T T Bui

M Bost

A Limam

J P Rajot

P Robit

Modular precast concrete facing for soil nailed retaining walls: Laboratory study and in situ validation

Keywords: retaining wall, soil nailing, precast concrete, experiment, numerical simulation

A new concept for constructing the facing of soil nailed walls is proposed and validated through experimental and numerical approaches. This new process uses precast concrete panels. For each excavation step, the soil reinforcements are first connected to the panels. A slightly cemented 4-6 mm crushed stone is then injected, through the weep holes, between the panels and the excavation vertical cut, applying a confining pressure to the ground and providing continuous high capacity drainage behind the facing. The structural design of the precast reinforced concrete panel, based on full scale loading tests in the laboratory and nonlinear numerical simulations, are presented in detail. Crack initiation and failure modes are properly predicted by the model. A full-scale experimental soil nailed wall 7.5 m high was also built to evaluate the environmental, economic and mechanical performances of the new construction technique compared to the conventional shotcrete technique. The new construction technique greatly improves worker's safety, significantly reduces construction duration and cost and improves the mechanical behavior of the soil nailing technique and drainage of the facing. Compared to the conventional shotcrete technique, the new technique reduces concrete consumption by 64% and greenhouse gases emission by 56%.

Introduction

Soil nailing is frequently used to stabilize natural slopes, temporary excavations, long term road widening, tunnel heads, underground parking facilities, etc. It may be viewed as a soil retention technique using grouted tension-resisting steel elements, attached to a facing, to support an excavation or stabilize steep cuts under static and seismic conditions [START_REF] Chavan | Seismic analysis of nail soil slope considering interface effects[END_REF]. In fact, the interaction between the soil and structural reinforcing elements installed in the ground is a complex topic in soil mechanic. This interaction involves combined axial and transverse efforts in the linear structural elements, whose values depend on local mobilization of soil strength and on global deformation behaviour of the entire reinforced soil mass. Case histories and postearthquake surveys show that soil nailed walls have performed extremely well during strong ground motions compared to most gravity retaining structures [START_REF] Colin | Field observations of reinforced soil structures under seismic loading[END_REF] [START_REF] Tatsuoka | Geotextile-reinforced soil retaining wall and their seismic bahaviour[END_REF].

Subsequent information for the design and construction of soil nailed walls may be found in [START_REF] Juran | Nailed-soil retaining structures: Design and practice[END_REF] and [START_REF] Bruce | Soil nailing: application and practice-part 1[END_REF], by [START_REF] Juran | Nailed-soil retaining structures: Design and practice[END_REF] and [START_REF] Bruce | Soil nailing: application and practice-part 1[END_REF] respectively. These recommendations were based on the study of measurements made on many full-scale trials and model tests during construction of real projects. In France, the design method recommended by CLOUTERRE for soil nailed walls relies on an ultimate limit state analysis with partial safety factors considering four potential failure modes [START_REF] Schlosser | Validation of calculation methods of soil nailed walls based on the experimentations of the National Project CLOUTERRE[END_REF]. Both of these design methods have been checked against experimental results in full scale structures at failure from the CLOUTERRE National Project.

After each excavation step and installation of sub-horizontal reinforcements in the natural soil, a drainage complex and a structural facing must be placed to mobilize a confining pressure in the ground before the next excavation step and transition to the long-term behaviour of the soil occurs. The conventional and widely used technique for constructing this facing consists of several passes of shotcrete on two reinforcing steel meshes. Examples of facing construction using shotcrete in recent projects are shown in [START_REF] Chavan | Seismic analysis of nail soil slope considering interface effects[END_REF] and [START_REF] Menkiti | Performance of soil nails in Dublin glacial till[END_REF]. During this construction process a significant amount of concrete is lost, bouncing off the vertical reinforcing steel meshes during projection. This method of construction is thus expensive and represents a waste for the natural resources and the environment.

An alternative process to shotcrete facing construction for soil nailed walls was therefore proposed. This process is based on using precast concrete panels, which are easier to install, cheaper and made of concrete with controlled mechanical properties. Various aesthetic surface texture may be added to the panels, whereas shotcrete is uniform, grey and rough. To gauge all aspects of this new concept, a real scale soil nailed wall had to be built and instrumented. In fact, for the purpose of objective comparison the conventional technique (shotcrete), two full scale instrumented walls were constructed adjacent to each other, both 7.50 m high, 10 m long, with identical soil reinforcements. As shown in an elevation view in Figure 1, however, the facing of first one (on the left in the figure) was constructed with the conventional shotcrete technique while the second was built with precast concrete panels. The first part of this study concerns the load capacity and the associated failure mode of a single precast reinforced concrete panel, in order to assess its structural design. An experimental study was performed in the laboratory [START_REF] Bui | Paroi Clouée Préfabriquée: Evaluation de la capacité portante d'une écaille constitutive: approche expérimentale et numérique[END_REF], subjecting a panel to unfavourable mechanical loading conditions compared to the field, where the reinforcement applies a load at the centre and the soil a distributed pressure on the opposite surface. A numerical model was developed to extend experimental test conclusions. Experiments and simulations allowed evaluating the load applied by the reinforcement corresponding to crack initiation, cracking behaviour (opening and propagation) and, finally, the ultimate state or load capacity as well as the associated failure mode. The reinforced concrete panel geometry and steel reinforcement were designed to sustain the soil reinforcement axial strength.

The second part of this study concerns the evaluation of the new construction process and the compared behaviors of the two full-scale soil nailed walls. Precast panels designed based on the experimental tests and numerical study in first part of this study, were used in the field.

Concrete precast panel of the modular soil nailed retaining wall: Laboratory tests

Precast elements and their connections

In view of the industrialization of the process, the different projects conducted these last years for different configuration sites and soil nailing designs were analysed. A standardization of the precast panel surface to 3m² appeared to be an optimal choice. In the geotechnical design of the soil nailing for the execution project, the vertical spacing between two horizontal rows of reinforcements was taken equal to 1.5 m and the horizontal spacing to 2 m. The length of the reinforcements and their tension depend on the geotechnical conditions and wall geometry.

Figure 2 shows the precast elementary panel, called "scale" of the facing. The reservation hole at the center of the scale will coincide with the reinforcement's head. The steel bar of the reinforcement will go through this hole and be tied to the facing with a bearing plate and a nut.

To reduce concrete material consumption, a linear variation of the thickness of the precast slabs was adopted. Since external load is concentrated at the center of the panel, where the anchorage is connected, a pyramidal shape was adopted. At each corner, a mechanical device is installed during construction to connecting adjacent panels to form the wall's facing. This device was designed to ensure continuity of the three displacements but to allow rotations between panels.

The precast panels and the connecting device were designed to withstand the maximum allowable tension force in the reinforcement considering the panel simply supported at its four corners only, which is unfavorable compared to its actual loading conditions in the facing of soil nailed walls, where tension in the reinforcement is balanced by earth pressure and shear forces between panels are limited.

Specimens characteristics and test setup

To evaluate the behavior of a precast panel subjected to representative mechanical loading conditions, two tests were carried out on identical specimens 2 m wide by 1.50 m high (see Figure 2). The objective was to quantify the onset of cracking as well as to evaluate the bearing capacity and the failure mode of the panel. The specimens were cast using ready-mixed concrete of class C40/50. The compressive strength of the concrete mixture used for the slabs was determined from compression tests on 11×22-cm cylindrical specimens performed during the day where the panel was tested. An average unconfined compressive strength value of 56.3 MPa was found.

As shown in Figure 3, the test on the panel specimens consisted in applying a quasi-static load at the center of the upper side of the pyramidal panel specimen, where the soil reinforcement is connected to the facing. The specimen was supported by four cylindrical

plates. An incremental quasi-static load was applied (imposed force) at the center by a hydraulic jack with a load speed of 1 kN/s (60kN/minute). The specimen displacements were observed using five LVDT gages. The failure modes observed for Specimen 1 may be defined by the pattern of cracks shown in Figure 6. On the bottom face, the failure due to bending in both directions has been found. The cracks correspond to the flexural rebar's positions. On the top face, shear cracks were also observed at the four corners of the specimen. The numerical investigations were conducted with Abaqus explicit software. The failure of RC slabs can be sudden, in particular for failure in shear. This could cause numerical instability and convergence problems in case of traditional quasi-static analysis (implicit incremental analysis). Therefore, the explicit method was used with very small load increments considering a so-called smooth-step function to develop quasi-static analysis. All calculations were made force controlled with an explicit solution technique. The computing time corresponding to the critical increment size is considerably shorter than the computing time under application of implicit solvers. But the total computing time associated with the explicit simulation is much lower to the one associated to the implicit simulation, knowing that implicit method suffers many convergence problems due to singularities of tangent stiffness matrix induced by cracks appearance along the equilibrium path. The concrete damaged plasticity model (CDP) in Abaqus was used for concrete material and the classic elastic-plastic model was applied to reinforcement steel.

The CDP model in Abaqus is based on the models initially proposed by Lubliner et al.

(1989) [START_REF] Lee | Plastic-damage model for cyclic loading of concrete structures[END_REF] for monotonic loading, and developed later by Lee and Fenves (1998) [START_REF] Lubliner | A plastic-damage model for concrete[END_REF] to consider the dynamic and cyclic loadings. The characteristics of C40/50 concrete are given in Table 1. For steel reinforcement type Fe500, an elastoplastic model with isotropic hardening is used. The characteristics of the reinforcements are specified in Table 2.

