# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 4 Bearing Capacity Considering the Self-Weight of the Rock Mass  
In this section, in contrast with the hypothesis adopted and the results presented in Section 3, the self-weight of the rock mass is considered. The same procedure is applied, the results obtained with the non-associative flow law with null dilatancy (\(P_{hNS}\)) are compared with those obtained with the associative flow rule (\(P_{hAS}\)). Figure 10 shows that the dispersion ranges observed in Section 3 (Figure 3) were not similar to the cases with self-weight. From this figure, it can be concluded that the parameters that influenced the \(\Delta p_{h}^{AS/NS}\) were \(GSI\), \(m_{i}\) and \(UCS\). Figure 9: The variation of the vertical displacements under the foundation (weightless rock mass) (\(m_{i}=12\); \(B=11\) m; \(UCS=100\) MPa; \(GSI=50\)). (**a**) Non-associative flow law (null dilatancy); (**b**) associative flow law. From Figure 10a, it can be observed that the difference between \(P_{hNS}\) and \(P_{hAS}\), as well as the dispersion of the results, grew with the increasing of \(m_{i}\). The cases with lower \(m_{i}\) (\(m_{i}=5\)) presented a variation below 20%, while for higher \(m_{i}\) (\(m_{i}=32\)), the variation could exceed 65%. Observing Figure 10b, it can be concluded that the width of the foundation (\(B\)) had little effect on the dispersion of the results. In relation to the _UCS_, Figure 10c shows that the variation between the results increased slightly with the increment of _UCS_. For low values of the _UCS_ (5-10 MPa), the dispersion was in the order of 0 to 40%, while the cases with _UCS_ = 100 MPa presented a range that varied between 0 and 65%. Figure 10d does not show a clear trend of influence in terms of the function of the _GSI_. In Figure 11, it is observed that the influence of the _UCS_ and \(m_{i}\) on the correlation between the results depended on the value of the _GSI_. Concluding that the _GSI_ was the most dominant parameter, in Figure 11a, it is observed that the dispersion range decreased with the increment of the _GSI_, and in Figure 11b, it can be seen that for greater values of the _GSI_ (_GSI_ = 85), the value of the _UCS_ had little effect on the dispersion of the results. Furthermore, it was observed that both the average value and the dispersion in the variation of the bearing capacity increased at higher values of the parameter \(m_{i}\). In Figure 11b, it is observed that for the combination of low _GSI_ and _UCS_, the variation of the flow law type had very little effect on the correlation of results for \(P_{hAS}\) and \(P_{hNS}\) (less than 10%). This was due to the fact that the majority of the bearing capacity was associated with the self-weight of the rock mass. The self-weight of the material tended to increase the normal stress, as can be seen in the diagram representing the failure criterion (Figure 12). In addition, the inclination of the Hoek and Brown failure envelope depended directly on the value of the _UCS_ (Figure 12); for this reason, in the cases of the low values of _UCS_, the envelope tended to become horizontal quickly (even for low stress states). Figure 10: Correlation of \(P_{hAS}\) and \(P_{hNS}\) as a function of: (**a**) \(m_{i}\); (**b**) \(B\); (**c**) _UCS_; (**d**) _GSI_. (The dots of each graph (**a–d**) correspond to the 192 calculation cases carried out). Figure 11: Correlation between \(P_{hAS}\) and \(P_{hNS}\) depending on \(GSI\), as a function of: (**a**) \(m_{i}\); (**b**) \(UCS\). (The dots of each graph (**a**) and (**b**) correspond to the 192 calculation cases carried out). Figure 12: Representation of the Hoek and Brown failure envelope (\(m_{i}=7\) and \(GSI=10\)). (**a**) Low stress state; (**b**) Mohr circle. Thus, when the self-weight of the material is considered, in cases with low _UCS_ and _GSI_, the normal stress was higher, and thus, the Mohr circle intercepted the Hoek and Brown failure envelope in the almost-horizontal sector (Figure 12b).