Sapienza Università di Roma

Transcription

Facoltà di Ingegneria
Dipartimento di Ingegneria Strutturale e Geotecnica
Dottorato di ricerca – XXVI Ciclo
Blast resistance assessment
of structures: explicit finite
element simulations and
fragility analyses
Candidate: Pierluigi Olmati
Advisor: Prof. Franco Bontempi
Co-Advisor: Prof. Clay J. Naito
Dissertazione presentata per il conseguimento del titolo di
Dottore di ricerca in Ingegneria delle strutture
THIS PAGE INTENTIONALLY BLANK
Blast resistance assessment of structures: explicit finite element simulations and fragility analyses
Acknowledgments
This Ph.D. Thesis is the result of the studies conducted during my Ph.D. program in
Structural Engineering at the Sapienza University of Rome. First I would like to thank
my advisor Professor Franco Bontempi. He constantly indicated to me the appropriate
approach in my study and the way to improve both my personal and my scientific
qualities.
Special thank goes to my co-advisor Professor Clay J. Naito of the Lehigh University
(Bethlehem, Pennsylvania, United State of America) where I had the opportunity to spend
six months as visiting scholar. Nowadays our collaboration is still ongoing on several
topics of study.
I would also like to acknowledge the help of Professors Charis Gantes and Dimitrios
Vamvatsikos of the National Technical University of Athens where I spent three months
as a visiting scholar. Their hospitality and their scientific contribution in parts of my
Ph.D. Thesis are much appreciated.
A would like to kindly acknowledge Dr. Francesco Petrini and Dr. Konstantinos
Gkoumas, without their advices I would not able to succeed in my studies. A thank is
also due to the rest of the research group of the Professor Bontempi and to Patrick
Trasborg of the Lehigh University.
A thank is for the Professor Giuseppe Rega, who coordinates the Ph.D. program.
I would like to express my gratitude is especially to my family that sustained me in my
decision to continue with my studies.
Pierluigi Olmati
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SUMMARY OF THE THESIS
1
Introduction ................................................................................................................ 13
2
The hazard mitigation ................................................................................................ 19
3
2.1
High detonations ................................................................................................. 23
2.2
Gas explosions..................................................................................................... 29
2.3
Computational Fluid Dynamic simulations ........................................................ 35
2.4
Blast load ............................................................................................................. 41
The local resistance .................................................................................................... 55
3.1
3.1.1
Blast load model .......................................................................................... 58
3.1.2
Cladding panel model .................................................................................. 60
3.1.3
Response parameters .................................................................................... 64
3.1.4
Fragility curves ............................................................................................ 65
3.2
4
The impulse density as intensity measure ........................................................... 73
3.2.1
Relation between the pressure-impulse diagram and the fragility surface .. 74
3.2.2
The fragility curve for impulse sensitive structures ..................................... 77
3.2.3
Application on a steel blast door .................................................................. 78
3.3
Slabs subjected to impulsive loads - The Blast Blind Simulation Contest ......... 99
3.4
Insulated panels under close-in detonations ...................................................... 109
3.4.1
Experimental Program ............................................................................... 111
3.4.2
Empirical assessment ................................................................................. 112
3.4.3
Experimental results................................................................................... 115
3.4.4
Numerical model ........................................................................................ 117
3.4.5
Summary .................................................................................................... 124
The global resistance................................................................................................ 127
4.1
The consequence factor ..................................................................................... 133
4.1.1
Tests on simple structures .......................................................................... 135
4.1.2
Application on a steel truss bridge ............................................................. 138
4.2
5
The scaled distance as intensity measure ............................................................ 57
The robustness curves ....................................................................................... 145
Conclusions .............................................................................................................. 154
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6
References ................................................................................................................ 158
7
Appendix A – Journal papers obtained from the Ph.D. Thesis ................................ 177
8
7.1
Published and accepted papers .......................................................................... 177
7.2
Submitted and ongoing papers .......................................................................... 177
Appendix B – Curriculum vitae ............................................................................... 179
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FIGURE INDEX
FIGURE 1-1: MASLOW’S HIERARCHY OF NEEDS [MASLOW 1943] ................................................................................ 13
FIGURE 1-2: COLLAPSE RESISTANCE DECOMPOSITION ................................................................................................ 15
FIGURE 2-1: CLASSIFICATION OF THE EXPLOSIVES [US ARMY 1992] ............................................................................ 21
FIGURE 2-2: BLAST LOAD HUMAN TOLERANCES ........................................................................................................ 22
FIGURE 2-3: CHARACTERISTICS OF EXPLOSIVES FROM [US ARMY 1992] ....................................................................... 24
FIGURE 2-4: CONTROL VOLUME AND SHOCK WAVE [US ARMY 1984] .......................................................................... 25
FIGURE 2-5: P-VS CHART [US ARMY 1984] ............................................................................................................ 28
FIGURE 2-6: DETONATION WAVE MOVING THROUGH EXPLOSIVE MATERIAL [US ARMY 1984] .......................................... 28
FIGURE 2-7: HEMISPHERICAL FUEL-AIR CHARGE BLAST FOR THE MULTI-ENERGY .............................................................. 31
FIGURE 2-8: MAIN FEATURES OF THE BLAST SCENARIO............................................................................................... 38
FIGURE 2-9: CONGESTED ROOM MODEL ................................................................................................................. 38
FIGURE 2-10: POSITION OF THE IGNITION POINTS ..................................................................................................... 38
FIGURE 2-11: MAX PRESSURES. EFFECT OF THE DOMESTIC CONGESTION; SIMULATION I ON THE LEFT AND SIMULATION II ON THE
RIGHT ..................................................................................................................................................... 39
FIGURE 2-12: MAX PRESSURES. EFFECT OF THE FRANGIBLE WALLS: SIMULATION III......................................................... 40
FIGURE 2-13: PRESSURE TIME HISTORY IN THE KITCHEN (THE GAS REGION) FOR THE THREE DIFFERENT SIMULATIONS: I, II, AND III
............................................................................................................................................................. 40
FIGURE 2-14: PRESSURE TIME HISTORY INSIDE THE KITCHEN FOR THREE ANALYSES WITH DIFFERENT IGNITION LOCATIONS ....... 40
FIGURE 2-15: FREE FIELD PRESSURE TIME HISTORY.................................................................................................... 43
FIGURE 2-16: REFLECTED PRESSURE TIME HISTORY ................................................................................................... 43
FIGURE 2-17: BLAST LOADS FOR FREE AIR BURST EXPLOSIONS, POSITIVE PHASE............................................................... 45
FIGURE 2-18: BLAST LOADS FOR SURFACE BURST EXPLOSIONS, POSITIVE PHASE .............................................................. 45
FIGURE 2-19: AIR BURST EXPLOSION SCENARIO [DOD 2008] ..................................................................................... 46
FIGURE 2-20: PARAMETERS DEFINING PRESSURE DESIGN RANGES [DOD 2008] ............................................................. 48
FIGURE 2-21: APPROXIMATE LOAD ON A SHELTER [BAKER 1983]................................................................................ 49
FIGURE 2-22: FRONT WALL LOADING [DOD 2008] .................................................................................................. 50
FIGURE 2-23: ROOF AND SIDE WALLS LOADING [DOD 2008] ..................................................................................... 51
FIGURE 2-24: REAR WALL LOADING ....................................................................................................................... 52
FIGURE 2-25: IMAGE CHARGE APPROXIMATION, FIGURE ADAPTED FROM [US ARMY 2008] ............................................. 53
FIGURE 3-1: UNCERTAINTIES PARAMETERS IN BLAST ENGINEERING PROBLEMS ................................................................ 58
FIGURE 3-2: BLAST LOADS (SURFACE EXPLOSIONS) BY THE ADOPTED MODEL (DOTTED LINES) AND THE SBEDS MODEL
(CONTINUOUS LINE)................................................................................................................................... 59
FIGURE 3-3: REINFORCING STEEL STRENGTH ENHANCEMENT VERSUS STRAIN VELOCITY ..................................................... 61
FIGURE 3-4: COMPONENT RESISTANCE - DISPLACEMENT RELATION .............................................................................. 64
FIGURE 3-5: FRAGILITY CURVES COMPUTING PROCESS ............................................................................................... 67
FIGURE 3-6: N° OF SAMPLES AND COVS FOR THE FC RELATIVE TO THE HF AND R EQUAL TO 20 METERS............................. 68
FIGURE 3-7: NUMERICAL AND LOGNORMAL INTERPOLATED FC ................................................................................... 68
FIGURE 3-8: PRESSURE - IMPULSE DIAGRAMS .......................................................................................................... 69
FIGURE 3-9: FROM TOP LEFT CLOCKWISE, FRAGILITY CURVES FOR THE HF, HD, SD, MD COMPONENT DAMAGE LEVELS ......... 70
FIGURE 3-10: LINES OF DEFENSE ........................................................................................................................... 71
FIGURE 3-11: BLAST LOAD PARAMETERS [DOD 2008] (A); DESIGN BLAST LOAD SHAPES (B) ............................................. 75
FIGURE 3-12: PROBABILISTIC DESCRIPTION OF THE BLAST RESPONSE FOR A STRUCTURAL COMPONENT. PRESSURE-IMPULSE
DIAGRAM (A); STRUCTURAL FRAGILITY (B) ...................................................................................................... 76
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FIGURE 3-13: CONCEPTUAL DEFINITION OF THE FRAGILITY CURVE FOR IMPULSE SENSITIVE STRUCTURES............................... 78
FIGURE 3-14: DETAILS OF THE CASE-STUDY BLAST DOOR. FRONTAL VIEW (A); SECTION ALONG THE WIDTH (B); SECTION ALONG
THE HEIGHT (C) ......................................................................................................................................... 79
FIGURE 3-15: PROBABILITY DENSITY FUNCTION OF RY AND DY ...................................................................................... 85
FIGURE 3-16: STRESS STRAIN RELATIONSHIP [KALOCHAIRETIS 2013] ........................................................................... 86
FIGURE 3-17: FINITE ELEMENT MODEL OF THE STEEL BUILT-UP DOOR ........................................................................... 86
FIGURE 3-18: STATIC RESISTANCE FUNCTION BY THE FEM AND THE SSM ..................................................................... 87
FIGURE 3-19: COMPARISON BETWEEN THE TIME HISTORIES OF THE SUPPORT ROTATION Θ OBTAINED WITH THE FE MODEL AND
THE SSM. 10 KG OF TNT (A); 15 KG OF TNT (B); 20 KG OF TNT (C); 25 KG OF TNT (D)....................................... 88
FIGURE 3-20: PLASTIC STRAINS ON THE DOOR OBTAINED BY THE FE MODEL. 10 KG OF TNT (A); 15 KG OF TNT (B); 20 KG OF
TNT (C); 25 KG OF TNT (D)........................................................................................................................ 89
FIGURE 3-21: FLOWCHART OF THE PROCEDURE FOR THE EVALUATION OF THE FRAGILITY CURVES. FC= FRAGILITY CURVE ........ 91
FIGURE 3-22: FRAGILITY CURVES OBTAINED BY THE SSM. SERVICEABILITY (A), OPERABILITY (B), AND LIFE SAFETY (C) ........... 91
FIGURE 3-23: NUMBER OF SAMPLES AND COV ........................................................................................................ 92
FIGURE 3-24: DESCRIPTION OF THE BLAST SCENARIO AND OF THE CONSIDERED VARIABLES ................................................ 93
FIGURE 3-25: LOGNORMAL PDF OF THE IMPULSE DENSITY AND FRAGILITY CURVES COMPUTED FOR THE CONSIDERED LIMIT
STATES .................................................................................................................................................... 94
FIGURE 3-26. DETERMINISTIC PRESSURE IMPULSE DIAGRAMS AND THE LOAD SAMPLES .................................................... 94
FIGURE 3-27: THE SAFETY FACTOR FOR THE LIMIT STATES ........................................................................................... 96
FIGURE 3-28: WINNERS’ ANNOUNCEMENT ........................................................................................................... 100
FIGURE 3-29: FE MODEL OF THE SLAB .................................................................................................................. 102
FIGURE 3-30: DETAIL OF THE BC ......................................................................................................................... 102
FIGURE 3-31: STRESS VS. PLASTIC STRAIN RELATIONSHIP FOR STEEL REINFORCEMENTS, .................................................. 103
FIGURE 3-32: STRESS VS. PLASTIC STRAIN RELATIONSHIP FOR STEEL REINFORCEMENTS, HARDENED SLAB........................... 103
FIGURE 3-33: DIF FOR STEEL .............................................................................................................................. 103
FIGURE 3-34: DIF FOR CONCRETE ....................................................................................................................... 103
FIGURE 3-35: APPLIED DEMANDS ........................................................................................................................ 103
FIGURE 3-36: CRACK PATTERN............................................................................................................................ 105
FIGURE 3-38: PREDICTED DEFLECTION HISTORY, .................................................................................................... 105
FIGURE 3-39: PREDICTED (NUMERICAL) VS. EXPERIMENTAL DEFLECTION, NORMAL SLAB ................................................ 106
FIGURE 3-42: PREDICTED DEFLECTION HISTORY, .................................................................................................... 107
FIGURE 3-43: SPALL/BREACH SCHEMATIC ............................................................................................................. 110
FIGURE 3-44: PLAN AND ELEVATION VIEWS OF TESTED PANELS .................................................................................. 111
FIGURE 3-45: TYPICAL GEOMETRY FOR SPALL AND/ BREACH PREDICTIONS ................................................................... 113
FIGURE 3-46: SPALL AND BREACH THRESHOLD CURVES ............................................................................................ 114
FIGURE 3-47: DAMAGE OBSERVED FROM CLOSE-IN DETONATIONS ............................................................................. 116
FIGURE 3-48: DYNAMIC INCREASE FACTOR (DIF) VERSUS STRAIN-RATE FOR CONCRETE ................................................. 119
FIGURE 3-49: STRESS VS. VOLUMETRIC STRAIN CHART OF THE USED EPS FOAM ............................................................ 120
FIGURE 3-50: MEASURED AND PREDICTED SPALL DIAMETER ON PROTECTED FACE ......................................................... 122
FIGURE 3-51: IMPACT FORCE DEMAND ON THE FRONT FACE OF THE INTERIOR WYTHE .................................................... 122
FIGURE 3-52: PARAMETRIC EXAMINATION OF INSULATION TYPE AND THICKNESS FOR SPALL ............................................ 123
FIGURE 4-1: STRATEGIES FOR SAFETY AGAINST EXTREME EVENTS AND CORRESPONDING REQUIREMENTS [GIULIANI 2012] .... 130
FIGURE 4-2: EXAMPLE SPRING STRUCTURE ............................................................................................................ 135
FIGURE 4-3: EXAMPLE TRUSS STRUCTURE (A) AND DAMAGE SCENARIO EVALUATION (B) ................................................. 137
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FIGURE 4-4: EXAMPLE STAR STRUCTURE (A) AND DAMAGE SCENARIO EVALUATION (B) ................................................... 138
FIGURE 4-5: EAST ELEVATION OF THE I-35W BRIDGE [NTSB 2007] ......................................................................... 138
FIGURE 4-6: 3D FE MODEL OF THE I-35 WEST BRIDGE ........................................................................................... 139
FIGURE 4-7: LATERAL TRUSS OF THE BRIDGE AND SELECTION OF DAMAGE SCENARIOS..................................................... 140
FIGURE 4-8: UPDATED LATERAL TRUSS OF THE BRIDGE AND SELECTION OF DAMAGE SCENARIOS ....................................... 141
FIGURE 4-9: DAMAGE SCENARIO EVALUATION IN TERMS OF CF FOR THE ORIGINAL CONFIGURATION OF THE BRIDGE ............. 141
FIGURE 4-10: DAMAGE SCENARIO EVALUATION IN TERMS OF CF FOR THE IMPROVED CONFIGURATION OF THE BRIDGE ......... 141
FIGURE 4-11: FE MODEL OF THE BUILDING ........................................................................................................... 146
FIGURE 4-12: EXAMPLES OF ROBUSTNESS CURVES UNDER BLAST DAMAGE SCENARIOS ................................................... 148
FIGURE 4-13: FLOWCHART OF THE PROCEDURE TO EVALUATE THE STRUCTURAL ROBUSTNESS AGAINST BLAST DAMAGE ........ 148
FIGURE 4-14: DAMAGE SCENARIOS (L; 1), AND DAMAGE SCENARIOS (L; 2) ................................................................ 149
FIGURE 4-15: COLUMN TYPES ............................................................................................................................ 150
FIGURE 4-16: LOAD FACTOR TIME HISTORY CHART .................................................................................................. 151
FIGURE 4-17: MOMENT-CURVATURE DIAGRAMS ................................................................................................... 151
FIGURE 4-18: RESPONSE TIME HISTORY FOR A NODE LOCATED AT THE TOP OF THE REMOVED KEY ELEMENT, D-SCENARIO (5; 1)
(LEFT, ARRESTED DAMAGE RESPONSE) AND D-SCENARIO (6; 1) (RIGHT, PROPAGATED DAMAGE RESPONSE) .............. 151
FIGURE 4-19: ROBUSTNESS CURVES UNDER BLAST DAMAGE ..................................................................................... 152
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TABLE INDEX
TABLE 2-1: PERFORMED CFD ANALYSIS .................................................................................................................. 38
TABLE 2-2: FRANGIBLE OBJECTS CHARACTERISTICS .................................................................................................... 40
TABLE 3-1: INPUT DATA....................................................................................................................................... 60
TABLE 3-2: COMPONENT DAMAGE LEVELS, AND THE ASSOCIATED THRESHOLDS IN TERMS OF RESPONSE PARAMETERS ............ 65
TABLE 3-3: RESULTS ........................................................................................................................................... 72
TABLE 3-4: LIMITS STATES ................................................................................................................................... 81
TABLE 3-5: PROBABILISTIC DISTRIBUTIONS OF THE STOCHASTIC VARIABLES ..................................................................... 84
TABLE 3-6: PARAMETRIC CARACHTERIZATION OF THE FRAGILITY CURVES FOR THE EXAMINATED LIMIT STATES ....................... 92
TABLE 3-7: EXCEEDING PROBABILITIES OBTAINED WITH THE CONDITIONAL AND UNCONDITIONAL APPROACHES ..................... 95
TABLE 3-8: INPUTS FOR MAT159,...................................................................................................................... 103
TABLE 3-9: INPUTS FOR MAT159,...................................................................................................................... 103
TABLE 3-10: PREDICTED RESULTS,....................................................................................................................... 105
TABLE 3-11: PREDICTED RESULTS,....................................................................................................................... 107
TABLE 3-12: TEST MATRIX ................................................................................................................................. 112
TABLE 3-13: SPALL AND BREACH THRESHOLD CURVE CONSTANTS............................................................................... 113
TABLE 3-14: EXPERIMENTAL SPALL AND BREACH RESULTS ........................................................................................ 115
TABLE 3-15: ASSUMED PHYSICAL PROPERTIES OF THE EPS INSULATING FOAM .............................................................. 120
TABLE 3-16: EXTERIOR (LEFT), INTERIOR (RIGHT) AND SECTION VIEW (BELOW) OF NUMERICAL RESULTS; DAMAGE PARAMETER
FROM 1.95 TO 2. ................................................................................................................................... 121
TABLE 4-1: ABNORMAL EVENTS THAT COULD THREATEN A STRUCTURE [STAROSSEK ET AL. 2012] .................................... 128
TABLE 4-2: COLUMN CROSS SECTION ................................................................................................................... 150
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ABSTRACT
A failure of the security system of a community leads to a socio-economic instability and
consequently to the decline of the community. Nowadays like in the past, a protective
design against man-made attacks is important, especially considering that the Free World
is constantly prone to destabilization by terrorism.
A protective construction should principally guarantee the maximum reasonable
survivability of the occupants. If the prevention strategies of defense fail (e.g.
intelligence and police activities), the design for blast offers the only possibility to limit
the consequences of an explosion. The resistance of a generic structure subjected to a
blast load is measured in terms of collapse resistance, defined as the exceeding of a
performance limit. The collapse resistance can be assessed directly by applying the blast
demand to the structure (un-decomposed approach) or by decomposing the collapse
resistance (decomposed approach) in three components: the hazard mitigation, the local
resistance, and the global resistance. In this Thesis the decomposed approach is preferred
and methods for a quantitative assessment of the collapse resistance’s components are
proposed and applied to case-study structures.
Concerning the hazard mitigation, deterministic computational fluid dynamic simulations
are carried out for assessing the influence of three crucial parameters determining the
severity of the blast load due to the deflagration of a gas cloud. The fragility analysis is
carried out in the framework of the performance-based blast engineering, in order to
quantify the local resistance of both precast concrete cladding wall panels and steel builtup blast resistant doors. Furthermore detailed finite element simulations are carried out
for investigating the behavior of concrete slabs and insulated panels subjected to far-field
and close-in detonations respectively. Finally, the global resistance is investigated by two
methods that take into account the consequences of extreme loads on structures, focusing
on the influence that the loss of primary elements has on the structural load bearing
capacity.
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1 INTRODUCTION
In the mid-twentieth century Abraham Maslow wrote a paper on the hierarchy of the
human needs [Maslow 1943]. Considering for instance the primaries needs (see Figure
1-1) the humans should have employment, health, and property; in other words the human
needs security. In general, security is guaranteed by a secure shelter, such as a secure
residence, a secure office, and a secure city. Without security a society cannot develop
and prosper.
Having security means to be protected by adverse environmental conditions, daily human
activities, animals, and man-made attacks. Moreover the perception of a risk due to each
one of the mentioned hazards is not perceived with the same intensity by the community
[Pidgeon 1998] and the perception of the risk is generally variable. In particular the
hazard due to man-made attacks is mostly perceived in the beginning of the twenty-first
century.
The protection of buildings and critical infrastructures against man-made attacks is a
priority for a stable and secure society. A failure of the security system of a community
leads to a socio-economic instability and consequently to a decline of the community.
Nowadays like in the past a protective design [Krauthammer 2008a] against man-made
attacks is imperative, especially considering that the Free World is constantly engaged by
terrorism that aims at the destabilization of the community.
Self-actualization
morale,
creativity,
openness
self-esteem,
confidence,
achievement, respect
Belonging / Love
Security
friendship, family, sexual intimacy
security, employment, health, property
Physiological
breathing, food, sex, sleep, excretion
SHELTER
Esteem
Figure 1-1: Maslow’s hierarchy of needs [Maslow 1943]
In fact, terrorism is the new kind of warfare. Records of the terrorism activities provided
by the U.S. Department of State report that the 85 % of the terrorist attacks is conducted
by explosive devices [DoS 2003, DoS 2004, DoS 2005, and DoS 2006]. This fact leads
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to the need to design buildings and critical infrastructures against explosions. Moreover,
also accidental explosions in explosive storage facilities (military or civil) and in the
urban contest are considered to be a serious threat for the security of a community.
Briefly designing structures for blast load is a security’s prerogative of the community.
Without an adequate level of protection a prosperous community cannot withstand the
threat of the terrorism and a socio-economic decline is inevitable.
A protective construction should principally guarantee the maximum reasonable
survivability of the occupants. If the prevention strategies of defense (e.g. intelligence
and police activities) fail, the design for blast is the only chance to limit the consequences
of an explosion.
Approaches to design for blast can be divided in either deterministic or probabilistic.
Generally a facility is designed based on a standard threat so a deterministic approach is
used. However if the statistics of the threat and of the mechanical properties of the
structure are known a probabilistic approach is preferred.
A generic structure is composed by several structural elements (components) forming an
organized structural scheme. The structural response of the components is identified as
“local response” (e.g. the structural response of a column); instead, the structural response
of the overall structural scheme is identified as “global response” (e.g. the stability of a
building after a column failure).
The resistance of a generic structure subjected to a blast load is measured in terms of
collapse resistance defined as the exceeding of a performance limit concerning the global
and/or the local response.
The collapse resistance can be assessed directly by applying the blast demand to the
structure (un-decomposed approach) or by decomposing the collapse resistance
(decomposed approach) in three components: the hazard mitigation (hazard), the local
resistance (vulnerability), and the global resistance (structural robustness). In the latter
case the collapse resistance is determined quantifying the three components by a
deterministic or probabilistic approach.
Looking at the probabilistic approach the collapse probability is given by the product of
each conditional probability of the collapse probability’s components. In probabilistic
terms commonly used in earthquake engineering (see [Bazzurro et al. 1998, Fragiadakis
et al. 2013, and Kennedy et al. 1984]) the decomposed approach is called conditional
approach and expressed formally by Eq. (1-1).
[ ]
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∑ [
] [
] [ ]
(1-1)
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where Hi is the hazard related to the blast scenario “i” (where the scenario is defined by
the parameters determining the intensity of the blast action), also known as Intensity
Measure (IM) in other engineering fields [Whittaker et al. 2003, Ciampoli et al. 2011, and
Reed et al. 1994], LD is the structural local damage, C is the collapse event, P[∙|∙]
indicates a conditional probability, P[∙] indicates a probability, and the summation ∑ is
extended to all scenarios.
Following the decomposed approach, the left part of Figure 1-2 shows the three
components of the collapse resistance. On the right part of Figure 1-2 instead, there are
the investigated methods for a quantitative assessment of the collapse resistance’s
components and practical applications are presented for each component of the collapse
resistance.
Section
Gas explosions in civil buildings
COLLAPSE RESISTANCE
HAZARD
MITIGATION
Fragility analysis
LOCAL
RESISTANCE
Explicit finite
element analysis
GLOBAL
RESISTANCE
The consequence
factor
Scaled distance as
intensity measure
3.1
Impulse density as
intensity measure
3.2
Concrete slabs
subjected to blast
loads
3.3
Spall & breach
resistance of
insulated panels
3.4
The robustness curves
Collapse
resistance
components
2.3
4.1
4.2
Methods for a quantitative assessment of the collapse
resistance’s components and applications
Figure 1-2: Collapse resistance decomposition
Concerning the hazard mitigation, this Thesis focuses on the gas explosions in civil
buildings. Deterministic Computational Fluid Dynamic (CFD) simulations are carried
out for assessing the influence of three crucial parameters determining the severity of the
blast load due to the deflagration of a gas cloud. These parameters are the room
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congestion, the failure of non-structural walls and the location of the ignition. Each of
these parameters can change drastically the blast demand on the structure. Usually the
room congestion is present in a building and it should be considered in the simulations;
the failure of non-structural walls should be taken into account for not overestimating the
overpressures generated by the explosion; instead the effect of the location of the ignition
should be investigated by performing several simulations with different locations of the
ignition (see section 2.3).
The local resistance is investigated both deterministically and probabilistically. As
mentioned previously, with the term local resistance is intended the resistance of the
single component of the structural system subjected to the blast load. The fragility
analysis, see [Bazzurro et al. 1998, Fragiadakis et al. 2013, and Kennedy et al. 1984], is
carried out with two different intensity measures, and two different applications are
proposed.
-
First a terroristic attack carried out by a probabilistically defined vehicle bomb is
considered and a method for computing the fragility curves is provided for a
concrete cladding wall panel subjected to blast loads. The probabilities of
exceeding the defined limit states are computed by both the uncoupled and
coupled approaches [Reed et al. 1994] for testing both the proposed method for
computing the fragility curves and the selected intensity measure (see section 3.1).
-
After that, the accidental explosion of mortar rounds in a military facility
engaging a steel built-up door is considered probabilistically. In addition, a safety
factor for carrying out deterministic analyses of steel built-up blast doors
subjected to accidental explosions of mortar ammunitions is proposed. Also for
this second application the fragility analysis is validated by confronting the
obtained results in terms of probability of exceeding a limit state with the results
obtained with the uncoupled approach [Reed et al. 1994] (see section 3.2).
The probabilistic study of the local resistance is developed in collaboration with the Prof.
Charis Gantes and the Prof. Dimitrios Vamvatsikos of the National Technical University
of Athens (NTUA) during the spring/summer 2013.
The local resistance is also investigated deterministically by detailed explicit finite
element simulations performed using LS-Dyna [LSTC 2012]. The blast generated
demands can be categorized into far design range and close-in design range. In the far
design range the blast generated pressure demands can be considered uniform on the
structure, while in the close-in design range blast pressures are non-uniform and the
pressure magnitudes can be very high.
-
Concerning the far design range the National Science Foundation (NSF) funded a
study made by the University of Missouri Kansas City (UMKC) to perform a
batch of blast resistance tests on reinforced concrete slabs. Based on these results,
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a Blast Blind Simulation Contest was being sponsored in collaboration with
American Concrete Institute (ACI) Committees 447 (Finite Element of Reinforced
Concrete Structures) and 370 (Blast and Impact Load Effects), and UMKC School
of Computing and Engineering. The goal of the contest was to predict, using
simulation methods, the response of reinforced concrete slabs subjected to a blast
load. The blast response was simulated using a Shock Tube (Blast Loading
Simulator) located at the Engineering Research and Design Center, U.S. Army
Corps of Engineers at Vicksburg, Mississippi. A team for participating at the
contest has been formed by the author of this Thesis Pierluigi Olmati (Sapienza
University of Rome), Patrick Trasborg (Lehigh University), Dr. Luca Sgambi
(Politecnico di Milano), Prof. Clay J. Naito (Lehigh University), and Prof. Franco
Bontempi (Sapienza University of Rome). The submitted prediction of the slab’s
deflection was declared The Winner of The Blast Blind Simulation Contest
(http://sce.umkc.edu/blast-prediction-contest/ - accessed August 2013) for the
concurring category (see section 3.3).
-
Regarding the close-in design range, concrete elements exhibit localized damage
in the form of spalling and/or breach. When the depth of the spall exceeds half of
the element thickness breach often occurs. The resistance to spall and breach in
concrete elements is an important design consideration when close-in detonations
of high explosives are possible. Spall on the interior face of the structural element
can result in the formation of small concrete fragments which can travel at
hundreds of feet per second [DoD 2008] causing serious injuries and equipment
damages. In this Thesis, the spall and breach resistance is investigated for
insulated concrete wall panels by detailed explicit finite element analyses
performed using LS-Dyna [LSTC 2012]. The spall and breach resistance is
assessed to be dependent by the thickness of the insulation that guarantees a gap
between the exterior and interior concrete wythes (see section 3.4). Moreover
experimental tests were conducted at the Air Force Research Laboratory in
Panama City, FL.
The study on the spall and breach resistance of insulated concrete cladding wall panels
was developed at the Lehigh University during the winter/spring 2012 in collaboration
with Prof. Clay J. Naito (Lehigh University).
Finally the global resistance of a structure is investigated by two methods. As mentioned
before, with global resistance is intended the resistance of a structural system against a
failure of one or more structural components.
-
The first method is based on the consequence factor obtained confronting the
elastic stiffness matrices of the damaged and undamaged structure. This
methodology takes into account the consequences of extreme loads on structures,
focusing on the influence that the loss of primary elements has on the structural
Pierluigi Olmati
Page 17 of 189
load bearing capacity. Briefly a method for the evaluation of the structural
robustness of skeletal structures is presented and tested in simple structures.
Following that, an application focuses on a case study bridge, the extensively
studied I-35W Minneapolis steel truss bridge. The bridge, which had a structural
design particularly sensitive to extreme loads, recently collapsed for a series of
other reasons, in part still under investigation. The applied method aims, in
addition to the robustness assessment, at increasing the collapse resistance of the
structure by testing alternative designs (see section 4.1).
-
The second proposed method take account the non-linear dynamic behavior of a
structure for assessing its structural robustness. The method is developed for
buildings and it is based on the hypothesis of the removal column scenario. The
column is suddenly removed and a non-linear dynamic analysis is carried out for
assessing if the disproportionate collapse occurs, if not a non-linear static
pushover is carried out on both the damaged and undamaged configuration of the
building for estimating the residual capacity of the building. This procedure is
repeated both increasing the number of the removed columns and for several
scenarios. The robustness is so assessed for a steel tall building (see section 4.2).
The following sections contain the carried out studies on the collapse resistance of
structures under man-made or accidental explosions following the decomposition of the
collapse resistance shows in Figure 1-2.
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2 THE HAZARD MITIGATION
There are many definitions of the explosion phenomenon. In general an explosion results
from a very rapid release of a large amount of energy within a limited space, or it can be
defined by a sudden conversion of potential energy into kinetic energy with the
production and release of gas under pressure [DoD 2008]. The sudden liberation of
energy causes a very rapid and considerable expansion of gases that initiate a shock wave.
An ideal shock wave is defined as a front where occur an instantaneous rise of pressure
followed by a gradual decrease of it. If the explosion is in air the shock wave is followed
by a strong wind.
Moreover there are different typologies of explosions. The most common are: physicals,
chemicals, nuclear, and electrical [Bjerketvedt et al. 1997].
A physical explosion [Baker et al. 1983] concerns a very rapid changing of phase of the
materials, generally from liquid phase to gaseous phase condition; so there is a rapid
changing of equilibrium conditions of the materials, as a quick release of gas initially at
high pressure conditions. Generally physical explosions occur by a very quick release of
gas at high pressure that generates a shock wave. A typical physical explosion is the
boiling liquid expanding vapors explosions (BLEVEs); the BLEVE is an explosion due to
flashing of liquids when a vessel with a high vapor pressure substance fails. A typical
BLEVE is a steam explosion, it can result from boiler failure that causes loss of
containment of the superheated water; a flash vaporization of the superheated water
occurs and this causes a shock wave.
A chemical explosion [Baker et al. 1983] is an exothermic reaction. The explosive charge
is converted in very hot gases that expanding rapidly provokes a shock wave. The
explosive materials can be solid, liquid, and also gaseous. A chemical reaction starts after
the ignition of a generic explosive. The speed of reaction respect to the unreacted
explosive (U) is different for various kind of explosives. If U is greater than the sound
velocity referred to the environmental conditions modified by the explosion (c'), the
phenomenon is a detonation regime, instead if U is less than c' the phenomenon is a
deflagration regime. The speed of the front of reaction respect a fix observer (S) is the
summation of U and u, where u is the velocity of the unreacted explosive. The passage
between deflagration to detonation regime is possible and it is called Deflagration to
Detonation Transition (DDT). In other hands a detonation is a directly molecular
decomposition; instead a deflagration is a rapid combustion (oxidation phenomena). The
shock wave made by a deflagration is less strong than the shock wave made by a
detonation. An example of deflagration is made by a rapid combustion in turbulent
regime of a mixture of air – fuel (gas cloud), as a matter of fact gas clouds generally
Pierluigi Olmati
Page 19 of 189
deflagrate, but a DDT event could occur.
trinitrotoluene, mainly detonate.
Instead the solid explosives, as the
A nuclear explosion is given by a mass fusion or fission of atomic nuclei. Usually fission
occurs in the nuclear civil facilities (nuclear power plant), so the uranium 235 or
plutonium 239 molecule splits in to smaller parts releasing energy. The fission was also
used in the nuclear weapon “The gadget”, that was the first nuclear test called “Trinity”
executed the 19th July 1945 in New Mexico (USA), and in the nuclear weapons “Little
Boy” and “Fat man” used the 6th and 9th August 1945 on Hiroshima and Nagasaki
(Japan) respectively. Instead in the fusion two or more nuclei joint together releasing
energy. An example is the fusion of the hydrogen isotope known as deuterium to form
helium nuclei, other particles, and energy. To have a fusion to utilize in civil facilities is
not feasible yet, a lot of energy to start fusion is necessary and conventional materials do
not support the resulting stress, about that some tests are ongoing. Otherwise some
weapons adopting the fusion exist and they are widely tested.
An electrician explosion occurs when there is a strong release of electrical energy; the
resulting electrical arc rapidly heats the surrounding gas that quickly expands causing a
shock wave. A dangerous scenario is when an electrical arc occurs inside a transformer,
the oil vaporizing so expand causing often the rupture of the transformer, hot gases and
oil are pushed out roughly and their ignition is probable.
So referring to the chemical explosions [Bjerketvedt et al. 1997], the term explosive is
applied to such solid or liquid substance as possess the faculty of undergoing
instantaneous decomposition, extending throughout their entire mass and accompanied by
a considerable disengagement of heat, the substance at the same time being partly or
wholly converted into gaseous decomposition products [Baker 1997]. The phenomenon
of explosion is a sudden and enormous expansion of gases and vaporous liberated from a
previous condition of chemical combination [Bjerketvedt et al. 1997]. The chemical
explosives can be classified by various criteria as:





The capacity to detonate or to deflagrate, so high explosives and low explosives.
The utilization of the explosive charge, for examples of demolition, as propellant,
etc.
The sensitivity to be ignited, so primary explosives and secondary explosive.
The chemical composition.
Condensed phase explosives and gaseous explosives.
Between the above criteria for classifying explosives generally using the classification
between primary and secondary explosives is more practice [US Army 1992]. The firsts
are extremely sensible at the ignition source, instead the seconds are stable namely they
are not sensible to weak ignition source [US Army 1992]. For this reason a primary
explosives are adopted as ignition source for the secondary explosives.
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However primary and secondary explosives are a subgroup of the high explosives [US
Army 1992]. Therefore there are low explosives and high explosives, the firsts only
deflagrate instead the seconds detonate [US Army 1984]. Resuming generally it is
possible classify explosives in low and high; and the seconds are divides between primary
and secondary explosives.
EXPLOSIVES
Low explosives
Black powder
Smokeless powder
Flash powder
High explosives
Primary high
explosives
Lead azide
Lead styphnate
Mercury fulminate
DDNP
Tetrazene
Secondary high
explosives
Boosters
PETN
RDX
Main charge
Dynamite
Binary explosives
Water gels
TNT
ANFO
Figure 2-1: Classification of the explosives [US Army 1992]
The effects of an explosion include the shock wave, thermal effects, projectiles, blast
wind, ground shocking, cratering, and electromagnetic disturbance [US Army 1992].






