Updating HEC-18 Pier Scour Equations

for Noncohesive Soils

PUBLICATION NO. FHWA-HRT-16-045 OCTOBER 2016

Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296
FOREWORD

Balancing safety and cost is critical to smart investment when estimating scour at bridge piers in 

noncohesive soils. This report summarizes a study to improve techniques for estimating scour 

under a broad range of conditions using quantitative measures of reliability and accuracy. Attention 

is focused on situations with higher uncertainty including sites with coarse bed materials and 

bridge designs with pier groups. This study will provide improved guidance to bridge engineers 

involved with foundation design. The study described in this report was conducted at the Federal 

Highway Administration Turner-Fairbank Highway Research Center J. Sterling Jones Hydraulics 

Laboratory.

Mayela Sosa

Acting Director, Office of Infrastructure

Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

in the interest of information exchange. The U.S. Government assumes no liability for the use 

of the information contained in this document. This report does not constitute a standard, 

specification, or regulation.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

manufacturers’ names appear in this report only because they are considered essential to the 

objective of the document.

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards 

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

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ensure continuous quality improvement.
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-16-045

2. Government Accession No. 3. Recipient’s Catalog No.

4. Title and Subtitle

Updating HEC-18 Pier Scour Equations for Noncohesive Soils

5. Report Date

October 2016

6. Performing Organization Code

7. Author(s)

Haoyin Shan, Roger Kilgore, Jerry Shen, and Kornel Kerenyi

8. Performing Organization Report No.

9. Performing Organization Name and Address

GENEX SYSTEMS, LLC

2 Eaton Street, Suite 603

Hampton, VA 23669

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Laboratory Report

January 2014–November 2015

14. Sponsoring Agency Code

15. Supplementary Notes

The Contracting Officer’s Technical Representative (COR) was Kornel Kerenyi (HRDI-50).

16. Abstract

A dataset of 594 bridge pier scour observations from two laboratory and three field studies was compiled. The dataset 

served as the testing ground for evaluating potential enhancements to the pier scour tools for noncohesive soils in

Hydraulic Engineering Circular 18 (HEC-18). In the current (fifth) edition of HEC-18, there are two primary equations 

for pier scour in noncohesive soils. One is the general equation applicable to most situations, including clear water and 

live bed conditions. The second is a coarse bed material equation recommended only for use under clear water 

conditions with coarse bed materials. The objective of this research was to determine if the coarse bed materials 

equation could be used for conditions beyond those under which it is currently limited. A framework for evaluating the 

two equations was developed using qualitative and quantitative tools.

The coarse bed equation is referred to as the Hager number/gradation coefficient (HN/GC) equation because it 

references the use of both in the equation formulation. After adjusting the HN/GC equation to a target reliability index 

of 2.0, it was evaluated on its ability to predict scour for a wide range of conditions in noncohesive soils. Partitioned 

subsets of the data based on key conditions—including the HEC-18 coarse bed criteria, clear water versus live bed 

transport conditions, gradation, and median grain size—were used for the evaluation. The equation performed 

reasonably consistently in all partitioned datasets, leading to the conclusion that it may be used for a broader range of 

conditions. A subgroup of pier group scour observations was assessed to determine if the equation could also be used 

for pier groups. The equation performed better for single piers but offered a basis for predicting local scour at pier 

groups. 

Considering these findings, the modified HN/GC equation is recommended for use on a broader range of noncohesive 

soil conditions for pier scour. Recommended limits for application of the equation are as follows: (1) clear water or live 

bed conditions (V1/Vc,50 < 5.2), (2) sands, gravels, and cobbles (0.0079 inches (0.2 mm) < D50 < 5 inches (127 mm)), 

(3) gradation coefficients ( ) less than 7.5, (4) Froude number less than 1.7, and (5) single piers.

17. Key Words

Bridge scour, local scour, pier scour, sediment gradation, nonuniform 

bed material, coarse bed material, reliability index, pier groups, Hager 

number.

18. Distribution Statement

No restrictions. This document is available 

to the public through the National Technical 

Information Service, Springfield, VA 

22161. 

http://www.ntis.gov/

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

29

22. Price

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized.

σ
ii

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius o

C 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit oF 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for the International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
iii

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION................................................................................................ 1

CHAPTER 2. DATA CHARACTERISTICS............................................................................. 3

CHAPTER 3. ANALYSIS OF PIER SCOUR............................................................................ 9

HEC-18 EQUATIONS ........................................................................................................... 10

Application of the HEC-18 Equations to the Data................................................................ 10

Error Analysis....................................................................................................................... 13

Limiting Predicted Scour ...................................................................................................... 17

CALIBRATING EQUATION COEFFICIENTS................................................................ 17

PIER GROUPS ....................................................................................................................... 19

CHAPTER 4. CONCLUSIONS AND RECOMMENDATIONS ........................................... 21

REFERENCES............................................................................................................................ 23
iv

LIST OF FIGURES

Figure 1. Equation. Pier scour in coarse bed materials (HEC-18).................................................. 1

Figure 2. Equation. Hager number.................................................................................................. 1

Figure 3. Equation. General pier scour equation in HEC-18.......................................................... 2

