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

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Framework for risk assessment Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( F not | D ) C ( F not ) } 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...

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Single column loss scenario Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( F not | D ) C ( F not ) } 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.

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HazardsP (D|H) (Vrouwenvelder, 2011) Explosion0.10 Fire0.10 Human Error0.10 HazardsP (H) [50 year] (Vrouwenvelder, 2011) Explosion2 x Fire20 x Human Error2 x Probability of single column loss Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( F not | D ) C ( F not ) } Probability of single column loss (somewhere in the building)

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Probability of Failure following Single column loss Risk = P ( H ) P( D | H ) { P ( F | D ) C ( F ) + P ( F not | D ) C ( F not ) }

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Probabilistic model for Capacity and Demand CapacityDistributionMean [μ]Std. Deviation [σ] Steel members yield stress (X 1 ) Lognormal1.2 x Nominal0.05 μ Joint component resistance (X 2 ) Lognormal1.2 x Nominal0.05 μ Joint component ductility (X 3 ) LognormalNominal0.15 μ DemandDistributionMean [μ]Std. Deviation [σ] Floor Dead Load (X 4 )NormalNominal0.10 μ Floor Live Load (X 5 )Lognormal0.70 kN/m μ

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where, F is the failure domain, μ i N and σ i N 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 X i to minimize β constrained by the limit state function: Structural Capacity (X i=1,2,3 ) = Structural Demand (X i=3,4 ) 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)

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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

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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

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- 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/m 2 - Floor Live Load: 5.0 kN/m 2 (factored 0.25)

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- Transverse beam: UB406X140X39 - Floor Live Load: 5.0 kN/m 2 (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/m 2

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

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Example Study : Modelling – Joints e.g.: edge beam partial depth flexible end-plate joint for peripheral column loss, EC4 reinforcement ratio Mean values (Rolle, 2011) Δ cr 0.05 mm Δ sl 0.76 mm ΔuΔu mm F cr kN FuFu kN Mean values (Rolle, 2011) K 0,tr kN/mm 2 F y,d kN F u,d kN ΔmΔm 23.7 mm Hogging concrete slab component Bolt-row 1 component

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(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 (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 (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 (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

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CapacityDistributionμσ Steel members yield stressLognormal1.2 x Nominal0.05 μ Joint component resistanceLognormal1.2 x Nominal0.05 μ Joint component ductilityLognormalNominal0.15 μ CapacityDistributionμσ Steel members yield stressLognormal1.2 x Nominal0.05 μ Joint component resistanceLognormal1.2 x Nominal0.05 μ Joint component ductilityLognormalNominal0.15 μ CapacityDistributionμσ Steel members yield stressLognormal1.2 x Nominal0.05 μ Joint component resistanceLognormal1.2 x Nominal0.05 μ Joint component ductilityLognormalNominal0.15 μ CapacityDistributionμσ Steel members yield stressLognormal1.2 x Nominal0.05 μ Joint component resistanceLognormal1.2 x Nominal0.05 μ Joint component ductilityLognormalNominal0.15 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

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where, α is the work-related factor β EB β IB1 β IB2 β IB3 β TB 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 α EB α IB1 α IB2 α IB3 α TB α ( )

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Example Study : First Order Reliability Method (FORM) Structural Capacity (X i=1,2,3 ) e.g.: peripheral column loss, EC4 reinforcement ratio X2X2 X3X3 Capacity (kN) 1-σ/μ σ/μ σ/μ1+σ/μ σ/μ σ/μ σ/μ 1-σ/μ σ/μ σ/μ Response Surface (second-order polynomial)

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

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

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ScenarioP (F|D)P (H)P (D|H)P(H) P (D|H) P (F|D) EC 4 slab solution Peripheral Column loss (EC4) E E-03 Corner Column loss (EC4) 5.776E-52.29E E-08 2 % reinforcement ratio solution Peripheral Column loss (2%) E E-4 Corner Column loss (2%) 1.580E E E-10 Example Study : Risk Assessment ScenarioP (F|D) EC 4 slab solution Peripheral Column loss (EC4) Corner Column loss (EC4) 5.776E-5 2 % reinforcement ratio solution Peripheral Column loss (2%) Corner Column loss (2%) 1.580E-6 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,

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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

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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

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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, 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.

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