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Q.B. Nguyen a,b,*, A. Mebarki a, R. Ami Saada a, F. Mercier b, M. Reimeringer b
aUniversit ParisEst, Laboratoire Modlisation et Simulb Institut National de lEnvironnement Industriel et des R
a r t i c l e i n f o
Article history:Received 22 November 2007Accepted 22 January 2009Available online 25 February 2009
Keywords:Domino effectExplosionImpactTanksRiskReliability
Many types of equipment under pressure may exist in industrialinstallations: tanks containing gas or highly pressurized liquids, forinstance.When reaching critical levels of overpressure, overheating
the tanks may suddenly explode and generate many fragments thatshould be considered as projectiles threatening the other equipments or installations in their neighbourhood [1]. They might impact other equipments, penetrate them partially or perforatethem. Depending on these degrees of perforation, new accidentsmay take place in the impacted objects, leading therefore to seriesof accidents known as domino effect. Many studies deal with theprevention or mitigation of the domino effect consequences [26].
The present paper develops a global methodology in order tostudy the domino effect, detailing in a probabilistic framework
* Corresponding author. Address: Universit ParisEst, Laboratoire Modlisationet Simulation Multi Echelle, MSME FRE3160 CNRS, 5 bd Descartes, 77454 MarnelaValle, France.
Email addresses: QuocBao.Nguyen@univparisest.fr (Q.B. Nguyen), Ahmed.
Advances in Engineering Software 40 (2009) 892901
Contents lists availab
Advances in Engin
sevMebarki@univparisest.fr (A. Mebarki).90 mm, target strengths ranging from 300 MPa up to 1400 MPa and incidence angles ranging from0 up to 70.
MonteCarlo simulations are ran in order to calculate the different probabilities: probability of impact,distribution of the penetration depth and probability of domino effect.
2009 Elsevier Ltd. All rights reserved.
1. Introduction or mechanical demand, a catastrophic sequence, may rise. Actually,09659978/$  see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.advengsoft.2009.01.002ation Multi Echelle, MSME FRE3160 CNRS, 5 bd Descartes, 77454 MarnelaValle, Franceisques (INERIS), Parc Technologique Alata BP2 60550 Verneuil en Halatte, France
a b s t r a c t
The present paper deals with domino effect analysis for industrial facilities. Actually, an explosion or accident may generate various sets of projectiles. In their trajectory, they may impact other existing facilities,such as tanks under highpressure or other strategic components or installations (headquarters, etc). Ifthe impacted targets fail, this may give rise to other sets of projectiles and so on. These potential seriesof accidents are known as domino effect. A probabilistic approach is developed by the authors. The probability of domino effect occurrence requires three main steps:
Probabilistic modelling of the source term (rst set of projectiles): probability of the rst explosionoccurrence and therefore number, masses, velocities, departure angles, geometrical shape and dimensions, constitutive materials properties are described with probabilistic distributions.
Probabilistic modelling of the target term (rst set of impacted targets): number of impacting projectiles, velocities, incidence angles and energy at impact, constitutive materials and the dimensions ofthe impacted targets, projectiles penetration depths into the targets are also described with probabilistic distributions.
Evaluation of the risks of second set of explosions that may take place in the impacted components.
Simulations (3D) are done within this probabilistic framework:
For the probabilistic description of the source term, the authors have collected existing models fromthe literature.
The authors propose new models for the impact (probability of impact which depends on the trajectory and geometry of both the target and projectile: ellipses, cylinders and planar plates, in a rst step)and the penetration depth when there is impact. A simplied mechanical model is actually developedin the case of cylindrical rods impacting rectangular plates, both are metal made. The estimated penetration depth into the target is compared to the experimental data (4 data sets) collected from theliterature with the following features: projectile masses ranging from 0.1 g up to 250 kg, projectilevelocities ranging from 10 m/s up to 2100 m/s, projectile diameters ranging from 1.5 mm up toIntegrated probabilistic framework for domino effect and risk analysisjournal homepage: www.elll rights reserved.le at ScienceDirect
eering Software
ier .com/locate /advengsoft
geometrical shape. . . The trajectory of the projectiles and their impact with surrounding objects are also investigated. Mechanical
mp fragment mass (kg)
gineemodels are developed in order to calculate the penetration depthand the mechanical damage caused to the impacted targets. Therisk or probability of domino effect occurrence is calculated byMonteCarlo simulations [814].
2. General framework
The overall domino effect sequences are described in Fig. 1a.This domino effect may be detailed by describing each of the elementary steps or cycles. Each cycle requires three elementaryall the elementary events of the catastrophic sequence. This paperis an updated and revised version of the conference paper [7]. Theprobabilistic distributions of the whole involved parameters arederived from existing bibliography and data or are postulatedotherwise: fragment number, initial velocity, initial angles, mass,
Nomenclature
R resistance of the impacted target (random variable having real values r)
S mechanical demand due to the projectile impact (random variable having real values s)
E limit state functionpX(x), fX(x) probability distribution (probability density function)
of the random variable X (values x)Pgen probability of fragment generationPimp probability of impactPrup probability of failurePf risk of the effect dominoC() Gamma cumulative density functionNsim number of simulations (MonteCarlo simulation)g gravity (m/s2)n fragment numberLVessel vessel length (m)RVessel vessel radius (m)LFragment cylinder length in type of the oblong endcap (m)
Q.B. Nguyen et al. / Advances in Encomponents (steps): a source term (explosion and generation ofthe fragments), the projectiles trajectory term (angles, velocitiesand displacements from the source), and the target term (impactand interaction between the projectile and the target), seeFig. 1b.
