Circular footings 2

I need mathematical methods for determining the bearing stress of a circular footing where the overturn moment is the dominant factor. If the foundation is starting "lift at the heel", ie the whole surface is not in contact with the soil, the geometry of the footing rapidly changes as the contact reduces, thereby constantly changing the contact area, hence bearing pressures increas sharply, but not linear. I am using empirical methods at the moment, but would like confirmation by way of mathematical methods.