Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

Hydrodynamic Forces on

Inundated Bridge Decks

Publication No. FHWA-HRT-09-028 MAY 2009
FOREWORD 

The Hydrodynamic Forces on Inundated Bridge Decks Study described in this report was 

conducted at the Federal Highway Administration’s (FHWA) Turner-Fairbank Highway 

Research Center (TFHRC) J. Sterling Jones Hydraulics Laboratory and at the Department of 

Energy’s Argonne National Laboratory’s (Argonne) Transportation Research and Analysis 

Computing Center (TRACC). The study was in response to a request of several State 

transportation departments asking for new design guidance to predict hydrodynamic forces on 

bridge decks for riverine conditions. The study included experiments (physical modeling) at the 

TFHRC J. Sterling Jones Hydraulics Laboratory and High Performance Computational Fluid 

Dynamics (CFD) modeling at the Argonne National Laboratory. This report will be of interest to 

hydraulic engineers and bridge engineers who are involved in estimating loads for bridge decks. 

This report is being distributed as an electronic document through the TFHRC Web site 

(www.tfhrc.gov). 

Cheryl Allen Richter 

Acting Director, Office of Infrastructure 

 Research and Development 

Notice 

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

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

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

regulation. 

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

names appear in this report only because they are considered essential to the object of the 

document. 

Quality Assurance Statement 

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

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

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

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement. 
TECHNICAL REPORT DOCUMENTATION PAGE 

1. Report No. 

FHWA-HRT-09-028 

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

N/A 

4. Title and Subtitle 

Hydrodynamic Forces on Inundated Bridge Decks 

5. Report Date 

May 2009 

6. Performing Organization Code 

N/A 

7. Author(s) 

Kornel Kerenyi, Tanju Sofu, and Junke Guo 

8. Performing Organization Report 

No. 

10. Work Unit No. (TRAIS) 

N/A 

9. Performing Organization Name and Address 

GKY and Associates, Inc. University of Nebraska 

4229 Lafayette Center Dr., Suite 1850 312 N. 14th Street 

Chantilly, VA 20151 Alexander Building West 

 Lincoln, NE 68588-0430 

11. Contract or Grant No. 

DTFH-04-C-00037 

13. Type of Report and Period 

Covered 

Laboratory Report 

November 2005–June 2008 

12. Sponsoring Agency Name and Address 

Office of Infrastructure Research and Development 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101-2296 14. Sponsoring Agency Code 

FHWA Task Order 16 

15. Supplementary Notes 

Contracting Officer’s Technical Representative (COTR): Kornel Kerenyi, HRDI-07 

Lars Hanson summarized the results and described the procedures used in this study. Afzal Bushra, Zhaoding 

Xie, and Oscar Berrios provided invaluable assistance with instrumentation, data collection, and analysis. 

16. Abstract 

The hydrodynamic forces experienced by an inundated bridge deck have great importance in the design of bridges. 

Specifically, the drag force, lift force, and the moment acting on the bridge deck under various levels of inundation and a 

range of flow conditions influence the design and construction of the bridge. This report explores the forces acting on bridges

in two ways. First, through physical experimentation on scaled-down bridge deck models tested in a flume and then with 

computational fluid dynamics (CFD) simulation models. Three bridge deck prototypes were used for the experimentation: a 

typical six-girder highway bridge deck, a three-girder deck, and a streamlined deck designed to better withstand the hydraulic 

forces. The forces (expressed as nondimensional force coefficients) on each of the bridge deck shapes were measured in the 

laboratory with an ultra-precise force balance under a range of inundation scenarios (including partial inundation) and at four

different velocities characterized by Froude numbers in the range of 0.16 to 0.32. 

CFD modeling was performed using both the Fluent® and STAR-CD® software packages. The CFD models were calibrated to 

the flow conditions of the six-girder bridge, and these same conditions were used for the other two bridge shapes. A range of 

model options were tested including two-dimensional versus three-dimensional models, different mesh resolutions, boundary 

conditions, and turbulence models; their effect on the accuracy of results and processing efficiency were noted. 

Fitting equations were generated to create an envelope around the experimental data and create design charts for each of the 

bridge types and force coefficients. 

Finally, the CFD models, though they can match some of the general behavior of experimental models in terms of the 

relationship between inundation ratio and force measured at the bridge, do not yet faithfully reproduce the critical values of 

the hydraulic forces and show very little response to velocity. The CFD simulations seem promising as a method to test bridge 

designs, but more research is needed before complex designs can be tested wholly in the CFD realm. However, the design 

charts from the experimental results should be a valuable tool for the bridge designer in a wide range of design applications. 

17. Key Words 

Hydrodynamic Forces, Force Balance, CFD, 

Inundated bridge decks 

18. Distribution Statement 

No restrictions. This document is available to the public 

through the National Technical Information Service 

(NTIS), Springfield, VA 22161. 

19. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 

44 

22. Price 

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized. 
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. Appropr e iate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
iii 

TABLE OF CONTENTS 

1. INTRODUCTION..................................................................................................................... 1

2. THEORY AND APPROACH.................................................................................................. 3

THEORY OF FORCES ACTING ON INUNDATED BRIDGE DECKS......................... 3

HYDRAULIC FORCE MEASUREMENT EXPERIMENTAL SETUP .......................... 5

PIV SETUP.............................................................................................................................. 8

COMPUTING AND TRACC ................................................................................................ 9

3. COMPUTATIONAL FLUID DYNAMICS SETUP AND VALIDATION ....................... 11

STAR-CD®............................................................................................................................. 11

Numerical Configuration .................................................................................................. 11 

FLUENT® .............................................................................................................................. 15

Numerical Configuration .................................................................................................. 15 

SIMULATION SUMMARY................................................................................................ 17

4. RESULTS ................................................................................................................................ 19

COMPARISON OF SIMULATED (FLUENT®) RESULTS WITH PIV 

EXPERIMENTAL DATA.................................................................................................... 19

RESULTS FOR THE SIX-GIRDER BRIDGE DECK ..................................................... 22

RESULTS FOR THE THREE-GIRDER BRIDGE DECK.............................................. 25

RESULTS FOR THE STREAMLINED DECK ................................................................ 28

FITTING EQUATIONS AND COEFFICIENT VALUES............................................... 31

5. DECK FORCE CALCULATION EXAMPLES.................................................................. 33

6. CONCLUSION ....................................................................................................................... 37
iv 

LIST OF FIGURES 

Figure 1. Diagram. Scaled six-girder bridge deck model used in assessments of drag and lift 

forces and the nomenclature for bridge deck and flume dimensions.............................................. 3 

Figure 2. Equation. Inundation ratio............................................................................................... 4 

Figure 3. Equation. Froude number................................................................................................ 4 

Figure 4. Diagram. Definition sketch of forces acting on bridge deck........................................... 4 

Figure 5. Equation. Drag coefficient............................................................................................... 4 

Figure 6. Equation. Lift coefficient................................................................................................. 5 

Figure 7. Equation. Moment coefficient......................................................................................... 5 

Figure 8. Diagram. Dimensions of the six-girder bridge deck ....................................................... 6 

Figure 9. Diagram. Dimensions of the three-girder bridge deck .................................................... 6 

Figure 10. Diagram. Dimensions of the streamlined bridge deck .................................................. 6 

Figure 11. Diagram. Side view of railings with dimensions for the six-girder and three-girder 

bridge decks .................................................................................................................................... 6 

Figure 12. Diagram. Test section of deck force analyzer showing strain gauge configuration...... 7 

Figure 13. Photo. Deck force analyzer system at the TFHRC hydraulics lab ................................ 8 

Figure 14. Image. Comparison of flow fields for the 2-D and 3-D models for the STAR-CD®

simulations .................................................................................................................................... 12 

Figure 15. Image. Segment of bridge used for 3-D model ........................................................... 13 

Figure 16. Diagram. Coarse meshes for STAR-CD® simulation ................................................. 14 

Figure 17. Diagram. Partially refined meshes for STAR-CD® simulation................................... 14 

