Elastic deformation of substrate due to rotation of rigid pillar

Consider a ridig pillar ontop of a elastic substrate. Applying a moment to the pillar will lead to elastic deformation of the substrate. If the pillar is infinitly large in diameter, then this problem is the same as an infinitely sharp crack, considering the symmetry of the crack problem, i.e. there are square root singularities. However, the infinitly large diameter assumtion does not hold if the global rotation of the substrate under the pillar is of interest, because the both sides of the pillar interact.

To identify a coordinate system: If the moment is applied along the y-axis and the z-axis is the axial direction of the pillar, then the maxium stresses in the pillar will be along the x-axis.  

In the x-direction: the stresses still form a singularity as found numerically: not square root but lower order (and arguments can be  made justifying that).

However, in the y-direction: the problem is less trivial: Two contradicting arguments can be made  1) Since the surface has to be flat also in the y-direction with a wedge like end: a singularity should be present.  2) Since it is not in the direction of the maximal deformation, the stresses have to be bound, i.e. they are not singular.

Any thoughts?