Table 1. Properties of concrete used in numerical model

Characteristics Value

Compressive strength fck (MPa) 40

Tensile strength fct (MPa) 3.5

Young's modulus (MPa) 35000

Table 2. Properties of steel used in numerical model

Characteristics Value

Young's modulus (GPa) 210

Yield strength fyk (MPa) 500

Rupture strength (MPa) 540

A perfect adhesion between steel rebar and concrete is assumed. This modeling is fairly faithful to the test.

Numerical results and analysis

Figure 5 shows that the load/deflection curve obtained numerically at the central point instrumented by sensors (Displacement transducer and load sensor) compares well with those observed during the tests. For the linear phase of the load-deflection curve, the rigidity obtained numerically is similar to that found experimentally, but also the non-linear behavior is correctly reproduced, showing the relevance of the numerical model. The ultimate load obtained by the numerical computation is of 452 kN. This value is 11 % higher than the maximum load observed in experimental test on Specimen 1 (Pexp-test1 = 408kN) and 9 % higher than that of experimental test on Specimen 2 (Pexp-test2 = 415kN). The elastic limit (beginning of cracking) is also quite well reproduced by the simulation (Pelas-FEM = 160kN, Pelas-exp = 140kN), with a difference of about 14 %. It is also important to emphasize that the bearing capacity of the concrete panel, as a prefabricated structural component of the facing of a soil nailed wall, is properly designed, since the maximum tensile load of the steel bar in the soil reinforcement, a

Gewi bar of diameter 25 mm with yield stress of 670MPa, is 329kN. These appear at the four corners of the panel and also on the long span (width of panel), whereas no crack appears on the smallest span (height of panel).This indicate a bending failure in only one direction, which confirms a beam-like behavior (one way direction of solicitation). The first cracking observed is the longitudinal crack due to bending. We recall that this test is very conservative compared to the actual loading of the panel in the facing of a soil nailed wall.

Experimental test Damage in tension

In situ full scale evaluation of the new construction technique

The second part of this study was focused on the evaluation of the performance of the new technique for constructing the facing of soil nailed walls, based on the precast panels whose structural design was presented in the preceding section. For that purpose, an excavation 20 m long and 7.5 m deep with vertical cut was created using soil nailing at a site located in Bevenais,

France. The design of the soil reinforcement was performed according to Eurocode 7, French

National Application Standard NF P 94270.

The usual construction process for soil nailed excavations was applied, with successive excavation stages, each one corresponding to a single horizontal row of reinforcements, then drilling for and grouting the reinforcements steel bars in the soil and, finally, constructing the facing for the current reinforcement row, before the next excavation stage.

However, as shown in the photo of the completed excavation in Figure 9, the facing was built using, for half of its length, on the left side, the conventional shotcrete technique and, on the right side, the new technique. for two years after the end of construction, the horizontal displacements of the facing and the tension forces distributions along the reinforcements were monitored and have been analyzed.

Construction of the experimental soil nailed wall in the field

The geotechnical soil profile at the construction site consisted of a clean and well graded gravel layer whose thickness was larger than 30 m. An unconfined aquifer with horizontal water table was observed at a depth of about 30 m, way below the bottom of the excavation project.

In situ total unit weight of the soil was evaluated to 22.5 kN/m 3 and natural water content to 4 %. Based grain size distribution and dry density, the effective friction angle was estimated to be about 41° and, for design, effective cohesion was considered to be negligible. A pressuremeter boring showed that, over the excavation depth, the Menard Limit Pressure was about During the tests, the maximum tension force applied at the reinforcement head turned out to be limited by the tensile strength of the steel bar, not the resistance of the anchored zone. Thus, for the design, the strength of the soilcement grout interface was assumed to be larger than 638 kPa, considering for design the theoretical borehole diameter of 115 mm, or larger than 452 kPa, considering for design a borehole diameter of 162 mm corresponding to the volume of slurry injected per meter.

The usual construction process for soil nailed walls was applied. For each of the successive excavation stages, a horizontal row of reinforcements was installed in the ground as described previously and a horizontal strip of facing was constructed, using shotcrete on the left side of the wall and precast panels on the right, before proceeding to the next excavation stage.

Figure 10 is a picture showing workers applying the first shotcrete layer to create the facing. Vertical trips of a draining complex have first to be placed against the vertical cut in the ground. The first reinforcing steel mesh is then attached to be maintained at the right location in the project during application of the first shotcrete layer, requiring several projections passes.

After about one day, the second steel mesh is also attached with spacers to ensure the specified distance to the first mesh and the second layer of shotcrete is applied.

Figure 10. Construction of the facing for one excavation stage using shotcrete

In addition to safety concerns for the workers applying shotcrete (eye injuries, sudden clogging of the projection pipe), this in situ construction of the facing using shotcrete presents several drawbacks. A gap may exist between the draining complex and the ground behind it, which may be filled by eroded fine soil particles. Finally, a very large amount of concrete is wasted for two reasons. First, to fill up the volume between the specified location of the facing in the project and the soil surface after vertical excavation, i.e. the so called "out of profile volume." This volume may be very large in granular soils, which require short term vertical support and are prone to soil disturbance during drillings to install the reinforcements.

Second, as illustrated in Figure 11, due to rebound of a significant amount of concrete on steel meshes and to poor adhesion to the draining complex or to the ground, since projection is impeded by the steel mesh. The picture on the right side in the figure shows that the thickness of wasted concrete due to rebound was of the order of 20 cm, close to the 25 cm specified wall thickness. As mentioned earlier, its actual thickness was even larger due to "out of profile" during the excavation stage and to disturbance of the soil during reinforcement installation. the soil reinforcements has to be precisely located with respect to the panels installed during the preceding stage. Using a drilling jig connected to the lower corners of the panel above, the 5 cm allowance thanks to a central hole in the panel larger than the bar diameter (Figure 2) turned out to be easily achieved during construction of the experimental wall.

Performance of the new construction technique

Environmental and economic performance

The construction of the experimental soil nailed wall was an opportunity for objective comparison, in terms of environmental and economic performance, of the two constructive techniques used for the facing. In such an analysis, the foremost aspect to be emphasized of course is the worker's safety, greatly improved with the new construction technique, which eliminates injuries related to shotcrete application.

The other aspects of environmental and economic performance considered were the amount of concrete and steel required for each technique, the overall duration for constructing each strip of facing (corresponding to one horizontal row of soil reinforcements) and the cost related to the use of equipments needed in the construction processes.

In the project, the part of facing to be built using each technique was 10 m long by 7.5 m high (75 m 2 ). During construction of the experimental soil nailed wall at the site, consumption of concrete and steel reinforcing rebars for in situ construction of the facing from several shotcrete layers was recorded. According to the design, the thickness of the facing was to be 25 cm. As indicated in Table 3, to build 100 m 2 of facing at the site with the shotcrete technique would require 52 m 3 of concrete. About 32 % of concrete was lost during shotcrete application.

Due to out of profile resulting from soil excavation or from ground disturbance when drilling for installing the soil reinforcements, the actual average thickness of shotcrete at the facing turned out to be rather of the order of 35 cm. With the new technique, the quantity of concrete needed to build 100 m 2 of facing is about 18.7 m 3 , corresponding to the volume of about 33 precast panels. Thus, compared to the shotcrete construction technique, the new technique reduces concrete consumption by 64 %.

Part of this difference in concrete consumption corresponds to the cemented 4/6 crushed stone injected through the weep holes to fill out the voids between the excavation surface and the back side of the panels. For a 100 m 2 facing with the new technique, about 15 m 3 of 4/6 cemented crushed stone and about 0.75 t of cement are also required.

As mentioned in Table 3, the quantity of steel rebars needed for constructing the facing is reduced by about 22 % with the new technique. The significant reduction in construction materials (concrete and steel) leads to a substantial construction cost reduction. Manufacturing of concrete (mainly the cement) and steel generates large amounts of greenhouse gases (GHG).

In Table 3, for the construction of a 100 m 2 soil nailed wall facing, the emission of greenhouse gases corresponding to the supply of concrete and steel have been evaluated for each construction technique in terms of equivalent CO 2 according to ADEME's Base Carbone® references [START_REF] Ademe | Documentation des facteurs d'émissions de la Base Carbone ®[END_REF]. The new technique cuts down the emission of GHG by at least 56 %.

In addition, the new construction technique for the facing of soil nailed walls based on precast panels eliminates the constraints on construction sequences related to the application of consecutive shotcrete layers. Usually, a delay of 24 hours is necessary for the first layer of shotcrete to set and to install the second welded mesh of the reinforced concrete wall. Then, the second shotcrete layer is applied and, again, a minimum delay of 24 hours is usually required for the facing to gain some strength before proceeding to the next excavation stage. With the new technique, construction time and constraints on work's schedule are thus significantly reduced, since installing one row of facing panels and injecting the cemented 4/6 crushed stone is much faster than applying only the first shotcrete layer at the construction site.

Mechanical performance

The experimental full-scale soil nailed wall has been instrumented to compare the influence of the two construction techniques on the mechanical behaviour. Both parts of the wall, the first one with shotcrete facing and the second one with precast panels, were equipped with the same instrumentation plan. Ground settlements behind the wall, vertical and horizontal displacements of the facing and distribution of tensile forces along the soil reinforcements have been observed during and after construction for almost two years.

