Load transfer mechanisms in geotextile-reinforced embankments overlying voids: Numerical approach and design

Geotextiles and Geomembranes 44 (2016) 381e395

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Load transfer mechanisms in geotextile-reinforced embankments overlying voids: Numerical approach and design Pascal Villard a, 1, Audrey Huckert b, 2, Laurent Briançon c, * a

Univ. Grenoble Alpes, 3SR, CNRS UMR 5521, Domaine Universitaire, BP 53, 38041 Grenoble Cedex 09, France EGIS Geotechnique, 3 rue du Dr Schweitzer, 38180 Seyssins, France c LGCIE INSA Lyon, 34 Avenue des Arts, 69621 Villeurbanne Cedex, France b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 October 2015 Received in revised form 15 January 2016 Accepted 20 January 2016 Available online xxx

A numerical model was used to investigate the mechanical behaviour of granular embankments reinforced by geosynthetics in areas prone to subsidence and to overcome the shortcomings of the current design methods. The ability of the numerical model to consider the load transfer mechanisms and the deflection of the geosynthetic was established by comparison with experimental data. By testing two numerical processes, it was demonstrated that the cavity opening modes have a great influence on the shape of the load distribution transmitted to the geosynthetic sheet above the cavity and on the expansion mechanisms of the soil. An approximate conical load distribution seems well adapted when considering a progressive cavity diameter opening process, whereas an inverted load distribution seems more suitable for a gradual settlement process. In both cases, the intensity of the load transfer mechanism can be approached by the Terzaghi's formulation using an appropriate value for the ratio between the horizontal and vertical stresses. Finally, recommendations based on the experimental and numerical results are proposed to promote a better design of such structures. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Geosynthetic reinforcement Localized sinkhole Granular material Numerical modelling Analytical design

1. Introduction In many countries, new environmental developments (housing, urban substructures, roads and railway infrastructures) increasingly occur in areas that present a high risk of localized sinkholes, such as karstic regions or former mining exploitation areas. To ensure the stability and longevity of the structures, various reinforcement methods such as piles, concrete slabs, nails, or geosynthetics are used. These reinforcements are then supposed to withstand the possible formation of a sinkhole with a determined design diameter after the construction of the infrastructure. This solution can be applied to reinforce platform bridging for buried utilities (El Naggar et al., 2015). The present study focuses on the geosynthetic solution and its behaviour to prevent surface damage. A particular attention was paid to railway and road infrastructures (new structures or

* Corresponding author. Tel.: þ33 472 438 370. E-mail addresses: [email protected] (P. Villard), audrey.huckert@ egis.fr (A. Huckert), [email protected] (L. Briançon). 1 Tel.: þ33 456 528 628; fax: þ33 476 827 043. 2 Tel.: þ33 476 484 748; fax: þ33 476 484 447. http://dx.doi.org/10.1016/j.geotexmem.2016.01.007 0266-1144/© 2016 Elsevier Ltd. All rights reserved.

rehabilitations of old structures) for which a low thickness of well graduated granular material is used above the geosynthetic in order to minimize the financial and environmental costs that represent the transport of material and the realization of the structure. In this kind of applications, commonly used in some European countries, the determination of the load transfer mechanisms within the granular embankments, very sensitive to the embankment thickness, is of primary importance. One of the most recent design methods of geosynthetic reinforcement over a cavity (Briançon and Villard, 2008) considers the friction mechanisms in anchorage areas or the change of orientation of the reinforcement sheet at the edges of the cavity. Despite this reformulation, the load transfer mechanisms within the embankment over a cavity are not yet fully understood: the distribution of the load on the geotextile sheet either in anchorage areas or over the cavity is considered to be uniform as a simplification, and the cavity opening mode is not considered. To better understand the load transfer mechanisms developed within the reinforced platform, a full-scale experimentation has been recently carried out to simulate the progressive opening of the cavity below a reinforced granular platform (Huckert et al., in press). From the experimental results obtained, it was concluded

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that the opening process of the cavity has a great influence on the intensity and the distribution of the load acting on the geosynthetic sheet. Nevertheless, the analysis of the experimental results does not allow exact determination of these load transfer mechanisms due to the high costs that represent the realization of numerous instrumented tests. Thus, a specific numerical code, coupling finite and discrete element methods, was used to consider two types of cavity openings that can be distinguished depending on the source of the underground initial movement and the nature of soil between the reinforced platform and the underground cavity: a gradual opening of the cavity diameter simulating a progressive subsidence (Process A) or a sinkhole development at a fixed opening diameter characterized by a sudden downward movement of the subsoil (Process B). Specific post-treatment procedures have been developed to achieve a full understanding of the mechanical and kinematic behaviour of the structure at both microscopic and macroscopic scales. Results for the case of non-cohesive granular embankments are then discussed and compared to experimental measurements. Based on this better understanding of mechanisms, the analytical design method developed by Briançon and Villard (2008) has been improved to consider the cavity opening mode and its consequences on the intensity and the distribution shape of the load acting on the geosynthetic. Other partially understood mechanisms such as the expansion of the soil or the load transfer mechanisms within the granular embankment are also investigated. 2. Background 2.1. Existing analytical methods The existing analytical design methods consider various mechanisms (Villard and Briançon, 2008; Huckert et al., in press) such as the load transfer within the granular embankment, the deflection of the geosynthetic sheet, the frictional mechanics and the elongation of the geosynthetic in the anchorage areas, and the expansion of the granular material above the cavity that allows limitation of the vertical surface settlement (Fig. 1). The two most commonly used European analytical methods are the British Standards BS 8006 (1995, 2010), and a method derived from the French research program “RAFAEL” (Giraud, 1997) based on full-scale experiments and numerical analysis. These methods both use the membrane effect theory developed in 2 dimensions for homogenous and isotropic sheets under simple load assumptions (Giroud, 1995). Moreover, they are both based on the assumption that the geosynthetic sheet is fixed at the edges of the cavity. The major difference between the BS 8006 and “RAFAEL”

