Section 4: Hydraulics of Bridge Openings
Anchor: #i1007580Bridge Modeling Philosophy
Numerous methods exist for estimating the hydraulic impact of bridge openings on water surface profiles. TxDOT recommends that computer programs be employed to perform such estimates. Generally, the documentation of the specific computer program should be referred to for the theory employed.
Note. Previously, TxDOT employed a single energy loss equation, (h = v^{2}/2g), to estimate the backwater effect of bridge openings. It is no longer used as the basis for design of TxDOT bridges.
Anchor: #i1124978Bridge Alignment
The discussions of bridge design assume normal lengths and alignments perpendicular to the flow at flood stage even though the low flow streambed may be at a skew to the bridge. In actuality, many bridges are skewed to some degree which causes the hydraulic length of the bridge to be longer than the plan width. For example, a 60foot wide bridge perpendicular to the stream has a hydraulic length of 60 feet. The same bridge at a 30º skew to the stream has a hydraulic length of 69.3 feet.
Hydraulic programs do not automatically account for the skew. The program operator is required to specifically account for the skew by putting in the correct hydraulic length.
Anchor: #i1007597Flow Zones and Energy Losses
Figure 910 shows a plan of typical cross section locations that establish three flow zones that should be considered when estimating the effects of bridge openings.
NOTE: These cross sections are related to the explanation of flow zones and energy losses and must not be confused with the cross sections required for analysis. The analysis cross sections may be further upstream and downstream.
Figure 910. Flow Zones at Bridge
Zone 1 – Downstream. Zone 1 represents the area between the downstream face of the bridge and a cross section downstream of the bridge within which expansion of flow from the bridge is expected to occur. The distance over which this expansion occurs can vary depending on the flow rate and the floodplain characteristics. No detailed guidance is available, but a distance equal to about four times the length of the average embankment constriction is reasonable for most situations. Section 1 represents the effective channel flow geometry at the end of the expansion zone, which is also called the “exit” section. Cross sections 2 and 3 are at the toe of roadway embankment and represent the portion of unconstricted channel geometry that approximates the effective flow areas near the bridge opening as shown in Figure 911.
Figure 911. Effective Geometry for Bridge (Section 2 shown, Section 3 similar)
Zone 2  Under Bridge Opening. Zone 2 represents the area under the bridge opening through which friction, turbulence, and drag losses are considered. Generally, consider the bridge opening by superimposing the bridge geometry on cross sections 2 and 3.
Zone 3 – Upstream. Zone 3 represents an area from the upstream face of the bridge to a distance upstream where contraction of flow must occur. A distance upstream of the bridge equal to the length of the average embankment constriction is a reasonable approximation of the location at which contraction begins. Cross section 4 represents the effective channel flow geometry where contraction begins. This is sometimes referred to as the “approach” cross section.
Anchor: #i1007641Extent of Impact Determination
The maximum effect of the bridge should occur at cross section 4. However, in order to determine the extent of the impact, continue water surface profile computations upstream until the water surface does not differ significantly from the estimated preconstruction conditions.
Anchor: #i1007653Water Surface Profile Calculations
Calculate the water surface profile through Zones 1 and 3 using the Standard Step Backwater Method (see Chapter 7) with consideration of expansion and contraction losses. The table 91 below provides recommended loss coefficients.
Transition Type 
Contraction (K_{c}) 
Expansion (K_{e}) 

No losses computed 
0.0 
0.0 
Gradual transition 
0.1 
0.3 
Typical bridge 
0.3 
0.5 
Severe transition 
0.6 
0.8 
Anchor: #i1007664
Bridge Flow Class
The losses associated with flow through bridges depend on the hydraulic conditions of low or high flow.
Low flow describes hydraulic conditions in which the water surface between Zones 1, 2, and 3 is open to atmospheric pressure. That means the water surface does not impinge upon the superstructure. (This condition should exist for the design frequency of all new onsystem bridges.) Low flow is divided into categories as described in Table 92 “Low Flow Classes”. Type I is the most common in Texas, although severe constrictions compared to the flow conditions could result in Types IIA and IIB. Type III is likely to be limited to steep hills and mountainous regions.