The results have been analysed in detail by Jean de Sauvage [START_REF] De | Etude du comportement des murs de soutènement par clouage des sols en place : application au dimensionnement du parement[END_REF] in his study on the design of the facing of soil nailed walls. The main conclusions, presented here, only pertain to the influence of the new facing construction technique on the mechanical behaviour of the wall.

Figure 14 shows an elevation view of the whole facing, symmetric with respect to the vertical profile at the centre of the wall, with shotcrete facing on the left and precast panels on the right.

The lateral displacements of the facing in the vertical profiles at the centre of each construction technique were found to be close to each other, slightly lower in the precast panels section.

Several grouted steel bars of the soil reinforcements were equipped with strain gages. The three lowest horizontal lines of soil reinforcements were considered. Each instrumented reinforcement was equipped with strain gages at different distances from the facing to study the distribution of tensile force along the reinforcement. At each point along the reinforcement bars, two strain gages were installed, along the upper generatrix of the steel bar and along the lower, in order to distinguish flexure from tension. This low value of maximum tensile force compared to the design value is believed to be due to an underestimation of the soil shear strength and to the choice of the soil reinforcement's length larger than required by the design. This choice was made to definitely eliminate a possible failure related to pull out of the soil reinforcements, in order to focus the study on the facing. The conventional design method, based on limit equilibrium calculations, thus without considering the influence of construction stages, from top to bottom, on the loading of soil reinforcements, also leads to an overestimation of the maximum tensile forces, predicted to occur in the lowest line of reinforcements, which is never the case. A complete analysis of the influence of reinforcement's length on tensile force magnitude and distribution along the reinforcements may be found in Jean de Sauvage [START_REF] De | Etude du comportement des murs de soutènement par clouage des sols en place : application au dimensionnement du parement[END_REF].

Nevertheless, tensile forces observed in the soil reinforcements were found to be lower in the precast panel facing part of the wall than in the shotcrete facing part. This indicates that the new facing construction technique, with faster application of a confining pressure to the soil during the construction process, leads to a lower soil disturbance within the volume of nailed soil and thus to lower efforts along the soil reinforcements.

For illustration, Figure 15 compares measurements in the strain gages installed on the grouted steel bars of the soil reinforcements at the same location in each part of the soil nailed wall, with shotcrete facing or with precast panels facing. The location of these strain gages was indicated in Figure 14. Both SG84 strain gages were installed in the central profile of the wall portion built with shotcrete facing, along the soil reinforcement in the next to last row, 3 m away from the facing. Extensions "-01" and "-02" refer to the upper and lower generatrix of the steel bar. Similarly, both SG94 strain gages were installed in the central profile of the wall portion built with precast panels facing, in the same row of soil reinforcements and at the same distance to the facing. These graphs show that the tensile force in both reinforcements, at that distance from the facing, increased linearly with time for about 300 days and then reached a plateau. It is interesting to notice that apparent cohesion produced by soil expansion during excavation of unsaturated clean gravel needs about one year to dissipate, progressively reducing the soil shear strength and, as a consequence, loading the soil reinforcements for equilibrium. The tensile force corresponding to the plateau is significantly lower at point SG94, in the part of the wall with precast panels facing, than at point SG84, in the part of the wall with shotcrete facing.

Comparing tensile forces in the soil reinforcements at other similar locations, behind precast panel facing and behind the shotcrete facing, values were found systematically lower behind the precast panels facing than behind shotcrete facing, however with differences not so significant. This indicates that the new construction technique for the facing of soil nailed walls improves the performance of the soil nailing technique compared to the conventional construction technique of the facing based on shotcrete.

Finally, it may be noticed in Figure 15 that tensile forces in the reinforcements at points SG84 and SG94 began to increase again after about 520 days. This corresponds to the moment where a trench 1.5 m deep was excavated at the toe of the soil nailed wall, in an attempt to produce and observe the wall failure. In fact, wall failure could not be reached, even when applying an additional surcharge at the top. To restore the site after experiment, the soil nailed wall had to be dismantled by excavation of the whole soil nailed volume.

Conclusion

In this study, a new technique for constructing the facing of soil nailed walls has been presented. Instead of the conventional technique based on shotcrete application, reinforced concrete precast panels are used. In the construction process, a slightly cemented 4/6 crushed stone is injected through weep holes, to fill the void between the back face of the panels and the ground excavation vertical wall. This new technique to build the facing greatly improves worker's safety, since it eliminates frequent injuries related to shotcrete application.

The structural design of the precast reinforced concrete panels has been presented in It was found that the new technique reduces concrete consumption by about 64 % compared to the shotcrete construction technique, in which a significant quantity of concrete is lost, due to rebound on steel meshes and to poor adhesion to the draining complex or to the ground, or wasted due to out of profile resulting from soil excavation and from ground disturbance when drilling for installation of the soil reinforcements. The new technique as well reduces by about 22 % the quantity of steel rebars needed for the facing.

Manufacturing of concrete (mainly the cement) and steel generates large amounts of greenhouse gases (GHG). As a consequence, according to ADEME's Base Carbone® references [START_REF] Ademe | Documentation des facteurs d'émissions de la Base Carbone ®[END_REF], the new technique cuts down the emission of GHG for construction of the facing of soil nailed walls by more than 56%.

From the economic point of view, the new construction technique significantly reduces the overall construction cost of soil nailed walls. The cost reduction is mainly due to the lower quantity of construction material needed and to the shorter construction duration, since the constraints on construction sequences implied by the application of consecutive shotcrete layers are eliminated. Simply, when switching from shotcrete to precast panels, the company's practice has to get adjusted to the new construction process. But the outcome is worth it.

Figure 1 .

 1 Figure 1. Full-scale soil nailed walls (Bevenais, France): shotcrete facingleft side, precast panelsright side (quoted distances in [mm]).

Figure 2 .

 2 Figure 2. Precast panel, 2 m wide and 1.50 m high, and connecting device at the four corners

Figure 3 .Figure 4 .

 34 Figure 3. Test setup for the specimen of precast concrete panel The rebar, amount, spacing, and the diameter of longitudinal and transverse steel reinforcement are given in Figure 4. For the lower reinforcement layout, the longitudinal and transversal reinforcement consisted of Ø8 mm with a spacing of 100 mm. For the upper reinforcement layout, the longitudinal and transversal reinforcement consisted of Ø6 mm with spacing of 200 mm. Two rebar circles placed at the centre of the slab with the diameters of Ø10 mm were added to reinforce the concrete about the concentrated load location. The cross longitudinal bars connecting two opposite corners of the panel were Ø10 mm in diameter

Figure 5 .

 5 Figure 5. Load/displacement curves: experimental vs numerical results

Figure 6 .

 6 Figure 6. Cracks observed on Specimen 1: bottom face and top face

  The constitutive theory aims to capture the effects of irreversible damage associated with the failure mechanisms that occur in concrete and other quasi-brittle materials under fairly low confining pressures (less than four or five times the ultimate compressive stress in uniaxial compression loading). The nonlinear behaviour of concrete is attributed to the damage and plasticity processes. The plastic behaviour can be characterized by several phenomena such as strain softening, progressive deterioration, etc. The damage process can be attributed to micro-cracking. Damage is associated with the concrete's failure mechanisms and therefore results in a reduction of the elastic stiffness. The model is based on the concept of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to represent the inelastic behaviour of concrete i.e. tensile cracking and compressive crushing. The relevance of CDP model to simulate the reinforced concrete structures has been validated by several authors in recent studies[13][14]. Due to the complexity of the selected geometric shape, a 3D finite element model developed with solid element has been chosen. The modeled geometry is identical to the real geometry and accounts for the presence of the steel reinforcements in the concrete panel. For symmetry reasons, only one quarter of the slab was modeled. Boundary conditions correspond to two plane of symmetry and one support (downwards displacement is zero) at the slab corner. Uniform pressure is applied to the ring in the central part of the top surface (Figure7), at the location of the loading jack in the experiment.

Figure 7 .

 7 Figure 7. Finite Element mesh of one fourth of the precast panel

Figure 8 .

 8 Figure 8. Failure modes: experimental vs numerical results (Specimen 2)

Figure 9 .

 9 Figure 9. In situ full-scale soil nailed walls built in Bevenais, France (left side, shotcrete facingright side, precast panels)

5

  MPa and the Menard Modulus 40 MPa. According to the geotechnical design, the soil reinforcement consisted of five horizontal rows of steel bars, with vertical spacing of 1.5 m and horizontal spacing of 2 m, installed into boreholes drilled with a 10° angle dip below the horizontal. From the top, the first two upper rows were 7 m long, the next two rows 6 m and the lowest row 5 m. All the bars were 25 mm diameter GEWI bars made of high yield strength steel (670 MPa).To install the soil reinforcements into such a soil, ODEX air drilling technique with continuous casing was employed. The theoretical diameter of the borehole (the boring diameter) was 115 mm. The steel bars with centering devices were then placed into the borehole and casing was progressively removed while cement slurry was injected from the lower end of the boring into the ground, up to the reinforcement head at the facing. The cement over water mass ratio of the slurry was equal to 2 to avoid cement settling.Several reinforcement pullout tests were performed to evaluate the strength of the soilcement grout interface. In those tests, the drilling, bar placement and grouting equipment and procedures were the same as those in the actual soil nailed wall, except for the orientation (pull out tests were vertical) and the length. In the tests, boring length was 3 m and only the lower 2 m of the bars were grouted to create the sealing bulb. The volume of cement slurry injected per meter of reinforcement corresponded to about twice the theoretical boring diameter, indicating irregular borehole walls and penetration of cement slurry into the granular soil.

