Enhanced Abutment Scour Studies

for Compound Channels 

PUBLICATION NO. FHWA-RD-99-156 AUGUST 2004

Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

ENHANCED ABUTMENT SCOUR STUDIES 

FOR COMPOUND CHANNELS

 Report No. FHWA-RD-99-156

 
FOREWORD 

This report describes a laboratory study of abutment scour for compound channels where the 

experiments simulated floodplains with defined channel and overbank flow areas at different 

elevations. A new abutment scour prediction equation was derived as part of this study and is 

presented in an appendix of the fourth edition of the Federal Highway Administration Hydraulic 

Engineering Circular No. 18 (HEC-18). This report will be of interest to bridge engineers and 

hydraulic engineers involved in bridge scour evaluations and to researchers involved in 

developing improved bridge scour evaluation procedures. It is being published as a Web 

document only. 

 T. Paul Teng, P.E. 

 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 

contents or the use thereof. This report does not constitute a standard, specification, or 

regulation. 

The U.S. Government does not endorse products or manufacturers. Trade or manufacturers’ 

names appear herein only because they are considered essential to the objective of this 

document. 
1. Report No. 

FHWA-RD-99-156 

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

5. Report Date 

August 2004 

6. Performing Organization Code 

4. Title and Subtitle 

Enhanced Abutment Scour Studies for Compound Channels 

7. Author(s) 

Terry W. Sturm 

8. Performing Organization Report No.

10. Work Unit No. (TRAIS) 

 

11. Contract or Grant No. 

DTFH61-94-C-00198 

9. Performance Organization Name and Address 

Georgia Institute of Technology 

School of Civil and Environmental Engineering 

Atlanta, GA 30332

13. Type of Report and Period Covered 

Final Report 

1994-1999

12. Sponsoring Agency and Address 

Office of Infrastructure Research and Development 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101-2296

14. Sponsoring Agency Code 

15. Supplementary Notes 

Contracting Officer’s Technical Representative (COTR): J. Sterling Jones, HRDI-07 

16. Abstract 

Experimental results and analyses are given in this report on bridge abutment scour in compound channels. 

Experiments were conducted in a laboratory flume with a cross section consisting of a wide floodplain adjacent to 

a main channel. The embankment length, discharge, sediment size, and abutment shape were varied, and the 

resulting equilibrium scour depths were measured. Water-surface profiles, velocities, and scour-hole contours were 

also measured. In the report, a methodology is developed for estimating abutment scour that takes into account the 

redistribution of discharge in the bridge contraction, abutment shape, sediment size, and tailwater depth. The 

independent variables in the proposed scour formula are evaluated at the approach-channel cross section and can 

be obtained from a one-dimensional water-surface profile computer program such as the Water-Surface Profile 

Program (WSPRO). The proposed scour evaluation procedure is outlined and illustrated, including consideration 

of the time required to reach equilibrium scour. The proposed methodology is applied to two cases of measured 

scour in the field. 

17. Key Words 

Bridge scour, abutment scour, compound channels. 

18. Distribution Statement 

No restrictions. This document is available to the public 

through the National Technical Information Service, 

Springfield, VA 22161. 

19. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 

144 

22. Price
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 oC 

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 th International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E e 380. 

(Revised March 2003)
iii

TABLE OF CONTENTS 

Page No.

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

BACKGROUND ................................................................................................................1 

RESEARCH OBJECTIVES.............................................................................................3 

CHAPTER 2. LITERATURE REVIEW.....................................................................................5 

COMPOUND-CHANNEL HYDRAULICS ....................................................................5 

ABUTMENT SCOUR .......................................................................................................8 

TIME DEVELOPMENT OF SCOUR...........................................................................15 

CHAPTER 3. EXPERIMENTAL INVESTIGATION.............................................................17 

INTRODUCTION............................................................................................................17 

COMPOUND-CHANNEL AND ABUTMENT GEOMETRY....................................17 

SEDIMENT......................................................................................................................19 

WATER-SURFACE PROFILE AND VELOCITY MEASUREMENTS ..................22 

SCOUR MEASUREMENTS ..........................................................................................24 

RESULTS .........................................................................................................................24 

Channel Roughness..............................................................................................24 

Discharge Distribution.........................................................................................27 

Water-Surface Profiles and Approach Velocity Distributions........................31 

Main-Channel Centerline Velocity Profiles.......................................................35 

Critical Velocity ...................................................................................................38 

Equilibrium Scour Depths ..................................................................................44 

CHAPTER 4. ANALYSIS OF RESULTS.................................................................................55 

EQUILIBRIUM SCOUR DEPTH..................................................................................55 

CLEAR-WATER SCOUR: FORMULATION I ..........................................................55 

Theoretical Clear-Water Contraction Scour for Setback Abutments............56 

Theoretical Clear-Water Contraction Scour for Bankline Abutments..........59 

Correlation of Results..........................................................................................61 

CLEAR-WATER SCOUR: FORMULATION II.........................................................67 

Dimensional Analysis...........................................................................................67 

Correlation of Results..........................................................................................68 

DISCUSSION...................................................................................................................70 
iv

APPLICATION TO LIVE-BED CONDITIONS..........................................................72 

WSPRO PREDICTIONS................................................................................................78 

TIME DEVELOPMENT RELATIONSHIP.................................................................87 

COMPARISON OF SCOUR FORMULAS WITH MEASURED DATA..................87 

PROPOSED IMPLEMENTATION PROCEDURE FOR ABUTMENT 

 SCOUR PREDICTION.......................................................................................93 

Example ................................................................................................................96 

Solution .................................................................................................................96 

COMPARISON WITH FIELD DATA........................................................................101 

Highway 22 Example .........................................................................................101 

Highway 12 Example .........................................................................................104 

LIMITATIONS..............................................................................................................106 

CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS ..........................................107 

APPENDIX A. WSPRO INPUT AND OUTPUT FOR BURDELL CREEK, 

HIGHWAY 22 BRIDGE OVER POMME DE TERRE RIVER, AND 

HIGHWAY 12 BRIDGE OVER POMME DE TERRE RIVER...............................109 

REFERENCES...........................................................................................................................123 

 
v

LIST OF FIGURES 

Figure No. Page No.

1. Compound-channel configurations used in scour experiments ...............................................18 

2. Abutment shapes used in scour experiments ...........................................................................20 

3. Sediment grain size distributions.............................................................................................21 

4. Manning’s n in the main channel and floodplain for compound channel B............................25 

5. Measured and computed normal depth for compound channel B ...........................................26 

6. Ratio of main-channel discharge to total discharge as a function of relative normal 

 depth in the floodplain .............................................................................................................29 

7. Dependence and use of discharge distribution factor M..........................................................30 

8. Effect of variable abutment length on water-surface profiles and velocity distributions 

 for constant discharge ..............................................................................................................32 

9. Effect of variable discharge on water-surface profiles and velocity distributions 

 for constant abutment length....................................................................................................33 

10. Effect of tailwater (yftw) on water-surface profiles and velocity distributions for 

 constant discharge....................................................................................................................34 

11. Water-surface profiles before and after scour for La/Bf = (a) 0.44 and (b) 0.88......................36 

12. Water-surface profiles before and after scour for La/Bf = (a) 0.97 and (b) 1.0........................37 

13. Main-channel centerline velocity from approach to contracted section for La/Bf = 0.88........39 

14. Main-channel centerline velocity from approach to contracted section for La/Bf = 0.97........40 

15. Main-channel centerline velocity from approach to contracted section for La/Bf = 1.0..........41 

16. Measured and calculated critical velocities at incipient motion ..............................................43 

17. Bed elevations for shorter VW abutments after scour.............................................................48 

18. Bed elevations for longer VW abutments after scour..............................................................49 

19. Bed elevations for shorter ST abutments after scour...............................................................50 

20. Bed elevations for longer ST abutments after scour................................................................51 

21. Bed elevations for VW, ST, and WW abutments after scour ..................................................52 

22. Bed elevations for VW abutments and sediments A, B, and C ...............................................54 

23. Definition sketch for idealized floodplain contraction scour in a laboratory compound 

 channel, compound channel A.................................................................................................57

24. Definition sketch for idealized main-channel contraction scour in a laboratory compound 

channel, compound channel B .................................................................................................60 

25. Scour-depth relationship based on approach hydraulic variables for VW 

 abutments with 0.17 ≤ La/Bf ≤ 0.66..........................................................................................62 

26. Scour-depth relationship based on approach hydraulic variables in main channel for VW 

 and ST abutments with 0.88 ≤ La/Bf ≤ 1.0 and d50 = 3.3 mm in compound channel B ...........63 

27. Scour-depth relationship based on approach hydraulic variables in floodplain for all 

 abutments and sediments, compound channel B .....................................................................65 

28. Scour-depth relationship based on local hydraulic variables for VW abutments 

with La/Bf ≤ 0.66 and d50 = 3.3, 2.7, and 1.1 mm ....................................................................69 

29. Scour-depth relationship based on local hydraulic variables for VW and ST 

 abutments with La/Bf ≥ 0.88 and d50 = 3.3 mm........................................................................71 

30. Relationship between local and approach hydraulic variables ................................................73 
vi

31. Live-bed contraction scour coefficient for La/Bf = 1.0 ............................................................76 

32. Calculated and measured water-surface profiles for compound channel A ............................79 

33. Calculated and measured approach velocity distributions for compound 

 channel A .................................................................................................................................80 

34. Calculated and measured resultant velocity distributions in the contracted section 

 for compound channel A..........................................................................................................81 

35. Calculated and measured water-surface profiles for compound channel B.............................82 

36. Calculated and measured approach velocity distributions for compound channel B ..............84 

37. Comparison of approach floodplain velocities ........................................................................85 

38. Comparison of approach floodplain depths .............................................................................85 

39. Comparisons of discharge distribution factor M .....................................................................86 

40. Comparisons of maximum velocity at abutment face..............................................................86 

41. Dimensionless representation of time development of scour ..................................................88 

42. Comparison of measured and predicted scour depths using the Melville formula..................90 

43. Comparison of measured and predicted scour depths using the Froehlich clear-water 

 scour formula ...........................................................................................................................90 

44. Comparison of measured and predicted scour depths using the Froehlich live-bed 

 scour formula ...........................................................................................................................91 

45. Comparison of measured and predicted scour depths using the G. K. Young (GKY) 

 formula.....................................................................................................................................91 

46. Comparison of measured and predicted scour depths using the Maryland formula................92 

47. Comparison of measured and predicted scour depths using the formula from the 

present study ............................................................................................................................92 

48. Cross sections and water-surface elevation for Burdell Creek bridge.....................................97 

49. Design 100-year hydrograph for Burdell Creek ......................................................................99 

50. Cross sections for Highway 22 over the Pomme de Terre River for the flood 

of April 9, 1997......................................................................................................................102 

51. Cross sections for Highway 12 over the Pomme de Terre River for the flood 

of April 9, 1997......................................................................................................................105 
vii

 LIST OF TABLES 

Table No. Page No.

1. Experimental parameters .........................................................................................................19 

2. Raw experimental results for fixed-bed hydraulic variables and equilibrium 

 scour depth...............................................................................................................................45 

3. Range of dimensionless variables in experimental relationships ..........................................106 

4. WSPRO input data file for Burdell Creek, Q100 ....................................................................109 

5. WSPRO water-surface profile output for Burdell Creek, Q100.............................................. 110 

6. Velocity distribution from HP record for unconstricted flow at approach section, 

 Burdell Creek, Q100 ................................................................................................................111 

7. Velocity distribution from HP record for constricted flow at approach section, 

 Burdell Creek, Q100 ................................................................................................................112 

8. WSPRO input data file for Highway 22 bridge.....................................................................113 

9. WSPRO water-surface profile output for Highway 22 bridge...............................................114 

10. Velocity distribution from HP record for unconstricted flow at approach section, 

Highway 22 bridge.................................................................................................................116 

11. Velocity distribution from HP record for constricted flow at approach section, 

Highway 22 bridge.................................................................................................................117 

12. WSPRO input data file for Highway 12 bridge.....................................................................118 

13. WSPRO water-surface profile output for Highway 12 bridge...............................................119 

14. Velocity distribution from HP record for unconstricted flow at approach section, 

 Highway 12 bridge.................................................................................................................121 

15. Velocity distribution from HP record for constricted flow at approach section, 

 Highway 12 bridge......................................................................................122
viii

 LIST OF SYMBOLS 

a = blocked flow area by abutment 

A = unobstructed flow area in approach channel 

Af1 = blocked flow area in approach-channel floodplain 

b = bridge pier width 

Bf = width of floodplain in compound channel 

Bm = width of main channel in compound channel 

Bm1 = width of approach main channel 

Bm2 = width of main channel in contracted section 

Cn = units conversion coefficient used in equation 23 

C0 = best-fit constant in proposed scour formula 

Cr = best-fit coefficient in proposed scour formula 

CLB = live-bed scour coefficient 

d50 = median diameter of sediment 

ds = local scour depth 

dsc = theoretical long contraction scour depth 

dst = unsteady scour depth at any time t 

F0 = approach value of densimetric grain Froude number in Lim scour formula 

F1 = approach-flow Froude number 

Fab = Froude number adjacent to abutment face 

k-ε = k-epsilon refers to a numerical modeling technique that uses the ratio of turbulent kinetic 

energy squared to turbulent energy dissipation rate to define the eddy viscosity term for 

computations 

kf = spiral-flow adjustment factor in Maryland scour formula 
ix

ks = equivalent sand-grain roughness 

kv = velocity adjustment factor in Maryland scour formula 

K1 = geometric shape factor for abutment in Froehlich scour formula 

K2 = embankment skewness factor in Froehlich scour formula 

Ks

*

 = abutment shape factor in Melville scour formula 

KG = channel geometric factor in Melville scour formula 

KI = flow-intensity factor in Melville scour formula 

KST = spill-through abutment shape factor for scour formula in present study 

Kθ

*

 = abutment alignment factor in Melville scour formula 

La = abutment/embankment length 

M = discharge distribution factor in approach section = unblocked Q over total Q 

 = discharge contraction ratio for a bankline abutment 

n = Manning’s n 

nfp = Manning’s n in floodplain of compound channel 

nmc = Manning’s n in main channel of compound channel 

nK = Manning’s n calculated from Keulegan’s equation 

Ns = sediment number (equivalent to densimetric grain Froude number) 

Nsc = critical value of sediment number 

q1 = flow rate per unit width in approach section in Maryland scour formula 

q2 = flow rate per unit width in contracted section in Maryland scour formula 

qbv = volumetric sediment discharge per unit width 

qf0 = flow rate per unit width in floodplain at normal depth (unconstricted) 

qf0c = critical flow rate per unit width in floodplain at normal depth (unconstricted) 

qf1 = flow rate per unit width in the approach floodplain (constricted) 
x

qf2 = flow rate per unit width in the contracted floodplain 

qm1 = flow rate per unit width in the approach main channel at the beginning of scour 

qm2 = flow rate per unit width in the contracted section in the main channel 

Q = total discharge in compound channel 

Qf1 = discharge in the approach floodplain 

Qm0 = discharge in main channel for uniform flow in a compound channel 

Qm1 = discharge in the approach main channel 

Qm2 = discharge in the contracted main channel 

Qobst1 = obstructed discharge in the approach section 

R = channel hydraulic radius 

SG = specific gravity of sediment 

t = time since beginning of scour 

te = time to reach equilibrium scour depth 

u*1 = approach value of shear velocity = (τ1/ρ)

0.5

u*c = critical value of shear velocity = (τc/ρ)

0.5

Vab = maximum resultant velocity near the upstream corner of the abutment face 

Vc = critical velocity for initiation of motion 

Vf0 = floodplain velocity at normal depth 

Vf0c = critical velocity for unconstricted floodplain depth of uniform flow 

Vf1 = bridge approach velocity in floodplain of compound channel 

Vf2 = floodplain velocity in the contracted floodplain after scour 

Vm0c = critical velocity for the unconstricted depth in the main channel 

Vm1 = bridge approach velocity in main channel of compound channel 
xi

Vm2c = critical velocity in the main channel at the contracted section at equilibrium scour 

Vx = mean contraction velocity in GKY scour formula 

VR = resultant velocity adjacent to tip of abutment in GKY scour formula 

V1 = bridge approach velocity in rectangular channel 

y = generic symbol for depth of flow in main channel or floodplain for calculating critical 

velocity 

yab = depth near the upstream corner of the abutment face 

yf0 = normal depth in floodplain 

yf1 = bridge approach depth in floodplain of compound channel, including backwater 

yftw = tailwater depth in floodplain 

ym0 = normal depth in main channel 

ym1 = bridge approach depth in main channel of compound channel, including backwater 

ym2 = depth of flow in main channel in contracted section at equilibrium scour 

ysc = total depth of flow after scour, including the contraction scour depth only 

(Maryland formula) 

y1 = bridge approach depth in rectangular channel 

y2 = total flow depth in contracted section after scour (Maryland formula) 

y’ = vertical coordinate measuring distance above the main-channel bed 

y’/ym= ratio of distance above the main-channel bed to depth of the flow in the main 

channel 

γ = specific weight of fluid 

γs = specific weight of sediment

μf = general discharge per unit width contraction ratio for the floodplain 

Φ = ratio of boundary shear force to streamwise component of weight 
xii

φ = angle of repose of sediment 

ρ = fluid density 

ρs = sediment density 

σ = ratio of Manning’s n in the main channel with compound-channel flow to the value for 

the flow in the main channel alone 

σg = geometric standard deviation of sediment grain size distribution 

τc = critical shear stress for initiation of sediment motion 

τ0 = average bed shear stress in uniform flow 

τ1 = bed shear stress in approach flow 

τ* = Shields’ parameter, τ/[(γs – γ) d50] 

τ*c = critical value of Shields’ parameter, τc/[(γs – γ) d50] 

τ*1 = Shields’ parameter in approach flow section 

τ*2 = Shields’ parameter in contracted flow section 

ξ = independent dimensionless ratio in proposed scour formula = qf1/Mqf0c
1

CHAPTER 1. INTRODUCTION 

BACKGROUND 

Recent bridge failures caused by local scour around piers and abutments have prompted a need 

for better technical information on scour prediction and scour-protection measures.(1-3) In 1987, 

the I-90 bridge over Schoharie Creek near Albany, NY, failed because of local scour around the 

pier foundations, resulting in the loss of 10 lives and millions of dollars for bridge 

repair/replacement.(2,4) During the 1993 upper Mississippi River basin flooding, more than 2400 

bridge crossings were damaged.(5) In 1994, tropical storm Alberto caused numerous bridge 

failures as a result of the 100-year flood stages being exceeded at many locations along the Flint 

and Ocmulgee Rivers in central and southwest Georgia.(6)

There are more than 480,000 bridges over water in the United States.(4) As demonstrated in the 

past decade, the potential for loss of life and serious disruption of a local economy in the event of 

a bridge foundation failure caused by an extreme flood is very significant. As a result, a 

comprehensive effort has been undertaken by the Federal Highway Administration (FHWA) to 

require all States to evaluate highway bridges for scour potential.(7) Approximately 17,000 

bridges have been identified by State departments of transportation (DOTs) as “scour critical” 

with the potential for scour-related failure of the foundations as a result of a flood disaster. 

Several technical publications have been developed by FHWA, including Hydraulic Engineering 

Circular Number 18 (HEC-18),(8) to provide guidance to the engineer on evaluating scour 

problems at bridges. Unfortunately, the alteration of flow patterns by bridge crossings and the 

concomitant scour process are quite complex and have defied numerical analysis for the most 

part. The scouring is the result of flow separation around a bridge pier or abutment and the 

formation of three-dimensional, periodic horseshoe and wake vortexes that interact with a 

movable sediment bed. In this situation, the engineer has been forced to rely on laboratory results 

that are numerous and sometimes conflicting because of idealized laboratory conditions that have 

been used in the past. In particular, the task of predicting local scour around bridge abutments 

remains problematic, with many engineers not trusting the current empirical formulas given in 

HEC-18 that are based on laboratory experiments in rectangular flumes.(9)

Previous laboratory experiments on abutment scour have emphasized the abutment length in a 

rectangular channel as one of the primary variables affecting scour. In an actual river consisting 

of a main channel and adjacent floodplains, an abutment terminating in the floodplain is not 

subject to the idealized, uniform approach velocity distribution obtained in previous laboratory 

experiments in rectangular flumes.(10) Instead, the scour is a function of the redistribution of flow 

between the main channel and the floodplain as flow through the bridge opening occurs. In other 

words, abutment length is certainly important; however, the same abutment length may result in 

different scour depths depending on the approach flow distribution in the compound channel and 

its redistribution as it flows through the contracted opening.(11)

Currently, FHWA recommends the prediction of abutment scour with a regression equation 

developed by Froehlich(12) that is based entirely on results from experimental investigators using 

rectangular laboratory channels. Laursen(13) has developed an equation for clear-water abutment 

scour that is based on contraction hydraulics, but that relies directly on abutment length. 
2

Melville(14) has proposed a methodology for predicting maximum abutment scour that also 

depends on abutment length for short- and intermediate-length abutments, and it does not include 

the effects of overbank flows or of flow distribution in compound channels. However, 

Melville(15) has also considered the case of compound channels under the condition that the 

abutment extends into the main channel rather than terminating on the floodplain. This case of 

encroachment into the main channel itself would be less common in practice than the abutment 

terminating on the floodplain or at the edge of the main channel. 

The concept of flow distribution in a compound channel depends on the interaction between 

main-channel flow and floodplain flow at the imaginary interface between the two where 

vortexes and momentum exchange occur. The net result is that less discharge occurs in the 

compound channel than would be expected from adding the separate main-channel and 

floodplain flows that would occur without interaction. The research by Sturm and Sadiq,(16) for 

example, suggests methods by which predictions of flow distribution between the main channel 

and the floodplain can be improved in the case of roughened floodplains. Wormleaton and 

Merrett(17) and Myers and Lyness(18) have also proposed techniques for predicting flow 

distribution in compound channels. A more detailed review of the literature on compoundchannel hydraulics can be found in chapter 2 of this report. 

Sturm and Janjua(19-20) have proposed a discharge contraction ratio as a better measure of the 

effect of abutment length, and the flow redistribution that it causes, on abutment scour. The 

discharge contraction ratio is a function of abutment length and compound-channel geometry and 

roughness. It can be obtained from the output of the water-surface profile program, WSPRO.(21)

The research reported herein, however, attempts to clarify the influence of bridge backwater on 

the flow redistribution and to improve the WSPRO methodology for computation of flow 

redistribution by incorporating more recent research results on compound-channel hydraulics.(22)

The abutment scour experiments by Sturm and Janjua(20) and Sturm and Sadiq(23) used a single, 

uniform sediment size of 3.3 millimeters (mm). The effect of sediment size on the equilibrium 

scour depth has been incorporated by including, as an independent variable, the ratio of the 

approach velocity in the floodplain to the critical velocity for the initiation of motion, which 

depends on sediment size. Experiments with three different sediment sizes at varying discharges 

and abutment lengths for a vertical-wall abutment were conducted in the present research to 

verify this method for quantifying the influence of sediment size on equilibrium scour depth. In 

all cases, the abutment length was large relative to the sediment size in order to remove any 

effects of energy dissipation caused by large, uniform sediment sizes in the bottom of the scour 

hole relative to a short abutment length.(24) 

The effect of abutment shape was considered in this research by conducting experiments in a 

compound channel on vertical-wall, wingwall, and spill-through abutment shapes. A single 

sediment size of 3.3 mm was used for this series of experiments, and the discharge and abutment 

lengths were varied. The abutment shape effect has been shown to be insignificant for long 

abutments in rectangular channels(15); however, this behavior has not been verified for long 

abutments in compound channels. Currently, FHWA procedures assume a reduction in scour for 

a spill-through abutment of 55 percent in comparison to the vertical-wall abutment.(8)
3

Additional experiments were also conducted on abutment lengths that approached the bank of 

the main channel for both the spill-through and vertical-wall abutments. The purpose of these 

experiments was to test the methodology developed for abutments that terminated on the 

floodplain for the more complicated three-dimensional flow field that occurs when the contracted 

flow joins the main channel near the abutment face. 

The live-bed scour case was considered analytically for the condition of sediment transport in the 

main channel with no sediment movement in the floodplain. This case would be of interest for 

the abutment located at or near the bank of the main channel with the scour hole occurring at 

least partially in the main channel rather than on the floodplain alone. Although experiments for 

this case were attempted, they were not successful because of the limitations of the present 

compound-channel geometry in the flume. 

Finally, a brief implementation procedure for the proposed methodology was developed and 

applied to a hypothetical example. It was also tested on two field cases of scour in Minnesota 

using data measured by the U.S. Geological Survey (USGS) in 1997. 

RESEARCH OBJECTIVES 

The objective of the proposed research is to develop better predictive equations for assessing the 

vulnerability of existing bridges to abutment scour for cases in which the abutment is located 

anywhere in the floodplain up to the bank of the main channel in a compound-channel geometry. 

Of primary interest is the effect of compound-channel hydraulics(16) on the redistribution of the 

main-channel and floodplain flows as the flow accelerates around the end of an abutment of 

varying shape. In addition, the effect of sediment size on abutment scour needs to be clarified as 

well as the effect of live-bed versus clear-water scour. 

The experimental research reported herein differs from most of the previous experimental studies 

of abutment scour that have not had a compound channel as the approach channel and that have 

not considered the effect of very long abutments in wide, shallow floodplain flow. The results of 

the experimental research have been used to develop a prediction equation for clear-water 

abutment scour that was tested on limited field data and was compared to the results of the other 

scour-prediction methods. 

The specific research objectives are: 

 Investigate the effects of flow distribution, as affected by abutment length, on clear-water 

abutment scour in a compound channel for abutment lengths that terminate on the floodplain 

as well as encroach on the bank of the main channel. 

 Quantify the effects of floodplain sediment size on abutment scour. 

 Explore the influence of abutment shapes, including wingwall, vertical-wall, and spillthrough shapes, on equilibrium scour-hole depth and scour-hole form. 

 Determine the relative importance of the live-bed scour case compared to the clear-water 
4

case when there is sediment transport in the main channel and the abutment encroaches on 

the main channel. 

 Combine the experimental results into a methodology for assessing field abutment scour and 

test it on available field data. 

This report provides a brief review of the literature on compound-channel hydraulics, and on 

clear-water and live-bed abutment scour in chapter 2. The experimental investigation is 

described in detail in chapter 3. Chapter 4 contains an analysis of the experimental results and a 

proposed abutment scour-prediction equation that addresses the effects of the alteration of the 

flow distribution by both short and long abutments, sediment size, abutment shape, time 

development, and live-bed conditions. A procedure for implementing the research results for the 

purpose of identifying scour-susceptible bridges in the field is then suggested in chapter 4 along 

with an example of a field application. The proposed procedure is tested with limited field data. 

Conclusions and recommendations are given in chapter 5.
5

CHAPTER 2. LITERATURE REVIEW 

In this chapter, the literature on compound-channel hydraulics and both clear-water and live-bed 

bridge abutment scour is reviewed. The connection between these two topics is an important 

contribution of this research, so a brief summary of previous research done on each topic is 

given. 

COMPOUND-CHANNEL HYDRAULICS 

A compound channel consists of a main channel, which carries base flow and frequently 

occurring runoff up to bank-full flow conditions, and a floodplain on one or both sides that 

carries overbank flow during times of flooding. Both channel roughness and the depth of the 

flow, and hence flow-channel geometry, can be drastically different in the main channel and the 

floodplains. In general, floodplain flows are relatively shallow, with slow-moving flow adjacent 

to faster-moving flow in the main channel, which results in a complex interaction that includes 

momentum transfer between the main-channel and floodplain flows. This phenomenon is more 

pronounced in the immediate interface region between the main channel and the floodplain, 

where there exists a strong transverse gradient of the longitudinal velocity. Because of the 

velocity gradient and anisotropy of the turbulence, there are vortexes rotating about both the 

vertical and horizontal axes along the main channel/floodplain interface.(25-27) These vortexes are 

responsible for the transfer of water mass, momentum, and species concentration from the mainchannel flow into the floodplain flow. The result is that for a given stage, the total flow in the 

compound channel is less than what would be calculated as the sum of the flows in the main 

channel and the floodplain assuming no interaction.(25)

Several attempts have been made at quantifying the momentum transfer at the main 

channel/floodplain interface using the concepts of imaginary interfaces that are included or 

excluded as the wetted perimeter and defined at varying locations with or without the 

consideration of an apparent shear stress acting on the interface (see references 17 and 28 

through 31). The resulting distribution of the discharge between the main channel and the 

floodplain caused by the interaction at the interface must be correctly predicted in water-surface 

profile computations and in calculations of approach floodplain velocities for the prediction of 

abutment scour. 

