Calibration of Partial Safety Factors Jochen Köhler NTNU
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.
Design equation: zr γ m zr γ m k * k γ G + γ Q G k Q k ( * * αγ ( ) ) G Gk α γq Qk + Different relations of GG and QQ Generalization with normalized variables and α = G G+ Q γ z G Q ( αγ ( ) ) G k α γq k * m = * + * * Rk Estimation of failure probability based on the limit state function: ( X) ξ α ( α) * * * * g = z R G Q ( ) ( F) g( X) Pr = Pr 3
Example Steel structure Variable Distribution COV p.s.f. yield strength lognormal.7.5 MU_steel lognormal.5 - dead load normal..35 snow load Gumbel..5 γ z G Q ( αγ ( ) ) G k α γq k * m = * + * * Rk ( X) ξ α ( α) * * * * g = z R G Q ( ) ( F) g( X) Pr = Pr
Example Steel structure Variable Distribution COV p.s.f. yield strength lognormal.7.5 MU_steel lognormal.5 - dead load normal..35 snow load Gumbel..5 5.5 Steel - NOT Calibrated 5.5 3.5 3.5...3...7.8.9 5
5.5 5 Calibration of Partial Safety Factors Example Steel structure Variable Distribution COV p.s.f. yield strength lognormal.7.5 MU_steel lognormal.5 - dead load normal..35 snow load Gumbel..5.5 3.5 3.5 Steel - NOT Calibrated - target - ij target...3...7.8.9 i ( ) i target Min w β β
Example Steel structure Variable Distribution COV p.s.f. p.s.f. (cal., w=) yield strength lognormal.7.5.35 MU_steel lognormal.5 - - dead load normal..35. snow load Gumbel..5.7 5.5 5.5 3.5 3.5 Steel - NOT Calibrated Steel - Calibrated - target...3...7.8.9 - ij target 7
Example Steel structure Variable Distribution COV p.s.f. p.s.f. (cal., w=) p.s.f. (cal., w=sp) yield strength lognormal.7.5.35. MU_steel lognormal.5 - - - dead load normal..35..97 snow load Gumbel..5.7. 5.5 5.5 3.5 3.5 Steel - NOT Calibrated Steel - Calibrated - target...3...7.8.9 - ij target 5.5 5.5 3.5 3.5 Steel - NOT Calibrated Steel - Calibrated - target...3...7.8.9 - ij target 8
Example Steel, Concrete, Timber structure Variable Distribution COV p.s.f. yield strength lognormal.7.5 MU_steel lognormal.5 - dead load normal..35 snow load Gumbel..5 concrete cap. lognormal.5 MU_concrete lognormal. - timber cap. lognormal.3.3 MU_timber lognormal. - 9
Example Steel, Concrete, Timber structure 5.5 5.5 3.5 3 Steel - NOT Calibrated Concrete - NOT Calibrated.5 Timber - NOT Calibrated...3...7.8.9
Example Steel, Concrete, Timber structure 5.5 5.5 3.5 Steel - Calibrated 3 Concrete - Calibrated Timber - Calibrated.5...3...7.8.9 target
Example Steel, Concrete, Timber structure Variable Distribution COV p.s.f. p.s.f. calibr. yield strength lognormal.7.5.7 MU_steel lognormal.5 - - dead load normal..35.33 snow load Gumbel..5.9 concrete cap. lognormal.5.33 MU_concrete lognormal. - - timber cap. lognormal.3.3.3 MU_timber lognormal. - -
Two variable loads (Eq..) 3
Example Steel, Concrete, Timber structure, two var. loads Variable Distribution COV p.s.f. yield strength lognormal.7.5 MU_steel lognormal.5 - dead load normal..35 snow load Gumbel..5 wind load Gumbel..5 concrete cap. lognormal.5 MU_concrete lognormal. - timber cap. lognormal.3.3 MU_timber lognormal. -
Example Steel, Concrete, Timber structure, two var. loads Steel - NOT Calibrated Concrete - NOT Calibrated Timber - NOT Calibrated G Q G Q G Q 5
Example Steel, Concrete, Timber structure, two var. loads Steel - Calibrated Concrete - Calibrated Timber - Calibrated G Q G Q G Q
Example Steel, Concrete, Timber structure, two var. loads Variable Distribution COV p.s.f. p.s.f. calibr. yield strength lognormal.7.5. MU_steel lognormal.5 - - dead load normal..35.8 snow load Gumbel..5.9 wind load Gumbel..5.9 concrete cap. lognormal.5.37 MU_concrete lognormal. - - timber cap. lognormal.3.3.7 MU_timber lognormal. - - 7
Two variable loads (Eq..a,b) { } z = max z, z, z 3 5 γ z ( α Q ) G γg Gk αg αq γq ψ k αq γq ψ Q k (Eq..a) ( ) ( ) m 3 = + + R k γ z G Q Q (Eq..b with Q leading) z ( α ) G ξ γg k αg αq γq k αq γq ψ k ( ) ( ) m = + + R k 5 γ m = αg ξ γg Gk + ( αg) αq γ + ( ) R k ( ) Q ψ k αq γq Q k Q (Eq..b with Q leading) 8
Two variable loads 5.5 5.5 Steel - Comparison Eq..a&b with &. Eq..a Eq..b Eq.. 3.5 3.5...3...7.8.9 G 9
Example Steel, Concrete, Timber structure Variable Distribution COV p.s.f. p.s.f. calibr. (Eq..a&b) yield strength lognormal.7.5.8 MU_steel lognormal.5 - - dead load normal..35.33 snow load Gumbel..5.7 Wind load Gumbel..5.7 concrete cap. lognormal.5.37 MU_concrete lognormal. - - timber cap. lognormal.3.3.7 MU_timber lognormal. - -
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.