlands)
Minimum values for the reliability index β with a minimum reference period
Updating
1) Updating distributions (eg concrete strength)
x x
2) Updating failure probability P{{{{F | I }}}}Example: I = {crack = 0.6 mm}
see JCSS document on Existing Structures en ISO13822
Observations
Information ( I )Rk at design
)(
)()(
IP
IF PIFP
∩=
)P(B)BP(AB)P(A =∩ Two types of information I:
equality type: h(x) = 0
inequality type: h(x) < 0; h(x) > 0
x = vector of basic variables
)I)P(IP(F I)P(F =∩
)0)((
)0)(0)(()(
1
12
>>∩<=
thP
t h tZPIFP
Reliability index β Reliability
classes Consequences for loss of human life, economical, social and environmental
consequences
βa for Ta= 1 yr
βd for Td= 50 yr
Examples of buildings and civil engineering works
RC3 – high High 5,2 4,3 Important bridges, public buildings
RC2 – normal Medium 4,7 3,8 Residential and office buildings
RC1 – low Low 4,2 3,3 Agricultural buildings, greenhouses
Target Reliabilities in EN 1990, Annex B
q)xL( q)fC)x|(qf QQ |ˆ|(ˆ ''' =
dq)x|(qfqxfxf QXUX ∫
∞
∞−
= ˆ)()( ''
Formal Updating formulas
Cost optimisation / design versus assessment
PF = 10 -β