Thus, each elementary branch of the domino effect requiresthree detailed steps, see Fig. 2:
The generation of the preliminary accident that gives rise tothe fragments projection: the factory site contains one ormore tanks under pressure of gas, heated liquids ormechanical aggression. Under the effect of the overpressuredue to the gas or the liquid, or the mechanical aggression,the tank may explode and generate fragments. This probabilistic term describes the probability of fragments generation, Pgen.
In their trajectory, the fragments may hit other equipment intheir neighbourhood. This possibility of impact is dened bythe probability of impact, Pimp.
The impacted targets may suffer partial damage or completepenetration of the projectile. Depending on the targets properties and their critical damage, an explosion may rise in thistargeted element. The probability of explosion after impact,Prup, denes the risk of another branch occurrence in thedomino effect sequence.3. Reliability analysis and probability of crisis
As a rst step, the present paper focuses mainly on the case ofan elementary sequence in the domino effect scenario. Accordingto the general framework shown in Fig. 2, this risk of failure depends on three main terms:
The risk of projectiles generation (Pgen) as result of the initiating accident.The risk of impact (Pimp), i.e. the risk that the generated projectiles impact surrounding targets. It depends on the kinematicsof the projectiles as well as the shape and the location of thesurrounding facilities.The risk of failure (Prup) for these impacted targets correspondsto:
Prup PE 0 ZE0
fee de fr;sdr ds and E R S
Mt tank mass (kg)/ vertical angle of departure (rad)h horizontal angle of departure (rad)q density of the fragment constitutive material (kg/m3)CL lift coefcientCD drag coefcientvp fragment velocity (m/s)et target thickness (m)hp penetration depth (m)ecr critical thickness (m)Ec kinetic energy (J)t time (s)VFragment fragment volume (m3)VTarget target volume (m3)fu ultimate strength of the target constitutive material
(N/m2)eu ultimate strain of the target constitutive materialring Software 40 (2009) 892901 8931With R is random resistance of the impacted target (which values are r) which marginal probabilistic distribution is denotedfr(), S is random mechanical demand due to the projectile impact (which values are s) which marginal probabilistic distribution is denoted fs (), fr,s () is the joint probability densityfunction (it is expressed as fr,s(r,s) = fr(r) fs(s) if R and S are statistically independent), Prup is probability of failure of the impacted target, E is limit state function (E < 0: denes thefailure domain, E > 0: denes the safety domain, E = 0: denesthe limit state surface). This probability depends on the interaction between the projectile and the impacted target, i.e. theform of the limit state function E() described in Eq. (1). Inthe present paper, a hard impact is assumed and a simpliedmodel is considered in order to calculate the penetration depthof the rigid projectile into the metal target.The general expression of the domino effect risk Pf might then
be expressed as:
Pf Pgen Pimp Prup Ppropa 2
With Pgen is probability of generation for a rst set of structuralfragments; Pimp is probability that the generated fragments impactsurrounding targets; Prup is probability that the impacted target is
nts
er
s
Tra
ineeSource
generate fragme
fragment numbshape mass velocity dimensions departure angle
Target Source
Source
y mi
z
x
vd,i fragment i
i
a
b
894 Q.B. Nguyen et al. / Advances in Engcritically damaged; Ppropa is probability of accident propagation, i.e.occurrence of a new set of explosions.
4. Source terms
4.1. Source term features
An industrial accident may generate several fragments havingvarious shapes, sizes, initial velocity, and initial departure angles.It is therefore required to dene correctly the distributions andthe features of the source term: fragments number, fragmentsshapes and size, their masses, their initials departure angles (horizontal and vertical angles), their initial velocities at departure,and their aerodynamic coefcients (lift and drag coefcients).
4.2. Fragment number
A tank explosion may generate one or many fragments according to the critical pressure, to the cracks propagation, to the tankconstitutive material and to the connection between the elementary mechanical components. According to scientic reports collected from INERIS [1518], the typical explosion (BLEVE) of acylindrical tank produces a limited number of massive fragments:generally two or three, and very seldom more than four or ve.
Fig. 1. Flowchart of the effect domino (a)
Identification of one accident
New accident
Determination of impact
Determination of risk
Probability of generation
Probability of impact
Probability of damage
+

Determination of penetration
Fig. 2. Domino effect sequences.Holden [5] quotes, for 31 BLEVEs having produced 76 fragments, an average number of 2.45 fragments, corresponding to2.87 without re production and 2.34 with re production. In fact,the accidents generate 1, 2, 3 or 4 fragments (except a case withoutre). A test of BLEVE, with a carriage of 45 m3 containing 5 tons ofpropane, produced 4 fragments from the tank envelop and theBLEVE tests on propane reserves of 400 l produced 1 up to 3 fragments [1618].
