Journal of Fluids

Volume 2016 (2016), Article ID 4835253, 19 pages

http://dx.doi.org/10.1155/2016/4835253

## Mathematical Modeling of Local Scour at Slender and Wide Bridge Piers

Royal Commission Yanbu Colleges and Institutes, Yanbu University College, P.O. Box 31387, Yanbu, Saudi Arabia

Received 11 January 2016; Accepted 12 April 2016

Academic Editor: Robert Spall

Copyright © 2016 Youssef I. Hafez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Most existing equations for predicting local scour at bridge piers suffer from overprediction of the scour depths which results in higher foundation costs. To tackle this problem, a mathematical model for predicting bridge pier scour is developed herein based on an energy balance theory. The present study equation was compared to commonly used bridge scour equations using scour field data in USA. The developed equation has several advantages among which we have the following: it adds to the understanding of the physics of bridge pier scour, is valid for slender and wide piers, does not suffer from overprediction of scour depths, addresses clear water and live bed scour, and includes the effects of various characteristics of the bed material such as specific gravity (or density), porosity, size, and angle of repose. In addition, the developed equation accounts for the debris effect and aids in the design of scour mitigation methods such as collars, side bars, slots, and pier protective piles.

#### 1. Introduction

According to a comprehensive collection of bridge failure data worldwide gathered by Imhof [1], natural hazard is the main cause of bridge collapse as it amounts to about 30% of total collected bridge collapse cases. Among the natural hazard listed causes, flooding or scour is responsible worldwide for around 60% of the collapse cases. The Federal Highway Administration (FHWA) has estimated that 60% of bridge collapse cases in the USA are due to scour [2, 3] and, on average, about 50 to 60 bridges fail each year in the USA [4]. Wardhana and Hadipriono [5] studied 500 failures of bridge structures in the United States between 1989 and 2000 and showed that the most recurrent causes of bridge failures were due to floods, scour, and impacts.

Brandimarte et al. [6] state that scour at bridge crossings is usually the result of the joint effects of three different scour processes (general scour, contraction scour, and local scour at piers) that may occur either independently or simultaneously, whose different origin suggests a different estimate of each individual scour contribution. Local scour usually results from the joint effect of contraction scour, due to the flow velocity increase associated with the reduction of the channel section, and the pier and abutment scour, due to the (local) alteration of the flow field induced by piers and abutments, Graf [7].

A system of vortices develops around a pier when the flow is obstructed by the pier. Brandimarte et al. [6] explain that depending on bridge geometry and flow conditions the system of vortices can be composed of all, any, or none of three individual basic systems acting at the pier: (a) the horse-vortex system at the base of the pier; (b) the wake-vortex system downstream of the pier; and (c) the surface roller ahead of the pier. Raudkivi [8], based on experimental observations of flow around piers, states that the horseshoe vortex is a consequence of scour, not the cause of it, although it becomes effective in transporting material away from the scour hole. The horseshoe vortex extends downstream, past the sides of the pier, for a few pier diameters before losing its identity and becoming part of general turbulence. In the scour hole, the horseshoe vortex pushes the maximum downflow velocity still closer to the pier. The downflow acts like a vertical jet eroding a groove in front of the pier. The eroded material is carried around the pier by the combined action of accelerating flow and the spiral motion of the “horseshoe vortex.” Melville and Coleman [9] report that the wake-vortex system acts like a vacuum cleaner sucking up stream bed material and transporting downstream of the pier the sediment moved by the downward flow and by the horse-vortex system.

Fischenich and Landers [11] report many useful observations about scour such as the following: generally, depths of local scour are much higher than general scour or contraction scour depths, often by a factor of ten, the wider the obstruction, the deeper the scour, the ratio of obstruction width to channel width is probably a better measure of scour potential than is the obstruction width alone, the limit on the increase in scour depth with an increase in projected length is when the projected length into the stream to the depth of the approaching flow is 25, the streamwise length of a structure has no appreciable effect on scour depth for straight sections, an increase in flow depth can increase scour depth by a factor of 2 or greater, scour depth also increases with the velocity of the approach flow, size of bed material may not affect the ultimate or maximum scour but only the time it takes to reach it, structures that cause flow convergence increase scour and vice versa, streamlining structures reduces the strength of horseshoe and wake vortices thus reducing ultimate scour depths, and ice and debris can increase both the local and general (contraction) scour.

To design safe bridges located on waterways under severe flooding conditions, many researchers have developed empirical formulae for predicting bridge pier scour depth. Review of the most-commonly used formulae is given in the next section.

#### 2. Review of Past Bridge Pier Scour Research

For more than half a century, a large number of equations have been suggested for estimating bridge pier scour. These equations are mostly empirical formulae which are often based on regression relations of laboratory and/or field scour data. Because the number of these equations is quite large, selection of the best performing equations is quite a difficult task. Comparison studies of scour formulae may help in selecting those formulae which have satisfactory performance.

