Structural Robustness and Optimization Seminar

Corso di Dottorato: Introduzione all'o mizzazione strutturale Roma, 21 Giugno 2014
Prof.-Ing. Franco Bontempi, Ing. Konstantinos Gkoumas, Ph.D.
“Structural robustness: definitions, examples and
consequence based assessment of structures”
Konstantinos Gkoumas, Ph.D., P.E.
Corso di Dottorato: introduzione all'ottimizzazione
strutturale
Prof.-Ing. Franco Bontempi
Dipartimento di Ingegneria Strutturale e Geotecnica
Dottorato di Ricerca in Ingegneria delle Strutture
Rome, June 21 2014

Index

Ottimizzazione Strutturale
franco.bontempi@uniroma1.it

Personal

Black
Swan
Vulnerability
Cause
Damage
Index
Robustness
Collapse
resistance
Progressive
collapse
Photo Credit: Wikipedia Commons.
Member
consequence
factor
• Significant collapse cases
• LPHC events and Black Swans
• Structural robustness in qualitative terms
• Structural robustness in civil engineering
design
• Collapse types
• Structural robustness and progressive
collapse definitions
• Measures against progressive collapse
• Quantification of robustness
• Robustness and optimization
• Member consequence factor
• Assessment of simple structures
• Assessment of complex structures
• References

Black
Swan
Vulnerability
Cause
Damage
Index
• Significant collapse cases
• LPHC events and Black Swans
• Structural robustness in qualitative terms
• Structural robustness in civil engineering
design
• Collapse types
• Structural robustness and progressive
collapse definitions
• Measures against progressive collapse
• Quantification of robustness
• Robustness and optimization
• Member consequence factor
• Assessment of simple structures
• Assessment of complex structures
• References
Robustness
Collapse
resistance
Progressive
collapse
Photo Credit: Wikipedia Commons.
Member
consequence
factor

Ronan Point Tower Block– May 16, 1968
Description:
- apartments building;
- built between 1966 and 1968;
- 64 m tall with 22 story;
- walls, floors, and staircases was precast
concrete;
- each floor was supported directly by the walls
in the lower stories, (bearing walls system).
The event:
- May 16, 1968 a gas explosion blew out an
outer panel of the 18th floor,
- the loss of the bearing wall causes the
progressive collapse of the upper floors,
- the impact of the upper floors’ debris caused
the progressive collapse of the lower floors.
Cause Damage Pr. Collapse

Description:
- apartments building;
- precast concrete wall and floor components
was the structural bearing system;
- ductile detailing and effective ties between
the precast components.
The event:
- June 25, 1996 9 tons of
TNTeq detonated in
front of the building;
- the exterior wall was
entirely destroyed;
- collapse did not
progress beyond areas
of first damage.
Khobar Towers Bombing – June 25, 1996

Description:
- office facility for the Deutsche Bank in
Manhattan;
- constructed in the early ‘70s in steel-framed
structure moment connected, 130 m tall, 40
story and 2 subterranean levels;
The event:
- On September 11, 2011, the WTC towers
debris impact on a building’s façade,
- heavy damage between the 9th and the 23rd
floor, the column was lost from the 9th and
the 18th floor;
- the framing system was able to support
and redistribute the loads.
Deutsche Bank Building – September 11, 2001

Probability of progressive collapse from an abnormal event
P(F) = P(D|H) P(F|DH)P(H) x x
damage is caused in
the structure
damage spreads in
the structure
occurrence of
critical event
occurrence of broad
or global collapse
STRUCTURAL INTEGRITY (ISO/FDS 2394)
COLLAPSE RESISTANCE (Starossek&Wolff 2005)
Faber (2006)
STRUCTURALNON STRUCTURAL
MEASURES
HAZARD
References: Ellingwood, B.R. and Dusenberry, D.O. (2005), “Building design for abnormal loads and progressive
collapse”, Comput-Aided Civ. Inf., 20(3), 194-205.

