 23 Oct 2022
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US BPR Bridge
 Updated on 23 Oct 2022
 11 Minutes to read

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The USBPR Bridge computes the afflux at bridges using the methodology developed by the US Bureau of Public Roads (USBPR).
Data
US BPR Bridge Data
Field in Data Entry Form  Description  Name in Datafile 

Upstream Label  Upstream node name  Label1 
Downstream Label  Downstream node name  Label2 
Upstream River Section  label of upstream RIVER section. This label is not required if the upstream RIVER section is Label1  Label3 
Downstream River Section  label of downstream RIVER section. This label is not required if the downstream RIVER section is Label2  Label4 
Calibration Coefficient  global calibration coefficient used to scale the calculated afflux if this is justified by observations. It should normally be set to 1. Setting cali to 0 removes the effect of the bridge (but not the effect of any flood culverts)  cali 
Skew Angle  skew angle of bridge (angle, in degrees, between the flow direction and the normal to the main axis of the bridge  usually set to zero)  skewb 
Bridge Width  width of bridge (ie distance between upstream and downstream faces of bridge) (m)  only used for modelling dual bridges  rdlen 
Dual Distance  distance from the upstream face of the upstream bridge to the downstream face of the downstream bridge (m)  only used for modelling dual bridges  duall 
Total Pier Width  total width of piers normal to flow direction (m)  pierw 
Abutment Type  abutment type identifier. Use a value of 3 except if the span of the bridge between the abutments is less than 60m and there is either a 90 degree wing or vertical wall abutment (iabut=1) or a 30 degree wing wall abutment (iabut=2)  iabut 
Number of Piers  number of bridge piers in line of flow at a typical pier location. Values of 3 and above give the same results  npier 
Soffit Shape & Pier Shape  if no piers present then enter shape of bridge soffit as either FLAT (if rectangular) or ARCH (if not rectangular). If piers are present then enter their cross sectional shape as either RECTANGLE, CYLINDER, SQUARE or I (for IBEAM piles). Select "Use Calibration number" (shape='COEF') if a pier coefficient is to be used instead of pier shape description. Single Ibeams and square piers are not covered by the theory, and are treated as rectangular. Twin Ibeams and rectangular piers are also not covered, and are treated as twin square piers.  shape 
Pier Faces  second description of bridge pier cross sectional shape (for possible use when there are 1 or 2 piers). Options are:
 diaph 
Calibration Number  calibration number of piers. Used if shape is set to COEF. This is a real number between 0.0 and 8.0 representing the streamlining of the piers. Zero represents no pier resistance, 8 gives the maximum resistance, representing several Ibeam piles. See US BPR report for full details.  prcoef 
Abutment alignment  indicator for abutments being aligned with normal direction of flow, ALIGNED or SKEW  altype 
Model surcharged bridge as orifice  Orifice flow flag  if checked (oflag ="ORIFICE") then switch to orifice flow when surcharging; otherwise, bridge equations are used for all flows.  oflag 
Lower Transition Distance  Lower transition depth (m below soffit). When the upstream water level is below (max soffit level  rlower), the full bridge equations are used. When this level is reached, it enters the transition phase between bridge and orifice flow. Constraint: rupper + rlower ≥ 0.  rlower 
Upper Transition Distance  Upper transition elevation (m above soffit). When the upstream water level exceeds (max soffit level + rupper), the bridge to orifice transition mode is left, and full orifice equations apply. Constraint: rupper + rlower ≥ 0.  rupper 
Orifice Discharge Coefficient  orifice discharge/calibration coefficient  cdorifice 
Channel Section at Bridge Data
Field in Data Entry Form  Description  Name in the Datafile 

 number of ensuing data sets describing the crosssection extending along the toe of any embankment on the left floodplain, across the channel, and along the toe of any embankment on the right flood plain. This can be thought of as the river and floodplain section prior to bridge construction  npts 
Crosschainage  cross chainage (m)  xp_{i} 
Elevation  elevation of bed or flood plain (mAD)  yp_{i} 
Mannings n  Manning roughness coefficient  rn_{i} 
Embankments  LEFT or RIGHT (chmain= 'L' or 'R' to indicate left or right embankments of main channel  chmain_{i} 
Bridge Arch/Opening Data
Field in Data Entry Form  Description  Name in the Datafile 

n/a  number of arches (openings between piers)  narch 
Start  horizontal coordinate of left side of arch (must coincide with a coordinate of cross chainage, xp) (m)  archxl_{i} 
Finish  corresponding horizontal coordinate of right side of arch (must coincide with a coordinate of cross chainage, xp) (m)  archxr_{i} 
Springing Level  springing level of arch (mAD)  spring_{i} 
Soffit Level  soffit level of arch (mAD)  aspfot_{i} 
Flood Relief Culvert Data
Flood relief culverts are assumed to be rectangular
Field in Data Entry Form  Description  Name in the Datafile 

