Civil Engineering Reference
In-Depth Information
was derived from the results:
Q
K
N
2
gH
1
+ β
A
=
by
=
,
(3.9)
V
u
where
b
is the effective width at the obstruction and is equal to
b
1
+
b
2
+
b
3
+
b
4
V
d
/
2
g
,
where
y
d
is the depth downstream of the pier or obstruction,
V
d
is the average veloc-
ity of the downstream channel for discharge,
Q
;
φ
is the adjustment factor (it has
been evaluated from experiments that
is generally about 0.3);
K
N
is the coeffi-
cient of discharge, which depends on the geometry of pier or obstruction and the
bridge opening ratio,
N
φ
is the correction
for upstream velocity,
V
u
, head (from experiments and depends on bridge opening
ratio,
N
) (for
N <
0.6,
=
b
/
B
;
H
1
is the downstream afflux;
β
2).
However, for subcritical flow, the Federal HighwayAdministration (FHWA, 1990)
recommendstheuseofenergyequation(Schneideretal.,1977)ormomentumbalance
methods (TAC, 2004) when pier drag is a relatively small proportion of the fric-
tion loss. When pier drag forces constitute the predominate friction loss through the
contraction, the momentum balance orYarnell equation methods are applicable. The
momentum balance method yields more accurate results when pier drag becomes
more significant.
The Yarnell equation is based on further experiments (summarized by Yarnell,
1934) with relatively large piers (typical of railway bridge substructures) that were
performed to develop equations for the afflux for use with Equation 3.10 (the
d'Aubuisson equation, which is applicable to subcritical flow conditions only):
β ∼
Q
K
A
2
gH
1
+
A
d
=
by
d
=
,
(3.10)
V
u
where
K
A
is the coefficient fromYarnell's experiments, which depends on the geome-
try of pier or obstruction and the bridge opening ratio,
N
=
b
/
B
, where
B
is the width
of the channel without obstruction.
The afflux depends on whether the flow is subcritical or supercritical (Hamill,
1999). For subcritical flow conditions:
Ky
d
F
d
K
0.6
(
1
N)
4
,
5
F
d
−
H
1
=
+
−
N)
+
15
(
1
−
(3.11)
where
K
isYarnell's pier shape coefficient (between 0.90 and 1.25 depending on pier
geometry) and
F
d
is the normal depth Froude number and is given by
Q
A
√
gy
d
≤
F
d
=
1.0.
The normal depth,
y
d
, is readily calculated from the usual open channel
hydraulics methods.
For supercritical flow conditions (which will cause downstream hydraulic jump),
theanalysisismorecomplexanddesignchartshavebeenmadetoassistinestablishing
the discharge past obstructions (Yarnell, 1934).