Reference: Bontempi, F. (2005) Frameworks for structural analysis, In: Innovation in Civil and Structural
Engineering Topping, BHV ed., pp. 1-24
HPLC
High Probability –
Low Consequences
LPHC
Low Probability –
High Consequences
Complexity
Non linear issues and
interaction mechanisms
Designapproach:
StochasticDeterministic
QUALITATIVE RISK
ANALYSIS
PROBABILISTIC
RISK ANALYSIS
PRAGMATIC
ANALYSIS OF
RISK SCENARIOS
Secondary
design
Primary
design
Low Probability – High Consequences Events

References: Taleb, Nassim Nicholas (April 2007). The Black Swan: The Impact of the Highly Improbable (1st ed.).
London: Penguin. p. 400. ISBN 1-84614045-5.
A Black Swan is an event with the following three attributes.
1. First, it is an outlier, as it lies outside the realm of regular expectations,
because nothing in the past can convincingly point to its possibility.
Rarity -The event is a surprise (to the observer).
2. Second, it carries an extreme 'impact'.
Extreme “impact” - the event has a major effect.
3. Third, in spite of its outlier status, human nature makes us concoct
explanations for its occurrence after the fact, making it explainable and
predictable.
Retrospective (though not prospective) predictability - After the first
recorded instance of the event, it is rationalized by hindsight, as if it could
have been expected; that is, the relevant data were available but
unaccounted for in risk mitigation programs. The same is true for the
personal perception by individuals.
Black Swans

QUALITY
DAMAGE or ERROR
REQUIRED PERFORMANCE
NOMINAL
PERFORMANCE
NOMINAL SITUATION
Structural Robustness

• Capacity of a construction to exhibit regular
decrease of its structural quality as a consequence
of negative causes.
• It implies:
a) some smoothness of the decrease of
structural performance due to negative
events (intensive feature);
b) some limited spatial spread of the
rupture (extensive feature).

Qualitative definitions of structural robustness
[EN 1991-1-7: 2006 ]: ability of a structure to withstand actions due
to fires, explosions, impacts or consequences
of human errors, without suffering damages
disproportionate to the triggering causes
[SEI 2007,
Beton Kalender 2008]: insensitivity of the structure to local failure
structure B
d
P
s
STRUCTURE B:
P
s
ROBUSTNESS CURVES
P (performance)
structure A
STRUCTURE A
damaged
integer
DP
damaged
more performant, less resistant
integer
(damage level)
DPDP
more performant, less robust less performant, more robust
A B

CommonULS&SLS
VerificationFormat
Assessment
1st level:
Material Point
2nd level:
Element
Section
3rd level:
Structural
Element
4th level:
Structural
System
Structural robustness in design

STRUCTURAL DESIGN
PRIMARY SECONDARY TERTIARY
LOADS
DEAD X
LIVE X
SNOW X
EARTHQUAKE X
FIRE X X
EXPLOSIONS X X
“BLACK SWAN” X
Member-based
structural design
Consequence-based
structural design
Black Swan event:
- unpredictable,
- large impact on community,
- easy to predict after its occurrence.
References:
Nafday, AM. (2011) Consequence-based structural
design approach for black swan events. Structural
Safety, 33(1): 108-114.

Uncertainty in the likelihood that
the harmful consequences of a
particular event will be realized
Uncertainty in the consequences
related to the specific event
Primary
design
Secondary
design
Tertiary
design

STRUCTURE
& LOADS
Collapse
Mechanism
NO SWAY
“IMPLOSION”
OF THE
STRUCTURE
“EXPLOSION”
OF THE
STRUCTURE
is a process in which
objects are destroyed by
collapsing on themselves
is a process
NOT CONFINED
SWAY
Bad VS Good collapse

