H OMO CORDIALIS EN RES PUBLICA

In order to demonstrate the shortcomings of empirical equations and to promote the use of numerical models for scour prediction, thirty of the better-known equations (listed in Appendix A) were evaluated for clear-water conditions in an alluvial bed. The results from the equations were compared to that of the experimental work for the small sample size of 24 tests. Because the empirical equations were developed for full-scale applications, the input parameters (other than sediment properties) and scour depth results were scaled by applying Froude similarity to the 1: 15 model (𝑛𝑛𝑦𝑦= 15 and 𝑛𝑛𝑣𝑣= √15).

Furthermore, critical velocities determined experimentally were used in the analysis (unless specified otherwise) to ensure the relative velocity ratio was maintained for both model and prototype scales. Several equations exist to determine critical velocity and most of the equations for bridge pier scour in this study fail to reference an appropriate method to calculate it. The threshold of sediment movement is an important parameter and yet literature neglects to address that different equations for critical velocity could produce different scour depth predictions.

The comparison of different empirical equations has been the topic of many studies and without exception, the conclusion is the same each time: empirical equations for bridge pier scour yield a wide range of varying and unreliable results (Johnson, 1995; Rooseboom, 2013). From Figure 6-10, it is evident that a wide range of scour depths were also produced in this study for each test or boxplot (each boxplot represents the range of scour depths predicted by 30 empirical equations for one test). The scour depth was predominantly overpredicted, as safe design equations intend to be conservative when they fail to be accurate. Nevertheless, the empirical equations are discordant and still predict scour depths varying up to 3 m from one another for the same test.

Since the equations are generally developed from a standard experimental setup with a cylindrical pier in a uniformly graded bed, the most accurate scour depths were predicted for the tests with the cylindrical pier as well as the crushed peach pips. It can be deduced that the scaling of the peach pips is a better representative of in-situ sediment than that of the sand. Moreover, larger flows yield larger scour depths, and consequently the equations yield less conservative predictions.

The statistical spread for each empirical equation is indicated by each boxplot in Figure 6-11 which can be evaluated in conjunction with the more detailed relative scour depth dataset in Figure 6-12 where the percentage error is taken as

Evidently, the equations are in weak agreement with one another and generally overestimate the observed scour depths with an average error of 78%. Hancu (1971) and Melville & Kandasamy (1998) are the most accurate methods while the safest equations for bridge pier design are Blench (1969), Shen et al. (1969) and Ali & Karim (2002), followed by the FDOT and HEC-18 equations. However, some of these equations tend away from the line of equality in Figure 6-12 and do not properly capture the processes responsible for local scour. This is in agreement with the literature study whereby the HEC-18 and Shen et al. (1969) equations are known to better resemble scour in the field. In addition, Shen et al. (1969) and Ali & Karim (2002) most likely performed better because they rely on the pier Reynolds number, a parameter which is significant in the formation of the horseshoe vortex (Roulund et al., 2005). The implication of this is that models taking the vortex formation into consideration could offer better scour depth predictions.

The simpler and older models of Breusers (1965), Laursen & Toch (1956), Blench (1969) and Melville & Kandasamy (1998) appear to be more accurate but they do not incorporate approach velocity or particle size. The equations are not considered applicable because they predicted the same scour depth for all the tests; only Melville & Kandasamy (1998) and Laursen & Toch (1956) differentiated between shapes. The simplest expression is that of Breusers which assumes that the maximum bridge pier scour can be taken as 1.4 times the pier size. Pier size is the most prevalent parameter emerging in all the equations except in Chitale (1962). Consequently, Chitale also performed deceptively well because only one pier size was tested. Instead, Chitale (as well as the HEC-18 equations) depend on the Froude number, which is related to the mode of sediment transport of the different bedforms (Graf, 1971). The relative flow depth is also prevalent in the HEC-18 models, which is related to the boundary layer thickness (Roulund et al., 2005).

Figure 6-11: Boxplot showing the distribution of scour depth as a percentage error for the different empirical equations from the experimental work

In contrast, formulae such as that of Coleman (1971) and Gao et al. (1993), also known as the simplified Chinese equation, are not fit for pier design due to underpredictions. In agreement with previous studies, Froelich (1988) also underestimated scour depth, and thus the overly conservative Froelich Design equation was developed (which simply adds the pier width to the predicted scour depth as a precautionary). Molinas (2004) also underestimated the scour depth especially for particle sizes < 2 mm as ascertained by Meuller & Wager (2005).

The most significant spread of errors was presented by Kothyari, Garde & Ranga (1992). It is the only identified scour model that accounts for sediment density and presumably overestimates scour depth due to the challenges posed by the scaling of physical models. Generally, formulae developed in affiliation with Melville overestimated the scour depth more than others. These formulae, as well as the HEC-18 equations, calculate the scour depth with a simplified approach employing dimensionless correction factors to account for time, channel geometry, sediment size, grade, pier shape, flow alignment, armouring, flow intensity or flow depth. The simplified approach demonstrates the effect of each parameter on the scour depth but by doing so neglects to acknowledge that the parameters are interrelated.

As illustrated in Section 6.3.3, it is difficult to mathematically describe the effect of pier shape on the horseshoe vortex and the scour depth with simply a constant shape factor. Half of the 30 empirical equations evaluated in this study account for pier shape by using different factors. Subsequently, the scour depths for the round nosed and sharp nosed were largely overestimated (refer to Figure 6-10). Figure 6-11 is supplemented with boxplots for the empirical equations applied only to the 8 tests with a cylindrical pier. The scour depths are better predicted but the average error of 50% is still fairly large. The HEC-18 equations consistently performed the best with the least underpredictions while the other equations were less inclined to overpredict the scour depth for a cylindrical pier.

Five different HEC-18 have been developed by the different FHWA manual revisions for scour at bridges by improving the armouring factor Ka, which is intended to be representative of the sediment material. With the exception of Mueller & Wagner (2005), Ka is determined by a dimensionless excess velocity intensity based on the critical velocity formulation introduced by Gao et al. (1993). The most recent FHWA manual discards the CSU’s HEC-18 approach for the Florida DOT (Arneson et al., 2012) which is based on the approach by Sheppard & Miller (2006) and Sheppard & Melville (2014) for wide piers with a new critical velocity calculation. While this method has a mean error percentage closer to zero, it also has a larger range of residuals (or higher SSR) with more underpredictions.

Figure 6-12: Comparison of relative scour depths observed from the experimental work and calculated by the different empirical equations