CONTRACTION OF THE WATER-WAY.
246. In building a bridge across a stream, we must be careful not to obstruct the water-way so as to prevent free passage to the highest floods. Regard must be had to this in fixing the size of the spans, and the thickness and number of the piers. By contracting the width of the stream the velocity is increased beneath the arches, the same amount of water being obliged to pass through a smaller space, and when the bottom is of such a nature as to yield to this action, there is danger of the foundation being undermined. If the form and size of the piers be so arranged as not to increase the velocity, such danger will be avoided and floods will pass without harm. In bridges crossing navigable streams, if the bottom is not destroyed the velocity may be made so great as to impede navigation.
247. The following table is from Gauthey, Construction des Ponts, showing the velocities which are just in equilibrium with the material composing the bottom of the stream.
| State of the water. | Velocity in feet per second. | Nature of bottom. |
|---|---|---|
| Torrents, | 10′ 0″ | Large rocks. |
| Floods, | 3′ 3″ | Loose rocks. |
| Common, | 3′ 0″ | Gravel and stones. |
| Regular, | 2′ 0″ | Fine gravel. |
| Moderate, | 1′ 0″ | Sand. |
| Slow, | 0′ 6″ | Clay. |
| Very slow, | 0′ 3″ | Common earth. |
248. If b represents the width of the natural water-way; c, that as reduced by the structure; V, the velocity of the stream in the natural state; then the augmented velocity is expressed by
W = mVb
c;
and c = mbV
W;
where m is a constant quantity expressing the contraction which takes place in passing the narrow place, which, according to Du-Buat, is 1.09; but depending somewhat upon the form of the bridge piers; adopting which value, we have
W = 1.09Vb
c;
and c = 1.09bV
W.
Example.—Let the bottom be gravel, the width of the natural water-way one hundred feet, the velocity one foot per second: now for a gravel bottom the velocity must not exceed two feet per second, whence
c = 1.09 × 100 × 1
2 = 54½ feet;
which is the width of the contracted water-way; and 100 – 54½, or 45½ feet may be occupied by piers or other obstructions.
The amount of fall which the water suffers in passing the pier is found by the following formula, the notation being the same,
fall = V2m2b2 – c2
64c2
Thus the velocity being one foot per second, m being 1.09 and b = 100; also, c = 54½, we have
fall = 1 × 1 × 1.09 × 1.09 × 100 × 100 – 54½ × 54½
64 × 54½ × 54½ = 0.047 ft.
The velocity of a river is greatest at its surface and at the centre of the stream. In the same river the velocity is nearly as the square root of the depth; thus the surface velocity being known, that for any other depth may be easily found. The velocity of streams should always be noted at the times of the highest floods. For measuring the velocity of running water a bottle enough filled with water to maintain an upright position, with a small rod placed through the stopper having a red flag upon the upper end, answers very well. Velocities of undercurrents may also be measured by so loading the bottle as to cause it to float two, four, six, or ten feet below the surface.