Bridge

General

Cross-section data

Summary: If there is only one cross-section available for the entire bridge, enter it as the Upstream cross-section – this will be automatically duplicated to the remaining bridge cross sections during the computational phase.

The bridge computations are by default based on the properties of the downstream bridge section; however, if the user feels that the data, e.g. especially with respect to constriction, is more applicable to the upstream face, then the option on the main form to switch to using this data for the computations. The data file format does allow for four cross-sections to be stored, with two of these (nominally upstream and downstream) being allowed to be directly entered in the User Interface. The representation of the cross-sections is in upstream to downstream order, as per below, with the two sections exposed to enter in the User Interface in bold:

• Section 1: Upstream river section

• Section 2: Upstream bridge section

• Section 3: Downstream bridge section

• Section 4: Downstream river section

Cross-section #1 is mandatory. The remaining cross-sections are all optional.

The hierarchy / rule is that if cross-section data does not exist, then it is effectively copied from the one above, e.g. if only cross-section #1 is entered, then all four cross-sections will assume this form. If cross-sections 1 and 3 are entered, then cross-section #2 will assume the form of cross-section #1, and cross-section #4 will assume the form of cross-section #3.

Therefore, if one only has one distinct cross-section available for the bridge, then it should be entered in cross-section #1.

Note that if the bridge orientation is not normal to the direction of flow, then the data should be entered along the face of the bridge, as opposed to normal to the flow direction, and also enter a skew angle (up to 45°, where zero indicates the orientation is normal). The cross-section data will then be projected during the computational phase to align with the flow direction, as will the Spill data and Flood Relief Culvert data (i.e. be projected).

Questions:

Why do we allow you to enter four different cross-sections, when only one is used?

This is currently true; however, we can envisage a time in the future where we may make use of all four cross-sections, e.g. we could possibly remove the need to schematise the BRIDGE and adjoining RIVER as separate entities. It is also feasible that the user may already have these four cross-sections available to them. Irrespective of whether they are currently used or not, it may be desirable from a data management viewpoint to be able to store all the available data.

Why are only two exposed in the User Interface?

Following on from the above point, there is a balance between allowing the user to store (and visualise) all available data and overloading the user with what may appear to be confusing requirements. One can envisage many situations where two different cross-section datasets (i.e. upstream and downstream) will be available, and fewer where all four are. The “power user” may need to know that this is available, the standard user less so. Furthermore, the existing Pier-loss bridge unit currently allows for two sections to be entered, and reducing the number of available cross-sections to one would mean the Bridge could contain less information than this.

Panels

In the bridge units, panel markers and bank markers are treated identically, namely to sub-divide the conveyance calculation in a lateral direction, i.e. the calculation of conveyance is performed separately left and right of the panel markers, and then summed. As per RIVER section theory, this is usually to avoid a drop in conveyance as the bed profile changes from steep to near-horizontal, e.g. when transitioning from in-bank to out-of-bank.

Other places which cause a similar effect to panel markers (imposed internally within the calculations), i.e. a break in the conveyance calculation occurs:

• At the beginning of each bridge opening

• When the water level reaches the springing point, i.e. the conveyance calculation occurs separately above and below the springing level

• When the bed level protrudes above the water level, i.e. for low flows where the bed level is above the water level to its left and right, a break in the conveyance calculation occurs.

NB The reason for including both panel markers and bank markers is to facilitate consistency between RIVER and BRIDGE sections, and also to maintain compatibility with previous bridge units.

Note: since cross-section conveyance is only used in the USBPR formulation (in order to calculate the blockage ratio), then the above is only pertinent to this type.

Calibration factors

The traditional calibration factors apply separately to bridge afflux and orifice discharge, the latter only applying when modelling surcharged flow as orifice flow. These have the effect of scaling the calculated quantity by the stipulated factor, e.g. to calibrate to observed data.

However, since the bridge afflux calibration factor applies to the afflux, and the orifice discharge factor applies to discharge, they effectively have the opposite effect, i.e. a factor of 1.1 for both will increase bridge afflux (reduce discharge) by 10%, and increase orifice discharge (reduce afflux/headloss) by 10%.

Take the following example; using the orifice equation

Where is the orifice discharge calibration factor and is a combination of A, root 2g and discharge coefficient. Then, we can rearrange this to read

Or, with the default = 1

Now, supposing then one must set to achieve this. Compare this, when posed as the bridge afflux equation, in any of its three forms (Arch, USPR or Yarnell), where the calibration factor is applied to the headloss, as opposed to the discharge:

Where is the bridge afflux calibration factor. Again, with the default then

Therefore, again if then one must set , or in general., for equivalence,.