Moreover, according to the analysis and the experiments ofBaum [19] rupture starts with the circumferential welding and ter
Target Target
impact, penetrate and damage
perforation (new accident)
random
Target
jectory
i
mi
va,i
global view and (b) elementary event.
ring Software 40 (2009) 892901minates with the split of bottom part of the tank (without or with apart of the ring). Therefore, it generates 2 up to 4 fragments: bottom of the tank (with or without the ring) and pieces of the ring.The size of the tank does not have an inuence on the fragmentsnumber.
For the cylindrical tank, Hauptmanns [3] nds out that thenumber of fragments follows a log normal distribution (see Figs.3 and 6).
However, with the accident data collected from Holden [6], themaximum entropy method should lead to the exponential distribution for the fragment number, see Fig. 4. The correspondingprobability density function might be actually expressed as:
pnn ek0k1 nk2 n2 3
With n is the fragment number (integer number) and the distribution parameters are k0 0:8145; k1 0:2252; k2 0:0321.
4.3. Fragments shape
The shape of the fragments depends mainly on both the rupturetype and the cracks propagation in the tank. However, the crackinitiates in the circumferential part and propagates until it cutsone of the two endcaps; then, it unfolds the ring [18,20]. It mayalso happen that the endcap does not split from the ring, the ringremains linked to the endcap as a rocket [18]. The ring generatessometimes 1 or 2 pieces.
On the other hand, the crack can start at the middle of the cylinder due to the manufacture defects. This crack unfolds the cylinder, then propagates towards the edges of the vessel. Finally it
gineeQ.B. Nguyen et al. / Advances in Enmight split the vessel along the circumference: one or two endcaps might then be projected. It happens also that the projectile
a
b
Fig. 6. (a) Horizontal constrained tank and (b) unconstrained tank [22].
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Ac
cid
ent n
um
ber
1 2 3 4 5 6 7 8 9Fragment number
AccidentModel
Fig. 4. Number of fragments ying as projectiles from the cylindrical tank undergas or liquid pressure [6].
y
z
Tg
Source
Fig. 3. Generation of various frag
(b) weld failure (a) bolt failure Fig. 5. Endcap of the vertical tank detached [26].is the whole set: an endcap and the cylinder (partly or completely)[17,21].
The fragment shape might have various forms. Holden [5] givesa mere classication of the fragments types according to the number of linked caps and number of circumferential cracks. The typesof fragments are similar to those collected from INERIS.
According to Gubinelli [2] some types of fragments due toBLEVE might be classied as: cylindrical, endcap, endcap attached to the ring, and the tube. Baum [22] has studied also theendcap (in case of constrained tanks) and the endcap attachedto the ring (in case of unrestrained tank).
Moreover, according to the accidental report of Holden [6], alarge ratio of the fragments are endcaps (44 pieces), attenedpieces of the ring (the plates) (57 pieces) and rings attached tothe endcaps (the oblong endcaps) (86 pieces). In the present paper, the generated fragments correspond to endcaps (23.5%),plates (30.5%) and oblong endcaps (46%), see Appendix B.
4.4. Fragment mass
For a cylindrical vessel, Hauptmanns [3] has shown that thefragment mass might be derived from tank mass as follows:
mp k Mt 4where mp is projectile (fragment) mass,Mt is tank mass and k is factor ranging within the interval [0,1] and following a Beta distribution with the parameters a = 0.41213 and b = 1.3926, i.e.:
pkk 10;1k1
Ba; b ka11 kb1 5
x
Target
rajectory
x
Fragment
ments due to explosion [14].
ring Software 40 (2009) 892901 89510;1k 1 if k 2 0;10 otherwise
6
With 1[0,1](k) is the indicator function; Ba; b CaCbCab and C() isthe Gamma function.
On the other hand, when the shape, dimensions and material ofthe fragment are known, the fragment mass, mp, is expressed by:
mp VFragment q 7
With VFragment is the fragment volume; q is the density of the fragment constitutive material.
Two patterns for the cracks propagation might occur: the crackmight start at the circular solder or at the middle of cylinder. As arst approach, we assume that the oblong endcap length and theplate dimensions follow a uniform distribution. The oblong endcap length might range between zero (case of an endcap) up tothe whole cylinder length (case of an endcap attached to thecomplete cylinder). The plate dimensions are positive: length andwidth remain depend on the cylinder length and cylinder
circumference, and the thickness corresponds to the cylinder thickness, see Appendix B.
4.5. Initial velocity
It is also required to dene the velocity of the fragments resulting from an industrial accident (explosion, etc...). Aquaro and Forasassi [23] has performed experimental studies on a reduction scalewith mass fragments ranging within the interval [20 up to 200] kgand velocity ranging within the interval [30 up to 100] m/s, respectively. The fragment velocity might therefore be derived accordingto the fragment mass and the pressure before combustion.
Actually, the initial velocity of the fragments requires theknowledge of the kinetic energy (Ek) [3,4]. For this purpose, Ek isdeduced from the total energy (E), on the basis of the methods suggested by Baker and Baum [2427].
