I. INTRODUCTION

Dynamics of articulated pipes conveying fluid has been first studied by Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF][START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Experiments[END_REF] as a discrete model of continuously flexible pipes. Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF] applied the Lagrangian method of analysis to a chain of n rigid pipe segments, connected by flexible joints. The inlet position of the system is fixed while the outlet, where the flow discharges in the atmosphere, is free of displacement. This system is a discrete form, with a finite number of degrees of freedom, of the so called cantilevered flexible pipe. When, as considered by Benjamin, the articulated pipe motion is confined in a plane its dynamics is governed by a system of n ordinary differential equations rather than a partial differential equation for its continuous counterpart. Then Benjamin focused on two-segment pipes, he analysed their linear stability [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF] and conducted experiments [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Experiments[END_REF]. He found both experimetally and theoretically that as the flow velocity is increased the static equilibrium position, corresponding to the aligned segments, can be destabilized at a critical value of the velocity. For a pipe freely hanging under gravity this instability manifests itself by an oscillatory motion of the pipe referred to as flutter and occuring through a Hopf bifurcation, or a static deflection, also called divergence, arising via a pitchfork bifurcation. In contrast, a horizontal pipe has been found to be subject only to flutter. Benjamin noted that the occurence of flutter, resulting from growth of small perturbations to attain limit cycle oscillations, requires a net transfer of energy from the fluid to the pipe during this phase. Later, Païdoussis et Deksnis [START_REF] Païdoussis | Articulated models of cantilevers conveying fluid: The study of a paradox[END_REF] generalized the linear stability analysis of Benjamin to an arbitrary number of segments to study the transition, as n is increased, between articulated pipe and continuous flexible pipe. They found that the critical velocities and frequencies for the flutter of articulated pipes converge to the values of the continuous system as n is increased. These authors also showed that the critical velocity for buckling tends to infinity as n → ∞, which could explained why freely hanging continuous pipes have been found to be only subject to flutter-type instability in a previous study of Païdoussis [START_REF] Païdoussis | Dynamics of tubular cantilevers conveying fluid[END_REF].

Rousselet and Herrmann [START_REF] Rousselet | Flutter of articulated pipes at finite amplitude[END_REF] were the first to address nonlinear aspects of this system. For a two-segment hanging pipe whith zero stiffness joints, they analysed the stability of the limit cycles and deduced that the Hopf bifurcation changes from sub-to supercritical as the ratio of the fluid to the pipe mass is increased. In contrary to others, this study did not assume that the flow velocity is a prescribed parameter independent of time but, to take into account the nonstationarity induced by the pipe flutter, a flow equation was coupled to the equation of the pipe motion. We can also mention, among others, the study of Sethna and Shaw [START_REF] Sethna | On codimension-three bifurcations in the motion of articulated tubes conveying fluid[END_REF] which revealed a very complex phase portrait around the point in space of the system parameters where pitchfork and Hopf bifurcations occur simultaneously.

Modified forms of two-segment pipes have also been considered, some examples concerning freely hanging pipe configurations are presented. Bohn and Herrmann [START_REF] Bohn | Instabilities of a spatial system of articulated pipes conveying fluid[END_REF] analysed the linear stability of a hanging pipe when the two segments are allowed to oscillate in two different vertical planes and found a transition from loss of stability from equilibrium by flutter to loss by divergence as the angle between the two planes is incresased from 0 to π/2. The effect of a harmonic component added to the mean flow has been analysed by Bohn and Herrmann [START_REF] Bohn | The dynamic behavior of articulated pipes conveying fluid with periodic flow rate[END_REF], they studied the resonances with the natural frequencies of the system and their combinations and concluded that even a small oscillation amplitude of the flow rate has a destabilizing effect. Later Jensen [START_REF] Jensen | Articulated pipes conveying fluid with high frequency[END_REF] found with a analytical approach that a high frequency harmonic component, when compared to the natural frequencies of the system, could be either marginally destabilizing or stabilizing depending on the value of the ratio of fluid to pipe mass, he also reported a change in the nature of the Hopf bifurcation from super-to sub-critical as the amplitude of the harmonic component is increased.

Flow-induced vibrations of pipes conveying fluid in general, including the particular case of articulated pipes, have generated an abundant literature because of their evident practical interests. The bibliography presented here is not exhaustive, the reader can refer to the comprehensive review presented by Païdoussis [START_REF] Païdoussis | Slender Structures and Axial Flow[END_REF]. The latter shows that the works devoted to articulated pipes have mainly focused on two-segment pipes and that the experimental results are fairly rare and incomplete. It should be pointed out that while the seminal works of Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF][START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Experiments[END_REF] and the study of PaÏdoussis and Deksnis [START_REF] Païdoussis | Articulated models of cantilevers conveying fluid: The study of a paradox[END_REF] were undertaken to deduce general statements about the stability of pipes conveying fluid, subsequent studies, including the present one, were mainly motivated by the fact the articulated pipes are examples of dynamical systems governed by ordinary differential equation system which can then be easily integrated and, moreover, that are amenable to experimental investigations. In the present study, we perfom experiments with articulated pipes made of 2 to 5 segments, our experimental setup is presented in §II. We also conduct a linear stability analysis accounting for the dissipation in the joints which was neglected in most of the previous studies, except for the references cited above in [START_REF] Bohn | Instabilities of a spatial system of articulated pipes conveying fluid[END_REF] and [START_REF] Jensen | Articulated pipes conveying fluid with high frequency[END_REF]. Details about this analysis are given in §III. Before conluding in §V, experimental and analytical results are presented together with comparisons in §IV.

A schematic of the experimental setup is shown in Fig. 1. An articulated pipe consists of n identical rigid segments of length L, mass per unit length M p , inner and outer radius R i and R e , respectively. They are connected by n -1 identical flexible joints. The highest segment is connected, in O, by a first joint to a vertical pipe of same radii and fixed in the laboratory frame (O, x, y, z). A water flow is supplied into the articulated pipe from its fixed top end O and discharges into the atmosphere by its free end E. The volume-flow rate Q is deduced from the mass-flow rate calculated by measuring the time to collect a set mass of water at the pipe exit, measured with a precision scale. The characteristic flow velocity is then calculated as U = Q/(πR 2 i ). With this method, the velocities U are determined within a 5% uncertainty.

In order to constrain the motion of the pipe in the vertical plane (O, x, z), two adjacent segments are connected with a pivot link of axis parallel to (O, y). For this purpose, the segment ends are sharpened and attached together with a stainless steel axis inserted through holes drilled into the sharpened segment ends as shown in pictures of Fig. 1. This pivot link is then fitted with a short length of flexible silicon tube, of thickness 1 mm and inner radius equal to R e , which is secured with four plastic collars. With this setup, the relative displacement of one segment relative to its neighbour is limited to about π/2 well above the amplitude of the observed motions.

Experiments are performed changing the flow velocity U . At low U the articulated pipe hangs statically under the action of gravity, whose acceleration is denoted by -→ G . At velocities above a critical value U c oscillatory displacement of the pipe in the vertical plane (O, x, z) occurs. This planar motion is fully described by the n angles φ r between the n segments and the vertical axis (O, z). The pipe motion is recorded with a video camera operating at 10 images per second and placed perpendicular to the plane of the pipe motion. The length of the recorded sequences is of 100 s. The n angles φ r and their temporal evolution are obtained by determinating from the successive images, the coordinates of two points of each segment axis, within a 2% uncertainty. Angular amplitude Φ r and frequency F r can then be deduced by simple analysis of the temporal signals φ r (T ), r = 1 to n.

Present experiments are conducted with segments made of copper of L = 0.195 m, R e = 0.004 m, R i = 0.003 m giving M p =0.180 kg/m. It should be pointed out that M p is calculated dividing the total pipe mass by its total length, in such a way that it includes the mass of the fitting tubes, plastic collars and axes of the links. This means that the mass concentration at the joints due to these elements is neglected in the following and in particular in the analysis presented in the next section, this mass represents about 6% of the total mass of the articulated pipe. The conveyed fluid is water whose density is taken to be equal to 1000 kg/m 3 resulting in a mass of fluid contained by unit length of pipe M f =0.028 kg/m. The stiffness K and damping C coefficients of the joints (all supposed identical) are estimated considering the damped oscillator made of a single freely hanging segment, without fluid, while the upper segment is fixed vertically. The segment is slightly displaced from its equilibrium position then released. The damped oscillations are recorded with the camera from which we deduced φ(T ) the temporal evolution of the angle made by the segment with the vertical axis. Estimations of K and C are then the values giving the best approximation of the small damped oscillations φ(T ) by the expression of the linear damped oscillator which writes

φ lin (T ) = φ 0 cos (Ω lin T + ψ) exp (-µT ) where Ω lin = 3G 2L + 3K MpL 3 -µ 2 1/2
is the pseudo-frequency and µ = 3C

III. LINEAR STABILITY ANALYSIS

To analyse the linear stability of the static equilibrium position of the hanging articulated pipe, we first derive the equations of motion by means of the Lagrange's formalism using the n angles φ r as generalized coordinates. As shown by Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF], the n Lagrange's equations, for the open system composed of the n segments and n joints of the articulated pipe and of the fluid it contains at any times, write

d dT ∂L ∂ φr - ∂L ∂φ r + ∂R ∂ φr = -M f U - -→ OE + U - → τ n • ∂ --→ OE ∂φ r (r = 1..n) (1)
where overdot denotes derivative with respect to time T . L = T -V is the system Lagrangian, T = T pipe +T f luid is the sum of the kinetic energies of the pipe and of the fluid. V = V gravity + V restor is the potential energy accounting for the gravity force on the pipe and on the fluid, V restor is associated to the restoring forces due to the stiffness of the joints. -→ τ r is the unit vector parallel to the segment r and then --→ OE + U -→ τ n is the fluid velocity at the pipe free end. For the calculation of the lagrangian we consider for simplicity the radii R e and R i as being small compared to the segment length L. The flow is supposed to be incompressible and normal to any section of the pipe. Moreover, because the present analysis is undertaken to investigate the characteristic of the system at the flutter threshold, the flow rate is assumed time independent and the velocity profile uniform in any section. This later assumption is supported by the fact that, at the threshold, the pipe Reynolds number is larger than 5 × 10 4 corresponding to a fairly flat turbulent velocity profile. With these assumptions, in any section of the segment r the flow velocity relative to this pipe segment writes U -→ τ r where U is constant. The restoring forces due to the joint stiffness are assumed linearly dependent on the deformation angle and characterized by the stiffness coefficient K.

