Drill Shaft Orientation

An engineer just thought of a new idea for a drilling shaft where the shaft is made from two different materials. His boss is not so sure of the idea and demands a stress analysis before any prototype is constructed.

What is known:

• The drill shaft is constructed from two materials, steel and aluminum.
• The drill motor will deliver a torque of 15 kip-ft. It is assumed that there is no friction along the length of the shaft.
• The outside section is made from steel and has an outside diameter of 4 in and an inside diameter of 3.5 in.
• The inside section is made from aluminum and has an outside diameter of 3.5 in and an inside diameter of 3.0 in.
• The two sections are rigidly attached (no slipping between the two materials).
• The aluminum material properties are E_A = 10,000 ksi and ν_A = 0.33.
• The steel material properties are E_S = 29,000 ksi and ν_S = 0.29.

What is the maximum stress in both materials? Does the aluminum significantly lower the stress in the steel?

• Since there is no friction, the torque on the shaft is constant along its length.
• While the total torque or moment is known, it is not known how much of this torque will be carried by each material section. Thus this is a indeterminate problem.
• Compatibility relationship needs to be determined to solve the problem. One possible condition is the angle of twist must be equal in both material sections since they are attached with no slipping. This gives
  
       θ_steel = θ_alum