From experimental studies, Baum [26] proposed the adequate
whereas the pieces belonging to the ring follow the minor axis.According to Holden [6], a uniform distribution might be considered in the case of a cylindrical tank; actually the ratios of fragments horizontal departures angles, see Fig. 7b, are so that 20%of the angles range within [30150], 30% of the angles rangewithin [150210], 20% of the angles range within [210330],and 30% of the angles range within [33030]. In the presentstudy, the initial horizontal angle is therefore assumed to followa uniform distribution such that: 20% of the angles range within[30150], 30% of the angles range within [150210], 20% ofthe angles range within [210330], and 30% of the angles rangewithin [33030].
4.7. Drag coefcient and lift coefcient
4.7.1. Lift coefcient, CL
rings. A uniform distribution for the drag coefcient, CD, might beconsidered within the given intervals, [3,4].
0
0
01
) S
896 Q.B. Nguyen et al. / Advances in Engineering Software 40 (2009) 892901formula for the endcap velocity of vertical cylindrical tank beingdetached and being accelerated by the exit of gas due to failureof the circumferential welding, see Fig. 5. They correspond to thezero mass of endcap model, the broad mass of endcap modeland the velocity limit of the endcap model.
From other tests, Baum [22] investigated the endcap velocityaccording to the kind of materials contained in the tank and thetemperature of the substances. Two types of fragment are investigated: endcap and the endcap linked to the ring (the rocket), [19].From the results of the tests performed by Baum [19,21,22,27], theinitial velocity is well described by a log normal distribution.
4.6. Initial departure angle
4.6.1. Vertical angleFrom the existing literature, there is no reliable information
about the horizontal angle. It is assumed therefore, as a rst step,that the initial vertical angle follows a uniform distribution withinthe interval [90; 90], [3,4].
4.6.2. Horizontal angleFrom analysis of reported accidents, Holden [6] has investigated
the distribution of the fragments initial trajectory within the tankprincipal axes system, see Fig. 7. According to the statistical analysis for 18 accidental BLEVEs having produced 52 fragments [6]approximately 60% fragments are projected in an angular sectorof 60 around the axis of the tank, and 40% fragments are projectedalmost perpendicularly to the tank major axis. Also, it has beenfound that endcaps are projected along the major axis of the tank
30 150
210 330
17 1812
16
6
135
225
45
315
8
15
18
2
30%
(a) Horizontal distribution of the fragments
(bFig. 7. Horizontal departure angles0
30
033
20%
20%
30%
implified horizontal distribution of the fragments In the present study, the following hypotheses are thenadopted:
For endcap fragments, the lift coefcient follows a uniform distribution within the interval [0.3510.468] and the drag coefcient follows a uniform distribution within the interval [0.81.1].
For the ring, the effect of the lift can be neglected. Thus the dragcoefcient follows a uniform distribution within the interval[1.11.8].The lift rises for dissymmetric faces of the fragment: upper surface and lower surface, [17]. Thus, the lift should be considered forthe endcap and the endcap attached to the ring. This lift coefcient is expressed as:
CL 0:351 0:6723a 8
With a is the inclination angle. CL is supposed to be equal to zero forthe angles ranging outside the interval [0, 10], [3].
In the present paper, it is assumed, as a rst step, that the liftcoefcient follows a log normal distribution.
4.7.2. Drag coefcient, CDThe drag phenomenon depends on several factors, i.e. the
geometry, the surface roughness and the orientation in respect tothe velocity direction. An acceptable range values for CD, in the caseof tank endcaps, is [0.81.1] and a range values [1.11.8] for theof the projected fragments [5].
5. Fragment trajectory and impact features
The generated fragments may impact potential targets on theirtrajectory, being therefore a mechanical threat for these targets, asit may damage them seriously. The trajectory of the projectiles hasthen to be exactly known. This kind of problem has already beenstudied in order to investigate the possible collision between afragment (the projectile) and a tank (the target) in its vicinity,[3,4]. The movement of the fragment has been analysed in bidimensional (2D) approach, see Fig. 8. The simplied form of the possible impact analysis supposes that the fragment trajectory can bedescribed by the trajectory of a point (its barycentre). For this purpose Gubinelli [2] presents also the model known as minimal distances. The simplied forms are the following: rectangular form inxy plane and circular form on xz plane, see Fig. 8.
The present study deals with the possible impact and requiresalso the detailed information in order to evaluate the mechanicaldamage caused to the targets by the impact. A three dimensionalanalysis is therefore considered. However, for simplication purposes at the present step, the rotation effect of the fragment duringthe movement is neglected. The equations of motion for fragmentcentre are as follows, see Fig. 9:
kD 1q kL x2 x:: 0
1q kD kL y2 y:: g 0
8>>>>>>>:With : kD 12
qair CD ADmp
; kL
12qair CL AL
mp; kD kL km; kD kL kp 10
And q = 1 at descending part; q = 2 at ascending part; g is gravity;mp is fragment mass; vp is fragment velocity; D is drag force; L is liftforce; CD is drag coefcient; CL is lift coefcient; AD is drag area andAL is lift area.
y
zo
xo
Tank
Trajectoryv
x
g
Fragment
Point Iz O
Fig. 8. Fragment trajectory.