methods is the use of a different geometry and behaviour for soil collapsing above the cavity. BS 8006 assumes a truncated geometry for the collapse without any soil expansion, whereas the “RAFAEL” method uses a cylindrical collapse over the cavity and considers the soil expansion with a global expansion factor. Amongst the most recently published standards, the German method EBGEO (1997, 2011) presents close similarities to the work of Schwerdt et al. (2004), who laid the principles of a method considering the isotropic or anisotropic structure of the geosynthetic reinforcement. In addition, the German standard also suggests the use of the “RAFAEL” method in most cases, with a slight modification of the computation of the expansion factor (Villard et al., 2000). Finally, the most recent work (Briançon and Villard, 2008) corrected some of the shortcomings in the existing “RAFAEL” method. This complementary approach considers the elongation and the friction behaviour of the geosynthetic sheet in anchorage areas by means of a Coulomb friction law. Another improvement consists of considering the localized mechanisms at the edges of the cavity such as the change in orientation of the sheet and the local increase of the vertical pressure. This phenomenon induces a decreasing tensile force in the geosynthetic at the vicinity of the edges of the cavity. These methods are nevertheless known for shortcomings due to their strong simplifying assumption (Villard et al., 2009); these are described by Huckert et al. (in press) and are summarized here. First, the load applied to the geosynthetic sheet above the cavity is computed using either a funnel shape or a cylindrical geometry of the collapsed embankment over the void. Despite the fact that this assumption remains an important design parameter because it determines the load transfer phenomenon within the embankment, this phenomenon was rarely studied. In fact, the cylindrical behaviour has been observed for full-scale experiments on ballast fills (Blivet et al., 2000) whereas the funnel shape, well adapted for granular non-reinforced embankments, is incompatible with the presence of a reinforcement at the base of the embankment from a kinematic perspective. However, Terzaghi's formulation (Terzaghi, 1943), which is used to compute load transfer along shearing bands for a cylindrical geometry, is associated in many design methods with the active earth pressure coefficient Ka, which does not necessarily integrate real mechanisms such as the rotation of the principal stresses. Actually, various other definitions (Marston and Anderson, 1913; Roscoe, 1970; Vardoulakis et al., 1981; Handy, 1985; Pardo and Saez, 2014) have been proposed in the literature, but none of them has been validated for application to sinkholes. Moreover, the load computed on the geosynthetic either on anchorage areas or

Fig. 1. Main physical mechanisms involved during the sinkhole development.

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above the sinkhole is considered as uniformly distributed, hence possibly ignoring localization effects on the edges of the sinkhole. The mechanical behaviour of the geosynthetic reinforcement is also simplified by the use of a linear elastic law, and most analytical design methods consider that the reinforcement is fixed above the edges of the cavity. Finally, another shortcoming is the global description of the embankment material expansion using an expansion factor defined by the RAFAEL method. This global factor is applied to the whole volume of the embankment localized above the cavity, whereas the expansion may not be a uniform phenomenon within the volume of soil affected by the sinkhole formation. Moreover, the lack of experience in determining the value of this expansion factor for design purposes hinders its usefulness within the analytical design method. 2.2. Numerical simulations The numerical simulation of geosynthetic-reinforced structures is a complex task that requires identification of the best types of methods needed to accurately describe:  the large deformation of the granular soil layer,  the mechanical behaviour of the geosynthetic,  the interaction between the geosynthetic and its surrounding soil. Currently, two kinds of models exist, using either finite or discrete elements methods. Finite models usually represent geosynthetic reinforcements by means of “cable”, “beam” or “plate” elements, usually associated with interface elements, to describe the geosynthetic/soil interface shearing or the deformation of the reinforcement. Sufficient displacements are then necessary to represent the membrane effect, although the model cannot reach high strain levels unless the mesh was regenerated. Another problematic technical point concerns the description of the fibrous structure of the geosynthetic and the direction of the reinforcement. The numerical model presented by Villard and Giraud (1998) answers these questions and considers both the discontinuous nature of the geosynthetic reinforcement, with specific reinforcement directions, and the possibility to attain large deformations and nonlinear tensile behaviour. Finite difference models can be useful to solve such problem in large strain but using this kind of model it remains difficult to define, in a simple way, an appropriate elasto-plastic law for the soil able to take account of the complex interactions between grains as a function of the loading mode history. In most applications, the limitations of these models are the description of the interface between the soil and the geosynthetic using interface elements with limited relative soil/geosynthetic displacements, the linear elastic behaviour of the geosynthetic reinforcement, or the use of a complex elasto-plastic law for the soil, which is sometimes difficult to calibrate to represent phenomena such as soil expansion, cracking, or collapse. Discrete element models enable precise description of the mechanical behaviour of the soil at the grain scale. Rheological phenomena can then be observed such as granular rearrangement, soil discontinuities, or ruptures. Large displacements, particles rotations, soil expansion and compaction, shearing and load transfers are then considered using a limited number of parameters. Different geometries can be assigned to the particles. Spheres or packs of indivisible spheres (clumps) are commonly used owing to the low cost of computation involved for the contact detection algorithm. The geosynthetic reinforcement can also be described using discrete elements joined together (Chen et al., 2012; Chareyre and Villard, 2005; McDowell et al., 2006), with limitations due to