With these assumptions, the calculation of the lagrangian does not present particular difficulties, details can be found in [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF] or [START_REF] Païdoussis | Articulated models of cantilevers conveying fluid: The study of a paradox[END_REF]. For convenience we introduce the constant angle φ 0 = 0 (then φ0 = φ0 = 0) in such a way that the different terms write

T pipe = M p L 3 2 n i=1    φ2 i 3 + φi i-1 j=0 φj cos (φ i -φ j ) +   i-1 j=0 φj cos (φ i -φ j )   2 +   i-1 j=0 φj sin (φ i -φ j )   2    , (2)
T f luid = M f L 3 2 n i=1    φ2 i 3 + φi i-1 j=0 φj cos (φ i -φ j ) +   i-1 j=0 φj cos (φ i -φ j )   2 +   i-1 j=0 φj sin (φ i -φ j )   2    + M f L 2 U n i=1 i-1 j=0 φj sin (φ i -φ j ) + n M f LU 2 2 , (3)
V gravity = -(M p + M f ) GL 2 n i=1   cos φ i 2 + i-1 j=0 cos φ j   + const. (4)
and

V restor = K 2 n i=1 (φ i -φ i-1 ) 2 + const. (5)
The dissipation in the joints, which is neglected in the seminal study of Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF], is taken into account here through the Rayleigh dissipation function R. The dissipative forces are supposed to be linearly dependent on the angular velocity and characterized by the damping coefficient C in such a way that

R = C 2 n i=1 φi -φi-1 2 . (6)
It should be noted that the conveyed fluid is not supposed inviscid but the system dynamics does not depend on the fluid friction because, as discussed by Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF], its overall effect on the articulated pipe is equivalent to a force parallel to the first segment, that is to -→ τ 1 , and therefore does not work.

Substituting Eqs. ( 2)-( 5) into Eq. ( 1) we can easily derive the n coupled non linear differential equations governing the motion of the articulated pipe. These equations are linearized about the equilibrium position (φ r = 0) r=1..n and nondimensionalized considering the dimensionless time t = ΩT where, following Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF],

Ω = 3K (Mp+M f )L 3 1 2 is
the frequency of the undamped (C = 0) oscillator made of a single horizontal (G = 0) segment filled with the fluid at rest (U = 0).

The linear nondimensional equations write, for r = 1 to n,

φr + 3 2   r-1 j=0 φj + (1 -δ rn ) n i=r+1   φi + 2 i-1 j=0 φj     + mu n i=r φi + (1 -δ rn ) n i=r+1 φi + c φr -φr-1 -(1 -δ rn ) φr+1 -φr + mu 2 (φ n -φ r ) + g [1 + 2 (n -r)] φ r + φ r -φ r-1 -(1 -δ rn ) (φ r+1 -φ r ) = 0 (7)
where δ rn is the Kronecker delta, overdot denotes now derivative with respect to dimensionless time t and the dimensionless parameters are

m = 3M f M p + M f , u = U LΩ , c = 3C (M p + M f ) L 3 Ω , g = 3G 2LΩ 2 . ( 8
)
To analyse the linear stability of the articulated pipe, we consider displacements of the segments of the form where the complex eigenvalue σ is nondimensionalized with Ω. By inserting in Eq. ( 7) we obtain a linear homogeneous system for the complex amplitude vector ϕ = (ϕ r ) r=1..n aϕ = 0

φ r (t) = ϕ r e σt (9)
where a is a n × n matrix. The matrix elements a i,j , whose expressions are given in Appendix, depend on n, m, g, c, u and σ. The displacement modes of the articulated pipe are the non-trivial solutions of Eq. ( 10) associated with σ roots of the determinant of a. The displacement is of buckling type, also referred to as divergence, if the circular frequency is zero (σ) = 0, or otherwise oscillatory or flutter. For a given configuration (n, m, g, c), as u is increased the static equilibrium state (φ r = 0) r=1..n becomes linearly unstable as soon as one of the displacement modes has a positive growth rate (σ) > 0. The critical velocity u c is given by (σ) = 0.

Figure 2 shows the locus of the roots σ on the complex plane for an articulated pipe made of n = 3 segments as u is varied from 0 to just above the critical velocity u c for two different configurations. In Fig. 2(a) the values of the parameters correspond to the present experiments (m = 0.4083, g = 0.1649, c = 0.3477) and the hanging equilibrium position loses stability by flutter at 2.6 < u c < 2.7 where a pair of complex roots cross the imaginary axis. For the configuration of Fig. 2(b), where m = 2, g = 5 and c = 0.3477, a single purely real root cross the imaginary axis, at u c between 2.1 and 2.2, that is indicative of divergence.

IV. RESULTS

The experimental results obtained for articulated pipes made of n = 2 to 5 segments are presented and compared to the predictions of the linear stability analysis performed with the experimental values of the dimensionless parameters, namely m = 0.4083, g = 0.1649 and c = 0.3477.

As shown in the previous section, when u is increased the destabilisation of the freely hanging articulated pipe can give rise to divergence-or flutter-type instability depending on the value of the parameters n, m, g and c. The boundary which separates the regions where the articulated pipe loses stability by flutter or by divergence, as calulated with the linear model, is plotted in Fig. 3 in the plane (g, m) for the experimental value c = 0.3477 and for n = 2, 3, 4 and 8. Flutter appears to be confined to the region of low values of g or m and from n = 4 all curves almost overlap for a given c. The boundaries calculated for n = 2 segments for joints without damping c = 0 and for c = 1 are also plotted to highlight that the effect of increasing c is to reduce the region of flutter. Note that the curve for c = 0 and n = 2 has already been presented by Lunn (reproduced in [START_REF] Païdoussis | Slender Structures and Axial Flow[END_REF]). The points labeled (a) and (b) in Fig. 3 correspond to the conditions of Fig. 2 The stability of a hanging articulated pipe is experimentally investigated by gradually increasing the flow velocity U . The behavior appears qualitatively similar for all the pipes and to the observations in [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Experiments[END_REF] for two-segment pipes. Without flow (U = 0), as the pipe is perturbed it returns to its equilibrium position after a few decaying oscillations. At low velocities, the sytem is overdamped that is the pipe settles back to the equilibrium without oscillations, while, when U is further increased, oscillations occur which decay over a time increasing with U . Then above a critical value U c of the flow, the oscillations grow to reach a limit cycle of frequency F c , corresponding to the flutter instability. The experimental dimensionless critical velocity u c = U c /(LΩ) and circular frequency ω c = 2πF c /Ω are plotted in Fig. 4(a) and (b), respectively, as function of n. For each n, the experiment is repeated several times. If the same pipe is used, the same critical velocity and frequency are found within the experimental uncertainty. In contrast, if different pipes are used, that is a pipe made with new joints and segments randomly changed, dispersion of the experimental results can reach almost 10% for both quantities. For example for n = 2 critical velocities are found between 11.5 and 12.3 m.s -1 (2.75 u c 2.95) and critical frequencies F c 2.08 s -1 (0.57 ω c 0.61). This dispersion is by error bars in Fig. 4. It is believed that this dispersion is the result of the dispersion of the measurements of the stiffness K and damping C coefficients for different joints mentionned in §II.

The behavior of the articulated pipes as u is increased can be interpreted by the analytical results previously presented, by looking at the trajectory of all the eigenvalues σ in the complex plane. Considering the example of the Fig. 2(a) where n = 3 (but it is similar whatever n) we see that all the eigenvalues move first towards smaller negative values of the growth rate (σ) then at least a pair of eigenvalues goes in the other direction up to get a positive growth rate at a critical value u c . critical velocity and frequency predicted by the linear analysis are also plotted in Fig. 4(a) and (b) for comparison; the analytical critical frequency is the value of (σ) at u c . For the small range 2 n 5 considered during the present experiments, the decrease of u c and ω c with n is qualitatively well described and quantitatively not very far from the experimental values although overestimated for n = 2 and 3.

The analytical results for the critical velocity u c of Fig. 4(a) reveal ranges in n with a local minimum. These ranges correspond to flutter modes of a different structure illustrated by the superimposed views of the pipe during one flutter period in the insets of the figure for n = 4, 8 and 16. For pipes made of n = 7 to 12 segments, the flutter mode has one node (marked by the arrow) and for larger n, mode with two nodes, and possibly more nodes for still more segments, are predicted. For shortest pipes (2 n 6), as considered in the experiments, flutter modes with zero node are obtained at the threshold. This is confirmed in Fig. 5 where the shape of the flutter modes observed during the experiments at the threshold is illustrated by superimposing views of the pipe during one flutter cycle, the flutter modes predicted by the linear stability analysis are also shown. Comparison reveals two main differences.