T = p.
v . mL
D W = g . mFig. 9. Forces applied to the fragment.1 V target\VFragmentt / k
1 if V target \ VFragmentt/
at the kth simulation0 otherwise
(12
where Nsim, the number of MonteCarlo simulations;1 V target\Vk;fragmentt/ k: the indicator function that indicates whetherthe projectile meets and impacts the target under study or not;Vtarget: the volume of target with a given location, dimensions andshape; VFragment(t): the volume of fragment at the kth simulationdepending on the dimensions, the shape, and the location on thetrajectory that depends on the time t (with t > 0).
6. Penetration and perforation
6.1. Simplied mechanical model
At the impact, the required informations are:
for the impacting projectile: its velocity of arrival vp, its massmp,its shape, its size and its constitutive material properties;
for the impacted target (metal plate in this study): its thicknessand its constitutive material properties.
The impact may cause a partial damage to the target. It may alsocause a total penetration or a critical damage that may causeexplosion of the target, see Fig. 10, where et is the plate thickness,hp is the penetration depth, dp is the fragment diameter, lp is thefragment length, vp is the fragment velocity, a is the fragment inclination and ecr is the critical thickness plate (threshold thicknessthat may generate an explosion of the impacted target if the residual thickness ethp after penetration is smaller than this threshold). The limit state function for this damaged target, see Eq. (1), isthen:
d
lphp
vp
Plate
Fragment
et
et  hp
etecrp ecr
Fig. 10. Impact of a projectile (fragment) on a target (a plate).
The last two cases also represent threat to the integrity of theimpacted target. For this purpose, mechanical models should berun in order to calculate the penetration depth and compare theresidual thickness to the threshold thickness.
For this purpose, various experimental data for this kind of impacts are collected from the existing literature [8,9,12,13]. For instance, a mechanical model is developed in order to calculate thepenetration depth for given ranges of the projectiles velocities:for instance, Bless [8], corresponds to a velocity of almost
penetration depth for the case a 0
vp
1 2
3
898 Q.B. Nguyen et al. / Advances in Engineering Software 40 (2009) 892901hpet(etecr) Fig. 11. Relationship between the projectile velocity and the penetration depth.
Ushape
1
0 0.235 0.54 1
P
0.235
0.54
EC
OEC
shapeUF
Fig. 12. Fragment shape distribution.ER; S R S 13where R = ethp and S = ecr.
Three scenarios might be considered, see Figs. 10 and 11:
Partial penetration in the critical domain: The fragment penetrates partially but the residual thickness of the plate at the zoneof impact remains larger than critical thickness of the plate, ecr(Curve 1, see Fig. 11).
Partial penetration beyond the critical domain: The fragment penetrates partially the target but the residual thickness rangeswithin the critical interval [0,ecr] (Curve 2, Fig. 11). Therefore,due to the effect of gas or liquid internal pressure, the targetmay not resist to this highpressure: cracks may appear withpossible explosion of this damaged target.
Complete perforation: The kinetic energy of the fragment is solarge that the projectile may penetrate completely the target,and might cause explosion of the target (Curve 3, Fig. 11).
The fragment may penetrate inside the target and creates a crater, reducing therefore the residual resisting thickness of the targetat the impacted zone.
Table 1Random variables.
Random variable Probabilistic distribution
n: Fragment number Exponential discreet distributionfp: Frequency of each fragment shape Uniform distribution for any shape amongmp: Fragment mass derived from the knowledge of the shapevp: Departure velocity of the fragments Gaussian distribution N(lu, ru)/: Departure vertical angle Uniform distribution within the interval [h: Departure horizontal angle Uniform distribution within each intervalCL: Lift coefcient Uniform distribution within the interval [0CD: Drag coefcient Uniform distribution within the interval [07. Applications and numerical simulations
In order to evaluate the risk of domino effect, when dealing withindustrial risks, MonteCarlo simulations has been performed withthe hypotheses and probabilistic distributions summarized inTable 1. This risk evaluation relies on the following models andset of hypotheses, see Tables 1 and 2:
endcap of tank, endcaps of tank attached to the ring and the attened ringand the size of each fragment; the size being supposed to be uniformly distributed
90; 90], i.e. 30150:20%; 150210:30%; 210330:20%; 33030:30%Two distinct random variables are involved in this limit statefunction: hp and et. The penetration depth hp depends on a setof random variables as shown in Eqs. (14) and (15). As a rst approach, it is supposed that the random variable et follows a truncated Gaussian distribution N(le,re) as et should always takepositive values, whereas the critical depth ecr is supposed to remain constant.hp dp cosa
dp cosa 2 4 tana Ecfu eu
2=3 1p
r2: tana
14
penetration depth for the case a = 0
hp Ecfu eu
2=3 1p dp 15
With the kinetic energy Ec mp v2p
2 , fu is the ultimate strength of thetarget constitutive material, eu is its ultimate strain, mp is fragmentmass.
