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11.12.2015
Calibration of Partial Safety Factors
1
Calibration of Partial Safety
Factors
Jochen KöhlerNTNU
11.12.2015
Calibration of Partial Safety Factors
2
Outline and scope The generic design equation.
Calibration of design equations with one variable load.
Calibration of design equations with two variable loads.
Notional reliability and hidden safety.
11.12.2015
Calibration of Partial Safety Factors
3
Representation of a generic design situation
( )( )* * ** 1 m
G k Q kk
z G QRγ α γ α γ= + −( )( )
** * 1 k
G k Q km
z R G Qα γ α γγ
≥ + −
kG k Q k
m
z R G Qγ γγ
≥ +
( ) ( )* * * * 1 0g z R G Qξ α α= − − − ≤X
Design equation:
Different relations of 𝐺𝐺 and 𝑄𝑄
Generalization with normalized variables and G
G Qα =
+
Estimation of failure probability based on the limit state function:
( ) ( )( )Pr Pr 0F g= ≤X
11.12.2015
Calibration of Partial Safety Factors
4
Representation of a generic design situation Example Steel structure
Variable Distribution COV p.s.f.
yield strength lognormal 0.07 1.05
MU_steel lognormal 0.05 -
dead load normal 0.1 1.35
snow load Gumbel 0.6 1.5
( )( )* * ** 1 m
G k Q kk
z G QRγ α γ α γ= + −
( ) ( )* * * * 1 0g z R G Qξ α α= − − − ≤X
( ) ( )( )Pr Pr 0F g= ≤X
11.12.2015
Calibration of Partial Safety Factors
5
Representation of a generic design situation Example Steel structure
Variable Distribution COV p.s.f.
yield strength lognormal 0.07 1.05
MU_steel lognormal 0.05 -
dead load normal 0.1 1.35
snow load Gumbel 0.6 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - NOT Calibrated
11.12.2015
Calibration of Partial Safety Factors
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - NOT Calibrated - target
ij -
target
Representation of a generic design situation Example Steel structure
Variable Distribution COV p.s.f.
yield strength lognormal 0.07 1.05
MU_steel lognormal 0.05 -
dead load normal 0.1 1.35
snow load Gumbel 0.6 1.5
( )2target i iMin w β β−∑
11.12.2015
Calibration of Partial Safety Factors
7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - NOT Calibrated
Steel - Calibrated - target
ij -
target
Representation of a generic design situation Example Steel structure
Variable Distribution COV p.s.f. p.s.f. (cal., w=1)
yield strength lognormal 0.07 1.05 1.35
MU_steel lognormal 0.05 - -
dead load normal 0.1 1.35 1.11
snow load Gumbel 0.6 1.5 1.76
11.12.2015
Calibration of Partial Safety Factors
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - NOT Calibrated
Steel - Calibrated - target
ij -
target
Representation of a generic design situation Example Steel structure
Variable Distribution COV p.s.f. p.s.f. (cal., w=1) p.s.f. (cal., w=sp)
yield strength lognormal 0.07 1.05 1.35 1.06
MU_steel lognormal 0.05 - - -
dead load normal 0.1 1.35 1.11 0.97
snow load Gumbel 0.6 1.5 1.76 2.44
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - NOT Calibrated
Steel - Calibrated
- target
ij -
target
11.12.2015
Calibration of Partial Safety Factors
9
Representation of a generic design situation Example Steel, Concrete, Timber structure
Variable Distribution COV p.s.f.
yield strength lognormal 0.07 1.05
MU_steel lognormal 0.05 -
dead load normal 0.1 1.35
snow load Gumbel 0.6 1.5
concrete cap. lognormal 0.15 1.5
MU_concrete lognormal 0.1 -
timber cap. lognormal 0.3 1.3
MU_timber lognormal 0.1 -
11.12.2015
Calibration of Partial Safety Factors
10
Representation of a generic design situation Example Steel, Concrete, Timber structure
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - NOT Calibrated
Concrete - NOT Calibrated
Timber - NOT Calibrated
11.12.2015
Calibration of Partial Safety Factors
11
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6
Steel - Calibrated
Concrete - Calibrated
Timber - Calibrated
target
Representation of a generic design situation Example Steel, Concrete, Timber structure
11.12.2015
Calibration of Partial Safety Factors
12
Representation of a generic design situation Example Steel, Concrete, Timber structure
Variable Distribution COV p.s.f. p.s.f. calibr.