There have been several studies in which comparisons were made of the performance of scour prediction equations. Mohamed et al. [12] selected four commonly used formulae for the validation process using both laboratory and field data. The selected equations were the Colorado State University (CSU), Melville and Sutherland, Jain and Fischer, and Laursen and Toch formulae. The experimental data were obtained from a laboratory model study at University Putra Malaysia while field data were obtained from 14 bridges from which 11 were in Pakistan, 2 in Canada, and 1 in India. They showed that the Laursen and Toch and the CSU formulae appeared to give a reasonable estimate when compared to laboratory and field data. The Melville and Sutherland and Jain and Fischer formulae appeared to overpredict the depth of the scour. The maximum absolute error between the field scour depths and the computed scour depths using the CSU formula was 3.15 m, while the minimum absolute error was 0.14 m.

Mueller and Wagner [13] in a detailed study used 266 pier scour measurements to investigate the performance of 27 pier scour equations. They reported that some equations (Ahmad, Breusers-Hancu, Chilate, Inglis-Poona I, Mellive and Sutherland, and Shen-Maza) show trends away from the line of equality (when graphing computed scour depths versus measured scour depths), indicating that those equations do not properly represent the processes responsible for local pier scour in the field. Several equations (Arkansas, Blench-Inglis I, Blench-Inglis II, Froehlich with no safety factor, Shen, and Simplified Chinese) underpredict the depth of scour for a significant number of observations and are not good candidates for design equations. The other equations have some trend along the line of equality with few underpredictions, but they display a broad scatter of data and often do not accurately predict the observed scour. Mueller and Wagner [13] stated that ranking the performance of scour prediction equations is difficult because of the tradeoff between accuracy and underpredictions. If only accuracy is considered, the sum of squared errors can be used to evaluate the equations’ performance. This statistic shows the Froehlich equation (no safety factor) to be the most accurate; however, the Froehlich equation is a regression expression and underpredicted the depth of scour for 129 of 266 field observations. If the smallest number of underpredictions is used, the Froehlich Design equation is the best because it underestimated only four observations. The Froehlich Design equation, however, ranked 19th based on the sum of squared errors criteria.

The author herein agrees with Mueller and Wagner [13] opinion that the magnitude of underpredictions is just as important (if not more important than) as the number of underpredictions; thus the sum of squared errors for those observations that were underpredicted is another important factor that should be considered. Mueller and Wagner [13] concluded that no single equation is conclusively better than the rest, but the top six equations for design purposes generally appear to be the Froehlich Design, HEC-18-K4, HEC-18-K4Mu, HEC-18-K4-Mo (>2 mm), Mississippi, and HEC-18 equations. The comparison of the scour depths predicted from these equations with measured scour depths shows processes in the field data not accurately accounted for in these equations. They report that the methodology for computing scour at bridges published in HEC-18 provides estimates that are generally conservative, in that the depth of scour is usually overpredicted.

Ghorbani [22] used data for 6 bridges on 3 rivers in Fars Province, Iran, to test several scour formulae. A comparison of scour equations with field measurements revealed that the Hanco, CSU, Viega, and Neill equations exhibited rather good agreement with field data; however, Indian and Inglis equations overestimated scour depth.

Lu et al. [23] collected field data, at the Si-Lo Bridge in the Lower Choshui River, Taiwan, comprising both general scour and total scour depths. They developed scour component separation methodology from which they obtained local pier scour. They used ten commonly used equilibrium pier scour equations, Neill [14], Shen et al. [15], Coleman [24], Breusers et al. [17], Jain and Fischers [25], Chiew and Melville [26], Froehlich [19], HEC-18 [27], Melville and Coleman [9], and Sheppard and Miller Jr. [28], to calculate local pier scour depths under peak flood conditions. They found that most formulae tend to overestimate the local scour depths.

Beg [29] selected fourteen commonly used and cited bridge pier scour predictors for testing against published laboratory and field data obtained from various sources and his experimental data. The study reveals that the predictors of Laursen and Toch and Jain and Fischer produce a reasonable estimate.

Recently, Sheppard et al. [21] used 569 laboratory and 928 field data for evaluation of existing equations for local scour at bridge piers. They started with twenty-three of the more recent and commonly used equilibrium scour equations for cohesionless sediments. Quality-control screening methods applied to both the data and the equations resulted in 441 laboratory and 791 field data and 17 predictive equations/methods. In their opinion, unknown maturity of the scour hole at the time of measurement for the field data resulted in use of field data only to evaluate underpredictions by the equations. They found that the regime equations of Inglis [30], Ahmad [31], and Chitale [32] yield negative scour depths in some cases, the Coleman [24] equation yields an unrealistic trend with increasing pier size, and Inglis [30], Ahmad [31], Chitale [32], Hancu [33], and Shen et al. [15] predict unreasonably high normalized scour depths. They also found that the predictive methods improve in accuracy over the years with those developed in recent years demonstrating the best performance. Sheppard et al. [21] concluded that Sheppard/Melville [21] method was found to be the most accurate method of those tested and is recommended for use in bridge design.