Figure 18. Model. A rendering of the 3-D six-girder bridge deck in Fluent® .............................. 16 

Figure 19. Image. Velocity profile from the 2-D Fluent® model.................................................. 16 

Figure 20. Image. Velocity profile from the 3-D Fluent® model.................................................. 17 

Figure 21. Diagram. Fluent® unstructured mesh in the vicinity of the bridge.............................. 17 

Figure 22. Image. Velocity profile from the Fluent® k-ε CFD model for the six-girder bridge... 20 

Figure 23. Image. PIV velocity profile for the six-girder bridge.................................................. 20 

Figure 24. Image. Velocity profile from the Fluent® k-ε CFD model for the three-girder 

bridge ............................................................................................................................................ 21 

Figure 25. Image. PIV velocity profile for the three-girder bridge .............................................. 21 

Figure 26. Image. Velocity profile from the Fluent® k-ε CFD model for the streamlined 

bridge ............................................................................................................................................ 22 

Figure 27. Image. PIV velocity profile for the streamlined bridge............................................... 22 

Figure 28. Graph. Drag coefficient versus inundation ratio for the six-girder bridge.................. 23 

Figure 29. Graph. Lift coefficient versus inundation ratio for the six-girder bridge.................... 24 

Figure 30. Graph. Moment coefficient versus inundation ratio for the six-girder bridge ............ 25 

Figure 31. Graph. Drag coefficient versus inundation ratio for the three-girder bridge............... 26 

Figure 32. Graph. Lift coefficient versus inundation ratio for the three-girder bridge................. 27 

Figure 33. Graph. Moment coefficient versus inundation ratio for three-girder bridge............... 28 

Figure 34. Graph. Drag coefficient versus inundation ratio for streamlined bridge deck ............ 29 

Figure 35. Graph. Lift coefficient versus inundation ratio for the streamlined bridge................. 30 

Figure 36. Graph. Moment coefficient versus inundation ratio for the streamlined bridge ......... 30 

Figure 37. Equation. Drag coefficient fitting equation for three- and six-girder bridges............. 31 

Figure 38. Equation. Lift coefficient fitting equation for three- and six-girder bridges............... 31 

Figure 39. Equation. Moment coefficient fitting equation for all bridge types ............................ 31 
v 

Figure 40. Equation. Drag coefficient fitting equation for the streamlined bridge....................... 31 

Figure 41. Equation. Lift coefficient fitting equation for the streamlined bridge......................... 31 

Figure 42. Equation. Upper fitting equation for drag coefficient as a function of h* .................. 34 

Figure 43. Equation. Total drag force per unit length on the example six-girder bridge for 

the 1,000-year flood...................................................................................................................... 34 

Figure 44. Equation. Velocity, v, at h*crit ..................................................................................... 35 

LIST OF TABLES 

Table 1. Fitting equation coefficient values for six-girder (6-g) and three-girder (3-g) bridges.. 32 

Table 2. Fitting equation coefficient values for the streamlined bridge ....................................... 32 

Table 3. Critical coefficient values by bridge type ....................................................................... 32 

Table 4. Bridge example dimensions............................................................................................ 33 

Table 5. Flow conditions for example design floods.................................................................... 33 

Table 6. High and low force coefficients for the two example floods.......................................... 34 
vi 

 LIST OF ACRONYMS AND ABBREVIATIONS 

2-D Two-dimensional 

3-D Three-dimensional 

ADV Acoustic Doppler Velocimeter 

Argonne Department of Energy’s Argonne National Laboratory 

CFD Computational fluid dynamics 

FHWA Federal Highway Administration 

LES Large Eddy Simulation 

PISO Pressure-Implicit Split-Operator 

PIV Particle Image Velocimetry 

PVC Polyvinyl Chloride 

R&D Research and development 

RANS Reynolds-averaged Navier-Stokes 

SIMPLE Semi-Implicit Method for Pressure Linked Equations 

TFHRC Turner-Fairbank Highway Research Center 

TRACC Transportation Research and Analysis Computing Center 

VoF Volume of fluid 

LIST OF SYMBOLS 

CD Drag coefficient 

CL Lift coefficient 

CM Moment coefficient 

FD Drag force 

FL Lift force 

Fr Froude number 

g Gravitational acceleration 

h* Inundation ratio 

h*crit Critical Inundation ratio resulting in maximum forces 

hb Height from bottom of flume to bottom of bridge 

hu Height of water from bottom of flume 

k-ε Turbulence model 

L Bridge length 

Mcg Moment about the center of gravity 

Re Reynolds number 

s Deck thickness 

V, v Free stream velocity 

W Bridge width 

ρ Fluid density (or water density)
1 

1. INTRODUCTION 

When a bridge crossing a waterway is partially or entirely submerged during a flood event, its 

deck may be subjected to significant hydrodynamic loading. The proper estimation of loading 

exerted by the flow on the structure is important for design and evaluation of vulnerability. This 

report uses a combination of reduced scale experiments and computer modeling to investigate 

the forces on inundated bridges. 

The analysts at Argonne National Laboratory (Argonne), a U.S. Department of Energy 

laboratory, are working in collaboration with the researchers at the Federal Highway 

Administration’s (FHWA) Turner-Fairbank Highway Research Center (TFHRC) to study 

computational fluid dynamics (CFD) techniques for simulating open channel flow around 

inundated bridges. The reduced scale experiments conducted at the TFHRC J. Sterling Jones 

Hydraulics Laboratory established the foundation of validated computational models of the same 

phenomena. The overall objectives of the study for which this report is based was to investigate 

the forces on inundated bridges and to establish validated computational practices to address the 

research needs of the transportation community in bridge hydraulics. 

Bridges are a critical component of the nation’s transportation network. Evaluation of a 

bridge’s stability during and after flooding events, including the structural response of the 

bridge, is critical to highway safety. During a flood or tsunami, highway bridges over large 

waterways may become partially or completely submerged. Flood flows add significant 

hydrodynamic loading on bridges, resulting in possible shearing or overturning of the bridge 

deck and failure of the bridge superstructures. Traditionally, bridge analysts and designers 

have relied on expensive scaled experiments to provide estimates of the flow field and 

structural response. With the rapid development of supercomputing technology, commercial 

CFD code provides a quick, economic way to study these systems. The availability of 

parallel computers and analysis capabilities of commercially available software provides an 

opportunity to shift these evaluations into the CFD domain. When validated using the broad 

experimental database, the use of CFD simulations will allow expanded parametric analysis 

and provide a means of directly evaluating the effects of scaling. 

The general external flow characteristics of a submerged body depend on the shape of the body. 

Generally, streamlined bodies (i.e., airfoils, streamlined cars) have little influence on the fluid 

around them compared with the effect from blunt bodies (i.e., triangle shapes and square-bodied 

bridge decks). The drag and lift on a bridge deck depend on many variables, notably the height 

of the free-surface level in relation to the bridge deck and the Reynolds number (Re) or Froude 

number (Fr), both of which describe the amount of turbulence in the flow and the degree to 

which the flow is critical. There are different mechanisms that have to be considered between the 

partially inundated case and completely inundated case. For a partially inundated deck, accurate 

estimates of hydrodynamic loading must take into account not only the forces from the upstream 

flow field but also the influence of free surface due to the hydrodynamic force from waves. For 

the completely inundated deck neglecting the effect of wave forces, the mean flow field 

upstream dominates the hydrodynamic loading on the bridge deck. 
2 

For a body moving through a fluid or a body immersed in a moving fluid, the interaction 

between the body and the fluid surrounding it produces forces at the fluid-body interface. The 

forces acting parallel to the free-stream direction due to the influence of viscosity may be called 

wall shear stresses, and the force acting normal to the free-stream direction due to pressure may 

be called normal stresses. The resultant force of shear stress and pressure distribution in the 

velocity’s direction is termed as drag, and the resultant force normal to the direction of velocity 

is termed lift. 