Figure 4. Graph. Distribution of grain size classification............................................................... 6

Figure 5. Equation. Laursen’s critical velocity equation ................................................................ 6

Figure 6. Equation. RI..................................................................................................................... 9

Figure 7. Equation. Ratio of measured to predicted scour depth.................................................... 9

Figure 8. Equation. RE.................................................................................................................... 9

Figure 9. Equation. RRMSE......................................................................................................... 10

Figure 10. Equation. Dimensionless scour depth.......................................................................... 10

Figure 11. Graph. Predicted versus measured scour: HEC-18 general pier scour equation......... 11

Figure 12. Graph. Predicted versus measured scour: HEC-18 coarse bed pier scour equation.... 11

Figure 13. Graph. Error versus bed load transport: general equation........................................... 14

Figure 14. Graph. Error versus bed load transport: coarse bed equation...................................... 14

Figure 15. Graph. Error versus y1/D50: general equation.............................................................. 15

Figure 16. Graph. Error versus y1/D50: coarse bed equation......................................................... 15

Figure 17. Graph. Error versus gradation coefficient: general equation....................................... 16

Figure 18. Graph. Error versus gradation coefficient: coarse bed equation ................................. 16

Figure 19. Equation. HN/GC equation for pier scour with RI = 2.0 ............................................ 17

Figure 20. Graph. Predicted versus measured scour: HN/GC equation with RI = 2.0 ................. 18

LIST OF TABLES

Table 1. Data sources...................................................................................................................... 3

Table 2. Noncohesive grain size classification for 594 pier scour observations............................ 5

Table 3. Ratio of velocity to critical velocity ................................................................................. 7

Table 4. Performance of HEC-18 general pier scour equation..................................................... 12

Table 5. Performance of HEC-18 coarse bed equation................................................................. 13

Table 6. Performance of the HN/GC equation with RI = 2.0 ....................................................... 19

Table 7. Performance of the HN/GC equation on pier type with RI = 2.0 ................................... 20
v

LIST OF SYMBOLS

a Pier diameter or width, ft (m)

D50 Median grain size of the sediment, ft (m)

D84 Grain size for which 84 percent (by weight) is smaller, ft (m)

Fr1 Froude number for approach flow, dimensionless

g Gravitational acceleration, ft/s2 (m/s2) 

H Hager number (densimetric particle Froude number), dimensionless

K1 Correction factor for pier nose shape

K2 Correction factor for angle of attack of flow

K3 Correction factor for bed condition

Kw Correction factor for wide piers in shallow flow

Sg Specific gravity of the sediment, dimensionless

V1 Approach flow velocity, ft/s (m/s)

Vc,50 Critical velocity based on D50, ft/s (m/s)

y1 Approach flow depth, ft (m)

ys Scour depth, ft (m)

ys,m Measured scour depth, ft (m)

ys,p Predicted scour depth, ft (m)

σ Sediment gradation coefficient (D84/D50), dimensionless

1

CHAPTER 1. INTRODUCTION

The fifth edition of Hydraulic Engineering Circular 18 (HEC-18) Evaluating Scour at Bridges

introduced a new equation for pier scour in coarse bed materials.(1) The equation, shown in 

figure 1, is based on work conducted by the Federal Highway Administration (FHWA) to 

develop more physics-based relations.

(2)

Figure 1. Equation. Pier scour in coarse bed materials (HEC-18).

Where:

ys = Scour depth, ft (m).

y1 = Approach flow depth, ft (m).

a = Pier diameter, ft (m).

K1 = Correction factor for pier nose shape, dimensionless.

K2 = Correction factor for angle of attack of flow, dimensionless.

σ = Sediment gradation coefficient (D84/D50), dimensionless.

H = Hager number (densimetric particle Froude number (Fr)), dimensionless.

As prescribed in HEC-18, this equation is only applicable to clear water flow conditions and to 

what is described as coarse bed materials. Coarse bed materials are defined as those with D50

greater than or equal to 0.79 inches (20.1 mm) and a gradation coefficient greater than or equal 

to 1.5.

The Hager number used in the equation was developed by Oliveto and Hager and is defined in 

figure 2.

(3,4)

Figure 2. Equation. Hager number.

Where:

V1 = Approach velocity, ft/s (m/s).

D50 = Median grain size, ft (m).

Sg = Specific gravity of the sediment, dimensionless.

g = Gravitational acceleration, ft/s2 (m/s2).

The equation in figure 1 was provided to supplement the primary scour equation in HEC-18, 

which is shown in figure 3.

yyss

aa0.62yy1

0.38 = 1.1KK1KK2 �tt tth � HH2

1.97σσ1.5��

HH = VV1

�gg�SSgg − 1�DD50
2

Figure 3. Equation. General pier scour equation in HEC-18.

Where:

K3 = Correction factor for bed condition, dimensionless.

Fr1 = Approach flow Fr, dimensionless.

The equation in figure 3 was modified from previous versions by dropping a coarse bed 

adjustment factor, K4, when the equation in figure 1 was introduced. Current guidance is to use 

the equation in figure 1 for conditions to which it is applicable and then to use the equation in 

figure 3 for most other conditions.