Type Designation 
Description 

I 
Subcritical flow through Zones 
IIA 
Subcritical flow Zones 1 and 3, flow through critical depth Zone 2 
IIB 
Subcritical Zone 3, flow through critical Zone 2, hydraulic jump Zone 1 
III 
Supercritical flow through Zones 1, 2 and 3 
High flow refers to conditions in which the water surface impinges on the bridge superstructure:
 Anchor: #RQBSHTYJ
 When the tailwater does not submerge the lowchord of the bridge, the flow condition is comparable to a pressure flow sluice gate. Anchor: #RAISGSWU
 At the tailwater, which submerges the lowchord but does not exceed the elevation of critical depth over the road, the flow condition is comparable to orifice flow. Anchor: #GEQCBXVL
 If the tailwater overtops the roadway, neither sluice gate flow nor orifice flow is reasonable, and the flow is either weir flow or open flow.
Zone 2 Loss Methods
Generally determine the losses in Zone 2 by one of the following methods depending on the flow characteristics and the engineer's judgment:
 Anchor: #BWNVMEAN
 Standard Step Backwater Method (based on balance of energy principle) Anchor: #KXWFLFHG
 Momentum Balance Method Anchor: #UMTHHILS
 WSPRO Contraction Loss Method [This is a method, not limited to the WSPRO program.] Anchor: #EGIPKEXK
 Pressure Flow Method Anchor: #AYGHSFWV
 Empirical Energy Loss Method (HDS 1).
Standard Step Backwater Method (used for Energy Balance Method computations)
Refer to Chapter 7 for the Standard Step Backwater Method. Figure 912 shows the relative location of section geometry for profile computations. B_{d} and B_{u} refer to the bridge geometry at the downstream and upstream inside faces, respectively.
Figure 912. Relative Location of Section Geometry
 Anchor: #FYWCQOCQ
 Solve the energy equation (step backwater) between cross section 2 and the downstream bridge face (B_{d}). Use the water surface at cross section 2 determined from the previous backwater profile computations. Anchor: #CIAKASXP
 Proceed with the standard step backwater calculations from the downstream bridge face to the upstream face. Use the bridge geometry superimposed on cross sections 2 and 3 respectively. Anchor: #FDWDVSEE
 Approximate the effects of piers and impingement of flow on the lowchord by reducing the section area and increasing the wetted perimeter accordingly. Anchor: #OHRPBLWM
 Similarly, consider roadway overflow as open channel flow. Proceed with calculations from the upstream bridge face (B_{u}) to cross section 3. Anchor: #JFFMFJCM
 As indicated in the previous Flow Zones and Energy Losses subsection, proceed with calculating the remainder of the bridge impact from cross section 3 upstream using step backwater calculations.
Under the right circumstances, you can consider the energy balance method for low flow and high flow.
Anchor: #LNFJJNIJMomentum Balance Method
This method computes the backwater through Zone 2 by balancing forces at three locations:
 Anchor: #JUJCJDAO
 between the inside, downstream face of the bridge (B_{d}) and cross section 2 Anchor: #BREVIOAJ
 between the downstream and upstream ends of the bridge (B_{d} to B_{u}) Anchor: #SJDDVFFW
 between the inside, upstream face of the bridge (B_{u}) and cross section 3.
Refer to Figure 910 and Figure 912 for zone and cross section locations. Assuming hydrostatic pressure conditions, the forces acting on a control volume between two cross sections (1 and 2) must be in balance and are generalized in Equation 92.
Equation 92.
where:
 Anchor: #JTKKDRNA
 F_{P1}, F_{P2 }= force due to hydrostatic pressure at cross section = Ay Anchor: #DNUDRIIP
 F_{m }= force causing change in momentum between cross sections = Qv Anchor: #RWBFCPFP
 F_{f }= force due to friction = (A_{1}+A_{2})LS_{f}/2 Anchor: #QNBAIEPL
 F_{d }= total drag force due to obstructions (e.g., for piers = C_{d}A_{o}v/2) Anchor: #KFXATUMP
 F_{w }= component of weight in direction of flow = (A_{1}+A_{2})LS_{o}/2.