The shock wave is a mechanical pressure wave that travels outward from the
source the explosion and interacts with the surrounding objects. Especially for
detonation it can be modeled by instantaneous increment of pressure followed by
an exponential decrement of the pressure. When the shock wave interacts with
some objects phenomena as reflections and refraction occur, whereby a
magnification of the pressure is manifested.
The thermal effect is due to the temperature of the gases, output of the explosion
reaction; it can be important near the charge and for nuclear explosions.
Projectiles or fragments are dangerous because they cause in general a lot of
victims and large damage at the equipment; they are fragments of materials
pushed out belonging at the bomb or at the objects destroyed by the explosion.
The blast wind is due by the expansion of the gases output of the explosion and by
the surrounding dislocated air. The effect of blast wind is generally called
dynamic pressure, and it follows the shock wave.
The ground shocking is due to the overpressure of the blast slapping on the
surface of the ground. This phenomenon is significant for strong explosion.
The cratering is a depression in the ground. The most significant cratering occurs
when the blast is near to the surface so the charge is in low elevation from the
ground. The formation of a crater dissipates explosive energy.
Pierluigi Olmati
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
The electromagnetic disturbance is observed especially for strong explosions and
for nuclear explosions. The electro equipment is out of order after this event.
The human tolerance to the blast load is relatively high [DoD 2008]. The pressure
tolerance for short duration blast loads is significantly higher than the pressure tolerance
for long duration blast loads. The critical human tissues are those contain air. The
release of air bubbles from disrupted alveoli of the lungs into the vascular system
probably accounts for most deaths. For short duration of shock, from 3 to 5 ms, the
lethality threshold is approximately 7 bar, but a severe lung hemorrhage can occur at 2.5
bar. For long duration loads a petechial hemorrhage can manifest approximately at 1 bar.
However the survival is dependent mainly by the mass of the human. The pressure levels
that humans can withstand are generally much lower than those causing eardrum or lunge
damage, so the loss of equilibrium and the impact with hard surfaces is a dangerous
threat; for this reason tolerable pressure level of humans would not exceed 0.18 bar;
however this pressure leads a temporary hearing loss.
100
1000
threshold eardrum rupture
threshold
99% survival
50% survival
1% survival
temporary threshold shift
10
P [psi]
P [psi]
50 % eardrum rupture
100
1
0.1
10
1
10
100
1000
0.01
i/Whuman1/3 [ms psi/w1/3]
(a) Survival curves for lung damage
[DoD 2008]
0.1
i [ms psi]
1
(b) Human eardrum damage [DoD 2008]
Figure 2-2: Blast load human tolerances
Pierluigi Olmati
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2.1 High detonations
The most famous and diffuse explosive especially in the past is the TNT. There are a lot
of experimental data about it. Therefore the common practice is to compare the other
explosive proprieties about the shock wave parameters with those ones of the TNT [US
Army 2008].
The heat of reaction (H) is defined as energy for unit mass so expressed in MJ/kg. The
potential energy of the explosive charge is the heat of reaction multiplied by the mass of
the charge. To compare a generic explosive charge with the TNT charge is necessary
comparing the two energies held by the two charges [US Army 1992]. The total mass of
the generic charge multiplied by the relative heat of reaction is posed equal at the TNT
heat of reaction multiplied by the equivalent mass of TNT; see Eq. (2-1).
Mex Hex  MTNT HTNT
(2-1)
The heat of reaction and the explosive charge are known and it is possible calculate the
equivalent mass of TNT by the ratio between the two heat of reactions called explosive's
TNT equivalent factor, or Equivalent Factor (EF). The TNT mass of the explosive charge
that has the same energy of a generic explosive charge is so obtained. Therefore a
generic explosive charge is considered as an equivalent charge of TNT [US Army 2008].
In reality the EF varies slightly in function of the standoff, the charge geometry, and the
atmospheric conditions [DoD 2008]. For the purposes of design a given EF of an
explosive is considered as constant. Use the TNT equivalent mass is convenient because
it is possible compute the blast load of an explosive charge (expressed by its TNT
equivalent mass) from the blast load of an experimental blast test. This is possible by the
application of the principle of similitude and the scaled distance.
By the principle of similitude [Baker et al. 1983] two charges made by the same explosive
and with the same shape but with different size proportional to a constant k, the peak
pressure Pso measured at any distance R1 from the center of the first charge will be equal
to those measured at distance R2, see Eq. (2-2), from the center of the second charge.
R 2  k * R1
(2-2)
The principle of similitude is widely experimental validated. If the sizes of the two
explosive charge are proportional to k, the mass of the explosive charge will be
proportional at the k3, resulting in Eq. (2-3).
1/3
M 
k   2 
 M1 
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(2-3)
Page 23 of 189
Where M1 is the mass of the first charge and M2 is the mass of the second charge. So by
coupling the Eq.s (2-2) and (2-3) it is possible write Eq. (2-4).
1/3
R2  R2 


R 1  R 1 
(2-4)
Remembering that at the distances R1 and R2 from the explosion source, the pressure Pso
developed by the detonation of any of the two charges made by the same explosive with
proportional shape is identical.
Name
Applications
Black Powder
Ammonium Nitrate
Amatol 80720
M1 Dynamite
Time fuse
Cratering charge
Bursting charge
Demolition charge
Detonating Cord
Priming
TNT
Tetrytol 75/25
Tetryl
Sheet Explosive
M118 and M186
Pentolite 50/50
Demolition charge,
composition explosive
Demolition charge
Booster charge,
composition explosive
Cutting charge
Booster charge,
bursting charge
Commercial dynamite
Nitroglycerin
Bangalore Torpedo
Demolition charge
M1A2
Shaped Charges
M2A3, M2A4, and
Cutting charge
M3A1
Composition B
Bursting charge
Composition C4 Cutting charge, bursting
and M112
charge
Booster charge,
Composition A3
bursting charge
Detonation cord,
PENT
blasting caps,
demolition charges
Blasting caps,
RDX
composition explosives
Detonation
m/sec ft/sec
400
1300
2700 8900
4900 16000
6100 20000
6100 20000
to
to
7300 24000
EF
0.6
0.4
1.2
0.9
1
Fume
Water
toxicity
resistan
Dangerous
Poor
Dangerous
Poor
Dangerous
Poor
Dangerous
Fair
Slight
Excellent
6900
22600
7000
23000 1.2
Dangerous Excellent
7100
23300 1.3
Dangerous Excellent
7300
24000 1.1
Dangerous Excellent
7450
24400
Dangerous Excellent
7700
25200 1.5
Dangerous
7800
25600 1.2
Dangerous Excellent
7800
25600 1.2
Dangerous Excellent
7800
25600 1.4
Dangerous Excellent
8040
26400 1.3
8100
26500
8300
27200 1.7
8350
27400 1.6
-
-
Dangerous Excellent
Good
Slight
Excellent
Dangerous
Good
Slight
Excellent
Dangerous Excellent
Figure 2-3: Characteristics of explosives from [US Army 1992]
Consequently the scaled distance (Z) is constant [Baker et al. 1983], see Eq. (2-5).
 R
Z  1
3 W
1

Pierluigi Olmati
  R2

 3 W
2
 




(2-5)
Page 24 of 189
As mentioned the ratio Z is called scale distance and by the validity of the principle of
similitude is adopted for obtaining the blast load parameters for any size of explosive
charge. Some analytical expressions and charts are provided in function of Z to predict
blast loads; besides they are referred to a spherical explosive charge. The most data
available about the scale distance is computed in imperial units and instead the charge
mass is considered as weight of the charge expressed in pound (lb).
The Jump Equation
In this chapter is highlighted the detonation phenomena occurring in a high explosive
charge [US Army 1984]. The objective is finding an expression to compute the
detonation pressure for different kinds of explosive.
Referring to Figure 2-4 and to the section:











The shock velocity is the velocity that the shock wave moves through the material.
The detonation process is stationary.
v is the velocity of the shock wave respect to a fix observer.
V1 and V0 are the explosive control volume, respectively ahead and behind the
shock wave.
u1 and u0 are the velocities of the particles of the material.
L1 and L2 are the distances that a particle travels in a time interval.
A is the constant sectional area of the control volume.
ρ1 and ρ0 are the explosive densities, unreacted and reacted respectively.
e1 and e0 are the internal energies of the V1 and V2.
t is the unit time.
P1 and P0 are the pressures acting on the two faces of the shock front.
v-u1
V1
L1
A
V0
v-u0
L0
Figure 2-4: Control volume and shock wave [US Army 1984]
Generally the mass can be expressed as:
Pierluigi Olmati
Page 25 of 189
m  V
(2-6)
The velocity of the shock wave moving through the material is equal at:
v0  (v  u 0 )
v1  ( v  u1 )
(2-7)
And the mass crossing the shock wave can be expressed as:
m 0  0 At( v  u 0 )
m1  1At( v  u1 )
(2-8)
Where the volume is expressed by the cross section multiplied per both the time and the
velocity.
By the principle of mass conservation m0=m1 and suppressing the product At in both the
members of the mass conservation equation it is possible to write the first jump equation:
0 (v  u 0 )  1 (v  u1 )
(2-9)
By the forces equilibrium equation the produced pressure is the difference of the force
ahead and behind the shock wave:
F  (P1  P0 )A
(2-10)
By the principle of momentum conservation written in terms of rate of momentum it is
possible write the changing rate of the momentum (M) of the system per unit time:
M / t  (mu1  mu 2 ) / t  1Atu1 (v  u1 )  0 Atu0 (v  u 0 ) / t
(2-11)
The changing rate of the momentum is equal at the force F of Eq. (2-10), therefore:
(P1  P0 )A  1Au1 (v  u1 )  0 Au0 (v  u 0 )
(2-12)
Suppressing the sectional area of the control volume and rearranging the Eq. (2-12) it is
obtained the second jump equation:
P1  1u1 (v  u1 )  P0  0u 0 (v  u 0 )
(2-13)
By the principle of the work conservation the rate of work (W) being done on the system
per unit of time is:
w / t  1Au1  0 Au0
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(2-14)
Page 26 of 189
By the principle of the energy conservation the increasing rate of the energy per unit of
time is the difference between the rate of change of the sum of the internal and kinetic
energies in the initial and final state:

 

(2-15)

(2-16)
E / t  1AL1e1  0.51AL1u12  0 AL0 e 0  0.50 AL1u 02 / t
Therefore the Eq. (2-15) can be expressed as:





E / t  At1 e1  0.5u12 v  u1   At0 e 0  0.5u 02 v  u 0  / t
Equating the rate of work done, Eq. (2-16), on the system with the increasing rate of
energy and suppressing the time t Eq. (2-17) is obtained:






1Au1  0 Au0  At1 e1  0.5u12 v  u1   At0 e0  0.5u 02 v  u 0 
(2-17)
Finally suppressing the sectional area A and rearranging the terms the third jump equation
is obtained:



1u1  1 v  u1  e1  0.5u12  0 u 0  0 v  u 0  e0  0.5u 02

(2-18)
The name “jump equations” [US Army 1984] is due because the state variables jump
from one value to another very rapidly across the shock. To solve this equation are
necessary relationships tying together the variable present in the three jump equations.
But a simpler relationship called the Hugoniot equation suffices.
The Hugoniot equation and the C-J pressure
The Hugoniot relationship [US Army 1984] written coherently with the adopted
symbolism is:
(v  u 0 )  C0  Su 0
(2-19)
Where C0 is the sound velocity in the medium, S is a constant related to the specific heat
and thermal expansively of the material, v-u0 is the shock velocity in the medium, and u0
is the particle velocity. The shock velocity in the medium is linearly related to the
particle velocity; by a set of experiments the parameter S and the sound velocity on the
medium can be computed.
Substituting the shock wave velocity of the Eq. (2-19) in both the three jump equations,
the pressure developed in the detonation can be estimated. For example, considering the
second jump equation, Eq. (2-13), generally the velocity of the unreacted explosive is
zero and its pressure is the atmospheric pressure (assumed as reference pressure).
Consequently the pressure immediately behind the shock wave is:
P0  0 (C0 u 0  su02 )
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(2-20)
Page 27 of 189
It is possible plot the Eq. (2-20) in the space of the pressure (P) and of the specific
volume (Vs), defined by the reciprocal of the density Vs=1/ρ, for both the unreacted (VS0)
and reacted (VSf) explosive.
Von Newman spike
Hugoniot of unreacted explosive
P
Von Neuman spike
P
Hugoniot of reacted explosive
C-J point
C-J pressure
Rayleigh line
Distance
Vsf
Vs0
Figure 2-5: P-Vs chart [US Army 1984]
Gas expansion,
Particle velocity
to 0 from u0
Reaction zone
Figure 2-6: Detonation wave moving
through explosive material [US Army
1984]
Champman and Jouquet (C-J) developed the theory of the shock wave propagation
through explosive materials. The CJ conditions for each explosive are unique, if the
initial density is changed, the CJ conditions are changed. The theory is based on these
assumptions: the pressure is constant from the shock wave to the C-J point, the pressure
decays in a Taylor wave beyond the C-J point, the reaction is complete and the products
are in equilibrium at the C-J point, and the energy in the Von Neuman spike is negligible
in comparison to the energy in the reaction zone and so it can be ignored.
From Figure 2-5 the C-J pressure is calculable by tracing the tangent of the Hugoniot of
reacted explosive from the point of initial state (VS0) of the Hugoniot of unreacted
explosive, the name of this tangent is Rayleigh line [US Army 1984]. The intersection of
the Hugoniot of unreacted explosive with the Rayleigh line is the Von Neuman spike
pressure [US Army 1984]. Figure 2-6 shows the pressure distribution of a detonation
wave that is moving through explosive material [US Army 1984].
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Page 28 of 189
2.2 Gas explosions
Generally a gas explosion is a deflagration but a transition to detonation is possible. The
pressure generated by the combustion wave depends how fast the flame propagates
respect the unburned gases, if it propagates at subsonic velocity the explosion is a
deflagration, instead if it propagates at supersonic velocity the explosion is a detonation.
A gas explosion occurs previously a release of inflammable gas or liquid. A released
substance can be a gas, an evaporating liquid, or a two phases gas and liquid flow. There
are two principal kinds of release of gases: the jet flow and the evaporating pool.
Between the two kinds of gas dispersion the jet release is the most effective, therefore the
most dangerous. A jet of gas (fuel) has a high momentum so the gas is in turbulent
regime, a rapid mixing with the air occurs, and the mixture of fuel and air can reach the
combustible level. An evaporating pool release has a low momentum, the fuel released is
generally in liquid phase, therefore the boundary conditions (opening, wind velocity, etc.)
are most important to form a flammable gas cloud. Experiments have highlighted a
formation of flammable gas cloud after a time of the beginning of the release in most
scenarios; moreover the ignition source and its power are relevant.
Every gas has an Upper and Lower Flammability Limit, respectively abbreviated with
UFL and LFL. These two flammability limits are function of the temperature; generally
the flammability limits expand with the increasing of the temperature. Furthermore when
a flammable mixture is heated up to a certain temperature, the chemical reaction will start
spontaneously; the minimum temperature of auto ignition is called Auto Ignition
Temperature (AIT).
The concentration of the fuel strictly necessary for combustion is called stoichiometric, at
this concentration the energy necessary for the ignition is the lower, and the maximum
pressure is obtained with a fuel concentration slightly higher at the stoichiometric
concentration.
The process of a gas cloud deflagration starts with the ignition, the energy heats the gas
mixture and the auto ignition region is reached, and the heat of this combustion auto
ignites the nearest gas mixture molecules. Initially the burning regime is laminar, but by
the instability of the combustion front the burning regime becomes turbulent. The
burning velocity increase so the turbulence increases again, it is a positive feedback loop
causing flame acceleration due to turbulence, therefore the pressure level increases.
Turbulence is generated also by the interaction of the flow with the objects; generally this
interaction is more influence to determine the pressure developed; obstruction made by
objects is crucial to the deflagration process. Another important issue to the pressure
development is the failure of the vent panels, this event change totally the explosion
development. When a deflagration becomes sufficiently strong, a sudden transition from
deflagration to detonation can occur.
Pierluigi Olmati
Page 29 of 189
There are same theories for predicting the gas explosions pressure, but because the
pressure is strictly related at the level of obstruction (environmental congestion) the
estimation of the pressure value is in general difficult. Same easy methods are developed,
as the TNT-Equivalent-Method and the Multi-Energy-Method, also exist a numerous
semi-empirical formulas to use generally for specific scenarios. Moreover a powerful
instrument to predict the blast loads is the Computational Fluid Dynamic (CFD)
numerical simulation.
TNT Equivalent Method
The TNT Equivalent Method is based on the wide data available for high explosives. The
principle is to make equal the energy of the gas cloud with the energy of an equivalent
mass of TNT. The blast load can be so obtained by charts or formulas for high
explosives. The method is useful for estimate blast load in the far field area and not in the
near field area because the extension of the gas region can be very wide, therefore to
estimate pressure inside the gas region the TNT-Equivalent-Method is not suitable.
For typical hydrocarbons the heat of combustion is 10 times higher than the heat of
reaction of the TNT, so the equivalent mass of TNT is equal at 10 times the mass of the
hydrocarbon multiplied a yield factor η estimated between the 3% and the 5%.
WTNT  10η0
HC
(2-21)
For natural gas, at atmospheric conditions, the equivalent mass of TNT can be estimated
as the 0.16 times the gas cloud volume expressed in cubical meters, fixing the energy of
the stoichiometric gas cloud per volume equal at 3.5 MJ/m3.
Multi Energy Method
The multi energy method is based on the only energy of the confined gas clouds. The
scaled distance is calculated as:
R
Z
3
E exp / Po
(2-22)
Where Eexp is the energy of the explosive cloud equal at 3.5 MJ/m3 multiplied by the
volume of the congested area of the gas cloud, and Po is the atmospheric pressure. If
there are more congested gas cloud areas there is a multiple deflagrations, and if do not
occurs a DDT event the flame velocity drops out of the congested area. By a set of
curves the peak pressure is tied at the scaled distance; there are ten curves, and for a
detonation the curve number 10 is adopted, instead for a deflagration the curves from the
number 1 to the number 9 are used; moreover there are three pressure time shape
associated at the ten curves. Similarly a chart exists to estimate the duration of the blast
pressure.
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Figure 2-7: Hemispherical fuel-air charge blast for the multi-energy
Semi empirical methods
This section reports some semi empirical methods to evaluate the peak pressure of a gas
explosion. Generally the semi empirical formulas are based on some parameters as: the
failure pressure of the vent panels Pv [mbar], the volume of the gas region V [m3], the
mass per surface unit of the vent panels W [kg/m2], the coefficient of vent K
dimensionless, the area Av of the vent panel surface [m2]; the laminar burning velocity SL
[m/s], and a multiplier of the SL. Furthermore their applicability is limited at specific
scenarios identified with: the volume V, the coefficient of vent K, the failure pressure of
the vent, and the mass per surface unit W. The peak pressure P computed is expressed in
[mbar]. The fuels of reference are often methane, propane or GPL. Following are
presented the most famous semi-empirical formulas.

Cubbage and Simmonds formula was made by a set of experimental tests in
industrial furnaces. The vent panel was light and not fixed at the structure, so the
Pv is zero. The volume of the test was of 14 m3, but the formula can be employed
for volumes until 200 m3. The authors provide the expression of the first and
secondary peaks of pressure. The limits of the formula are: a) the ratio ρ of the
minimum and maximum edge of the volume V between 1 and 3; b) weigh of the
vent panel less than 24 kg/m2; c) the coefficient K=V2/3/Av less than 5.
P1  SL (4.3KW  28) / V1/ 3
P2  SL K

(2-23)
Cubbage and Marshall formula was developed by tests with a fix vent panels
with a finite failure pressure Pv. Moreover is considered the eventuality that the
volume V is not full of gas by a coefficient α (the expression of this coefficient is
not here reported). The formula is limited by: a) the value of the ratio ρ between 1
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and 3; b) W less than 24 kg/m2; c) K less than 5; d) opening pressure of the vent
panel Pv less than 490 mbar.
 23KW 
P  Pv  S2L  1/ 3 
 V


(2-24)
Rasbash formula was made by experimental data about domestic volumes, so it is
a formula for civil buildings. The fuel adopted for the tests was GPL. The
formula is valid for: a) the value of the ratio ρ between 1 and 3; b) W less than 24
kg/m2; c) K less than 5; d) Pv less than 70 mbar.
 4.3KW  28
P  1.5Pv  SL 
  77.7SL K
V1/ 3

(2-25)
To adapt these semi empirical formulas at different scenarios that have more vent panels
it is possible to determine a (K W)average as following:
n
 1 
1

  
KWav i1  KWi 
(2-26)
Instead the failure pressure of the vent panels is the weighted average of the single failure
pressure:
n
Pv 
A
i 1
vi
A v tot
Pvi
(2-27)
In the previous formulas, for predicting the peak pressure, is present the laminar burning
velocity of the gas and not the turbulent burning velocity, therefore the effect of the
turbulence is implicitly accounted in these formulas. Moreover the turbulence accounted
is that of the experimental test scenarios. It is validated that the implicit effect of the
turbulence (accounted in these formulas) on the laminar burning velocity is of a factor
equal at 3. In other hands in these semi empirical formulas is implicit a multiplier of the
laminar burning velocity equal at 3.
If the congestion level of the scenario is different of the formula's experimental scenario
(in the test the volume V was empty of domestic furniture or industrial equipment), it will
be necessary multiply SL by a turbulent factor β for taking account the increased
turbulence due at the increased congestion. The factor β is taken equal at:


1.5 for congestion due at obstacles inside the volume V.
From 1.5 to 5 for the gas region V built by more volumes, for the propagation of
the deflagration from a volume to another, and for very high congested volume.
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So the increasing of the burning velocity due to the turbulence is considered explicitly.
The total factor multiplying the SL is obtained by the product of the implicit turbulent
factor with the explicit turbulent factor (β). A total turbulent factor of value from 3 to 15
can be obtained.
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2.3 Computational Fluid Dynamic simulations
By a CFD simulation, starting from a fuel release, the deflagration phenomena can be
simulated. Therefore it is a complete analysis of the explosion event from its cause of
origin, that makes possible a very accurate assessment and mitigation of the associated
risk.
These simulations solve the famous Navier-Stokes equation, other than the continuity and
transport equations. Both the three set of equations take account of the burning
mathematical model.
 
D
v  P  T  f
dt
(2-28)
The first member of Eq. (2-28) is the substantial derived of the momentum, ρ is the fluid
density, and v is the velocity vector; at the second members there are respectively: both
the divergence of the average and deviatory component of the stress tensor, and the
matrix of the volume force.
For the Newtonian fluids the deviatory stress tensor is a function of the flow velocity and
of both the viscosity μ and bulk viscosity λ; δij is the Kronecke delta.
 u u j 
    v
Tij   i 
 x x  ij
i 
 j
(2-29)
The principal issue is how consider turbulence in the Navier-Stockes equation. The
velocity is assumed as the summation of the average velocity vm and of the turbulent
fluctuation velocity v'.
(2-30)
v  v m  v'
Therefore the Eq. (2-29) considering the Eq. (2-30) makes the expression of the deviatory
stress tensor in turbulent regime. The effect of the turbulence is accounted for the so
called Reynolds stress (-ρv'jv'i). Now by numerical methods it is feasible directly solve
the problem with the Reynolds stresses as one set of unknowns (Raynolds Average
Navier-Stokes, RANS), but a lot of computational power is necessary, almost it is
impossible use the RANS method for practical applications. Therefore other methods are
developed. Especially for simulate deflagration is diffuse the so called k-ε method. By
the Bussinesq assumption the Reynolds stresses are tie at a fictitious viscosity: the
turbulent viscosity μt. Thereafter the effective viscosity μeff has the following expression:
 eff     t    C
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k2

(2-31)
Page 35 of 189
The turbulent viscosity is an unknown in the problem and varies both in the space and in
the time. Moreover μt depends both from the square of the kinetic energy k and from its
dissipation ε, namely the conversion of energy from kinetic to internal energy. The
deviatory stress assumes the form:
 u u j  2 
    k   eff u k
Tij   eff  i 
 x x  3 ij 
x k
i 

 j



(2-32)
For computing the kinetic energy k and its dissipation ε the transport equations of both k
and ε are necessary. The output of CFD simulations are the complete time histories of the
explosion loads for all scenarios without limit of complexity.
Gas explosions (occurring when there is a very rapid combustion or reaction of chemical
elements) are either due to deflagration or due to detonation. The “gaseous explosives”
(i.e. cloud of gas – air mixture) usually explode in a deflagration regime. Yet, in some
circumstances, they develop in a detonation regime, depending on the blast scenario (gas
type; ignition power, location and type; geometry and congestion of the environment).
As stated before the CFD simulation is a powerful tool for obtaining an accurate
evaluation of the blast pressure on the structural elements. Several modern CFD codes
allow taking into account accurately the effect of some fundamental phenomena:



The congestion in the environment. This issue has been extensively studied in
industrial facilities and partially explored in civil structures (domestic congestion).
The congestion is caused by the presence of all the objects inside a room. In the
CFD codes the domestic congestion is implemented in the simulation by modeling
solid objects whose effects (e.g. turbulence generation) can be considered as an
interaction between flow and objects.
The failure of non-structural walls. When a gas explosion occurs, the failure of
non-structural walls causes the modification of the geometrical scenario and
consequently the development of the entire explosion. The non-structural walls
are modeled in the CFD simulation by special objects having a cut off pressure
level.
The ignition type and position of the gas cloud ignition, which can vary on a caseby-case basis, and have an important role in the development of the explosion.
The aleatory uncertainty related to the above-mentioned issues is one of the principal
reasons causing the variability of the intensity and direction of the blast action. Of
course, a significant dispersion of results, especially in cases of complex numerical
models such as the ones used for the simulation of gas explosions, is due to the epistemic
uncertainty (e.g. model uncertainty).
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In the case of gas explosions, the “gas region” is defined as the volume containing the
cloud of gas, usually assumed as homogeneous. This gas region can be characterized by
the so-called “Equivalent Ratio” (ER):
 Vfuel 


 VO 
 2  actual
ER 
 Vfuel 


 VO 
 2  stoichiometric
(2-33)
Where O2 indicates the Oxygen molecule, Vi (i =”fuel” or “O2”) indicates the volume of
the cloud component “i” at normal atmospheric conditions and the term “fuel” is referred
to the flammable component of the gas cloud (e.g. methane, hydrogen, etc.). The term
“actual” refers to the exploding cloud, while the term “stoichiometric” refers to the
quantities of fuel and Oxygen needed for a balanced chemical reaction. In Eq. (2-33), the
value of the denominator can be derived from literature, while the value of the numerator
in the design phase can be chosen with reference to experimental data, in order to
maximize the initial (laminar) burning velocity, which represents an important aspect for
the determination of the severity of the explosion.
With reference to the three fundamental phenomena previously outlined (congestion of
the environment, failure of non-structural walls, ignition type and position), a number of
CFD analyses is carried out in order to evaluate the effects of the scenario parameters on
the blast load. The analyses are carried out using the CFD commercial code Flacs®. A
set of gas explosions at the ground floor are modeled where the presence of some
commercial activities and one restaurant are hypothesized, including the kitchen (where
the ignition point and the gas region are located). Methane is assumed to be the fuel and
the equivalent ratio (ER), see Eq. (2-33), is assumed equal to 1.12, in order to maximize
the initial laminar burning velocity. The main features of the blast scenario are shown in
Figure 2-8. Different CFD models are considered and they are resumed in Table 2-1.
The room congestion is realized by rigid furniture, modeled by still filled blocks in the
uncongested room. Only two room congestion cases have been considered, indicated
respectively as “free room” (where the room is considered without furniture) and as
“congested room” (where furniture is present, see Figure 2-9)
When the failure of non-structural walls is considered (“frangible wall” cases), the walls
are modeled by cut-off pressure panels that are able to increase the porosity of the walls
from 0 (undamaged wall) to 1 (completely damaged wall) after the crossing of a threshold
pressure level (wall strength). In the “non-frangible walls” cases the walls are
undamaged (porosity equal to 0) in all the simulation. The parameters of the cut-off
pressure panels are reported in Table 2-2 [Lees 1980]. Six different ignition locations
inside the kitchen have been considered and the ignition locations are show in Figure
2-10.
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In order to obtain realistic results in a CFD explosion analysis, the adoption of an
appropriate mesh grid is fundamental. The details of the mesh grid are the following:
inside the building ground floor the edge of the cubical cells is always 0.20 meters, while
outside the building the mesh grid is stretched in order to reduce the total number of cells.
The max aspect ratio (the longest side of the control volume divided by the shortest one)
is equal to 5.59, thus lower than 10, as recommended by [Bjerketvedt et al. 1997], while
the total number of cells is about one million. By using a 3.6 GHz CPU computer with 4
Gigabytes of RAM the analysis time of the single scenario is approximately 12 hours.
Closed
windows or doors
gas region
shop
restaurant
elevators
15 m
shop
15 m
kitchen
bar
5m
Boiler
room
5m
Figure 2-8: Main features of the blast scenario
Room
Ignition
congestion (Fig. 12)
Non-frangible
none
a
Non-frangible
yes
a
Frangible
yes
a
Frangible
yes
b
Frangible
yes
c
Frangible
yes
d
Frangible
yes
e
Frangible
yes
f
Frangible
yes
g
Figure 2-9: Congested room model
Simulation Type of walls
Table 2-1: Performed CFD analysis
quote [m]
a
6m
I
II
III
IV
V
VI
VII
VIII
IX
planimetry
f
d
c
g
1.5
f
1.3
1.2
abc
0.1
hd e
e
b
g
5m
furniture
walls
ignition
Figure 2-10: Position of the ignition
points
Figure 2-11 shows the effect of the domestic congestion. Simulations I and II have a
different pressure peak due to the interaction between the flow and the objects inside the
building. In the congested room case (simulation II) the turbulence of the flows increases,
consequently both the burning velocity and the pressure increase as well, inducing more
turbulence. By referring to a certain monitoring point inside the kitchen, the domestic
congestion (simulation II) causes a 43% increase of the pressure peak and a 28% increase
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of the pressure impulse (area under the curves reported in Figure 2-13) with respect to the
uncongested case (simulation I).
In simulation III both failing walls and room congestion are considered. Comparing the
results obtained in simulations II and III the percentage decrements in terms of pressure
peak and pressure impulse (see Figure 2-13) are equal to 66% and 77% respectively
(Figure 2-12). Moreover, the two explosions are completely different in the spatial
development. These results indicate that the failure of the walls significantly modifies the
explosion development.
Figure 2-13 shows a comparison between the pressure time histories, in the same
monitoring point, obtained by simulations I, II and III, where the previously mentioned
differences can be appreciated. Moreover, Figure 2-13 shows that both the pressure
gradient and the time instant corresponding to the pressure peak are highly influenced by
the congestion level. The increase of the congestion level produces an increase in the
pressure gradient, thus anticipating the occurrence of the pressure peak.
0.5 barg
0.3 barg
0.05 barg
Figure 2-11: Max pressures. Effect of the domestic congestion; simulation I on the left
and simulation II on the right
All these results obtained by the CFD computations, clearly shown that the structural
design against such type of explosions cannot be conducted carefully without a specific
evaluation of the action. In terms of design practice, or design standards for civil
buildings, these kinds of simulations can be useful for defining parametric equations with
the aim of appropriately defining the blast load on structural elements, also by taking into
account the above described phenomena.
For risk assessment purposes, due to the high variability of the action with the considered
parameters, specific studies aiming in assessing plausible probability distributions for
those parameters are needed for a correct evaluation of the hazard.
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Panel
Mass
[kg/m2]
exterior walls
interior walls
windows
doors
250
100
20
2
Pressure of
opening
[barg]
0.05
0.03
0.015
0.001
0.5 barg
Table 2-2: Frangible objects characteristics
0.3 barg
0.05 barg
Figure 2-12: Max pressures. Effect of the
frangible walls: simulation III
Figure 2-14 highlights the influence of the ignition location. In this figure the pressure
time histories obtained by simulations III, VII and VIII are shown (all data are referred to
the same monitoring point, considering a congested room and failing walls, but with
different ignition positions). The resulting curves are different both in terms of pressure
peak and in terms of time development (e.g. in terms of pressure gradient and peak time).
0.4
Simulation I
Simulation II
0.15
Simulation III
0.10
Pressure [barg]
Pressure [barg]
0.3
Simulation III Simulation VII Simulation VIII
II
0.2
0.1
VII
0.00
I
0.0
VIII
III
0.05
III
-0.05
-0.1
0.1
0.3
Time [s] 0.5
0.7
Figure 2-13: Pressure time history in the
kitchen (the gas region) for the three
different simulations: I, II, and III
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0.1
0.3
Time [s]
0.5
0.7
Figure 2-14: Pressure time history inside
the kitchen for three analyses with different
ignition locations
Page 40 of 189
2.4 Blast load
A detonation is a very rapid and stable chemical reaction which proceeds through the
explosive material at a supersonic speed in the unreacted explosive [Krauthammer
2008a]. The products of the detonation expand forcing the surrounding air out of the
space that it previously occupied [Ballantyne et al. 2010]. This produces the shock wave
that propagates away from the explosion source. When the shock wave expands it decays
in strength and in velocity but increase its duration. This behavior is caused by so called
spherical divergence. The shock wave is followed by the gas molecules that move at a
lower velocity than the velocity of the shock wave; at this movement of molecules is
associated the dynamic pressure. Generally only one third of the total explosive mass
involve in detonation reaction, the other two thirds deflagrate or simply burn.
Blast loads on structures can be classified into two main groups [DoD 2008]: unconfined
and confined explosions. The confined explosions can be subdivided into [DoD 2008]:
fully vented, partially confined, and fully confined; instead unconfined explosions can be
subdivided into: free air burst, air burst, and surface burst.
Unconfined explosions:



