R Boulaud

C Douthe

A comparative assessment of ASM4 rockfall barrier modelling

Keywords: rockfall barrier, ASM4, intrinsic characteristic, dynamic relaxation, structural analysis 1. State-of-art on the modelling of ASM4 ring nets

One key element of exible rockfall barriers is the net which intercepts the block trajectory. The ASM4 ring net is one of the widely spread net technology for medium to high energy impacts. The accurate modelling of its mechanical behaviour has thus been the aim of many research programs, most of which use the discrete elements method and tension members with material non-linearities. This paper proposes hence a comparison of the three main families of models: linear spring models, tensile ring models and exural ring models. First, homogenisation techniques are used to characterize intrinsic properties at local scale. Then, the question of the experimental identication of these properties is addressed. Afterwards, the relevance of the identied behaviour is investigated through two case studies. The rst one consists in a simple hyperbolic paraboloid structure which has a closed form solution involving large displacements.

The second one is based on an experiment on a reduced scale barrier with centered impact and extend the comparison to various kind of data and loading cases. The conclusion presents the main results of the various comparisons, it gives insight on the features that each model can capture and provides some guidelines for designers at early stage design.

The ASM4 ring net

During the second world war, the ASM4 ring nets were used as Anti-Sub-Marines protective fence (and their name comes from this original use). Since then, part of the stocks were reused for the conception of protection structures like exible rockfall barriers. Because of its low cost and good mechanical properties, this type of net is still one of the most used in this eld. Schematically, these structures can be described as follows: a wire net (usually formed by an ASM ring net, but not necessarily) intercepts the rock trajectory. Then, it deforms, sliding along the supporting cables which are attached to the posts or the cli through dissipating devices or brakes. Because it has a large capacity of deformation, the ring net is a complex object which has been studied for a long time and lead to many models. The elementary pattern of this net is formed by one ring interlaced with its four neighbours (see Figure 1).

Review of existing ring models

Even if traditional nite element approaches are sometimes used to accurately simulate rockfall barriers [START_REF] Escallón | Parameter identication of rockfall protection barrier components through an inverse formulation[END_REF][START_REF] Escallón | Mechanics of chain-link wire nets with loose connections[END_REF], these models often focus on cable nets [START_REF] Castanon-Jano | Use of explicit FEM models for the structural and parametrical analysis of rockfall protection barriers[END_REF][START_REF] Gentilini | Design of falling rock protection barriers using numerical models[END_REF][START_REF] Gentilini | Three-dimensional numerical modelling of falling rock protection barriers[END_REF] and double-twisted hexagonal net [START_REF] Cazzani | Dynamic nite element analysis of interceptive devices for falling rocks.pdf[END_REF][START_REF] Thoeni | Discrete modelling of hexagonal wire meshes with a stochastically distorted contact model[END_REF][START_REF] Mentani | A New Approach to Evaluate the Eectiveness of Rockfall Barriers[END_REF] where sliding within the net are less preponderant. In the case of ASM4, the natural topology of the net and its nonlinearities encourage the use of discrete element methods [START_REF] Coulibaly | Sliding cable modeling: An attempt at a unied formulation[END_REF][START_REF] Grassl | Steel-net rockfall protection: Experimental and numerical simulation[END_REF][START_REF] Nicot | Design of rockfall restraining nets from a discrete element modelling[END_REF][START_REF] Volkwein | Protection from Landslides and High Speed Rockfall Events: Reconstruction of Chapman's Peak Drive[END_REF][START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF]. Among the various discrete models, dierent strategies have been used to reproduce the typical two stages behaviour of these rings: large deformation by bending during the rst stage and then high stiness due to the axial stiness of the cable during the second stage. One of the rst ASM net model was developed by F. Nicot [START_REF] Nicot | Design of rockfall restraining nets from a discrete element modelling[END_REF]. He proposed a discrete model in which the rings are represented by a node at their center and interact with each-others by means of an axial spring. This link is described by a non-linear law depending on the elongation of spring. This approach has the advantage of facilitating the geometrical description but the behaviour's determination under any loading condition is complex and needs to t a multi-linear law. Another disadvantage of this model is that it can only transmit axial forces (in the direction given by the spring) and cannot reproduce any transverse strain orthogonally to the spring strain. This kind of spring model was adopted recently in [START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF] in a DEM framework as well as FEM in [START_REF] Gentilini | Three-dimensional numerical modelling of falling rock protection barriers[END_REF][START_REF] Gentilini | Design of falling rock protection barriers using numerical models[END_REF]. In its principle, it is similar to the one used for twisted wire net [START_REF] Bertrand | Modelling a geo-composite cell using discrete analysis[END_REF][START_REF] Bertrand | Fullscale dynamic analysis of an innovative rockfall fence under impact using the discrete element method: From the local scale to the structure scale[END_REF][START_REF] Volkwein | Rockfall characterisation and structural protection -A review[END_REF][START_REF] Thoeni | Discrete modelling of hexagonal wire meshes with a stochastically distorted contact model[END_REF].

A few years after F. Nicot's works, H. Grassl proposed a new discrete element model of a ASM4 ring. The shape of his elementary cell is an octagon where one in two vertices represents a contact node between two rings. The eight nodes are connected by truss elements with a variable length. The bending behaviour is conferred by rotational springs on the eight angles [START_REF] Grassl | Steel-net rockfall protection: Experimental and numerical simulation[END_REF]. A. Volkwein proposed a simplication of the previous model based on the experimental tensile tests carried out by H. Grassl.

He transformed the eight nodes ring by removing the intermediate nodes, keeping only the contact points. Furthermore, he replaced the bending behaviour by the axial strain of two diagonal trusses [START_REF] Volkwein | Numerische Simulation von exiblen Steinschlagschutzsystemen[END_REF]. By halving the num-ber of degrees of freedom, this geometrical simplication signicantly reduces the computation time. Doing so, one breaks the link between the original nature of non linearity (the shape change) and the behaviour of the model, which consequently made the identication of member stiness less straightforward.

Besides, A. Volkwein' ring model presents some deformation modes with no energy (similarly to hourglass effects in nite elements [START_REF] Flanagan | A uniform strain hexahedron and quadrilateral with orthogonal hourglass control[END_REF]). Therefore J. Coulibaly used a similar geometry to propose a new model in which links between two consecutive nodes are added to prevent such kinematics [START_REF] Coulibaly | Nonlinear Discrete Mechanical Model of Steel Rings[END_REF][START_REF] Coulibaly | Toward a generic computational approach for exible rockfall barrier modeling[END_REF]. This model also describes very precisely the plastic behaviour of rings, and therefore requires the calibration of many material parameters.