The current version of WSPRO (as well as the Hydrologic Engineering Center River Analysis 

System (HECRAS) divides the compound channel into subsections using a vertical interface 

between the main channel and the floodplain, but neglects any contribution of the interface to the 

wetted perimeter of the subsections. In effect, the interaction between the main channel and the 

floodplain is neglected because this procedure is equivalent to assuming no shear stress at the 

interface. Wright and Carstens(28) proposed that the interface be included in the wetted perimeter 

of the main channel, and that a shear force equal to the mean boundary shear stress in the main 

channel be applied to the floodplain interface. Yen and Overton,(29) on the other hand, suggested 

the idea of choosing an interface on which shear stress was, in fact, nearly zero. This led to 

several methods of choosing an interface, including a diagonal interface from the top of the 

main-channel bank to the channel centerline at the free surface, and a horizontal interface from 

bank to bank of the main channel. Wormleaton and Hadjipanos(32) compared the accuracy of the 
6

vertical, diagonal, and horizontal interfaces in predicting the separate main-channel and 

floodplain discharges measured in an experimental 1.21-meter (m) flume, having a fixed ratio of 

floodplain width to main-channel half-width of 3.2.(30) The wetted perimeter of the interface was 

either fully included or fully excluded in the calculation of the hydraulic radius of the main 

channel. The results showed that even though a particular choice of interface might provide a 

satisfactory estimate of total channel discharge, nearly all of the choices tended to overpredict 

the separate main-channel discharge and underpredict the floodplain discharge. It was further 

shown that these errors were magnified in the calculation of the kinetic energy flux correction 

coefficients used in water-surface profile computations. 

In a modification of the earlier method, Wormleaton and Merrett(17) applied a correction factor 

called the Φ index to the main-channel and floodplain discharges calculated by a particular 

choice of interface (vertical, diagonal, or horizontal), which was either included or excluded 

from the wetted perimeter. The Φ index was defined as the ratio of the boundary shear force to 

the streamwise component of fluid weight(33) as a measure of apparent shear force. If the Φ index 

is less than unity on the main-channel interface, for example, then the apparent shear force 

resulting from the main channel/floodplain interaction resists the fluid motion in the main 

channel. This modified method was applied to the experimental results from the very large 

compound channel at Hydraulics Research, Wallingford Ltd. (Wallingford, Oxfordshire, U.K.). 

The channel is 56 m long by 10 m wide, with a total flow capacity of 1.1 cubic meters per second 

(m3

/s). In the experiments, the ratio of the floodplain width to the main-channel half-width varied 

from 1 to 5.5, and the ratio of the relative floodplain depth to the main-channel depth varied from 

0.05 to 0.50. The calculated main-channel and floodplain discharges, when multiplied by the 

square root of the Φ index for each subsection, showed considerable improvement when 

compared to the measured discharges, regardless of the choice of interface. The difficulty of the 

method is in the prediction of the Φ index. A regression equation was proposed for this purpose, 

with the Φ index given as a function of the velocity difference between the main channel and the 

floodplain, the floodplain depth, and the floodplain width. The regression equation is limited to 

the range of experimental variables observed in the laboratory. 

Ackers(34) has also proposed a discharge calculation method for compound channels using the 

Wallingford data. He suggested a discharge adjustment factor that depends on coherence, which 

was defined as the ratio of the full-channel conveyance (with the channel treated as a single unit 

with perimeter weighting of boundary friction factors) to the total conveyance calculated by 

summing the subsection conveyances. Four different zones were defined as a function of the 

relative depth (the ratio of the floodplain to the total depth), with a different empirical equation 

for the discharge adjustment for each zone. 

Myers and Lyness(18) have proposed a two-step method for predicting the distribution of flow 

between the main channel and the floodplains for a compound channel in overbank flow. First, 

they propose a power relationship between the ratio of the total discharge to the bank-full 

discharge and the ratio of the total depth to the bank-full depth. They show that from both 

laboratory and field data that such a relationship is independent of bed slope and scale, and is 

dependent only on geometry for similar roughnesses in the main channel and the floodplain. 

Second, they divide the total discharge obtained from the first step into the main-channel and 

floodplain components, assuming that the ratio of main channel-to-floodplain discharge is an 
7

inverse power function of the ratio of the total depth to the floodplain depth. In this case, the 

relationship was not shown to be universal for different geometry, scale, and roughness. This 

approach is, in reality, a conveyance weighting method that does not account explicitly for the 

interaction between the main channel and the floodplain. 

The variation of Manning’s n with depth further complicates the problem of water-surface 

profile computations and velocity predictions in compound channels.(16,35) In addition, the 

existence of multiple critical depths can lead to difficulties in both computing and interpreting 

water-surface profiles. Blalock and Sturm(36) and Chaudhry and Bhallamudi(37) have suggested 

the use of a compound-channel Froude number in computational procedures for determining 

multiple critical depths in compound channels. Blalock and Sturm(36) used the energy equation, 

while Chaudhry and Bhallamudi(37) employed the momentum equation to define a compoundchannel Froude number. Blalock and Sturm(38) showed from their experimental results that the 

energy and momentum approaches resulted in the same values for critical depth. Yuen and 

Knight(39) have confirmed from their experimental results that the compound-channel Froude 

number suggested by Blalock and Sturm(36) gives values for critical depth that are reasonably 

close to the measured values. Sturm and Sadiq(40) have demonstrated the usefulness of their 

compound-channel Froude number for computing and interpreting water-surface profiles in 

compound channels having two values for critical depth: one in overbank flow and one in mainchannel flow alone. 

Numerical analysis of flow characteristics in compound open channels, particularly for the case 

of an obstruction, such as a bridge abutment on the floodplain, have received rather limited 

attention in comparison to experimental studies. Most numerical models have been developed for 

the parabolic flow situation, which has a predominantly longitudinal flow direction with no 

reverse flow (e.g., uniform or gradually varied flow). There are, however, a few studies applied 

to an elliptical flow situation (with flow separation and recirculation allowed), but these are 

applicable only to a simple rectangular channel. The effect of compound-channel hydraulics on 

flow characteristics in the region close to obstructions is in need of further investigation. 

One- and two-dimensional numerical models have been used to compute complex flow fields, 

such as those in compound open-channel flow. Some of the two-dimensional models(41-42) are 

applicable to the boundary-layer type of flow, in which the flow can be described by a set of 

differential equations that are parabolic in the longitudinal (flow) direction.(43) The flow situation 

in a compound open channel can become even more complex if structures, such as bridge piers 

or abutments, are placed in the floodplain. In this situation, at least for the regions near the 

structure, the parabolic flow assumption becomes invalid. This is because there is no longer any 

predominant flow direction in the region close to the structure. The flow will separate 

downstream from the structure and a recirculating region with a reverse flow and an adverse 

pressure gradient will be formed that violates the parabolic flow assumption. In addition, the 

water-surface elevation no longer varies only in the streamwise coordinate. Because of the 

presence of the obstruction, there exists a rapid variation in water depth near a bridge abutment, 

both in the longitudinal and transverse directions. The water-surface elevation upstream from the 

abutment increases and forms a backwater profile. These flow characteristics are associated with 

an elliptical flow field that has additional complexity because of the intercoupling of the velocity 

and pressure (water depth) fields. 
8

Three-dimensional models have been developed and applied primarily to the case of uniform 

flow in a compound channel.(44-46) These models allow the simulation of turbulence-driven 

secondary motion in the transverse plane of a compound open channel, as well as the Reynolds’ 

turbulent shear stresses. Experimental measurements by Tominaga and Nezu(27) using a fiberoptic laser Doppler anemometer showed a pair of longitudinal vortexes at the main 

channel/floodplain interface, with an inclined secondary current directed from the intersection of 

the main channel and the floodplain bed toward the free surface. The numerical model of Naot, 

et al.,(46) was able to reproduce this behavior. The numerical results of Krishnappen and Lau(44)

using an algebraic stress model demonstrated good agreement with measured divisions of flow 

between the main channel and the floodplain using data from Knight and Demetriou(31) and 

Wormleaton, et al.(30) Pezzinga’s nonlinear k-ε turbulence model(47) was used to compare the 

effects of various main channel/floodplain subdivisions (vertical, horizontal, diagonal, and 

bisector) on computed subsection discharges. The diagonal interface and the interface formed by 

the bisector of the corner angle between the main channel and the floodplain gave the best results 

for both the discharge distribution and the kinetic energy correction coefficients. 

In this research, the velocity field is sought in the bridge approach section and near the face of a 

bridge abutment that terminates on the floodplain where the width-to-depth ratio of the flow is 

large and the vertical mixing is strong. Under these conditions, the velocity field can be 

computed adequately from the depth-averaged equations of motion, including the regions of 

adverse pressure gradient and flow separation, especially if a k-ε turbulence closure model is 

used.(42, 48-53) Biglari(54) and Biglari and Sturm(55) applied a depth-averaged k-ε turbulence model 

in elliptical form to the problem of flow around a bridge abutment on a floodplain. 

ABUTMENT SCOUR 

Early experimental research on scour around abutments was, in some cases, motivated by a 

desire to predict local scour around spur dikes; however, the results of these studies have been 

applied to the problem of abutment scour. Ahmed(56) proposed a scour-depth relationship for spur 

dikes that used the “flow intensity,” or flow rate per unit width in the contracted section, as the 

independent variable in agreement with previous regime formulas. Laursen and Toch(57) argued 

that in live-bed scour around bridge piers and abutments, the approach flow velocity has no 

effect on equilibrium scour depth because increases in velocity not only increase the sediment 

transport into the scour hole from upstream, but also increase the strength of the vortex 

responsible for transporting sediment out of the scour hole. This argument applies in cases where 

there is appreciable sediment in bed-load transport. If the sediments in transport are fine and 

mostly carried as suspended loads, the assumptions of live-bed scour behavior may not be 

appropriate. 

Garde, et al.,(58) studied experimentally the scour around spur dikes in a rectangular channel and 

related the nondimensional ratio of scour depth to approach depth, ds/y1, to the approach Froude 

number, F1, and to the geometric contraction ratio, m, which is defined as the ratio of the width 

of the contracted opening to the approach channel width. In addition, the coefficient of 

proportionality and the exponent in the Froude number were found to depend on sediment size. 

Some sediment transport occurred in the approach flow for these experiments. 
9

Liu, et al.,(59) considered the scour around bridge constrictions caused by abutment models in 

1.2-m- (4-foot (ft)-) wide and 2.4-m- (8-ft-) wide flumes. Their experimental results for live-bed 

scour indicated that the ratio of the abutment length to the normal depth, La/y0, and the uniformflow Froude number were the most important influences on the dimensionless scour depth. The 

normal depth was determined with sediment in equilibrium transport before the abutment was 

placed in the flume. They proposed an equation for equilibrium live-bed scour depth ds: 

where La = abutment length (vertical wall), y0 = normal depth of flow, and F0 = Froude number 

of uniform flow. The experimental values of La/y0 varied from approximately 1 to 10, and the 

Froude numbers varied from 0.3 to 1.2. In a separate series of experiments, clear-water scour 

was studied by pre-forming the scour hole and determining the flow conditions necessary to just 

initiate sediment motion in the bottom of the scour hole. In this case, the dimensionless clearwater scour depth was found to be directly proportional to F0/m, where F0 = uniform-flow 

Froude number and m = geometric contraction ratio defined by the ratio of the constricted 

channel width to the approach channel width. The coefficient of proportionality was 

approximately 12.5. 

Gill(60) argued from his experimental results on the scour of sand beds around spur dikes that the 

distinction between clear-water and live-bed scour is unimportant for the design determination of 

maximum scour depth. He proposed that the maximum scour depth be based on the geometric 

contraction ratio, m, and on the ratio of the sediment size to the flow depth based on both clearwater and live-bed scour experiments having a duration of 6 hours. His proposed equation is: 

where y0 = initial uniform-flow depth, d50 = median sediment grain size, and m = geometric 

contraction ratio given by the ratio of the contracted width to the full channel width. 

Laursen(61) developed scour-depth relationships for bridge abutments that were based on treating 

the abutment as a limiting case of scour through a long flow constriction. Live-bed scour was 

considered to be a function only of the ratio of the abutment length to the approach flow depth, 

La/y1, and the ratio of the discharge per unit width in the overbank flow region to the discharge 

per unit width in the scour region. The scour region was assumed to have a constant width of 

2.75 times the scour depth. In a subsequent analysis of relief-bridge scour,(13) which was 

considered to be a case of clear-water scour, the same approach was taken in relating the 

abutment scour to that which would take place in a long constriction. The contracted width was 

assumed to be equal to a scour-hole width of 2.75 times the scour-hole depth. This assumption 

F0

1/ 3

0.4

2.15 ( ) y

L = y

d

0

a

0

s (1) 

) 1 1 8.38 ( ) (

1/ 4 6 / 7

 

m y

d + = y

d

0

50

0

s (2) 
10

resulted in an implicit relationship for scour depth: 

where La = abutment length, y1 = approach flow depth, ds = maximum scour depth, τ1 = bed shear 

stress in the approach flow, and τc = critical shear stress for initiation of sediment motion. 

In a comprehensive experimental study of the effect of flow depth on clear-water scour around 

abutments, Tey(62) held constant the ratio of approach shear stress to critical shear stress, τ1/τc, at 

a value of 0.90, while varying the flow depth and the abutment length and shape. The results 

showed an increasing scour depth with increasing flow depth, but at a decreasing rate as the 

depth became larger. The length of the abutment obstructing the approach flow and the abutment 

shape were also found to influence the scour depth. Longer abutments and blunter abutment 

shapes caused deeper scour holes. 

Froehlich(12) completed a regression analysis of 164 laboratory experiments from 11 separate 

sources on clear-water scour around abutments or spur dikes. His proposed regression equation 

is: 

where ds = scour depth; y1 = approach flow depth, K1 = geometric shape factor for abutment and 

embankment, K2 = embankment skewness factor, La = abutment length, F1 = approach Froude 

number, d50 = median sediment grain size, and σg = geometric standard deviation of the sediment 

size distribution. Froehlich further proposed that a factor of safety (FS) of 1 should be added to 

the value of ds/y1 obtained from the regression analysis, and it has been included on the righthand side of equation 4. Currently, HEC-18 recommends the live-bed scour equation obtained by 

Froehlich’s regression analysis of other investigators’ results for this case because equation 4 

seems to greatly overestimate abutment scour. The live-bed scour equation is: 

with FS = 1 included on the right-hand side. In both equations 4 and 5, Froehlich calculated the 

approach Froude number based on an average velocity and depth in the obstructed area of the 

approach-flow cross section. While this worked well for the experimental results that he used, 

which were for rectangular channels, it is not clear what the representative approach velocity and 

depth should be in natural channels subject to overbank flow. 

Melville(14) summarized a large number of experimental results on clear-water abutment scour 

1]

( )

1) 11.5

1 (

2.75 [

1 2

1

1

6

7

1 1

 

 + y

d 

 y

d = y

L

c

s

a s −

τ

τ

 (3) 

0 78 ( ) ( ) 1 1.87

50

1

0 43

1

116

1

0 63

1 2

1

σ + 

d

y y

L

 = . K K y

d

g

.

a .

.

s −

F (4) 

2 27 [ ] 1 1 0 61

1

0 43

1 2

1

 + 

y

L

 = . K K y

d a .

.

s

F (5) 
11

from rectangular channels and proposed a design method for maximum scour depth that depends 

on empirical correction factors for flow intensity, abutment shape, alignment, and length. He 

classified abutments as short (La/y1 < 1) or long (La/y1 > 25), and suggested that the maximum 

clear-water scour depth was 2La for the former case and 10y1 for the latter case. For intermediate 

abutment lengths, the equilibrium clear-water scour depth was given as: 

where KI = flow-intensity factor (V1/Vc), V1 = approach velocity, Vc = critical velocity for 

initiation of sediment motion, Ks

*

 = abutment shape factor, Kθ

*

 = abutment alignment factor, y1 = 

approach flow depth, and La = abutment length. 

Subsequently, Melville(24) suggested that the same methodology could be applied to both bridge 

piers and abutments, albeit with slightly different equations. He further showed that sediment 

size effects appear only in the flow-intensity factor for clear-water scour as long as La/d50 > 25. 

Abutment shape effects were reported to be important only for shorter abutments (i.e., Ks

*

 = 1.0 

for vertical-wall abutments; 0.75 for wingwall abutments; and from 0.45 to 0.60 for spill-through 

abutments), depending on the sideslope, but only if La/y1 ≤ 10. For long abutments (La/y1 ≥ 25), 

Ks

*

 = 1.0 with a linear relationship between Ks

*

 and La/y1 for the intermediate range of La/y1 = 10 

to 25. It must be emphasized that all of Melville’s integrated abutment/pier results(24) are 

considered to be for abutments that are significantly shorter than the floodplain width itself so 

that flow contraction effects are not important. 

Live-bed abutment scour results were also summarized by Melville(24) for short abutments using 

Dongol’s data.(63) Under clear-water conditions (V1 < Vc), the scour depth increased to a 

maximum at V1/Vc = 1. For live-bed conditions (V1 > Vc), the scour depth decreased slightly as 

V1 increased, but then increased again to a value equal to the maximum clear-water scour. Piers 

showed the same behavior under live-bed conditions. 

Melville and Ettema(64) and Melville(15) reported research results on abutment scour in a 

compound channel, but for the case of an abutment terminating in the main channel rather than 

on the floodplain. It was proposed that the scour depth in this instance could be calculated as the 

scour in an equivalent rectangular channel of the same width and with a depth equal to the mainchannel depth by multiplying equation 6 by a geometric factor given as: 

where La = abutment length extending into main channel, Bf = width of floodplain, yf1 = depth of 

flow in floodplain, ym1 = depth of flow in main channel, nmc = Manning’s n in main channel, and 

nfp = Manning’s n in floodplain. 

2 ( ) 1

* * 0 5

d = K K K y L a

.

s I s θ (6) 

1 [ 1 ( ) ]

1

1

5 3

n

n

y

y L

B

K = fp

mc

m

f

/

a

f

G − − (7) 
12

Sturm and Janjua(19-20) conducted a series of experiments in a compound channel consisting of a 

main channel and a floodplain with the abutment terminating in the floodplain, and showed that 

the approach flow distribution and its readjustment by the abutment in the contracted section are 

important factors that should be included in equations for predicting scour depth in natural 

channels. On the basis of a dimensional analysis and the application of Laursen’s(13) analysis of 

relief bridge scour in a long contraction to compound channels, they proposed a relationship 

given as: 

where ds = equilibrium scour depth, yf1 = approach floodplain flow depth, Vf1 = approach 

floodplain flow velocity, Vc = critical velocity, and M = discharge contraction ratio defined as 

the fraction of the total discharge in the bridge approach section over a width determined by 

extending the bridge opening upstream to the approach section. Melville(15) postulated that these 

experimental results gave smaller scour depths than his equation for intermediate-length 

abutments because the experiments had not reached equilibrium; however, subsequent 

experimental results given by Sturm and Sadiq(23) for a different compound-channel geometry, 

but at much longer scour durations, were similar to those of Sturm and Janjua.(20)

Young, et al.,(65) developed a regression equation for clear-water as well as live-bed abutment 

scour using the calculated contraction scour as a nondimensionalizing parameter for the 

abutment scour depth. More recently, Young, et al.,(66) suggested an abutment scour equation: 

where y1 = approach depth in meters, ds = scour depth in meters, n = Manning’s resistance factor, 

τ*c = critical value of Shields’ parameter, SG = specific gravity of sediment, d50 = median 

particle size in meters, and VR = resultant velocity adjacent to the tip of the abutment in meters 

per second. The resultant velocity is calculated from Vx/cos θ, where Vx = mean contraction 

velocity from continuity and θ = 69.85 (a/A)0.2425 (with a coefficient of determination of r2

 = 

0.54), where a = blocked flow area by the abutment and A = total unobstructed flow area 

including the main channel to the median flow bisector. Equation 9 is not dimensionally 

homogeneous and is meant for the International System of Units (SI), known as the metric 

system. It was tested on an experiment by Lim(67) with a very short abutment (La/y1 = 1) and 

showed good agreement for this case. 

Chang(68) has applied Laursen’s long contraction theory to both clear-water and live-bed scour. 

He suggested a velocity adjustment factor, kv, to account for the nonuniform velocity distribution 

in the contracted section, and a spiral-flow adjustment factor, kf, at the abutment toe that depends 

on the approach Froude number. The value of kv was based on potential flow theory, and kf was 

determined from a collection of abutment scour experiments in rectangular laboratory flumes.(69)

The resulting scour equation was: 

7 70 [ 0 35 ] 1

1

 . M

V / V

 = . y

d f c

f

s − (8) 

] ( ) ( 1) ( ) 137 [ 1

6 7

50

2 3 7

1 y V

τ SG d

n

y + d = . R

/

*c

/

s − (9) 
13

where y2 = flow depth in contracted section after scour, y1 = approach flow depth, q1 = flow rate 

per unit width in approach section, q2 = flow rate per unit width in contracted section, and θ = 

0.857 for clear-water scour. The value of kv = 0.8 (q1/q2)

1.5 + 1 and kf = 0.1 + 4.5F1 for clearwater scour, while kf = 0.35 + 3.2F1 for live-bed scour. The approach Froude number F1 = 

V1/(gy1)

0.5. Equation 10 does not include the effect of sediment size on clear-water abutment 

scour. It has since been modified(70) to the form: 

where y2 = total depth of flow at the abutment including scour depth, ysc = total depth of flow 

including the contraction scour depth only, and kf and kv are unchanged from the previous 

formulation. The value of ysc is calculated from q2/Vc, where q2 = unit discharge at the 

contraction and Vc = critical velocity obtained from the expressions given by Neill.(71) The 

evaluation of q2 is unclear for the case of the contracted section having a compound section with 

a variable q2 across the cross section. 

Lim(67) has derived an equation for predicting clear-water abutment scour: 

where ds = scour depth, y1 = approach flow depth, Ks

*

 = abutment shape factor, and X is 

expressed as: 

where F0 = approach value of the densimetric grain Froude number, τ*c = critical value of 

Shields’ parameter, d50 = median grain size, and La = abutment length. Equations 12 and 13 are 

derived on the basis of satisfying continuity before and after scour, evaluating the velocity before 

and after scour in the contracted section from a power law with an exponent of 1/3, and using an 

θ ( )

1

2

1

2

q

k q

 = k y

y v

f (10) 

y = k k y v sc

.

f ( )

0 857

2 (11) 

( 0 9 2) *

1

 = K . X - 

y

d s

s (12) 

[ 0 9 ( ) 1 ]

( )

1

0 5

0 375

1

50

0 25

0

0 75

 + 

y

L . τ

y

d 

X = a

.

*c

.

.

.

F

 (13) 
14

expression for the shear velocity in the contracted expression proposed by Rajaratnam and 

Nwachuku.(72) The latter expression is limited to values of La/y1 ≤ 1. Lim tested equations 12 and 

13 on his own abutment scour data as well as on data from Dongol,(63) Rajaratnam,(73) and Liu, et 

al.,(59) that were, for the most part, limited to very short abutments with La/y1 ≤ 1. 

Lim(74) has also proposed an abutment scour-prediction equation for live-bed scour. He assumed 

that the sediment transport in a strip of the approach section, with a width equal to the abutment 

length plus the scour-hole width, is carried completely through the scour-hole width in the 

contracted section. The scour-hole width is estimated as ds/tan φ, where φ = angle of repose of 

the bed material. Then, by using a sediment transport relationship similar to that of Meyer-Peter 

and Muller (Julien(75)), and by making the same assumptions as for his clear-water scour 

equation, the resulting equation given by Lim for live-bed scour is: 

where u*1 = approach value of shear velocity and u*c = critical value of shear velocity. When 

u*c/u*1 > 1, the term [1 – (u*c/u*1)

2

] is taken as equal to zero and equation 14 reduces to the clearwater scour case according to Lim. This equation still suffers from the dependence on abutment 

length and the limitation on the expression for shear stress for very short abutments as pointed 

out for the clear-water scour case by Richardson.(76)

An equation in Highways in the River Environment(77) was developed from the U.S. Army Corps 

of Engineers’ field data for scour at the end of spur dikes on the Mississippi River. It is 

recommended in HEC-18 for predicting scour around long abutments with La/y1 > 25. The 

equation is: 

where yab = depth of flow at the abutment and Fab = Froude number based on the velocity and 

depth adjacent to the abutment. 

TIME DEVELOPMENT OF SCOUR 

Within the context of this report, the time to equilibrium of the scour process is important in 

establishing the necessary duration of the experiments. In addition, as discussed in a previous 

1 ) ( 1 ) tan (

1 12

) 2 (1

*1

2

*

2 3 2

*1

2

*

2

1

1

4 3

u

u + - d

L φ + 

u

u

y

L + .

 = 

y

d + 

c

s

a

/

c

a

s

/

 (14) 

Fab

.

ab

s = 

y

d 0 33 4 (15) 
15

report by the author,(78) one consequence of developing scour equations from laboratory data for 

the equilibrium condition is that in the field case for small watersheds, the duration of the design 

discharge may be considerably shorter than the time to reach equilibrium. The result is an 

overestimate of field scour for a given design event. 

Time development of scour was first studied in Laursen’s pioneering research on scour by 

jets.(79) Laursen argued that clear-water scour is an asymptotic process in which scour depth 

increases linearly with the logarithm of time. Theoretically, equilibrium would never be reached 

under these circumstances except at infinity. Thus, as a practical matter, some limiting depth of 

scour is determined in laboratory experiments during long durations when the rate of change in 

the scour depth is very small. Most of the research on the time development of scour has been 

done for situations other than abutment scour (e.g., jet scour, pier scour, or hydraulic structure 

scour). Carstens(80) has proposed a general method for determining the time development of 

scour by combining: (1) the sediment continuity equation, (2) a sediment pickup function, and 

(3) an assumed scour-hole shape that is geometrically similar at all times. He showed that the 

simultaneous solution of these three equations agreed with Laursen’s experimental results for the 

time development of scour from a horizontal jet, although the experimental data were used to 

develop the sediment pickup function. 

Raudkivi and Ettema(81) showed from their experimental results on the development of pier scour 

that dimensionless scour depth (ds/b) was linearly proportional to the logarithm of td50/b3

, where 

ds = scour depth, b = cylindrical pier diameter, d50 = median sediment grain diameter, and t = 

time. Yanmaz and Altinbilek(82) applied Carstens’ method to the problem of time development of 

scour around bridge piers. The resulting differential equation was solved numerically to obtain a 

curve of ds/b as a function of dimensionless time: 

where SG = specific gravity of the sediment. The function was linear with the logarithm of 

dimensionless time up to a large value of time after which it leveled off in agreement with their 

experimental results. Kothyari, et al.,(83) proposed a different approach in which the temporal 

variation of scour around a bridge pier was calculated in successive time steps governed by: (1) 

the time required for removal of one sediment grain by the primary vortex and (2) the resulting 

gradual decrease of shear stress in the scour hole as it grew larger. 

Chiew and Melville(84) suggested from their experimental results on scour development around 

bridge piers an empirical relationship for a dimensionless equilibrium time, t* = V1te/b, as a 

function of V1/Vc, where V1 = approach velocity, Vc = critical velocity, b = pier diameter, and te

= equilibrium time. The equilibrium time was defined as the time at which the rate of increase of 

scour depth was less than 5 percent of the pier diameter in 24 hours. The experimental results 

were then presented as the ratio of scour depth at a given time to equilibrium scour depth (dst/ds) 

as a function of the ratio of elapsed time to equilibrium time (t/te) for various constant values of 

V1/Vc. 

] ( 1) [ 2

50 50

b

t d SG - gd = f b

d s (16) 
16

A relationship for time development of scour is proposed in this report based on the experimental 

results for compound channels. It is used to better describe the effect of event duration on the 

scour depth, and it is incorporated into the proposed scour-prediction procedure. However, it 

must be used carefully, because scour can be cumulative over many events. 
17

 CHAPTER 3. EXPERIMENTAL INVESTIGATION 

INTRODUCTION 

Experiments were conducted in a 4.2-m-wide by 24.4-m-long flume of a fixed slope in the 

Hydraulics Laboratory of the School of Civil and Environmental Engineering at the Georgia 

Institute of Technology. Scour depths were measured as a function of discharge, sediment size, 

and abutment shape and length for two different compound-channel configurations constructed 

in the flume at two different fixed slopes. Velocity distributions at the bridge approach section 

and complete water-surface profiles at the beginning and end of scour were measured. The 

details of the experiments and some typical results are described in this chapter, and the results 

are analyzed in chapter 4. 

COMPOUND-CHANNEL AND ABUTMENT GEOMETRY 

Compound-channel sections were constructed inside the 4.2-m-wide by 24.4-m-long flume for 

two different series of experiments and are shown in figure 1. Compound channel A was 

constructed for a series of experiments reported by Sadiq(22) and Sturm and Sadiq.(23) Compound 

channel A was replaced by compound channel B for use in this study as well as in a separate 

Georgia DOT study.(78) Compound channel A had a total inside width of 2.13 m and a fixed-bed 

slope of 0.0050, while compound channel B was 4.21 m wide with a fixed-bed slope of 0.0022 

as shown in figure 1. 

