Cyril Douthe
email: cyril.douthe@univ-eiffel.fr

Chloé Girardon

Romain Boulaud

Sensitivity analysis of the global response of flexible rockfall barriers

Keywords: Rockfall barrier, Structural Engineering, Brake activation threshold, geometric stiff-9 ness.) 10

Protection barriers against the fall of boulders and rocks are structures with non-linear 1 mechanical behaviour that make the study particularly complex. In this paper, the understanding 2 of an experimentally observed variability is investigated numerically using a non-linear spring-3 mass equivalence. First, key figures of the experiments on which this study is based are detailed. 4 Then, the numerical model for the dynamic simulation of the barrier deformation under impact is 5 presented. Finally the variability due to block-related parameters and then net-related parameters 6 are explored and evidence the role of the cables geometric stiffness in the global response of the 7 fence.

Introduction 11

The risk due to falling rocks and boulders is all the greater because it is difficult 12 to predict it, as well as to anticipate the trajectory or energy of the block [START_REF] Volkwein | Rockfall characterisation and structural protection -a review[END_REF]. Passive 13 protections must therefore be designed to be able to stop the fall of blocks alongside more 14 or less large areas of instability. These protections often take the form of nets supported 15 by edge cables and posts attached to the cliff [START_REF] Yang | A review of flexible protection in rockfall protection[END_REF]. [START_REF] Boulaud | A comparative assessment of ASM4 rockfall barrier modelling[END_REF] Mechanical understanding of these rockfall protections is difficult: non-linear and 17 dynamic behaviours, in a context of large displacements and irreversible mechanisms 18 to dissipate energy, make the study of these barriers all the more complex. A series of 19 test campaigns were conducted: by rolling rocks along the slope (Muraishi et al. [START_REF] Muraishi | Development of a flexible low-energy rockfall protection fence[END_REF], [START_REF] Boulaud | Modelling of curtain effect in rockfall barrier with the dynamic relaxation[END_REF] McCauley et al. [START_REF] Mccauley | Rockfall Mitigation[END_REF]); by moving them along a cable (Peila et al. [START_REF] Peila | Evaluation of Behaviour of Rockfall Restraining Nets by Full Scale Tests[END_REF], who proposed semi-21 empirical design methods based on their measurements of the maximum deceleration) [START_REF] Day | An introduction to dynamic relaxation[END_REF] or launching them vertically (Gerber et al. [START_REF] Gerber | Flexible rockfall barriers development, standardisation and type-testing in switzerland[END_REF] and Muraishi and Sano [START_REF] Muraishi | Full-scale rockfall test of ring net barrier and components[END_REF]) as well as 23 the experiments of Grassl et al. [START_REF] Grassl | Steel-net Rockfall Protection -Experimental And Numerical Simulation[END_REF] in partnership with the Swiss company Geobrugg.

24

Experimental tests for a better understanding at the net scale were conducted by Trad 25 [START_REF] Trad | Analyse du Comportement et Modélisation de Structures Souples de Protection: le cas des Ecrans de Filets Pare-Pierre sous Sollicitations Statique et Dynamique[END_REF] and Bertrand et al. [START_REF] Bertrand | Full-Scale Dynamic Analysis of an Innovative Rockfall Fence Under Impact Using the Discrete Element Method: from the Local Scale to the Structure Scale[END_REF].

26 Some of these experiments may be realistic but cannot be controllable and thus 27 difficult to repeat. Others are well repeatable, but the rotating component of the speed 28 is missing. Therefore, a standardisation of experimental test protocols was carried out 29 (ETAG [START_REF] Eota | ETAG-027: guideline for European technical approval of falling rock protection kits[END_REF]) which sets out a 1:1 test protocol and a series of results to be achieved 30 in order to validate a barrier and allow its commercialisation in the European Union.

31

Impacts are defined by their kinetic energy E c . A MEL impact ("Maximum Energy Level") corresponds to an energy E c ; a SEL impact ("Service Energy Level") corresponds 33 to two successive energy impacts of E c /3. These two types of impact are centred in the 34 central module of the barrier. The ETAG standard also defines the shape of the module 35 and requires an impact velocity of at least 25 m/s. A review of approval of flexible 36 rockfall protection systems was recently provided by Volkwein et al. [START_REF] Volkwein | Review of Approval of Flexible Rockfall Protection Systems According to ETAG 027[END_REF] and outlined 37 some general features on the response of such structures over a large variety of kits. to the net, and free at the pole heads. Thus, with this assembly, the net is held by two 89 cables in its module centres and by one cable near to the posts.

