Chapter4-How to calculate discharge and afflux

• Sites with wide, heavily vegetated floodplains. With wide valleys the flow approaching a bridge has to move from the floodplain into the main channel in order to pass through the constriction. Laursen (1970) termed this a zone of accretion. Similarly, having passed through the opening, the flow expands from the main channel back onto the floodplain. This was termed a zone of abstraction. For flow to expand laterally in the zone of abstraction it follows that the water surface must slope away from the main channel (this is the reverse of the normal situation). If the floodplain is heavily covered with trees and brush, a large transverse gradient is required to overcome the resistance to lateral flow, so backwater can occur downstream of a bridge as well as upstream. Consequently Laursen concluded that in wide valleys the upstream backwater can be much greater than that predicted by either the USGS or USBPR method, so it is not surprising that under these conditions the backwater measured by Kaatz and James (1997) was roughly double that expected (computer models also had significant errors; see Section 4.6). Laursen presented a numerical approach to this problem, but a drawback was that the accretion and abstraction rates had to be assumed.
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bridges. However, all four methods can sometimes be applied to arches, as shown in Section 4.6 (Matthai, 1967; Hamill, 1993). Which method is adopted depends to some extent upon the data available, the ease of application, the type of flow experienced (e.g. subcritical, supercritical, abnormal stage), the waterway geometry (e.g. rectangular, arched, single or multispan), and the purpose of the investigation. The USGS and the USBPR methods are the most frequently employed. Both have some restrictions on their use (Sections 4.2.3 and 4.3.7), which should be understood before undertaking an analysis. With any of the methods large errors may be incurred under adverse conditions or when conditions differ significantly from those of the original research (Skogerboe et al., 1973; Barret and Skogerboe, 1973; Fiuzat and Skogerboe, 1983; Kaatz and James, 1997). Adverse conditions include the following.
The methods described below are relatively simple, most being devised before the computer era. Thus they provide a method of analysis for engineers without access to today’s specialist computer software and, for those who do, a means of quickly checking the results. Since there have been relatively few major investigations of bridge hydraulics, some of the principles and equations described in Chapters 4–6 will be included in the software.
• US Geological Survey (USGS); • US Bureau of Public Roads (USBPR); • Biery and Delleur; • Hydraulics Research (HR).
The first two listed were developed almost entirely for rectangular openings and ‘deck type’ bridges, while the last two were mainly concerned with arch
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4 How to calculate discharge and afflux
4.1 Introduction
Usually bridge hydraulics is concerned with determining either the stage-discharge relationship of a structure or the afflux, or possibly both. This chapter shows how to analyse the flow through rectangular and arched bridge waterway openings with either single or multispans, before and after the waterway becomes drowned. The abutments provide the primary contraction, with the piers being a secondary influence: often the gross area of the waterway opening (AN2, which includes the area occupied by any piers) is employed initially in the calculations, and then a correction factor (J) is introduced to allow for the secondary obstruction of the piers. Typically J =AP/AN2 where AP is the cross-sectional area of the submerged parts of the piers projected onto section 1. On the other hand, Chapter 5 considers long crossings where it is the piers that control the flow while the effect of the abutments is negligible.
• Sites where the flow is non-uniform and abnormal stages exist. Paradoxically, Barret and Skogerboe (1973) found that the USBPR method was more accurate with an abnormal stage, but conjectured that a greater range of data would reveal greater errors. However, it is certainly true that an abnormal profile complicates the calculations, which are much simpler when uniform flow exists and normal depth provides a convenient datum (Figs. 2.3 and 2.4). Generally, with an abnormal stage the bridge exerts a relatively weak control on the flow, reducing the accuracy of an analysis.