Steel rails on the top of the flume walls were adjusted to serve as a level track for the instrument 

carriage, which is driven by an electric motor. A pumping system provided the water supply to 

the flume from a recirculating sump into which the flume discharges. The flume discharge was 

measured by calibrated bend meters with an uncertainty in discharge of ±0.0003 cubic meters per 

second (m3

/s), and the total capacity of the pumping system was approximately 0.20 m3/s. The 

tailwater depth in the flume was adjusted with a motor-driven tailgate at the downstream end of 

the flume. The flume entrance included a head tank and a series of stilling devices to remove the 

flow entrance effects. The compound-channel section terminated 3.1 m upstream of the tailgate 

in order to provide a sedimentation tank for the settling of any sediment scoured from the 

channel. As a result of the length occupied by the head tank and the sedimentation tank, the 

actual length of the compound channel was 18.3 m. 

The main-channel bed of compound channel A consisted of concrete having a longitudinal slope 

of 0.005 with a tolerance of ±0.001 m from the required elevation at any station along the 

channel. Fine gravel with a mean diameter of 3.3 mm and a geometric standard deviation of 1.3 

was used to provide the roughness on the main-channel bed and walls. Gravel was affixed to the 

main-channel bed and walls with varnish. Roughness in the floodplain was also provided by 

gravel of the same size. To prevent movement of the gravel in the floodplain for the uniformflow experiments, the upper 5-centimeter (cm)-thick layer was stabilized by mixing portland 

cement in the gravel in a ratio of 1:6 by volume. Water was then added until a reasonable 

workability was achieved to form a fixed bed consisting predominantly of gravel roughness in 

the resulting matrix. In the case of compound channel B, a lean concrete mix that used the 3.3-

mm gravel as aggregate was poured to a constant slope of 0.0022 with a standard error of 
18

±0.002 m in bed elevation. In both compound channels, the bed surfaces were relatively rough as 

determined by the protruding grains of gravel. 

Compound Channel A

Compound Channel B

3.66 0.393 0.158

0.154

0.152

0.267 0.933

(All dimensions in meters)

Slope = 0.0050

Slope = 0.0022

Bf

Bf

Bm

Bm

yf

y

0.933

Figure 1. Compound-channel configurations used in scour experiments. 
19

In the case of compound channel A, the bridge abutments were formed by a row of rectangular 

concrete blocks measuring 0.15 m high by 0.15 m wide (in the flow direction) that were fixed to 

the floor of the flume with cement mortar at station 9.75 located 9.75 m (32 ft) downstream of 

the flume entrance. In the text that follows in this report flume station refers to the distance from 

the flume entrance in meters. The abutments for compound channel B were constructed as a row 

of concrete blocks poured into custom-built forms to create vertical-wall (VW), spill-through 

(ST), and wingwall (WW) abutment shapes as shown in figure 2. The sideslopes for the 

wingwall and spill-through shapes were 2:1 (horizontal:vertical) and the wingwall angle was 30 

degrees as shown in figure 2. The centerline of the abutment for compound channel B was also 

located at station 9.75 as it was for compound channel A. The abutment lengths, La, and relative 

abutment lengths, La/Bf, for both compound channels are summarized in table 1. 

Table 1. Experimental parameters. 

Shape Compound 

Channel 

Lengths 

(La), m 

La/Bf Sediment 

d50,

mm 

Vertical-Wall (VW) A 0.152, 0.305, 

0.457 

0.17, 0.33, 

0.50 

A 3.3 

Vertical-Wall (VW) B 0.80, 1.60, 2.40, 

3.23, 3.55, 3.66 

0.22, 0.44, 

0.66, 0.88, 

0.97, 1.0 

A, B, 

C 

3.3, 2.7, 

1.1 

Spill-Through (ST) B 1.17, 1.97, 2.37, 

3.23, 3.55, 3.66 

0.32, 0.54, 

0.65, 0.88, 

0.97, 1.0 

A 3.3 

Wingwall (WW) B 1.61, 2.22 0.44, 0.61 A 3.3 

SEDIMENT 

Three sediments having median sediment grain sizes (d50) of 3.3, 2.7, and 1.1 mm were used in 

this research as indicated in table 1 where they are referred to as sediments A, B, and C, 

respectively. The measured size distributions of all three sediments are shown in figure 3. The 

geometric standard deviation σg (= [d84/d16]

0.5) of sediments A, B, and C was approximately 1.3. 

These sediments can be considered to be uniform in terms of the size distribution effects on 

scour. 
20

0.15 m

L a

0.15 m

0.15 m

2:1

2:1

2:1

2:1

30o

VW

ST

WW

2:1

Figure 2. Abutment shapes used in scour experiments. 

VW = vertical wall, ST = spill-through, WW = wingwall. 
21

0

20

40

60

80

100

0.1 1 10

Sieve Size, mm

Percent Passing

Sediment A Sediment B Sediment C

Figure 3. Sediment grain size distributions. 
22

WATER-SURFACE PROFILE AND VELOCITY MEASUREMENTS 

Both water-surface profiles and velocities were measured in the fixed-bed compound channels 

just described, first for uniform flow to determine the bed roughness, and then with the 

abutments in place to obtain the hydraulic conditions at the beginning of scour. In the uniformflow experiments, stations for measuring the water-surface elevations were fixed at 1.22-m 

intervals along the length of the channel. The desired discharge was obtained by adjusting the 

inlet valve, and the tailgate was raised or lowered to achieve the required M1 or M2 profile with 

no abutments in place, where M1 and M2 are customary textbook symbols for water surface 

profiles in open channel flow on a mild slope. M1 profile flow decelerates in the downstream 

direction; M2 profile flow accelerates in the downstream direction. At each station, the watersurface elevation was measured at three points across the compound channel for compound 

channel A, and at eight points for compound channel B. The mean of these elevations was then 

used to determine the mean elevation of the water surface at each station. The normal depths 

were determined by taking the mean of the flow depths at a point where both M1 and M2 watersurface profiles asymptotically approached one another in the upstream direction, which was 

generally about 6 to 9 m downstream of the channel entrance. The relative depth ratio (yf0/ym0), 

where yf0 = normal depth in the floodplain and ym0 = corresponding normal depth in the main 

channel (see figure 1), varied from 0.26 to 0.35 for compound channel A, and from 0.13 to 0.32 

for compound channel B. 

A miniature propeller current meter with a diameter of 1.5 cm was used to measure point 

velocities averaged over 60 s. In the uniform-flow experiments, station 9.75 was selected for 

measuring the point velocities across the channel cross section because at this station there were 

negligible entrance effects, and also because the water surface could be adjusted easily to the 

normal depth at this station. Depth-averaged velocity distributions were obtained from a point 

velocity measurement at an elevation above the channel bottom of 0.37 times the flow depth in 

each vertical profile in the floodplain, and from six to eight point measurements spaced out over 

each vertical profile in the main channel. Fourteen to 19 vertical profiles were measured at a 

single cross section to establish the depth-averaged velocity distribution. The resulting measured 

velocity distributions when integrated over the cross section produced discharges that were 

within ±4 percent of the discharges measured by the calibrated bend meters in the flume supply 

pipes for the uniform-flow experimental runs. 

The separate discharges in the main channel and the floodplains were determined from the point 

velocities by integration. This information was combined with the measured normal depths to 

determine the separate main-channel and floodplain roughnesses in the uniform compoundchannel flow from Manning’s equation. Normal depth was also determined with flow in the main 

channel alone in order to determine the equivalent sand-grain roughness of the main channel. 

Measurement of water-surface profiles and point velocities was also conducted with the 

abutments in place in the fixed-bed channel. Away from the abutments, the water-surface 

elevations were measured at stations 1.22 m apart as in the uniform-flow case. Close to the 

abutments, water-surface elevations were measured at stations 0.3 m to 0.6 m apart. The depth of 

the flow at the downstream end of the channel was set at the normal depth for each discharge by 

adjusting the tailgate. For the longer abutments, the tailwater was set higher than the normal 
23

depth in some instances to limit the maximum scour that could occur. 

With the abutments in place, and for both a fixed bed at the beginning of scour and a movable 

bed at the end of scour, detailed point velocity measurements were made across the entire 

channel cross section at the bridge approach section, which was station 8.5 for compound 

channel A, and either station 6.7 or station 7.3 for compound channel B, depending on the 

abutment length. This station was located in the region where the maximum backwater occurred 

as well as where the floodplain velocities were not retarded by the abutment. In general, this 

location varied between 67 and 133 percent of the bridge opening width measured upstream of 

the bridge. Depth-averaged velocities were determined at 17 to 19 positions across the cross 

section. These data were used to determine the discharge contraction ratio M at the beginning of 

scour as well as the approach velocity and depth upstream of the end of the abutment. Velocity 

and depth measurements were also made at the downstream face of the abutment, mostly for 

compound channel A. At this location, the depth of flow in the floodplain was very small; 

therefore, the velocities were measured in the main channel only and the discharge was obtained 

from integration. The discharge in the floodplain was obtained from continuity as the difference 

between the measured total discharge and the main-channel discharge. 

Resultant velocities were measured at the upstream face of the bridge for both compound 

channels A and B for the fixed-bed case. The resultant velocity direction was determined either 

by flow visualization or by rotating the velocity meter until a maximum reading was obtained. In 

the case of compound channel B, additional resultant velocities were measured in the bridge 

opening to find the maximum velocity near the face of the abutment. 

For the long abutments (La/Bf = 0.88, 0.97, 1.0), an acoustic Doppler velocimeter (ADV) was 

used to measure three-dimensional velocity components in the main channel at stations 

beginning at the approach section and extending to the contracted section. The measuring 

volume for this instrument is a cylinder 9 mm high by 6 mm in diameter located a distance of 5 

cm below the probe. The ADV transmits two sound pulses separated by a time lag and receives 

the reflected signals from particles in the flow at the location of the sampling volume. The 

instrument measures the Doppler shift between the frequencies of the transmitted and received 

signals based on the change in phase of the return signal from each pulse, which can be 

correlated with velocity in the flow. Some limitations associated with this probe include: (1) a 

minimum water depth of approximately 7 cm is required for submergence of the probe with 

allowance for the 5-cm distance to the sampling volume; (2) when operating in highly turbulent 

environments, the signal-to-noise ratio (SNR) may drop below accepted levels; (3) the probe 

cannot sample a location too close to a boundary because of the size of the sampling volume and 

possible interference with the boundary; and (4) the maximum sampling frequency is 25 hertz 

(Hz). The first limitation means that velocities near the free surface cannot be measured and that 

the probe is not useful for very shallow flows over the floodplain in this investigation. Thus, the 

ADV was used only in the main channel where depths were on the order of 20 to 25 cm. The 

center of the sampling volume was limited to a minimum distance of 5 mm from the channel 

bottom. The SNR varied from 5 to 15 decibels (dB), which is adequate for measuring mean 

current speed; however, for SNR < 15, the manufacturer does not recommend high-resolution 

measurements (i.e., sampling at 25 Hz, which is required for turbulence measurements). 
24

SCOUR MEASUREMENTS 

After the fixed-bed measurements were completed, a movable-bed section was constructed in the 

vicinity of the abutments. The cement-stabilized floodplain surface was removed between 

stations 8.5 and 11.0 for compound channel A, and between stations 6.7 and 12.8 in the case of 

compound channel B. This resulted in a total length of movable bed for compound channel A of 

2.5 m and a length of 6.1 m for compound channel B. The movable-bed floodplain consisted of 

sediments A, B, or C, with sizes as indicated previously in table 1. In both compound channels A 

and B for relative abutment lengths less than 0.66, the main channel remained as a fixed bed; for 

the longer abutments in compound channel B, the main channel was also made into a movable 

bed. It was formed using a template to produce the same slope as for the fixed bed. The 

abutments were sealed with cement mortar to the floor of the flume. 

Scour measurements were made for several discharges at each of the abutment lengths given in 

table 1. At the start of each experimental scour test, the loose gravel in the movable-bed test 

section was carefully leveled to the floodplain elevations corresponding to the constant-bed 

slope. Water was then introduced into the channel very gradually from the upstream and 

downstream ends of the flume with the tailgate raised so that the movable bed remained 

undisturbed. Once the whole channel was flooded, the desired discharge was obtained by 

adjusting the valve on the main inlet pipe. The tailgate was then lowered slowly until the 

corresponding normal depth was reached at the downstream end of the channel. Scour was 

allowed to continue for 12 to 16 hours for compound channel A, and for 24 to 65 hours for 

compound channel B. After equilibrium had been reached, the water-surface profile and the 

velocity distributions in the approach section and at the downstream face of the abutment were 

measured in the case of compound channel B. Then the channel was carefully drained and the 

bed elevations throughout the scour and deposition areas were measured with a point gauge 

having a scale uncertainty of ±0.03 cm. As a practical matter, the uncertainty in the scour-depth 

measurements was about ±0.09 cm. In general, bed elevations in the vicinity of the scour hole 

were measured at approximately 100 spatial locations from which scour contours could be 

plotted. 

The time rates of scour measurements were made for sediment A and for the vertical-wall 

abutment for some experiments. For these runs, scour depth was measured from a scale inscribed 

on a Plexiglas® block that formed the face of the abutment. The scale could be read from above 

by the use of a mirror. Only those cases where the maximum scour depth occurred at the 

upstream corner of the abutment face were suitable for this measurement technique. In general, 

these cases occurred for the greater discharges for each abutment length. Scour-depth 

measurements were generally taken at times of 0.08, 0.17, 0.25, 0.50, 0.75, 1, 2, 4, 8, 16, 24, and 

36 hours after the beginning of the experiment. 

RESULTS 

Channel Roughness 

The experimental results for Manning’s n in the main channel and the floodplain with 

compound-channel flow are shown in figure 4 for sediment A and compound channel B. First, 
25

0.01

0.10

1.00 10.00 100.00

R / ks

n/ks1/6

Floodplain Data Main Channel Data

Floodplain Eq. 17 Main Channel Eq. 17

Main Channel

ks = 0.004 m

σ = 1.23

Floodplain

ks = 0.004 m

Figure 4. Manning’s n in the main channel and floodplain for compound channel B. 
26

0.15

0.20

0.25

0.00 0.05 0.10 0.15 0.20

Q, m3

/s

Depth, m

Calculated Normal Depth Measured Normal Depth

Calculated Critical Depth

Figure 5. Measured and computed normal depth for compound channel B. 
27

the value of the equivalent sand-grain roughness in the main channel was determined from the 

normal depth measurements for flow in the main channel alone (without overbank flow) with the 

result that ks = 0.004 m (4.0 mm or 1.2d50). Then the separate discharges were determined in the 

main channel and the floodplain for overbank flow at several discharges as described previously, 

and the data points are shown in figure 4 in comparison with Keulegan’s equation, given as: 

where R = hydraulic radius in meters and ks = equivalent sand-grain roughness in meters. For the 

separate flow in the floodplain, ks was found to be 0.004 m as in the main-channel flow alone, 

which is consistent with the fact that the roughness surfaces were identical. In the case of mainchannel flow, an additional drag caused by the main channel/floodplain interaction was 

quantified in terms of a coefficient σ, defined as: 

where nmc = effective Manning’s n in the main channel with overbank flow and nK = Keulegan’s 

value for Manning’s n in the main channel with ks = 0.004 m and no interaction with the 

floodplain (i.e., zero contribution of the interface to the wetted perimeter). The best-fit value of σ

was 1.23. In effect, the main-channel conveyance was decreased by 23 percent because of the 

main channel/floodplain interaction. Similar results were found previously for compound 

channel A by Sadiq(22) and Sturm and Sadiq(40) with σ = 1.19. Shown in figure 5 are the 

measured values of normal depth in comparison with the calculated values for compound 

channel B using Keulegan’s equation and the best-fit values of ks and σ. The standard error in the 

normal depth is ±0.076 cm. While it is true that the values of Manning’s n were first calibrated 

with the same data as in figure 5, the agreement shown in figure 5 is indicative of how well this 

calculation approach can reproduce the discharge Q over the full range of experimental values. 

Also shown in figure 5 is the calculated critical depth curve, which is below the normal depth 

curve, demonstrating that the channel is mild over the experimental range of discharges. 

Discharge Distribution 

The purpose of considering the main channel/floodplain interaction in developing Manning’s n 

values for compound-channel flow is illustrated in figure 6 in which the ratio of the mainchannel discharge to the total discharge is given as a function of relative depth in the floodplain. 

The data in figure 6 were measured in uniform flow without the presence of the abutment. In 

both compound channels A and B, the calculated discharge distribution agrees well with the 

measured values. For comparison, the standard WSPRO method of assuming a constant 

Manning’s n and no interaction between the main channel and the floodplain overestimates the 

relative proportion of main-channel discharge as shown in figure 6 for compound channel A. 

) 12 2 2 03 log (

0113 ( )

1 6

1 6

k

. R . 

k

R . 

 = 

k

n

s

s

/

s

/ (17) 

nmc = σ nK (18) 
28

The correct proportioning of the main-channel and floodplain discharge is necessary to predict 

the discharge distribution factor M for which experimental values are given in figure 7(a) as a 

function of relative floodplain depth in the bridge approach section (indicated by a subscript of 1 

on the depth).1

 The factor M is defined as: 

where Qm1 = discharge in the approach main channel, Qf1 = discharge in the approach floodplain, 

Qobst1 = obstructed floodplain discharge over a length equal to the abutment length in the 

approach cross section, and Q = total discharge, which is equal to Qm1 + Qf1. While the values of 

M are primarily a function of the abutment length and the compound-channel geometry, figure 

7(a) shows that they decrease slowly with increasing relative depth in the approach cross section. 

The abutment shape seems to have relatively little influence on M in figure 7(a). 

Sturm and Janjua(19) have shown that for small depth changes from the approach section to the 

bridge section, M represents the ratio of discharge per unit width in the approach floodplain to 

that in the contracted floodplain in the bridge section, qf1/qf2. This discharge ratio has a 

significant effect on the equilibrium abutment scour depth as will be shown in chapter 4. 

Alternatively, it can be shown by the same reasoning that M is indicative of the value of the ratio 

of main-channel discharge in the approach section to that in the contracted section, Qm1/Qm2. The 

experimental results in figure 7(b) confirm that such a relationship exists. The value of M, and 

hence qf1/qf2, is not the same as the geometric contraction ratio m, which is defined as the bridge 

opening width over the total channel width. This is because of the characteristic behavior of 

compound-channel flow in which some of the approach floodplain flow joins the main-channel 

flow in the bridge section. The redistribution of flow between the main channel and the 

floodplain in the contracted bridge section has a significant influence on the scour depth as will 

be discussed further in chapter 4. 

Raw experimental results for water-surface profiles, velocity distributions, critical velocity, and 

equilibrium scour depths, as well as scour contours for compound channel B, are summarized in 

the following sections of this chapter. 

________________________________________ 

1

In the remainder of this chapter and the rest of the report, the subscript “0” refers to depths, velocities, and discharges measured 

in uniform flow at the bridge location without the presence of an obstruction. The subscript “1” refers to the approach section

upstream of the bridge with the obstruction in place where bridge backwater is at a maximum and where floodplain velocities are

not retarded by the embankment. The subscript “2” represents the contracted flow section at the downstream face of the bridge. 

Main-channel variables are indicated by the subscript “m” and floodplain variables are denoted by the subscript “f.” For example, 

yf1 is the floodplain depth in the approach section with the bridge in place. See figure 23.

Q

Q + Q Q M = m f obst ( ) 1 1 1 − (19) 
29

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Qm0/Q

yf0/ym0

Measured, Compound Channel A Measured, Compound Channel B

Calculated, Variable n, A Calculated, Variable n, B

Calculated, Constant n, A

Figure 6. Ratio of main-channel discharge to total discharge as a function 

 of relative normal depth in the floodplain. 
30

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

yf1/ym1

M

A, VW, 0.17 B, VW, 0.22 A, VW, 0.33 B, VW, 0.44 A, VW, 0.50

B, VW, 0.66 B, VW, 0.88 B, VW, 0.97 B, VW, 1.0 B, ST, 0.32

B, ST, 0.52 B, ST, 0.66 B, ST, 0.88 B, ST, 0.97 B, ST, 1.0

KEY: Compound Channel (A or B), Abutment Shape, La/Bf

(a) Dependence of M on approach depth and abutment length. 

0.5

0.6

0.7

0.8

0.9

1.0

0.5 0.6 0.7 0.8 0.9 1.0

Qm1 / Qm2

M

B, VW B, ST A, VW 1:1 Line

KEY: Compound Channel (A or B), Abutment Shape

(b) Use of M to reflect main-channel discharge ratio. 

Figure 7. Dependence and use of discharge distribution factor M. 
31

Water-Surface Profiles and Approach Velocity Distributions 

Measured water-surface profiles and corresponding approach-section velocity distributions are 

shown in figures 8 and 9 for the vertical-wall abutment and the fixed-bed case (beginning of 

scour). In figure 8, the discharge is held constant at 0.0850 m3

/s and the relative abutment length 

is varied from 0.22 to 0.66. The downstream boundary condition is the same for each profile 

because it is the normal depth for the given discharge as set by the tailgate. The increasing 

backwater upstream of the abutment that is caused by increasing abutment length is apparent in 

figure 8(a). In figure 8(b), the main-channel velocities are higher than those in the floodplain as 

expected for small floodplain depths in a compound channel. The increasing backwater for 

longer abutments shown in figure 8(a) causes a general decrease in both the floodplain and mainchannel velocities across the full width of the compound channel as shown in figure 8(b). 

In figure 9, the relative abutment length is held constant at 0.44; however, the discharge is varied 

from 0.0708 m3

/s to 0.0991 m3/s. In this case, the downstream boundary condition of normal 

depth causes an increasing tailwater depth with increasing discharge as can be seen in figure 

9(a). The backwater upstream of the bridge abutment is also observed to increase with increasing 

discharge as seen in figure 9(a). In contrast to the effect of increasing abutment length at constant 

discharge, increasing discharge for the vertical-wall abutment of constant length causes an 

increase in the floodplain velocities as shown in figure 9(b), even though the backwater is 

increasing. The main-channel velocities also increase, but this occurs chiefly in the sideslope 

region of the main channel. The increase in the floodplain velocities with increasing discharge is 

a consequence of a greater proportion of the discharge being carried in the floodplain relative to 

the main channel as the relative depth increases. The change in flow distribution is also apparent 

in figure 7(a), in which M decreases with increasing relative floodplain depth because of the 

decrease in the unobstructed discharge for the same abutment length. Results similar to those of 

figures 8 and 9 were observed for the spill-through abutments. The relative degree of upstream 

backwater for vertical-wall and spill-through abutments is most easily summarized in terms of 

the ratio of the floodplain normal depth, yf0, which would occur with no abutments, to the 

approach floodplain depth, yf1. This backwater ratio varied from 0.6 to 1.0 as a function of 

abutment length and shape, and discharge. 

For the larger abutment lengths, the scour depth increased even at the lowest discharges to the 

point that it was possible for the scour hole to bottom out on the concrete floor of the flume. To 

avoid this condition, the tailwater was raised above the normal depth so that the tailwater level 

became the independent variable for the scour depth while holding the discharge constant for a 

given abutment length. The effects of variable tailwater on the water-surface profiles and the 

approach velocity distributions are shown in figure 10. This is the case of the abutment 

terminating at the edge of the main channel (La/Bf = 1.0) for a constant discharge of 0.0567 m3

/s. 

It is clear from figure 10(a) that the higher the tailwater level, the less backwater upstream of the 

abutment, where backwater is defined relative to an upstream depth that would occur without the 

contraction. Beginning at the tailgate, the floodplain depth decreases in the upstream direction as 

an M1 profile because of boundary resistance both downstream and upstream of the abutment; 

however, the largest drop in depth occurs through the contracted section. The higher tailwater 

also causes less difference between the floodplain and main-channel velocities with the velocity 

distribution becoming nearly uniform for the highest tailwater as shown in figure 10(b). The 
32

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

La/Bf = 0.22 0.44 0.66

Abutment

(a) Compound Channel B: Water-Surface Profiles, 

 VW Abutment, Q = 0.0850 m3

/s. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Transverse Station, m

Velocity, m/s

La/Bf = 0.0 0.22 0.44 0.66

(b) Compound Channel B: Approach Velocity Distributions, 

 Station 7.3 m, VW Abutment, Q = 0.0850 m3

/s. 

Figure 8. Effect of variable abutment length on water-surface profiles 

 and velocity distributions for constant discharge. 
33

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

Q, m 0.0708 0.085 0.0991 3/s =

Abutment

yf03

yf02

yf01

(a) Compound Channel B: Water-Surface Profiles, 

 VW Abutment, La/Bf = 0.44. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Transverse Station, m

Velocity, m/s

Q, m 0.0708 0.085 0.0991 3/s =

(b) Compound Channel B: Approach Velocity Distributions, 

 Station 7.3 m, VW Abutment, La/Bf = 0.44. 

Figure 9. Effect of variable discharge on water-surface profiles 

 and velocity distributions for constant abutment length. 
34

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

yftw, m = 0.262 0.210 0.194 0.177

Abutment

yf0

(a) Compound Channel B: Water-Surface Profiles, 

 VW Abutment, La/Bf = 1.0, Q = 0.0567 m3

/s. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Transverse Station, m

Velocity, m/s

yftw, m = 0.262 0.210 0.194

(b) Compound Channel B: Approach Velocity Distributions, 

 VW Abutment, La/Bf = 1.0, Q = 0.0567 m3

/s. 

Figure 10. Effect of tailwater (yftw) on water-surface profiles and 

 velocity distributions for constant discharge. 
35

highest tailwater in figure 10 corresponds to a relative floodplain depth of yf/ym = 0.27, at which 

the floodplain/main-channel velocity interactions become minimal, but the discharge per unit 

width must still be significantly larger in the main channel than the floodplain because of the 

larger depths there. Hence, even at these larger depths, the compound channel in these 

experiments cannot be treated as a rectangular channel in which the discharge per unit width is 

uniform across the channel. 

As scour occurs over time, the hydraulic conditions at the contracted section change to the extent 

that the scour hole is large enough to alter the channel geometry. Water-surface profiles were 

measured before and after scour and some representative results are compared in figures 11 and 

12. The water-surface profiles before scour were measured for a fixed bed of the same sediment 

as the movable bed, while the water-surface profiles after scour were measured after the scour 

hole had reached equilibrium. In figure 11, water-surface profiles before and after scour are 

compared for relative abutment lengths of 0.44 and 0.88. In figure 11(a), the tailwater is equal to 

the normal depth, while for the longer abutment shown in figure 11(b), the discharge is less and 

the tailwater is higher than the normal depth. The scour depths in figures 11(a) and 11(b) are 

13.4 cm and 20.1 cm, respectively. Thus, the combination of a larger scour depth and wider 

scour hole in a narrower contracted section causes a marked decrease in the water depths 

upstream of the abutment after scour as shown in figure 11(b), while the change in the watersurface profile shown in figure 11(a) is much smaller. A similar comparison is shown in figure 

12 for abutments that encroach on the main channel (La/Bf = 0.97 and 1.0). In comparison with 

figure 11(b), the water-surface profile drops much more after scour as shown in figure 12(a), 

even though the water discharge and the scour depths are the same (20.1 cm). This is probably 

caused by the increased backwater and narrower contraction as shown in figure 12(a). Upon 

comparison of figures 12(a) and 12(b), the scour depth in figure 12(b) is 18.3 cm, which is just 

slightly less than that in figure 12(a). The tailwater has been raised in figure 12(b) and the 

abutment is slightly longer. Apparently, these differences are sufficient to cause a greater watersurface drop after scour, almost to the level of the tailwater elevation downstream as shown in 

figure 12(b). All of the scour and all of the flow are forced to occur in the main channel of the 

contracted section for the conditions of figure 12(b). 

Main-Channel Centerline Velocity Profiles 

The ADV was used to measure centerline velocity profiles in the main channel starting from the 

approach section and continuing downstream to the contracted section. Velocities were measured 

in three dimensions, with the x-coordinate positive in the longitudinal flow direction, the ycoordinate positive upward, and the z-coordinate positive to the right when looking downstream. 

The x and z components of the velocity, Vx and Vz, are shown in figures 13, 14, and 15 for 

relative abutment lengths of 0.88, 0.97, and 1.0 as a function of the relative depth y’/ym, where 

ym = local depth in the main channel and y’ = vertical coordinate taken positive upward from the 

bottom of the main channel. The vertical components of the velocity were positive downward as 

drawdown occurred toward the contracted section; however, they were an order of magnitude 

smaller than the z components (lateral components) of the velocity and are not shown in the 

figures. The stations (STA.) shown in figures 13, 14, and 15 refer to the distance in meters 

downstream from the channel entrance. The centerline of the abutment is located at STA. 9.8. 
36

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

Before Scour After Scour Normal Depth

Abutment

(a) Tailwater equal to normal depth, 

 Compound Channel B: Water-Surface Profiles, 

 VW Abutment, La/Bf = 0.44, Q = 0.0850 m3

/s. 

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

Before Scour After Scour Normal Depth

Abutment

(b) Tailwater greater than normal depth, 

 Compound Channel B: Water-Surface Profiles, 

 VW Abutment, La/Bf = 0.88, Q = 0.0567 m3

/s (B). 