Therefore, if we want to apply a blanket calibration factor, so that we want to apply the same relative adjustment to bridge and orifice formulations, there is the option to apply a “universal” discharge calibration factor (i.e. effectively applying to the discharge for the bridge in question). Thus one only needs to provide one calibration coefficient (the [orifice] discharge), which works in the “same direction” for afflux and discharge calculations. Effectively this sets the afflux calibration factor to the inverse of the square of the discharge calibration (1/c²) to be of an equivalent magnitude, since approximately headloss is proportional to the square of discharge.

Additional Outputs (.zzx file) – spill flow and structure coefficients

There are two options for displaying additional output to the [supplementary] results (*.zzx) file. These can be subsequently accessed graphically by viewing a Time Series plot , or numerically via the Tabular CSV tool (select “Supplementary Results” from “Results Module”). These are as follows:

• Overtop and through bridge flow – this separates the discharge within the bridge structure as a whole into its constituent components as applicable –

  • Bridge - Aperture flow: Discharge through the bridge arches, excluding relief culverts

  • Bridge - Spill flow: Discharge over the designated spill section

  • Bridge - Culvert flow: Discharge through the relief culverts; note that this is an approximation when the flow is in orifice transition mode (i.e. between full afflux and full orifice modes)

  thus the total of the three values will be equal to the nodal flow value, i.e. that obtained when selecting “Flow” for the Bridge node.

• Structure Coefficients . This outputs intermediate coefficients used in the calculations, which enables the user to gain a deeper understanding of the resultant afflux; for instance, one particularly useful parameter is the total headloss coefficient, K, used in the USBPR method, which can be used to compare with similar headloss equations used in orifice flow, blockage losses, etc. Since the calculation methods for each formulation are appreciably different, then (some of) the outputs for each are also different, as follows (NB the variables are output as Structure Coefficient #1, #2, etc.):

Tabulate Cross-section properties

Available from the Flood Modeller Toolbox, the Tabulate Cross-Section properties tool can be applied to Bridge units, in addition to River cross-section units.

To activate this:

1. select one or more BRIDGE units from the map or network table (one can also select any number of RIVER sections concurrently)

2. From the Toolbox, select Model Review Tools > 1D River Models > Tabulate Cross-Section Properties (hint: one can start typing “Tabulate…” in the search box and the menu item will become exposed and highlighted)

3. By default, the River section properties table will be active, whether RIVER sections have been selected or not

4. Select the Bridge properties tab – a table similar to the following will appear

One can therefore visualise, without running a simulation, the relationship between water elevation and various properties used in calculating the afflux, such as respective obstructed [i.e. by the bridge constriction] and unobstructed areas and conveyances, and blockage ratio (the measure of constriction used to calculate afflux).

One can also view these graphically via the Bridge property plots tab:

Conveyance plots

When opening a bridge property form, the cross-section property calculations are automatically performed in the background , therefore the conveyance is available as an additional series on the cross-section graph. When selecting “Plot”, the conveyance is off by default, but can be viewed by checking the appropriate box on the plot legend. Note that since conveyance is used only for the USBPR method only, then this is only of relevance to calculations for a USBPR-type bridge. However, it can also be useful in determining appropriate orifice transition distances for all formulations.

ARCH

See also ARCH Bridge theory and guidance for Arch Bridge theory.

The opening for an Arch-type bridge is modelled as a parabola above vertical sprung walls, thus in theory three coordinates are used to define it (NB for a flat soffit, this is equivalent to setting all three elevations to soffit level). However, this is simplified by the constraints that the arch is assumed symmetrical about the mid-point between left and right springing points (i.e. the soffit chainage coordinate is halfway between left and right springing chainages, and the right springing elevation is the same as the left), thus only four values are required as follows for each arch:

1. The left-springing chainage

2. The right-springing chainage

3. The springing elevation

4. The soffit elevation

Note that the network data file requires all three coordinates (six values) to be entered, but the two remaining of these are calculated automatically.

A vertical wall is assumed between the springing point and the bed (cross-section); it is permissible for the springing level to be below ground level (in which case, the opening will start at the intersection of parabola and the bed). The springing horizontal (x-chainage) coordinates are not required to exist within the underlying cross-section data.

USBPR

See also USBPR Bridge theory and guidance for USPBR Bridge theory.

The base coefficient, K_B, is dependent on the blockage ratio and abutment type. A slightly higher coefficient for the same blockage ratio occurs for abutments with a 30° wingwall (type 2), and further for ab abutments with a 90° wingwall or vertical wall (type 1), both so long as the bridge span does not exceed 60m. For all other situations, abutment type 3 (the default) should be chosen. For further details, see Figure 6, Hydraulics of Bridge Waterways (1978).