First, contrary to the analytical futter modes, the experimental ones present a more or less pronounced asymmetry with respect to the vertical axis. This asymmetry seems to be reminiscent of an assymmetry of the equilibrium states, indeed, the hanging articulated pipe without flow is inevitably not perfectly straight in the experiments in such a way that the flow gives rise to lateral forces leading to the increase of the alignment defect at the joints, as already mentionned by Benjamin [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Experiments[END_REF]. We illustrate this behavior choosing a configuration (n = 3 in Fig. 5) for which the flutter is strongly asymmetric and we present in Fig. 6 the static equilibrium states that precede this flutter for different flow velocity u < u c . The pipe is seen to be more and more deflected from the vertical straight position as u is increased and the flutter mode is shifted in the same direction as the preceding static equilibrium states. Another difference we can note by comparing experimental and analytical mode shapes is the distribution of the flutter amplitude of each segments along the pipe. The amplitudes predicted by linear stability analysis being arbitrary, we compare in Fig. 7 the angular amplitudes Φ r of the oscillatory motion of each segment r normalized by the amplitude of the last segment of the pipe Φ n . Experimental values in Fig. 7 correspond to the flutter motions of Fig. 5, the dispersion of these quantities as the experiments are repeated is small of the order of the symbol size. For all n, the model predicts a gradual increase of the amplitude from the first (highest) segment to the last (lowest). In contrast, the experiments show a flutter amplitude more and more confined to the last segments for n above 2 and the amplitude of a segment appears smaller and smaller, when compared to the analytical results, as the number of segments below it increases.

This discrepancy is due to assumption made in the analysis that all the joints are identical. Indeed, damping in a joint comes on one hand from the dissipation in the joint material and, on the other hand, from the friction of the segments on the axis of the pivot link. The latter depends on the force applied on the axis that is mainly due to the weight and to the fluid friction. Both effects increasing with the number of segments below the joint. It results that the higher the segment, the larger the damping coefficient C and C is maximum for the highest joint. In contrast in the model C is supposed to be constant and equal to the mean value experimentally measured for a single segment without flow (see §II).

V. CONCLUDING DISCUSSION

Experiments have been conducted to investigate the stability of articulated pipes conveying a fluid as the flow velocity is increased. For the value of the experimental parameters considered in the present study, the pipes lost stability by flutter. The critical velocities have been measured and the resulting flutter modes have been characterized for a number of segments n between 2 to 5. The configuration with only one segment (n = 1) is not discussed because its equilibrium position (φ 1 = 0) is always stable as experimentally observed and confirmed by the linear analysis. This can be understood by considering the energy transfer from the fluid flow to the pipe that writes over a time ∆T (see [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Theory[END_REF])

∆W = - T +∆T T M f U - -→ OE 2 + U - → τ n • - -→ OE dT. (11)
The first term in the integrand is always positive and the second term is zero when n = 1, it results that ∆W is always negative for the configuration with only one segment in such a way that flutter can not occur. Configurations comprising of more segments were initially going to be considered but, due to the increase of the head loss with the pipe length, during all the attempts with n = 6 segments we get a fast swelling then explosion of the first joint. This occured sometimes during the experiments with 5 segments.

Linear stability analysis of an idealized system has also been performed extending previous works. Comparisons with the experimental results have shown the ability of the model to describe at least qualitatively the flutter instability at the threslold. But two complementary experimental observations must be mentionned. First, the critical velocity u c for the occurence of flutter has been experimentally determined by increasing flow velocity. If the velocity was then decreased the flutter persisted up to u d u c . These hysteresis loops, already reported in [START_REF] Benjamin | Dynamics of a system of articulated pipes conveying fluid. I. Experiments[END_REF], could be also reponsible for the dispersion of the experimental values of u c in Fig. 4(a), their wideness defined as (u c -u d )/u c have been found to be smaller than 9% for all pipes. Second, for increasing as well as for decreasing velocities only finite amplitude flutter has been observed, in other words flutter such that Φ r 1 for all r = 1 to n has never been observed. These two points suggests that to go further than the characterization of the flutter at the threshold it would be necessary to consider the system of nonlinear equations governing the pipe dynamics and obtained by substituting Eqs. ( 2)-( 5) into Eq. ( 1). Investigations of the nonlinear dynamics of articulated pipes are currently underway both experimentally and numerically integrating the resulting system of ordinary differential equations using a Runge-Kutta method.

2MpL 3 the

 3 damping rate. Repeating this for 30 different joints, we get the mean values K = 0.235 N.m and C = 0.00382 N.m.s with the fairly large standard deviations σ K = 0.033 N.m and σ C = 0.00061 N.m.s showing the difficulty of making identical joints.

FIG. 1 .

 1 FIG.1. Schematic of the experimental setup, pictures show two perpendicular views of a segment joint.

6 FIG. 2 .

 62 FIG. 2. Root locus diagram as the velocity u is varied from 0 to the critical value uc for n = 3, c = 0.3477 and, (a) m = 0.4083 and g = 0.1649, (b) m = 2 and g = 5.

  (a) and (b), respectively. Point (a) represents the present experimental conditions m = 0.4083 and g = 0.1649 and, in agreement with our observations where loss of stability only occurs by flutter, this point is clearly in the flutter region whatever the number n of segments.

FIG. 3 .

 3 FIG. 3. Boundaries between regions of loss of stability of the freely hanging articulated pipe by flutter and divergence for c = 0.3477 and n = 2, 3, 5 and 10, and for c = 0 and 1 and n = 2. The points labeled (a) and (b) correspond to the conditions of Fig. 2(a) and (b), respectively.

8 ;FIG. 4 .

 84 FIG. 4. Dimensionless critical velocity uc (a) and circular frequancy ωc (b) as function of the number of segments n as predicted by the linear stability analysis ( ) and as obtained during the experiments (•), the error bars represent the experimental value dispersion. The shape of the flutter modes at the threshold calculated for n = 4, 8 and 16 are shown.

FIG. 5 .

 5 FIG. 5. Superimposed views of the articulated pipe during one flutter cycle at the threshold for n = 2, 3, 4 and 5 (from left to right) as recorded during the experiments (top) and predicted by the linear analysis (bottom).

FIG. 6 . 1 FIG. 7 .

 617 FIG.6. Superimposed views of the equilibrium states for U = 0, 6.8 and 7.8 m.s -1 (u = 0, 1.63 and 1.87 from left to right) preceding the flutter shown in Fig.5for n = 3.

Appendix

The elements of the n × n matrix a (where n is the number of segments into the pipe) of the linear system Eq. [START_REF] Païdoussis | Slender Structures and Axial Flow[END_REF] depend on the dimensionless quantities m, g, c, u and σ, themselves defined as functions of the physical parameters of the system in §III. Expressions of these elements are given hereafter: For all n a n,n = σ 2 + (mu + c) σ + g + 1, (A.1)

and for i = 1 to n -1

and for j = 1 to i

If n > 3 we have for i = 1 to n -3 and j = i + 2 to n -1 a i,j = [9 + 6 (n -1 -j)] 2 2 + 2muσ. (A.8)


Mehdi Saqalli
email: mehdi.saqalli@univ-tlse2.fr

Nicolas Maestripieri
email: nicolas.maestripieri@univ-tlse2.fr

Melio Saenz
email: msaenz47@gmail.com

Mahamadou Belem
email: mahamadou.belem@gmail.com

The need of tearing down the walls between disciplines and emphasize transdisciplinary approach within a multidimensional framework

Introduction

« We have not enough data about that ». This is often the answer one modeler may encounter while he/she is asking about one process an issue and by then a model to build needs to construct. The common answer the modeler think about is "ok, but then? We can do something ugly, fuzzy, and not precise but clearly, the mistake will be lower than not putting the process because of the lack of data". Even more, the "not-enough-data" assertion is theoretically always valid: whatever the issue, data will be lacking. Where to position the limit after which one can say "ok, let us combine what we know to see it has a meaning", i.e. try to build a theory, a model of the reality for answering a question? More generally, our purpose in this communication is to argue that asking "how can we model without all the necessary information?" means actually ""how can we model?".

One may consider that modelling is the last avatar of the less known and less visible and valorizing second part of research: the construction and creation of theories from data and information scientists have previously collected. This demarche, often called "inductive" in the most accepted hypothetico-deductive research approach, i.e. deduction, induction and abduction (Blecic & Cecchini, 2008, p. 539-540), implies to combine altogether "elements", pieces of science along a plan that may enlighten an issue. This may have been assessed in the past along an intuitive process, such as described by [START_REF] Bergson | L'Intuition philosophique[END_REF]: he described this capacity of connecting processes, patterns and dynamics along sometimes illogical analogies, comparisons and consistencies as a purely human action, impossible to formalize. Nowadays, modelling tools, especially when implemented to integrate both qualitative and quantitative information and rules, do have this ability if and only if they are used along a formalized and epistemologically relevant methodology. They can then be used as a way to explicit relations along a systemic approach. We do consider that they should be used as a way to fulfill the second fraction of research: connecting data.

Interdisciplinarity: when do we need it? Actually, astrophysics is not a scientific discipline that feels an interest for neither interdisciplinarity, nor for instance fundamental chemistry and finally, nor the combination of the two sciences. So, one may ask: "when do we need "interdisciplinarity"? Even more important, how can we characterize it? Better than using a philosophic background we do consider as relevant but beyond our competency and our level of abstraction, let us use the vox populi argument: When, where and in what domain can we see a plea for interdisciplinarity?