90

The barrier works as follows: a block is intercepted by the net, which deforms. The 91 net slides along the edge cables (Boulaud et al. [START_REF] Boulaud | Modelling of curtain effect in rockfall barrier with the dynamic relaxation[END_REF]) which deform. When the stress in 92 these cables reaches a threshold force F s , friction brakes (so-called "snake brake" by the 93 provider NGE fondation which are shown in Figure 3) activate. In practice, the stress 94 in the edge cables are not uniform and one cannot foresee which end will be the most 95 solicited. Therefore, one brake is set at each end for all the cables, except the lateral 96 cables 9 and 10 which form closed loop and require thus only one brake. The block stops 97 finally when all the kinetic energy is dissipated.

Measuring devices

99

For each test of the two campaigns, a number of force sensors were attached to the 100 cables between the brake and the anchor (each sensor corresponds to a canal indicated in 101 Figure 2). All the central cables (labelled 4, 5, 6 and 7 in Figure 2) were hence monitored, plus one strut between the cliff and the head of the second posts (see Figure 2). The An impact centred on a structure composed of a single net instead of three, with an 3. A log-normal distribution of parameters µ = 1.60 and σ 2 = 0.25 is defined (x > 0) :

f X (x; µ, σ) = 1 xσ √ 2π • exp - (lnx -µ) 2 2σ 2 (1)
The average value of this distribution is 6.2kN, slightly higher than the value obtained For left to right, and top to bottom : Off-centred impact, cable 3; Off-centred impact, cable 1; Lateral impact, cable 2; Consecutive impact, cable 8 -Pre-stress is taken into account.

The brake threshold distribution in Figure 7 with the characteristics displayed The starting point of the numerical model is a code that was already used for 214 rockfall barrier in two previous publications by the authors [START_REF] Boulaud | A comparative assessment of ASM4 rockfall barrier modelling[END_REF][START_REF] Boulaud | Modelling of curtain effect in rockfall barrier with the dynamic relaxation[END_REF] and which is therefore not recalled here. This tool is based on quasi-static calculations conducted by the 216 dynamic relaxation method (Day [START_REF] Day | An introduction to dynamic relaxation[END_REF]), an algorithm which is suited for the study of 217 static equilibrium of structures with large displacements (Douthe and Baverel [START_REF] Douthe | Design of nexorades or reciprocal frame systems with the dynamic relaxation method[END_REF]). second order perturbation. The system becomes hence equivalent to a mass (the block)

226
interacting with a complex spring (the net with its supporting structures), the reaction 227 force of the spring being calculated from a quasi-static force applied by the net on the 228 actual position of the block. A similar approach had been implemented in Grassl [START_REF] Grassl | Experimentelle und numerische Modellierung des dynamischen Trag-und Verformungsverhaltens von hochflexiblen Schutzsystemen gegen Steinschlag[END_REF].

229

The algorithm goes hence as follow (see Figure 8):

(ii)

A small displacement is imposed δ i (0) = ui (0)dt, and the equilibrium solution 235 is calculated according to the quasi-static analysis of the barrier.

(iii)

From R i the reactions of the net on the block, the new position is calculated:

üi (t) = R i m ui (t + dt 2 ) = ui (t - dt 2 ) + üi (t) × dt u i (t + dt) = u i (t) + ui (t) × dt + üi (t) × dt 2 2
(2) with i = x, y or z the three dimensions of the block 237 (iv)

The "brakes+cables" system following an elasto-plastic behaviour, their proper-

238
ties are updated at each iteration.

(v)

The algorithm stops when all edge cables start unloading.

240

This last issue is questionable as, when the cables start unloading, the structure is In a quantitative approach, the total brake elongation for experimental and numeri-286 cal MEL tests is shown in x-direction (resp. y-direction) were there would be direct impact of the block on the The goal is to evaluate an average braking force, as if the braking were constant 317 over the duration of the total energy dissipation (even if this braking is by no means 318 constant). The amplitude of this braking is calculated as an equivalent static force F s,eq 319 based on the initial mechanical energy of the block E M,ini over the braking distance l(t e ).

