Marwen Bouasker

Pierre Mounanga
email: pierre.mounanga@univ-nantes.fr

Philippe Turcry

Ahmed Loukili

A Khelidj

Chemical shrinkage of cement pastes and mortars at very early age: Effect of limestone filler and granular inclusions

Keywords: Chemical shrinkage, Cement paste, Mortar, Early age, Filler limestone, Hydration

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Chemical shrinkage of cement pastes and mortars at very early age: effect of limestone filler and granular inclusions

Introduction and context of the study

Early age shrinkage of cementitious matrices is the result of several complex physico-chemical phenomena. Those phenomena are related to the hydration reactions between cement and water and to the progressive hardening of the mineral skeleton. It is generally considered that at mature age, in autogenous conditions, the autodesiccation of the material, i.e. the progressive desaturation of its porosity, generates compressive forces high enough to cause volume changes called autogenous shrinkage. These forces are par-ticularly intense in cement-based matrices with low W/C ratio [START_REF] Bazant | Creep and shrinkage in concrete structures[END_REF].

At very early age and during setting, two phenomena are directly responsible for the shrinkage of material in sealed conditions: capillary depression in the porous space and Le Chatelier's contraction. Le Chatelier's contraction is explained by the density difference between hydrates on the one hand and cement and water on the other hand. At complete hydration, this contraction can represent approximately 10% of the initial volume of the material.

The material rarely exhibits such a high deformation because the progressive hardening resists to this chemical shrinkage. However at very early age and before setting, when the cementitious matrix does not reach its full maturity and resistance, Le Chatelier's contraction represents a significant part of the deformations. The comprehension of the autogenous shrinkage mechanism at the first stage of hydration thus requires to better quantify the influence of both mix and curing parameters on the amplitude and the kinetics of Le Chatelier's contraction.

This study, in the following of a previous research work on the influence of W/C ratio and curing temperature on the chemical shrinkage of cement pastes [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF], focuses on the effect of limestone filler and granular inclusions on Le Chatelier's contraction of mortars and cement pastes. These two mineral components can be regarded as chemically inert at early age. However, they have an indirect action on the hydration and the microstructure development and thus on the deformations of the cementitious matrices. The limestone filler can particularly play the role of nucleation and growth sites for the hydrates. Several studies revealed that the hydration process of C 3 S and C 3 A, two major components of Portland cements, was affected by the presence of limestone filler: the hydration kinetics is accelerated, the hydration reactions release more heat and the hydrate composition is modified [START_REF] Chloup-Bondant | Tricalcium aluminate and silicate hydration. Effect of limestone and calcium sulfate[END_REF][START_REF] Jiang | Influence of fillers on textural and mechanical properties of C 3 S pastes[END_REF][START_REF] Pe ´ra | Influence of finely ground limestone on cement hydration[END_REF][START_REF] Poppe | Effect of limestone filler on the cement hydration in self-compacting concrete[END_REF], particularly beyond one day of hydration [START_REF] Tsivilis | A study of the hydration of Portland limestone cement by means of TG[END_REF][8][START_REF] Bentz | Modelling the influence of limestone filler on cement hydration using CEMHYD 3D[END_REF]. Aggregates induce a better dispersion of the cement grain clusters during mixing and are usually at the origin of a porous interface zone with the binding phase [START_REF] Zimbelmann | A contribution to the problem of cement-aggregate bond[END_REF][START_REF] Bentz | Experimental and simulation studies of the interfacial zone in concrete[END_REF][START_REF] Ollivier | Interfacial transition zone in concrete[END_REF].

In this work, two cement types, CEM 1 and CEM 2, are investigated. The filler limestone incorporated is largely used for the making of self-compacting concrete [START_REF] Turcry | Cracking tendency of self-compacting concrete subjected to restrained shrinkage: experimental study and modelling[END_REF]. Granular inclusions are of two origins: a natural siliceous river sand and glass beads, in order to study the influence of the inclusion shape on the chemical shrinkage of mortars. The period of investigation is limited to the first 24 h of hydration. Indeed, beyond a few hours, the main mechanism of deformation becomes the autodesiccation of the matrix and the volume changes result from the competition between the gain of stiffness of the solid skeleton and the mechanical stresses generated by the progressive desaturation of the capillary space.

The article firstly analyzes the previous research works published on the evolution of the chemical shrinkage of cementitious systems. Then, the studied materials and the testing methods used in the present research task are detailed. The results obtained are discussed by comparing parallel measurements of chemical shrinkage and hydration degree.

Bibliographical analysis

Chemical shrinkage and autogenous shrinkage

At very early age, the autogenous shrinkage is a physicochemical phenomenon, coupled to mechanical capillary actions due to the consumption of pore water by the cement hydration. During the first hours of hydration (from the water-cement contact up to the beginning of the setting), the capillary actions can be regarded as negligible. During this phase, the autogenous shrinkage mainly corresponds to Le Chatelier's contraction (Fig. 1). Le Chatelier's contraction is measured by ensuring a constant water saturation of the porous space of the material. On the basis of this principle, many experimental studies [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF] have been carried out and some analytical and numerical models have been proposed [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF][START_REF] Holt | Early age autogenous shrinkage of concrete[END_REF][START_REF] Bentz | CEMHYD3D: A Three-Dimensional Cement Hydration and Microstructure Development Modeling Package[END_REF] in order to determine the chemical shrinkage of cement pastes.

Measurement methods

Justnes et al. [START_REF] Justnes | Chemical shrinkage of cement pastes with plasticizing admixtures[END_REF] listed, in 2000, three principal measurement methods of Le Chatelier's contraction: dilatometry, pycnometry and gravimetry. The principles of the three methods are recalled in Fig. 2.

Among these test methods, the most widely used remains the dilatometry. Developed by the inventor of Le Chatelier's contraction, it was sophisticated by Powers [START_REF] Powers | Absorption of water by Portland cement paste during the hardening process[END_REF] and by Buil [START_REF] Buil | Contribution a `l'e ´tude du retrait de la pa ˆte de ciment durcissante[END_REF]: Buil proposed for example an automatic follow-up of the water level in the capillary tube. Many researchers [START_REF] Justnes | An evaluation of methods for measuring chemical shrinkage of cementitious paste[END_REF][START_REF] Justnes | Total and external chemical shrinkage of low W/C ratio cement pastes[END_REF][START_REF] Justnes | The chemical shrinkage of pozzolanic reaction products[END_REF][START_REF] Justnes | The influence of cement characteristics on chemical shrinkage[END_REF][START_REF] Justnes | Chemical shrinkage of cementitious pastes with mineral additives[END_REF][START_REF] Justnes | Chemical shrinkage of cement pastes with plasticizing admixtures[END_REF][START_REF] Justnes | Correlating the deviation point between external and total chemical shrinkage with the setting time and other characteristics of hydrating cement paste[END_REF][START_REF] Tazawa | Chemical shrinkage and autogenous shrinkage of hydrating cement paste[END_REF][START_REF] Persson | Experimental studies of the effect of silica fume on chemical shrinkage and self-desiccation in Portland cement mortars[END_REF][START_REF] Wild | Chemical shrinkage and autogenous shrinkage of Portland cement -metakaolin pastes[END_REF][START_REF] Hammer | Early age chemical shrinkage and autogenous deformation of cement pastes[END_REF][START_REF] Boivin | Retrait au jeune a ˆge du be ´ton: De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF][START_REF] Beltzung | Dissolution of cement and early chemical shrinkage of cement paste[END_REF][START_REF] Charron | Comparative study of the effects of water/binder ratio and silica fume on the volume instability of hydrating cement pastes at early-age[END_REF][START_REF] Charron | Early-age deformations of hydrating cement systems: comparison of linear and volumetric shrinkage measurements[END_REF][START_REF] Bentur | Autogenous deformation and internal curing of concrete[END_REF] have used the dilatometry to study the influence of the cement composition, the W/C ratio, the presence of pozzolanic additions and polymer additives on the evolution of chemical shrinkage. This method is the object of a measurement protocol defined by the Japanese Concrete Institute (JCI) in 1999 [START_REF]Technical committee on autogenous shrinkage of concrete[END_REF].

In 1999, Garcia-Boivin [START_REF] Boivin | Retrait au jeune a ˆge du be ´ton: De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF] showed theoretically and experimentally the equivalence of the results obtained by dilatometry and gravimetry. The gravimetry, which is used in the present study, has been initiated by Rey [START_REF] Rey | Nouvelle me ´thode de mesure de l'hydratation des liants hydrauliques[END_REF] and improved by Geiker and Knudsen [START_REF] Knudsen | Chemical shrinkage as an indicator of the stage of hardening[END_REF][START_REF] Geiker | Chemical shrinkage of Portland cement pastes[END_REF][START_REF] Geiker | Measurements of chemical shrinkage and a systematic evaluation of hydration curves by means of the dispersion model[END_REF][START_REF] Knudsen | Obtaining hydration data by measurement of chemical shrinkage with an archimeter[END_REF], Paulini [START_REF] Paulini | A weighing method for cement hydration[END_REF][START_REF] Paulini | Chemical shrinkage as indicator for hydraulic bond strength[END_REF] and Garcia-Boivin [START_REF] Boivin | Experimental assessment of chemical shrinkage of hydrating cement paste[END_REF][START_REF] Boivin | Retrait au jeune a ˆge du be ´ton: De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF]. It allows, with a simple experimental device (see Fig. 2), to record automatic and continuous measurements of chemical shrinkage. CEM 2 one (Airvault). Note that for the CEM 1 cement, two batches, manufactured in the same factory but at different times, have been used: the CEM 1-a cement for the study on filler effect and the CEM 1-b cement for investigating the influence of granular inclusions. They have a similar particle size distribution (PSD) but slight differences in their Bogue's composition.

Experimental program

The composition and the Blaine specific surface of the cements and the mass fractions of the principal phases C 3 S, C 2 S, C 3 A and C 4 AF are given in Table 1. Fig. 3 provides the PSD of the cements.