Figure 11. Water-surface profiles before and after scour for La/Bf = (a) 0.44 and (b) 0.88. 
37

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

Before Scour After Scour Normal Depth

Abutment

(a) Tailwater equal to normal depth, 

 Compound Channel B: Water-Surface Profiles, 

 VW Abutment, La/Bf = 0.97, Q = 0.0567 m3

/s. 

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

X-Station, m

Depth, m

Before Scour After Scour Normal Depth

Abutment

(b) Tailwater greater than normal depth, 

 Compound Channel B: Water-Surface Profiles, 

 VW Abutment, La/Bf = 1.0, Q = 0.0567 m3

/s (B). 

Figure 12. Water-surface profiles before and after scour for La/Bf = (a) 0.97 and (b) 1.0. 
38

The longitudinal velocities, Vx, in figures 13(a), 14(a), and 15(a), show that while there is 

significant acceleration in the main channel from STA. 6.1 to STA. 8.5, most of the acceleration 

occurs between STA. 8.5 and STA. 9.8. The flow from the floodplain joins the flow in the main 

channel as the contracted section is approached. This lateral flow is apparent in figures 13(b), 

14(b), and 15(b), where its velocity, Vz, is observed to continuously increase as the contracted 

section is approached. The lateral velocities vary from 10 to 15 percent of the longitudinal 

velocities. The corresponding water-surface profiles have already been shown as figures 11(b), 

12(a), and 12(b), and they show a very gradual increase in depth from STA. 6.1 to STA. 8.5 

followed by a leveling off up to the upstream side of the embankment and then an abrupt drop 

into the contracted section itself. The three-dimensional velocities shown in figures 13, 14, and 

15 make it clear that the lateral flux of flow from the floodplain into the main channel creates a 

driving mechanism for scour in addition to the horseshoe vortexes shed at the upstream edge of 

the abutment. 

The magnitudes of the velocities in figures 13, 14, and 15 relative to the critical velocity for the 

initiation of motion suggest a dilemma for conducting live-bed scour experiments with long 

abutments in which sediment transport occurs only in the main channel upstream of the 

abutment. If the critical velocity of the sediment is approximately 50 cm/s, for example, then no 

sediment transport can occur in the approach section where the velocities in the main channel are 

approximately 20 cm/s; however, large scour depths can occur in the contracted section where 

the initial velocities are higher than 50 cm/s. An increase in discharge will increase the velocity 

in the contracted section and cause more scour; however, it will result in relatively small 

increases in the approach velocity in the main channel as shown previously in figure 9(b). On the 

other hand, increases in tailwater to limit the scour depth at the same discharge only serve to 

increase the critical velocity while decreasing the approach flow velocities. This leaves the 

alternative of a drastic reduction in sediment size to achieve a critical velocity of less than 20 

cm/s, but then the scour depth will become significantly larger if maximum clear-water scour has 

not been reached, with the danger of bottoming out on the flume floor. Thus, the compound cross 

section, sediment size, and sediment thickness must be specifically designed for the case of livebed scour. 

Critical Velocity 

Critical velocity for the initiation of sediment motion at the abutment face was measured for all 

three sediments. The tailwater was raised higher than the normal depth for a given flow rate, the 

flow rate was set, and then the tailgate was gradually lowered until the normal depth for that flow 

rate was reached. If sediment motion did not occur, then the tailgate was raised again and the 

flow rate was increased. This process was repeated until sediment motion had just begun at the 

face of the abutment. The determination of the conditions for the initiation of sediment motion 

necessarily involved some qualitative judgment; hence, there is scatter in the data points shown 

in figure 16. Only for sediment C was it possible to initiate motion on the floodplain with no 

abutments in place and for uniform flow. This experimental point is also shown in figure 16. 

Shown for comparison in figure 16 are the various relationships that can be used to calculate the 

critical conditions in terms of the critical value of the sediment number Nsc: 
39

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90

Vx, cm/s

y'/y

m

STA. 6.1 STA. 7.3 STA. 8.5 STA. 9.8 STA. 10.9

STA. = Longitudinal Station, m

Vx = Longitudinal Velocity

(a) Main-Channel Centerline Velocity-x, 

 VW Abutment, La/Bf = 0.88, Q = 0.0567 m3

/s (B). 

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90

Vz , cm/s

y'/y

m

STA. 6.1 STA. 7.3 STA. 8.5 STA. 9.8 STA. 10.9

STA. = Longitudinal Station, m

Vz = Transverse Velocity,

 Positive to the right

 looking downstream

(b) Main-Channel Centerline Velocity-z, 

 VW Abutment, La/Bf = 0.88, Q = 0.0567 m3

/s (B). 

Figure 13. Main-channel centerline velocity from approach 

 to contracted section for La/Bf = 0.88. 
40

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90

Vx, cm/s

y'/y

m

STA. 6.1 STA. 7.3 STA. 8.5 STA. 9.8

STA. = Longitudinal Station, m

Vx = Longitudinal Velocity

(a) Main-Channel Centerline Velocity-x, 

 VW Abutment, La/Bf = 0.97, Q = 0.0567 m3

/s. 

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90

Vz , cm/s

y'/y

m

STA. 6.1 STA. 7.3 STA. 8.5 STA. 9.8

STA. = Longitudinal Station, m

Vz = Transverse Velocity,

 Positive to the right

 looking downstream

(b) Main-Channel Centerline Velocity-z, 

 VW Abutment, La/Bf = 0.97, Q = 0.0567 m3

/s. 

Figure 14. Main-channel centerline velocity from approach 

 to contracted section for La/Bf = 0.97. 
41

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90

Vx, cm/s

y'/y

m

STA. 6.1 STA. 7.3 STA. 8.5 STA. 9.8

STA. = Longitudinal Station, m

Vx = Longitudinal Velocity

(a) Main-Channel Centerline Velocity-x, 

 VW Abutment, La/Bf = 1.0, Q = 0.0567 m3

/s (B). 

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90

Vz , cm/s

y'/y

m

STA. 6.1 STA. 7.3 STA. 8.5 STA. 9.8

STA. = Longitudinal Station, m

Vz = Transverse Velocity,

 Positive to the right

 looking downstream

(b) Main-Channel Centerline Velocity-z, 

 VW Abutment, La/Bf = 1.0, Q = 0.0567 m3

/s (B). 

Figure 15. Main-channel centerline velocity from approach 

 to contracted section for La/Bf = 1.0. 
42

] 2

12.2

: 5.75[ ] log[

50

1/ 2

* d

y KEULEGAN Nsc c = τ (22) 

where Nsc = Vc/[(SG – 1)gd50]

1/2, Vc = critical velocity, SG = specific gravity of sediment, d50 = 

median sediment grain diameter, y = flow depth, and τ*c = critical value of Shields’ parameter. 

Based on the results in figure 16 and the best fit of the scour data, which will be discussed in the 

next chapter, the uniform-flow critical velocity Vc0 was calculated from Keulegan’s relationship, 

with τ*c determined from Shields’ diagram for the given sediment size.(75) However, Laursen’s 

relationship was found to give a better fit to the data for critical velocity at the abutment face 

(Vc), with τ*c varying from 0.035 for sediment C to 0.039 for sediment A. (The value of 0.039 

corresponds to τc = 4d50 in U.S. units.) 

Equations 20, 21, and 22 can be used in either SI or U.S. units. Equation 20 is an empirical fit of 

the data taken by Parola on rock riprap in a laboratory channel as reported by Pagan-Ortiz,(85)

while equation 22 follows directly from Keulegan’s equation, with ks = 2d50 and the definition of 

the critical shear velocity in terms of Shields’ parameter. However, the assumptions embedded in 

equation 21 can be clarified by outlining its derivation. As shown by Sturm and Sadiq,(23)

expressing shear stress in terms of slope from the assumption of uniform flow and substituting 

for the slope from Manning’s equation results in: 

where Cn = 1.486 in U.S. units and 1.0 in SI units; kn = constant in Strickler-type relationship for 

Manning’s n (n = knd50

1/6), which is equal to 0.0340 in U.S. units and 0.0414 in SI units; SG = 

specific gravity of the sediment; τ*c = critical value of Shields’ parameter; d50 = median grain 

diameter; and y = depth of uniform flow. Equation 21 follows directly from equation 23. The 

coefficient of 7.70 (= Cn/kng0.5) in equation 21 is the same regardless of the system of units; 

however, it assumes the use of the Strickler constant. If some other constant is used, then the 

coefficient in equation 21 would change accordingly. 

[ ]

161

:

50

0 135 d / y

. PAROLA N = . sc (20) 

[ ]

7 70

:

50

1 6

d / y

. τ LAURSEN N = /

*c

sc (21) 

 SG - τ d y k

C V = 

/ /

*c

n

n

c

1 6

50

1 3 ( (23) 1)
43

1

10

0.01 0.1 1

d50/y

Nsc

VW-Sed. A VW-Sed. B

VW-Sed. C ST-Sed. A

Uniform-Sed. C Laursen (SP = 0.039)

Laursen (SP = 0.035) Keulegan (SP = 0.047)

Parola

Figure 16. Measured and calculated critical velocities at incipient motion. 

 (SP = Shields Parameter). 
44

Equilibrium Scour Depths 

Raw data for the full set of equilibrium scour experiments are given in table 2. Several depths 

and velocities were measured prior to the beginning of scour for the fixed-bed case, and they are 

summarized in the table. The normal depth in the floodplain is yf0 and it varies with the discharge 

according to the uniform-flow measurements described previously in this chapter. In some 

instances, for the longer abutments, the tailgate was raised to submerge normal depth and limit 

the maximum scour. Under these conditions, the depth in the floodplain just upstream of the 

tailgate was measured and is reported as yftw to indicate a tailwater different from the normal 

depth. The approach velocity and depth, Vf1 and yf1, were measured in the floodplain upstream of 

the end of the abutment at the approach cross section, which was taken either as station 6.7 or 7.3 

for La/Bf ≤ 0.66, or as station 6.1 for La/Bf ≥ 0.88, with the centerline of the abutment located at 

station 9.8. These distances were chosen so that the backwater was near the maximum value and 

the velocity distribution was unaffected by the contraction. The approach velocity in the 

floodplain was measured at a relative depth of 0.4 as an estimate of the depth-averaged velocity. 

The resultant velocity and depth near the upstream corner of the abutment face were measured 

and are given in table 2 as Vab and yab, respectively. The measured approach velocity distribution 

was measured as described previously and integrated to produce values of the discharge 

contraction ratio M defined previously by equation 19. The measured velocity distributions were 

also integrated to determine the discharge in the main channel in ratio to the total discharge from 

the integration, and this ratio is given as Qm1/Q in column 6 of table 2. The critical velocity, V0c, 

was calculated from Keulegan’s equation using the floodplain normal depth yf0 for relative 

abutment lengths, La/Bf, less than or equal to 0.88. For all longer abutments that encroached on 

the main channel, the depth used to obtain the critical velocity was the normal or tailwater depth 

in the main channel. The scour depths, ds, shown in table 2 were all measured downward from 

the floodplain elevation at the abutment. They vary from 1.2 cm to 31.7 cm, and show a 

consistent increase with discharge. The smaller the sediment size, the larger the scour depth, as 

expected for clear-water scour. It should be noted that these scour experiments cover a range of 

values of La/yf1 from 12 to 97, and thus cover both intermediate-length and long abutments 

according to Melville’s classification. The last column in table 2 shows the scour test duration, 

which varied from 9.5 to 65.6 hours with a mean value of 34 hours. In all but one case, the 

duration equaled or exceeded 24 hours. 

Typical scour contours are shown in figures 17 through 22. The contours are given in terms of 

bed elevation in centimeters. The undisturbed floodplain and main-channel elevations at the 

station corresponding to the centerline of the abutment (station 9.75) are approximately 32.3 cm 

and 16.9 cm, respectively. A typical scour hole is shown in figure 17(a) for an intermediate 

vertical-wall abutment length (La/Bf = 0.44) and a discharge of 0.0992 m3

/s. The bottom of the 

scour hole is displaced laterally from the abutment face and the long axis of the scour hole is 

skewed from the approach flow direction as an indication of the deflected streamlines in the 

contracted section. A prominent bar deposit can be seen on the downstream side of the abutment 

in the region where the flow is expanding. An increase in discharge to 0.117 m3

/s is shown in 

figure 17(b), and the shape and location of the scour hole have changed. The maximum depth of 

scour occurs at the upstream corner of the abutment face, and the scour-hole shape is more 
45

Table 2. Raw experimental results for fixed-bed hydraulic variables and equilibrium scour depth. 

Shape La, 

m 

Q, 

m3

/s 

d50, 

mm 

M Qm1/Q yf0, 

cm 

yftw, 

cm 

yf1, 

cm 

Vf1, 

cm/s 

yab, 

cm 

Vab, 

cm/s 

ds, 

cm 

time, 

hrs 

VW 0.80 0.0700 3.3 0.900 0.51 3.29 3.29 3.23 25.63 2.65 49.7 1.2 9.5 

VW 0.80 0.0850 3.3 0.895 0.45 3.87 3.87 4.11 29.35 3.14 57.5 3.7 29.3 

VW 0.80 0.0994 3.3 0.883 0.40 4.42 4.42 4.60 30.75 3.66 60.3 11.0 51.0 

VW 0.80 0.1133 3.3 0.868 0.37 4.94 4.94 5.21 33.16 4.05 64.9 16.5 32.5 

VW 0.80 0.1278 3.3 0.865 0.35 5.39 5.39 5.88 34.62 4.72 68.4 19.5 29.8 

VW 0.80 0.1425 3.3 0.859 0.33 5.85 5.85 6.46 35.84 5.36 71.1 22.9 34.0 

VW 1.60 0.0637 3.3 0.790 0.51 2.97 2.97 3.96 23.71 2.53 61.3 6.1 30.4 

VW 1.60 0.0714 3.3 0.785 0.47 3.29 3.29 4.36 24.81 2.74 63.5 10.1 31.4 

VW 1.60 0.0705 3.3 0.785 0.47 3.29 3.29 4.36 24.81 2.74 63.5 8.8 37.7 

VW 1.60 0.0850 3.3 0.764 0.41 3.87 3.87 5.30 26.21 3.05 71.0 13.4 35.5 

VW 1.60 0.0992 3.3 0.749 0.37 4.42 4.42 6.13 28.22 3.60 77.9 19.8 32.2 

VW 1.60 0.0992 3.3 0.749 0.37 4.42 4.42 6.13 28.22 3.60 77.9 17.7 55.4 

VW 1.60 0.1028 3.3 0.744 0.36 4.69 4.69 6.49 28.86 3.69 78.1 17.7 29.4 

VW 1.60 0.1130 3.3 0.739 0.34 4.94 4.94 6.86 29.59 3.81 78.6 22.9 35.0 

VW 1.60 0.1173 3.3 0.729 0.33 5.18 5.18 7.35 30.27 4.02 83.0 28.3 28.8 

VW 2.40 0.0564 3.3 0.701 0.50 2.63 2.63 4.05 19.38 1.22 69.9 8.5 50.1 

VW 2.40 0.0637 3.3 0.679 0.46 2.97 2.97 4.85 20.15 1.55 76.2 15.5 33.2 

VW 2.40 0.0708 3.3 0.657 0.43 3.29 3.29 5.88 20.79 1.92 83.1 18.3 35.0 

VW 2.40 0.0850 3.3 0.629 0.38 3.87 3.87 6.61 22.65 2.59 91.3 23.8 36.4 

VW 2.40 0.0856 3.3 0.629 0.38 3.87 3.87 6.61 22.65 2.59 91.3 20.7 35.7 

VW 2.40 0.0994 3.3 0.607 0.34 4.42 4.42 7.68 24.99 2.90 98.5 29.0 33.1 

VW 3.23 0.0499 3.3 0.530 0.46 2.26 2.32 4.69 16.82 2.26 69.5 20.1 33.0 

VW 3.23 0.0567 3.3 0.452 0.36 2.50 6.46 7.16 15.39 5.49 76.8 18.0 34.0 

VW 3.23 0.0567 3.3 0.458 0.37 2.50 4.88 6.28 16.55 3.87 69.6 21.6 27.5 

VW 3.23 0.0567 3.3 0.442 0.35 2.50 5.52 6.67 16.37 4.42 74.2 25.3 27.0 

VW 3.55 0.0496 3.3 0.453 0.42 2.26 2.26 5.09 15.82 1.22 83.0 17.1 27.2 

VW 3.55 0.0565 3.3 0.389 0.36 2.50 2.50 6.46 16.95 1.46 94.7 20.1 24.0 

VW 3.55 0.0671 3.3 0.347 0.31 3.05 3.32 8.20 15.88 1.65 108.3 29.3 24.0 

VW 3.66 0.0569 3.3 0.264 0.26 2.50 10.91 11.49 10.39 10.67 63.0 15.5 28.5 

VW 3.66 0.0569 3.3 0.305 0.31 2.50 5.79 8.35 13.59 6.71 77.3 18.3 26.8 

VW 3.66 0.0567 3.3 0.318 0.32 2.50 4.21 7.56 15.70 5.49 83.9 27.4 27.0 

VW 0.80 0.0856 2.7 0.895 0.45 3.87 3.87 4.11 29.35 3.14 57.5 10.7 31.8 

VW 0.80 0.0992 2.7 0.883 0.40 4.42 4.42 4.60 30.75 3.66 60.3 14.3 55.5 

VW 0.80 0.1144 2.7 0.868 0.37 4.94 4.94 5.21 33.16 4.05 64.9 19.5 32.1 

VW 0.80 0.1278 2.7 0.865 0.35 5.39 5.39 5.88 34.62 4.72 68.4 21.3 26.7 

VW 1.60 0.0711 2.7 0.785 0.47 3.29 3.29 4.36 24.81 2.74 63.5 12.2 65.6 

VW 1.60 0.0853 2.7 0.764 0.41 3.87 3.87 5.30 26.21 3.05 71.0 14.6 33.0 

VW 1.60 0.0992 2.7 0.749 0.37 4.42 4.42 6.13 28.22 3.60 77.9 20.1 33.8 

VW 1.60 0.1065 2.7 0.744 0.35 4.69 4.69 6.49 28.86 3.69 78.1 27.1 35.3 

VW 2.40 0.0567 2.7 0.701 0.50 2.63 2.63 4.05 19.38 1.22 69.9 12.5 35.5 
46

Table 2. Raw experimental results for fixed-bed hydraulic variables and equilibrium scour depth (continued). 

Shape La, 

m 

Q, 

m3

/s 

d50, 

mm 

M Qm1/Q yf0, 

cm 

yftw, 

cm 

yf1, 

cm 

Vf1, 

cm/s 

yab, 

cm 

Vab, 

cm/s 

ds, 

cm 

time, 

hrs 

VW 2.40 0.0637 2.7 0.679 0.462 2.97 2.97 4.85 20.1 1.55 76.2 18.6 35.2 

VW 2.40 0.0711 2.7 0.657 0.426 3.29 3.29 5.88 20.8 1.92 83.1 21.3 33.0 

VW 2.40 0.0853 2.7 0.629 0.381 3.87 3.87 6.61 22.6 2.59 91.3 27.4 33.8 

VW 0.80 0.0567 1.1 0.909 0.595 2.63 2.63 2.68 23.6 2.10 45.5 11.6 26.4 

VW 0.80 0.0708 1.1 0.900 0.506 3.29 3.29 3.23 25.6 2.65 49.7 16.2 28.5 

VW 0.80 0.0856 1.1 0.895 0.447 3.87 3.87 4.11 29.4 3.14 57.5 20.4 29.7 

VW 1.60 0.0569 1.1 0.798 0.557 2.63 2.63 3.54 22.6 2.32 57.9 18.6 28.1 

VW 1.60 0.0708 1.1 0.785 0.469 3.29 3.29 4.36 24.8 2.74 63.5 21.6 26.8 

VW 1.60 0.0850 1.1 0.764 0.412 3.87 3.87 5.30 26.2 3.05 71.0 26.8 24.0 

VW 2.40 0.0419 1.1 0.726 0.640 1.85 1.85 3.08 16.8 0.82 59.5 14.3 23.4 

VW 2.40 0.0499 1.1 0.711 0.562 2.26 2.26 3.66 18.0 1.04 65.4 21.6 28.7 

VW 2.40 0.0569 1.1 0.701 0.504 2.63 2.63 4.05 19.4 1.22 69.9 25.6 21.3 

ST 1.17 0.0708 3.3 0.852 0.490 3.29 3.29 3.57 25.2 1.89 50.1 1.2 9.5 

ST 1.17 0.0850 3.3 0.837 0.434 3.87 3.87 4.11 29.6 2.62 68.8 2.7 21.3 

ST 1.17 0.1113 3.3 0.822 0.364 4.94 4.94 5.70 32.4 4.18 71.2 8.5 56.4 

ST 1.17 0.1419 3.3 0.812 0.327 5.85 5.85 7.01 34.7 5.18 80.2 15.8 37.1 

ST 1.97 0.0708 3.3 0.733 0.459 3.29 3.29 4.48 22.6 2.10 62.8 4.3 49.0 

ST 1.97 0.0853 3.3 0.711 0.403 3.87 3.87 5.61 25.6 2.62 70.1 13.4 45.0 

ST 1.97 0.0997 3.3 0.693 0.366 4.42 4.42 6.43 27.2 3.38 73.6 19.5 50.5 

ST 1.97 0.1136 3.3 0.683 0.341 4.94 4.94 7.19 27.9 3.63 89.2 22.6 50.8 

ST 2.37 0.0561 3.3 0.720 0.520 2.63 2.63 4.05 19.3 2.35 56.0 4.0 60.6 

ST 2.37 0.0714 3.3 0.685 0.450 3.29 3.29 5.09 20.9 2.07 73.4 12.8 56.8 

ST 2.37 0.0847 3.3 0.665 0.401 3.87 3.87 6.37 22.5 3.08 85.2 19.2 50.0 

ST 2.37 0.0983 3.3 0.653 0.369 4.42 4.42 7.13 24.4 2.80 95.6 22.9 52.6 

ST 3.23 0.0637 3.3 0.470 0.385 2.86 3.20 6.34 18.4 1.77 90.2 31.1 27.5 

ST 3.23 0.0705 3.3 0.437 0.351 3.35 3.63 6.98 19.7 2.32 96.1 31.7 31.0 

ST 3.23 0.0847 3.3 0.414 0.323 3.93 3.81 8.38 19.9 3.20 100.2 31.4 27.0 

ST 3.55 0.0493 3.3 0.471 0.453 2.26 1.77 4.82 16.8 1.40 73.7 23.5 36.0 

ST 3.55 0.0555 3.3 0.400 0.378 2.50 2.29 6.07 17.7 1.65 88.1 22.6 27.5 

ST 3.55 0.0637 3.3 0.371 0.349 2.86 2.99 7.13 17.4 1.43 94.5 30.5 28.0 

ST 3.55 0.0697 3.3 0.302 0.321 3.20 3.29 7.86 17.6 1.77 102.6 30.5 33.5 

ST 3.66 0.0493 3.3 0.420 0.420 2.26 2.10 5.15 17.4 1.65 80.2 22.2 53.5 

ST 3.66 0.0569 3.3 0.360 0.360 2.50 2.44 6.64 17.3 1.95 94.3 26.2 31.0 

ST 3.66 0.0697 3.3 0.315 0.315 3.17 3.14 8.53 16.9 2.19 110.3 28.3 26.0 

WW 1.61 0.0850 3.3 0.764 3.87 3.87 5.30 26.2 4.9 25.2 

WW 1.61 0.0992 3.3 0.749 4.42 4.42 6.13 28.2 14.0 26.7 

WW 1.61 0.1122 3.3 0.739 4.94 4.94 6.86 29.6 15.2 31.8 

WW 2.22 0.0705 3.3 0.684 3.29 3.29 5.55 21.6 13.7 32.9 

WW 2.22 0.0841 3.3 0.658 3.87 3.87 6.34 23.4 19.8 28.0 

WW 2.22 0.0994 3.3 0.637 4.42 4.42 7.35 25.7 23.2 30.2 
47

nearly conical. The effect of increasing the abutment length is shown in figure 17(c), with 

essentially the same discharge as in figure 17(a). The scour hole is obviously larger in both depth 

and volume. The point of deepest scour is again displaced laterally from the face of the 

abutment, and the scour hole is elongated as in figure 17(a). 

Scour-hole contours for the longer vertical-wall abutment lengths (La/Bf = 0.88, 0.97, and 1.0) 

are illustrated in figure 18. The discharge of 0.0567 m3

/s is the same for all three abutment 

lengths in figure 18; however, the tailwater varies. It is equal to the normal floodplain depth (8.2 

cm) in figure 18(b), and has a ratio to the normal floodplain depth of 1.95 and 2.32 in figures 

18(a) and 18(c), respectively. The scour hole in figure 18(a) retains the laterally displaced 

elongated shape observed in figures 17(a) and 17(c); however, it extends into the sideslope of the 

main channel. As the abutment encroaches on the main channel in figures 18(b) and 18(c), there 

is a definite contraction-scour effect; however, multiple scour holes are apparent in figure 18(b) 

that combine local and contraction scour. 

Scour-hole shapes for the spill-through abutment can be observed in figures 19 and 20. At the 

lower discharge of 0.0850 m3

/s in figure 19(a) as compared to figure 19(b), the scour hole is 

laterally displaced from the centerline of the abutment face. As the discharge increases, it moves 

to the upstream quadrant of the abutment face, and the deepest point of the scour hole is located 

immediately adjacent to the abutment. Comparing figure 19(b) with figure 18(c), for which the 

discharge and abutment length are essentially the same, the scour hole for the vertical-wall 

abutment is deeper and remains laterally displaced from the centerline of the abutment face. 

Apparently, all other things being equal, the discharge required to localize the scour hole to the 

upstream corner of the abutment is lower for the spill-through abutment. It is interesting to note 

from figure 19(c) that an increase in relative abutment length La/Bf to 0.88 does not change the 

essential shape of the scour hole. Although there is some minor encroachment of the scour hole 

into the main channel, it is much less than in the case of the vertical-wall abutment of the same 

length in figure 18(a), having a smaller discharge. However, comparing figures 19(c) and 20(a) 

for the same length of spill-through abutment (La/Bf = 0.88), but for a smaller discharge in figure 

20(a), the scour hole moves almost completely into the main channel in figure 20(a). In figures 

20(b) and 20(c), the scour process appears to be dominated by contraction scour that moves well 

upstream of the abutment as a headcut and extends considerably downstream as well. 

The influence of the abutment shape at the same discharge and at essentially the same abutment 

length is illustrated in figure 21 where scour contours are shown for the vertical-wall, spillthrough, and wingwall abutments. The maximum scour depths in figures 21(a), 21(b), and 21(c) 

are 29.0 cm, 22.9 cm, and 23.2 cm, respectively. The laterally displaced and elongated scour hole 

for the vertical-wall abutment shown in figure 21(a) moves next to the abutment face in the 

upstream quadrant in figure 21(b). In figure 21(c), the scour hole is also in the upstream quadrant 

of the abutment face; however, it is more conical in shape and has been displaced laterally. The 

dependence of the maximum scour depth on the abutment shape is related to the degree of flow 

blockage so that for long abutments with a high degree of blockage, the local effect of the 

separation zone on the scour as affected by shape diminishes. 
48

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m)

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm) 

Main

Channel

(a) Q = 0.0992 cms

La/Bf = 0.44

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m)

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m)

Main

Channel

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm) 

(b) Q = 0.117 cms

La/Bf = 0.44

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m) 

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m) 

Main

Channel

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

(c) Q = 0.0994 cms

La/Bf = 0.66

Figure 17. Bed elevations for shorter VW abutments after scour (cms = m3/s). 
49

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf = 0.88

(a) Q = 0.0567 cms (B)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf=0.97

(b) Q = 0.0567 cms

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf=1.00

(c) Q = 0.0567 cms (B)

Figure 18. Bed elevations for longer VW abutments after scour (cms = m3/s). 
50

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

Main

Channel La/Bf=0.65

(a) Q = 0.0850 cms

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

Main

Channel La/Bf=0.65

(b) Q=0.0983 cms

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf=0.88

(c) Q=0.0850 cms

Figure 19. Bed elevations for shorter ST abutments after scour (cms = m3/s). 
51

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf=0.88

(a) Q=0.0708 cms (B)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf=0.97

(b) Q = 0.0697 cms

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

La/Bf=1.00

(c) Q = 0.0697 cms

Figure 20. Bed elevations for longer ST abutments after scour (cms = m3/s). 
52

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m) 

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m) 

Main

Channel

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

(a) Q = 0.0994 cms

La/Bf = 0.66

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

Main

Channel La/Bf=0.65

(b) Q=0.0983 cms

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z-STA. (m)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X-STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm)

Main

Channel

La/Bf=0.61

(c) Q = 0.0992 cms

Figure 21. Bed elevations for VW, ST, and WW abutments after scour (cms = m3/s). 
53

Sediment size effects on clear-water abutment scour are shown in figure 22 for sediments A, B, 

and C for the conditions of the same discharge and same length of a vertical-wall abutment. The 

scour-hole shapes and depths, as well as the downstream bar deposits for sediments A and B, are 

very similar (as shown in figures 22(a) and 22(b)), probably because of their small difference in 

size. For sediment C, however, the scour hole is significantly enlarged both in lateral extent and 

depth. In addition, the velocities exiting the main scour hole are sufficient to erode an elongated 

trench downstream of the main scour hole. There can be no doubt that sediment size as reflected 

by critical velocity has an important effect on the depth of the clear-water scour. 
54

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m)

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm) 

Main

Channel La/Bf=0.44

(a) Q = 0.0850 cms

 Sediment A

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m)

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m)

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm) 

Main

Channel

La/Bf=0.44

(b) Q = 0.0850 cms

 Sediment B

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Z - STA. (m) 

9.5

10.0

10.5

11.0

11.5

12.0

12.5

X - STA. (m) 

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

36.0

40.0

Elevation (cm) 

Main

Channel

La/Bf=0.44

(c) Q = 0.0850 cms

 Sediment C

Figure 22. Bed elevations for VW abutments and sediments A, B, and C (cms = m3/s). 
55

CHAPTER 4. ANALYSIS OF RESULTS 

EQUILIBRIUM SCOUR DEPTH 

Clear-water scour at a bridge abutment located on the floodplain of a compound open channel is 

influenced by the flow distribution between the floodplain and the main channel in the approach 

and contracted bridge cross sections. It has been shown that the effect of the flow contraction can 

be accounted for by the discharge contraction ratio M, which depends on both the abutment 

length and the discharge distribution in the approach section of a compound channel.(20) In 

addition, previous equations for predicting clear-water scour depths at abutments have depended 

on the ratio of floodplain velocity V1 to critical velocity Vc in the bridge approach section,(14,20)

with maximum clear-water scour occurring when V1 approaches Vc (see chapter 2). In this 

formulation, the independent variables for scour prediction are determined from one-dimensional 

numerical models such as WSPRO. Sturm and Chrisochoides(86) have explored the suitability of 

these one-dimensional estimates of reference velocities and depths for scour prediction, and their 

results will be given in this chapter. For the case of significant backwater caused by the bridge 

opening, Sturm and Sadiq(23) have shown that the appropriate reference depth is the unconstricted 

floodplain depth rather than the approach floodplain depth with the bridge constriction in place. 