Sensitivity analysis: block-related parameters

320

At the time t e when all the energy is absorbed (either plastically by the brakes or elastically by the cables and the net), the block has moved a distance: l(t e ) = (x(t e )x 0 ) 2 + (y(t e )y 0 ) 2 + (z(t e )z 0 ) 2 (3)

The equivalent static force is then defined as follows:

321

F s,eq = E M,ini l(t e ) (4)
Where E M,ini is the mechanical energy of the block at t = 0 and with the static force 322 oriented according to the block's displacement. The equivalent static force is represented according to the point of impact in Figure Figure 14 shows the block's displacement according to time. The error on the 338 maximal displacement z max is 3.6 %. However, the impact duration is significantly lower x direction, the less demanding configuration for the brakes is the centred impact at 360 x = 7.5m. In the y direction, this statement needs to be qualified: as they approach the 361 edges, cables dissipate less energy. The impact centred in x and y is reduced from 87.6% 362 to 83.4% for (x, y) = (7.5, 2.24) and 84.1% for (x, y) = (7.5, 0.5).

363

The impact which appears to be the most demanding for the brakes (i.e. where 364 most energy is dissipated by them) is an impact very close to the left or right extremities.

376

To go more into details, the percentage of energy dissipated plastically by each 377 cable is represented in Figure 16. For each impact location, one can hence analyse the 378 distribution of the energy between the various brakes and have a picture of the load 379 paths from the impact point to the anchorages throughout the net. One remarks that:

• there are characteristic stripes on the maps, which denotes an invariance along y 381 and the fact the the problem varies essentially according to x; During the experiment with a lateral impact, no cliff-side cables ran. For an un-407 known reason, the brakes seemed to be completely blocked. There are therefore signifi-408 cant differences between numerical simulation and real testing. These differences are less 409 pronounced when summing the elongations of the brakes of symmetric cables according 410 to y, as shown in Table 8. A proportion of the energy that could not be dissipated by the 411 blocked brakes is found in their symmetric brakes. Globally the differences observed 412 for the brake elongations for the lateral test are comparable to those observed for the 413 off-centred test, except for cable 10. These differences might thus be attributed to the To explain the very large difference in cable 9, the detailed arrangement of the net, cables and brakes is studied. It shows that these cables are looped (see Figure 2). To build these loops, the lateral cables pass through shackles with a relatively sharp angle which hinders their sliding by friction. A more realistic numerical model is thus built by introducing friction at the head and bases of the post for these lateral cables. The force in the cable will then depend on the angle α of this cable over time and of the activation threshold F s of the brake (see Figure 17).

F s • δ + F s • (e µα -1) • δ = F s • e µα • δ (5)
with µ = 0.3 the friction coefficient and δ the brake elongation.

416

Adding this adjustment to the cable behaviour of the side cables reduces the total 417 brake elongation of cable 9 by 60 %. The difference between numerical and experimental As expected, the residual height is greater in the most rigid areas (i.e. where the net 435 deforms less). However, there is a local maximum for the centred impact. This can be 436 explained by the symmetry of the system: the cables on each side are less stressed at one 437 time, and therefore deform less plastically, while for the off-centre impact in y, the cables Indeed, during the impact, the cables rearrange and the final equilibrium is the same.

470

However, for cables with very low prestressing (typically below 3kN), the amount of 471 energy dissipated for these cable decreases significantly. Considering that the cable 472 stiffness under lateral load at the time of impact is directly linked with their geometric 473 stiffness: the higher the initial prestress, the higher the cable stiffness. Considering then 474 that the structure is hyperstatic, it is well-known that, in hyperstatic structures, stiffness 475 attracts forces. It seems thus natural that when the structure is unequally prestressed, 476 forces be attracted by the most prestressed cables, resulting in some small variations of Globally, the Figure 24 represents the average of the thresholds for a simulation as 499 a function of the total energy dissipated during the impact. Here, it can be seen that 500 the higher the average brake activation threshold, the less dissipation there is, and vice 501 versa: that is, when all the brakes are triggered later, the amount of energy dissipated is

  32

Figure 1 .

 1 Figure 1. Top view of the experimental set-up (by courtesy NGE fondations).

Figure 2 .

 2 Figure 2. Architecture of the experimental flexible-barrier. Top: Technical drawing from above of the barrier by cortesy NGE fondations. Middle: Decomposition of the supporting cable and of their weaving with the ASM4 ring net. The circle at cables end represent the brakes ("snake" brake provided by NGE fondation), the black square the force sensors. Bottom: Supporting structures made of post and struts.

  98

Figure 3 .

 3 Figure 3. Image of the so-called snake brake provided by NGE foundations: it is made of two steel plates connected by a series of bolts through which a cable is braided at both end of the loop (by courtesy NGE fondations).