1. The need facing an "epistemological obstacle" [START_REF] Bachelard | La Formation de l'esprit scientifique. Contribution à une psychanalyse de la connaissance objective[END_REF]: This is a necessary but not sufficient condition for an interdisciplinary need. It only means that one scholar from one discipline faces an issue that goes beyond the "normal" science he/she belongs to.

The background need:

The first criterion is obvious: "when a scientist cannot borrow data and information from other disciplines he/she may need without having knowledge of the mechanisms that create such information. For instance, using prices of goods collected from economic census for better characterizing the needs of a social group into a socio-anthropologic study does not mean it is an interdisciplinary study because some economics is used. It means that each discipline having its own point of view and paradigms on the reality, having the corresponding background that gives access to this point of view and these paradigms is thereby a sine qua non condition for interdisciplinarity.

3. The need for a paradigm on dynamics and variables: The second criterion is related to the previous one and come naturally when thinking about interdisciplinarity: it concerns combination of disciplines very different one from one another. Actually, three groups can be considered:

• The Physics group: It belongs to disciplines where refutation [START_REF] Popper | Realism and the Aim of Science: From the Postscript to The Logic of Scientific Discovery[END_REF]) can be assessed with fully controlled experiences, i.e. where the experience is considered as ceteris paribus i (here "considered" means that the related scientific community accepts that it is ceteris paribus and has tried to refute it).

Relations and dynamics are defined according to a limited range of variables. Full determinism is not reached only because of the principle of Heisenberg and the importance of the knowledge of initial conditions which ignorance leads to chaos. Only quantitative data are considered.

• The Biology group: it belongs to disciplines where refutation can be assessed with observations and samples, with the support of some controlled experiences. Relations and dynamics are defined according to a medium range of variables. Along the effects of Heisenberg and initial conditions, non-determinism is brought by the lack of information on differential biological functions (for instance, the absorption capacity regarding water or other elements differ from one plant to another) but also emergence, i.e. the property of a phenomenon which is a combination of several elements and which behavior cannot be reduced to each element [START_REF] Müller | Des systèmes autonomes aux systèmes multi-agents : Interaction, émergence et systèmes complexes (HDR Habilitation à Diriger les Recherches en Informatique)[END_REF]. However, only quantitative data are considered, but dynamics may be described along qualitative schemes without determinism (for instance, animals do not pilot their fate in absence of self-reflexivity).

• The group of social sciences: Practically, no experiences can be assessed except some well-famous but questionable very controlled behavior experiments such as the one of Milgram. Because it is far more difficult to characterize which variables are not part of the studied system (Why social psychology is not fully part of economics?), relations and dynamics are defined according to a large range of variables.

Hypotheses are proposed with far more conditions and many anthropologists do tend to avoid theorizations. Refutation can be assessed only with observations and samples. Along the effects of Heisenberg, initial conditions, lack of differential knowledge and emergence, non-determinism is everywhere: no human groups ever behave in the same manner twice. To face the diversity of variables and the structural non-determinism, dynamics are mainly described and characterized along qualitative manners supported by quantitative data. Both qualitative and quantitative tools and data are used.

These three groups do have, because of differences between paradigms, principles, methods and practices large difficulties for inter-dialogue. We do consider that interdisciplinarity is needed if and only if it concerns sciences that belong to at least two of these groups.

4. Finally, another factor does act as an incentive for interdisciplinary: applicability. For instance, the Apollo program did have to include psychology and sociology counter-intuitively for space scientists once the program did raise the application level, i.e. the time when the group of Physics sciences did have to be built, i.e. integrate the oekoumen, the human community.

Interdisciplinarity: Why are socio-ecological sciences its most prominent supporter?

Let us consider a macro-scenario where Nature, Life and Society coexist. This scenario has the three properties which define a system: a universe with elements which have a position and leads to identify the system's structure, a set of properties and attributes on which basis it is possible to assign qualitative and quantitative variables to describe the states of the system and, among its elements we can find or establish a set of relationships which determines the behavior of the system.

1. Socio-ecological systems need a formalism for apprehending a fluctuant, complex system, i.e. a model: We have seen that any system, whatever the discipline it belongs cannot be fully known, observed and described.

That is impossible because the quantity of variables needed to do so is infinite and the variability of them in time. Then, we have to choose a subset of relevant variables in agreement with our interest and needs. At this time we are not describing the reality but a part of it. We call this, a model. Value of models can be established by conceptual validation, accurate representation of the reality, stability of the simulation processes and convergence of solutions.

2. Once the world of academy finds that its own existence is only justified in the measure that it gets community needs, social life would involve a very near and strong relationship with scientific and fundamental knowledge and with solutions which can be provided from research and development. At this moment, the interdependence between Science and Society encounters an amazing growth which is completed with the integration of Nature as the main source of resources for life.

Reality cannot be explained in terms of an isolated thinking discipline. We must take into account the consequences produced by interactions. Knowledge models need to reflect at least those consequences which are enough to explain the behavior of the macro-system and which will be useful to solve the community problems. Models need to be fed with qualitative and quantitative data which could lead to increment the value of the second step in scientific research: the construction and creation of theories from data and information that scientists have previously collected and qualified.

Nowadays, modelling tools, especially when implemented to integrate both qualitative and quantitative information and rules, have this ability if and only if they are used along a formalized and epistemologically relevant methodology.

They can then be used as a way to establish explicit relations along a systemic approach. We do consider that they should be used as a way to fulfill the second fraction of research: connecting data.

Along with this necessity of a formalization of information as the second mandatory part of the research process, formalizing elements is a cure for self-focusing scientific approaches.

Interdisciplinarity: what for?

Several points are supporting the plea for interdisciplinarity:

• "the better complexity": Because interdisciplinary research objects such as socio-ecological systems include de facto many disciplines, eliminating some (and sometimes because of the lack of data on them, which we deeply deplore as in the introduction) is an important error. More exactly, we consider that taking into account all the components of a socio-ecological system (including society, environment and economy) and simplifying each component is more relevant and less harmful than neglecting some activities by focusing on the one which seems to be the most important.

• Holism: More globally and following [START_REF] Verburg | Land use change modelling: current practice and research priorities Peter[END_REF] our observations are bound to the extent and resolution of measurement causing each observation to provide only a partial description of the whole multiscale land use system. Beyond the scale of analysis of for instance land use change, it means that scientists and stakeholders have to tear down the walls of disciplinary approach and cultural context which lead to misinterpretation (subjective interpretation) about these phenomena. For instance, analyzing socioecological processes, one of the most complex interdisciplinary scientific objects, combining social and biophysical sciences, the complexity of it must be tackled with the analysis of the holistic character of reality.

• "for the greater good": Indeed, the main quality of social and environmental formalization, and by then a "loud and clear" formalized interdisciplinarity, is to "force" scientists and scholars to represent reality in itself:

the need to achieve the model, computerized or not, required for every scholar to sacrifice a portion of the complexity of its own theme on the altar of the combination of disciplines. The importance of encouraging the success of the object as a whole and not its task as oneself is crucial as is the legitimacy of the objective and the people in charge.

• Exactitude: Following the previous point, interdisciplinarity is a test for scientific rigor and more precisely, exactitude. Exactitude is often confounded with precision and precision do have this shiny power of data with several figures after the comma. But what do means such figures for cases such as, for instance, demographic analyses without including migration? It often happens that scientists do present their datasets without justifying the origins of the variables on which the data were collected: why these lasts were chosen and not others? One should then differentiate variables and data and by then characterize the scope of the object of research, i.e. its exactitude before looking after precision. Variables allow getting relationships and dynamics while data parameterize these relationships.

• Epistemological formalism: Again following the previous point, these variables have to be identified and hierarchized according to a paradigm or a principle (local perception? Experts' common agreement?) which legitimizes their selection to the detriment of other variables. We do plea for a formalization of the criteria, the principle on which variables are designated as relevant. Along the go & return flux between induction and deduction, they should be de jure also the criteria for "validation".

• Plausibility: Along with this necessity of a formalization of information as the second mandatory part of the research process, formalizing elements is a cure against self-focusing scientific approaches. Along the traditional self-checking loop mistake (a set of data is used to characterize a function describing a phenomenon and for "validating this function, it is compared to the very same data which created it), one may consider that "validating" a function coming from one scientific discipline with data from the same discipline has much more risks to create a similar loop. We then plea for "validation" steps that implies using sets from other disciplines to lower the risks of such loops.

The complexity of socio-ecological systems is not inherent to a specific discipline or a domain. Interdisciplinary must avoid all reproduction of domination and should -ideally -transcend frontiers of each discipline. Dawn [START_REF] Youngblood | Interdisciplinary Studies and the Bridging Disciplines: A Matter of Process[END_REF] explains that "what interdisciplinary studies can therefore learn from the bridging disciplines is the importance of not becoming a domain, as domain creates territory and territory creates niche dominance […]".

As a human animals, researchers act -in a certain way -like non-human animals. Ethological studies talk about social animals in a hierarchic community. We have the alpha, the beta and the omega which interact (and fight) for a social position and/or recognition of liability within a territory. Maybe researchers should think about "adisciplinarity" which is located elsewhere in another space outside of discipline's boundaries.