A free air burst explosion occurs in free air and the shock wave propagates away
from the source striking the structure without intermediate interactions.
An air burst explosion occurs when the explosive charge is located in air at a
distance from the structure so that an interaction generally with the ground surface
occurs before of strike the structure. It is the shock wave modified by the
interaction with the surface that engages the structure.
A surface burst explosion occurs when the charge is located very near or on the
ground surface; the resulting shock wave is influenced by this interaction.
Confined explosions:


A fully vented explosion occurs when the explosive charge is adjacent to a nonfrangible wall as a barrier. The initial shock wave interacts immediately with that
structure and the products of detonation are vented to the surround air forming a
leakage pressures which propagate out of the structure of confinement.
A partially confined explosion occurs when the non-frangible structure of
confinement has a limited size opening and/or frangible surface. The initial shock
wave interacts with the confinement structure and the detonation products are
vented relatively slowly, hence more than the shock wave a quasi-static pressure
called gas pressure acts on the structure of confinement.
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
A fully confinement explosion occurs when the structure of containment is full or
quite full closed. The blast loads consist in the shock wave and in a long duration
of the gas pressure.
The explosive outputs of interest are principally the shock wave, the dynamic pressure,
and their impulse.
Free air burst explosion
The explosive charge is positioned so that the shock pressure strikes the structure without
any previous interaction with other structures (ground surface included) [NCHRP 2010].
This shock wave propagated directly from the source is called incident shock wave. The
pressure function of the time is called Ps, and its peak pressure is called incident pressure
(indicated with the abbreviation Pso). Especially for detonations the incident shock wave
is an immediate rising of the pressure at Pso, after it is decay exponentially [Gantes et al.
2004] with time under the atmospheric pressure (positive phase). After it become
negative until reaches the atmospheric pressure (negative phase). The atmospheric
pressure is indicated with the abbreviation Po. The duration time of the positive phase of
the shock wave is called to and the duration of the negative phase is called t-o. The arrival
time of the shock wave is indicated with ta.
The decay law can be described by the exponential following expression:

t  α
PS  PS0 1  e t 0
 t0 
t
(2-34)
Where α is the rate of decay parameter for the specific explosive and scaled distance
[Baker et al. 1983]. The shock wave velocity is indicated with Us and result equal at:
US  C0 1 
PS0
7P0
(2-35)
Where Co is the sound velocity in air for normal conditions at sea level and atmospheric
pressure (Co=340 m/s and Po = 0.1 N/mm2 = 10 bar = 145 psi). The shock wave front
speeds are usually stated relative to the speed of sound modified to account for the
compression of the air; this ratio is called Mach number and indicated with the
abbreviation M.
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Prα
Reflected pressure
Pso
Positive specific impulse
Ps
Incident pressure
Pso
Negative specific impulse
PoP-so
P rα
Po
P-so
ta
t-o
to
ta
Figure 2-15: Free field pressure time
history
t-o
to
Figure 2-16: Reflected pressure time
history
To estimate Pso there are same experimental formula and charts in function of the scale
distance Z. Same expression for free air explosion incident peak pressure expressed in
bar are report following.
6.7
1
Z3
if Pso > 10 bar
(2-36)
0.975 1.455 5.85

 3  0.019
Z
Z2
Z
if 0.1 bar < Pso < 10 bar
(2-37)
PS0 
PS0 
The air behind the shock front is moving outward from the source at lower velocity than
the shock wave. The dynamic pressure depends from the air particles movement. The
peak of dynamic pressure abbreviated with qso is depending from the incident pressure
and can be written as [Zipf et al. 2007]:
q S0 
5PS20
2PS0  7P0 
(2-38)
Now the drag and lift pressure on the structures is given by qso multiply respectively buy
the drag and lift coefficients relatives at the specific structure. Moreover the dynamic
pressure always remains positive because that pressure is determined using the square of
the wind velocity, making it positive regardless of the direction of the wind.
When the incident shock wave impinges a structure, in this case a wall of infinite
extension, the air particles behind the incident shock wave stop to respect the boundary
conditions on the wall (normal velocity on the wall surface equal at zero). Result of this
fact is an amplification of the shock pressure acting on the structure that it is called
reflected pressure, its peak pressure value is abbreviated with Prα; the subscript α indicate
Pierluigi Olmati
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the angle of incidence of the incident shock wave with the wall, therefore α is the angle
that the vector from the explosion source to the wall's point form with the normal vector
at the wall surface from the same wall's point. The reflected pressure depends to the
angle α. The duration of the reflected pressure acting on the surface depends of the real
extension of the wall. The duration time of the Prα on a wall indefinitely extended is
abbreviated with trα. The reflected shock wave propagates away from the surface of
reflection; it moves into an atmosphere modified by the incident shock pressure and full
by the detonations products, consequently the reflected shock pressure travelling at a
greater speed than the incident wave. Moreover it does not expand radially, as the
incident shock wave, at cause of the modified atmosphere.
The value of the reflected shock pressure can be computed by the available charts in
function of the scale distance Z, or by the following expressions valid only for an angle of
incidence of zero degree (the wall is orthogonal at the propagation direction of incident
shock wave). There are same difference about the value of Prα using the above two
approaches especially for strong incident pressure. In the expression of Prα [Zipf et al.
2007] the subscript α is omitted because the equation is valid for an angle of incidence
equal at zero.
Pr  2PS0  (  1)qS0
(2-39)
Where γ is specific heat ratio of the combustion products approximately equal at 1.4. By
the Eq. (2-38) it is possible to have the expression of Prα in function of the Pso.
 7P  4PS0 

Pr  2PS0  0
 7P0  PS0 
(2-40)
Moreover the Prα and its duration trα can be calculated by the charts of Figure 2-17, the
same for compute both to and t-o. Furthermore if the time of a shock load is hypothesized
as a triangular pulse it can be compute by the simple expression below
t
2i
P
(2-41)
Where I is the impulse of the shock wave.
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100000
Z [ft/lb1/3]
P [psi]
Pr [psi]
ir/w1/3 [ms psi/lb 1/3]
is/w1/3 [ms psi/lb 1/3]
t o/w1/3 [ms/lb 1/3]
t a/w1/3 [ms/lb 1/3]
U [ft/s]
Lw/w1/3 [ft/lb1/3]
Pr
10000
P
1000
ir/w1/3
100
is/w1/3
10
1000000
100000
Z [ft/lb1/3]
P [psi]
Pr [psi]
ir/w1/3 [ms psi/lb1/3]
is/w1/3 [ms psi/lb1/3]
t o/w1/3 [ms/lb1/3]
t a/w1/3 [ms/lb1/3]
U [ft/s]
Lw/w1/3 [ft/lb1/3]
Pr
10000
P
ir/w
1000
100
1/3
is/w1/3
U
U
10
1
Lw/w1/3
0.1
1
t o/w1/3
t a/w1/3
0.01
0.001
Lw/w1/3
0.1
t o/w1/3
0.01
t a/w1/3
0.001
0.1
1
Z = R/W1/3
10
100
Figure 2-17: Blast loads for free air burst
explosions, positive phase
0.1
1
Z = R/W1/3
10
100
Figure 2-18: Blast loads for surface burst
explosions, positive phase
Surface burst explosion
This kind of explosion occurs when the explosive charge is on or very near the ground
surface [NCHRP 2010]. Blast loads from a surface burst explosion of a given charge
mass are more intense than from a free air explosion. This fact is due at the confinement
of the explosion energy in a hemispherical shape for the case of surface burst explosion
instead than a spherical shape for the case of free air explosion. Generally if the ground
surface was perfectly rigid, the blast loads for a surface burst explosion would be the
same of a free air explosion due to a double mass of explosive. But the ground surface is
not perfectly rigid, therefore a factor of 1.8 for the mass can be used [Army 2008],
moreover to evaluate the blast loads for a surface burst explosion it is possible to utilize
the same formulas and the same chart of a free air explosion assuming a fictitious mass of
1.8 W. Otherwise it is possible adopting the chart of Figure 2-18 for evaluating the blast
load, or the following formula.
W
W
PS0  6784 3  93 3 
R
R 
0.5
(2-42)
Air burst explosion
An air burst explosion is produced by an explosive charge positioned sufficiently above
the ground [DoD 2008], the high of the charge position is called HC. The incident shock
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wave impacts the ground surface and the phenomenon of shock wave reflection occurs
(as previously explicated). The reflected shock wave originated on the ground surface
propagates into an atmosphere modified by the incident shock wave and full of the
detonation products; consequently its speed is generally greater and then the speed of the
incident shock wave. Furthermore the velocity of the reflected shock wave is not
constant and so it does not propagate radially. The intersection of the incident shock
wave and reflected wave is no longer on the ground surface; a new shock wave (mach
stem, abbreviated M) connects the ring intersection point of incident, reflected, and M
shock wave to the ground surface. The point of this intersection is called triple point. A
simple representation of the mach stem front is in Figure 2-19.
The high of the triple point is abbreviated HT, the projection point of the explosive charge
on the ground surface is called ground zero, the distance from the base of the structure to
the ground zero is called ground distance, and the Z of the ground distance is called
ground scaled distance ZG.
HT increase as the M propagates away from the ground zero. The structure will be
considered subjected loaded by M hypothesized as a plane wave if HT is greater than the
high of the structure.
Reflected
wave
α
Hc
Incident
wave
Mach front
Path of triple
point
Ground zero
HT
Protective
structure
Ground surface
RG
Figure 2-19: Air burst explosion scenario [DoD 2008]
Confined explosion
A confined explosion occurs when the explosive charge is near at the structure or
enclosed between more structures [DoD 2008]. The effects of the confinement are a high
reflection of the incident wave, the confinement of the product of the detonation, the
leakage of both the shock wave and detonation's products outward the structure of
confinement. The exterior pressure is called “leakage pressure”, the pressure due at the
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incident wave and reflected is called “shock pressures”, and the pressure due at the
accumulation of the detonation's products is called “gas pressure”.
The actual distribution of the blast loads is irregular because a multiple reflections occur.
Whereby the blast loads are averaged on the wall surface, so it is hypothesized that a
structure is able to distribute the peaks of loads acting on its surface.
The concept of structures of confinement is strictly referred at their resistance and mass.
If a structure is able to reflect the shock wave it will be a non-frangible structure,
although it fails after the shock reflection event. If a structure is not able to reflect the
shock wave it is a frangible structure. In other hands a structure of confinement is
considered frangible or not in function at its dynamic behavior under the blast loads.
Therefore an element which is not considered frangible for the shock pressure may be
frangible for gas pressure. In general for structure with a closure's resistance to outward
motion (resistance of the frangible element) less than 0.0012 MPa the frangibility can be
considered related only at its mass; instead for the structure with a resistance greater than
0.0012 MPa the frangibility must be evaluated considering both its mass and resistance.
However if the blast pressure is very large in comparison to the resistance of the element,
the effects of the resistance can be neglected without introducing significant errors.
The structures of containment considered are a single wall or a set of wall forming a
cubical structure with same sides open. To determine blast loads it is necessary know the
configuration of the structure and obviously the mass of the explosive charge. The wall
parallel and opposite to the surface in question has a negligible contribution to the shock
loads for the range of parameters used, so it is not considered.
From a set of charts [DoD 2008] depending from the previous geometrical ratios and
scaled distance is it possible compute the peak of the reflected pressure and its impulse.
The shape of the shock wave can be considered triangular, whereby duration of the shock
wave can be estimated as the double impulse divided by the peak of the shock wave.
However these charts do not take account the increased blast effects produced by contact
charges with the surface in question, so in this case they cannot be applied. A minimum
distance between the explosive surface and the wall surface must be applied.
Blast regions
Coupling the blast loads and the dynamic response of the structure three range of design
are defined: high pressure design range, low pressure design range, and very low pressure
design [DoD 2008]. The occurring time of the maximum structural response is
abbreviated with tm and the duration time of the blast load is abbreviated with to, so the
ratio tm/to can established the limit of the pressure region [Biggs et al. 1964].
At the high pressure range the duration of the applied load is short in comparison to the
response time of the structure. The structure can be designed only for the impulse, (tm/to
> 3).
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At the low pressure range generally the peak pressure is smaller than at high pressure
range, however the duration of the load is comparable with the response time of the
structure that can be designed for both pressure and impulse loads, (3 > tm/to > 0.1).
At very low pressure range the duration of the load is greater than the response time of
the structure that can be designed only for the maximum pressure, (tm/to > 0.1).
If both a same structure and explosive charge are considered the design range depend
from the distance from the source of explosion.
Figure 2-20: Parameters defining pressure design ranges [DoD 2008]
Blast loads on structures
Evaluate the blast load on structures is a complex issue. To understand the phenomenon a
simple rectangular structure on the ground surface is take in account. Moreover it is
feasible extend the procedure to include structures with other shapes and above or under
the ground surface. For the typical explosion scenarios of free air burst, air burst and
surface burst the peak of the incident, reflected, dynamic pressures are established with
their duration and impulse. For designing purpose the pressure time curve can be
simplify with an equivalent multi linear path. Thence the duration time of the incident
positive phase is a fictitious time computed as:
t of 
2iS
PS0
(2-43)
The fictitious duration of the dynamic pressure can be assumed equal at the fictitious
duration of the incident pressure Ps. The equivalent negative pressure time curve has a
time of rise equal to 0.25 to whereas the fictitious duration t-of is given as tof formula but
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the impulse and peak pressure is relative at the negative phase. The effect of the dynamic
pressure in the negative phase region usually is neglected.
Therefore with refer at a rectangular structure it is necessary define the blast loads on the
front, side and back walls.
Figure 2-21: Approximate load on a shelter [Baker 1983]
Front wall
As previously mentioned when a shock wave engages a surface normal or with an
incident angle the reflection phenomenon occurs; a uniformly average pressure on the
walls is assumed [DoD 2008]. The fictitious duration time of the reflected pressure is
given as:
t r 
2i r
Pr
(2-44)
If the wall is infinitely extended and non-frangible. However the walls of the structure
have finite dimensions, so at the corners of the wall exists a discontinuity about the
pressure value. There is no physical means to maintain this pressure imbalance, whereby
a “clearing” wave propagates towards the center of a reflecting surface from all the free
edges that reduces the reflected pressure to an incident and dynamic pressure. The
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average time required for the clearing wave to propagate to the edges of a surface from
the center of the surface is the clearing time tc.
tC 
4S
(1  R )C r
(2-45)
where with referring at the figure Figure 2-22:




S is the clearing distance, equal at the smaller between H or W/2
H height of the structure
R ratio of S/G where G is the greater between H or W/2
Cr sound velocity in reflected region.
Figure 2-22: Front wall loading [DoD 2008]
If tC is less than trf the pressure time curve is affected by clearing. It is necessary to adopt
the curve gives the smallest value of the impulse. After the reflection phase on the front
wall acts the incident pressure added with the dynamic pressure, the last one is affected
by the drag coefficient depending from the pressure range; however it can be assumed
equal at 1. The negative phase is possible to compute assuming a time of rise equal to
0.25 to.
Roof and side walls
The roof and the side walls are loaded only by the incident and the dynamic pressure
because are in side-on position respect the propagation of the shock wave. The shock
wave propagates on the roof and side walls and it decays with the distance from the
source increases, therefore the walls are not uniformly loaded.
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For designing purpose a uniformly pressure on the walls is hypothesized, the peak
pressure is the sum of contribution of the equivalent uniform pressure and drag pressure:
PR  CE Psf  CDq sf
(2-46)
Where Psf and qdf are computed at the nearest point of the walls to the explosive charge
(point f in the Figure 2-23), in other hands for a rectangular structure they are computed
respectively at the front wall position. The drag coefficient CD for the roof and side walls
is a function of the peak dynamic pressure.
Instead the coefficient CE is the equivalent load factor equal at the ratio between the PR
and Pof function of the ratio between the wave length LW and the length of the wall L. LW
is defined as length at a given distance from the detonation which, at a particular instant
of time, is experiencing positive pressure. There is a wave length for a positive and
negative phase.
Figure 2-23: Roof and side walls loading [DoD 2008]
Rear wall
The rear wall is loaded by a set of waves made by a reflection of both the incident and
rarefaction wave. Whereby the effective load on the rear walls is complex to estimate, an
average pressure time curve is adopted similar at that of the roof and side walls; the
coefficient CE is computed considering the high of the rear wall HS instead the length of
the wall L.
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Figure 2-24: Rear wall loading
Rectangular structure with opening
The attention is focused on the front wall because the effect of opening limits the
reflection phenomenon. If the windows and doorway are frangible the cleaning time will
be reduced, whereby the pressure time curve of the applied blast load acting on the front
wall of a structure with openings is the same as that of a front wall of a closed rectangular
structure except the clearing time will be reduced. In other hand a t'C is evaluated by
compute a fictitious cleaning distance S', that is a weighted average cleaning distance of
each sub panel of the front wall; the weights are the nearest distance from an edge and the
cleaning factor. A detailed description of the theory is illustrated in the [DoD 2008].
The image charge method
Due to the walls delimiting the testing site multiple reflections of the original shock wave
occurred; consequently the blast load on the specimens is greater than the blast load on a
specimen tested in an open space. For taking account the phenomenon of the
reverberated shock waves [US Army 2008] the Arbitrary Lagrangian Eulerian method
[Bontempi et al. 1998] is the most appropriate method, but it is very computationally
expensive. Using the uncoupled approach [NCHRP 2010] the image charge method
provides acceptable results without increasing the computational effort.
The image charge method predicts the pressure pulses from a reverberating shock wave.
The image charge method consists in taking account the pressure pulse from a
reverberating shock wave by a pressure pulse due to a spherical free-air detonation of a
fictitious (image) charge with the same weight of the actual charge but located at a standoff distance from the target equal to the full path length (see for example both the paths B
and C in Figure 2-25) of the shock wave to the reflecting wall and then to the target; and
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hitting the target with the angle of incidence of the reverberating shock wave. Figure
2-25 is adapted from [US Army 2008] and it shows an elementary scenario of
reverberating shock waves. The path A is the direct shock wave path; instead the paths C
and B are the reverberating shock wave paths on the target.
Reflecting surface
C
A
B
Reflecting surface
Figure 2-25: Image charge approximation, figure adapted from [US Army 2008]
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3 THE LOCAL RESISTANCE
The local resistance is the resistance against a blast load of the single structural element
(component) of the structure. The blast demand is applied directly to the component,
which can be load bearing or non-load bearing.
Generally the non-load bearing components are loaded directly by the blast load. The
load is consequently transferred to the load bearing components like columns, walls or
girders. Instead if the load bearing components are isolated the blast demand is directly
applied to such components. Exception is the case of the slabs and roofs where the blast
load is generally applied directly.
Non-load bearing components like cladding wall panels are crucial for protecting the
inside of a building. The fatalities and the equipment damages depend by the ability of
the non-load bearing component to resist to the blast load.
In the following the local resistance of two kinds of cladding wall panel system, of one
kind of slab, and of a steel built-up blast resistant door is investigated.
Two approaches are adopted here for investigating the local resistance: the fragility
analysis and the detailed explicit finite element analysis. The fragility analysis is adopted
for the case study of the concrete cladding wall panel system and of the steel built-up
blast door; while the detailed explicit finite element analysis is adopted for the case study
of the insulated cladding wall panels and of the concrete slab.
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3.1 The scaled distance as intensity measure
The wall under investigation is a non-load bearing precast concrete wall panel used as
exterior cladding for buildings. Typically the length and the width of these walls are
subject to specific architecture requirements while their thickness is approximately 15 cm.
The steel reinforcements are generally placed in the middle of the cross section. This
wall is assumed to be designed to protect inhabitants and equipment within from external
detonations.
The non-linear dynamic analyses are carried out by the well-established method of the
equivalent non-linear SDOF system, where the precast concrete wall is modeled by an
equivalent non-linear SDOF on the basis of energetic considerations. Both the FCs and
the probability of failure of the cladding wall are computed using MC simulations.
The FCs are evaluated for the investigated cladding panel referring to each Component
Damage Level (CDL) defined in a PBD prospective. Then the FCs are used to estimate
the failure probability of the cladding panel subjected to blast load scenarios (vehicle
bomb). Finally, the probability of failure of the wall panel subjected to the same
scenarios is estimated by with MC simulation and compared to the results obtained with
the FCs.
As a first step, the uncertainties characterizing blast engineering problems need to be
properly individuated and addressed (see Figure 1). These uncertainties can be divided
into three main groups:



Hazard uncertainties (e.g. explosive, stand-off distance).
Structure uncertainties (e.g. stiffness, dimensions, damping, material
characteristics, damping, etc.).
Interaction mechanism uncertainties (e.g. the reflected pressure, pressure duration,
etc.).
This classification of the uncertainties in three groups (load, structure, interaction
mechanisms) is generally valid for many engineering fields.
The IM in general should be chosen among the first group of uncertain parameters or as a
combination of those parameters, while the entity of the blast action is determined by the
parameters characterizing the interaction between the IM and the structural parameters.
In probabilistic terms, hazard and structural parameters can be characterized as
unconditional with respect to parameters belonging to one of the other two groups, while
parameters representing the interaction mechanisms must be usually characterized in
conditional probabilistic terms with respect to the hazard and the structural parameters
[Ciampoli et al. 2011].
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Vehicle bomb
Cladding wall
p [W]
p [Θi]
w [kgf]
θi
Fence barrier
Stand-off distance
p [R]
r [m]
Figure 3-1: Uncertainties parameters in blast engineering problems
3.1.1 Blast load model
The side-on blast pressure Ps0 [MPa] can be estimated by the expression given in [Mills
1987] (Eq. (3-1)). While the side-on specific impulse i0 [Pa∙sec] is estimated by the
expression given [Held 1983] (Eq. (2-2)).
(
)
(
)
( √ )
√
( )
(3-1)
(3-2)
(3-3)
Where Z is the scaled distance, W is the explosive (charge) weight (here in kgf of TNT),
instead R is the stand-off distance (here in m).
Both Eq. (3-1) and Eq. (3-2) are valid for free-air explosions. In this study detonations
occurring on a surface are considered (surface explosions); therefore the energy of the
detonation is confined by the ground surface. This phenomenon, referred to as reflective
pressure, is taken into account by using the same equations for the free-air explosions but
assuming that a given W on a deformable ground produces the same load as a free-air
explosion of a charge weight equal to 1.8 W.
The reflected pressure Pr [MPa] for a normal angle of incidence is computed using [Mills
1987]:
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(
(3-4)
)
Where Patm is the atmospheric pressure (0.1 MPa).
Without loss of generality the negative pressure phase is neglected from the blast load
time history [DoD 2008], while the duration of the positive phase of the blast load is
computed by assuming a triangular shape of the load function, given by:
(3-5)
The atmospheric pressure is obtained starting from the reflected pressure by means of the
Friedlander pulse shape [US Army 1985] as shown in Eq. (3-6).
()
(
(3-6)
)
Where ta is the arrival time of the blast load, taken here equal to zero, and β is the decay
coefficient. In this study a value of 1.8 for β is assumed.
The clearing effect is neglected in this study since the cladding wall is part of a building
façade; and thus no conditions are satisfied for the clearing of the reflected shock wave
[Chang et al. 2010].
In Figure 3-2 some blast pressure time histories computed for different values of W (in
kgf) and R (in m) are shown. The obtained curves are found to be in good agreement with
the curves obtained by SBEDS [US Army 2008].
The blast load is considered uniformly distributed on the cladding wall, which is typical
for values of the scaled distance Z higher than approximately 1.2 to 2.0
.
Figure 3-2: Blast loads (surface explosions) by the adopted model (dotted lines) and the
SBEDS model (continuous line)
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3.1.2 Cladding panel model
The cladding panel examined as case study is part of a frame structure. A 3500 mm long
and 1500 mm wide panel is used, with a cross sectional thickness of 150 mm. The
supports of the panel are placed on the external frame beams of the building. Length,
width and cross sectional thickness are assumed as stochastic variables. The mean values
and Coefficients of Variations (COVs) used in the analysis are shown in Table 3-1. The
longitudinal reinforcements consist of 10 rebar with a diameter of 10 mm, and they are
placed in the middle of the cross section. The reinforcement strength mean value and
coefficient of variation are shown in Table 3-1 as well. The panel under investigation
does not have shear reinforcement as transversal reinforcement is not significant for the
considered (flexural type) behavior and for the purpose of this study.
Symbol
Description
Mean
COV Distribution
fc
Concrete strength
28MPa
0.18
Lognormal
fy
Steel strength
495 MPa
0.12
Lognormal
L
Panel length
3500 mm
0.001 Lognormal
H
Panel height
150 mm
0.001 Lognormal
b
Panel width
1500 mm
0.001 Lognormal
c
Panel cover
75 mm
0.01
Lognormal
W
Explosive weight
227 kgf
0.3
Lognormal
R
Stand-off distance 15 m 20 m 25 m 0.05
Lognormal
1
Table 3-1: Input data
3.1.2.1 Concrete
The concrete compressive strength fc is assumed as random variable, while the Young’s
modulus of the concrete Ec and the concrete density ρ are expressed as deterministic
functions of fc. The mean value of fc is 28 MPa, with a COV of 0.18 as adopted in
[Enright et al. 1998] with a lognormal probability density function (see Table 3-1).
The Young’s modulus is computed by Eq. (2-7) [EN 1990] while the concrete density is
computed by Eq.(2-8) [ASCS 1988]. Both Ec and fc are expressed in MPa while ρ
expressed in kg/m3.
(
(
(
)
(3-7)
)
(3-8)
)
The compressive strength enhancement of the concrete due to strain velocity is
considered in this study. This strength enhancement is taken into account by means of the
Dynamic Increasing Factor (DIF): a multiplicative coefficient of fc. Since the
compressive strength enhancement of the concrete varies slightly in case of ductile
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flexural response over the range of the considered strain velocity, the DIF is assumed as
constant and equal to 1.19 times the static compressive strength.
This hypothesis is in accordance with the compressive strength enhancement proposed in
[US Army 2008], and increases computational efficiency by avoiding additional cyclic
iterations in the algorithm of the SDOF equation solver. However, cyclic iterations are
necessary for computing the strength enhancement on the reinforcement, which for
ductile flexural response is more important than the compressive strength enhancement of
the concrete.
3.1.2.2 Reinforcing steel
For this case study, grade 450 MPa steel is used. For estimating the mean value of the
yielding strength, an average strength factor equal to 1.1 provided by the [US Army 2008]
is adopted. Instead the COV is of 0.12 as proposed in [Enright et al. 1998] for a
lognormal probability density function (see Table 3-1: Input data). The Young’s modulus
is assumed as deterministic and taken equal to 210 GPa.
The steel strength enhancement due to the strain velocity is taken into account by the
Cowper and Symonds model [Cowper et al. 1957]. Thus, the DIF is evaluated by Eq.
(3-9):
(
(3-9)
)
Where dε/dt is the strain velocity of reinforcement, q is taken equal to 500 sec-1 and ξ is
taken equal to 6. Both C and p are estimated by fitting the strength increasing trend
versus the strain velocity given in [US Army 2008]. Eq. (3-9) is shown in Figure 3-3. By
solving the SDOF equation, the DIF is iteratively updated until the convergence threshold
is reached.
2
DIF [-]
1.8
1.6
1.4
1.2
1
0.001 0.01
0.1
1
10
Strain-rate [1/sec]
100
Figure 3-3: Reinforcing steel strength enhancement versus strain velocity
The strain velocity of the steel reinforcement (dε/dt) in Eq. (3-9) is evaluated by the Eq.
(3-10).
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(
)
(3-10)
Where L is the length of the cladding panel, Ec is the Young’s modulus of the concrete, Jc
is the moment of inertia of the cracked cross section, d is the distance from the external
compressed fiber of the cross-section to the centroid of the tensile reinforcement, and
dS/dt is the rate of the resistance force (S) developed by the panel when subjected to the
demand load. Eq. (3-10) is valid for simply supported elements, when the response is
governed by the flexural behavior without shear failure.
3.1.2.3 Mechanical model for the cladding panel under blast load
In order to model a structural element subjected to a blast load with an equivalent SDOF,
the latter is defined as a system that has the same energy of the original structural element
(in terms of work energy, strain energy, and kinetic energy) when the last one, if
subjected to a blast load, deflects in a given deformed shape. The displacement field of
the component can be expressed as u(t)=Φ(x)ymax(t), where Φ(x) is the assumed deformed
shape of the component under the blast load. Furthermore, displacement of the
component is obtained by the SDOF equation:
̈( )
̇( )
( ( ))
()
(3-11)
Where y(t) is the displacement of the component and M is the total mass of the
component, S(y(t)) is the resistance of the component as a function of the displacement
expressed in unit force (see Figure 3-4), F(t) is the blast pressure multiplied by the
impacted area (A) expressed in force units, C is the damping (the percentage of the
critical damping is assumed to be 1 % in all the analyses), KLM is the load-mass
transformation factor equal to the ratio of KM and KL (the mass transformation factor and
the load transformation factor respectively). The last two are evaluated by equating the
energy of the two systems (in terms of work energy and kinetic energy respectively).
L
KL 
 p(x)(x)dx
0
L
 p(x)dx
0
L
KM 
 m ( x )
2
( x )dx
0
(3-12)
L
 m(x)dx
0
Referring to Eq. (3-12) and Figure 3-4, p(x) is the blast load shape on the component,
m(x) is the distributed mass, and r is the resistance of the element in terms of pressure.
The load-mass transformation factor KLM is different at each deformation stage of the
component response; for a bilinear resistance function two values of the KLM can be
defined: one for the elastic response and one for the plastic response. More details on the
equivalent SDOF method are provided in [Biggs et al. 1964] and [US Army 2008].
In order to obtain the bilinear resistance function of the simply supported concrete
cladding wall, it is necessary to compute the resisting moment (Mr) in the mid-span of the
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panel. The yielding point of the resistance function (Sy) is obtained by Eq (3-13) (a),
where the loaded surface (A) is equal to the cladding wall length (L) times the cladding
wall width (b).
( )
(3-13)
( )
The resisting moment is evaluated by Eq.(3-14) [US Army 2008]:
(
)
(3-14)
Where, As is the reinforcements area, fdy is the dynamic yield strength of the reinforcing
steel, fdc is the dynamic compressive strength of the concrete, b width of rectangular
section.
It is also necessary to evaluate the yielding displacement of the resistance function. For a
simply supported component the yielding displacement (δe) is given by Eq. (3-15).
(3-15)
(
)
(3-16)
Where J is the moment of inertia of the cross section as evaluated by Eq.(3-16), while Jc
and Jg are computed by Eq. (3-17) and Eq. (3-18) respectively.
(3-17)
(3-18)
The coefficient F in Eq. (3-17) is evaluated starting by the design chart provided in [US
Army 2008]. In this study an analytical formula (see Eq. (3-19)) is determined by fitting
the curves of the original chart.
(
)( )
(3-19)
Where p is the percentage of reinforcements in the cross section of the panel as evaluated
by neglecting the reinforcements cover, and n is the ratio between the steel Young’s
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modulus and the Ec. Eq. (3-19) is valid for singly reinforced cross-sections. Since Sy and
δe are evaluated, the resistance function of the simply supported cladding wall can be
defined. Implementing Equations from (3-12) to (3-19), the central difference method is
used to solve Eq. (3-10) [Chopra 1995]; the latter presents three non-linarites: firstly due
to the bilinear shape of the resistance function, then due to the load mass transformation
factors, and finally due to the dynamic strength enhancement of the reinforcing steel
(which affects the resistance function).
Sel = rel A
-rel
r
L
Elastic Plastic
Φelastic
δel
S=rA
-r
L
δ
Φplastic
Tension
membrane
effect (tm)
δel δlim δtm
Mplastic
Figure 3-4: Component resistance - displacement relation
3.1.3 Response parameters
For structural components subjected to blast loads in flexural response regime, generally
two response parameters are of interest: the support rotation angle (θ) and the ductility
ratio (μ). These parameters are defined in Eq. (3-20) and Eq. (3-21):
(
(3-20)
)
(3-21)
Where δmax is the maximum displacement of the component.
A structural component subjected to a blast load is generally expected to yield (ductility
greater than 1), as it is economically impractical to design a member to remain in elastic
range. While other significant response parameters can be defined, for example [Low et
al. 2001] considers the strain on reinforcements. This study focuses to the response
parameters usually adopted for antiterrorism design [US Army 2008].
In a performance-based blast design prospective, five Component Damage Levels (CDLs)
are considered: Blowout (BO), Hazardous Failure (HF), Heavy Damage (HV), Moderate
Damage (MD), and Superficial Damage (SD). Following the [US Army 2008], the above
mentioned levels are defined as follows:
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




Blowout (BO): the component is overwhelmed by the blast load causing debris
with significant velocities.
Hazardous Failure (HF): the component has failed, and debris velocities range
from insignificant to very significant.
Heavy Damage (HD): the component has not failed, but it has significant
permanent deflections causing it to be un-repairable.
Moderate Damage (MD): the component has some permanent deflection. It is
generally repairable, if necessary, although replacement may be more economical
and aesthetic.
Superficial Damage (SD): the component has no visible permanent damage.
The thresholds corresponding to these CDLs are defined in terms of the response
parameters θ and μ. For a non-structural concrete cladding wall without shear
reinforcement, neglecting tension membrane effect, the CDL thresholds are those
reported in Table 3-2 below. The FCs computed in the following for each of the
mentioned CDLs.
Component damage levels θ [degree]
Blowout
>10°
Hazardous Failure
≤10°
Heavy Damage
≤5°
Moderate Damage
≤2°
Superficial Damage
none
1
μ [-]
none
none
none
none
1
Table 3-2: Component damage levels, and the associated thresholds in terms of response
parameters
3.1.4 Fragility curves
As early described the blast load on the panel depends by the peak pressure and the
impulse density (Eq. (3-2) and Eq. (3-4)). The pressure is inversely proportional to Z
(Eq. (3-3)), while the impulse density depends on both the Z and the W (Eq. (3-1) and Eq.
(3-2)). Consequently, two detonations with the same Z can have different impulse
density, depending on the amount of explosive. Thus, the two explosions have the same
peak pressure but different duration.
Summarizing, the blast load depends on both the Z and the W. Therefore the choice of
the IM for computing the FCs is a crucial issue. In this study, the Z is taken as the IM
(higher values of the Z the cladding wall has an inferior structural response than for lower
Z values), some aspects related with this choice are discussed in the next section.
FCs are developed for each of the CDL, the algorithm implemented in MATLAB® for
FCs evaluation is shown in Figure 3-5. With reference to the same figure, “i” indicates
the i-th point of the FC, “j” indicates the “j”-th CDL, and “k” indicates the k-th stand-off
distance R for which the FC is computed. “N” is the maximum value for “i” and therefore
the total number of points forming the FC. “M” is the maximum value for “j” and
Pierluigi Olmati
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therefore the total number of the CDLs. Finally “L” is the maximum value of “k”
therefore the total number of the Rs for which the FC related to the j-th CDL is computed.
The name “FC-CDL (j, k)” indicates the FC computed for the k-th R, the j-th CDL, by
varying the W. The i-th point of the FC (named FC-CDL (i, j, k)) is computed by
considering the blast load at the k-th R and the i-th W. The minimum and maximum
amount of W should be enough for computing the values of the FC-CDL (j, k) ranging
from 0 to 1 (or from 0 to 100 %).
The FC-CDL (i, j, k) is obtained by a MC simulation, the complete (cyclic) procedure of
Figure 3-5 is hereby described.