The objective of the research described in this report is to determine if the equation described in 

figure 1 can be used for conditions beyond those to which it is currently limited.

yyss

aa0.65yy1

0.35 = 2.0KK1KK2KK3[FFFF1

0.43]
3

CHAPTER 2. DATA CHARACTERISTICS

The development of the pier scour equation for coarse bed materials in figure 1 was based on

datasets from Colorado State University (CSU), the FHWA Turner-Fairbank Highway Research

Center, and the U.S. Geological Survey (USGS), which are listed as the first four datasets in 

table 1.

(2) These data included a range of conditions with observations of clear water and live bed 

scour. For the previous study, only the clear water data were used for the development of the 

equation in figure 1.

Table 1. Data sources.

Source

Total Number of 

Scour Observations Data Type

CSU(5) 184 Laboratory

FHWA(2) 20 Laboratory

USGS-2011(6) 103 Field

USGS-2004(7) 42 Field

USGS-1995(8) 384 Field

Total* 694 Laboratory/Field

*Thirty-nine observations are included in both the USGS-2004 and 

USGS-1995 datasets.

In addition, the last dataset shown in the table (USGS-1995) will be used for the analyses in this 

study. Combined, these datasets provide 694 unique pier scour observations.

Selected data observations were removed from the dataset either because of incomplete data or 

because of inappropriate data for the study. In total, 12 observations were filtered out of the CSU 

laboratory dataset, and 88 observations were filtered out of the USGS-1995 dataset for the 

following reasons:

• USGS-1995 field data. The following changes were made:

o Fifty-eight observations were actually 29 pairs of measured scour depths upstream 

and downstream of the same pier. The smaller of the two measurements was 

deleted (29 observations).

o Scour depths that were minimal (measured scour depth less than 0.2 inches

(5.1 mm) or measured scour depths less than the reported error of field 

observation) were deleted (38 observations).

o Soils that were cohesive were deleted (7 observations).

o Observations with unusually high gradation coefficients (5 with a gradation 

coefficient of 12.1 and 2 with a gradation coefficient of 20.4) were deleted 

(7 observations).

o Soils with no gradation information were deleted (7 observations).

• CSU laboratory data. Scour depths that were minimal (measured scour depth less than 

0.2 inches (5.1 mm)) were deleted (12 observations).
4

After filtering, 594 pier scour observations remained for further analysis. Of these, 402 were 

from the 3 USGS field datasets, and 192 were from the CSU and FHWA laboratory datasets.

The remaining data included a variety of pier types. Within the field data, there were 

60 observations with pier groups and 342 observations with single piers. The single pier 

types/nose types were cylindrical (8), round nose (158), sharp nose (141), square nose (27), and 

unknown (8). The unknown observations were treated as round nose. All of the laboratory data 

were for single cylindrical piers.

The data contained a broad range of grain size classes for noncohesive soils. Table 2 summarizes 

the grain size classification and gradation according to the system of Krumbein and Aberdeen.

(9)

The data range from fine sand to cobble sizes based on the D50. The most numerous size range 

represented in the data is coarse sand (215 observations). 

The gradation coefficient ranges from 1.15 to 7.22. For this study, a noncohesive soil is 

considered uniform when the gradation coefficient is less than 1.5. With the exception of the 

very fine gravel, nonuniform gradations make up at least half of the observations for each soil 

class.

The definition of soil gradation coefficient used in this study and in HEC-18 is D84/D50. Another 

definition used is (D84/D16)

0.5.(2) These two definitions are equivalent when D84/D50 = D50/D16. 

Landers and Mueller reported gradation coefficients for the 384 data observations included in 

that study.(8) Each gradation coefficient was recomputed using the definition for this study and 

compared with the reported values. For 123 observations, the recomputed values matched the 

reported values. However, for 132 observations, the recomputed values exceeded the reported 

values, but for 129 observations, the recomputed values were less than the reported values. In all 

cases, the recomputed values were used for this study because they are supported by the data.
5

Table 2. Noncohesive grain size classification for 594 pier scour observations.

D50

Range 

(mm)

Grain Size 

Classification Observation

D50 (mm) Gradation Coefficient,

Minimum Maximum Median Mean

Standard 

Deviation Minimum Maximum Median Mean

Standard

Deviation

0.125–

0.25 Fine sand 13 0.17 0.18 0.18 0.18 0.003 1.47 2.06 2.06 2.01 0.16

0.25–

0.5 Medium sand 39 0.28 0.45 0.38 0.36 0.06 1.33 2.47 2.10 1.96 0.46

0.5–1 Coarse sand 215 0.50 0.94 0.75 0.72 0.11 1.28 5.22 2.43 2.53 0.70

1–2 Very coarse sand 74 1.00 1.87 1.80 1.59 0.37 1.15 7.22 3.70 3.56 2.11

2–4 Very fine gravel 6 2.00 2.85 2.00 2.14 0.35 1.35 1.65 1.35 1.40 0.12

4–8 Fine gravel 34 4.00 7.60 6.90 6.67 1.00 2.17 6.50 2.17 2.84 1.11

8–16 Medium gravel 23 8.00 15.00 9.63 10.38 2.60 1.40 4.14 3.75 3.61 0.74

16–32 Coarse gravel 70 16.70 31.30 27.00 24.15 5.41 1.28 4.14 1.94 2.08 0.75

32–64 Very coarse gravel 61 32.00 60.70 49.30 47.18 9.56 1.18 3.98 1.99 2.11 0.52

64–256 Cobble 59 64.70 108.00 73.00 81.68 13.72 1.29 2.56 2.05 1.96 0.38

1 inch = 25.4 mm.

