The influence of the void shape in the bearing capacity of rock mass with a superficial void

In this study is proposed a chart that allow estimating the reduction of bearing capacity due to the existence of a superficial void in a rock mass. It is contemplated the influence of different shapes and size of the void in relation to the foundation. The reduction of the bearing capacity is analyzed through the Finite Difference Method. In the numerical model, six possible cavity shapes are considered, with three different sizes and depths. Calculations are made considering eighteen types of rock masses, modelled by the H&B failure criterion. The results demonstrate that the critical depth where the cavity no longer influences the bearing capacity varies considerably depending on the geometric variables (size, shape and location of the void). In addition, it is observed that the relation between the bearing capacity obtained considering the rock mass with and without void is not affected by the geotechnical parameters of the H&B failure criterion. According to the results analyzed and graphic output, the variation in the stress distribution in the rock mass and the change of the failure mechanism under the different hypotheses studied can be observed. The results are presented in chart that facilitate to directly estimate the reduction of the bearing capacity of a rock mass with void based on geometric parameters.


Introduction
The existence of an underground void (associated with a cave, tunnel or mining works) changes the mechanical behavior of a rock mass.Therefore, it is expected that when the void is located close to the surface the bearing capacity of a shallow foundation on a rock mass is reduced.However, in the field of rock mechanics, few studies quantify this reduction in a systematic way.
In the study conducted by Wang et al. (1987), a theoretical development of potential failure mechanisms is presented, and some cases were analyzed by the finite element method.The rock mass was modelled using the Mohr-Coulomb failure criterion, which is not the most suitable model to simulate a rock mass (Merifield et al. 2006; Clausen 2013; Qin and Chian 2018).Waltham et al. (2005) showed some cases of rock collapse over a cave, describing some possible failure mechanisms and claimed, according to evidence from the various available sources, that a roof thickness of about half the cave width is stable and safe under most conditions of loading.Xiao et al. (2018) elaborated charts that allow an estimation of the reduction of the bearing capacity of a rock mass with one or two rectangular voids, depending on the size and location of the cavities.They also analyzed the influence of the geotechnical parameters of the Hoek and Brown failure criterion (Hoek and Marinos 2007) in the reduction of the bearing capacity of rock mass due to the void.Despite few studies related to the bearing capacity of shallow foundations on rock mass with voids, there is an extensive literature concerning the roof stability of the caves in a rock mass.Barton (1976) proposed a chart to evaluate if the excavation is stable without support depending on the Q index and the width of the cavity.Carter and Miller (1995) and Carter (2000Carter ( , 2014) ) analyzed the safety factor of the crown pillar, proposing the acceptable risk exposure guidelines.Jordá-Bordehore et al. (2016) analyzed the stability of the lava tubes of the Galapagos Islands applying the Q index to evaluate if the cave is stable or unstable, although the authors observed that even a stable cave could show a factor of safety that does not meet the minimum required.
Jordá-Bordehore et al. ( 2016) also studied the stability of the karstic caves, concluding that the stability graph method (Mathews et al. 1981;Potvin 1988;Nickson 1992) (that is widely applied to the pre-design of large mine stopes (Potvin 1988;Hoek et al. 1995)) seemed to be more precise and more realistic than the Q index itself (bidimensional).
It must be pointed out also the studies conducted to evaluate the roof stability of the cavities in rock mass that have special shapes.Hatzor et al. (2002) investigated the stability of bell-shaped caverns at Bet Guvrin, Israel.Cristescu and Paraschiv (1995) developed an elastic exact solution for stress distribution around a rectangular cavern with rounded corners in a relatively unfractured rock mass and studied the optimal shape that minimizes the volume of rock exposed to failure.Fava and Ricca (1997) studied the stresses and strains behavior of cavities excavated in the Italian Alps.This study led to changes in the excavation section in order to improve the redistribution of tensions in the rock mass.
It is supposed that the stress distribution in the rock mass varies with the shape and the size of the void.Therefore, the present study examines the reduction of the bearing capacity of a shallow foundation on a rock mass with a void considering six shapes (circle, horizontal ellipse, vertical ellipse, square, horizontal rectangle and vertical rectangle), three sizes and different combinations of geotechnical parameters (mi, UCS and GSI).

Problem statements
In the present study, the influence of the geotechnical and geometric parameters on the reduction of the bearing capacity of shallow foundations on rock mass with a void is analyzed.
Combining the geotechnical parameters of the modified Hoek and Brown failure criterion (2002), i.e. rock type (mi), overall geotechnical quality of the rock mass (GSI) and uniaxial compressive strength (UCS), eighteen types of rock mass were generated, with the values listed in Table 1.The values of mi and GSI try to cover a great range of types of rock mass, and the UCS values were chosen considering a usual characteristic strength of concrete (40 MPa), so was adopted a rock mass less and more resistant than the concrete.
This study included a great range of shapes, depth and sizes of the void (see Figure 1a).Figure 1b shows the shapes used in the models: circle, horizontal and vertical ellipse, square, horizontal and vertical rectangle.Three different void sizes for each geometry are considered (Table 2), however, it is emphasized the adoption of other values for the foundation width (eccentricity/slenderness) is only a change of scale.All void are located under the footing (X=0 -Figure 1a), and vertically the depth of the void is varied from very shallow (R/B = 0.6) to the depth at which the void no more affects the bearing capacity.Table 2. Summary of the geometrical parameters adopted.
In addition, it was checked that the percentage of the reduction of the bearing capacity of the rock mass with a void is depending on the ratio W/B, instead of the absolute foundation (B) or void´s widths (Wc, Wh or Wv).Simulations using different footing widths (maintaining the W/B and R/B ratios) verified that the reduction percentages of the bearing capacity did into change.This confirms that the adoption of other values for the foundation width is only a change of scale.In the study a value of B = 0.9 m is adopted and the cavity width is denominated W, independent of the shape (Wc, Wh or Wv).
Numerical calculations were carried out using 2D models in the finite difference method employing the commercial code FLAC version 7. The simulations applied the plane strain condition (strip footing) with a symmetrical model, where only half of the strip footing is represented.The boundaries of the meshes are located at a distance that does not interfere in the result.In all simulations the self-weight of rock mass is considered, the associative flow-rule and the rough interface at the base of the foundation are adopted.The Modified Hoek-Brown constitutive model (that adopted the modified Hoek and Brown failure criterion (2002)), available in FLAC v.7, was used to model the rock mass.

