stabilized soils subjected to a uniaxial flexural test

International Journal of Fatigue 77 (2015) 41–49

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Mechanical fatigue behavior in treated/stabilized soils subjected to a uniaxial flexural test Mathieu Preteseille, Thomas Lenoir ⇑ Institut Français des Sciences et Technologies des Transports, de l’Aménagement et des Réseaux, Centre de Nantes, Route de Bouaye, CS4 44344 Bouguenais Cedex, France

a r t i c l e

i n f o

Article history: Received 21 August 2014 Received in revised form 3 March 2015 Accepted 8 March 2015 Available online 14 March 2015 Keywords: Transport infrastructures Mechanical fatigue Flexural test Stabilized/treated soils Cemented soils

a b s t r a c t The use of in situ fine-grained soils treated with lime and/or hydraulic binders as subgrade in common civil engineering infrastructures is a sustainable upgrading process for natural materials with low mechanical performances. In the case of land transport projects, the lack of knowledge on mechanical fatigue behavior in these materials leads either to empirical oversized design of the layers made with these materials or to their rejection. However, the development of a relevant test now enables us to accurately measure the mechanical fatigue performances of treated soils. First, sample preparation appears to explain most fatigue performances, not sample mineralogy. Second, based on original results on three treated soils and previous results from the literature, it seems that a behavior law governs these performances. Finally, a simple classification tool shows that these materials can be considered within the entirety of transport infrastructures from subgrade layers to subbase layers. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In situ soils present in right-of-way of civil engineering projects generally have mechanical characteristics inconsistent with stress rates generated by civil engineering infrastructures. To enhance their engineering and mechanical properties for a use in subgrade layers, it is common to mix them with a few percent of hydraulic binders [16,22,36]. This process has the advantage of minimizing environmental impact and reducing the cost of the infrastructures [43,49,30,44,45,35,32]. This approach is widely used in numerous civil engineering applications, for instance, embankments, foundations, slabs and pile and pavement construction. It is based on the measurement of monotonic mechanical performances [51,14,13,29]. However, for transport structures, the service life of infrastructures leads to a number of loadings superior to 107 cycles for pavements and 108 for high speed rail lines. The result is that, for these infrastructures, fatigue is one of the main failure modes [26,39,57]. Currently, difficulties in measuring mechanical fatigue performances of treated/stabilized natural soils with a high number of cycles at the laboratory scale has led to empirical design of pavement layers made with these materials [51,36,33]. Consequently, those layers are oversized and these materials are restricted to subgrade layers. They are seldom used in pavement subbase layers ⇑ Corresponding author. E-mail address: [email protected] (T. Lenoir). http://dx.doi.org/10.1016/j.ijfatigue.2015.03.010 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.

in which standardized materials treated with hydraulic binders, such as lightly cemented granular mixtures [9,10,11,12], are preferred [50,6]. In the railway sector, in classic High Speed Rail (HSR) projects, this lack of knowledge has purely and solely led to them not being used. The in situ materials that do not have sufficient characteristics are stripped, landfilled and substituted by materials from quarries [47]. Therefore, to rationalize the costs of the 25-million kilometers of new transport infrastructures anticipated by 2050 [28,38], understanding mechanical fatigue mechanisms for stabilized/treated natural soils and defining design rules in relation to these mechanisms are major technical, economic and environmental challenges. Two kinds of tests are used to study the fatigue behavior of hydraulically bound materials. Large scale tests aim to reproduce a part of the structure [34,31] or even the whole structure [40] in a tank. These tests are generally costly and need a complicated procedure that does not allow parametric studies. Consequently, smaller-scale tests must first be carried out. These are more practical for taking into account scattered results of fatigue tests. This second category of tests simulates the repetition of stress states due to traffic loadings into a specimen. The main tests used are flexural beam tests with three existing different configurations. Three- and four-point bending test configurations subject a beam resting on two support brackets to a compressive load with one or two points of load application [32,46]. The compressive load

