 22 Sep 2022
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Pierloss Bridge
 Updated on 22 Sep 2022
 9 Minutes to read
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The pier loss bridge computes the afflux at bridges where the dominant cause of losses is the friction from piers. It utilizes the empirical equation derived by Yarnell (1934).
Data
Pierloss Bridge Data
Parameter name  Description  Label name in data file  

Upstream label  Upstream node name – immediately upstream node in network  Label1  
Downstream label  Downstream node name immediately downstream node in network  Label2  
Upstream river section  Label of upstream RIVER section. Optional: This label is not required if the upstream RIVER section is Label1  Label3  
Downstream river section  Label of downstream RIVER section. Optional: This label is not required if the downstream RIVER section is Label2  Label4  
Loss model method  YARNELL. Keyword referring to the modelling method: currently only option is YARNELL, hence this is not included as a setting in the pier loss bridge interface. The software will set this value automatically in your network file.  Keyword  
Calibration coefficient  Global calibration coefficient: Can be any nonnegative real number. 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  
alternative method  Alternative method. Defines calculations to apply when bridge is surcharged. The default option is ORIFICE, set by the 'Model surcharged bridge as orifice flow' checkbox; if unchecked (not recommended, since the Yarnell equation is not validated for surcharged flow), this field is left blank and the Yarnell equation will apply for surcharged flow also.  altmethod  
Orifice Discharge Coefficient  orifice discharge/calibration coefficient – only used when calculations transition to ORIFICE alternative method (i.e. as bridge becomes surcharged). Must be real, nonnegative number.  cdorifice  
Lower Transition Distance  Lower transition depth (distance below soffit in m or ft). When the upstream water level is below [max soffit level  rlower], the Yarnell bridge equations only are used. When this level is reached, it enters the transition phase between bridge and orifice flow where flow is calculated using both Yarnell and orifice equations and a weighted average is used. Constraint: rupper + rlower ≥ 0  rlower  
Upper Transition Distance  Upper transition elevation (distance above soffit in m or ft). When the upstream water level exceeds [max soffit level + rupper], the bridge to orifice transition ends and full orifice equations apply. Constraint: rupper + rlower ≥ 0  rupper  
pier coefficient  Pier (or Yarnell) coefficient, K: can be any real value in the range 0.7 to 2.5. This coefficient accounts for differences in friction caused by pier shape. Suggested values for different standard pier shapes are provided in Table 1 below (note this list may not be exhaustive): Table 1: Suggested values of pier coefficient
Note: Adjustment of this coefficient can help with model calibration.  K  
bridge width  Width of bridge (in m or ft), i.e. distance between upstream and downstream faces of bridge. Not currently used by computation, entry is for reference only (in case you want to change to a different bridge type in future that requires this) Should be nonnegative real number (can just be left as default = 0.0)  rdlen 
Data for Channel Section at Upstream Face of Bridge
Parameter name  Description  Label name in data file 

No. of data points  Number of data entries describing the upstream face of the bridge crosssection. Not a data entry  solver sets this based on number of data rows defined in upstream section table. Minimum of 3 data points should be defined, i.e. a triangular section. OR can be set to zero, in which case the downstream section data will be adopted for both bridge faces (and so must be defined) The upstream bridge crosssection extends 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_us 
Crosschainage  Crosschainage (m or ft). Chainage is expected to be a real number and increasing  usxpi 
Elevation  Elevation of bed or flood plain (m or ft AD). Must be a real number  usypi 
Mannings n  Manning roughness coefficient. Must be a positive, real number  usrni 
Embankment indicator  Mandatory field of 1 character; only options are ‘L’ or ‘R’ to indicate left or right embankments of main channel Interface presents options “Left” and “Right” for you to select to set these indicators. The section can only have one left and one right marker.  uschmaini 
Top level  The top level represents the soffit elevation at your specified left and right embankment points (note that any toplevel entries on rows that are not embankment indicators will be ignored by the solver). The toplevel values should be measured using the same reference datum as the section elevation data.  ustoplev 
Data for Channel Section at Downstream Face of Bridge
Parameter name  Description  Label name in data file 

No. of data points  Number of data entries describing the downstream face of the bridge crosssection. Not a data entry – solver sets this based on number of data rows defined in downstream section table. Minimum of 3 data points should be defined, i.e. a triangular section. OR can be set to zero, in which case the upstream section data will be adopted for both bridge faces (and so must be defined) The downstream bridge crosssection extends 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_ds 
Crosschainage  Crosschainage (m or ft). Chainage is expected to be a real number and increasing  dsxpi 
Elevation  Elevation of bed or flood plain (m or ft AD). Must be a real number  dsypi 
Mannings n  Manning roughness coefficient. Must be a positive, real number  dsrni 
Embankment indicator  Mandatory field of 1 character; only options are ‘L’ or ‘R’ to indicate left or right embankments of main channel Interface presents options “Left” and “Right” for you to select to set these indicators. The section can only have one left and one right marker.  dschmaini 
Top level  The top level represents the soffit elevation at your specified left and right embankment points (note that any toplevel entries on rows that are not embankment indicators will be ignored by the solver). The toplevel values should be measured using the same reference datum as the section elevation data.  dstoplev 
Bridge Pier Data
Parameter name  Description  Label name in data file 