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the interface roughness directly linked to the particle size or the possible spacing between particles under high axial strain and stresses. Moreover, the rigid link between particles does not exactly reproduce the tension and deformation of the geosynthetic. To avoid these issues, different authors (Villard et al., 2009; Elmekati and Shamy, 2010; Dang and Meguid, 2013; Tran et al., 2015) have proposed various coupling techniques to use DEM for the granular material and FEM to describe the geosynthetic behaviour. Villard et al. (2009) developed a specific code including both the discrete and finite elements in the same numerical process, and Tran et al. (2015) provided a coupling between two different codes using either discrete elements or finite elements. In the last case, interface elements allow the transmission of the contact forces between the finite and discrete domains. 3. Numerical model The 3D numerical model used in this study is a coupling method that combines the finite and discrete element methods (Villard et al., 2009) and allows consideration of the discrete nature of the granular material, the fibrous and continuous nature of the geotextile sheet and the complex frictional interaction at the interface between the soil particles and the finite elements used to describe the geotextile behaviour (rolling sliding and friction). The finite elements used to describe the 3D geotextile behaviour are thin, triangular elements (Villard and Giraud, 1998). An anisotropic tensile behaviour of the geosynthetic (no compression or flexion) can be considered according to the definition of specific stiffness values of the fibres in particular directions e for example, in reinforcement directions (Gourc and Villard, 2000). The discrete nature of the granular embankment is described by means of elementary rigid particles of various sizes interacting through contact points (molecular dynamics method, Cundall and  and Strack, 1979). A classical linear elastic contact law (Donze Magnier, 1995) characterized by a friction coefficient m and two stiffness coefficients kn and ks in the normal and tangential directions, respectively, is used for this study. The normal contact stiffness (or tangential contact stiffness) between two spheres of radius Ri and Rj is a function (Eq. (1)) of the normal rigidity Knij (or tangential rigidity Ksij) of the two constitutive materials in contact expressed in N m2.



 Kn ¼ Knij Ri *Rj = Ri þ Rj

(1)

The macroscopic behaviour of the numerical sample is a function of shape and grading of the particles, the numerical bulk density of the granular assembly and the intrinsic microscopic contact parameters (Salot et al., 2009; Szarf et al., 2011). Owing to the fact that there is no direct relation between the microscopic parameters (normal and tangential stiffness, contact friction coefficient, grading, particle shape and density of the numerical sample) and macroscopic parameters (Young's modulus, Poisson's ratio and internal friction angle), a specific calibration process is required. A classical way to obtain the micromechanical parameters is to simulate laboratory tests performed in well-controlled conditions and compare the numerical results to the experimental ones. For this study, a numerical process based on comparisons between numerical and experimental triaxial test results (Salot et al., 2009) is used to calibrate the numerical parameters. The main advantage of the DEM method compared to the others lies to the possibility to simulate large displacements, collapse, compaction or expansion and to easily consider the change of the granular behaviour consecutively with the variation of its density (due to collapse, for example) or of the loading process (loading, unloading, cycles).

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The numerical procedure needed to solve the problem is an iterative process that successively alternates the resolution of Newton's second law of motion (applied to the discrete elements and the nodes of the finite elements) and the actualization of the interaction forces at each contact point. The contact forces between two neighbouring elements are calculated considering the overlapping and the relative displacement between the two jointed elements. Soil/geotextile interaction is described thanks to a specific contact law similar to that between soil particles by means of normal and tangential stiffness and the soil/geosynthetic friction angle. The normal contact stiffness (or tangential contact stiffness) between a soil particle of radius Ri and a finite triangular element is a function (Eq. (2)) of the normal rigidity Kni (or tangential rigidity Ksi) of the two materials in contact expressed in N m3. The coupling between the finite and discrete models enables compatibility between the two methods, which allows a good description of the interaction between the geosynthetic and its surrounding soil. Applications of the model to reinforced earth structures were proposed by Le Hello et al. (2006), Le Hello and Villard (2009), Chevalier et al. (2011) and Villard et al. (2009).

Kn ¼ Kni ð4*Ri *Ri Þ

(2)

The numerical model (Fig. 2) used for this study include, from top to bottom:  an assembly of clumps of two spheres describing the behaviour of the granular embankment,  thin, finite, triangular elements to consider the geotextile behaviour,  a layer of spheres regularly distributed in a square mesh at the base of the model to mimic the action of an elastic supporting soil and well simulate the sinkhole opening process. Frictionless lateral walls are used to consider the symmetry so that the cavity is located on a corner of the numerical model. Two numerical opening processes are defined:  Process A e a progressive opening of the cavity by an increase of the cavity diameter D,  Process B e a gradual downward movement of all spheres under the geosynthetic included in a circular area of diameter D. The velocity of the opening processes is low enough to avoid dynamic effects. From an engineering point of view, the size of the model and the number of particles need to be adapted to avoid excessive calculation durations and to preserve an accuracy precision. For this purpose, three different numerical geometries of the numerical model (Table 1) are used for this study to adapt the size

of the discrete and finite elements to the diameter of the cavity. Note that the size of the discrete elements for model B is rather similar to the size of the grains of the experimental granular material and that the lower part of model C (half lower part) is adapted to small-diameter cavities. Before applying the cavity opening process, the static stability of the numerical embankment under gravity was reached. To roughly mimic the shape of the granular soil elements and well approach the mechanical macroscopic behaviour of the experimental soil (particularly the internal friction angle at the peak), the granular elements are modelled by means of clusters of two juxtaposed and unbreakable particles of same diameter d. The grading of the numerical material is defined, as the experimental one, by a size ratio between the large and small clumps of 2. From a macroscopic point of view, more complex shapes or a perfect similitude with the experimental material is not needed with respect to the present calibration process. The density of the discrete numerical assembly was controlled using the ERDF method (Enlarge Radius and Decrease Friction e Chareyre and Villard, 2005). In the present case, a loose state of the numerical assembly was retained to fit the real macroscopic mechanical behaviour. The values of the microscopic contact parameters between the spheres of the granular embankment are: a normal rigidity of 100 MN m2, a tangential rigidity of 100 MN m2 and a microscopic contact friction angle of 28 . The results of three numerical triaxial tests are presented Fig. 3. The average macroscopic mechanical parameters of the numerical sample are a Young's modulus of 19 MPa, a Poisson's coefficient of 0.3, a friction angle at peak of 36.5 and a threshold friction angle of 31. The numerical parameters used to restore the mechanical behaviour of the geosynthetic (two reinforced yarn directions defined in the x and y axes are considered) and the soil/geosynthetic interface (Table 2) are deduced from experimental measurements (Huckert et al., in press). The interface parameters between the geosynthetics and the lower ground gravel or the upper embankment material were obtained by shearing tests using a shearing box 0.3  0.3 m2 in size while the tensile parameters of the geosynthetic in x and y direction are deduced from traction tests performed by the geosynthetic manufacturer. The results expected by the coupling model are: the displacements of the discrete particles, the geosynthetic strain and tensile forces in both reinforced directions (x and y), the contact forces between soil particles and at the soil/geosynthetic interfaces. The stress tensor within a volume V of the granular assembly can be computed by Eq. (3) considering the contact forces acting at all contact points included in the volume V (Weber, 1966). Nc is the number of contact points in V, fi is the projection of the contact force f on the i-axis, and l j is the projection of the branch vector l on

