E L PARADIGMA DE LA MOVILIDAD

Most scour depth equations traditionally used in bridge designs have been developed on the basis of experimentation, dimensional analyses and simplified theoretical models (Chadwick, 2013). The equations have been derived by assuming dominant parameters, reducing them to simplified relationships and then calibrating them by means of a coefficient from laboratory and field data. Of all the relationships developed to date, Appendix A lists 30 of the better known prediction formulas for clear-water bridge pier scour in an alluvial bed. Only equations which are dimensionally correct and in SI units are presented.

In short, the formulas describe the resulting equilibrium scour depth 𝑑𝑑𝑠𝑠 as a function of a combination of one or more of the following parameters:

𝜌𝜌 = Fluid density

𝜌𝜌𝑠𝑠 = Sediment density

𝑠𝑠 = Relative density

υ _{= Kinematic viscosity}

𝑡𝑡 = Time

𝐷𝐷 = Pier diameter or width

𝐿𝐿 = Pier length

𝑑𝑑 = Median sediment size

𝜎𝜎𝑔𝑔 = Particle size distribution

𝐵𝐵 = Channel width

𝑆𝑆 = Energy slope

𝑣𝑣1 = Approach flow velocity 𝑦𝑦1 = Approach flow depth 𝑔𝑔 = Gravitational acceleration 𝑣𝑣𝑐𝑐 = Sediment critical velocity 𝐶𝐶𝑡𝑡∗ = Shear velocity threshold

𝑅𝑅𝑅𝑅 = Reynolds number

𝐹𝐹𝐹𝐹 = Froude number

𝛼𝛼 = Angle of flow in radians

𝐾𝐾𝑠𝑠 = Shape factor

Some interesting remarks on the given equations are briefly discussed:

• _{The simplest expression is that of Breuers (1965) which assumes that the maximum} bridge pier scour can be estimated as 1.4 times the pier size.

• Pier size is by far the most predominant parameter appearing in all the formulas except Chitale (1962) which is based on the Froude number. The HEC-18 equations also depend on the Froude number, which describes the gravity effect at the free water surface, thus it is only valid for low flow depths (Guo, 2012).

• _{Simplified expressions have been proposed to represent debris accumulation and} complex pier geometries in terms of a single equivalent or effective pier width to allow them to be incorporated into equations such as Melville & Coleman (2000) and Amini et al. (2011). Arneson et al. (2012) recommends that complex pier configurations are divided into three substructural elements namely the pier stem, pile cap / footing and pile group, and that the scour for each component that obstructs flow is superimposed. While this falls beyond the scope of the study, it has potential for further investigation in numerical modelling.

• Another prevalent parameter is the relative flow depth appearing in all but three equations (Breusers, 1956; Shen et al., 1969; Coleman, 1971). The equations generally yield similar trendlines as shown in Figure 2-25 within the relative scour depth envelope of 1 and 3 as observed by Chiew (1984) in Section 2.7.3.1.

Figure 2-25: Comparison of empirical equations relative to flow depth (Richardson & Davis, 2001) • _{Five of the earlier models, namely Laursen & Toch (1956), Breusers (1965), Blench}

(1969), Mississippi (1995) and Melville & Kandasamy (1998) do not incorporate the approach velocity. However, Koen (2014) found that they estimate the maximum possible scour depth for clear-water scour fairly well in the laboratory.

• Guo (2012) uses a novel approach by employing the densiometric particle Froude number, a parameter dependent on sediment density. The only other models that account for relative sediment density are Hancu (1971), Ali & Karim (2002) and Kothyari, Garde & Ranga (1992) which are derived from the sediment transportation theory. According to Koen (2014), Kothyari, Garde & Ranga (1992) produces significantly different results from the other equations, overestimating the scour depth by the same proportion in every instance, most likely owing to the challenges posed by physical model scaling.

• _{Typically, empirical equations are developed to describe local scour. However,} ambiguities exist whereby equations could fail to distinguish between total scour, contraction scour or local scour. Kothyari, Garde & Ranga (1992) attempt to address contraction scour by incorporating an opening ratio.

• Generally, the equations were developed from an experimental setup with a cylindrical pier in a uniformly graded bed. Fifteen of the equations presented can be calibrated by factors for different shapes such as those listed in Table 2-1. The difference in shape factors demonstrate the conflict in literature regarding the extent of the impact of a bridge pier shape on scour depth.

