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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 ( 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...

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**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.

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**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

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**Probability of Failure following Single column loss**

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

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**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

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**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)

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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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**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

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**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

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**Example Study : Modelling - Beam**

EC4 Effective Width Structural steel S355 Shear Connectors d=20mm Concrete: C30 Reinforcement steel 460B

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**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

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**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

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**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

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**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

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**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)

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**Example Study : First Order Reliability Method (FORM)**

Structural Demand (Xi=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) e.g.: peripheral column loss, EC4 reinforcement ratio for,

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**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

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**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

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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)

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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, 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.

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