Hollow cylinder apparatus for testing unbound granular materials of pavements

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Hollow cylinder apparatus for testingunbound granular materials ofpavementsBernardo Caicedo a , Manuel Ocampo b , Luis Vallejo c & JuliethMonroy aa Department of Civil and Environmental Engineering, Universidadde los Andes, Bogotá, Colombiab Department of Civil Engineering, Pontificia UniversidadJaveriana, Bogotá, DC, Colombiac Department of Civil and Environmental Engineering, University ofPittsburgh, Pittsburgh, Pittsburgh, PA, 15261, USAPublished online: 26 Jun 2012.

To cite this article: Bernardo Caicedo , Manuel Ocampo , Luis Vallejo & Julieth Monroy (2012)Hollow cylinder apparatus for testing unbound granular materials of pavements, Road Materials andPavement Design, 13:3, 455-479, DOI: 10.1080/14680629.2012.694093

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Road Materials and Pavement DesignVol. 13, No. 3, September 2012, 455–479

Hollow cylinder apparatus for testing unbound granular materialsof pavements

Bernardo Caicedoa*, Manuel Ocampob, Luis Vallejoc and Julieth Monroya

aUniversidad de los Andes, Department of Civil and Environmental Engineering, Bogotá, Colombia;bPontificia Universidad Javeriana, Department of Civil Engineering, Bogotá DC, Colombia; cUniversity ofPittsburgh, Pittsburgh, Department of Civil and Environmental Engineering, Pittsburgh PA 15261, USA

Unbound granular materials used in pavement structures are subjected to a complex stress pathwhich includes rotational stresses. Hollow cylinder apparatuses (HCAs) are suitable laboratorydevices for reproducing stress paths found in the field. However, as the size of the HCAdepends on the size of the granular particles, their use for testing unbound granular materialsfor pavements has not been reported in the literature. This paper presents the development of anew large size hollow cylinder apparatus designed to study the response of unbound granularmaterials subjected to stress rotation. This large HCA has hydraulic actuators for verticalmovement and torsion while shear and vertical stresses are servo-controlled to reproduce stresspaths produced by a heavy vehicle moving on a pavement structure. Confining stress is appliedthrough the use of rings with controllable stiffness. This paper highlights the role of ring stiffnesson stress paths. The results show the capabilities of the large HCA for studying the behaviourof unbound granular materials used in pavement structures.

Keywords: granular materials; hollow cylinder apparatus; rotation of stresses; pavements

1. IntroductionIncreasing interest in studying the performance of flexible pavements through the use of numericalmodels is heightening the need for improvements in our basic understanding of the response ofunbound granular materials subjected to repeated loading. Moving wheel loads constitute one ofthe main characteristics of pavements. These moving loads produce continuous rotation of theprincipal stresses in pavements. Consequently, the rate of shear and volumetric strains duringcyclic loading increases in a way that is similar to increases earlier demonstrated for sands (Sayaoand Vaid, 1989; Wong and Arthur, 1986). Despite the evident effect of rotation of principal stresseson the response of unbound granular materials in pavements, their effects on pavement structureshave been neglected in the usual studies based on tri-axial tests. For years tri-axial apparatuseshave been the most important devices used to characterise resilient behaviour and plastic strains ofunbound granular materials. Tri-axial tests can be performed under constant or variable confiningpressures, CCP or VCP respectively. Nevertheless, while these tests allow application of a widerange of stress paths and an abrupt rotation of principal stress by 90◦, none of these paths includecontinuous rotation of principal stresses (Tutumluer and Seyhan, 1999).

The directional shear cell (Arthur, Chua, & Dunstan, 1977) and the hollow cylinder apparatus(Chan and Brown, 1994) are alternative apparatuses for reproducing stress rotation. The latterapparatus allows reproduction of pavement field conditions by controlling both axial and shear

*Corresponding author. Email: [email protected]

ISSN 1468-0629 print/ISSN 2164-7402 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/14680629.2012.694093http://www.tandfonline.com

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456 B. Caicedo et al.

stresses (Brown, 1996). Since most HCAs have been developed to study the influences of inter-mediate stresses and continuous rotation of principal stress in clays and sands, their sizes areinappropriate for the study of the unbound granular materials used in pavements. Consideringthis limitation, the development of a new HCA apparatus which reproduces field stresses inunbound granular materials is crucial for a better understanding of the performance of flexiblepavements.

This paper presents the development of a large new HCA adapted for unbound granular mate-rials used in pavements. This HCA has a set of hydraulic servo controlled actuators which allowfor reproduction of a wide range of stress paths while rotating the principal stresses. Rings ofcontrolled stiffness are proposed as a new alternative for application of variable confining pres-sure. The paper demonstrates that this option allows application of different stress paths in the q,p plane. The good performance of the new HCA presented in this paper indicates that it could bean encouraging alternative for predicting the behaviour of unbound granular materials (UGMs)under more realistic stress paths.