No. of piers  Number of piers in bridge: must be integer value >0 Not a data entry – solver sets this based on number of data rows defined in pier data table. Minimum of 1 pier must be defined. Pier data used to define blockage ratio (α).  npiers 
Left x  left xlocation of pier  must be a positive real number Coordinates of pier edges assumed to use the same reference datum as crosssection chainage. Values should be defined left and right embankment indicators  xleft 
Right x  right xlocation of pier  must be a positive real number Coordinates of pier edges assumed to use the same reference datum as crosssection chainage. Values should be defined left and right embankment indicators  xright 
Left height  height of pier at left xlocation. Must be positive real number  htleft 
Right height  height of pier at right xlocation. Must be positive real number  htright 
Check  (Optional) check for duplicate/overlapping entries 

Theory and Guidance
Aim
The pier loss bridge unit provides an improved calculation of flows through bridges where the dominant cause of energy loss is the friction from bridge piers. This is particularly useful when low flows are more frequent. The method utilizes the empirical equation derived by Yarnell (1934).
This option can be considered for any bridge that incorporates one or more piers.
Theory
The formula utilized was derived by analyzing results from a large number of experiments and onsite measurements. It is known as the Yarnell equation (1) and relates the water stage (H) at the upstream and downstream areas of the bridge.
The subscripts US and DS denote the upstream and downstream sections respectively. The formula shows that the afflux is dependent on the shape of the piers (coefficient K), the ratio (α) of the area blocked by the structure to the total effective area upstream of the bridge, and the flow characteristics (velocity and depth) at the downstream end of the structure. In particular, V_{DS} the velocity downstream and ω is the ratio of the velocity head (V^{2}_{DS}/2g) to the downstream water depth.
Plan view of a bridge showing the relative position of cross sections (numbered) needed to apply Yarnell equation. The embankments are also shown.
Based on the plan view of a bridge shown in the above figure, without the actual piers, the river cross sections (2) and (3) are located at the very end of bridge embankments at the upstream and downstream faces of the structure. Sections (1) and (4) are further up or down of the bridge, i.e. the adjacent river cross sections in your network. Note that if your structure is not attached directly to river sections, e.g. attached to a junction to enable incorporation of spill units with bridge, then sections (1) and (4) in the above diagram will be the closest “remote” sections to your bridge.
Limitations
This method has been validated through multiple experiments and onsite measurements and only applies to bridges where losses are dominated by the friction on the piers. Furthermore, there are some additional limitations to consider when applying this unit. The most important condition that needs to be satisfied is that the flow through the opening has the following criteria:
There are some limitations when it comes to applying this modelling technique. The most important condition that needs to be satisfied is that the flow through the opening is a class A, low flow. Specifically, the flow should be:
 Not surcharged: the water stage should be below the soffit of the bridge opening
 Subcritical: the Froude number should be below 1 (0.9 for tolerance)
 Not choked: i.e. specific energy is above the critical value in the constriction
The unit can still accommodate surcharging, but in this case the calculated hydraulics will assume an orifice type flow through the defined bridge openings (see below).
The other limitation is your bridge must include one or more piers, since a fundamental part of this unit is to calculate the blockage caused by them. Thus, this method will not be suitable for all bridges.
Orifice Flow
The pier loss bridge unit will switch to orifice flow at a given depth, which is defined in 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.
You need to 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 you can adjust the coefficient by changing the orifice discharge coefficient within the bridge unit.
Datafile Format
Line 1: BRIDGE (keyword)
Line 2: PIERLOSS (keyword to distinguish from other types)
Line 3: Label1, Label2, Label3, Label4 (first two: directly connected, second two: remote units)
Line 4: YARNELL (keyword referring to the modelling method)
Line 5: cali, altmethod, cdorifice, rlower, rupper
Line 6: K, rdlen
Line 7: npts_us
Line 8 to line 8+npts_us: usxpi, usypi, usrni, uschmaini, ustoplev
Line 8+npts_us+1: npts_ds
Line 8+npts_us+2 to Line 8+npts_us+2+npts_ds: dsxpi, dsypi, dsrni, dschmaini, dstoplev
Line 8+npts_us+2+npts_ds+1: npiers
Line 8+npts_us+2+npts_ds+2 to line 8+npts_us+2+npts_ds+2+npiers: xleft, xright, htleft, htright.