Through out the years, the ASM4 model has thus been enriched toward more accuracy, but also toward larger complexity in the identication process of constitutive parameters which in return makes their use less easy.

Purpose and methodology of the present study

In this quick literature review, it appears that there are three typical discrete ring models: linear spring models later called model A (after Nicot and following authors), tensile ring models later called model B (after Volkwein and following authors), exural ring models later called model C(after Grassl and following authors). When ne tuned, all three models show good agreement with experimental results. A structural engineer in early stage design might therefore ask: how do they compared? what features they can capture? and how easy they are to calibrate?

The aim of the present paper is to answer these three questions and to present a comparative assessment of ASM4 models. The proposed methodology is progressive: from local to global behaviour, from small deformation to large rotations.

Identication of intrinsic characteristics with homogeneisation techniques

Comparing these models at net scale may require very often the identication of relevant characteristics at local scale, recalling a continuous description of the net.

Some authors have even proposed complete continuous representation of the net. For example N. Sasiharan [START_REF] Sasiharan | Numerical analysis of the performance of wire mesh and cable net rockfall protection systems[END_REF] and S. Dhakal [START_REF] Dhakal | Experimental, numerical and analytical modelling of a newly developed rockfall protective cable-net structure[END_REF] have used both a shell element with a special membrane formulation in a general nite element framework for pocket-type rockfall structures. Recently, A. Mentani et al. [START_REF] Mentani | An equivalent continuum approach to eciently model the response of steel wire meshes to rockfall impacts[END_REF] have also proposed an orthotropic elasto-plastic equivalent membrane model, for a chain-link wire net. Such approaches require the denition of the net local behaviour a priori and thus a knowledge of its intrinsic characteristics.

A rst idea for comparing these three typical ASM4 models consists hence in identifying these intrinsic properties by homogeneisation techniques. Such methods are widely used in the eld of granular materials or heterogeneous media to identify macroscopic properties [START_REF] Florence | A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams[END_REF]. Although often limited to small displacements, this method provides insight on the local behaviour of repetitive structures. A rst application to ASM4 ring nets in tension had been made in Ghoussoub [START_REF] Ghoussoub | Analyse de quelques éléments du comportement des écrans de lets pare-blocs[END_REF] and was later completed by the rst author [START_REF] Boulaud | Etudes et modélisations du comportement d'un écran de let pare-blocs à diérentes échelles[END_REF]. Detailed calculations of closed form solutions being purely technical and somehow standard, they will not be recalled in this paper. However the yet unpublished results will be presented in section 2 and allow for a rst comparison of the three models.

Calibration on experimental results from the National Project C2ROP

Discrete like continuous models require the identication of parameters in order to understand how they compared to reality and to be used in a predictive manner.

One of the major aim of the C2ROP National Project (www.c2rop.fr) was hence to provide a wide experimental range for the calibration of numerical models that is not limited to standard ETAG test. A working group has formed around the modelling and testing of exible barriers. A 1:2 scaled barrier was developed with documented separated tests of each components which will be used in section 3 for the calibration of the ring net models at local scale and intrinsic characteristic identication. The results of the quasi-static tests on the complete barrier will then be used in section 5 for the investigation of the inuence of the ASM4 model on the global response of the structure in large deformation [START_REF] Olmedo | Extended experimental studies on rockfall exible fences[END_REF].

The whole set of non-standard experiments realised in this framework has been published in [START_REF] Olmedo | étude expérimentale multi-échelle sur les écrans pareblocsomportement des ouvrages hors cadre etag27[END_REF]. They show rst that there is a signicant spreading of results due to the spreading of components characteristics and inherent imperfections induced by on-site rope work. Then, through the comparison of quasi-static and dynamic loading tests on the barrier, they show that qualitatively, on the tested barrier, the two behaviours were very similar. This can be explained by the fact that 85% of the system mass is concentrated in the block which therefore concentrates all the kinetic energy.

Simulation with the dynamic relaxation method

Building on these observations, quasi-static analysis of barriers should be sucient to evidence the dominant phenomena inuencing the net response at reasonable calculation time [see detailed analysis of this hypothesis in 5, chap 5]. Therefore, it was decided to focus on equilibrium congurations which will be investigated by the dynamic relaxation method in the sections 4 and 5. In this method developed by A.S. Day [START_REF] Day | An Introduction to Dynamic Relaxation[END_REF], the equilibrium of the system is regarded as the result of a highly damped dynamic process. This simple method has proved very interesting results in structural application and tensile structure [see for example the state-of-art presented by 34].

All numerical computations presented in this paper are thus conducted with a numeric tool developed in laboratory [START_REF] Douthe | Design of nexorades or recip-rocal frame systems with the dynamic relaxation method[END_REF][START_REF] Douthe | Form-nding of a grid shell in composite materials[END_REF][START_REF] Peloux | Modeling of bending-torsion couplings in active-bending structures[END_REF]. This tool is implemented in the frame-work of the commercial software Rhinoceros3Dand more specically in its plug-in Grasshopper.

Summary of contributions

The present study proposes thus a comparison of three net models in three steps.

In Section 2, the intrinsic mechanical properties of the nets formed by the innite repetition of three discrete models of ASM4 are characterized at local scale by an homogenisation method in small perturbation.

Afterwards in Section 3, the stiness parameters related to the homogenised elasticity tensors are identied thanks to a plane tensile experiment carried out on a square ring net. The experiment being carried out well beyond small perturbations, it provides local properties for small and large displacements.

Then, in Section 4, the relevance of the identied membrane behaviours is investigated through the transformation of a square net, initially at, into a paraboloid hyperbolic shape, a transformation involving smooth large displacements with closed form solution at structural scale.

Finally in the last section, the inuence of the net model on the whole rockfall barrier is investigated through the simulation of an experiment with full non-linearities.

Identication of ASM4 models intrinsic properties

Description of the three net models

In this section, the intrinsic mechanical properties of the net, formed by three dierent discrete models of ASM4 ring net, are identied with help of homogenisation techniques. The considered ring models are strongly inspired by those presented in Section 1.2. The rst one (Model A in Figure 2) is an adaptation of the Nicot's model in which each particle is linked to its neighbours by axial springs [START_REF] Nicot | Etude du comportement mécanique des ouvrages souples fr protection contre les éboulements rocheux[END_REF][START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF] (in the initial model by Nicot, each ring was connected to six neighbour instead of four in the present paper). The second model (Model B in Figure 2) used the geometry proposed both by A. Volkwein [START_REF] Volkwein | Numerische Simulation von exiblen Steinschlagschutzsystemen[END_REF] and J. Coulibaly [START_REF] Coulibaly | Nonlinear Discrete Mechanical Model of Steel Rings[END_REF].