Initial load-bearing element failure that
triggers the rigid fall of a part of the
structure onto another and leads to a
sequential impacts on the rest of the
structure, that collapses on itself.
Characteristic feature is the force
redistribution into alternative paths,
impulsive loading due to sudden element
failure and force concentration in elements
to fail next.
Zipper Domino
Section Instability Mixed
Pancake
Initial cross-section cut and stress
concentration that cause the rupture of
further cross-sectional parts (fast fracture)
and failure progression throughout the
entire section.
Initial element rigid overturning and
falling over another element, that, by
means of transformation of potential into
kinetic energy, trigger the overturning of
the following element.
The destabilization of some load-carrying
elements in compression due to an initial
failure of stabilizing elements can trigger a
failure progression throughout the
structure.
Some collapses are less amenable to
generalization because the relative
importance of the contributing basic
categories of collapse can vary and
combine in progression of failures.
- DOMINO + PANCAKE
(e.g. A.P.Murrah Building, Building
during Izmit Earquake)
- ZIPPER + INSTABILITY
(e.g. cable-stayed bridges)
Reference: Betoncalendar, 2008 (adapted from “Structural integrity: robustness assessment and progressive collapse
susceptibility”, Luisa Giuliani, PhD Thesis, Sapienza University of Rome, Dipartimento di Ingegneria Strutturale e Geotecnica)
Collapse types

Initial load-bearing element failure that
triggers the rigid fall of a part of the
structure onto another and leads to a
sequential impacts on the rest of the
structure, that collapses on itself.
Characteristic feature is the force
redistribution into alternative paths,
impulsive loading due to sudden element
failure and force concentration in elements
to fail next.
Zipper Domino
Section Instability Mixed
Pancake
Initial cross-section cut and stress
concentration that cause the rupture of
further cross-sectional parts (fast fracture)
and failure progression throughout the
entire section.
Initial element rigid overturning and
falling over another element, that, by
means of transformation of potential into
kinetic energy, trigger the overturning of
the following element.
The destabilization of some load-carrying
elements in compression due to an initial
failure of stabilizing elements can trigger a
failure progression throughout the
structure.
Some collapses are less amenable to
generalization because the relative
importance of the contributing basic
categories of collapse can vary and
combine in progression of failures.
- DOMINO + PANCAKE
(e.g. A.P.Murrah Building, Building
during Izmit Earquake)
- ZIPPER + INSTABILITY
(e.g. cable-stayed bridges)
Reference: Betoncalendar, 2008 (adapted from “Structural integrity: robustness assessment and progressive collapse
susceptibility”, Luisa Giuliani, PhD Thesis, Sapienza University of Rome, Dipartimento di Ingegneria Strutturale e Geotecnica)
Collapse types
Islamabad Earthquake 2005
Münsterland, 2005
Viaduct after earthquake
Izmit Earthquake
1999
Tanker S.S. Schenectady, 1941

The Boeing B-17 Flying Fortress collided with another aircraft during World War II and, although
sustaining large amount of structural damage, landed safely, due to the high redundancy of the
fuselage connections.
Design Strategy #1: Continuity (robust behavior-redundancy)

On July 1945 a B-25 bomber crashed into the Empire State Building, The impact of the plane
created a 5.5x6 m hole in the side of the tower. This crash caused extensive damage to the
masonry exterior and the interior steel structure of the building.
The 278 m building was rocked by the impact but resist the impact in consequence of the
intrinsic redundancy of its framed system.
Plane crash on the Empire
State Building, 1945
Design Strategy #1: Continuity (robust behavior-redundancy)

Design Strategy #2: Segmentation (Compartmentalization)
A service-induced damage led to explosive decompression and loss of large portion of fuselage
skin when small fatigue crack suddenly linked together. The subsequent fracture was eventually
arrested by fuselage frame structure and the craft landed safely.
Aloha Boeing 737, April 1988
(compartmentalization by strengthening)

Design Strategy #2: Segmentation (Compartmentalization)
The partial collapse, started in the roof and due design and execution errors, stoped at the two joints
which separated the collapsing section from the adjacent structures.
A higher continuity could have unlikely sustained the forces during collapse, since the construction
deficiencies affected also adjacent sections.