First a k-th R is selected.
Then the j-th CDL is selected.
Consequently the “i” index is increased by solving the previously introduced
equations for each “i” in order to evaluate the i-th points of the FC-CDL (j, k)
until tracking the complete FC-CDL (j, k).
After that a new j-th CDL is considered with the same value of “k”. When j=M a
different R is selected and the previous two described cycles are repeated until
k=L.
Finally the piecewise curves obtained point by point with the above steps they are
interpolated by a lognormal cumulative function, see Figure 3-7.
As said, the FCs describe the conditional probability of failure (Pf(X>x0|Z)) of the
response parameter X (most critical between the values of θ and μ, see Table 3-2) with
respect to the threshold x0 (identifying the CDL). As expected, for a constant number of
samples at each i-th point, the COV of Pf(X>x0|Z) increases with the decreasing of
Pf(X>x0|Z). In order to obtain an acceptable COV, the number of samples adopted in the
analysis is increased with decreasing Pf (this means that the number of samples increase
with increasing Z). In this work, an exponential law has been set for this increasing trend.
In Figure 3-6 the number of the samples and the relative COVs are plotted in function of
the Pf(X>x0|Z) for the FC related to the HF CDL and for R equal to 20 meters.
For better understanding the sufficiency of the adopted intensity measure (scaled distance
Z), some considerations can be made with reference to the pressure-impulse diagrams
[Krauthammer 2008] related to the case study panel.
For this purpose, reference is made to the mean values of both materials and geometrical
parameters (see Table 3-1), and the DIFs for the concrete and steel are taken as constant
and equal to 1.19 and 1.20 respectively. The pressure-impulse diagrams referred to
different values of θ are shown in Figure 3-8: the impulse density expressed in kPa∙ms is
related to the pressure peak measured in KPa. Generally three regions can be
individuated in the pressure-impulse diagrams, each related with a different regime of
structural response subjected to a load time history. These are defied as: impulsive (I),
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dynamic (D), and pressure (P) region, depending on the characteristics of the load time
history with respect to the dynamical proprieties of the structure [DoD 2008].
R=k
•
•
•
•
CDL (j)
•
j=1 i=1 k=1
i=i+1
Z=i
MC analysis
FC-CDL (i, j, k)
NO
i=N ?
•
•
•
•
•
•
•
CDL: Component Damage Level
R: Stand-off distance
Z: Scaled distance
FC-CDL: numerical Fragility Curve
of the CDL
i: the i-th point, of the j-th FC-CDL
corresponding to the k-th R
j: the j-th CDL
k: the k-th R
MC analysis: Monte Carlo analysis
N: number of FC-CDL points, or
number of the Zs
M: number of the CDLs
L: number of the Rs
Interpolated FC-CDL: lognormal
interpolated FC-CDL
j=j+1
YES
FC-CDL (j,k)
NO
j=M ?
k=k+1
YES
FC-CDL (k)
NO
k=L ?
YES
FC-CDL
Lognormal
Interpolation
Interpolated
FC-CDL
Figure 3-5: Fragility curves computing process
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0.10
N° of samples
80000
0.08
C.O.V.
0.06
60000
40000
0.04
20000
0.02
0
C.O.V.
N of samples
100000
0.00
0.1
0.9
3.3
9.0
22.4 40.4 59.5 77.9 90.1 96.6 98.8 100.0
Pf(X>x0|Z) [%]
Figure 3-6: N° of samples and COVs for the FC relative to the HF and R equal to 20
meters
100
Numerical
FC
Pf piecewise
Pf (X> x0|Z) [%]
80
Pf smooth FC
Interpolated
60
40
20
0
3.7
3.9
4.1
4.3
4.5
Z
Figure 3-7: Numerical and lognormal interpolated FC
Two blast loads are now taken into account, these can be chosen in order to being
characterized by the same IM (and consequently the same peak pressure) but by different
W and R. As matter of fact, the two blast loads are consistent with two different demands
on the pressure-impulse diagrams, having the same peak pressure but different values of
the impulse density.
As it can be observed in the Figure 3-8, the difference between the structural responses of
the panel subjected to two different blast demands (again, characterized by the same
pressure peak but by different impulse densities) depends on where these demands are
located in the pressure-impulse diagram: if these blast demands are located in the
impulsive region (I), a certain value for this difference will be observed, while if blast
demands are located in the dynamic (D) or pressure (P) region, then this difference will
be lower than the previous case.
Considering the above, it can be concluded that the sufficiency of chosen IM is greater in
the D and P regions than in the I region. In the I region a fragility surface made by
considering both R and W as independent elements of a vectorial IM would be more
appropriate.
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For taking account this approximation, as explained in the previous section, the FCs are
computed for different values of the R (R=Z∛W). In what follows, when the FCs are
used for estimating the failure probability of a component damage level, the specific FC
corresponding to the mean value of R is used for this purpose. This increases
considerably the sufficiency of the chosen IM.
1000
I: impulsive region
D: dynamic region
P: pressure region
I
100
P [kPa]
D
P
θ=2
10
θ=5
θ=10
1
100
1000
10000
i [kPa ms]
100000
Figure 3-8: Pressure - Impulse diagrams
In Figure 3-9 the computed FCs are shown for different values of R. Focusing on the
considered CDLs, from the figure can be observed that, as expected, the FCs of the SD
level have a different slope compared to that of the other three CDLs (HF, HD, and MD).
It should be noted that the SD level is based on the ductility (μ) of the component while
HF, HD, and MD levels are based on the support rotation (θ). The SD level for a concrete
cladding panel prescribes the elastic response of the component, and for the case study
panel it appears to be more sensitive to the considered uncertainties compared to the HF,
HD, and MD levels. By varying the number of samples the maximum obtained COV for
the lower probability of failure (close to zero), is about 9%; see Figure 3-6. This value is
considered acceptable for the specific case, and it is consistent with other studies on blast
applications (see [Stewart et al. 2008]).
Hazardous Failure
100
80
Pf (X> x0|Z) [%]
Pf (X> x0|Z) [%]
100
60
40
20
Heavy Damage
80
60
40
20
0
0
2.4
2.6
2.8
Z
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3.2
3.4
2.8 3.0 3.2 3.4 3.6 3.8 4.0
Z
Page 69 of 189
Moderate Damage
Superficial Damage
100
80
Pf (X> x0|Z) [%]
Pf (X> x0|Z) [%]
100
60
40
20
0
80
60
40
20
0
3.0
3.5
Z
4.0
4.5
5.0
5
6
7
Z
8
9
10
11
Figure 3-9: From top left clockwise, fragility curves for the HF, HD, SD, MD component
damage levels
For computing the probability of failure from the FCs it is necessary to stochastically
characterize the blast scenario and integrate Eq. (3-23). In this study a vehicle bomb is
considered. The amount of explosive (W) in the vehicle depends among else on the
security measures in place. These security measures can be structured in different lines
(see Figure 3-10) and for each line of security a different mean value of W is expected.
The expected value expected value of W decrease with the decreasing of R from the
target, since the line of security system reduces progressively the severity of the possible
attacks.
In the example of Figure 3-10, level 1 prevents trucks entering the target zone, so no truck
bomb should be expected. Level 2 in Figure 3-10 (for example a fence barrier) prevents
vehicles entering. Finally, Level 3 prevents pedestrians approaching the target.
With this in mind, in the specific application a scenario concerning a truck bomb (with
about 4000 to 27000 kgf of TNT or equivalent) is unreasonable (e.g. by assuming that the
intelligence service is able to prevent this threat). Instead a vehicle bomb (with about 27
to 454 kgf of TNT or equivalent) is considered. The mean amount of TNT or equivalent
in the vehicle is assumed equal to 227 kgf with a COV equal to 0.3 (see Table 3-1). This
assumption is in line with [FEMA 2003]. Instead the R is assumed to have a mean value
equal to 20 m with a COV of 0.05, considering that the vehicle could impact and overpass
the fence barrier but without being able to move further.
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Street
Target
Level 3
Level 2
Level 1
Figure 3-10: Lines of defense
The conditional probability of failure of the CDL with respect to the event (P f(X>x0)) is
evaluated by Eq. (3-23). As previously stated, X is the most critical between the response
parameters θ and μ, assumed here as uncorrelated. Consequently Pf(X>x0) is the union of
the failure probabilities evaluated by considering only one of the two response parameters
characterizing the component damage level (see Table 3-2 and Eq. (3-22)). The
probability density function of the Z (p(Z)) is computed by fitting both W and R with a
lognormal distribution. As mentioned above, the FC (Pf(X>x0|Z)) used for evaluating Eq.
(3-23) is the one corresponding to the mean value of the R (Table 3-1).
(
(
)
∫
)
(
(
) ( )
)
(
∑
)
(
(3-22)
) ( )
(3-23)
The obtained results are shown in Table 3-3. The first column reports the CDLs, while
the second and third the Pf(X>x0) for each blast scenario obtained by Eq. (3-23) and by
the MC analysis respectively.
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CDL
Mean W=227 kgf COV=0.3 lognormal distribution
Mean R, COV=0.05 lognormal distribution
FC analysis
1
SD
MD
HD
HF
100.0 %
96.6 %
55.7 %
13.6 %
SD
MD
HD
HF
100.0 %
74.6 %
14.2 %
1.02 %
SD
MD
HD
HF
100.0 %
97.9 %
93.6 %
67.8 %
MC analysis
R = 20 m
100.0 %
97.5 %
55.5 %
12.1 %
R = 25 m
100.0 %
77.3 %
12.6 %
1.02 %
R = 15 m
100.0 %
99.9 %
96.9 %
72.6 %
Difference Δ%
0.0 %
0.9 %
0.3 %
11.0 %
0.0 %
3.5 %
11.2 %
0.0 %
0.0 %
2.0 %
3.4 %
6.6 %
Table 3-3: Results
From these results the maximum percentage difference between the Pf(X>x0) computed
by the FCs and the MC simulation is 11%. Further studies are necessary to confirm
whether this difference is acceptable or not.
However it is also necessary to consider that the W in the vehicle has an elevated
dispersion, something that amplifies this difference due to the dependence of the impulse
density to both the R and the W. Thus, the difference between the Pf(X>x0) computed by
the fragility analysis method and by the MC simulation method increases with the
increase in the difference between the R with which the FC is computed (men value of R)
and the R of the MC samples.
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3.2 The impulse density as intensity measure
In this section an accidental explosion of ammunitions is considered, and the probability
to exceed a limit state (exceeding probability) for a steel built-up blast door is estimated
by both the conditional and unconditional approach for three different limit states.
Moreover a safety factor is proposed, evaluated as a function of an acceptable exceeding
probability. Such a safety factor can be used in order to design the built-up blast door by
making use of deterministic analyses carried out by the equivalent Single Degree of
Freedom (SDOF) method [DoD 2008].
For achieving the above mentioned exceeding probability by the conditional and the
unconditional approaches, the Monte Carlo (MC) simulation method is adopted; with the
conditional approach the MC simulation is used in computing the fragility curves for each
limit state, while with the unconditional approach the MC simulation is used for
computing directly the exceeding probability. In order to efficiently carrying out the
numerical analyses for MC simulations, a Simplified Stochastic Model (SSM) is
proposed. The SSM is a SDOF-based model and it is validated by the avail of a detailed
Finite Element model (FE model) of the case-study built-up door.
Referring to significant literature regarding the design and the deterministic assessment of
the structural response of blast doors subjected to impulsive loads, the [DoD 2008]
furnishes the methodology and the practice for designing structural elements against
accidental explosions, a specific section of the [DoD 2008] is focused on special
considerations about blast doors, where basic procedures and performance requirements
are defined. More detailed design procedures for such a kind of elements are in the
[USACE 2009]. In [Chen et al. 2012] a new kind of blat door consisting in a multi-arch
double-layered blast-resistance panel is developed and the blast performances of such a
system are investigated by both numerical and experimental studies. In [Xingna et al.
2012] the performances of a refuge chamber door under blast loading are investigated; an
accurate analysis is performed by numerical simulations for optimizing the configurations
of the stiffeners.
Also if traditionally the deterministic approach is preferred in the design of structures
under blast, a number of works can be useful in order to calibrate probabilistic models
and bounding the uncertainties affecting the design of blast resistant elements. In
[Stewart et al. 2008] two kind of window glazing systems are studied, and the crucial
issue of selecting an appropriate intensity measure for computing the fragility curves for
blast loaded structures is investigated. The fragility curves are developed in function of
two different intensity measures (the explosive weight and the stand-off distance), and
several fragility curves are computed for the specific cases of study. In the work of
[Netherton et al. 2009] the accuracy of the blast loading prediction model is investigated,
resulting that the reliability analysis is sensitive to the uncertainties about the blast load
model. An example regarding the complexity of the blast load modeling is shown in
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[Ballantyne et al. 2010] where the clearing effect for finite width surfaces is investigated.
In [Wu et al. 2009] a series of different kinds of concrete slabs are tested in order to both
compare their blast resistance and evaluate the uncertainty affecting the pressure
estimation procedures provided in [DoD 2008]. In [Chang et al. 2010] Monte Carlo
simulations are performed in order to estimate the probability of failure for windows
subjected to a blast load made by a vehicle bomb. [Low et al. 2001] presents results of a
parametric investigation on the reliability of reinforced concrete slabs under blast loading
in order to establish appropriate probabilistic distributions of the resistant parameters.
[Whittaker et al. 2003] proposes the extension of probabilistic approaches from the
performance-based earthquake engineering to the blast design problems, also by
suggesting appropriate variables for the intensity measures, the damage measures, and the
response parameters definition. In [Galiev 1996] experimental observations are provided
for impulsively loaded structural elements in order to calibrate theoretical models. The
counterintuitive phenomenon of elastic–plastic beam dynamics studied by [Symonds et
al. 1985] is investigated by [Li et al. 2003] with a probabilistic approach following the
experimental evidences provided in [Li et al. 1991] and [Kolsky 1991]. In [Guillaumat et
al. 2005] a composite structure made by glass fiber and impregnated with polyester
subjected to accidental (low velocity) impacts is investigated in probabilistic terms; a
polynomial expression of the force occurring during the impact is developed by
experimental data and statistical distributions are provided for the coefficients of the
expression. [Choudhury et al. 2002] shows that the reliability index of a buried concrete
structure subjected to missile impacts decreases significantly with the increasing of the
uncertainty affecting the problem. [de Béjar et al. 2008] develops probabilistic models to
predict the probability of the residual velocities of mortar round fragments after the
perforation of a wall. The fragment effect on fiber panels is investigates also in [Jordan et
al. 2010] by developing an empirical equation for calculating fragment impact velocity
from penetration data. [Fyllingen et al. 2007] performs numerical simulations of square
aluminum tubular elements subjected to axial loading introducing stochastic geometric
imperfections in order to reproduce the experimental evidences provided in [Jensen et al.
2004].
3.2.1 Relation between the pressure-impulse diagram and the fragility
surface
The blast load on structures depends mainly by two parameters: the scaled distance (Z)
and the amount of explosive or charge weight (W). Figure 3-11 (a) shows the
dependence from Z and W of the most important blast load parameters for the case of
surface explosions [DoD 2008]. The stand-off distance is the distance from the target to
the explosive source, Z is obtained by dividing the stand-off distance by the cube root of
the explosive charge weight, p0 is the side-on pressure, pr is the reflected pressure, and
finally i0 and ir are the side-on and reflected impulse densities respectively.
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Based on the [DoD 2008] the blast load can be defined as an equivalent triangular pulse
as indicated in Figure 3-11 (b), where td is the duration time of the equivalent triangular
pulse.
By the functional relations shown in Figure 1 (a) in terms of Z and W, a direct
dependence of the blast load to both the peak pressure ppeak (pr in the case of Figure 3-11
(b)), and the impulse density (i) can be extrapolated, being the last one defined by the half
of the product between the peak pressure and the equivalent time td. .
1000000
100000
Z
p0
pr
ir/w1/3
is/w1/3
pr
10000
[ft / lb1/3]
[psi]
[psi]
[ms psi / lb1/3]
[ms psi / lb1/3]
p0
pr
1000
ir/w1/3
100
is
10
p0
/w1/3
tr
td
t0
1
0.1
0.1
1
(a)
10
100
(b)
Figure 3-11: Blast load parameters [DoD 2008] (a); design blast load shapes (b)
The Figure 3-12 (a) represents an iso-response curve, that is a constant structural response
defined in terms of a certain parameter (in this case the support rotation θ) plotted in
function of both the peak pressure and the impulse density of the blast load. The chart
shown in Figure 3-12 (a) is called pressure-impulse diagram and it is very common in the
blast engineering for designing structural elements.
The pressure-impulse diagram indicates that the structural response depends by both the
peak pressure and the impulse density; therefore the intensity measures to adopt in the
blast engineering should be these two. Considering the above, the intensity measure
should be a vector of dimension two and, consequently, the fragility should be
represented by a fragility surface instead of a fragility curve.
Deterministic design procedures are based on a single pressure-impulse diagram for a
component, as well known in the literature, see for example the works of [Scherbatiuk
2008, Krauthammer 2008a, Shi et al. 2008, Yim et al. 2009, Ma et al. 2008, and Fallah
2007]. But if the design is probabilistic, there are infinite pressure-impulse diagrams,
each one corresponding to a value of the conditional exceeding probability between 0 and
1. Each pressure-impulse diagram is a cross section of the above mentioned fragility
surface, and it is defined by a plane at constant conditional exceeding probability, while
each fragility curve defined choosing the impulse as intensity measure is a cross-section
defined by a plane at constant pressure of the fragility surface. Figure 3-12 (b) shows
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these cross sections of the fragility surface that define the fragility curves, each one for a
constant value of the pressure. This direct relationship between the fragility curves made
at a constant pressure (Figure 2 (b)) and the pressure-impulse diagram (Figure 3-12 (a)) is
crucial because from the fragility curves of Figure 3-12 (b) is immediate to compute the
pressure-impulse diagram for a constant conditional exceeding probability, see Figure
3-12 (a). In this case the points of a certain pressure-impulse diagram can be viewed as a
series of iso-probability impulse values, each one belonging to a different fragility curve,
as shown in the illustrative example of Figure 3-12 (a). This evidence can be useful in
defining a stochastic pressure-impulse diagram, and it could be a future development of
this work. Moreover, in Figure 3-12 (a) the different regions of the pressure-impulse
diagram are shown: a) the impulsive region (I) where only the impulse density is relevant
for the structural response of a component; b) the dynamic region (D) where the structural
response of the component is governed by the load shape and the pressure magnitude; and
finally c) the quasi-static region (S) where only the peak pressure is relevant for the
structural response of a component.
P[ Θ>θ | i, p ]=P0
I: impulsive region
D: dynamic region
S: quasi-static region
a
I
D
b
d
e
P[•]=P0
S
c
f
P[ Θ>θ | i ]
Pressure
The fragility curves of Figure 3-12 (b), being the cross sections of the fragility surface
and being each one defined by a plane at constant pressure, are pretty coincident until the
pressure value belongs to the impulsive region. This is because in the impulsive region a
variation of the pressure does not imply a significant variation of the structural response
and, consequently, of the structural fragility. On the other hand, when the cutting plane
(pressure value) is moving toward the dynamic and quasi-static regions the fragility
curves became substantially different each other (a variation of the pressure implies a
significant variation of the structural response and of the structural fragility).
p1
a
p2 p
3 p
4
b c d
e
p5
f
Θ=θ
Impulse
(a)
p6
Impulse
(b)
Figure 3-12: Probabilistic description of the blast response for a structural component.
Pressure-impulse diagram (a); structural fragility (b)
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3.2.2 The fragility curve for impulse sensitive structures
For most structural elements the load conditions due to detonations of high potential
explosives are associated with the impulsive region of the pressure-impulse diagram.
Therefore the impulse density is selected as intensity measure because in this case the
structural response depends by the impulse characterizing the blast load. However for
very stiff structures, the load conditions can be on the dynamic or quasi-static region of
the pressure-impulse diagram; for that kind of structures the impulse density is not
appropriate as intensity measure and it can lead to erroneous estimations of the exceeding
probability.
In Figure 3-13 (a) a general pressure-impulse diagram is shown. A number of load
samples can be obtained by extracting couple of values of the explosive charge and the
stand-off distance; each load sample is defined by a peak pressure and an impulse density
in order to characterize a triangular or exponential load shape, in the last case the decay
coefficient should also be defined. If the load samples are belonging to the region show
in Figure 3-13 (a), this is the case where the impulse density (i) is a good intensity
measure for the structural element under investigation, as generally (as mentioned above)
verified for the most common structural elements. As shown in Figure 3 (a), the choice
of i as intensity measure means that the pressure-impulse curve is approximated only with
its impulsive asymptote.
Under these premises, the fragility curve for impulse sensitive structures can be defined
as follow: a pressure value p0 belonging to the impulsive region of the pressure-impulse
curve is considered; if the uncertainties (both aleatory and epistemic) are taken account, a
range of intensity measure where the conditional exceeding probability of the response
parameter (P[Θ>θ Ι i] in figure) assumes the values from 0 to 1 is identified, for example,
by the points a (P[Θ>θ Ι i] =0) and c (P[Θ>θ Ι i] =1) in Figure 3-13 (a). This trend
defines a curve representing the structural fragility for impulse sensitive structural
elements as shown in Figure 3-13 (b).
In the following application the impulse density in selected as intensity measure and the
fragility curve is built numerically selecting a pressure value (p0) belonging to the
impulsive zone of the pressure-impulse diagram of structural element.
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p0
P[ Θ>θ | i ] = 1
Load (p, i)
samples region
abc
P[ Θ>θ | i ]
Pressure
c
p0
b
P[ Θ>θ | i ] = 0
Θ=θ
a
Impulse
Impulse
(a)
(b)
Figure 3-13: Conceptual definition of the fragility curve for impulse sensitive structures
3.2.3 Application on a steel blast door
A blast door is conceived to contain an explosion and therefore prevent the propagation of
pressure, fireball leakage and fragment inside the protected area. Generally a blast door is
designed to protect personnel [Mayorga et al. 1997, and Stuhmiller et al. 1996] and
equipment from the effects of external explosions.
There are different typologies of blast doors, classified on the basis of their structure (e.g.
single leaf or double leaf) and on the basis of the opening mode (e.g. vertical lift and
horizontal sliding). There are also several kinds of standard performance requirements for
categorizing the blast doors according to their function. Performance requirements
include:
–
protection of personnel and equipment from external blast pressures resulting
from an accidental explosion;
–
prevention of accidental explosion propagation into an explosive storage area;
–
maintain complete serviceability for doors designed as part of a containment cells
commonly used in the repeated testing of explosives;
–
maintain integrity for doors designed as part of a containment structures
commonly used to protect nearby the personnel and structures in the event of an
accidental explosion.
In this study a built-up steel door with a single leaf is considered. The door is made by
welding steel plates to a steel beam grid, and the exterior plate (thicker than the interior
plate) is designed as a continuous member supported by the grid. The grid is made by
different beams; the beams on the boundaries support the spandrel beams that support
directly the exterior plate. Such a built-up door can be considered as an orthotropic plate.
Pierluigi Olmati
Page 78 of 189
The case-study door is 2500 mm high and 1400 mm wide, the exterior plate is 5 mm thick
and the interior plate is 1 mm thick. The beams on the boundaries and the four spandrels
have UPN 80 and L 80x60/7 cross-sections respectively. In Figure 3-14 details and
sections of the door are shown. The steel is the S275 and by a traction test the actual
yielding stress is assessed to be 340 MPa.
1400
A
SECTION B-B’
1400
Blast side
Blast side
5
2500
6
UPN 0
B’
B
1
Side away
from blast
A’
(a)
(b)
SECTION A-A’
2500
5
Blast side
6
UPN 0
L 0x60/
1
Side away from blast
(c)
Figure 3-14: Details of the case-study blast door. Frontal view (a); section along the width
(b); section along the height (c)
The door is located on the exterior side of a building belonging to a military facility zone.
The army in this facility is equipped with 60 mm mortars and exercitations are frequents.
Along the side of the building there is a street passed through by military vehicles
carrying on 60 mm mortar rounds. The accidental scenario considered in this study
implies the detonation of ammunitions on a vehicle passing through the street.
Adopting a performance-based philosophy, four limit states are considered, related to the
Serviceability, the Operability, the Life Safety, and the Critical Failure of the blast door
(Table 1). For this purpose one or more response parameters and appropriate threshold
values of these parameters need to be defined. The selected response parameters are: the
support rotation (θ) and the ductility ratio (m) both defined in Eqs. (3-24), where ymax is
the maximum displacement of the component, dy is the yielding displacement of the
resistance function of the component, and L is the span of the component, in this case Ly
is considered because, being shorter than Lx, it leads to a greater support rotation; more
details are provided in next sections.
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Since a general consensus concerning the threshold values for the different limit states
has not been reached in the scientific community, here these values have been chosen by
means of a critical examination of both the literature and the physics of the problem, also
with the support of appropriate numerical analyses described below. These values are
shown in Table 3-1. Note that different threshold values of a response parameter are
expected for different typologies of the blast resisting doors. In order to give an idea of
the uncertainty affecting the threshold values, a Qualitative Confidence Index (QCI) is
also provided in the table, ranging from “high” (low level of uncertainty) to “low” (high
amount of uncertainty). Finite element analyses and experimental investigations should
be conducted to clarify this point.
In the table, with the term “failure” is intended a structural response of the door causing
the projection into the protected space of the door itself or parts of it.
(
)
(a)
(3-24)
(b)
The first limit state is about the Serviceability. The blast door after the event should be
able to be fully operable without repairs. Damages to both the door structure and door
accessories (like the panic opening system) are not allowed. Thus the door should remain
integer and fully operable after the event. The ductility ratio m is the response parameter
selected for the assessment of this limit state. To satisfy the Serviceability limit state the
ductility ratio is limited to 1.
The second limit state is about the Operability. The door after the event should be able to
be opened. Damages to the door structure are allowed, but damages to the door
accessories are not allowed. Thus the door should remain operable after the event, even if
permanent deformations are present. The support rotation is selected as response
parameter for this limit state. To satisfy the Operability limit state the support rotation is
limited to 2 degrees [DoD 2008] [Chen et al. 2012]. The Operability LS is important for
avoiding failure and/or blockage of the panic opening system of the door, in a way that
both the evacuation of the building and the police/fireman operations can be easily
conducted.
The third limit state is about the Life Safety. The inoperability of the door after the event
is accepted but the structure must not fail; significant permanent deflections of the door
are allowed. For this limit state the support rotation is selected as response parameter. To
satisfy the Life Safety limit state the support rotation is limited to 10 degrees. This value
is chosen on the basis of the results obtained by a static pushover analysis of the casestudy the built-up door (see next sections).
Pierluigi Olmati
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The last limit state is about the Critical Failure and it occurs when the Life Safety limit
state is exceeded. In Table 3-4 the considered limit states are resumed. The fragility
curves and the exceeding probability are computed for the first three limit state.
Limit State
Serviceability
Operability
Life Safety
Critical Failure
The door has no
permanent
deflections.
The door is
operable, but it
has permanent
deflections.
The door has
failed.
Response
Parameter
Threshold
values
ductility ratio
(m)
support rotation
(θ)
The door has
not failed, but it
has significant
permanent
deflections.
support rotation
(θ)
support rotation
(θ)
<1
<2°
<10°
>10°
QCI
High
Medium
Low
Low
Damage
Level
Table 3-4: Limits States
3.2.3.1 The Simplified Stochastic Model (SSM)
As said above, a Simplified Stochastic Model (SSM) has been used in evaluating the
fragility of the blast door. The SSM is the equivalent Single Degree of Freedom (SDOF)
model of the steel built-up door, taking into account for both the aleatory and epistemic
uncertainties are taken into account.
An equivalent SDOF system is obtained by evaluating appropriate transformation factors
for the system’s mass, damping, load and resistance. Furthermore, inherent with a SDOF
analysis is the assumption that the system behaves only in a single mode shape. As the
system begins to deflect under the blast load, it eventually yields and forms plastic hinges
at various locations depending on the applied boundary conditions. Thus in reality, the
system’s mode shape changes with the progression of plastic hinges. Therefore, the
transformation factors are adjusted to take into account for the change of the mode shape.
For a simply supported one way panel under uniform loading, it is assumed that a single
plastic hinge is formed at the center of the span. The resistance-deflection relationship
for such a panel is assumed to have an elastic-perfectly plastic shape. Thus, at a certain
yielding deflection, the component will continuously deform at near constant resistance
until an ultimate deflection limit is reached, at that point the component will fail. This
resistance-deflection relationship (resistance function) serves as constitutive relation for
the non-linear stiffness in the equation of motion.
The displacement field of the component can be expressed as u(t)=Φ(x)y(t), where Φ(x)
is the assumed deformed shape of the component under the blast load and y(t) is the
displacement of the component in the location of maximum deflection. Furthermore,
displacement of the component is obtained by the SDOF equation:
Pierluigi Olmati
Page 81 of 189
̈( )
̇( )
( ( ))
()
(3-25)
where M is the total mass of the component, S(y(t)) is the resistance of the component as
a function of the displacement expressed in unit force, F(t) is the blast pressure multiplied
by the loaded area (A) expressed in force units, C is the damping (the percentage of the
critical damping is assumed to be 1 % in the analyses), KLM is the so-called “load-mass
transformation factor”, that is equal to the ratio of KM and KL (the mass transformation
factor and the load transformation factor respectively). The last two are evaluated by
equating the energy of the two systems (in terms of work energy and kinetic energy
respectively).
The load-mass transformation factor KLM is different at each deformation stage of the
component response; for a bilinear resistance function two values of the KLM can be
defined: the first for the elastic range of the response and the second for the plastic range
of the response. These two coefficients are well established in literature, more details on
the equivalent SDOF method are provided in [US Army 2008] and [DoD 2008].
The build-up blast door considered in this study is a two-dimensional orthotropic
structure; and it is make equivalent to a SDOF model with a bilinear resistance function.
For obtaining such resistance function, the yielding point (Py) needs to be defined, this is
characterized by the yielding pressure (ry) and the yielding displacement (dy) of the
component. In order to define the ry and the dy both aleatory and epistemic uncertainties
are introduced.
In Eq.s (3-26) the formulas for computing both the ry and dy are shown.
(
)(
)
( )
(a)
(b)
(3-26)
(c)
Where, Ke is the stiffness of the SDOF, Lx and Ly are the longer and the shorter
dimension of the door respectively, E is the Young’s modulus of the steel. The
coefficients XA and XB are taken equal to 1.374 and 198.6 respectively, and they are valid
for orthotropic plates with Ly/Lx equal to 1.78 [Biggs et al. 1964, US Army 2008, DoD
2008]. Mpx and Mpy are the plastic moments of the two orthogonal cross sections of the
built-up door.
Pierluigi Olmati
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Lx, Ly, and E are assumed as deterministic parameters; instead both J y and Jx, and
consequently Mpx and Mpy are assumed as stochastic variables to take into account for the
epistemic uncertainties. Moreover the aleatory uncertainty affecting the yielding stress of
the steel is considered.
Also with reference to Figure 3-14 the moments of inertia are computed by Eq.s (3-27) by
assuming valid the hypothesis of plane sections.
)⁄
(
[(
[(
)(
(
)⁄
) ((
) ]
(a)
)
(
)⁄
(3-27)
) ]
(
(b)
)⁄
Where, NL is the number of the spandrels orthogonal to Lx, JL and JUPN are the moments
of inertia of the spandrels and of the external frame respectively, t1 and t2 are the
thicknesses of the blast side plate and of the side away from blast side plate respectively,
and dG is the center of mass of the composite section. For computing J x the transport
moment of inertia due to the plates is not considered because there are not spandrels
along the Lx direction.
The stochastic coefficient α is introduced for taking into account the uncertainty on the
moments of inertia, with a mean value equal to 1 and a COV equal to 0.1. The sample
values Jxi and Jyi of Jx and Jy are evaluated as shown in Eq.s (3-28).
(a)
(3-28)
(b)
The Mpxi and Mpyi are computed by Eq.s (3-29) and considering Eq.s (3-30).
(a)
(3-29)
(b)
(a)
(b)
(3-30)
(c)
Pierluigi Olmati
Page 83 of 189
Where bx and by are the longest distance between the center of mass and the two external
sides of the cross sections, σydi is the sample value of the dynamic yielding stress of the
steel, σyi is the sample value of the static yielding stress of the steel, assumed as stochastic
variable. The sample value DIFi of the Dynamic Increasing Factor, is obtained by adding
to the unit the decimal part DIF0i, and the last one is assumed as stochastic variable to
consider the epistemic uncertainty. Finally ϕi is the sample value of the plastic coefficient
obtained adding to the unit the decimal part ϕ0i; also ϕ0i is assumed as a stochastic
variable affected by the epistemic uncertainty. The considered stochastic variables are
shown in Table 3-5 with indications on their mean value, COV, and distribution function.
The mean value of α is set equal to the unity. The mean value of the σy is estimated by
assuming a strength factor equal to 1.1 as provided by [US Army 2008], for a steel grade
of 450 MPa this leads to a mean value of σy equal to 302.5 MPa. The mean value of ϕ is
estimated by fitting the resistance of the SSM with the static pushover curve computed by
the FE model as described in what follow. Finally the mean value of the DIF is provided
by [US Army 2008] for an equivalent grade of steel.
The COV of the σy is referenced by [Enright et al. 1998]; the COV of both α and ϕ are
estimated for obtaining a reasonable dispersion of the Py with respect to the static
pushover curve obtained by the FE model. The COV of the DIF is estimated by the
values of the DIF provided by [US Army 2008] for several strain rate velocities.
Parameter Mean value C.O.V. Distribution
σy
302.5 MPa
0.12
log-normal
α
1
0.1
log-normal
ϕ0
0.3
0.1
log-normal
DIF0
0.19
0.2
log-normal
Table 3-5: Probabilistic distributions of the stochastic variables
By substituting the sample values obtained in Eq.s (3-28), (3-29), and (3-30) into the Eq.s
(3-27), the sample Pyi of the yielding point Py of the resistance function is computed. In
Figure 3-15 the log-normal distribution functions of the computed values of both ry and dy
are shown. The resulting average value of ru and dy are 306 kPa and 6.8 mm respectively,
their COV are 0.16 and 0.125 respectively.
Pierluigi Olmati
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8
0.4
6
0.3
f (dy)
0.5
f (ru)
10
4
0.2
0.1
2
0
0
0
0.2
0.4
ru [MPa]
(a)
4
0.6
6
8
10
dy [mm]
(b)
12
Figure 3-15: Probability density function of ry and dy
3.2.3.2 Validation of the SSM by the Finite Element Model (FEM)
In order to validate the SSM a detailed Finite Element Model (FE model) is built. The FE
model is developed by using the commercial FE solver LS-Dyna® [LS-Dyna 2012]. The
FE model is a three dimensional model consisting of shell elements. The support frame
of the door is explicitly modeled with its geometrical features in order to accurately take
into account for the unilateral boundary conditions by making use of contact elements.
Moreover, other contacts are provided in correspondence to the door opening hinges and
door locking system for allowing the rebound response.
The model is made by a total of 84794 shell elements and 85062 nodes. The shell
elements are of Belytschko-Tsay type [LS-Dyna 2012] and the contact algorithm is the
automatic surface to surface one [LS-Dyna 2012].
Regard to the steel, a piecewise linear plasticity model [LS-Dyna 2012] is adopted, with
the true stress-strain relationship obtained by experimental tests. Figure 3-16 shows this
stress-strain relationships: the engineering stress-strain curve is directly obtained in
[Kalochairetis 2013] by an experimental test considering the length and the initial cross
sectional area of the specimen, instead the true stress-strain curve is obtained analytically
by assuming logarithmic strains [LS-Dyna 2012], consequently the true plastic stressstrain curve is computed as required input by the piecewise linear plasticity model [LSDyna 2012].
Moreover, a fracture criterion is implemented without taking into account for the effect of
the stress triaxiality on the metal fracture; the fracture occurs when the effective plastic
strain reaches 0.2473. The strain rate effect is taken into account by the Cowper and
Symonds model shows in Eq. (3-31) [LS-Dyna 2012], where ξ is equal to 500 [1/s] and γ
is equal to 6.
̇
( )
Pierluigi Olmati
(3-31)
Page 85 of 189
It is crucial to highlight that the steel yielding stress shown in Figure 3-16 does not match
with the mean value of the steel yield stress of Table 3-5; in order to validate the SSM by
the FE model, the input parameters are assumed to have the mean values and the steel
yielding stress-strain relationship shown in Figure 3-16. Furthermore a DIF equal to 1 and
1.19 for the case of the static resistance and dynamic response respectively has been
assumed for the SSM.
600
Stress [MPa]
500
400
300
200
True stress-strain
Engineering stress-strain
True plastic stress-strain
100
0
0
0.1
0.2
ε [-]
0.3
0.4
Figure 3-16: Stress strain relationship [Kalochairetis 2013]
In Figure 3-17 (a) the FE model and details of the built-up door are shown, a detail of the
FE model is presented in Figure 3-17 (b), where the mesh refinement can be appreciated.
The characteristic dimension of the single rectangular finite element is 15 mm and it is
quite constant for all the mesh.
Unilateral BCs
and hinge system
UPN 80
UPN 80
2500 mm
Blast side plate t=5mm
L 80x60/7
L 80x60/7
Unilateral BCs
and hinge system
Blast side plate
t=5mm
Unilateral BCs
and hinge system
Away from blast plate t=1mm
(not in view)
(a)
(b)
Figure 3-17: Finite element model of the steel built-up door
For obtaining the static resistance function of the built-up blast door a static pushover
analysis is carried out by applying an uniform load to the blast side plate. The uniform
pressure is applied quasi-statically by a ramp load function until the collapse of the door
is reached. In Figure 3-18 the static resistance functions computed by the SSM (by
assuming the mean values of the input parameters) and the FE model are shown. In
Figure 3-18 (a) the static resistance function is plotted as a function of the mid-span
displacement δ, while in Figure 3-18 (b) it is plotted as a function of the support rotation
θ defined in Eq. (3-24). Especially for the range of support rotation from 0 to 2 degrees
Pierluigi Olmati
Page 86 of 189
0.5
0.5
0.4
0.4
0.3
r [MPa]
r [MPa]
there is a good agreement between the two predictions. However an experimental test
should be performed to definitely confirm the results.
0.2
0.3
0.2
FEM
FEM
0.1
0.1
SSM
0
0
10
20
30 40
δ [mm]
50
60
70
SSM
0
0
1
2
(a)
3
4
θ [deg]
5
6
(b)
Figure 3-18: Static resistance function by the FEM and the SSM
The FE model and the SSM are compared in terms of dynamic response. The built-up
door is subjected to four detonations and the structural response is computed by both the
SMM and the FE model. All the detonations occur at 500 mm from the ground and at 6 m
away from the built-up door, then these detonations are surface burst explosions. The
explosive charges of the four detonations are assumed to consist in 10, 15, 20, 25 kg of
TNT. The blast pressure is assumed as uniformly distributed in the SSM, but it is
properly evaluated as non-uniformly distributed in the FE model by the LS-Dyna®
function named load blast [LS-Dyna 2012]. Only the positive phase of the shock wave is
taken into account by the equivalent triangular pulse, alternatively an exponential decay
law can be adopted [Gantes et al. 2004].
Values adopted for the parameters characterizing the SSM for comparison purposes with
the FE model are: σy=340 MPa, ϕ=1.3, α=1, and DIF=1.19. It is important to remember
that, as mentioned above, for the successive computation of the fragility curves and of the
safety factor, the mean value of σy is assumed to be 302.5 MPa. The motion equation of
the SSM is solved by using SBEDS® [US Army 2008]. In Figure 3-19 the comparison
between the time histories of the support rotation θ obtained with the FE model and the
SSM are reported for all the four detonations.
1.5
1.5
FEM
1
θ [deg]
1
θ [deg]
SDOF
SDOF
FEM
0.5
0.5
0
0
-0.5
-0.5
0
0.01
Time [sec]
0.02
(a) W=10 kg TNT - R=6 m
Pierluigi Olmati
0
0.01
Time [sec]
0.02
(b) W=15 kg TNT - R=6 m
Page 87 of 189
2
2.5
1
θ [deg]
θ [deg]
1.5
1.5
0.5
0.5
SDOF
FEM
0
SDOF
FEM
-0.5
-0.5
0
0.01
Time [sec]
0.02
(c) W=20 kg TNT - R=6 m
0
0.01
Time [sec]
0.02
(d) W=25 kg TNT - R=6 m
Figure 3-19: Comparison between the time histories of the support rotation θ obtained
with the FE model and the SSM. 10 kg of TNT (a); 15 kg of TNT (b); 20 kg of TNT (c);
25 kg of TNT (d).
In Figure 3-20 the plastic strains on the door obtained by the FE model are plotted.
Plastic strains are represented in black color while in grey is the elastic steel (in the black
zones the dynamic yielding stress of the steel was reached). In Figure 3-20 the boundary
conditions, and the away from blast side plate are removed from the view for allowing the
checking of the spandrels.
With reference to Figure 3-19, it can be appreciated that generally there is a good
agreement between the predictions of the support rotations made by the SSM and the FE
model in the initial loading phase. However the SSM looks little more conservative with
respect to the FE model. This acceptable difference in the prediction can be attributed to
the fact that in the SSM a constant value of the DIF is assumed, on the other hand the DIF
in the FE model is computed at the scale of the single finite element.
Due to the non-linear boundary conditions implemented in the FE model, the rebound
response is pretty different between the two models, and also the time of the max support
rotation is slightly different. But this is not relevant for the purpose of the SSM that is to
estimate only the maximum support rotation of the built-up door.
Pierluigi Olmati
Page 88 of 189
(a)
(b)
(c)
(d)
Figure 3-20: Plastic strains on the door obtained by the FE model. 10 kg of TNT (a); 15
kg of TNT (b); 20 kg of TNT (c); 25 kg of TNT (d).
Pierluigi Olmati
Page 89 of 189
On the basis of the results in terms of plastic strains shown in Figure 3-20, it can be stated
that the non-uniform distribution of the blast load does not lead to a particularly nonuniform structural response of the built-up door: the plastic strains on the spandrels are
quite uniform. Moreover, from the results can be argued that the door develops a
resistant mechanism that is of flexural type since only a limited plasticity is developed at
the connection of the spandrels with the external frame. The flexural-type behavior is
something that is at the basis of the hypothesis made by the SSM.
In the case of 25 kg of TNT (see Fig. Figure 3-20(d)), the blast side plate shows spread
plasticity but it maintains the ability to transfer the load on the spandrels; note that a
fracture criterion is implemented in the FE model and an eventual fracture of the blast
side plate would be detected.
3.2.3.3 The fragility analysis of the built-up door
In this section the fragility curves for the built-up blast door are developed for each limit
state previously defined. Particularly important is the Operability because this limit state
should avoid the failure and/or blockage of the panic opening system of the door, in a
way that both the evacuation of the building and the police/fireman operations can be
easily conducted [DoD 2008, and Chen et al. 2012], as previously mentioned.
The fragility curve is computed point by point using MC simulations [Olmati et al. 2013],
then the points are interpolated by a log-normal cumulative function for obtaining a
smooth curve to use in computing the probability of exceeding a limit state and the safety
factor.
A flowchart representing the steps in computing the fragility curves is shown in Figure
3-21. Looking at the flowchart, N is the number of the points in which the fragility curve
is numerically evaluated, j is the loop counter identifying the MC simulation which is
performed to evaluate the single point FC(j) of the fragility curve, corresponding to the jth value IMj of the intensity measure (impulse density). For j=1 a MC simulation is
carried out and the conditional exceeding probability is estimated. The next step is to
compute the next point of the fragility curve, so for the new value of the intensity
measure a new MC is performed and the conditional exceeding probability is estimated.
This cycle is repeated until j=N.
Pierluigi Olmati
Page 90 of 189
j=1
j=j+1
IM=IMj
MC analysis
NO
j=N?
FC(j)
• IMj: impulse density
• FC: numerical Fragility
Curve
• FC(j): the jth point of the FC
• MC analysis: Monte Carlo
analysis
• N: number of FC points
• Interpolated FC: lognormal
interpolated FC
YES
FC
Lognormal
Interpolation
Interpolated
FC
Figure 3-21: Flowchart of the procedure for the
evaluation of the fragility curves. FC= fragility curve
In the flowchart of Figure 3-21 the fragility curve obtained with this algorithm is called
numerical fragility curve. The final step consists in the interpolation of the points of the
numerical fragility curve for obtaining the interpolated fragility curve (log-normal shape)
defined by the mean value and the standard deviation.
1
1
0.8
0.8
0.8
0.6
0.4
0.2
P[ Θ>θ | i ]
1
P[ Θ>θ | i ]
P[ Θ>θ | i ]
In Figure 3-22 the fragility curves obtained for the Serviceability, Operability, and Life
Safety limit states are shown. Their mean values (μln) and COVs (βln) are shown in Table
3-6.
0.6
0.4
0.2
ymax = dy
0.1
0.3
i [kPa sec]
(a)
0.5
0.4
0.2
θ=2
0
0
0.6
θ = 10
0
0.6
0.8
1
i [kPa sec]
(b)
1.2
1.1
1.6 2.1 2.6
i [kPa sec]
3.1
(c)
Figure 3-22: Fragility curves obtained by the SSM. Serviceability (a), Operability (b), and
Life Safety (c)
Pierluigi Olmati
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Limit State
Serviceability Operability Critical Fail
Response Parameter
y < dy
θ < 2°
θ < 10°
μln [kPa sec]
0.3080
0.8700
1.9800
FC
COVln
0.1518
0.0748
0.0785
Table 3-6: Parametric carachterization of the fragility curves for the examinated limit
states
The number of samples Nsj used in the MC simulation performed for computing the
single point FC(j) of the numerical fragility curve is not constant; Nsj is chosen for each j
in order to maintain the COVj under an acceptable value. With the decreasing of the
conditional exceeding probability [
]. The COVj is quantified by the Eq. (3-32).
[
]
√
[
[
]
(3-32)
]
With regards to the fragility curve associated with the Operability LS, the variation of
both Ns and COV with [
] is shown in Figure 3-23. The number of samples
decreases exponentially from the lowest to the highest probability of the numerical
fragility curve. However, since the number of samples should not decrease under a
threshold limit, in this application the maximum and minimum number of samples is 105
and 103 respectively. As shown in Figure 3-23 the maximum COV is less than 0.1 for a
conditional exceeding probability of 0.001 and it decreases quickly, for example it is less
than 0,02 for a conditional exceeding probability of 0.1.
0.10
N° of samples
80000
0.08
COV
0.06
60000
40000
0.04
20000
0.02
0
COV
N of samples
100000
0.00
0.001
0.016
0.110
0.352
0.661 0.880
P [ Θ>θ | i ]
0.967
0.994
0.999
Figure 3-23: Number of samples and COV
3.2.3.4 Scenario-based evaluation of the probability of exceeding the limit
states
In this section the Probability Density Function (PDF) of the intensity measure is defined.
A scenario consisting on the accidental explosion of ammunitions is considered. As
previously mentioned, the door is located on the exterior side of a building belonging to a
military facility zone. The army in this facility zone is equipped with 60 mm mortars and
exercitations are frequents. Along the side of the building there is a street passed through
by military vehicles carrying on 60 mm mortar rounds.
Pierluigi Olmati
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In order to define the probability density function of the impulse i (the intensity measure),
it is necessary to know the PDF of both the explosive charge and stand-off distance.
Then the impulse density is computed by the blast load model provided in [DoD 2008]
(see also Figure 3-11). The PDF of the explosive charge is assumed as log-normal, while
the PDF of the stand-off distance is assumed as rectangular (non-informative) over the
road cross section.
A common High Explosive (HE) 60 mm mortar round contains 160 g (0.34 lb) of TNT
[Krauthammer et al. 2008b] and a metal ammunition box contains four mortar rounds.
The vehicle (jeep or van) adopted in this military facility generally carry on about twelve
ammunition boxes leading an average explosive weight of 7.7 kg of TNT. The 25 % and
the 75% of the cases, 3 and 5 ammunition boxes respectively are carried on by the
vehicles. Consequently a log-normal distribution with the COV of 0.31 fits well with the
amount of explosive on the military vehicle. Moreover the properties of the log-normal
PDF allow taking into account samples where the vehicle is loaded over the allowed
maximum number of ammunition boxes.
The stand-off distance has a rectangular PDF, and the detonation point is supposed to be
over the road section used by the vehicles. With reference to Figure 3-24, R1 is the
distance between the door and the edge of the road, R2 is R1 plus the sidewalk of the road,
and finally R3 is the distance from the detonation point to the sidewalk. R2 is known and