The four contact points of the ring with its neighbours are connected with a same link. The last one (Model C in Figure 2) is an adaptation proposed by the authors of the model initially developed by H. Grassl [START_REF] Grassl | Steel-net rockfall protection: Experimental and numerical simulation[END_REF]. The geometry of the ring is approached by an octagon which is not regular. The eight edges have the same length, the sum of which corresponds to the perimeter of the real ring. The angles formed by two consecutive edges are however not equal. The eight vertices are connected by a strain energy associated to the perimeter of the ring and by a bending energy associated to the local curvature variations [START_REF] Douthe | Design of nexorades or recip-rocal frame systems with the dynamic relaxation method[END_REF].

Presentation of the homogenisation method

The homogenisation, sometimes also called the asymptotic analysis, includes a set of methods enabling the evaluation of the inuence of the micro-structure on the whole medium behaviour. The method presented in this section has been developed by C. Florence and K. Sab [START_REF] Florence | A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams[END_REF]. This method enables the identication of the mechanical properties of an innite periodic lattice from the resolution of an auxiliary problem of an elementary cell in the framework of the linear elasticity. There are two ways to solve this elementary problem. The rst one consists in enforcing a kinematic admissible strain eld to the elementary cell. This method is called "kinematic method". The second one consists in enforcing a statically admissible eld and is called "static method". Only the kinematic method is presented here and used in this paper for conciseness.

It can however be proved that these two methods are dual and give equivalent results [START_REF] Florence | A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams[END_REF].

To respect the framework of this method, it is assumed that the ASM4 ring net is a periodic and innite lattice.

This assumption is very strong and will fail when the solution of the problem will be dominated by boundary eects.

The elementary cell, whose periodic repetition forms the whole media, is composed of a set of particles of dierent types (two dierent particles have the same type if they have an identical neighbourhood: same connectivity, same geometry and same spring characteristics and orientations). Figure 3 shows the formation of the Model C lattice by periodic repetition of its elementary cell: all the particles drawn in red are of the same type (the particles in the other corner points have dierent spring orientations and, by there, are of dierent type with dierent periodic displacements). The purpose of this method is to identify the local strain eld and the elastic strain energy of an elementary cell. The eective local eld is the one which minimises the elastic energy of the cell. This minimum corresponds to the strain energy in the homogeneous material.

Let E be the symmetric second order tensor of the global strain, i the index of a particle belonging to the elementary cell and X i its position vector. A kinetic displacement of the particle i is obtained by superposing its linear homogeneous displacement E • X i and its periodic given by: u i = E • X i + u per i . Noting that the particles don't have degrees of freedom in rotation, the set of the kinematic admissible displacement elds K.A.(E) is hence dened by:

K.A.(E) = {u/u i = E • X i + u per i ∀i ∈ cell} (1)
Then, the elastic strain energy of the discrete elementary cell ψ hom (E) is computed as the sum of the elastic strain energies provided by the interactions between particles in the cell ψ i (u) divided by its area A Ω . The relation of homogenization is thus given by:

ψ hom (E) = 1 2 E : A : E = min u ∈ K.A.(E) 1 A Ω i ψ i (u) (2)
where A is the overall elasticity tensor. A part of the cell deformation is xed by the global behaviour of the lattice E. The other part, associated with u per , requires solving a minimization problem which is quadratic in u per .

In order to simplify the writing in the further developments, the Voigt's Notation is introduced as in [START_REF] Ghoussoub | Analyse de quelques éléments du comportement des écrans de lets pare-blocs[END_REF]:

A : E =      A 1111 A 1122 A 1112 A 2211 A 2222 A 2212 A 1112 A 2212 A 1212      :      E 11 E 22 2E 12      (3)

Calculation of the elastic strain energy

The purpose of this section is to describe the dierent interactions between particles used in the three considered models of net. Three types of interaction are studied: a spring interaction between a couple of particles, a tensile ring interaction involving a closed set of particles and a exural beam interaction between three particles (see Fig-

Bar interaction

The bar interaction simply refers to the spring interaction between a couple of particles. The associated elastic strain energy is thus given, for small strains, by the following expression:

ψ ba ( ba ) = 1 2 E ba S ba l ba d 2 ba ( 4
)
where

E ba S ba l ba
is homogeneous to a stiness in N.m -1 and l ba and d ba are the rest length (the distance between the two particles in the stress free conguration) and the relative displacement of the interacting particles. This relative displacement is dened by d i = u i+1 -u i . Introducing the particle displacements induced by the global strain, the relative displacement of the couple c i can be rewritten as:

d i = E • l i + ∆u per i with l i = X i+1 -X i (5)

Ring interaction

The second interaction is of ring type. It refers to an ordered set of N particles. Ordered means that each particle has always the two same neighbours. In other words, each particle belongs to two consecutive couples of particles and the ring interaction acts on the ordered series of these N couples (for instance in Figure 4 the particle i belongs to the couples

[i -1, i] and [i, i + 1]
). The form of the elastic strain energy related to the ring interaction is the same as the one used in the case of the bar interaction:

ψ r ( r ) = 1 2 E r S r l r d 2 r ( 6
)
The ring elongation is the sum of the elongations of each couple c i :

d r = N k=1 d i (7)
As previously, the expression of the overall rest length is also derived from the rest length of each couple by:

l r = N k=1 l i (8)

Bending interaction

Experimental results on ring tensile tests have shown a stage of the behaviour which is due to the ring bending (see Figure 7). To model it with only the assumptions of the cable theory, many authors have chosen to add material non-linearities to reproduce the stiness changes [START_REF] Nicot | Design of rockfall restraining nets from a discrete element modelling[END_REF][START_REF] Volkwein | Protection from Landslides and High Speed Rockfall Events: Reconstruction of Chapman's Peak Drive[END_REF][START_REF] Coulibaly | Nonlinear Discrete Mechanical Model of Steel Rings[END_REF][START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF]. However, there exist simple models to reproduce bending forces in axi-symmetric sections [START_REF] Adriaenssens | Tensegrity spline beam and grid shell structures[END_REF][START_REF] Douthe | Design of nexorades or recip-rocal frame systems with the dynamic relaxation method[END_REF][START_REF] Douthe | Form-nding of a grid shell in composite materials[END_REF][START_REF] Peloux | Modeling of bending-torsion couplings in active-bending structures[END_REF]. These models have the advantage of not requiring rotational degrees of freedom to compute the bending moment. An elastic energy depending only on the nodal translation can thus be associated with this bending interaction. In standard strength of material, the bending moment is computed by:

M i = EI r i (9)
where r i is the local radius of curvature associated with the particle i and calculated from the osculating circle passing Using the same notations as in section 2.3.1, the radius of curvature is dened by:

r i = l i-1,i+1 + d i-1,i+1 2 sin θ i ( 10
)
where θ i is the angle formed by the vectors

(l i-1 + d i-1 )
and

(l i + d i ).
In the framework of the proposed homogenization method, energies are quadratic functions of the strain eld. To this end, the sine function must be expressed as a function of the particle couple :

sin θ i • e z = l i-1 + d i-1 ∧ (l i + d i ) l i-1 + d i-1 l i + d i (11)
In the case of the ring, the initial curvature is not equal to zero. The calculation of the bending moment must be thus slightly modied to account for the curvature variation:

M i 2EI = l i-1 + d i-1 ∧ (l i + d i ) l i-1 + d i-1 l i + d i l i-1,i+1 + d i-1,i+1 - l i-1 ∧ l i l i-1 l i l i-1,i+1 (12)
Expression ( 12) is used in this form for the implementation in the custom code cited in Section 1.3.3. To simplify notations, the variables referring to the couple of particles (i -1, i + 1) will be indexed in the rest of the paper with the subscript i and the superscript c (for chord). To remain within the framework of the homogenisation method [START_REF] Florence | A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams[END_REF], the particle displacements must remain small and the expression of the bending moment (12) can hence be linearised according to the relative displacements of each couple of particles:

d i-1 , d i and d i-1,i+1 = d c i . M i 2EI l i-1 l i l c i = l i-1 ∧ d i + d i-1 ∧ l i + l i ∧ l i-1 l c i • d c i l c i 2 + l i-1 • d i-1 l i-1 2 + l i • d i l i 2 (13)
The elastic bending energy associated with the particle i is then given by the formula:

ψ i be = 1 2 M i 2 EI l i-1 + l i 2 (14)
Even if in this section, the elementary cells are considered planar, it can be remarked that the expressions of the bending moment and of the elastic bending energy presented above ((12) ( 14)) remain valid for the study of 3D problems [START_REF] Douthe | Design of nexorades or recip-rocal frame systems with the dynamic relaxation method[END_REF].

These three interactions enable the calculation of the strain energy of each elementary cell introduced in Section 2.1. It can be reminded that in Figure 2, the cell perimeters are limited by the square drawn in blue. Only Model C has particles whose displacements do not only depend on the overall strain eld E. Therefore, the identication of the homogeneous elasticity tensor requires the resolution of a minimisation problem only in this last case.

Identication of the stiness matrices

For this part of the study, it is assumed that the three models consist in innitely periodic material under uniform strain undergoing small deformation.

Model A

In this model, each ring is represented by one particle in its center and four links of bar-type interaction connecting it with its four neighbours (Model A in Figure 2).

The identication of the homogeneous elasticity tensor is straight forward and lead to:

A A =   K A ba 0 0 K A ba   (15)
with:

K A ba = E A ba S A ba 2R (16)
where superscript A refers to model A, R is the radius of the ring, which means that 2R is here the length between two neighbour particles and E A ba S A ba is the stiness of the bar interaction in N . The equivalent homogeneous net has no shear stiness and a null Poisson's ratio.

Model B

In this model, each ring is formed by four particles located at the contact points with the adjacent rings (see Figure 1). The loads are transmitted between the particles through only one ring interaction. Using the relation ( 6) and ( 2), the derivation of the stiness matrix is straightforward again and leads:

A B = √ 2 2   K B r K B r K B r K B r   (17)
with:

K B r = E B r S B r 2R (18)
where superscript B refers to model B, R is the ring radius and K B r is the stiness of the ring perimeter. In Model B the shear stiness is still zero, but the Poisson ratio is now equal to 1 which introduces a signicant dierence with Model A in which stresses in orthogonal directions were uncoupled.

Model C

Unlike the two previous elementary cells, Model C has inner nodes which do not belong to the boundary of the cell and whose displacements are not determined by periodicity conditions as in Model A (see Figure 3). The elastic energy expression thus depends on the periodic displacements of the inner nodes. The identication of the elasticity tensor is, this time, a little more complex and requires solving an auxiliary minimisation problem. The detail of the calculations is proposed in Boulaud's Phd thesis [START_REF] Boulaud | Etudes et modélisations du comportement d'un écran de let pare-blocs à diérentes échelles[END_REF].

It is proved that the elasticity tensor of Model C can be numerically expressed as:

A C = 2.3K C be      1 0.8 0 0.8 1 0 0 0 0.4      (19)
where superscript C refers to model C and

K C be ∼ E C I C R 3 (20)
K C is homogeneous to a exural stiness with I C the inertia of the ring. The comparison of the intrinsic elastic properties of these three models indicates thus, that, for small strains, the way they transfer loads is dierent. The micro-structure of Model C being closer to the real ring structure, it is expected that its behaviour is closer to the reality. Nonetheless, it will be shown in the next section that, when trying to identify the characteristic stinesses of the three models, Model C is also the simplest.

3. Identication of the intrinsic equivalent behaviour through a large deformation tensile test

Experimental set-up

The experiment presented below, carried out in the framework of the french national project "C2ROP" [START_REF] Olmedo | Etude experimentale multiechelle sur les ecrans pare-blocs: comportement des ouvrages hors cadre etag27[END_REF][START_REF] Olmedo | étude expérimentale multi-échelle sur les écrans pareblocsomportement des ouvrages hors cadre etag27[END_REF], consists of a square and rigid support frame on which a three by three ASM4 ring net is xed (see Figure 6). Force sensors are placed on the moving edge and on one of the lateral edges so that the whole resultant forces applied by the net on the frame are recorded. Considering then that the deformation in the net is uniform, the force applied on a single ring is assumed to be equal to one third of the total force applied on the edge.

Furthermore, the order of magnitude of displacements applied in this test was chosen in order to investigate also a deformation range beyond that of previous section. Equivalent material characteristics identied by this test will thus cover the domain of small perturbations (for strains smaller than 4% of the specimen size) and also allow for the identication of a second linear domain (for strains larger than 25%) where the rings have become diamonds and work in pure tension. The choice was made to identify only two parameters for each model to ease comparisons: one for small strains (and bending energy in the ring), one for large strains(and strain energy in the ring). Other choices with more parameters could also be made, like in [START_REF] Coulibaly | Nonlinear Discrete Mechanical Model of Steel Rings[END_REF]).