Limestone filler

The limestone filler comes from Erbray (France). Its carbonate content is 98.4%. Its specific surface and bulk density are 3970 cm 2 /g and 2714 kg/m 3 , respectively. Fig. 3 displays its PSD.

Natural and artificial granular inclusions

The natural aggregate used is siliceous river sand. Before the mortar preparation, it was filtered with sieves of 1.25 and 0.80 mm (sand noted ''1 mm'') and sieves of 2.5 and 1.63 mm (sand noted ''2 mm''). The sand water content is 1.27%. The artificial aggregates correspond to completely non-porous and dry glass beads with a diameter of 2 mm.

Preparation of the cement pastes and mortars

The cement pastes and mortars are prepared by mixing the solid components with water, according to the procedures and mixing times detailed on Fig. 4. The 5 L-mixer used is in conformity with European standards . Finally, the different investigated mixtures are recapitulated in Table 2.

Test methods

Vicat setting time

Initial and final setting times of cement pastes are determined at 20 ± 1 °C according to the procedure described in the EN 196-1 European standards. The precision of the measurement is ±10 min. For each cement paste, three simultaneous Vicat tests are carried out to determine the average setting time.

Chemically-bound water and hydration degree

The hydration degree of the cementitious matrices is calculated from the measurement of the quantity of chemically-bound water (W n ). In the case of the cement pastes, the amount of W n , defined here as the mass loss recorded between 145 and 950 °C, is determined by thermogravimetric analysis (TGA). The detailed test procedure can be found in [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF][START_REF] Bouasker | Influence of limestone filler on chemical shrinkage and hydration of cement pastes at early age[END_REF].

The TGA method is not easily applicable to mortar because of the very little volume of the crucible containing the sample. The amount of the chemically-bound water within the mortars is thus measured by loss on ignition (LOI). This testing method indeed makes it possible to study larger material samples. The various stages of the method are recapitulated in Table 3.

The principal experimental parameters of TGA and LOI methods are finally summarized in Table 4. For each studied mixture, two samples are tested in order to calculate the average hydration degree of the material.

W/C = 0.4 A/C = 0 CEM 1-b 0.40 - - - - Glass (2 mm) W/C = 0.

Chemical shrinkage

The chemical shrinkage is measured by gravimetry. The test procedure is detailed in [START_REF] Mounanga | Predicting Ca(OH) 2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF]. Table 5 points out the main experimental parameters of the testing method. Each studied mixture is prepared and tested twice in order to determine the average curve of chemical shrinkage.

Several researchers [START_REF] Geiker | Chemical shrinkage of Portland cement pastes[END_REF][START_REF] Boivin | Retrait au jeune a ˆge du be ´ton: De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF][START_REF] Sant | Measurement of volume change in cementitious materials at early ages: review of testing protocols and interpretation of results[END_REF] were interested in quantifying the influence of bleeding and ponding fluid on the rate and amplitude of chemical shrinkage.

Geiker and Knudsen [START_REF] Geiker | Chemical shrinkage of Portland cement pastes[END_REF] measured the amount of bleeding water 2 h after the first water-cement contact and corrected the W/C ratio of the cement pastes subjected to this bleeding. The higher difference of W/C ratio before and after bleeding observed on cement pastes with W/C = 0.8 was about 5%. This difference was not significant for the pastes with W/C = 0.40.

Sant et al. [START_REF] Sant | Measurement of volume change in cementitious materials at early ages: review of testing protocols and interpretation of results[END_REF] tested various concentrations and compositions of the ponding fluid on the sample and recorded a difference in chemical shrinkage during the very first hours. Garcia-Boivin [START_REF] Boivin | Retrait au jeune a ˆge du be ´ton: De ´veloppement d'une me ´thode expe ´rimentale et contribution a `l'analyse physique du retrait endoge[END_REF] carried out the same experiment with distilled water and water saturated with lime as the ponding fluid. By readjusting the shrinkage curves at 100 min after the first water-cement contact, she obtains almost superimposed evolutions of chemical shrinkage.

In the continuation of the present study, the liquid of immersion is distilled and deaerated water and no particular precaution is taken to remove the bleeding water of the cementitious matrices, considering their W/C ratio (W/ C = 0.30 and 0.40) and in accordance with Geiker and Knudsen's results [START_REF] Geiker | Chemical shrinkage of Portland cement pastes[END_REF].

Results and discussion

Repeatability of the tests

As previously indicated, each test is achieved twice for the measurement of the hydration degree and the chemical shrinkage and three times for the Vicat setting times. The maximum variations recorded between two tests carried out on the same mortar or cement paste formulation are given in Table 6. In the following of this article, each curve or value presented is an average result calculated with two or three test results.

Effect of limestone filler

Hydration degree

Figs. 5 and 6 present the evolution of the hydration degree of the various cement pastes. The presence of limestone filler is at the origin of an increase of hydration rate of the two cement types. This filler effect, already described in previous studies [START_REF] Chloup-Bondant | Tricalcium aluminate and silicate hydration. Effect of limestone and calcium sulfate[END_REF][START_REF] Jiang | Influence of fillers on textural and mechanical properties of C 3 S pastes[END_REF][START_REF] Pe ´ra | Influence of finely ground limestone on cement hydration[END_REF][START_REF] Poppe | Effect of limestone filler on the cement hydration in self-compacting concrete[END_REF][START_REF] Tsivilis | A study of the hydration of Portland limestone cement by means of TG[END_REF][8][START_REF] Bentz | Modelling the influence of limestone filler on cement hydration using CEMHYD 3D[END_REF], is generally explained by the additional solid surface brought by the filler particles, which act like sites of nucleation and growth for the hydrates and induce an acceleration of the hydration reactions. Figs. 5 and6 show that the filler effect is significant since the very first hours of hydration. In addition, during the first 12 h, the increase in limestone filler proportioning from 20% to 40% does not cause additional acceleration of the cement hydration. Beyond 12 h of hydration, the filler effect depends on the cement type. Indeed, it can be observed on Fig. 7 that in the case of CEM 2 cement, the hydration degree at 24 h of hydration is an increasing linear function of the filler proportioning, whereas for the CEM 1-a cement, the value of hydration degree at 24 h exhibits a maximum for LF/C = 0.25.

Vicat setting time

Table 7 displays the results of the Vicat needle tests. For both cement types, the larger the degree of substitution LF/C, the shorter the final Vicat setting time.

The acceleration of setting is a consequence of the cement hydration rate increase previously observed (section 4.2.1; Figs. 5 and6). The acceleration of the hydrate production causes a faster percolation of the solid network. It is noted however that the effect of the filler is much more significant on the setting time than on the hydration rate. In fact, it is necessary to keep in mind that the Vicat needle test is a mechanical one, whose result depends largely on the consistency of the cement paste and thus on the water/solid ratio W/(LF + C). Indeed, the hydrate volume required to reach the percolation threshold decreases when the water/solid ratio decreases. The reduction of the setting time consecutive to the limestone filler addition should be interpreted as the combined result of the hydration rate increase and the augmentation of the solid fraction.

Chemical shrinkage

Figs. 8 and 9 represent the evolution of the chemical shrinkage as a function of the cement paste age up to 24 h of hydration. The curves are initialized at 1 h after the first water-cement contact.

For a given age, the chemical shrinkage is higher for the cement pastes containing limestone filler. At least two explanations can be advanced: on the one hand, as the chemical shrinkage is a direct consequence of the hydration reactions, the acceleration of the cement hydration due to the presence of the filler induces an increase of the chemical shrinkage rate. In addition, the presence of filler modifies cement hydration products. For example, the monosulfoaluminate (Afm) and the limestone filler react slowly to form monocarboaluminate (Afmc) (see the bibliographical analysis in [START_REF] Bentz | Modelling the influence of limestone filler on cement hydration using CEMHYD 3D[END_REF]), which has a higher density than the Afm phase [START_REF] Renaudin | Order and disorder in the lamellar hydrated tetracalcium monocarboaluminate compound[END_REF]. This latter effect is quantified by representing the chemical shrinkage evolution as a function of cement hydration degree. On Figs. 10 and11, it can be observed that, for each cement type, there is a quasi-linear relation between the two parameters, whatever the limestone filler proportioning. This result shows that the filler effect on the cement paste chemical shrinkage can be regarded as a mainly kinetic effect at very early age.

Effect of granular inclusions

The second part of this article is dedicated to the study of the influence of granular inclusions on the rate and the amplitude of the cementitious matrix chemical shrinkage at very early age. The effect of shape, size and concentration of aggregates is investigated. Two aggregate types are used: glass beads with a diameter of 2 mm and natural river sand filtered to obtain two tightened particle size distributions. Sand denoted ''2 mm'' corresponds to the fraction of aggregates recovered between the sieves 1.6 and 2.5 mm and sand called ''1 mm'' is the fraction obtained between the sieves 0.8 and 1.25 mm.

Influence of aggregate shape, concentration and size

Fig. 12 displays the evolution of chemical shrinkage between 1 and 24 h for different W/C-ratio mortars (W/ C = 0.30 and 0.40) prepared with aggregates of different shape (sand and glass beads), size (1 and 2 mm) and with different concentrations A/C (A/C = 0.5 and 1).

The curves obtained are almost superimposed up to 8 h of hydration; beyond, they form a tightened spindle until the end of the test. Keeping equal all other parameters, a modification of W/C ratio, aggregate concentration or shape causes only weak or very weak variation (lower than 3%) of the chemical shrinkage rate of the studied mortars.

At 24 h, the maximum difference, equal to 11%, is observed between the final amplitudes of the curves of mortars prepared with the same W/C and A/C ratios but with a different aggregate size: these are the curves ''Sand (1 mm) W/C = 0.4 A/C = 0.5'' and ''Sand (2 mm) W/C = 0.4 A/C = 0.5''. The difference between the two curves can be explained by a more effective mixing in presence of ''1 mm'' aggregates. Indeed, for a constant granular mass fraction, the finer aggregates are more numerous and their granular specific surface per gram of sample is larger: the number of cement clusters broken by shearing or impact with sand grains is then more important and the cement fraction directly in contact with water increases. It results in higher chemical shrinkage amplitude for the mortar containing ''1 mm'' aggregates.