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Nomenclature Fatigue parameters rf maximum bending stress (MPa) Nfail. number of loadings leading to failure rlog10(Nfail.) tensile stress that leads to rupture after Nfail. cycles (MPa) r6 tensile stress that leads to rupture after Nfail. = 106 cycles (MPa) S = rlog10(Nfail.)/rf stress ratio e = r6/rf endurance b slope of the fatigue curve DS uncertainty on the measurement of the stress ratio S (MPa) Dlog10(Nfail.) uncertainty on the measurement of the number of cycles that leads to failure Nfail. Mechanical considerations e strain tensor r stress tensor E Young’s modulus of materials (MPa)

generates a tensile stress in the lower part of the beam. The fourpoint bending test is most commonly used because between the two loading points, the specimen is subjected to pure bending. This test is largely used to evaluate the fatigue behavior of stabilized recycled aggregates [53,59,27] and has also been somewhat used to characterize the fatigue behavior of treated soils [18,52,55,17]. The third configuration is the two-point bending test. It consists in applying an alternative load to the top surface of a trapezoidal specimen embedded at its base. This test was also used to evaluate the fatigue behavior of cement-treated aggregates or sands [2,3,29]. For the same maximum tensile stress, this test has several advantages compared to the common four-point configuration test on a 400  100  100 mm beam: – the requisite load is about four times lower; – the deflexion is about four times higher. As a result, for the modulus measurement, two-point bending is more appropriate. But in the field of treated/stabilized soils, the difficulties in preparing laboratory samples with the same dimensions as for cement-treated aggregates has led to testing materials on smaller samples with non-relevant dimensions for soils with aggregates larger than 2 mm [25]. Based on an analytical approach, numerical calculations and experimental results on three natural fine-grained soils treated with hydraulic binders, the aims of this paper are to:

G

m Fe A(z) h Iy(z) v(z) g Mfy(z) u(z)

shear modulus of materials (MPa) Poisson’s ratio of materials applied load (N) area of the specimen (m) height of the specimen (m) moment of inertia (m4) distance from neutral axis (m) gap between the top of the specimen and the point of the load application bending moment (N m) displacement of the specimen along the x-axis (m)

Geotechnical considerations C2lm cumulated undersize for particles with a diameter less than 2 lm (%) WL Liquid limit of materials (%) WP Plastic limit of materials (%) Ip = WL  WP plasticity index of materials A = Ip/C2lm clay activity of materials

structure so that the material quality indexes (QI) could be determined. The three indexes were finally compared with the indexes of classic hydraulically bound materials. 2. Materials and methods 2.1. Fatigue test 2.1.1. Principle For the design of land transport infrastructures, the sizing criterion for layers made with hydraulically bound materials is the maximum tensile stress rmax resulting from the loading on the whole structure [50]. This worst tensile stress is located at the bottom of the stabilized layers [24,54,47]. To be suitable, the material must have a fatigue strength, rlog10(N), defined as its cyclic tensile stress that can be withstood for N loadings, superior to rmax. [37,48,25,42,50,51,58]. The number of loadings N is determined in relation with the expected traffic and the service life of the structure. The fatigue test is a two-step procedure applied on pseudotrapezoidal specimens clamped at their base (Fig. 1). First, the monotonic flexural test consists in applying a steadily increasing load to the top surface of the specimen to measure the maximum

– Demonstrate that the mechanical fatigue behavior of common treated soils can be accurately measured at the laboratory scale with the two-point bending test; – discuss the resulting performances; – show that these performances are relevant for structure design compared to other usual hydraulically bound materials. First, we present the fatigue test that we used. Experimental results on a reference polyvinyl chloride (PVC) specimen were compared with analytical and numerical calculations to validate the test configuration. Second, three natural soils treated with hydraulic binders were tested. Results are first discussed with regards to the geological nature of matrixes and to sample preparation, then compared with data from the literature to propose a general behavior law, and finally implemented on a simple bilayer

Fig. 1. Picture of experimental device.

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bending stress rf and the Young’s modulus E. The monotonic test was performed on three specimens for repeatability. Then, a sinusoidal cyclic load ‘‘rlog10(Nfail.)’’ with a constant amplitude and a zero mean value at 30 Hz was applied to the top surface to measure the number Nfail. of cycles before the failure. Eleven specimens were tested in cyclic mode. Only results where the failure occurs at least 3 cm from extremities of the specimen were considered. Results were then presented in a Wohler’s diagram and fitted with a straight line [25,50,29].

S ¼ 1 þ b logðN fail: Þ

2.1.2. Analytical approach The analytical approach is based on the Euler–Bernoulli beam theory (Fig. 2a). The stress and strain tensors within the specimen are given by (2) and (3).