Model A

In Model A, the non-linear behaviour of the ring is modelled by means of material non-linearity. It is assumed that the tensile behaviour of the ring can be described by means of a bi-linear law in traction with a null stiness in compression (similarly to what is made by F. Nicot et al [START_REF] Nicot | Design of rockfall restraining nets from a discrete element modelling[END_REF] or by L. Dugelas et al [START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF]. The left part of Figure 7 shows the pattern of this model drawn in red as well as

U a + 6R 0 6R 0 2 -1 (21)
The two stinesses of the bi-linear behaviour which are noted K A ba,1 and K A ba,2 respectively, are tted from the experimental results by means of least squares tting on the two asymptotic behaviours. The low stiness is tted for axial strains lower than 4% whereas the second stiness is tted for strains higher than 25%. The parameter values are given below. A lim is the strain for which the change of stiness occurs.

         K A ba = 0 kN.m -1 if ≤ 0 K A ba = 31 kN.m -1 if ≤ A lim = 23% K A ba = 2160 kN.m -1 if > A lim = 23% (22)

Model B

In Model B, the non-linear behaviour is modelled thanks to one ring with a bi-linear behaviour in traction, in the same way as for the Model A, with a low stiness for the The tting of the ring's characteristics requires to calculate the strain of the diamond ring from the displacements of the moving edge U a and its tensile stress from the axial force. The Green-Lagrange strain of the diamond ring, B r , is given by the following expression:

B r = 1 2 4R 2 0 + 2R 0 + Ua 3 2 8R 2 0 -1 (23)
Referring to the left scheme in Figure 8 the tensile force in the diamond ring is calculated by:

N B r = 1 2
F axial 3 4R 2 0 + 2R 0 + Ua 3 2 2R 0 + Ua 3 ( 24
)
The stinesses are still calculated by tting the real behaviour for the same limit of axial strains in model A (lower than 4% and higher than 25% of the specimen length), which correspond to strains of the ring lower than 2% and higher than 12.5%. The parameter of Model B are hence: Model C behaviour being richer than the previous ones, the characterization of its mechanical property is directly performed from the expression of the elasticity tensor [START_REF] Florence | A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams[END_REF] and refers to the notations used in [START_REF] Boulaud | Etudes et modélisations du comportement d'un écran de let pare-blocs à diérentes échelles[END_REF]. For small axial displacements applied during the experiment, the following equation is valid:

         K B r = 0 kN.m -1 if ≤ 0 K B r = 43 kN.m -1 if ≤ B lim = 11.5% K B r = 2750 kN.m -1 if > B lim = 11.5% (25)
F axial 2R 0 = A 1111 a = 2.3K C be a ( 26
)
where a is still the axial strain. Moreover, in the case of Model A (see section 3.2), it has been already shown that: F axial 2R0 a = 31 kN.m -1 , which enables to straightforward deduce the value of K C be . Then, for large axial strain, the behaviour of Model C becomes similar to the one of Model B, because the stiness K C r is large compared to K C be (see [START_REF] Boulaud | Etudes et modélisations du comportement d'un écran de let pare-blocs à diérentes échelles[END_REF]), the octagonal ring deform into a diamond shape. To identify the value of the ring stiness, the same approach as in Section 3.3 can thus used. However, the Green-Lagrange strain needs to be modied by taking into account the actual perimeter of the octagonal ring:

C r = 1 2 4R 2 0 + 2R 0 + Ua 3 2 4π 2 R 2 0 -1 (27)
The tting of this last parameter is presented in Figure 9 as well as the pattern of the Model C. Finally, the parameters of the Model C are:

K C r = 1140 kN.m -1 and K C be = 13 kN.m -1 (28)
Unlike the previous models, this one does not require the identication of a third parameter linked to the material non-linearity. The transition from one behaviour to the other is smoothly done through the geometric deformation of the ring.

For the sake of completeness, it must be added here As mentioned earlier, more complex behaviours could be implemented, using for instance polynomial laws as in [START_REF] Coulibaly | Nonlinear Discrete Mechanical Model of Steel Rings[END_REF][START_REF] Boulaud | Modelling of curtain eect in rockfall barrier with the dynamic relaxation[END_REF], but that would require the identication of more parameters with dicult physical interpretation.

Conclusion of the experimental identication process

The plane behaviours of the three net models are now completely characterized and give each one a good approximation of the real net behaviour. The experiment allowed for an identication which goes beyond small perturbations and covers the whole range of uni-axial deformations of the rings in real structures. The mechanical properties, identied in this section for the three models, will be those used in the rest of this paper. At this stage of the comparison, Model C seems to be more relevant to describe accurately the behaviour of a ring net. In a rockfall bar- the displacements in the three directions, the transformation called hypar transformation and presented in Figure 11 is characterised by:

u 3 (x, y) = x 1 x 2 c ( 29
)
Considering that H is the vertical displacement of each corner (see Figure 11), the parameter c is given by: c = In the following, the equilibrium state of the membrane under the Von Karman assumptions will be rst established, then a convergence study with various mesh den-sity of the three models will be conducted before a nal comparative assessment is presented.

Equilibrium state of the membrane undergoing an hypar transformation

According to the assumptions of the Von Karman plate theory: large out-of-plane displacements but small plane strains [START_REF] Von Kármán | Festigkeitsproblem im Maschinenbau[END_REF], the components of the Green-Lagrange strain ( 11 , 12 and 22 ) of the equivalent membrane can be deduced from the derivative of displacements in the tangent plane (u 1 and u 2 ) and the square of the displacement in the normal direction (u 3 ) by:

11 = u 1,1 + 1 2 u 2 3,1 22 = u 2,2 + 1 2 u 2 3,2 12 = 1 2 u 1,2 + u 2,1 + u 2 3,1 u 2 3,2 (30)
Assuming then a linear elastic behaviour, the constitutive law of the membrane takes the following form:

     N 11 N 22 N 12      =      A 1111 A 1122 0 A 1122 A 2222 0 0 0 A 1212           11 22 2 12      (31)
where N 11, , N 22 and N 12 represent the components of membrane forces in the shell and A ijkl the components of the stiness matrix. For the three equivalent homogeneous membranes, the equivalent stiness matrix has been identied in [START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF], ( 17) and [START_REF] Florence | A rigorous homogenization method for the determination of the overall ultimate strength of periodic discrete media and an application to general hexagonal lattices of beams[END_REF]. They are all orthotropic with the same stiness in the two principal directions so that A 1111 = A 2222 . So, introducing the Poisson ratio (ν = A1122 A1111 ), two cases have to be distinguished: Membrane A and B where the shear stiness is zero and Membrane C where the shear stiness is non zero.