Extension of this methodology to abutments that encroach on the banks of the main channel in a 

compound channel has been reported by Sturm and Chrisochoides(87) and will be discussed in 

this chapter along with a consideration of the live-bed scour case. 

A possible alternative to parameterizing scour depth in terms of approach velocity and discharge 

distribution is to relate it directly to local hydraulic conditions near the abutment face, although 

these conditions can only be predicted by two- or three-dimensional numerical models. Possible 

advantages of this alternative formulation include: (1) it may provide a means of unifying scourprediction equations obtained from experiments in rectangular channels with those based on the 

more realistic compound-channel geometry, and (2) it may offer greater sensitivity of scour 

predictions to changes in local depth and velocity at the location of scour initiation near the 

abutment face. Theoretically, local scour is initiated when the ratio of local flow velocity to 

critical velocity exceeds unity, and it continues at an ever-decreasing rate until equilibrium is 

reached. The initial rate of scour and limiting equilibrium depth of scour might be expected to 

depend on the excess of the local velocity ratio in comparison to unity among other variables. 

The implementation procedure for the prediction of clear-water abutment scour that is described 

herein is based on laboratory experiments in two different compound-channel cross sections as 

described in chapter 3. Abutment length, abutment shape, discharge, and sediment size were 

varied, and the resulting equilibrium scour depths were measured. In this chapter, a relationship 

for the prediction of equilibrium scour depth is formulated. 

CLEAR-WATER SCOUR: FORMULATION I 

The first formulation is based on Laursen’s long contraction scour theory for clear-water scour(13)

as modified by Sturm and Sadiq(23) for abutments ending on the floodplain of a compound 

channel (setback abutments). The long contraction theory is also extended to the case of the 

abutment encroaching on the bank of the main channel (bankline abutments) for both clear-water 

and live-bed scour. The theoretical development for clear-water scour is given first, followed by 

correlation of the scour data for both compound channels A and B. 
56

Theoretical Clear-Water Contraction Scour for Setback Abutments 

With reference to figure 23 for equilibrium scour conditions in a long contraction, it is assumed 

that the approach conditions tend toward the unconstricted depth and velocity on the floodplain, 

yf0 and Vf0, respectively. This assumption is consistent with the assumption of an idealized 

contraction with negligible head loss and velocity-head changes made by Laursen(13) in his 

analysis of clear-water contraction scour. The continuity equation for the floodplains from the 

approach section to the contracted section at equilibrium is: 

where Vf0 and yf0 = average unconstricted velocity and depth in the approach floodplain, 

respectively; Vf2 and yf2 = average velocity and depth in the contracted floodplain, respectively; 

and μf = generalized discharge contraction ratio. If the channel is rectangular with a width equal 

to the floodplain width, then μf = Bf2/Bf0, the ratio of contracted floodplain width to approach 

floodplain width. However, in a compound channel, some of the approach floodplain flow joins 

the main-channel flow in the contracted section as shown previously in figure 7(b). Under these 

circumstances, μf becomes an empirical factor that depends on the flow distribution in the 

approach channel and its redistribution between the main channel and the floodplain in the 

contracted section. For equilibrium clear-water scour conditions, the velocity in the contracted 

floodplain section, Vf2, is set equal to the critical velocity, Vf2c. If this substitution is made in 

equation 24, and it is divided by the critical velocity in the approach section, Vf0c, the result is: 

Now from equation 23 for critical velocity, the ratio of critical velocities, Vf2c/Vf0c, is 

proportional to the ratio of the corresponding depths to the 1/6 power (i.e., (yf2/yf0)

1/6). If this 

substitution is made in equation 25 and it is solved for yf2/yf0, the resulting equation is: 

An equation of this form has been suggested by Richardson and Davis(8) in HEC-18 for 

estimating clear-water contraction scour, except that μf is evaluated as a geometric contraction 

ratio. Using the assumption that velocity-head changes and head losses at equilibrium 

contraction scour are small,(13) it can be shown that yf2 = dsc + yf0 (see figure 23). Then equation 

26 can be written in terms of the theoretical contraction scour depth, dsc, as: 

V y = μ V y f f f f f 0 0 2 2 (24) 

 y

V

V y = μ V

V

f

f c

f c

f f

f c

f

2

0

2

0

0

0 (25) 

[ ] 0 0

6 7

0

2

μ

V / V

 = 

y

y

f

f f c

/

f

f (26) 
57

Plan

Sec. A-A Sec. B-B

A A

B

B

1 2

La

B

B

f

m

y

f1 y

f0 y

f0 y

Main Channel

Floodplain

Abutment

Q

y

f2 y

f0

Vf1

Vab

m0

dsc

Approach Bridge

Figure 23. Definition sketch for idealized floodplain contraction scour 

 in a laboratory compound channel, compound channel A. 
58

Laursen’s(13) assumption is that local abutment scour ds = rdsc, where r is some constant greater 

than 1 that multiplies the theoretical contraction scour depth given by equation 27. Although the 

contraction scour predicted by equation 27 is for a fictitious long contraction with several 

restrictive assumptions, including the assumption that the approach depth and velocity tend 

toward their unconstricted values as scour approaches equilibrium, it suggests that an equation 

for abutment scour might take the form: 

where Cr and C0 are constants to be determined by experiment, and the exponent in equation 27 

has been taken to be approximately unity as found in several experimental investigations.(14,19)

There are two dimensionless ratios of interest on the right-hand side of equation 28. The velocity 

ratio in the numerator is the unconstricted floodplain velocity in the approach section in ratio to 

the critical velocity under the same conditions. Theoretically, once this velocity ratio reaches the 

value of unity, maximum clear-water scour occurs and live-bed scour begins, assuming that the 

approach depth and velocity tend toward their unconstricted values as equilibrium is reached. 

The other independent dimensionless variable in equation 28, which is especially relevant in the 

present experiments, is the discharge contraction ratio at equilibrium, μf. Sturm and Janjua(20)

suggested the use of the discharge contraction ratio M, which is defined as the ratio of 

unobstructed discharge in the approach channel to total discharge at the beginning of scour, as an 

estimate of μf. They showed that M was a good estimate of μf for small depth changes in the 

contraction, which was indeed the case for their experiments with a contraction experiencing 

negligible changes in depth through the contraction because the tailwater was high with respect 

to critical depth. However, if there is choking in the contraction and/or significant upstream 

backwater at the beginning of scour, then an improved estimate of μf is needed. If it is assumed 

that M is an estimate of qf1/qf2 as shown by Sturm and Janjua,(20) then μf can be estimated as: 

where M is evaluated for the approach depth at the beginning of scour and can be obtained from 

the WSPRO output,(21) qf0 = Vf0yf0 at the end of scour as estimated by the unconstricted approach 

floodplain velocity and depth, and qf1 = Vf1yf1 at the beginning of scour determined by the 

approach floodplain velocity and depth obtained from the constricted water-surface profile in the 

WSPRO output. Substituting equation 29 into equation 28 results in: 

1 [ ] 0 0

6 7

0 μ

V / V

 + = 

y

d

f

f f c

/

f

sc (27) 

[ ] 0

0 0

0

 C

μ

V / V

 = C y

d

f

f f c

r

f

s − (28) 

q

q = M 

q

q

μ = 

f

f

f

f

f

1

0

2

0 (29) 
59

where qf0c = Vf0cyf0 and Vf0c is the critical velocity in the floodplain for the unconstricted 

floodplain depth yf0. Equation 30 provides a working equation for fitting the experimental 

results. 

Theoretical Clear-Water Contraction Scour for Bankline Abutments 

For the abutment encroaching on the main-channel banks, the floodplain portion of the 

contracted section no longer exists, and the formulation of equation 30 has to be revised for this 

case. As shown in figure 24, the idealized contraction scour is assumed to occur in the contracted 

main-channel section, and by definition for La = Bf, M is exactly equal to qm1/qm2 for a constantwidth main channel. Thus, in the contracted section, continuity for critical conditions at 

equilibrium scour results in: 

where Vm2c = critical velocity in the main channel at the contracted section at equilibrium, ym2 = 

depth of flow in the contracted section in the main channel for equilibrium scour, qm1 = flow rate 

per unit width in the approach main-channel section at the beginning of scour, and M = discharge 

contraction ratio. Dividing equation 31 by Vm0cym0 gives: 

where Vm0c = the critical velocity in the main channel for the unconstricted flow depth in the 

main channel of ym0. Using equation 23 to replace the ratio of critical velocities on the left side 

of equation 32 with the ratio of depths to the 1/6 power, (ym2/ym0)

1/6, and solving for ym2/ym0

produces: 

To be consistent with the case of the abutment terminating on the floodplain, it seems convenient 

to define the depth of scour relative to the floodplain elevation in both cases. Then, as 

C M q

q

 = C y

d

f c

f

r

f

s [ ] 0

0

1

0

− (30) 

M

q V y = m

m c m

1

2 2 (31) 

M V y

q

 = 

y

y

V

V

m c m

m

m

m

m c

m c

0 0

1

0

2

0

2 (32) 

6 / 7

0 0

1

0

2 ] ( ) [ M V y

q

 = 

y

y

m c m

m

m

m (33) 
60

Plan

Sec. A-A

A A

1 2

L

a B

B

f

m

y

m1

d

m0

Main Channel

Floodplain

Abutment

Q

ym2 yf0

Vf1

Water Surface

Floodplain

Main Channel

y

sc

Figure 24. Definition sketch for idealized main-channel contraction scour 

 in a laboratory compound channel, compound channel B. 
61

shown in figure 24, ym2 = dsc + yf0. Again, for consistency with equation 30, equation 33 can be 

multiplied by ym0/yf0 to yield: 

Finally, if it is assumed that the 6/7 power can be approximated by unity as in the previous 

floodplain scour analysis, equation 34 can be placed in a form similar to equation 30: 

where the coefficients Cr’ and C0’, in general, would not be the same for floodplain scour as 

given by equation 30. 

Correlation of Results 

The correlation of scour-depth measurements according to equation 30 is presented in figure 25 

for the shorter abutment lengths with significant setbacks from the bank of the main channel 

(setback abutments). The results for vertical-wall abutments are given for both compound 

channels (A and B) and for relative abutment lengths La/Bf varying from 0.17 to 0.66. Sediment 

sizes of 1.1 mm, 2.7 mm, and 3.3 mm are included in the results in figure 25. Shown for 

comparison in figure 25 is the proposed relationship suggested by Sturm and Janjua(20) from a 

much more limited data set in a short, horizontal laboratory compound channel. In the latter data 

set, the tailwater essentially submerged the contraction so that there was very little backwater 

from the contraction; however, these data agree with the present data if the unconstricted 

floodplain depth is substituted for yf0, even though it is not the normal depth. The best-fit 

relationship for all of the data is also given in the figure for which Cr = 8.14 and C0 = 0.40 in 

equation 30. For this relationship, the coefficient of determination r2

 is 0.86, and the standard 

error of estimate in ds/yf0 is 0.68. 

In figure 26, the scour data for longer abutments that approach the bank of the main channel 

(bankline abutments) are correlated according to equation 35. These additional data comprise 

relative abutment lengths La/Bf of 0.88, 0.97, and 1.0. The major differences between equations 

30 and 35 are the replacement of qf1 in equation 30 with qm1 in equation 35, and the evaluation of 

the critical velocity in the main channel for the unconstricted depth ym0 in equation 35 instead of 

the critical velocity in the floodplain in equation 30. The value qm1 was evaluated approximately 

as the measured discharge in the approach main channel divided by the main-channel topwidth. 

For those cases in which the tailwater depth exceeded normal flow depth, the tailwater depth yftw

was substituted for the normal depth yf0. It can be observed from figures 11 and 12 that the 

tailwater depth is representative of the depth in the contracted section as was the normal depth 

y

y M V yq + = 

y

d

f

m

m c m

m

f

sc

0

6 / 7 0

0 0

1

0

] ( ) 1 (34) [

] ( ) [ 0

0 0

1

0

C M V y

q

 = C y

d

m c f

m

r

f

s ′ − ′ (35) 
62

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ξ = qf1/(Mqf0c)

ds/yf0

B, 3.3 mm A, 3.3 mm B, 2.7 mm B, 1.1 mm Best Fit

KEY: Compound Channel (A or B), Sediment d50

Sturm & Janjua 1994

Figure 25. Scour-depth relationship based on approach hydraulic variables 

 for VW abutments with 0.17 ≤ La/Bf ≤ 0.66. 
63

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 2.0 4.0 6.0 8.0 10.0

qm1/(MVm0cyf0)

ds/yf0

Vertical Wall Spill-through Best Fit

Figure 26. Scour-depth relationship based on approach hydraulic variables 

 in main channel for VW and ST abutments with 0.88 ≤ La/Bf ≤ 1.0 

 and d50 = 3.3 mm in compound channel B. 
64

for the unsubmerged cases. Strictly speaking, the derivation of equation 35 applies only to the 

case of La/Bf = 1.0; however, the additional abutment length data seem to follow the same trend 

in figure 26. The scour hole clearly extends from the floodplain into the main channel for La/Bf = 

0.88 in figures 18(a) and 20(a), while it is localized near the abutment face on the floodplain in 

figure 19(c). In all cases of La/Bf = 0.97 and 1.0 in figures 18, 19, and 20, the scour hole is 

located in the main channel. The best-fit values of the coefficients in equation 35 are Cr’ = 2.0 

and C0’ = 0.47, with an r2

 = 0.92 and a maximum value of ds/yf0 of approximately 10. 

The data shown in figure 26 represent the maximum total scour depths regardless of their 

classification as local or contraction scour. Indeed, these two types of scour occur simultaneously 

at different locations, as in figure 18(b), while interfering and combining as in figure 18(c), for 

example. Furthermore, the lengthening of the scour hole in the upstream direction in figures 

20(b) and 20(c) seems to reflect the interaction of the floodplain and main-channel flows well 

upstream of the abutment in a contraction-scour effect. Based on these results, the artificial 

addition of contraction and local abutment scour when the abutment is near the bank of the main 

channel may be responsible for the overestimates of scour in these cases.

If the scour is a combined local and contraction effect for abutments near the bank of the main 

channel, it would seem that either equation 30 or 35 might serve as scour predictors because both 

are based on acceleration of the flow caused by the contraction (i.e., the floodplain flow in the 

case of equation 30 and the main-channel flow in the case of equation 35). Accordingly, the data 

for both short and long abutments are combined in figure 27, with the critical velocity calculated 

in the main channel for La/Bf = 0.97 and 1.0, and in the floodplain for all other cases. In other 

words, equation 30 is modified as: 

 

where Vxc = Vf0c for abutments located on the floodplain (setback abutments), and Vxc = Vm0c for 

abutments near the bank of the main channel (bankline abutments). In addition, the data for spillthrough and wingwall abutments are incorporated into figure 27. It is apparent that the relative 

scour depth levels off to a constant value for large values of the independent variable just as in 

figure 26. The actual mean value of the relative scour depth is 9.4 ±1.0 after the point of leveling 

off; however, considering the uncertainty involved and remaining on the conservative side, the 

maximum value is shown as 10 in figure 27. The scour data for the linear portion of the graph are 

compared with the best-fit relationship given previously in figure 25 for the setback abutments 

only. The data for compound channel A are not shown in the figure to retain clarity; 

nevertheless, the best-fit regression analysis remains essentially unchanged with the addition of 

the long abutment data (Cr = 8.14, C0 = 0.40), except that the standard error of estimate for the 

relative scour depth increases from 0.68 to 0.75. In all cases, the scour depth is defined as the 

maximum depth in the scour hole measured below the undisturbed floodplain elevation. The fact 

that equations 30 and 35 both predict the scour depth reasonably well for the long abutments 

seems to be indicative of the contraction process controlling scour, whether it is measured by the 

main-channel flow acceleration or the floodplain flow acceleration. Both equations 30 and 35 

( ) 0

0

1

0

C M V y

q

 = C

y

d

xc f

f

r

f

s − (36) 
65

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

qf1/(MVxcyf0)

ds/yf0

 VW, Setback VW, Bankline ST, Setback ST, Bankline

 WW, Setback Best Fit - VW & ST Best Fit - ST Setback

Figure 27. Scour-depth relationship based on approach hydraulic variables 

 in floodplain for all abutments and sediments, compound channel B. 
66

require an estimate of average floodplain depth in natural channels, and it is recommended that 

the average depth of flow blocked by the abutment and embankment on the floodplain in the 

approach section be used for the unconstricted flow (yf0) and the constricted flow (yf1) for 

consistency. 

Also shown in figure 27 are data for the spill-through and wingwall abutments. The spill-through 

data appear in two different regions, depending on the magnitude of the independent variable in 

figure 27. For shorter abutments, or more precisely, abutments that block less floodplain flow, 

the scour depth is smaller for the spill-through than for the vertical-wall abutment. The 

difference becomes less and less as the independent variable increases, which corresponds to an 

increasing discharge or an abutment that blocks more flow. The spill-through data seem to merge 

with the vertical-wall data for qf1/(MVxcyf0) greater than about 1.2 and remain comingled with 

the vertical-wall data as the best-fit line levels off at a maximum value of relative scour depth of 

about 10. Apparently, the local flow effects at the face of the abutment caused by the differences 

in the abutment shape become unimportant as the degree of contraction becomes large. A scourdepth correction for the spill-through shape can be derived from the data in figure 27. A leastsquares best fit for the spill-through data produces a linear relationship, as shown in figure 27, 

having a coefficient of determination of r2

 = 0.93, with a standard error of estimate for ds/yf0 of 

0.45. If the relative scour depth for a spill-through abutment is taken as a ratio to the value for a 

vertical-wall abutment, the result is a shape correction factor for the spill-through abutment KST: 

where ξ = qf1/(MVxcyf0), and KST = 1.0 for ξ ≥1.2 and 0 for ξ ≤ 0.67. The shape factor of 0.55 

given in HEC-18(8) for spill-through abutments corresponds to a ξ value of approximately 0.82, 

which would correspond to short setback abutments for which M approaches 1, yf0 = yf1, and 

Vf1/Vc = 0.82. In other words, the shape factor of 0.55 given in HEC-18 should be viewed as an 

average value for short abutments with no backwater and conditions approaching threshold livebed scour. Melville(24) has also observed that the shape factor approaches a value of 1 for very 

long abutments, although his values are given in terms of the abutment length rather than the 

discharge contraction ratio. The wingwall data shown in figure 27 are not significantly different 

from the spill-through data. This is a result of the fact that the approach velocity and depth were 

not measured for this case, but rather estimated from the vertical-wall abutment data. Although 

there are only small differences in M for the spill-through versus vertical-wall abutment data in 

figure 7(a), apparently the combined differences in approach depth, velocity, and M for the 

wingwall abutment are sufficient to obscure the shape effects when the independent variables are 

estimated rather than measured. Thus, the wingwall shape data are inconclusive. 

Because of the influence of the discharge contraction ratio M and the backwater effects, it should 

not be assumed that the maximum clear-water scour depth of 10 times the unconstricted 

floodplain depth as shown in figure 27 is always the appropriate estimate of equilibrium scour 

depth. In other words, the scour depth can be less than 10 times the unconstricted floodplain flow 

depth measured below the floodplain even though the approach velocity has reached critical 

velocity and the threshold of live-bed scour. For example, as the abutment length decreases and 

0 67 12

0 40

0 67 152 for . < ξ . ξ - .

ξ - . K = . ST ≤ (37) 
67

less flow is blocked, M increases and approaches 1. In addition, backwater effects become less 

likely so that yf1 approaches yf0. So even if Vf1 = Vc, equation 29 gives a maximum relative scour 

depth ds/yf0 of approximately 5. Considered from a different viewpoint, large main channels with 

very rough floodplains tend to have larger values of M, which reduce the relative scour depth 

even if threshold live-bed conditions are reached in the approach channel. In addition, for 

increasing tailwater depths at the same discharge and abutment length, or for increasing the 

abutment length at the same discharge, the approach floodplain velocity is reduced, making 

threshold live-bed scour conditions much less likely to occur. In conclusion, it should not be 

assumed from figure 27 that a scour depth of 10 times the unconstricted floodplain depth is 

always the maximum equilibrium scour depth expected. 

CLEAR-WATER SCOUR: FORMULATION II 

Clear-water scour at a bridge abutment located on the floodplain of a compound open channel 

occurs at an ever-decreasing rate from the initiation of scour until equilibrium is achieved. 

Theoretically, local scour is initiated when the ratio of the local bed shear velocity U* to its 

critical value U*c, or the ratio of local flow velocity V to critical velocity Vc, exceeds unity. 

Furthermore, the initial rate of scour and limiting depth of scour have been shown to increase 

with the value of V/U*c in experiments on scour by jets of water.(79) Thus, although the local 

depth-averaged velocity near the abutment face is continually decreasing as the scour hole grows 

larger with time, it seems possible to relate the maximum depth of scour to the maximum depthaveraged velocity near the abutment face Vab at the beginning of scour as defined in figure 23. 

This velocity cannot be predicted by one-dimensional models. Biglari and Sturm(55) have applied 

a depth-averaged k-ε turbulence model to predict the flow field and Vab around an abutment on 

the floodplain of a compound channel using compound channel A and vertical-wall abutments. 

For compound channel B, Vab was measured for several discharges for each abutment length. 

Dimensional Analysis 

As a test of the hypothesis that scour depends on the local depth-averaged velocity near the 

abutment face, a relationship for the equilibrium clear-water scour depth is sought in the form: 

where ds = equilibrium scour depth, yf0 = unconstricted flow depth in the floodplain set by the 

uniform flow downstream of the bridge, yab = floodplain flow depth at the location of Vab in the 

contracted section, Vab = maximum velocity near the upstream corner of the abutment face, ρ = 

fluid density, ρs = sediment density, g = gravitational acceleration, and d50 = median sediment 

diameter. Dimensional analysis of equation 38 results in: 

where Fab = Vab/(gyab)

0.5 = Froude number in the contracted floodplain near the abutment face, 

and Ns = Vab/[(SG – 1)gd50]

0.5 = sediment number in the contracted section as defined by 

[ , , , , ( ), ] 0 d50 d = f y y V ρ ρ ρ g, s f ab ab s − (38) 

[ ] 50

0 0 y

d , N , F , y

y = f 

y

d

ab

s ab

f

ab

f

s (39) 
68

Carstens,(80) with SG = specific gravity of the sediment. The value of the relative roughness 

(d50/yab) can be replaced by the critical value of the Froude number Fc, or the critical value of the 

sediment number Nsc, for the case of fully rough turbulent flow(85) as in these experiments. For 

quartz sediment and water, either Fab or Ns can be considered redundant with respect to the other. 

If it is further assumed that scour is related to the excess of velocity (or sediment number) with 

respect to its critical value as suggested by several sediment transport formulas,(80, 90-92) a 

relationship for dimensionless scour can be presented as: 

 

Alternatively, the scour relationship could be given as: 

Correlation of Results 

Figure 28 illustrates the correlation of scour data in terms of the excess velocity ratio suggested 

by equation 41. The experimental scour data for compound channel A were measured by 

Sadiq(22) and reported by Sturm and Sadiq,(23) while the scour data for compound channel B were 

collected as part of this study. The values of Vab and yab were predicted by Biglari’s numerical 

two-dimensional turbulence model for compound channel A,(54) while they were measured for 

compound channel B. For compound channel A, the left and right floodplains were slightly 

asymmetrical because of finite construction tolerances so that slightly different scour depths 

were measured for the left and right abutments. These two scour depths in each experiment were 

averaged for comparison with the numerical model runs, which assumed perfect symmetry. The 

value of Vc was calculated from Keulegan’s equation as a function of d50/yab, with the 

appropriate values of Shields’ parameter. 

The results in figure 28 include experimental data for compound channels A and B for six 

different values of La/Bf up to a maximum of 0.66, and for three different sediment sizes (1.13 to 

3.30 mm). The value of La/yf0 varied from 3 to 90, which includes both intermediate- and largescale abutment lengths according to Melville’s classification.(14) The variable yab/yf0 in equation 

41 is an indication of relative local water-surface drawdown near the upstream corner of the 

abutment; however, it was found not to be a significant explanatory variable for the observed 

scour depths in these experiments. The coefficient of determination for the least-squares best-fit 

relationship in figure 28 is r2

 = 0.88, and the best-fit equation is: 

[ ]

0 0 y

y = f N N , y

d

f

ab

s sc

f

s − (40) 

[ 1 ]

0 0 y

y , V

V = f 

y

d

f

ab

c

ab

f

s − (41) 
69

0.1

1

10

100

0.01 0.1 1 10

(Vab/Vc) - 1

ds/yf0

A, 0.17 A, 0.33 A, 0.66 B, 0.22 B, 0.44 B, 0.66 Best Fit

KEY: Compound Channel, La/Bf

Figure 28. Scour-depth relationship based on local hydraulic variables for 

 VW abutments with La/Bf ≤ 0.66 and d50 = 3.3, 2.7, and 1.1 mm. 
70

for [(Vab/Vc) – 1] ≤ 1.94 with ds/yf0 = 10 for greater values. Correlation of the data according to 

the excess sediment number as given in equation 40 was not as successful as the result given by 

equation 42. 

If the data for longer abutments are also plotted according to equation 42, it can be observed in 

figure 29 that the best-fit relationship does not compare well with these data. It is plausible that 

the measurement of Vab as a depth-averaged, resultant velocity is no longer sufficient to explain 

the complex, three-dimensional flow pattern that occurs at the face of the abutment as the 

abutment encroaches on the main channel. Clearly, an additional flow factor is needed to explain 

better the scour data for this case, barring the use of a three-dimensional numerical model. 

DISCUSSION 

A comparison of figures 25 and 28 shows relatively little difference in using formulations I or II 

for scour prediction for relative abutment lengths less than 0.66. Formulation I is based on 

variables easily determined by WSPRO, while formulation II requires output from at least a twodimensional, depth-averaged numerical model. It is not clear that the finite elements surface 

water modeling system (FESWMS) will satisfy this need with its assumption of constant eddy 

viscosity. Biglari’s model(54-55) used a k-ε turbulence closure to capture the interaction between 

the main-channel and floodplain flow, and even higher order turbulence closure models may be 

needed to model the large-scale eddies of separation at the abutment. Only three-dimensional 

numerical models can be used to even attempt a prediction of the action of the horseshoe vortex 

and its effect on scour depths. This seems to be borne out by the relatively large deviation in the 

scour data for longer abutments in figure 29 from the best-fit relationship obtained for setback 

abutments that terminate in the floodplain well back from the main channel. Nevertheless, 

formulation II is attractive because of the expected rapid development and use of twodimensional and three-dimensional numerical models in the next few years, and also because of 

the possibility of unifying the experimental results on abutment scour from rectangular channels 

with those from compound channels. 