Table 2-1: Coefficients for the impact of pier shape on scour depth relative to a cylindrical pier Pier shape Cylindrical Round

nosed Sharp nosed Square nosed

Froelich (1988) 1 1 0.7 1.3

Melville & Sutherland (1988) 1 1 0.9 1.1

• Formulas developed in affiliation with Melville are usually given in terms of the pier width multiplied by 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 illustrates the effect of each parameter on the scour depth but by doing so neglects to acknowledge that the parameters are interrelated.

• The only equations that attempt to model the temporal evolution of the scour depth is that of Melville & Coleman (2000) and Ali & Karim (2002) which employ exponential functions. The time effects are significant when considering the poor correlation between results from the field and the laboratory (Melville & Chiew, 1999).

• _{The equations of Hancu (1971), Breuser et al. (1977), Sheppard & Miller (2006) and} Sheppard & Melville (2014) have biases of zero for the condition 𝑣𝑣1/𝑣𝑣𝑐𝑐 < ~0.5. The assumption implies that local scour does not occur at velocities less than half the 𝑣𝑣𝑐𝑐 although literature exists that observes scour at such low velocities (Johnson, 1995). • _{Most bridge pier scour equations are a function of critical velocity but models such as}

Breuser at al. (1977), Jain & Fisher (1979), Jain (1981) and Sheppard & Miller (2006) do not reference an appropriate method for the calculation thereof. The threshold of sediment movement is clearly an important parameter in scour calculations and yet literature neglects to address that different equations for critical velocity could yield different scour depth predictions.

• _{Unlike the other models, Ali & Karim (2002) developed an equation for bridge pier} scour from a numerical model for the junction flow field. Furthermore, Shen et al. (1969) is the only other equation that acknowledges the pier Reynolds number, a parameter which has been identified as significant in vortex formation by recent numerical models. The implication of this is that the empirical equations are developed by directly describing the effect of the parameters on the scour hole, without considering the vortices.

• _{Five different HEC-18 models have been developed by FHWA manual revisions that} modify the armouring factor for sediment, typically based on dimensionless excess velocity intensity. The form of the equation resulted from a series of studies by Shen et al. and Richardson & Davis (2001) that came to be known as the Colorado State University (CSU) equation.

The comparison of different empirical equations has been the topic of many studies. Johnson (1995) used field data to evaluate the accuracy of seven pier scour equations. Landers & Mueller (1996) analysed five selected equations with field data. Gaudio et al. (2010) tested six formulas by using synthetic and original field data. In more recent studies, Koen (2014) and Toth (2015) both evaluated ten different equations. The outcome of Toth’s comparative study is shown in Figure 2-26. The comparative studies are based on statistical analyses using, amongst others, percentage difference or percentage error, standard deviation, bias or rankings. One of the most comprehensive studies is that of the US Federal Highway Administration (FHWA) by Mueller & Wagner (2005) who compiled a database of scour at 79 bridges to evaluate 26 published pier scour equations. Sheppard et al. (2014) evaluated 23 equations for under-prediction using compiled laboratory and field databases and then combined the equations to produce the Sheppard & Melville Model. The most recent FHWA manual discards the HEC-18 approach for the Florida DOT (Arneson et al., 2012) based on the Sheppard & Melville Model.

Figure 2-26: Boxplot representing the distribution of scour depth prediction errors over a test set for different empirical equations (Toth, 2015)

Throughout the research, the following conclusions are recurring:

• Different formulas produce significantly different predictions from the field.

• Furthermore, they are in weak agreement with one another. The equations are not universal and only yield good results under conditions similar to those from which they were derived.

• _{Most of the equations overestimate observed scour depths and may perform better in} conservative designs. However, this may lead to uneconomical designs of unnecessarily expensive foundations or countermeasures.

• On the other hand, some of the formulas are not fit for pier design due to underpredictions, for example Froelich (1988). As a result, the Froelich Design equation came about, which adds the upstream flow depth to the predicted scour depth as a precautionary.

• _{No single equation is conclusively superior. Ranking the performance of equations is} difficult due to the tradeoff between accuracy and underpredictions.

• Nevertheless, the HEC-18 design equations are commonly favoured for results that most closely resemble the field and rarely underpredict, as illustrated in Figure 2-26. Interestingly, the Shen et al. (1969) model, which relies on the pier Reynolds number, also performed well.

• Further research and improved prediction models are recommended.