Problem 72

The pressure on the culvert varies with depth, as shown in the diagram below.

The most convenient way to define the pressure distribution is with polar coordinates:

p = \gamma (h - r \sin\theta)

The vertical component of this pressure is

p_v = p \sin\theta = \gamma h \sin\theta - \gamma r \sin^2\theta

The total vertical force on the culvert (per unit length perpendicular to the paper/screen) is twice the integral of this pressure from base to crown:

2 \int_0^{\pi/2} p_v r\, d\theta = 2 \left[ \gamma rh\int_0^{\pi/2} \sin\theta\, d\theta - \gamma r^2 \int_0^{\pi/2} \sin^2\theta\, d\theta \right]

Working out the integral, we get

2 \gamma\,r (h - \frac{\pi r}{4})

\gamma\,d (h - \frac{\pi d}{8})

which is the answer to part a).

Part b) is just plug and chug:

2.5 \cdot 62.4 \cdot 6 ( 6 - \frac{6 \pi}{8}) = 3410\,\rm{lb/ft}

Problem 73Problem 71

Last modified: January 22, 2009 at 8:32 PM.

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Problem 72