One may then ask why, if it is so interesting and relevant, it is not so practiced? The answer may come from this clue: Our own observations support the idea that interdisciplinarity is harmful for scholars [START_REF] Turner | Contested identities: Human-environment geography and disciplinary implications in a restructuring academy[END_REF].

So, why it is so hard and boring?

• Interpreting complex phenomena, especially when one looks after a socio-ecological system, is related de facto to the way it is interpreted, i.e. how it is perceived and what part is seen as interesting and relevant: it does have all to do with notions of subjectivity and objectivity, even in science. We do plea for the inclusion of the affective part of each scientist (why do you like this aspect of the object of research?) not as anthropology dealt with it in the past, with a high level of honesty but with a low efficiency in the dialogue with other disciplines, but more as a factor among others.

In a way, we find complexity (different scientific point of view and/or approach) inside complexity (socioecological systems). It could be "hard and boring" because one phenomena generates various complex questions: where and when ends a complex phenomenon? Consider the whole hypothetical variables would allow optimum accuracy of prediction? [START_REF] Maestripieri | Validation spatiale de deux modèles de simulation : l'exemple des plantations industrielles au Chili[END_REF]. Are the limit the same between physical sciences, mathematics and social and human sciences? [START_REF] Deffuant | Visions de la complexité. Le démon de Laplace dans tous ses états[END_REF] questioned different visions of complexity and have questions about -among others -time and spatial scales, the relevance of Big

Data. What about the cognitive status of the individual, unexpected phenomena, uncertainty, multiplicity and instability of itself?

• Again and pursuing on the emotional part that is often neglected, interdisciplinarity implies that one scholar from other sciences may discuss one's own methodologies and results. However, this may have harmful effects and we also plea for a gentleman's agreement for more clarity. Game theory supports the idea that any group, including scholars, does have conflictual relations. Such conflicts are "sociologically normal" and we consider that open formal oppositions are less harmful than hidden agendas. Such a gentleman's agreement may include the necessity for each scholar to involve a moment in the discipline of others with whom he/she wants to collaborate, the formalization of the research construction along management methods such as ARDI [START_REF] Michel | ARDI: a co-construction method for participatory modelling in natural resources management[END_REF] or the use of a model which should "force" the dialogue.

• Following the above statement, the creation of an object is like the establishment of a research project. One seeks to place his discipline to have more assets before the project itself. This is reinforced by the powers that be (who owns the management? The financing? The institutional connections). The interdisciplinary problem lies in the use of method. The methods remaining to build, they are most easily influenced by the context and there is no project that is not "the plaything" of extra-scientific aspects. The interdisciplinary is the least armed, because it is naturally under construction. The use of recognized tools (ZADA, ADM, ARDI, and UML) can reduce this confusion but not make it disappear.

• The principle of « each one at home and the sheep will be well kept » is easy, much more than interdisciplinary. Hence the need to seek a compromise and not a consensus. The latter is becoming more inaccessible as partners include a research project (the more people are, the more emergence of conflict is high). According to Mohamed Nachi ( 2006) "a compromise is a process that develops between partners seeking to reach agreement at the price of some accommodations, modifications, and reciprocal concessions between competing interests." This relates to the question of domination discussed earlier.

Conclusion

There is a palpable cognitive impasse. Many researchers (reinforced by the media image) see science as a collection of data. Otherwise, they are inventors who press the buttons. We, modelers, seeking to make connections, are seen as technicians. This comes back to the original idea of cultural cognition. According to the Cultural Cognition Project (http://www.culturalcognition.net/) it "refers to the tendency of individuals to conform their beliefs about disputed matters of fact (e.g., whether global warming is a serious threat; whether the death penalty deters murder; whether gun control makes society more safe or less) to values that define their cultural identities".

The difficulty is to build a true transdisciplinary approach beyond the emotional aspect. The latter has an impact on the question of the research question, the subject and the object of research (two different things). In short, to question everything.


F Tena-Chollet
email: florian.tena-chollet@mines-ales.fr

J Tixier

G Dusserre

J-F Mangin

Metl

Development of a spatial risk assessment tool for the transportation of hydrocarbons: methodology and implementation in a Geographical Information System

Keywords: GIS, risk assessment, dangerous goods transportation, decision making

The transportation of dangerous goods is a complex issue involving various potential consequences for a wide range of high-stake elements. Most particularly, hydrocarbon transportation requires the carrying out of a global study in order to assess the risks involved. The aim of this study is to develop a prediction code for analyzing different possible hydrocarbon supply routes in order to determine whether modifying the flow of hydrocarbon transportation significantly increases the risk (for people, infrastructure and the environment). On the one hand, this paper details the methodology proposed for assessing risk levels using hazard scenarios and the vulnerability of high-stake elements.

On the other hand, it presents the modeling tool developed (CARTENJEUX), based on an existing geographical information system (MapInfo), through a case study (Paris, France). Several maps (severity of accident, vulnerability and risk levels) generated using CARTENJEUX are presented in order to illustrate how stakeholders can determine preferential routes at regional scale.

INTRODUCTION

In the event of a Dangerous Goods Transportation (DGT) accident, the nature and quantity of the products transported can generate various physicochemical effects (thermal flux, overpressure, toxic, radiological and corrosive effects, etc.). Another major cause of complexity is the multiplicity of circulation routes (highways, national, local, etc.), and the multiple causes of accidents (technical, organizational, human, natural). These factors make it difficult to define specific emergency plans, hence the necessity of proposing predictive methods to assess the risk level of these transportation activities.

Although the number of road accidents involving DGT is very low -1.5% of total truck accidents (BARPI, 2005) -they can have a significant impact on the population and environment, especially regarding health, ecological, psychological and economic factors.

The main hazards associated with DGT accidents, which are similar to accidents involving fixed facilities, include the following dangerous phenomena: explosions, fires, emission of toxic or radioactive clouds, and atmosphere, water or soil pollution [START_REF] Barpi | Bilan des accidents technologiques 1992-2004[END_REF].

Indeed, the hazard level of DGT can be catastrophic, and can occur at multiple points over a given territory. According to the French Ministry of Environment [START_REF] Ddt | Risque transport de matières dangereuses[END_REF], among the 58 identified industrial accidents that have caused over 50 deaths since 1900, 23 of them concerned DGT. And among the most lethal, we can note: the crash of a propylene tanker at Los Alfaques (Spain) in 1978, which resulted in 216 deaths and dozens of injuries, and the pipeline leak at Ufa (USSR) in 1989, which caused 192 deaths and 706 injuries [START_REF] Barpi | Bilan des accidents technologiques 1992-2004[END_REF].

Although there were no deaths, the Mississauga train derailment (Toronto) in Canada (10-16 November 1979) is of great interest, because it required the long-term evacuation of about 240,000 people, with several accidental phenomena, such as the BLEVE of a propane wagon and the toxic dispersion of chlorine.

In this context, the French authorities carried out a global study of DGT, more specifically hydrocarbon transportation and its associated risks, in the Ile de France region (including Paris). The objective was to determine whether increasing the flow of hydrocarbon transportation in Ile de France significantly increased the risk.

To respond to this question, it is necessary to assess both the DGT accident risk level and the vulnerability of high-stake sites. To reach this goal, a methodology of risk level assessment over a given territory was developed, based on the severity of the accident and the vulnerability of the area located close to the accidental phenomena in terms of the stakes involved [START_REF] Dandrieux | An integrated methodology research to determine consequences of a vessel destruction[END_REF]. This approach provides a risk level assessment for each point in the studied area so as to choose an optimized route for DGT. In order to optimize a set of routes (that is to say one or several roads) between two locations, risk maps can be produced. Furthermore, this spatial approach was implemented on the geographical information system (GIS) MapInfo, and a specific tool, named CARTENJEUX, was developed for stakeholders.

MATERIAL AND METHODS

2.1

Examples of DGT risk analysis DGT is a very important topic in European countries and several researchers have proposed different approaches. General studies of DGT accidents have been carried out, for example [START_REF] Oggero | A survey of accident occurring during the transport of hazardous substances by road and rail[END_REF], a survey of accidents involving the transportation of hazardous goods by road and rail. They identify the causes, consequences, material involved, severity and frequency of each one. They provide generic scenarios (event tree), generic causes of accident and accumulated probabilities of road/rail accidents based on the MHIDAS database in order to obtain an overview of DGT accidents.

Some risk analyses have also been developed at local level. These local studies are generally based on thorough analysis of the hazard, either using a probabilistic approach to quantify an accident rate in function of road features (tunnel, bridge, highway, etc.) [START_REF] Fabiano | A framework for risk assessment and decision-making strategies in dangerous good transportation[END_REF], or by means of F/N curves [START_REF] Milazzo | A study of land transport of dangerous substances in Eastern Sicily[END_REF] and accident impact areas [START_REF] Bubbico | Risk analysis for road and rail transport of hazardous materials: a GIS approach[END_REF]. Stakes are generally taken into account by the average population density near the road or the area of interest, in function of accident impact area [START_REF] Bubbico | Risk analysis for road and rail transport of hazardous materials: a GIS approach[END_REF][START_REF] Bubbico | Risk management of road and rail transport of hazardous materials in Sicily[END_REF]. In the same way, Milazzo used population density and vulnerable elements (shopping centers, hospitals, schools, etc.) to describe the spatial distribution of human stakes [START_REF] Milazzo | HazMat transport through Messina town: from risk analysis suggestions for improving territorial safety[END_REF]. All these methods are generally implemented in a geographical information system to produce risk maps.

Concerning the assessment of the stakes attached to vulnerable elements located in the territory concerned, we can note that generally only human stakes are considered, in the form of average population densities [Verma, 2011]. A major limit is the fact that few approaches take natural and infrastructural stakes into account.

