3-10. A shaft is supported by a smooth bearing at  and has an outer diameter  mm and a thickness  mm.  The motor operates at an angular velocity  rad/s and delivers kW of power.  What is the probability the shaft will fail if the allowable torsion shear stress is ? Assume  and  are independent.

Solution:

Torque:

 N·m                                                                                                   (1)

N·m                                                                                                       (2)

 N·mm due to unit balance.

Torsion shear stress: The polar moment of inertia is  mm⁴.

 MPa                                                                                                     (3)

 MPa                                                                                                       (4)

MPa      

Probability of failure:

                                                                                       (5)

Since  kW, and  MPa are independent,  also follows a normal distribution of .

MPa                                                                                  (6)

MPa                                                                             (7)

Equation (5) can be written as

Ans.