Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

PUBLICATION NO. FHWA-HRT-22-097 AUGUST 2022 

Aerodynamic Stability of Bridge Stay 

Cables—Dynamic Tests and Simulations
FOREWORD

Cable-stayed bridge configurations have become more common over the past several decades for 

bridges in the medium- to long-span range. The stay cables on these bridges are slender and 

flexible elements that are vulnerable to wind-induced vibration, potentially leading to safety, 

serviceability, and user comfort issues. Two of the significant types of wind-induced vibration 

are dry galloping and galloping in icing conditions. The Federal Highway Administration

(FHWA) initiated this study to further investigate the effect of small deviations of the nominally 

circular cable cross section, a factor identified in recent research work as one of the important 

factors in galloping. This study is a significant step toward better prediction and prevention of 

aerodynamic issues on cable-stayed bridges. The results shed light on potential mitigation 

methods not only in design and retrofitting, but also in cable manufacturing, erection, and quality 

control. While additional research is recommended to develop appropriate models for analyzing 

and predicting vibration potential, the stability threshold demonstrated in this study may be an 

interim reference for reducing the risk of significant cable vibration before final design and 

analysis methods are available. 

Cheryl Allen Richter, P.E., Ph.D. 

Director, Office of Infrastructure

Research and Development

Notice

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

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

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

specification, or regulation. 

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

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

objective of the document. 

Quality Assurance Statement 

The Federal Highway Administration 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. 

Recommended citation: Federal Highway Administration, Aerodynamic Stability of Bridge 

Stay Cables—Dynamic Tests and Simulations (Washington, DC: 2022) 

https://doi.org/10.21949/1521926.

TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-22-097

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

4. Title and Subtitle

Aerodynamic Stability of Bridge Stay Cables—Dynamic Tests and 

Simulations

5. Report Date

August 2022

6. Performing Organization Code

7. Author(s)

Guy L. Larose*, Stoyan Stoyanoff*, and Brian Dunn* 

8. Performing Organization Report No.

9. Performing Organization Name and Address

Genex Systems

11848 Rock Landing Dr. Suite 303

Newport News, VA 23606

*Rowan Williams Davies & Irwin (RWDI)

600 Southgate Drive

Guelph, ON, Canada N1G 4P6

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH61-13-D-00011

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Laboratory Report

2017

14. Sponsoring Agency Code

HRDI-40 

15. Supplementary Notes

The Contracting Officer’s Representative: Harold R. Bosch, HRDI-40.

16. Abstract

This project was a part of a multiyear, multipartner experimental investigation of the effects of wind on inclined stay 

cables that was initiated by the Federal Highway Administration (FHWA) in 2000. Between 2015 and 2017, the 

investigation on the galloping of inclined stay cables has focused on characterizing the influence of the cross-sectional 

shapes of stay cables, which have been observed to deviate from a perfect round shape, on the aerodynamics of the 

stays. This report documents the second phase of a project for this purpose. The first phase of the project, which started 

in 2015, included field measurements of cross-sectional shapes of stay cables and the measurements of their static 

aerodynamic force coefficients in a wind tunnel. The second phase covers dynamic tests on stay-cable replicas and the 

numerical simulation of their response to wind actions. Experimental results showed that for an inclination of 60 

degrees at 0-degree yaw angle, large wind-induced oscillations were observed for most of the models and angles of 

attack investigated that could not be predicted solely based on the variations of the static force coefficients with 

Reynolds number and angles of attack. For these cases, increasing structural damping had only a small influence on the 

amplitude of the vibrations. For other orientations, inspection of the variations of the static coefficients with Reynolds 

number provided a good indication of the onset of the instability. For a cable model with helical fillets, increasing the 

Scruton number mitigated the vibrations. Comparison between experimental results and results from proposed 

numerical model showed that the three-dimensional, nonlinear, quasi-static modeling method can predict vibration 

vulnerability in some cases. Differences were observed in a cable-wind angle of 60 degrees and in cables with spatially 

variable cross-sectional shapes. Findings from numerical study included the following. Turbulence might be able to 

suppress the vibrations predicted in smooth flow. Numerically predicting aerodynamic vulnerability sometimes

required a high resolution in parametric variation that would be difficult to achieve experimentally. One or more 

driving mechanisms for cable vibration might not have been identified, evidenced by an unexpected lack of influence 

from damping in some cases.