Results and analysis
The results obtained are analyzed to know the influence void shape in the reduction of the bearing capacity of rock mass with void applying the plane strain condition.The correlation coefficient VF (void factor) was used, which is defined as the value that must be multiplied to the bearing capacity of rock mass without a void (Ph), to obtain the bearing capacity of the rock mass with the same geotechnical characteristics but with a void (Phv).
According to Xiao et al. ( 2018) the reduction of bearing capacity of a rock mass with one or two rectangular voids are independent of the geotechnical parameters.It was observed in the cases studied, being the cases generated by the combination of geotechnical parameters listed in Table 1.This happens because the geotechnical parameters tend to affect in equal proportion the value of the bearing capacity of a rock mass with and without void, so the ratio between these two values of bearing capacity (VF) is independent of them.
To determine the influence of the void shape in the correlation coefficient VF, models with voids with round angles (circle, horizontal ellipse and vertical ellipse) and right angles (square, horizontal rectangle and vertical rectangle) were modeled (see Figure 1).The analyses were carried out adopting the centered load (X = 0) and the sizes of the cavities are described in Table 2.
Figure 3 shows the graphs with the results obtained as a relation between VF and R/B for different rectangular and round voids (Wv/B, Wh/B, Wc/B).It can be observed that with the increase in the value of W/B (considering the same value of R/B) the value of VF is lower, meaning that the larger the void in relation to the foundation (keeping other geotechnical and geometrical variables constant) the bearing capacity is smaller.With an increment of W/B it can be observed (see Figure 3) that the curves follow almost parallel trajectories.In those cases, the trend is that the rock masses with rounded voids (with the same geotechnical and geometric conditions) support more load than those with a void with straight edges.It must be pointed out that with the increment of W/B ratio, the effect of the edges shape becomes more pronounced.For examples, in the cases studied, the variation of VF depending on the shape reaches up to 20%, for the highest ratio Wv/B = 2.2 (vertical ellipse and rectangle) and for R/B = 2. Furthermore, in Figure 3 it can be seen that the value of VF depends on the shape of the edge, but not on the shape of the cavity, whereby the nine lines plot on top of each other (see Figure 4).The critical depth (i.e., depth where the void no longer influences the bearing capacity, being VF = 1) also varies considerably as a function of W/B ratio and depends on the edges shape.It is observed that the thickness of the critical depth tends to be greater in the rectangular voids than in the round ones, and deeper as the width of the void increases.
Figure 5 shows the graphic output of the maximum principal stress, where it can be observed that if the void width varies (keeping constant B and R) the stress distributions changes.It is noted that the stresses directions rotate horizontally to deflect the cavity, being biggest the rotation, the wider is the void.It is also observed a concentration of stress in the upper corner of a square void, which explains why VF values are smaller in cases associated with rectangular voids in comparison with voids with rounded edges.Furthermore, it is noted that in cases where the void is small (compared to the footing width, e.g.W/B = 0.6) a stress concentration at the edge does not occur, and therefore, VF value does not follow a clear trend depending on the void shape, as shown in Figure 3.The graphs shown previously can be summarized in Figure 6, that allows the estimation of the reduction of the bearing capacity of a rock mass due to the presence of a rounded or rectangular cavity in a generic way (independent of the slenderness of the cavity).

Conclusions
Taking into account the results of bearing capacity of rock mass with one void described in previous sections obtained through the finite difference method under the hypothesis of plane strain condition, with associated flow rule, the rough interface at the foundation base and considering the self-weigh of rock mass, the following can be concluded:  It is observed that the shape of the cavity, rectangular or rounded, interferes in the VF value.Thus, the rounded voids allow a better stress distribution in the rock mass.The rounded cavities decrease less the bearing capacity of a rock mass compared to a cavity of similar width but with straight edges.It is also noted that the greater the W/B (void width / foundation width) ratio the greater is the influence of the shape on the value of VF.  The graphic output of maximum horizontal and vertical displacements are consistent with the results obtained.Regarding the void shape it is noted that stress is concentrated in the upper corner of a square void, resulting in a weak point that does not occur in the round void.It is also observed that the stress rotates towards the horizontal direction to deflect the cavity, being bigger for wider voids. Finally, a chart is proposed that allows the estimation of the value of VF considering a centred load.

Figure 1 .
Figure 1.Geometry of the model, representing a footing on the rock mass with one void.(a) Model geometry; (b)Void geometry.

Figure 2 .
Figure 2. 2D model used in the calculation through FDM.

Figure 4 .
Figure 4. Critical depth (VF = 1) as function of the R/B and W/B.

Table 1 .
Summary of the geotechnical parameters adopted.