103

  acceleration of the block was recorded by an accelerometer within the block. After 104 impact, the residual height and final brake elongation were measured on the deformed 105 structure. 106 Then, each block fall was filmed with high speed cameras: one with a top view, one 107 with frontal view. When the quality of the video allows it, digital image correlation (DIC) 108 could be used to retrieve the block position through time as well as the brake elongations 109 as a function of time. This manual operation introduce some intrinsic variability which 110 has been quantified performing the same measurement 4 times for 4 different videos. 111The resulting relative deviation of the maximum block displacement is 4.5% which seems 112 acceptable considering the variability on the other variables.

113 2 . 3 .

 23 First campaign 114 The first set of tests during this campaign consists of two impacts centred on the net, 115 with the same impact energy E MEL = 270 kJ. For the second set of tests, three consecutive 116 impacts (with the removal of the block from the net before the next impact) occurred, 117 with an impact energy for each of E SEL = 90 kJ, i.e. a third of E MEL . The structure was 118 not repaired between the three impacts.119The main objectives of this campaign were first to investigate the reproducibility of 120 experimental results and second to investigate the hypothesis that the effect of successive 121 impacts compared with the effect of one impact with an energy equal to the sum of 122 successive impacts (3SEL ≡ MEL?).

123 2 . 4 .

 24 Second campaign 124 During the second test campaign, five different impacts took place (the first one on 125 an isolated module, the other four on the complete barrier): 126 1.

Figure 4 .

 4 Figure 4. Location of impacts in the two experimental campaign, as well as zone mapped by the numerical simulations (a margin of 0.5m (resp. 1m) is taken along the long edges (resp. lateral edges) to avoid direct interaction of the block with edge cables).

180

  consists in an hyperstatic structure, with a relatively high level of hyperstaticity: when 181 one pulls on one cable, the tensions in the others vary. It is therefore preferable to take 182 the pre-stress values for each cable individually from each test (these values are indeed 183 independent), to identify for each cable the distribution law characteristics (through the 184 various experiments). Then from the identified prestress distributions, one can determine 185 an average distribution law which is shown in Figure 5 with the characteristics displayed 186 in Table

  187

Figure 5 .

 5 Figure 5. Distribution of experimentally measured prestressing forces in all the cables.

188 3 . 4 . 201 Figure 6 .

 342016 Figure6. Four Force-Displacement diagrams used to determine brake activation thresholds. For left to right, and top to bottom : Off-centred impact, cable 3; Off-centred impact, cable 1; Lateral impact, cable 2; Consecutive impact, cable 8 -Pre-stress is taken into account.

212 4 .

 4 Non-linear spring mass system213

Figure 7 .

 7 Figure 7. Distribution of experimentally measured brake activation thresholds.

218 4 .

 4 1. A dynamic model based on finite differences 219 The mass of the net is distributed and about 7 times smaller than the mass of the 220 block which is concentrated in one point. In a first approximation, one could thus 221 considered that the kinetic energy stored by the net is negligible compared to the kinetic 222 energy of the block (which is completely true in the beginning where only the block 223 moves). By there, it was decided to concentrate dynamical aspects of the barrier response 224 in the dynamic of the block and, consequently, to consider the dynamics of the net as a 225

  Start from a position u i (0), speed ui (0) and initial acceleration üi (0) of the 231 block.In all the simulation, the initial velocity and initial acceleration are always 232 taken purely vertical. Horizontal movement is allowed but is only caused by 233 the reaction of the structure.

248 Figure 8 .

 2488 Figure 8. Equivalent spring-mass dynamic algorithm of the barrier.

4. 2 .

 2 Limitation of the structural model 249 The support cables are modelled using the sliding cable element developed by 250 Boulaud et al. [20] which has no friction between the cables and the net. In this model, 251 the only sources of dissipation are thus the brakes. This assumption is relatively strong 252 because, in practice, part of the impact energy is dissipated by friction as documented 253 in Coulibaly et al. [25] on a simple cable example or by the net itself [9,10]. However, 254 the friction coefficient is a parameter which is independent from other parameters 255 concerned by the present sensitivity analysis (location of impact, prestress and brake 256 threshold). There might be couplings or interactions between parameters, but, in a 257 first approximation, it was considered that interactions can be neglected and that the 258 influence of parameters can be investigated independently. 259 Concerning the ASM4 ring net model, the simplest model was chosen with a spring 260 representation of each ring interaction similarly to what is done in Nicot et al. [14] or 261 Dugelas et al. [26]. The influence of this modelling choice was investigated in Boulaud 262 and Douthe [16] who demonstrated that the ring model had negligible influence on the 263 global response of the barrier which is driven by the deformation of supporting cables. 264 Concerning contact between the block and the net, non-penetration interactions 265 are defined on the basis of proximity conditions between net nodes and the bottom 266 of the block. Friction between the block and the net is neglected. High speed camera 267 images of the experiments show that this hypothesis is reasonable for impacts in the 268 central module where the block exhibits almost no rotation movement. However, during 269 the impact on the lateral module, one observes significant rotation of the block which 270 necessarily implies that tangential forces are applied on its surface. The presented 271 sensitivity analysis is thus less relevant for impact in outer part of the barrier.