It should finally be noted that a difference of 11% also exists between the curves ''Sand (1 mm) W/C = 0.4 A/ C = 0.5'' and ''Glass (2 mm) W/C = 0.3 A/C = 1'', but the simultaneous variation of several mix parameters makes the interpretation of this difference very complex.

Effect of granular surface area

In order to specify the aggregate effect on the chemical shrinkage of sand mortars, the chemical shrinkage amplitude reached at 23 h is represented on Fig. 13 as a function of the granular specific surface area per gram of sample. This specific surface is calculated by assuming the aggregates ''1 mm'' and ''2 mm'' as spheres of radius 0.5 and 1 mm, respectively. Fig. 13 shows a tendency of the chemical shrinkage amplitude to increase as the developed aggregate surface area grows. The influence of aggregates on Le Chatelier's contraction was already discussed by Holt [START_REF] Holt | Early age autogenous shrinkage of concrete[END_REF]: she supposes that this effect is mainly due to the improvement of mixing, inducing a better cement cluster destruction, and the formation of a very porous interfacial zone around the aggregates. It should be noted that this double effect remains weak: on Fig. 13, the multiplication by 3 of the specific surface area of aggregates does only cause an increase of 10.5% of the chemical shrinkage at 23 h of hydration.

Chemical shrinkage vs. hydration degree

The analysis of Figs. 12 and 13 revealed that the presence of aggregates only causes a slight increase of the mortar chemical shrinkage. This result is in accordance with the hydration degree measurements whose results are provided in Fig. 14.

In order to quantify the relation between Le Chatelier's contraction and the degree of hydration of mortars, the evolution of the chemical shrinkage has been plotted as a function of the hydration degree on Fig. 15. It is observed that for the various studied mortars, the amplitude of the chemical shrinkage increases quasi-linearly with the progress of hydration and the slope value of this linear function can be considered as unique for the whole mixtures.

Conclusions

The results of this study on the influence of limestone filler and granular inclusions on the evolution of chemical shrinkage and hydration degree of cementitious matrices between 0 and 24 h showed that:

• The presence of limestone filler causes an acceleration of the chemical shrinkage and the hydration process since the very first hours. This filler effect is generally explained by the supplementary surface area provided by the filler particles, playing the role of nucleation sites for hydrates. Consecutively, the Vicat setting time is also accelerated in presence of limestone fillers. The increase of chemical shrinkage and hydration rate depends on the cement type: it is more important for CEM 2 cement than for CEM 1 cement. • The presence of granular inclusions with diameter between 1 and 2 mm, only has a weak influence on the evolutions of Le Chatelier's contraction and hydration degree of mortars at very early age. • Limestone fillers and granular inclusions do not cause significant modification of the quasi-linear relation linking the chemical shrinkage to the hydration degree of the cementitious matrices.

This last information is particularly important in the aim of predicting the early age volume changes of cement-based systems: the chemical shrinkage is the first phase of deformation (Fig. 1) and its expression as a function of the hydration degree can be obtained, for a given mortar, directly and with a good approximation from the corresponding W/C-ratio cement paste (Figs. 10, 11 and 15).

3. 1 .Fig. 1 .

 11 Fig. 1. Chemical and volumetric autogenous shrinkage of a CEM 1 cement paste with W/C = 0.40. Inspired by Bouasker et al. [42].

Fig. 2 .

 2 Fig. 2. Chemical shrinkage measurement methods.

Fig. 3 .

 3 Fig. 3. Particle size distribution of cements and limestone filler.

  3 A/C = 1 0.30 Glass beads (B 2 mm) 1.00 Sand (2 mm) W/C = 0.3 A/C = 0.5 Natural sand (B 2 mm) 0.50 Glass (2 mm) W/C = 0.4 A/C = 0.5 0.40 Glass beads (B 2 mm) 0.50 Glass (2 mm) W/C = 0.4 A/C = 1 1.00 Sand (1 mm) W/C = 0.4 A/C = 0.5 Natural sand (B 1 mm) 0.50 Sand (2 mm) W/C = 0.4 A/C = 0.5 Natural sand (B 2 mm) 0.50 Sand (2 mm) W/C = 0.4 A/C = 1 1.00 C, cement; W, water; LF, limestone filler; A, aggregates.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Hydration degree of CEM 1-a cement pastes (W/C = 0.40).

Fig. 7 .

 7 Fig. 7. Hydration degree of cement pastes at 24 h.

Fig. 8 .CEM 2 : 4 Fig. 9 .

 8249 Fig. 8. Chemical shrinkage of CEM 1-a cement pastes (W/C = 0.40).

Fig. 10 .

 10 Fig. 10. Chemical shrinkage vs. hydration degree of CEM 1-a cement pastes (W/C = 0.40).

5 Fig. 12 .

 512 Fig.12. Effect of aggregate shape.

4 Fig. 13 .

 413 Fig.[START_REF] Turcry | Cracking tendency of self-compacting concrete subjected to restrained shrinkage: experimental study and modelling[END_REF]. Effect of granular surface area on chemical shrinkage measured at 23 h of hydration.

Fig. 11 .

 11 Fig. 11. Chemical shrinkage vs. hydration degree of CEM 2 cement pastes (W/C = 0.40).

Fig. 14 .

 14 Fig. 14. Hydration degree of cement paste and mortars.

Fig. 15 .

 15 Fig. 15. Chemical shrinkage vs. hydration degree of CEM 1 cement paste and mortars.

Table 1

 1

	Chemical composition of the cements used
	Composition	CEM I 52.5 N	CEM II/A-LL 42.5 R CE
	(mass percentage)	CE CP2 NF		PM-CP2 NF
		CEM 1-a	CEM 1-b	CEM 2
	Clinker	97		92
	Limestone	2		6
	Filler	1		2
	Bogue's cement composition (mass percentage)
	C 3 S	62.89	62.01	53.31
	C 2 S	9.12	11.13	17.69
	C 3 A	6.76	8.11	8.11
	C 4 AF	9.54	8.45	8.45

Table 3

 3

	Stages of the loss of ignition (LOI) method
	Stages			Time
	Sampling of 50 ± 10 g of mortar			10 min
	Crushing of the sample			10 min
	Impregnation of the sample with methanol (to stop hydration)	10 min
	Drying at 145 °C			2 0 h
	Placing in a desiccator			15 min
	Mass measurement (accuracy: ±0.01 g)		2 min
	Heating at 950 °C			4 h
	Placing in a desiccator			15 min
	Mass measurement (accuracy: ±0.01 g)		2 min
	Table 4
	Test parameters for the measurement of chemically-bound water
	Parameters	Test methods
		TGA	LOI
	Mass of sample	40-70 mg	15-20 g
	Age of sample	3, 6, 9, 24 h 4, 8, 12, 16, 20, 24 h
	Type of crucible	Platinum	Porcelain
	Heating rate	10 °C/min	-
	Temperature range for the	145-950 °C 145-950	°C
	calculation of W n
	Environment	Nitrogen	Air (1 bar)
		(1 bar)
	Pre-treatment of the sample	None	(1) Crushing
			(2) Impregnation
			with methanol
			(3) Drying at 145	°C

Table 5

 5 Test parameters for the measurement of Le Chatelier's contraction

	Test parameters	Value
	Mass of sample	20-30 g
	Exchange surface area between sample and bath water	20 ± 1 cm 2
	Thickness of sample	9 ± 1 mm
	Liquid of immersion	Distilled water
	Temperature of the liquid of immersion	20 ± 0.1	°C

Table 7

 7

	Results of Vicat setting tests
	Designation	LF/C	Initial Vicat	Final Vicat
			setting time (h)	setting time (h)
	CEM 1:CaCO3-1:0	0	3.0	7.4
	CEM 1:CaCO3-0.8:0.2	0.25	2.3	4.8
	CEM 1:CaCO3-0.6:0.4	0.67	1.2	3.2
	CEM 2:CaCO3


Jean-Marc Collin
email: jeanmarc.collin@yahoo.fr

Gérard Mauvoisin

Olivier Bartier

Rochdi El Abdi

Philippe Pilvin

Experimental evaluation of the stress-strain curve by continuous indentation using different indenter shapes

Keywords:

Experimental applications of the methodology developed for spherical indentation are proposed in this paper. Two quasi-spherical indenters with different shapes were used in order to evaluate the stress-strain curve of five steels. Although the shape of the indenter was not perfectly spherical, it was shown that models developed for spherical indentation can be used with an adequate correction. The results are in good agreement with those obtained by tensile tests. Moreover, the case of the austenitic alloy (AISI 316L) revealed the importance of sample preparation for the experimental results.

Introduction

The indentation test was developed several decades ago in order to determine certain mechanical properties. The determination of the elastic properties is based on equations deduced from the Hertz theory [1][2][3][4][5][6][7][8][9]. An overview of the elastic theories applied to indentation is proposed in the paper of Borodich and Keer [10]. Concerning the determination of plastic behaviour, most of the proposed methods are commonly called reverse analysis. In such an approach, the determination of the mechanical properties is based on the inversion of models deduced from numerical studies of the indentation test. Most of these methods are based on Tabor's theory [START_REF] Tabor | Hardness of Metals[END_REF], which introduced the representative strain [START_REF] Dao | [END_REF][13][14][15]. We can however quote other methods which are not based on the Tabor theory; Kucharski and Mróz [16] use several partial unloadings in order to determine the plastic part of the indentation depth. The determination of the yield stress and the hardening parameters can be calculated from this plastic part. Nayebi et al. [17] propose to determine the yield stress and the work hardening exponent from indentation loading. However, this method can only be applied to steels with a Young's modulus of 210 GPa and a Poisson ratio of 0.3. Another way of determining mechanical properties by indentation is by the inverse analysis approach [18,19]. In this approach, an adequate algorithm is used in order to minimize the difference between experimental and simulated observable variables.

