example of a probabilistic robustness analysis

Example of a probabilistic robustness analysis

M. Pereira, B.A. Izzuddin, L. Rolle, U. Kuhlmann

Contributors: T. Vrouwenvelder and B. Leira

Framework for risk assessment

Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( Fnot | D ) C ( Fnot ) }

Probability of Hazard – gas explosions, fire, human error, ...Probability of Damage given certain Hazard – Single column loss (Vlassis et al. 2008), multiple column loss (Pereira & Izzuddin, 2011), failed floor impact (Vlassis et al. 2009), partial column damage (Gudmundsson & Izzuddin, 2009), transfer beam loss, infill panels loss, ...

Probability of Failure given certain Damage Scenario – Progressive Collapse

Cost of Failure – Material and human losses, ...

Probability of avoiding Failure given certain Damage Scenario – Safety against Progressive Collapse

Cost of Local Damage – Material and human losses...

Single column loss scenario


• Restrict risk assessment to two damage scenarios in the example study: - Single Peripheral Column loss - Single Corner Column lossComment: for illustration purposes the single internal column loss scenario was not considered

• Given a specific hazard, these damage scenarios are more likely to occur, i.e., P (D | H ) is higher, when compared to failed floor impact (Vlassis et. al, 2009) or multiple column loss (Pereira & Izzuddin, 2011) scenarios.

However, they are less demanding in terms of structural performance, i.e., P ( F | D ) is lower.

Hazards P (D|H) (Vrouwenvelder, 2011)

Explosion 0.10

Fire 0.10

Human Error 0.10

Hazards P (H) [50 year] (Vrouwenvelder, 2011)

Explosion 2 x 10-3

Fire 20 x 10-3

Human Error 2 x 10-3

Probability of single column loss


Probability of single column loss (somewhere in the building)

Probability of Failure following Single column loss


Probabilistic model for Capacity and Demand

Capacity Distribution Mean [μ] Std. Deviation [σ]

Steel members yield stress (X1)

Lognormal 1.2 x Nominal 0.05 μ

Joint component resistance (X2)

Lognormal 1.2 x Nominal 0.05 μ

Joint component ductility (X3)

Lognormal Nominal 0.15 μ

Demand Distribution Mean [μ] Std. Deviation [σ]

Floor Dead Load (X4) Normal Nominal 0.10 μ

Floor Live Load (X5) Lognormal 0.70 kN/m2 0.05 μ

where, F is the failure domain, μi

N and σiN are the equivalent

normal mean and standard deviation obtained for each variable, based on Normal Tail Approximation, R is the correlation matrix, simplified to be the identity matrix

• Solve Xi to minimize β constrained by the limit state function:

Structural Capacity (Xi=1,2,3) = Structural Demand (Xi=3,4)

1min

TN Ni i i i

N Nx Fi i

X XR

First Order Reliability Method (FORM)

P ( F | D ) = Ф ( - β )

where, Ф is the cumulative standard Gaussian distribution β is the reliability index:

Failure Probability in a Single Column Loss scenario

Simplified Assessment Framework for Progressive Collapse due to Sudden column loss (Izzuddin et al. 2008)

First-order approximation in standard normal space (from Beck & da Rosa, 2006)

Example Study : Overview

• Seven-storey steel-framed composite structure

• Designed as a simple structure according to UK steel design practice

• Joint detailing and design based on BCSA/SCI: “Simple connections” code

• BS5950 robustness provisions based on minimum tying force requirements are satisfied

• Two solutions studied for slab reinforcement ratio:

- EC4 minimum ratio (0.84%) - 2 % reinforcement ratio

Assessment framework multi-level application

(a) Floor systems vertically aligned with lost column and surrounding frame modelled by means of boundary conditions

(b) Multiple floors above lost column, subject to surrounding columns stability

(c) Individual floor system, for structures with regular load and configuration in height

(d) Individual beams system, for negligible slab membrane effects

- Edge beams: UB406X140X39

Example Study : Floor systems and Loading

Peripheral floor area affected by column loss

• Structural configuration:

- Internal beams: UB305X102X25

- Transverse beam: UC356X368X153

• Service Load configuration:

- Facade load: 8.3 kN/m

- Floor Dead Load: 4.2 kN/m2

- Floor Live Load: 5.0 kN/m2 (factored 0.25)

- Transverse beam: UB406X140X39 - Floor Live Load: 5.0 kN/m2 (factored 0.25)

- Edge beams: UB406X140X39

Example Study : Floor systems and Loading

Corner floor area affected by column loss

• Structural configuration:

- Internal beams: UB305X102X25

• Service Load configuration:

- Facade load: 8.3 kN/m

- Floor Dead Load: 4.2 kN/m2

Example Study : Modelling - Beam

• EC4 Effective Width

• Reinforcement steel 460B• Concrete: C30

• Structural steel S355• Shear Connectors d=20mm

Example Study : Modelling – Joints

e.g.: edge beam partial depth flexible end-plate jointfor peripheral column loss, EC4 reinforcement ratio

Mean values (Rolle, 2011)

Δcr 0.05 mm

Δsl 0.76 mm

Δu 17.74 mm

Fcr 335.16 kN

Fu 775.63 kN

Mean values (Rolle, 2011)