2. Stress paths produced by moving wheel loadsTo characterise the behaviour of UGMs in pavements it is important that the actual loading inducedby moving wheels be reproduced in the laboratory. This type of loading imposes variable vertical,horizontal and shear stresses whose combined effects produce continuous rotation of the principalstresses as shown in Figure 1 (Brown, 1996). The variation of the magnitude of the stress path andits direction, Figure 2, can be characterised by the mean stress p, the differential stress q, and theangle αzy which represent the rotation of the stress path in the plane (z, y). Furthermore the roleof the intermediate principal stress can be characterised by the parameterb as defined by Habib

Figure 1. Direction of principal stress under the symmetry axis of a double wheel load, analysis made ina linear visco-elastic isotropic behaviour, using ALIZE (Balay, 2009).

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Road Materials and Pavement Design 457

Figure 2. Principal stresses and rotation angle in the z, y plane.

Figure 3. Trajectories in the (b, α) plane achieved in various devices.

(1951). These parameters are defined in a three dimensional stress path as follows:

p = σ1 + σ2 + σ3

3(1)

q =√

12[(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2] (2)

αzy = 12

Tan−1(

2τzy

σz − σy

)(3)

b = σ2 − σ3

σ1 − σ3(4)

The continuous rotation of the principal stresses cannot be simulated using CCP or VCP tri-axial tests (Tutumluer et al., 1999). In fact, according to Saada and Townsend (1981), Figure 3, theprincipal stresses in tri-axial tests of compression and extension (TC and TE) remain in the axialand radial directions respectively (α = 0◦ or α = 90◦). In true tri-axial tests (TT) the directionof principal stresses remains in line with the loads applied. Continuous rotation of principalstresses is feasible using directional shear cells (DSCs) or HCAs. In the case of HCAs there areample possibilities for trajectories of stresses in the plane of principal stress direction and in theintermediate principal stress parameter (α, b). These can be achieved by varying the internal andexternal pressures (pi, pe). When internal and external pressures are the same, the intermediatestress factor depends on the principal stress direction through the relationship b = sin2 α (Saadaet al., 1981).

Neglecting the effect of the continuous rotation of the principal stresses on unbound granularmaterials has an important consequence for predictions of pavement behaviour since, according

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Figure 4. Idealised flexible pavement structure.

to Wong and Arthur (1986) and to Sayao and Vaid (1989) continuous rotation of principal stressesincreases the magnitude of shear and volumetric strains.

An accurate evaluation of stresses in the granular layers of flexible pavements requires theinclusion of all the complexities of UGM’s behaviour in a three dimensional finite element model.These complexities include non-linearity, anisotropy, non-saturation, compaction induced residualstresses, and the role of rotation of stresses (Akou, Heck, Kazai, Hornych, Odeon, & Piau, 1999;Aubry, Baroni, Clouteau, Fodil, & Modaressi, 2001; Balay, Gomes-Correia, Hornych, Jouve,& Paute 1997; Blanc, 2011; Blanc, Di Benedetto, & Tiouajni, 2011; Chazallon, 2000; Dutine,Di Benedetto, Pham Van Bang, & Ezoui, 2007; Hicher, Daouadji, & Fedghouche, 1999; Hoff,Nordal, & Nordal, 1999; Tutumluer et al., 1999). However, a rough idea of the range of stressvariations in granular layers of flexible pavements under wheel loading is sufficient for establishingwhether or not it is feasible to reproduce these variations in an HCA. This kind of estimate can beobtained with a conventional program that assumes the pavement structure is a system of elasticlayers having infinite extent in the horizontal direction over a semi-infinite sub-grade. Figure 4represents the idealised flexible pavement structure we used for calculating stress variation. Thisstructure is composed of a 0.1 m bituminous layer with a Young’s modulus of 4000 MPa, auniform 1 m granular layer with a resilient Young’s modulus (MR) alternating among 50, 200and 400 MPa, and a sub-grade with a resilient Young’s modulus of 50 MPa. This structure isexposed to 130 kN double wheel loading. Stresses are calculated using ALISE (Balay, 2009) andthen analysed in two critical sections as recommended in routine design procedures (Gomes-Correia, 2001).

Figure 5 represents the stress trajectories calculated in Section 1 of Figure 4 for a moving wheelload on the pavement structure represented in Figure 4. The stress trajectories are calculatedat different depths in the base layer (points A to I ) and are represented in the plane of meanstress, differential stress and rotation angle (p, q, α), and in the plane of the rotational angle andintermediate principal stress parameter (α, b). Effects with a base layer resilient Young’s modulus(MR) of 50 MPa are represented in Figures 5(a) and 5(d). Effects when MR = 200 MPa are shownin Figures 5(b) and 5(e), while those for MR = 400 MPa are shown in Figures 5(c) and 5(f).