Fig. 2. Geometry of the numerical samples.

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Table 1 Characteristic values of the numerical samples. Model

A

B

C

Width of the numerical model (m) Thickness of the granular embankment (m) Number of clumps of the granular embankment

4.0 1.0 32,000

1.0 1.0 32000

Size of clumps of the granular embankment (mm)

48e96

19e38

Numerical porosity of the granular assembly of clumps Numerical density of clumps (kg/m3) Bulk density of the numerical assembly of clumps (kg/m3) Number of finite triangular elements Size of the finite triangular elements (m) Thickness of the finite triangular elements (m) Number of spheres of the supporting soil Diameter of the spheres of the supporting soil (m) Radius of the cavity (m)

0.42 2734.7 1586 12,800 0.05 0.02 25,600 0.025 0.5e2.5

0.42 2734.7 1586 3200 0.025 0.02 1600 0.025 0.2e0.5

0.5 1.0 Lower part 50,000 Lower part 8.5e17 0.42 2734.7 1586 5000 0.01 0.01 2500 0.01 0.1e0.2

Upper part 4000 Upper part 19e38

Fig. 3. Results of three numerical simulations of triaxial test (confining pressure of 10 kPa).

Table 2 Numerical parameters describing the tensile behaviour of the geotextile and the micromechanical parameters of the soil/geosynthetic interfaces. Tensile stiffness of the geotextile in the x direction (kN m1) Tensile stiffness of the geotextile in the y direction (kN m1) Normal rigidity at the contact between discrete particles and geotextile elements (MN m3) Tangential rigidity at the contact between discrete particles and geotextile elements (MN m3) Contact friction angle between the clump of the granular embankment and the upper interface of the geotextile ( ) Contact friction angle between the clump of the granular embankment and the lower interface of the geotextile ( )

the j-axis with i ¼ x, y, z and j ¼ x, y, z. The branch vector l is defined by the vector linking the centres of the clumps in contact.

sij ¼

1 XNc i i f l a¼1 a a V

(3)

4. Comparison between experimental and numerical results The experimental results used for comparison with the numerical model are those presented in Huckert et al. (in press) concerning the formation of a sinkhole under a granular embankment of 1 m height by progressively increasing the cavity diameter. For these experiments, a specific instrumentation has been set up to determine the surface settlement, geosynthetic deflection, strain of the geosynthetic sheet and vertical stresses around the cavity. The most relevant and representative results are those obtained at the end of the opening process (D/H ¼ 2.2) in the case of granular platforms (g ¼ 15.65 kN/m3) reinforced by geosynthetic sheets, for which the tensile stiffness is equal to Tx ¼ 3000 kN/m in the

3000 250 200 15 23 40

longitudinal direction and Ty ¼ 250 kN/m in the transversal direction. The granular material is made of rounded grains with sizes between 20 and 40 mm. Note that the shape of the grains and the grading of the granular material are rather comparable to those used in model B of the numerical study. Moreover, the interface friction parameters, tensile characteristics of the geosynthetics and macroscopic parameters of the soil (Huckert et al., in press) are similar to those of the numerical model. The experimental results of the geosynthetic deflection and the surface settlements are compared in Fig. 4 to the numerical results obtained with the two cavity opening processes. The numerical results corresponding to Process A match well with the experimental results established with a similar opening process, whereas the results of Process B significantly underestimate the geosynthetic deflection and the surface settlement. This shows that the opening process is of primary importance to the design of such reinforced structures and should be considered in a proper estimation of the surface settlement. Note that the good agreement between the experimental and numerical results of the surface settlements in the case of the progressive cavity opening

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Fig. 4. Surface settlement and geosynthetic deflection.

process shows that when roughly similar shape and grading for the numerical and experimental soils are used, it is possible to well describe the expansion mechanism involved within the granular embankment. Fig. 5 compares the geosynthetic strain measured in the axis of the cavity by optical fibres and the numerical strain obtained using the progressive cavity opening process: experimental and numerical geosynthetic strains are close, providing new evidence of the relevance of the numerical tool.

Concerning the expansion mechanism, the numerical average values of the expansion coefficient are computed by considering the volume resulting from the surface settlement Vs, the volume due to the deflection of the geosynthetic sheet Vg and the initial volume V0 of the column of soil sited above the cavity (Ce ¼ (V0 þ Vg  Vs)/V0). It can be noted that the numerical average value of the expansion coefficient of 1.048, computed for the progressive cavity opening process, has the same magnitude order than the average experimental value of 1.037 obtained by

Fig. 5. Comparison between experimental and numerical geosynthetic strain.