References:
(EN 1991-1-7 2006): "Eurocode 1 – Actions on structures, Part 1-7: General actions – accidental actions." Comité
European de Normalization (CEN).
(Bontempi F, Giuliani L, Gkoumas K, 2007): "Handling the exceptions: robustness assessment of a complex structural
system.“, Invited Lecture, Structural Engineering, Mechanics and Computation (SEMC) 3, 1747-1752.
(Starossek U, 2009): “Progressive collapse of structures.” London: Thomas Telford Publishing, 2009.
Definitions:
1- "The ability of a structure to withstand events like fire, explosions,
impact or the consequences of human error without being damaged to an
extent disproportionate to the original cause." (EN 1991-1-7 2006)
2- "The robustness of a structure, intended as its ability not to suffer
disproportionate damages as a result of limited initial failure, is an
intrinsic requirement, inherent to the structural system organization."
(Bontempi F, Giuliani L, Gkoumas K, 2007)
3- “Robustness is defined as insensitivity to local failure." (Starossek U,
2009)

References:
(ASCE 7-05 2005): "Minimum design loads for buildings and other structures." American Society of Civil Engineers
(ASCE).
(GSA 2003): "Progressive collapse analysis and design guidelines for new federal office buildings and major
modernization projects." General Services Administration (GSA).
(UFC 4-010-01 2003): "DoD minimum antiterrorism standards for buildings." Department of Defense (DoD).
Progressive Collapse
Definitions:
1-"Progressive collapse is defined as the spread of an initial local failure
from element to element resulting, eventually, in the collapse of an entire
structure or a disproportionate large part of it." (ASCE 7-05 2005)
2- "A progressive collapse is a situation where local failure of a primary
structural component leads to the collapse of adjoining members which, in
turn, leads to additional collapse. Hence, the total collapse is
disproportionate to the original cause." (GSA 2003)
3-"Progressive collapse: a chain reaction failure of building members to an
extent disproportionate to the original localized damage." (UFC 4-010-01
2003)

References:
Arup (2011), Review of international research on structural robustness and disproportionate collapse, London,
Department for Communities and Local Government.
Starossek, U. and Haberland, M. (2010), “Disproportionate Collapse: Terminology and Procedures”, J. Perf. Constr.
Fac., 24(6), 519-528.
Observations:
− A progressive collapse is one which develops in a progressive manner akin to the collapse
of a row of dominos.
− A disproportionate collapse is one which is judged (by some measure defined by the
observer) to be disproportionate to the initial cause. This is merely a judgement made on
observations of the consequences of the damage which results from the initiating events.
− A collapse may be progressive in nature but not necessarily disproportionate in its extents,
for example if arrested after it progresses through a number of structural bays. Vice versa, a
collapse may be disproportionate but not necessarily progressive if, for example, the
collapse is limited in its extents to a single structural bay but the structural bays are large.
− The terms of disproportionate collapse and progressive collapse are often used
interchangeably because disproportionate collapse often occurs in a progressive manner
and progressive collapse can be disproportionate.
Progressive Collapse VS Disproportionate Collapse

Robustness and collapse resistance in a dependability framework
Sgambi, L., Gkoumas, K. and Bontempi, F. (2012), “Genetic
algorithms for the dependability assurance in the design of a long-
span suspension bridge”, Comput-Aided Civ. Inf., 27(9), 655-675.

The currently available design strategies and methods to
prevent disproportionate collapse are as follows:
− Prevent local failure of key elements (direct design)
− Specific local resistance
− Non-structural protective measures
− Presume local failure (direct design)
− Alternative load paths
− Isolation by segmentation
− Prescriptive design rules (indirect design)
Reference:
Starossek, U. 2008. Collapse resistance and robustness of bridges. IABMAS’08: 4th International Conference on
Bridge Maintenance, Safety, and Management Seoul, Korea, July 13-17, 2008
Measures against disproportionate collapse

