Torsion in Shafts

Abstract

Torsion in shafts is found in many industrial applications, especially in drive shafts of vehicles.

This chapter first explains the variation of shear stress in a shaft under the action of a torque. Mathematical expressions have been derived for the shear stress, shear strain, and the angle of twist. Additionally, an expression for shaft diameter in terms of the torque and shear stress has been derived. In industrial practice, an electric motor is used to transmit power through a connected solid shaft. This is why, the relationship between power and the torque is also presented. Torsional flexibility and torsional stiffness have been defined and the formulae to calculate them is also presented. This chapter contains 13 worked examples, 14 formulae, and 10 exercise problems.

Questions and Problems

• 10.1. Draw a sketch of a shaft subjected to torque, showing the angle of twist, shear strain, and the dimensions.

• 10.2. Derive an expression for the diameter of a solid shaft in terms of torque and shear stress.

• 10.3. A torque of 1000 N ⋅ m is acting on a hollow cylindrical shaft with length 1.5 m, outer diameter of 40 mm, and inner diameter of 25 mm. The shaft is made in steel with modulus of rigidity of 80 GPa. Calculate the maximum shear stress produced in the shaft.

• 10.4. A torque of 800 N ⋅ m is acting on a solid cylindrical shaft with diameter 40 mm and length 1.4 m. The shaft is made in steel with modulus of rigidity of 82 GPa. Calculate the: (a) maximum shear stress, (b) angle of twist, and (c) shear strain produced in the shaft.

• 10.5. A 10 kW electric motor shall be used to transmit power through a solid connected shaft. The shaft rotates with 2000 rev/min. The machine design requires the maximum allowable shear stress in the shaft to be 70 MPa. Calculate the torque in the shaft.

• 10.6. By using the data in Problem 10.5, calculate the diameter of the solid cylindrical shaft.

• 10.7. A solid steel drive shaft is to be capable of transmitting 55 hp. at 500 rpm. What should its diameter be if the maximum torsional shear stress is to be kept less than 2/3 of the tensile yield strength?

• 10.8. A solid cylindrical shaft is designed for torque of 1200 N ⋅ m. The shaft is made in steel with modulus of rigidity of 81 GPa. The allowable (maximum) shear stress is 50 MPa, and the allowable angle of twist is 1.1°/m. What should be the diameter of the shaft?

• 10.9. A torque of 800 N ⋅ m is acting on a solid cylindrical shaft with diameter 40 mm and length 1.4 m. The shaft is made in steel with modulus of rigidity of 82 GPa. Calculate the: (a) torsional flexibility and (b) torsional stiffness.

• 10.10. A torque is to act on a solid cylindrical shaft with diameter 50 mm and length 1.6 m. The shaft is made in steel with modulus of rigidity of 83 GPa. The machine design requires the maximum allowable shear stress in the shaft to be 60 MPa. The allowable angle of twist is 2°. Determine the allowable torque in the shaft.