The incremental effect on K due to piers, ∆Kp, is dependent on the shape and number of piers. For zero piers, one needs to select the soffit type as FLAT (no effect; pier coefficient=0) or ARCH (pier coefficient=5). When modelling piers, one needs to specify the total pier width, as well as the number of piers (in line in the direction of flow, up to a maximum of 3) and the shape of piers. For further details, see Figure 7, Hydraulics of Bridge Waterways (1978), which relates the increment to pier shape, number of piers and pier ratio, J. Note in this diagram, the curves equate to the pier coefficient number from 1 (lower [flatter] curve) to 8 (higher [steeper] curve), with zero equating to no increment. One can also specify a pier coefficient (in the range 0-8) instead of shape, relating to these curves (and interpolating between for non-integer values).

The opening for an USBPR-type bridge is modelled as a parabola above vertical sprung walls, thus in theory three coordinates are used to define it (NB for a flat soffit, this is equivalent to setting all three elevations to soffit level). However, this is simplified by the constraints that the arch is assumed symmetrical about the mid-point between left and right springing points (i.e. the soffit chainage coordinate is halfway between left and right springing chainages, and the right springing elevation is the same as the left), thus only four values are required as follows for each arch:

1. The left-springing chainage

2. The right-springing chainage

3. The springing elevation

4. The soffit elevation

Note that the network data file requires all three coordinates (six values) to be entered, but the two remaining of these are calculated automatically.

A vertical wall is assumed between the springing point and the bed (cross-section); it is permissible for the springing level to be below ground level (in which case, the opening will start at the intersection of parabola and the bed). The springing horizontal (x-chainage) coordinates are not required to exist within the underlying cross-section data .

Yarnell

See also Pier-loss Bridge theory and guidance for Yarnell theory.

The opening for a Yarnell-type bridge is modelled as a single straight line soffit, thus two coordinates are used to define it. When adding a new opening for a Yarnell-type bridge, one is therefore presented with two rows in which to enter cross-chainage and elevation coordinates, which represent:

1. The leftmost soffit coordinate

2. The rightmost soffit coordinate

A vertical wall is assumed between the soffit ends and the bed (cross-section). The horizontal (x-chainage) coordinates of the soffit ends are not required to exist within the underlying cross-section data.

Since the internal piers are fundamental to this formulation, then a Yarnell-type bridge requires at least two openings (i.e. one pier) to be defined.

Orifice transition

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 is particularly recommended when modelling surcharged/overtopping flow. 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 flows

Transition smoothing factor

The smoothing factor is provided for the orifice transition, in order to preserve the continuity of gradient between bridge afflux and orifice flow calculations, and therefore aid stability. By default, this is zero (off) and therefore assumes a linear transition between the linearised bridge and orifice coefficients.

The transition uses a tanh curve, reprojected from (-∞, ∞) to (0,1), with the smoothing applied as the revised weighting f, where:

f = u (for T=0, i.e. no smoothing)

f = (1+tanh(Tx))/2 (for T>0)

where:

• T is the smoothing factor

• f is the weighting towards orifice coefficient ultimately applied (vs (1-f) towards bridge)

• x = tan(π(u- ½))

• u = non-smoothed weighting; proportion of depth within transition zone, i.e. 0 at lower transition, z₀; 1 at upper transition, z₁; varies linearly with depth in between; i.e. u=(h-z₀)/(z₁-z₀₎, for z₀ ≤ h ≤ z₁

Note that T=0 is a special, independent case, and not the limiting case as T->0, i.e. setting the coefficient T close to zero (but nonzero) does not tend to a linear transition and is therefore not recommended (as T->0, f -> 0.5, i.e. tends to a constant 50% split between bridge and orifice). As T -> ∞, then the function tends to a step function with the step at u=0.5

A recommended value is between 0.5 and 2. An illustration of smoothed proportions against unsmoothed for various values of the smoothing factor, T, is given below (note that setting T=0 applies a straight line between (0,0), and (1,1) ; u=0 being the lower transition level and u=1 the upper transition level)

• See also guidance on applying appropriate transition levels.

Spills

Flow over the top of a bridge may be modelled by including spill data within a Bridge unit. This defines the geometry of the surface of the ground, including the bridge deck above the river bed.

Note that if the bridge contains a skew angle, the data must be entered along the face of the bridge top. The software will internally project the horizontal chainage so that it is normal to the flow direction.

