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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
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fragment numbshape mass velocity dimensions departure angle
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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
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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...