A classical approach was used to model the system and to qualify the question of risk assessment [START_REF] Tixier | Review of 62 risk analysis methodologies of industrial plants[END_REF]. It is based on following the three elements: the source of the hazard, the hazard propagation flux, and the potential targets.

Characterization of the study area

The city of Paris covers an area of 2,723 km², much smaller than the urban area of which it is the core. Suburban development has accelerated in recent years with an estimated total of 11.4 million inhabitants in 2005.

Paris' closest ring includes three adjoining districts which are fully saturated with urban growth. The ring of the four outside districts is only covered in their inner areas by Paris' urbanization. These eight districts form the larger administrative region "Ile-de-

France", which includes a large number of high-stake economic, historical, touristic, cultural and natural sites.

Paris is located in the north-bending arc of the river Seine (Figure 1), includes two islands which form the oldest part of the city, and is bordered by two outlying parks ("Bois de Boulogne" and "Bois de Vincennes").

Figure 1: map of Paris.

Finally, because it is a capital and also the largest French city, Paris features many important and heterogeneous high-stake sites, whose sensitivity must therefore be considered in the risk studies.

2.3

The proposed modeling procedure

Generalities

In order to assess the risk, it is necessary to describe the studied system, which consists of three main elements (Figure 2): the source of hazard (a hydrocarbon tanker), the propagation of the effects in the environment (here only thermal radiation effects were considered), and the high-stake elements (classified in a typology).

Figure 2: schematization of the studied system.

The risk level thus results from the contribution of two indexes:

-The accident severity, which takes into account the intensity of consequences, the accident frequency of a DGT and the rate of road accidents;

-The vulnerability of the high-stake elements along the hydrocarbon supply routes [START_REF] Tixier | Assessment of the environment vulnerability in the surroundings of an industrial site[END_REF].

So, the definition of the risk level (RL) is:

RL = f (Severity, Vulnerability)

In order to calculate these two main indexes, which are necessary to assess the risk level of a territory, spatial discretization must be performed. This principle can be applied using a geographic mesh composed of a set a cells (Figure 3) which have a specific size determined in function of the precision of the data used and the impact areas of accidental phenomena. For each cell, each index can be calculated in order: the severity, then the vulnerability, and finally the risk level. The first two indexes are normalized using a specific scale in order to assess the risk.

Figure 3: General approach to assess the risk level

The main advantage of this approach is that it provides visual mapping of the results and helps to understand the elements (severity and vulnerability) which contribute to the risk level.

Assessment of accident severity

The aim of the severity index is to include two dimensions: one deterministic, the other probabilistic. It is therefore described by means of two variables:

-The intensity of consequences of a DGT accident;

-The accident frequency on a specific route.

Hence, the severity index is defined as:

Severity index = f (intensity of consequences, accident frequency)

The following parts explain the way to assess the intensity of consequences and accident frequency indexes.

Intensity of consequences

The intensity of consequences index (IC) takes into account the deterministic dimensions and the characteristics of accident scenarios. The IC of an accident is evaluated by the definition of effect areas which are based on specific thresholds defined in function of the type of high-stake element.

In order to define these effect areas, two sets of thresholds are used [START_REF] Medd | Arrêté du 29 septembre 2005 relatif à l'évaluation et à la prise en compte de la probabilité d'occurrence, de la cinétique, de l'intensité des effets et de la gravité des conséquences des accidents potentiels dans les études de dangers des installations classées soumises à autorisation[END_REF]:

one for human stakes and another one for environmental and infrastructural stakes. It is important to call to bear in mind that DGT fire accidents involving hydrocarbons will impact the territory over no more than some hundreds of meters in most cases.

The first set for human stakes contains the following thresholds (Figure 4):

-3kW/m², irreversible effect threshold for the definition of the area of significant hazard to life; -5kW/m², lethal effect threshold for the definition of the area of great hazard to life; -8kW/m², significant lethal effect threshold for the definition of the area of very great hazard to life. The second set for environmental and infrastructural stakes consists of the following thresholds:

-5kW/m²: threshold of significant broken windows; -8kW/m²: threshold of serious damage to infrastructures; -16kW/m²: threshold of very serious damage to infrastructures, apart from concrete;

-20kW/m²: threshold of serious damage to concrete infrastructure; -200kW/m²: threshold of catastrophic damage to concrete for at least 10 minutes.

For each of these areas, an index value between 0 and 5 is assumed, as summarized in the following table: Beyond the distances obtained for human stakes with a threshold of 3kW/m² and for environmental and infrastructural stakes of 5kW/m², the value of severity is null (NA)

by assumption.

In order to finalize the severity index, it is necessary to incorporate the accident frequency dimension.

Accident frequency

The accident frequency index is based on the flow rate of DGT per year (Flux_DGT), the length of the road (L in km) and the accident rate (RA in % per km).

The accident frequency index (that is to say the number of DGT accidents per year), for one section of road, is calculated as:

Accident frequency = L * RA * Flux_DGT

Range of accident frequency Index of accident frequency 10 -8 > accident frequency 1 10 -7 < accident frequency < 10 -8 2 10 -6 < accident frequency < 10 -7 3 10 -5 < accident frequency < 10 -6 4 accident frequency > 10 -5 5

Assessment of the severity index of a route

In order to obtain the value of the severity index of one mesh, it is necessary to take into account all the contributions provided by each road. So, the severity index of one mesh is calculated by the sum of each contribution from each road, knowing that a road is defined by its length, its rate of accidents and finally its DGT flow.

Severity index mesh = ∑ (1 to n) Consequence intensity i * accident frequency i

with i representing the number of roads influencing the mesh. In another words, the mesh undergoes the influence of n roads in term of consequence intensity and accident frequency in function of areas of effect defined by thresholds.

Assessment of the vulnerability of high-stake elements

Research question

The aim of the vulnerability index is to assess the danger to high-stake elements if exposed to a potential hazard. To reach this goal, the critical or sensitive elements within the territory must be identified, geolocated and weighted to assess their relative importance (Tixier et al. 2012). The idea developed is to identify and characterize the vulnerability of the elements located in the surroundings of a DGT route. The following figure summarizes the problem, which aims to answer the following question: is Area #1, which contains human, environmental and infrastructural high-stake elements, more or less vulnerable than Area #2, which also contains human, environmental and infrastructural high-stake elements, but different in quantity and nature?

Figure 5: question of the definition of vulnerability.

To answer this question, it is necessary to characterize the study area (see "3.1. Data and Databases used"), then to define the stakes. A specific methodology for the assessment of the vulnerability, based on the work performed during the European project ARAMIS (Tixier, 2006), can be used. This methodology includes a semi-quantitative approach (a multicriteria decision method named the "Saaty Method") which depends on expert judgments. The analytical hierarchical process (AHP) can be used for multicriteria assessment problems [START_REF] Marinoni | Water planning in a changing climate: Joint application of cost utility analysis and modern portfolio theory[END_REF] in order to score global objectives in an easy way by including the judgments of stakeholders or other key people (Zerger et al., 2011). This provides a broad-based representative view of expert opinion concerning a studied domain [START_REF] Chen | Uncertainty analysis in a GIS-based multi-criteria analysis tool for river catchment management[END_REF]. The main criticism of the AHP approach is due to the "rank reversal phenomenon" which can lead to a change in the ranking of two options when a new one is include (DCLG, 2009;Zerger et al., 2011).

We must therefore ensure that the choice of options used to describe the risk ranking system is exhaustive.

Saaty multicriteria decision method

The purpose of the Saaty method is to assess priorities. To this end, the first point is to have a consensus regarding the objective, and secondly to break down the complex and unstructured situation into its main components. The types of results can be a classification, the allocation of numerical values regarding subjective judgments, or the aggregation of judgments to determine the criteria with the highest priorities. The multicriteria hierarchical method enables decision-making by a group in a consensual way due to a better coherence of judgments, through the construction of a linear additive model [START_REF] Chen | Uncertainty analysis in a GIS-based multi-criteria analysis tool for river catchment management[END_REF]).

Saaty's multicriteria hierarchical method [START_REF] Saaty | Décider face à la complexité : une approche analytique multicritère d'aide à la décision[END_REF] is based on four main steps:

-A description of the studied system (some elements and criteria are proposed in order to characterize the situation);

-The construction of hierarchies (to organize the elements and the criteria to respond to the question);

-An assessment of priorities (based on expert judgments);

-Validation of coherence.

The construction of a hierarchical structure requires the creation or identification of links between the various levels of that structure.

Each element or criteria of a hierarchical structure is situated at a given level of the structure. The upper level corresponds to the global objective (or dominant element).

The elements of a given level are ranked by binary comparison according to their impact on an element at the next highest level, in order to the elements. The various levels of a hierarchy are, consequently, interconnected.

A complex situation can be analyzed by a systematic approach with the help of such a hierarchical structure. The priorities have to be assessed. This process is performed by comparing elements two-by-two (binary comparison). This ranks the elements according to their relative importance. Finally, the logical coherence of the structure is validated to confirm the whole applied process. To make the binary comparisons, it is necessary to use a scale based on standard numerical variables or more qualitative variables that help to take into account qualitative aspects.

From this definition and from hierarchical structures, the matrices and functions of the vulnerability index are deduced. The matrices are translated into a questionnaire which enables experts' judgments to be gathered for the evaluation of each coefficient of the vulnerability functions.