Figure 11 .

 11 Figure 11. Wasted concrete due to rebound during shotcrete application

Figure 13 .

 13 Figure 13. Temporary suspension brackets to avoid vertical sliding of the first panel row During construction of the facing for the following rows of soil reinforcements, a mechanical device (see Figure 2) will connect the upper corners of each new panel to the lower corners of the above panel and its lower corner to the adjacent panel installed just before in the same row. As a consequence, in the new construction technique, the location of the borings for

Figure 14 .

 14 Figure 14. Location of strain gages SG84 and SG94 in the elevation view and cross section of the experimental wall For both facing construction techniques, in all levels of soil reinforcements and at all distances from the facing, the tensile forces in the soil reinforcements were observed to be lower than expected according to the geotechnical design. The long term (observed 640 days after construction) maximum tensile force in the soil reinforcements, of the order of 60 kN, was observed in the second line of reinforcements from the base of the wall.

Figure 15 .

 15 Figure 15. Strain in soil reinforcements at the same location in the shotcrete section of the soil nailed wall (SG84) and in the precast panel section (SG94)

  detail. It was based on laboratory tests performed on a single facing panel subjected to unfavourable mechanical loading conditions compared to the field, where the reinforcement applies a load at the centre and the soil a distributed pressure on the opposite surface. A numerical model was also developed to simulate the laboratory tests. Predictions of crack initiation and failure modes were in good agreement with the behaviour observed in laboratory tests. The reinforced concrete panel geometry and steel reinforcement, designed to sustain the ultimate axial load of the soil reinforcements, have been presented in detail.A full-scale experimental soil nailed wall 7.5 m high was built at a site where the soil profile consisted of a very thick layer of dense clean gravel. Half of the facing was constructed using the conventional shotcrete technique and the other half with the new technique based on precast concrete panels. The purpose was to evaluate the environmental, economic and technical performance of the new construction technique compared to the conventional shotcrete method.

Finally, the construction

  of the instrumented full-scale soil nailed wall demonstrated the mechanical performance of the new technique for facing installation compared to the conventional shotcrete technique. Horizontal displacements of the facing and surface settlement behind the wall facing were observed slightly lower in the precast panels section of the experimental wall. Tensile forces in the soil reinforcements were systematically observed lower in the precast panels facing section than in the shotcrete facing section.This indicates that the new construction technique for the facing of soil nailed walls improves the overall mechanical performance of the soil nailing technique compared to the conventional shotcrete construction technique. This may be due to the shorter delay before applying back a confining pressure to the excavation face, thus reducing soil disturbance in the ground. Besides, the new construction technique includes a continuous and high capacity drainage behind the facing, while in practice poor drainage of the facing is the main cause of soil nailed walls failure.

Table 3 .

 3 Concrete and steel rebars quantities for construction of a 100 m 2 soil nailed wall facing using either the conventional shotcrete technique or the precast panels.

	Supplies	Shotcrete	Precast panels
	Concrete	52 m 3	17.9 t of CO 2	18.7 m 3	6.4 t of CO 2
	Steel Rebars	1920 kg	4.1 t of CO 2	1500 kg	3.2 t of CO 2
	Total	22 t of CO 2	9.6 t of CO 2

Acknowledgments

This research project was funded by NGE FONDATION, saint-Priest, France.


Visou Ady

David Dubayle

Pascale Le Blanc

Valery Shestopalov

Claude Meunier

Carole Levenes
email: carole.levenes@gmail.com

Patterned expression of Pannexin 1 channels in the adult cerebellar Purkinje cells

Pannexin1 (PanX1) are recently discovered proteins that can form large pore channels at the cell surface. They have been implicated in ATP-dependent cell-to-cell communication and in several pathophysiological processes such as inflammation, cell death and epilepsy. Using immunohistochemistry in the adult mouse, we describe the presence of PanX1 in the deep cerebellar nuclei, in large cells of the granular layer, presumably Golgi interneurones, in some Bergmann glia radial processes as well as in the soma and dendrites of Purkinje cells. In the latter, PanX1, like many other proteins, distribute heterogeneously. Only Zebrin II-positive Purkinje cells express PanX1, in accordance with the so-called "zebra-striped" modular architecture of the cerebellum. This distribution in zebra-stripes suggest that PanX1 may contribute to the control of ensemble activity within cerebellar microdomains or to the response of Purkinje cell to excitotoxicity and cell-death messages.

Introduction

Pannexins are membrane proteins ubiquitously present in vertebrate organisms. They were discovered in 2000 by sequence homology with the invertebrate innexins [START_REF] Panchin | A ubiquitous family of putative gap junction molecules[END_REF]. They show structural similarities with connexins [START_REF] Panchin | A ubiquitous family of putative gap junction molecules[END_REF] but do not seem to form gap-junctions in native cells ( [START_REF] Sosinsky | Pannexin channels are not gap junction hemichannels[END_REF] and refs in [START_REF] Penuela | The biochemistry and function of pannexin channels[END_REF]. Among the three types of pannexins described to date, pannexin 1 (PanX1) is by far the most studied.

It is heavily expressed in the central nervous system, in both neurons and glial cells.

Its mRNA has been detected in the retina, the cerebellum, the neocortex and hippocampus, in the amygdala, substantia nigra and the olfactory bulb, [4-6, 7 , 8, 9, 10 , 11]. PanX1 can be activated by cytoplasmic Ca 2 + , membrane depolarization, extracellular ATP and K + , by mechanical stretch and caspase cleavage ( [12, 13 , 14] and refs in [START_REF] Penuela | The biochemistry and function of pannexin channels[END_REF]). They can form large conductance channels of up to 550 pS [START_REF] Bao | Pannexin membrane channels are mechanosensitive conduits for ATP[END_REF] that carry non-selective ion fluxes and are permeable to molecules up to 1 kDa, including ATP itself [START_REF] Locovei | Pannexin 1 in erythrocytes: function without a gap[END_REF][START_REF] Macvicar | Non-junction functions of pannexin-1 channels[END_REF], [START_REF] Chiu | Intrinsic properties and regulation of Pannexin 1 channel[END_REF]. Once released, ATP can act on purinergic receptors or can be further degraded into adenosine by ectonucleotidases. Thus pannexins mediate auto-and paracrine communication, making them possible actors of retrograde neuronal communication and modulators of neuronal activity, for example through presynaptic adenosine receptors [START_REF] Kawamura | Metabolic autocrine regulation of neurons involves cooperation among pannexin hemichannels, adenosine receptors, and KATP channels[END_REF]. However, their exact physiological role remains unclear. Evidence shows that they can initiate Ca 2+ waves, for instance after mechanical stretch [START_REF] Bao | Pannexin membrane channels are mechanosensitive conduits for ATP[END_REF], control the vascular tonus, regulate differentiation, cell death and immune function [START_REF] Penuela | The biochemistry and function of pannexin channels[END_REF]. In pathological conditions, they may activate the inflammasome ( [START_REF] Silverman | The pannexin 1 channel activates the inflammasome in neurons and astrocytes[END_REF] reviewed in [START_REF] Adamson | The role of pannexin1 in the induction and resolution of inflammation[END_REF]), contribute to caspase-mediated apoptosis [START_REF] Chekeni | Pannexin 1 channels mediate 'find-me' signal release and membrane permeability during apoptosis[END_REF] and ischemic cell-death [START_REF] Bargiotas | Functional outcome of pannexindeficient mice after cerebral ischemia[END_REF], and enhance epileptic seizures. They could also play a role as tumor suppressors (refs in [START_REF] Penuela | The biochemistry and function of pannexin channels[END_REF]).

In the cerebellum, PanX1 proteins are heavily expressed, particularly in Purkinje cells [7 , 10], but their contribution to cerebellar physiology or pathology remains totally unknown. Despite its apparent homogeneous crystal-like organization, the cerebellum is, in fact, highly compartmentalized. It displays genetically determined and reproducible topographic domains: the transverse zones and parasagittal stripes, including the cortical so-called zebra-stripes [START_REF] Apps | Cerebellar cortical organization: a one-map hypothesis[END_REF]. The latter can be revealed by the patterned expression of numerous molecules, particularly in Purkinje cells. Zebrin II, now known to be the respiratory enzyme Aldolase C, was the first example of such a protein [START_REF] Brochu | Zebrin II: a polypeptide antigen expressed selectively by Purkinje cells reveals compartments in rat and fish cerebellum[END_REF]. It is expressed in parasagittal bands of Purkinje cells that extend throughout the cerebellar cortex, separated by bands of Zebrin IInegative Purkinje cells. Many other proteins match Zebrin II Purkinje cells patterns.

