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Here we note that the principal normal stresses σ₁ = σ₂ = σ_r = σ_θ since the out-of-plane shear stresses, τ_rz = τ_θz = 0 and σ₃ = σ_z.  And we can deduce that τ_max = |τ₁|=|τ₂|=|(σ₁-σ₂) / 2|.  The effective stress (using the Von-Mises criterion) along the y-axis can be computed as the following:

Lastly, we also confirm that the applied load at the top vertex of the sphere matches our numerical contact pressure, integrated along the interface.

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