The first case, shown in Figures 5(a) and 5d, represents the extreme situation of a homogeneouspavement structure in which the resilient Young’s modulus of the base is the same as that of the sub-grade. In this case the trajectories of the stresses in the plane (p, q, α) follow an inclined bell curve

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Figure 5. Rotation angle and b parameter in Section 1 of Figure 4 for double wheel loading and differentresilient Young’s moduli: MR = 50 MPa, (a), (d); MR = 200 MPa, (b), (e); MR = 400 MPa, (c), (f). Linearvisco-elastic “isotropic” analysis.

with a slightly curved projection in the p − q plane. In the α − −b plane the intermediate stressfactor begins at a non-zero value when the load is on the point where the stresses are calculated(b = 0.2, α = 0 at point A, and b = 0.1, α = 0 at point B). This non-zero value appears as a resultof a three dimensional stress state at the central point between the two wheels. For deeper points(C to I ) the role of the intermediate principal stress decreases when the load is directly over thepoint. As the angle relating the principal stress direction grows from α = 0 to α = 30, parameterb at points A and B decreases, but then increases from α = 30 to α = 60 and approaches the curveb = sin2 α for α = 50.

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As the resilient Young’s modulus of the base layer grows, Figures 5(b), 5(c), 5(e), 5(f), tensionstresses appear in the deeper points of the base layer. Consequently, the intermediate principalstress parameter b and the slope of the p − q trajectory become negative. At points higher inthe layer, the b parameter begins at a higher value than in the case of a homogeneous layer,(b = 0.35 and b = 0.18 for α = 0 and MR = 200 MPa, and b = 0.5 and b = 0.22 for α = 0 andMR = 400 MPa at points A and B). For deeper points (C to I ) the role of the intermediate stress isless important when the load is on top of these points, with the b parameter less than 0.1. Whenthe angle of the principal stress direction grows, the trend is similar to that of the homogeneouscase, but the trajectories are closer to the line b = sin2 α and even cross this line at α = 50.

Figure 6 represents the stress trajectories for the same cases considered in Figure 5, but calcu-lated in Section 2 of Figure 4 (at the centre of one of the wheels). The stress trajectories in thisposition are similar to those presented in Figure 5. However, there are several slight differences.The magnitude of the differential stress is higher, and the p − q paths are closer to straight lines.In addition, the intermediate principal stress parameter is closer to zero when the load is on topof the points, even for the upper points (A and B). It is important to note that these results wereobtained considering linear visco-elastic isotropic behaviour; clearly this is a simplification of theactual behaviour of granular and bituminous materials, however it is adequate for the purpose ofhaving an idea about the range of stress variations in granular layers of flexible pavements.

3. Stress path in a hollow cylinder apparatus with deformable ringsThe stress path in the granular layers of flexible pavements which can be reproduced by a hollowcylinder apparatus is shown in Figures 3, 5 and 6. Nevertheless, to achieve this goal, two majorchallenges must be overcome. First, the size of the HCA must be in agreement with the grainsize distribution of unbound granular materials used in pavements. Second, the apparatus mustapply variable internal and external confining pressures in order to better reproduce the stress pathinduced by moving wheels.

The consequence of these two requirements is a huge apparatus with a large sized cylinder andconfinement device. One way to reduce the complexity of the apparatus is to use deformable ringsas confinement devices. The benefits of using deformable cylinders to study the friction angle ofgranular materials including large particles have been shown by Biarez and Hicher (1994).

Stresses and strains within a hollow cylinder made of an isotropic elastic material and exposed toinside pressures, outside pressures and torsional shear stress can be calculated using the equationsgiven by Lade (1988) (Figure 7):

∂2uρ

∂r2 + 1r

∂uρ

∂r− uρ

r2 = 0 (5)

ερ = −∂uρ

∂r(6)

εθ = −uρ

r(7)

σρ = (λ + 2G)ερ + λεθ + λεz (8)

σθ = λερ + (λ + 2G)εθ + λεz (9)

σz = λερ + λεθ + (λ + 2G)εz (10)

In these equations uρ is the radial displacement, r is the radial distance to a point in the hollowcylinder, σρ is the radial normal stress, σθ is the tangential normal stress, σz is the vertical stress,ερ , εθ and εz are the strains in each direction, λ is Lame’s constant, and G is the shear modulus.

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Figure 6. Rotation angle and b parameter in Section 2 of Figure 4 for double wheel loading and differentresilient Young’s moduli: MR = 50 MPa, (a), (d); MR = 200 MPa, (b), (e); MR = 400 MPa, (c), (f). Linearvisco-elastic “isotropic” analysis.

As the hollow cylinder is confined by deformable rings, Equation (5) must be solved consideringthe following boundary conditions:At r = rext

uρ−ext = Kn−extrext

σρ−ext(11)

And for r = rint

uρ−int = −Kn−intrint

σρ−int(12)

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Figure 7. Stresses acting on hollow cylinder specimen during torsional shear tests.

Figure 8. Radial and tangential stresses in a hollow cylinder specimen are a function of ring stiffness,linear elastic analysis.

In the above equations Kn is the stiffness of the ring. In the case of a thin walled elastic ring,Kn is given by:

Kn = E1 − υ2

eR

(13)

In this equation E and ν are the elastic constants of the ring, e is the thickness of the wall andR is the radius of the ring.

