US BPR Bridge
    • 23 Oct 2022
    • 11 Minutes to read
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    US BPR Bridge

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    Article summary

    The USBPR Bridge computes the afflux at bridges using the methodology developed by the US Bureau of Public Roads (USBPR).


    US BPR Bridge Data

    Field in Data Entry Form


    Name in Datafile

    Upstream Label

    Upstream node name


    Downstream Label

    Downstream node name


    Upstream River Section

    label of upstream RIVER section. This label is not required if the upstream RIVER section is Label1


    Downstream River Section

    label of downstream RIVER section. This label is not required if the downstream RIVER section is Label2


    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)


    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)


    Bridge Width

    width of bridge (ie distance between upstream and downstream faces of bridge) (m) - only used for modelling dual bridges


    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


    Total Pier Width

    total width of piers normal to flow direction (m)


    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)


    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


    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 I-BEAM piles). Select "Use Calibration number" (shape='COEF') if a pier coefficient is to be used instead of pier shape description. Single I-beams and square piers are not covered by the theory, and are treated as rectangular. Twin I-beams and rectangular piers are also not covered, and are treated as twin square piers.


    Pier Faces

    second description of bridge pier cross sectional shape (for possible use when there are 1 or 2 piers). Options are:

    • STRMLINE - streamline pier faces
    • SEMICIRCLE - semi-circular pier faces
    • TRIANGLE - triangular pier faces
    • DIAPHRAGM - diaphragm wall between piers. A diaphragm with only one pier is not possible and is ignored


    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 I-beam piles. See US BPR report for full details.


    Abutment alignment

    indicator for abutments being aligned with normal direction of flow, ALIGNED or SKEW


    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.


    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.


    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.


    Orifice Discharge Coefficient

    orifice discharge/calibration coefficient


    Channel Section at Bridge Data

    Field in Data Entry Form


    Name in the Datafile


    number of ensuing data sets describing the cross-section 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



    cross chainage (m)



    elevation of bed or flood plain (mAD)


    Mannings n

    Manning roughness coefficient



    LEFT or RIGHT (chmain= 'L' or 'R' to indicate left or right embankments of main channel


    Bridge Arch/Opening Data

    Field in Data Entry Form


    Name in the Datafile


    number of arches (openings between piers)



    horizontal coordinate of left side of arch (must coincide with a coordinate of cross chainage, xp) (m)



    corresponding horizontal coordinate of right side of arch (must coincide with a coordinate of cross chainage, xp) (m)


    Springing Level

    springing level of arch (mAD)


    Soffit Level

    soffit level of arch (mAD)


    Flood Relief Culvert Data

    Flood relief culverts are assumed to be rectangular

    Field in Data Entry Form


    Name in the Datafile


    number of culverts in bridge structure


    Invert Level

    invert level of culvert (mAD)


    Soffit Level

    soffit level of culvert (mAD)


    Section Area

    cross section area of culvert (m2)


    Cd Part Full

    part full discharge coefficient


    Cd Full Flow

    full flow discharge coefficient


    Drowning Coeff

    drowning coefficient for part full flow (must be <1)


    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:

    1. the river bed of the un-obstructed river channel along the toe of the bridge and extending into the floodplain (if appropriate)
    2. arch/opening and pier details

    Other data include the upstream and downstream node labels, pier, skew and abutment details.


    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 re-established.

    Backwater expression

    The expression for computation of backwater upstream from a bridge constricting flow is:





    h1* = total backwater (or afflux)

    K* = total backwater coefficient

    a1 = kinetic energy coefficient at the upstream section

    a2 = kinetic energy coefficient in the constriction

    VB = average velocity in constriction

    AB = gross water area in constriction

    A4 = water area in downstream section

    A1 = 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:

    1. Stream constriction as measured by the bridge opening ratio M
    2. Type of bridge abutments
    3. Number, size, shape and orientation of piers in the constriction
    4. Eccentricity, or asymmetric position of bridge with the floodplains
    5. Skewness of bridge and floodplains

    It was demonstrated that K* consists of a base curve coefficient, Kb, to which is added incremental coefficients to account for the effect of piers (DKp), eccentricity (DKe) and skew (DKs). The value of K* is nevertheless primarily dependent on the degree of constriction of flow at a bridge.

    Base Curve (Kb)

    The backwater coefficient base curve, Kb, 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 un-obstructed river.

    The base curve is valid for M in the range 0.2 to 1.0 which yields Kb values within the range 2.8 to 0.0 (as M decreases Kb increases).

    Effect of Piers, DKp

    Backwater caused by introduction of piers in a bridge constriction has been treated as an incremental backwater coefficient designated DKp, which is added to the base curve coefficient Kb when piers are present in the waterway. The value of DKp 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, DKs

    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, DKs, 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 broad-crested 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.



    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 sub-critical 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 - xpi , ypi , rni [,chmaini]

    (subsequent line numbers depend on npts, narch and nculv)

    Line 11 - narch

    Line 12 to Line 11+narch - archxli , archxri , springi , asofiti

    Line 13 - nculv

    Line 14 - cinvrti , csofiti , careai , dispti , disfuli , cdrowni

    (lines 14 and 15 are repeated 'nculv' times)


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