PLAIN AND CIVIL: EXAMPLE 1.4.3. HOOP STRESS

Monday 16 September 2019

A large pipe called a penstock in hydraulic work is 1.5m in diameter. It is composed of wooden staves bound together by steel hoops, each $300mm^{2}$ in cross-sectional area, and is used to conduct water from a reservoir to a powerhouse. If the maximum tensile stress permitted in the hoops is 130MPa, what is the maximum spacing between hoops under a head of water of 30m? (Use mass density of water = $1000\frac{kg}{m^{3}}$ .

For one hoop, the FBD shows two F for the hoop against the hydrostatic force caused by pressure.

$P=pressure\left ( area \right )=pressure\left ( diameter \right )\left ( length \right )$

And the force of the hoops is computed as:

$\sigma =\frac{Force}{Area}=\frac{F}{A}$

$F=\sigma A$

From the FBD, use statics to attain equilibrium (prevent the staves to explode).

$P=2F$

$pDL=2\left [ \sigma A \right ]$

$L=\frac{2\left [ \sigma A \right ]}{pD}$

where pressure, $p=\rho gh$

$p=1000\frac{kg}{m^{3}}\left ( 9.81\frac{m}{s^{2}} \right )\left ( 30m \right )=294,300Pa$

So, the required length becomes:

$L=\frac{2\left ( 130\frac{N}{mm^{2}}\left ( 300mm^{2} \right ) \right )}{0.2943\frac{N}{mm^{2}}\left ( 1500mm \right )}={\color{Red} \mathbf{176.69mm}}$

PLAIN AND CIVIL

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Monday 16 September 2019

EXAMPLE 1.4.3. HOOP STRESS

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