n/a  number of culverts in bridge structure  nculv 
Invert Level  invert level of culvert (mAD)  cinvrt_{i} 
Soffit Level  soffit level of culvert (mAD)  csofit_{i} 
Section Area  cross section area of culvert (m_{2})  carea_{i} 
Cd Part Full  part full discharge coefficient  dispt_{i} 
Cd Full Flow  full flow discharge coefficient  disful_{i} 
Drowning Coeff  drowning coefficient for part full flow (must be <1)  cdrown_{i} 
Click here to view Broad Crested Weir equations
Click here to view Orifice equations
Theory and Guidance
The US BPR Bridge computes the afflux at bridges using the methodology developed by the US Bureau of Public Roads (US BPR).
The bridge afflux is calculated using the methods described in Hydraulics of Bridge Waterways (1978). You are advised to read the report in order to gain a full understanding of the methodology and limitations of the approach.
The US BPR Bridge requires that the network has a River section upstream of the bridge (or a Replicated river section), ideally at the point of maximum backwater, and a River section downstream of the bridge, ideally where normal water level has been achieved.
Input data for the US BPR Bridge principally comprises two cross sections:
 the river bed of the unobstructed river channel along the toe of the bridge and extending into the floodplain (if appropriate)
 arch/opening and pier details
Other data include the upstream and downstream node labels, pier, skew and abutment details.
Equations
A practical expression for backwater has been formulated by applying the principle of conservation of energy between the point of maximum backwater upstream from the bridge, and a point downstream from the bridge at which normal stage has been reestablished.
Backwater expression
The expression for computation of backwater upstream from a bridge constricting flow is:

where: h_{1}^{*} = total backwater (or afflux) K^{*} = total backwater coefficient a_{1} = kinetic energy coefficient at the upstream section a_{2} = kinetic energy coefficient in the constriction V_{B} = average velocity in constriction A_{B} = gross water area in constriction A_{4} = water area in downstream section A_{1} = total water area in upstream section including that produced by the backwater 
The value of the overall backwater coefficient, K^{*}, which was determined experimentally, varies with:
 Stream constriction as measured by the bridge opening ratio M
 Type of bridge abutments
 Number, size, shape and orientation of piers in the constriction
 Eccentricity, or asymmetric position of bridge with the floodplains
 Skewness of bridge and floodplains
It was demonstrated that K^{*} consists of a base curve coefficient, K_{b}, to which is added incremental coefficients to account for the effect of piers (DK_{p}), eccentricity (DK_{e}) and skew (DK_{s}). The value of K^{*} is nevertheless primarily dependent on the degree of constriction of flow at a bridge.
Base Curve (K_{b})
The backwater coefficient base curve, K_{b}, is dependent on the bridge opening ratio, M, and to a lesser extent the abutment type.
The bridge opening ratio, M, defines the degree of river channel constriction involved. It is defined as the ratio of the flow which can pass unimpeded through the bridge constriction to the total flow of the unobstructed river.
The base curve is valid for M in the range 0.2 to 1.0 which yields K_{b} values within the range 2.8 to 0.0 (as M decreases K_{b} increases).
Effect of Piers, DK_{p}
Backwater caused by introduction of piers in a bridge constriction has been treated as an incremental backwater coefficient designated DK_{p}, which is added to the base curve coefficient K_{b} when piers are present in the waterway. The value of DK_{p} is dependent on the ratio of the area of the piers to the gross area of the bridge opening (known as J), and the type of piers and the bridge opening ratio, M. For skewed crossings, J is determined using the areas projected normal to the general direction of flow.
A problem arises here, with the distinction between pier and arch. The pier coefficient curves have been determined for the pier ratio, J , in the range 0.0 to 0.18 and M between 0.4 to 1.0. The upper range of J is probably less reliable, being determined by extrapolation, therefore as a guide, to keep J in the range 0.0 to 0.10, the total width of piers normal to flow should not exceed 10% of the unobstructed channel width. If the sum of pier widths exceed 10% of the unobstructed channel, it is recommended that they be treated as arches.
Defining Piers and Arches
You have options when defining a bridge with more than one arch. You should bear in mind the points above when deciding how to define the bridge within the model. The bridge could be defined as:
 a single arch with intermediate piers
 several arches with no intermediate piers
 a combination of the two
For example, take a look at the example bridge below.
You could define this bridge in several ways:
 as a single span (AF) with Total Pier Width = w + x
 as two spans (AD and EF) with Total Pier Width = w
 as three spans AB, CD and EF with Total Pier Width = 0
Effect of Eccentricity, DKe
The eccentricity, e, is defined as 1 minus the ratio of the lesser to the greater discharge outside the main bridge opening.
For channels with an eccentricity of 0.80 or greater, DK e values have been determined for M in the range 0.2 to 1.0. For channels with an eccentricity of less than 0.80, DKe is deemed negligible i.e. DKe = 0.
Effect of Skew, DK_{s}
The method of computation for skewed crossings differs from that of normal crossings in that the bridge opening ratio, M, is computed on the projected length of the bridge rather than on the length along the centerline.
The incremental coefficient, DK_{s}, varies with the opening ratio, M, the angle of skew of the bridge q, with the general direction of flood flow, and the alignment of the abutment faces. The angle of skew, q, is defined as the angle between the flow direction and the normal to the main axis of the bridge. Abutments are aligned if the abutment faces are parallel with the flow direction (otherwise they are skew).
Dual Bridges
The backwater produced by dual bridges  two bridges of essentially identical design, placed parallel and only a short distance apart  is greater than that for a single bridge, yet less than the value which would result from considering the two bridges separately.
The dual bridge afflux is determined as a function of the single bridge afflux and the ratio Ld/l where Ld is the distance from the upstream face of the upstream bridge to the downstream face of the downstream bridge, and l is the single bridge width. The formulation is applicable for Ld/l between 3 and 11. Dual bridges for which Ld/l is less than 3 may be treated as a single bridge with an increased pier number.
Flood Culverts
Flow through flood culverts is calculated using a broadcrested weir equation, or an orifice equation if the culvert is flowing full. An iterative method is used to calculate the afflux for the remaining flow through the main bridge arches/openings. This method is not included in the USBPR report.
Click here to view Broad Crested Weir equations
Click here to view Orifice equations
Road Flow
Flow over the top of the bridge is not considered. If this type of flow is likely, the user should place a structure such as a Spill in parallel with the bridge, using junctions as necessary.
General
The backwater expression is reasonably valid if:
 the channel in the vicinity of the bridge is essentially straight
 the cross sectional area of the river is fairly uniform
 the gradient of the bottom is approximately constant
 the flow is free to contract and expand
 there is no appreciable scour of the bed in the constriction
 the flow is in the subcritical range.
Outside these conditions the USBPR method should be used with care. In addition, as the USBPR method is a design procedure based on American rivers, even with the "correct" coefficients and dimensions, the predicted afflux may not agree with observed values of water level when used in other countries.
Friction losses within the bridge are not modelled by the US BPR Bridge . Flow over the bridge parapet (road flow) is not modelled.
If the bridge is skewed, the bridge section data should be specified along the axis of the bridge  its projection normal to the flow direction is performed internally within the software.
The unit state for this unit is the flow through the flood relief culvert (if present).
Orifice Flow
The bridge units in Flood Modeller may switch to orifice flow at a given depth if the user selects this option from the unit form. This has the benefits of representing surcharged flow as an orifice, which may be more representative, whilst retaining the bridge afflux calculations when not surcharged.
The user can specify a lower level (specified as distance below highest arch soffit) at which the transition from bridge flow to orifice flow commences, and an upper level (specified as distance above highest arch soffit) at which the transition to orifice flow is complete. This allows a smooth transition from bridge to orifice flow to occur.
The orifice equation used is the standard orifice equation in Flood Modeller, although the user may adjust the coefficient by changing the orifice discharge coefficient within the bridge unit.
The unit mode for a bridge is as follows:
Mode 1  bridge flow
Mode 2  transition flow (between bridge and orifice)
Mode 3  orifice flow
Datafile Format
Line 1  keyword 'BRIDGE'
Line 2  keyword 'USBPR1978'
Line 3  Label1, Label2, [Label3, Label4]
Line 4  keyword 'MANNING'
Line 5  cali, skewb, [rdlen, duall], pierw,oflag,rlower,rupper,cdorifice
Line 6  iabut
Line 7  npier, shape [,diaph] [,prcoef]
Line 8  altype
Line 9  npts
Line 10 to Line 9+npts  xp_{i} , yp_{i} , rn_{i} [,chmain_{i}]
(subsequent line numbers depend on npts, narch and nculv)
Line 11  narch
Line 12 to Line 11+narch  archxl_{i} , archxr_{i} , spring_{i} , asofit_{i}
Line 13  nculv
Line 14  cinvrt_{i} , csofit_{i} , carea_{i} , dispt_{i} , disful_{i} , cdrown_{i}
(lines 14 and 15 are repeated 'nculv' times)