Without external forces and assuming that the bending stiness is negligible in comparison to the in-plane stiness, the solution of von Karman's problem [START_REF] Von Kármán | Festigkeitsproblem im Maschinenbau[END_REF] can be sought in such a way that u 3 (x 1 , x 2 ) = x1x2 c everywhere and reduces hence to:

         u 1,11 + A1122 A1111 u 2,12 + x1 c 2 + A1212 A1111 u 1,22 + u 2,12 + x1 c 2 = 0 u 2,22 + A1122 A2222 u 1,12 + x2 c 2 + A1212 A2222 u 1,12 + u 2,11 + x2 c 2 = 0 A 1212 u 1,2 + u 2,1 + x1x2 c 2 = 0 (32)
First case: A 1212 = 0. In this case, the in-plane displacements u 1 and u 2 can be expressed each one as a single variable function. The boundary conditions impose that the in-plane displacements are null in the four corners. Moreover the symmetry conditions impose that

u 1 (0, x 2 ) = u 2 (x 1 , 0) = 0.
Finally it is established that:

   u 1 (x 1 ) = ν 6c 2 x 1 l 2 4 -x 2 1 u 2 (x 2 ) = ν 6c 2 x 2 l 2 4 -x 2 2 ( 33
)
Without shear stiness, it is possible to nd a solution of the problem (32) which satises the condition:

u 3 (x 1 , x 2 ) = x1x2 c
by imposing only a vertical motion to the boundary ∂Ω. It can be also noted that in the trivial case for which the Poisson ratio ν is equal to zero, the equilibrium equations are satised without in-plane displacements.

Second case: A 1212 = 0. This time, the third equation of [START_REF] Sasiharan | Numerical analysis of the performance of wire mesh and cable net rockfall protection systems[END_REF] imposes that the shear strain is null. The displacement eld solution is then given by:

   u 1 (x 1 , x 2 ) = x1 4c 2 l 2 4 -x 2 2 + ν 3 l 2 4 -x 2 1 u 2 (x 1 , x 2 ) = x2 4c 2 l 2 4 -x 2 1 + ν 3 l 2 4 -x 2 2 ( 34
)
Under this condition, it is hence observed that the in-plane displacements satisfying the conditions:

u 3 (x 1 , x 2 ) = x1x2 c
are maximum in the middle of each edge.

Behaviour of the net models for an hypar loading

In this section, the three equivalent membranes are loaded in displacement by imposing the conditions provided by [START_REF] Veenendaal | An overview and comparison of structural form nding methods for general networks[END_REF] to their boundary. This specic shear-free state of strain allows to compare the behaviour of the three models.

Behaviours of the homogeneous equivalent materials

Without shear strain and for small displacements, the dierences between the behaviour of the three homogeneous materials are only due to the value of the Poisson ratio (see Section 2.4). From the expression (34) of the displacement eld, the strains are calculated using [START_REF] Olmedo | étude expérimentale multi-échelle sur les écrans pareblocsomportement des ouvrages hors cadre etag27[END_REF] and then the stress eld, using [START_REF] Peloux | Modeling of bending-torsion couplings in active-bending structures[END_REF]:

N =   N (x 2 ) 0 0 N (x 1 )   ( 35
)
with

N (x) = A 1111 4c 2 l 2 (3 + ν)(1 + ν) 12 + (1 -ν) 2 x 2 (36)
It can be remarked that, for a Poisson ratio equal to 1, which is the case for Model B, the stress state is uniform and isotropic in the equivalent material. The geometry of the membrane is therefore a minimal surface. Otherwise the stress evolves quadratically.

To go further and easily compare analytic and numerical results, the resultant force applied along an edge of the membrane provides an interesting global measure of forces in the membrane, it is given by: In this convergence study, a 1 m square net is deformed in an hypar form with a ratio H l equal to 1 4 . The choice of the ratio value is deliberately made here beyond the hypothesis of small displacements. The results are presented in Figure 12. The relative gap between the values obtained for the highest density and the current density is plotted as a function of the mesh count (the number of rings in the net. It can be noticed that for the three models, errors of less than 1 % are achieved for a net of 9x9 rings, which is comparable to the real mesh density in a typical rockfall barrier unit (for example the net shown in Figure 14 Now that the mesh density to be used for numerical simulations has been investigated, the behaviour of the models can be compared with the one of their equivalent homogeneous membrane on a 9 × 9 net, which ensures hence less than 1% error. The computations are carried out by varying the ratio H l and comparing the resultant applied to an edge [START_REF] Volkwein | Rockfall characterisation and structural protection -A review[END_REF]. The results are presented in Fig- For model C, the numerical model predicts an additional stiening which is not included in the closed form solution. This stiening corresponds to geometrical changes at local scale: the ring geometry varies signicantly with the ratio H l from octagonal shape toward diamond shape. The equivalent homogeneous elastic properties of the membrane increase thus with the deformation and cannot be considered constant. Interestingly, this should also be the case for model B where the ring shape also changes from square to diamond. In the deformation range studied here, the order of magnitude of local stiness increases induced by these changes in model B are thus one order of magnitude lower than the one induced by changes in model C.

F = l 2 -l 2 N (x)dx = 1 + ν 3 A 1111 4H 2 l ( 37

Second case-study: complex rockfall barrier under large displacements

The simple academic case study proposed in Section around the impacted zone. This phenomenon, whose many modellings can be found in the literature [START_REF] Coulibaly | Sliding cable modeling: An attempt at a unied formulation[END_REF][START_REF] Volkwein | Numerische Simulation von exiblen Steinschlagschutzsystemen[END_REF][START_REF] Ghoussoub | Analyse de quelques éléments du comportement des écrans de lets pare-blocs[END_REF][START_REF] Boulaud | Modelling of curtain eect in rockfall barrier with the dynamic relaxation[END_REF], is usually called the "curtain eect".

The characteristic behaviour of a brake is elasto-plastic and since a support cable is linked by one brake at each end, the set formed by the three elements has also a bilinear elasto-plastic behaviour. For the modelling, the three elements (one cable with two brakes) are merged into one single element which is called "sliding cable" in the rest of this section. In the discrete modelling, a sliding cable represents a set of n ordered nodes which are linked to their neighbours by forces of same intensity [START_REF] Boulaud | Modelling of curtain eect in rockfall barrier with the dynamic relaxation[END_REF]. The uniform tensile tension in the sliding cable is the consequence of the hypothesis that the contact points slide along it without friction. The idealised mechanical behaviour of the sliding cables is shown in Figure 16.