There is ample precedent for taking the “hydraulic approach” and developing a simple method 

for predicting Vab along with a velocity adjustment factor for longer abutments until more 

advanced numerical models are readily available and usable. For example, one possible reference 

velocity is the mean cross-sectional velocity in the contracted cross section that can be obtained 

from WSPRO. However, relating Vab to the average cross-sectional velocity for the whole cross 

section is complicated by the fact that WSPRO predicts the average cross-sectional velocity at 

the downstream face of the bridge, whereas Vab is the resultant, depth-averaged velocity at the 

upstream corner of the abutment face. In addition, the average cross-sectional velocity is not at 

all representative of the floodplain velocity for setback abutments that are located on the 

6 55[ 1 ]

0 64

0 V

V

 = . y

d

c

ab

.

f

s − (42) 
71

0.1

1

10

100

0.01 0.1 1 10

(Vab/Vc) - 1

ds/yf0

VW, 0.88 VW, 0.97 VW, 1.0 ST, 0.88 ST, 0.97 ST, 1.0 Best Fit

KEY: Abutment Shape, La/Bf

Figure 29. Scour-depth relationship based on local hydraulic variables for 

 VW and ST abutments with La/Bf ≥ 0.88 and d50 = 3.3 mm. 
72

floodplain. The floodplain velocity in the contracted section is, in reality, determined by both the 

local acceleration near the abutment face and the entrainment of the floodplain flow into the 

main channel. Furthermore, the practice of taking the WSPRO streamtube velocity nearest the 

abutment face in the floodplain is bound to fail (as will be shown subsequently) because the flow 

distribution in WSPRO is determined by the conveyance distribution, which does not reflect the 

flow acceleration around the abutment face. As the abutment encroaches on the main channel, 

the entrainment of the floodplain flow into the main-channel flow occurs near the abutment face 

so that again a simple estimate of mean cross-sectional velocity cannot properly predict the scour 

potential in this highly three-dimensional flow region. In spite of these difficulties, there remains 

one other possibility suggested by the fact that the independent dimensionless variables in 

figures 25 and 28 provide a similar degree of explanation of scour depth so that they must be 

related to each other. The correlation between the two for the shorter abutments (La/Bf ≤ 0.66) is 

shown in figure 30(a), with r2

 = 0.88 and the best-fit equation as: 

Equation 43 has the advantage that it depends on hydraulic variables on the right-hand side that 

can be predicted well by a one-dimensional model such as WSPRO. However, the data for 

longer abutments (La/Bf ≥ 0.88) are plotted in figure 30(b), and the correlation of figure 30(a) no 

longer holds. This is the reason for the disagreement between measured and predicted scour in 

figure 29, where the measured [(Vab/Vc) – 1] is used as the independent variable for the longer 

abutments. The resultant velocity was measured near the upstream corner of the abutment face 

for both the shorter and longer abutments. In the latter case, it is likely that a significant 

downward vertical component of the velocity existed that was not measured by the miniature 

propeller meter, and so the resultant velocity was underestimated. Only a three-dimensional laser 

Doppler velocimeter (LDV) with a fiber-optic probe located inside a Plexiglas abutment could be 

used to measure the resultant velocity near the abutment in these experiments, or it could be 

computed using a verified three-dimensional numerical model. Both of these methods were 

outside the scope of the present study. In conclusion, formulation I is recommended as the best 

overall clear-water scour-prediction method for both short and long abutments. The limitations 

of the recommended scour-prediction equation are discussed at the end of the chapter. 

APPLICATION TO LIVE-BED CONDITIONS 

Live-bed scour is the condition in which sediment transport occurs in the channel approaching 

the bridge contraction so that sediment continuously passes through the scour hole. Equilibrium 

is reached when the sediment transport rate out of the scour hole equals the sediment transport 

rate into the scour hole. For a compound channel in riverbank flow, the floodplain velocities are 

quite likely to be less than the critical velocity so that live-bed conditions can be realized only in 

the main channel. Thus, as a practical matter, abutment scour under live-bed conditions is often 

limited to the case of the abutment’s encroaching on the bank of the main channel with sediment 

transport occurring only in the approach main channel. Even this condition may be difficult to 

1 156 ( 0 4 )

0

1

1 78

. M q

q

 = . V

V

f c

f

.

c

ab − − (43) 
73

0.1

1

10

0.1 1 10

[qf1/(Mqf0)] - 0.4

(Vab/Vc) - 1

A, VW B, VW B, ST Best Fit

KEY: Compound Channel, Abutment Shape

(a) La/Bf ≤ 0.66 

0.1

1

10

0.1 1 10

[qf1/(Mqf0)] - 0.4

(Vab/Vc) - 1

B, VW B, ST Best Fit

KEY: Compound Channel, Abutment Shape

(b) La/Bf ≥ 0.88 

Figure 30. Relationship between local and approach hydraulic variables. 
74

find in practice if there is significant upstream backwater, which leads to a rise in stage and a 

reduction in approach velocity below the value required to carry sediment. 

The live-bed scour analysis is based initially on contraction scour. The basic condition that must 

be satisfied for equilibrium live-bed scour comes from a continuity of sediment transport rates 

between the approach main channel and the main channel in the contracted section, which may, 

in general, have different widths. If we use the Meyer-Peter and Mueller bedload function,(75) for 

example, then it must be true that: 

where subscripts 1 and 2 refer to the approach and contracted main channel, respectively, and 

q*

bv is the dimensionless volumetric sediment transport rate given as: 

where qbv = volumetric sediment transport rate per unit width, SG = specific gravity of the 

sediment, d50 = median sediment grain size, τ* = Shields’ parameter, and τ*c = critical value of 

Shields’ parameter. Substituting the sediment transport relationship given by equation 45 into 

equation 44 and rearranging it, we have: 

Note that for Bm2 = Bm1, τ*2 = τ*1 regardless of the value of τ*c/τ*1, although it must be less than 

or equal to 1 for live-bed scour. If τ*c/τ*1 is exactly equal to 1, which must be true at the inception 

of live-bed scour (threshold live-bed scour), we must again have τ*2 = τ*1. 

Now Shields’ parameter can be expressed in terms of the shear stress formula and Manning’s 

equation for uniform flow in a wide channel in SI units as: 

Then, taking the ratio of Shields’ parameter between the approach and contracted sections, and 

q B = q Bm bv m bv 2 *

2 1 *1 (44) 

8 ( )

( 1) * *

3 2

50

3

* = τ τ

SG g d

q

q = c

bv /

bv −

−

 (45) 

τ

τ + 

τ

τ

B

B

 = 

τ

τ c c

m

m

/

*

*

*1

*

*1

*

2

1

2 3

1

2 ( ) (1 − ) (46) 

SG d y

n V

 = 

γ γ d

τ

τ = /

s

1 3

50

2 2

50

0

*

( − ) ( − 1) (47) 
75

replacing the velocity, V, with Q, using continuity for a rectangular main channel, the result is: 

where it has been assumed that Manning’s n remains constant from the approach to the 

contracted section in the main channel. Solving for the depth ratio, we have: 

Now, equations 46 and 49 can be solved simultaneously for the scour depth ym2 relative to the 

water surface. The solution is shown graphically in figure 31 in the form of a live-bed scour 

coefficient, CLB, defined as: 

where 

From equation 51 and figure 31, it is apparent that the coefficient CLB depends on the ratio of 

approach shear stress to critical shear stress in the main-channel τ*1/τ*c, and on the ratio of 

approach to main-channel widths Bm1/Bm2. As τ*1/τ*c becomes large, the curves approach the 

horizontal and the coefficient for live-bed contraction scour depth depends only on the mainchannel width ratio to the 4/7 power, which is close to the power of 0.59 given in HEC-18. On 

the other hand, the effect of τ*1/τ*c has to be included when it is small, as pointed out by 

Froehlich,(88) although he used a different sediment transport relationship than employed herein. 

If the main-channel width is constant, the ratio τ*1/τ*c has no influence and the coefficient CLB

has a value of 1. 

( ) ( ) ( )

1

2

2

1

2

7 3

1

2

*2

*1

B

B

y

y

Q

Q

 = 

τ

τ

m

m

m

m

/

m (48) 

[ ] [ ]

1

1 2

6 7

*2

*1

3 7

1

2

Q /Q

B / B τ

τ

 = 

y

y

m

m m

/ /

m

m (49) 

[ ]

1

6 7

1

2

Q

Q

 = C

y

y

m

/

LB

m

m (50) 

[( ) ( 1 ) ]

( )

*1

*

*1

*

2

1

2 3 3 7

2

1

6 7

τ

τ + 

τ

τ

B

B

B

B

C = 

c c

m

m

/ /

m

m

/

LB

−

 (51) 
76

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

τ*1/τ*c

CLB

Bm1/Bm2

3

2

1

0.5

0.25

Figure 31. Live-bed contraction scour coefficient for La/Bf = 1.0. 
77

It is useful at this point to compare the theoretical clear-water contraction scour given by 

equation 33 and the live-bed contraction scour given by equations 50 and 51. Both equations are 

for bankline abutments for which the abutment is at the edge of the main channel (La = Bf). In 

the limiting case where there is no backwater (ym0 = ym1) and the approach and contracted mainchannel sections have equal widths (Bm1 = Bm2), the threshold live-bed scour occurs for Vm1 = 

Vm0c in equation 33. Under these conditions, equation 33 becomes: 

Upon comparison with the live-bed scour equation given by equations 50 and 51, it is clear that 

equations 50–52 are identical for CLB = 1, which is true for Bm1 = Bm2 and from the definition of 

M = Qm1/Q. Furthermore, the live-bed scour coefficient remains equal to 1 as τ*1/τ*c increases. 

Thus, for these conditions, and assuming that local abutment scour is proportional to the 

theoretical contraction scour, it seems reasonable to estimate live-bed scour for bankline 

abutments as the threshold of live-bed scour from the clear-water scour equation given by 

equation 35, in which Cr’ = 2.0 and C0’ = 0.47. In other words, if Vm1 exceeds Vm0c, then Vm1 is 

set equal to Vm0c in equation 35 for bankline abutments with Bm1 = Bm2. By the same reasoning, 

equation 36 with Vf1 = Vf1c could be used for the unlikely case of live-bed scour around setback 

abutments in the floodplain. This is the approach that is recommended in this report. 

For live-bed scour in the main channel and Bm1 is not equal to Bm2, the data in this report are not 

directly applicable. However, it has been shown theoretically that live-bed and clear-water 

contraction scour equations are identical for Bm1 = Bm2 and CLB = 1.0 at the threshold of live-bed 

scour. Furthermore, the experimental data taken in this study resulted in equation 35, which 

predicts a combined abutment and contraction scour of approximately twice (Cr’ = 2) the 

theoretical contraction scour for the abutment at or near the bank of the main channel. This 

suggests that an equation like equation 35 could be combined with the theoretical live-bed 

coefficient of equation 51 to predict total contraction and abutment live-bed scour for bankline 

abutments even if Bm1 is not equal to Bm2. Further experiments are needed to confirm this 

hypothesis. 

This theoretical analysis does not account for the observation that a second live-bed scour peak 

different from the threshold live-bed scour peak (τ*1/τ*c = 1) can occur for bridge piers at a 

relative shear stress ratio τ*1/τ*c > 1.(24,89) The second live-bed scour peak occurs at the bedform 

transition from dunes to flat bed. Melville(24) showed that for short bridge abutments, the second 

live-bed scour peak is less than or equal to the threshold live-bed scour peak for uniform 

sediments, except in the case of ripple-forming sediments (d50 < 0.6 mm). For nonuniform 

sediments that armor the streambed by larger particles as the finer particles are washed out of the 

top bed zones, the threshold live-bed scour peak is less than the second live-bed scour peak, 

which is itself smaller than the threshold peak for uniform sediments. Whether these conclusions 

hold for bridge abutments at the edge of the main channel in a compound-channel cross section 

requires additional experiments. 

] 1 [

6 7

1

2

M

 = 

y

y /

m

m (52) 
78

WSPRO PREDICTIONS 

To determine the adequacy of one-dimensional hydraulic computations to be used in formulation 

I scour predictions, WSPRO simulations of the water-surface profiles and approach velocity 

distributions for both compound channels A and B are compared to measured laboratory results 

in this section. Some comparisons are also made with the results from Biglari’s two-dimensional 

numerical model. (54, 55) More extensive comparisons and discussion can be found in a previous 

report by the author(78) and in a paper by Sturm and Chrisochoides.(86)

In the WSPRO runs for compound channel A, Manning’s n was assumed to be constant with 

depth, with values of 0.020 for the main channel and 0.0174 for the floodplain based on the 

experimental data for uniform flow. Computed and measured water-surface profiles for 

compound channel A are compared in figure 32 for a discharge of 0.057 m3

/s and for a verticalwall abutment having a relative length of La/Bf = 0.33. The two-dimensional numerical model 

tracks the experimental results for depth very closely, both upstream and downstream of the 

abutment, including the expansion zone immediately downstream of the abutment where 

WSPRO merely produces a constant depth. This is clearly a limitation of the one-dimensional 

model. Of more interest, however, are the comparisons upstream of the abutment where WSPRO 

consistently overestimates floodplain depths by about 10 percent. 

Predicted velocity distributions in the bridge approach section are compared with measured 

values in figure 33 for the same experimental conditions as in figure 32. Both WSPRO and the 

two-dimensional numerical model predict approach floodplain velocities reasonably well. The 

two-dimensional model seems to capture well the velocity distribution in the interaction zone 

between the main channel and the floodplain, which was the reason for choosing a k-ε turbulence 

model with variable turbulent eddy viscosity vt. However, there are some difficulties with the 

WSPRO results at the main channel/floodplain interface and at the main-channel centerline. The 

WSPRO results are computed from the cross-sectional areas of 20 streamtubes having equal 

conveyances and assumed equal discharges. The vertical wall of the artificial main channel 

where the depth changes abruptly is apparently the reason for the very low velocity computed by 

WSPRO near the interface. The WSPRO data points are very close together near the centerline 

of the main channel because of the large conveyances there. 

Comparisons between measured and computed velocity distributions in the bridge contraction 

section are shown in figure 34. In this section, WSPRO cannot predict the velocity distribution 

because of the accelerating nonuniform flow in the vicinity of the abutment face. The numerical 

model, however, shows computed resultant velocities at the upstream face of the bridge opening 

that agree well with the measured resultant velocities. 

Computed water-surface profiles for compound channel B are compared with measurements in 

figure 35 for a relative abutment length of La/Bf = 0.44 and a discharge of Q = 0.085 m3

/s. In 

spite of the change in cross-sectional shape and channel roughnesses, the WSPRO comparisons 

are very similar to those for compound channel A. Downstream of the abutment, the WSPRO 

depth is nearly constant for uniform flow, while upstream of the abutment, the computed 

WSPRO depths are larger than the measured values, resulting in a relative error in floodplain 

depth of about 10 percent. In general, the computed WSPRO depths at the downstream face of 
79

0.150

0.175

0.200

0.225

0.250

0 2 4 6 8 10 12 14 16 18

X-Station, m

Depth, m

Calculated by 2D Model Measured Calculated by WSPRO Normal Depth

Normal Depth

Figure 32. Calculated and measured water-surface profiles for compound channel A. 

 (VW abutment, La/Bf = 0.33, Q = 0.057 m3

/s). 
80

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Transverse Station, m

Longitudinal Velocity, m/s

0.0

0.2

0.4

0.6

0.8

1.0

Water-surface and Bed 

Elevation, m

Velocity (2D model) Velocity (measured)

Velocity (WSPRO) Water-surface elevation (2D model)

Bed

Figure 33. Calculated and measured approach velocity distributions for compound channel A. 

 (VW abutment, La/Bf = 0.33, Q = 0.057 m3

/s). 
81

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Transverse Station, m

Longitudinal Velocity, m/s

0.0

0.2

0.4

0.6

0.8

1.0

Water-surface and Bed 

Elevation, m

Velocity (2D model) Velocity (measured)

Water-surface elevation (2D model) Bed and Abutments

Figure 34. Calculated and measured resultant velocity distributions in the contracted section 

 for compound channel A. (VW abutment, La/Bf = 0.33, Q = 0.057 m3

/s). 
82

0.150

0.175

0.200

0.225

0.250

0 2 4 6 8 10 12 14 16 18

X-Station, m

Depth, m

Measured Depth Normal Depth Calculated by WSPRO

Figure 35. Calculated and measured water-surface profiles for compound channel B. 

 (VW abutment, La/Bf = 0.44, Q = 0.085 m3

/s). 
83

the bridge agree rather well with the measured values. Computed and measured approach 

velocity distributions for compound channel B are shown in figure 36 for the same experimental 

conditions as in figure 35. There is good agreement between measured and computed WSPRO 

velocities in the floodplain. However, computed velocities near the interface between the 

floodplain and the main channel are underpredicted by WSPRO because of the limitations of a 

one-dimensional model in this region. 

More detailed comparisons of computed and predicted values of Vf1, the approach floodplain 

velocity upstream of the end of the abutment; yf1, the approach floodplain depth; and M, the 

discharge distribution factor, are given in figures 37, 38, and 39 for both cross sections and all 

discharges with La/Bf ≤ 0.88. The results for both vertical-wall and spill-through abutment shapes 

are included in these figures. The approach velocity Vf1 is predicted well by WSPRO for 

compound channel B as shown in figure 37, with a root mean square deviation between the 

measured and calculated values of about 0.015 m/s. The WSPRO comparisons for Vf1 in 

compound channel A are somewhat scattered; however, the two-dimensional numerical 

turbulence model performs reasonably well for compound channel A. The approach depth yf1 is 

consistently overestimated by WSPRO in both compound channels A and B as shown in figure 

38. The mean percent error is about 12 percent. The two-dimensional numerical model slightly 

underpredicts the approach floodplain depths. 

In figure 39, computed and predicted values are shown for the discharge distribution factor M, 

which is defined as the ratio of the discharge, in that portion of the approach cross section having 

a width equal to the bridge opening width, to the total discharge. Thus, the prediction of M 

depends on the prediction of the approach depth and velocity distributions. It can be observed 

that M is predicted equally well by WSPRO and the two-dimensional numerical model in 

compound channel A; however, WSPRO underestimates M for the smaller values in compound 

channel B. This problem is caused by the differences in the velocity distributions noted earlier 

for compound channel B, as well as in the overprediction of yf1 by WSPRO. The results for M 

show that at least some of the effect of the main channel/floodplain interaction has been 

accounted for by using measured Manning’s n values; however, this achieves only the correct 

split between the main-channel and floodplain flow, not necessarily the correct detailed velocity 

distribution near the interface. This is a problem with the one-dimensional analysis afforded by 

WSPRO that can be overcome by a two-dimensional numerical model. 

As a final comparison, the calculated and measured velocities in the bridge contraction section 

near the abutment face are shown in figure 40 for compound channel B. The measured velocities 

Vab are the measured maximum resultant velocities found near the abutment face. The velocities 

calculated by WSPRO for this case necessarily come from the bridge section at the downstream 

face of the bridge opening and are taken as the mean velocity in the streamtube nearest the 

abutment face. WSPRO clearly cannot predict velocities in this part of the accelerating flow field 

and would not be expected to do so. Based on these results, it is inadvisable to use WSPRO to 

predict velocities near the abutment face for use in scour-prediction equations.
84

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0

Transverse Station, m

Longitudinal Velocity, m/s

0.0

0.4

0.8

1.2

1.6

2.0

Water-surface and Bed Elevation, 

m

Velocity (measured) Calculated Velocity (WSPRO)

Bed Water-surface elevation (measured)

Figure 36. Calculated and measured approach velocity distributions for compound channel B. 

 (VW abutment, La/Bf = 0.44, Q = 0.085 m3

/s). 
85

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Measured Vf1, m/s

Calculated Vf1, m/s

WSPRO: CC-A, VW 2D: CC-A, VW WSPRO: CC-B, VW WSPRO: CC-B, ST

Figure 37. Comparisons of approach floodplain velocities (CC = compound channel). 

0.00

0.02

0.04

0.06

0.08

0.10

0.00 0.02 0.04 0.06 0.08 0.10

Measured yf1, m

Calculated yf1, m

WSPRO: CC-A, VW 2D: CC-A, VW WSPRO: CC-B, VW WSPRO: CC-B, ST

Figure 38. Comparisons of approach floodplain depths (CC = compound channel). 
86

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Measured M

Calculated M

WSPRO: CC-A, VW 2D: CC-A, VW WSPRO: CC-B, VW WSPRO: CC-B, ST

Figure 39. Comparisons of discharge distribution factor M (CC = compound channel). 

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Measured Vab, m/s

Calculated Vab, m/s

WSPRO: CC-B, VW WSPRO: CC-B, ST

Figure 40. Comparisons of maximum velocity at abutment face 

 (CC = compound channel). 
87

TIME DEVELOPMENT RELATIONSHIP 

Based on the dimensional analysis result given by equation 45, time can be added as an 

independent variable in the scour development process to produce an expected relationship of the 

form: 

where dst = scour depth at any time t. The influence of yab/yf0 has not been included in equation 

48 based on the experimental results. The experimental measurements of scour depth with time 

are presented according to equation 53 for three sediment sizes (d50 = 3.3 mm, 2.7 mm and 1.1 

mm) in figure 41. These results for the time development of scour include only those cases for 

which the maximum scour depth developed near the upstream corner of the abutment as 

discussed previously in chapter 3. However, the location of the scour hole at the upstream corner 

occurred for the larger discharges so that the worst cases are included in these results. The curves 

for different values of Vab/Vc have a functional form that begins with a linear development of 

scour depth with the logarithm of time, followed by an abrupt leveling off to a nearly constant 

value equal to the equilibrium scour depth that depends only on Vab/Vc. 

The functional behavior and collapse with respect to sediment size shown in figure 41 suggest 

the possibility of a universal set of time-development curves that can be applied to field cases 

provided that the dimensionless variables fall in the same range as in the laboratory experiments. 

Accordingly, a least-squares regression analysis was applied to the data and suggested 

interpolated curves were developed and plotted as solid lines in figure 41. Thus, for a given 

sediment size, which determines Vc; a given abutment velocity Vab, which is determined by the 

abutment shape, degree of floodplain contraction, and flow velocity distribution; and a given 

time corresponding to the design flood duration, an estimated depth of scour can be obtained. For 

example, if Vab/Vc = 1.5, approximately two-thirds of the equilibrium scour depth is reached in a 

flow duration of only about 10 percent of the equilibrium time. Figure 41 shows very clearly the 

interplay of time, flow distribution, and sediment size in determining a design value of abutment 

scour depth. 

COMPARISON OF SCOUR FORMULAS WITH MEASURED DATA 

Several formulas for clear-water abutment scour depth, in addition to the one formulated herein, 

have been proposed in the literature. These were discussed in detail in chapter 2 and the formulas 

will be referred to as: (1) Melville,(14,24) (2) Froehlich clear-water scour (CWS),(12) (3) Froehlich 

live-bed scour (LBS),(12) (4) GKY,(66) and (5) Maryland.(68,70) The formulas proposed by Melville 

and Froehlich both include data collected in Auckland, New Zealand, in rectangular flumes. The 

GKY formula relies on data from Lim(67) and Sturm and Janjua,(20) while the Maryland procedure 

is based on experimental data in rectangular flumes assembled by Palaviccini.(69) Because the 

scour data collected in this study include both vertical-wall and spill-through abutments, a shape 

factor was applied for each formula. The shape factor for vertical-wall abutments was taken to be 

1.0 for all formulas. The shape factor KST for spill-through abutments in the Melville formula 

[ ]

0 0

 y

V t , V

V = f 

y

d

f

c

c

ab

f

st (53) 
88

0

1

2

3

4

5

6

7

8

9

10

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07

Vct/yf0

dst/yf0

1.25

Vab/Vc = 2.50 

1.50

2.00

Vab/Vc d50, mm

1.25 3.3

1.25 2.7

1.50 3.3

1.50 1.2

2.0 1.2

2.5 1.2

Figure 41. Dimensionless representation of time development of scour. 
89

varies linearly from 0.45 (for 1.5:1 sideslopes) to 1.0 as La/y1 varies from 10 to 25 and is 

constant outside of these limits with KST = 0.45 for La/y1 ≤ 10 and KST = 1.0 for La/y1 ≥ 25. 

(Melville does not give a shape factor for 2:1 sideslopes, so the closest value of 1.5:1 was used in 

the predictions to follow.) For the other formulas, a standard constant shape factor of 0.55 for 

spill-through abutments was used based on the guidance given by HEC-18. The formula from the 

present study (equation 36 with Cr = 8.14 and C0 = 0.40) is also compared with the experimental 

data using the shape factor for spill-through abutments derived herein as equation 37.

The comparisons between the measured and predicted scour depths for the five formulas listed 

above are given in figures 42 through 46, for a total of 74 data points measured in the present 

study in a compound channel (compound channel B). The line of perfect agreement is shown as 

a solid line, and the two dashed lines indicate a variation of ±30 percent from the line of perfect 

agreement in order to gauge the degree of scatter in the comparisons. 

The Melville formula overpredicts scour depths in the lower range of the data by several hundred 

percent as shown in figure 42. Only about one-half of the data points fall within the ±30 percent 

range of variation relative to perfect agreement. The Froehlich CWS formula shown in figure 43 

shows a similar pattern to the Melville formula, which is to be expected because they are based 

on a similar data set. However, several larger values of scour are underpredicted by the Froehlich 

CWS formula because of the constant spill-through abutment shape factor of 0.55, whereas the 

Melville formula has a shape factor of 1.0 for the larger values of La/yf1. The Froehlich LBS 

formula, which is recommended by HEC-18, even for clear-water scour, is given in figure 44. In 

general, it shows similar data scatter in comparison to the Froehlich CWS formula, but with 

greater overprediction of nearly all scour depths. The GKY and Maryland formulas in figures 45 

and 46, respectively, display a trend opposite to that of the first three formulas. They both tend to 

underpredict the measured scour depth by considerably more than the lower 30-percent range. In 

order to maximize the predicted values for both formulas, the entire approach floodplain flow 

was assumed to pass through the contracted-section floodplain for La/Bf ≤ 0.88, while the total 

flow was assumed to pass through the main channel in the contracted section for La/Bf = 0.97 

and 1.0. In general, the Maryland formula predicts greater scour depths than the GKY formula, 

and both formulas suffer from the use of the spill-through shape factor of 0.55 when the present 

study as well as the experiments by Melville(24) show that it approaches 1.0 as the relative 

abutment length, or the fraction of blocked flow, becomes large. The GKY formula was applied 

in SI units with the leading constant factor of 1.37 as recommended in reference 66; however, a 

value approximately twice as large would greatly improve the performance of the formula. 

Finally, the formula recommended in this study is compared with the experimental data in figure 

47. While the formula itself is based on the data used in the comparison, figure 47 is indicative of 

the overall performance of the formula. Only 6 of the 74 data points lie completely outside the 

±30 percent range. Four of these outliers occur for measured scour depths on the order of 5 cm or 

less, which are more difficult to measure and reproduce because of the inherent uncertainty in the 

experiments. In conclusion, it seems that the formulas based primarily on experiments in 

rectangular flumes cannot reproduce the physics of the flow and the coupled flow distribution 

and scour behavior associated with the compound channel. 
90

0

10

20

30

40

50

0 10 20 30 40 50

Measured ds, cm

Predicted ds, cm

Melville Perfect agreement +30% -30%

Figure 42. Comparison of measured and predicted scour depths 

 using the Melville formula. 

0

10

20

30

40

50

0 10 20 30 40 50

Measured ds, cm

Predicted ds, cm

Froehlich CWS Perfect agreement +30% -30%

Figure 43. Comparison of measured and predicted scour depths 

 using the Froehlich clear-water scour formula. 
91

0

10

20

30

40

50

0 10 20 30 40 50

Measured ds, cm

Predicted ds, cm

Froehlich LBS Perfect agreement +30% -30%

Figure 44. Comparison of measured and predicted scour depths 

 using the Froehlich live-bed scour formula. 

0

10

20

30

40

50

0 10 20 30 40 50

Measured ds, cm

Predicted ds, cm

GKY Perfect agreement +30% -30%

Figure 45. Comparison of measured and predicted scour depths 

 using the G. K. Young (GKY) formula. 
92

0

10

20

30

40

50

0 10 20 30 40 50

Measured ds, cm

Predicted ds, cm

Maryland Perfect agreement +30% -30%

Figure 46. Comparison of measured and predicted scour depths 

 using the Maryland formula. 

0

10

20

30

40

50

0 10 20 30 40 50

Measured ds, cm

Predicted ds, cm

Present Study Perfect agreement +30% -30%

Figure 47 . Comparison of measured and predicted scour depths 

 using the formula from the present study. 
93

PROPOSED IMPLEMENTATION PROCEDURE FOR ABUTMENT SCOUR 

PREDICTION 

As a result of the experimental results on the equilibrium depth of clear-water abutment scour 

given in figure 27 and on the time development of scour shown in figure 41, a prediction 

procedure that accounts for discharge distribution, sediment size, and time can be developed. 

Based on the extensive comparisons between WSPRO estimations of scour parameters and 

parameter estimation by a two-dimensional numerical turbulence model, it is assumed that the 

WSPRO predictions provide a reasonable estimation of the approach-flow hydraulic parameters 

for the prediction of clear-water abutment scour. It is further assumed that the abutment 

velocities to be used in the scour time development determination in figure 41 can be obtained 

from the relationship developed in figure 30, which depends only on parameters calculated from 

the WSPRO results. First, the formulas recommended in the scour-prediction procedure are 

repeated below. Bridge abutments that terminate at the edge of the main channel are referred to 

as bankline abutments, while abutments located on the floodplain are called setback abutments. 

