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Imperial CollegeLondon
Structural Systems Analysisfor Robustness Assessment
Bassam A. Izzuddin
Department of Civil & Environmental Engineering
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)
Disproportionate: No
Murrah Building (1995)
Disproportionate: ?
Ronan Point (1968)
Disproportionate: Yes
Setúbal, Portugal (2007)
Robust structure
4 July 2012 Robustness Summer School - COST Action TU0601 2
Structure
Structural Design – Predictability3
ActionsResponse
Codified properties
Statistical data
Site supervision & QA
…
Codified calculations
Simplified analysis
Detailed analysis
…
Codified loads
Statistical analysis
Event modelling
…
Malicious/terrorist
actions
Acceptable?
4 July 2012 Robustness Summer School - COST Action TU0601 3
• 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
Codified Design for Robustness
Invoked if notional member removal leads to excessive damage (15% of floor area or 70m2)No clear guidance
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
More performance based
However, conventional design checks (ignoring large deformations)
Ignores dynamic effects
Unrealistic designs and damage assessment
4 July 2012 Robustness Summer School - COST Action TU0601 4
Probabilistic Risk Assessment
Risk = ∑ P(H) P(D|H) P(F|D) C(F)
Hazard Local damage System failure
Vulnerability Damage tolerance Consequences
Structural robustness
Involve similar types of structural analysis
May be considered together depending on event resolution
1. Deterministic evaluation of failure | D = sudden column loss
2. Material related issues (steel and composite structures)
3. Application in probabilistic risk assessment
4 July 2012 Robustness Summer School - COST Action TU0601 5
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
4 July 2012 Robustness Summer School - COST Action TU0601 6
• 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
Sudden Column Loss (D)
Too complicated for practical application in design
Excessive dynamic amplification factor equal to 2
Conventional design checks
4 July 2012 Robustness Summer School - COST Action TU0601 7
• 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
• 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 (F)
4 July 2012 Robustness Summer School - COST Action TU0601 8
• 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
Load Factor Approaches
Dynamic response Static analysis4 July 2012 Robustness Summer School - COST Action TU0601 9
• DIF in terms of ductility for use with nonlinear static analysis in new GSA/DoD guidance
• Monotonic reduction in DIF with ductility
Load Factor Approaches
0.12
0.45DIF 1.04 (Marchand et al. [2008]: Concrete Structures)
m 0.480.76
DIF 1.08 (Marchand et al. [2008]: Steel Structures)m 0.83
DIF 1.44m (Stevens et al. [2008]: Steel Structures
f
y
)
uplastic deformationm 1
yield deformation u
4 July 2012 Robustness Summer School - COST Action TU0601 10
Load Factor Approaches
Elastic-plastic static response
Dynamic resistance
Static resistance
DIF = 2
DIF = 1.67 DIF = 1.04
Monotonic reduction of DIF with ductilityConsistent with load factor approaches for new GSA/DoD guidance
4 July 2012 Robustness Summer School - COST Action TU0601 11
Load Factor Approaches
Elasto-plastic static response with hardeningDIF increases with ductility after initial reductionProposed load factor approaches very unsafe for ductile structures
4 July 2012 Robustness Summer School - COST Action TU0601 12
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
4 July 2012 Robustness Summer School - COST Action TU0601 13
Failure | Sudden Column Loss
• Maximum gravity load sustained under sudden column loss
• Applicable at various levels of structural idealisation
• Reduced model where deformation is concentrated
• Columns can take re-distributed load
• Floors identical in components and loading
• Planar effects are neglected
4 July 2012 Robustness Summer School - COST Action TU0601 14
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
4 July 2012 Robustness Summer School - COST Action TU0601 15
Nonlinear Static ResponseFailure | Sudden Column Loss
• Sudden column removal similar to sudden application of gravity load– Maximum dynamic response can be approximated using
amplified static loading (ld P)
4 July 2012 Robustness Summer School - COST Action TU0601 16
4 July 2012 Robustness Summer School - COST Action TU0601 17
Failure | Sudden Column LossNonlinear 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
4 July 2012 Robustness Summer School - COST Action TU0601 18
Failure | Sudden Column LossNonlinear 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
4 July 2012 Robustness Summer School - COST Action TU0601 19
Failure | Sudden Column LossNonlinear 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
4 July 2012 Robustness Summer School - COST Action TU0601 20
Failure | Sudden Column LossNonlinear 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
i i ii
i i s,i s
P
P , P P (u ) P P(u )
4 July 2012 Robustness Summer School - COST Action TU0601 21
Failure | Sudden Column LossNonlinear 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
i ii
i i s,i s
P
P , P P (u ) P P(u )
4 July 2012 Robustness Summer School - COST Action TU0601 22
Failure | Sudden Column LossNonlinear 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
• 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
Simplified Dynamic Assessment
ld<<2l
Failure | Sudden Column Loss
4 July 2012 Robustness Summer School - COST Action TU0601 23
Failure | Sudden Column LossSimplified 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
4 July 2012 Robustness Summer School - COST Action TU0601 24
• Pmax corresponds to (ud=uf) for monotonic 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