As flow separates from the leading edge corner for a bridge deck, large numbers of vortices form 

at different scales along the surface of the bridge deck. Eventually, they shed from the trailing 

edge in a process called the vortex shedding phenomenon. The center of a vortex, or a vortex 

core, has a local minimum pressure. Thus, the formation and development of the vortices tend to 

dominate the progression of the drag, lift, and moment on the surface of the bridge deck. 

Different deck geometries submerged by the flow have different flow fields and distributions of 

vortices along the surface of the decks leading to the alterations of forces or the force coefficients 

with the change in the geometry. 

The applicability of commercial CFD software to predict flow field and evaluate drag and lift 

forces is being investigated. CFD provides a prediction of fluid flow by means of numerical 

modeling and software tools. It enables scientists and engineers to perform experiments (i.e., 

computer simulations) in virtual flow laboratory and significantly reduces the amount of 

experimentation and overall cost. CFD is a highly interdisciplinary research area that lies at the 

interface of physics, applied mathematics, and computer science. 

The CFD-based simulations can be used for a range of hydraulics research, including the 

assessment of lift and drag forces on flooded bridge decks, shape optimizations to minimize 

pressure scour and pier erosion, analysis of sediment transport and its influence on scour, 

evaluation of active or passive countermeasures for damage mitigation, and consideration of 

environmental issues such as fish passage through culverts. Currently, the applicability of the 

commercial CFD software for prediction of these phenomena is being investigated, and the 

agreement between the code predictions and experimental data from TFHRC flumes is presented 

in this report. 
3 

2. THEORY AND APPROACH 

The study for this report investigated the forces acting on inundated bridge decks through 

physical experimental testing and CFD modeling with two commercial software packages. Three 

types of bridges were tested for their reaction to drag forces, lift forces, and moment-generating 

forces in relation to the degree of inundation (inundation ratio). The bridge types included 

traditional six-girder and three-girder shapes, and prototype streamlined deck shapes designed to 

reduce the force load during inundation. 

The first technical task performed for this study was the analysis of drag and lift forces and 

moments for inundated bridge decks as a function of immersion height and flow rate. The 

analysis results for these prototypical open channel flow configurations were compared to the 

force balance and Particle Image Velocimetry (PIV) measurements from TFHRC tests to validate 

the corresponding CFD models. Both two-dimensional (2-D) and three-dimensional (3-D) 

models of the bridge decks with six- and three-girders were studied using Fluent® and STARCD® CFD software. A set of simulations were completed to assess the impact of various 

modeling options, including the following, while always considering accuracy of results and 

computational efficiency: 

• The influence of the free surface. 

• The effects of channel wall. 

• Mesh sensitivity study for grid independent solutions. 

• Turbulence model evaluations. 

THEORY OF FORCES ACTING ON INUNDATED BRIDGE DECKS 

In determining the structural response of the bridge decks, both the properties of the fluid and the 

configuration of the bridge were important. Consistent with the experimental methodology, the 

nomenclature for bridge and flume dimensions are illustrated in figure 1. 

W

Figure 1. Diagram. Scaled six-girder bridge deck model used in assessments of 

drag and lift forces and the nomenclature for bridge deck and flume dimensions. 

In determining the forces on bridges, the properties of the flowing stream that had the greatest 

impact were the height of the water and its velocity. The height of the water surface in relation to 

the flume dimension, the position of the bridge deck, as well as the flow velocity can be 
4 

expressed as the inundation ratio, h*. Shown in figure 2, the inundation ratio is a measure of the 

difference between the height of the free surface over the low chord of the bridge deck (hu – hb) 

divided by the deck thickness, s. Higher values of h* mean that the bridge is more submerged. 

The flow velocity can be characterized by the dimensionless Fr. Fr (figure 3) is expressed as the 

free stream velocity, v, over the square root of the depth of flow multiplied by the gravitational 

acceleration, g 

Figure 2. Equation. Inundation ratio. 

Figure 3. Equation. Froude number. 

There were three primary responses of the bridge deck to the flowing water. Figure 4 illustrates 

the forces acting on a submerged bridge deck. The drag force, FD, acting parallel to flow, affects 

the bridge deck horizontal stability and tends to “push” the bridge off its piers and abutments. 

The lift force, FL, acts vertically and perpendicular to flow and can lift the bridge deck off of its 

piers and abutments. Unevenly distributed forces on the bridge decks lead to moments about the 

center of gravity, Mcg, of the bridge deck, which can cause the bridge to overturn. 

Figure 4. Diagram. Definition sketch of forces acting on bridge deck. 

The forces of drag and lift and moments acting on the bridge deck are usually expressed in 

nondimensional coefficient forms. The definition of the coefficient of drag, CD, depends on h* 

and is shown in figure 5. The two cases correspond to a partially inundated bridge (h* < 1) and a 

fully submerged bridge (h* > 1). 

, if h* > 1; and , if h* < 1 

Figure 5. Equation. Drag coefficient. 
5 

The lift coefficient is defined in figure 6. The moment coefficient with respect to the center of 

gravity of the bridge deck is defined in figure 7. 

Figure 6. Equation. Lift coefficient. 

Figure 7. Equation. Moment coefficient. 

Here, FD and FL are the forces integrated over the surface of the bridge deck along the flow and 

perpendicular directions, respectively, where ρ is the density of water, and L is the length of the 

bridge (orthogonal to the plane of figure 1). When calculating the integrated vertical force, FL, its 

component associated with buoyancy force is excluded to be consistent with the experiments in 

which force balances were calibrated for zero lift under no-flow conditions. 

HYDRAULIC FORCE MEASUREMENT EXPERIMENTAL SETUP 

The physical experimental setup was designed to empirically determine the effect of the 

inundation ratio on the lift, drag, and moment forces for three different bridge prototypes. 

The experiments were conducted in a 12.8 m long by 0.4 m wide by 0.5 m high Plexiglas®

rectangular flume. The flume was set horizontally, and the depth of flow was controlled with an 

automatic, adjustable tailgate located at the downstream end of the flume. Flow was supplied by 

a 0.054-m3/s pumping system. The mean free-surface level was measured using ultrasonic 

sensors at two cross sections along the flume. An electromagnetic flow meter measured the 

discharge. An Acoustic Doppler Velocimeter (ADV) probe was used to measure the velocity 

distribution of the flow. 

The following three models of bridge decks were tested in the experiments: 

 Six-girder bridge deck (shown in figure 8). 

 Three-girder bridge deck (shown in figure 9). 

 Streamlined bridge deck (shown in figure 10). 
6 

Figure 8. Diagram. Dimensions of the six-girder bridge deck. 

Figure 9. Diagram. Dimensions of the three-girder bridge deck. 

Figure 10. Diagram. Dimensions of the streamlined bridge deck. 

Both the six-girder and three-girder bridge decks had railings on both sides with the dimensions 

shown in figure 11. 

Figure 11. Diagram. Side view of railings with dimensions for the six-girder 

and three-girder bridge decks. 

The six-girder bridge, representing a typical U.S. highway bridge deck shape, was constructed 

out of Polyvinyl Chloride (PVC) at a 1:40 geometrical reduction scale based on the depths, 

maximum discharges, and inundations possible in the flume. This scaled bridge deck was 

optimally sized to investigate a range of force values in flow conditions ranging from low to high 

flows, all at subcritical Fr in the upstream flow. The six-girder and three-girder bridge decks had 

the same width, but the three-girder deck had a 22 percent larger frontal area (s = 0.70 m). 
7 

The three bridges were evaluated in the same flume and under the same experimental conditions 

in order to reduce the experimental error. The experiments were conducted for four approach 

velocities ranging from 0.25 to 0.5 m/s. The flow depths, hu, in the experiments were always kept 

at 0.25 m. Under these conditions, the Fr varied within the range of 0.16 to 0.32 in response to 

the changing velocities. The submergence of the bridge deck within the 0.25-m flow depth was 

varied from slight submergence of the deck girders to complete overtopping of the bridge deck, 

corresponding to an inundation ratio of h* = 0.29 to roughly 5.0 (for the streamlined bridge). 

