Cuap. XXV.

ON STONE BRIDGES.

1481

as to present a resistance more considerable than the force with which they are attacked.If the rapidity of the river has, on the contrary, undergone a diminution, the bed willincrease from the effect of the successive deposits, until the size of the section becomesequal to what had been taken away, according to the relation marked out by naturefor the stability of rivers.

It results from these principles, that if the width of the river be diminished by worksgoing on in its bed, the rapidity being increased, a shoal or island will be formed, and sucha fall that the bed will be deepened until the increase of the section combined with that ofthe velocity will compensate for the diminution in the width of the river, and the velocitybe again in cquilibrio with the resistance of the bottom. If, on the contrary, the widthof the bed be augmented, the rapidity being diminished, it will have a tendency to till up.

It is very essential in constructing a bridge to pay attention to the velocity with whichthe water will move under the arches, and which must not be allowed so to augment asto force the current to attack the bed of the river and injure the foundations of the piersand abutments ; it must not either be sensibly diminished, because in that case a con-siderable additional length would be required for the bridge, and the deposits which wouldtake place might become dangerous. The nature of the ground of which the bottom of theriver is composed must be considered: if excessively compact and tenacious, and approach-ing to rock, it will not yield sensibly to the action of the water, and whatever then maybe its velocity, there is nothing to fear for the safety of the bridge ; but as the waterscannot acquire a great velocity, an eddy more or less considerable will be formed above thebridge, which, in the case of the mouth being extremely narrow, might produce inundationsin the upper part of the river, and render the navigation difficult. If the bottom becomposed of matter which can easily be acted on by the water, care must be taken toprevent the usual velocity from being sensibly augmented: if the ground yield to theaction with such a facility that there is reason to fear the increase of the currentwill be carried under some arches and hollow out the bottom, whilst it would deposit,at the same time, sand under others; this would be a case for constructing a general frame-work of ground timber, by which the bottom, presenting uniformly the same resistance tothe action of the current, would oblige it to distribute itself equally through the wholewidth of the river; and so long as this framework should subsist, there would be noreason to fear that any of the piers would be injured. From the preceding observationsit is evident, that the velocity with which the water will flow under the bridge should bein all cases determined before-hand, either from the nature of the ground which composesthe bed of the river, or by the height of the eddy formed above bridge, and which dependson the value of the velocity ; and as the quantity of water which the river dischargesis also known, the surface of water-way which the bridge should leave can be easily es-tablished. The question then resolves itself into the following problem : the section of thebed of the river, and the velocity of the water being given, to determine the new velocitywhich the current will take, and the height of the eddy which will be formed, supposingthat the bed is narrowed by the construction of the piers and abutments of a bridge.This problem cannot be very rigorously solved, but weshall, neglecting some circumstances, the effects of whichare not very sensible, and which compensate for them-selves in a great degree, give after Dubuat an approxi-mate solution, which can be usefully applied.

Let us suppose A C, D B to represent the lateralface of a pier, and GE I' the natural inclination or fallof the river before the bridge is constructed ; the cur-rent being confined in the interval E F, the velocity, andconsequently the fall, will be there more considerable ;and the surface of the water, making allowance for theparticular resistances produced by the starlings, will take an inclination represented by theline I F; the surface above bridge will necessarily rise above the point I, represented bythe line II K, which for a small length is horizontal.

Let us call H the area of the section natural to the river;

w, the area of the section after the bridge is constructed, or the area of thesurface of the water-way ;

V, the medium velocity of the river;

t?, the mean velocity of the water under the arches, after the construction ofthe bridge;

I, the fall by metre of the river ;

s, the length of the piers and abutments = A B, or E F;

II, the height of the eddy = EK;

< 7 , the accelerated force of the weight = 9 *809 metres.

We shall have v = — V, since in a river the velocities arc in an inverse ratio to the area

H_

I

K

T

O

li

A

Fig. 2702.