Reference:
Giuliani, L., 2012. Structural safety in case of extreme actions. International Journal of Lifecycle Performance Engineering
IJLCPE Special Issue on: "Performance and Robustness of Complex Structural Systems", Guest Editor Franco Bontempi, ISSN
(Online): 2043-8656 - ISSN (Print): 2043-8648.
Design strategies against progressive collapse

RISK-BASED
[Faber, 2005]
R
I
inddir
dir
rob
R
R
+
=
direct risk
indirect riskDAMAGE-BASED
∑
=
=
n
1i
'
i
i
)K(tr
)K(tr
.Deg.Stiff
ithelement stiffness matrix
(integer state)
damaged
elements
ithelement stiffness
matrix (damaged state)
[Yan&Chang, 2006] [Biondini &
Frangopol, 2008]
1
0
∆Φ
∆Φ
ρ =Φ
∆ energy between intact
and damaged system
(backward pseudo-loads)
∆ energy between intact
and damaged system
(forward pseudo-loads)
Indirect
Risk
Direct
Risk
Indirect
Risk
Direct
Risk
Reference:
Olmati, P., Brando, F., Gkoumas, K. “Robustness assessment of a Steel Truss Bridge”, ASCE/SEI Structures Congress,
Pittsburgh, Pennsylvania, May 2-4, 2013.
B
A Withstand actions, events
Withstand damages
Structural Robustness assessment
TOPOLOGY-BASED ENERGY-BASEDOther:

[Baker et al. 2008]
R
I
inddir
dir
rob
R
R
+
=
direct risk
indirect risk
Reference:
Baker J.W., Schubert M., Faber M.H., (2008). On the Assessment of Robustness, Journal of Structural Safety, Volume
30, Issue 3, pp. 253-267, DOI:10.1016/j.strusafe.2006.11.004
“A robust system is considered to be one where indirect
risks do not contribute significantly to the total system
risk”
Rdir˃˃Rind
Rdir: associati con il danni iniziali
Rind: associati con danni successivi
EXBD: Exposure before damage
D : Damage
D : No Damage
F : Probability of system failure
Cdir : Direct consequences
Cind: Indirect consequences
Risk Based Structural Robustness assessment

− ≥ 0
member-based design
− ≥ 0limit state design
Resistance (probabilistic) Solicitation (probabilistic)
Resistance (design values) Solicitation (design values)
(1 − ) − ≥ 0
Member consequence factor based design
0 ≤ ≤ 1
• Cf quantifies the influence that a loss of a structural element has on the load carrying capacity.
• Cf provides to the single structural member an additional load carrying capacity, in function of the
nominal design (not extreme) loads that can be used for contrasting unexpected and extreme loads.
• Essentially, if Cf tends to 1, the member is more likely to be important to the structural system;
instead if Cf tends to 0, the member is more likely to be unimportant to the structural system.
Member consequence factor and robustness assessment
0EγγRγγ kEMEk
1
Rd
1
MR ≥− ∑−−
0E)R(*)C1( kEdMEk
1
Rd
1
MRf ≥γγ−γγ− ∑−−

• The structure is subjected to a set of damage scenarios and the consequence of the
damages is evaluated by the member consequence factor ( ) that for
convenience can be easily expressed in percentage.
• For damage scenario is intended the failure of one or more structural elements.
• Robustness can be expressed as the complement to 100 of , intended as the
effective coefficient that affects directly the resistance.
• is evaluated by the maximum percentage difference of the structural stiffness
matrix eigenvalues of the damaged and undamaged configurations of the structure.
=
−
100
!"#
where, and are respectively the i-th eigenvalue of the structural stiffness
matrix in the undamaged and damaged configuration, and N is the total number of the
eigenvalues.

• The corresponding robustness index ( ) is therefore defined as:
=1 -
• Values of Cf close to 100% mean that the failure of the structural member most
likely causes a global structural collapse.
• Low values of Cf do not necessarily mean that the structure survives after the failure
of the structural member: this is something that must be established by additional
analysis that considers the loss of the specific structural member.
• A value of Cf close to 0% means that the structure has a good structural
robustness.
The proposed method for computing the consequence factors, for different reasons,
should not be used for:
1. Structures that have high concentrated masses (especially non-structural masses) in
a particular zone; and,
2. Structures that have cable structural system (e.g., tensile structures, suspension
bridges).