 Anchor: #THXKWOKM
 For subcritical flow, determine the water surface elevation and average velocity at section 2 from step backwater computations. Anchor: #MCAYAPJJ
 Determine the water surface elevation and average velocity at Section B_{d} by applying successive assumed water surface elevations to Equation 93 until equality is achieved within a reasonable tolerance. Anchor: #HPEAFDVQ
 Determine the momentum correction factor (B), which accommodates natural velocity distributions similar to the energy correction factor, , using Equation 94. Anchor: #WLNEPRSE
 Using the resulting water surface elevation at B_{d}, determine the water surface elevation and average velocity at Section B_{u} by applying successive assumed water surface elevations at Section B_{u} to Equation 95 until achieving equality within a reasonable tolerance. B_{u} refers to the upstream face of the bridge. Anchor: #EMSXVDYW
 Determine the final momentum balance between the upstream face of the bridge and cross section 3 using Equation 96. Table 93 “ Suggested Drag Coefficients for Bridge Piers” presents suggested drag coefficients for different pier types. Anchor: #UFMACGEB
 As discussed in the above Flow Zones and Energy Losses section, proceed with the remainder of the bridge impact computations from cross section 3 upstream using step backwater calculations.
Equation 93.
where:
 Anchor: #YAOGIXJW
 Subscripts 2 and Bd refer to section 2 and the downstream bridge face, respectively. Anchor: #ONQAFFOA
 A = effective flow area at cross sections (sq.ft. or m^{2}) Anchor: #MRWAXJUU
 = height from water surface to centroid of effective flow area (ft. or m) Anchor: #RTYJOISI
 g = acceleration due to gravity (ft./s^{2} or m/s^{2}) Anchor: #GWTXYMVC
 Q = discharge (cfs or m^{3}/s) Anchor: #AQQDEUQO
 A_{pd }= obstructed area of pier at downstream side (sq. ft. or m^{2}) Anchor: #LSNOFWIT
 L = distance between cross sections (ft. or m) Anchor: #AJRYCEFX
 S_{f }= friction slope (ft./ft. or m/m) (see Chapter 6) Anchor: #IUXNDSPG
 S_{o }= channel bed slope (ft./ft. or m/m) Anchor: #RHFXVMXY
 β = momentum correction factor. Anchor: #XANPSRMV

Equation 94.
where:
 Anchor: #OBQLYGPY
 Ki_{ }= conveyance in subsection (cfs or m^{3}/s) Anchor: #VPOEESBS
 Ai= area of subsection (sq. ft. or m^{2}) Anchor: #MXMNVPXA
 K_{T }= total conveyance of effective area section (cfs or m^{3}/s) Anchor: #JCFNMKFJ
 A_{T }= total effective area (sq.ft. or m^{2}).
Equation 95.
Equation 96.
where:
 Anchor: #DSRXXPWQ
 Subscript 3 refers to cross section 3 Anchor: #ALFUWOJR
 A_{pu }= Obstructed area of piers at upstream side (sq.ft. or m^{2}) Anchor: #MRTTLXCK
 C_{d }= drag coefficient
Pier Type 
Drag Coefficient, C_{d} 

Circular 
1.20 
Elongated with semicircular ends 
1.33 
Elliptical (2:1 aspect ratio) 
0.60 
Elliptical (4:1 aspect ratio) 
0.32 
Elliptical (8:1 aspect ratio) 
0.29 
Square nose 
2.00 
Triangular nose (30^{o} apex) 
1.00 
Triangular nose (60^{o }apex) 
1.39 
Triangular nose (90^{o }apex) 
1.60 
Triangular nose (120^{o} apex) 
1.72 
WSPRO Contraction Loss Method
The Water Surface Profile (WSPRO) method is a contraction model that uses step backwater calculations and empirical loss coefficients. This method is not limited to the WSPRO program.
 Anchor: #IUSISWJO
 Base the model on providing approach and exit cross sections (cross sections 1 and 4) at distances from the downstream and upstream faces approximately equal to the bridge opening length. Anchor: #TGXMNWBR
 Compute the flow in Zones 1 and 3 using step backwater computations with a weighted flow length based on 20 equal conveyance tubes. Refer to Bridge Waterways Analysis Model (Shearman et al., 1986) for details on this method. (See Reference for details on obtaining this document.)
