# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 5 Example  
The correlation factor (\(D_{F}\)) proposed in Equation (3) makes it possible to estimate the percentage of the variation of the bearing capacity according to the flow law adopted; thus, for the estimation of the bearing capacity with null dilatancy, this factor can be applied to the bearing capacity calculated with other methods that use the associative flow rule. In this section, an example is given to illustrate the use of factor \(D_{F}\) to estimate the bearing capacity. A rock mass with the following geomechanical properties is considered: \(m_{i}=5\), \(B=4.5\) m, \(UCS=5\) MPa and \(GSI=10\), under the assumption of the plane strain, horizontal ground and the weightless rock material. It is first necessary to estimate the bearing capacity considering the associative flow law (\(P_{hAW}\)) by a method that can be numerical (FDM, FEM) or analytical [18]. The method proposed in [18] applies the theory of characteristic lines, by adopting the plane strain hypothesis, the associated flow rule, the coaxiality, the perfectly plastic yield surface and the weightless rock mass (therefore, independent of width). This analytical formulation of the ultimate bearing capacity introduced a bearing capacity factor that makes the failure pressure proportional to the uniaxial compressive strength of the rock (\(UCS\)). According to this analytical formulation, the ground surface that supports the foundation is composed of two sectors (Figure 15): boundary \(1\) (free) with the inclination \(i_{1}\) where the load acting on a surface is known (for example, the self-weight load on the foundation level or the load from installed anchors); and boundary \(2\) (foundation), where the bearing capacity of the foundation should be determined (acting with the inclination of \(i_{2}\)). The solution based on the characteristic line method requires the equation of the Riemann invariants (\(I_{a}\)) to be fulfilled along the characteristic line:  
\[I_{a}(\rho_{1})+\psi_{1}=I_{a}(\rho_{2})+\psi_{2} \tag{4}\]  
\[I_{a}(\rho)=\frac{1}{2\cdot k}\cdot\left[cot(\rho)+ln\left(cot\left(\frac{\rho }{2}\right)\right)\right] \tag{5}\]  
In this equation, the instantaneous friction angle at boundary \(2\) (\(\rho_{2}\)) is the only unknown, because the other variables can be defined at boundary \(1\), namely the instantaneous friction angle at boundary \(1\) (\(\rho_{1}\)) and the direction of the principal stress in this sector (\(\Psi_{1}\)),  
Figure 15: [PERSON] et al.’s [18] model of the bearing capacity of the strip footing. In this Figure, a diagram of the configurations that can be solved using the analytical formulation is shown, although the solved calculation example is particularized for a horizontal ground. thus expressing \(\Psi_{2}\) as a function of \(\rho_{2}\). Finally, knowing \(\rho_{2}\), the ultimate bearing capacity can be estimated. Through the analytical method, the bearing capacity is obtained using Equation (6). \[P_{h}=\beta_{a}\cdot\left(N_{\beta}-\zeta_{a}\right) \tag{6}\]  
The resistant parameters \(\beta_{a}\) and \(\zeta_{a}\) are applied to make the calculation of the Hoek and Brown failure criterion dimensionless. \(\beta_{a}\) represents the characteristic strength that has the same units as _UCS_ and is used to make pressures dimensionless, while \(\zeta_{a}\) (the \"tenacity coefficient\") is a dimensionless coefficient that, when multiplied by \(\beta_{a}\), corresponds to the tensile strength. \[\beta_{a}=A_{a}\cdot\textit{UCS};\quad\zeta_{a}=\frac{s}{\left(m\cdot A_{a} \right)};\quad A_{a}=\left(\frac{m\cdot\left(1-a\right)}{2^{1/a}}\right)^{1/k}; \quad k=\frac{\left(1-a\right)}{a} \tag{7}\]  
\(A_{a},k\) and the exponent \(a\) are constants for the rock mass, and depend on the rock type (\(m\)), _UCS_ and _GSI_. \(N_{\beta}\) is the bearing capacity factor and it can be calculated, according to the problem statement, as follows. The angle of internal friction (\(\rho_{1}\)) can be obtained by iteration from the load at boundary 1.