reliability considerations for steel frames …reliability considerations for steel frames designed...
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![Page 1: Reliability Considerations for Steel Frames …Reliability Considerations for Steel Frames Designed with Advanced Analysis Stephen G. Buonopane Benjamin W. Schafer Takeru Igusa Dept](https://reader030.vdocuments.net/reader030/viewer/2022040521/5e7c11d5d0400328e0156aa9/html5/thumbnails/1.jpg)
Reliability Considerations for Steel Frames Designed with Advanced Analysis
Stephen G. BuonopaneBenjamin W. SchaferTakeru Igusa
Dept. of Civil EngineeringJohns Hopkins University
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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
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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
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Frames for Study • Steel Frames from Ziemian et al. (1992)• Designed by both LRFD (1986) and AA• AA Design saves ~12% by weight
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Member Size ComparisonW
12x1
9W
12x1
4
W14
x132
W14
x99
W14
x109
W14
x82
W10
x12
W14
x109
W14
x109
W27x84 W36x135
W18x40 W27x94
Design by: LRFDAA
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Frames for Study
BaseFixity
GravityLoad
Geometry
3.
4.
5.
1. Member Sizes: LRFD or AA
2. Yield Strength: Uncorrelated and Correlated
Total = 32 Frames Analyzed
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Frame Analysis Details
• OpenSees (http://opensees.berkeley.edu)
– 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
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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
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Frame Simulation Details
• Each Frame of 32– 10,000 Samples with Random Fy
• Each Sample – Load Increased Until Failure– Strength Limit States considered
• 1st Plastic Hinge • Plastic Collapse
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OverviewNon-Linear Analyses
with RandomYield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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OverviewNon-Linear Analyses
with RandomYield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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Load-Deflection Behavior
Lateral Deflection of Top Story (in.)
Nor
mal
ized
Tot
al G
ravi
ty L
oad
Uncorrelated FyAA Design
1st Plastic Hinge Plastic Collapse
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Load-Deflection Behavior
Lateral Deflection of Top Story (in.)
Correlated FyAA Design
1st Plastic Hinge Plastic Collapse
Nor
mal
ized
Tot
al G
ravi
ty L
oad
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Member vs. System Limit State
Uncorrelated FyAA Design
Plastic Collapse Strength
1st P
H S
treng
th
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Uncorrelated FyAA Design
Uncorrelated FyLRFD Design
Plastic Collapse StrengthPlastic Collapse Strength
1st P
H S
treng
thMember vs. System Limit State
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Strength Distributions
1st Plastic Hinge
Plastic CollapseLoad
StrengthUncorrelated FyLRFD Design
Rel
ativ
e Fr
eque
ncy
Normalized Strength and Load
10,000 samples
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1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
Mean Strength at Plastic CollapseM
ean
Stre
ngth
LRFDAA
Design by
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OverviewNon-Linear Analyses
with RandomYield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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Reliability and LRFDQ = Load R = Strength
ln R / Q( )
βFO =ln Rm / Qm( )
VR2 +VQ
2
βFO VR2 +VQ
2( )
ln Rm / Qm( )
First-Order Reliability
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Reliability by Sampling
Pf = I r, q( )fR r( )fQ q( )dr dq−∞
∞
∫−∞
∞
∫
Load Strength
fQ(q) fR(r)= 1 for R≤Q= 0 for R>Q
I r,q( )
ˆ P f =1N
I ri ,qi( )i=1
N∑
Monte Carlo Estimate
β = −Φ−1 ˆ P f( )
Probability of Failure
ri sampled from fRqi sampled from fQ
Reliability
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2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
Reliability at Plastic CollapseR
elia
bilit
y, β
LRFDAA
Design by