Equations (5) to (12) can be solved for different values of the relationship between the stiffness ofthe ring and the resilient Young’s modulus of the material in the hollow cylinder in order to obtainthe radial and tangential stresses at any point. Figure 8 shows the radial and tangential stressesnormalised by the external stresses and calculated for a hollow cylinder having rext = 359 mm andrint = 226.5 mm. The results show that for rings with high degrees of stiffness (50 < Kn/MR <

200) the radial and tangential stresses are close together, and there is a slight variation along theradial distance. On the other hand when the rings have a low degree of stiffness (Kn = MR), there isa significant variation of the radial and tangential stresses along the radial distance of the hollowcylinder. There is a clear relationship between the stiffness ratio, Kn/MR, and the relationship

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Figure 9. Relationship between the stiffness ratio (Kn/MR) and the ratio of external and internal radialstresses (σρint/σρext), linear elastic analysis.

Figure 10. Variation of the mean and differential stresses (p, q) acting in a hollow cylinder specimen, linearelastic analysis.

between internal and external radial stresses, σρ int/σρ ext. This relationship is described with goodaccuracy with a hyperbolic equation as indicated in Figure 9.

Although there is a significant variation of the radial and tangential stresses as the ring stiffnessdecreases, the mean stress p and the differential stress q remain approximately constant alongthe radial distance even when the ring stiffness is low (Kn = MR). In fact, in this case the mainstress p differs by less than 0.3% between the external and internal radii, while the difference indifferential stress is less than 5%, Figure 10.

Variation of the relationship between the external radial stress and the vertical stress is pre-sented in Figure 11. As observed, this proportion increases as the stiffness ratio Kn/MR increasesand approaches the stress relationship corresponding to an elastic material, Kh = ν/(1 − ν), forstiffness ratios higher than 50.

When shear stress is applied in the tangential direction (τzθ�= 0), the principal stress becomes

σ1 = 12

[(σz + σθ ) +

(σz − σθ

cos 2α

)](14)

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Figure 11. Relationship between the external radial stress and the vertical stress, linear elastic analysis.

The results of the calculated stresses in a hollow cylinder under vertical loading confined bydeformable rings show that the major principal stress follows the vertical direction (σ1 = σz), theintermediate principal stress acts in the tangential direction (σ2 = σθ ), and the minor principalstress’s direction is radial (σ3 = σρ) (Figure 12(a)).

σ2 = max{σρ ,

12

[(σz + σθ ) −

(σz − σθ

cos 2α

)]}(15)

σ3 = min{σρ ,

12

[(σz + σθ ) −

(σz − σθ

cos 2α

)]}(16)

The direction of the major principal stress is

α = 12

tan−1(

2τZθ

σZ − σθ

)(17)

As a result of the torsion shear stress, the major principal stress grows and the intermediateprincipal stress decreases (Figures 12(b) and 12(c)). This causes the difference between the inter-mediate and minor principal stresses to decrease to a limit when the position of these stressesshifts. The intermediate principal stress becomes the radial stress. At this point the minor principalstress is in the tangential plane. All these changes in stresses lead to an increase in the differentialstress q.

Assuming that the material in the hollow cylinder is linear elastic and is confined by deformablerings, the magnitudes of the radial and tangential stresses will increase proportionally to the verticalstress. The trajectory in the (p, q) plane will be linear, corresponding to a variable confining stresspath. The application of torsion shear stress will lead to an increase in the q/p slope (Figure 13).

The slope of the stress path in the (p, q) plane is related to the rotation angle α and to the stiffnessof the rings. Figure 14 shows the different paths projected in the p − q plane corresponding todifferent ring stiffnesses. As observed, by using rings of different stiffnesses it is possible to coverstress paths similar to those recommended in the European standard, EN 13286-7: “Unbound andhydraulically bound mixtures – Part 7: Cyclic load tri-axial test for unbound mixtures” (procedureA with variable confining pressure).

The stress paths in the (b, α) plane also depend on the stiffness of the rings. As shown in Figure 8,when any shear stress is applied, the difference between the intermediate principal stress and minorprincipal stress decreases as the stiffness of the rings grows. The b parameter for α = 0 grows

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Figure 12. Mohr circles for torsion loading.

Figure 13. Stress paths in the (p, q, α) plane during torsion loading, MR = 200 MPa, Kn = 200 MPa, linearelastic analysis.

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Figure 14. Stress paths in the plane (p, q) for different relationships Kn/MR, linear elastic analysis.

Figure 15. Stress paths in the (b, α) plane for different Kn/MR relationships, linear elastic analysis.

as the stiffness of the rings decreases. When the shear stress grows, the b parameter decreasesnear zero because the intermediate and minor principal stresses approach each other. Then the bparameter grows again when the intermediate and minor principal stresses shift (Figure 15).