1. Definition of discharge contraction ratio, M (equation 19 from chapter 3): 

where Qm1 = discharge in the approach main channel, Qf1 = discharge in the approach floodplain, 

Qobst1 = obstructed floodplain discharge over a length equal to the abutment length in the 

approach cross section, and Q = total discharge = Qm1 + Qf1. The value of M as defined by 

equation 54 can be calculated from the WSPRO output as 1 – M(K). For road overflow, a 

consistent definition of M would require the denominator to include only that portion of the total 

Q going through the bridge opening. 

2. Keulegan’s equation for critical velocity (equation 22 from chapter 3): 

where Vc = critical velocity, SG = specific gravity of sediment, d50 = median sediment grain 

diameter, y = flow depth, and τ*c = critical value of Shields’ parameter. If the channel is not very 

wide, then the depth, y, should be replaced by the hydraulic radius, R. Equation 55 only applies 

to fully rough turbulent flow. For sediment sizes of less than about 1.2 mm, the Einstein 

correction factor given on p. 259 of reference 75 should be used to multiply the logarithmic 

argument. 

3. Scour-prediction formula for clear-water scour around setback and bankline abutments, 

including shape correction KST and safety factor FS (modified equation 36 from chapter 

4): 

Q

Q + Q Q M = m f obst ( ) 1 1 1 − (54) 

] 2

12 2 5 75 log [ ( 1) 50

*

50 d

. y = . τ

SG g d

V c

c

− (55) 
94

where KST = spill-through abutment shape factor from equation 57 below; Cr = 8.14; C0 = 0.40; 

qf1 = Vf1yf1; Vf1 and yf1 = average floodplain velocity and depth, respectively, in the obstructed 

portion of the floodplain in the bridge approach-flow section for constricted flow; Vxc = Vf0c for 

abutments located on the floodplain and Vxc = Vm0c for abutments near the bank of the main 

channel, where Vf0c and Vm0c are critical velocities for the unconstricted flow in the floodplain 

and main channel, evaluated for depths of yf0 and ym0, respectively; and yf0 and ym0 are the 

unconstricted flow depths in the floodplain and main channel at the approach section. A value of 

FS = 1.0 is recommended because this is greater than the standard error of 0.75 for ds/yf0 for the 

best fit of the experimental data. If Vf1 is greater than the critical velocity Vf1c in the approach 

floodplain flow, then it should be set equal to the critical velocity. Finally, if qf1/(MVxcyf0) > 1.6, 

then ds/yf0 should be set equal to the maximum value of 10. The elevation of the bottom of the 

scour hole is taken to be a distance of ds + yf0 below the unconstricted water-surface elevation at 

the bridge. 

4. Spill-through abutment shape factor (equation 37 from chapter 4): 

 

where ξ = qf1/(MVxcyf0c) and KST = 1.0 for ξ ≥ 1.2 and 0 for ξ ≤ 0.67. 

5. Scour-prediction formula for clear-water and live-bed scour around bankline abutments 

(equation 35 from chapter 4): 

where qm1 = Vm1ym1; Vm1 and ym1 = mean velocity and depth, respectively, in the approach main 

channel; Cr’ = 2.0; C0’ = 0.47; and FS = 1.0. Equation 58 is based on clear-water scour data 

around bankline abutments for which the abutment shape factor was equal to 1.0. In this report, 

equation 56 is recommended for clear-water scour around bankline abutments instead of 

equation 58. However, equation 58 is suggested as an interim equation for live-bed scour around 

bankline abutments with Vm1 = Vm1c until the data for the live-bed scour case are available. 

C + FS

M V y

q

 = K C

y

d

xc f

f

ST r

f

s ( ) 0

0

1

0

− (56) 

0 67 12

0 40

0 67 152 for . < ξ < . ξ .

ξ . K = . ST

−

− (57) 

C + FS

M V y

q

 = C

y

d

m c f

m

r

f

s ] ( ) [ 0

0 0

1

0

′ − ′ (58) 
95

6. Conversion from approach hydraulic variables to local hydraulic variables for use in 

figure 41(equation 43 from chapter 4): 

where qf0c = Vxcyf0c; Vab = resultant depth-averaged velocity near the upstream corner of the 

abutment face; and Vc = critical velocity at the same location, which is determined from 

Keulegan’s equation for the flow depth in the bridge at the toe of the abutment. The values of 

Vct/yf0 and Vab/Vc are required to use figure 41. 

The steps in the proposed procedure are: 

1. From the field data, obtain at least one surveyed cross section, and preferably three 

sections (bridge exit, downstream face of the bridge, and bridge approach). Also estimate 

Manning’s n for the floodplains and the main channel, and obtain bridge geometry data 

and sediment size d50. 

2. Determine the 100-year and 500-year design discharges based on drainage area and 

regional frequency estimates (available from the National Flood Frequency (NFF) 

program in version 6 of the highway drainage integrated computer system (HYDRAIN) 

software which was funded by 31 State highway agencies. 

3. Run WSPRO and obtain M, Vf1, yf1, and yf0 from the results. Calculate Vf1 and yf1 as the 

average values in the obstructed portion of the floodplain in the approach section for the 

constricted flow. The value of yf0 is obtained in a similar manner for the unconstricted 

flow. Estimate the value of Vf0c from yf0 and d50 for setback abutments, or Vm0c from Rm0

and d50 for bankline abutments, using Keulegan’s equation (equation 55), where Rm0 = 

hydraulic radius of the main channel. Also calculate the value of V1c in the approach flow 

from Keulegan’s equation for either the floodplain or the main channel, depending on 

whether the abutments are setback or bankline abutments, respectively. 

4. Calculate the dimensionless clear-water scour depth ds/yf0 from equation 56, including 

the abutment shape factor from equation 57 and FS = 1.0. The maximum value of ds/yf0

should not be taken as being any greater than 10 based on the experimental results. In 

addition, if Vf1 ≥ Vf1c for setback abutments, then threshold live-bed scour occurs, so set 

Vf1 = Vf1c. The elevation of the bottom of the scour hole is taken to be a distance of ds + 

yf0 below the unconstricted water-surface elevation at the bridge. 

5. If it is a bankline abutment, check for the possibility of live-bed scour by determining 

whether Vm1 ≥ Vm1c. If so, then set Vm1 = Vm1c in equation 58 and solve for the live-bed 

scour depth. For the bankline abutment, regardless of whether the scour is clear-water or 

live-bed, the scour depth calculated from the equations recommended herein includes 

1 156 ( 0 4 )

0

1

1 78

. M q

q

 = . V

V

f c

f

.

c

ab − − (59) 
96

both contraction and local abutment scour. 

6. From the watershed size and a hydrologic estimate of the lag time of the watershed, use 

NFF in HYDRAIN to generate a hydrograph corresponding to the peak design discharge 

from which a flood duration can be estimated. 

7. For the duration obtained from step 6, determine the percentage of the equilibrium scour 

depth that will occur from figure 41. To estimate Vab/Vc as needed in figure 41, use 

equation 59. This time-dependent scour-depth calculation is only intended to temper 

engineering judgment and should not necessarily be used to reduce the estimated 

equilibrium scour depth, because scour can be cumulative. 

Example 

The following example has been adapted and modified from HEC-18, and it was first given in a 

report by the author.(78) However, it has been converted to SI units and modified to include the 

spill-through abutment shape factor in comparison to the presentation in that report. 

A bridge with a 228.6-m (750-ft) opening length spans Burdell Creek, which has a drainage area 

of 971 square kilometers (km2

) (375 mi2). The exit and bridge cross sections are shown in figure 

48 with three subsections and their corresponding values of Manning’s n. The slope of the stream 

reach at the bridge site is constant and equal to 0.001 m/m. The bridge has a deck elevation of 

6.71 m (22.0 ft) and a bottom chord elevation of 5.49 m (18.0 ft). It is a type 3 bridge with 2:1 

abutment and embankment slopes, and it is perpendicular to the flow direction (zero skew). The 

tops of the left and right spill-through abutments are at X-stations of 281.9 m and 510.5 m, 

respectively, and the abutments can be considered to be setback abutments. There are six bridge 

piers, each with a width of 1.52 m (5.0 ft). The sediment has a median grain diameter d50 of 2.0 

mm (0.00656 ft). The requirement is to estimate the clear-water abutment scour for the 100-year 

design flood. 

Solution 

From HYDRAIN, the NFF program can be used to estimate the design flood flows. For the given 

drainage area and for a given region of the country (coastal plain in Georgia), the predicted Q100

= 397 m3

/s (14,000 ft3/s) and the predicted Q500 = 567 m3/s (20,000 ft3/s). Calculations are done 

in this example for Q100. 

The WSPRO input data file is shown as table 4 in appendix A for Q100 = 397 m3

/s. The program 

was actually run twice, first to obtain the water-surface elevations for both the unconstricted and 

the constricted cases at the approach cross section, and second, with the HP 2 data records 

(hydraulic parameter data record for a particular cross section of the WSPRO output file) to 

compute the velocity distribution in the approach section for the unconstricted (undisturbed) 

water-surface elevation of 4.038 m (13.25 ft) and the constricted water-surface elevation of 4.157 

m (13.64 ft). These elevations can be extracted from table 5, where the results of the watersurface profile computations are given. Tables 6 and 7 give the results of the velocity distribution 

computations for the unconstricted and constricted cases, respectively. 
97

0.0

2.0

4.0

6.0

8.0

0 200 400 600 800

Transverse Station, m

Elevation, m

Exit Bridge Water surface elevation Top of road

n = 0.032 n = 0.042 n = 0.032

Bottom Chord

Figure 48. Cross sections and water-surface elevation for Burdell Creek bridge. 

 (Drainage area = 971 km2

, Q100 = 397 m3/s). 
98

Now the scour parameters can be calculated from the WSPRO results. The value of M(K) from 

table 5 is 0.189, which by definition gives M = 1 – M(K) = 0.811. For consistency with the 

current FHWA methodology, the unconstricted and constricted floodplain depths and velocities 

are determined by the procedure given in HEC-18 for Froehlich’s equation. First, for the 

constricted case, the abutment length is determined from table 5 as the difference between the 

left edge of the water (LEW) for the BRDG and APPR sections: 

L LEW LEW 232.8m (764 ft) a = BRDG − APPR = (60) 

Then, from table 7 for the constricted flow, the blocked flow in the approach section up to X 

STA. = LEWBRDG is 1.97 streamtubes by interpolation. Therefore, the blocked flow Qf1 is 

(1.97/20)397 = 39.1 m3

/s (1380 ft3/s), because each of the 20 streamtubes carries 1/20 of the total 

flow. Also, the blocked flow area Af1 can be interpolated from table 7 to be 106.8 m2

 (1150 ft2). 

Now we can calculate Vf1 = Qf1/Af1 = 39.1/106.8 = 0.366 m/s (1.20 ft/s), and yf1 = Af1/La = 

106.8/233 = 0.458 m (1.50 ft). In a similar manner, the value of yf0 is found for the equivalent 

blocked flow area in the unconstricted cross section from table 6 to be 0.357 m (1.17 ft). 

The critical velocities for coarse sediments are determined by substituting into equation 55 

(Keulegan’s equation). For the constricted approach section, we have for a floodplain depth of 

0.458 m (1.50 ft): 

where Shields’ parameter was taken to be 0.035 for this sediment size.(75) Because Vf1 < Vf1c, it 

is apparent that we have clear-water scour. In a similar manner, the value of Vf0c for an 

unconstricted floodplain depth of 0.357 m (1.17 ft) is 0.59 m/s (1.93 ft/s). 

To compute the scour depth for the setback abutments, substitute into equation 56 to obtain: 

where shape factor KST = 0.77 from equation 57. Finally, the scour depth is 4.44 x 0.357 = 1.59 

m (5.22 ft). In general, this calculation would be repeated for the right abutment; however, this 

example has an essentially symmetrical cross section. 

The NFF program can be used to develop a design flood hydrograph. Assuming that the 

watershed is 48.3 km (30 mi) long with a lag time of 15 hours, the resulting hydrograph can be 

computed and is shown in figure 49. As a conservative assumption, the flood duration is 

estimated as the time required for a constant discharge equal to the peak discharge to give the 

 = . m/s

* .

. . V = . . . . * . * c f 0 61

2 0 002

(12 2)(0 458) (2 65 1)(9 81)(0 002)(0 035) 5 75 log 1 − (61) 

0.4] 10 4 44

(0 81)(0 61)(0 357)

(0 366)(0 457) 814 [

0

 + . = . . . .

. . = . K

y

d

ST

f

s − (62) 
99

0

100

200

300

400

500

600

0 5 10 15 20 25 30 35 40

Time, hrs

Q, m3/s

Figure 49. Design 100-year hydrograph for Burdell Creek. 

 (Drainage area = 971 km2

, lag time = 15 hours). 
100

same volume of direct runoff as the original hydrograph. This results in a duration of 13.4 hours. 

The critical velocity at the toe of the abutment is computed from a depth equal to the watersurface elevation in the bridge opening, which is 3.80 m (12.47 ft) from table 5 minus the ground 

elevation at the toe (3.15 m) to give yab = 0.646 m (2.12 ft). Then, using equation 55 again for 

consistency, the critical velocity Vc = 0.64 m/s (2.1 ft/s), and Vct/yf0 = (0.64)(13.4)(3600)/(0.357) 

= 9 x 104

. From equation 59, Vab/Vc = 1.54, and figure 41 indicates that approximately 90 

percent of the equilibrium scour will occur over the estimated flood duration of 13.4 hours. 

Under these circumstances, the reduction in scour caused by equilibrium not being reached is 

small so that the final result for the clear-water abutment scour depth is left at the value of 1.59 

m (5.22 ft) that was previously estimated. 

Because scour holes can develop as a result of several floods over time, the time analysis 

proposed herein is intended as a judgment factor in reducing the estimated scour only when: (1) 

the estimated scour is significantly higher than that actually observed on an existing bridge, or 

(2) the watershed is so small that only a small fraction of equilibrium scour is reached in a 

typical design flood. It is strongly recommended that a whole range of discharges be considered 

and that the percentage of equilibrium scour be investigated in each case. Figure 41 provides the 

first method for estimating the percentage of equilibrium scour for floods of differing magnitude, 

and it should greatly enhance the engineer’s judgment in making a final evaluation of scour 

susceptibility. 

It should be noted that the estimated equilibrium scour has been determined for a spill-through 

abutment with a shape factor of 0.77 for this particular example, which is greater than the value 

of 0.55 currently recommended by HEC-18. However, the shape factor varies and can be as large 

as 1.0 for long spill-through abutments as indicated by equation 57 (shape has no influence in 

some cases). In addition, the estimated scour could be increased by a skewness factor for 

abutments not perpendicular to the flow as recommended by HEC-18; however, for this 

example, the skewness factor was taken to be 1.0 for a perpendicular abutment. 

Finally, for the purposes of comparison, it is reasonable to calculate the abutment scour for this 

case by using Froehlich’s live-bed scour equation or the HIRE (Highways in the River 

Environment) equation as recommended by HEC-18. Substituting into Froehlich’s equation, 

given previously as equation 5, and using the values of La, yf1, and Vf1 already determined, we 

have: 

so that ds = (7.24)(0.458) = 3.32 m (10.9 ft), which is approximately twice as large as the value 

estimated by the procedure proposed herein. 

For the HIRE equation, which is recommended by HEC-18 for long abutments, the value of yab

is taken to be 0.646 m (2.12 ft) as determined previously from the water-surface elevation in the 

bridge section. However, the value of Vab is estimated to be equal to 0.98 m/s (3.2 ft/s) as 

) 1 7 24

(9 81)(0 458)

0 366 ) ( 0 458233 (2 27)(0 55)(10)( 0 43 0.61

1

 + = . . .

.

. = . . . y

d .

f

s (63) 
101

determined from figure 30 or equation 59 developed in this research. The use of the WSPRO 

results from the bridge section to estimate Vab is not recommended because conveyance ratios 

cannot predict the local acceleration occurring near the abutment face. Now, substituting into 

the HIRE equation given by equation 15 in chapter 2, we have: 

Then, if we add an FS of 1.0 for consistency with the other methods used, ds = (3.9)(0.646) = 

2.52 m (8.3 ft). This estimated scour depth lies between the one predicted from the results of this 

research (1.59 m) and the one from Froehlich’s live-bed scour equation (3.32 m). The HIRE 

equation necessarily assumes live-bed scour, because there is no dependence on sediment size. In 

this example, live-bed scour is not predicted to occur. 

COMPARISON WITH FIELD DATA 

Field data for the flood of April 1997 on the Pomme de Terre River in Minnesota were measured 

by Mueller and Hitchcock(93) of the USGS and were provided to the author by Sterling Jones(94)

of FHWA. A bridge scour investigation report for the bridge over Highway 12, including a 

WSPRO data file, was also made available.(95) Xibing Dou(96) provided his spreadsheets of the 

cross-sectional data for Highways 12 and 22 to the author. 

New WSPRO runs were made by the author for each bridge, and scour calculations were carried 

out according to the methodology presented in this report. First, the downstream water-surface 

elevation at the exit section was adjusted for the measured discharge until the measured watersurface elevation at the bridge during the flood was reproduced. The bridge cross section after 

scour was used in these computer runs. Then, WSPRO runs were made using the cross sections 

before scour to obtain the necessary information for the scour predictions. The cross-section data 

for Highway 22 were more limited than that for Highway 12, so a template cross section was 

used to provide all of the necessary cross-section data in agreement with a measured river bed 

profile before scour.

Highway 22 Example 

The Swift County Highway 22 bridge crosses the Pomme de Terre River near Artichoke Lake in 

Minnesota, about 10 km upstream from the Highway 12 bridge. The abutments are set at the 

edge of the main channel, and there are two bridge piers with a spacing of approximately 12.2 m 

(40.0 ft) as shown in figure 50. (The horizontal stations were established as increasing from right 

to left; however, the distances are given as negative numbers in the figure so that they are 

increasing from left to right while looking downstream.) The abutments are protected by rock 

riprap on a 2:1 sideslope. On April 5, 1997, a discharge of 129.5 m3

/s (4570 ft3/s) was measured, 

and scour began to develop near the right abutment. On April 9, the discharge was measured as 

145.9 m3

/s (5150 ft3/s) and the bridge was temporarily closed until riprap could be placed to 

) 2 9

(9 81)(0 646)

0 98

(4 0)(

1 3

 = . . .

. = . y

d

/

ab

s (64) 
102

305

310

315

320

325

330

-700 -600 -500 -400 -300

Station, m

Elevation, m

Upstream edge of bridge, 4-09-97 Full valley

Road Upstream water surface, 4-09-97

Figure 50. Cross sections for Highway 22 over the Pomme de Terre River 

 for the flood of April 9, 1997. 
103

protect the bridge. The minimum bed elevation reached on that date was 311.3 m (1021.5 ft) as 

shown in figure 50. The bed material from elevation 312.4 m (1025.0 ft) downward is shown on 

the Minnesota Department of Highways bridge survey as being fine gravel. 

From the after-scour WSPRO run for the measurements taken on April 9, it was determined that 

a water-surface elevation of 317.21 m (1040.96 ft) at the exit cross section gave an elevation of 

317.25 m (1040.90 ft) at the downstream face of the bridge for Q = 145.9 m3

/s (5150 ft3/s). 

Adding a drop of 0.09 m (0.30 ft) in the water surface through the bridge measured on April 4 at 

nearly the same discharge resulted in a water-surface elevation on the upstream face of the 

bridge of 317.34 m (1041.20 ft) in agreement with the measured value on April 9. The beforescour WSPRO run was then made with the same tailwater elevation at the exit section as for the 

after-scour run. The input data file is given as table 8, and the results are given in tables 9 

through 11, from which the necessary information is obtained for the scour computations. 

The value of the discharge distribution factor from table 9 is M = 1 – M(K) = 0.336.1

 The 

computations are made for the right abutment where the deepest scour occurred because of the 

wider floodplain on that side. The detailed procedure for making the computations has already 

been given in the previous example, so only the main results are highlighted in this field 

application. In table 9, the length of the abutment on the right side is determined to be 174.6 m 

(573 ft), and the water-surface elevation at the approach section can be obtained. From tables 10 

and 11 for the velocity distributions at the approach section for the unconstricted and constricted 

water-surface elevations, respectively, it is determined that yf0 = 1.55 m (5.08 ft), Vf1 = 0.24 m/s 

(0.79 ft/s), and yf1 = 1.67 m (5.48 ft). The median grain diameter d50 of the sediment is taken to 

be 2.0 mm, from which the critical velocity in the main channel of the bridge section is 0.87 m/s 

(2.85 ft/s) using Keulegan’s equation, but substituting the hydraulic radius for the depth. From 

table 10, the velocities are all less than this critical velocity so that no sediment of the size being 

scoured could be carried into the scour hole (clear-water scour). The scour depth is then 

determined as clear-water scour around a bankline abutment from equation 56 using the shape 

factor for a spill-through abutment of 0.67, which is calculated from equation 57. The result is: 

The scour depth measured from the unconstricted water surface at the bridge is then ds + yf0 = 

(2.64)(1.55) + 1.55 = 5.6 m (18.4 ft) so that the bottom elevation in the scour hole is the watersurface elevation of 317.3 m (1041.0 ft) at the full valley section minus 5.6 m, or an elevation of 

311.7 m (1022.7 ft). For comparison, the lowest measured bed elevation is 311.3 m (1021.4 ft), 

which is well within the standard error of the scour-prediction formula. Also, no FS has been 

included in the computation. If it is included, the predicted bottom elevation is 310.1 m (1017.4 

ft), which provides a reasonable margin of safety. 

___________________________________________

1

Alternatively, the value of M could be determined from the centerline of the main channel to the right edge of water in the 

approach section using the velocity distribution in Table 11, but it makes only a small difference in the result in this case. 

0 4 ] 2 64

(0 336)(0 87)(155)

(0 24)(167) 814 (0 67) [

0

. = . . . .

. . = . . y

d

f

s − (65) 
104

Highway 12 Example 

The second field example comes from the same flood on the Pomme de Terre River, but at the 

bridge on Highway 12 near Holloway, MN. The bridge was closed because of scour at the right 

abutment that extended below the abutment footing. The bridge is a single span with a length of 

26.8 m (88.0 ft), and it has vertical abutments with 45-degree wingwalls. The channel and bridge 

cross sections are shown in figure 51. The measured discharge on April 9, 1997, was 162.9 m3

/s 

(5750 ft3

/s), with a measured water-surface elevation on the upstream side of the bridge of 302.6 

m (993.0 ft). The minimum bed elevation was measured to be 293.5 m (963.0 ft) near the right 

abutment. The surficial sediment in the streambed is described as “organic, silty sand” with a d50

of 0.15 mm; however, the boring log shows that the material below about an elevation of 305 m 

(1000 ft) is “sand and gravel, gray, saturated.” 

From the after-scour WSPRO run, the exit tailwater elevation was determined to be 302.6 m 

(992.8 ft) for a discharge of 162.9 m3

/s (5750 ft3/s). The input data file for the before-scour 

WSPRO run is given in table 12, and the WSPRO results are shown in tables 13 through 15. The 

value of the discharge distribution factor from table 13 is M = 1 – M(K) = 0.335. The 

computations are made for the right abutment (west) where the deepest scour occurred. From 

table 13, the length of the abutment on the right side is determined to be 121.3 m (398.0 ft), and 

the water-surface elevations at the approach section can be obtained. From tables 14 and 15 for 

the velocity distributions at the approach section for the unconstricted and constricted watersurface elevations, respectively, it is determined that yf0 = 1.12 m (3.67 ft), Vf1 = 0.19 m/s (0.62 

ft/s), and yf1 = 1.22 m (4.00 ft). If the median grain diameter d50 of the sediment is taken as 2 mm 

in the scour-hole region as done for Highway 22, the critical velocity in the main channel of the 

approach section is 0.88 m/s (2.89 ft/s) using Keulegan’s equation. However, the WSPRO results 

in table 15 show maximum approach velocities of approximately 0.84 m/s (2.76 ft/s). For 

practical purposes, this means that the scour is approaching threshold live-bed scour. Thus, for 

live-bed scour around a bankline abutment, equation 58 is used with Vm1 = Vm1c. Because M is 

relatively small, equation 58 indicates that the maximum value of ds/yf0 = 10 has been reached. 

The scour depth measured from the water surface in the bridge before scour is then ds + yf0 = 

(10)(1.12) + 1.12 = 12.3 m (40.4 ft), so that the bottom elevation in the scour hole is the 

unconstricted water-surface elevation of 302.6 m (992.9 ft) at the full valley section minus 12.3 

m, or an elevation of 290.3 m (953 ft). For comparison, the lowest measured bed elevation is 

293.5 m (963 ft), which is approximately 3 m higher than predicted. Given the presence of both 

sand and gravel, it is likely that armoring occurred, which limited the scour-hole depth in 

comparison to the predicted value; however, there is simply not enough information available on 

the sediment size distribution for this example, so these results are somewhat inconclusive. 
105

290

295

300

305

310

315

100 200 300 400 500 600 700

Station, m

Elevation, m

Exit Full valley

Bridge Road

After scour Approach

Upstream water surface, 4-09-97 

Figure 51. Cross sections for Highway 12 over the Pomme de Terre River 

 for the flood of April 9, 1997. 
106

LIMITATIONS 

The experimental results developed herein are for the case of clear-water abutment scour in a 

compound channel with a wide floodplain. The ranges of variables covered in the experiments 

reported herein are given in table 3. 

Table 3. Range of dimensionless variables 

in experimental relationships. 

Variable Range 

M 0.25-0.95 

Vf1/Vf1c 0.30-0.80 

Vf0/Vf0c 0.40-1.00 

yf0/yf1 0.40-1.00 

La/Bf 0.17-1.00 

qf1/(Mqf0c) 0.40-2.30 
107

CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS 

The results of this research on bridge abutment scour in compound channels show that the 

discharge distribution factor is the appropriate variable to use rather than abutment length to 

measure the effects of flow contraction and flow redistribution in the contracted section on local 

scour depth. Sediment size is incorporated into the proposed scour-prediction technique in terms 

of the critical velocity required to initiate motion of the sediment. It is shown that the 

unconstricted flow depth in the floodplain should be used as the reference depth in the scourprediction formula when bridge backwater is significant. The abutment shape is demonstrated to 

be important for shorter abutments, while no experimental differences in the abutment shape 

effects could be detected as the abutment increased in length and caused more contraction with 

encroachment on the main channel. Attempts to relate local abutment scour to local hydraulic 

variables near the abutment face were successful only for the shorter abutments; however, a 

relationship was developed between the approach hydraulic variables and the local contraction 

hydraulic variables. It is recommended that the proposed scour formula (equation 56) (which 

depends on the approach hydraulic variables predicted by WSPRO) be used. The suggested scour 

formula is for clear-water scour for both setback and bankline abutments. A second formula 

(equation 58) was proposed for bankline abutments that experience threshold live-bed scour 

conditions in the main channel; however, experiments in a compound section specifically 

designed for live-bed scour are needed. Testing of five current abutment scour formulas has 

shown that they all significantly overestimate or underestimate the scour data measured in this 

research. Field evaluation of the proposed method provided good agreement on one bridge, but 

an overestimate of scour on a second bridge. Armoring may have occurred in the latter case; 

however, inadequate data on the field sediment size distribution precluded a definite conclusion. 

It must be emphasized that the experimental results for bankline abutments that were used to 

develop the proposed scour formulas herein do not distinguish between contraction and abutment 

scour. Thus, the method of superposition of contraction and abutment scour for bankline 

abutments as though they were independent may be overly conservative. 

It is concluded that the experimental results and methodology developed from this research can 

be used to estimate the depth of bridge abutment scour. It has been determined that for a 

compound channel over a much wider range of variables than was previously available that the 

effects of discharge distribution, sediment size, and time development on scour depth can be 

predicted from the relationships proposed herein. The comparisons of the WSPRO results, the 

experimental results, and the results from a two-dimensional numerical turbulence model have 

shown that the results from WSPRO are adequate for estimating the independent parameters 

needed for abutment scour prediction as long as bridge approach hydraulic conditions are used as 

predictor variables. It is recommended that the proposed procedure for abutment scour 

estimation be used alongside current FHWA procedures subject to the limitations on the ranges 

of dimensionless variables given in table 3, and that it be tested on additional field data as it 

becomes available. It is emphasized that sediment properties and their changes with depth must 

be known to adequately evaluate clear-water scour depths. 

It is recommended that further research be conducted on abutment scour in order to evaluate and 

protect scour-critical bridges that are subject to possible failure. Suggested areas for research are: 
108

1. Laboratory study of scour countermeasures is needed in order to design the most efficient 

abutment scour-protection schemes. The study should consider: (a) the extent, size, and 

placement of riprap at abutments, and (b) the effectiveness of spur dikes. 