For example, phospholipase C β4 (PLC β4; [START_REF] Marzban | Phospholipase Cbeta4 expression reveals the continuity of cerebellar topography through development[END_REF]), the metabotropic glutamate receptor 1b (mGluR1b; [START_REF] Mateos | Parasagittal compartmentalization of the metabotropic glutamate receptor mGluR1b in the cerebellar cortex[END_REF]) and the early B-cell factor 2 (EbF2; [START_REF] Croci | A key role for the HLH transcription factor EBF2COE2,O/E-3 in Purkinje neuron migration and cerebellar cortical topography[END_REF]) are expressed in Zebrin II-negative Purkinje cell stripes. On the other hand, PLCβ3 [START_REF] Sarna | Complementary stripes of phospholipase Cbeta3 and Cbeta4 expression by Purkinje cell subsets in the mouse cerebellum[END_REF], the excitatory amino-acid transporter 4 (EAAT4; [START_REF] Nagao | EAAT4, a glutamate transporter with properties of a chloride channel, is predominantly localized in Purkinje cell dendrites, and forms parasagittal compartments in rat cerebellum[END_REF]), and the GABA-B 1b receptor [START_REF] Fritschy | GABAB-receptor splice variants GB1a and GB1b in rat brain: developmental regulation, cellular distribution and extrasynaptic localization[END_REF] are expressed in Zebrin II-positive Purkinje cell. These striped distributions of neurotransmitter receptors and their effectors indicate that Zebrin II-negative andpositive Purkinje cells respond differentially to their inputs. For instance, the glutamatergic climbing fibers that form synapses onto Zebrin II-positive Purkinje cells release more glutamate per action potential than those terminating on Zebrin IInegative ones [START_REF] Paukert | Zones of enhanced glutamate release from climbing fibers in the mammalian cerebellum[END_REF]. Zebrin II-positive Purkinje cells seem to be adapted to respond to increased glutamate inputs and to better resist death and neurodegeneration of various origins [30, 31 , 32 , 33]. Sphingosine kinase, a lipid metabolism protein that plays an important role in apoptosis, is also found selectively in Zebrin II-positive Purkinje cells [START_REF] Terada | Compartmentation of the mouse cerebellar cortex by sphingosine kinase[END_REF]. Overall, the proteins expressed in Zebrin II-positive Purkinje cells, including Zebrin II itself, seem to confer these cells capacities to respond to sustained neuronal activity and to the associated increase in metabolic demand.

In the present study, we use a type of antibodies directed against the extreme C-terminal domain of PanX1 to investigate the distribution of PanX1 in the cerebellum. These antibodies that had never been used in this structure reveal a zebra-striped pattern of PanX1 in Purkinje cells. This differential rostro-caudal expression,typical of the cerebellum, had never been described before for PanX1. It suggests that PanX1s contribute to the adaptation of Purkinje cells to the increased activity observed in the zebra stripes, and thus that pannexins may play important functions in the physiology and pathophysiology of the cerebellum.

Materials and methods

Animals

The animals used were all adult (n=25) C57BL/6JRj males and females C57BL/6JRj mice (from 8 to 12 weeks of age), purchased at Janvier Laboratory (Le Genest-St-Isle, France). Mice were housed at the central animal facility of Les Saints-Pères (Paris, France). Housing and all procedures were performed in accordance with the guidelines of the French Ministry of Agriculture and the European Community and have been approved by the ethical committee of Paris Descartes University (agreement: Paris Descartes # 14-013). A minimal number of animals was used and they were handled with maximum care in order to minimize their suffering.

Tissue preparation

The animals were deeply anaesthetized with sodium pentobarbital (175 mg/Kg i.p.). The intracardiac perfusion consisted of 50 ml 0.01 M phosphate-buffered saline (PBS, pH: 7,4), followed by 50 ml of 4% paraformaldehyde in 0.1 M PBS. The cerebellum was removed, postfixed for 1 day in 4% paraformaldehyde solution and cryoprotected in a 30% sucrose phosphate-buffered solution at 4°C for 2 days before cutting. The cerebellum was cut into longitudinal or sagittal 80 mm-thick sections using a freezing sliding microtome (Frigomobil Reichert-Jung) and collected in plastic-wells containing PBS. Serial sections over the entire rostrocaudal length of the cerebellum were treated and analyzed. The two types of PanX1 antibodies that we used were rabbit polyclonal antibody raised against the C-terminus of the human PanX1. The first one came from Chemicon (USA). The second one was made by V. Schestopalov [START_REF] Dvoriantchikova | Expression of pannexin family of proteins in the retina[END_REF]

Antibody Characterization

Immunostaining for light microscopic analysis

Distribution of Pannexin1 labeling matches that of Zebrin II.

The rostro-caudal heterogeneous expression pattern of PanX1 that we observed in sagittal sections is reminiscent of that of the several tens of proteins expressed in zebra stripes in the cerebellum (reviewed in [START_REF] Apps | Cerebellar cortical organization: a one-map hypothesis[END_REF]), the most studied being Zebrin II [START_REF] Brochu | Zebrin II: a polypeptide antigen expressed selectively by Purkinje cells reveals compartments in rat and fish cerebellum[END_REF], identified as the Aldolase C enzyme [START_REF] Ahn | The cloning of zebrin II reveals its identity with aldolase C[END_REF]. The very name "Zebrin" comes from the zebra-like pattern that clearly appears in longitudinal sections., In conclusion, the distribution of PanX1-positive Purkinje cells matches that of Zebrin II-positive cells either in rostral, medial or caudal parts of the cerebellar cortex.

Labelings of

Discussion

In the present study, we detect PanX1 in Purkinje cells, DCN cells, Bergmann glia processes and in scattered cells in the granular layer (likely Golgi interneurons) [START_REF] Vogt | Pannexin1 and Pannexin2 expression in the developing and mature rat brain[END_REF][START_REF] Ray | Pannexin expression in the cerebellum[END_REF][START_REF] Ray | Site-specific and developmental expression of pannexin1 in the mouse nervous system[END_REF][START_REF] Bunse | The potassium channel subunit Kvbeta3 interacts with pannexin 1 and attenuates its sensitivity to changes in redox potentials[END_REF][START_REF] Zappala | Expression of pannexin1 in the CNS of adult mouse: cellular localization and effect of 4-aminopyridineinduced seizures[END_REF]. Our results are in agreement with previous literature for all cell types expressing PanX1. However, we evidence for the first time a rostro-caudal gradient in the distribution of PanX1 in cerebellar Purkinje cells. In our experiments, PanX1 were almost undetectable in the anterior lobules but strongly labeled in the caudal lobules, particularly in the posterior zone. This distribution matches that of the standard in the subject, the Zebrin II protein, which is expressed according to a striped pattern, highly reproducible and highly conserved across evolution [START_REF] Apps | Cerebellar cortical organization: a one-map hypothesis[END_REF]. To our knowledge, the antibodies that we used here had only been tested before in the retina [START_REF] Dvoriantchikova | Expression of pannexin family of proteins in the retina[END_REF], but never in the cerebellum. In our western-blots, made on adult cerebellum protein extracts, the anti-PanX1 antibodies detect a 43 kDa band as expected for PanX1. In addition, we used the corresponding immunizing PanX1 for PanX1a, 1c, and 1d respectively [START_REF] Li | Characterization of novel Pannexin 1 isoforms from rat pituitary cells and their association with ATP-gated P2X channels[END_REF]. PanX1 can also be glycosylated and Snitrosylated, which modulates their surface expression and activity [START_REF] Cone | A comparative antibody analysis of pannexin1 expression in four rat brain regions reveals varying subcellular localizations[END_REF][START_REF] Lohman | Snitrosylation inhibits pannexin 1 channel function[END_REF]. This could explain discrepancies between our study and the others published so far. Cone et al. [START_REF] Cone | A comparative antibody analysis of pannexin1 expression in four rat brain regions reveals varying subcellular localizations[END_REF] made an elegant comparative analysis of labelings obtained with four different antibodies in the adult mouse cerebellum, in which they show that different polyclonal antibodies actually detect distinct species of PanX1. These antibodies revealed various sub-cellular and cellular localization of PanX1, depending on the epitope design. Our antibodies likely target a specific species that is restricted to Zebrin II-positive Purkinje cells. PanX1 isoforms with molecular weight around 43 kDa, like ours, likely corresponds to non -or weakly -glycosylated forms that tend to be more confined to intracellular compartments [START_REF] Cone | A comparative antibody analysis of pannexin1 expression in four rat brain regions reveals varying subcellular localizations[END_REF] but can also reach the membrane in some conditions [START_REF] Cone | A comparative antibody analysis of pannexin1 expression in four rat brain regions reveals varying subcellular localizations[END_REF][START_REF] Penuela | Pannexin 1 and pannexin 3 are glycoproteins that exhibit many distinct characteristics from the connexin family of gap junction proteins[END_REF]. The species that we detect here seems to reach the membrane because dendrites and spines were heavily stained in our immunohistochemistry experiments.

The distribution of PanX1 proteins in zebra stripes is intriguing,. PanX1 have been so far essentially implicated in pathologies. They have been shown to be activated by pathological conditions such as epileptic seizures or infection [START_REF] Thompson | Activation of pannexin-1 hemichannels augments aberrant bursting in the hippocampus[END_REF][START_REF] Aquilino | Pannexin-1 channels in epilepsy[END_REF]. In turn, once activated, they contribute to worsen these conditions. Their activation can engage the inflammasome and trigger aptototic processes [START_REF] Dahl | Pannexin: from discovery to bedside in 11+/-4 years?[END_REF][START_REF] Crespo Yanguas | Pannexin1 as mediator of inflammation and cell death[END_REF] (see also [START_REF] Shestopalov | Molecular pathways of pannexin1-mediated neurotoxicity[END_REF]).

