## Hoop Stress

If a thin-walled cylinder pressure vessel in the figure below has an inner radius of 0.25”, what is the tangential (hoop) stress?

##
__
__**Hint**

**Hint**

Tangential (hoop) stress of a thin-walled cylinder:

$$$\sigma_t=\frac{P_ir}{t} $$$

where
$$P_i$$
is the internal pressure,
$$r$$
is the average radius, and
$$t$$
is the wall thickness.

##
__
__**Hint 2**

**Hint 2**

$$$r=\frac{r_i+r_o}{2}$$$

where
$$r_i$$
is the inner radius, and
$$r_o$$
is the outer radius.

Remember, a cylinder can be considered thin-walled if the wall thickness is about 1/10 or less of the inside radius. If so, the internal pressure is resisted by both the axial and hoop stress. Tangential (hoop) stress of a thin-walled cylinder:

$$$\sigma_t=\frac{P_ir}{t} $$$

where
$$P_i$$
is the internal pressure,
$$r$$
is the average radius, and
$$t$$
is the wall thickness. To find
$$r$$
:

$$$r=\frac{r_i+r_o}{2}$$$

where
$$r_i$$
is the inner radius, and
$$r_o$$
is the outer radius. Thus,

$$$r=\frac{0.25inch+(0.25inch+0.03inch)}{2}=\frac{0.53inch}{2}=0.265\:inch$$$

Therefore,

$$$\sigma_t=\frac{500\frac{lb}{in^2}(0.265in)}{0.03in}=\frac{132.5}{0.03}psi=4,417\:psi$$$

4,417 psi