Although all the analyses of stresses in a hollow cylinder apparatus presented in this sectionwere performed using a linear elastic material (MR variable and Poisson ratio ν = 0.35), theyshow the feasibility of reproducing stress paths in the granular layer of pavements using a hollowsample confined by deformable rings. In particular they show that it is possible to reproduce thecontinuous stress rotation and variation of the intermediate stress parameter b produced by wheelloading. The following sections of this paper describe the design of this large new HCA and someresults obtained with this equipment.

4. Description of the new HCAThe small sizes of most of the HCAs developed over the last 60 years do not allow testing ofunbound granular materials for pavements. In fact, the sample thicknesses within the hollowcylinders of these apparatuses, which are shown in Table 1, restrict testing to clays and sands. Thedevelopment of a new HCA designed specifically to test UGMs for pavements had to meet the

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Table 1. Review of HCAs developed in the last 60 years.

Reference Year Di (mm) De (mm) H (mm)

Habib 1953 – 35 87Kirpatrick 1957 64 102 152Broms and Casbarian 1965 76 127 254Sulkje and Drnovsek 1965 40 64 80Saada and Baah 1967 51 71 127–177Lomize et al. 1969 250 310 180Ishibashi and Sherif 1974 51 102 13–25Ishihara et al. 1975 60 100 70Lade 1975 180 220 50Ishihara et al. 1980 60 100 106Lade 1981 180 220 100–400Tatsuoka et al. 1982a 60 100 200Tatsuoka et al. 1982b 30 50 200Robinet et al. 1983 150 180 180Hight et al. 1983 203 254 254Chehade et al. 1989 150 180 180Towhata and Ishihara 1985 60 100 104Miura et al. 1986 60 100 200Alarcon et al. 1986 38–71 71–100 200Dutine 2005 160 200 120Khemissa 1992 70 100 150Crockford and Sousa 1992 178 229 203Di Benedetto et al. 2001 160 200 120O’Kelly et al. 2005 70 100 200Brown and Richardson 2004 224 218 500Nishimura et al. 2007 71 100 195HongNam and Koseki 2005 120 200 300Reiffsteck et al. 2007 70 100 150Kiota et al. 2008 90 150 300Ocampo 2009 453 718 500

following challenges. First, the thickness of the sample must match the grain size distribution ofthe material. Second, the size of the hollow cylinder must be large enough to avoid disturbances inthe distribution of stresses. Third, the confining stress must be variable in order to better replicatethe stresses produced by wheel loading.

The new HCA presented in this paper was designed to perform tests on samples of well gradedmaterials in which the maximum dimension of the largest particles is 25 mm and the minimumthickness of the sample is 125 mm. This is in accordance with Japanese standards for laboratoryshear tests using hollow cylinders (Kuwano, Katagiri, Kita, Nakano and Kuwano, 2001). Thosestandards recommend that hollow cylinder apparatus tests using well graded materials have aminimum cylinder thickness five times the size of the largest particle, whereas they recommend aminimum cylinder thickness 10 times the size of largest particle when uniformly graded samplesare used. These recommendations overlap with the results of Gourves in Biarez et al. (1994)which concluded that sample size should be at least 10 times larger than the largest particle inorder to reduce the variation of stresses at the microscopic level.

The geometry of the sample within the HCA affects the uniformity of the distribution of stressessince shear stress in an elastic hollow cylinder subjected to torsion increases linearly in relationto the radius of the cylinder. According to Saada et al. (1981), a proportion of 0.65 between theinternal and external radius is sufficient for reducing the relative difference in the distribution ofshear stresses to 16%.

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Table 2. Sample dimensions.

Dimensions Values

Internal radius (mm) 226.50External radius (mm) 359.00Average radius (mm) 292.75Thickness (mm) 132.50Height (mm) 500.00Internal/external radii ratio 0.63Ratio height/average radius 1.70

Lade (1981) studied the influence of height in hollow cylindrical samples. He demonstratedexperimentally that the effect of the plates on a sample diminishes as the height of the sampleincreases and becomes negligible when the height is four times the average of the radius. Saadaet al. (1981) concluded that increases in the precision of measurement of the stress distributionbecome insignificant when a sample is 1.5 or more times higher than its average radius. Similarresults were obtained by Hight, Gens and Symes (1983) in a finite element analysis of this problem.The dimensions of the HCA presented in this paper are shown in Table 2. These dimensions resultin a relationship between internal and external radii of 0.63, a relationship between the heightof the sample and the average radius of 1.71, and a relationship between the thickness of thesample and the size of the largest particle of 5.3. These relationships are in agreement with theones recommended in the literature.

Figure 16. Confining rings; (a) assembled rings, (b) internal and external rings, (c) loading frame.

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Table 3. Dimensions of the rings used in the hollow cylinder apparatus.

External diameter Internal diameter Wall thickness Height StiffnessRing (mm) (mm) (mm) (mm) MPa

Internal 453 379 37 50 225External 758 718 20 50 65

Applying variable confining stresses in a large hollow cylinder having the described dimensionsrequires a large confining chamber. As described in Section 3 of this paper, an alternative forreproducing variable confining stresses is to use deformable rings as confining devices. Thissolution was adopted for the design of the new HCA. The sample is confined by steel ringsseparated by ball bearings so that the rings can rotate freely (Figures 16(a) and 16(b)). Thissolution significantly reduces the torsion direction friction. Vertical direction friction was reducedthrough the use of silicon grease covered by a 0.20 mm plastic sheet as proposed by Fang, Chen,Holtz and Lee (2004).