Zebrin II-positive Purkinje cells better resist to infectious and chemical injuries than their Zebrin II-negative sisters and resist better to cell-death. Therefore, the preferential localization of PanX1 in Zebrin II bands seems rather paradoxical, unless the specific species of PanX1 we have evidenced there display a higher threshold for activation. In this case, they would be less activated in pathological condition than their Zebrin-II negative counterparts, thus rendering the Zebrin-II positive cells more resistant to insults. However, this remains an unsubstantiated hypothesis.

Another picture emerges if we consider instead the role of pannexins in normal conditions [START_REF] Penuela | The biochemistry and function of pannexin channels[END_REF], rather than focus on their deleterious effects. PanX1 exert some of their paracrine functions as ATP channels. ATP, which in turn activates PanX1, can be hydrolyzed into ADP, AMP, and eventually in adenosine by ectonucleotidases.

These molecules are agonists of purinergic and adenosine receptors, known to modulate neuronal activity in the cerebellar cortex [START_REF] Deitmer | Modulation of synaptic activity in Purkinje neurons by ATP[END_REF][START_REF] Donato | GABA release by basket cells onto Purkinje cells, in rat cerebellar slices, is directly controlled by presynaptic purinergic receptors, modulating Ca2+ influx[END_REF]. For example, adenosine receptors A1R, expressed at both pre-and post-synaptic sides glutamatergic afferences on Purkinje cells, decrease glutamate release [START_REF] Kocsis | Adenosine selectively blocks parallel-fiber-mediated synaptic potentials in rat cerebellar cortex[END_REF] and inhibit long-term depression [START_REF] Kamikubo | Functional cooperation of metabotropic adenosine and glutamate receptors regulates postsynaptic plasticity in the cerebellum[END_REF]. Thus pannexins could trigger the activation of presynaptic A1R by releasing ATP -further converted into adenosine -and reduce glutamate release. In this scheme, PanX1 would participate to a negative feedback control of synaptic transmission. Such a mechanism would provide a homeostatic regulation of neuronal transmission in the cerebellum, as has recently been suggested elsewhere [START_REF] Shestopalov | Pannexins Are Potential New Players in the Regulation of Cerebral Homeostasis during Sleep-Wake Cycle[END_REF]. The In conclusion, we can hypothesize that the PanX1 species we have evidence here provide Zebrin II-positive Purkinje cells with additional tools to adapt to the specific physiology of this zone characterized by increased glutamate release, enhanced metabolic demand and better resistance to cell-death. All these possibilities remain hypotheses for the moment, but our data open new perspectives that call for further investigation. 319

  as follows. A rabbit polyclonal antibody against the carboxyl terminus of human PanX1 cDNA encompassing the entire coding region was synthesized by PCR amplification of the cDNA insert from clone IMAGE: 4390851 using two gene-specific primers (Px1F: 5′-). It was then inserted into the BamHI and HindIII sites of pET-23a to yield the pETPx1 plasmid. The resultant 18 kDa pETPx1 protein fused to the His-tagged C-terminus was purified from E. coli and used for immunization of the rabbit. The affinity purified rabbit serum containing specific anti-human PanX1 activity and lacking any significant non-specific activities was used for the immunohistochemistry. In the present study, we verified the specificity of the two PanX1 antibodies we used with their immunogenic peptide in western-blot and immunohistochemistry experiments (see results). The sequence of the peptide corresponds to the 135 last amino-acids (C-ter) of the human PanX1 protein. Blasting this peptide sequence shows its high specificity for PanX1. The specificity of PanX1 antibody was tested by peptide pre-adsorption: 1/2000 anti-PanX1 was pre-incubated for 2 h at room temperature with 1/400 of immunizing peptide and then used for western-blot of 30 µg of protein extract from adult mice cerebellum. The same peptide was used in immunohistochemistry with the same preincubating procedure except that the PanX1 antibody was used at 1/750 and blocking peptide dilutions were 1/750, 1/150 and 1/75. Both anti-PanX1 antibodies gave similar results. Therefore, data obtained with these to different antibodies were pooled.

  Cerebellar sections were processed for PanX1, calbindin (CaBP), Zebrin II expression analysis. Sections were first incubated in blocking buffer containing PBS-T-G (PBS 0.1%, Triton X-100 0.25%, Glycine 0.75% and gelatine from porcine skin 0.25%, Sigma, UK) for 1 hour at room temperature. Following three washes in PBS-G (PBS-T-G, without Triton), sections were incubated for 72 hours at room temperature in a mixture of the primary antibodies. Following three washes in PBS-G, the slices were incubated at room temperature for 2 hours with a mixture of secondary antibodies containing: anti-mouse Alexa fluor (AF) 546 (1:250) and antirabbit AF488 (1:250), both from Molecular Probes (USA). Following a final wash in PBS-G, the sections were air-dried and mounted in a homemade Mowiol-based medium. Images were acquired with a LSM510 or LSM710 microscope (Zeiss) and processed with the ImageJ software (public domain: http://rsbweb.nih.gov/ij/). Final montages were made with Adobe Photoshop.ResultsDistribution of PanX1 immunoreactivity: rostrocaudal gradient and parasagittal stripes.We first verified the specificity of the PanX1 antibodies using western blot and immunohistochemistry. In western blots of mouse cerebellar extracts, the PanX1 antibodies recognize a single ~ 43 kDa band as expected for PanX1 in brain extracts[START_REF] Dvoriantchikova | Expression of pannexin family of proteins in the retina[END_REF]. This band disappeared after incubating the antibodies with a saturating concentration of their immunogenic peptide (Fig1A). In the immunohistochemistry experiments, the pattern of PanX1 labeling observed in sagittal cerebellar sections vanished with increasing concentrations of the blocking peptide in a dose-dependent manner (Fig 1B-E). We also verified that the blocking peptide did not interfere with the immunolabeling procedure by testing it with an antibody directed against parvalbulmin (Fig1F). As expected, the peptide did not affect the parvalbumin labeling.We next used the anti-PanX1 antibodies to assess the distribution of PanX1 in the cerebellum. As a marker of Purkinje cells, we used the calcium binding protein calbindin-D28k (CaBP). CaBP labeling also allowed us to verify the aspect of these cells and the quality of intracardial fixation. Confocal images of sagittal cerebellar slices made in the vermis revealed that PanX1 antibodies strongly label the soma and dendrites of Purkinje cells (Fig 2A), as previously observed [10, 34 ]. The labeling was spotty and more concentrated in proximal dendrites (Fig 2A,B). In distal branches, it was also detectable in the vicinity of spines at the level of the thin spiny branchlets (Fig 2B top insets). Remarkably, numerous PanX1 spots seemed to be located outside Purkinje cells themselves, in close apposition to their dendrites. This is suggestive of a possible PanX1-mediated communication between Purkinje cells spines and their afferences (Fig 2B top insets). Molecular layer interneurons were negative for PanX1 labeling (Fig 2B). The radial processes of some -but not all -Bergman glial cells were immunoreactive for PanX1 (Fig 2B). Granule cells were devoid of labeling, but longitudinal sections revealed the presence of PanX1 in round cell bodies of the granular layer, presumably from Golgi cells (Fig 2C). In the deep cerebellar nuclei (DCN), Purkinje cell axons were strongly immunoreactive for PanX1, and some large cells immunopositive for CaBP were also reactive for PanX1 antibodies (Fig 2D). All these observations are in agreement with previous literature [7, 10, 35, 36]. Intriguingly, and in contrast with previous data, PanX1 labeling of Purkinje cells was almost absent in the anterior cerebellar lobules, although well-shaped Purkinje cells were present, as attested by CaBP labeling (Fig 3A). Mosaic parasagittal images of entire cerebellar sections (Fig 3B) revealed a clear rostrocaudal gradient of PanX1 labeling, in contrast to the homogeneous CaBP labeling. This was particularly obvious in the vermis, when comparing the rostral lobule III (Fig 3A) with the caudal lobule IX (Fig 3C). In the central and posterior lobules, PanX1 immunofluorescence was discontinuous, as PanX1-immunopositive zones were intermingled with regions of PanX1-immunonegative Purkinje cells. Staining in the lobule X was constant among slices and more pronounced than in the other lobules.

