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**Structural Systems Analysis for Robustness Assessment**

Bassam A. Izzuddin Department of Civil & Environmental Engineering

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**Progressive Collapse… But Is It Disproportionate?**

Structures cannot be designed to withstand unpredictable extreme events But should be designed for structural robustness: the ability of the structure to withstand the action of extreme events without being damaged to an extent disproportionate to the original cause WTC (2001) Setúbal, Portugal (2007) Ronan Point (1968) Murrah Building (1995) Disproportionate: No Robust structure Disproportionate: ? Disproportionate: Yes Design is based on evaluating the acceptability of what can be predicted Designers cannot be expected to design against all types of progressive collapse, but should design against disproportionate collapse Robustness Summer School - COST Action TU0601 4 July 2012

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**Structural Design – Predictability3**

Structure Response Actions Codified properties Statistical data Site supervision & QA … Codified calculations Simplified analysis Detailed analysis … Codified loads Statistical analysis Event modelling … Acceptability of the predictable No uncertainty (CoV very small) – deterministic design Uncertainty in reality -> codified methods (can be either too conservative or unsafe for certain types of scenario) Uncertainty in reality -> performance based design using probabilistic assessment Uncertainty of terrorist action (intelligent malicious action, statistical data inapplicable, even if mean is defined CoV would be very large) -> Robustness design Use of probabilistic risk assessment as a decision support tool for cost-benefit analysis Acceptable? Malicious/terrorist actions Robustness Summer School - COST Action TU0601 4 July 2012

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**Codified Design for Robustness**

Structural robustness (UK Building Regulations, Eurocode EN 1990) ability to withstand extreme events without being damaged to an extent disproportionate to the original cause UK Building Regulations: A3 Disproportionate Collapse Class 2B buildings (up to 15 storeys) Prescriptive tying force requirements Notional member removal Key element design Class 3 buildings (more than 15 storeys) Systematic risk assessment More performance based However, conventional design checks (ignoring large deformations) Ignores dynamic effects Unrealistic designs and damage assessment Do not guarantee robustness Implicit reliance on tensile catenary action, while ignoring ductility issues Do not allow comparison between alternative designs utilising redundancy, ductility and energy absorption Invoked if notional member removal leads to excessive damage (15% of floor area or 70m2) No clear guidance Robustness Summer School - COST Action TU0601 4 July 2012

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**Probabilistic Risk Assessment**

Risk = ∑ P(H) P(D|H) P(F|D) C(F) Deterministic evaluation of failure | D = sudden column loss Material related issues (steel and composite structures) Application in probabilistic risk assessment Hazard Local damage System failure Vulnerability Damage tolerance Consequences Use of probabilistic risk assessment as a decision support tool for cost-benefit analysis Structural robustness Involve similar types of structural analysis May be considered together depending on event resolution Robustness Summer School - COST Action TU0601 4 July 2012

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**Robustness Summer School - COST Action TU0601**

Sudden Column Loss (D) Event-independent scenario More than just a standard test of robustness Sudden column loss (SCL) vs column damage by blast Comparison of deformation demands in upper floors SCL presents an upper bound on floor deformations SCL can be scaled to correspond to intermediate levels of blast SCL realistic for multi-storey buildings even considering blast uplift and extended local damage Can be assessed without full nonlinear dynamic analysis Sudden column loss Robustness Summer School - COST Action TU0601 4 July 2012

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**Robustness Summer School - COST Action TU0601**

Sudden Column Loss (D) From prescriptive to performance-based design Recent GSA (2003) and DoD (2005) guidance Consider sudden column loss as a design scenario Detailed nonlinear dynamic analysis Simplified equivalent static approach Move to modify and unify GSA/DoD guidance Important changes in equivalent static approach Excessive dynamic amplification factor equal to 2 Conventional design checks Too complicated for practical application in design Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure Limit State (F)**

Collapse of above floors and considering resistance of lower structure Impact and debris loading on lower structure Top floors sacrificed Even collapse of one floor is too onerous on lower floor, causing progressive collapse Failure state Failure limit state at point of collapse of above floors Allowing large deformations Outside conventional strength limit, but within ductility limit Ductility limit Maximum dynamic deformed configuration Demand supply Failure in this case is a binary outcome (true or false) Robustness Summer School - COST Action TU0601 4 July 2012

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**Load Factor Approaches**

Basis of equivalent ‘pushdown’ static approach proposed for new GSA/DoD guidance Nonlinear static analysis DIF equal to 2 applicable only for linear elastic response Recognition of influence of available ductility on DIF But not the characteristics of nonlinear static response Dynamic response Static analysis Robustness Summer School - COST Action TU0601 4 July 2012

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**Load Factor Approaches**

DIF in terms of ductility for use with nonlinear static analysis in new GSA/DoD guidance Monotonic reduction in DIF with ductility Robustness Summer School - COST Action TU0601 4 July 2012