σ
6

The distribution of the 594 bed materials by grain size classification is shown in figure 4. While 

the field data include observations in all 10 grain size classifications, the laboratory data from 

CSU and FHWA include observations in five of the classifications: medium sand, coarse sand, 

very coarse sand, very fine gravel, and coarse gravel. Among all grain size classes, coarse sand is 

the most frequently represented with more than 35 percent of the observations. The least 

represented is very fine gravel, with only six observations, including five from the laboratory 

data and one from the field. 

Figure 4. Graph. Distribution of grain size classification. 

The dataset includes observations of both clear water and live bed scour. If the approach velocity 

is greater than the critical velocity, live bed conditions exist; if not, clear water conditions 

govern. The critical velocity is calculated by Laursen’s critical velocity equation, as shown in 

figure 5.

(1)

Figure 5. Equation. Laursen’s critical velocity equation.

Where:

Vc,50 = Critical velocity based on D50, ft/s (m/s).

Ku = Unit conversion constant, 11.17 for U.S. customary units (6.19 for SI units).

Table 3 summarizes the ratio of V/Vc,50 for determining clear water versus live bed conditions for 

the dataset. The maximum ratio of the average velocity to the critical velocity is 5.13 in the 

medium sand classification. The ratios range from 0.29 to 2.61 for the other grain size 

classifications.

0%

5%

10%

15%

20%

25%

30%

35%

40%

Fine

sand

Medium

sand

Coarse

sand

Very

coarse

sand

Very

fine

gravel

Fine

gravel

Medium

gravel

Coarse

gravel

Very

coarse

gravel

Cobble

Percentage

VVcc,50 = KKuu yy1

1/6

DD50

1/3
7

Table 3. Ratio of velocity to critical velocity.

D50 Range 

(mm)

Grain Size 

Classification Observation

V/Vc,50

Minimum Maximum Median Mean

Standard 

Deviation

0.125–0.25 Fine sand 13 0.49 1.54 0.78 0.84 0.28

0.25–0.5 Medium sand 39 0.45 5.13 1.51 2.21 1.53

0.5–1 Coarse sand 215 0.34 2.31 0.83 0.97 0.45

1–2 Very coarse sand 74 0.29 2.58 0.83 1.04 0.60

2–4 Very fine gravel 6 0.82 1.32 0.82 0.91 0.20

4–8 Fine gravel 34 0.36 2.61 1.22 1.18 0.40

8–16 Medium gravel 23 0.61 1.56 1.15 1.07 0.29

16–32 Coarse gravel 70 0.38 2.02 0.90 0.95 0.30

32–64 Very coarse gravel 61 0.43 1.46 0.79 0.82 0.21

64–256 Cobble 59 0.37 1.35 0.74 0.74 0.17

1 inch = 25.4 mm.

9

CHAPTER 3. ANALYSIS OF PIER SCOUR

As previously stated, the objective of this research is to determine if the equation described in 

figure 1 can be used for conditions beyond those for which it is currently limited. The pier scour 

equations are evaluated using the scour observations from the field and laboratory datasets. The 

evaluation will compare the performance of the equations on data within the ranges specified in 

HEC-18 as well as beyond those ranges.

Performance is assessed by visual inspection of graphs of predicted and measured scour and by 

quantitative measures of reliability and accuracy. Reliability is defined by the reliability index 

(RI), which measures the risk of underpredicting the actual value of some property, which, in this 

case, is scour depth. A higher RI indicates a lower risk of underpredicting scour depth. RI is 

defined in general terms in figure 6.

Figure 6. Equation. RI.

Where:

Mx = Mean of a set of x values (dimension is that of the x values).

Sx = Standard deviation of a set of x values (dimension is that of the x values).

For this analysis, x is defined as shown in figure 7.

Figure 7. Equation. Ratio of measured to predicted scour depth.

Where:

x = Ratio of measured to predicted scour depth, dimensionless.

ys,m = Measured scour depth, ft (m).

ys,p = Predicted scour depth, ft (m).

To facilitate direct comparison between the laboratory and field data, accuracy is characterized 

by the relative error (RE) and the relative root mean square error (RRMSE). RE is defined in 

figure 8, and RRMSE is defined in figure 9.

Figure 8. Equation. RE.

RRRR = 1 − MMxx

SSxx

xx = yyss,mm

yyss,pp

RRRR = yyss,pp − yyss,mm

yyss,mm
10

Figure 9. Equation. RRMSE.

Where:

n = Number of observations.

HEC-18 EQUATIONS

The equations for general pier scour (figure 3) and pier scour for coarse bed materials (figure 1) 

both employ the following adjustments described by HEC-18:

• K1.

• K2.

• The correction factor (Kw) for wide piers in shallow water.

• The rules of thumb for maximum scour depth for round nose piers.

In addition, the general pier scour equation employs K3 as described in HEC-18.