  PanX1 in longitudinal sections displayed a striking zebra-type distribution, while that of CaBP appears homogeneous attesting the actual presence of Purkinje cells in the overall slice (Fig 4). In those longitudinal sections, the floculi, parafloculi and Crus II regions consistently appeared PanX1 positive (Fig 4A,D), while lobules I to V displayed several little stripes of few immunopositive Purkinje cells (Fig 4 B,C). The profile of PanX1 distribution in rostral longitudinal section being typical of that of Zebrin II, we made double labelings of these two proteins to compare their respective distributions using large scale mosaic imaging (Fig 5). In parasagittal sections, the overall distribution of PanX1 matched that of Zebrin II (Fig 5A). This was clear from lobule VIb to the posterior tip of lobule VII, and in lobules VIII to X, where Purkinje cells where almost uniformly labeled. In other lobules, there was sometimes a difference in the intensity of staining, which was likely due to different subcellular distributions of Zebrin II and PanX1. However, examination at higher magnification showed that the boundaries of strongly immunoreactive bands and weakly immunoreactive (or unreactive) bands were very similar for these two markers. We made longitudinal sections in the rostral, medial and caudal parts of the cerebellum to verify if PanX1 and Zebrin II distributions matched over the whole antero-posterior structure (Fig 5B-D). In rostral regions, most Purkinje cells were Zebrin II-negative and PanX1-negative (Fig 5B). In the vermis, the midline and the two lateral Zebrin II-positive bands on each side (presumably P1+, P2 +, P3+ according to [38]) were PanX1-positive and were separated by broad PanX1negative bands (lobule III or IV). In medial and caudal sections, staining in lobules VIb, VII and X was uniform and dense across Purkinje cells (Fig 5C and D). In lobule IX, the midline and the two lateral Zebrin II-positive bands were also PanX1-positive (Fig 5D). Sections through the posterior and the nodular cerebellum showed perfect match between PanX1-and Zebrin II-positive Purkinje cells (Fig 5D).

  peptide pre-adsorbed and incubated with the PanX1 antibody to verify specificity, both in western-blot and in immunohistochemistry. In this latter set of experiments, the dose-dependent blocking effect of the immunizing peptide further confirmed the validity of our data. Why the specific localization of PanX1 in Zebrin II-positive Purkinje cells has never been reported before is an intriguing question. It may be that our antibodies recognize a different subset of PanX1 isoform(s) compared to previous studies. Several protein modifications explain the various banding patterns observed in western-blots, such as pre-and post-translational modifications (glycosylation, Snitrosylation or caspase-mediated cleavage). PanX1 seems to be subjected to numerous splicing modifications. Besides the full-length variant named PanX1a (48 kDa), alternative splicing of the human or rat PanX1 encoding genes generates at least four variants (PanX1a-d), of predicted molecular weights of 48, 40, and 34 kDa

  presence of PanX1 in Zebrin II-positive regions could thus help neurons adapt to the increased activity observed in these microzones. It is striking that the majority of the molecules present in parasagittal bands are directly involved in synaptic transmission, control of neuronal activity, or cell responses to the associated metabolic demand. The specific species of PanX1 present in zebrin II-bands may well contribute to this adaptation.

Fig 1 .

 1 Fig 1. Specificity of the anti-PanX1 antibody.

Fig 2 .

 2 Fig 2. Cellular distribution of PanX1 in the cerebellar cortex.

Fig 3 .

 3 Fig 3. Localization of PanX1 in the cerebellum: dense labeling in Purkinje cells

Fig 4 .Fig 5 .

 45 Fig 4. Longitudinal cerebellar sections reveal striped PanX1 labeling.

Table 1 . Primary antibodies used in this study. Antigen Immunogen Source, host species, RRID Dilution Pannexin 1 18

 1

		kDa C-terminal portion of	Chemicon, Millipore,	1:750
		human Pannexin 1 protein	USA ; rabbit polyclonal ;
			affinity purified.
			RRID : AB_2236635
	Pannexin 1 18 kDa C-terminal portion of	Dr. V. Schestopalov,	1:750
		human Pannexin 1 protein,	Miami, USA ; rabbit
		affigel-conjugated	polyclonal ; affinity
			purified.
			RRID : AB_2315054
	Calbindin Calbindin D-28k protein	Swant, Switzerland ;	1:7500
			mouse monoclonal.
			RRID : AB_10000347
	Zebrin II	Recognizes a single polypeptide	Dr. R. Hawkes, Calgary,	1:50
	(Aldolase C)	band of apparent molecular	Canada ; mouse
		weight 36 kDa on cerebellum	monoclonal.
		homogenates from all vertebrate
		classes studied with this Ab.	RRID : AB_2315622

Acknowledgements

We thank Richard Hawkes for the kind gift of the anti-Zebrin II antibody as well as precious recommendations. We also thank David Hansel for critical reading and comments on the manuscript. We sincerely acknowledge the imaging facility SCM (Service Commun de Microscopie -Faculté des Sciences Fondamentales et Biomédicales -Paris) and specially Jean-Maurice Petit.


F Bareille

S Franco

De Rosa

R P Singh

M Ichchou

O Bareille

F Franco

S De Rosa

Uncertainties in wave

Keywords: WFEM, uncertainty modeling, periodic media, bandgap, fuzzy analysis. 1

To predict the wave characteristics of the periodic media in the presence of fuzzy uncertainties, the wave finite element method in conjunction with fuzzy logic and algebra has been applied. For one-dimensional wave propagation, firstly, the most significant input parameters such as Young's modulus and mass density are identified and then fuzzified using the membership functions. Then, the fuzzy variable is propagated through the numerical model of interval analysis. The dispersion curves for flexural and longitudinal waves with fuzzy parameters have been used to illustrate the generality of the proposed approach also looking at the possibility to have frequency region in which those waves cannot propagate (frequency band gaps). The triangular membership functions have been used in the numerical examples and the obtained results are compared against the classical Monte Carlo simulations (MCS). The approach was presented for very simplified test-cases but it is found to be more efficient when compared with the conventional MCS approach in terms of computational cost.

INTRODUCTION

In recent years the propagation of elastic waves in periodic media with periodic material and geometrical properties has been extensively studied. These periodic media offer property known as band gaps where the propagation of the elastic wave is restricted in all the directions. Such properties are demanded for the vibroacoustic performance and dynamics of the structure and especially sought in aeronautics, transport, energy and space for vibration reduction, acoustic blocking, acoustic channeling and acoustic cloaking [START_REF] Singh | A study of structured uncertainties in wave characteristic of one-dimensional periodic structures[END_REF]. Three different frequency ranges identified: at the low-frequency range, because of the low order modes and low sensitivity to variability generally, the finite element approaches are applicabe; and high frequency, energy-based methods (e.g. Statistical Energy Analysis (SEA)) are well suited because of the involvement of many high order modes; finally the midfrequency range affected by the variability of the model parameter and wave-based approaches has been developed by increasing computational efficiency [START_REF] Singh | A Literature Review for the Analysis of Structured and Unstructured Uncertainty Effects on Vibroacoustic[END_REF]. The design of periodic media is generally based on deterministic models without considering the effect of intrinsic uncertainties existing in these media. In general, the design is aimed at controlling as much as possible the mechanical waves; however, inherent uncertainty may affect their characteristics. Periodic media are diffused in all the transportation engineering and demand a high level of robustness, which can be ensured with the careful consideration of the presence of uncertainty in the numerical models. The uncertainties, in terms of material properties and geometrical parameters, are mostly exhibited in both the manufacturing and assembly processes. To address this unavoidable actuality, the effects of uncertainties need to be considered when analyzing band structures (pass and stop bands). Generally, the stochastic characteristics of the periodic media can be determined by studying the design parameter uncertainties which are often modeled by random variables with consideration for spatial variability of the material and geometrical properties.

Manohar et al. studied the randomness in the wave propagation in waveguides using spectral element analysis [START_REF] Manohar | Axial vibration of a stochastic rod[END_REF]. Ichchou et al. proposed a numerical approach using the Wave Finite Element Method (WFEM) considering spatially homogeneous variability in waveguides using a first and second order perturbations are proposed [START_REF] Ichchou | Stochastic wave finite element for random periodic media through first-order perturbation[END_REF] [START_REF] Bouchoucha | Stochastic wave finite element method in uncertain elastic media through the second order perturbation[END_REF]. Recently Rayleigh quotient method introduced to investigate the symplectic eigenvalue problem of random symplectic matrix [START_REF] Zhao | Symplectic Approach on the Wave Propagation Problem for periodic structures with uncertainty[END_REF]. The material or geometrical uncertainty often displays spatial correlation, in layered structures. There, random fields theory can be used to model spatially distributed variability with probability measure [START_REF] Fabro | Wave propagation in onedimensional waveguides with slowly varying random spatially correlated variability[END_REF]. Also, this would be an approximate representation for the spatially varying system [START_REF] Fabro | Wave propagation in slowly varying waveguides using a finite element approach[END_REF].

The probabilistic models to account for these uncertainties employ probabilistic methods which require a wealth of data on probabilistic parameters. The exact sources of uncertainty are rarely found since their identification represents a difficult task. Furthermore, even small inaccuracies in the data can lead to large errors in the computed probability of response [START_REF] Sarkar | Mid-frequency structural dynamics with parameter uncertainty[END_REF].

When faced with incomplete information such as, "Young's modulus of the material lies between 70 GPa to 72.4 GPa," "mass ration is about 4%," and "lenght of side slightly larger than 1.2m," cannot be accomdated by deterministic model [START_REF] Liu | Fuzzy Finite Element Approach for Analysis of Fiber-Reinforced Laminated Composite Beams[END_REF]. In fact, statements as ''the mean are approximately equal to.. ." and ''the variance lies in the range.. ." are typical when handling real mechanical data and by virtue of their subjective nature they deal with fuzzy uncertainties [START_REF] Quaranta | Finite element analysis with uncertain probabilities[END_REF]; the adoption of the probabilistic approach can result in very challenging evaluations. In this scenario, the fuzzy set theory offers a way to approximating the uncertainty distribution in the form of the confidence interval through fuzzy membership functions. These are equivalent representations for the characterization of the linguistic, vague and missing data uncertainties.

The work reported so far in the literature with the best of the author's knowledge, is limited to wave propagation in elastic media and random media with probability method. In this work, a possibilistic method developed for the fuzzy uncertainty modeling and propagation in conjunction with WFEM and named as FWFEM (fuzzy wave finite element method). The approach was presented for a very simplified test-case 1D periodic bar and beam model with fuzzy uncertainty. It is modeled using triangular membership function and uncertainty bound are computed with fuzzy arithmetic steps and α-cut method. The results of FWFEM formulation are compared to those obtained from MCS.

