# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 3 Bearing Capacity for Weightless Rock Mass  
With the results obtained by varying the flow law, we can observe that the variation of the dilatancy angle made it possible to modify the bearing capacity by up to 100% (Figure 3). The results were obtained in the extreme condition, in which an associative flow law (\(P_{hAW}\)) and a non-associative flow law with a null dilatancy angle (\(P_{hNW}\)) were compared, under the assumptions of weightless rock mass. Figure 2: Estimation of bearing capacity by FDM (\(m_{i}=5\), \(B=4.5\) m, \(UCS=5\) MPa and \(GSI=10\)): (a) stress–strain curve at a point of the ground; (b) convergence analysis. The comparison ratio (\(\Delta P_{h}^{AW/NW}\)) between the results \(P_{hAW}\) and \(P_{hNW}\) is expressed in Equation (1). \[\Delta P_{h}^{AW/NW}=\frac{P_{hAW}}{P_{hNW}}-1 \tag{1}\]  
From Figure 3a-d it can be concluded that the two parameters that most influenced the dispersion of the ratio between \(P_{hAW}\) and \(P_{hNW}\) were \(m_{i}\) and \(GSI\). On the other hand, Figure 3b,c, show that the dispersion range was independent of the \(B\) and \(UCS\) values. In Figure 3, it can be observed that the influence of dilatancy angle value on the bearing capacity grew with the increasing of \(m_{i}\). It is recalled that due to the numerical instability problems, it was not possible to obtain the results of the bearing capacity for the combination of \(m_{i}=32\) and \(GSI=10\) for the weightless rock mass, which would be the cases that correspond to greater dispersion according to Figure 3a-d. Due to the absence of these results in the mentioned graphs, the variation between \(P_{hAW}\) and \(P_{hNW}\) for \(m_{i}=20\) ranged from 25 to 105%, while for \(m_{i}=32\), the variation ranged from 30 to 75%, and this difference would probably be much greater, confirming the tendency for \(\Delta P_{h}^{AW/NW}\) to increase with the increasing of \(m_{i}\). Figure 4 presents the values of \(\Delta P_{h}^{AW/NW}\) as determined in relation to the values of the bearing capacity obtained with non-associative flow rule with null dilatancy (\(P_{hNW}\)) as a function of the value of \(GSI\). From this figure, it can be concluded that the cases of low bearing capacity were the most conditioned by the variation of the dilatancy angle, and that the values obtained with the associative flow rule could exceed, in the most unfavorable cases (generally in the cases of low \(GSI\)), twice the value obtained under the hypothesis of the non-associative flow law with zero dilatancy angle. It is emphasized that the value of 25 MPa is the medium value of the compressive strength of concrete, as well as being the limit between soft and hard rock according to the ISRM [40]. Therefore, as the low values of bearing capacity were especially sensitive to the flow law according to Figure 4, it can be concluded that the non-associative law has a singular importance in soil-structure interaction studies. Figure 4 shows that the bearing capacity value and the dispersion with the angle of dilatancy were highly conditioned by the value of the _GSI_. Figure 5 shows that the range of dispersion depended on the combination of the _GSI_ with the \(m_{i}\), varying from 5% for lower \(m_{i}\) and great _GSI_, to more than 100% for a larger \(m_{i}\) and a small _GSI_. Therefore, the increase in the value of \(m_{i}\) implies the increase in the dispersion range in the results. As said previously, due to numerical instability problems, it was not possible to obtain results for the combination of \(m_{i}=32\) and \(GSI=10\), and for that reason, in Figure 5, for _GSI_ = 10, only three instead of four columns are presented. It is important to note that the cases that showed the greatest variation were those of the low _GSI_ values, corresponding to cases for which [PERSON] and [PERSON] [13] recommend the adoption of null dilatancy, since it corresponds to a very poor quality rock mass.