The measurement of forces on the bridge deck was performed by a deck force analyzer system, 

which was constructed for this study (figure 12 and figure 13). This transducer system was 

designed to measure forces induced by the bridge deck model simultaneously in the x-direction 

(Drag) and z-direction (Lift) by use of electric strain gauges. The bridge deck model was 

mounted on a bracket and positioned in the flume. The bracket was attached at the platform of 

the system by four ball-beared pendulums which were connected with the platform by four flat 

springs. This allowed it to move back and forth in the direction of flow as well as vertically. 

Motion in each direction was resisted by two pairs of flat springs. The tension in these springs 

was detected by strain gauges, which generated an electrical signal proportional to the forces 

upon the bridge deck model in the x-direction (Drag) and z-direction (Lift). 

Figure 12. Diagram. Test section of deck force analyzer showing strain 

gauge configuration. 
8 

Figure 13. Photo. Deck force analyzer system at the TFHRC hydraulics lab. 

The transducer was dead-load calibrated prior to use and was checked periodically during data 

collection. The calibration was performed by running an experiment with still water for each 

inundation ratio h*. The output of the acquisition data was set as the zero-reference value. The 

mean drag and lift forces of an experimental run were evaluated by subtracting the corresponding 

zero-reference value. 

PIV SETUP 

A secondary experimental setup was built to examine the flow field around the bridges in detail 

using PIV, which is a noninvasive measurement technique to visualize flow distributions. The 

technique involves adding microscopically small particles that are highly reflective to the flow 

and then using a laser to illuminate a thin layer of the flow so that only the particles in that light 

sheet reflected the laser’s light. Using cameras pointed with different angles toward the light 

sheet, researchers can capture images of the moving particles. In order to measure the velocity, at 

least two exposures were needed. They could be recorded on one or several frames. The frames 

were split in a large number of interrogation areas, or tiles. It was then possible to calculate a 

displacement vector for each tile with the help of signal processing (an auto correlation/cross 

correlation algorithm). This was converted to a velocity using the time between image exposures. 
9 

The PIV experiment was conducted in a smaller flume with bridge models scaled down by a 

factor of 1.5. The experiments were conducted for V = 250 mm/s with hb = 50 mm, 101 mm, and 

116 mm and the depth of the flow, hu, as 170 mm. 

COMPUTING AND TRACC 

Besides establishing credible experimental data on the response of bridge decks to various 

forces, a major objective of this study was to establish validated computational practices to 

address the transportation community’s research needs in bridge hydraulics. Traditionally, bridge 

hydraulics work relies on scaled experiments to provide measurements for flow field, which is 

the velocity and turbulence of a fluid as functions of position and time. Now, however, parallel 

computers and commercially available software provide an opportunity to shift the focus of these 

evaluations to the CFD domain. After being validated using the data from a limited set of 

experiments, high-fidelity CFD simulations can be used to expand the range and scope of 

parametric studies. The CFD simulations may be used in the future to predict the effects of 

scaling by studying differences between the reduced-scale experiments and full-scale bridges. 

In this study, the reduced-scale experiments conducted at the TFHRC hydraulics laboratory 

established the foundations of a CFD-based simulation methodology. Researchers at Argonne 

and TFHRC worked together to study CFD techniques for simulating open channel flow around 

inundated bridges. The computationally intensive CFD modeling programs require enormous 

computing power. 

Argonne is leading the initiative to establish the high-performance computing center, 

Transportation Research and Analysis Computing Center (TRACC), to pursue research and 

development (R&D) programs. TRACC is a general purpose advanced computing and 

visualization facility available to the transportation community for a broad spectrum of 

applications. The TRACC components include high-performance computing, visualization, and 

networking systems. 

Argonne analysts provided technical support to researchers at TFHRC on CFD simulations for 

this study. Two CFD software packages were investigated for their utility in modeling open 

channel flow around bridges for this study. The STAR-CD® and Fluent® programs were used in 

this study to build models of the six- and three-girder bridges and to run simulations. The models 

were calibrated using the results of the six-girder reduced scale experiments and PIV results. The 

three-girder bridge deck was simulated using the same calibration settings as the six-girder 

models to serve as a “blind-test” of validity. The following chapter describes the two models and 

their numerical configuration and evaluation for this study. 

11 

3. COMPUTATIONAL FLUID DYNAMICS SETUP AND VALIDATION 

Two commercial CFD programs were configured to model the forces on the bridge decks in 

simulations: STAR-CD® and Fluent®. Building the CFD model involved selecting the mesh 

resolution, simulation algorithm, boundary conditions including the air-water interface 

properties, and turbulence model. Additionally, the model simulations investigated the effect of 

Fr and inundation ratio on the three bridge decks in the same manner as the experimental tests. 

In addition to testing the ability of the models to replicate the experimental model results, the 

CFD simulation had a secondary objective to determine the robustness and efficiency of the 

models. Due in part to scheduling constraints, the CFD model was only calibrated using the sixgirder bridge deck experimental results. The same flow configurations were applied to the 

models of the three-girder and streamlined bridge decks. By comparing these “blind” simulation 

results with the experimental results, an effective test of the CFD models’ general applicability 

was created. The CFD simulations were also conducted with attention to the efficiency of 

computation. Even with the TRACC facility, the high computational intensity of CFD modeling 

demands an efficient model setup to ensure a manageable runtime. 

The setup and chosen configuration of the two models are described below. 

STAR-CD® 

Cd-adapco’s STAR-CD® was used by researchers at the University of Nebraska at Omaha to 

simulate the forces acting on the three bridge decks. Like many commercial CFD models, 

STAR-CD® computes the overall flow of the fluid and associated forces by breaking the 

modeling environment into a mesh of many tiny cells. Then, the enormously complex problems 

of continuity and energy can be solved at the individual cell level. This project’s STAR-CD®

approach was based primarily on the structured hexahedral mesh option for modeling. 

The STAR-CD®’s solver choices include well established pressure-based solution algorithms 

like the Pressure-Implicit Split-Operator (PISO) and the Semi-Implicit Method for Pressure 

Linked Equations (SIMPLE). Although PISO is a more robust scheme with predictor/corrector 

steps, the alternative SIMPLE scheme can also predict accurate results for approximately half the 

computational cost. Therefore, the SIMPLE solver was used for the bridge deck simulations. 

Numerical Configuration 

To set up the numerical configuration of the simulation, the bridge dimensions and the flow 

conditions had to be specified. 

Since water flows perpendicular to a bridge and the modeled bridge decks had regular cross 

sections along their length, several simplifications of the CFD model were possible and were 

used to test for their effectiveness in matching the experimental results and reducing 

computational time. 

Instead of modeling the whole bridge deck at the exact dimensions of the experimental setup, the 

CFD model could be constructed with just a small portion of the bridge deck. The simplest 
12 

model would be a true 2-D model with the bridge modeled in cross section with the thickness of 

only a single computational cell. To assess the validity of this approach, the 2-D model’s flow 

field was compared to the flow field of a 3-D model that replicated the experimental setup. 

Instead of a full 3-D model, a half model of the bridge deck with one side along a flume wall and 

what was once the middle of the bridge along a symmetry boundary was used. The flow field 

results for the 3-D model along this symmetry plane and the 2-D model are displayed in figure 

14. The flow fields indicate sufficient agreement that a 2-D model could be sufficient for the 

computation of the forces and moments. 

The single-cell 2-D model shown in figure 14 does not provide a perfect representation of the 

six- and three-girder bridge due to the patterns of posts supporting the railings and voids. To 

account for these differences, an extended 2-D model was created so that one edge of the deck 

was at the midpoint of a post, and the other edge was at the midpoint of one of the void spaces. 

This situation is indicated by the yellow stripe in figure 15. By placing symmetry boundaries on 

both edges, a quasi-infinite bridge deck was simulated. The railings were simulated, as in the 

physical experiments, on both the leading and trailing edge of the bridge deck. 

Figure 14. Image. Comparison of flow fields for the 2-D and 3-D models for 

the STAR-CD® simulations. 
13 

Figure 15. Image. Segment of bridge used for 3-D model. 