Comparison between measured and predicted scour estimates will employ dimensionless scour 

depths so that field and laboratory data may be directly compared. The dimensionless scour 

depth is defined in figure 10.

Figure 10. Equation. Dimensionless scour depth.

Application of the HEC-18 Equations to the Data

Application of the HEC-18 general pier scour equation to the full dataset resulted in predicted 

scour estimates that were compared to measured scour estimates in figure 11. Because the 

HEC-18 equation is intended for design, it is not surprising that predicted scour is predominately 

greater than measured scour. A limited number of data points are not shown in the figure because

the dimensionless predicted scour goes as high as 8.5.

Using the HEC-18 coarse bed material equation in the same manner results in the comparison 

summarized in figure 12. As a design equation, it also generally estimates scour depths 

exceeding measured values, though the patterns are not identical. For example, this equation 

results in a larger number of underpredicted scour values. As before, a limited number of data 

points are not shown in the figure because the dimensionless predicted scour for this equation 

goes as high as 10.2.

RRRR RR = ����

yyss,pp − yyss,mm

yyss,mm

�

2

� /( − 1)

dddd dd dddd dddd sscc = yyss

aa
11

Figure 11. Graph. Predicted versus measured scour: HEC-18 general pier scour equation.

Figure 12. Graph. Predicted versus measured scour: HEC-18 coarse bed pier scour 

equation.
12

Table 4 provides detailed information on the performance of the HEC-18 general pier scour 

equation. The reliability of the equation reflected in the RI and the accuracy of the equation 

reflected in the RRMSE are provided for the data as a whole and partitioned into various groups. 

Results for the field and laboratory data are also differentiated.

Table 4. Performance of HEC-18 general pier scour equation.

Data Description

HEC-18 Coarse 

Bed Criteria

Bed Load 

Transport Gradation

D50 ≥ 

0.79 Inches

Source Parameter All Yes No

Clear

Water

Live 

Bed Uniform Nonuniform Yes No

Field n 402 120 282 218 184 31 371 168 234

RI 2.56 6.22 2.16 2.42 2.75 2.29 2.58 1.10 1.24

RRMSE 3.42 4.28 2.98 3.56 3.25 2.95 3.46 1.92 3.02

Lab n 192 0 192 163 29 52 140 0 192

RI 2.25 0.00 2.25 2.40 2.12 1.72 2.56 0.00 2.25

RRMSE 5.10 0.00 5.10 5.52 1.18 3.80 5.51 0.00 5.10

Total n 594 120 474 381 213 83 511 168 426

RI 2.46 6.22 2.20 2.41 2.68 1.93 2.58 1.10 1.54

RRMSE 4.04 4.28 3.98 4.50 3.05 3.51 4.12 1.92 4.09

1 inch = 25.4 mm.

The RI and RRMSE measures for the combined dataset of 594 scour observations are 2.46 and 

4.04, respectively. In the columns labeled HEC-18 Coarse Bed Criteria, the data are partitioned 

as follows based on the criteria described in HEC-18 for use of the coarse bed equation:

• D50 ≥ 0.79 inches (20.1 mm).

• σ ≥ 1.5 (nonuniform gradation).

• Clear water scour: V1/Vc,50 < 1. 

When all three criteria are met, HEC-18 recommends that the coarse bed pier scour equation be 

used rather than the general pier scour equation. For the field data, the RI of the general pier 

scour equation is 6.22 for the 120 observations meeting the criteria as coarse bed materials, while 

it is only 2.16 for the 282 observations not meeting the criteria. These results show that the 

HEC-18 general pier scour is more reliable when applied to data meeting the coarse bed material 

criteria than it is for data not meeting the criteria. However, the reverse is true in terms of 

accuracy where the equation is more accurate for data not meeting the criteria (RRMSE = 2.98) 

compared with data meeting the criteria (RRMSE = 4.28).

Table 4 also summarizes the data partitioned by nonuniform ( ≥ 1.5) versus uniform gradations 

as well as by median grain size greater than or equal to 0.79 inches (20.1 mm). None of the 

laboratory observations included data with D50 greater than 0.79 inches (20.1 mm).

Table 5 provides the performance information for the HEC-18 coarse bed material equation. The 

RI and RRMSE measures for the combined dataset of 594 scour observations are 1.19 and 2.66, 

respectively. Compared to the general pier scour equation, the coarse bed material scour equation 

is less reliable (more prone to underpredicting) but more accurate (lower RRMSE). Both of these 

observations are apparent when comparing figure 11 with figure 12.

σ
13

With respect to the HEC-18 coarse bed criteria, the HEC-18 coarse bed equation is significantly 

more accurate for the field data meeting the criteria (RRMSE = 1.60) than for those data not 

meeting the criteria (RRMSE = 2.94). However, it is also less reliable (RI = 0.88 versus RI = 

1.34) when considering the field data only.

Table 5. Performance of HEC-18 coarse bed equation.