272 4 . 3 .

 43 Convergence study 273The determination of the time step dt was done through a convergence study for 274 an off-centred impact at (x, y) = (10.5, 1.37). Considering a time step of 0.01 seconds 275 and comparing it to a time step 10 times smaller, the difference in the calculation of the 276 new positions is, at most, less than 2% of the value (for the position according to z). The 277 value dt = 0.01 seconds is therefore considered appropriate, as shown in Figure9.

278 4 . 4 .

 44 Validation by comparison with a few characteristic experimental tests 279 A few tests are performed to compare the accuracy of the model over time with 280 the experimental data. The block trajectory for the MEL test is shown in Figure 10. The 281 numerical solution fits the MEL 2 trajectory. Qualitatively, the dropping profile is the 282 same. Quantitatively, the comparison is difficult because of the variability between the

Figure 9 .Figure 10 .

 910 Figure 9. Position in the x, y and z directions as a function of time for dt = 0.01 s and dt = 0.001 s.

Figure 11 Figure 11 .

 1111 Figure11shows the numerically obtained elongations of the brakes over time

297 5 . 1 .Figure 4 .

 514 Figure 4. It runs through the three units and cover the inner part of the net. The outer 304

  305

323Figure 12 .

 12 Figure 12. Mapping of the static force as a function of the impact position on the net [kN] for an impacting energy E M,ini of 270kJ.

324 12 .Figure 13 Figure 13 .

 121313 Figure13shows the map illustrating the maximum displacement reached by the

Figure 14 .

 14 Figure 14. Off-centred impact: block's displacement according to time for an impacting energy of 270kJ.

339 5 . 3 .Figure 15 Figure 15 .

 531515 Figure15shows the ratio of the plastic energy dissipated by all the brakes at cables

365

  Indeed, compared to the central module, the lateral modules or extremities can mobilised 366 the elastic deformation capabilities of only neighbour module (instead of two for the 367 central module). Their capability to spread the load through the net is thus lower and 368 their rigidity higher. This boundary case aside, another demanding case is the one where 369 the impact is made near the poles (around x = 5 m or x = 10 m) which are indeed other 370 rigid area of the barrier. Beside, one remarks that, approaching the edges (cliff or edge 371 side) causes less dissipation in the edge cables and brakes wherever one is on the net. In 372 general, the maximum variation in energy dissipated is just over 15% and the most rigid 373 areas dissipate the most.

374 5 . 4 .

 54 Distribution of the energy dissipated between the various brakes 375 5.4.1. Analysis of numerical results

Figure 16 .

 16 Figure 16. Percentage of the energy dissipated plastically in each cable on the initial mechanical energy, supposing constant threshold of brakes.

418Figure 17 .Figure 18 .

 1718 Figure 17. Detailed scheme of lateral cables (9 or 10) showing characteristic angles at shackles.An attempt was then made to reproduce properly the lateral test as it is experi-

5. 5 .

 5 Residual height h r 429 The residual height as a function of the impact position of the block is plotted 430 on Figure 19. It represents the smallest height of the barrier taken perpendicularly to 431 the cliff when all the energy has been dissipated. To obtain these results, the previous 432 calculations are conducted one step further and the equilibrium configurations under 433 gravity load with the computed brake elongations are calculated.

Figure 19 .

 19 Figure 19. Mapping of the residual height as a function of the impact position on the net [m].

Table 9 : 450 Figure 20 .

 945020 Figure 20. Side hole in the barrier after the lateral impact during the second campaign.

477

  load paths through the net which has time to deform significantly before tensions in the 478 less prestressed cables reach the threshold value and start dissipating energy. 479 Notice that 480 This result is also found at global scale as shown in Figure 22: the higher the average 481 prestress level, the larger the amount of energy dissipated by the brakes (about 20% 482 increase approximately). A high global prestress level provides a higher initial stiffness 483 of the barrier which will start dissipating earlier. The global response of the barrier is 484 not modified, only the share between plastic energy dissipated by the brakes and elastic 485 energy stored by the components changes.

Figure 21 .

 21 Figure 21. Locally dissipated energy according to prestress in each cable.

Figure 22 .

