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Reliability Considerations for Steel Frames Designed with Advanced Analysis
Stephen G. BuonopaneBenjamin W. SchaferTakeru Igusa
Dept. of Civil EngineeringJohns Hopkins University
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
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
Frames for Study • Steel Frames from Ziemian et al. (1992)• Designed by both LRFD (1986) and AA• AA Design saves ~12% by weight
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
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
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
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
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
OverviewNon-Linear Analyses
with RandomYield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
OverviewNon-Linear Analyses
with RandomYield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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
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
Member vs. System Limit State
Uncorrelated FyAA Design
Plastic Collapse Strength
1st P
H S
treng
th
Uncorrelated FyAA Design
Uncorrelated FyLRFD Design
Plastic Collapse StrengthPlastic Collapse Strength
1st P
H S
treng
thMember vs. System Limit State
Strength Distributions
1st Plastic Hinge
Plastic CollapseLoad
StrengthUncorrelated FyLRFD Design
Rel
ativ
e Fr
eque
ncy
Normalized Strength and Load
10,000 samples
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
OverviewNon-Linear Analyses
with RandomYield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
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
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
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
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
OverviewNon-Linear Analyses
with Random Yield Strength
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
Resistance Factorsφ = Rm / Rn( )exp −0.55βtVR( )
Nominal Strength
Mean of “True”Strength
TargetReliability
Variationof “True” Strength
True StrengthRmRnφRn
VR
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( )
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
0.98
1.00
1.02
1.04
1.06
1.08
Mean Bias FactorsM
ean
Bia
s Fac
tors
CorrelatedUncorrelated
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
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
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
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
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
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
Reliability by Sampling
Uncorrelated FyAA Design
Load
Stre
ngth
Safe Failure
Reliability by Sampling
Number of Samples
P f(×
10-6
)
Uncorrelated FyAA Design
Corr.Uncorr.
LRFDAA
BaseFixity
GravityLoad
Geometry
3.
4.
5.
1. Member Sizes: LRFD or AA
2. Yield Strength: Uncorrelated and Correlated
LRFDAA
Design by
LRFDAA
Design by
Corr.Uncorr.
LRFDAADesign by
Corr.Uncorr.
LRFDAADesign:
First-OrderMonte Carlo
CorrelatedUncorrelated
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
Reliability and Advanced Analysis
• What Reliability is Associated with Design by AA?
• Implications for Resistance Factor, phi
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
ApproachNon-Linear Analyses
with Random Properties
Strength Distributions
Load Distributions
Reliability Estimates
Resistance Factors
Advanced Analysiswith
Nominal Properties
System vs. Member Limit State
Uncorrelated FyAA Design
Uncorrelated FyLRFD Design
Plastic Collapse StrengthPlastic Collapse Strength
1st P
H S
treng
th
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:
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 =
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