THEORY AND PRACTICE OF ENGINEERING.
1482
Book II.
of the corresponding sections, and the heights due to the rapidities V and v will berepresented by
v 8 .n*v*
2 g an
The part I K of the height of the eddy correspondingwill be then
IK =
yvn’
')•
to the increase of the rapidity
and as, on account of the contraction which the current will undergo in general at thepassage of the bridge, the section o> is diminished, we must substitute in this expressionthe area w by m a>, designating by m a coefficient, the value of which depends principallyon the width of the piers, as well as the form of the starlings and springing of the arches."\Ve shall have then
EK = V(-?- 8 -l).
2g\7n 2 co* )
"We must now seek the value of the fall, which will form itself on the length of the piers :before the construction of the bridge this fall was equal to si; and as the falls increasenearly in the relation of the heights due to the correspondent velocities, we shall have forthe value of the new fall,
n 8
Tims the part of the height of the eddy which arises from the increase of the fall underthe arches of the bridge will be represented by
EI =
and we shall have
EI + IK = H=(^ + S l)(Jl i - 1 )
for the height to which the waters of the river rise above bridge, since their level mustremain the same below bridge. To apply this result it will be sufficient to put in the
place of V the expression v and to give afterwards to r the value which will have
been fixod before, according to the principles referred to, for the velocity which the waterstake under the bridge. As to the value of the coefficient m, we are unable to fix it veryexactly ; the experiments relative to this subject have been made on orifices pierced inwalls of different thicknesses, to which pipes have sometimes been adapted, and throughwhich the water flowed under weights more or less considerable ; independently of thecircumstance that the dimensions of these orifices, even in experiments made on the greatestscale, are not to be compared with those of the arches of bridges, it must be rememberedthat the flowing is different from that now under consideration.
The form given to the starlings of the piers of bridges has a sensible influence, as wellas the thickness of the piers, on the manner in which the contraction operates, bywhich the natural flow of the river is modified on meeting the bridge. The value of thecoefficient m, as we have seen, depends principally on the form of the starlings andthe springing of the arches, when they are plunged in the water. We have not fixed withexactness that which it would take in different cases; but we should not be far fromthe truth in supposing wi = 0*95, when the piers are terminated in a half circle, of by acuteangles; m = 0’90, when terminated by obtuse angles; ?n = *85, when terminated square,supposing the arches to be large: in the most disadvantageous cases, that is to say, forsmall arches, and when their springing is under water, the value of the coefficient mmay be about 0*7. In putting in the preceding equation for H and t? the values whichwe have given, according to the nature of the bed and other local circumstances, we shalleasily deduce from it that of the relations of the two sections w and fl.
To determine the height of the eddy which may be formed above bridge, we mustconsider the changes it will occasion in the height of the river, and the inundations winchmay hence result on the banks. According to Dubuat, it is generally admitted thatthe surface of the waters, which, before the construction of the bridge and in the hypo-thesis of a uniform fall, was an inclined plane, becomes after the construction a concavesurface, touching the primitive surface at the point where the exhaustion of the waterceases. This author has also given rules for the calculation of the eddy, on the suppo-sition of the profile of this surface being the arc of a circle of great amplitude ; butmuch confidence cannot be placed in them. When the bed of a river is irregular,and presents considerable variations in the fall and in the width,-—when, in the case of an