Then, i think the FEM solution should be updated as calculation of vertical and horizontal stress at the center due to application of point load. and the analytic solution can be used as described in the primer.

The tensile and compressive stresses that develop in the specimen as a result of the point loads P are a function of the diameter (D) and thickness (t) of the cylindrical specimen. At the centre of the specimen, the compressive stress is 3 times the tensile stress and the analytical formulation for these are, respectively:

(1)\sigma_t=\frac{2P}{\pi tD}

(2)\sigma_c=\frac{6P}{\pi tD}