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considering the experimental results of true scale experiments (Huckert et al., in press). To compare the numerical results to the experimental data of the Earth Pressure Cells (Huckert et al., in press), the increase of stresses (Ds/s0) on an annular area 0.4 m wide (radius ranging between 1.1 m and 1.5 m) is computed by means of the vertical contact forces at the upper soil/geotextile interface. The obtained value of the numerical increase in vertical stresses of 34% can roughly be compared to the experimental value of 49% (Huckert et al., in press), measured by Earth Pressure Cells (0.4 m  0.4 m in size). The numerical model provides a satisfying approximation of the experimental values, which are strongly influenced by the size of the sensor, the size of the granular elements (between 20 mm and 40 mm), the position of the Earth Pressure Cell (under the geosynthetic sheet), the proximity of the edge of the cavity and a possible tilting of the sensors. Nevertheless, from the results presented in this section, it is assumed that the numerical model and the opening processes are validated and can be used for a more detailed analysis. 5. Numerical analysis of load transfer mechanisms above the cavity The load transfer mechanism results in a change in the contact force network within the granular embankment due to the formation of the cavity. To appreciate the influence of the opening process on the redistribution of forces and the reinforcement process, several numerical simulations have been performed based on the two numerical processes described previously: a progressive increase of the cavity diameter or a gradual downward movement of the supporting soil for a fixed value of the cavity diameter D. For this purpose, the DEM is well adapted by considering the progressive change in the contact forces in accordance with the opening process. Another advantage of DEM is the possibility to restore, using a single limited set of parameters, different macroscopic behaviours of soil depending on the change of density of the granular material during the opening processes. The load transfer mechanism within the granular embankment can be easily characterized by the efficiency E, which can be defined by the ratio between the load reported on the sides of the cavity and the weight Ws of the cylindrical part of the soil sited over the cavity. From the load acting on the geosynthetic sited above the cavity Fg, the efficiency of the load transfer within the granular embankment can be defined by:


 E ¼ Ws  Fg Ws

(4)

Results obtained from the different numerical models (models A, B and C) are the curves of the efficiency versus the maximum vertical displacement of the geosynthetic sheet (fg) during the gradual settlement process (Fig. 6a) for various values of the ratio D/H or during the progressive opening of the cavity diameter (Fig. 6b) for a ratio D/H varying between 0 and 4. As seen in Fig. 6a, maximum efficiency is obtained for very small displacements of the soil particles, highlighting that the load transfer mechanism can be quickly mobilized. We can also observe a decrease in efficiency with increasing deformation of the embankment. Note that the contact between the geosynthetic and the trapdoor is progressively opened during the gradual settlement process such that the maximum vertical displacement of the geosynthetic is obtained at the end of the numerical simulation. These results are consistent with those obtained by Chevalier et al. (2012) and Iglesia et al. (2014) concerning the trapdoor problem. Results of the three numerical simulations concerning the cavity opening process are plotted in Fig. 6b. These results do not depend on the size of the

387

numerical model or the number of particles as long as the size of the particles is small with respect to the cavity diameter. We can also observe that the maximum vertical displacement of the geosynthetic sheet is, at the end of the progressive opening of the cavity diameter (Process A, Fig. 6a), greater than the vertical displacement obtained at the end of the gradual settlement process (Process B, D/H ¼ 4, Fig. 6b). The results of 15 computations performed for the two opening processes are presented Fig. 7 for various values of the cavity diameter or different size of the model. For each cavity diameter (Process B), two points were plotted: the first one corresponds to the maximum value of the efficiency obtained for small settlements, and the other corresponds to the efficiency obtained at the end of the settlement process (no contact between the support and the geotextile sited over the cavity). As can be seen, the difference between these two points increases with the ratio D/H and with the intensity of the soil deformation. Comparing the two opening processes, it can be concluded that the efficiencies are equivalent for small ratios D/H and rather similar for larger ratios (the values obtained for Process A are included between the two values obtained for Process B). A comparison between the two opening processes can be made for a given value of the cavity diameter (D/H ¼ 2.2) via the vertical displacements of the geosynthetic fg, the surface settlements s and the expansion coefficients Ce. As seen in Table 3, a comparison between the numerical average expansion coefficients for the two opening processes indicates a significant difference in the expansion mechanisms within the granular embankments as a function of the kinematic of the opening process. The maximum vertical displacements of the geotextile (in x and y directions) and the maximum surface settlements are presented Fig. 8 for the two opening processes. As it can be seen on this figure, the vertical displacements in x and y directions are rather similar so that it can be assumed that the load distribution on the geosynthetic, above circular cavities, is axisymmetric. On the other hand, the loading is sustained by the reinforced yarns of the geosynthetic in two orthogonal directions as a function of their tensile stiffness so that it can be concluded that 92% of the load acting on the geosynthetic sheet is sustained in the x direction (J ¼ 3000 kN/m) and 8% in the y direction (J ¼ 250 kN/m). On Fig. 8, we can also note that the maximum vertical displacements of the geotextile are very different from one opening process to another.8 Considering that the total vertical loads acting on the geotextile are quite similar from the two opening processes (equivalent efficiency), the differences obtained for the vertical displacement of the geotextile and the surface settlement can appear to be strange unless the load distributions acting on the geosynthetic are different from one case to another. To answer to this, the increase of stresses (Ds/s0 where s0 represents the initial vertical stresses) acting on the upper face of the geosynthetic are computed in both opening processes by means of the vertical contact forces at the upper soil/geotextile interface. Stresses are computed on annular areas centred on the cavity. Results are presented Fig. 9 for 3 cavity diameters (D/H ¼ 1, D/H ¼ 2.2 and D/H ¼ 4). Note that similar results are obtained for each cavity diameter considered. The load distribution (Fig. 9) obtained in the case of the increasing diameter process (Process A) takes an approximate conical shape whereas the load distribution is rather constant or slightly inverted in the case of the gradual settlement process (Process B). The load transfer mechanisms occurring within the granular embankment at the beginning of the gradual settlement process are not disturbed during the opening process, and the load transfer mechanisms are continually modified during the increase of the cavity diameter. In both cases, at the end of the opening

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Fig. 6. Efficiency versus maximum vertical displacement of the geosynthetic for both processes and different values of the ratios D/H.

geosynthetic and thus on the vertical displacements of the geosynthetic and the surface settlement. Moreover, the soil expansion mechanism is not uniform within the granular material and is also greatly influenced by the opening process. Before introducing these new results in the analytical design models, a comparison between the numerical results and full-scale experimental data was proposed. 6. Design rule improvement The increase in knowledge concerning the mechanisms involved during sinkhole formation may lead to improved design methods developed for granular reinforced embankments, to a better definition of the bias introduced by simplifying assumptions and to a definition of lower values of the safety coefficients; these are not considered in the present study owing to the large number of design strategies or standards used in each country. From the results presented above, three major points can be discussed or integrated in design methods: Fig. 7. Efficiency versus D/H for the two cavity opening processes.

