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 =