 22 Figure 22. Global energy variation according to the average prestress in all cables.

6. 2 .Figure 23 .

 223 Figure 23. Local dissipated energy variations according to brake activation threshold in each cable.

Table 1 :

 1 Characteristics recorded for each of the two MEL and three SEL impacts -first campaign.

		test	MEL 1	MEL 2	SEL 1	SEL 2	SEL 3	SEL tot
		theor. energy	270 kJ	270 kJ	90 kJ	90 kJ	90 kJ	3*90kJ
		total brake elong.	828 cm	1136 cm	348 cm	288 cm	256 cm	892 cm
		(eq. energy 25kN )	(207 kJ)	(284 kJ)	(87 kJ)	(72 kJ)	(64 kJ)	(223 kJ)
		maximal displace-	4.44 m	5.05 m
		ment
		residual height	1.4 m	1.2 m	1.5 m	1.1 m	0.9 m
	141	It is observed that the resulting data differs significantly for the two MEL tests:
	142	there is a 13 percent difference between the two maximum displacements reached by the
	143	block, as well as a 31 percent difference in the total brake elongation. Considering that
	144	the brake threshold is constant and equal to 25kN (as specified by the manufacturer),
	145	this difference in the brake elongation can be interpreted in terms of energy dissipated
	146	by the brakes. Following this reasoning for MEL2, this equivalent energy is higher than
	147	the impacting energy, which is not possible and indicates that the average threshold
	148	values must have been lower than 25kN in this case.
	149	Considering then the three successive SEL tests, it appears that the energy dissipated
	150	by each test diminishes progressively with the number of tests. This can be easily
	151	understood by the fact that, test after test, the geometry of the fence changes and
	152	becomes more efficient to resist vertical forces. Indeed, while the net deforms, the edge
	153	cables incline so that consequently the vertical component of the forces in the edge cables,
	154	which are given by the brake threshold, increases. And if the vertical component of the
	155	cable forces increase, the vertical reaction on the block increases and the kinetic energy
	156	dissipated for a given elongation also. Therefore, the additional brake elongation (or
	157	energy dissipated by the brakes) diminishes with each impact increment. Notice that
	158	the total energy dissipated by the three SEL test is comparable to MEL1 and MEL2. This
	159	confirms the idea that what drives the barrier response is the impacting energy.
	160	3.2. Second campaign
	161	Table 2 then shows the results of the second campaign. It is observed that the impact
	162	characteristics vary greatly from one test to another: a twofold variation in the total
	163	brake elongation and a maximum block displacement variation of nearly 40%.
	164	This variability may depend on many factors. Non-exhaustively, one can mention
	165	the location of the impact of the block, the pre-stress in the edge cables or the uncertainty
	166	on the effective brake activation threshold. Yet, force sensors had been installed on the
	167	main cables during the tests and pre-stress measurement had been conducted before
	168	the impact. Before conducting the sensitivity analysis, experimental data on these two
		characteristics are presented.

Table 2 :

 2 Characteristics recorded for each of the five test impacts for the second campaign.

		test	indiv. mod-	pre-loaded	off-centred	lateral	consecutive
			ule
		location	x = 7.5m	x = 7.5m	x = 6.1m	x = 13.4m	x = 7.5m
			y = 1.37m	y = 1.37m	y = 1.8m	y = 1.37m	y = 1.37m
		total brake elong.	581 cm	579 cm	1210 cm	474 cm	670 cm
		(eq. energy 25 kN)	(145 kJ )	(145 kJ)	(303 kJ)	(119 kJ)	(168 kJ)
		max. displ	4.74 m	4.19 m	5.08 m	4.18 m	3.48 m
		res. height	1.36 m	no data	1.01 m	0.94 m	no data
	170	3.3. Analysis of the prestress distribrution in the cables
	171	The ring net and their supporting cables have no bending rigidity. Therefore, to
	172	resist gravity load, they need to be prestressed during the set-up of the fence in order
		to build a quasi horizontal barrier perpendicular to the cliff. The curvature of the edge

173

cables (that can be seen in Figure

2

for example) is directly associated with the ratio 174 between the distributed gravity load induced by the ring net and the prestress in the 175 cable. The level of this prestress is relatively low for two reasons: first to be easily set by 176 workers and second to let the highest possible margin with brake thresholds.