yield strength lognormal 0.07 1.05 1.17
MU_steel lognormal 0.05 - -
dead load normal 0.1 1.35 1.33
snow load Gumbel 0.6 1.5 1.91
concrete cap. lognormal 0.15 1.5 1.33
MU_concrete lognormal 0.1 - -
timber cap. lognormal 0.3 1.3 1.63
MU_timber lognormal 0.1 - -
11.12.2015
Calibration of Partial Safety Factors
13
Representation of a generic design situation Two variable loads
(Eq. 6.10)
11.12.2015
Calibration of Partial Safety Factors
14
Representation of a generic design situation Example Steel, Concrete, Timber structure, two var. loads
Variable Distribution COV p.s.f.
yield strength lognormal 0.07 1.05
MU_steel lognormal 0.05 -
dead load normal 0.1 1.35
snow load Gumbel 0.6 1.5
wind load Gumbel 0.6 1.5
concrete cap. lognormal 0.15 1.5
MU_concrete lognormal 0.1 -
timber cap. lognormal 0.3 1.3
MU_timber lognormal 0.1 -
11.12.2015
Calibration of Partial Safety Factors
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Representation of a generic design situation Example Steel, Concrete, Timber structure, two var. loads
1
Steel - NOT Calibrated
Q
0.500
0.5
G
2
4
6
1
Concrete - NOT Calibrated
1
Q
0.500
0.5
G
2
4
6
11
Timber - NOT Calibrated
Q
0.500
0.5
G
2
4
6
1
11.12.2015
Calibration of Partial Safety Factors
16
1
Q
Steel - Calibrated
0.500
0.5
G
2
4
6
11
Concrete - Calibrated
Q
0.500
0.5
G
6
2
4
11
Timber - Calibrated
Q
0.500
0.5
G
2
4
6
1
Representation of a generic design situation Example Steel, Concrete, Timber structure, two var. loads
11.12.2015
Calibration of Partial Safety Factors
17
Representation of a generic design situation Example Steel, Concrete, Timber structure, two var. loads
Variable Distribution COV p.s.f. p.s.f. calibr.
yield strength lognormal 0.07 1.05 1.20
MU_steel lognormal 0.05 - -
dead load normal 0.1 1.35 1.28
snow load Gumbel 0.6 1.5 1.49
wind load Gumbel 0.6 1.5 1.49
concrete cap. lognormal 0.15 1.5 1.37
MU_concrete lognormal 0.1 - -
timber cap. lognormal 0.3 1.3 1.67
MU_timber lognormal 0.1 - -
11.12.2015
Calibration of Partial Safety Factors
18
Representation of a generic design situation Two variable loads
(Eq. 6.10a,b)
{ }
( ) ( )( )( ) ( )( )( )
3 4 5
3 01 1 02 2
4 1 02 2 1
5
max , ,
1 Q 1 (Eq. 6.10a)
1 1 Q (Eq. 6.10b with Q leading)
1
mG G k G Q Q k Q Q k
k
mG G k G Q Q k Q Q k
k
mG G k G Q
k
z z z z
z G QR
z G QR
z GR
γ α γ α α γ ψ α γ ψ
γ α ξ γ α α γ α γ ψ
γ α ξ γ α α γ
=
= + − + −
= + − + −
= + − ( )( )01 1 2 2 Q 1 (Eq. 6.10b with Q leading)Q k Q Q kQψ α γ + −
11.12.2015
Calibration of Partial Safety Factors
19
Representation of a generic design situation Two variable loads
G
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.5
3
3.5
4
4.5
5
5.5
6Steel - Comparison Eq. 6.10a&b with &6.10
Eq. 6.10a
Eq. 6.10b
Eq. 6.10
11.12.2015
Calibration of Partial Safety Factors
20
Representation of a generic design situation Example Steel, Concrete, Timber structure
Variable Distribution COV p.s.f. p.s.f. calibr.(Eq. 6.10a&b)
yield strength lognormal 0.07 1.05 1.18
MU_steel lognormal 0.05 - -
dead load normal 0.1 1.35 1.33
snow load Gumbel 0.6 1.5 1.67
Wind load Gumbel 0.6 1.5 1.67
concrete cap. lognormal 0.15 1.5 1.37
MU_concrete lognormal 0.1 - -
timber cap. lognormal 0.3 1.3 1.67
MU_timber lognormal 0.1 - -
11.12.2015
Calibration of Partial Safety Factors
21
Summary and Discussion Partial safety factors can be calibrated in order to reach
target reliability “as best as possible” for generalized design equations.
The results are conditional on the assumptions. Caution is necessary, since engineering models contain a
multitude of conservative assumptions.
All considerations are only valid for linear design equations.
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