Table 3 Comparison between both processes.

Maximum vertical displacements of the geotextile (m) Maximum surface settlements (m) Ce

Process A

Process B

0.207 0.126 1.048

0.13 0.086 1.036

process, the shearing forces at the periphery of the soil cylinder sited over the cavity are rather similar such that the efficiencies and the total loads acting on the part of the geosynthetic located above the cavity are equivalent. The difference in behaviour for the two opening processes can also be seen (Fig. 10) by the comparisons of the principal stress tensors within the granular embankments for various ratios D/H. The load transfer mechanism can be seen as a change in orientation of the main stresses. The major differences between the two opening processes can mainly be observed above the cavity, especially for high values of the ratio D/H, whereas the load transfers to the edges of the cavity are rather similar. This is consistent with the previous results showing a change in the load distribution on the sheet over the cavity and a rather constant efficiency value. As demonstrated, the cavity opening process has a great influence on the shape of the load distribution acting on the

 the load transfer mechanism acting inside the granular embankment,  the shape of the load distribution acting on the geosynthetic above the cavity,  the determination of the expansion coefficient.

6.1. Load transfer mechanism inside the granular embankment The assumption proposed by Terzaghi of a moving cylinder of soil between fixed areas seems, from a kinematic point of view, relevant for the two opening processes as long as the ratio D/H is sufficiently large. Terzaghi's formulation (Eq. (5)) of the load acting on the geosynthetic sheet (assumed to be uniform) is obtained by studying the static equilibrium of any layers of soil above the cavity.

q¼

  D*g * 1  eK* tan f*4*H=D 4*K* tan f

(5)

where K is the stress ratio between the horizontal and vertical stresses sh and sv, respectively. The main difficulty in the use of this formulation lies in the definition of K. Numerous design methods propose consideration of the active earth pressure coefficient Ka ¼ (1  sinf)/(1 þ sinf) as proposed by Marston and Anderson (1913). Owing to the fact that Ka tan f is rather constant for the usual values of f, the load acting on the geosynthetic is quasi-

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Fig. 8. Vertical geosynthetic displacements and surface settlements (m) for both process and for H/D ¼ 2.2 (Model A).

Fig. 9. Increase in the vertical stresses acting on the upper face of the geosynthetic for Model A and B and for three ratios D/H.

389

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Fig. 10. Principal stresses within the granular embankment for Process B or Process A.

independent of the frictional soil characteristics, which is unsatisfactory. Numerous authors have proposed various definitions for the value of K to consider different mechanisms such as shearing bands (Vardoulakis et al., 1981; Roscoe, 1970), rotation of the principal stresses and the bending of the soil layer above the geosynthetic (Handy, 1985), but none of them was validated with experimental and numerical results. The representation of the horizontal and vertical stresses on different parts of the numerical model shows that from a far distance, the cavity values of K are approximately 0.3, whereas on the upper part of the soil cylinder sited over the cavity, some values greater than 3 were obtained (Fig. 11). Above the edges of the moving soil cylinder, owing to the load transfer mechanism and the change in orientation of the principal stresses, a clear definition of K is problematic. The comparison (Fig. 12) of the efficiencies given by the numerical model and resulting from Terzaghi's formulation (using friction angles of 36.5 and 31 at the peak and threshold, respectively) shows that a value of K equal to 1.3 is required to obtain the best fit between the two curves. Note that the analytical curve corresponding to the threshold friction angle (31 ) is more adapted for a large value of D/H for which numerically large displacements of the embankment are reached (Fig. 12, right part), whereas the analytical curve corresponding to the peak friction angle (36.5 ) is more adapted for low values of D/H (Fig. 12, left part). A similar analysis of the literature performed by Chevalier et al. (2012), dealing with the numerical and experimental studies of the trapdoor problem, shows that values of K of 1.17 and 1.46 for sands and coarse granular materials, respectively, are needed to fit Terzaghi's formulation. All of these results show that the usual value of Ka is not well adapted to evaluate the load transfer mechanisms.

6.2. Shape of the load distribution acting on the geosynthetic above the cavity The shape of the load distribution acting on the geosynthetic above the cavity has been clearly noted by the numerical model. This shape is not uniform and takes an approximate conical shape for Process A, whereas the load distribution is rather constant in the centre of the cavity and slightly inverted at the edges in the case of Process B (i.e., concentration of loads at the edge of the cavity). An improvement of the usual analytical design methods consists of the implementation of new load distribution shapes above the cavity. This involves an iterative process to resolve the equations of equilibrium of both horizontal and vertical efforts on any part of the geosynthetic sheet (above the cavity and in the anchorage areas). In the present case, the original analytical formulation proposed by Villard and Briançon (2008) is adapted for approximate conical and inverted parabolic distributions (Fig. 13). The main equation to be solved (Eq. (6)) is obtained by equalizing both geometric and constitutive elongation of the reinforcement overlying the cavity (Villard and Briançon, 2008). J is the tensile stiffness of the geosynthetic; z(x) is the vertical displacement of the geosynthetic; TH is the horizontal tensile force in the geotextile, which is constant over the cavity and can be obtained by resolving (Eq. (6)); and UA the geosynthetic displacement in anchorage areas, defined as a function of the variable b (Villard and Briançon, 2008). x¼D=2 Z

DL ¼ x¼0

With:

D vs  ¼ UA þ 2

x¼D=2 Z

εðxÞ:vs x¼0

(6)

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391

Fig. 11. Horizontal and vertical stresses inside the granular embankment.

