Question

Deformation of soil under circular or square footing is how many dimensional

Answer 1

If soil support of a continuous footing yields, all the soil particles move parallel to the plane which is perpendicular to the center line of the footing. On the other hand, if the soil support of a square or circular footing yields, the soil particles move in radial and not in parallel. So bearing capacity increases. It is probably related to the volume of the mass under the footing that needs to be displaced to lead to failure and it's moment of inertia relative to the footing centerpoint. 

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Answer 2

If soil deform due to circular footing the displacement of soil form a cylindrical foot print. A cylinder have height and circular base. The dimension of cylindrical footprint have height and radius of the circular base.

If soil deform due to square footing the displacement of soil form a cuboid foot print. A cuboid have length, width and height. The dimension of cuboid  footprint have length, width and height.