# Assessment of the Bearing Capacity of Bridge Foundation on Rock Masses
## 2 Analytical Formulation for the Ultimate Bearing Capacity  
As is generally known, in rock mechanics, the non-linear Hoek-Brown failure criterion [13] is the most used and it is applicable for the rock mass and is formulated in function of the major principal stress (\(\sigma_{1}\)) and minor principal stress (\(\sigma_{3}\)) according to the following equation:  
\[\frac{\sigma_{1}-\sigma_{3}}{\sigma_{c}}=\left(m\cdot\frac{\sigma_{3}}{\sigma _{c}}+s\right)^{a} \tag{1}\]  
The uniaxial compressive strength (\(UCS\)) is \(\sigma_{c}\), while the parameters \(m\), \(s\), and \(a\) can be evaluated following [13] by Equations (2)-(4) and depend on the rock type (\(m_{l}\)), geotechnical quality index of the rock mass (\(GSI\)), and damage in the rock mass due to human actions (\(D\)) that in shallow foundations is usual equal to zero. \[m=m_{i}\cdot e^{\frac{GSI-100}{3-14\cdot 10}} \tag{2}\]  
\[s=e^{\frac{GSI-100}{9-3D^{0}}} \tag{3}\]\[a=\frac{1}{2}+\frac{1}{6}.\left(e^{\frac{-GSI}{15}}-e^{\frac{-20}{3}}\right) \tag{4}\]  
[PERSON] et al. [15] proposed an analytical formulation for estimating the ultimate bearing capacity of the shallow foundation independently of width, based on the modified Hoek-Brown failure criterion [13], taking into account the associated plastic flow rule, and strain plane. A brief summary of the analytical formulation is presented below, since it allows obtaining a closed solution to the result, for which in this research it is intended to incorporate a corrective coefficient that directly allows considering the effect of the width of the foundation. According to this analytical formulation the ground surface that supports the foundation is composed of two sectors (Figure 1): boundary 1 (free) with the inclination \(\alpha\) where the load acting on a surface \(f_{1}\) acting with the inclination of \(i_{1}\) is known (for example, the self-weight load on the foundation level or the load from installed anchors); and the boundary 2 (foundation), where the bearing capacity of the foundation \(P_{h}\) should be determined (acting with the inclination of \(i_{2}\)). The analytical solution [15] based on the characteristic line method needs the equation of the Riemann invariants (\(l_{a}\)) fulfilled along the characteristic line:  
\[I_{a}(\rho_{1})+\psi_{1}=I_{a}(\rho_{2})+\psi_{2} \tag{5}\]  
\[I_{a}(\rho)=\frac{1}{2.k}.\left[cot(\rho)+ln\left(cot\left(\frac{\rho}{2} \right)\right)\right] \tag{6}\]  
In this equation, the instantaneous friction angle at the boundary 2 (\(\rho_{2}\)) is the only unknown, because the other variables can be defined at the boundary 1: instantaneous friction angle at the boundary 1 (\(\rho_{1}\)) and the direction of the principal stress in this sector (\(\Psi_{1}\)), thus expressing \(\Psi_{2}\) in function of \(\rho_{2}\). Knowing \(\rho_{2}\) the ultimate bearing capacity can be estimated. Through the analytical method [15] the bearing capacity is obtained by (7). \[P_{h}=\beta_{a}.\left(N_{\beta}-\zeta_{a}\right) \tag{7}\]  
The resistant parameters \(\beta_{a}\) and \(\zeta_{a}\) defined by [PERSON] et al. [15] are applied to make dimensionless the calculation of the Hoek-Brown failure criterion. \(\beta_{a}\) represents the characteristic strength that has the same units as _UCS_ and is used to make pressures  
Figure 1: Mathematical model of the bearing capacity of the strip footing. dimensionless, while \(\zeta_{a}\) (\"tenacity coefficient\") is a dimensionless coefficient that multiplied by \(\beta_{a}\) corresponds to the tensile strength. \[\beta_{a}=A_{a}\cdot UCS;\ \zeta_{a}=\frac{s}{(m\cdot A_{a})};\ A_{a}=\left(\frac{m \cdot(1-a)}{2^{\frac{1}{a}}}\right)^{\frac{1}{t}};\ k=\frac{(1-a)}{a} \tag{8}\]  
\(A_{a}\), \(k\) and the exponent \(a\), are constants for the rock mass, and depend on 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 \(N_{\beta}\) can be obtained by iteration from the load at the boundary 1.