1508

THEORY AND PRACTICE OF ENGINEERING.

Book IT.

altered, it is impossible that any damage can take place, and this would be true ifthe mean velocity were distributed uniformly throughout the mass of the waters to whichthe bridge gives passage. But the obstacles which the piers and the springing of thearches oppose to the current injure that equal partition of the velocity, and whilst veryrapid currents are formed, in other places the water turns and returns on itself; it isthen important to give to piers the form the most proper to prevent these effects, the con-sequences of which are often dangerous.

In the piers of ancient bridges the starlings are generally half a circle, or a rectilinealtriangle, but modern engineers have rounded the angles formed by the faces of the star-lings and the body of the pier. There are some bridges built in the middle ages wherethe starlings have been entirely suppressed, and the piers terminated parallelly at the heads ;this latter arrangement is most effective: when the current strikes against the stallingof a, pier, the water rises to meet it, the dash which tends to shake the starling is obligedto divide, and produces two oblique currents, which turn aside from the lateral faces ofthe pier, against which there only remains a dead water; these two currents narrow theprincipal run, by forming with those produced by the neighbouring piers little shoals,where the water flows less quick towards the edges than in the centre, because it has lostsome of its rapidity against the obstacle which it meets with, and although the rapidityof the current be generally greater in the middle of the arch than at the shoulder ofthe pier, that part is not less exposed to it, on account of the falls and eddies which areformed there, and which attack the ground and cause the wearing away : all theseeffects, which are produced whatever be the form of the pier, are more evident as the facesstruck by the current are more directly opposed to it. When water is introduced into a,canal narrower than that where it first ran, the rapidity must augment by the effect ofthe diminution of the section, if the entrance into the narrow canal be not widened; and

if the section diminish suddenly there is necessarily a contraction, which obliges thecurrent to take a velocity greater than the diminution of the section requires: it is thenimportant to render this contraction as small as possible, and this is the more essential asby it may be avoided the fall which it occasions to the shoulder of the pier, and which isthe cause of the undermining: with this view we have endeavoured to compare the effectsof the contraction for piers of different forms, in the experiments of which we are aboutto give an account, after having referred to some researches made on the subject.

We find in “ Recherches sur la Construction la plus avantageuse des Digues,” by MM.Bossut and Viallet, that problem resolved which has for its object to determine theproper form for the head of a jetty, and the starling of a pier subject to the sameconditions ; it is equally necessary to defend these structures against the action of thefluid which continually strikes against them, and it appears that the form proper to the oneis equally so to the other. M. Bossut supposes that each face of the head of the jetty isstruck by threads of the fluid (small currents), the directions of which are parallel, andthe rapidity equal ; he then seeks, according to a general theory of the shock of fluids,the form for the base of each face, in order that this shock may be the least possible; hefinds that the right line resolves the problem, and that if the base be a triangle, it shouldbe an isosceles, the two equal angles of which would be 45°. This solution is subjectto many difficulties, as it appears unnecessary that the head of a jetty or the starling ofa pier should receive any shock from the current, because there is no fear that it woulddisplace the mass entirely: this must be taken into consideration if the subject were theprow of a vessel, but when the facing of the head of a jetty is well constructed, unlessthe water undermines it, it never can be carried away, and it is only essential to guardagainst this latter effect.

M. Garipuy, in a manuscript preserved in the Ecole des Ponts et Chaussees, has

endeavoured to modify the triangular form generally given to starlings, so as to obtain

from the fluid an equal pressure over the whole length of their faces; as the fluidaccelerates its movement a little, going round the length of the starling, it followsfrom the ordinary theory on the resistance of fluids, that each face should be slightlyconvex; the form which is obtained in this manner differs very little from the triangle,

and it is not under that point of view we must consider the question. We will pass

over similar theories, to arrive at that which M. Dubuat has given in his Principles ofHydraulics: he thought that the essential object was to prevent the contraction and theundermining which is its consequence, and which always manifests itself when a currentundergoes a sudden change in its section.

Call V the primitive mean rapidity of the current, v the mean rapidity resultingfrom the diminution of the section, A and a the corresponding widths of the bed, I theinclination by metre of the river, if no contraction be formed, we shall evidently have

A .V 2

= — V. The height due to the rapidity V is —— ; and if the water had acquired this

rapidity in flowing over an inclined plane, where it had not me't with any obstacle, it