The spill component behaves like the conventional Spill unit, by calculating a series of discharges

via a weir equation over a sloped crest between each pair of entered spill points, based on the water levels upstream and downstream of the bridge. This is then added to the discharge through the bridge to obtain a composite (nodal) discharge of the complete structure (for the same upstream and downstream water levels).

Separate discharge through the bridge and over the top, as well as any through relief culverts, can be obtained/viewed by selecting the “Overtop and through bridge flow” Additional Output option at run-time and the individual components (“Bridge – Aperture Flow” and “Bridge - Spill Flow”) from the results.

Note that the user-entered spill coefficient should be inclusive of the ✓(2g) term, discharge coefficient, etc., so that for a horizontal weir, this deforms to

Where c_sp is the spill coefficient, b is the breadth of spill/weir segment, h is the water elevation and z_c is the elevation of spill crest, therefore to approximate a round-nosed, broad-crested weir, for instance, c_sp should be set to 1.7, assuming a discharge coefficient of 1.

Blockages

The blockage integrated within a bridge is calculated by applying Bernoulli’s principle (unless implicitly applying the bridge equations by using the bridge equations with a reduced area) to a fixed percentage blockage throughout the simulation. Compared with the individual Blockage unit, there are three main differences, although the underlying principle and basic equations are the same:

1. Three different formulations are allowed with the bridge’s integrated blockage, namely to obtain the velocities for the constriction using either the upstream or downstream depth at the bridge, or to represent using the standard bridge methods with a reduced percentage opening.

2. For the separate blockage unit, the blockage is schematised in series, generally in front of the bridge, meaning that the headloss is calculated as an addition to the bridge afflux, i.e. upstream water level = bridge headloss + blockage headloss, with the water levels available at the intermediate steps. For the integrated blockage, the blockage calculation is treated as an integrated blockage, and therefore no intermediate water level (hypothetical, between blockage and bridge) exists. Thus the option is available to use velocities based on upstream or downstream depth.

3. At present, there is no facility to represent a time-varying blockage within the integrated BRIDGE unit; if such is required, then the user is referred to the standalone BLOCKAGE unit, which allows this to be applied in series.

Flood Relief Culverts

Flow through separate flood relief openings, up to a maximum of twenty per bridge, can be modelled. Typically these are of a smaller aperture than a bridge opening, and raised at a higher level than the bed. These are defined by the user specifying the following physical quantities:

• Elevation of flood relief opening invert above datum

• Elevation of flood relief opening soffit above datum

• Total area

The opening is assumed rectangular in shape, and therefore the breadth can be calculated as area/total depth (total depth = soffit level – invert level).

An orifice equation is assumed when the relief opening is running full, whereas a weir equation applies when running partially full, which can be free or drowned, according to the downstream water level. A round-nosed broad-crested weir equation is used by default, therefore the “weir discharge coefficient” can be used to take into account or adjust for coefficients of discharge and velocity (otherwise assumed unity) or different weir types. The modular limit also applies to the transition between free and drowned weir flow. An orifice coefficient is provided to enable scaling of the discharge from the standard orifice equation, e.g. to match calibration or account for less-efficient orifice flow.

A cross-chainage coordinate is also available, although this has no effect on the computations and is purely to place appropriately along the cross-section plot for visualisation purposes.

The flood relief opening can work in two different ways: when the bridge calculations are operating in bridge afflux mode, and when the bridge is operating in orifice mode (either can include simultaneous overtopping). In bridge-orifice transition mode, a hybrid between the two applies.

In bridge afflux calculation mode, the bridge afflux and flood relief flow are calculated separately, and an iterative solution is found whereby the combined discharge of the bridge and flood relief openings and upstream water levels both match the total bridge inflow.

In orifice mode, the Flood Relief Opening area is simply added to the bridge opening area. Note that if partially full, only the wetted portion is added.

The flood relief opening area is assumed to be based on the plane of the bridge face. Thus if the bridge has a skew angle defined, the full area along the face should be entered, which is factored (by cosine of the skew angle) internally by the software.

The Flood Relief Openings are available with all three bridge formulations. Previously, this was only the case for the USBPR Bridge. Note that previous concerns about the performance with the USBPR Bridge have been addressed for Flood Modeller v7.3.

The flow through the Flood Relief Culverts (combined across multiple openings within the same bridge) can be visualised by switching on the “Overtop and through bridge flow” Additional Output option prior to running a simulation, and viewed via the “Bridge – culvert flow” variable; note that for consistency with previous software releases, the flood relief culvert flow whilst in bridge afflux flow mode is also output to the Unit State variable.