6.2. Resistance
As stated in Eq. (13), the target resistance against critical damage after impact corresponds to the following limit state function:
Eet;hp; ecr et hp ecr 162.15 km/s. Many other models have also been developed andcompared to the experimental data, Neilson [13], BRL and SRI models, [9,12]. The authors have recently issued a mechanical modelthat evaluates the penetration depth, hp, in the case of metal targets and metal rods as projectiles, Mebarki [14,2830] The penetration depth depends on the incidence angle a of the projectileat the impact, see Fig. 10:.351; 0.468]
.8; 1.1]
Impact analysis: the possible impact is analysed for the case ofvarious shapes of the projectiles and the targets (ellipsoids, cylinders, rectangles) [31]. The probability of impact is evaluatedby MonteCarlo simulations.
The mechanical damage at the impact: the penetrationdepth is evaluated thanks to a simplied analytical model,[14,2830]. The probability of critical damage for theimpacted tank is also calculated by MonteCarlosimulations.
A sensitivity analysis is performed in order to evaluate the relative effect of both target position and fragment shape on risk ofdomino effect, as described in Table 3. The corresponding resultsare given in Table 4. It can be drawn that:
The reduction of residual energy or kinetic energy (as an amountof the energy is lost during the movement) depends on bothrelative distance from the fragments source to the target, and
the fragment shape (the oblong form causes an important reduction of the kinetic energy).
The probability of impact Pimp decreases rapidly according to thedistance from the fragments source to the target. Indeed, thisprobability increases with the fragments size as observed forthe plate type.
Obviously, the angle at the impact and the edge of the fragment that impacts the target inuence signicantly the penetration depth. However, in the present study, the penetrationmodel does not consider explicitly the type of edge thatimpacts the target. Under the set of hypotheses and data considered in the present study, one might notice that the oblongendcap fragments have the deepest penetration into the target. The corresponding probability of failure is then important.The kind of fragments that may impact and/or cause targetfailure depends on the angle between the source axes andthe target. This kind of fragments does not depend on the distance between the source and the target, as observed for cases1 and 2.
Table 2Data and MonteCarlo simulations results.
Fragments Target
Endcap Oblong endcap Plate Tank cylindrical horizontal form
Radius (or dimensions for plates) (m) 2 2 2 2 2 2 Width < 5 2 2 2Length (m)
The fragment follows a trajectory with two distinct parts:ascending or descending part. The corresponding movements are
ineethat may cause a new sequence of projectiles generation leading toa domino effect, Prup).
The number of fragments, their size, their mass, their velocityat departure, and their departure angles are considered as independent random variables. During industrial accidents, the fragments can have a large set of shapes. However, this studyfocuses on three typical shapes of fragments: endcap type, oblong endcap type and plate type. Furthermore, the study is limited to the case of one projectile that can impact only onetarget.
The velocities and incidence angles at the impact of potentialtargets are also considered as independent random variables. Themechanical features of the impacted targets are described as random variables.
The mechanical models developed for this purpose evaluate theresidual depth of the targets and the penetration depth of the projectiles after impact. Sometimes, the constitutive materials of thetargets and the fragments can be metal, concrete, masonry, glassand so on. The present study focuses on the case of metal targetsand metal fragments.
MonteCarlo simulations are run and the risk of domino effect isevaluated. The domino effect is thus analyzed within the probabilistic framework. It is shown that:
The reduction of kinetic energy during the movement dependson both relative distance from the fragments source to the target, and the fragment shape.
The probability of impact Pimp depends on both distance fromthe fragments source to the target and fragments size.
The kind of fragments that may impact and/or cause target failure depends on the angle between the source axes and the target. It does not depend on the distance between the source andthe target.
Obviously, the methodology presented in this paper can also beextended to other natural or industrial hazards: the source termcan represent any hazard effect (seismic effects andmaximal accelerations, hydraulic pressures due to rain ows or debris ows, volcanic lavas and their thermal, mechanical and hydraulic effects,shock waves, heat and re effects, differential soil settlementsdue to drought, etc.), the mechanical effect (or damage) dependson the mechanical behaviour or interaction of the structural component. The reliability and risk of failure of the component understudy depend intimately on the collected data, the mechanicaland physical models, as well as the probabilistic hypothesesadopted for each of the source and mechanical parts. The component failure can cause a new set of mechanical failures to othercomponents existing in its vicinity. The probabilistic methodologydeveloped for the domino effect analysis, in case of industrial acci8. Conclusions
The present paper deals with the domino effect in industrialfacilities. The risk representing the probability of a domino effectoccurrence as a consequence of an explosion that generates projectiles is investigated. These generated projectiles may impactother industrial components (such as tanks) at their neighbourhood. The features of the projectiles as well as the features ofthe impacted targets are described within a probabilisticframework.
Three main steps are required, i.e. a probabilistic analysis of thesource event (probability of explosion and generation of fragments,Pgen), a probabilistic analysis of the projectiles movement and theirpossible impact on targets (probability of impact, Pimp) and a probabilistic analysis of the target damage (probability of target rupture
900 Q.B. Nguyen et al. / Advances in Engdents, can then be easily adapted for any other hazard and systemunder study (mechanical, electronic, and so on).With ux, uy, uz initial velocities at the departure point O and xP,yP, zP, tP, coordinates and time at the maximum point of the fragment barycentre trajectory.