K0,tr 99.73 kN/mm2

Fy,d 80.76 kN

Fu,d 199.00 kN

Δm 23.7 mm

• Hogging concrete slab component• Bolt-row 1 component

(i) Nonlinear static response of the damaged structure under gravity loading

(ii) Simplified dynamic assessment to establish the maximum dynamic response under column loss scenarios

(iii)Ductility assessment of the connections/structure










Example Study : Sudden Column Loss Assessment

e.g.: edge beam, EC4 reinforcement ratio

Capacity Distribution μ σ

Steel members yield stress Lognormal 1.2 x Nominal 0.05 μ

Joint component resistance Lognormal 1.2 x Nominal 0.05 μ

Joint component ductility Lognormal Nominal 0.15 μ













Example Study : Probabilistic model for Structural Capacity

• No change in nonlinear response since composite beams remain elastic up to connection failure (partial-strength connected frames)

e.g.: edge beam, EC4 reinforcement ratio

• Nonlinear static FEA required per variation of joint component resistance, considered simultaneously for all joint components of the individual beam

• Simple assessment of deformation level at critical component from nonlinear analysis: assumption of system ductility limit equal to first component failure

Total number of FEA required for μ – σ , μ and μ + σ of all Capacity variables: 3

where, α is the work-related factor

βEB βIB1 βIB2 βIB3 βTB

1.00 0.152 0.456 0.759 1.00

Example Study : Probabilistic model for Structural Capacity

e.g.: peripheral column loss, JCR = μ-σ, JCD = μ+σ, EC4 reinforcement ratio

1floor i i i

i

P P

where, β is the compatibility factor

αEB αIB1 αIB2 αIB3 αTB α

0.5 0.5 0.5 0.5 1.00.287(0.25-0.292)

Example Study : First Order Reliability Method (FORM)

Structural Capacity (Xi=1,2,3)

e.g.: peripheral column loss, EC4 reinforcement ratio

X2 X3 Capacity (kN)

1-σ/μ 1-σ/μ 523.9268

1-σ/μ 1 560.3771

1-σ/μ 1+σ/μ 593.6995

1 1-σ/μ 526.9997

1 1 565.8955

1 1+σ/μ 598.2537

1+σ/μ 1-σ/μ 561.459

1+σ/μ 1 575.7152

1+σ/μ 1+σ/μ 575.7152

Response Surface (second-order polynomial)

2 21, 2 3 2 2 3

2 2 22 3 2 3 2 3

23 2 3

Capacity( , ) 20696.104 685.63

4466.64 19150.230 35827.504

16822.539 41861.234 3252.92 21348.245

X X X X X X

X X X X X X

X X X


Structural Demand (Xi=4,5)

4 5 4 5593.865Demand( , ) 82.95X X X X


First-order polynomial


Probability of Failure P (F|D)

1.121

for, 2 3 4 51.068, 0.910.999 .474, 2, 0X X X X

( | ) 0.868P F D


Scenario P (F|D) P (H) P (D|H) P(H) P (D|H) P (F|D)

EC 4 slab solution

Peripheral Column loss (EC4) 0.868 21.7E-3 0.10 1.88E-03

Corner Column loss (EC4) 5.776E-5 2.29E-3 0.10 1.32E-08

2 % reinforcement ratio solution

Peripheral Column loss (2%) 0.217 21.7E-3 0.10 4.71E-4

Corner Column loss (2%)

1.580E-62.29E-3 0.10 3.61E-10

Example Study : Risk Assessment

Scenario P (F|D)

EC 4 slab solution

Peripheral Column loss (EC4) 0.868

Corner Column loss (EC4) 5.776E-5

2 % reinforcement ratio solution

Peripheral Column loss (2%) 0.217

Corner Column loss (2%)

1.580E-6

( ) ( )i spatiali

P H P SGas explosions, fire and human errorSpatial probability of event: peripheral/corner hazard

which, assuming equal probability for each column to be subjected to the studied hazards,

spatial

no. of corner / peripheral columnsP (S ) =

no. of total columns

• Multiple independent damage scenarios, with different P (D| H) associated: e.g. separate levels of single column damage, single column loss, two adjacent column losses,...

• Spatial distribution in terms of event and material/loading values

• Structural irregularity

• Accuracy of FORM analysis versus Monte Carlo simulations

• Dissociation of structural performance between blast-induced damage scenarios and fire-induced damage scenarios

Issues in real design application

• The simplified assessment framework offers a practical basis for performing a structural risk assessment based on a damage scenario commonly considered in design codes

• The information on the probability of failure can be used in a richer Risk Assessment framework where an Acceptance Criteria is established (Working Group 1) and Costs are quantified (Working Group 3)

Conclusions

References

B.A. Izzuddin, M. Pereira, U. Kuhlmann, L. Rölle, T. Vrouwenvelder, B.J. Leira,“Application of Probabilistic Robustness Framework: Risk Assessment ofMulti-Storey Buildings under Extreme Loading”, Structural EngineeringInternational, Vol. 1, 2012.

U. Kuhlmann, L. Rölle, B.A. Izzuddin, M. Pereira, “Resistance and response ofsteel and steel-concrete composite structures in progressive collapseassessment”, Structural Engineering International, Vol. 1, 2012.

example of a probabilistic robustness analysis

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