1. Example of a probabilistic robustness analysis
M. Pereira, B.A. Izzuddin, L. Rolle, U. Kuhlmann
Contributors: T. Vrouwenvelder and B. Leira

2. 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...

3. Single column loss scenario
Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( Fnot | D ) C ( Fnot ) }
Restrict risk assessment to two damage scenarios in the example study:
- Single Peripheral Column loss
- Single Corner Column loss
Comment: 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.

4. Probability of single column loss
Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( Fnot | D ) C ( Fnot ) }
Probability of single column loss (somewhere in the building)
Hazards
P (D|H) (Vrouwenvelder, 2011)
Explosion
0.10
Fire
Human Error
Hazards
P (H) [50 year] (Vrouwenvelder, 2011)
Explosion
2 x 10-3
Fire
20 x 10-3
Human Error

5. Probability of Failure following Single column loss
Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( Fnot | D ) C ( Fnot ) }

6. Probabilistic model for Capacity and Demand
Distribution
Mean [μ]
Std. Deviation [σ]
Steel members yield stress (X1)
Lognormal
1.2 x Nominal
0.05 μ
Joint component resistance (X2)
Joint component ductility (X3)
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

7. First Order Reliability Method (FORM)
Failure Probability in a Single Column Loss scenario
P ( F | D ) = Ф ( - β )
where, F is the failure domain, μiN 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)
where, Ф is the cumulative standard Gaussian distribution β is the reliability index:
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)

8. 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

9. 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

10. Example Study : Floor systems and Loading
Peripheral floor area affected by column loss
Service Load configuration:
Structural configuration:
- Edge beams: UB406X140X39
- Facade load: 8.3 kN/m
- Floor Dead Load: 4.2 kN/m2
- Internal beams: UB305X102X25
- Floor Live Load: 5.0 kN/m2 (factored 0.25)
- Transverse beam: UC356X368X153

11. Example Study : Floor systems and Loading
Corner floor area affected by column loss
Service Load configuration:
Structural configuration:
- Facade load: 8.3 kN/m
- Edge beams: UB406X140X39
- Floor Dead Load: 4.2 kN/m2
- Internal beams: UB305X102X25
- Floor Live Load: 5.0 kN/m2 (factored 0.25)
- Transverse beam: UB406X140X39

12. Example Study : Modelling - Beam
EC4 Effective Width
Structural steel S355
Shear Connectors d=20mm
Concrete: C30
Reinforcement steel 460B

13. Example Study : Modelling – Joints
Hogging concrete slab component
Bolt-row 1 component
e.g.: edge beam partial depth flexible end-plate joint for peripheral column loss, EC4 reinforcement ratio
Mean values (Rolle, 2011)
Δcr
0.05 mm
Δsl
0.76 mm Δu
17.74 mm
Fcr kN Fu kN
Mean values (Rolle, 2011)
K0,tr
99.73 kN/mm2
Fy,d
80.76 kN
Fu,d kN Δm
23.7 mm

14. Example Study : Sudden Column Loss Assessment
e.g.: edge beam, EC4 reinforcement ratio
Nonlinear static response of the damaged structure under gravity loading
Simplified dynamic assessment to establish the maximum dynamic response under column loss scenarios
Ductility assessment of the connections/structure
Nonlinear static response of the damaged structure under gravity loading
Simplified dynamic assessment to establish the maximum dynamic response under column loss scenarios
Ductility assessment of the connections/structure
Nonlinear static response of the damaged structure under gravity loading
Simplified dynamic assessment to establish the maximum dynamic response under column loss scenarios
Ductility assessment of the connections/structure
Nonlinear static response of the damaged structure under gravity loading
Simplified dynamic assessment to establish the maximum dynamic response under column loss scenarios
Ductility assessment of the connections/structure

15. Example Study : Probabilistic model for Structural Capacity
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
No change in nonlinear response since composite beams remain elastic up to connection failure (partial-strength connected frames)
Capacity
Distribution μ σ
Steel members yield stress
Lognormal
1.2 x Nominal
0.05 μ
Joint component resistance
Joint component ductility
Nominal
0.15 μ
Capacity
Distribution μ σ
Steel members yield stress
Lognormal
1.2 x Nominal
0.05 μ
Joint component resistance
Joint component ductility
Nominal
0.15 μ
Capacity
Distribution μ σ
Steel members yield stress
Lognormal
1.2 x Nominal
0.05 μ
Joint component resistance
Joint component ductility
Nominal
0.15 μ
Capacity
Distribution μ σ
Steel members yield stress
Lognormal
1.2 x Nominal
0.05 μ
Joint component resistance
Joint component ductility
Nominal
0.15 μ
Total number of FEA required for μ – σ , μ and μ + σ of all Capacity variables: 3

16. Example Study : Probabilistic model for Structural Capacity
e.g.: peripheral column loss, JCR = μ-σ, JCD = μ+σ, EC4 reinforcement ratio
where, β is the compatibility factor
where, α is the work-related factor
αEB
αIB1
αIB2
αIB3
αTB α
0.5
1.0
0.287
( )
βEB
βIB1
βIB2
βIB3
βTB
1.00
0.152
0.456
0.759

17. 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
1+σ/μ
Response Surface (second-order polynomial)

18. Example Study : First Order Reliability Method (FORM)
Structural Demand (Xi=4,5)
e.g.: peripheral column loss, EC4 reinforcement ratio
First-order polynomial

19. Example Study : First Order Reliability Method (FORM)
Probability of Failure P (F|D)
e.g.: peripheral column loss, EC4 reinforcement ratio
for,

20. Example Study : Risk Assessment
Gas explosions, fire and human error
Spatial probability of event: peripheral/corner hazard
which, assuming equal probability for each column to be subjected to the studied hazards,
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
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
1.32E-08
2 % reinforcement ratio solution
Peripheral Column loss (2%)
0.217
4.71E-4
Corner Column loss (2%)
1.580E-6
3.61E-10

21. Issues in real design application
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

22. Conclusions
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)

23. References
B.A. Izzuddin, M. Pereira, U. Kuhlmann, L. Rölle, T. Vrouwenvelder, B.J. Leira,
“Application of Probabilistic Robustness Framework: Risk Assessment of Multi-Storey Buildings under Extreme Loading”, Structural Engineering International, Vol. 1, 2012.
U. Kuhlmann, L. Rölle, B.A. Izzuddin, M. Pereira, “Resistance and response of steel and steel-concrete composite structures in progressive collapse assessment”, Structural Engineering International, Vol. 1, 2012.