A series of three-dimensional finite element analyses was conducted to investigate the effects of the embedment depth, the non-homogeneity of clay, and combined loads on the undrained bearing capacities of bucket foundations. The undrained shear strength and Young's modulus of clay were assumed to vary linearly with depth. Meanwhile, the stress-strain response of clay was simulated using the Tresca criterion. The numerical modeling adopted in this study was verified by comparing the calculated capacities with those from previous studies. Based on the results of the finite element analyses over 1400 cases, new equations were proposed to calculate the vertical, horizontal, and moment bearing capacities as well as to define the capacity envelopes under general combined loads. Comparisons with the capacity envelopes of previous studies showed that the proposed equations properly predicted the bearing capacities of the bucket foundation by considering the effects of the non-homogeneity of clay and embedment depth.