In addition to the bridge model specification, the general simulation mesh had to be specified. 

Increasing the mesh resolution increased computational time, so the mesh size had to be chosen 

carefully. Two mesh sizes were simulated. The “coarse grid,” shown in figure 16 and figure 17, 

had a cell size of 3.5 mm by 2.5 mm by 2.5 mm. The “partially refined” mesh increased the mesh 

resolution near the bridge by dividing each coarse cell into eight smaller cells by halving the cell 

along each dimension. 
14 

Figure 16. Diagram. Coarse meshes for STAR-CD® simulation. 

Figure 17. Diagram. Partially refined meshes for STAR-CD® simulation. 

The parameters of the fluid itself made up the final component of the simulation. The most 

important considerations were in the treatment of air-water interface and in the turbulence model 

selected. Advances in computational fluid mechanics have provided the basis for insight into the 

dynamics of multiphase flows. Since the free surface in open channel flow could be an important 

feature of the flow field influencing the parameters of interest defined above, the Volume of 

Fluid (VoF) computational technique was pursued to capture its form accurately. The VoF 

solution methodology was well suited for free-surface flow problems where the two fluids were 

separated by a well defined, sharp interface. 

The VoF methodology deals with a single continuum whose properties vary according to its 

composition as derived from solution of transport equations for the component fluids. These 

component fluids are assumed to be immiscible and labeled as the base, light fluid, (in this study, 

air) and the heavy fluid (water). The density of each fluid component is assumed constant, and, 

when both components are present in a computational cell, they share the field variables 

(velocity and pressure) computed for the fluid mixture. The result is that water depth can vary 

slightly along the flow path, as it would in reality. 

Since the form of the free surface was generally stable and reasonably flat in this study, 

especially for high inundation ratios (h* > 1), a separate set of calculations in which the top 
15 

surface is at h = hu, was modeled as a flat “fixed surface” in STAR-CD®. In these single-phase 

(water-only) simulations, the top surface was modeled either as a slip-wall or free-stream 

boundary. As expected, the results of these steady-state “fixed-surface” simulations were 

generally consistent with the results of the “free-surface” simulations. Since the calculations with 

the fixed-surface models did not require a long computation time, many of the turbulence model 

evaluations and grid sensitivity studies were performed with the fixed-surface models in 

STAR-CD®. 

These turbulence models were the final major consideration in the modeling setup. Undoubtedly, 

the high Re flows approaching the bridge decks could be considered turbulent. Turbulent flow is 

highly unsteady and irregular. The available computer power was not sufficient to represent all 

of the eddies ranging from the smallest scale (corresponding to dissipative motions) to the largest 

scale (responsible for most of the momentum transport in a high speed flow). Therefore, 

turbulence models were used to describe turbulence based on some simplifying assumptions. 

STAR-CD® provides a range of turbulence modeling options, including several based on the two 

equation Reynolds-averaged Navier-Stokes (RANS) model and even higher order models. 

Ultimately, the high Re variant of the k-ε model was selected for the detailed simulations. 

FLUENT® 

ANSYS, Inc.’s ® Fluent® CFD modeling software was also used for modeling the deck forces on 

the bridge decks. Though Fluent® and STAR-CD® have many similar capabilities, they were 

used differently for this study. The following section describes the numerical configuration for 

Fluent®. 

Numerical Configuration 

Though there are many mesh options, the Fluent® experiments for this study used an 

unstructured mesh as opposed to the structured hexahedral mesh in STAR-CD®. The 

unstructured mesh captures the flow around complex shapes like the bridge decks. The 

Tet/Hybrid mesh was chosen, which is composed primarily of tetrahedrons. Figure 18 shows 

a 3-D model of the six-girder bridge rendered in this mesh. 
16 

Figure 18. Model. A rendering of the 3-D six-girder bridge deck in Fluent®. 

The simulations using Fluent® were performed for three different shapes of bridge deck (i.e., sixand three-girder bridge deck models with railings on both sides and streamlined bridge deck). As 

in the STAR-CD® simulations, 2-D and 3-D models were built and tested for the six-girder 

bridge deck. The 2-D model was simplified to have a solid wall instead of a railing, but the 3-D 

model was modeled exactly as in the experiments. The velocity maps in figure 19 and figure 20 

show that there was general agreement between the 2-D and 3-D models but that the 

3-D model captured more detail and was less affected by the railings. 

Figure 19. Image. Velocity profile from the 2-D Fluent® model. 
17 

Figure 20. Image. Velocity profile from the 3-D Fluent® model. 

The simulation mesh was also unstructured and Tet/Hybrid. All simulations were performed with 

a mesh that was finer in the vicinity of the bridge and coarser farther away, as shown in 

figure 21. All of the simulations were also performed using the VoF transient multiphase model 

with both air and water. Since the free surface was an important feature of open channel flow, all 

of the Fluent® simulations were run for the two-phase model with air and water. 

Figure 21. Diagram. Fluent® unstructured mesh in the vicinity of the bridge. 

The Fluent® software also required selection of a turbulence model to complete the simulation. 

Two different turbulence models were used in parallel for the Fluent® simulations. The unsteady 

k-ε model was selected since it was also used in the STAR-CD® simulations. Additionally, the 

Large Eddy Simulation (LES) model was selected since it simulates large eddy motion explicitly 

based on flow direction and does not perform time-averaging as the RANS models do. The LES 

could more faithfully resolve the turbulent flow in the model but at greater computational cost. 

SIMULATION SUMMARY 

The first objective of the simulations was to provide estimates of the drag, lift, and moment 

coefficients using the same variables as in the physical experiment. That is, the simulations were 
18 

conducted for approach velocities ranging from 0.25 to 0.5 m/s (Fr varied within the range of 

0.16 to 0.32), and the simulations were run for various inundation ratios. 

The STAR-CD® simulations modeled only the six- and three-girder bridge decks, while the 

Fluent® k-ε simulations modeled all three types. The Fluent® LES simulation considered only the 

six-girder deck. The experimental data for six-girder bridge deck model were only available 

during the numerical simulations, whereas the three-girder and streamlined bridge deck 

experimental data were provided once the simulations were completed.
19 

4. RESULTS 

Through both physical experiments and CFD simulation, the forces exerted on the three bridge 

deck types were analyzed in detail. Before analyzing the performance of the CFD models in 

estimating the force values, it was important to verify qualitatively that the flow fields generated 

in the CFD models matched those produced in the PIV experiments. The first section of this 

chapter describes these comparisons. 

The following sections display the results of the experimentation and the CFD simulation on 

design charts showing the relationship between the inundation ratio (h*) and the unitless 

coefficients for drag, lift, or moments. The experimental results represent averages of three to 

five trials performed for each Fr at each inundation ratio. To further expand results, the data 

were used to calculate parameters for fitting equations which roughly bound the high- and lowforce coefficients values. These equations are shown in the drag, lift, and moment plots as blue 

lines and are also referred to as envelope curves since they generally form and envelope around 

the experimental results. The form and coefficient values of these equations are laid out in the 

final section of this chapter. 

COMPARISON OF SIMULATED (FLUENT®) RESULTS WITH PIV EXPERIMENTAL 

DATA 

Figure 22 through figure 27 show comparisons of the simulated velocity contours around the 

bridge deck to the PIV measurements from the experiments for each bridge type. 

The PIV measurements were conducted with smaller scale bridge decks, which were smaller by 

the factor of 1.5 from the bridge deck used in the evaluations of hydrodynamic forces. The flow 

conditions compared were for V = 250 mm/s with hu = 50 mm, 101 mm, and 116 mm and the 

depth of the flow, hu, as 170 mm. Each PIV image was matched with a velocity map from the 

CFD model simulated with a similar value of h*. The velocity profiles show the velocity of the 

water in the direction of flow across the bridge cross section. Red indicates high velocity, while 

orange, yellow, green, and blue represent progressively slower velocities. Darker blue areas 

indicate where the flow was not moving or moving in reverse during the experiment, possibly 

indicating the presence of a stable vortex. The flow field vectors were drawn in both maps, but 

the PIV vectors were not connected into streamlines as the CFD results were. 