0

6

Fe AðzÞ

0

3

0

0

0

F e ðhþgzÞv ðzÞ Iy ðzÞ

7 7 5

Fe r ¼ 6 4 AðzÞ

0

ð2Þ

where Fe: applied load (N); A(z): area of the specimen (m2); h: height of the specimen (m); g: gap following the z-axis between the top of the specimen and the point of the load application (m); Iy: moment of inertia (m4); and v(z): distance from neutral axis (m).

2

t rEzz 6 e ¼ 4 ryx G 0

ryx G

t rEzz 0

0

3

7 05

ð3Þ

rzz E

where: m: Poisson’s ratio; E: Young’s modulus (MPa); G: shear modulus (MPa). In addition to the stresses and strains expressions, these tensors lead to the Young’s modulus expression that can be determined with:

E¼

M fy ðzÞ Iy ðzÞ:u00 ðzÞ

Part

Material

E (MPa)

m

Flexible joint Load sensor Tie Glue (1 mm thick) Specimens

Steel Steel Aluminum AralditeÒ Treated soils PVC

193,000 193,000 68,900 2478 1000–10,000 3300

0.3 0.3 0.33 0.3 0.25 0.39

ð1Þ

where: S: stress ratio (applied stress on flexural strength); b: slope of the fatigue curve and Nfail.: number of cycles leading to failure.

2

Table 1 Test characteristics for numerical modeling.

ð4Þ

where u(z): displacement of the specimen along the x-axis (m); Mfy(z): bending moment (N m).

2.1.3. Numerical approach The test configuration is modeled with commercial finite-element code ABAQUS. The modeling is performed in linear elasticity with 20-node quadratic brick. Meshing has been refined until the result converges (Fig. 2b). The model is constituted of 6323 elements. Parameters used are given in Table 1. Positive and negative loads of 50 daN are applied on the flexible joint along the x-axis. A positive load means the actuator is pulling the trapezoid; a negative load means the actuator is pushing. The left part of the flexible joint can only be displaced along the x-axis because this part is fixed to the actuator. Motion is only possible in this direction. On the other side of the flexible joint, a tie is glued on the top of the specimens. Several cases of specimens were calculated. First, to compare analytical and numerical approaches, calculations were performed for two specimens with moduli E = 1000 MPa and E = 10,000 MPa. This range matches treated soils’ ranges [47]. Then to compare analytical, numerical and experimental approaches, calculations were carried out on a specimen with a modulus E = 3300 MPa, which is the modulus of the reference PVC specimen. In all cases, bases of the trapezoid specimen are clamped with glue on the support block, i.e. the base of the trapezoid cannot be displaced in the modeling.

2.1.4. Experimental approach A reference PVC specimen (Fig. 3) was instrumented with strain gages (120 O, 6 mm measuring grid). The reference specimen had an elastic modulus of 3300 MPa [7,8] and a Poisson’s Ratio of 0.39 [5]. Five strain gages were set on each face of the specimen. The validation procedure consisted in first subjecting the specimen to a load of 50 daN. Second the specimen was cyclically loaded with a sinusoidal signal at 30 Hz and a zero mean load. The amplitude of the imposed load was 42 daN.

Fig. 2. (a) Schematic representation of the two-point bending test (unit is mm). (b) Numerical stress distribution along the z-axis within the specimen for a modulus of 1000 MPa.

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Fig. 3. (a) Position of strain gages placed on the reference PVC specimen for the experimental validation of the fatigue test. (b) Picture of the reference PVC specimen.

Fig. 4. Sketching of the HSR project Bretagne – Pays de la Loire. Black dots point out sampling areas – Degré (AD), Coulans sur Gée (CSG), Etrelles (ASE).

2.2. Soils Three soils were sampled from the right of way of the French HSR project ‘‘Bretagne Pays de la Loire’’. This railway link will connect the cities of Le Mans and Rennes, 160 km apart (Fig. 4). The sampling was performed with an excavator equipped with a ditch bucket at a depth between 1.5 m and 3 m. From the geological point of view, the two sampling points near Le Mans (AD and CSG) belong to the western margin of the Parisian Basin bordering the Armorican Massif, and the Etrelles zone (ASE) is located in the central zone of the Armorican Massif. The AD and CSG materials were deposited during the Albian and Cenomanian stages (Cretaceous period) at the end of a marine regression (Early Cretaceous) followed by a transgression period (Cenomanian). These detritic formations are therefore composed of marine sediments and terrigenous materials. Since the land emerged at the end of Cretaceous era, the formations have been altered by climatic conditions such as Quaternary glaciations. The ASE soil arises from the alteration of Brioverian formations in central Brittany (Proteozoic). 2.2.1. Soil no. 1: AD 2.2.1.1. Geological formation. According to geological data [20], this formation (C1) is described as a rich-iron ore glauconitic clay deposited during the albian–cenomanian stage (112 to 94 million years). Interlaying sands are observed within decalcified and ferruginous yellow clays. They can be cemented by iron oxides.

