In the absence of the RG, the opening of the cavity created a complete void at the surface. The presence of the RG limits the soil collapse and induces subsidence with a maximum occurring in the centre of the cavity equal to 27.8 mm (Fig. 2).

The subsidence surface is slightly larger than the area of the cavity. The deformation of the RG is flattened, mainly in the central part above the cavity (Fig. 3). This is explained by the fact that the load acting on the RG is not uniform. The displacements of the soil surface are smaller than those of the RG thanks to the expansion of the granular soil volume. Figure 4 presents the measurements of the vertical stresses before and after the formation of the cavity. The initial stress measured is of the order of 40 % compared to γH. The difficulty of measuring the stresses in that the sand is very well identified (Hassoun et al., 2018). The final stresses increase by 80 % after the formation of the cavity close to its boundary.

The vertical displacements of the RG were calculated analytically (Briançon and Villard, 2008). The two friction angles of the interface (geosynthetic/soil and geosynthetic/wood plate friction) in the anchor zones are estimated at 30 and 22∘ respectively. The analytical solution is developed for 2D conditions and the RG is assumed to be unidirectional (Briançon and Villard, 2008). The load transfer hypothesis proposed by Terzaghi for circular cavities (Eq. 1, 1946) leads to a theoretical value of the uniformly distributed load σvg acting on the RG of 1.82 kN m-12. The soil characteristics are: K=0.27, γ=15.3 kN m-1 and ϕ=35∘. K is assumed to be the active earth pressure coefficient (K) but its value is not well established. Following 2D laboratory experiments, Chevalier et al. (2012) has proposed values of K equal to 0.55 and 1.2 for coarse sand and gravel respectively. P is the overload applied on the surface. 1σZ=σvg=Bγ4Ktan⁡φ1-e4Ktan⁡φHB+pe4Ktan⁡φHB The 2D design method (Briançon and Villard, 2008) is applied to the geosynthetic strip, sited in the axis of the cavity. Load distribution is assumed to be uniform or to have an inverse triangular shape in 3D as presented in Fig. 5b. Figure 6 presents the analytical and experimental results. As it can be seen, the proposed load distribution, in form of an inverse triangle, allows to obtain a better concordance between the experimental and analytical deformations of the geosynthetic.

Effect of overload: after the opening of the cavity, the soil surface is overloaded in the centre of the cavity on a 15 cm wide square surface. The overload was applied progressively in three phases: 15, 35 and 55 kg. Figure 7 shows the vertical displacements of RG. The application of a localized overload in the axis of the cavity increases the RG deflection mainly in the central part of the cavity. The unloading phase resulted in an elevation of the RG. The RG deflection obtained at the end of the unloading remains greater than the value before loading, due to the sliding of the geosynthetic in the anchorage areas and the presence of the soil layer that collapsed on the GR and whose behaviour is not reversible (Fig. 7).

To compare the analytical deformation of the RG to the experimental ones, three geometries of stress distribution on the RG have been studied (Fig. 9). The analytical displacement is calculated on the geosynthetic strip, sited in the axis of the cavity. The different load distributions proposed respect the total load calculated using Terzaghi equation 1 (equivalent loads of p respectively 0.6, 1.4 and 2.2 kN m-1 were used to take into account the overloading of 15, 35 and 55 kg). Based on this assumption the maximal value of the overloading for the parabolic distribution is Q=8/πσvg and for the inverted triangular distribution is Q=3/2σvg instead of for the inverted triangular distribution (Q=3/2σvg).

Table 1 compares the analytical and experimental results. The stress distribution acting on the RG, for localized loading, is no longer inverse triangular. Therefore, loading the soil surface modifies the geometry of the load distribution on the RG, from initially inverse triangular to a distribution whose maximum value is in the centre of the cavity.

