# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 2 Numerical Analysis
It was proven that the boundaries of the model took on negligible stresses, which implied that, in some FLAC models, the width of the model was adapted according to the width of the footing (load application). These checking and adaptation procedures were undertaken for each model. In the numerical model, the vertical load was applied directly on the nodes (ground surface) via velocity increments; therefore, the characteristics of the foundation and the interaction with the ground surface did not influence the result. In the numerical analysis, it was assumed that the bearing capacity was reached when the continuous medium could not withstand more load because this indicated that the internal failure mechanism had formed. The hypothesis of a rigid or flexible foundation depended on the relative deformability of the shallow foundation regarding the supporting ground. This consideration could not be included in the analytical solution since the bearing capacity was independent of the ground deformability. The conditions of uniform stresses and displacements at the contact points could not be simultaneously imposed, and therefore, it was necessary to previously investigate how to consider the rigidity of the foundation using the FDM method. The best approximation of the numerical result with the analytical solution (for cases with associated dilatancy where such a theoretical solution existed) was observed when assigning a constant deformation velocity for the contacting points of the foundation and the rock. Thus, the load in FLAC was applied through velocity increments, and the \begin{table}
\begin{tabular}{c c c c} \hline \hline \(m_{i}\) & \(B\) (m) & \(UCS\) (MPa) & \(GSI\) \\ \hline
5 (claystone) & 4.5 & 5 & 10 \\
12 (gypsum) & 11 & 10 & 50 \\
20 (sandstone) & 16.5 & 50 & 85 \\
32 (granite) & 22 & 100 & 85 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the adopted parameters. Each column indicates the adopted values of each parameter; in this way, their combination results in 192 calculation cases (4 of \(m_{i}\times 4\) values of \(B\times 4\) of \(UCS\times 3\) of \(GSI\)). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Models Type** & **Weight** & **Flow Law** \\ \hline Plane-strain & Weightless & Non-associative \\ Plane-strain & Weightless & Associative \\ Plane-strain & Self-weight & Non-associative \\ Plane-strain & Self-weight & Associative \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hypothesis adopted for numerical analysis. bearing capacity was determined from the relation between the stresses and displacements of one of the nodes; in this case, the central node of the foundation was considered. In Figure 2a, the displacement of the central node of the foundation (horizontal axis) with respect to the load applied to the ground from the foundation is represented. In this figure, it can be observed that the maximum load that the ground supported was limited to the asymptotic value of the represented curve. Figure 1: The 2D model employed in the numerical calculation. A convergence study was carried out that involved controlling the value of the bearing capacity obtained under different increments of the applied velocity. Figure 2b shows the dependence of the results of the bearing capacity in relation to the velocity increments applied on the nodes, and how, with the decrease in the value of velocity increments, the result converged towards the final value. This convergence study was carried out for each case with a different combination of geometrical and geotechnical parameters. In this study, both the associative flow rule (with an angle of dilatancy equal to the instantaneous friction angle) and the non-associative flow rule (with an angle of dilatancy equal to zero) were used. The commercial program FLAC has two constitutive models implemented that combine the generalized Hoek and Brown failure criterion [14] with different flow rules, allowing the adoption of different values of dilatancy.