Simple torsion equation

twisting moment, polar section modulus, torsional rigidity, torsional stiffness, solid shaft, hollow shaft

1. Torsion of Shafts

2. Torsion A shaft is said to be in torsion, when equal and opposite torques are applied at the two ends of the shaft. Due to the application of the torques at the two ends, the shaft is subjected to a twisting moment. This causes the shear stresses and shear strains in the material of the shaft. The torque is equal to the product of the force applied (tangentially to the ends of a shaft) and radius of the shaft.

3. Circular & Non-circular shafts When subjected to torsion, every cross-section of circular (solid or hollow) shaft remains plane and undistorted. This is due to axisymmetry of cross section. Cross-sections of noncircular shafts are distorted when subjected to torsion – since no axisymmetry.

4. Simple torsion equation Derivation of shear stress produced in a circular shaft subjected to torsion

5. The shear stress is maximum at the outer surface and shear stress is zero at the axis of the shaft. R  

6. MAXIMUM TORQUE TRANSMITTED BY A CIRCULAR SOLID SHAFT

7.       Simple Torsion Equation

8. POLAR MODULUS Polar modulus is defined as the ratio of the polar moment of inertia to the radius of the shaft. It is also called torsional section modulus . It is denoted by Z p Z p =  

9. Polar section modulus for a solid shaft Polar section modulus for a hollow shaft

10. STRENGTH OF A SHAFT AND TORSIONAL RIGIDITY The strength of a shaft means the maximum torque or maximum power the shaft can transmit. Torsional rigidity or stiffness of the shaft is defined as the product of modulus of Rigidity (G) and polar moment of inertia of the shaft(J). Torsional Rigidity = G. J Torsional rigidity is also defined as the torque required to produce a twist of one radian per unit length of the shaft.

11. T orsional stiffness The resistance offered by the loaded member to torsional deflection or twisting is referred to as torsional stiffness.

12. Q2. A solid shaft of 150 mm diameter is used to transmit torque. Find the maximum torque transmitted by the shaft if the maximum shear stress induced to the shaft is 45N/mm 2  

13. Q2. The shearing stress in a solid shaft is not to exceed 40 N/mm 2 when the torque transmitted is 20000 N-m. Determine the minimum diameter of the shaft.

14. Q3. In a hollow circular shaft of outer and inner diameters of 20 cm and 10 cm respectively, the shear stress is not to exceed 40 N/mm 2 . Find the maximum torque which the shaft can safely transmit.

15. Assumptions in simple torsion equation: 1.The material of the shaft is uniform throughout. 2. The twist along the shaft is uniform. 3. The shaft is of uniform circular section throughout. 4. Cross-sections of the shaft, which are plane before twist remain plane after twist. 5. All radii which are straight before twist remain straight after twist.