Reliability Aspects

Milan HolickýCTU in Prague

Leonardo da VinciAssessment of existing structuresProject 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 engineeringSome 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

Eurocode EN 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

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

Guσσμμ

σμβ

+−

==−=

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 caseZ = 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 from the origin

Relaibility margin and index β

Prubab. Failure

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

Probabilistic models

40 60 80 100 120 140 0 .00

0 .02

0 .04

0 .06

P rob ab ility d ensity ϕ E (x ), ϕ R (x )

R an d o m v ariab le X

L oad effec t E , G um b el d is trib u tion , μ E = 50 , σ E = 5 R esis tan ce R

lo g-n o rm al d is trib u tion , μ R = 1 0 0 , σ R = 1 0

12

Eurocode concepts of partial factors

Reliability index β

Action reliability indexβE = αE β

Resistance reliability indexβR = αR β

Main actionβE = - 0,7 β

AccompanyingβE = - 0,28 β

ResistanceβR = 0,8 β

Partial factor

E

R

distribution mean sd R resistance Lognormal 100 10 E load effect Gumbel 50 5

Partial factor approach

40 60 80 100 120 140 0 .00

0 .02

0 .04

0 .06

P rob ab ility d ensity ϕ E (x ), ϕ R (x )

R an d o m v ariab le X

L oad effec t E , G um b el d is trib u tion , μ E = 50 , σ E = 5 R esis tan ce R

lo g-n o rm al d is trib u tion , μ R = 1 0 0 , σ R = 1 0

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+Δt) > 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]

POD

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 = 350Then mR has sm = 17,5/√3 = 10.If the load is to  304 then:

mZ= 350‐304=46sZ =√(17,52 +102) = 20,2β=2,27Pf=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=2,35Pf=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

relia

bilit

y in

dex

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 = π d2 fy / 4Load effect: E = Vρ

Failure if E>R or: Vρ > π d2 fy / 4Limit state: Vρ = π d2 fy / 4

Limit state function: Z = R-E = π d2 fy / 4 - Vρ

Reliability index

Probability of Failure = Φ(‐β) ≈ 10 ‐ β

β 1.3 2.3 3.1 3.7 4.2 4.7

P(F)=Φ(-β) 10-1 10-2 10-3 10-4 10-5 10-6

Relation Partial factors and beta‐level:

α = 0.7‐0.8β = 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 βfor a one year reference period 

Consequences of failure ⇒

Minor Moderate Large

Large β=3.1 (pF≈10‐3) β=3.3 (pF≈ 5 10‐4) β=3.7 (pF≈ 10‐4)

Normal β=3.7 (pF≈10‐4) β=4.2 (pF ≈ 10‐5) β=4.4 (pF ≈ 5 10‐6)

Small β=4.2 (pF≈10‐5) β=4.4 (pF≈ 5 10‐5) β=4.7 (pF≈ 10‐6)

Cost toincrease 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]

times

0,0

01

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 andcivil 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