Drilling Pipe Angle of Twist

The drilling shaft is constructed from two different materials, steel and aluminum. Each material section is 0.25 inches thick with the steel on the outside. The applied torque is 15 kip-ft (equal to 15,000 lb-ft) and is constant along the 100 foot pipe length.

As the diagram at the left shows, the angle of twist, θ, will be the same for both materials at any given location along the pipe. This condition is critical in solving for the stresses.

Before developing the basic equations, there are several constants that are needed. The first is the shear stiffness, G. Recall, G can be calculated from E and ν, giving

The polar moment of inertia call also be calculated for each material section.

Shaft Cross Section

The drilling shaft maintains a torque or moment of 15 kip-ft but the amount of the torque in each of the two sections is not known. Summing the moments around the pipe axis gives

     T_A + T_S = T_total = 15 kip-ft

Thus, there are two unknowns but only one equation from static equilibrium. A second equation can be developed from the angle of twist compatibility relationship. Each section must twist the same, or

      θ_steel = θ_alum

Recall, the angle of twist for any circular shaft is

Using this in the compatibility equation gives

Rearranging and substituting values gives

     T_S = 4.586 T_A

This indicates the steel section takes almost 80% of the torque. Thus, the aluminum will not help much. This result can be substituted back into the moment equilibrium equation, giving

     T_A + 4.586 T_A = 15,000 lb-ft

     T_A = 2,685 lb-ft

     T_S = 12,310 lb-ft

The stresses in each of the two sections can now be determined using the torsion stress equation.

The maximum stress occurs at the outer edge of each section

     t_S-max = 14.2 (2) = 28.40 ksi

     t_A-max = 4.752 (1.75) = 8.316 ksi

To better understand the stress distribution, the diagram at the left shows the shear stress as a function of the radius. Notice, there is a discontinuity at the interface between the material sections. The angle of twist and strain is equal at the interface, but the stresses are not.