An interpretation of spherical indentation experiments and the determination of hardening law parameters was the challenge of previous papers by Collin et al. [20,21]. The first paper concerned the determination of the contact radius by using equations deduced from the Hertz theory [1]. A methodology was then proposed to deduce both the indenter deformation and the contact radius from the knowledge of indenter and sample elastic properties. The difference ı between the measured displacement and the desired one (Fig. 1) (linked to indenter deformation) is quantified by the following formula which can be applied to spherical indenters:

ı = R A + B P R 2 1 -2 i E i + C P a 2 1 -2 i E i + D P R 2 1 -2 i E i 2 +E P a 2 1 -2 i E i 2 + F P R 2 1 -2 i E i 2 P a 2 (1)
In Eq. ( 1), R is the indenter radius, P is the indentation load, a is the contact radius and coefficients A-F are given by A = -1.6817.10 -5 ; B = 3.6242; C = 0.2003; D = -7784.7622; E = -15.3689 and F = 649.4548. Eq. ( 1) is different to the one proposed in [20] in order to take into account the indenter Poisson ratio value. The polynomial equation which links the contact radius a and the elastic properties of both the indenter (denoted E i and i ) and the sample (denoted E s and s ) can be written as

˛E2 s + ˇEs + = 0, ( 2
)
˛= 1 -2 i E i 1 -2 i E i S -2a , ˇ= 2(1 -2 s ) 1 -2 i E i S -a - 4a 2 (1 -2 s )(1 + s ) 3 R , = (1 -2 s ) 2 S.
In these equations, S is the unloading slope given by S = dP/dh. Eqs. ( 1) and ( 2) only depend on the elastic properties of both the indenter and the sample. However, these equations can be applied in the presence of plastic deformations of the sample. Indeed, the contact radius a or the unloading slope S depend on the plastic properties of the sample. Thus, Eqs. (1) and (2) are indirectly linked to the plastic properties of the sample by the appearance of the contact radius or the unloading slope.

The second paper concerned the study of the uniqueness of the solution in the case of spherical indentation. Several models were proposed in order to describe the changes of four pieces of indentation data for power hardening law parameters. It was then shown that it is possible to deduce a quite good evaluation of the power law parameters by inversion of models. It was shown in [20,21] that isotropic hardening behaviour was not sufficient to describe the whole unloading part of the indentation curve. Then the data which can be used in the reverse analysis are the total energy W t and the indentation load P. Our numerical analysis showed that P/E*R 2 and W t /E*R 3 are well described by a power equation (E* is the reduced Young modulus). As an example, the indentation loading curve is given by Eq. ( 3):

P E * R 2 = h R A exp(-B) (3)
In Eq. ( 3) coefficients A and B depend on the plastic properties of the sample. The reader can refer to [21] in order to find more details about A and B. As an example, of the Hollomon law parameters ( y and n) and for the P(h) curve, A and B are given by the following formulas using * y = y /E * : In this paper, we propose the application of the methodology developed in [20,21] on five steels which present different kinds of microstructures. Moreover, it is shown that spherical indentation theories can be used even if the indenter has not a perfectly spherical shape.

A = 0.

Materials

In this study, five different steels were considered. Among them, four are low carbon steels: AISI 1035, 1080, 1100 and 35CrMo4. In these steels, two are composed of pearlite and ferrite phases (1035 and 35CrMo4) (Fig. 2) and two have been treated in order to transform the pearlite phase into cementite globules and ferrite (1080 and 1100) (Fig. 3). These four steels have a body-centred cubic crystal structure. The last sample is the well-known austenitic steel, AISI 316L (Fig. 4) with a face-centred cubic crystal structure.

Figs. 234show the indentation imprint obtained for a load of 200 N. At this applied load, the number of metallic grains which are affected by the indentation is sufficient for considering the tested domain as representative.

The elastic properties of these samples were measured by an ultra-sound method and are close to E = 210 GPa and = 0.29 for 1035, 1080, 1100 and 35CrMo4. Concerning 316L, the measured values are E = 195 GPa and = 0.28. Tensile tests, realized on normalized samples issued from the same rod as the indentation samples, led to the stress-strain curve of the five steels.

Indenters

Indentation tests were realized on the five samples with two different indenters made of Tungsten Carbide. The first one, was previously presented in [21], and does not have a perfectly spherical shape. Thus, its profile was measured in order to simulate its behaviour. The second indenter has a sphere shape of radius R = 1.25 mm. The elastic properties of these two indenters were measured by ultra-sound. The Young modulus was E i = 590 GPa and the Poisson ration was i = 0.22.

We realized indentation tests with these two indenters at 200 N and 10 loading, unloading and reloading cycles. In order to apply the spherical indentation theories to the first indenter, an equivalent radius function was determined. An indentation test was simulated with the true profile of this indenter on a fictive sample (with fixed Hollomon's law parameters). From the simulated P(h) curve, the model (Eq. ( 3)) and the mechanical properties of the simulated sample, it was possible to determine which equivalent radius can give the same P(h) curve as the simulated one. In other words, Eq. ( 3) was inversed with R considered as an unknown function. This led to an equivalent radius function (quoted R eq ) which is determined as a function of the contact radius a. This procedure can be applied to other indenters which are not perfectly spherical in order to apply spherical indentation theories. In this case, the indenter radius R has to be replaced by the R eq function.

Experimental evaluation of the stress-strain curve by continuous indentation

It was observed that depending on the points which are considered for minimization, we can obtain a minimum with a low yield stress and a very high work hardening exponent. This phenomenon occurs if only the first points of the P(h) curve are considered. It can be explained by the low deformation level at this stage which is not sufficient to describe the material's behaviour well. Fig. 5 presents the comparison between experimental and numerical P(h P0 ) curves which correspond to the measured indentation curves before correction. This figure shows that the simulated P(h) curve with this minimum is not close to the experimental one. We can say that, although several minima can be found, this is not a problem of non-uniqueness of the solution. This is the reason why we suggest simulating the found minimum in order to compare the experimental and numerical curves. If the two curves are close, the minimum can be considered as correct.

The procedure described in [20] has been applied to the experimental indentation curves in order to remove the displacement due to the indenter's deformations. It enabled the P(h P1 ) curve (see Fig. 1) to be obtained from the P(h P0 ) measured curve together with knowledge of the elastic properties of both the indenter and the sample. This procedure also allowed the contact radius changes during the indentation test to be determined. From the P(h P0 ) curves, it was possible to determine the indentation data W t (h), W e (h) and W p /W t (h). As explained in [21], W e (h) and W p /W t (h) cannot be used in the minimization procedure as a consequence of the kinematic hardening influence. We observed that minimization with W t (h) or P(h) gave almost the same results. Then, the presented results are obtained after minimization on the P(h) curve.

The identification procedure was applied on the five steels and the solutions found were simulated. Figs. 6789present the comparison between the contact radius changes which were experimentally determined and the simulated ones for the two studied indenters. In Fig. 7, which concerns AISI1080, we can see that, for the non-spherical indenter, a difference occurs between the experimental and the numerical contact radius at the end of the indentation test. Experimental displacement is higher than that of the simulated one and the experimental contact radius is less than that of the simulated one. This is in concordance with Fig. 5 where we can observe a difference between the experimental and numerical unloading part. This difference can be explained by a variation of the elastic properties of the sample with plastic deformation. However, this aspect cannot be taken into account in the present work. Figs. 10-13 present the comparison between the identified stress-strain curves and the experimental ones for AISI1035, AISI1080, AISI1100 and 35CrMo4. In these figures, the identified parameters were obtained after minimization on the whole P(h) curve. It is then shown that, for these four samples, we obtained a quite good evaluation of the stress-strain curve by continuous indentation. Moreover, the use of a non-spherical indenter with the adequate correction also gave a good evaluation of the stress-strain curve.

Concerning the austenitic steel 316L, Fig. 14 shows that this steel is very sensitive to the way the surface was polished. If this was only manually polished, the indentation displacement was much lower than if the surface was prepared with electrolytic polishing. Among the five studied samples, only 316L presents this phenomenon. This can be explained by several aspects. First of all, as this steel is not magnetic, it is impossible to rectify the indentation sample before polishing in order to remove the deformed coat induced by the sample machining. Moreover, Fig. 15 shows that the stress increased very quickly at the beginning of the stress-strain curve. Then, on the one hand, if we consider the yield stress as the end of the elastic area (area 1 in Fig. 15) of the curve, we obtain approximately 100 MPa. On the other hand, if we consider the commonly used yield stress at a plastic strain equal to 0.2% (area 2 in Fig. 15), we obtain approximately 400 MPa. This is the reason why electrolytic polishing was realized on the 316L indentation samples. However, this method is limited and it is not possible to remove a great depth without damaging the surface, particularly the planarity.

Fig. 16 shows the application of our methodology on 316L steel with manual and electrolytic polishing. We can see that the results obtained from the sample which was manually polished are higher than the experimental stress-strain curve. Concerning the electrolytic polished sample, we can observe that the results are nearer to the experimental stress-strain curve. However, electrolytic polishing seemed not to have removed the entire deformed coat on the surface of the sample. This example, which does not work as well as the four other samples, shows the importance of the preparation of the indentation sample. As a conclusion, the investigation on the five studied materials shows that the indentation test can provide a quite good evaluation of the stress-strain curve. The results are in good agreement with the tensile tests for low carbon steels which can be rectified and do not present a high hardening level for low deformations. Concerning the 316L steel, the results obtained by indentation cannot be considered as incorrect. Indeed, the simulation is in good agreement with the experimental indentation curve (Fig. 17 for the spherical indenter). In this case, the indentation test shows that the surface of the sample is not representative of the tensile test sample.

Thus, if the aim of the indentation test is to have a good assessment of the strain-stress curve, an adequate preparation of the surface is required. However, if the aim of the test is to characterize the material's properties at a localized area, Figs. 14 and16 show that different indentation curves lead to a different stress-strain curve. A comparison with the stress-strain curve obtained by a tensile test can determine how much the surface was hardened.