A simple example of the application of the Saaty method is presented below. If we define the vulnerability as a function of criteria A, B and C and of elements w, x, y and z, the following hierarchical structure (Figure 6) can be built.

Figure 6: example of hierarchical structure.

If we consider the top of the hierarchical structure in Figure 6, the following matrix is deduced in order to make binary comparisons (and to define the priorities): The vulnerability (V) is thus a function of the vulnerability criteria A, B and C.

V A B C A 1 X1 X2 B 1 X3 C 1
V =   A +   B +   C
With ,  and , the vulnerability factors of the vulnerability function.

A specific processing operation was required to aggregate the experts' judgments. Each judgment was aggregated by the use of the geometrical mean. A single questionnaire was thus obtained, which was an aggregation of the judgments of all the experts consulted. All the judgments are added to the matrices and the vulnerability factors assessed as presented below.

In order to validate the vulnerability factors, the Consistency Ratio (CR) is assessed and must be less than 10%.

CR = (CI) / (RI).

Where CI is the Consistency Index, with CI = (λmax -n) / (n-1) and n the number of compared elements.

λmax. is the maximum eigenvalue (Perron root) of the matrix. RI = Random Index. For each matrix of size n, Saaty's team generated random matrices and computed their mean CI value and called it the Random Index.

Study area and high-stake element typologies

The aim of this section is to define the territory subject to DGT hazard in order to determine the risk level. It is therefore necessary to propose a set of types of high-stake element so as to accurately characterize the environment. It is necessary to find the right balance between the number of site-types to be taken into account and the limitations due to the multicriteria decision method.

First of all, sites were divided into three categories in function of the stakes involved, and each of them detailed in a list of generic types of high-stake element:

- o Public structure (I4)

As described above, Saaty's multicriteria hierarchical method [START_REF] Saaty | Décider face à la complexité : une approche analytique multicritère d'aide à la décision[END_REF] is based on three main steps, once a system has been described:

-construction of hierarchies;

-assessment of priorities;

-validation of coherence.

The construction of a hierarchical structure requires the creation or the identification of links between the various levels of this structure. In order to have a good understanding of the situation, the description of the environment is completed by a typology of impacts due to exposure to thermal radiation. Three impacts due to thermal radiation are considered to characterize the effects of major accidents on high-stake element:

-Integrity impact, which respectively qualifies the human, environmental and infrastructural damage;

-Economic impact which qualifies an effect in terms of loss of production or of rehabilitation;

-Psychological impact which qualifies an effect in terms of influence on a group of people.

It is then necessary to organize these typologies in order to respond to the vulnerability question. The next step therefore consists in structuring the information in accordance with the following definition of the vulnerability: the vulnerability of each type of high-stake element in function of thermal radiation effect is evaluated by binary comparison using the three impact types as characterization criteria.

The result is the vulnerability of one class of high-stake element to a thermal radiation effect. The associated hierarchical structure for human vulnerability is presented in Figure 7. In this hierarchical structuration, each category of human element is compared in function of each criterion (integrity, economic and psychological impact). Thirty eight experts had already been individually consulted during the ARAMIS European project [START_REF] Tixier | Environmental vulnerability assessment in the vicinity of an industrial site in the frame of ARAMIS European project[END_REF]. The breakdown of experts by type is presented in Figure 8: about 60 % were risks experts (from public or private organizations), while the remaining 40% were industrialists or local authority employees. The tool uses a batch processing operation (Step 3, Figure 9) to schedule and automate a set of operations and computations which form its kernel. Several difficulties occurred during the software modeling and development: the main ones are detailed in the following section through some of the implemented algorithms.

Quick survey of the main algorithms implemented in the kernel

Grid creation process

The method selected for creating the grid is recursive construction of each cell, as

shown in Figure 10. The size choice for the cells depends on the following two intermediates sets of meshes. The created cells can be unified when all segments of all sections of all studied roads are processed. The following figure summarizes the mechanism as it was conceived during this design phase.

Figure 11: first algorithm tested.

Coverage problem and clipping solution

However, a coverage problem occurred in the meshing of many routes due to their selfstructure. In fact, these objects are defined as polylines in order to describe the road network. Figure 12 shows the benchmarks, i.e. the starting and the ending points, used in the meshing process of a road section,.

Figure 12: coverage problem of the first meshing algorithm.

This problem was solved by improving the first algorithm in order to browse (temporary clipping) each unitary segment (single line) of each road section (polyline). For the rest, the initial principle was retained (Figure 13).

Figure 13: the two main mechanisms of the second meshing algorithm.

Homogeneity problem and magnetic arrangement

Nevertheless, new unexpected results concerning the grids' homogeneity occurred.

With the method employed, each mesh was generated by following the direction of its associated road segment, which meant that the cells were not drawn using the same reference points (Figure 14). In order to solve this problem, we created a specific method named "magnetic arrangement" in order to set all meshes in the same orthonormal basis. But this approach resulted in a final problem, as shown in Figure 15 below.

Figure 15: continuity problem between magnetized grids.

Continuity problem and enhanced magnetic arrangement

To correct this lack of continuity, we had to set up a unified magnetic grid (we developed a method named "enhanced magnetic arrangement") to calibrate each intermediate mesh with its neighbors in order to finally obtain a uniform continuous set of meshes. Figure 16 shows the mechanism used for the grid calibration.

Figure 16: "enhanced magnetic arrangement" of two sets of meshes.

Figure 16 highlights furthermore the inevitable presence of repetitions (grayed cells):

the ending point of a segment may also be the starting point of another one. These specific cells located at the ends of road segments can be removed by a method which we developed to verify the unicity of the individual elements making up the grid.

SOFTWARE

Data and Databases used

The first step in the work consisted in describing the expected data. It aim was to define which inputs will be used in the risk assessment model and the index calculation processes for one or several routes selected by the user. The cartographic data required are:

-A description of the typology of the human, infrastructural, environmental high-stake elements present in the study area;

-A recent census of the population;

-A road network with traffic data specific to DGT, and accident rates.

Three cartographic database tables were used in order to describe the required data (Figure 17). The data used were calculated with average values over a year (2003). The structure and content of these tables are confidential, but it is important to say that the information precision is 20 meters.

Figure 17: an orthophoto and its vectorial layer of population densities.

The main constraint relates to the type of existing map objects included in the database used: surface objects and polylines.

Indeed, the methodology takes as input the length of each route section: that is why it is necessary to manipulate line segments (polylines in a GIS) for the roads. But it is also essential to include each high-stake element (described by surface objects) prorated to their surface area in order to integrate territorial disparities into the meshing structure.

For this project, the French stakeholders provided very detailed databases of the territory of interest, but it is important to say that it may be difficult to obtain such quality for another region.

General algorithm

The main process implemented is based on four steps:

-The manual selection of a set of routes (one or more roads) by the user; -The automatic generation of an analysis grid along this route; -The cross-analysis between the grid and the GIS layers; -The assessment of the risk for DGT.

The risk assessment step uses at the same time the high-stake element typology defined by the methodology and a grid. Figure 18 illustrates the cross-analysis operation between a grid and GIS layers in order to extract the expected data.

Figure 18: illustration of the cross-analysis between a set of meshes and GIS layers.

The process is designed like a transverse mechanism associating information about the human, infrastructural and environmental stakes into each mesh. All data are calculated taking into account the size and the area of meshes (densities of surface objects, proportional lengths of polylines). Subsequently, the results are stored in the GIS table which contains the analysis grid (MapInfo file format).

GIS-based tool "CARTENJEUX"

The final tool implemented to study hydrocarbon supply routes was based on a Geographical Information System. This software named CARTENJEUX, which means "stakes-based mapping" in French (figure 19), is in the form of an add-on [START_REF] Ayral | Les Systèmes d'Information Géographique : mise en oeuvre[END_REF] which aims to assess human, infrastructural and environmental risk.

Figure 19: welcome page of the software.

Route selection

In order to study a route through grid analysis, it is necessary first to select linear objects directly on the GIS road layer. This solution has the advantage of being visual and concrete for the user since it physically indicates the road segments with which he/she wants to work. That is why the selection tools available in MapInfo are very useful because they offer a varied panel of possible selections of linear objects (for example, road sections).

Grid generation

Once a set of road segments has been made up, the user can configure a mesh size (in meters) for the grid building and a dimension (also in meters) on both sides of the axis, indicating the spreading of the grid around the road axis. This last distance is obtained in function of effects thresholds and can be entered by leans of a specific user-friendly interface for the grid generation settings. Users hence have the capability of changing the size of the grid at run time.

It is important to say that the tool allows the user to manage a set of grids. Indeed, it is possible to display two routes currently loaded in memory in order to compare them.

When several grids are shown simultaneously, the user can select one of them in order to choose the set of meshes on which the next operations will be done (statistics for example).

At this stage, the generation parameters are validated and the program starts the construction of the grid. As a result, the meshes created are aggregated in a unified grid, ready to be used to assess the risk of the covered territory.

Cross-analysis of the grid with stakes layers

A dynamic grid is now created and saved in a specific MapInfo table, whose fields are empty. This table has both upper and lower bounds, and all the cells of the grid are contiguous within those bounds.

So, the risk assessment process consists of cross-analysis, iterating all the cells objects in order to aggregate them with the data layers opened in MapInfo.

In the end, all the table's fields are assigned by quantification factors about each class of stakes.

Risk assessment and thematic analysis

At this stage, the functions of calculations specified in the methodology and implemented in the tool are launched in order to exploit the aggregated stakes data.