177

During these two test campaigns, 47 prestress values were thus measured thanks 178 to an arbalest with an accuracy of ±2%

[START_REF]Mesure de la tension des câbles et armatures de précontrainte au moyen de l'arbalète principe et description sommaire[END_REF]

. To establish a distribution law for the 179 prestress, it is necessary to ensure independence between values. However, the barrier

Table 4 :

 4 Brake activation threshold distributions.

		Standard deviation Average Expected value
		10.8 kN	27.0 kN	27.0 kN
	209
	210	of the barrier is the purpose of the propose paper. However, before conducting the
		numerical sensitivity analysis, it is necessary to present the numerical model used in the

202

in Table

4

is determined from these values. A log-normal distribution of parameters 203 µ = 3.22 and σ 2 = 0.15 is defined.

204

The average value is close to the theoretical threshold defined by the manufacturer 205 (25kN). One observes however that the dispersion is very large (around 40%) which 206 means that the hypothesis, currently admitted in numerical models, that all threshold 207 values are equal, suffers some serious limitation as already suspected from the equivalent 208 energy deduced in Table

1 and 2

. How this dispersion precisely influence the response 211 study, which is done in the coming section.

Table 5 .

 5 The numerical value is between the two experimental

287

values. The order of magnitude is thus verified. 288

Table 5 :

 5 Total sum of brakes elongation comparison between MEL 1, MEL 2 and numerical simulation.

	test	MEL 1	MEL 2	num. simulation
	total elongation	828 cm	1136 cm	1004 cm

Table 6 :

 6 This assumes that the model is sufficiently stable to be able to interpolate between the Difference (in percentage) between the calculated values and the values obtained by interpolation.

	306
	supporting cables which are phenomena not considered in the present study.

307

Then, as the number of measurements is limited, value interpolation is necessary: it 308 must be ensured that this interpolation is possible. The interpolation (performed with 309 the software MATLAB©) is verified by interpolating according to the natural neighbour 310 interpolation method between some computed values and comparing the result of 311 the interpolation with actual computed values. The maximum error is below 3 %.

314

Table 7 :

 7 The elongations of brakes by cable for the off-center test, in the experimental 393 case and in the numerical case, are compiled in Table7. These elongations are first 394 calculated by using, in the numerical simulation, the theoretical 25 kN brake threshold. To to more into detail, one observe that there is a significant 397 difference for cables 2 and 3, while this difference is reduced for all other cables. Looking 398 at the activation thresholds measured during the experiment, for cable 2, there is a 399 brake threshold at F s,2 = 14.5 kN and for cable 3, a brake threshold at F s,3 = 12.8 kN. Brake elongations by cable number for the off-centred test.

	382
	383	•	the energy dissipation in the brakes is almost symmetrical according to y;
	384	•	on average, impact closer to the cliff result in higher dissipation ( nearly 5% more)
	385		than impact on the opposite side which can be explained by the fact that the cliff
	386		side is more rigid because it is closer to the supports;
	387	•	when approaching the edge at x =1m or x =14m, the brakes of the edge cables
	388		dissipate more than half of the block's initial mechanical energy;
	389	•	all brakes activate and dissipate significant part of the impact energy for some
	390		impact location, meaning that, in the considered structure, all brakes are necessary
	391		and will be each the most solicited brake for a given impact location.
	392	5.4.2. Comparison with experimental results

395

Qualitatively, one observed that the numerical model predicts well the brakes that are 396 solicited during the impact.

400

By adjusting the threshold values (see last line of the table), the difference between 401 the experimental and numerical values for those two cables falls down to comparable 402 level. Unfortunately, the number of force sensors in the structure did not allow for the 403 measuring of all the thresholds and it was not possible to proceed in the same way with 404 all the cables. Nevertheless, it appears clearly that the knowledge of the exact threshold 405 is key to the prediction of the brake elongation. 406

Table 8 :

 8 Brake elongations by cable number grouping cable by pairs in the y direction for the lateral impact.

	414
	effective threshold values of brakes.
	cable couple	1+3	2+8	4+5	6+7	9	10
	exp. [cm]	0	173	0	230	71	0
	num. [cm]	29	236	12	344	269	2

415

6. Sensitivity analysis: net-related parameters 451

  After studying the influence of the impact position of the block, the effects related 452 to the dispersion of some of the supporting cable parameters are examined.

	453
	454	6.1. Prestress variability
	455	In order to explore the influence of prestress level on the main impact characteristics
	456	(energy dissipation, maximum displacement and residual height), a prestress distribu-
	457	tion was randomly applied on the cables. This prestress distribution follows the law
	458	determined experimentally (see section 3.3). Practically, it takes the form of a length
	459 460	default: ∆l = prestress at iteration i = 1 resulting from these length defaults is not exactly the desired F pre-load •l 0 . The barrier being hyperstatic, the effective distribution of the E s S
	461	one. However, one can verify on a single example that the prestress only rearranged
	462	marginally in the structure (see Table 10) and that the order of magnitude and statistical
	463	distribution are similar. Especially, the larger the prestress level, the lower the difference
		between the target value and the equilibrium state.