Conclusion

An experimental evaluation of the spherical indentation theories developed in [20,21] was performed on five different steels. It is shown that these theories can be applied in order to obtain a good evaluation of the stress-strain curve even if the indenter does not have a perfectly spherical shape. We then propose the methodology in order to correct the indentation curve obtained with non-spherical indenters. This procedure is not restricted to our indenter, as it can be applied to other indenters. However, it required a good knowledge of the indenter profile. Moreover, the results presented in this paper are not restricted to our models. Methods cited in the first paragraph can be applied with nonspherical indenters with the adequate correction function.

It is also shown that, for some cases, the preparation of the sample surface has a great influence on the results. For samples which can be polished manually after machining quite accurate results can be obtained. However, for the steel AISI 316L, electrolytic polishing gives better results.

Fig. 1 .

 1 Fig. 1. Difference between measured displacement hP0 and wanted displacement hP1.

Fig. 2 .

 2 Fig. 2. Microstructure of AISI 1035.

Fig. 3 .

 3 Fig. 3. Microstructure of AISI 1100.

Fig. 4 .

 4 Fig. 4. Microstructure of AISI 316L.

Fig. 5 .

 5 Fig. 5. Comparison between experimental and numerical indentation curves for AISI 1080.

Fig. 6 .

 6 Fig. 6. Comparison between experimental and numerical a(h) curves for AISI 1035.

Fig. 7 .

 7 Fig. 7. Comparison between experimental and numerical a(h) curves for AISI 1080.

Fig. 8 .

 8 Fig. 8. Comparison between experimental and numerical a(h) curves for AISI 1100.

Fig. 9 .

 9 Fig. 9. Comparison between experimental and numerical a(h) curves for 35CrMo4.

Fig. 10 .

 10 Fig. 10. Determination of stress-strain curve for AISI 1035.

Fig. 11 .

 11 Fig. 11. Determination of stress-strain curve for AISI 1080.

Fig. 12 .

 12 Fig. 12. Determination of stress-strain curve for AISI 1100.

Fig. 13 .

 13 Fig. 13. Determination of stress-strain curve for 35CrMo4.

Fig. 14 .

 14 Fig. 14. Influence of polishing on the indentation curve for AISI 316L.

Fig. 15 .

 15 Fig. 15. Beginning of the stress-strain curve of AISI 316L.

Fig. 16 .

 16 Fig. 16. Determination of stress-strain curve for AISI 316L.

Fig. 17 .

 17 Fig. 17. Comparison between experimental and numerical indentation curve for the steel AISI 316L.

Acknowledgment

This research was sponsored by the Region of Brittany.


Patrick Lemonnier

Abdul-Hamid H Soubra

Richard Kastner

Variational displacement method

Keywords: Slopes, Reinforced soil, Analysis, Stability

published or not. The documents may come

Introduction

The problem of internal and external stability is one of the most important in the design and analysis of geosynthetic reinforced soil slopes. The internal stability includes the rupture of reinforcement and pullout failure. The external stability includes sliding, overturning, bearing capacity and overall stability. The majority of methods available in literature concerning the determination of the safety factor against failure are based on lateral earth pressure considerations [START_REF] Steward | Guidelines for use of fabrics in construction and maintenance of low-volume roads[END_REF]Broms, 1978;Collin, 1986;[START_REF] Bonaparte | Soil reinforcement design using geotextiles and geogrids. Geotextile testing and the design engineer[END_REF] or employ the approach commonly used in conventional slope stability analysis, modified to account for the inclusion of the tension in the reinforcements [START_REF] Blanchier | Etude de la stabilite´d'un talus renforce´par des nappes ge´otextiles[END_REF][START_REF] Leshchinsky | Stability of membrane reinforced slopes[END_REF][START_REF] Delmas | Le dimensionnement des ouvrages renforce´s par ge´otextiles programme CARTAGE[END_REF][START_REF] Hird | Stability charts for reinforced embankments on soft ground[END_REF][START_REF] Huisman | Design guideline for reinforced embankments on soft soil using Stabilenka reinforcing mats[END_REF][START_REF] Low | Slip circle analysis of reinforced embankments on soft ground[END_REF][START_REF] Kaniraj | Rotational stability of unreinforced and reinforced embankments on soft soils[END_REF]. The latter methods have used different assumptions concerning the magnitude and the inclination of the reinforcement tension at the failure surface. The different directions that have been assumed by some investigators are (1) horizontal [START_REF] Leshchinsky | Stability of membrane reinforced slopes[END_REF][START_REF] Delmas | Le dimensionnement des ouvrages renforce´s par ge´otextiles programme CARTAGE[END_REF][START_REF] Hird | Stability charts for reinforced embankments on soft ground[END_REF], (2) tangential to the failure surface [START_REF] Delmas | Le dimensionnement des ouvrages renforce´s par ge´otextiles programme CARTAGE[END_REF], (3) in the direction of the bisector to the horizontal and the tangential directions [START_REF] Huisman | Design guideline for reinforced embankments on soft soil using Stabilenka reinforcing mats[END_REF][START_REF] Low | Slip circle analysis of reinforced embankments on soft ground[END_REF][START_REF] Kaniraj | Rotational stability of unreinforced and reinforced embankments on soft soils[END_REF] and ( 4) orthogonal to the radius of the log-spiral slip surface [START_REF] Leshchinsky | Stability of membrane reinforced slopes[END_REF].

Furthermore, there is a number of other design methods available for geosynthetic reinforced slopes based on a displacement approach. These methods have considered a prescribed displacement at the top of the slope in order to mobilise the tensions in the reinforcements. The original method, called the 'displacement method' [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF], is being widely used in France and in several other countries for the design of such structures. The use of this method to estimate the failure limit state of such structures can now be considered to give satisfactory results. Geotechnical engineers have built up this reliance from several experimental investigations and theoretical validations all over the world (GourGourc et al., 1992;[START_REF] Gourc | Pull-out behaviour of reinforcements centrifuge tests and theoretical validations[END_REF][START_REF] Yoshioka | Validation of the 'Displacements Method' on an experimental reinforced wall at failure[END_REF][START_REF] Delmas | Two full size structures reinforced by geotextiles[END_REF][START_REF] Fidler | Evolution of the 'Displacement Method' applied to soil reinforced structures[END_REF]. This method is based on the principle of soil limit equilibrium, associated with the anchored membrane behaviour of the geosynthetic reinforcement. The original method ('Cartage' French software) considered a simplified mechanism for this membrane (rectilinear shape). Later on, a more rigorous mechanism (circular shape) was proposed. The advantage of this method is its ability to take into account the extensibility of the reinforcements in the design, which is a major characteristic of geosynthetics. Also, it allows one to take into account the complex soil-geotextile interaction mechanism that actually occurs, and one can check the soil-reinforcement strain compatibility.

In this paper, an extension of the 'displacement method' presented by [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF] is proposed. It allows the determination of the geometry of the membrane, which gives the minimum value of the tension in the reinforcement at the failure surface and simultaneously satisfies the three static equilibrium equations of this membrane using a variational approach. In the following section, the 'displacement method' given by [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF] is briefly reviewed, and then the present variational formulation is presented.

Determination of the tension in the reinforcement

This study takes place in the general framework of geotextile reinforced slope stability analysis. Fig. 1 shows a geosynthetic reinforced slope inclined at (with the Fig. 2. Anchored membrane concept for geotextile reinforcement [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF]. horizontal direction). This paper only considers the determination of tension ¹ in the reinforcement i at the intersection with the failure surface ½(X) (see Fig. 1).

As mentioned above, [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF] have suggested a displacement approach and have considered an anchored membrane in the neighbourhood of the failure surface (see Fig. 2). They have made assumptions concerning the magnitude and the direction of the uniform pressure distribution, p, applied to this membrane. These authors have shown that the assumed distributions lead to a membrane that has the form of a circle or a hyperbola, and to a constant tension in the membrane. In this paper, a new method for the determination of the reinforcement tensions is proposed. It is based on the following assumptions:

E The total membrane B B (see Fig. 3) is assumed to be symmetrical with respect to point A. This is in conformity with published results from model tests [START_REF] Galera | Approche the´orique du comportement du complexe sol-ge´osynthe´tique[END_REF]. Thus, tension ¹ in the geotextile sheet at point A, is determined by means of the equilibrium of a half membrane, E The location of maximum tension in the reinforcement coincides with the failure surface, E Tension ¹ is assumed to be tangential to the membrane since the reinforcement has no flexion rigidity, E The mechanical behaviour of the soil-reinforcement interface is characterised by a friction angle and an adhesion c that are assumed to be constant on both sides of the reinforcement. These parameters should be determined by soil-geosynthetic direct shear tests.

The present method includes the four following points (see Fig. 3):

1. Application of the variational calculus to the equilibrium of upstream membrane AB (upstream refers to the failure surface). It allows the determination of the extreme shape of the membrane AB that (1) minimises tension ¹ , and ( 2) satisfies simultaneously the three static equilibrium equations of the membrane. This will lead to a relationship between the tension forces at points A and B (i.e. ¹ and ¹

).

Determination of tension ¹

, using the tension-displacement relationship of the anchorage B F. This relationship allows the determination of the tensions ¹ and the relative soil-geotextile displacements u at any point of the anchorage as functions of u $ (relative displacement at point F). This calculation scheme is based on the work of [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF]; 3. Determination of unique solution for u $ value, and consequently for ¹ , using the displacement compatibility equation of the upstream zone. 4. Equilibrium of the downstream membrane in order to check the validity of the solution with respect to the failure by breakage or sliding of the reinforcement.

Note that points 1, 3 and 4 are new considerations, whereas point 2 is part of the existing 'displacement method'. These four points are detailed in the following sections.

