A generalized nonlinear Baker failure criterion is employed with the upper bound limit analysis to study the surrounding rock stability of underground cavities. A three-dimensional (3D) failure mode is established by extending the two-dimensional (2D) failure mode, which offers an upper bound expression of the surrounding rock pressure. This method is validated with a series of examples before the influence of four parameters of scale parameter, curvature parameter, shift parameter and lateral pressure coefficient, on the surrounding rock pressure is analyzed. According to these results, failure ranges of the underground cavities are determined. The following conclusions are reached: (1) the proposed approach is more accurate to predict surrounding rock pressure than the Mohr−Coulomb failure criterion; (2) the surrounding rock with large scale parameter, curvature parameter, shift parameter, and lateral pressure coefficient can lead to a more stable underground cavity; (3) the failure range in 3D mode can be predicted according to the upper bound solutions.