Each ring has a height of 50 mm. They can be installed one above the other, so that it is possibleto test samples with different heights. Dimensions and stiffnesses of the rings are presented inTable 3.

The rings are placed inside a loading frame, 1.76 m high by 1.12 m wide (Figure 16(c). Thisframe was designed to test UGM samples under vertical stresses up to 600 kPa, and shear stressesup to 400 kPa. Vertical and shear loading are transmitted to the sample through a rigid plate havinginserts that guaranty good transmission of shear stresses. This plate is attached to one verticalhydraulic actuator and two horizontal hydraulic actuators attached to the plate and the frame byaxial and radial bearings respectively.

4.1. Instrumentation, data acquisition system, and control systemEach hydraulic actuator has a load cell located at the front and an LVDT sensor, Figure 16c. Thesesensors allow calculation of vertical and shear stresses and external vertical and shear strains. Tomeasure the vertical and shear strains directly on the sample, two sets of two resistive displacementsensors are located on the external cylinder in direct contact with the material. These sensors arepositioned in a triangular configuration. Each vertex of the triangle has a screw inserted in thesample which passes through the external wall of the HCA through holes in two rings (Figure 17).As shown in Figure 17(d), four strain gauges installed on the central external ring measure radialstress which indicates deformation of the external rings.

4.2. Idealisation of the loading path applied in the HCAThe HCA presented in this paper allows application of infinite combinations of vertical andshear stresses. However, to use this HCA to study the behaviour of UGMs materials in pavementstructures, it is useful to choose a combination of stresses as close as possible to the stressesproduced by moving loads. As described in Section 2, the stresses produced by a moving load ata point in a pavement structure vary in accordance with numerous variables. These include theexternal load, the thickness of the layers, the depth of the point, and the stiffness of the differentlayers of the pavement. For this reason defining a laboratory stress path based on the results of theanalysis of a particular pavement structure could be non useful for practical purposes. Instead afirst approximation of the relationship between shear and vertical stress in simplified semi-infinitespace can be obtained using Boussinesq’s solution. In the case of a point load P, the state of stressis described by Equations (18) and (19) in which σz is the vertical stress and τrz is the shear stress

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Figure 17. Triangular frame of resistive displacement sensors and strain gauges.

Figure 18. (a) Stresses generated by a moving point load. (b) Vertical and shear stresses assuming a sinefunction for the vertical stress.

produced by a moving point load on the surface of the pavement.

σz = 3P2π

z3

R5 (18)

τrz = τzr = 3P2π

rz2

R5 (19)

The location of the point whose stresses are calculated is described by z, r, and R (Figure 18(a)).The maximum vertical stress σz_ max can be obtained from Equation (18) by setting r = 0 (whenthe load is directly over the element):

σz_ max = 3P2π

1z2 (20)

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Then the vertical and shear stresses can be represented as a function of maximum vertical stressas follows:

σz = σz_ maxz5

R5 = σz_ max

[( rz

)2 + 1]−5/2

(21)

τrz = σz_ maxrz4

R5 = σz_ max

[( rz

)2 + 1]−5/2 ( r

z

)(22)

Following from Equations (18) and (20), the relationship r/z is

rz

= ±√(

σz_ max

σz

)2/5

− 1 (23)

Finally, following from Equations (21), (22) and (23) the shear stress can be expressed as afunction of the vertical stress σz and maximum vertical stress σz_ max:

τrz = ±σz

√(σz_ max

σz

)2/5

− 1 (24)

The derivation of Equation (22) indicates that the maximum shear stress appears at z = 2r. Asa consequence, the maximum shear stress as a function of the maximum vertical stress becomes

τrz_ max = σz_ max16

55/2 (25)

The stresses required to reproduce the continuous stress rotation produced in pavements bymoving loads in the HCA can now be defined. The vertical stress σz can be defined using anycyclic function (e.g. sine function). This function in turn is defined by the maximum vertical stressσz_ max and the frequency, ω. Shear stress τrz is defined as a function of the vertical stress usingEquation (24). For example, Figure 18(b) shows the vertical and shear stresses corresponding tothe function σz = σz_ max| sin(ωt)|. For this case τrz in Equation (24) is positive for ωt < π/2, andnegative for ωt > π/2.

4.3. Derivation of stresses and strains from the experimental measurementsThe external radial stress is obtained from the measurements taken from the strain gauges locatedin the middle external ring as in Equation (26) in which σρ_ext is the external radial stress, Es isYoung’s Modulus, e is the thickness of the external ring, Re is the external radius of the HCA andεr is the strain measured by the strain gauges.