Variational approach applied to the membrane

As shown in Fig. 3, a geosynthetic reinforcement is placed at a depth h below the ground surface, and subjected to a known vertical displacement y at point A. Note that, for the purpose of clarity, the representation of y in Fig. 3 is exaggerated. B , B are the unknown points of the reinforcement that separate the anchored and membrane zones. Thus, the two zones GB and B F, which behave as anchorages, remain horizontal, whereas the zone B B , which behave, as a membrane, gets an unknown shape. In the following, we are looking for the shape of this membrane that gives the minimum value of the reinforcement tension at the failure surface. The justification of such a statement is that when a state of limit equilibrium is reached, the upstream membrane AB and the neighbouring soil deform in such a manner to attain the least lateral pressure as possible by developing the active zone, that is to mobilise the minimum value of ¹ . Thus, the problem is to assess the unknown tension force ¹ (magnitude and direction: ¹ , ( ). As shown in Fig. 4, upstream membrane AB is described by an unknown function y(x) in the (A, x, y) coordinate system. It is subjected to tension ¹ at point A, to tension ¹ at point B and finally to normal and tangential stress distributions and (resp. and ) on the upper (resp. lower) face of the membrane. The equilibrium equations are given as follows:

E Horizontal equilibrium:

¹ cos "¹ # l ( cos ! sin ) ds# l ( cos # sin ) ds (1.1)
E Vertical equilibrium:

¹ sin " l ( sin ! cos ) ds# l ( sin # cos ) ds (1.2) E Moment equilibrium around point A: ¹ y " l [(( ! ) sin !( ! ) cos ) y #(( # ) sin #( ! ) cos )x] ds, (1.3)
where " (s) represents the inclination of the membrane at point M to the horizontal direction, (x , y ) are the coordinates of point B in the local (A, x, y) reference system, s is the curvilinear coordinate of point M, and l is the length of the membrane (i.e. AB

).

Note that tension ¹ , which appears in these equations, is determined through the equilibrium of the upstream anchored zone B F (see Fig. 3) as will be shown later. Furthermore, one assumes that a state of limit equilibrium is reached for the soil-membrane interface and that the Mohr-Coulomb criteria are satisfied on both sides of the membrane.

" tg #c ( " tg )#c (2)
The quantity ¹ required to bring the soil-membrane interface to a state of limit equilibrium depends on the choice of the three functions y(x), (x) and (x). Note that the friction on this membrane is mostly mobilised on the concave face (i.e. on the lower face of this membrane). Therefore, the authors think that it would be dangerous to make any assumptions concerning the normal stress distribution on this face [i.e.

(x)]. However, the normal stress distribution on the upper face of the membrane is essentially due to the soil weight above this membrane. Hence, ¹ necessarily depends on depth h (see Fig. 3). Now, this term does not appear in the equilibrium equations (Eqs. (1.1,1.2,1.3,2), (1.2,1.3,2), (1.3,2) and ( 2)). Thus, in order to relate ¹ to h , four different assumptions (namely models 1-4) concerning the distribution are made. They are defined as follows:

1. Model 1: One assumes that , the resultant of the elementary stresses and tg equals soil weight above the membrane (see Fig. 5):

Q " h#q
where represents the effective unit weight of the soil, h the height of soil above point M, and q the uniformly distributed normal surcharge at the top of the reinforced slope. Thus, stress can be written as follows: "( h#q).cos

(3) 2. Model 2: The magnitude of the normal stress is expressed as follows:

"( h#q) cos (4)
This expression of is sometimes called the 'Fellenius stress' in the slope stability analysis. Indeed, it refers to the equilibrium of a vertical slice of soil whose base is inclined at to the horizontal direction, assuming that interslice forces are parallel to the base of each slice. Fig. 6 shows the normal stress and the resultant stress as defined in model 1.

Model 3:

This model is the one proposed by [START_REF] Faure | Analyse des contraintes dans un talus par la me´thode des perturbations[END_REF] in his slope stability analysis method. The principal directions are assumed to be the horizontal and vertical ones. The principal stresses are thus expressed as follows:

" h#q on a horizontal facet unknown on a vertical facet Then, the normal stress that acts on a facet inclined with an angle to the horizontal direction, can be assessed by a simple construction of the Mohr's circle. It can be written as follows:

"

h#q 2 (1#cos 2 )# 1!cos 2 tg 4 # 2 # c (cos 2 !1) tg 4 # 2 (5)
4. Model 4: This model is based on the assumption of Bishop concerning inter-slice forces in his slope stability analysis method. The normal stress , which acts on a base of a slice inclined with an angle to the horizontal direction, corresponds to the equilibrium of a vertical slice of soil whose inter-slice shear forces are neglected. Thus, the following expression is obtained:

" h!c tg #q 1#tg tg (6)
In the following, the variational calculus is applied to the equilibrium of the upstream membrane AB considering model 1, then the obtained results will be extended to the three other models.

Variational approach-Model 1

The normal stress distribution is given by Eq. ( 3). The depth h of soil above M can be expressed in the local (A, x, y) coordinate system (see Figs. 3 and4) as follows:

h"h #y !y (7.1)
Furthermore, one can write y"tg (7.2) and ds" dx cos .

(7.3) Substituting Eqs. ( 2) and ( 3) into Eqs. (1.1,1.2,1.3), (1.2,1.3) and (1.3), taking into account Eqs. (7. 1,7.2,7.3), (7.2,7.3) and (7.3), three limit equilibrium equations of the upstream membrane can be expressed as follows:

¹ cos "¹ # V 2c # (tg !y)#k [ (h #y !y)#q] ; sin ( # E ) cos dx (8.1) ¹ sin " V 2c y# (tg .y#1)!k[ (h #y !y)#q] ; cos ( # ) cos dx (8.2) !¹ y " V k.[ .(h #y !y)#q] sin ( # ) cos y# cos ( # ) cos x #[2c # .(tg !y)] y![2c y#r.(tg !y#1)] x dx. (8.3)
Note that ¹ appears in the two equilibrium Equations (8.1) and Eq. (8.2). This tension may be expressed from one of these equations, in terms of the two functions y (x) and (x). The quantity ¹ , for which the membrane attains a state of limit equilibrium, depends in general on the kinematical and the statical functions y(x) and (x), respectively. Therefore, ¹ is a functional of two functions. This functional is termed the tension functional, and the critical tension ¹ is the minimum value of ¹ :

¹ "min WN ¹ +y(x); (x),"¹ [y C (x), ( C (x)],
where y C (x) [resp. C (x)] defines the critical shape of the membrane (resp. the critical normal stress distribution). The problem of the equilibrium of the membrane can now be stated as follows: find a pair of functions y C (x) and C (x) that realises the minimum value ¹ (critical tension) of the tension functional ¹ , such that the three limit equilibrium equations (Eqs. (8.1,8.2,8.3),(8.2,8.3) and (8.3)) are satisfied. This problem is solved using variational calculus as proposed by [START_REF] Baker | Variational approach to slope stability[END_REF] in the case of unreinforced slope stability analysis. The calculus is briefly presented here. The reader can find more details elsewhere [START_REF] Lemonnier | Application de la me´thode variationnelle au proble`me de la stabilite´des talus renforce´s par des nappes ge´osynthe´tiques[END_REF].

The terms cos and sin that appear, respectively, in Eq. (8.1) and (8.2), give the tension functional a particular property: its dependence on y , which represents the slope of the membrane at point A. Indeed, if we choose Eq. (8.2) to define the functional, the problem can be expressed as follows:

min ¹ " V 1#y y 2c y# (tg y#1)![ (h #y !y)#q] ; cos ( # ) cos dx (9.1)
subject to the two following constraints:

x 2c # C (tg !y C )#[ (h #y !y C )#q] sin ( # ) cos dx "¹ cos K !¹ B (9.2) x [ (h #y !y C )#q] sin ( # ) cos y C # cos ( # ) cos x # [2c # C (tg !y C )] y C ![2c y C # C (tg E !y C #1)] x dx "!¹ B y , (9.3)
where the angle K corresponds to the inclination of ¹ , that is to the critical position. Eqs. (9.2) and Eq. ( 9.3), which are the integral constraints, are derived from Eqs. (8.1) and (8.3), respectively. The tension functional can be reduced to the following form:

J" x ¹ (x, y, y , y, ) dx
Note that one endpoint of the functional is null, and the other (i.e. x ) is variable. Furthermore, the two integral constraints are of the following type:

x k G (x, y, y, y , )" G ∀i 3 +1, 2, 3,,
where k G and G are obtained by identification. Thus, this problem is a standard isoperimetric one with a variable endpoint. The solution to this problem can be obtained using the method of Lagrange's undetermined multipliers. Following the same procedure as in [START_REF] Baker | Variational approach to slope stability[END_REF], an auxiliary function R may be introduced as follows: R"F#( K # K , where F, K , K are intermediate functions.

Therefore, the functions y C (x) and C (x) that constitute the solution to the problem have to satisfy the following conditions:

Eq. ( 11) represents the equation of log-spirals of angle , with pole at (x A , y A ) (see Fig. 7).

As shown in Fig. 7, r decreases when angle increases, for O0. In case of "0, r remains constant when varies; thus, one obtains, in this case, a circle whose radius is r

. At point B , the tangent to the membrane remains in the horizontal direction. This condition implies that angle is constant and equals ( /2! ). Consequently, the family of possible critical shapes of the membrane is the one of log-spirals, which are defined by only one parameter, i.e. angle . Furthermore, note that the log-spiral has a particular geometrical property, that the resultant of the elementary forces ( ds) and (tg ds) passes through the pole C of the spiral. Hence, the moment equation about the pole is independent of the stress distribution (x) and may be used for the determination of the critical tension ¹ . The two remaining equilibrium equations may be satisfied by every (x) distribution that has two degrees of freedom. Thus, one may find the critical value that simultaneously satisfies the moment equation (Eq. (12.1)) and gives the minimum value of ¹ (Eq. (12.2)). This is done by a one-dimensional minimisation procedure of ¹ with respect to .

m A ( )"0P¹ "¹ ( , ¹ ) (12.1) ¹ "min (¹ ).
(12.2)

Second Euler and transversality equations.

