Using the generalized Hoek and Brown yield criterion, the stress characteristics method (SCM) was used to evaluate in a very accurate manner the bearing capacity of smooth and rough ring foundations placed over rock media. Failure mechanisms, involving a combination of outward and inward shear zones, with consideration of the stress singularities at the outer and inner edges of the ring, were generated. The complete computational procedure is elaborated, and the computer code, written in MATLAB version 2018a, is available free upon request. To compare the results obtained, the solution is also provided making use of finite-element limit analysis (FELA) using OPTUM-G2 version 2018.07.28. The bearing capacity obtained using the SCM compares very well with the solution on the basis of the FELA, which confirms the rigor of the analysis performed. The failure patterns obtained from the SCM and the FELA were found to be almost identical.