σρ_ext = EsεreRe

(26)

As shown in Figure 9, the internal radial stress is related to the external radial stress throughthe relationship between the stiffness of the ring and the resilient Young’s modulus of the materialin the HCA:

σρ_int

σρ_ext= 1

1 + 3.29(

KnMR

)−1.44 (27)

All the stresses at the mean radius of the hollow cylinder (σz, σρ , σθ , τθz), can be derived whenthe internal stresses, external radial stresses, vertical load Fz and torque T applied to the material

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in the HCA are all known (Reiffsteck, Tacita, Mstat and Pilniere, 2007):

σz = Fz + π(R2eσρ_ext − R2

i σρ_int)

π(R2e − R2

i )(28)

σρ = Reσρ_ext + Riσρ_int

Re + Ri(29)

σθ = Reσρ_ext − Riσρ_int

Re − Ri(30)

τθz = 3T2π(R3

e − R3i )

(31)

In the previous equations Re and Ri are the external and internal radius of the hollow cylinder.The principal stresses are derived as follows:

σ1 = σz + σθ

2+

√(σz − σθ

2

)2

+ τ 2θz (32)

σ2 = σρ (33)

σ3 = σz + σθ

2−

√(σz − σθ

2

)2

+ τ 2θz (34)

Finally, the mean and differential stress (p, q) as well as the stress rotation angle and theintermediate stress parameter (α, b) can be calculated using Equations (1) to (4).

Vertical and shear strains (εz, γ ) can be calculated externally using measurements from theLVDTs located on the vertical and horizontal actuators, or they can be calculated internally usingmeasurements from the triangular frame of resistive sensors located mid height on the hollowcylinder. The external strains are expressed in Equations (35a) and (35b) in which w is thedisplacement of the vertical actuator, s is the displacement of the horizontal actuators, Ra is theradius of the points where the horizontal load is applied (Figure 17), Rm is the mean radius ofthe hollow cylinder, and H is the height of the sample.

εz = wH

, γ = Rm

RaHs (35a and 35b)

The strains at mid-height of the sample can be calculated through the deformation of thetriangular frame of resistive sensors placed at mid height of the sample (Figure 17). First theangle β2 of the deformed triangular frame is calculated using the length of the sides of the triangle(L1 + �L1, L2 + �L2), the position of the frame (radius Rs and angle θ ), and εr , the tangentialstrain of the middle ring as measured by the strain gauges:

β2 = cos−1[(L1 + �L1)

2 + (2Rs sin(θ/2)(1 + εr))2 − (L2 + �L2)

2

4(L1 + �L1)Rs sin(θ/2)(1 + εr)

](36)

Then �H , the vertical displacement, is calculated as the variation of H0, the initial height ofthe triangular frame, as follows:

�H = H0 − (L1 + �L1) sin(β2) (37)

�s, the tangential displacement of the frame, is calculated as follows:

�s = Rs sin(θ/2)(1 + εr) − (L1 + �L1) cos(β2) (38)

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Finally the vertical and shear strains at mid height and mid radius of the hollow cylinder (εzi,γi) are calculated using Equations 39a and 39b.

εzi = �HH0

, γi = Rm

Rs

�sH0

(39a and 39b)

5. Results and discussionA trial run of the apparatus was performed using a dry sample of a granular material havingparticles whose sises varied from 1.19 mm to 25.4 mm. The sample material was weatheredsandstone from Soacha, Colombia (Caicedo, Coronado, Fleureau, & Gomes-Correia, 2009). Therings confining the sample were covered with silicon grease and with 0.20 mm plastic sheeting tominimise vertical and tangential friction (Fang et al., 2004).

Five layers of dried material were placed in the apparatus. Since the material was in a dry state,the sample was compacted according to the ASTM C 29/C 29M – 07 – Standard Test Method forBulk Density and Voids in Aggregates. To compact each layer of the sample, 150 strokes from asteel tamping rod with a rounded tip were evenly applied over the surface. After compaction thesample height was 50 centimetres.

The test consisted of six series of 20,000 load cycles. The first series was conducted with avertical stress of 40 kPa. In each subsequent series the vertical stress was increased an additional40 kPa until 240 kPa was reached in the final series (Ocampo, 2009). Cyclic vertical stress wasapplied following a half sine function with a minimum vertical stress of 6 kPa to avoid anyseparation between the load plate and the sample. The test frequency was one hertz. To simulatea moving wheel the imposed shear stress was calculated using Equation (25).

Vertical loads, horizontal loads, the displacement of the hydraulic cylinders, and the tangentialstrain of the middle external ring were all measured during the test. After the whole series of loadcycles, the grain size distribution was measured. Finally, the distribution of crushed particles andthe initial grain size distribution were compared and analyzed.

