Energy Method. The energy method, described in detail in Chapter 2, is applied in a similar manner for bridges. When an energy solution is obtained, losses through the bridge opening sections (2, BD, BU, and 3) are computed as if each were an unob-structed cross section. The friction losses between sections and losses due to expan-sion or contraction are computed and summed. If piers are present, the additional lengths on both sides of each pier are included in the wetted perimeter calculation and the area of the piers is removed from the cross-sectional flow area of sections BD and BU.
For bridge analysis with the energy method, expansion and contraction losses domi-nate through the bridge opening (between sections 2 and 3), as they are much larger than the friction losses through this same area. This is a reversal of the situation for normal valley cross sections. This is due to the short reach length between the upstream and downstream bridge face (minimizing the friction loss) and the (usually)
Section 6.2 Low Flow Through Bridges 171
large velocity heads experienced within the bridge opening, which often result in sig-nificant expansion or contraction losses. Water surface elevations and energy losses through a bridge are computed with Equations 2.43 through 2.46 (see page 57), apply-ing the standard-step method as if the bridge sections represented normal valley cross sections.
The energy method is also used to compute losses between sections 1 and 2 and between sections 3 and 4 for all of the other methods of bridge computations. Bridge computations differ only in the way the water surface elevations are computed between sections 2 and 3.
Momentum Method. With the momentum method, momentum balances are com-puted through the four cross sections (2, BD, BU, and 3, respectively) that define the bridge opening. The equations used by the program for a momentum solution are presented in the following paragraphs.
Modified from FHWA (HDS-1)
Figure 6.3 Low flow classification at bridges.
From section 2, just outside the downstream face, to section BD, just inside the down-stream face of the bridge, the momentum equation is written as
(6.1)
where ABD and A2= the active flow areas at the respective cross sections (ft2, m2)
= the obstructed area of the piers (ft2, m2)
Y2 and YBD= the vertical depths from the water surface to the centroid of the cross-sectional area at the indicated sections (ft, m)
βBD and β2= the momentum coefficients at the indicated locations (dimensionless)
Q2 and QBD= the discharges at the indicated sections (ft3/s, m3/s) g = the gravitational constant (32.2 ft/s2, 9.81 m/s2)
Ff = the frictional resistance force acting from section 2 to section BD (lb, N)
Wx= the weight component acting from section BD to section 2 in the direction of flow (lb, N)
The forces Ff and Wx act in opposite directions and, when the distance between sec-tions 2 and BD is limited, these forces are quite small compared to the other terms in the equation. These two terms are often neglected in hand computations, without sig-nificant error.
In HEC-RAS, Ff and Wx can be toggled on or off, together or independently. The default in HEC-RAS is for Ff to be included and Wx not. The Wx term requires an esti-mate of the channel slope, s0, between adjacent sections. Around bridges, s0 can be difficult to accurately determine and the slope may even be adverse (negative). In addition, the section just inside the bridge may have the same elevation as the section just outside the bridge, resulting in the value s0 = 0. Large errors in momentum can result from a poor estimate of the slope term.
The momentum equation, from section BD to section BU, is written as
(6.2)
From section BU to section 3 the momentum equation is
(6.3)
where CD = the drag coefficient used to estimate the drag force on the piers.
ABU and A3 = the active flow areas at the respective cross sections (ft2, m2)
= the obstructed area of the piers (ft2, m2)
Y3 and YBU= the vertical depths from the water surface to the centroid of the cross-sectional area at the indicated sections (ft, m)
βBU and β3 = the momentum coefficients at the indicated locations (dimensionless) Q3 and QBU= the discharges at the indicated sections (ft3/s, m3/s)
Section 6.2 Low Flow Through Bridges 173
Drag forces are caused by the flow splitting around the piers, flowing along the piers, and then creating a downstream pier wake. The drag coefficient represents the effect of pier shape or streamlining. Common drag coefficients for piers are listed in Table 6.1.
The energy and momentum equations can both be used for Class A low flow at bridges with or without piers. If the water surface, or the energy grade line (if selected), exceeds the highest value of the bridge low chord elevation, the momentum solution is no longer valid. HEC-RAS reverts to a pressure flow, pressure/weir flow, or energy/weir flow solution. These combinations are discussed in Section 6.3.
Yarnell Equation. The Yarnell equation (Yarnell, 1934) is another valid approach to examine Class A low flow through bridges with piers. The Yarnell equation is an empirical solution, developed in the 1920s from more than 2600 laboratory model tests. It evaluates the effect of bridge piers on the water surface elevation upstream of the bridge. The equation is most applicable for bridges that have many piers, with the piers causing the majority of the energy losses through the bridge. The Yarnell equa-tion is concerned only with the pier shape, the pier obstructed area, and the velocity of the water. However, this method does not include any effects of the shape of the bridge opening, the shape of the abutments, or the width of the bridge. Yarnell’s experiments were conducted for rectangular and trapezoidal channel shapes, so these shapes are most appropriate for application of the Yarnell method. Figure 6.4 shows a bridge that can be appropriately modeled with the Yarnell method. This bridgeʹs width is about 300 ft (90 m) and there are 15 to 20 trestle bents supporting the road-way.
A railroad trestle is often best modeled with the Yarnell equation, as follows:
(6.4) where H3–2= the drop in the water surface elevation from section 3 (immediately upstream) to section 2 (immediately downstream) of the bridge (ft, m) K = the Yarnell pier shape coefficient (dimensionless)
ω = the ratio of the velocity head to the depth at section 2 (ft/ft, m/m) Table 6.1 Drag coefficients for selected pier shapes.
Pier Shape CD
Circular 1.20
Elongated with semicircular ends 1.33 Elliptical with 2:1 length-to-width ratio 0.60 Elliptical with 4:1 length-to-width ratio 0.32 Elliptical with 8:1 length-to-width ratio 0.29
Square nose 2.00
Triangular nose with 30° angle 1.00 Triangular nose with 60° angle 1.39 Triangular nose with 90° angle 1.60 Triangular nose with 120° angle 1.72
No piers 0.00
H3–2 2K K( +10ω 0.6– ) α 15α( + 4)V22 ---2g
=
α = the obstructed area of the piers divided by the total unobstructed area at section 2 (dimensionless)
V2 = the velocity at section 2 (ft/s, m/s)
Only pier losses (no friction losses) are considered in Equation 6.4. H3–2 is simply added to the downstream water surface elevation of cross section 2 to obtain the water surface elevation at cross section 3, immediately upstream of the bridge. To use the Yarnell method, a K value must be assigned. This coefficient is based on the pier shape, as is the drag coefficient for the momentum method. Table 6.2 lists commonly used values for the Yarnell K.
Energy, momentum, or WSPRO may be more appropriate solutions for bridges for which significant losses are expected from bridge abutments or from the shape of the bridge opening. WSPRO is described in Section 6.8.
Table 6.2 Values of Yarnell K for selected pier shapes.
Pier Shape Yarnell K
Semicircular nose and tail 0.90
Twin-cylinder pier with connecting diaphragm 0.95 Twin-cylinder pier without diaphragm 1.05
90º triangular nose and tail 1.05
Square nose and tail 1.25
10-pile trestle bent 2.50
Figure 6.4 Flood flow passing through a railroad trestle bridge, St. Charles County, Missouri.
Section 6.2 Low Flow Through Bridges 175