Den Hartog’s Mechanics

A web-based solutions manual for statics and dynamics

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})

or

\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.