2.2.1.2. Geotechnical properties. The maximum diameter of larger elements in this material is less than 50 mm. It can be classified as fine grained soil [4]. Through a 2-mm sieve, the cumulated undersize is 94%. The cumulated undersize less than 80 lm is 49%, which means that its mechanical behavior is controlled through its fine fraction. It is made up of less than 5% silt and has C2lm = 32% clays (Fig. 5). Therefore it is a sandy clay material [15]. Mineral identification shows that this clay fraction is composed of quartz, phyllosilicates with mica sheets and iron oxides. For information, the Liquid limit WL is 43% and Plastic limit WP to 25% [1]. The plasticity index IP (=WL  WP) is 18. The clay activity

Fig. 5. Grading curves of studied soils (AFNOR, 1992, [4]).

M. Preteseille, T. Lenoir / International Journal of Fatigue 77 (2015) 41–49

is thus A = IP/C2lm = 0.56. These results mean that the clays may be inactive and should have a low impact on the soil behavior when soil is wetted or dried. 2.2.2. Soil no. 2: CSG 2.2.2.1. Geological formation. Based on geological data [20], these materials are sedimentary marine deposits from the Cenomanian stage (100 to 94 million years). The formation (C2a) is composed of ferruginous and decalcified coarse and gravelly yellow sands. 2.2.2.2. Geotechnical properties. The maximum diameter of larger elements in this material is less than 50 mm. It can be classified as fine-grained soil. Through a 2-mm sieve, the cumulated undersize is 85%. The cumulated undersize less than 80 lm is 35%. This means its mechanical behavior is controlled both by its fine fraction and by its stone skeleton, for which the mineralogical nature is quartz. It is made up of less than 5% of silt and of 21% of clays (Fig. 5). Mineral identification shows that this clay fraction is composed by quartz, phyllosilicates with mica sheets, phyllosilicates with kaolinite layers and iron oxides. All these considerations show that this soil can be considered as clayey sand and highlights the importance of the grains in the matrix. The Liquid limit WL is 40% and the Plastic limit WP is 19%. The plasticity index IP is 21. The clay activity is A = IP/C2lm = 1. These results highlight that the clay’s activity is normal and should have a low impact on the soil behavior when soil is wetted or dried. 2.2.3. Soil no. 3: ASE 2.2.3.1. Geological formation. According to geological data [21], these materials have been sampled within a sandy clayey regolith resulting from the alteration of indurated clayey silts (b2cS) deposited during the Brioverian stage (670 to 540 million years). 2.2.3.2. Geotechnical properties. As before, the maximum diameter of larger elements in this material is less than 50 mm and it can be classified as fine-grained soil. Through a 2-mm sieve, the cumulated undersize is 84%. The cumulated undersize less than 80 lm is 56%. As for CSG, this result means that its mechanical behavior is controlled both by its fine fraction and by its stone skeleton. Interestingly, mineralogical identification of this skeleton shows, in addition to quartz, a large occurrence of micas. It is also made up of 20% of silts and 18% of clays (Fig. 5). The clay fraction is composed of quartz, phyllosilicates with kaolinite layers, and a large occurrence of phyllosilicates with mica sheets, and iron oxides as specified by mineral identification. All these considerations show that this soil is at the edge between sandy and coarse-grained material and shows the importance of the grains in the whole matrix. The Liquid limit WL is 41% and the Plastic limit WP is 22%. The plasticity index IP is 19. The clay activity is A = IP/C2lm = 1.05. As before, these results highlight that clay’s activity is normal and that it should have a low impact on the soil behavior when soil is wetted or dried. 2.2.4. Treatment and sample preparation Both soils CSG and ASE were treated with 5% (by dry weight) of cement CEM II/B-M (LL-V) 42.5 R [7,8]. Due to its high clay activity, soil AD was treated with 1% of lime and 5% of cement. Specimens for fatigue tests were compacted in a double static mode in one layer. Compaction rates corresponded to the optimum standard Proctor density for the AD and CSG mixes (1.69 g/cm3 for a water content of 21.5% and 1.91 g/cm3 for a water content of 13.8%, respectively) and at 103% of the optimum standard Proctor density for the ASE mix (1.88 g/cm3 and 14.3%). Sample density homogeneity was checked on a gamma bench. Specimens were stored in a