Cost of robustness measures ≤ Reduction of failure consequences
• The objective function for optimization may be very complex
and depend on the type of the structural system, robustness
measures, characteristics of failure consequences and
probabilities of occurrence and intensities of various hazards.
• If the total cost of robustness measures exceeds the reduction
in failure consequences, then the system may be considered as
robust but uneconomic. In such a situation, probabilistic
methods of risk assessment may be effectively used
Reference:
COST Action TU0601 Robustness of Structures STRUCTURAL ROBUSTNESS DESIGN FOR PRACTISING
ENGINEERS. EUROPEAN COOPERATION IN SCIENCE AND TECHNOLOGY, Editor T. D. Gerard Canisius.
Robustness in Optimization

Reference:
Casciati, S. and Faravelli, L. (2008) Building a Robustness Index. Robustness of Structures COST Action TU0601,
1st Workshop, February 4-5, ETH Zurich, Switzerland.
Robustness in Optimization
Example: Hierarchy of the failure modes (“weak beam/strong column”)
...develop the less catastrophic failure
modes first.
...ranking the failure modes in terms of
a hierarchy in such a way that the less
harmful ones are generated at lower
loading levels
Objective function:
Robustness term:
Pfi: probability of the i-th failure mode
m: number of failure modes
A robust structure requires the plastic moment of the column, MPc, being larger than the
one of the beam, MPb; that is, Z = MPc– MPb≥ 0
µc, σc, µb, σb: means and the standard deviations of the plastic moments of the columns and
of the beam, respectively.
To ensure robustness, the index I needs to be kept positive. The objective is, therefore, to
minimize FI=-I.

Stiffness matrix
Kun λi
un
Eigenvalues
Kdam λi
dam
Consequence factor
Robustness indexRscenario= 100 - Cf
scenario
N1i
un
i
dam
i
un
iscenario
f 100
)(
maxC
−=






λ
λ−λ
=
Structural Robustness assessment - overview

ka
kb
x
y
K%& =
10 0
0 10
C(!
1 = 0% C(*
1 = 30%
R1 = 70%
R1 = 100 − C(
1N: total eigenvalues number
i: single eigenvalue number
a and b: elements
a
b
N1i
un
i
dam
i
un
iscenario
f 100
)(
maxC
−=






λ
λ−λ
=
K./0
=
10 0
0 7
Scenario 1
Single damage – analytic calculation

• Single bay frame structure with a diagonal beam brace, composed in total of 5
members
• IPE 300, S235 steel, one meter length, pinned boundary conditions.
The evaluated scenarios consist in the removal of elements 1, 2 and 3 sequentially, so the
damage is cumulative: this means that the each scenario includes the damage from the
previous one.
Cumulative damage – numerical assessment
DSj = Σi=(1-j) di

Cumulative damage – numerical assessment
• star-shaped structure – 8 members - pipe cross section - 20 centimeters outside
diameter - 20 millimeters thickness - S235 steel.
• members 1, 3, 5, and 7 are 0.5 meters long and members 2, 4, 6, and 8 are 0.7
meters long.
All the members are connected to each other by a fixed type connection. Also the boundary
conditions are of the fixed type and the structure is plane.
DSj = Σi=(1-j) di

• Built 1967
• 3 spans, 1067 feet long
• 1977 – new wearing surface
• 1998 – curbs and railings replaced
I-35 West Bridge, Minneapolis, MN

North
Downtown
I-35 West Bridge, Minneapolis, MNPhotofromaircraftflyingoverhead.
• At 6:05 pm on August 1st 2007 Bridge Collapsed
• 13 People killed & approximately 145 Injured