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**Load Factor Approaches**

Monotonic reduction of DIF with ductility Consistent with load factor approaches for new GSA/DoD guidance Elastic-plastic static response Static resistance Dynamic resistance DIF = 1.67 DIF = 1.04 DIF = 2 Robustness Summer School - COST Action TU0601 4 July 2012

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**Load Factor Approaches**

Elasto-plastic static response with hardening Proposed load factor approaches very unsafe for ductile structures DIF increases with ductility after initial reduction Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss**

Limit state: dynamic failure of floors above Two stages of assessment Nonlinear static response accounting for ductility limit Simplified dynamic assessment Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss**

Maximum gravity load sustained under sudden column loss Applicable at various levels of structural idealisation Floors identical in components and loading Planar effects are neglected Columns can take re- distributed load Reduced model where deformation is concentrated Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss**

Benefits of multi-level approach Low level models can be used to assemble response at higher levels Realised even if conditions of model reduction are not applicable Beam models assemble a grillage approximation of floor Floor model assembles SDOF response of multiple floors, assuming rigid column Robustness Summer School - COST Action TU0601 4 July 2012

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**Nonlinear Static Response**

Failure | Sudden Column Loss Nonlinear Static Response Sudden column removal similar to sudden application of gravity load Maximum dynamic response can be approximated using amplified static loading (ld P) Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Nonlinear Static Response**

Proposed framework supports detailed / simplified models Detailed and simplified modelling may be combined Detailed at lower levels to capture complex nonlinear response (connections, composite action, …) Simplified assembly at higher levels Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Nonlinear Static Response**

Detailed models, largely based on NLFE At beam level: geometric and material nonlinearity, connection nonlinearity using component-based approach, composite action, … At floor level: additionally membrane action, geometric orthotropy, … At higher levels: additional sophistication, but excessive computational demands Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Nonlinear Static Response**

Simplified modelling Facilitates practical application in design Applicable at various levels of structural idealisation At lowest beam level More sophisticated simplified models needed Can be substituted by detailed models Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Nonlinear Static Response**

Simplified floor grillage model Assumed SDOF mode, realistic at large deflections Assume load distributions, but not intensities, on component beams Accuracy of load distribution unimportant at large deflections Nonlinear load-deflection response of floor system obtained as weighted sum of individual beam responses Simplified / detailed beam models may be used Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Nonlinear Static Response**

Simplified multi-floor model Assume SDOF mode, realistic if load redistribution between floors well within column capacity Assume load distributions on floors For practicality, ignore force transferred via line of columns above failed column Nonlinear load-deflection response of overall system as weighted sum of individual floor responses Simplified / detailed individual floor models Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Nonlinear Static Response**

Ductility limit Ductility demands in connections and their components related to system displacements Smallest displacement at which ductility demand exceeds supply in one of the connections Importance of accounting for rotational and axial deformations Need for extensive experimental data on connection ductility Proposed framework accommodates refined data Robustness Summer School - COST Action TU0601 4 July 2012

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**Simplified Dynamic Assessment**

Failure | Sudden Column Loss Simplified Dynamic Assessment Applicable at various levels of structural idealisation Based on conservation of energy Work done by suddenly applied load equal to internal energy stored Leads to maximum dynamic displacement (also to load dynamic amplification) Definition of “pseudo-static” response ld<<2l Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Simplified Dynamic Assessment**

Dynamic “pseudo-static” (P,ud) response constructed from corresponding nonlinear static (P,us) response Represents response to sudden application of gravity load (P) Provides valuable information about influence of different levels of gravity load under sudden column loss Dynamic analysis would require excessive runs to obtain similar information Robustness Summer School - COST Action TU0601 4 July 2012

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**Failure | Sudden Column Loss Simplified Dynamic Assessment**

Pmax corresponds to (ud=uf) for monotonic static response Not necessarily for softening static response ‘Pseudo-static capacity’ as a rational performance-based measure of structural robustness Emphasis not on dynamic amplification of static loads with conventional design, but on dynamic demand within ductility limit Combines redundancy, ductility and energy absorption within a simplified framework Robustness Summer School - COST Action TU0601 4 July 2012

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**Robustness Summer School - COST Action TU0601**

Sudden Component Loss Pseudo-static approach also applicable to sudden loss of other components Provided dynamic response is dominated by a single mode Pseudo-static response obtained from nonlinear static response of damaged structure, as before Robustness Summer School - COST Action TU0601 4 July 2012

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**Robustness Summer School - COST Action TU0601**

Sudden Component Loss Static analysis unsafe and load amplification based on a factor of 2 grossly conservative Truss subject to sudden brace failure (e.g. due to sudden connection failure) Nonlinear dynamic response under 3 levels of gravity loading Excellent comparison between pseudo-static approach and nonlinear dynamic analysis Accounting for initial deflections in pseudo- static response Nonlinear static response and pseudo-static response Maximum dynamic displacements from pseudo-static response at three load levels Robustness Summer School - COST Action TU0601 4 July 2012