Appendix B. Fragments shape and dimensions
 Probability distribution of the fragment shape:
According to the analysis of fragment shape, the generated fragments may have any type among endcaps (23.5%), plates (30.5%)and oblong endcaps (46%). A random variable Ushape is used togenerate the type of shape. This random variable is chosen withinthe interval [0,1] with a uniform distribution as follows, see Fig. 12:
s Ushape e [0,0.235], the fragment type is endcap (EC),
s Ushape e ]0.235,0.54], the fragment type is plate (P),
s U e ]0.54,1], the fragment type is oblong endcap OEC).described as follows:
For the ascending parts Formulas x, z
km = 0: x ux t ! t xux km 0, ux = 0: x = 0, no movements on axis X km 0, ux > 0: x lnux km t1km ! t e
x km1km ux
km 0, ux < 0: x ln ux km t1 km ! t 1exkm
km uxs Formulas y
kp = 0: y 12 g t2 uy t ! t uy
u2y2gy
pg
kp > 0: y 12kp lnuykpg
q sin g kpp t cos g kpp t2
kp < 0: y 1kpkp gp t ln2 12 ln
g
p
e2kp g
pt uy
kp
p gp gp uy kpp
2 For the descending parts Formulas x, z
kp = 0 : x uPx t tP xP ! t xxPuPx tP kp 0, uPx = 0: x = xP, no movements on axis x kp 0, uPx > 0: x xP lnuPxkp :ttP1kp ! t e
xxP kp1uPxkp tP
kp 0, uPx < 0: x xP lnuPxkpttP1kp ! t 1exxP kpuPxkp tP
s Formulas y Km 0 : y 12 g t2 g tP t 12 tP
yP ! t tP 2gyPy
pg
km < 0 : y yP 12km cosg km
p
2 Km > 0 : y yP
1kmkm g
pt tP ln 2 12 ln e2
kmgt
p e2
km gtP
p 2 contribution of the research project IMFRA (IMpacts of FRAgments)which nancial contributors are European Community (FEDER andFSE), French national partners (FNADT, Conseil Rgional Centre,Conseil Gnral du Cher), NEXTER munitions, French Departmentfor Ecology and Sustainable Development (MEDAD), Centre National des Risques Industriels (CNRI).
Appendix A. Analytical solutions of motion equations systemAcknowledgements
This research work has received the support of Institut Nationalde lEnvironnement Industriel et des Risques (INERIS, France), as aring Software 40 (2009) 892901shape
Probability distribution of the fragment dimensions:
Depending on the shape of the generated fragment, the dimensions of the fragment as calculated as follows:
s Endcap type: the fragment radius and thickness are the samethan for the exploded vessel (source term),
s Oblong endcap type: the fragment radius and thickness are thesame than for the exploded vessel. A random variable ULength isused to generate the length of the fragment. This random variable is chosen within the interval [0,1] with a uniform distribution. Thus, the length of the fragment, LFragment, is derived from
[4] Hauptmanns U. A MonteCarlo based procedure for treating the ight ofmissiles from tank explosions. Probabilist Eng Mech 2001;16(4):30712.
[11] Khan FI, Abbasi SA. An assessment of the likelihood of occurrence, and thedamage potential of domino effect (chain of accidents) in a typical cluster ofindustries. J Loss Prevent Process Indus 2001;14(4):283306.
[12] Lepareux M, Jamet PH, Matheron PH, Lieutenant JL, Couilleaux J, Duboelle D,et al. Experimental and numerical studies of impacts on stainless steelplates subjected to rigid missiles at low velocity. Nucl Eng Des 1989;115(1):10512.
[13] Neilson AJ. Empirical equations for the perforation of mild steel plates. Int JImpact Eng 1985;3(2):13742.
[14] Mbarki A, Ami Saada R, Nguyen QB, Mercier F, Meftah F, Reimeringer M.Mechanical behavior of metallic plates under metallic rods impact andindustrial risks. Keynote lecture. In: International conference oncomputational and experimental engineering and sciences, Chennai, India;2005.
[15] Ineris. Analyse du choc de projectiles sur un rservoir calorifuge de GPL.Dpartement Scurit Environnement Sret; 1995. No. 9318699SES/JPLHN.
Q.B. Nguyen et al. / Advances in Engineering Software 40 (2009) 892901 901[8] Bless SJ, Barber JP, Bertke RS, Swift HF. Penetration mechanics of yawed rods.Int J Eng Sci 1978;16(11):82934.
[9] Bukharev YI, Zhukov VI. Model of the penetration of a metal barrier by a rodprojectile with an angle of attack. Combust Explo Shock Waves 1995;31(3):1049.
[10] Khan FI, Abbasi SA. Estimation of probabilities and likely consequences of achain of accidents (domino effect) in Manali Industrial Complex. J Clean Prod2001;9(6):493508.[5] Holden PL, Reeves AB. Fragment hazards form failures of pressurised liqueedgas vessels. Inst Chem Eng Symp Series 1985;93:20520.