Figure 19 shows an example of the results obtained for a 200 kPa vertical stress. The measuredvertical, shear and radial stresses on the external ring are presented in Figures 19(a) to 19(c).These figures show that the vertical stress, σz, follows the imposed half sine signal with goodaccuracy. The shear stress, τrz, follows a signal that fulfils the requirements for simulating a movingwheel: zero shear stress for zero vertical stress and for maximum vertical stress. However, thestress signal presented in Figure 19(b) approaches a triangular signal more than does the signalpresented in Figure 18(b). This discrepancy is probably related to the accuracy of the controlsystem of the horizontal hydraulic cylinders. Figure 19(c) shows the radial stress on the externalring calculated using Equation (26) and based on the measure of the tangential strain in the middlering. This figure shows proportionality between radial and vertical stresses, although the stressproportionality becomes blurred when approaching the vertical maximum. Further research isnecessary to explain this behaviour.

Since the relationship between the rigidity of the rings and the resilient Young’s modulus inthis test (Kn/MR) is higher than 50, the radial stress on the internal ring σρ_int can be assumed to beequal to the radial stress on the external ring σρ_ext (Figures 8 and 9). Based on this assumption,equations for the whole stress state can be calculated. These include equations for principalstresses, mean stress p, differential stress q, the angle α relating the rotation of stresses and theintermediate stress parameter b (Equations (14), (16), (1), (2), (3), and (4)). The variations ofthese stress parameters are presented in Figures 19(d) to 19(g). It is important to note that theinternal stress is calculated based on Equation (27), this equation can be validated experimentallyincluding strain gages in the internal rings.

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Figure 19. Example of results of the vertical, shear and radial stresses measured and the calculated p, q,α and b parameters.

Figure 20 presents the stress path in the p, q, α plane and the relationship between the inter-mediate stress parameter b and the rotation angle α during the cycles presented in Figures 19(a)to 19(c). The stress path in the p, q plane follows a linear variation representing proportionalitybetween p and q as happens in the granular layers of pavements (Figures 5 and 6). The stress pathwhich includes the rotation angle α has a rectangular shape instead of the bell shaped paths ofFigures 5 and 6. Despite this discrepancy the stress path obtained in the hollow cylinder representsa good approximation of the stress path induced by a moving wheel.

Figure 21 represents the relationship between the radial and vertical stresses (Figure 21(a)), andbetween radial and principal stress σ1 (Figure 21(b)). The ratio of the horizontal normal stress atthe mid-height of the specimen to the vertical stress at the top of the specimen decreases when thevertical stress becomes larger than about 150 kPa (Figure 21(a)), this behaviour appears becauseradial stress σr remains approximately constant as a result of the simultaneous increase of verticalstress σz and decrease of shear stress τrz (Figures 19(a), 19(b) and 19(d)). On the other handthe ratio of the horizontal normal stress at the mid-height of the specimen to the principal stressbecomes much more linear since the stress σ1 remain approximately constant in the same rangeof stresses than the radial stress is approximately constant (Figure 19(c)). The Poisson ratio canbe obtained from the relationship between the vertical and horizontal stresses. The Poisson ratiocalculated using the results of Figure 21(a) is ν = 0.24.

Figure 22 shows the hysteretic cycles relating vertical strain and vertical stress during loading.The permanent strain and the resilient strain during loading are clearly identified in this figure

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Figure 20. Example of results of the stress path in the p, q, α plane and variation of the b parameter.

Figure 21. Relationship between the external radial stress and the vertical stress.

Figure 22. Hysteretic cycles during loading and unloading.

demonstrating that the new HCA presented in this paper is useful for measuring the resilientYoung’s moduli and permanent strains of unbound granular materials.

The grain size distributions of the Soacha material used in this study exhibits large variationsduring loading (Caicedo et al., 2009). As expected, crushing of some particles was evident after

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Figure 23. Crushing of particles after loading and evolution of grain size distribution after testing.

loading (Figures 23(a) and 23(b)). As a consequence, the grain size distribution shows an evolutiontowards smaller particles after loading (Figure 23(c)).

6. ConclusionsThis paper presents the development of a new HCA whose size is adapted for unbound granularmaterials used in pavements. The HCA has a set of hydraulic servo-controlled actuators allowingreproducing a wide range of stress paths with continuous rotation of the principal stresses.

Tri-dimensional stress paths on schematised pavement structure are analysed to choose thecharacteristics of the new HCA necessaries to provide the capacity of reproducing pavementstresses under moving wheels.

Applying variable confining stresses in a large size HCA requires huge technological com-plexities, for this reason the use of rings of controlled stiffness is proposed as a new alternativeto apply variable confining pressure. Analysis of stresses in the HCA is carried out assuming anelastic behaviour of unbound granular materials (UGMs), this study shows that using differentrelationships between the rigidity of the rings and the resilient Young’s modulus of UGMs it ispossible to reproduce different stress paths in the p, q plane, these paths have a good agreementwith the paths followed by UGMs in pavement layers.

Preliminary results show the good performance of the HCA apparatus presented in this paper,this new apparatus appears as an encouraging alternative to predict the behaviour of UGMs undermore realistic stress paths.

AcknowledgementsThe authors want to particularly acknowledge the Department of Science, Technology and Innovation ofCOLCIENCIAS who supported this work.

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Documents

Hollow cylinder apparatus for testing unbound granular materials of pavements