deterministically defined, instead the stochastic variable R3 is defined by the above
mentioned rectangular PDF. Finally the stand-off distance consists in R2 plus R3. In this
work R1 is 2 m and R3 ranges between 0 and 7m.
Building
950
R1
Door
125
R2
350
R3
350
125
Detonation
Road cross section
Figure 3-24: Description of the blast scenario and of the
considered variables
The lognormal PDF of the impulse density is obtained by extracting 105 samples of both
the explosive charge and the stand-off distance from their PDF respectively, and it is
shown in Figure 3-25 together with the fragility curves of the limit states. The equivalent
triangular pulse defined above is computed as shows in Figure 3-11 (b). As said in
section 2, pr is the reflected pressure, p0 is the side-on pressure, and tr and t0 are their
duration time respectively, instead td is the duration time of the equivalent triangular pulse
with pr as peak pressure. All the terms in Figure 3-11 (b) are computed by the [DoD
2008] procedure. The resulting mean value and COV of the impulse density (the
intensity measure) are equal to 0.616 kPa sec and 0.601 respectively.
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1
0.012
0.01
0.008
0.6
0.006
p(i)
P[ Θ>θ | i ]
0.8
0.4
0.004
f (i)
0.2
0.002
0
0
0
0.5
1
1.5
i [kPa sec]
2
2.5
Figure 3-25: Lognormal PDF of the impulse density and fragility curves computed for the
considered limit states
By adopting the mean values of the above mentioned stochastic parameters (see Table
3-5, and Eq. (3-30)), the pressure-impulse curves corresponding to each limit states
(average pressure-impulse curves) are obtained by the SSM and shown in Figure 3-26. In
the same figure the load samples used in the evaluation of the exceeding probabilities are
plotted.
As appreciable in Figure 3-26, in this case-study, the load samples fall in the impulsive
region for the Operability and Life Safety limit states, while they fall close to the dynamic
region at least for the Serviceability limit state. In the next section the exceeding
probability is computed by both the conditional and unconditional approach (CA and UA
respectively). Then, the reliability of the fragility analysis is evaluated by computing the
difference between the exceeding probabilities estimated by the two approaches, under
the assumption that the exceeding probabilities evaluated by the unconditional approach
are exact. In the case of the Serviceability limit state being the load conditions over the
dynamic region an overestimation of the exceeding probability by the conditional
approach with respect to the unconditional approach is expected.
8
Pressure [MPa]
Load
6
θ=2
4
θ = 10
y = dy
2
0
0
0.5
1
1.5
impulse [kPa sec]
2
2.5
Figure 3-26. Deterministic pressure impulse diagrams and the load samples
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3.2.3.5 Comparison of the exceeding probability evaluated by conditional and
unconditional approaches
Since all the terms in Eq. (3-33) are provided, the exceeding probability can be computed
by both the conditional and unconditional approaches and compared each other.
Following the conditional approach, Eq. (3-33) is numerically solved by making use of
the fragility curves P[Θ>θ Ι i] and of the PDF of the impulse density f(i) shown in Figure
3-25. The unconditional approach, which consists in a single MC simulation, is made by
105 samples.
[
]
∫
[
] ()
∑ [
] ()
(3-33)
Table 3-7 provides the obtained values for the exceeding probabilities computed by both
the conditional and unconditional approaches.
̅ = 7.7 kg COV=0.3
R2 = 2 m 0 ≤ R3 ≤
Limit State
Serviceability
Operability
Life Safety
CA
0.8303
0.1830
0.0195
lognormal distribution
rectangular distribution
UA
0.6343
0.2065
0.0078
Δ=CA-UA
0.1960
-0.0230
0.0117
Table 3-7: Exceeding probabilities obtained with the conditional and unconditional
approaches
For both the Operability and the Life Safety limit states the conditional and the
unconditional approaches provide quite the same exceeding probability with a slightly
difference due to the differences in the COVs of the computed exceeding probabilities,
see for example Figure 3-23; on the other hand, for the Serviceability limit states the
difference between the two estimations is greater than in the previous cases and the
exceeding probability computed by the conditional approach can be considered as
erroneous because the hypothesis of impulsive loading is not respected (see both Figure
3-26 and Figure 3-13 (a)). Therefore the exceeding probability estimated by both the
conditional and unconditional analysis methods substantially match if the structure is
impulsively loaded being the impulse density the intensity measure.
3.2.3.6 The safety factor for SDOF models of the built-up doors
In order to design sensitive structures a safety factor is provided. The proposed safety
factor is intended to be used for the design of structural elements by the SDOF method.
In particular for each limit state a safety factor expressed as function of an acceptable
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exceeding probability is provided, and the use of this safety factor in the design process is
discussed.
Referring to the Eq. (3-34) the subscript c indicates the capacity, instead the subscript d
indicates the demand. In Eq. (3-34) (a) Vc and Vd are the COVs of the capacity and
demand respectively, assumed to have a log-normal PDF, used in the evaluation of the
dispersion measure βZ (with Z=c-d) of the difference between the capacity and the
demand [Schultz 2010]. In Eq. (3-34) (b) AEP is the Acceptable Exceeding Probability,
Φ-1 is the inverse of the standardized cumulative Gaussian distribution, and Kx is the
standardized Gaussian variate associated to the fact that the probability 1-AEP is not
exceeded.
√ (
)
(
)
(a)
(3-34)
(
)
(b)
The safety factor (λ) is defined by Eq. (3-34) (c) [Cornell et al. 2002]; where ̅̅̅̅ and ̅̅̅̅
are the average values of the intensity measure corresponding to the capacity and to the
demand respectively.
In Figure 3-27 the safety factor obtained for the case-study blast resistant door, and by
assuming the impulse density as intensity measure is plotted as function of the acceptable
exceeding probability for the Serviceability, Operability, and Life Safety limit states.
5
4
λ
y=dy
2.1
1.9
3
θ=2
θ=10
1.7
0.09
0.11
0.13
0.15
2
1
0
0.2
APF
0.4
Figure 3-27: The safety factor for the limit states
From the Eq. (3-34) (a) it can be appreciated that the dispersion measure βZ depends by
the COVs of both the capacity and demand. In this case-study, looking also at Table 3-6,
the COV of the capacity obtained from the fragilities related to the Operability and Life
Safety limit states is substantially the same; this leads to a quite identical safety factor in
function of the acceptable exceeding probability for these two limit states, as shown in
Figure 3-27.
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Concerning the Serviceability limit state, the COV of the capacity is greater than the one
of the Operability and Life Safety limit states (see Table 3-6), but it remain quite small
with respect to the COV of the demand (which is equal to 0.601 as said in section
3.2.3.4), therefore the dispersion of the capacity is predominant in determining the
dispersion measure βZ.
The proposed safety factor is intended to be applied as multiplier of the intensity measure
for the design of blast resistant elements without carrying out probabilistic analyses. An
example of its use is shown below with the avail of an analytical approximate model for
predicting the maximum deflection of a component under blast.
In Eq. 12 (a), an approximate formula for predicting the maximum deflection ymax of a
component in case of impulse sensitive structures is shown [Krauthammer et al. 2008b].
The component must be designed in order to maintain ymax as lower than the threshold
value yLS of the considered limit state. In this case, the safety factor can be used as shown
in Eq. (3-35) (b), where it is intended as a factor of the demand intensity I.
(
(
(
)
(
(
)
(a)
(3-35)
)
)
)
(b)
In Eqs. (3-35), A is the loaded area of the door, I is the impulse density of the demand
multiplied by A (demand intensity), M is the total mass of the door, and Sy is the yielding
resistance of the door multiplied by A. For utilizing the safety factor in the design of the
structural element, the Eq. (3-35) (a) should be replaced by the Eq. (3-35) (b).
In alternative to Eqs. (3-35), when the SDOF motion equation is solved in time domain,
and the time history of the deflection is computed, the increment of the intensity measure
can be made directly by using the safety factor in increasing the peak pressure of the blast
demand as shown in Eq. (3-36).
(3-36)
Since the equation is associated to a variation on the pressure-impulse diagram of the
design load point, specific care must be placed in checking that the new design solutions
(driven from this operation) are in the same dynamic regime of the original one (obtained
without the safety factor).
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3.3 Slabs subjected to impulsive loads - The Blast Blind
Simulation Contest
In this section the structural response of slabs subjected to impulsive load is investigated.
The prediction of the structural response of one of the two kinds of slab tested at the
Engineering Research and Design Center, U.S. Army Corps of Engineers at Vicksburg,
Mississippi is declared the winner of The Blast Blind Simulation Contest
(http://sce.umkc.edu/blast-prediction-contest/ - accessed August 2013). In Figure 3-28 is
the winners’ announcement of The Blast Blind Simulation Contest.
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Figure 3-28: Winners’ announcement
The structural response assessment of Reinforced Concrete (RC) slabs subjected to
impulsive loads due to a detonation of an explosive is a crucial task for the design of blast
resistant concrete structures. When used properly, nonlinear dynamic finite element
methods and analytical modeling provide a valuable tool for predicting the response and
assessing the safety of a RC component. Finite Element (FE) analysis and Analytical
Modeling (AM) approaches are validated using a series of shock tube tests conducted on
normal and high strength RC slabs by the University of Missouri Kansas City (UMKC) at
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the Engineering Research and Design Center, U.S. Army Corps of Engineers in
Vicksburg, Mississippi.
The National Science Foundation (NSF) funded a study by University of Missouri Kansas
City (UMKC) to perform a batch of blast resistance tests on reinforced concrete slabs
(Award # CMMI 0748085, PI: Ganesh Thiagarajan). Based on these results, a Blast
Blind Simulation Contest is being sponsored in collaboration with American Concrete
Institute (ACI) Committees 447 (Finite Element of Reinforced Concrete Structures) and
370 (Blast and Impact Load Effects), and UMKC School of Computing and Engineering.
The goal of the contest is to predict, using simulation methods, the response of reinforced
concrete slabs subjected to a blast load. The blast response was simulated using a Shock
Tube (Blast Loading Simulator) located at the Engineering Research and Design Center,
U.S. Army Corps of Engineers at Vicksburg, Mississippi.
The objective of this Blind Simulation Contest is to highlight the efficacy of available
material models and promote the development of material models that can predict the
response of reinforced concrete structures subjected to highly dynamic loading such as
blast. Several factors contribute to the prediction of the response of a structure when
subjected to shock/blast loading. These factors include boundary conditions, complexity
of material properties available, material models used and finite element parameters such
as element type selection, mesh size sensitivity, material model rate effects amongst
others. There are a number of concrete material models developed by several researchers
over the past few decades for both static and dynamic loading and the primary objective
of this contest it to evaluate their effectiveness under blast/shock loading.
In this section is presented the modeling techniques adopted in the FE approach in order
to properly conduct the structural response assessment of RC slabs subjected to impulsive
loads due to detonations. Two concrete slabs are investigated: a Normal Slab, and a
Hardened Slab. Both the slabs are subjected to two impulsive loads.
The prediction of the structural response for the Normal Slab is resulted to be the most
accurate and it is declared the winner of its category.
Given the double symmetry of the experimental setup and loading the FEM consists of a
quarter of the slab; see Figure 3-29 and Figure 3-30. The nodes belonging to the
symmetry planes are constrained in the orthogonal direction to the pertinent symmetry
plane; moreover the nodes of the reinforcements on the symmetry planes are also
constrained against bending rotation. The finite element mesh is comprised of constant
stress solid elements for the concrete slab and the Boundary Conditions (BC) and
Hughes-Liu beam elements for the steel reinforcement. The complete model consists of
270,960 solid elements and 130 beam elements with a total of 290,628 nodes. The beam
elements used for modeling the reinforcements are embedded in the solid concrete
elements through the LS-Dyna® keyword “Constrained Lagrange in Solid”. The shock
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load is modeled as a pressure demand on the blast side face of the slab. The analyses are
dynamic, and both material and geometric non-linarites are taken into account. The
central difference method is used for solving the dynamic equations. Damping effects
and friction on the BC are not taken into account. A gap of 0.25 in. (6.35 mm) is
assumed to exist between the slab and the upper support (see Figure 3-30). This
dimension was illustrated but not explicitly provided in the contest information. The
measurement was scaled from the setup details provided.
Shock load
Upper support
Upper support
Gap 0.25”
Down support
Contact
surfaces
Contact surfaces
Down support
Figure 3-29: FE model of the slab
Figure 3-30: Detail of the BC
The constitutive models used for the concrete and for the steel are in the following
summarized. The steel constitutive model is the Piecewise Linear Plasticity Model,
MAT024 in LS-Dyna®. This is an elastic-plastic material model with a user defined
stress-strain curve and strain rate dependency. The stress-strain curve used is shown in
Figure 3-31 for the Normal Slab and in Figure 3-32 for the Hardened Slab. The elastic
modulus is 29,000 ksi (200,000 N/mm2), the Poisson coefficient is 0.3, and the yielding
stress is 70 ksi (482.6 N/mm2) for the Normal Slab and 82 ksi (565.4 N/mm2) for the
Hardened Slab. The strain rate effect is taken into account by the Cowper and Symonds
model (C= 500 1/sec, p=6), see Figure 3-33. The concrete utilizes the Continuous
Surface Cap Model (CSCM), MAT159 in LS-Dyna®. The yield stresses are defined by a
three-dimensional yield surface based on the three stress invariants. The intersection
between the failure surface and the hardening cap is a smooth intersection. The softening
behavior of the concrete is taken into account by a damage formulation that affects both
the concrete strength and a reduction in the unloading/loading stiffness. The increase in
concrete strength with increasing strain rate is taken account by a visco-plastic
formulation. The Dynamic Increase Factor (DIF) relation used for the concrete is shown
in Figure 3-34. The CSCM input parameters are listed in Table 3-8 and in Table 3-9 for
the Normal and hardened Slab respectively. Parameters not listed use the default values.
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4 2
4 2
Density
2.248 lbf/in s
3
3
2.4*10 kg/m
Density
2.248 lbf/in s
3
3
2.4*10 kg/m
fc
5400 psi
2
37 N/mm
fc
11600 psi
2
80 N/mm
Cap
retraction
Rate
effect
Erosion
Cap
retraction
Rate
effect
Erosion
active
active
none
Table 3-8: Inputs for MAT159,
Normal Slab
120
100
Stress [kpsi]
Stress [kpsi]
none
140
120
80
60
True Stress
Stress
40
100
80
60
True Stress
Stress
40
20
20
0
0
0.05
0.1
0.15
Plastic strain [-]
0
0.2
Figure 3-31: Stress vs. Plastic strain
relationship for steel reinforcements,
Normal Slab
0
0.05
0.1
0.15
Plastic strain [-]
0.2
Figure 3-32: Stress vs. Plastic strain
relationship for steel reinforcements,
Hardened Slab
2
8
1.8
Compressive
Tensile
6
1.6
DIF [-]
DIF [-]
active
Table 3-9: Inputs for MAT159,
Hardened Slab
140
1.4
4
2
1.2
1
0.001 0.01
0.1
1
10
Strain-rate [1/sec]
0
0.001
100
Figure 3-33: DIF for steel
0.1
10
Strain-rate [1/sec]
1000
Figure 3-34: DIF for concrete
60
60
50
PH-Set 1a
40
PH-Set 1b
30
Load 1
Load 2
20
Pressure [psi]
Pressure [psi]
active
50
PH-Set 2a
40
PH-Set 2b
30
Load 1
Load 2
20
10
10
0
0
0
20
40
60
Time [msec]
80
100
0
(a)
20
40
60
Time [msec]
80
100
(b)
Figure 3-35: Applied demands
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The shock loads “PH-Set 1a”, “PH-Set 1b”, “PH-Set 2a”, and “PH-Set 2b”, the firsts two
and the seconds two for the Normal and Hardened slab respectively, defined for the
contest are used for the FE analysis. Both pressure time histories applied to the model are
simplified as linear piecewise curves and renamed respectively “Load 1” and “Load 2”,
see Figure 3-35.
The Normal Slab model exhibits distributed cracking and inelastic response. For both the
applied demands, the dynamic yielding threshold of the reinforcement is reached resulting
in a residual deflection of the slab. Cracking is concentrated near the mid-span of the slab
on the rear surface and is distributed over the center 1/3 of the slab. For Load 1 and Load
2 the maximum displacement at the slab center point is predicted to be 4.05 in. (103 mm)
and 2.74 in. (69.6 mm) respectively. The time of occurrence of the maximum
displacement is 0.025 sec and 0.024 sec respectively for Load 1 and Load 2. Residual
deflections of 3.27 in. (83.0 mm) and 2.28 in. (58 mm) are observed for Load 1 and Load
2, respectively.
As is typical with reinforced concrete slabs the modeled system does not have
supplemental shear reinforcement. Under the applied pressure demand the shear stresses
at the supports are high. To ensure stability of the model during the response history the
use of beam elements over truss elements for the reinforcement is imperative. Another
crucial issue is the gap between the slab and the upper supports.
When the slab contacts the upper support the BC changes from “simple-simple” to
“fixed-fixed”, resulting in a sudden increase in slab resistance. This resistance is
provided by the dynamic tensile resistance of the concrete and the dynamic flexural
resistance to negative moment. The negative flexural resistance is low due to the single
plane of reinforcement. Crack patterns for both load cases are shown in Figure 3-36 and
in Figure 3-37. Predicted deflections (δ) are shown in Figure 3-38, and summarized in
Table 3-10. It is important to note that an error in the assumed gap of 0.25 in. will affect
the deformations predicted.
The predicted slab deflection is compared with the experimental measured deflection in
Figure 3-39.
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33.75 in. (857 mm)
64 in. (1625 mm)
64 in. (1625 mm)
33.75 in. (857 mm)
Figure 3-36: Crack pattern
on the rear side - Load 1, Normal Slab
on the rear side - Load 2, Normal Slab
5
Load 1
δ [inch]
4
3
Max. Def. [in. (mm)]
4.05 in.
103 mm
Time of occurrence of
max Deformation [sec]
0.025 sec
Residual Def. [in. (mm)]
3.27 in.
83.0 mm
2
Load 2
1
Load 1
Load 2
0
0
0.05
Time [sec]
0.1
0.15
Figure 3-38: Predicted Deflection History,
Normal Slab
Pierluigi Olmati
2.74 in.
69.6 mm
Time of occurrence of
max Deformation [sec]
0.024 sec
2.28 in.
58 mm
Table 3-10: Predicted Results,
Normal Slab
Page 105 of 189
Figure 3-39: Predicted (numerical) vs. experimental deflection, Normal Slab
The Hardened Slab model exhibits minimal cracking and inelastic response due to tensile
yielding of the reinforcement for both the applied loads. Minimal distributed cracking
occurs in the mid-span of the slab on the rear surface (see Figure 3-40 and Figure 3-41).
For Load 1 and Load 2 the maximum displacement at the slab center point is predicted to
be 2.43 in. (61.7 mm) and 1.46 in. (37.0 mm) respectively. The time of occurrence of the
maximum displacement is 0.020 sec and 0.0168 sec respectively for Load 1 and Load 2.
Residual deflections of 2.17 in. (55.1 mm) and 1.17 in. (29.7mm) are observed for Load 1
and Load 2, respectively. The response and summary prediction are presented in Figure
3-42 and Table 3-11.
During the development of the CSCM concrete model, the input parameters were
obtained by fitting experimental data for concrete with unconfined compression strength
between 2900 psi (20 MPa) and 8410 psi (58 MPa). The concrete examined in this
predictive contest has unconfined compression strength of 11600 psi (80 MPa), which is
outside of the bounds of the model. The original CSCM model parameter data is utilized
to develop estimation curves allowing for extrapolation for the higher unconfined
compression strength of 11600 psi (80 MPa). The parameters used and the extrapolation
techniques are not included here for brevity. For comparison purposes the model is run
with both strength of 7900 psi and the tested strength of 11600 psi (see Figure 3-42). The
11600 psi model is submitted as our official estimate of response. As previously
mentioned, a gap of 0.25 in. was assumed between the blast face and the support. As the
panel deforms, the support could contact the panel resulting in additional fixity. The
results of the analysis indicate that contact does not occur under both Load 1 and Load 2.
It is important to note that an error in the assumed gap will affect the deformations
predicted.
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33.75 in. (857 mm)
64 in. (1625 mm)
64 in. (1625 mm)
33.75 in. (857 mm)
on the rear side - Load 1, Hardened Slab
δ [inch]
3.5
on the rear side - Load 2, Hardened Slab
Load 1
3
2.43 in.
61.7 mm
2.5
Time of occurrence of max
Deformation [sec]
0.02 sec
2
2.17 in.
55.1 mm
1.5
1
(fc=7.9 ksi) Load 1
(fc=7.9 ksi) Load 2
(fc=11.6 ksi) Load 1 Prediction
(fc=11.6 ksi) Load 2 Prediction
0.5
0
0
0.05
0.1
0.15
Time [sec]
Figure 3-42: Predicted Deflection History,
Hardened Slab
Pierluigi Olmati
Load 2
Time of occurrence of max
Deformation [sec]
1.46 in.
37.0 mm
0.0168 sec
1.17 in.
29.7 mm
Table 3-11: Predicted Results,
Hardened Slab
Page 107 of 189
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3.4 Insulated panels under close-in detonations
Generally blast generated demands can be categorized into the far field design range and
close-in design range. In the far field design range blast generated pressure demands can
be considered uniform on the structure and basic single degree of freedom approximate
analysis is often implemented. In the close-in design range blast pressures are nonuniform and the pressure magnitudes can be very high [DoD 2008]. These ranges are
categorized by the scaled distance of the detonation relative to the structure. The scaled
distance is measured in terms of distance, R, divided by the weight of explosive, W, in kg
(lbf) of TNT to the 1/3 power. A close-in detonation is often considered to exist when the
scaled distance is less than 1.2 m/kg1/3 (3.0 ft/lb1/3).
Following the detonation of a high explosive at a small scaled-distance from a concrete
wall a shock wave is generated. Part of the shock wave that strikes the wall surface is
transmitted to the concrete, resulting in a compressive wave. When the transmitted shock
wave reaches the back surface, it reflects resulting in a tensile wave. If the tensile stress
on the back face is greater than the dynamic concrete tensile resistance the concrete will
fragment, i.e., spall [DoD 2008]. The front zone can also spall by excessive compressive
stress, or if it is subject to a sufficiently strong tensile shock wave as with the back face.
The failure of both back and front face to a depth of at least half the wall thickness each
will produce a breach. A breach can also form if the shock front contains enough energy
to completely fragment a localized zone through the depth of the wall, or if the tensile
waves surpass the tensile capacity of the concrete, creating a void through the entire
member.
The amount of spall may vary from the exterior to interior face of a panel depending on
the mechanics of the shock wave propagation through the material. Since breach
represents the void generated, a singular breach diameter is measured on the wythe. A
schematic of spall and breach are shown for an insulated wall panel in Figure 3-43.
Resistance to spall and breach in concrete elements is an important design consideration
when close-in detonations of high explosives are possible. Spall on the interior face of
the structural element can result in the formation of small concrete fragments that can
travel at hundreds of feet per second [DoD 2008]. These fast moving fragments in the
protected space of the building can result in fatalities and damage to equipment. When
breach occurs, the protected space of the building becomes accessible which can be
undesirable for secure facilities. The formation of spall or breach can be predicted for
solid elements using empirical methods developed by McVay [Mc Vay 1988], Marchand,
Woodson, Knight [Marchand 1994] and DoD [DoD 2008]. While these methods have
been validated for solid concrete elements minimal research has been conducted on multiwythe concrete panels.
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Figure 3-43: Spall/breach schematic
Multi-wythe insulated concrete panels are a popular form of exterior building cladding
used by the precast concrete industry for residential and commercial buildings. Insulated
wall panels have been produced in the United States for more than 50 years PCI [PCI
1997]. These wall systems consist of an exterior concrete wythe, an interior insulation
layer, and an interior concrete wythe (Figure 3-43). These systems can be configured
with the interior and exterior wythe connected via shear ties to provide composite action
to out-of-plane loads. It can also be configured as non-composite with an interior
structural wythe, an exterior architectural wythe, and nominal number of shear ties.
Insulated panel systems lend themselves to precast construction allowing for expedited
onsite erection of the building envelope. The insulation layer typically consists of
expanded polystyrene (EPS), extruded polystyrene (XPS), or polyisocyanurate (Polyiso).
The type and thickness of the insulation materials depends on the energy efficiency
requirement for the building envelope. The most common use of insulated panels is for
exterior walls, but they can also be adopted as internal partition walls, especially when
thermal transmission within the facility is restricted.
The aim of this study is to assess the behavior of insulated panels subjected to close-in
explosions through experimental evaluation and numerical modeling. The behavior of a
conventional 6 in. (152 mm) precast concrete wall is compared with the behavior of
insulated wall panels. Several insulated panel configurations are considered for
investigating both the influence of the foam layer and the performance sensitivity to foam
thickness. Since only a localized region near the explosive charge is affected by a closein explosion, similar to a localized impact [Ozbolt et al. 2011], the global behavior of the
wall is not considered in this study.
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3.4.1 Experimental Program
The experimental program consisted of an assessment of conventional insulated wall
panels subjected to close-in detonations of high explosives. The explosive charge and
standoff distance are the same for all the simulations. The main goal of the study is to
assess the behavior of insulated panels subjected to close-in detonations in comparison to
the behavior of the conventional solid RC panels. As the demand is the same for all
simulations and the goal is to compare the results of the various models, conclusions can
be made without referring explicitly to the explosive weight. The panels are subjected to
a detonation of high explosives at a stand-off distance of 5 in. (127 mm) which is
estimated to produce a reflected pressure of 43,000 psi (296 MPa). All the insulated
panels are comprised of an exterior and interior reinforced concrete wythe with a
thickness (t1 and t2) of 3 in. (76 mm). The foam is varied with thicknesses (tf) of 2 in., 4
in., and 6 in. (51, 101, 152 mm) as well as a case where two concrete panels are tested
with no foam. The panels have a planar dimension of 64 in. by 64 in. (1626 x 1626 mm)
and are reinforced with #4@10 in. (Φ No.13 12.7 mm diameter, spaced at 254 mm) and 6
x 6 W4.0 x W4.0 (Φ 5.7 mm diameter, spaced at 152 mm). A solid 6 in. thick panel is
also examined. This panel was tested separately as part of an earlier study and therefore
has a smaller planar dimension. The details for the two panel types are illustrated in
Figure 3-44.
Figure 3-44: Plan and elevation views of tested panels
The panels represent standard construction details used by the Precast Concrete industry
in the United States. The specified concrete design strength for all panels was 5000 psi
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(34.5 MPa). The concrete strength was measured in accordance with ASTM C39 [ASTM
2012a] within 7 days of the detonation experiments and was found to be 5560 +/- 150 psi
(38.3 MPa) and 7160 +/- 110 psi (49.4 MPa) for the solid and insulated panels
respectively. The bar reinforcement met the requirements of ASTM A615 [7] Grade 60
(420 MPa) and the welded wire reinforcement (WWR) met the requirements of ASTM
A1064 [8]. Yield stress for both materials is assumed to be 60 ksi (420 MPa).
The research study focused on the most economical insulation option, EPS foam. The
foam has a specific weight of 1.4 lbf/ft3 (220 N/m3) [PCI 2011], elastic modulus of 250
psi (1.72 MPa) and Poisson’s coefficient of 0.05 [Widdle 2008 and Masso-Moreu 2003].
Wythes were connected via 0.5 in. (12.5 mm) diameter bolts 6 in. (152 mm) from each
corner. Fender washers with a diameter of 3 in. (76 mm) were applied to mitigate
concentrated load effects at the corners. Insulation layers were formed by stacking
individual 2 in. (50 mm) thick EPS sheets to meet the prescribed foam thickness. The test
matrix is summarized in Table 3-12.
ID
Description
Thickness of
exterior wythe,
t1 [in. (mm)]
Thickness of EPS
insulation,
tf [in. (mm)]
Thickness of
interior wythe,
t2 [in. (mm)]
6C
6 in. solid
6.0 (152)
0.0
Not Applicable
3C-0F-3C
Stacked 3 in.
panels
3.0 (76)
0.0
3.0 (76)
3C-2F-3C
3 in. panels with 2
in. EPS
3.0 (76)
2.0 (51)
3.0 (76)
3C-4F-3C
3 in. panels with 4
in. EPS
3.0 (76)
4.0 (102)
3.0 (76)
3C-6F-3C
3 in. panels with 6
in. EPS
3.0 (76)
6.0 (152)
3.0 (76)
Table 3-12: Test matrix
3.4.2 Empirical assessment
An empirical approach for predicting spall or breach of solid concrete elements is
provided in the UFC 3-340-02 [DoD 2008]. In this section, the occurrence of spall and
breach is empirically examined for the panels evaluated experimentally. The empirical
formulas, provided by the UFC 3-340-02 [DoD 2008] are used for the solid panel and are
adapted to the case of the insulated panels.
The spall and breach threshold curves are extrapolated by experimental tests and are
plotted as functions of the spall parameter (ψ) and the ratio between the height of wall
section (h) in feet and the stand-off distance (R) in feet. More details about these curves
are provided by UFC 3-340-02 [DoD 2008] in Chapter 4.55. Experimental tests
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comprised of a cylindrical charge in contact with the ground, oriented side-on at a
prescribed stand-off distance from a wall as shown Figure 3-45. Various contact charges
with spherical and hemispherical shape were also tested. An empirically derived spall
threshold curve (Eq. (3-37)) and the breach threshold curve (Eq. (3-38)) were developed
as functions of the ratio h/R and the spall parameter ψ.
h
R
Typical cylindrical
cased charge, W
D
Concrete wall
Equivalent
hemispherical
surface charge, Wadj
L
Figure 3-45: Typical geometry for spall and/ breach predictions
(3-37)
(3-38)
Where a, b and c are constants per UFC 3-340-02 [DoD 2008] listed in Table 3-13; and
the spall parameter ψ is a function of both the stand-off and contact charges, as given in
Eq. (3-39) for non-contact and Eq. (3-40) for contact detonations.
(
)
(3-39)
(3-40)
Constant
a
b
c
Spall
-0.02511
0.01004
0.13613
Breach
0.028205
0.144308
0.049265
Table 3-13: Spall and breach threshold curve constants
Where f’c is the concrete compressive strength expressed in psi; Wc is the steel casing
weight expressed in lbf; and Wadj is the adjusted charge weight expressed in lbf. Wadj,
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given in Eq. (3-41), is the weight of a hemispherical surface charge that applies an equal
explosive impulse as that of the actual charge.
(3-41)
Where W is the equivalent TNT charge weight expressed in lbf; Bf is the burst
configuration factor, equal to 1.0 for surface bursts, and to 0.5 for free air bursts; and Cf is
the cylindrical charge factor given in Eq. (3-42) and Eq. (3-42).
(
(
)(
)
√
)
(3-42)
√
; all other cases
(3-43)
Where L and D are the charge length (in.) and diameter (in.) respectively. A specific
threat scenario provides the h/R and ψ values. Once the ratio h/R and the spall parameter
ψ are known, the response of a concrete panel in terms of spall, breach, or neither can be
determined. The threshold curves for spall and breach are illustrated in Figure 3-46. The
figure is divided into three sections, each region corresponding to breach, spall, or neither
(safe).
10
h/R
SAFE REGION
1
BREACH REGION
Spall
6C
3C-2F-3C
3C-6F-3C
Breach
3C-0F-3C
3C-4F-3C
0.1
1
ψ
10
Figure 3-46: Spall and breach threshold curves
In Figure 3.46 the expected results for tested concrete panels are presented relative to the
spall and breach threshold curves. The charge is assumed to be a free air blast explosion.
The burst configuration factor is taken as 0.5 and the charge shape factor as 1.0 for all
cases. The performance of the insulated panels is assessed with the conservative
assumption that the exterior wythe and insulation are not present. For example the 3C4F-3C is analyzed for a panel thickness, h, of 3 in. (76 mm) at a standoff distance, R, of
12 in. (305 mm). Recall that the specimens are tested with a stand-off from the exterior
front face of 5 in. (127 mm). Based on the empirical formulations the solid panel is
expected to spall and the 3C-0F-3C panel, neglecting the protection provided by the
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exterior wythe, is expected to breach. Using the adaption of the empirical formulas,
provided by the UFC 3-340-02 [DoD 2008], to the insulated wall panels the addition of
foam layers, and consequently standoff-distance to the front of the interior wythe, results
in a marginal improvement in breach resistance; however, for all cases spall is expected.
3.4.3 Experimental results
The results of the experimental program are summarized in this section. Each panel was
subjected to one detonation as previously discussed. The results of the damage incurred
on each panel are illustrated in Figure 3-47. Damage photos are taken for each panel face
that sustained damage. In the discussion, exterior refers to the wythe on the exterior of
the wall closest to the detonation while the interior refers to the wythe on the interior of
the building furthest from the detonation. The face locations correlate to the designation
used in Figure 3-43 (i.e., 1-front, 1-rear, 2-front, 2-rear). The diameter of spall was
measured on each face and the breach was measured on each panel if it occurred. The
effective diameter of the spall or breach was determined graphically from high-resolution
images. The area of the damaged region on each image was used to determine an
equivalent circular area and subsequently equivalent diameter. The results are
summarized in Table 3-14. For cases where no spall or breach occurred a value of 0.0 is
reported.
Panel
3C-0F-3C
3C-2F-3C
3C-4F-3C
3C-6F-3C
6C
Exterior Wythe
1 - Front 1 - Rear
Breach
Spall Dia. Spall Dia. Diameter
[in. (mm)] [in. (mm)] [in. (mm)]
7.9 (200) 9.9 (252)
0.0
8.3 (211) 11.7 (297) 8.3 (211)
7.7 (196) 11.0 (279) 7.7 (196)
7.6 (193) 15.5 (394) 7.6 (193)
5.6 (142) 22.4 (569)
0.0
Interior Wythe
2 - Front 2 - Rear
Breach
Spall Dia. Spall Dia. Diameter
[in. (mm)] [in. (mm)] [in. (mm)]
9.4 (239) 21.3 (541) 2.5 (64)
6.3 (160) 21.1 (536) 6.3 (160)
0.0
0.0
0.0
0.0
0.0
0.0
N.A.
N.A.
N.A.
Table 3-14: Experimental spall and breach results
The greatest amount of spall occurred on the solid 6 in. thick concrete panel (6C). A
comparable level of damage was observed on the panel composed of two 3 in. thick
concrete panels with no insulation (3C-0F-3C), however the failure mechanism changed.
The damage differed in that the stacked arrangement resulted in a breach of the interior
section. This interior breach however would not change the protection level since the
exterior wythe was not breached and access would not be possible. The breach of the
interior wythe on the stacked arrangement, however, may result in a greater quantity of
ejecta than that of the 6 in. solid.
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Figure 3-47: Damage observed from close-in detonations
The use of insulation foam provided mixed results as the thickness of foam was
increased. A small amount of insulation (3C-2F-3C) resulted in the lowest performance.
A full breach of both wythes occurred on the 3C-2F-3C panels with a similar amount of
interior spall diameter to that of the 6C and 3C-0F-3C panels. This indicates that small
separations created by insulation may provide enough space to allow for the damage to
the exterior wythe to eject and impact the interior wythe. This is further supported by
comparing the damage to the exterior wythe of the 3C-0F-3C and the 3C-2F-3C panels.
The damage levels are similar with the exception that no breach occurs when the exterior
wythe is bearing against the hard surface of the interior concrete wythe.
The use of a greater amount of foam on the 3C-4F-3C and the 3C-6F-3C panels resulted
in a complete protection of the interior concrete wythe. For both cases, no damage was
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observed on either the front or rear of the interior wythe. The amount of spall on the
exterior wythe remained comparable between the 3C-2F-3C and 3C-4F-3C but increased
for that of the 3C-6F-3C. This may indicate that larger amounts of foam may result in
less containment of the exterior wythe.
The empirical prediction of spall and breach was in line with the measured values for the
solid concrete panel (6C). The spall occurred as expected (compare the 6C panel in
Figure 3.46, Table 3-14 and Figure 3-47). Nevertheless, utilizing the empirical
formulation assuming that the exterior wythe is not present is not accurate. The stacked
3C-0F-3C was expected to have a breach; however, only spall was present. The 3C-2F3C was expected to have spall however a breach occurred and the larger foam thicknesses
was expected to produce spall but no damage was observed. Based on these observations
it is clear that the mechanics of the shockwave propagation through insulated panels is
complex and consequently a numerical evaluation is conducted.
3.4.4 Numerical model
Numerical analyses are carried out in order to both design the experimental tests and
further investigate the response of the insulated wall panels subjected to close-in
detonations. The numerical investigation is valid for all three types of insulated wall
panels (non-composite, composite, and partially-composite) as the shear connectors,
which provide coupling between the two concrete layers, are significant for the global
response of the insulated panel [Naito et a. 2011]. For the analyses performed, only the
local effect of the insulation is of concern, the ties [Naito et a. 2012] are therefore not
included.
Several studies have focused on high load demands on slabs and/or protective metal
plates. The research indicates that numerical simulations can accurately predict the
response of structures loaded by both close-in detonations [Zhou et al. 2008] and impact
loads [Flores-Johnson et al. 2011].
Zhou et al. [Zhou et al. 2008] conducted numerical and experimental studies on concrete
slabs, comprised of conventional and steel fiber reinforced concrete, subjected to two
consecutive detonations. Furthermore, Zhou et al. [Zhou et al. 2008] adopted a damage
model for the concrete using the erosion algorithm [LS-Dyna 2012a] in order to model
the fracture in the concrete. A similar set of numerical methods was conducted by FloresJohnson, Saleh, and Edwards [Flores-Johnson et al. 2011] who presented an investigation
on the ballistic performance of monolithic, double- and triple-layered metallic plates.
Finite element models, examining high rate loading effects, are further validated by the
experimental works of Børvik, Dey and Clausen [Børvik et al. 2009] and Forrestal,
Borvik, and Warren [Forrestal et al. 2010], and many of the physical characteristics of the
penetration process observed experimentally were numerically reproduced, allowing for a
reduction in the number of experiments required.
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Furthermore, about the study of high load demands on structures, analytical formulations
were developed in order to both estimate the damage of concrete pavement slabs under
close-in explosions [Luccioni et al. 2006] and the penetration of projectiles into concrete
panels [Li et al. 2006]. Several experimental tests were also carried out in order to assess
the behavior of panels fabricated with various types of concrete under close-in
detonations [Ohkubo et al. 2008]. Concerning the this study, a comparable effort was
made by [Yamaguchi et al. 2011] on the use of thin shock absorbing materials for using
between concrete panels. The results of their work indicated that the adoption of thin
layers of foam and rubber does not improve the resistance to the spall. The examined
thickness however, was 15 mm, much lower than insulation thicknesses used in
conventional construction in the United States.
Many numerical solution techniques can be utilized for this evaluations including the
“Lagrangian”, “Eulerian”, “Eulerian-Lagrangian” methods [Bontempi et al. 1998 and 23],
and the “Smoothed Particle Hydrodynamics” method [LSTC 2012b and Manenti et al.
2012]. Furthermore, two methods exist to take into account the interaction between the
shock wave and the structural component: the coupled and the uncoupled approach
[NCHRP 2010]. In this study the “Lagrangian” method and the un-coupled approach are
utilized [Davidson et al. 2005] in order to reduce the computational effort, the blast load
is computed and applied independently from the structural response of the insulated
panel. Consequently, the Load Blast Enhanced keyword [LS-Dyna 2012a] is used to
provide the blast load demand [Coughlin et al. 2010].
The finite element models have constant solid stress elements for modeling the concrete
and foam materials, and truss elements for modeling the reinforcement [LS-Dyna 2012a].
To bond the truss and solid elements, the LS-DYNA keyword Constrained Lagrange in
Solid is used.
The material model of the reinforcement is provided by the kinematic hardening plasticity
model [Chen et al. 2012] and the strain rate effect is accounted for by the Cowper and
Symonds strain-rate model [Cowper et al. 1957, and Su et al. 1995]. The parameters
selected for this model are: D=40 s-1 q=5 [Su et al. 1995].
The adoption of an appropriate constitutive model for the concrete [Bontempi et al. 1997]
is imperative to accurately model the response of concrete structures under close-in
explosions. In this study, the Karagozian & Case (K&C) Concrete Damage Model
[Malvar et al. 1997] (Material Type 72R3) with automated parameter generation [LSDyna 2012a] was selected for modeling the concrete.
The implementation of the strain-rate effects in the concrete model is also crucial to
properly simulate the behavior of concrete wall panels subjected to impulsive loads [Xu
et al. 2006]. In fact, the concrete has different strain-rate effects in tension and
compression; furthermore, the hydrostatic component of the stress tensor is important for
the concrete behavior [Li et al. 2006], which complicates the experimental assessment of
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the strain-rate effects [Grote et al. 2001]. The results of numerical simulations are
influenced by the strain-rate dependence and several strain-rate curves are proposed in the
literature. However, the research by [Williams et al. 2011] suggested that dynamic
increase factors for the concrete constitutive model are not necessary because the finite
element models are able to capture the strain-rate effects by the inertia confinement only.
In this study the strain-rate curves developed experimentally by [Tedesco et al. 1997] are
adopted for the numerical models, as shown in Figure 3-48.
6
5
Compressive
Tensile
DIF
4
3
2
1
0
1E-7
1E-5
1E-3
1E-1 1E+1 1E+3 1E+5
Figure 3-48: Dynamic Increase Factor (DIF) versus strain-rate for concrete
The foam is modeled using the Modified Crushable Foam (Material Type 63) [LS-Dyna
2012a]. The disadvantage of this material model is the elastic unloading; however, since
the study is focused on the max inbound effects the unloading is not critical to the
analysis. The foam constitutive law is characterized by the stress versus volumetric strain
curves for each strain-rate deformation regime.
The stress versus volumetric strain relationship of the EPS foam is taken from
comparative experimental tests conducted on different foam types by Croop and Lobol
[Croop et al. 2009]. The values are obtained for the insulating foam at many load rates
and the stress axis is normalized by the static yield stress. Data on the yielding stress of
EPS foam is taken from PCI recommendations [PCI 1997], from which the stress versus
volumetric strain chart is obtained via the previously derived normalized chart. This
approach assumes that the two foams have approximately the same chemical and
morphologic characteristics, resulting in the same behavior at high load rate. The two
foams have only a different specific weight, a parameter which mainly influences the
foam resistance [Di Landro et al. 2002]. Table 3-15 and Figure 3-49 summarize the foam
characteristics used.
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200
1.4
1.2
100/s
0.01/s
1
120
0.8
80
0.6
0.4
40
Stress [MPa]
Stress [psi]
160
0.2
0
0
0
0.2
0.4
0.6
Vol. strain [-]
0.8
1
Figure 3-49: Stress vs. volumetric strain chart of the used EPS foam
Density
Water absorption
Compressive strength
Tensile strength
Linear coefficient of expansion
Shear strength
Flexural strength
Thermal conductivity
Maximum use temperature
1.4 lbf ft3
<3%
15 psi
25 psi
40·106 / °F
35 psi
40 psi
0.26 Btu-in /hr/ft2/°F
165 °F
220 N/m3
<3%
103 kPa
172 kPa
72·106 / °C
241 kPa
276 kPa
0.037 Wm/m2/°C
74 °C
Table 3-15: Assumed physical properties of the EPS insulating foam
In order to capture the interaction between the two concrete wythes the LS-DYNA
Contact Eroding Single Surface parameter was used. Furthermore, in order to avoid both
numerical instability and excessively short time steps (Δt<10-7 second), the foam is
allowed to erode through the use of the LS-DYNA Mat Add Erosion when the volumetric
strain reaches 0.95.
Table 3-16 illustrates the predicted concrete damage patterns on the front of the exterior
wythe, the rear of the interior wythe, and on a section view respectively. The damage
parameter of the concrete model has been used in order to illustrate both the cracking and
the spall patterns on the concrete wythes, mainly because the damage parameter is
cumulative so no time dependent. The threshold values for evaluating the failure of the
concrete were adopted from [Wang et al. 2008] and used to capture the failure of the
concrete. The failure patterns in Table 3-16 are illustrated using the damage parameter
fringe level from 1.95 to 2.
As illustrated in Table 3-16, the numerical analysis provides a sufficiently accurate
estimation of the insulated wall panels behavior. The occurrence of spall and breach on
the panels is the same as the experiment. The solid panel and the 3C-2F-3C panel resulted
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3C-2A-3C
3C-6F-3C
3C-0F-3C
3C-2F-3C
3C-4F-3C
6C
in breach while the 3C-4F-3C and 3C-6F-3C panels had damage only on the exterior
wythe. The stacked panel 3C-0F-3C resulted in damage primarily to the interior wythe.
Table 3-16: Exterior (left), interior (right) and section view (below) of numerical results;
damage parameter from 1.95 to 2.
The predicted spall diameter on the rear face of the internal wythe is compared with the
experimental data in Figure 3-50. The horizontal axis denotes the insulated wall panel
under investigation (with the amount of foam increasing from left to right), while the
vertical axis is the spall diameter (in the rear face of the interior wythe). As mentioned
previously the spall diameter is measured by the plot of the damage parameter. Two
ranges of the damage parameters are considered in order to provide a minimum and a
maximum threshold for the predicted spall diameter. The minimum value is assessed by
the plot of the primary damage into the concrete, while the maximum value is assessed by
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the plot of the primary plus the secondary damage into the concrete. The primary and
secondary damage are represented by the damage parameter in the range from 1.95 to 2
and from 1.8 to 1.95 respectively [Wang et al. 2008]. The numerical model matches the
response of the insulated panels; however, the stacked and solid panels are marginally
under-predicted.
64
20
48
15
32
10
Experiment
16
Numerical Minimum
5
Numerical Maximum
3C-6F-3C
3C-5F-3C
3C-4F-3C
3C-3F-3C
3C-2F-3C
3C-1F-3C
3C-0F-3C
0
6C
0
Diameter [cm] (+/- 2.5 cm)
Diameter [in.] (+/- 1 in.)
25
Figure 3-50: Measured and predicted spall diameter on protected face
250
3C-2F-3C
200
1
3C-4F-3C
3C-6F-3C
150
3C-2A-3C
0.5
3C-4A-3C
100
3C-6A-3C
50
0
0
0
1
2
3
time [ms]
4
Force on Center of 2-front [MN]
Force on Center of 2-front [kips]
The mechanism of damage of the interior wythe can be characterized by the impact of the
concrete debris of the exterior wythe on the front of the interior wythe. For the blast
demand examined, the exterior 3 in. (76 mm) thick wythe spalls for all foam thicknesses.
When the concrete spalls the debris impacts the interior wythe. The impact force is
measured in the numerical model over a 4 in. (102 mm) diameter region on the center of
the 2-front surface by the means of a circular control section in the finite element models.
The force demand is illustrated in Figure 3-51. The impact force decreases considerably
as the foam thickness increases from the base level of 2 in. (51mm) to the insulated wall
panel with 6 in. (152 mm) of foam.
5
Figure 3-51: Impact force demand on the front face of the interior wythe
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The numerical simulations are also used to investigate the effect of the insulation material
on the resistance of the insulated wall panel to spall and breach. The use of air over that
of three foam types (A, B and C) is examined. The foam A is the material used in the
experimental investigation. As mentioned, this material has a density of 1.4 lb f/ft3 (22.4
kg/m3), a compressive strength of 15 psi (103 kPa) and represents the medium grade of
EPS foam [PCI 1997]. Foam B represents denser EPS foam and has a density of 1.8
lbf/ft3 (28.8 kg/m3) and a compressive strength of 25 psi (172 kPa). Foam C represents
XPS with a density of 1.8 lbf/ft3 (the same as Foam B) and a compressive strength of 40
psi (276 kPa).
The results of the parametric numerical analyses are summarized in Figure 3-52. As
illustrated, the damage to the interior wythe of the insulated wall panels increases as the
density of the foam increases. Air provides the best defense against the transfer of the
demand from the exterior wythe while dense and strong XPS foam provides the lowest
resistance to damage. Evidently, the air provides improved resistance by spreading the
damage over a larger portion of the exterior wythe. This is illustrated in a comparison of
the fringe plots for the 2 in of air versus 2 in of foam A panels in Table 3-16. The forces
imparted to the front of the interior wythe are comparable between the air and foam
(Figure 3-51) however the force is spread over a larger area thus decreasing the stresses
imparted and associated damage.
Insulation thickness [cm]
5
10
0
15
25
Air Minimum
64
20
48
15
32
10
16
5
Spall diameter [cm]
Spall diameter [in.]
Air Maximum
Foam A Minimum
Foam A Maximum
Foam B Minimum
Foam B Maximum
Foam C Minimum
0
0
0
1
2
3
4
Insulation thickness [in.]
5
Foam C Maximum
6
Figure 3-52: Parametric examination of insulation type and thickness for spall
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3.4.5 Summary
A research program was conducted to assess the viability of insulated reinforced concrete
wall panels in mitigating spall and breach from close-in detonation of high explosives.
The performance was assessed relative to experimental tests, existing empirical
formulations, and numerical analyses. The experimental tests were conducted on full
scale panels built in accordance with the standard practice of the United States
Precast/Prestressed Concrete Industry. The panels consisted of conventional geometries
and included an internal and external concrete wythe 3 in. (76 mm) thick and EPS
insulation varying from 0 to 6 in. (152 mm). The empirical formulations developed by
the DoD [DoD 2008] were used as a basis of comparison. The numerical simulations
were conducted using LS-DYNA finite elements program [LS-Dyna 2012a]. The
following conclusions are drawn from the results presented:

The solid 6 in. concrete panel (6C panel) and the panel composed of two stacked 3 in. wythes
(3C-0F-3C panel) provide a comparable level of resistance to close-in detonations. The
mechanism of failure however is altered in that the stacked panel prevents the occurrence of a
complete breach with minimal damage on the exterior wythe and breach only on the interior
wythe.

The use of EPS insulation foam resulted in mixed performance as a function of the insulation
foam thickness.
Small amounts of insulation, 2 in. (51 mm), resulted in a full breach
(similarly as the experimental results of Yamaguchi et al. [Yamaguchi et al. 2011]]) while
the case with no insulation (3C-0F-3C panel) had no breach. Greater thicknesses of insulation
resulted in full protection of the interior wythe with no damage on either the front or rear face
of the interior wythe.

The empirical formulations for spall and breach matched the data for the solid panel (6C
panel).

The use of empirical formulations for predicting the spall and breach on the insulated wall
panels was made by assuming that the exterior wythe was not present and the stand-off
distance was increased. This approach was found to be inaccurate as it does not represent the
complex behavior that occur, as the shockwave propagates through the various panel
materials and the external wythe debris impacts the interior wythe.

The numerical simulations are able to predict the occurrence of the spall and breach for
insulated panels subjected to close-in detonations; the breach diameters on the rear face of the
interior wythe were found to be marginally unconservative for small foam thickness but
bound the response at higher thicknesses.

