kbbeing the conveyance of the gross contracted section with the same
normal depth and roughness characteristics as the upstream approach
section whose conveyance is kB.
For rectangular unflumed sections the conveyance ratio (contraction
ratio, 1 ) becomes b/B,bbeing the clear width of the stream (of
normal width, B) under the bridge (Fig. 10.10).
The bridge loss coefficient is also a function of the geometry of the
bridge, its skew and eccentricity, and the submergence of the superstruc-
ture (i.e. the deck).
V 2 is the velocity just downstream of the piers, using the gross area
under the bridge with the same upstream normal depth, and 1 is the
energy correction coefficient of the approach section. Lis assumed to be
equal to the bridge length (abutment to abutment), and S 0 is the normal
bedslope of the unobstructed steam.
LONG CONTRACTIONS
In the case where the bridge has a number of large piers and/or long
approach embankments contracting the water width, the backwater effect
is considerable. Referring to the flow profile shown in Fig. 10.10, through
such a long contracting section, ∆yis the afflux entirely created by the
presence of piers and channel contraction.
Momentum and continuity equations between sections 1 and 3
(assuming hydrostatic pressure distribution with a negligible bed slope and
frictional resistance) result in
∆y/y 3 {A[A^2 12 CD(b/B)Fr^23 ]1/2}/6 (10.10)
where
A{CD(b/B)2}Fr^23 2 (10.11)
Fr 3 being the Froude number (V 3 /(gy 3 )1/2) at section 3.
Equation (10.10) should give good results if the drag coefficient CD
can be accurately estimated. The pier drag coefficient has been found to
be a function of the velocity gradient of the approach flow, b/B, and the
pier shape; however, owing to the non-availability of reliable drag coeffi-
cient values, the use of equation (10.10) is limited.
Yarnell’s (1934) experimental data on the flow through bridge piers
resulted in the following empirical equation:
∆y/y 3 KFr^23 (K 5 Fr^23 0.6)( 15 ^4 ) (10.12)
where
1 1 b/B (10.13)
andKis a function of the pier shape according to Table 10.3.
436 CROSS-DRAINAGE AND DROP STRUCTURES