2. A three-dimensional numerical model with advanced turbulence closure schemes needs 

to be applied to the laboratory model used in this research for several selected cases of 

abutment scour. This effort should include three-dimensional velocity and turbulence 

measurements in the scour-hole area at different stages of scour-hole development with 

respect to time by temporarily fixing the bed in the scour-hole area. 

3. A well-planned, detailed field study of a bridge subject to abutment scour in cooperation 

with the USGS is needed. The bridge should be instrumented and scour determined over 

a 3-year period with detailed field measurements of velocity and bed elevation changes. 

A laboratory model of the instrumented field bridge should be constructed and tested to 

compare scour predictions based on the laboratory data with those actually measured in 

the field to finally resolve laboratory scale-up issues. 

4. Further experimental investigation of the live-bed scour case with the abutment at the 

edge of the main channel is warranted. This will require special design of the compoundchannel geometry so that a careful selection of a combination of flow rates, sediment 

size, and discharge distribution will result in sediment transport in the main channel 

without scour exceeding the physical limits imposed by the finite depth of the sediment. 
109

APPENDIX A. 

WSPRO INPUT AND OUTPUT FOR BURDELL CREEK, HIGHWAY 22 BRIDGE 

OVER POMME DE TERRE RIVER, AND HIGHWAY 12 BRIDGE OVER POMME DE 

TERRE RIVER 

Table 4. WSPRO input data file for Burdell Creek, Q100. 

______________________________________________________________________________ 

*F 

SI 0 

T1 ABUTMENT SCOUR EXAMPLE 

* 

Q 14000 

SK 0.001 

* 

XS EXIT 750 0.0 0.5 0.0 0.001 

GR 0, 19 100, 15 200, 11 500, 10.75 900, 10 1100, 9.0 

GR 1215, 5.5 1250, 4.9 1300, 3.05 1350, 4.85 1385, 5.1 

GR 1500, 9.0 1700, 10 2100, 10.75 2400, 11 2500, 15 

GR 2600, 19 

N 0.042 0.032 0.042 

SA 1100 1500 

* 

XS FULV 1500 0.0 0.5 0.0 

* 

BR BRDG 1500 * 0.0 0.5 0.0 * 

BL 0 750 1100 1500 

BC 18 

CD 3 50 2 22 0.0 0.0 0.0 

AB 2 2 

PD 0 5.65 30 6 

N 0.042 0.032 0.042 

SA 1100 1500 

* 

XS APPR 2300 0.0 0.5 0.0 

* 

HP 2 APPR 13.25 0 13.25 14000 

HP 2 APPR 13.64 0 13.64 14000 

* 

DA BRDG 1 1 * * 14000 1 

* 

EX 0 

ER 
110

Table 5. WSPRO water-surface profile output for Burdell Creek, Q100. 

______________________________________________________________________________ 

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 ABUTMENT SCOUR EXAMPLE 

 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: EXIT 11.696 .268 14000.000 4721.088 ********* 182.604 

 Header Type: XS 11.964 ****** 2.965 442473.00 ********* 2417.396 

 SRD: 750.000 9.447 ****** .504 ****** 1.960 ****** 

 Section: FULV 12.449 .267 14000.000 4728.060 750.000 182.526 

 Header Type: FV 12.716 .750 2.961 443183.70 750.000 2417.474 

 SRD: 1500.000 10.197 .000 .502 .0010 1.959 .003 

 <<< The Preceding Data Reflect The “Unconstricted” Profile >>> 

 Section: APPR 13.250 .267 14000.000 4730.357 800.000 182.500 

 Header Type: AS 13.517 .798 2.960 443418.00 800.000 2417.500 

 SRD: 2300.000 10.997 .000 .502 .0010 1.959 .003 

 <<< The Preceding Data Reflect The “Unconstricted” Profile >>> 

 <<< The Following Data Reflect The “Constricted” Profile >>> 

 <<< Beginning Bridge/Culvert Hydraulic Computations >>> 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: BRDG 12.470 .470 14000.000 3038.370 750.000 936.685 

 Header Type: BR 12.940 .921 4.608 386306.30 750.000 1663.315 

 SRD: 1500.000 10.668 .055 .474 ****** 1.424 -.004 

 Specific Bridge Information C P/A PFELEV BLEN XLAB XRAB 

 Bridge Type 3 Flow Type 1 ------ ----- -------- -------- -------- ------- 

 Pier/Pile Code 0 .8380 .067 18.000 750.000 940.750 659.250 

 --------------------------- ------ ----- -------- -------- -------- ------- 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: APPR 13.635 .186 14000.000 5595.480 750.000 172.865 

 Header Type: AS 13.822 .780 2.502 536819.40 776.571 2427.135 

 SRD: 2300.000 10.997 .100 .387 .0010 1.914 -.017 

 Approach Section APPR Flow Contraction Information 

 M( G ) M( K ) KQ XLKQ XRKQ OTEL 

 -------- -------- --------- -------- -------- -------- 

 .675 .189 438924.8 938.577 1661.368 13.635 
111

Table 6. Velocity distribution from HP record for unconstricted flow 

 at approach section, Burdell Creek, Q100. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 ABUTMENT SCOUR EXAMPLE 

 

 *** Beginning Velocity Distribution For Header Record APPR *** 

 SRD Location: 2300.000 Header Record Number 4 

 Water-Surface Elevation: 13.250 Element # 1 

 Flow: 14000.000 Velocity: 2.96 Hydraulic Depth: 2.116 

 Cross-Section Area: 4730.38 Conveyance: 443419.90 

 Bank Stations -> Left: 182.500 Right: 2417.500 

 X STA. 182.5 803.7 1045.3 1141.7 1181.9 1209.7 

 A( I ) 628.6 454.8 279.4 184.1 156.0 

 V( I ) 1.11 1.54 2.51 3.80 4.49 

 D( I ) 1.01 1.88 2.90 4.58 5.62 

 X STA. 1209.7 1232.9 1253.5 1271.9 1288.1 1302.5 

 A( I ) 146.1 138.2 133.4 128.5 121.2 

 V( I ) 4.79 5.07 5.25 5.45 5.77 

 D( I ) 6.30 6.69 7.27 7.91 8.46 

 X STA. 1302.5 1317.4 1333.9 1352.7 1373.2 1394.8 

 A( I ) 124.2 127.1 133.8 138.2 141.8 

 V( I ) 5.64 5.51 5.23 5.07 4.94 

 D( I ) 8.29 7.73 7.10 6.76 6.55 

 X STA. 1394.8 1421.9 1458.5 1558.5 1799.0 2417.5 

 A( I ) 157.2 172.9 290.7 449.8 624.5 

 V( I ) 4.45 4.05 2.41 1.56 1.12 

 D( I ) 5.81 4.73 2.91 1.87 1.01 
112

Table 7. Velocity distribution from HP record for constricted flow 

 at approach section, Burdell Creek, Q100. 

______________________________________________________________________________ 

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 ABUTMENT SCOUR EXAMPLE 

 

 *** Beginning Velocity Distribution For Header Record APPR *** 

 SRD Location: 2300.000 Header Record Number 4 

 Water-Surface Elevation: 13.640 Element # 1 

 Flow: 14000.000 Velocity: 2.50 Hydraulic Depth: 2.487 

 Cross-Section Area: 5605.83 Conveyance: 537996.80 

 Bank Stations -> Left: 172.750 Right: 2427.250 

 X STA. 172.7 684.7 945.3 1101.6 1161.2 1195.4 

 A( I ) 658.8 506.4 423.1 241.3 187.3 

 V( I ) 1.06 1.38 1.65 2.90 3.74 

 D( I ) 1.29 1.94 2.71 4.05 5.47 

 X STA. 1195.4 1223.0 1246.6 1268.0 1286.0 1302.5 

 A( I ) 176.6 163.5 159.7 147.8 144.8 

 V( I ) 3.96 4.28 4.38 4.74 4.83 

 D( I ) 6.40 6.93 7.47 8.19 8.81 

 X STA. 1302.5 1319.1 1338.0 1359.2 1382.3 1407.8 

 A( I ) 143.9 151.0 156.4 163.7 169.5 

 V( I ) 4.87 4.64 4.48 4.28 4.13 

 D( I ) 8.65 8.01 7.35 7.09 6.64 

 X STA. 1407.8 1441.5 1497.0 1658.1 1917.6 2427.3 

 A( I ) 190.1 229.2 435.6 502.4 655.0 

 V( I ) 3.68 3.05 1.61 1.39 1.07 

 D( I ) 5.64 4.13 2.70 1.94 1.29 
113

Table 8. WSPRO input data file for Highway 22 bridge. 

______________________________________________________________________________

*F 

SI 0 

T1 BRIDGE 76518 

T2 HIGHWAY 22 OVER POMME DE TERRE RIVER 

T3 EXISTING BRIDGE, 1997 FLOOD ANALYSIS, BEFORE SCOUR 

J1 0.25 

Q 5150 

WS 1040.96 

* TEMPLATE SECTION AT BRIDGE, FULL VALLEY 

XT FV 1020 

GR -2106.1,1042.1 -2094.9,1041.8 -2090.5,1041.5 -2080.7,1041.0 -2040.1,1039.3 

GR -2003.4,1037.5 -1916.3,1037.0 -1911.3,1037.0 -1907.3,1037.1 -1905.9,1036.3 

GR -1904.1,1035.8 -1903.5,1035.1 -1903.4,1034.4 -1903.3,1033.7 -1901.3,1032.9 

GR -1900.5,1032.0 -1900.3,1031.2 -1897.1,1031.1 -1891.6,1030.8 -1885.6,1030.6 

GR -1878.8,1030.4 -1875.8,1030.2 -1871.2,1029.9 -1866.8,1029.7 -1861.7,1029.4 

GR -1855.9,1029.3 -1849.3,1029.7 -1845.9,1030.1 -1841.9,1030.4 -1834.4,1031.2 

GR -1829.6,1031.6 -1825.4,1032.0 -1819,1032.5 -1813.8,1033 -1778.3,1033.1 

GR -1757.4,1033.4 -1730.9,1033.3 -1700.4,1033.8 -1651.6,1035.1 -1603.5,1036.4 

GR -1546.7,1036.6 -1503.1,1036.9 -1405.6,1037.4 -1351.8,1038.2 -1329.4,1038.5 

GR -1318.1,1038.7 -1307.3,1039 -1298.7,1039.3 -1289.8,1039.6 -1278.8,1040 

GR -1271.4,1040.3 -1259.1,1040.6 -1248.1,1040.9 -1238.3,1041.2 -1229.4,1041.5 

GR -1219.6,1041.8 -1211.8,1042.1 -1206.6,1042.6 -1197.7,1042.9 -1188.1,1043.6 

* 

XS EXIT 900 

GT +0.60 

N 0.08 0.035 0.08 

SA -1910 -1790 

* 

XS FULV 1020 * * * -0.005 

* 

BR BRGE 1020 1041.21 

GR -1910,1041.2 -1908,1039.2 -1903,1037.7 -1897,1034 -1891.6,1030.8 

GR -1885.6,1030.6 -1878.8,1030.4 -1875.8,1030.2 -1871.2,1029.9 -1866.8,1029.7 

GR -1861.7,1029.4 -1855.9,1029.3 -1849.3,1029.7 -1845.9,1030.1 -1841.9,1030.4 

GR -1834.4,1031.2 -1829.6,1031.6 -1825.4,1032 -1819,1032.5 -1813.8,1033 

GR -1803,1033.5 -1798,1035.7 -1793,1039.2 -1790,1041.2 -1910,1041.2 

N 0.035 

* 

PW 0 1029.8,1.5 1031.7,1.5 1031.7,3 

* TYPE BRWDTH EMBSS EMBELEV 

CD 3 40 2 1045.4 

KQ * * * -1910 -1790 

* 

HP 2 BRGE 1040.82 * * 5150 

* 

XS APPR 1180 

N 0.05 0.035 0.05 

SA -1910 -1790 

* 

HP 2 APPR 1041.08 * * 5150 

HP 2 APPR 1041.56 * * 5150 

* 

EX 

ER 
114

Table 9. WSPRO water-surface profile output for Highway 22 bridge. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 BRIDGE 76518 

 

 HIGHWAY 22 OVER POMME DE TERRE RIVER 

 EXISTING BRIDGE, 1997 FLOOD ANALYSIS, BEFORE SCOUR 

 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: EXIT 1040.960 .097 5150.000 3511.259 ********* -2065.417 

 Header Type: XS 1041.057 ****** 1.467 298703.30 ********* -1268.937 

 SRD: 900.000 1035.804 ****** .210 ****** 2.894 ****** 

 Section: FULV 1041.014 .073 5150.000 4045.370 120.000 -2080.970 

 Header Type: FV 1041.087 .030 1.273 352758.00 120.000 -1244.383 

 SRD: 1020.000 1035.204 .000 .174 .0003 2.908 .000 

 <<< The Preceding Data Reflect The “Unconstricted” Profile >>> 

 ===135 CONVEYANCE RATIO OUTSIDE OF RECOMMENDED LIMITS AT “APPR”. 

 KRATIO: 1.54 

 Section: APPR 1041.079 .030 5150.000 4787.963 160.000 -2097.845 

 Header Type: AS 1041.109 .022 1.076 544559.30 160.000 -1217.549 

 SRD: 1180.000 1034.376 .000 .106 .0001 1.684 .000 

 <<< The Preceding Data Reflect The “Unconstricted” Profile >>> 

 <<< The Following Data Reflect The “Constricted” Profile >>> 

 <<< Beginning Bridge/Culvert Hydraulic Computations >>> 

 ===220 FLOW CLASS 1 ( 4 ) SOLUTION INDICATES POSSIBLE PRESSURE FLOW. 

 WS3, WSIU, WS1, PFELV: 1040.82 1041.52 1041.55 1041.21 

 ===245 ATTEMPTING FLOW CLASS 2 ( 5 ) SOLUTION. 

 ===250 INSUFFICIENT HEAD FOR PRESSURE FLOW. 

 YU/Z, WSIU, WS: 1.06 1041.70 1041.76 

 ===270 REJECTED FLOW CLASS 2 ( 5 ) SOLUTION. 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: BRGE 1040.825 .673 5150.000 1014.368 120.000 -1909.625 

 Header Type: BR 1041.498 .063 5.077 174703.30 120.000 -1790.563 

 SRD: 1020.000 1035.708 .378 .397 ****** 1.678 .000 
115

Table 9. WSPRO water-surface profile output for Highway 22 bridge (continued). 

______________________________________________________________________________

 Specific Bridge Information C P/A PFELEV BLEN XLAB XRAB 

 Bridge Type 3 Flow Type 1 ------ ----- -------- -------- -------- ------- 

 Pier/Pile Code 0 .7719 .030 1041.210 ******** ******** ******* 

 --------------------------- ------ ----- -------- -------- -------- ------- 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: APPR 1041.555 .025 5150.000 5212.262 120.000 -2106.100 

 Header Type: AS 1041.580 .038 .988 611618.10 149.347 -1209.151 

 SRD: 1180.000 1034.376 .044 .093 .0001 1.647 .000 

 Approach Section APPR Flow Contraction Information 

 M( G ) M( K ) KQ XLKQ XRKQ OTEL 

 -------- -------- --------- -------- -------- -------- 

 .864 .664 205746.2 ******** ******** 1041.555 

 -------- -------- --------- -------- -------- -------- 

 <<< End of Bridge Hydraulics Computations >>> 
116

Table 10. Velocity distribution from HP record for unconstricted flow 

 at approach section, Highway 22 bridge. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 BRIDGE 76518 

 

 HIGHWAY 22 OVER POMME DE TERRE RIVER 

 

 EXISTING BRIDGE, 1997 FLOOD ANALYSIS, BEFORE SCOUR 

 

 *** Beginning Velocity Distribution For Header Record APPR *** 

 SRD Location: 1180.000 Header Record Number 4 

 Water-Surface Elevation: 1041.080 Element # 1 

 Flow: 5150.000 Velocity: 1.08 Hydraulic Depth: 5.440 

 Cross-Section Area: 4788.92 Conveyance: 544702.60 

 Bank Stations -> Left: -2097.886 Right: -1217.521 

 X STA. -2097.9 -1951.7 -1898.1 -1885.5 -1874.3 -1864.0 

 A( I ) 439.4 286.6 138.3 129.1 124.7 

 V( I ) .59 .90 1.86 1.99 2.07 

 D( I ) 3.01 5.34 11.04 11.48 12.07 

 X STA. -1864.0 -1854.1 -1843.7 -1832.0 -1817.9 -1800.8 

 A( I ) 123.3 125.6 129.5 139.2 151.8 

 V( I ) 2.09 2.05 1.99 1.85 1.70 

 D( I ) 12.50 12.08 11.05 9.87 8.91 

 X STA. -1800.8 -1780.6 -1753.6 -1725.3 -1694.6 -1658.4 

 A( I ) 178.8 232.2 241.8 252.8 269.2 

 V( I ) 1.44 1.11 1.06 1.02 .96 

 D( I ) 8.81 8.62 8.54 8.23 7.44 

 X STA. -1658.4 -1611.9 -1551.5 -1484.9 -1404.4 -1217.5 

 A( I ) 294.6 327.1 339.1 376.9 488.9 

 V( I ) .87 .79 .76 .68 .53 

 D( I ) 6.34 5.42 5.09 4.68 2.62 
117

Table 11. Velocity distribution from HP record for constricted flow 

 at approach section, Highway 22 bridge. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 BRIDGE 76518 

 

 HIGHWAY 22 OVER POMME DE TERRE RIVER 

 

 EXISTING BRIDGE, 1997 FLOOD ANALYSIS, BEFORE SCOUR 

 

 *** Beginning Velocity Distribution For Header Record APPR *** 

 SRD Location: 1180.000 Header Record Number 4 

 Water-Surface Elevation: 1041.560 Element # 1 

 Flow: 5150.000 Velocity: .99 Hydraulic Depth: 5.816 

 Cross-Section Area: 5217.09 Conveyance: 612424.90 

 Bank Stations -> Left: -2106.100 Right: -1209.095 

 X STA. -2106.1 -1958.7 -1899.9 -1886.7 -1874.7 -1863.8 

 A( I ) 476.5 327.6 152.2 142.2 137.3 

 V( I ) .54 .79 1.69 1.81 1.88 

 D( I ) 3.23 5.57 11.46 11.93 12.54 

 X STA. -1863.8 -1853.3 -1842.5 -1830.0 -1814.5 -1796.9 

 A( I ) 135.7 134.8 142.0 157.3 164.5 

 V( I ) 1.90 1.91 1.81 1.64 1.57 

 D( I ) 12.98 12.48 11.38 10.12 9.34 

 X STA. -1796.9 -1774.0 -1745.5 -1716.7 -1684.8 -1645.8 

 A( I ) 212.0 257.4 258.9 272.6 296.9 

 V( I ) 1.21 1.00 .99 .94 .87 

 D( I ) 9.28 9.04 8.99 8.53 7.62 

 X STA. -1645.8 -1595.5 -1536.5 -1470.1 -1392.8 -1209.1 

 A( I ) 324.2 343.6 363.4 393.1 524.8 

 V( I ) .79 .75 .71 .66 .49 

 D( I ) 6.44 5.83 5.48 5.08 2.86 
118

Table 12. WSPRO input data file for Highway 12 bridge. 

______________________________________________________________________________ 

*F 

SI 0 

T1 BRIDGE 5359 

T2 TH 12 OVER POMME DE TERRE RIVER 

T3 **** EXISTING BRIDGE BEFORE SCOUR, 1997 FLOOD ANALYSIS **** 

J1 0.25 

Q 5750 

WS 992.8 

* FIELD SURVEYED X-SEC. 75’ DOWNSTREAM FROM BRIDGE 5359 

XS DS100 25 

GR 640,1004.09 640,994.49 675,989.09 

* 750,987.19 800,1005.0 800,989.29 850,989.19 900,989.89 

GR 940,988.79 953,988.19 

GR 954,983.36 965,982.56 985,983.06 1000,982.16 1025,980.56 

GR 1036,981.06 1046,983.26 1047,986.09 1053,986.69 1060,988.09 

GR 1100,989.09 1200,987.89 1300,987.49 1312,991.79 1320,1005.0 

N 0.08 0.03 0.08 

SA 953 1047 

* FULL VALLEY X-SEC. COMPOSED FROM PROPAGATING XS-DS100 UP TO 

* BRIDGE AND SUPERIMPOSING THE SURVEYED BRIDGE OPENING 

XS FULLV 100 

GR 600,1004.09 640,994.49 675,989.09 

GR 940,988.79 953,988.19 

GR 954,983.26 956,981.48 980,981.85 1000,981.54 1020,981.39 

GR 1044,980.19 1046,983.26 1047,986.09 1053,986.69 1060,988.09 

GR 1100,989.09 1200,987.89 1300,987.49 1312,991.79 1320,1005.0 

* BRIDGE DEFINITION RECORDS FOLLOW 

BR BRDGE 100, 994.86 

GR 956,994.86 956,981.48 980,981.85 1000,981.54 

GR 1020,981.39 1044,980.19 1044,994.67 956,994.86 

N 0.035 

* BRTYPE BRWIDTH EMBSS EMBELV WWANGL WWWID 

CD 4 51 3 998 45 

KQ *,*,*, 956, 1044 

* 

XR ROAD 125, 50 

GR 454,1018.39 554,1012.83 654,1007.87 754,1004.34 854,1001.74 

GR 954,999.09 1045,998.18 1146,997.44 1246,996.98 1346,996.03 

GR 1446,994.89 1546,994.09 1646,993.71 1746,993.45 1846,993.50 

GR 1946,993.55 2046,993.83 4689,1003.83 

* IHP SEC ELMIN YINC ELMAX Q 

HP 2 BRDGE 992.63 * * 5750 

AS APPR 325 

GR 500,1005.36 570,990.45 600,988.35 

GR 700,987.75 785,987.85 890,987.8 960,987.8 966,986.95 

GR 976,983.09 1024,983.09 1035,986.35 1270,987.15 1380,988.85 

GR 1520,989.75 2000,991.75 2300,1004.0 

N 0.080 0.05 0.030 0.05 0.080 

SA 950 966 1035 1100 

* IHP SEC ELMIN YINC ELMAX Q 

HP 2 APPR 993.05 * * 5750 

HP 2 APPR 993.40 * * 5750 

EX 

ER 
119

Table 13. WSPRO water-surface profile output for Highway 12 bridge. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 BRIDGE 5359 

 

 TH 12 OVER POMME DE TERRE RIVER 

 **** EXISTING BRIDGE BEFORE SCOUR, 1997 FLOOD ANALYSIS   

 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: DS100 992.800 .245 5750.000 2281.291 ** ******* 940.000 

 Header Type: XS 993.045 ****** 2.521 295141.50 ********* 1312.612 

 SRD: 25.000 987.030 ****** .283 ****** 2.483 ****** 

 Section: FULLV 992.923 .145 5750.000 3492.329 75.000 650.158 

 Header Type: FV 993.068 .023 1.646 373264.60 75.000 1312.686 

 SRD: 100.000 986.366 .000 .234 .0003 3.435 .000 

 <<< The Preceding Data Reflect The “Unconstricted” Profile >>> 

 Section: APPR 993.048 .048 5750.000 6355.054 225.000 557.801 

 Header Type: AS 993.097 .046 .905 437889.90 225.000 2031.796 

 SRD: 325.000 989.048 .000 .150 .0002 3.791 -.017 

 <<< The Preceding Data Reflect The “Unconstricted” Profile >>> 

 <<< The Following Data Reflect The “Constricted” Profile >>> 

 <<< Beginning Bridge/Culvert Hydraulic Computations >>> 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: BRDGE 992.637 .647 5750.000 989.807 75.000 956.017 

 Header Type: BR 993.284 .052 5.809 180129.70 75.000 1044.086 

 SRD: 100.000 986.485 .185 .339 ****** 1.233 -.017 

 Specific Bridge Information C P/A PFELEV BLEN XLAB XRAB 

 Bridge Type 4 Flow Type 1 ------ ----- -------- -------- -------- ------- 

 Pier/Pile Code ** .9006 .000 994.860 ******** ******** ******* 

 --------------------------- ------ ----- -------- -------- -------- ------- 
120

Table 13. WSPRO water-surface profile output for Highway 12 bridge (continued). 

______________________________________________________________________________

 *** Roadway Section Located at SRD 125.000 *** 

 Section: ROAD Header Type: XR 

 <<< Embankment Is Not Overtopped >>> 

 WSEL VHD Q AREA SRDL LEW 

 EGEL HF V K FLEN REW 

 CRWS HO FR # SF ALPHA ERR 

 --------- ------ ---------- ---------- --------- --------- 

 Section: APPR 993.401 .040 5750.000 6876.729 174.000 556.146 

 Header Type: AS 993.441 .093 .836 487502.70 222.813 2040.433 

 SRD: 325.000 989.048 .066 .131 .0002 3.638 .013 

 Approach Section APPR Flow Contraction Information 

 M( G ) M( K ) KQ XLKQ XRKQ OTEL 

 -------- -------- --------- -------- -------- -------- 

 .940 .665 162782.2 971.035 1059.104 993.377 

 -------- -------- --------- -------- -------- -------- 

 <<< End of Bridge Hydraulics Computations >>> 
121

Table 14. Velocity distribution from HP record for unconstricted flow 

 at approach section, Highway 12 bridge. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 BRIDGE 5359 

 

 TH 12 OVER POMME DE TERRE RIVER 

 **** EXISTING BRIDGE BEFORE SCOUR, 1997 FLOOD ANALYSIS   

 

 ** * Beginning Velocity Distribution For Header Record APPR *** 

 SRD Location: 325.000 Header Record Number 5 

 Water-Surface Elevation: 993.050 Element # 1 

 Flow: 5750.000 Velocity: .90 Hydraulic Depth: 4.313 

 Cross-Section Area: 6357.50 Conveyance: 438116.30 

 Bank Stations -> Left: 557.794 Right: 2031.836 

 X STA. 557.8 668.9 742.2 819.3 893.2 961.6 

 A( I ) 463.3 384.6 402.5 386.9 359.0 

 V( I ) .62 .75 .71 .74 .80 

 D( I ) 4.17 5.25 5.22 5.23 5.25 

 X STA. 961.6 977.9 987.5 997.2 1006.8 1016.5 

 A( I ) 124.5 96.1 96.8 95.7 95.7 

 V( I ) 2.31 2.99 2.97 3.00 3.00 

 D( I ) 7.64 9.96 9.96 9.96 9.96 

 X STA. 1016.5 1026.4 1047.5 1079.6 1121.4 1182.3 

 A( I ) 98.3 152.2 211.7 270.9 383.8 

 V( I ) 2.93 1.89 1.36 1.06 .75 

 D( I ) 9.87 7.21 6.60 6.48 6.30 

 X STA. 1182.3 1249.8 1329.2 1451.5 1635.2 2031.8 

 A( I ) 410.3 442.1 517.5 593.6 772.2 

 V( I ) .70 .65 .56 .48 .37 

 D( I ) 6.08 5.57 4.23 3.23 1.95 
122

Table 15. Velocity distribution from HP record for constricted flow 

 at approach section, Highway 12 bridge. 

______________________________________________________________________________

 ************************* W S P R O *************************** 

 Federal Highway Administration - U.S. Geological Survey 

 Model for Water-Surface Profile Computations. 

 Input Units: English / Output Units: English 

 *---------------------------------------------------------------* 

 BRIDGE 5359 

 

 TH 12 OVER POMME DE TERRE RIVER 

 **** EXISTING BRIDGE BEFORE SCOUR, 1997 FLOOD ANALYSIS   

 

 ** * Beginning Velocity Distribution For Header Record APPR *** 

 SRD Location: 325.000 Header Record Number 5 

 Water-Surface Elevation: 993.400 Element # 1 

 Flow: 5750.000 Velocity: .84 Hydraulic Depth: 4.632 

 Cross-Section Area: 6875.20 Conveyance: 487355.20 

 Bank Stations -> Left: 556.150 Right: 2040.409 

 X STA. 556.2 666.3 740.6 816.6 892.3 960.4 

 A( I ) 488.6 415.4 423.2 422.7 381.0 

 V( I ) .59 .69 .68 .68 .75 

 D( I ) 4.43 5.59 5.57 5.58 5.60 

 X STA. 960.4 978.1 988.2 998.4 1008.5 1018.5 

 A( I ) 139.4 104.3 105.1 103.8 103.8 

 V( I ) 2.06 2.76 2.74 2.77 2.77 

 D( I ) 7.87 10.31 10.31 10.31 10.31 

 X STA. 1018.5 1029.4 1056.2 1089.4 1139.5 1202.8 

 A( I ) 107.8 192.5 229.9 339.5 417.2 

 V( I ) 2.67 1.49 1.25 .85 .69 

 D( I ) 9.91 7.19 6.92 6.78 6.59 

 X STA. 1202.8 1272.2 1356.7 1489.1 1673.2 2040.4 

 A( I ) 441.1 470.3 568.1 626.2 795.2 

 V( I ) .65 .61 .51 .46 .36 

 D( I ) 6.36 5.56 4.29 3.40 2.17 
123

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