45

room with temperature regulated at 20 °C and were protected with thin plastic films to keep constant water content. Before being tested, specimens were stored 100 days for AD, 200 days for CSG and 120 days for ASE. 3. Results 3.1. Analytical and numerical study of the stress distribution Fig. 6 shows the stress distribution along the z-axis of the analytical and numerical approaches for both faces of the trapezoid for a modulus of 10,000 MPa. From the top (h = 48 cm) to a height h = 7 cm of the specimen, both curves, analytical and numerical, are superimposed. In absolute values, stresses increase from the top of the specimen to a height of 25.2 cm (0.14 MPa to 0.59 MPa for analytical results and 0.03 MPa (pulling) and 0.21 MPa (pushing) to 0.60 MPa for numerical results). Then stresses decrease to 0.51 MPa at the clamping for analytical results. At this low end of the specimen, results are different (0.57 MPa). This difference may be due in particular to the different boundary conditions imposed in both models and to Finite Element Method meshing. Even so, both models give close results for the maximum tensile stress (difference of 0.2%). For a specimen with a modulus of 1000 MPa, results are identical for analytical calculations because Eq. (1) shows that stresses are modulus-independent. For numerical results, they slightly differ, because the support condition at the top of the specimen (blade) is modulus-dependent (0.01 MPa (pulling) and 0.02 MPa (pushing) at h = 48 cm to 0.58 MPa at h = 25.2 cm), and the difference for the maximum tensile stress is less than 2% (1.7%). In conclusion, this study shows that numerical and analytical representations of the entire fatigue-testing setup are similar for treated soils with a modulus between 1000 and 10,000 MPa. 3.2. Experimental validation of the two-point bending test configuration 3.2.1. Strain In order to validate the configuration of the test for treated soils, a reference PVC specimen (Fig. 3) was instrumented with 10 strain gages. Both monotonic and cyclic tests were studied. The strain distribution as a function of the height in the specimen of the monotonic test is given in Fig. 7a. The actuator pushes the specimen; consequently the actuator side of the specimen works in tension whereas the laser side works in compression. During the monotonic test, the measured strain is superimposed with the numerical modeling and the analytical solution, apart from near the clamping where it is only close to the numerical modeling. As before, this difference is mainly due to

Fig. 6. Comparison of stresses along z-axis between analytical and numerical approaches for a modulus of 10,000 MPa. The insert highlights values that are close of the maximal tensile stress.

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Fig. 7. Strain distribution (absolute values) on the reference specimen of the (a) monotonic test and (b) cyclic test.

Fig. 8. Results of fatigue tests expressed in a Wohler’s diagram of the three treated soils.

support conditions. The analytical calculation takes into account theoretical support conditions: free movement at the top of the specimen, perfect clamping at the base. The maximum tensile strain is located at the strain gage J8 and is 177 lm/m, compared with 180 lm/m for the numerical modeling and 181 lm/m for the analytical solution. In conclusion, experimental, numerical and analytical results give close values for the maximum tensile stress. During the cyclic test, both faces of the specimen (actuator and laser side) work alternatively in tension and in compression. Because the failure mode is due to tension, only the tensile strain distribution as a function of the height of the specimen is discussed (Fig. 7b). Experimentally, the tensile strain distribution on the actuator side is located at strain gage J8 (141 lm/m) and is slightly weaker than numerical modeling and analytical solution (150 lm/m and 149 lm/m respectively). On the laser side the tensile strain distribution is close to the analytical result (152 lm/m) and higher by 1.2% for the maximum strain. As before, the experimental strain distribution is close to the numerical modeling and the analytical solution except near the clamping where numerical modeling and experimental measurements are very close. We note that with this experimental test, the failure may occur on the laser side.

