Integrated probabilistic framework for domino effect and risk analysis

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  • Q.B. Nguyen a,b,*, A. Mebarki a, R. Ami Saada a, F. Mercier b, M. Reimeringer b

    aUniversit Paris-Est, 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 equip-ments or installations in their neighbourhood [1]. They might im-pact 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 Paris-Est, Laboratoire Modlisationet Simulation Multi Echelle, MSME FRE3160 CNRS, 5 bd Descartes, 77454 Marne-la-Valle, France.

    E-mail addresses: Quoc-Bao.Nguyen@univ-paris-est.fr (Q.B. Nguyen), Ahmed.

    Advances in Engineering Software 40 (2009) 892901

    Contents lists availab

    Advances in Engin

    sevMebarki@univ-paris-est.fr (A. Mebarki).90 mm, target strengths ranging from 300 MPa up to 1400 MPa and incidence angles ranging from0 up to 70.

    Monte-Carlo 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,0965-9978/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.advengsoft.2009.01.002ation Multi Echelle, MSME FRE3160 CNRS, 5 bd Descartes, 77454 Marne-la-Valle, 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 acci-dent may generate various sets of projectiles. In their trajectory, they may impact other existing facilities,such as tanks under high-pressure 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 prob-ability 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 dimen-sions, constitutive materials properties are described with probabilistic distributions.

    Probabilistic modelling of the target term (rst set of impacted targets): number of impacting projec-tiles, 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 probabi-listic 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 trajec-tory 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 pen-etration 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 im-pact 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 byMonte-Carlo 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 ele-mentary 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 hav-ing real values r)

    S mechanical demand due to the projectile impact (ran-dom 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 (Monte-Carlo simulation)g gravity (m/s2)n fragment numberLVessel vessel length (m)RVessel vessel radius (m)LFragment cylinder length in type of the oblong end-cap (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 proba-bilistic term describes the probability of fragments genera-tion, 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 prop-erties 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 de-pends on three main terms:

    The risk of projectiles generation (Pgen) as result of the initiat-ing accident.The risk of impact (Pimp), i.e. the risk that the generated projec-tiles 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 val-ues are r) which marginal probabilistic distribution is denotedfr(), S is random mechanical demand due to the projectile im-pact (which values are s) which marginal probabilistic distribu-tion 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 sta-tistically independent), Prup is probability of failure of the im-pacted 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 inter-action 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

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    generate fragme

    fragment numbshape mass velocity dimensions departure angle

    Target Source

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    y mi

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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 (hor-izontal 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 accord-ing to the critical pressure, to the cracks propagation, to the tankconstitutive material and to the connection between the elemen-tary mechanical components. According to scientic reports col-lected 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 frag-ments, 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 frag-ments [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: bot-tom 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 distri-bution 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 distribu-tion 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 end-caps; then, it unfolds the ring [18,20]. It mayalso happen that the end-cap does not split from the ring, the ringremains linked to the end-cap as a rocket [18]. The ring generatessometimes 1 or 2 pieces.

    On the other hand, the crack can start at the middle of the cyl-inder due to the manufacture defects. This crack unfolds the cylin-der, 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 end-caps 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. End-cap of the vertical tank detached [26].is the whole set: an end-cap 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 num-ber 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, end-cap, end-cap at-tached to the ring, and the tube. Baum [22] has studied also theend-cap (in case of constrained tanks) and the end-cap attachedto the ring (in case of unrestrained tank).

    Moreover, according to the accidental report of Holden [6], alarge ratio of the fragments are end-caps (44 pieces), attenedpieces of the ring (the plates) (57 pieces...

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