Fig. 12. Comparison between numerical and analytical efficiencies.

4

vs ¼ x¼0

changes in the definition of q(x) and z(x) linked together by the following relation (Villard and Briançon, 2008):

2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 x¼D=2  2 Z

x¼D=2 Z

1þ

dz dx

5dx

(7)

x¼0

x¼D=2 Z

εðxÞ:vs ¼

TH J

x¼0

x¼D=2 Z

! dz 2 1þ dx dx

qðxÞ d2 z ¼ TH dx2



(8)

x¼0

dz ðx ¼ D=2Þ b¼ dx

Assuming that the total loads applied to the geosynthetic are the same in each case results in:  For a uniform load:

(9)

Thus, the tensile force T(x) in the geotextile is given by:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 dz TðxÞ ¼ TH 1 þ dx

(11)

(10)

The consideration of new load distributions requires only

8 qðxÞ ¼ q > <
 q 4x2  D2 > : zðxÞ ¼ 8TH  For an approximately conical load (Fig. 13):

(12)

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Fig. 13. Comparison between analytical and numerical geosynthetic deflection.

8 qðxÞ ¼ q2 þ q1  2q1 x=D > > > > > > > < q1 ¼ 3gq=3 þ g q2 ¼ 3q=3 þ g > > > > > > ðq þ q1 Þx2 q x3 ð3q2 þ 2q1 ÞD2 > : zðxÞ ¼ 2  1  2TH 3DTH 24TH

(13)

 For an inverted parabolic load (Fig. 13):

8 > > > > > > > <

. qðxÞ ¼ q2 þ 4q1 x2 D2 q1 ¼ 2gq=2 þ g

q2 ¼ 2q=2 þ g > > > > > 2 2 4 > > : zðxÞ ¼ q2 x þ q1 x  ð6q2 þ q1 ÞD 2 2TH 48TH 3D TH

anchorage areas is 15.65 kN/m2. The geosynthetic sheet is 8 m long and fixed at each extremity (as in the numerical model). The analytical geosynthetic deflections are compared to the numerical results obtained using the two numerical opening processes. We observe (Fig. 13) that the analytical assumption of an approximate conical load distribution (q1/q2 ¼ 4 for D/H ¼ 2.2), clearly identified with the numerical model, matches very well with the progressive diameter cavity opening process (Process A). Likewise, an inverted parabolic load distribution (q1/q2 ¼ 4 and D/H ¼ 2.2), leading to a greater load near the edges of the cavity, leads to a very good fit of the numerical results (Process B). Thus, the analytical assumption of a uniform load distribution, usually used in design methods, seems inadequate regardless of the cavity opening mode. 6.3. Determination of the expansion coefficient

(14)

q1 and q2 are defined as a function of q in order that the total loads applied on the geosynthetic are the same in each case. q1 is the maximum value of the non-uniform part of the load distribution on the geosynthetic, q2 the value of the uniform part of the load distribution on the geosynthetic and g the ratio q1/q2 allowing to precisely define the shape of the load distribution. Eq. (6) can be resolved easily by an iterative procedure and a numerical integration process given the unknown value of TH and thus the displacements and the tensile forces in the geotextile. The deflection of the geosynthetic is dependent on the longitudinal axis of the cavity for uniform, conical and inverted parabolic load distributions. The total load transfers q are calculated using K ¼ 1.3 as proposed previously. On the basis of the numerical results, ratios g ¼ q1/q2 ¼ 4 were retained for D/H ¼ 2.2. The values of the analytical parameters (cavity diameter, tensile stiffness of the geosynthetic and frictional interface parameters) are derived from the experimental and numerical values. The maximum relative displacement U0 that is needed to reach the maximum frictional forces is assumed to be 5 mm, and the vertical load acting on the

The dependence of the expansion mechanism of soil with the opening process by means of the change in local porosities within the granular embankment has been clearly identified by the numerical model. The numerical increases or decreases of porosities have been computed by considering the change in volume of the solid fraction of the particles included in elementary cubes regularly distributed within the granular embankment. Fig. 14 shows that the shearing areas of soil affected by the sinkhole are not uniform within the granular embankment and are greater in the case of the progressive cavity diameter opening process. These local variations of the increase in porosity can be correlated to the values of the average expansion coefficients of 1.036 and 1.048 obtained previously for the two numerical processes. It can be concluded that when current design methods are used, a greater average value of the expansion coefficient is needed when considering the case of an increase in the cavity diameter rather than a gradual settlement process. 6.4. Procedure to follow to design the geosynthetic From the results presented in this paper, we propose a procedure for designing geosynthetic reinforcements in the case of noncohesive granular layers. The first step of this procedure consists of

P. Villard et al. / Geotextiles and Geomembranes 44 (2016) 381e395

estimation with respect to the cavity opening process, the geometry (circular, trench) and the size of the cavity on the basis of geologic data or experimental field measurements. As we have highlighted in this study, the cavity opening process has a considerable influence on the shape of the load distribution above the cavity, the deformed shape of the geosynthetic sheet and the surface settlement. Moreover, without any information about the cavity opening process, we propose the use of the progressive cavity diameter opening process (Process A), which leads to greater surface settlements to obtain a safe design. The second step of the design procedure is to estimate or measure the mechanical parameters needed for the design. Table 4 presents all required parameters, their significance on the design and one method to evaluate them. Of course, when possible, it seems essential to evaluate the main design parameters with laboratory tests. Among the parameters for which no experimental method exists, the ratio K between the horizontal and vertical stresses in the vicinity of the collapsed soil cylinder has a great influence on the geosynthetic design. On the base of the results of this work and previous studies, we propose as the value of K 1.2 for sand and 1.5 for coarse granular materials. The expansion coefficient has also a great influence on the geosynthetic design when it is necessary to respect a settlement surface criterion. Without any information on the expansion coefficient, it is required to design the geosynthetic with a safety approach by using an expansion coefficient equal to 1.03. The third step of the design procedure leads to the design of the tensile stiffness J of the geosynthetic on the basis of an appropriate value of the granular layer height (if it is not fixed by the constraints of the civil engineering structure). The analytical method proposed by Villard and Briançon (2008) can be used considering the improvements proposed in this paper related to:  the shape of load distribution acting on the geosynthetic above the cavity (approximately conical load for Process A and inverted parabolic load for Process B),  the definition of an average value of the expansion coefficient as a function of the opening process,  the definition of the stress ratio between the horizontal and vertical stresses. The analytical results must be compared to the design criteria related to the surface settlement and the permissible tensile force in the geosynthetic sheet. This calculation must be performed until finding the tensile stiffness J for a fixed granular layer H or the pair

