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6. Torsion

• If we consider a cylindrical shaft with torques applied it will tend to rotate an angle proportional to the torque.

• We can find the torque exerted using simple stress equations for a plane cutting through the cylinder,

• We can describe the twist, or angle of deflection, using two angles.

Problem 6.1 What is the maximum shear stress in the bar? If the bar is made of cold-rolled yellow brass, what is the angle of deflection along the bar?

6.1 The Relationship Between Stress and Strain in Torsion
• Although we are applying a torque (moment) it creates stresses and strains in the material.

• Consider the surface of a rod in torsion.

• Consider the example problem below. [Gere and Timoshenko, 3.7-4 Pg. 256]

Problem 6.2 If the bar is made of cold-rolled yellow brass, what are the maximum axial and shear stresses and strains? How do these compare to given values?

#### 6.1.1 Summary

• If we draw a stress element we can derive the relationship between the normal and shear stresses.

• The basic torsion relationships are given below.

6.2 Problems
Problem 6.3 We need to design a stepped shaft to fit between a 1” diameter and 1/2” diameter holes. The shaft is to transmit 100W of power at 10 rpm. 3” of the 1” diameter shaft is in torsion, and 2” of the 0.5” diameter shaft is in torsion. If the shaft is made of brass,

a) What is the minimum radius for a fillet between the shaft sections?

b) What is the angle of deflection?

c) What are the maximum stresses and strains?

Problem 6.4 a) Given the pipe below find the torsion about the centerline of member AB (AB lies in the x-z plane). The round members from A to B, and B to C are solid with a diameter of 1/8in. Assume that the material has the properties; E = 50 Mpsi and G = 20Mpsi. (marks 25)

b) If the torsion is found to be 100 lb then what is the angle of twist for AB? (marks 25)

6.3 References
6.1 Soustas-Little, R.W. and Inman, D.J., Engineering Mechanics Statics, Prentice-Hall, 1997.