The numerical models indicate that the density and strength of the insulation foam is the main
factors in transfer of demand to the interior wythe.
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
The numerical models supported the experimental data and could be used to further develop
semi empirical spall and breach curves for insulated wall panels subjected to close-in
detonations.
In conclusion, the insulated wall panels have enhanced spall and breach performance
against close-in blast demands when adequate foam thickness is used. This is due to the
exterior concrete wythe acting as a sacrificial wythe, allowing the gap and foam to
dissipate much of the concrete fragment kinetic energy and mitigate the incipient
shockwaves from the initial shock.
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4 THE GLOBAL RESISTANCE
Structural robustness is a research topic particularly relevant both in the design of new
structures, and also for the safety assessment of existing structures. Behind this attention,
there is the interest from a society that cannot tolerate death and losses as in the past.
This is more evident after:




Recent terrorist attacks (a series of terror attacks in America and beyond, the
deadliest being the September 11, 2001 attacks in New York at the World Trade
Centre).
Recent bridge collapses due to deterioration or bad design or bad construction (for
example, the De la Concorde overpass collapse in Montreal, 2006 and the I-35
West Bridge in Minneapolis in 2007).
Recent difficult to foresee multiple hazard events from natural sources (wind,
earthquake, flooding, wildfire, etc.) and from human sources (terrorism, fire, etc.)
that lead to dramatic consequences, the most significant of which is the 2011
earthquake, off the Pacific coast of Tōhoku, that triggered powerful tsunami
waves.
Among all other steel structures, many steel truss bridges in their various forms,
very common worldwide, are now aged, not often optimally maintained, and need
to be checked equally for safety and for serviceability. In this sense, also the
optimal cost effective allocation of resources and the prioritization in the
retrofitting phase is a very important issue.
Even though a variety of terms have been used in literature, robustness is commonly
defined as the “insensitivity of a structure to initial damage” and collapse resistance as the
“insensitivity of a structure to abnormal events” [Starossek et al. 2010].
Similarly [ASCE 2005] defines progressive collapse as the spread of an initial local
failure from element to element, eventually resulting in collapse of an entire structure or a
disproportionately large part of it. [Starossek et al. 2010] focus on the differences of
progressive and disproportionate collapse, concluding that 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.
From a historical perspective, progressive collapse came up as the first structural
engineering concern, just after the collapse of the Ronan Point Tower, a residential
apartment building in Canning Town, London, UK, in May 1968, two months following
initial occupancy of the building. Ronan Point was a 22-story building, with precast
concrete panel bearing wall construction. An explosion of natural gas from the kitchen of
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a flat on the 18th floor failed an exterior bearing wall panel, which led to loss of support
of floors above and subsequent collapse of floors below due to impact of debris
[Ellingwood et al. 2005].
Concerning the above mentioned topics, there has been a lot of research in the recent
years. [Starossek et al. 2010] provide a terminology. A review of international research
on structural robustness and disproportionate collapse is provided in [Arup 2011].
Regarding the quantification of robustness related issues, [Canisius et al. 2007] provide
an overview of methods. [Starossek 2009] covers issues related to progressive collapse.
[Bontempi et al. 2007] and [Sgambi et al. 2012] provide a dependability framework,
adapted from the electronic engineering field, where dependability attributes are either
related to structural safety or serviceability. Focusing on structural safety, the attributes
of structural integrity, collapse resistance, damage tolerance and structural robustness are
investigated. Strategies and methods for the robustness achievement are discussed in
[Bontempi et al. 2008b], together with the robustness assessment of a very long span
suspension bridge.
That said, and even though many robustness research topics focus on explosions and
terrorist attacks, as Table 1 suggests, there is a variety of reasons or events that could
endanger a structure, eventually leading to a progressive collapse [Starossek et al. 2012].
Potential failure scenarios specific for bridges are also provided in [FHWA 2011], within
a framework aiming at the resilience improvement.
Faults
External
Man-made
(accidental or
intentional)
Impact (car, train,
ship, aircraft, and
missile)
Explosion (gas,
explosives)
Fire
Excessive loading
(liveload)
Errors
Intrinsic
Environmental
(natural)
Earthquake
Extreme wind
Heavy snowfall (excessive
roof loads)
Scour
Impact (avalanche,
landslide, rock fall, floating
debris)
Volcano eruption
Lack of
strength
Cracks
Deterioration
Design errors
Construction
errors
Usage errors
Lack of
maintenance
Table 4-1: Abnormal events that could threaten a structure [Starossek et al. 2012]
The collapse likelihood of a structure is typically characterized in probabilistic terms.
When an unexpected or critical event occurs, [Ellingwood et al. 2005] describe, in
probabilistic terms, the probability of a collapse in a structure as the product of the
probabilities of three sub events:

The extreme action associated with the event hits the structure;
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

The structure is damaged in the area directly affected by the action;
The local damage causes failures of other structural elements and leads to the
collapse of a significant part of the structure.
The assessment of the risk associated with the event (commonly defined as the product of
a probability of occurrence and of the corresponding consequence) can be performed
using standard risk techniques. Several authors have focused on aspects of risk analysis
and assessment in the civil engineering field - see for example [Faber et al. 2003], and,
more recently, [Gkoumas 2008]. Risk related special issues include the risk aversion for
low-probability, high-consequence events [Cha et al. 2012] and the risk consistency in
multihazard design for frame structures [Crosti et al. 2011].
Focusing on disproportionate collapse in probabilistic terms the probability of
disproportionate collapse P[C] as a result of an abnormal event can be decomposed into
three constituents: abnormal event, initial damage, and disproportionate failure spreading.
Decomposition also adopted in [Ellingwood et al. 2007]. This is represented as the
product of partial probabilities:
P[C] = P[C|D] P[D|E] P[E]
(4-1)
Where, P[E] is the probability of occurrence of the abnormal event E that affects the
structure; P[D|E] is the conditional probability of the initial damage D, as a consequence
of the abnormal event, and P[C|D] is the conditional probability of the disproportionate
spreading of structural failure, C, due to the initial damage D. The safety of structures
with regards to the single elements contained in the equation, each characterizing the
single sub-event mentioned above, is pursued in modern structural codes by the
introduction of partial safety factors.
According to this approach, [Giuliani 2012] identifies these three design strategies for
obtaining robustness:



Prevention or mitigation of the effects of the event (increase collapse safety);
Prevention or mitigation of the effects of the action (increase structural integrity);
Prevention or mitigation of the effects of the damage (increase structural
robustness).
These strategies are schematically depicted in Figure 4-1.
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Figure 4-1: Strategies for safety against extreme events and corresponding requirements
[Giuliani 2012]
The assessment of structural robustness is also strongly related to the degradation state of
the structures, caused by environmental agents: concrete carbonation, steel reinforcement
corrosion, alkali aggregate reaction, freeze-thaw cycles can lead, over time, to an
assessment of structural strength that is very different from that provided in the design
phase [Biondini et al. 2009]. The effect of the above factors could compromise the
structural response under a localized event.
Furthermore, different structural systems exhibit different degrees of robustness [Wolff et
al. 2010], something neglected even in modern design procedures that use partial safety
factors. Another issue very important in determining structural robustness for bridges is
redundancy. Bridge redundancy, is defined in the [Ghosn et al. 1998] as the capability of
a bridge to continue to carry loads after incurring damage or the failure of one or more of
its members. This capability is due to redistribution of the applied loads in transverse
and/or longitudinal directions.
Moreover, the inherent uncertainty associated with actions and mechanical, geometric and
environmental parameters cannot be ignored since they affect the structural response
[Biondini et al. 2004, Ciampoli et al. 2011, Garavaglia et al. 2012, and Petrini et al.
2012].
Steel truss structures and bridges have been the subject of recent research on what
concerns their ultimate strength and progressive collapse susceptibility. [Choi et al. 2009]
focus on the vertical load bearing capacity of truss structures, using a sensitivity index
that accounts for the influence of a lost element to the load bearing capacity. [Miyachi et
al. 2012] focus on how the live load intensity and distribution affect the ultimate strength
and ductility of different steel truss bridges. [Malla et al. 2011] conduct nonlinear
dynamic analysis for the progressive failure assessment of bridge truss members,
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considering their inelastic post-buckling cyclic behavior. [Saydam et al. 2011] use FE
skills to investigate the vulnerability, redundancy and robustness of truss bridges, taking
into account the stochastic time-dependent deterioration of the structure.
Progressive collapse literature indicates extensive research has been performed for the
past few years on steel moment frames possibly owed to the fact that different design
guidelines are issued in the US by the General Service Administration [GSA 2003] and
the Department of Defense [DoD 2009]. [Kim et al. 2009]) conduct nonlinear dynamic
analysis on benchmark buildings (3, 6 and 15-story) and compare the results with more
straightforward linear static step-by-step analysis. Using nonlinear dynamic finite
element simulations, [Kwasniewski 2010] investigates the collapse resistance of an 8story steel framed structure, and inquires on the uncertainties affecting the problem.
[Izzuddin et al. 2008a] provide a framework for progressive collapse assessment of multistory buildings, considering as a design scenario the sudden loss of a column. Using this
framework, the same authors [Izzuddin et al. 2008b] investigate possible scenarios, in the
form of the removal of either a peripheral or a corner column, in a typical steel-framed
composite building. [Yuan et al. 2011] investigate the progressive collapse of a 9-story
building, at a global level, using a numerical spring-mass-damper model. [Hoffman et al.
2011] investigate different column loss scenarios on 3 and 4-story steel buildings,
focusing on different aspects of the problem, among else, the load redistribution and the
column lost location. [Galal et al. 2010] compare retrofitting strategies for 18-story
buildings with different spans using 3D nonlinear dynamic analyses.
An important issue is the model complexity in the progressive collapse assessment.
[Alashker et al. 2011] deal with approximations in the numerical modeling, using a 10story steel building as a case study, and compares four models of different levels of
complexities (planar and 3D). Their conclusion is that, under restricted conditions, planar
models can lead to reasonable results regarding the progressive collapse characterization,
however, a full 3D analysis, in spite of its computational cost, may be the only sure way
to rigorously investigate this aspect. [Rezvani et al. 2012] conduct different non-linear
static and dynamic analyses, among else, on an 8-story building, aiming at the progressive
collapse assessment, and compare the results from the different analysis methods.
A relevant issue related to the structural robustness evaluation, is the choice of proper
synthetic parameters describing the sensitivity of a damaged structure in suffering a
disproportionate collapse.
Recently [Nafday 2011] discusses the usefulness of
consequence event design (as opposed to using a probabilistic approach), for extremely
rare, unforeseen, and difficult to characterize statistically events (black swans). In this
view, the author, with reference to truss structures, proposes an additional design phase
that focuses on the robustness, the damage tolerance and the redundancy of the structure.
This proposed metric is based on the evaluation of the determinants of the normalized
stiffness matrixes for the undamaged and damaged structure.
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Concerning extreme loads on structures, a scientific debate takes place nowadays on the
appropriate design methodology to adopt. To this point, the member-based design is not
efficient for contrasting extreme loads on structures that in general are unpredictable and
not probabilistically characterized [Nafday 2011]. Following the approach of [HSE 2001]
in the case of high uncertainties regarding the extreme loading likelihood, it is necessary
to put emphasis on the consequences of the event.
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4.1 The consequence factor
The method applied in this section aims at increasing the collapse resistance of a
structure, by focusing on the resistance of the single structural members, and accounting
for their importance to the global structural behavior consequently to a generic extreme
event that can cause a local damage. Moreover, the method is particularly helpful for
unpredictable events that by definition are not possible to take into account in the design
phase. This does not mean that the collapse resistance [Starossek 2009] is accounted only
for the single member resistance, because the authors intend, as a design philosophy, to
increase the resistance of the single members in addition to the structural stability analysis
that provide the assessment of the global structural behavior. In other words, if the
collapse resistance of a structure is identified by: the “load characterization”, the “local
resistance”, and the “insensitivity to a local damage” [Starossek 2009], this method
focuses on the issue of “local resistance”. Thus, it neglects the “load characterization” of
the extreme load since it is considered unpredictable, and it is complementary to the socalled threat independent stability analyses.
Focusing on skeletal structures (e.g. trusses), current member-based design in structural
codes does not explicitly consider system safety performance during the structural design,
while the level of safety in new designs is usually provided on the basis of intuition and
past experience [Nafday 2008]. On the other hand, the Ultimate Limit State (ULS) of the
Performance-Based Design (PBD) requires (see for example [EN 1990]) that individual
structural members are designed to have a resistance (R) greater than the load action (E),
where both R and E are probabilistically characterized [Stewart et al. 1997].
The member-based design is summarized in the following design expression, valid for a
single structural member:
R dundamaged  Edundamaged  0
(4-2)
Where Rdundamaged and Edundamaged are the design values respectively of the resistance and
of the solicitation [EN 1990] in the undamaged configuration of the structure.
Concerning the commonly implemented standards this equation is not respected with a
probability of 10-(6÷7). The method applied here aims to introduce an additional
multiplicative coefficient in the first term of the Eq. (4-2): this is identified as the member
consequence factor (Cf), takes values within a range from 0 to 1, and quantifies the
influence that a loss of a structural element has on the load carrying capacity. Essentially,
if Cf tends to 1, the member is likely to be important to the structural system; instead if C f
tends to 0, the member is likely to be unimportant to the structural system. Cf provides to
the single structural member an additional load carrying capacity, in function of the
nominal design (not extreme) loads. This additional capacity can be used for contrasting
unexpected and extreme loads.
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(1  Cscenario
) * R dundamaged  E dundamaged  0
f
(4-3)
[Nafday 2011] provides Eq. (4-3) in a similar manner, with the only difference being on
the range mean of Cf that is the inverse of the proposed one, so the first term of Eq. (4-3)
is multiplied directly by Cf. Thus, in this study the equation proposed by [Nafday 2011]
has been slightly revised in order to fit with the here proposed expression of the C f, see
both Eq. (4-3) and Eq. (4-4).
The structure is subjected to a set of damage scenarios and the consequence of the
damages is evaluated by the consequence factor (Cfscenario) that for convenience can be
easily expressed in percentage. For damage scenario is intended the failure of one or
more structural elements. Moreover, the robustness can be expressed as the complement
to 100 of Cfscenario, intended as the effective coefficient that affects directly the resistance see Eq. (4-3).
Cfscenario is evaluated by the maximum percentage difference of the structural stiffness
matrix eigenvalues of the damaged and undamaged configurations of the structure.
dam
 (un

i  i )

Cscenario

max
100 
f
un

i

i1 N
(4-4)
Where, λiun and λidam 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 (Rscenario) related to the damage scenario is therefore
defined as:
R scenario  100  Cscenario
f
(4-5)
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 a non-linear dynamic 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.
Some further considerations are necessary. The proposed method for computing the
consequence factors should not be used 1) for structures that have high concentrated
masses (especially non-structural masses) in a particular zone, and 2) for structures that
have cable structural system (e.g. tensile structures, suspension bridges).
The first issue is related to the dynamic nature of a structural collapse, since Eq. (4-4)
does not take into account the mass matrix of the system that is directly related to the
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inertial forces. It is possible to accept this limitation only if the masses are those of the
structural members, thus distributed uniformly. Moreover there is no way to consider any
dynamic magnification phenomena with Eq. (4-4).
The second issue is related to the geometrical non-linearity of cable structures. For such
structures the stiffness matrix is a function of the loads, something not accounted for in
the elastic stiffness matrix. Moreover for the nature of the elastic stiffness matrix,
eventual structural dissipative behaviors and non-linear resistive mechanisms (e.g.
catenary action) are not taken into account.
In the authors’ opinion the above limitations can be accepted if the desired outcome is a
non-computational expensive method, since the Cf value provides an indication of the
structural robustness in a quick and smart manner. Additional research could focus on the
development of criteria that a Robustness index should have to take into account the
previous issues that Eq. (4-4) does not account for.
With this in mind the Cf as expressed in Eq. (4-4) can be used primarily as an index to
establish the critical structural members for the global structural stability, or to compare
different structural design solutions from a robustness point of view. The latter
implementation of Cf can be very helpful for the robustness assessment of complex
structures, for example wind turbine jacket support structures [Petrini et al. 2011], since it
provides an indication on the key structural elements that in a complex structure are of
difficult evaluation.
In the following the member consequence factor is computed for the structural elements
of a steel truss bridge. Before that, the method is applied to simple structural systems.
4.1.1 Tests on simple structures
In this section, first, a simple example is shown, in order to provide insight on the method
proposed for computing the consequence factor and the structural robustness index.
Figure 4-2 shows a linear spring system.
y
ka
x
a
b
kb
Figure 4-2: Example spring structure
In a two-dimensional space there are two single degree of freedom translational springs.
Spring “a” has stiffness ka and spring “b” has a stiffness kb. The stiffness matrix of the
system is given by Eq. (6).
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k
K a
0
0
k b 
(4-6)
To make a numerical example, assuming ka=kb=10 kN/m, the obtained undamaged
stiffness matrix is:
K
undamaged
10 0 


 0 10
(4-7)
A hypothesis is made that a damage scenario (called scenario 1) reduces the stiffness of
the spring “b”: kbdamaged = 7 kN/m < kbundamaged = 10 kN/m. Consequently, the damaged
stiffness matrix takes the form of:
K
damaged
10 0


 0 7
(4-8)
At this point, applying Eq. (4-4), the following values for the consequence factors are
obtained:
C1f 1  0%
C1f 2  30%
(4-9)
The maximum consequence factor of the two, for the scenario 1, is Cf2. Consequently for
this scenario the consequence factor is the Cf2 equal to 0.3. Finally applying Eq. (4-5) the
robustness index obtained is 70%.
This method, previously applied analytically, is now applied numerically to two
additional examples (two simple structures). First, a single bay frame structure with a
diagonal beam brace, composed in total of 5 members, is considered (Figure 4-3 (a)). All
the cross sections of the structural members are the European IPE 300 (similar to a W
12x30) in European S235 steel (comparable to the ASTM A36), while both the frame
span and the frame height are one meter. The structure is plane and the boundary
conditions are of the pinned type. 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. Cf is computed by Eq. (4-4) and the
results in terms of Cf and robustness are indicated on the right side of Figure 4-3 (b).
After the failure of members 1 and 2 the structure is still redundant so the Rscenario has a
non-zero value; instead after the failure of members 1, 2, and 3 the structure is a
mechanism and consequently the Rscenario is zero (Cf is equal to 100%).
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1 meter
Cf max
IPE 300 steel S235
Robustness
Robustness %
IPE 300 steel S235
IPE 300 steel S235
1 meter
100
80
60
2
3
92
100
40
20
1
47
53
8
0
1
2
3
Damage Scenario
(a)
(b)
Figure 4-3: Example truss structure (a) and damage scenario evaluation (b)
The second structure considered is a star-shaped structure (Fig. Figure 4-4 (a)). In totally
there are 8 members with a pipe cross section: the outside diameter is of 20 centimeters,
and the thickness is of 20 millimeters. The steel is the European S235 one. With respect
to Figure 4-4 (a), 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.
The evaluation consists in removing elements 1 through 8, and the damage is intended as
cumulative like in the previous example. The results in terms of Cf and robustness are
indicated on the Figure 4-4 (b). Until reaching damage scenario 6 the Rscenario has a nonzero value. After that for damage scenario 7 the structure is reduced to a cantilever and
the Rscenario is 0.4%. Finally, Rscenario is equal to zero when the final structural member is
eliminated (Cf in this case is equal to 100%).
It is possible to observe from both Figure 4-3 and Figure 4-4 that the proposed method
captures the structural robustness reduction with the increase of the damage level. On the
other hand, Cf increases with the damage level.
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1 meter
Cf max
Robustness
100
27
1 meter
6
7
5
8
1
Robustness %
80
40
3
52
60
79
92
40
60
51
7
8
21
8
0
1
(a)
100
48
2
Sections: Pipe 200/20 steel S235
99.6
73
20
4
49
2
3
4
5
6
Damage Scenario
(b)
Figure 4-4: Example star structure (a) and damage scenario evaluation (b)
4.1.2 Application on a steel truss bridge
This section focuses on the robustness assessment of a steel truss bridge using the
member consequence factor method.
The bridge used as a case study is the I-35 West Bridge in Minneapolis. The I-35 West
Bridge was built in the early 1960s and opened to traffic in 1967. The bridge spanned
across the Mississippi River, Minneapolis. The bridge was supported on thirteen
reinforced concrete piers and consisted of fourteen spans. Eleven of the fourteen spans
were approach spans to the main deck truss portion. The total length of the bridge
including the approach and deck truss was approximately 580 meter (1,907 feet). The
length of the continuous deck truss portion which spanned over four piers was
approximately 324 meter (1,064 feet). The elevation of the entire bridge is shown in
Figure 4-5 (data and figure from [NTSB 2007]).
Figure 4-5: East elevation of the I-35W Bridge [NTSB 2007]
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The deck truss portion of the bridge was supported on a pinned bearing at Pier 7 and
roller bearings at the other three supports. The main bridge trusses were comprised of
built-up welded box and I-sections sandwiched by gusset plates at each panel point.
Steel truss bridges, like the I-35 West Bridge, had longer and lighter spans than their
contemporaries. The innovations, which facilitated the reduction in weight, include the
efficiencies inherent in statically determinant trusses, new stronger steels, thin gusset
plate connections, and welded box sections.
The catastrophic collapse which occurred on August 1st 2007 was probably due to a
combination of the temperature effect, roller bearings condition, and increased gravity
loads on the bridge prior to collapse. For this functionally non-redundant bridge the
initial buckle at the lower chord member close to the pier and local plastic hinges in the
member resulted in global instability and collapse [Malsch et al. 2011].
The bridge has been thoroughly studied by [Brando et al. 2010] focusing on the issues of
redundancy, progressive collapse and robustness, studies have been conducted in order to
assess the effect of the collapse of specific structural components [Crosti et al. 2012].
For computing the consequence factors and the robustness index of the structure for the
selected damage scenarios a FE model of the structure is necessary. Figure 4-6 shows the
three-dimensional FE model of the I-35 West Bridge built using the commercial FE
solver Sap2000® [Brando et al. 2010].
Figure 4-6: 3D FE model of the I-35 West Bridge
Both shell and beam finite elements are used in the FE model. The bridge superstructure
and both the deck girders and beams are built using beam elements, while, the concrete
deck is modeled using shell elements. Moreover, contact links connect the deck with
both the deck girders and beams. In accordance to the original blueprints of the I-35
West Bridge [MDT 2012], standard and non-conventional beam cross sections are
implemented in the model.
From this model a simplified (plane) FE model is extracted and is adopted for computing
the structural stiffness matrix in both the damaged and undamaged configurations. This
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choice has mostly to do with computational challenges in computing the stiffness matrix
for the full model. Regarding the structural decomposition of complex structures it is
possible to refer to the [Bontempi et al. 2008a] and [Petrini et al. 2011].
The expression of the consequence factor provided by Eq. (4-4) refers to the eigenvalues
of the elastic stiffness matrix. The choice to use a simplified model is also justified and
feasible since Eq. (4-4) is independent from the mass of the structure. Eq. (4-4) is also
independent from the loads, so the loads in the FE model are not considered. The
concrete deck is only simply-supported by the bridge superstructure, so the concrete deck
is not considered in the analyses and it is omitted in the model, consequently, the contact
links are deleted as well. The deck girders and beams are also omitted since they do not
have a strong influence to the load bearing capacity of the bridge.
The two trusses of the bridge superstructure are similar and connected by a transverse
truss structure, so the analyses focus on a single truss; at this point one plane truss is
obtained from the three-dimensional model, in order to have a two-dimensional FE
model, implemented for computing the stiffness matrix in both the damaged and
undamaged configurations.
Concluding, only a single lateral truss of the bridge is considered, and a set of damage
scenario is selected (Figure 4-7). The damage scenarios for this application are not
cumulative, so only a single member is removed from the model for each damage
scenario [Brando et al. 2012]. In this application the scenarios chosen focus on the area
recognized as initiating the collapse according to forensic investigations in different
researches [NTSB 2007, and Malsch et al. 2011].
6
3
2
1
5
7
4
Figure 4-7: Lateral truss of the bridge and selection of damage scenarios
With the aim of increasing the structural robustness of the bridge, and in order to test the
sensitivity of the method proposed, an improved variation of the structural system is
considered. In this case (Figure 4-8) the updated bridge truss is a hyper-static steel truss
structure.
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The results of both the original and the enhanced structural schemes, under the same
damage scenarios, are shown in Figure 4-9 and Figure 4-10.
6
3
2
1
5
7
4
Figure 4-8: Updated lateral truss of the bridge and selection of damage scenarios
Cf max
Robustness
100
Robustness %
80
41
58
63
55
65
62
35
38
3
4
5
Damage Scenario
6
77
60
40
59
20
42
37
45
23
0
1
2
7
Figure 4-9: Damage scenario evaluation in terms of Cf for the original configuration of
the bridge
Cf max
100
17
13
12
80
Robustness %
Robustness
14
47
36
40
60
40
83
87
88
86
53
20
64
60
0
1
2
3
4
5
Damage Scenario
6
7
Figure 4-10: Damage scenario evaluation in terms of Cf for the improved configuration of
the bridge
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The proposed robustness index (based on the member consequence factor Cf) captures
both the lack of robustness of the I-35 W Bridge, and its robustness enhancement as a
consequence of increasing the redundancy of the structure.
Generally speaking, it can be observed that the case-study bridge shows a low robustness
index. This is due to the fact that it is (internally) statically determined. In order to better
understand the use of the proposed consequence factor, it is useful to focus the attention
on the Damage Scenario number 7 (DS7), since it is particularly critical for the robustness
requirement of this structure. It has to be noted moreover that the proposed method
highlights the sensitivity of the bottom chord member, which was pinpointed from the
investigation on the causes of the collapse performed by [Malsch et al. 2011]. From the
analysis of the bridge in its original configuration and for the chosen damages
configurations, a consequence factor of 0.77 has been computed for the DS7 and,
consequently, a robustness index of 0.23 is obtained. This result can be used to design or
improve the bridge structure by means of different strategies:



The consequence factors obtained by the analysis of the various damage scenarios
can be used, as shown in Eq. (4-3), for the re-sizing of the structural elements
(each element is linked to the specific Cf obtained from the analysis considering
its failure). In this case the structural scheme of the bridge does not change with
respect to the original one. This option can be considered as a local (elementbased) improvement of the structural system;
The consequence factor can be used only as a robustness performance index,
without making use of Eq. (4-3). More than one structural configuration can be
examined in order to assess which is the best solution in terms of Cf. An example
of this strategy is given in the previous application of Figure 4-8. In this case the
scheme of the bridge has been modified by inserting additional structural elements
in order to obtain a redundant truss bridge. In the examined case the consequence
factor obtained by the DS7 decreases from 0.77 to 0.36; this appreciable result is
probably due to the position of the failed element in the DS7 which being a lower
element of the truss plays an important role in the load carrying capacity of the
original system. Generally speaking, the redundant bridge configuration (Figure
4-8) shows certain insensitivity to the internal damage scenarios (number 1, 2 and
3). This option can be considered as a global improvement of the structural
system;
The previous strategies can be adopted simultaneously: i) the designer-sizing of
the elements can be affected by the robustness index by using Eq. (4-3); and ii) the
structural scheme can be changed (also on the basis of the Cf values) in order to
increase the robustness. In this case, both local and global solutions provide
improvements to the structural system.
In this section, the robustness of structures is inquired using a metric based on the
member consequence factor. The application of this metric seems to be promising for the
Pierluigi Olmati
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robustness assessment of a complex structural system, such as the I-35 Bridge used as a
case study, by identifying critical damage scenarios (scenarios involving the loss of
elements) associated with low values of this metric. This method could be used as tool in
the design, analysis and investigation processes, for localizing critical areas.
Furthermore, comprehensive assessments that consider a larger set of damage scenarios
can be performed by implementing this method using appropriate search heuristics.
Limitations of the implemented method arise from the fact that in the analyses a reduced
structural system is used. In this sense, findings can be considered preliminary, and have
to be verified using complete models and advanced numerical analyses.
Some indications for further research can be identified. A better expression for the Cf
could be obtained by considering both the stiffness and mass matrix of the structure.
Moreover the plastic resources of the structure could be take account in the Cf expression.
Future studies could also focus on the establishment of a threshold value for the member
consequence factor that does not lead to a structural collapse. Furthermore, nonlinear
dynamic analyses could be performed on complete model for the critical cases identified
with this method.
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Pierluigi Olmati
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4.2 The robustness curves
A way to characterize the behavior of buildings subjected to explosion is to compute the
dynamic structural response due to a local damage (assumed to be caused by a blast) and
consequently assess the robustness of the structure. The structural robustness can be
assessed by evaluating the residual load bearing capacity of the damaged configurations
as illustrated in [Giuliani 2009], where the analysis is based on the assumption of
different levels of damage in various locations. The robustness evaluation procedure
presented in the following is based on the assumption of a certain damage level caused by
a generic load, which is able to instantaneously cut off the contribution of a structural
element to the load bearing capacity of the system [Yagob et al. 2009, and CPNI 2011].
Therefore, the method proposed can be used for the design against actions generated both
by intentional and accidental explosions and by hazards of different type (e.g. impact).
Focusing on steel frame building systems (such as the one studied in this study, shown in
Figure 4-11), whose key structural elements are the columns at the ground floor
[Almusallam et al. 2010, and Valipour et al. 2010], the local damage level can be
identified by the number of the destroyed key elements. It is assumed that the columns
directly acted upon by the blast wave are instantaneously destroyed, thus the case of
partially damaged key elements is neglected in this study. Following these assumptions,
the first set of damage scenarios is defined by the removal of a single key element
(column), the second set by the removal of two key elements, and so on.
Considering the above, two parameters identify the single damage scenario: the location
of the first destroyed key element (L), and the local damage level (N) of the scenario (i.e.
the number of the removed key elements). The specific local damage scenario is then
identified as “D-scenario (L=i; N=j)”, where capital letters indicate parameters and lower
case letters indicate the specific value assumed by the parameters. This means that the
generic scenario is obtained by removing a total number of key elements equal to “j”, and
that the first of these elements was the one positioned in the location number “i”. A set of
initial NL damaged key elements defines the D-scenario (L; 1) with (L=1,…, NL; local
damage level N=1). These scenarios (L; 1) can be chosen a priori by considering the
explosion type (e.g. it is realistic to assume that gaseous explosions take place in kitchens
or boiler rooms, while intentional explosions in the external perimeter of the building).
Moreover the generic D-scenario (i; j+1) corresponding to the local damage level N=j+1,
is heuristically obtained from the previous D-scenario (i; j) by removing the most stressed
key element (critical element for the D-scenario (i; j)) obtained by the caring out a
Nonlinear structural Dynamic Analysis (NDA) (Vassilopoulou and Gantes 2011) of the
D-scenario (i; j). The non-linearity is due to both the inelastic behavior of the steel and
the large displacements; the analyses are conducted by considering the effects of live and
permanent loads with the associated masses.
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70 m
Y
Z
X
Figure 4-11: FE model of the building
As an outcome of the NDA to the D-scenario (i; j) the damaged structural configuration
may have two kinds of response: a) the critical element for the D-scenario (i; j) does not
collapse (i.e. it remains under a certain conventional response threshold - “arrested
damage response”) or, b) the critical element for the D-scenario (i; j) collapses (i.e. it
exceeds the conventional response threshold - “propagated damage response”). The
sequential steps of the procedure are the following:


For damaged configurations leading to a “propagated damage response”, the
progression of the collapse is presumed, and the computation proceeds by
changing the damage location L=i+1. This assumption should need to be verified
since, in general, the fact that another element is failing in consequence of an
initial damage, it does not necessary mean that a progressive collapse is triggered.
However the Authors’ opinion is the indirect failure of a key element should be
avoided.
For damaged configurations leading to an “arrested damage response”, two steps
are carried out:
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-
-
The residual strength is computed, this can be estimated in different ways.
Here the so-called “pushover analysis” [Pinho 2007, Kalochairetis et al.
2011, and Lignos et al. 2011] is used, and the residual strength for the Dscenario (i, j) is identified by the ratio λ%(i,j) between the ultimate load
multiplier of the damaged structure and the one corresponding to the
original undamaged one. The residual strength (pushover) analysis is
carried out under horizontal loads having a triangular distribution along the
height of the building. This choice is made with two motivations in mind:
i) horizontal loads can activate both horizontal and vertical load bearing
structural systems of the building and, ii) direct reference is made to the
unlikely eventuality that a seismic aftershock occurs after an explosion.
This event is possible in the case that the explosion occurs after a seismic
main shock (e.g. hydrogen explosions caused by the Japan 2011
earthquake main shock).
The increase of the local damage level (N=j+1) as stated before (i.e. by
removing the critical element for the D-scenario (i; j)), and performing a
new NDA.
After each “propagated damage response” a robustness curve is obtained, defined by the
variation of the ratio λ%(i,j) with the local damage level (N). Once all the NL locations
have been analyzed a set of curves describing the robustness of the structure under the
considered damage scenarios are obtained (see example of Figure 4-12). The whole
procedure is summarized in the flowchart of Figure 4-13. The outcome of the analysis
gives a representation of the structure robustness when it is damaged by a blast in the
considered locations. These robustness curves under blast damage scenarios are useful
for:


