# Bearing Capacity of Footings on Rock Masses Using Flow Laws
## 1 Introduction
Concerning the influence of the flow law on the bearing capacity, no technical literature was found that quantifies how it conditions the bearing capacity of the rock mass. In general, in everyday practice, the dilatancy is usually not taken into consideration in rock engineering. The papers related to the influence of the dilatancy angle or the flow law on the rock mass engineering problems are mainly focused on the slope stability analysis and tunneling. Considering that the mechanical behavior of the rock masses is non-linear, [PERSON] (2018) claimed that the assumption of a constant dilatancy angle is believed to be unrealistic because the dilation should be a function of the plastic strain (damage) and the confining stress. Thus, the definition of the value of the dilatancy angle is very complex, and the adoption of the null dilatancy (\(\psi=0^{\circ}\)) or the associative flow rule (\(\psi=\varphi\)) is common on both soils and rocks. The adoption of the null dilatancy in some cases underestimates, but never overestimates, the bearing capacity of the rock mass, so this option is on the conservative side. Authors such as [PERSON] [PERSON] and [PERSON] (2020), and [PERSON] and [PERSON] (2018), claim that the associative flow rule overestimates the strength of the rock mass; the value of the dilatancy angle recommended by these authors is much lower than the instantaneous friction angle. According to [PERSON] and [PERSON] (2018), the soil mechanics concept of a dilatancy angle is useful for concrete and rock, and the dilatancy angle is at least \(20^{\circ}\) less than the internal friction angle (\(\varphi\)). [PERSON] and [PERSON] [PERSON] and [PERSON] (2020) proposed correlations depending on the overall geotechnical quality of the rock mass (geological strength index, \(GSI\)): (a) \(\psi=\varphi/4\) for very good quality hard rock mass (\(GSI=75\)) that presents an elastic-brittle behavior; (b) \(\psi=\varphi/8\) for average quality rock mass (\(GSI=50\)), considering the behavior as strain softening; (c) \(\psi=0^{\circ}\) for very poor quality rock mass (\(GSI=25\)) that behaves perfectly plastically. In the field of tunnels, [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) proposed a method for the estimation of the variation of the dilatancy angle, depending on the stress state, in a way that allows a better adjustment of the tunnel support (shotcrete and reinforcement) and the estimation of deformations during underground excavations. In [PERSON] and [PERSON] (2018), convergence curves were developed for circular tunnels subjected to hydrostatic pressures based on the theory of plasticity. The solution is valid for any failure criterion and for any plastic flow law, considering that the dilatancy varies from one point to another in the plastic area as the result of being the function of failure stresses and the instantaneous friction angle. [PERSON] and [PERSON] [35], also focusing on support design, proposed an empirical dilation model, based on the dilation model of [PERSON] and [PERSON] [36], as a function of the loading history and confinement of the rock mass. [PERSON] et al. [37] emphasized that the hypothesis of the associative flow law is broadly accepted in applications related to slope stability analysis, resulting in values of the safety factor that are evidently higher than the values obtained under the assumption of the non-associative flow law. For this paper, the upper and lower limits of the bearing capacity in terms of the function of the flow law are estimated by numerical analysis through the finite difference method, employing the FLAC commercial code (Version 7, Itasca) [38]. The correction factor (\(D_{F}\)) is defined to estimate the decrease in the bearing capacity due to the adoption of the non-associative flow rule with null dilatancy under the assumptions of a weightless rock mass. This factor can be applied to the bearing capacity value obtained by analytical formulation, as exemplified in the work of [PERSON] et al.