Does anybody know the equations for calculating stresses in a sheave? I know the P=2*T/D*d equation for sizing a sheave diameter for the wire rope. I have what I believe is a single piece steel rim pulley and for use at low speeds.
Shigley's "Mechanical Engineering Design" givees the same equation that you gave, for calculating the bearing pressure of the rope acting on the sheave groove. It also gives a table (sourced from AISI Wire Rope Users Manual, 1979) that illustrates the maximum allowable bearing pressures of ropes on sheaves of various materials. For Cast Steel (HB = 160 min.):
6 x 7 rope, reg. lay, 550 psi
6 x 19 reg lay, 900 psi
6 x 37 reg lay, 1075 psi
8 x 19 reg lay, 1260 psi
6 x 7 lang lay, 600 psi
6 x 19 lang lay, 1000 psi
6 x 37 lang lay, 1180 psi
Wire rope manufacturers could probably offer you more insight, if required.