**Shear Stress Based on Circular Bar Theory**
Shear stress based on circular bar theory is considered exact for circular bars and shafts. It will be interesting to see how well thin-walled theory compares.
The shear stress for a circular bar is given by
τ = Tr_{o}/J
where J is the polar moment of inertia and r_{o} is the outside radius. For this example, the polar moment of inertia is
J = π r_{o}^{4}/2 - π r_{i}^{4}/2 = π [(1+0.2/2)^{4} - (1-0.2/2)^{4}]/2
= 1.269 in^{4}
Thus, the shear stress is
τ = (5 kip-ft)(12 in/ft)(1.1 in) / (1.269 in^{4})
= 52.01 ksi
This is about 9% higher than the value calculated from thin-walled theory. Thus, for thin-walled theory to be accurate, the tube thickness should be less than 10% of the overall dimension. |