3.2.2. Modulus An experimental Young’s modulus can also be determined with Eq. (4). It is 3360 MPa for the monotonic case and 3277 MPa for the cyclic case. The difference is 1.2% and 0.7%, respectively, in comparison with the real modulus of the PVC material (E = 3300 MPa). In conclusion, the experimental study of the two-point bending test gives results (strain distribution and modulus) that are very close to both analytical solution and numerical modeling. Therefore this test is relevant to measure mechanical fatigue performances of treated soil specimens. 3.3. Fatigue of treated soils For the treated soil AD, the bending strength or modulus of rupture rf is 0.71 (0.05) MPa and the elastic modulus E is 4230 (419) MPa. The values between parentheses correspond to the standard deviation. Results of fatigue expressed in a Wohler’s diagram (Fig. 8) lead to a slope b = 1/15.8. For a given number of cycles Nfail. = 106, the endurance e, which is defined as the ratio r6/rf, is 0.62 and r6 = 0.44 MPa. The test uncertainties are Dlog10(Nfail.) = 1 and DS = 0.06 MPa. This uncertainty DS is directly linked to Dlog10(Nfail.) by geometrical considerations and is given for clarity.

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For the treated soil CSG, bending strength is higher, with a value

rf = 1.10 (0.04) MPa for a Young’s modulus E of 5704 (193) MPa. In the Wohler’s diagram, the slope is 1/14.8. For one million cycles, this results leads to endurance e = 0.59 and r6 = 0.65 MPa, with test uncertainties Dlog10(Nfail.) = 0.6 and DS = 0.04 MPa. For the treated soil ASE, bending strength is close to that of CSG, with rf = 1.13 (0.05) MPa for a Young’s modulus E of 5068 (237) MPa. In the Wohler’s diagram, the slope is 1/14.2. For one million cycles, this data leads to endurance e = 0.58 and r6 = 0.66 MPa, with test uncertainties Dlog10(Nfail.) = 0.7 and DS = 0.05 MPa.

4. Discussion 4.1. Fatigue performances of the three treated soils We note that fatigue performances of AD are weaker than performances of CSG and ASE. This probably comes from the different geotechnical nature of the materials. AD is a sandy clay matrix with its mechanical behavior controlled through its fine fraction, while CSG and ASE are clayey sands. It follows first, that distribution of sand grains during preparation of AD specimens is more unpredictable than for CSG and ASE ones. Second, AD’s density is lower than the densities of CSG and ASE (1.69 g/cm3 vs 1.91 g/cm3 and 1.88 g/cm3 for both clayey sands). This higher uncertainty on grain distribution probably leads to higher uncertainty on the number of cycles Dlog10(Nfail.), which is 1 for AD vs 0.6–0.7 for CSG and ASE. The lower density probably leads to lower mechanical performances. Indeed, even if both mineralogical nature and granularity of CSG and ASE are different, these two matrixes exhibit very close density (around 1.9 g/cm3) and very close fatigue performances.

4.2. Comparison with literature results on hydraulically bound materials Some measurement of fatigue performances of hydraulically bound material with low (less than 15% dry matter) cement and/ or lime content can be found in the literature. They are supplied in supplementary information Tables S1 and S2. The original matrixes are diverse. They can be natural soils [55,56,25,17], natural or standardized sands, and natural or standardized gravel [50,41,29,9,10,11,12]. Test frequencies vary from 5 Hz to 50 Hz. Sample preparation differs widely, so it is not possible to follow on from the discussion from the previous paragraph about density. Nevertheless, interestingly, even though the fatigue tests differ, they all generate a uniaxial tensile stress in the specimens [46]. It is therefore relevant to compare the results. The means of fatigue parameters are given in Table 2. The monotonic parameters, rf and E, show on average a large standard deviation compared to their mean values (1.35 vs 0.72 MPa for rf and 10,698 vs 7697 MPa for E). This reflects the important diversity of the samples. Parameters derived from the representation with a straight line, 1/b and endurance e show smaller standard deviation compared to their mean (respectively 11.73 vs 2.21 and 0.47 vs 0.1). The standard deviation of r6 is also high compared to its mean, due to the diversity of tested materials. Results on Dlog10(Nfail.) and DS are not discussed because of the lack of data (see Table S2). Experimental results obtained on the three treated soils are consistent with the mean values of literature results. Especially, the parameters of slope 1/b and endurance e are close to the mean values. In agreement with this observation, the rf values (including new experimental values and literature values) are highly correlated (R2 = 0.92) to the corresponding r6 values (Fig. 9).