0.02
3.17×10-5
9.90×10-10
Pf =
1.35×10-3
2.87×10-7
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0.0
1.0
2.0
3.0
4.0
5.0
Reliability at 1st Plastic HingeR
elia
bilit
y, β
LRFDAA
Design by
1.35×10-3
0.16
3.17×10-5
Pf =
0.02
0.50
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OverviewNon-Linear Analyses
with Random Yield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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Resistance Factorsφ = Rm / Rn( )exp −0.55βtVR( )
Nominal Strength
Mean of “True”Strength
TargetReliability
Variationof “True” Strength
True StrengthRmRnφRn
VR
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Resistance Factors of LRFD
LRFD Bias Factors: P = ProfessionalM = MaterialF = Fabrication
Means
Rm / Rn = Pm MmFm = 1.07
COVs
VR = VP2 +VM
2 +VF2 = 0.15
φ = Rm / Rn( )exp −0.55βtVR( )
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Resistance Factors for AA
VB AA = VR AA2 +VF
2RmAA / Rn
AA = BmAA
φ = Rm / Rn( )exp −0.55βtVR( )
AA StrengthDistribution
AA StrengthDistribution
AA NominalStrength
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0.98
1.00
1.02
1.04
1.06
1.08
Mean Bias FactorsM
ean
Bia
s Fac
tors
CorrelatedUncorrelated
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0.86
0.88
0.90
0.92
0.94
Resistance Factors for AA at PCR
esis
tanc
e Fa
ctor
s, φ
CorrelatedUncorrelated
βt = 3.00
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0.78
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
Resistance Factors for AA at 1st PHR
esis
tanc
e Fa
ctor
s, φ
CorrelatedUncorrelated
βt = 3.00
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Summary of Resistance Factors
0.900.860.900.91Max
0.880.830.880.89Mean
0.860.800.870.86Min
Corr.Uncorr.Corr.Uncorr.Fy
1st Plastic HingePlastic CollapseLimit State
Target Reliability, βt = 3.00
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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
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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
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Reliability by SamplingLoad Strength
Pf = I r,q( )fR r( )fQ q( )dr dq−∞
∞
∫−∞
∞
∫fQ(q) fR(r)
= 1 for R≤Q= 0 for R>Q
I r,q( )
ˆ P f =1N
I ri ,qi( )i=1
N∑
Monte Carlo Estimate
β = −Φ−1 ˆ P f( )
Probability of Failure
ri sampled from fRqi sampled from fQ
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Reliability by Sampling
Uncorrelated FyAA Design
Load
Stre
ngth
Safe Failure
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Reliability by Sampling
Number of Samples
P f(×
10-6
)
Uncorrelated FyAA Design
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Corr.Uncorr.
LRFDAA
BaseFixity
GravityLoad
Geometry
3.
4.
5.
1. Member Sizes: LRFD or AA
2. Yield Strength: Uncorrelated and Correlated
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LRFDAA
Design by
LRFDAA
Design by
Corr.Uncorr.
LRFDAADesign by
Corr.Uncorr.
LRFDAADesign:
First-OrderMonte Carlo
CorrelatedUncorrelated
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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
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Reliability and Advanced Analysis
• What Reliability is Associated with Design by AA?
• Implications for Resistance Factor, phi
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Member Size ComparisonW
12x1
9W
12x1
4
W14
x132
W14
x99
W14
x109
W14
x82
W10
x12
W10
x12
W14
x109
W14
x109
W14
x109
W14
x109
W27x84W27x84
W36x135W36x135
W18x40W18x40
W27x94W27x94
Member Sizes by:LRFDAdvanced Analysis
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ApproachNon-Linear Analyses
with Random Properties
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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System vs. Member Limit State
Uncorrelated FyAA Design
Uncorrelated FyLRFD Design
Plastic Collapse StrengthPlastic Collapse Strength
1st P
H S
treng
th
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0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
COV of Strength at Plastic CollapseC
OV
of S
treng
th Corr.
Uncorr.LRFDAADesign:
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3.0
3.5
4.0
4.5
5.0
Reliability at Plastic CollapseR
elia
bilit
y, β
First-OrderMonte Carlo
1.35×10-3
2.33×10-4
3.17×10-5
3.40×10-6
Pf =