Reliability implications of advanced analysis in design of steel frames
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1 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
2 Advanced Analysis Non-Linear Structural Analysis e.g. Inelastic materials, P- effects, fiber elements System Behavior e.g. Frame stability, moment redistribution No member checking, etc., in design Analysis code: OpenSees (opensees.berkeley.edu )
3 Reliability & Advanced Analysis (AA) Compare Reliability of Steel Frames Designed by the American AISC-LRFD vs. AA Compare Limit States Member (AISC-LRFD) vs. System (AA) Calculate Resistance Factors for AA Loads (D, L) and F y random in our analysis
4 Frames for Study Steel Frames from Ziemian et al. (1992) Designed by both AISC-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 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors
8 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors
9 Load-Deflection Behavior Normalized Total Gravity Load 1st Plastic Hinge Plastic Collapse Uncorrelated F y AA Design Lateral Deflection of Top Story (in.)
10 Load-Deflection Behavior Normalized Total Gravity Load 1st Plastic Hinge Plastic Collapse Correlated F y AA Design Lateral Deflection of Top Story (in.)
11 Mean Strength at Plastic Collapse Mean Strength Design by LRFD AA
12 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors
13 Strength Distributions Relative Frequency Load Strength Plastic Collapse 1st Plastic Hinge Uncorrelated F y LRFD Design 10,000 samples Normalized Strength and Load
14 Reliability at Plastic Collapse Reliability, β Design by LRFD AA P f =
15 Reliability at 1st Plastic Hinge Reliability, β Design by LRFD AA P f =
16 Member vs. System Limit State 1st PH Strength Plastic Collapse Strength Uncorrelated F y AA Design
17 Member vs. System Limit State 1st PH Strength Plastic Collapse Strength Plastic Collapse Strength Uncorrelated F y AA Design Uncorrelated F y LRFD Design
18 Overview Non-Linear Analyses with Random Yield Strength Advanced Analysis with Nominal Properties Load Distributions Strength Distributions Reliability Estimates Resistance Factors
19 Resistance Factors for AA at PC Resistance Factors, φ Correlated Uncorrelated β t = 3.00
20 Resistance Factors for AA at 1st PH Resistance Factors, φ Correlated Uncorrelated β t =
21 Conclusions Probabilistic Basis for Advanced Analysis Resistance Factors No Simple Transformation from Member Design [Reliability] to System Design [Reliability] Increased Probability of 1st PH with Design by AA. Serviceability controls?
22
23 Frame Analysis Details Analysis code: OpenSees (opensees.berkeley.edu) Random Properties Yield Strength 50 ksi (345 MPa) 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
24 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
25 Random Properties Yield Strength 50 ksi (345 MPa) 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
26 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
27 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
28 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
29 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 ( )
30 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
31 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
32 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
33 Mean Bias Factors Mean Bias Factors Correlated Uncorrelated 0.98
34 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
35
36 Reliability by Sampling Safe Failure Uncorrelated F y AA Design Strength Load
37 Reliability by Sampling Uncorrelated F y AA Design P f ( 10-6 ) Number of Samples
38 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
39 COV of Strength at Plastic Collapse COV of Strength Design: Uncorr. Corr. AA LRFD
40 Reliability at Plastic Collapse Reliability, β First-Order Monte Carlo P f =
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