Assuming that the only common parameter for all these low treated/stabilized materials is the fatigue mode with uniaxial flexion, a behavior law for this test can be proposed, where r6 = rf/2. 4.3. Influence on infrastructure design To compare the mechanical properties of different pavement materials, it is usual to define the Quality Index (QI) of the material [19]. To determine this indicator, a simple model of a pavement structure is defined. That means different materials can be compared on a figure. This simple model is a two-layer structure constituted of a pavement subbase layer made with the treated material being considered, resting on an infinite elastic bed with a modulus of 100 MPa. For simplicity, it is assumed that this modulus is independent of the traffic, consequently there is no support modification for the studied material. The interface between the subbase layer and the bed is embedded. This structure is loaded with a 2  6.5 tons per axle. The stress at the bottom of the subbase layer (rT) depends on its modulus and on the thickness of the layer (Fig. 10). In this chart, it is possible to compare mechanical properties of different materials, especially the measured properties of the three treated soils and the properties of standardized materials [50]. These standardized materials with mastered specifications [23] are commonly used in subgrade layers of road infrastructures. The QI for treated gravels vary from 20 cm to 40 cm. Considering the uncertainty on r6 leads to a QI = 50 cm for one material. The QI for treated sands vary from 40 cm to unclassed (i.e. QI > 50 cm) for two materials. Experimental results on the three treated soils give QI of 40 cm for AD and 30 cm for CSG and ASE. Results on the three treated soils highlight at least comparable mechanical performances compared to standardized treated gravels and sands. This result is mainly due to smaller moduli. Indeed, the three treated soils have moduli five to six times inferior than treated gravels. Treated soils are less stiff, and in the structure, the tensile stress at the bottom of the layer is directly related to the stiffness of the material. Higher modulus leads to higher tensile stress for a given thickness.

Table 2 Mean parameters of uniaxial fatigue tests performed in the literature.

rf Mean Std

E (MPa)

1/b

(MPa) 1.35 0.72

10,698 7697

11.73 2.21

e

r6

Dlog10(Nfail.)

DS (MPa)

0.80 0.15

0.10 0.05

(MPa) 0.47 0.10

0.65 0.41

Fig. 9. Relation between flexural strength rf, and r6, the tensile strengths obtained after Nfail. = 106 loadings.

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Réseau Ferré de France (RFF) for their interest in and support of this work. They also would like to thank Dr. B. Bechet for her help on geological aspects, Drs. J.M. Balay, V. Ferber and T. Sedran for fruitful scientific discussions, and the Lafarge Company for supplying the cement. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ijfatigue.2015.03. 010. References

Fig. 10. Chart for determining the quality indexes (QIs) of materials as a function of the couple (E, r6). QIs of treated soils and standardized materials [50].

Moreover, experimental values of r6 can be compared with values obtained from the behavior law (r6BL = rF/2). The interest of the behavioral law is to forecast the range of r6. The calculation leads to:  AD: r6 = 0.44 MPa vs r6BL = 0.36 MPa  CSG: r6 = 0.65 MPa vs r6BL = 0.55 MPa  ASE: to r6 = 0.65 MPa vs r6BL = 0.56 MPa. In all cases, the QI decrease, highlighting the interest of measuring mechanical fatigue performances to rationalize infrastructures. A complete fatigue test also allows the determination of DS, which is a fundamental parameter for integrating risk determination during the design process [50,6]. 5. Conclusions In conclusion, analytical, numerical and experimental results have shown that the two-point bending test is relevant for measuring mechanical fatigue performances in treated/stabilized soils in uniaxial flexion. For materials treated with a low amount of lime and/or hydraulic binders, a behavior law for uniaxial tests governs the results of mechanical fatigue performances, linking flexural strength rf, and fatigue strength for a million loadings r6. In addition, with lower stiffness, naturals soils treated with hydraulic binders can compete against standardized materials such as cement-stabilized aggregates for a use in subgrade layers and also in subbase layers of roads, to rationalize the global costs of infrastructures. Finally, a complete fatigue test enables the integration of risk determination during the design process. Acknowledgements The manuscript benefited from comments and suggestions by two anonymous reviewers. The authors would like to thank

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