Data Description

HEC-18 

Coarse Bed 

Criteria

Bed Load 

Transport Gradation

D50 ≥ 

0.79 Inches

Source Parameter All Yes No

Clear

Water

Live 

Bed Uniform Nonuniform Yes No

Field n 402 120 282 218 184 31 371 168 234

RI 1.18 0.88 1.34 0.91 1.70 1.61 1.16 1.10 1.24

RRMSE 2.61 1.60 2.94 2.05 3.16 2.59 2.62 1.92 3.02

Lab n 192 0 192 163 29 52 140 0 192

RI 1.23 0.00 1.23 1.38 0.66 0.75 1.51 0.00 1.23

RRMSE 2.75 0.00 2.75 2.97 0.59 2.79 2.74 0.00 2.75

Total n 594 120 474 381 213 83 511 168 426

RI 1.19 0.88 1.30 1.06 1.58 1.03 1.22 1.10 1.23

RRMSE 2.66 1.60 2.87 2.49 2.94 2.72 2.65 1.92 2.90

1 inch = 25.4 mm.

Error Analysis

Prediction errors for both equations were evaluated against a variety of variables to expose trends 

that might exist. Figure 13 and figure 14 display the RE versus the ratio of the approach velocity 

to the critical velocity for the general and coarse bed equations, respectively. For ratios less than 

1, clear water conditions exist, while for ratios greater than 1, live bed conditions exist. There is 

no apparent trend in RE with increasing bed transport capacity for either equation. In fact, it 

appears that for both equations, the distinction between live bed and clear water is not 

meaningful. A few observations (6 for the general pier scour equation and 2 for the coarse bed 

pier scour equation) exhibited REs exceeding 14 and are not shown on the figures.

Figure 15 and figure 16 present the REs versus the ratio of approach depth to median grain size 

for the general and coarse bed equations, respectively. For the general equation, a trend of 

decreasing error with increases in the ratio seems apparent, especially if the observations with 

the greatest errors are removed. Conversely, for the coarse bed equation, a trend of increasing 

error with increases in the ratio seems apparent.

Figure 17 and figure 18 show RE versus the gradation coefficient. Again, there is no discernable 

pattern in the error distributions for either equation between uniform versus nonuniform 

gradations.
14

Figure 13. Graph. Error versus bed load transport: general equation.

Figure 14. Graph. Error versus bed load transport: coarse bed equation. 
15

Figure 15. Graph. Error versus y1/D50: general equation.

Figure 16. Graph. Error versus y1/D50: coarse bed equation.
16

Figure 17. Graph. Error versus gradation coefficient: general equation.

Figure 18. Graph. Error versus gradation coefficient: coarse bed equation.
17

Limiting Predicted Scour

For round nose piers aligned with the flow direction, HEC-18 recommends that predicted scour 

be limited to 2.4 times the pier width when the Fr is less than 0.8 and limited to 3.0 times the pier 

width when the Fr is greater than 0.8. These limits governed for 22 observations when using the 

general equation and for 20 observations when using the coarse bed equation. (Nine additional 

predicted scour estimates were limited by the wide pier adjustment when using both equations.)

The maximum measured dimensionless pier scour values were 2.6 in the field data and 1.8 in the 

laboratory data. Limiting the dimensionless predicted scour to 3.0 was evaluated relative to its 

effect on the RI and RRMSE. Such a limit only affects the field data because all predicted 

dimensionless scour values from both equations are less than 3.0 in the laboratory data. The 

results of this evaluation were mild increases in the accuracy and mild decreases in the reliability 

of both equations. Because development of such an adjustment was not an objective of this 

research, no further consideration was given. However, additional research into such a limit may 

be useful.

CALIBRATING EQUATION COEFFICIENTS

Bridge designers seek reliable (low risk of underdesign) and cost-effective (accurate) estimates 

of pier scour. Because a higher reliability generally means a lower accuracy, finding the most 

appropriate balance is a challenge. The trade-off is further complicated because although the 

study dataset is large and includes a wide variety of situations, it may not be representative for 

specific bridge design situations.

To address these challenges and to achieve the study objective, the coefficient from the HEC-18 

coarse bed material equation (figure 1) was adjusted to a target RI. The selection of an 

appropriate target RI is best accomplished within the context of a larger risk analysis. For this 

study, it is noted that, for the general pier scour equation (figure 3) applied to the current dataset 

for the data not meeting the coarse bed criteria, the RI is 2.20 for 474 field and laboratory 

observations. For the coarse bed pier scour equation (figure 1), the RI is 0.88 for 120 field 

observations. Therefore, a target reliability of 2.0 is used that roughly reflects the composite 

reliability for pier scour estimates under current HEC-18 guidance.

To achieve an RI of 2.0 for the Hager number/gradation coefficient (HN/GC) based pier scour 

equation (figure 1), the coefficient required is 1.32. The resulting equation is shown in figure 19. 

Note that the adjustment for bed condition, K3, has been reintroduced to the HN/GC equation to 

evaluate its use for clear water (K3 = 1.1) and live bed (1.1 ≤ K3 ≤ 1.3) conditions. Because the 

previous analyses showed equivalent performance under a wide range of conditions, the equation 

is referred to as the HN/GC equation rather than as a coarse bed pier scour equation.

