# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 2 Numerical Analysis  
The numerical analysis was developed by the finite difference method, using FLAC v.7 [38]. In the finite differences method, every derivative in the set of governing equations is directly replaced by an algebraic expression written in terms of the field variables (stress or displacement) at discrete points in space; these variables are undefined within elements. Using the approach by [PERSON] [39], boundaries can be of any shape, and any element can have any property value. FLAC uses an explicit time-marching method to solve the algebraic equations. Although a static solution is needed, the dynamic equations of motion are included in the formulation. Because of instability in the physical system model, this approach is used to make the numerical scheme stable: physical instability can be present with nonlinear materials. In contrast, schemes without inertial terms must use some numerical procedure to treat physical instabilities. This procedure starts with the equations of motion to derive new velocities and displacements from stresses and forces. Then, the strain rates are derived from velocities and new stresses are derived from the strain rates. The numerical implementation of the Hoek-Brown model uses a linear approximation, whereby the nonlinear failure surface is continuously approximated by the Mohr-Coulomb tangent at the current stress level indicated by the minor main stress. The use of tangent linear approximation in geotechnical software is usually considered to be independent of the numerical method adopted for the calculation. However, its use could be the main limitation in terms of obtaining adequate numerical solutions when the failure criterion curve is very steep. Thus, in cases of rock masses of low geomechanical quality (low \(GSI\)), the Hoek and Brown criteria estimate a very low tensile strength, close to zero. Furthermore, the failure criterion always responds with a steeper slope in the vicinity of the origin for higher values of the parameter \(m_{i}\). This means that in the absence of load (a weightless rock mass or a rock mass without external overload in the boundary adjacent to the foundation), the slopes of the failure criterion are close to \(90^{\circ}\) and are very sensitive to stress variations and, therefore, highly unstable. In particular, among the calculation cases that were performed in this research, for values of \(GSI=10\) and \(m_{i}=32\), at times, the unbalanced force was extremely high and the calculation did not converge to a valid result. Several numerical calculations were developed involving the application of the finite difference method to define and quantify the influence of the application of different flow laws on the estimation of the bearing capacity. A total of 192 cases were analyzed, resulting from the combination of four influential parameters (rock type (\(m_{i}\)), foundation width (\(B\)), uniaxial compressive strength of the rock (\(UCS\)) and geological strength index (\(GSI\))). With the values given in Table 1, a wide variety of types and states of rock masses are covered. The widths were varied between 4.5 and 22 m in an attempt to represent foundation widths in bridges, and dams or foundation slabs in high buildings, where the bearing capacity of a rock can be restrictive. Theses 192 cases were studied using the hypothesis described in Table 2. The flow law (associative and non-associative with null dilatancy angle) was varied and the self-weight of the rock mass was also considered, assuming a weightless rock mass or adopting the unit weight of 26 \(\mathrm{kN/m^{3}}\) up to a total of 768 calculation models. In all cases, a plane strain model and perfectly rough interface were used. A symmetrical model was used, where only half of the strip footing was represented, and the boundaries of the model were located at a distance that did not interfere with the result (Figure 1).