In general, the velocity profiles show strong agreement between the flow fields. The three bridge 

types show noticeable differences. The three-girder bridge easily produced the most substantial 

vortex zones, with a particularly strong one just downstream of the third-girder bridge. 

Additionally, the three-girder bridge had a larger zone of no-flow than the six-girder bridge just 

downstream of the first girder. The six-girder bridge appeared to create a larger disturbance on 

the top of the bridge, however. Both of these phenomena were most easily seen in the PIV 

profiles. 

The streamlined bridge deck had a much smaller impact on the velocities than the other two deck 

shapes. As flow encountered the first lobe, there was evidence of contraction-induced 

acceleration as the velocity above and below increased slightly. The CFD and PIV results for the 

streamlined bridge differed somewhat in the location of the primary flow obstructions. The CFD 
20 

results place two small but noticeable vortices (shown in figure 26) on the downstream edge of 

both short triangular guardrails. There was virtually no reduction in flow velocity beneath the 

bridge. The PIV results, however, display a significant zone of very slow flow in the cavity on 

the underside of the streamlined deck (shown in figure 27). 

Overall, the velocities in the flow field seemed quite comparable between the CFD models and 

PIV experiments. Even minute differences in the flow fields or non-visible flow parameters, 

however, could lead to significant differences in the simulated force values. The next sections 

highlight the deck force results. 

Figure 22. Image. Velocity profile from the Fluent® k-ε CFD model for the 

six-girder bridge. 

Figure 23. Image. PIV velocity profile for the six-girder bridge. 
21 

Figure 24. Image. Velocity profile from the Fluent® k-ε CFD model for the 

three-girder bridge. 

Figure 25. Image. PIV velocity profile for the three-girder bridge. 
22 

Figure 26. Image. Velocity profile from the Fluent® k-ε CFD model for the 

streamlined bridge. 

Figure 27. Image. PIV velocity profile for the streamlined bridge. 

RESULTS FOR THE SIX-GIRDER BRIDGE DECK 

The six-girder bridge deck represented the dimensions of a typical highway bridge deck. The 

1:40 scale of the model allowed the range of Fr used in the experimentation to correspond well 

with realistic Fr for actual flood flows interacting with bridges. 

Figure 28 depicts the relationship of the inundation ratio with the drag coefficient, CD, for the 

six-girder bridge. The experimental data are shown in four series of points. The fitting equations 

are shown as lines, and the calibrated results of the STAR-CD® simulation and two Fluent®

models are presented as points. 
23 

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

Inunudation ratio, h*

Drag Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equation

STAR-CD

Fluent LES 

Fluent k-epsilon

Figure 28. Graph. Drag coefficient versus inundation ratio for the six-girder bridge. 

The drag coefficient plot shows that the drag coefficient was positive at all values of h* but that 

there was a major dip in the drag coefficient at h* around 0.5–0.8. This corresponds to a case 

when the bridge was slightly more than halfway inundated, perhaps as the water level was 

reaching the top of the girders and was beginning to transition to overtopping the deck. As the 

bridge became more inundated (h* > 1.5), the drag coefficient values leveled off to around 2. 

The fitting equations enclose the experimental data generally well over the whole range of 

inundation ratios. 

The CFD models did not show any dependence on Fr, so agreement with experimental data 

should not be overstated. The Fluent® k-ε results showed very good agreement with the 

experimental data, replicating both the shape and magnitude, but slightly overestimating the 

maximum values. The Fluent® LES results generally provided low estimates of CD and did not 

increase quickly enough in the 1.0 to 2.0 range of h*. The STAR-CD® results showed good 

agreement at high inundation ratios but did not match the shape of the experimental data below 

h* = 2. 

Figure 29 displays the lift coefficient plot for the six-girder bridge. Experimental data, fitting 

equations, and CFD simulation results are displayed in the same manner as in the drag 

coefficient plot. The experimental results revealed that the lift coefficient was negative for all the 

cases tested. A negative lift coefficient means that the flow was actually exerting a pull-down 

force on the bridge. While the effect was quite small when the water level was just barely 
24 

reaching the bottom of the girders, the lift coefficient rapidly became more negative until h*

roughly equaled 0.65. The lift coefficient slowly returned to 0 as the inundation ratio exceeded 3. 

As shown in figure 29, the fitting equations are generally representative of the experimental data 

but break down at higher inundation ratios and higher Fr. 

-2

-1.5

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5

Inundation Ratio, h*

Lift Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equations

STAR-CD

Fluent LES

Fluent k-(epsilon)

Figure 29. Graph. Lift coefficient versus inundation ratio for the six-girder bridge. 

The CFD model results did not closely follow the experimental results. The STAR-CD® results 

roughly captured the gradual increase in lift coefficient but fell below the bottom envelope curve. 

The two Fluent® models showed major departures from the experimental results’ behavior and 

showed comparatively little influence from the inundation ratio. As shown in the figure, the 

Fluent® LES results are mostly flat when the experimental results show a considerable drop near 

h* = 1. Meanwhile, the Fluent® k-ε start above the upper envelope curve but then roughly follow 

the experimental results at higher values of h*.

The moment coefficient plot is shown in figure 30. The experimental results demonstrate a 

similar shape no matter the Fr but are intriguing for their relative positions on the moment 

coefficient axis. There is little difference between the Fr = 0.16 and Fr = 0.22 cases, and both 

stay almost completely above 0. This corresponds to a moment rotating the bridge 

counterclockwise (rotating the upstream side of the bridge down and the downstream side up). 

The peak moment coefficient came during the study when the bridge was roughly halfway 

inundated, and the flow was pushing almost entirely on the first girder and thus below the center 

of gravity. At higher Fr, the effect was less pronounced. Additionally, in the 1.5–2 range for h*,
25 

the moment coefficient became negative and then stabilized for the Fr = 0.28 and Fr = 0.32 

cases, which meant the bridge would turn over in the clockwise direction. 

As shown in the figure, the CFD results demonstrate only fair agreement with the experimental 

results for the moment coefficient. Interestingly, the STAR-CD® and Fluent® k-ε models show 

moderate agreement with each other. They also show reasonable agreement with some of the 

experimental results at high inundation ratios, but the moment coefficients are near 0 anyway. 

The Fluent® LES model has results consistently lower than the experimental Fr = 0.16 case. The 

model represents the reduction in the moment coefficient and stabilization but somewhat more 

aggressively than in the experimental results. 

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.5 1 1.5 2 2.5 3 3.5

Inundation Ratio, h*

Moment Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

STAR-CD

Fluent LES

Fluent k-(epsilon)

Figure 30. Graph. Moment coefficient versus inundation ratio for the six-girder bridge. 

RESULTS FOR THE THREE-GIRDER BRIDGE DECK 

The three-girder bridge deck represented another common highway bridge design. The bridgedeck height was somewhat larger than the height for the six-girder, and the girders were 

rectangular and proportionally more massive. The increased deck height meant that the bridge 

could be tested for a narrower range of inundation ratios since the water depth remained at 

0.25 m for all of the laboratory experiments. 

Figure 31 shows the drag coefficient plot for the three-girder bridge. The experimental results 

display a shape similar to the results of the six-girder bridge, and the minimum point occurs at 
26 

roughly the same inundation ratio. The two higher Fr cases (0.28 and 0.32) tend to group fairly 

closely, as do the two lower Fr cases (0.16 and 0.22). Additionally, the drag coefficient stabilizes 

at a higher value for Fr = 0.28 than for Fr = 0.32. 

The CFD results for the three-girder bridge are also displayed on the plot in figure 31. Unlike the 

simulations for the six-girder bridge, the flow conditions in the CFD model are not calibrated 

based on the experimental results. Instead, the configuration of the six-girder model is used, and 

only the bridge deck model is changed. The STAR-CD® model results show only a slight 

response to Fr. The results, as plotted, seems to agree most closely with the higher Fr

experimental results. The Fluent® results for the unsteady, 3-D k-ε model display a general 

agreement with experimental results at lower h* values but somewhat overestimate the drag 

coefficient at higher values. 

