## Torsion

In the below diagram, section OA is a solid titanium cylinder that is connected to section AB, which is a hollow steel pipe. What is the max shear stress in section AB?

##
__
__**Hint**

**Hint**

$$$\tau =\frac{Tr}{J}$$$

where
$$\tau$$
is torsion,
$$T$$
is the torque at the section of interest,
$$r$$
is the radius to the point of interest, and
$$J$$
is the section's polar moment of inertia

##
__
__**Hint 2**

**Hint 2**

Because we are interested in the polar moment of inertia of a hollow cylinder shaft:

$$$J_{hollow}=\frac{\pi (D_{outer}^{4}-D_{inner}^{4})}{32}$$$

where
$$D$$
is the diameter.

Torsion stress in circular solid or thick-walled (t<0.1r) shafts:

$$$\tau =\frac{Tr}{J}$$$

where
$$\tau$$
is torsion,
$$T$$
is the torque at the section of interest,
$$r$$
is the radius to the point of interest, and
$$J$$
is the section's polar moment of inertia

Because we are interested in the polar moment of inertia of a hollow cylinder shaft:

$$$J_{hollow}=\frac{\pi (D_{outer}^{4}-D_{inner}^{4})}{32}$$$

where
$$D$$
is the diameter.

$$$J=\frac{\pi ((0.75m)^{4}-(0.5m)^{4})}{32}=\frac{\pi(0.3164-0.0625)m^4}{32}=0.025\:m^4$$$

Thus,

$$$\tau=\frac{10N\cdot m(\frac{0.75}{2}m)}{0.025\:m^4}=150\frac{N}{m^2}=150\:Pa$$$

150 Pa