I-35 W bridge
I-35 West Bridge, Minneapolis, MN
NTSB 2007

Undamaged
Damaged
scenario
I-35 West Bridge, Minneapolis, MN – damage scenarios

I-35 West Bridge, Minneapolis, MN – damage scenarios
3D
2D

d1
d2d3
d4
d5
d7
d6
37
59
42 45
35 38
23
63
41
58 55
65 62
77
0
20
40
60
80
100
1 2 3 4 5 6 7
Robustness%
Scenario
Cf max Robustness
1 2 3 4 5 6 7
Scenario
Cf max Robustness
Damage scenario
d1 d2 d3 d4 d5 d6 d7
DSj = di
I-35 West Bridge, Minneapolis, MN – single damage

d1
d2d3
d4
d5
d7
d6
83 87 88
53
60
86
64
17 13 12
47
40
14
36
0
20
40
60
80
100
1 2 3 4 5 6 7
Robustness%
Scenario
Cf max Robustness
Damage scenario
d1 d2 d3 d4 d5 d6 d7
I-35 West Bridge, Minneapolis, MN/ enhanced– single damage
DSj = di

References
Alashker, Y., Li, H. and El-Tawil, S. (2011), “Approximations in Progressive Collapse Modeling”, J. Struct. Eng.- ASCE, 137(9), 914-924.
Arup (2011), Review of international research on structural robustness and disproportionate collapse, London: Department for
Communities and Local Government.
ASCE 7-05 (2005), Minimum design loads for buildings and other structures, American Society of Civil Engineers (ASCE).
Biondini, F. and Frangopol, D. (2009), “Lifetime reliability-based optimization of reinforced concrete cross-sections under corrosion”,
Struct. Saf., 31(6), 483-489.
Biondini, F., Frangopol, D.M. and Restelli, S. (2008), “On structural robustness, redundancy and static indeterminancy”, Proceedings of
the 2008 Structures Congress, April 24-26, 2008, Vancouver, BC, Canada.
Bontempi, F. and Giuliani, L. (2008), “Nonlinear dynamic analysis for the structural robustness assessment of a complex structural
system”, Proceedings of the 2008 Structures Congress, April 24-26, 2008, Vancouver, BC, Canada.
Bontempi, F., Giuliani, L. and Gkoumas, K. (2007), “Handling the exceptions: dependability of systems and structural robustness”, Invited
Lecture, Proceedings of the 3rd International Conference on Structural Engineering, Mechanics and Computation (SEMC), Cape Town,
South Africa, September 10-12.
Brando, F., Testa, R.B. and Bontempi, F. (2010), “Multilevel structural analysis for robustness assessment of a steel truss bridge”, Bridge
Maintenance, Safety, Management and Life-Cycle Optimization - Frangopol, Sause and Kusko (eds), Taylor & Francis Group, London,
ISBN 978-0-415-87786-2.
Canisius, T.D.G., Sorensen, J.D. and Baker, J.W. (2007), “Robustness of structural systems - A new focus for the Joint Committee on
Structural Safety (JCSS)”, Proceedings of the 10th Int. Conf. on Applications of Statistics and Probability in Civil Engineering
(ICASP10), Taylor and Francis, London.
Casciati, S. and Faravelli, L. (2008) Building a Robustness Index. Robustness of Structures COST Action TU0601, 1st Workshop,
February 4-5, 2008, ETH Zurich, Zurich, Switzerland.
Cha, E. J. and Ellingwood, B. R. (2012), “Risk-averse decision-making for civil infrastructure exposed to low-probability, high-
consequence events”, Reliab. Eng. Syst. Safe., 104(1), 27-35.
Choi, J-h. and Chang, D-k. (2009), “Prevention of progressive collapse for building structures to member disappearance by accidental
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Acknowledgements
- Luisa Giuliani, PhD. Associate Professor, DTU, Denmark.
- Francesca Brando, PhD. Senior Engineer, Thornton Tomasetti, NY.
- Pierluigi Olmati, PhD. Post-doc, Surrey University, UK.

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