• Not necessarily for softening static response
Failure | Sudden Column LossSimplified Dynamic Assessment
4 July 2012 Robustness Summer School - COST Action TU0601 25
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
4 July 2012 Robustness Summer School - COST Action TU0601 26
• Nonlinear dynamic response under 3 levels of gravity loading
• Nonlinear static response and pseudo-static response
• Maximum dynamic displacements from pseudo-static response at three load levels
• Accounting for initial deflections in pseudo-static response
• Excellent comparison between pseudo-static approach and nonlinear dynamic analysis
• 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)
Sudden Component Loss
4 July 2012 Robustness Summer School - COST Action TU0601 27
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
4 July 2012 Robustness Summer School - COST Action TU0601 28
• … for instance following a compressive arching stage
• Maximum load at intersection between pseudo-static and descending static curves
• Residual pseudo-static capacity after second component loss
• …but not with more severe second component loss
• …unless system ductility and static resistance picks up
Static responseof undamagedstructure
Successive Component Losses
• Structural system subject to initial damage followed by second component loss
Static response ofinitially damagedstructure
Secondcomponentloss
Completesystemfailure
• Maximum pseudo-static capacity may not even be related to a specific ductility limit
4 July 2012 Robustness Summer School - COST Action TU0601 29
Application to Composite Buildings
304 July 2012 Robustness Summer School - COST Action TU0601
7-storey steel framed composite building with simple frame design
6000 6000
2375
3000
3000
1500
Removed column
504070
70
7040
Supporting column305 x 305 x 118 UCGrade S355
Supported beam406 x 140 x 39 UB
Grade S355
Supported beam406 x 140 x 39 UB
Grade S355
60
7080
50
Anti-crack mesh Shear stud
130
Sudden loss of peripheral columnAssuming identical floors assessment at floor level of idealisationGrillage approximation:edge beaminternal secondary beamstransverse primary beamEdge beam connections
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
314 July 2012 Robustness Summer School - COST Action TU0601
Application to Composite Buildings
324 July 2012 Robustness Summer School - COST Action TU0601
0
2
4
6
8
10
12
14
16
0 100 200 300 400 500 600 700 800 900
Displacement, δ (mm)
Dyn
amic
Loa
d (kN
/m)
0%
25%
50%
75%
100%
Per
cent
age
of S
ervi
ce L
oads
(%
)
ρ = 0.87%, w/ axial restraintρ = 2.00%, w/ axial restraintBare-steel frame, w/ axial restraintρ = 0.87%, w/o axial restraintρ = 2.00%, w/o axial restraintBare-steel frame, w/o axial restraint
• Pseudo-static response curves of internal beams
• Pseudo-static response curves of transverse beam
0
100
200
300
400
500
0.000 0.030 0.060 0.090 0.120 0.150
Rotation, φ (rad)
Dyn
amic
Mom
ent (kN
m)
ρ = 0.45%ρ = 0.87%ρ = 2.00%Bare-steel frame
• Pseudo-static response curves of edge beam
0
5
10
15
20
0 100 200 300 400 500 600 700 800 900
Displacement, δ (mm)
Dyn
amic
Loa
d (kN
/m)
0%
25%
50%
75%
100%
125%
150%
Per
cent
age
of S
ervi
ce L
oads
(%
)
ρ = 0.87%, w/ axial restraint
ρ = 2.00%, w/ axial restraint
Bare-steel frame, w/ axial restraint
ρ = 0.87%, w/o axial restraint
ρ = 2.00%, w/o axial restraint
Bare-steel frame, w/o axial restraint
• Static and pseudo-static curves for edge beam with ρ = 1.12%
0
5
10
15
20
0 100 200 300 400Displacement, δ (mm)
Stat
ic/D
ynam
ic L
oad
(kN/m
)
0%
25%
50%
75%
100%
125%
150%
Per
cent
age
of S
ervi
ce L
oads
(%)
Static resposne, w/ axial restraint
Static response, w/o axial restraint
Pseudo-static response, w/ axial restraint
Pseudo-static response, w/o axial restraint
Gap closure
Application to Composite Buildings:Individual Beam Responses
Application to Composite Buildings
334 July 2012 Robustness Summer School - COST Action TU0601
Application to Composite Buildings:Assembled Floor Grillage
Deformation profile Case No.
φd,TB (rad) ud,IB1 (mm) ud,IB2 (mm) ud,IB3 (mm) ud,EB (mm)
1 0.0364 54.6 163.7 272.9 359.3
2 0.0381 57.2 171.6 286.0 376.5
3 0.0359 53.8 161.3 268.9 354.0
4 0.0623 93.5 280.5 467.6 615.6
δSB3
δSB1
δSB2
δMB
φj
ρmin, EC4, w/ axial restraintρ = 2%, w/ axial restraintρ = 2%, w/ο axial restraintBare-steel frame,w/ axial restraint
Case No. Capacity P
(N)
Demand Po
(N)
Capacity/Demand
ratio
1 598729 741990 0.81
2 774358 741990 1.04
3 709675 741990 0.96
4 148530 741990 0.20
• Assumed deformation mode defines ductility limit
• Case 2 (r=2% with axial restraint) is just about adequate
• Inadequacy of prescriptive tying force requirements
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
344 July 2012 Robustness Summer School - COST Action TU0601
~ 4º
> 8º
Application to Composite Buildings:General Observations
Probabilistic Risk Assessment
Risk = ∑ P(H) P(D|H) P(F|D) C(F)
1. Deterministic evaluation of failure | D = sudden column loss
2. Material related issues (steel and composite structures)
3. Application in probabilistic risk assessment
4 July 2012 Robustness Summer School - COST Action TU0601 35
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. 1308-1318.
• 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. 1424-1438.
• 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. 1522-1534.
• 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. 22-26.
• Izzuddin, B.A., "Robustness by Design – Simplified Progressive Collapse Assessment of Building Structures“, Stahlbau, Vol. 79, No. 8, August 2010, pp. 556-564.
4 July 2012 Robustness Summer School - COST Action TU0601 36
Imperial CollegeLondon
Structural Systems Analysisfor Robustness Assessment
Bassam A. Izzuddin
Department of Civil & Environmental Engineering