A fatigue testing machine used to determine fatigue life under rolling contact consists of...

A fatigue testing machine used to determine fatigue life under rolling contact consists of a steel toroid (body 2) rolling on a steel cylinder (body 1), where R1 = 32 mm, R'1 = ∞, R2 = 32 mm, and R'2 = 20 mm. For steel, E = 200 GPa and v = 0.29.

a. Determine an expression for σmax in terms of P.

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b. Fatigue test results indicate that fatigue failure occurs at approximately N = 109 cycles with σmax = −2758 MPa. Determine the applied load P. Since α = 0 and R1, and R2 lie in the x-z plane (see Figure 17.5), b (the minor semiaxis of the contact eclipse) is in the direction of rolling.

FIGURE 17.5 Toroids in contact.