Figure 19. Equation. HN/GC equation for pier scour with RI = 2.0.

yyss

aa0.62yy1

0.38 = 1.32KK1KK2KK3 �tt tth � HH2

1.97σσ1.5��
18

Application of the HN/GC pier scour equation to the full dataset results in predicted 

dimensionless scour estimates that are compared to measured dimensionless scour estimates in 

figure 20. Because the coefficient was increased, the predicted scour estimates also increased.

The maximum scour predictions using the equation were limited by the following:

• Wide pier adjustment using Kw.

• ys/a ≤ 2.4 for round nose piers aligned with the flow and Fr ≤ 0.8.

• ys/a ≤ 3.0 for round nose piers aligned with the flow and Fr > 0.8.

For the HN/GC equation, the limits were enforced for 76 field observations and 9 laboratory 

observations.

Figure 20. Graph. Predicted versus measured scour: HN/GC equation with RI = 2.0.

Table 6 provides the performance information for the HN/GC equation. The RI and RRMSE 

measures for the combined dataset of 594 scour observations are 2.0 (by design) and 3.67, 

respectively. The original development of the HN/GC equation was accomplished on a subset of 

the current dataset that generally (though not exclusively) met the requirements for application as 

the HEC-18 coarse bed equation. Therefore, when partitioning the data between those that meet 

the criteria and those that do not, it is not surprising that the equation is more accurate (lower 

RRMSE) for those data that meet the criteria. As shown in table 6, the RRMSE for data meeting 

the criteria is 2.34 while for the remaining data it is a much higher 3.94. However, because of the 

typical trade-off between accuracy and reliability, the equation is less reliable for the data 

meeting the criteria (RI = 1.68) than for the data not meeting the criteria (RI = 2.12).
19

For the bed load partitioning, the same ambiguity between accuracy and reliability exists. The 

equation is more accurate in estimating clear water scour but more reliable in estimating live bed 

scour.

The gradation partition reveals virtually no distinction between uniform and nonuniform 

gradation. For the uniform and nonuniform partitions, the RIs are 2.01 and 2.02, respectively, 

and the RRMSEs are 3.75 and 3.66, respectively.

For D50, the picture is slightly different. The reliability is virtually identical between the two

partitions: 1.96 for D50 ≥ 0.79 inches (20.1 mm) and 2.03 for D50 < 0.79 inches (20.1 mm). 

However, for the same level of reliability, the equation is more accurate for the larger D50

(RRMSE = 2.75) compared with the smaller D50 (RRMSE = 3.97).

Table 6. Performance of the HN/GC equation with RI = 2.0.

Data Description

HEC-18 

Coarse Bed 

Criteria

Bed Load 

Transport Gradation

D50 ≥ 

0.79 Inches

Source Parameter All Yes No

Clear

Water

Live 

Bed Uniform Nonuniform Yes No

Field n 402 120 282 218 184 31 371 168 234

RI 1.93 1.68 2.08 1.61 2.55 2.74 1.89 1.96 1.93

RRMSE 3.61 2.34 4.03 2.87 4.32 3.62 3.60 2.75 4.11

Lab n 192 0 192 163 29 52 140 0 192

RI 2.24 0.00 2.24 2.40 2.00 1.65 2.60 0.00 2.24

RRMSE 3.80 0.00 3.80 4.11 0.89 3.82 3.79 0.00 3.80

Total n 594 120 474 381 213 83 511 168 426

RI 2.00 1.68 2.12 1.85 2.49 2.01 2.02 1.96 2.03

RRMSE 3.67 2.34 3.94 3.46 4.02 3.75 3.66 2.75 3.97

1 inch = 25.4 mm.

Recall that the objective of this study is to determine if the equation described in figure 1 can be 

used for conditions beyond those to which it is currently limited. The previous observations 

suggest that the equation, modified with a revised coefficient to achieve a RI of 2.0, can be used 

for a broader range of conditions. These conditions are summarized as follows:

• Clear water or live bed conditions (V1/Vc,50 < 5.2).

• Sands, gravels, and cobbles (0.0079 inches (0.21 mm) < D50 < 5.0 inches (127 mm)).

• Gradation coefficients ( ) less than 7.5.

• Fr less than 1.7.

PIER GROUPS

The HEC-18 pier scour methodology includes guidance to estimate scour for piers that are 

composed of groups of cylinders. Of the 402 field scour data observations, 60 involve pier 

groups. None of the laboratory observations include pier groups.

σ
20

Table 7 provides the performance metrics for pier groups with the HN/GC equation from figure 

19. The pier group observations show a significantly higher RI than for the single pier 

observations but with significantly lower accuracy. The HN/GC equation overpredicts all 

60 scour observations.

Table 7. Performance of the HN/GC equation on pier type with RI = 2.0.

Data Description Pier Type

Source Parameter All Group Single

Field n 402 60 342

RI 1.93 3.36 1.81

RRMSE 3.61 5.79 3.07

Although the HN/GC could be used for pier groups, the addition of an adjustment factor 

reflecting pier groups might be appropriate. For the HN/GC equation, the factor would be less 

than one. It is recommended that this be investigated further for both this equation and the 

equation in figure 3. The 60 data observations in the current dataset could be used as a basis for 

further evaluation.
21

CHAPTER 4. CONCLUSIONS AND RECOMMENDATIONS

A dataset of 594 pier scour observations from 2 laboratory and 3 field studies was compiled. The 

dataset served as the testing ground for evaluating potential enhancements to the pier scour tools 

for noncohesive soils in HEC-18. In the current (fifth) edition of HEC-18, there are two primary 

equations for pier scour in noncohesive soils. The first is the general equation applicable to most 

situations including clear water and live bed conditions. The second is a coarse bed material 

equation recommended only for use under clear water conditions with coarse bed materials. 