Once the territory is evaluated, a module of thematic analysis assists the user to enable him or her to quickly visualize the severity index, the index of human, infrastructural and environmental vulnerability or risk level (Figure 20).

Figure 20: the three data layers mappable by CARTENJEUX.

In fact, this module is equivalent to a thematic map browser which proposes standard scales of visualizations (relative or absolute) in a few mouse-clicks. The user is thus able to obtain several maps about the severity of a potential accident, the vulnerability of high-stake elements, and the different levels of risk of the initial DGT route.

RESULTS AND DISCUSSION

It is important to say that the exact location of hydrocarbon tanks and the precise supply routes are sensitive and therefore confidential. The examples presented in the following sections are fictive, and any similarity to actual routes would be purely coincidental.

Overview

In order to illustrate the results given by the cartographic tool CARTENJEUX, this part

shows an example (Figure 21) of two possible supply routes (named route 1 and route 2) between an hydrocarbon tank and a gas station (respectively named A and B). The objective is to compare the risk of the two routes from a human, environmental and infrastructural point of view, thanks to spatial and statistical results. This first example is just theoretical because the real supply routes chosen by French stakeholders are confidential.

Figure 21: two possible routes between A and B (left) with their analysis grids (right).

Two grids are automatically created by the developed routines around each road axis.

The different assessments will be run later, in order to obtain a map of risks. Figure 21 shows the two study areas before the cross-analysis.

In a last step, the analysis begins and displays the spatial results. Figure 22 indicates the human vulnerability in the surroundings of the two chosen routes. It is possible to see directly many disparities thanks to the use of cells which enable the consideration of one particular high-stake element group. A function is also available to specify the transparency of the grid in order to consult the subjacent layers and the thematic analysis simultaneously.

Figure 22: human vulnerability around two routes.

The dark gray cells indicate the presence of critical human high-stake elements which are due to highly urbanized areas.

In order to compare more precisely the two routes, the cartographic tool produces a statistical index of risk, as shown in the following figure. Indeed, the user can compare the evolution of the human, infrastructural and environmental risk levels (RL).

Route #1  Route #2 Human RL  23 % Human RL Environmental RL  65% Environmental RL Infrastructural RL  3 % Infrastructural RL
Figure 23: evolution of the statistical index of risk between two supply routes.

In this example, the first route presents the lowest environmental risk level, whereas the second one leads to a decrease in the human and infrastructural risk levels. Thereby, this synthesis approach helps the user to understand the impacts of the two possible supply routes: in a highly urbanized area and if the environmental stakes are not overriding, the second route may be preferred.

In the previous figure, the main differences between the two routes were the human and environmental risks. The move from 1 to 2 would involve a reduction of 23% for the human stakes risk and a rise of 65% for the environmental ones. With regard to the infrastructural stakes, the analysis shows there is a light reduction in vulnerability of 3%.

Further details

The accuracy of the cartographic data led to an analysis of a cell size in order to obtain a fair compromise for mitigating the error factor during the meshing process of the road network. Accordingly to the geometric precision of the database (about 20 meters), a cell width of 50 meters was validated.

With the help of CARTENJEUX, different kind of maps can be obtained. The following examples (figures 24, 25, 26) concern several routes of supply for which the severity, the vulnerability and the global risk has been quantified. This analysis concerns all the territory of "Ile de France", but only an extract of the results is presented for reasons of confidentiality.

Figure 24: map of the severity index. As we can see, the human vulnerability layer presents some disparities due to urbanized areas which are heterogeneously distributed on either side of the highway. Although the severity index is uniform along the route section, we notice an area (North West on the bottom right map) where the level of human risk is low. Indeed, the human vulnerability is the lowest of this study area at this point due to the presence of a large park (so it is not a very urbanized area). Thus, the same average level of risk can be measured either by combination of a high severity index with a very low vulnerability in one area or by combination of a low severity index with high vulnerability in another.

On the one hand, the weighting of the vulnerability by severity of accident gives an integrated overview of the risk encountered in the case of an accident in this area. On the other hand, it is possible to see that the visual interpretation of results may be a source of errors because the use of color is not a robust way to provide rich information.

That is why, the statistical module of CARTENJEUX (Figure 28), which produces index of vulnerability, severity of accident and risk for human, infrastructural and environmental high-stake elements, is a good additional source of information. point is the same gas station. The initial route is named "#1" and the new one is named "#2". The delivered quantity of hydrocarbon is the same but there is necessarily a variation in the severity index because the route (and thus the hazard) changes ("#2"

follows a new highway). Figure 29 also shows the vulnerability levels of newly assigned high-stake elements, and the impact on the risk levels. The different stakes located along the two routes lead to heterogeneous levels of vulnerability. But, because "#2" is more secure than "#1", this strategy reduces the risks of a road accident and thus leads to a decrease in the severity index.

Route

We see with this hypothetical case that level of human risk decreases while the risk levels for the environment increases. The weighting factors of the methodology, which integrate the rise in infrastructural vulnerability and the reduction in the severity index for the same stakes, indicate that there is no change in the infrastructural levels of risk.

CONCLUSIONS

The project consists in the development of a predictive model for determining whether increasing the flow of hydrocarbon transportation (consequence of relocations) significantly increases risk (for people, infrastructure and the environment) and by how much.

The major contribution of this work is to provide a structured proposition for assessing risk levels using severity and vulnerability indexes depending on the location of a DGT accident. A GIS-based tool implements a structured methodology which is proposed to assess the risk levels of supply routes by determining the severity index, the territorial vulnerability and the risk level. The quantification of the vulnerability of the high-stake elements, combined with the severity index of the hydrocarbon transportation gives the associated risk level. In order to implement the proposed methodology, a development scheme is proposed. On this basis, the identified functionalities are implemented by way of specific routines to build a grid analysis and to run optimized GIS queries. The result is an integrated user friendly tool named CARTENJEUX. Although the methodology was calibrated for studies in "Ile de France" (including Paris), this operational software, based on a geographical information system, might be applied to another territory or further studies. Decision makers can produce three kinds of thematic maps (vulnerability, severity of accident and risk level) for human, infrastructural and environmental stakes. A statistical module gives also another vision of the results in order to manage risks. CARTENJEUX thus helps to determine the most preferable route for hydrocarbon DGT.

Using the developed GIS-tool, French stakeholders were able to consider the issues globally and achieve better management of DGT in the "Ile de France" region.

Figure 4 :

 4 Figure 4: Intensity of consequences of a DGT on a road section.

  X2 and X3 are the weighting factors provided by experts through a questionnaire. The questionnaires are analyzed by inverse methods to retrieve the weighting factors. From this matrix, eigenvectors can be calculated and named ,  and . In another words, each of these values gives the relative importance of each criterion.

o

  Public utility or infrastructure (I2) o Private structure (I3)

Figure 7 :

 7 Figure 7: hierarchical structure for human vulnerability.

Figure 8 :

 8 Figure 8: Breakdown of experts by type.

Figure 9 :

 9 Figure 9: dynamic context diagram of the tool.

Figure 14 :

 14 Figure 14: homogeneity problem of the second meshing algorithm.

Figure 26 :

 26 Figure 26: map of environmental risk.

Figure 27 :

 27 Figure 27: severity index (top right), human vulnerability (middle right) and human risk (bottom right) maps of the same area (south of the Bois de Vincennes).

Figure 28 :

 28 Figure 28: statistical summary of the study area.

Figure 29

 29 Figure 29 below shows a former and a new supply strategy. This example aims to study the relocation of a hydrocarbon depot 20km away (new route departure). The arrival

ACKNOWLEDGMENTS

The authors would like to thank the French ministry of industry (DGEMP) for supporting this research.

The quantification factors determine the level of stakes in the study area. A quantification factor is defined as a dimensionless variable, assuming values in the range 0-1, where 0 indicates the absence of the stake in the area and 1 indicates that the quantity of this stake in the area reaches its expected maximum.

2.4

Development scheme

Global use cases

In a preliminary study, the actors and interactions of the system were defined in order to construct the messages intended to be exchanged with the users. These specifications were further refined using various UML (Unified Modeling Language) models. As shown in Figure 9, the following four fundamental steps can be identified as being required:

-The manual selection of a road route (1): the user first selects a sequence of road segments which can be considered as a route for the hydrocarbon transportation;

-The generation of an analysis grid to cover the study area (3): square cells are created in order to divide the environment into a set of meshes. The spatial analysis can be made more or less precise by adjusting the size of each mesh; constraints: the accuracy of the geography database and the distances of thermal radiation effects for the studied hazard scenarios (see Section 2.3.2). By assumption, we set up a size of 50 meters for the cells of each grid.

Figure 10: generation grid algorithm.

The primary focus is in fact to choose between the two main possible strategies in order to cover a territory with an analysis grid made up of square meshes.

The first one consists in the creation of a simple rectangular grid for all the studied territory (here, the city of Paris) thanks to a basic algorithm based on a variable which increments the rows/columns indexes to add each cell. This common approach can be implemented simply and quickly but the high number of meshes potentially created for an extended territory and which have to be crossed with GIS layers may cause significant time problems in the subsequent computational processes.

The second one only deals with the geographical areas of interest (here, the road sections plus the distances of thermal effects). It is thus possible to improve the processing time, but this approach requires a specific algorithm to cover only the targeted parts of the territory.

Therefore, it can be suggested to operate all road sections with the following sequential process: creation, georeferencing, union into a unified grid. In the existing data, each road section is a polyline object which can be composed of several linear segments.

Each section of a given road is split into unitary segments which are covered by