Since the aim of this study is the determination of the critical tension and its inclination, the results obtained so far are sufficient to solve the problem. Indeed, it has been shown by [START_REF] Baker | Variational approach to slope stability[END_REF], in the case of unreinforced slope stability analysis, that the second Euler and the transversality equations give the normal stress distribution (x). In the present analysis, it is also the case, but since the (x) distribution is not necessary to assess ¹ and K , we will not express these equations.

Variational approach-Other models

Let us note G the distribution corresponding to model i (i"1, 2, 3 or 4), then Eqs. ( 3)-( 6) may be expressed as follows:

G "s G ( h#q) cos E , where s "1 s " cos cos s " 1 2 cos 1#cos 2 # 1!cos 2 tg 4 # 2 # c (cos 2 !1) tg 4 # 2 ( h#q) s " ( h#q)!c tg cos ( h#q) (1#tg tg ) .
Thus, by multiplying corresponding to model 1, by a coefficient s G defined above, one obtains G corresponding to model i. However, these coefficients depend on y(x) and y(x) (through h and , respectively), but are independent of the (x) distribution. Consequently, since the first Euler equation concerns the partial derivative of the functional with respect to (x), one can easily show that the variational calculus applied to the three other models gives the same conclusions obtained for model 1. Nevertheless, the critical tension is not the same for each model.

Equilibrium of the downstream membrane

As mentioned above, membrane AB is assumed to be symmetrical to AB with respect to point A. Thus, membrane AB is subjected to (see Fig. 8): critical tension ¹ ; tension ¹ , which remains horizontal; normal and tangential stress distributions and (resp. and ) on the lower (resp. upper) face of the reinforcement. Furthermore, the following assumptions for zone AB will be adopted here:

1. A state of limiting equilibrium is assumed to occur on both faces of downstream membrane AB , that is: 2. The interface friction is mostly mobilised on the concave face of the membrane, that is on its upper face. Thus, no assumption will be made for the distribution.

" tg #c " tg #c (13)

Concerning

distribution, an assumption is presently made in order to simplify the resolution: it is assumed to be null, that is the membrane decollates at its lower face.

Note that the two first points are the same as for zone AB , whereas point 3 is different.

Thus, writing the only moment equilibrium equation about the pole of the known log-spiral defining AB zone, one obtains ¹ without any assumptions concerning normal stress distribution (x). For the same reason as for the upstream membrane, the two other force equilibrium equations are implicitly satisfied.

Tension-displacement relationship of the anchorage

To determine the tensions in the anchorage, one considers the model of the soil-geotextile interaction proposed by [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF]. It is based on the following two relationships (see Fig. 9):

1. A linear elastic tension-strain relationship for the geotextile:

¹"J , ( 14
)
where ¹ is the tension force, J the tensile stiffness and the strain in the geotextile; 2. An elasto-plastic relationship for the friction behaviour of the soil-geotextile interface:

" where is the mobilised shear stress at any point M of the interface, u the relative soil-reinforcement displacement at this point, the slope of the straight line that characterises the elastic behaviour of the interface, the ultimate shear stress and u the minimum value of u that mobilises (see Fig. 9(b)). The relative displacement u should be determined by a laboratory friction test.

u if u(u if u*u (15)
Thus, considering that a state of limiting equilibrium is reached on both soil-geotextile interfaces, the Mohr-Coulomb criterion has to be satisfied:

" tg #c (16)
Combining Eq. ( 14) with Eqs. ( 15) and ( 16) for each anchorage zone, one obtains two relationships giving the displacement u and the tension ¹ at any point of the anchorage for a prescribed boundary condition at point G (see Fig. 3). This boundary condition may be either: (1) free anchorage where ¹ % "0 and u % O0, or (2) fixed anchorage where ¹ % O0 and u % "0. Let us consider the free anchorage; then these relationships may be reduced as

¹ "f (u $ ), ¹ "f (u % ) (17.1) (17.2)
Note that zones GB and B F (see Fig. 3) are characterised by the rates of linear behaviour of these zones (called ta % and ta $ , respectively). These rates are defined as the ratio of the length of the portion that behaves linearly (called x % and x $ , respectively) over the total length of the anchorage for each zone (B F and GB , respectively):

ta $ " xpF B F ta % "
x N% GB These computed rates (ta % and ta $ ) have to be larger than a minimum value ta prescribed by the designer:

ta $ 'ta ta % 'ta
These conditions assure the non-failure of the geotextile reinforcement by lack of adherence (sliding of the sheet).

Displacement compatibility

The analysis done so far leads to a problem with one degree of freedom. That is, the critical tension and its inclination depend on the relative displacement u $ (cf. Eqs. (12.1,7.1) and (17.1)). The missing equation is the one that allows the satisfaction of the displacement compatibility for the zone AF of the geotextile. Indeed, one must verify that the original length of the geotextile reinforcement in the resistant zone plus the corresponding change in length due to the extension must equal the total length after deformation. This condition allows the determination of a unique u $ value and thus (¹, u) distributions along the reinforcement.

Since the displacement of point A of the reinforcement is led by the global failure surface of the slope, the displacement compatibility condition depends on the location of this surface. To express this condition, it has been assumed that the global slip line is a log-spiral characterised by two angular parameters and , as shown in the companion paper. Let us note l and l the respective changes in length of zones A B and B F . Thus, the displacement compatibility equation may be written as follows:

A B #B F "A F # l # l . ( 18
)
Length B F may also be expressed as follows:

B F "B F !u $ "A F !A B !u $ . ( 19
)
Substituting Eq. ( 19) into Eq. ( 18), one obtains the displacement compatibility condition:

u $ #A B !A B # l # l "0 (20)
It has been shown [START_REF] Lemonnier | Application de la me´thode variationnelle au proble`me de la stabilite´des talus renforce´s par des nappes ge´osynthe´tiques[END_REF] that l may be expressed as a non-linear function in terms of u $ . Therefore, the unique theoretical u $ value, satisfying the compatibility condition, is reached by means of incremental calculation until Eq. ( 20) is satisfied. Thus, for any u $ value, one has to determine four terms: A B , A B , l and l

. These terms may be expressed with respect to (1) the angular parameters of both the global and the local log-spirals; ( 2) the polar co-ordinates of point A (i.e. G , r G , see Fig. 10); ( 3) the length of the upstream anchorage zone l "B F ; ( 4) the mechanical parameters of the soil-geotextile interface and of the inclusion: , J, ; ( 5) the relative soil-geotextile displacement u $ [START_REF] Lemonnier | Application de la me´thode variationnelle au proble`me de la stabilite´des talus renforce´s par des nappes ge´osynthe´tiques[END_REF]. In the following, for the purpose of clarity, Eq. ( 20) is reduced to v(u $ )"0.

Computational procedure

Fig. 11 represents the flow chart giving the computational procedure to obtain the critical tension ¹ and its inclination K . The design data are the soil-geotextile friction angle and adhesion ( , c ), the soil unit weight , the depth of soil over the reinforcement h , the vertical local displacement y , the tensile stiffness J of the geotextile, the initial length of the reinforcement in the upstream zone ¸, the minimal value of the rate of linear behaviour of the anchorage ta and finally the parameters defining a particular failure surface.

Note that each u $ value corresponds, if it exists, to ¹ and K values. Tension ¹ has to satisfy the two following conditions (see step 5, Fig. 11):

E It should not exceed the allowable tensile force to avoid failure of the reinforcement; E It should not exceed the available adhesive and frictional forces along the reinforcement length A F to avoid sliding of the reinforcement. Now, a unique u $ value satisfies the displacement compatibility condition (see step 6, Fig. 11): v(u $ )"0. Thus, steps 2-6 have to be executed for different u $ values until this condition is satisfied.

Numerical results

The present theoretical analysis is now completely defined. In this section, one firstly presents a parametric study concerning a unique geotextile sheet reinforcing a vertical wall (see Fig. 12). The influence of u $ , ¸S and y on the determination of the reinforcement tension ¹ , in case of model 4 of the stress distribution, is presented and discussed. Then, the influence of the different assumptions (i.e. models 1-4) concerning the stress distribution on the determination of the reinforcement tension, is presented. Secondly, a comparative case study, with results obtained from other current design methods and with some data measured in situ, is presented and discussed. the variational calculus to the upstream membrane. This minimum is the critical tension ¹ relative to this u $ value, and is reached at an angle u $ , which gives the optimal inclination EK of ¹ as follows:

K " EK # E ! 2 .
This minimum value ¹ increases, and decreases when u $ increases. That means that a greater u $ value corresponds to a lesser K value, that is the membrane tends to become flatter. increase significantly with u $ and ( 4) tension ¹ is always greater than tension ¹ . Note that point ( 4) concurs with the assumption concerning the location of maximum tension in the reinforcement (cf. Section 2).

Influence of the anchorage length in the upstream zone

Fig. 15 shows the variation of the critical position of the upstream membrane for a fixed u $ value (5 mm), the total anchorage length in the upstream zone ¸ varying from 2 to 3.5 m. It shows that when ¸ increases, the membrane tends to become flatter and tensions ¹ and ¹ increase significantly.

Influence of the vertical displacement of point A

In this section, the length ¸"2.5 m is considered. Fig. 16 shows the variation of the critical position of the upstream membrane for a fixed u $ value (5 mm), the vertical displacement y of point A, varying from 1 to 4 cm. It shows that when y increases, ( 1) the critical tension ¹ increases slightly;

(2) tension ¹ decreases slightly; (3) K increases; and (4) the length of the membrane increases.

Influence of the model of the

stress distribution Fig. 17 represents the variation of ¹ and K versus u $ for the four proposed models of the stress distribution. It shows that these distributions have a negligible effect on the determination of ¹ for all u $ values considered. Concerning K , there is a significant effect for small u $ values. From a particular u $ value (approximately 7 mm), the four models give quite the same results. The influence on the moment of tension ¹ about the pole of the given global log-spiral has also been considered. The results, which are not presented in this paper, are very similar to these ones.

