So that, in all useful cases, it is likely to be nearly counterbalanced by the great rate of contraction we as sume.
In order that the Table should be complete, we must divide it into two parts, one referring to the velocity, and the other to the difference of level of the river's surface, for a space equal to the breadth of the bridge. The depth, too, is a material consideration in discover ing the acquired velocity. But we do not see the im portance of these minutia', for the requisite data are not to be obtained with similar exactness.
As an example of the purposes to which this Table may be applied, let us suppose that a bridge is to be built over a river of 100 feet wide, the usual velocity of which is S feet per second, and, of course, the bed is in all likelihood composed chiefly of round pebbles. Let these pebbles and gravel be supposed to extend to the depth of S feet, and under that a stratum of fine firm clay. Let it be proposed to give the bridge a water-way of 75 feet, that is to say,two abutments projecting 41 feet each, and two piers of 8 feet thick each, a centre arch of 35 feet, and two side arches of 20 feet span each. It is only proposed to lay the foundations two feet below the bed, and to spring the centre arch two feet above the usual giving it a rise of one-third of the span.
Let us enquire whether such a structure is likely to be durable.
From the table it appears, that the obstruction being one-fourth, and the velocity 3 feet, the head will be .2484, or about S inches, and is therefore not likely to encroach on the crown. But the velocity under the bridge will be 5 feet per second, and, of course, would require boulder stones or rock to withstand it ; the gravel bed will therefore he cut up under the bridge, and to a depth which, although not easily predicted, is likely to be that which will make the area of the section of the current, allowing for contraction, as great as where the river is free. For this will restore the original velocity, and prevent farther damage, provided the pebbly stratum holds to that depth ; for should the strata below be hard er or coarser the damage will be less, and if softer the contrary.
Suppose, again, the depth of the river, in its usual tenors, to be S feet at the left, and 4 feet at the right pier. Nothing is more common than such a difference of depth ; and it is to be observed, that, whatever may be the cause of the inequality, the erection of the bridge does little or nothing to remove it. We may therefore suppose the inequality of depth as likely to continue, whatever other changes are produced.
The depth cut in the uniform stratum will not, in deed, be quite so great as this ; for the matter excava ted will be thrown tip as a bar across the river below the bridge, and will add to the depth by heightening the surface of the water.
The left pier, then, which is only founded two feet under the bed, may stand well enough, but the right pier is in manifest danger, being undermined nearly eight inches. It must therefore be laid deeper. It will not be safe, however, in proceeding deeper with the foundation, to expose the smallest part of the clay ; for that will move off with a less velocity of current than the gravel or pebbles, and the pier will be still fur then endangered. Our Table slims us, that it will not bear one-third of the velocity of this stream, and, consequent ly, runs the risk of being excavated to a great depth in deed. The only safety is in the gravel rolling into the hole thus formed, and ultimately stopping it, not, how ever, without leaving the pier in a dangerous situation.
Suppose further, that the river is liable to floods, and that, from observations of its higher marks, it is thought that the channel may be in that case 200 feet wide and 6 feet deep, and the progress of the frcshes about 31 miles per hour. What will be the consequence of such an accident happening after the bridge is built over it ? If we take the depth of the river at 6 feet on an ave rage, the water-way undcr the bridge is only 4, and it is probable that the diminution of depth towards the shores will be made up by a greater depth in the chan nel, suppose 9 feet : This would encroach on the crown, and place the bridge in a still more dangerous predica ment. Yet adhering to the supposition of an obstruc tion of 4, we find, that for a velocity of five feet (3.4 miles,) the head is 3.950, or about 4 feet, and the ac quired velocity 161 feet per second. This will produce an absolute cataract, and will 'sweep out stones, gravel and clay, to such a depth, if continued even for a short time, as will undoubtedly destroy the structure. A pavement, or even an inverted arch, will be an ineffec tual preventative, in a case like this. But that we may sec the result more distinctly, But as there is only 3 feet of pebbles, it passes to the clay ; and as this will not bear more than 1 of the com mon velocity, the river will cut in it until the depth be 60.84, which is far below any security that can be given to the structure, without a total change of the foundation.