Risk assessment analysis, if the uncertainties affecting the structural response after
a local damage LD (e.g. due to the uncertain structural characteristics) are
considered (e.g. by a Monte Carlo analysis, see [Petrini et al. 2012]).
Risk mitigation analysis, for planning the optimal strategy against the hazard
[FEMA 2003], for example by adopting adequate structural or non-structural
measures [DoD 2010] focusing on the most critical scenario indicated by the
robustness curves (the scenario producing the less robust structural response).
Pierluigi Olmati
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Residual strength λ%(i;
j)
100
80
60
40
20
0
0
1
2
3
Local Damage Level
D-scenario (1;N)
D-scenario (3;N)
4
D-scenario (2;N)
D-scenario (4;N)
Figure 4-12: Examples of robustness curves under blast damage scenarios
Select NL
locations
Key elements: columns at
the ground floor.
Damage level (N): number
D-scenario (i; j=1)
N=j+1
Increase damage level
by removing the critical
element for the
D-scenario (i;j)
L=i+1
Arrested
Damage
Response
(ADR)
NO
D-scenario (i;j)
Structural
response
evaluation by
NDA
Does failure
spontaneously
occur to another
key element?
ADR ?
YES
Propagated
damage
response
Progressive collapse is
presumed
(no residual strength)
λ %(i;j) = 0
Residual strength
(pushover) analysis
λ%(i;j) >0
YES
of key elements instantly
removed.
Location (L): position of the
first key element removed (≡
blast location).
NL: number of locations.
D-scenario (i; j): location
(i) and damage level (j).
NDA: non linear dynamic
analysis implementing large
displacements and inelastic
materials.
λ%(i;j) : ratio between the
damaged and undamaged
ultimate load multiplier
(pushover analysis).
ADR: arrested damage
response.
(i,j) Robustness
curve point under
blast damage
NO
NO
i = NL ?
YES
Set of Robustness
curves under
blast damage
Figure 4-13: Flowchart of the procedure to evaluate the structural robustness against blast
damage
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The procedure introduced above for evaluating structural robustness under blast damage
has been applied to a case study building. The case study building is an office structure
70 meters high for a total of 20 story, each one being 3.5 meters high. The layout is
rectangular with two protruding edges on the longest side and is globally delimited in a
45 x 25 square meters area (see Figure 4-11 and Figure 4-14). Columns and beams have
European HE cross-sections. The beam-column and beam-beam connections are made by
double angle cleat connections (shear-resisting), while the column-column connections
are moment-resisting welded and bolted connections. In addition to the previous subsystems, appropriate braced walls are present in order to support the horizontal loads,
these having beam-column moment resisting connections, and diagonal tension members
that consist in 2L 100x50/8 profiles; the position of the bracing systems is shown in
Figure 4-15. The floors have a horizontal braced system that is formed by a set of
members having an L 100x50/8 profile. The column cross-section shapes are shown in
Figure 4-15 and classified and grouped in Table 4-2; for each column type (A, B, C, D)
the size of the cross section decreases through the building height. The slab is a steel
ribbed slab, spanning North to South, simply supported by girders and beams (see Figure
4-14). The girders cross-sections are HEA 240, spanning North to South, while the floor
beams cross-sections are HEA 200, spanning West to East. The girders and beams
belonging to the braced walls have a HEB 300 shape. The girders and beams have a span
of 5 meters and are placed at 5 meters and 2.5 meters steps respectively. A grade S235
steel is adopted, with a yielding (fyk) and ultimate (fuk) stress equal to 235 and 360 N/mm2
respectively.
DS(8;1)
DS(3;1)
5m
DS(8;2)
DS(3;2)
1
1
DS(1;1)
y
1
DS(5;1)
15 m
DS(2;1)
DS(2;2)
DS(4;2)
DS(4;1)
1
x
DS(6;1)
15 m
15 m
DS(7;1)
15 m
Braced wall
Key element instantly removed
DS(i;j) Blast Damage Scenario:
(L= i location; N= j local damage level)
DS(i;j)
1
1
Blast Damage Scenario:
(L= i location; N= j local damage level)
DS(L;1)
Figure 4-14: Damage scenarios (L; 1), and Damage scenarios (L; 2)
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N
15 m
W
C
C
B
C
C
B
D
B
A
D
A
A
A
C
B
A
A
A
A
B
A
A
A
B
D
A
A
A
A
A
A
A
A
D
B
B
D
B
A
A
B
D
B
B
C
C
5m
15 m
C
E
C
15 m
15 m
S
Figure 4-15: Column types
Column type
Quote [m]
0 - 12
12 - 33
33 - 39.5
39.5 - 70
A
HEM 550
HEB 550
HEB 320
HEB 300
B
HEX 700x356
HEX 700x356
HEX 700x356
HEB 550
C
HEB 300
HEB 260
HEB 240
HEB 240
D
HEM 550
HEM 550
HEM 550
HEB 550
Table 4-2: Column cross section
In what follows, direct reference is made to the flowchart of Figure 4-13. Eight locations
(L) have been considered (NL=8) defining the blast damage scenarios, as indicated in
Figure 4-14. As previously stated, the columns at the ground floor have been considered
as key elements and the numerical investigations are carried out removing
instantaneously the key element by NDA. In Figure 4-11 the finite element model of the
structure developed with Straus7® [G+D Computing HSH 2004] is shown. Only the
frame system is explicitly modeled, both the floors and the live load are taken into
account by considering additional (fictitious) mass density on the beams. The building is
subjected to gravity and the structural properties of the cladding system are not
considered. The structural response to the D-scenario (i; j) is evaluated by carrying out
non-linear Lagrangian [Bontempi et al. 1998] dynamic implicit FE analyses. Explicit FE
solver (more capable in evaluating triggering effects due to local collapses) has been
avoided in order to limit the computational efforts. The use of implicit method is also
justified by the fact that, in this specific structure, the failure (assessed by implicit
analyses) of some structural key elements can be conceptually associated to the
propagation of the collapse to other structural parts supported by the key elements.
An initial damage is considered in the FE model by replacing a column by its reaction
force (computed with the Dead + Permanent + 0.3 Live load combination). In order to
minimize inertial effects caused by this loading phase, a sufficiently slow load ramp is
provided. Moreover, a successive oscillation extinction phase is added (see Figure 4-16)
where the load factor is maintained equal to 1. After that the reaction of the key element
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is suddenly removed to simulate the damage with a time interval (Δt) smaller than 1/10 of
the fundamental time period associated with the pertinent vertical modal shape of the
damaged structure [DoD 2009]. The implemented load factor time history is shown in
Figure 4-16. Concerning the material and geometrical nonlinearities, the distributed
plasticity model along the length of the beams and the large displacements assumption are
adopted. In Figure 4-17 some moment-curvature diagrams for girders and beams are
shown, the softening behavior is not implemented.
500
400
Loading phase
Oscillation
extinction
phase
0.5
Moment [kN m]
Load factor [-]
1.0
Dead + 0.3 Live
Δt
Reaction force of
the key element
200
100
0
0.00
0.0
0
10
20
30
time [sec]
40
50
Figure 4-16: Load factor time history chart
HEB 300
HEA 240
HEA 200
300
0.01
0.02 0.03 0.04
Curvature [-/m]
0.05
0.06
Figure 4-17: Moment-Curvature diagrams
The typical vertical displacement time history for a node located on the top of the
removed key element is shown in Figure 4-18 for both the “arrested damage response”
and “propagated damage response” with local damage level N=1. The first one, after the
extinction of the initial high frequency oscillation, shows a decaying response (damped
oscillation), while the second one shows an unbounded response. The computed
robustness curves are shown in Figure 4-19. It results that for the selected scenarios, with
the local damage level equal to two (D-scenario (L; 2)), the structure always shows a
“propagated damage response” (i.e. a progressive collapse is presumed).
Time [sec]
22
24
26
28
Time [sec]
30
32
24
34
-8
-12
-16
High
frequency
oscillations
response
extinction
-20
-15
Displacement [m]
-12
Max displacement
-4
25.5
Residual displacement
Displacement [mm]
25
26
0.0
0
-9
25
-0.6
-1.2
Displacement under collapse
20
-1.8
Figure 4-18: Response time history for a node located at the top of the removed key
element, D-scenario (5; 1) (left, arrested damage response) and D-scenario (6; 1) (right,
propagated damage response)
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27
Residual strength λ%
(i; j)
100
75
50
25
0
0
1
2
Local Damage level
D-scenarios (i;j):
D-scenario (1;N)
D-scenario (2;N)
(1;N)
(3;N)
(4;N)
D-scenario(2;N)
(3;N)
D-scenario
(4;N)
(5;N)
D-scenario(6;N)
(5;N)
(7;N)
(8;N)
D-scenario
(6;N)
D-scenario (7;N)curves under
D-scenario
(8;N)
Figure 4-19: Robustness
blast
damage
In some cases (D-scenarios (6; 1) and (7; 1)) the collapse progression occurs even at the
first level of local damage. This behavior occurs in all cases where the local damage is
located in the external columns of the building that are not part of braced walls (see
Figure 4-14). In fact, in all other D-scenarios, for local damage level N=1 the load
originally carried by the removed key element is re-distributed to a number of adjacent
key elements positioned in both floor directions (x and y in Fig. 4) with respect to the
removed one, this allows the development of a double catenary effect (in x and y
directions), something that is not realized in the critical D-scenarios (6; 1) and (7; 1).
The robustness curves obtained in this section form a suitable tool that can be helpful for
risk management and assessment. The procedure can be employed to handle different
hazards, such as terrorist attacks or accidental explosions. In particular, the proposed
procedure for robustness assessment is based on the assumption that the structural
members directly involved in the blast fail instantaneously, without any prior evaluation
on the blast intensity. Since in the recent years a number of intentional explosions were
caused by truck bombs near the buildings, leading to the failure of some columns, the
previous mentioned assumption seems particularly reliable in case of intentional
explosions. The same approach could be extended for computing the robustness curves in
case of structures subjected to impact of ships and vehicles that engage key structural
elements.
The approach based on the removal of key structural elements and on the subsequent
investigation of the dynamic structural nonlinear behavior has been adopted by different
authors (see for example [Yagob. 2009, Purasinghe et al. 2012, Weerheijm et al. 2009,
and Sasani et al. 2011]), and has been implemented in guidelines (e.g. [DoD 2009]). To
this regard, the novelty offered by this study consists in describing the results of this
analysis in a synthetic and fruitful manner, provided by the computation of the robustness
curves. Moreover, the proposed method of evaluation by means of robustness curves,
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takes into account the dynamic effects of the structural initial damage by evaluating the
structural behavior under impulsive loads.
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5 CONCLUSIONS
The protection of buildings and critical infrastructures against man-made attacks is a
priority for a stable and secure society. For this reason a security organization against
man made attacks is necessary. The 85% [DoS 2003, DoS 2004, DoS 2005, and DoS
2006] of the attacks involve explosives devices. As a consequence, the resistance of a
structure to explosions is crucial for having an adequate level of protection of the
structure. For better understanding and assessing the capacity of a structure to withstand
loads from an explosion, the collapse resistance of a structure has been decomposed in
three components: the hazard mitigation, the local resistance, and the global resistance.
In this Thesis, all three components of the collapse resistance have been deeply
investigated proponing methods for their quantitative assessment.
Furthermore,
applications of these methods have been carried out.
The hazard mitigation has been investigated for gas explosions in a residential building.
Three crucial parameters determining the severity of the blast load due to the deflagration
of a gas cloud have been identified. These parameters are the room congestion, the
failure of non-structural walls and the location of the ignition. Each of those parameters
can change drastically the blast demand of the structure.
The local resistance (intended as the resistance of the single component of the structural
system subjected to the blast load) has been investigated both probabilistically by means
of the fragility analysis and deterministically by means of detailed explicit numerical
simulations.
The fragility analysis has been carried out by two different intensity measures: the scaled
distance and the impulse density. The first one (the scaled distance) has been applied on
precast concrete cladding wall panels subjected to a vehicle bomb. This intensity
measure shows good results in terms of exceeding probability compared with the
exceeding probability obtained with the unconditional approach. However the adopted
fragility curve for carrying out the fragility analysis must be the one corresponding with
the mean value of the stand-off distance of the scenario. The second proposed intensity
measure (the impulse density) has been applied to a built-up blast door subjected to an
accidental detonation of mortar rounds. Also this intensity measure shows good results
for impulse sensitive structures in terms of exceeding probability compared with the
exceeding probability obtained with the unconditional approach, however, it is
completely inopportune for pressure sensitive structures. That said, for structures loaded
in the dynamic region of the pressure impulse diagram conservative results are obtained.
The advantage of the impulse density as intensity measure compared to the scaled
distance is that only one fragility curve is necessary for carrying out the fragility analysis.
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Moreover, a safety factor for designing steel build-up blast doors with the equivalent
single degree of freedom method has been proposed.
As mentioned the local resistance has been also investigated by detailed explicit
numerical simulations.
Numerical investigations have been made concerning batch tests on reinforced concrete
slabs founded by the National Science Foundation (NSF). A Blast Blind Simulation
Contest was sponsored in collaboration with American Concrete Institute (ACI)
Committees 447 (Finite Element of Reinforced Concrete Structures) and 370 (Blast and
Impact Load Effects), and UMKC School of Computing and Engineering. The goal of
the contest was to predict, using simulation methods, the response of reinforced concrete
slabs subjected to a blast load. The blast response was simulated using a Shock Tube
(Blast Loading Simulator) located at the Engineering Research and Design Center, U.S.
Army Corps of Engineers at Vicksburg, Mississippi. A team for participating at the
contest has been formed by the author of this Thesis Pierluigi Olmati (Sapienza
University of Rome), Patrick Trasborg (Lehigh University), Dr. Luca Sgambi
(Politecnico di Milano), Prof. Clay J. Naito (Lehigh University), and Prof. Franco
Bontempi (Sapienza University of Rome). The submitted prediction of the slab’s
deflection was declared The Winner of The Blast Blind Simulation Contest
(http://sce.umkc.edu/blast-prediction-contest/ - accessed August 2013) for the concurring
category.
Finite element analyses have been carried out also for assessing the spall and breach
resistance of insulated panels against a close-in detonation. Moreover experimental tests
have been conducted at the Air Force Research Laboratory in Panama City, FL.
Generally insulated panels have shown an improved response to close-in detonations
respect classic concrete wall panels; however to avoid concrete spall at the interior
concrete wythe, a sufficient insulation foam thickness is necessary. Furthermore, the
insulation foam layer does not constitute a major component in dissipating energy from
the detonation, but rather the thickness of the gap between the exterior and interior
concrete wythes is crucial. In every insulated panel numerical model, the exterior
concrete wythe breached. The exterior wythe can be considered to act as a sacrificial
element, maintaining a larger stand-off distance from the interior concrete wythe. In
conclusion the insulated panels in addition to a superior energy performance have
displayed enhanced spall and breach performance to close-in blast demands. This is due
to the exterior concrete wythe acting as a sacrificial layer, allowing the gap and foam to
dissipate much of the concrete fragment kinetic energy and mitigate incipient shock
waves from the initial blast.
Finally two methods for assessing the global resistance (intended as the resistance of a
structural system subjected to a failure of one or more structural components) of a
structure have been proposed.
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In the first method the structural robustness has been inquired using a metric based on the
member consequence factor. The application of this metric seems to be promising for the
robustness assessment of a complex structural system, such as the I-35 Bridge used as a
case study, by identifying critical damage scenarios (scenarios involving the loss of
elements) associated with low values of this metric. This method could be used as tool in
the design, analysis and investigation processes, for localizing critical areas.
Furthermore, comprehensive assessments that consider a larger set of damage scenarios
can be performed by implementing this method using appropriate search heuristics. In
the second method the structural robustness has been studied using both non-linear
dynamic and static analyses. The method has been developed for buildings and it is based
on the hypothesis of the removal column scenario. The column is suddenly removed and
a non-linear dynamic analysis is carried out for assessing if the progressive collapse
occurs, if not a non-linear static pushover is carried out on both the damaged and
undamaged configuration of the building for estimating the residual capacity of the
building. This procedure is repeating both increasing the number of the removed columns
and for several scenarios. This method has been applied on a steel tall building and the
structural robustness has been assessed quantitatively showing the behavior of the
building under the failure of one and more columns.
Concluding, the collapse resistance, composed by the contribution of the hazard
mitigation, the local resistance, and the global resistance, has been thoroughly assessed in
this Thesis. The assessment was carried out by both deterministic and probabilistic
methods and by the means of explicit finite element simulations and fragility analyses.
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6 REFERENCES
[Alashker et al. 2011]
Alashker Y, Li H, El-Tawil S. Approximations in
Progressive Collapse Modeling. Journal of Structural
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Stewart MG, Netherton MD. Security risks and probabilistic
risk assessment of glazing subject to explosive blast
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[Stuhmiller et al. 1996]
Stuhmiller JH, Ho KH, Vander Vorst MJ, Dodd KT,
Fitzpatrick T, Mayorga M. A model of blast overpressure
injury to the lung. Journal of Biomechanics 1996; 29(2):
227–234.
[Su et al. 1995]
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Valipour HR, Foster SJ. Nonlinear analysis of 3D reinforced
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[Wang et al. 2008]
Wang F, Wan YKM, Chong OYK, Lim CH, Lim ETM.
Reinforced concrete slab subjected to close-in explosion.
Proceedings: 7th German LS-DYNA Forum, Bamberg,
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[Wang et al. 012]
Wang W, Zhang D, Lu F, Wang S, Tang F. Experimental
study on scaling the explosion resistance of a one-way
square reinforced concrete slab under a close-in blast
loading. International Journal of Impact Engineering 2008;
doi: 10.1016/j.ijimpeng.2012.03.010
[Weerheijm et al. 2009]
Weerheijm J, Mediavilla J, van Doormaal JCAM. Explosive
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[Whittaker et al. 2003]
Whittaker AS, Hamburger RO, Mahoney M. Performancebased engineering of buildings and infrastructure for
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Williams GD, Williamson EB. Response of reinforced
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Wolff M, Starossek U. Cable-loss analyses and collapse
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analysis on structure safety of refuge chamber door under
explosion load. Procedia Engineering 2012; 45: 923-929.
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Xu K, Lu Y. Numerical simulation study of spallation in
reinforced concrete plates subjected to blast loading.
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Yim HC, Krauthammer T. Load–impulse characterization
for steel connection. International Journal of Impact
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Yuan W, Tan KH. Modeling of progressive collapse of a
multi-storey structure using a spring-mass-damper system.
Structural Engineering and Mechanics 2011; 37(1): 79-93.
[Zhang et al. 1998]
Zhang J, Kikuchi N, Li V, Yee A, Nusholtz G. Constitutive
modeling of polymeric foam material subjected to dynamic
crash loading. International Journal of Impact Engineering
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Zhou XQ, Kuznetsov VA, Hao H, Waschl J. Numerical
prediction of concrete slab response to blast loading.
[Zineddin et al. 2007]
Zineddin M, Krauthammer T. Dynamic response and
behavior of reinforced concrete slabs under impact loading.
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Pierluigi Olmati
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Pierluigi Olmati
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7 APPENDIX A – JOURNAL PAPERS OBTAINED FROM THE PH.D.
THESIS
This section wants collect the journal papers obtained from the Ph.D. Thesis. The section
provides: i) the published and the accepted papers for publications; and ii) the submitted
and ongoing papers.
7.1 Published and accepted papers
-
-
-
Olmati P, Trasborg P, Naito CJ, Bontempi F. Blast resistance of reinforced precast
concrete walls under uncertainty. International Journal of Critical Infrastructures
2013; in press.
Olmati P, Gkoumas K, Brando F, Cao L. Consequence-based robustness
assessment of a steel truss bridge. Steel and Composite Structures 2013; 14(4):
379-395.
Olmati P, Petrini F, Bontempi F. Numerical analyses for the structural assessment
of steel buildings under explosions. Structural Engineering and Mechanics 2013;
45(6): 803-819.
7.2 Submitted and ongoing papers
-
-
-
-
Giovino G, Olmati P, Garbati S, Bontempi F. Blast vulnerability assessment of
precast concrete cladding wall panels for using in police stations: experimental
and numerical investigations. Submitted to the International Journal of Impact
Engineering November 2013.
Olmati P, Trasborg P, Naito C, Sgambi L. Bontempi F. Finite element and
analytical approaches for predicting the structural response of reinforced concrete
slabs under blast loading. Invited paper for a Special Issue on the American
Concrete Institute (ACI) journal.
Olmati P, Petrini F, Vamvatsikos D, Gantes CJ. Safety factor and fragility analysis
for structures subjected to accidental explosion of ammunitions: the case of steel
built-up blast doors. Ongoing, September 2013.
Olmati P, Petrini F. Development of fragility curves for cladding panels subjected
to blast load. Submitted to the Structural Safety journal, April 2013.
Olmati P, Naito CJ, Davidson J, Trasborg P. Assessment of insulated concrete
walls to close-in blast demands. Submitted to the International Journal of Impact
Engineering, January 2014.
Pierluigi Olmati
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Pierluigi Olmati
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8 APPENDIX B – CURRICULUM VITAE
Curriculum Vitae
Personal information
First name(s) / Surname(s)
Address
Telephone(s)
E-mail(s)
Nationality
Date of birth
Gender
Pierluigi Olmati
Loc. Palombella, (Italy)
+390644585072
Mobile
Skype
+393409092382
pierluigi.olmati
pierluigi.olmati@uniroma1.it
pierluigi.olmati@francobontempi.org
Italian
03 August 1984
Male
Education and training
Dates
Title of qualification awarded
Name and type of organisation
providing education and training
Dates
01 November 2010 →
Ph.D. Student in Structural Engineering
Sapienza University of Rome (Structural Engineering)
Via Eudossiana, 18, 00184 Rome
20 May 2013 – 02 September 2013
Visiting Scholar
Principal subjects / occupational
skills covered
Performing research on the probabilistic blast design applied to blast
doors. Under the supervision of Prof. Charis Gantes and Prof.
Dimitrios Vamvatsikos
Department of Structural Engineering, National Technical University
of Athens (NTUA), Zografou Campus, Iroon Polytechneiou 9, 15780
Zografou, Athens, Greece
Dates
01 February 2012 - 31 July 2012
Visiting Scholar
skills covered
Performing research on closed-in detonation on sandwich panels and
assisting with an on-going research program sponsored by the NSF
(National Science Foundation)
Department of Civil and Environmental Engineering at the Lehigh
University, Bethlehem, PA, USA. Under the supervision of Prof. Clay
Naito
Pierluigi Olmati
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Dates
01 October 2007 - 29 July 2010
MS in Structural Engineering
skills covered
Simulation of explosions, and mitigation of the associated risk
(Awarded grade: 110/110)
Dates
01 October 2003 - 21 September 2007
BS in Civil Engineering
skills covered
Awarded grade: 103/110
Personal skills and
competences
Mother tongue(s)
Italian.
Other language(s)
English, French.
Self-assessment
European level (*)
Understanding
Listening
Reading
Speaking
Spoken
interaction
Writing
Spoken
production
English
B1 Indep. User B1 Indep. User B1 Indep. User B1 Indep. User B1 Indep. User
French
A1 Basic User A1
Basic User
A1 Basic User A1 Basic User A1 Basic User
(*) Common European Framework of Reference (CEF) level
Technical skills and competences
Keywords and research topics
- Blast Engineering.
- Protective Structures.
- Structures subjected to impulsive loads (e.g. blast, impact).
- Finite Element modelling.
- Structural analysis.
- Structural robustness and progressive collapse.
- Computational fluid dynamics simulation of gas explosions.
Computer skills and competences
Finite Elements Analysis:
- LS-DYNA (www.lstc.com)
- Code_Aster (www.code-aster.org).
- Strand7 (www.strand7.com).
- NeiFusion (www.nenastran.com).
- Sap2000 (www.csiberkeley.com).
- Impact (impact.sourceforge.net).
Pierluigi Olmati
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- Abaqus/CAE (www.simulia.com/products/abaqus_cae.html).
CFD for explosions:
- FLACS (www.gexcon.com).
Other:
- Matlab (www.mathworks.it)
- SolidWorks (www.solidworks.com).
- Salome-Meca (www.salome-platform.org).
- DraftSight (www.3ds.com/it/products/draftsight/free-cad-software).
- AutoCad (usa.autodesk.com).
Driving licence(s)
B
Additional information and PROFESSIONAL MEMBERSHIPS
Awards July 2011: Professional Engineer, “Ordine degli Ingegneri - Provincia di
Viterbo, A-925”, CEng-Equivalent
AWARDS
April 2013: Winner in Category 1 of the “Blast Response of
Reinforced Concrete Slab Blast Blind Simulation Contest”.
Sponsored in collaboration with American Concrete Institute (ACI)
Committees 447 (Finite Element of Reinforced Concrete
Structures) and 370 (Blast and Impact Load Effects), and UMKC
School of Computing and Engineering. http://sce.umkc.edu/blastprediction-contest
-
October 2011: Best MS Thesis award in Steel Structural Design:
“10/11 PREMI TESI DI LAUREA” by “Associazione fra i Costruttori
in Acciaio Italiani (ACAI)”, journal “Costruzioni Metalliche”,
“Collegio dei Tecnici dell’Acciaio (CTA)”, “Fondazione Ingegneri di
Padova”, and “Fondazione Promozione Acciaio”.
-
October 2010: Public examination for the admission to the Ph.D.
course in Structural Engineering at Sapienza University of Rome
(“XXVI cycle”), Department of Structural and Geotechnical
Engineering.
Annexes A. Journal papers
B. Conferences proceedings
C. Presentations
D. Research overview
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Annex A. Journal papers
-
Olmati P, Trasborg P, Naito CJ, Bontempi F. Blast resistant design of precast
reinforced concrete walls for strategic infrastructures under uncertainty.
International Journal of Critical Infrastructures 2013; in press.
-
Trasborg P, Naito C, Bocchini P, Olmati P. Fragility analysis for ballistic design.
Submitted to the International Journal of Impact Engineering.
-
Naito C, Olmati P, Trasborg P, Davidson J, Newberry C. Assessment of insulated
concrete walls to close-in blast demands. Submitted to the International Journal of
Impact Engineering.
-
Olmati P, Petrini F, Vamvatsikos D, Gantes CJ. Safety factor and fragility analysis
for structures subjected to explosions: the case of steel built-up blast resistant
doors. On-going.
-
Trasborg P, Nickerson J, Naito C, Olmati P, Davidson J. Forming a predictable
flexural mechanism in reinforced wall elements. Submitted to the ACI Structural
Journal.
-
Olmati P, Petrini F. Development of fragility curves for cladding panels subjected
to blast load. Submitted to the Structural Safety journal.
-
Olmati P, Gkoumas K, Brando F, Cao L. Consequence-based robustness
assessment of a steel truss bridge. Steel and Composite Structures 2013; 14(4):
379-395.
-
Olmati P, Petrini F, Bontempi F. Numerical analyses for the structural assessment
of steel buildings under explosions. Structural Engineering and Mechanics 2013;
45(6): 803-819.
-
Olmati P. Simulazione di esplosioni e metodologie progettuali per la mitigazione
del rischio associato. Costruzioni Metalliche 2012; 1: 59-60.
Annex B. Conferences proceedings
-
Olmati P, Giovino G, Bontempi F. “Probabilistic performance assessment of a
precast concrete wall subjected to blast load”. Proceedings of The 15th
International Symposium on the Interaction of the Effects of Munitions with
Structures, Potsdam, Germany, 17-20 September 2013.
-
Giovino G, Olmati P, Bontempi F. “Vulnerability assessment of precast concrete
cladding wall panels for police stations: experimental and numerical
investigations”. Proceedings of The 15th International Symposium on the
Pierluigi Olmati
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Interaction of the Effects of Munitions with Structures, Potsdam, Germany, 17-20
September 2013.
-
Olmati P, Petrini F, Gkoumas K. “Blast resistance assessment of a reinforced
precast concrete wall under uncertainty”, Proceedings of The 11th International
Conference on Structural Safety & Reliability, Columbia University, New York,
June 16-20, 2013.
-
Olmati P. “Monte Carlo analysis for the blast resistance design and assessment of
a reinforced concrete wall”, Proceedings of The of 4th ECCOMAS Thematic
Conference on Computational Methods in Structural Dynamics and Earthquake
Engineering, Kos Island, Greece, 12–14 June 2013.
-
Olmati P, Giuliani L. “Progressive Collapse Susceptibility of a Long Span
Suspension Bridge”, Proceedings of The 2013 Structures Congress, Pittsburgh,
PA, May 2-4, 2013.
-
Olmati P, Trasborg P, Naito CJ, Bontempi F. “Blast resistance of reinforced
precast concrete walls under uncertainty”, Proceedings of The 2013 Critical
Infrastructure Symposium, April 15-16, Thayer Hotel, West Point, New York,
2013.
-
Saviotti A, Olmati P, Bontempi F (2012), “Finite element analysis of innovative
solutions of precast concrete beam-column ductile connections”, Proceedings of
The Bridge Maintenance, Safety, Management, Resilience and Sustainability
Conference, CRC Press, Taylor & Francis Group, Italy, Stresa, 2012.
-
Crosti C, Olmati P, Gentili F, “Structural response of bridges to fire after
explosion”, Proceedings of The Bridge Maintenance, Safety, Management,
Resilience and Sustainability Conference, CRC Press, Taylor & Francis Group,
Italy, Stresa, 2012.
-
Brando F, Cao L, Olmati P, Gkoumas K, “Consequence-base robustness
assessment of bridge structures”, Proceedings of The Bridge Maintenance, Safety,
Management, Resilience and Sustainability Conference, CRC Press, Taylor &
Francis Group, Italy, Stresa, 2012.
-
Trasborg P, Olmati P, Naito C, “Increasing the ductility of reinforced concrete
panels to improve blast response”, Proceedings of The 2012 Critical Infrastructure
Symposium, Washington DC, 2012.
-
Sgambi L, Olmati P, Petrini F, Bontempi F, “Seismic performance assessment of
precast element connections”, Proceedings of 2011 PCI Convention and National
Bridge Conference, US, Salt Lake City, 22-26 October 2011.
Pierluigi Olmati
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-
Olmati P, Petrini F, Bontempi F, “Effect of explosions on steel buildings”,
Proceedings of XXIII Conference CTA, Italy, Ischia, 9-12 October 2011,
ISBN/ISSN 978-88-89972-23-6.
-
Olmati P, Petrini F, Giuliani L, Bontempi F, “Blast design for structural
elements”, Proceedings of XXIII Conference CTA, Italy, Ischia, 9-12 October
2011, ISBN/ISSN 978-88-89972-23-6.
-
Olmati P, Bontempi F, Petrini F, “Structural Robustness of Buildings and Design
for Structural Elements under Explosions”, Proceedings of First World Congress
on Advances in Structural Engineering and Mechanics (ASEM 11plus), Korea,
Seoul, 18-23 September 2011.
-
Olmati P, Petrini F, Bontempi F, “Design and analysis of steel structures for
explosions”, Proceedings of the 6th European Conference on Steel and Composite
Structures (EUROSTEEL 2011), Hungary, Budapest, 31 August - 2 September
2011, ISBN/ISSN 978-92-9147-103-4.
-
Olmati P, “Gas explosion simulation by CFD tools”, Proceedings of the 2th
Handling Exceptions in Structural Engineering: structural system, accidental
scenarios, design complexity (HE10), Italy, Rome, 8-9 July 2010, DOI:
10.3267/HE2010.
Annex C. Presentations
-
The 15th International Symposium on the Interaction of the Effects of Munitions
with Structures, Potsdam, Germany, 17-20 September 2013. Title of the
presentation: “Probabilistic performance assessment of a precast concrete wall
subjected to blast load”.
-
The 4th ECCOMAS Thematic Conference on Computational Methods in
Structural Dynamics and Earthquake Engineering, Kos Island, Greece, 12–14 June
2013. Title of the presentation: “Monte Carlo analysis for the blast resistance
design and assessment of a reinforced concrete wall”.
-
The 2013 Structures Congress, Pittsburgh, PA, May 2-4. Title of the presentation:
“Progressive Collapse Susceptibility of a Long Span Suspension Bridge”.
-
The 2013 Structures Congress, Pittsburgh, PA, May 2-4. Title of the presentation:
“Structural robustness assessment of tall buildings”.
-
The 2013 Critical Infrastructure Symposium, April 15-16, Thayer Hotel, West
Point, New York, 2013. Title of the presentation: “Blast resistance of reinforced
precast concrete walls under uncertainty”.
Pierluigi Olmati
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-
Bridge Maintenance, Safety, Management, Resilience and Sustainability
Conference, Italy, Stresa, 9-12 July 2012. Title of the presentation: “Consequencebase robustness assessment of bridge structures”.
-
Class lecture, 13-14 March 2012, Lehigh University, class: CEE-467-043-SP12 of
the Prof. Clay J. Naito. Title of the lecture: “Progressive Collapse”.
-
XXIII Conference CTA, Italy, Ischia, 9-12 October 2011. Title of the
presentation: “Blast design for structural elements”.
-
XXIII Conference CTA, Italy, Ischia, 9-12 October 2011. Title of the
presentation: “Design and analysis of steel structures for explosions”.
-
Eurosteel 2011, 6th European Conference on Steel and Composite Structures, 31
August – 2 September 2011. Title of the presentation: “Design and analysis of
steel structures for explosions”.
-
FLUG Meeting by GexCon, Rome Italy, 16-17 November 2010. Title of the
presentation: “Flacs in gas explosion simulation”.
-
Workshop “Handling Exceptions in Structural Engineering: Sistemi Strutturali,
Scenari Accidentali, Complessità di Progetto”, University of Rome “La
Sapienza”, Rome, Italy, -9 July, 2010. Title of the presentation: “Gas explosion
simulation by CFD tools”.
Annex D. Research overview
I began the three years Ph.D. program in Structural Engineering in November 2010.
During the first year I focused on the behavior of buildings subjected to severe structural
damages like the loss of one or more columns at the ground floor due to accidental or
man-made explosions (for an example of such events, see the collapse of the Ronan Point
tower - 16 May 1968). A procedure to evaluate the capacity of a structural system (e.g.
building) to withstand structural damages (structural robustness) is proposed in a first
publication together with a study on the simulation of gas explosions (for an example of
such events, see the case of Buncefield, London - 11 December 2005): "Olmati P, Petrini
F, Bontempi F. Numerical analyses for the structural assessment of steel buildings under
explosions. Structural Engineering and Mechanics 2013; 45(6): 803-819". A second
publication regards the structural robustness of steel truss bridges (see the collapse of the
I-35W Minneapolis Bridge): "Olmati P, Gkoumas K, Brando F, Cao L. Consequencebased robustness assessment of a steel truss bridge. Steel and Composite Structures 2013;
14(4): 379-395".
Pierluigi Olmati
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During the second year I spent five months (from February 2012 to July 2012) at the
Lehigh University (Bethlehem, PA, USA) under the supervision of Prof. Clay Naito,
performing research on the performance assessment of insulated panels subjected to
close-in detonations. Detailed numerical simulations (using the LS-Dyna software) were
carried out in order to assess the advantages in terms of scabbing and breach resistance of
the insulated panels versus the classic concrete panels. Experimental tests were
conducted at the Air Force Research Laboratory in Panama City, FL, USA.
In the autumn of 2012 I participated at the "Blast Blind Simulation Contest 2012 assessment of the deflection of reinforced concrete slabs subjected to a blast demand
(http://sce.umkc.edu/blast-prediction-contest)".
The contest was sponsored in
collaboration with American Concrete Institute (ACI) Committees 447 (Finite Element of
Reinforced Concrete Structures) and 370 (Blast and Impact Load Effects), and UMKC
School of Computing and Engineering. The team was composed by Mr. Olmati (myself),
Mr. Trasborg (Lehigh University), Dr. Sgambi (Politecnico di Milano), Prof. Naito
(Lehigh University), and Prof. Bontempi (Sapienza University of Rome). The performed
simulation was declared the Winner of the concurring category and the team is invited to
publish a paper in a special publication of the American Concrete Institute (ACI) journal.
During the third year I focused on the probabilistic aspects of the design for blast resistant
structures. In particular, I implemented in blast engineering the probabilistic theory
developed in earthquake engineering. The fragility analysis was carried out for precast
concrete cladding wall panels subjected to a terroristic vehicle bomb attack, and the
proposed approach was verified and validated by reliability analyses performed by Monte
Carlo simulations. A paper on this study has been accepted for publication: "Olmati P,
Trasborg P, Naito CJ, Bontempi F. Blast resistance of reinforced precast concrete walls
under uncertainty. International Journal of Critical Infrastructures, 2013". Moreover a
second paper has been submitted to the Structural Safety journal: "Olmati P, Petrini F.
Development of fragility curves for cladding panels subjected to blast load".
The study on the performance-based blast engineering continued at the National
Technical University of Athens where I spent about three months (from mid-May to
September 2013) under the supervision of Prof. Charis J. Gantes and Prof. Dimitrios
Vamvatsikos. The fragility analysis was carried out for a steel built-up blast resistant
door subjected to an accidental explosion of ammunitions, and in a similar manner to the
previous studies, the proposed approach was verified and validated by reliability analyses
performed by Monte Carlo simulations. Moreover, a safety factor was provided in order
to design the steel built-up door with the common state of the practice method. The
preparation of a journal paper on this topic in ongoing: "Olmati P, Petrini F, Vamvatsikos
D, Gantes CJ. Safety factor and fragility analysis for structures subjected to explosions:
the case of steel built-up blast resistant doors".
Pierluigi Olmati
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In July 2013 I was involved in an experimental test on concrete cladding wall panels
subjected to detonation of explosive. The test was executed at the testing site of the
R.W.M. ITALIA s.p.a. (www.rwm-italia.com). Moreover, detailed numerical simulations
were carried out in order to reproduce the experimental evidence. The work was
presented at the 15th International Symposium on Interaction of the Effects of Munitions
with Structures (ISIEMS) in Potsdam, Berlin and the journal version of the conference
proceeding is ongoing.
Pierluigi Olmati
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Pierluigi Olmati
Page 188 of 189

Sapienza Università di Roma

Transcription

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