Reliability Considerations for Steel Frames Designed with Advanced Analysis

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1 Reliability Considerations for Steel Frames Designed with Advanced Analysis Stephen G. Buonopane Benjamin W. Schafer Takeru Igusa Dept. of Civil Engineering Johns Hopkins University

2 Features of Advanced Analysis Non-Linear Structural Analysis e.g. Inelastic materials, P- effects System Behavior e.g. Frame stability Advantages over Existing Code Individual Member Checking Not Required Adjustment Factors Not Required e.g. Effective length, Second-order effects, Interaction equations

3 Reliability & Advanced Analysis Compare Reliability of Steel Frames Designed by LRFD vs. AA Compare Member (LRFD) vs. System (AA) Limit States Calculate Resistance Factors for AA

4 Frames for Study Steel Frames from Ziemian et al. (1992) Designed by both LRFD (1986) and AA AA Design saves ~12% by weight

5 Member Size Comparison W12x19 W12x14 W14x132 W14x99 W10x12 W14x109 W14x109 W14x82 W14x109 W18x40 W27x94 W27x84 W36x135 Design by: LRFD AA

6 Frames for Study 1. Member Sizes: LRFD or AA 2. Yield Strength: Uncorrelated and Correlated Base Fixity Gravity Load 5. Geometry Total = 32 Frames Analyzed

7 Frame Analysis Details OpenSees ( Geometric Non-Linear Fiber-Element Cross-Section Elastic-Plastic Material Out-of-Plumb Column Imperfection of H/400 Out-of-Plane Behavior Restrained No Residual Stresses Random Yield Strength and Gravity Loads

8 Random Properties Yield Strength 50 ksi Nominal Normal Distribution with COV=0.10 Members Uncorrelated and Correlated Gravity Loads Dead Load ~ Normal Distribution Live Load ~ Extreme Type I Distribution Both COV=0.10 Consistent with LRFD Assumptions

9 Frame Simulation Details Each Frame of 32 10,000 Samples with Random F y Each Sample Load Increased Until Failure Strength Limit States considered 1st Plastic Hinge Plastic Collapse

10 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors

11 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors

12 Load-Deflection Behavior Normalized Total Gravity Load 1st Plastic Hinge Plastic Collapse Uncorrelated F y AA Design Lateral Deflection of Top Story (in.)

13 Load-Deflection Behavior Normalized Total Gravity Load 1st Plastic Hinge Plastic Collapse Correlated F y AA Design Lateral Deflection of Top Story (in.)

14 Member vs. System Limit State 1st PH Strength Plastic Collapse Strength Uncorrelated F y AA Design

15 Member vs. System Limit State 1st PH Strength Plastic Collapse Strength Plastic Collapse Strength Uncorrelated F y AA Design Uncorrelated F y LRFD Design

16 Strength Distributions Relative Frequency Load Strength Plastic Collapse 1st Plastic Hinge Uncorrelated F y LRFD Design 10,000 samples Normalized Strength and Load

17 Mean Strength at Plastic Collapse Mean Strength Design by LRFD AA

18 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors

19 Reliability and LRFD Q = Load R = Strength ln( R m / Q m ) ln( R / Q) β ( FO V 2 2 R + V ) Q First-Order Reliability β FO = ln ( R m / Q m ) V 2 2 R + V Q

20 Reliability by Sampling Probability of Failure P f = Ir, ( q)f R ()f r Q ()dr q dq I( r,q) = 1 for R Q = 0 for R>Q Load f Q (q) Strength f R (r) Monte Carlo Estimate ˆ P f = 1 N N i=1 Ir ( i,q i ) r i sampled from f R q i sampled from f Q Reliability β = Φ 1 ˆ P f ( )

21 Reliability at Plastic Collapse Reliability, β Design by LRFD AA P f =

22 Reliability at 1st Plastic Hinge Reliability, β Design by LRFD AA P f =

23 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors

24 Resistance Factors φ = ( R m / R n )exp( 0.55β t V R ) Mean of True Strength Nominal Strength Target Reliability Variation of True Strength V R φr n R n R m True Strength