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**Successive Component Losses**

Further component losses could occur during dynamic response, without necessarily defining overall dynamic system resistance Pseudo-static approach can still be applied: Single dominant mode Nonlinear static response of initially damaged structures Reduction in nonlinear static response due to component failure Robustness Summer School - COST Action TU0601 4 July 2012

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**Successive Component Losses**

Maximum load at intersection between pseudo-static and descending static curves Maximum pseudo-static capacity may not even be related to a specific ductility limit … for instance following a compressive arching stage Structural system subject to initial damage followed by second component loss …but not with more severe second component loss …unless system ductility and static resistance picks up Residual pseudo-static capacity after second component loss Static response of initially damaged structure Second component loss Complete system failure Static response of undamaged structure Robustness Summer School - COST Action TU0601 4 July 2012

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**Application to Composite Buildings**

edge beam internal secondary beams transverse primary beam Edge beam connections Grillage approximation: Assuming identical floors assessment at floor level of idealisation Sudden loss of peripheral column 7-storey steel framed composite building with simple frame design Robustness Summer School - COST Action TU0601 4 July 2012

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**Application to Composite Buildings**

Pseudo-static response of individual beams Simplified assembly to obtain pseudo-static capacity of floor slab Importance of connection ductility, additional reinforcement and axial restraint Inadequacy of prescriptive tying force requirements Robustness Summer School - COST Action TU0601 4 July 2012

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**Application to Composite Buildings**

Application to Composite Buildings: Individual Beam Responses Static and pseudo-static curves for edge beam with ρ = 1.12% Pseudo-static response curves of edge beam Pseudo-static response curves of internal beams Pseudo-static response curves of transverse beam Robustness Summer School - COST Action TU0601 4 July 2012

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**Application to Composite Buildings**

Application to Composite Buildings: Assembled Floor Grillage Application to Composite Buildings Assumed deformation mode defines ductility limit Case 2 (r=2% with axial restraint) is just about adequate Inadequacy of prescriptive tying force requirements φj δSB3 δSB1 δSB2 δMB ρmin, EC4, w/ axial restraint ρ = 2%, w/ axial restraint ρ = 2%, w/ο axial restraint Bare-steel frame, w/ axial restraint Robustness Summer School - COST Action TU0601 4 July 2012

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**Application to Composite Buildings**

Application to Composite Buildings: General Observations Application to Composite Buildings Response of composite beams with partial strength connections dominated by compressive arching in the presence of axial restraint Ductility of partial strength connections typically insufficient to mobilise full catenary action Increasing ‘tying force capacity’ is helpful but not necessarily via catenary action, unless rotation capacity exceeds 8º Infill panels can double resistance of composite buildings to progressive collapse Material rate-sensitivity is another potentially significant parameter ~ 4º > 8º Robustness Summer School - COST Action TU0601 4 July 2012

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**Probabilistic Risk Assessment**

Risk = ∑ P(H) P(D|H) P(F|D) C(F) Deterministic evaluation of failure | D = sudden column loss Material related issues (steel and composite structures) Application in probabilistic risk assessment Use of probabilistic risk assessment as a decision support tool for cost-benefit analysis Robustness Summer School - COST Action TU0601 4 July 2012

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**Robustness Summer School - COST Action TU0601**

References Izzuddin, B.A., Vlassis, A.G., Elghazouli, A.Y., and Nethercot, D.A., "Progressive Collapse of Multi-Storey Buildings due to Sudden Column Loss — Part I: Simplified Assessment Framework“, Engineering Structures, Vol. 30, No. 5, May 2008, pp Vlassis, A.G., Izzuddin, B.A., Elghazouli, A.Y., and Nethercot, D.A., "Progressive Collapse of Multi-Storey Buildings due to Sudden Column Loss — Part II: Application“, Engineering Structures, Vol. 30, No. 5, May 2008, pp Vlassis, A.G., Izzuddin, B.A., Elghazouli, A.Y., and Nethercot, D.A., "Progressive Collapse of Multi-Storey Buildings due to Failed Floor Impact“, Engineering Structures, Vol. 31, No. 7, July 2009, pp Gudmundsson, G.V., and Izzuddin, B.A., "The ‘Sudden Column Loss’ Idealisation for Disproportionate Collapse Assessment“, The Structural Engineer, Vol. 88, No. 6, 2010, pp Izzuddin, B.A., "Robustness by Design – Simplified Progressive Collapse Assessment of Building Structures“, Stahlbau, Vol. 79, No. 8, August 2010, pp Robustness Summer School - COST Action TU0601 4 July 2012

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**Structural Systems Analysis for Robustness Assessment**

Bassam A. Izzuddin Department of Civil & Environmental Engineering

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