FORMULATION OF FWFEM

Firstly the definition of the fuzzy membership function is revisited [START_REF] Singh | Vibration and Bending Behavior of Laminated Composite Plate With Uncertain Material Properties Using Fuzzy Finite Element Method[END_REF]. Let U is a classical set of objects, it is called the universe. The generic elements of U are denoted u. Then the membership in a classical subset A of U can be viewed as a characteristic function A from U to {0, 1} such that

1, if ( ) 0, if A u A u u A     =      (1)
The set {0, 1} is called a valuation set. A set A is called a fuzzy set if the valuation set is allowed to be the real interval [0, 1]. The fuzzy set A is characterized by the set of pairs

  [ , ( )],

A A u u u U  =  where ( ) A u 
is called the grade of the membership function.

  ( ), ( ) , [0 1] a b c d d b    + - - - 
(2) To propagate the fuzzy uncertainty in the dynamic model, the α-cut method is adopted. In this method, fuzzy membership function is descritized to different iInterval using α-cut level concept. For each α-cut, the interval analysis is performed with Sobol sequence to get the máximum and mínimum bound of the response.

One dimensional periodic media obtained by formulating the unit cell and then repeating in the propagation direction. Then, the study of this media can be converted into a study of unit cell based on the Floquet-Bloch theorem [START_REF] Mencik | Multi-mode propagation and diffusion in structures through finite elements[END_REF]. The schematic representation is shown in Fig. 2.

The WFEM is used to study the behavior of periodic media. This is extended to apply the WFEM for the fuzzy uncertainty modeling. This starts with consideration of the fuzzy membership function as a supplementary dimension through the fuzzy arithmetic using fuzzy set theory. Which start from the discretization of one sub-element of length (d). This discretization leads to fuzzy dynamic equilibrium of any substructure. The imprecise or vague variables are modeled using the triangular membership function, such that the dynamical equilibrium is expressed as:

(3)

The (.) symbol denotes the fuzzy arithmetic. The dynamic stiffness matrix is condensed into the left and right side denoted by L and R respectively.

(4) By defining two state vectors in the left and right side of each element and are related by symplectic matrix S as:

(5) Where Then using the periodicity and Bloch's theorem, spectral eigenvalue problem is expressed as

(6)
The solution of the above equation leads to identify the propagation constant and mode shapes. The considered structure is dissipative and the wave classified as incident ( ) and reflected ones (

). Considering the above classification the wave basis 𝜱𝜱 composed by eigenvectors in following form:

A B A B Cell k Cell k+1
Figure 2: Schematic representation of periodic structure [START_REF] Fabro | Wave propagation in onedimensional waveguides with slowly varying random spatially correlated variability[END_REF] Where q and F are the displacement and force component of eigenvectors with , , , of the size of matrices. To obtain the harmonic responce, The amplitude of wave at the first element obained using Projection of kinematic variable on wave basis and compute the displacement at the m th element by relation [START_REF] Fabro | Wave propagation in slowly varying waveguides using a finite element approach[END_REF]

RESULTS

This section shows the validation of FWFEM formulation presented in the previous section. The analysis is performed for the periodic rod and beam. Periodic rod and beam consist of section A of length and section B of length with different materials as depicted in the Fig. 3. Firstly, the longitudinal wave studied for the band gap and frequency response function (FRF) then flexural wave in the beam is studied.

Periodic rod with fuzzy uncertainty

In this subsection, the FWFEM is used to study the effect of fuzzy parametric uncertainty on the band diagram and FRF of a longitudinal wave in the periodic rod. The section of periodic rod namely A and B made of aluminum and steel respectively. The length of section A is and section B is with 1m each with circular cross section of radius of 0.0644m. To validate FWFEM results, a comparison made between the result from the proposed method and upper and lower bound obtained from MCS. The uncertainty effect is studied considering fuzzy uncertain material properties modeled with triangular fuzzy membership function with lower and upper variation fixed at ±10 percent of the nominal value. The considered frequency range for the numerical experiment from 1 Hz up to 5500 Hz. This frequency range considered to get the four complete band gap. The MCS In FWFEM formulation, stiffness and mass matrix of the two-node rod elements are considered.

The global stiffness and mass matrices formed with 100 elements in the unit cell of periodic rod in the MATLAB environment. The input material and geometric properties are shown in Table 1. It is to be mentioned that the material properties are choosen artibiary for the test case.

1: Geometric parameters and material properties of periodic rod

The variation of fuzzy Young's modulus of section A (material 1) and section B (material 2) is considered as furnished in Fig. 4. FWFEM result at α-cut=0 (largest interval) in terms of upper and lower bounds of the wave number is compared with sampling upper and lower bound with 5000 samples. The comparison is shown in Fig. 5. The results are in good agreement. The band gap envelops for each value of the fuzzy elasticity in every α-cut can be extracted shown in Fig. 6. It is visible that the effect of uncertainty (at the fixed level of uncertainty) on band gap bound is increasing with increasing frequency. The variation bound of the wave number at α-cut=0 is shown in Fig. 7 and it can be inferred that the response membership function is not always symmetric about mean/crisp value. The upper bound of wave number is more sensitive to the uncertain elasticity. The FRF of a periodic rod of the finite extent of 20 m excited by unit force on the left end is furnished in Fig. 8 indicating the upper and lower bounds comparison at α-cut=0 (maximum bound). In Fig. 9 fuzzy bound of the FRF for the different value of α-cut is plotted. It can be seen that as the value of α-cut increase the fuzzy bound of FRF also decreases and finally at α=1, without any simulation bound as expected, α=1 represent the deterministic value.

Geometrical and Material property Nominal value

Periodic beam with fuzzy uncertainty

In this subsection, the study of the fuzzy uncertainty effect on the flexural wave in the periodic beam is performed. For numerical simulation, Euler-Bernoulli beam theory is considered. The input material and geometrical properties are listed in Table 2.

For the fuzzy uncertainty propagation, the elasticity of material 1and material 2 is fuzzified using triangular membership function and shown in Fig. 4.

A two-node beam element with two degrees of freedom per node is considered. The global matrices (stiffness and mass) formed with 20 elements in the unit cell of the periodic beam in the MATLAB environment with input material and geometric properties given in Table 2.

FWFEM result at α-cut=0 is computed and compared with upper and lower bound obtained with MCS sampling (10000 samples) of the WFEM result and shown in Fig. 10. The frequency range of computation is 1-2000 Hz. The results are in good agreement. The variation bound of the wave number at a different frequency for the 11 α-cut levels is shown in Fig. 11. it showed that at the same level of the α-cut upper bound of wave number is more sensitive to the uncertain elasticity. The FRF bound is shown in Fig. 12 representatives the bound at the different α-cut level. As α-cut value increase the fuzzy bound of FRF is expected to decrease. same can be seen from Fig. 13 as the value of membership increase the fuzzy bound is decreasing and lowest at α=0.9.

For the computation cost comparison, elapsed time for the MCS and FWFEM obtained using homemade WFEM code which exploits the resources of the workstation with following characteristics, Intel® Core™i7 7820 HQ CPU@2.90GHz with 32 GB RAM and data storage using solid state drive. The comparison is presented in Table 3. Table 3: Elapsed time comparison rod and beam with fuzzy variations. The uncertainty quantification of wave number and frequency response function with fuzzy variables are derived using WFEM.

The approach was presented for very simplified test-cases, but it is found to be more efficient when compared with the conventional MCS approach in terms of computational cost. The present study can be extended for the future investigation for the complex unit cell and a large number of fuzzy variables.

Figure 1 :

 1 Figure 1: Triangular membership function α -Cuts: When an element . The α-cut Aα of A, which is a fuzzy set of crisp values U and is the original set of membership values greater than some threshold α [0 1]. If triangular fuzzy set A shown in Fig.1 is denoted by A= < a, b, d > then the α-cuts of the A can be shown as:

Figure 3 :

 3 Figure 3: Symmetric unit cell

Figure 4 :Figure 5 :

 45 Figure 4: Membership function for the fuzzy parameter

Figure 6 :Figure 7 :

 67 Figure 6: Band gap bound

Figure 8 :Figure 9 :

 89 Figure 8: Upper and lower bound comparison of FRF

Figure 10 :

 10 Figure 10: Upper and lower bound comparison

Figure 11 :

 11 Figure 11: Variation bound of wave number at 11 α-cut level

Figure 12 :

 12 Figure 12: Fuzzy bound of FRF for the different value of alpha-cut

Table 2 :

 2 Geometric parameters and material properties of the periodic beam

	Geometry/Property	Value
	Beam length	0.5m
	Height of beam	0.003m
	Width of beam	0.003m
	Young's modulus (A)	210 Gpa
	Young's modulus (B)	70 Gpa
	Mass density (A)	7800 kg/m 3
	Mass density (B)	2700 kg/m 3
	Loss factor (A) and (B)	0.01

R. P. Singh, M. Ichchou, O. Bareille, F. Franco, S. De Rosa

ACKNOWLEDGMENTS

The research of R.P. Singh was funded by an Early Stage Researcher grant within the European Project VIPER Marie Curie European Joint Doctorates (GA 675441).