# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 1 Introduction  
For the analysis of the bearing capacity of foundations, several procedures have been developed: the limit equilibrium method [1, 2], the characteristic line method [3], the limit analysis method [4, 5], the kinematical approach (as a part of limit analyses) [6], the numerical method [7, 8, 9] and artificial intelligence techniques [10, 11]. Each method adopts a series of hypotheses and failure criterion; currently, the most used is the non-linear method proposed by [PERSON] (in its original 1980 version [12], and modified versions from 1997 [13] and 2002 [14]), which is applicable for a rock mass with homogeneous and isotropic behavior, meaning that through the inexistence or the abundance of discontinuities, they have the same physical properties present in all directions. Traditional formulations of the bearing capacity are based on the Mohr-Coulomb parameters (cohesion and friction angle). The usual practice in rock mechanics is to obtain the equivalent Mohr-Coulomb parameters from the non-linear Hoek and Brown failure criterion according to the stress level of the rock mass. Thus, for example, some Mohr-Coulomb based approaches considered to be non-associative flow rules [6]. The flow law (also called flow rule) is not an intrinsic hypothesis, neither in the Hoek and Brown failure criterion nor in the Mohr-Coulomb approach; it should be adopted for the method chosen for the estimation of bearing capacity. The associativity of the plastic deformation of the soil has a significant influence on the result of the bearing capacity. [PERSON] [6] and [PERSON] [15] studied this effect in soils with the Mohr-Coulomb criterion, concluding that these differences become very substantial for large friction angle values and the bearing capacity results aresignificantly reduced with decreases in the angle of dilatancy. In this way, it is necessary to consider the influence of the non-associative flow rule especially for rocks, where high values of the instantaneous friction angle are expected. In this regard, works such as those by [PERSON] and [PERSON] [PERSON] and [PERSON] (2016) consider the influence of dilatancy in their specific studies applied to rock mechanics. In the past thirty years, some analytical methods were developed to estimate the bearing capacity of rock masses based on the parameters that define the non-linear failure criterion. The analytical method that solves the internal equilibrium equations combined with the failure criterion was proposed by [PERSON] and [PERSON] (2018) and [PERSON] et al. [PERSON] and [PERSON] (2018); [PERSON] (2019) and the modified [PERSON] and [PERSON] failure criterion [PERSON] and [PERSON] (2020), respectively. The method was based on the characteristic line method ([PERSON] and [PERSON], 2019), adopting the plane strain hypothesis, the associative flow law, the coaxiality, the perfectly plastic yield surface and the weightless rock mass. Other formulations proposed for the estimation of the bearing capacity on rock masses, such as ([PERSON] and [PERSON], 2019; [PERSON] and [PERSON], 2019; [PERSON] and [PERSON], 2020), recommended by [PERSON] (2019), and those presented in the regulations outlined in The Canadian Foundation Engineering Manual ([PERSON] and [PERSON], 2020) and Eurocode 7 ([PERSON] and [PERSON], 2020), do not define the flow law that is adopted. Recent studies have focused on the Hoek and Brown criteria, and the associated flow law is usually presented ([PERSON] and [PERSON], 2020; [PERSON] and [PERSON], 2020). The use of numerical methods and the progress of computational geomechanics have allowed access to the practical calculation of more sophisticated bearing capacity problems, involving anisotropic rock masses ([PERSON] and [PERSON], 2020), and the shape of the foundation ([PERSON] and [PERSON], 2020; [PERSON] and [PERSON], 2020) or the bilayer rock under the footing ([PERSON] and [PERSON], 2020).