Comparative case study

In this section, the results obtained from the present model are compared with those obtained from some of the most current design methods available for geosynthetically reinforced slope stability analysis in the case of a large-scale model geogrid reinforced wall [presented in a paper by [START_REF] Bathurst | Laboratory study of Geogrid Reinforced Walls[END_REF]]. Fig. 18 shows the geometry of this wall, which is referred to as the RMC model wall (Royal Military College of Canada). The surcharge of 50 kN/m, acting on top of the wall, was the one corresponding to failure. The above-mentioned current design methods were already applied to this model wall by [START_REF] Claybourn | A comparison of design methods for geosynthetic-reinforced earth walls[END_REF]. These methods are: These methods may be classified into two main groups:

1. The first one includes methods ( 1)-( 4) (referred to as the tied-back wedge methods), which are based on lateral earth pressure considerations. That is the reinforcement tensions, which are assumed to act in the horizontal direction, balance the horizontal forces due to lateral earth pressures tending to cause instability. In these methods, a planar failure surface through the reinforced mass described by a Rankine active failure condition is typically presumed. 2. The second one includes methods ( 5) and ( 6), which are based on the approach commonly used in conventional slope stability analysis (i.e. analysis of stresses on a failure surface) modified to take into account the effect of the reinforcement tensions. Method ( 5) is based on a variational approach applied on limiting equilibrium analyses. Method ( 6) is based on limiting equilibrium analysis using wedge failure models.

Because of the significant variations among the methods, [START_REF] Claybourn | A comparison of design methods for geosynthetic-reinforced earth walls[END_REF] has chosen to consider the ultimate wide width reinforcement tensile strengths and set all safety factors to 1.0. This author has then presented the results, in terms of reinforcement tensions, obtained from the different methods along with the measured reinforcement stress reported by [START_REF] Bathurst | Laboratory study of Geogrid Reinforced Walls[END_REF]. These results, together with those obtained from the present model, are presented in Fig. 19. The latter results correspond to a particular value of the vertical displacement on top (y "19.7 mm) for which the minimum factor of safety F equals 1, considering a log-spiral as failure surface (see companion paper). Furthermore, the limit soil/geotextile displacement u (cf. Fig. 9) that is required for the present model is not available in the RMC wall report. Nevertheless, the influence of this parameter on the determination of the reinforcement tension is not significant (cf. [START_REF] Ratel | Mode´lisation d'un sol reinforce´par ge´osynthe´tique: application de la 'me´thode en de´placements[END_REF]. For the present case, the value of 2.5 cm for u has been selected, which is a realistic value according to several experimental studies on the determination of this parameter considering granular material reinforced with geogrids [START_REF] Ratel | Mode´lisation d'un sol reinforce´par ge´osynthe´tique: application de la 'me´thode en de´placements[END_REF][START_REF] Gotteland | Reinforcement des sols par ge´osynthe´tiques; dimensionnement et validation[END_REF].

The results presented in Fig. 19 show that the present model gives similar results to those using Schmertmann et al.'s method in terms of the magnitude of the reinforcement tension distribution along the failure surface. It should be noted that, among these six methods, Schmertmann et al.'s method is (1) the only one that is specific to geogrid-reinforced soil slopes and ( 2) except for the lowest sheet, the one that gives the best prediction of the actual behaviour of the structure. However, the RMC Model Wall report indicated that the low bottom layer stress was probably influenced by the model set-up [START_REF] Claybourn | A comparison of design methods for geosynthetic-reinforced earth walls[END_REF]. Therefore, the comparison with the experimental data, for the fourth layer, is meaningless.

In terms of acting direction of the critical tensions, methods ( 1)-( 6) consider it as horizontal. Furthermore, method ( 5) considers also the reinforcement tensions acting in the direction of the failure plane (for the Rankine active failure plane, the inclination equals 71.5°with respect to the horizontal direction). The values of these inclinations, obtained from the present model, are given in Table 1, in which the maximum inclination values correspond to the direction of the failure plane. Since the inclination of the membrane at the failure surface was not measured in this project, no comparison can be made with measured data. Nevertheless, the results from the present model show that the inclination tends to increase with the depth, and that the values are clearly smaller than the maximum one.

The obtained critical failure surface (log-spiral), shown in Fig. 18, is close to the plane described by a Rankine active failure condition, and describes a bigger sliding mass.

Finally, in the RMC wall report, an effective reinforcement length is given (¸+1.189 m). This is the length beyond which the performance of the wall is presumed (by the current design methods) to be unaffected by further lengthening of the reinforcement for the loading condition at failure. The effective reinforcement length obtained from the present model is slightly smaller ¸+1.07 m (a difference of 11.6%).

Consequently, this result and the one concerning the inclination of the critical tensions show that, for this particular case, our model seems to be less conservative than the current design methods.

Conclusions

All methods currently used for the design of geosynthetic reinforced slopes are based on assumptions regarding the determination of reinforcement tension. The present model is a more rational one since it allows the determination of these tensions with a reduced number of assumptions. This reduction has been made possible by the use of a variational approach together with the concept of the anchored membrane [START_REF] Gourc | Design of fabric retaining walls: the displacements method[END_REF]. Thus, the variational calculus has been applied to the equilibrium of the upstream membrane for a given stress distribution on its upper face (model 1). The following results have been obtained: E the shape of the upstream membrane that realises the minimum value of ¹ , is a log-spiral of angle with only one variable angular parameter ; E the critical tension ¹ and its inclination K are independent of the normal stress distribution (x) along the lower face of the upstream membrane y(x); E the only moment equilibrium equation about the pole of the log-spiral is sufficient to assess the tension ¹ and its inclination associated with this slip line. The two remaining force equilibrium equations are implicitly satisfied, because every function of the normal stress distribution (x) that has at least two degrees of freedom is a solution to the problem; E ¹ is obtained by an unidimensional minimisation procedure with respect to .

Three other stress distributions have been proposed. It has been shown that (1) the above results are also valid for these three models, and (2) a numerical study has shown that these four models give quite the same results for the critical tension ¹ and its inclination . Furthermore, the influence of the anchorage length and the vertical local displacement of the membrane on the critical position of the membrane have been shown. Finally, a comparative case study, with results obtained from other current design methods and with some data measured in situ, has shown that, in that particular case, our model gives (1) a satisfactory prediction of the behaviour of the reinforced structure in terms of magnitude of the reinforcement tensions, and (2) seems to be less conservative than the current design methods.

Fig. 1 .

 1 Fig. 1. Geotextile reinforced slope stability analysis.

Fig. 3 .

 3 Fig. 3. Anchored membrane. Notations.

Fig. 4 .

 4 Fig. 4. Free body diagram of upstream membrane AB .

Fig. 5 .

 5 Fig. 5. Model 1 for distribution.

Fig. 6 .

 6 Fig. 6. Model 2 for distribution.

Fig. 7 .

 7 Fig. 7. Shape of the upstream membrane from the present variational analysis.

Fig. 8 .

 8 Fig. 8. Free body diagram of downstream membrane AB .

Fig. 9 .

 9 Fig. 9. Tensile and friction behaviour of the inclusion.

Fig. 10 .

 10 Fig. 10. Displacement compatibility.

E

  Fig. 10 shows such a line in a wall reinforced by a geotextile sheet. The reinforcement is represented before (dashed line) and after (bold-faced) deformation of soil. The upstream zone of the reinforcement includes two different parts: E Curvilinear zone A B corresponding to the deformed membrane. Before deformation, this zone was rectilinear and of length A B ; Rectilinear zone B F corresponding to the deformed anchorage. Before deformation, this zone was of length B F .

Fig. 11 .

 11 Fig. 11. Flow chart showing the calculation scheme of tension ¹ .

4. 1 .

 1 Parametric study 4.1.1. Influence of relative soil-geotextile displacement u $ Fig. 13 shows the variation of ¹ versus for several u $ values. It appears that for each u $ value, there is one value that realises an absolute minimum for function ¹. These numerical results confirm the theoretical ones concerning the application of

Fig. 12 .

 12 Fig. 12. Case study.

  Fig. 14 shows the influence of u $ on the location of the critical position of the upstream membrane. It clearly shows that (1) the membrane tends to become flatter as the u $ value increases; (2) ¹ variation is similar to that of ¹ ; (3) these tensions

Fig. 14 .

 14 Fig. 14. Critical positions of the membrane for u $ "1-5 mm (Model 4).

Fig. 15 .

 15 Fig. 15. Critical positions of the membrane for ¸"2 to 3.5 m (model 4).

Fig. 16 .

 16 Fig. 16. Critical positions of the membrane for y "1-4 cm (model 4).

Fig. 17 .

 17 Fig. 17. Influence of the stress distribution on ¹ and as given by the present four models.

Fig. 18 .

 18 Fig. 18. RMC model wall configuration [after Claybourn (1990)], and failure surfaces considered in the design methods.

Fig. 19 .

 19 Fig. 19. Reinforced tension calculated for RMC wall configuration.

  1. Two Euler differential equations for the function R, namely: . For this type of end-point, a variational boundary condition, known as the transversality condition, is to be satisfied.

	First equation:
	R NC	! dx d	R	C	"0	(10.1)
	Second equation:
	R WC	! dx d Ry C "0	(10.2)
	2. Two integral constraints (Eqs. (9.2,9.3) and (9.3)).
	3. Two boundary conditions for each end-point:
	2.1.1.1. First Euler equation.
	R is independent of Y C the first Euler equation (Eq. (10.1)) is a first-order differential equation in y (cf. Eqs. (9.1,9.2,9.3), (9.2,9.3) and (9.3)), and dependent on C only. C Solving this equation in terms of polar co-ordinates (r , ), one obtains
	r "r		.e F \F E( .

E fixed end-point A: x"0 and y"0. E variable end-point B :

x"x

Table 1

 1 Inclinations of the reinforcement tensions as given by the present model

	Layer number	1	2	3	4
	(deg)	48.71	50.47	51.69	52.52
	max (deg)	72.31	70.14	68.04	66.09