Two tests with the cohesive soil (SK1 and SK2) were carried out. Under the self-weight of the soil, the cohesive soil layers bridge the cavity area and behave as a flexural slab which limits the vertical displacement to values smaller than 2 mm. The maximal vertical displacement of the RG in the axis of the cavity is 7 mm due only to its self-weight. After the opening of the cavity, the stress measured by the sensors closest to the cavity C7, C8 and C13, C10 increase (Fig. 10). The furthest sensors from the cavity (C2, C3 and C4) recorded a small stress increase. The comparison of stress measurements between granular and cohesive soil is presented in Fig. 11. The load of C2, C3 and C4 cells, the furthest away from the cavity, are greater in the case of the granular soil than those obtained with the cohesive soil, for which very small increases in stress are recorded.

As a result, in the case of granular soils, the transferred loads are less intense and more diffuse, whereas they are, due to the beam-like behaviour of the cohesive soil layer, more intense and localized at the edge of the soil.

Effect of overload: a progressive overload was applied on the soil surface in the axis of the cavity. Sudden collapse occurred at a load of 55 kg (breakage stress of 24.5 kPa). The collapse of the soil layer leads to a localized subsidence restricted to the loading surface. The volume of collapsed soil is approximately a truncated pyramid of height H with a base equal to the size of the cavity (50cm×50cm, Fig. 12). The collapse of the soil layer leads to a vertical displacement of the RG of 54.6 and 58.6 mm for the two tests SK1 and SK2. The average horizontal slippages of the geosynthetic at the right and left edges of the cavity for SK1 and SK2 are 2 and 0.75 mm, respectively.

The comparison of the RG deformations between the experiments with Fontainebleau sand and the sand-kaolin mixture (performed under the same conditions), shows (Figs. 7 and 14) that the deformation of the RG is flattened in the case of the cohesive soil. This flattened shape is a consequence of the breakage in blocks acting in a specific way onto the geosynthetic. Figure 15 compares the maximum RG deflection for both soils. There is a gradual evolution of the geosynthetic deflection in the case of the sand soil layer whereas a sudden soil movement is obtained in the case of cohesive soil. In addition, the deformations of the RG in the central part of the cavity in the case of the cohesive soil are always smaller than those obtained with the sand, even after unloading.

Analysis of stresses evolution: After the cavity opening and for a load of less than 55 kg, the sensors C2, C3 and C4, furthest away from the cavity, did not detect any increase in stress. The largest load transfer increase was measured by the C8 sensor for which the stress is increased by 720 %. The two sensors C7 and C13, sited symmetrically to the axis of the cavity, have measured three times the initial value obtained before the cavity opening, i.e. 331 % and 301 % respectively. For an overload of 55kg for which the soil layer was broken, the collapse of the soil layer lead to a sudden drop in stress on various sensors, particularly those closest to the collapse zone.

Analysis of load transfer mechanisms: the collapse mechanism obtained in Fig. 12 is quite similar to the one proposed by Huckert (2014), who assimilates the action of the collapsed blocks to punctual loadings. Based on experimental observations, a truncated pyramid shape has been proposed to describe the geometry of the collapsed soil zone (Fig. 13). The 2D design method is applied to the geosynthetic strip, 0.15 cm in width, sited in the axis of the cavity that is assumed to be submitted to the maximum loading (self-weight of the soil and overloading). We assumed that the weight of the collapsed soil (block A) and the surface overload (2P) after collapse are totally transmitted to the RG (i.e., a total load per meter of width of (55+4.5)×9.81/0.15=3.9 kN m-1). The weight of the two parts B and C on the RG at the edges of the cavity is assumed negligible compared to the intensity of the stress induced in the central part of the cavity by the overload at failure (55 kg). The input parameter values, deduced from observations and experimental measurements, for the use of the Huckert model (2014) are summarized in Table 2. UA and UB are the average horizontal displacements of the geosynthetic at the left and right edges of the cavity due to geosynthetic slippage, respectively. As presented in Fig. 15, the analytical results give a reasonable approximation that can be used to design a GR in the case of a cohesive soil layer prone to subsidence.