464

Table 10 :

 10 Prestress in supporting cables: target level (i = 0) and practical equilibrium state (i = 1). About ten numerical simulations are performed, the results of which are presented 465 in Figures21 and 22. Figure21illustrates the variation in the amount of energy dissipated 466 by a cable for different prestress values applied to this cable in the case of a centred test.

	cable	input at i = 0 [kN] equilibrium at i = 1 [kN] difference [kN]
	1	8.6	9.5	0.9
	2	4.7	6.3	1.6
	3	0.7	2.9	2.2
	4	7.0	7.3	0.3
	5	5.3	5.9	0.7
	6	10.4	10.9	0.5
	7	1.8	3.1	1.3
	8	6.0	6.9	0.8
	9	4.4	5.2	0.8
	10	8.6	8.9	0.4
	average	5.7	6.7	1.0
	469

467

It appears that the increase in the prestress of a cable has almost no influence on the 468 amount of plastic energy dissipated by this cable (dots are almost vertically aligned).

Version January 12, 2022 submitted to Geosciences https://www.mdpi.com/journal/geosciences

website : https://www.c2rop.fr/

Acknowledgements

lower. Considering that the global deformation increases with time and, by there, that 503 the strain energy increases with time, one might suppose that the later the triggering of 504 the brakes, the higher the share of elastic strain energy (or equivalently the smaller the 505 dissipated energy). Moreover, as the barrier deforms, the cables incline and the vertical 506 component of the braking forces increase: the more vertical the braking force, the more 507 efficient the dissipation. The dissipation being more effective, the dissipation needs are 508 also reduced. This shows that the efficiency of the structure is closely linked with its geometric 510 stiffness which combines the large deformation with the intensity of the forces in the 511 cables, i.e. with the brake thresholds. Indeed, the geometric stiffness allies the working 512 force in the cable with its direction change: the higher the brake threshold, the stiffer the 513 cables; the stiffer the cables, the lower the elastic strain energy in the net and the higher 514 the dissipated energy by the brakes.

Discussion

516

The Table 11 compiles, in percentage terms, the standard deviations and maximum 517 deviations obtained during the series of measurements analysed in the previous sections.

518

It can be noted that pre-stressing is indeed not a dominant factor for the variability of 519 the results, since the standard deviations are systematically below 3%. On the contrary, 520 the position of the block impact is significant, especially for the values of residual height 521 and maximum displacement. The variation in the thresholds is significant as well: the 522 maximum observed differences in the amount of energy dissipated are almost 40%.

523

Table 11: Variability due to impact position, prestress and brake activation threshold uncertainties.

547

The first phase of the sensitivity analysis concerned the influence of the impact 548 position of the block. It was shown that, for the studied fence, each cable is important 549 and each brake is activated for some impact position (see Figure 16), so that all the 550 brakes are necessary at some points. The centred impact is, according to the study, the 551 most unfavourable (the one which exhibits the highest deformation), so that the ETAG is 552 indeed safe.

553

The second phase of the sensitivity analysis investigated: first the influence of 554 prestress applied to the edge cables, then that of brake thresholds. Concerning the 555 prestress, even if the dispersion of the values was large (more than 50%), the influence 556 was found of secondary order on the global response of the fence. Concerning the 557 variation in brake thresholds, their dispersion was smaller (less than 40%) but by far had 558 more significant influence, with a standard deviation on dissipated energy of 13.5% and 559 a maximum deviation among the simulations performed of nearly 40%.

560 Some general comments on the fence structural behaviour can hence be drawn. The 561 key issue in the design of flexible rockfall barrier is its architecture: it is the arrangement 562 of the supporting cables which drives the way the structure deforms from planar to 563 three dimensional network. The modelling choices of the net have few influence on 564 the response, provided that the sliding of the cable is correctly taken into account [START_REF] Boulaud | A comparative assessment of ASM4 rockfall barrier modelling[END_REF].

565

It is the architecture of the fence and the sliding that guarranty its ability to adapt its 566 geometry to the position of the impact so that it can resist the load with reasonable 567 internal forces in the system.

568

The efficiency of the structure is then principally linked with its geometric stiffness 569 which combines the working force in the cable (given by the brake threshold) with