The bearing capacity of shallow foundations is a classical problem with widespread applications, particularly in modern offshore engineering. Design is typically based on limit solutions for plane strain situations and vertical loading, with semi-empirical adjustments to allow for shape, inclination and eccentricity of the applied load, and for situations where the soil strength varies with depth. The ultimate accuracy of this approach is questionable, and indeed has been shown to be non-conservative where moment loading is involved. Although numerical techniques such as the finite element method are capable, in principle, of providing accurate solutions to specific situations, they are cumbersome to use for parametric studies, and also can lead to unacceptable errors for inexperienced users. The present paper presents new upper bound solutions for circular foundations on clay under general loading conditions. Computation of the three-dimensional mechanisms has been simplified through recent theoretical developments, and the derived failure envelopes offer good accuracy in comparison with finite element solutions. The approach not only provides a relatively simple computational basis for foundation design, regardless of the strength profile with depth, but also opens the door to novel upper bound solutions for a variety of geotechnical problems.