25 Resistance Factors of LRFD φ = ( R m / R n )exp( 0.55β t V R ) R m / R n = P m M m F m = 1.07 Means V R = V P 2 + V M 2 + V F 2 = 0.15 COVs LRFD Bias Factors: P = Professional M = Material F = Fabrication

26 Resistance Factors for AA φ = ( R m / R n )exp( 0.55β t V R ) R m AA / R n AA = B m AA 2 V B AA = V R AA + V F 2 AA Strength Distribution AA Nominal Strength AA Strength Distribution

27 Mean Bias Factors Mean Bias Factors Correlated Uncorrelated 0.98

28 Resistance Factors for AA at PC Resistance Factors, φ Correlated Uncorrelated β t = 3.00

29 Resistance Factors for AA at 1st PH Resistance Factors, φ Correlated Uncorrelated β t =

30 Summary of Resistance Factors Limit State Plastic Collapse 1st Plastic Hinge F y Uncorr. Corr. Uncorr. Corr. Min Mean Max Target Reliability, β t = 3.00

31 Conclusions Probabilistic Basis for AA Resistance Factors No Simple Transformation from Member to System Design Approach Increased Probability of 1st PH with Design by AA

32 Future Work Serviceability Concerns due to Increased Probability of 1st PH with AA Is a Single φ Appropriate for ALL Steel Frames? How to Apply φ in AA? To System Strength To Member Properties

34 Reliability by Sampling Load f Q (q) Strength f R (r) Probability of Failure P f = Ir,q ( )f R ()f r Q ()dr q dq I( r,q) = 1 for R Q = 0 for R>Q Monte Carlo Estimate ˆ P f = 1 N β = Φ 1 N Ir ( i,q i ) i=1 ˆ P f ( ) r i sampled from f R q i sampled from f Q

35 Reliability by Sampling Safe Failure Uncorrelated F y AA Design Strength Load

36 Reliability by Sampling Uncorrelated F y AA Design P f ( 10-6 ) Number of Samples

37 Uncorr. Corr. AA LRFD 1. Member Sizes: LRFD or AA 2. Yield Strength: Uncorrelated and Correlated Base Fixity Gravity Load 5. Geometry

38 Design: Uncorr. Corr. AA LRFD Uncorr. Corr. AA Design by LRFD Design by LRFD AA Design by LRFD AA First-Order Monte Carlo Correlated Uncorrelated

39 Simulation Results Field of F-D curves Histograms of Strength, 1st PH Plots of mean, COV Estimates of Pf, beta Plots of Pf, beta Correlation of Strength, 1st PH

40 Reliability and Advanced Analysis What Reliability is Associated with Design by AA? Implications for Resistance Factor, phi

41 Member Size Comparison W18x40 W18x40 W27x94 W27x94 W14x109 W14x82 W12x19 W12x14 W14x132 W14x99 W14x109 W14x109 W10x12 W10x12 W14x109 W14x109 W27x84 W27x84 W36x135 W36x135 Member Sizes by: LRFD Advanced Analysis

42 Approach Advanced Analysis with Nominal Properties Non-Linear Analyses with Random Properties Strength Distributions Load Distributions Resistance Factors Reliability Estimates

43 System vs. Member Limit State 1st PH Strength Plastic Collapse Strength Plastic Collapse Strength Uncorrelated F y AA Design Uncorrelated F y LRFD Design

44 COV of Strength at Plastic Collapse COV of Strength Design: Uncorr. Corr. AA LRFD

45 Reliability at Plastic Collapse Reliability, β First-Order Monte Carlo P f =

Reliability implications of advanced analysis in design of steel frames

Reliability implications of advanced analysis in design of steel frames Stephen G. Buonopane Benjamin W. Schafer Takeru Igusa Dept. of Civil Engineering Johns Hopkins University Advanced Analysis Non-Linear