For the four upstream and downstream sliding cables, the values of the parameters introduced in Figure 16 are: Concerning the global behaviour, it is observed that, due to vertical deformation of the net and its sliding along its boundary, the loads tend to concentrate around the four points where the edge cables leave the net (see Figure 15). For a block displacement of 3m (Figure 19, graph (c)), the sliding of the net is almost achieved and the node distribution along the cable no longer changes. This load distribution is the same for the three models, and seems to depend only on the architecture of the fence. However some dierences in the intensity of forces be-

Non-centred impacts

This last paragraph is devoted to the exploration of the states of equilibrium of the fence when the loading position changes in the plane of the net. For modelling simplicity, this time, perfect contacts between the block and the net are considered in this last section. Although very interesting from the designer point of view, direct interaction of the block with the posts is also neglected, the phenomena at stake would require a dedicated study which is too far from the scope of the present study. Indeed, the goal here is not to reproduce extreme loading cases in a realistic manner, but to study the inuence of the net model on the behaviour of the structure. The loading is hence applied by imposing the position of the center of the block, which is here set to 3m in order to activate signicantly the brakes (as observed in previous section). Now that the inuence of the position of the block on the total energy of the structure has been identied, the part of this energy which is due to the strain of the edge cables can be studied for each of the three models. The ratio of the strain energy stored in the cables and the total strain energy is hence shown in Figure 22. For the three models, at least 70% of the energy is stored in the edge cables (except in the central module of model C). The share of energy only due to the deformation of the edge cables seems thus to follow the evolution of the total energy: the structure is softer in the area where sliding is allowed (in the centre of the module) and solicit less the edge cables. To be more specic, the strain energy of edge cables splits into elastic and plastic energy dissipated by the brakes. Considering the high elastic stiness of the cables and the low threshold the brakes, it can be assumed that the strain energy in the edge cables approximates the energy dissipated by the brakes. Therefore, the higher the local stiness of the net, the higher the share of the total energy dissipated by the brakes on the edge cables. Following this interpretation, model A is stier than model B which is also stier than model C (which apparently is in contradiction with observations made in section 4).

Conclusion

In this paper, the inuence of the discrete ring model in view of modelling the behaviour of ASM4 barriers was assessed. Three dierent modelling strategies have been compared. These strategies were adapted from existing models in the literature in order to built three typical characteristic modelling choice and to compare their consequence.

In the rst stage, the intrinsic mechanical properties of the net models have been identied thanks to a robust homogenisation method, which allows to establish the expression of an equivalent elastic membrane tensors. This This conrms hence one conclusion of Dugelas et al [START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF] who demonstrate that DEM models with dierent complexity level may result in similar abilities in predicting the barrier impact response. On the contrary, when looking at global deformations and residual height, the cable model (model A) was found much stier than the two others. This has been interpreted as a consequence of the nullity of the Poisson ratio which does not allow for a diusion of the forces in both directions of the net and hence limits the rearrangement of the net around the loaded area. This comparative assessment can further be used as guidelines for designers in order to chose their models, to interpret their results and to separate which aspect is intrinsecaly linked with the numerical model and which one is linked with the real structure behaviour.

Figure 1 :

 1 Figure 1: Typical arrangement of rings in ASM4 ring nets where each ring is in contact with four neighbours.

Figure 2

 2 Figure 2 shows the geometry and the interactions of each discrete ring model. The blue squares represent the elementary cells boundary which form the whole net by periodic repetition. For simplicity reasons, the models will be called A, B and C respectively in the rest of the paper.

Figure 2 :

 2 Figure 2: Patterns of the three elementary cells : A. Truss model, B. Tensile ring, C. Bent ring

Figure 3 :

 3 Figure 3: Periodic repetition of Model C elementary cell

ure 4 )

 4 . The rst type is used in the three models, the second reects coupling between strain in orthogonal directions and is used in Model B and C. The third one is only used in Model C. It is important to remember that, in this section, all interactions are elastic, and that all vertices motions are assumed small compared to the characteristic size of the elementary cell.

Figure 4 :

 4 Figure 4: Illustration of the three types of interaction: spring, ring and beam interactions. Note that for the ring interaction, the main characteristic is the loop. The number of nodes in the loop might vary: it is of 8 in model C and 4 in model B.

Figure 5 :

 5 Figure 5: Calculation of the radius of curvature and of the bending moment

2. 5 .

 5 Comparison of the elasticity tensors of the three models The three models have been presented by order of complexity. The Model A composed of two bar interactions has a null Poisson ratio and no shear stiness. A ring interaction is added in Model B and its Poisson ratio is now equal to 1 but the cell has still no shear stiness. This Poisson ratio equal to 1 corresponds to a kind of plane strain incompressibility condition which reects an important physical property of ASM4 rings. Furthermore the introduction of a bending stiness in Model C has slightly lowered the Poisson ratio to 0.79 and conferred shear stiness to the net.

  The rings are considered to be perfectly circular in their relaxed geometry with a radius R 0 = 13.5 cm. They are made of 7.5 mm strand of galvanised steel. A uni-axial tensile loading is applied to the ring net by pulling one edge of the support frame. The normal displacement of this edge is imposed while on the other edges tangential displacements are free and normal displacements are blocked.

Figure 6 :

 6 Figure 6: Experimental set-up of the plane tensile test on a ASM4 ring net (by courtesy of national project C2ROP).

Figure 7 :

 7 Figure 7: Pattern of Model A and tting of its bi-linear behaviour

Figure 8 :

 8 Figure 8: Pattern of Model B and tting of its behaviour

  that an additional geometric constraint has been introduced in model C insuring that the nodes, which do not belong to the cell boundary, move along the bisector of the chord formed by their two neighbour nodes. For numerical computation this geometric constraint is changed to a mechanical constraint by adding eight springs along the edges of the octagonal ring. In order to minimize the inuence of these springs on the behaviour of Model C, their stiness must be very small compared to the stiness of the ring K C r . Computation will hence be performed with a value of 10 -2 K C r .

Figure 9 :

 9 Figure 9: Pattern of Model C and tting of its behaviour

Figure 10 :

 10 Figure 10: Comparison of the behaviour of the three models (continuous line: experimental results with 3x3 rings specimen; dots and stars: numerical results with real density of rings (3x3) and 100 times density (30x30) respectively).

Figure 11 :

 11 Figure 11: Transformation of the at membrane in a hyperbolic paraboloid

) 4 . 2 . 2 .