[6] Holden PL. Assessment of missile hazards: review of incident experiencerelevant to major hazard plant. Safety Reliab Directorate, Health SafetyDirectorate 1988.
[7] Nguyen QB, Mbarki A, Mercier F, Ami Saada R, Reimeringer M. The Dominoeffect and integrated probabilistic approaches for risk analysis. In: ToppingBHV, Montero G, Montenegro R, editors. Proceedings of the eighthinternational conference on computational structures technology.Stirlingshire, United Kingdom: CivilComp Press; 2006. Paper 31.the total length of the exploded vessel, LVessel, as follows:
LFragment ULength:LVesselThe particular values correspond to: ULength = 0 (case of an end
cap) and ULength = 1 (case of an endcap attached to the completecylinder).
s Plate type: the fragment length and width.
The length follows a uniform distribution within [0,LVessel].The width follows a uniform distribution within [0,2pRVessel]
(where RVessel is the exploded vessel radius).The thickness corresponds to the cylinder thickness of the ex
ploded vessel.
References
[1] Mercier F, Boulanger R, Mbarki A, Brunet L, Reimeringer M. Risques lis auximpacts de fragments lors des accidents industriels. La lettre TI RisquesIndustriels 2006;16:56.
[2] Gubinelli G, Zanelli S, Cozzani V. A simplied model for the assessment of theimpact probability of fragments. J Hazardous Mater 2004;116(3):17587.
[3] Hauptmanns U. A procedure for analyzing the ight of missiles fromexplosions of cylindrical vessels. J Loss Prevent Process Industries2001;14(5):395402.[16] Ineris. Mthode pour Identication et Caractrisation des effets Dominos.Rapport nal; 2002. No. DRA200225472.
[17] Ineris. Les clatements de rservoirs, Phnomnologie et modlisation deseffets. Direction des Risques Accidentels; 2004. No. InerisDRA200446055.
[18] Ineris. Etude de la tenue mcanique de talus une onde de soufe et unimpact. Rapporttude; 2006. No. InerisDRA/REST72677.
[19] BaumMR. The velocity of endcap and rocket missiles generated by failure of agas pressurised vessel containing particulate material. J Loss Prevent ProcessIndus 1999;12(4):25968.
[20] Ineris. Effets mcaniques de projectiles impactant des sphres de gazliqu. Dpartement Scurit Environnement Sret; 1993. No. 931934.0/PCL/PCL.
[21] Baum MR. The velocity of large missiles resulting from axial rupture of gaspressurised cylindrical vessels. J Loss Prevent Process Indus 2001;14(3):199203.
[22] Baum MR. Failure of a horizontal pressure vessel containing a hightemperature liquid: the velocity of endcap and rocket missiles. J LossPrevent Process Indus 1999;12(2):13745.
[23] Aquaro D, Forasassi G. An experimental set up for intermediate velocity impactsimulation studies related to the safety of NPP structures andcomponents. Italy: Dipartimento di Costruzioni Meccaniche e Nucleari,University of Pisa; 1987.
[24] Baker QA, Baker WE. Pros and cons of TNT equivalence for industrial explosionaccidents. In: Proceedings of the international conference and workshop onmodelling and mitigating the consequences of accidental releases ofhazardous materials; 1991. p. 58597.
[25] BaumMR. Disruptive failure of pressure vessels: preliminary design guidelinesfor fragment velocity and the extent of the hazard zone. J Press Vessel Technol1988;110(2):16877.
[26] Baum MR. Rupture of a gaspressurized cylindrical vessel: the velocity of adetached endcap. J Loss Prevent Process Indus 1995;8(3):14961.
[27] Baum MR. Rocket missiles generated by failure of a high pressure liquidstorage vessel. J Loss Prevent Process Indus 1998;11(1):1124.
[28] Mbarki A. Modle pour attnuation sismique : prdiction probabiliste despics des acclrations. RFGC Revue Franaise de Gnie Civil, Herms Ed.2004;8(910):107186.
[29] Mbarki A, Nguyen QB, Mercier F, Ami Saada R, Meftah F, Reimeringer M.A probabilistic model for the vulnerability of metal plates under theimpact of cylindrical projectiles. J Loss Prevent Process Indus 2007;20(2):12834.
[30] Mbarki A, Nguyen QB, Mercier F, Ami Saada R, Reimeringer M. Reliabilityanalysis of metallic targets under metallic rods impact: towards a simpliedprobabilistic approach. J Loss Prevent Process Indus 2008;21(5):51827.
[31] Wang W, Wang J, Kim M. An algebraic condition for the separation of twoellipsoids. Comput Aided Geom Des 2001;18(6):5319.
Integrated probabilistic framework for domino effect and risk analysisIntroductionGeneral frameworkReliability analysis and probability of crisisSource termsSource term featuresFragment numberFragments shapeFragment massInitial velocityInitial departure angleVertical angleHorizontal angle
Drag coefficient and lift coefficientLift coefficient, CLDrag coefficient, CD
Fragment trajectory and impact featuresPenetration and perforationSimplified mechanical modelResistance
Applications and numerical simulationsConclusionsAcknowledgements Analytical solutions of motion equations system Fragments shape and dimensionsReferences