Gaudio et al. [34] state that existing scour equations were derived in small scale conditions and therefore the application to practical cases is uncertain. They selected six design equations, namely, Breusers et al. [17], Jain and Fischer [18], Froehlich [19], Kothyari et al. [35], Melville [20], and FHWA (HEC-18, [36]), for testing using synthetic data (obtained from Monte Carlo simulation technique) and the original field data set for uniform sediments. They found that the selected equations performance is not satisfactory when predicting maximum scour depth at equilibrium conditions.

The following review presents selected pier scour equations based on the foregoing review of past investigations. These equations represent scour research efforts spanned over nearly 50 years. These equations include the most influencing scour variables based on either laboratory or field data and are thought to perform at a satisfactory level.

Almost fifty years ago, Neill [14] used Laursen and Toch’s [37] design curve to obtain the following explicit formula for the scour depth:where is the equilibrium scour depth, is the obstruction width (or pier width), and is the approach water depth. This equation does not include the Froude number or in other words the velocity of the attacking stream.

Shen et al. [15] used the Froude number in their scour depth prediction in addition to the pier width aswhere is the Froude number and the other variables are as defined before.

The Colorado State University or CSU formula [16] is developed as a best fit to the data (laboratory) available at the time. The formula is given asThe CSU [16] formula is similar in form to Shen et al. [15] equation. Later on correction factors were added for effects of flow angle of attack, pier shape, and bed sediment conditions.

Breusers et al. [17] investigated clear water and live bed scour conditions. They included the critical velocity for incipient motion as follows:where is the average approach stream flow velocity and is the critical velocity for sediment motion computed with the Neill [38] equation in SI units aswhere the Shields mobility parameter, , can be computed based on sediment size as given in Mueller and Wagner (p. 20 in [13]), and is the median grain size. Therefore this equation includes implicitly the sediment size through the critical velocity.

Jain and Fischer [18] developed a set of equations based on laboratory data. For , in live bed conditions the formula reads aswhere and are the Froude number and critical Froude number, respectively. For in clear water conditions, the formula is For the larger value which is obtained from (6) and (7) is to be taken. This formula provides separate expressions for each of the clear and live bed conditions.

Froehlich’s [19] design equation for live bed scour at bridge crossings based on field data of about 170 live bed scour measurements iswhere is projected pier width with respect to the direction of the flow and is coefficient based on the shape of the pier nose. This equation includes a safety factor () added to the right hand side of (8) that accounts for contraction scour in most cases and this equation will be called herein Froehlich equation with safety. To test this formula as a prediction formula, this factor is omitted as in (9), as only local bridge pier scour is considered. Equation (9) is named herein Froehlich’s local scour equation with no safety:Melville [20] formula for clear water and live bed scour conditions iswhere , , , , , and are coefficients taking into account the depth scale, the flow intensity, the sediment size, pier shape, pier alignment, and channel geometry effects on scour depth, respectively.

Among the few analytically based equations is the one developed by Hafez [10] based on his energy balance theory. The energy balance theory assumes that at the equilibrium geometry of the scour hole the work done by the attacking fluid flow upstream of the bridge pier is equal to the work done in removing the volume of the scoured bed material out of the scour hole. In other words the energy contained in the fluid flow attacking the bridge pier is converted to an energy consumed in removing or transporting the bed material, thus forming a scour hole. When all the flow energy is consumed in transporting the sediment out of the scour hole, scour ceases and the scour hole becomes stable and at its maximum or equilibrium scour depth.

The following assumptions or postulates were made in Hafez [10] and repeated here for completeness and for comparison with the present study: the shape of the upstream slope of the scour hole in the stagnation vertical symmetry plane is linear; that is, the scour hole has a triangular shape, Figure 1(a); the equilibrium scour hole has an upstream slope that is equal to the angle of repose of the bed material; the scour hole is formed due to the conversion of the horizontal momentum of flow coming to the pier to downward or vertical momentum attacking the bed surface, Figure 1(a); the downflow component is responsible for transferring the momentum of the attacking flow to the bed material particles which are raised or transported to the original bed level and carried away by the horseshoe vortices; the analysis is done for a jet thickness of one sediment particle diameter (also unit width) which is similar to working in the stagnation symmetry plane (two-dimensional analysis); the resultant horizontal force of the attacking flow travels downward a distance equal to half the water depth plus half the scour depth; and the volume of the scoured bed which is assumed to be triangular in shape and as a mega sediment particle is moved to the original bed level out of the scour hole. Its center of mass is located at a distance equal to 1/3 of the scour depth, Figure 1(b). After using the above postulates the following equation was obtained by Hafez [10] aswhere is the maximum or equilibrium scour depth, is the water depth, is the upstream slope of the scour hole in the symmetry plane (assumed to be equal to the bed material angle of repose), is the bed material porosity, is the bed material specific gravity, is the pier width, is the channel width in case of one pier or the bridge span or pier centerline-to-centerline distance in case of multiple piers, is a transfer coefficient of the horizontal momentum into a vertical momentum in the downward direction, is the local longitudinal flow velocity just upstream of the pier of the jet attacking the bridge in the direction normal to the pier ( is assumed to be equal to the approach average velocity, , in case of no data), and is the gravitational acceleration.