393

of values H and J that meet the surface criterion and the permissible tensile force in the geosynthetic. Naturally, this procedure could be adapted to Ultimate Limit State (ULS) design and to Serviceability Limit State (SLS) design considering the reduction factors for installation damage, material creep and durability. 7. Conclusion A numerical tool based on the coupling between discrete and finite element methods is used to study the behaviour of granular embankments reinforced by geosynthetics in areas with high sinkhole risk. It has been shown by comparison with experimental results that the numerical model provides relevant information on the surface settlement, the geosynthetic deflection, the soil expansion mechanism and the shape of the load distribution acting on the geosynthetic sheet. It can also be noted that the discrete nature of the numerical model for soil allows a good description of the evolution of the load transfer and contact forces within the granular material considering the kinematic of the cavity opening process. The numerical model has been used both for a better understanding of the load transfer mechanisms and for validating or improving the existing design methods. Two cavity diameter opening processes were tested: a progressive cavity diameter opening process and a gradual settlement process. It has been established that:  The area of soil affected by the sinkhole is a cylinder of soil sited above the cavity. The load transfer mechanism from the passive zone to the active area (i.e., the shearing forces at the periphery of the soil cylinder) is independent of the cavity opening process.  For a ratio D/H ranging between 0.5 and 4, the intensity of the load transfer mechanism can be estimated by the Terzaghi's formulation by using an adequate value of K. For rounded granular materials (Dmin ¼ 20 mm and Dmax ¼ 40 mm) a value of 1.3 allows good concordance between the experimental and numerical results.  The shape of the load distribution is an approximate cone when considering a progressive cavity diameter opening process (D/ H < 4). This leads to greater values of the geosynthetic deflection and the surface settlements by comparison to uniform or inverted parabolic load distributions.  Depending of the ratio D/H, a gradual settlement process involves a more or less inverted parabolic load distribution that

Fig. 14. Change in porosity within the granular embankment for both processes.

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Table 4 Significance of the parameters needed for the design. Parameter

Notation Significance to the geosynthetic design

Method to evaluate the parameters

Friction angle of the granular material Friction angle between the soil and the geosynthetic Expansion coefficient

f ( ) d ( )

High Medium

Laboratory tests (shearing box or triaxial tests) Laboratory tests (shearing box or inclined plane) or fixed value at 0.8  f

Ce

High

Unit weight of the soil

gd

No standard method. Depending on the material, values ranging between 1.03 and 1.15 can be found in the literature. Laboratory tests

Displacement to fully mobilize the friction between GSY and soil Stress ratio between the horizontal and vertical stresses

High (kN/m3) Low u0 (m) K

High

leads to concentrated loads on the geosynthetic near the edge of the cavity.  The expansion mechanisms of soil are not uniform within the granular embankment and depend strongly on the sinkhole formation process. An average value of the expansion coefficient can be used in the analytical methods. For the case where D/ H ¼ 2.2, similar to the experimental test (Huckert et al., in press), values of 1.048 and 1.036 are obtained for rounded gravel for the progressive cavity diameter opening process (Process A) and the gradual settlement process (Process B), respectively, whereas a value of 1.037 has been deduced from experimental results. From these results, it seems important in a design approach to carefully investigate the geological context that leads to the formation of the cavity to best estimate the cavity opening process and the maximum diameter of the cavity. It is possible that the geosynthetic can have an influence on the cavity opening process, depending on the geosynthetic stiffness and the mechanical characteristics of the subsoil. In such complex cases, the numerical model, which has been validated successfully through comparison with experimental data, remains the best way to catch the influence of very complex phenomena. From a practical point of view, the results presented here refer to rounded granular materials and cavity diameters ranging between D/H ¼ 0.5 and D/H ¼ 4. The extension of the results to other geometries or to other soil natures must be conducted carefully. Acknowledgements The authors would like to thank all organisations that made this work possible, including competitiveness clusters Techtera and Fibres, and the geosynthetic producer Texinov. List of symbols a adhesion between the soil and the geosynthetic (kPa) c0 cohesion intercept (kPa) Ce expansion coefficient f maximum geosynthetic deflection (m) D cavity diameter (m) dmin and dmax minimum and maximum sizes of the grains (m) H embankment height (m) J tensile stiffness per unit width of the geosynthetic fabric (kN/m) K stress ratio between the horizontal and vertical stresses q1 maximum value of the non-uniform part of the load distribution on the geosynthetic (kPa) q2 value of the uniform part of the load distribution on the geosynthetic (kPa).

No current method. We propose u0 ¼ 0.005 m No current method. We propose 1.2 for sand and 1.5 for coarse granular materials

s T u0 w

d ε

f0 g gd

surface settlement (m) tensile force per unit width of the geosynthetic fabric (kN/m) displacement to fully mobilize the friction between GSY and soil (m) moisture content (%) friction angle between the soil and the geosynthetic ( ) geosynthetic strain (%) friction angle of the granular material ( ) the ratio q1/q2 unit weight of the soil (kN/m3)

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