 422 Numerical behaviour of the net modelsThe common modelling choice in exible rockfall barrier is to use one numerical element for one physical element. However, considering the numerical model as a pure discretisation problem of a continuous mechanical problem, it is common knowledge that the size of elements relatively to the characteristic size of the structure has an inuence on the quality of the approximated solution given by the numerical model. One might therefore ask if, in the present study, the size of the ring elements is suciently small to reproduce the behaviour of the net, especially in this case-study where a closed form solution exists.To investigate this question a convergence study is thus conducted by varying the size of the ring elements and keeping the size of the hypar structure constant, as well as the net stiness per unit length. To this end, the individual stiness of the ring element have to be adjusted for each calculation. Indeed, the size of the ring element appears in all the expressions of the stiness identied in section 3 (see16, 18 and 20). To keep the stiness per unit length constant, it was thus decided to modify the Young Modulus of the ring inversely to R the radius of the ring element for model A and B and to R 3 for model C.

  has

17 × 23

 23 rings between two successive posts). With a mesh density of 15 × 15 rings, the errors falls down to 1.5for the models A and B and 2.5for Model C.

Figure 12 :

 12 Figure 12: Convergence study in terms of mesh density

ure 13 ,

 13 where closed-form solutions of each model is referred as "analytic" curve and discrete element model as "numerical" curve. It is observed that for the models A and B, the analytic behaviour of the equivalent membrane is very close to the one of the discrete net. The stiening induced by the geometric changes of the underlying surface (the more Gaussian curvature in the surface, the stier the structure) is clearly captured by the models. The inuence of the Poisson ratio, predicted by the theoretical expression of the resultant force (37), is also veried for the discrete nets (F A ≈ 2F B ).

Figure 13 :

 13 Figure 13: Comparison of the three net models

4

  has shown that, at structural scale, the response of the three models have signicant dierences, even in the global response to simple loading. In this second case-study, a realistic application is chosen with the aim of investigating which features of the behaviour of a rockfall barrier the three models can capture. Large deections will still be evaluated with the dynamic relaxation method. Numerical results obtained with the three models are rst compared to experimental results on a centred impact and then between them on non-centred impacts.5.1. Experimental set-up5.1.1. Barrier architectureA full-scale quasi-static experiment was carried out in the framework of the french national project C2ROP (previously mentioned in Section 3.1) on a rockfall barrier composed of three units, each one 5 meters wide and 2.75 meters high[START_REF] Olmedo | Extended experimental studies on rockfall exible fences[END_REF][START_REF] Olmedo | étude expérimentale multi-échelle sur les écrans pareblocsomportement des ouvrages hors cadre etag27[END_REF][START_REF] Dugelas | Assessment of the predictive capabilities of discrete element models for exible rockfall barriers[END_REF]. A picture of the whole structure which can be seen as a reduced scale of a high energy barrier is shown in Figure14.

Figure 14 :

 14 Figure 14: Front view of the tested rockfall barrier prototype (by courtesy of national project C2ROP)

Figure 15 : 5 . 2 .

 1552 Figure 15: Detailed wiring between the net and the support cables

k 1 =

 1 1140 kN, k 2 = 35 kN, and T lim = 25 kN, while for the two lateral sliding cables they are: k 1 = 1190 kN, k 2 = 35 kN, and T lim = 25 kN. Those values are calibrated on separate experiments on brakes which were conducted in the framework of the national project C2ROP [29, 30]. Compared to the plane tensile test presented in Section 3.1, the net installed in the barrier is turned with an angle of π 4 (indeed one can see in Figure6 that the axes of the

Figure 16 :

 16 Figure 16: Numerical behaviour of the sliding cable elements

Figure 17 :

 17 Figure 17: Triangular rings along the edge cable in Model B

Figure 18 :

 18 Figure 18: Comparison between the experiment and the three numerical force-displacement curves.

5. 3 . 2 .

 32 Resultant forces applied to the net boundary After comparing the overall behaviour of the structure, the distribution of the forces applied along the boundary of the net is investigated. The forces applied by the net to the two lateral cables are not considered here because they are relatively low compared to that applied to the upper and lower cables. Furthermore, the force distributions along the upper and lower edges of the net are close due to the strong symmetry properties of the structure and the small displacements of the post heads. Only the one applied to the upper sliding cables is thus plotted in Figure 19. Five levels of the block vertical displacements are considered: from 1m to 5m. The forces applied to the net boundary are plotted according to their application points along the edge.

  tween the three models can be noticed. During the rst stage where the sliding occurs (graphs (a) and (b) of Figure 19), Model C concentrates forces in the central zone where deformations of rings are high and therefore where the net has become sti. Model A presents low force everywhere, the lack of Poisson eect directs forces only to the supports points. Model B is uniformly sti and therefore pulls everywhere on the edges. When the sliding is completely achieved (graphs (c), (d) and (e)), the behaviours of the three models become similar. The forces increases at the support points and more quickly at the two lateral points, located in x = -4 m and x = 4 m. However, the rates of force increase are dierent between the models and cor-

Figure 19 :

 19 Figure 19: Force distribution on the upper edge of the net

Figure 20 : 5 Y ∈ -h 4 ; h 4 .

 2054 Figure 20: Position of the block used for the spatial characterisation of net behaviours.

Figure 21 :

 21 Figure 21: Evolution of the strain energy as a function of the block position for the three models.

Figure 22 :

 22 Figure 22: Evolution of the share of the strain energy stored in the edge cables as a function of the block position for the three models.

Figure 23 :

 23 Figure 23: Evolution of the share of the strain energy stored in the cable B ing as a function of the block position for the three models.

Figure 24 :

 24 Figure 24: Evolution of the residual height as a function of the block position for the three models.

  preliminary result gives an objective criterion to compare the behaviour of the discrete models and thus identify their intrinsic limitations. For instance, it has been deduced that the equivalent membrane of the spring model (model A) has a null Poisson ratio whereas that of the exural ring model (model C) has a shear stiness contrary to the others. Then, using the form of the elasticity tensors, the mechanical property of the net models have been calibrated in such way that their axial stiness was the same. The diculty to reproduce the characteristic non-linear behaviour of a ring was hence observed, as well as the dierent strategy to model it. A ctitious material non-linearity must be introduced for model A and B, while consistent geometric non linearities are naturally present in model C, so that calibration is only based on initial axial and bending stiness. The third stage of the comparison was dedicated to the study of a particular transformation, involving large space displacements and for which a closed form solution exists. This transformation has on one hand highlighted the advantage of the geometric non-linearity in model C, and on the other hand, allowed to validate the performance and the accuracy of the numerical tools. Finally, the behaviours of the three models were studied on a whole rockfall barrier. All of them present remarkable agreement with the experimental results at global scale, despite the intrinsic dierence in their local behaviour.

Acknowledgements

The authors wish to thank all the members of the national project and particularly the members of the working group on exible barriers for their invaluable support, namely M-A. Chanut and J. Coulibaly from CEREMA, P. Robit, I. Olmedo from GTS, C. Galandrin and M. Verdet from CAN, L. Dugelas form IRSTEA and D. Bertrand from INSA Lyon.