0

0.5

1

1.5

2

2.5

0 0.511.522.5 3

Inundation Ratio, h*

Drag Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equation

STAR-CD

Fluent k-epsilon

Figure 31. Graph. Drag coefficient versus inundation ratio for the three-girder bridge. 

The lift coefficient plot in figure 32 shows a generally similar pattern to figure 29. The critical 

lift coefficient is slightly larger (more negative) for the three-girder deck. The three-girder deck 

shows a slightly more regular response to Fr as the experimental results do not cross each other 

nearly as much as in the six-girder experiments. The CFD results for both models replicate the 

lift forces fairly well, but the simulations do not capture the critical lift values. 

The moment coefficient plot (figure 33) shows a similar shape to the moment plot for the sixgirder bridge. The maximum positive moment occurs at an inundation ratio of approximately 0.8. 

The maximum moment coefficient value shown in the fitting is approximately 0.24, a reduction 
27 

from the 0.30 coefficient for the six-girder bridge. This difference may be partially explained by 

the first girder being set back further from the leading edge of the bridge deck. At high 

inundation ratios, the direction of the moment reverses and attains a maximum value of negative 

0.11 according to the fitting equations. Neither of the CFD models seems to capture the positive 

moment coefficients. The STAR-CD® model fits the experimental data reasonably well at high 

inundation ratios, but the Fluent® k-ε model shows only weak agreement and a different shape. 

-2

-1.5

-1

-0.5

0

0.5

0 0.511.522.5 3

Inundation Ratio, h*

Lift Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equations

STAR-CD

Fluent k-epsilon

Figure 32. Graph. Lift coefficient versus inundation ratio for the three-girder bridge. 
28 

-0.2

-0.1

0

0.1

0.2

0.3

0 0.511.522.5 3

Inundation ratio, h*

Moment Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equations

STAR-CD

Fluent k-epsilon

Figure 33. Graph. Moment coefficient versus inundation ratio for three-girder bridge. 

RESULTS FOR THE STREAMLINED DECK 

This section presents the results for the streamlined bridge deck. Only one CFD model, a Fluent®

k-ε model, was created for the streamlined bridge deck. As was the case with the three-girder 

deck, the flow configuration settings of the six-girder model were used for the streamlined deck 

because modeling results were not available for calibration. 

Figure 34 shows the drag coefficient plot for the streamlined bridge deck. In contrast to the 

three- and six-girder bridge decks, the drag coefficients are significantly lower for the 

streamlined decks by a factor of roughly 50 percent. Additionally, the streamlined deck appears 

much less influenced by Fr, as all four sets of experimental results are tightly bunched. The 

experimental results have difficulty capturing data between 0.5 and 1 due to the model bridge 

deck’s small profile. Nonetheless, the minimum drag coefficient value appears to occur at an 

inundation ratio of 1.2 versus roughly 0.8 for the other deck shapes. 

The CFD results match the experimental results fairly well for the drag coefficient only. The 

CFD results for the lift and moment coefficient, however, do not match very well probably 

because the flow conditions were calibrated for the larger and differently shaped six-girder deck. 

Figure 35 displays the lift coefficient plot. The data take a similar shape to the results for the 

other bridges. The critical value of roughly -1.2 represent a 20 to 25 percent reduction in pulldown force compared to the other two bridge shapes. The experimental results’ variation by Fr is 
29 

dramatic, as there appears to be a strong grouping among the Fr = 0.22, 0.28, and 0.32 scenarios 

but then a wide gap to the Fr = 0.16 scenario. 

The streamlined bridge’s moment coefficient plot (figure 36) exhibits different behavior than the 

other moment plots. Notably, the shape of the experimental results does not follow a smooth 

curve up to a peak and then a smooth decline but rather abruptly go to a peak. The peak location 

is not known precisely due to a lack of data at h* < 1, but the peak seems to occur at about h* = 1 

or less, which is interesting because the critical points of the lift and drag plots occur at 

somewhat higher values of h*. The down slope of the experimental values also appears as two 

fairly linear segments (with a change to a flatter slope at roughly h* = 2.25), rather than the 

smooth curve exhibited by the six-girder data. Overall, though, it is evident that the peak moment 

coefficient on the upper envelope curve of roughly 0.2 is about a 33 percent reduction from the 

six-girder coefficient. 

Figure 34. Graph. Drag coefficient versus inundation ratio for streamlined bridge deck. 
30 

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4

Inundation ratio, h*

Lift Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equations

Fluent k-epsilon

Figure 35. Graph. Lift coefficient versus inundation ratio for the streamlined bridge. 

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4

Inundation ratio, h*

Moment Coefficient

Exp. Fr=0.16

Exp. Fr=0.22

Exp. Fr=0.28

Exp. Fr=0.32

Fitting Equations

Fluent k-epsilon

Figure 36. Graph. Moment coefficient versus inundation ratio for the streamlined bridge. 
31 

FITTING EQUATIONS AND COEFFICIENT VALUES 

This section provides the fitting equations for the curves shown on the plots and the necessary 

coefficient values to generate each fitting curve. The equation for each force or moment 

coefficient is given as a function of h*. The upper and lower fitting curves for each individual 

plot always have the same equation form but have different coefficients. The fitting equations for 

the six-girder and three-girder bridge decks are of the same form and only need different 

coefficient values to describe their differences. The streamlined deck has a similar equation form 

to the other two bridge types for the moment equation but has a very different form for the drag 

and lift coefficient equations. Figure 37 to figure 41 show the forms of the fitting equations for 

the various bridges. 

Figure 37. Equation. Drag coefficient fitting equation for three- and six-girder bridges. 

Figure 38. Equation. Lift coefficient fitting equation for three- and six-girder bridges. 

Figure 39. Equation. Moment coefficient fitting equation for all bridge types. 

Figure 40. Equation. Drag coefficient fitting equation for the streamlined bridge. 

Figure 41. Equation. Lift coefficient fitting equation for the streamlined bridge. 

The coefficients in the fitting equations (A, B, a, etc.) are given in table 1 for the six- and threegirder bridges and in table 2 for the streamlined bridge. With these coefficient values inserted 

into the proper equation, the envelope of force coefficients can be calculated for any desired 

value of h*. The coefficient values allow the fitting equations to provide the high and low 

estimates of the force coefficient for each bridge type. In many cases, however, the designer 

needs only a single critical value of drag, lift, or moment to design the bridge. 

To estimate the critical values, the designer can simply look at the design charts for a reasonable 

guess. Table 3 lists the critical force coefficient values as calculated by the fitting equations and 

the associated value of h* (h*crit). The drag coefficient is critical at its maximum positive value. 

The fitting equations give one maximum at h* = 0, but, in reality, drag is highest when h* is 

roughly greater than 2.5 or 3. The fitting equation for drag for the streamlined deck increases 

steadily as h* increases, but, since h* would rarely exceed 5, the drag coefficient value at h* = 5, 

is reported (CD = 1.1). As in the physical experimentation results, the lift coefficient reaches its 
32 

critical value at its most negative. Unlike the drag coefficient, the critical CL value occurs near 

the transition from the partially inundated to completely inundated case (i.e., h* is close to 1). 

Table 1. Fitting equation coefficient values for six-girder (6-g) and 

three-girder (3-g) bridges. 

Equation A B a b c d f g α

6-g upper 2.7 2.7 2.15 2 0.65 1 1.4 0.03 2

6-g lower 2.7 2.7 1.75 3 0.62 0.8 1.6 -0.07 2

3-g upper 1.8 2 1.95 2 0.6 1 1.5 0 2

3-g lower 1.8 2 1.62 3 0.5 0.8 1.5 -0.11 2

Table 2. Fitting equation coefficient values for the streamlined bridge. 

Equation m n j d f g α

Streamlined upper 1.2 0.25 1.4 0.25 0.8 0.05 3

Streamlined lower 1.7 0.11 1.0 0.25 0.8 0 3

Table 3. Critical coefficient values by bridge type. 