Coarse bed materials are defined as those where the D50 is greater than or equal to 0.79 inches

(20.1 mm) and the gradation coefficient, , is greater than or equal to 1.5.

The original objective of this research was to determine if the coarse bed materials equation 

described in figure 1 could be used for conditions beyond those under which it is currently 

limited. A framework for evaluating the pier scour equations was developed using qualitative 

and quantitative tools. Qualitative tools included visual inspection of comparative graphs. 

Quantitative tools included the RI as a measure of equation reliability and the RRMSE as a 

measure of equation accuracy.

The HN/GC equation was adjusted to a target RI of 2.0. It was evaluated by partitioning the 

dataset into two groups based on the key conditions, including the HEC-18 coarse bed criteria, 

clear water versus live bed transport conditions, gradation, and median grain size. The equation 

performed reasonably consistently in all partitioned datasets; this result leads to the conclusion 

that it can be used for a broader set of conditions.

A subset of the dataset included pier group scour observations. The equation performed better for 

single piers but offered a basis for predicting local scour at pier groups. The HN/GC equation 

tended to provide predictions higher than desired for pier groups and could be adjusted 

downward.

Considering all of these findings, the HN/GC equation in figure 19 is recommended for a broad 

range of conditions. It is further recommended that the application be generally limited to the 

following conditions:

• Clear water or live bed conditions (V1/Vc,50 < 5.2).

• Sands, gravels, and cobbles (0.0079 inches (0.21 mm) < D50 < 5.0 inches (127 mm)).

• Gradation coefficients ( ) less than 7.5.

• Fr less than 1.7.

• Single piers.

Designers may choose to extend these limits after site-specific evaluation of the risks and 

alternatives. The evaluation of pier groups in the previous chapter may be a starting point for 

considering the application of the equation to pier groups. 

In addition to these primary conclusions and recommendations, this study of pier scour equations 

revealed the following issues, which warrant further investigation:

σ

σ
22

• Trends in errors did not exist for most variables involved with scour prediction. The one 

exception was that for both equations (HEC-18 general scour and HN/GC), trends in 

error were observed with increases in the ratio of approach depth to median grain size 

(y1/D50). These trends should be further investigated and possibly eliminated.

• Refinements either to the HEC-18 method for analyzing pier groups or an adjustment to 

the recommended equation should be provided to improve scour predictions for pier 

groups.

• The RI was employed as a quantitative measure of reliability indicating the likelihood of 

overpredicting actual scour. A target value of 2.0 was selected for this study based on an 

assessment of the reliability of current practice. Further evaluation of the appropriate 

reliability level would inform trade-offs between conservativism and costs.

• Investigation of other quantitative means of estimating reliability and accuracy could 

improve the development and testing of scour estimation methods.
23

REFERENCES

1. Arneson, L.A., Zevenbergen, L.W., Lagasse, P.F., and Clopper, P.E. (2012). Evaluating 

Scour at Bridges, Fifth Edition, Report No. FHWA-HIF-12-003-HEC-18, Federal Highway 

Administration, Washington, DC.

2. Guo, J., Suaznabar, O., Shan, H., and Shen, J. (2012). Pier Scour in Clear-Water Conditions 

with Non-Uniform Bed Materials, Report No. FHWA-RD-12-022, Federal Highway 

Administration, Washington, DC.

3. Oliveto, G. and Hager, W.H. (2002). “Temporal Evolution of Clear-Water Pier and 

Abutment Scour,” Journal of Hydraulic Engineering, 128(9), 811–820.

4. Oliveto, G. and Hager, W.H. (2005). “Further Results to Time-Dependent Local Scour at 

Bridge Elements,” Journal of Hydraulic Engineering, 131(2), 97–105.

5. Molinas, A. (2003). Bridge Scour in Nonuniform Sediment Mixtures and in Cohesive 

Materials: Synthesis Report, Report No. FHWA-RD-03-083, Federal Highway 

Administration, Washington, DC.

6. Holnbeck, S.R. (2011). Investigation of Pier Scour in Coarse-Bed Streams in Montana, 2001 

through 2007, USGS Scientific Investigations Report 2011–5107, U.S. Geological Survey, 

Reston, VA.

7. Chase, K.J. and Holnbeck, S.R. (2004). Evaluation of Pier-Scour Equations for Coarse-Bed 

Streams, USGS Scientific Investigations Report 2004-5111, U.S. Geological Survey, Reston, 

VA.

8. Landers, M.N. and Mueller, D.S. (1995). Channel Scour at Bridges in the United States, 

Report No. FHWA-RD-95-184, Federal Highway Administration, Washington, DC.

9. Krumbein, W.C. and Aberdeen, E. (1937). “The Sediments of Baratatia Bay,” Journal of 

Sedimentary Research (SEPM), 7(1), 3–17.
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