Pressure Flow Method
By definition, pressure flow methods represent high flow conditions. Figure 913 shows a high flow condition in which the water surface at the upstream face of the bridge has impinged the lowchord but the downstream face is not submerged. You may approximate this condition as a sluice gate using Equation 97. You need to assume successive elevations at cross section 3 (y_{3}) until the calculated discharge in Equation 97 is equal to the design discharge within a reasonable tolerance.
Equation 97.
where:
 Anchor: #WTHGCCIO
 Q = calculated discharge (cfs or m^{3}/s) Anchor: #NYQSLPKO
 C = discharge coefficient (0.5 suggested) Anchor: #UMUJOYFW
 A_{b }= net area under bridge (sq. ft. or m^{2}) Anchor: #TBYJAAJC
 y_{3 }= depth of flow at cross section 3 (ft. or m) Anchor: #QQGTULWG
 D_{b }= height of lowchord from mean stream bed elevation (ft. or m).
Figure 913. Sluice Gate Type Pressure Flow
Figure 914 shows a submerged bridge opening with a tailwater lower than the overtopping elevation Equation 98 represents orifice flow. You need to assume successive elevations at cross section 3 (y_{3}) until the calculated discharge in Equation 98 is equal to the design discharge within a reasonable tolerance.
Equation 98.
where:
 Anchor: #DMRMRJPA
 C = discharge coefficient (0.8 typical) Anchor: #YYCSINHS
 H = difference between energy grade at cross section 3 and water surface at cross section 2 (ft. or m), Equation 99.
Equation 99.
where:
 Anchor: #DTFQKNET
 α_{3} = kinetic energy correction coefficient Anchor: #BHLOFDIQ
 C_{d }= coefficient of discharge, Equation 912.
Equation 910.
where:
 Anchor: #JBEVRVLV
 b = width of top of embankment at bridge abutment (ft. or m) (see Figure 915) Anchor: #NSIRMQBH
 L_{c }= length of bridge opening between abutment faces (ft. or m).
Figure 914. Orifice Type Pressure Flow
Figure 915. Bridge Dimensions for Pressure Flow Analysis
Empirical Energy Loss Method (HDS 1)
Although hand computations are used rarely, the FHWA publication Hydraulics of Bridge Waterways ( HDS 1) presents methods for estimating bridge backwater effects. The HDS1 Empirical Loss Method presents a summary of the method that is appropriate for low flow Type I. (This flow type should predominate for the design of bridges over Texas streams.)
Anchor: #i1008257Twodimensional Techniques
Twodimensional (2D) horizontal flow, depthaveraged techniques are highly specialized. Contact the Design Division’s Hydraulics Branch for consultation.
Anchor: #i1008267Roadway/Bridge Overflow Calculations
Consider flow over the bridge or roadway in one of two ways:
 Anchor: #JVSDLQLQ
 Weir flow if the tailwater does not drown out critical depth of flow in the overtopping section  the approach is similar to that in the Roadway Overtopping subsection of Chapter 8, except that the design engineer must use the bridge loss methods above instead of culvert head loss computations. That is, apportion flow between the bridge and the weir such that the head at cross section 3 results in a flow apportionment that sums to equal the design flow within a reasonable tolerance. Anchor: #SNJUDNUH
 Open channel flow if the tailwater is too high  As the depth of flow over the road increases and the tailwater submerges the road, the design engineer considers the flow over the road as open channel flow and use step backwater computations across the road.
Backwater Calculations for Parallel Bridges
The backwater calculation for parallel bridges (depicted in Figure 916) requires the application of a coefficient. The chart in Figure 917 relates the value of the backwater adjustment coefficient (μ) to the ratio of the outtoout dimension of the parallel bridges to the width of a single embankment (see Figure 918). Determine the backwater head calculation for a parallel bridge with Equation 911.
Equation 911.
where:
 Anchor: #BPPDHELV
 h = total backwater head (ft. or m) Anchor: #BRGLRGCD
 μ = backwater adjustment coefficient (see Figure 917) Anchor: #OCIKQWQN
 h_{1 }= backwater head for one bridge as discussed in the Bridge Flow Class subsection above.
Figure 916. Parallel Bridges
Figure 917. Parallel Bridges Backwater Adjustment
Figure 918. Definition of Parameters