Bridge type CD h*crit CL h*crit CM h*crit

6-g 2.15 0, > 3 -1.644 0.8025 0.2927 hi 

-0.07 lo 

0.8452

0, > 2.5

3-g 1.95 0, > 3 -1.815 0.8312 0.2453 hi 

-0.11 lo 

0.8165

0, > 2.5

Streamlined ~ 1.1 ~ 4 -1.219 1.367 0.1932 1.369
33 

5. DECK FORCE CALCULATION EXAMPLES 

This section describes how to use the results of this study to estimate the force and moment 

coefficients on a proposed bridge design. 

To calculate the force coefficient, the dimensions and elevation of the proposed bridge, the 

dimensions of the channel, and the velocity and inundation elevation of the stream at the flood of 

interest must be known. 

Suppose a small six-girder bridge is planned with dimensions as in table 4. 

Table 4. Bridge example dimensions. 

Dimension Value 

Deck thickness (s) 2.5 m

Length (L) 280 m

Width (W) 11.2 m

Low chord elevation 106 m 

(NAVD88)

The bridge is planned over a small stream with an enormous range of flow. Suppose that the 

stream has a channel bottom elevation of 102 m at the crossing. The 50-year flood rises to an 

elevation of 105.9 m and is passed by the bridge. The 100-year flood has an elevation of 107.5 m 

and has an average stream velocity of 2.3 m/s. The 1,000-year flood rises to elevation 110 m and 

has an average velocity of 3.2 m/s. 

Using these dimensions, hu, hd, h*, and Fr can be calculated. Table 5 calculates these flow 

conditions for the two floods. 

Table 5. Flow conditions for example design floods. 

100-year flood 1,000-year flood 

hu = 107.5 – 102 = 5.5 m hu = 10 m 

hb = 106 – 102 = 4 m hb = 4 m 

h* = (hu – hd)/s = (5.5 – 4)/2.5 = 0.6 h* = 2.4 

Fr = 2.3/(9.8 x 5.5) 0.5 = 0.313 Fr = 0.323 
34 

Supposing the bridge is a six-girder bridge deck, the force coefficients can be calculated for the 

two floods using the equations in figure 37 to figure 39 and the appropriate coefficients in 

table 3. 

For example, to compute the drag coefficient for the 100-year flood, take the equation in 

figure 37 and apply the six-girder, high estimate coefficients to get CD as a function of h* as in 

figure 42. 

Figure 42. Equation. Upper fitting equation for drag coefficient as a function of h*.

For the 100-year flood, substitute h* = 0.6 into the 6-g upper equation (figure 42) to find the high 

estimate of CD equals 1.403. A similar process is followed to get the force coefficient values in 

table 6 for both example floods. 

Table 6. High and low force coefficients for the two example floods. 

Force coefficient 100-year high 100-year low 1,000-year high 1,000-year low 

CD 1.403 1.003 2.114 1.714

CL -0.955 -1.462 -0.112 -0.192

CM 0.247 0.092 0.0318 -0.0695

Table 6 shows that the 1,000-year flood has the higher force coefficient when the drag force is 

considered, but, for the lift force and moment, the 100-year flood values are more important. It 

should be noted that the h*crit value occurs between the two floods for the lift and moment 

coefficients, so for design purposes, it is preferable to use the critical coefficient values in 

table 3. 

Figure 5 through figure 7 allow the total forces and moments to be calculated. For instance, 

figure 5 can be rewritten to express the total drag force per unit length, as shown in figure 43, 

which inserts the bridge dimensions and flow values for the 1,000-year flood and solves the force 

as approximately 27.1 kilonewtons per meter (kN/m). 

Figure 43. Equation. Total drag force per unit length on the example six-girder bridge for 

the 1,000-year flood. 

The designer may calculate the maximum drag force by knowing the velocity at the critical value 

of h*, which can be computed if the Fr is assumed to be constant. By combining figure 2 and 

figure 3 and keeping the Fr constant, velocity equals 3.43 m/s (at h* = 3), as shown in figure 44. 
35 

Figure 44. Equation. Velocity, v, at h*crit. 

Using the velocity from figure 44 and the critical value of CD (2.15), the total drag force is 

31.6 kN/m. Now, suppose a streamlined bridge is considered instead of the six-girder bridge. 

While L, W, and hb remain the same, the bridge thickness is reduced by the same proportion as 

the experimental prototypes to s = 1.64 m. The critical value of CD for the streamlined bridge is 1.1 

at roughly h* = 5, or 4.57 if the velocity remains the same. Solving the equation in figure 43 with 

the streamlined parameters, the total drag force is 11.2 kN/m, which is a sizeable reduction from 

the six-girder bridge. Similar calculations may be followed to analyze the lift force and moment. 

37 

6. CONCLUSION 

Understanding the forces acting on inundated bridge decks is a worthwhile pursuit to improve 

the durability of bridges which can, even if infrequently, be overtopped in flooding events. The 

study described in this report investigated the properties of three common bridge deck types by 

measuring the response of drag force, lift force, and overturning moment to changes in the 

inundation ratio. 

Experiments on scale models of the three bridge deck types were performed with an ultra-precise 

force balance. These experiments defined the general response of the force coefficients to 

different flow conditions, inundation ratios, and bridge types. The following key conclusions can 

be made about the forces on bridge decks from the experiments: 

 The drag, lift, and moment coefficients showed a definite response to the inundation 

ratio, especially near the partially inundated to completely inundated transition (h* = 1). 

The lift and moment coefficients increased in overall magnitude and reached their critical 

values in this zone. The drag coefficient, however, was reduced in overall magnitude in 

this transition zone and reached a stable critical value as the inundation ratio increased. 

 The Fr had an influence on the force and moment coefficients. The Fr did not greatly 

influence the shape of the response to the inundation ratio but instead acted as a roughly 

constant offset. The total variation between the highest and lowest Fr (or the upper and 

lower fitting equations) depended on the type of bridge. The streamlined bridge had a 

much smaller envelope of coefficient values than the six- or three-girder bridges. 

 The bridge type had an influence on the coefficient values. The six-girder bridge had the 

highest critical values for the drag and moment coefficients while the three-girder had the 

most critical lift coefficient. In general, the streamlined bridge deck showed a reduction 

in all three force coefficients under the same flow conditions as the other bridge types. 

Overall, the experimental results provide the bridge designer with a wealth of information on the 

bridges’ response to hydrodynamic forces when inundated. The design charts, fitting equations, 

and critical coefficient values provide a great deal of useful information for designing more 

durable bridges. In the future, however, more research will be needed to investigate the forces on 

the bridge deck in the region of transition from the partially inundated to completely inundated 

case as the critical lift (or pull-down) and moment values occur in this region. 

CFD modeling was also performed as an alternative method to estimate the forces acting on 

bridge decks. Two CFD software packages, STAR-CD® and Fluent®, were used to model the 

bridge deck. A wide range of simulation options were explored, and the models reproduced the 

flow conditions of the experiments reasonably well, as judged by velocity map comparisons with 

the PIV results. 

The CFD models performed reasonably well at estimating the force coefficient for the six-girder 

bridge for certain ranges of inundation coefficients. The models performed fairly poorly in 

reproducing the critical coefficient values and, in some cases, failed to match the behavior of the 

coefficients’ response to h* in general. The three-girder and streamlined bridge CFD results did 
38 

not match the experimental values because they were calibrated using flow conditions from the 

six-girder experiments. Recalibrating the flow conditions for these models could lead to 

substantial improvements. 

The CFD modeling showed great promise for estimating the performance of inundated bridge 

decks. While the models did not capture the full range of behavior shown in the experimental 

results, the estimates of forces showed enough similarity that further refinements may make the 

CFD models capable of producing a reliable projection of force coefficients for design purposes. 

The flexibility of CFD models to represent almost any scenario means that they have a definite 

upside over physical experimentation. 

HRDI-07/05-09(WEB)E