17. Key Words

Cable-stayed bridges, cables, vibrations, wind, rain 

18. Distribution Statement

No restrictions. This document is available to the public 

through the National Technical Information Service, 

Springfield, VA 22161. http://www.ntis.gov

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

64

22. Price

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

TABLE OF CONTENTS

EXECUTIVE SUMMARY .......................................................................................................... 1

Experimental campaign.......................................................................................................... 1

Numerical simulations ............................................................................................................ 2

CHAPTER 1. INTRODUCTION ................................................................................................ 3

Background ............................................................................................................................. 3

Objectives................................................................................................................................. 5

CHAPTER 2. EXPERIMENTAL APPROACH ........................................................................ 7

Scope of the Dynamic Tests.................................................................................................... 7

Description of the Experiments ............................................................................................. 7

Wind Tunnel Facility and Flow Conditions........................................................................ 7

Dynamic Rig ....................................................................................................................... 9

Model Fabrication, Geometry, and Properties .................................................................. 10

Test Program ..................................................................................................................... 14

CHAPTER 3. RESULTS OF THE EXPERIMENTS ............................................................. 17

Flat-face Reference Model ................................................................................................... 17

Cable Model IR113E-HF ...................................................................................................... 19

Cable Model IR118E............................................................................................................. 22

Cable Model with Ice/wet-snow Accreted Shape ............................................................... 25

Conclusions of the experiments ........................................................................................... 30

CHAPTER 4. NUMERICAL SIMULATIONS ....................................................................... 31

MOTIVATION ..................................................................................................................... 31

Methodology .......................................................................................................................... 31

Natural Wind Effects ........................................................................................................ 31

Coordinate Framework ..................................................................................................... 32

Linear Quasi-Static Formulation....................................................................................... 34

Nonlinear Quasi-Static Formulation ................................................................................. 35

Experiments ........................................................................................................................... 36

Results .................................................................................................................................... 37

Aerodynamic Damping ..................................................................................................... 37

Discussions and conclusions ................................................................................................. 45

APPENDIX. TEST LOGS.......................................................................................................... 47

REFERENCES............................................................................................................................ 54
iv

LIST OF FIGURES

Figure 1. Photo. View of the 2- by 3-m wind-tunnel test section with the false floor and 

ceiling installed. .................................................................................................................... 8

Figure 2. Photo. View of the lower dynamic rig anchored to the wind-tunnel turntable. .............. 9

Figure 3. Images. Sketches of the (a) ceiling and (b) floor dynamic rigs. .................................... 10

Figure 4. Image. Side-view sketch of the dynamic rig. ................................................................ 10

Figure 5. Image. Cross section of the Indian River Bridge, cable 118E, smooth surface. ........... 11

Figure 6. Image. Cross section of the Indian River Bridge, cable 113E, helical fillets. ............... 12

Figure 7. Illustration. Flat-face reference shape. .......................................................................... 12

Figure 8. Illustration. Simplified ice/wet-snow accreted shape. ................................................... 13

Figure 9. Photos. (a) View of cable model IR113E with helical fillets in the test section of 

the 2- by 3-m wind tunnel and (b) close-up view of the helical fillets. .............................. 15

Figure 10. Graph. Variations of the critical reduced velocity to exceed 0.5D amplitude as a 

function of Scruton number for the flat-face reference model at a cable-wind angle of 

60 degrees. ........................................................................................................................... 17

Figure 11. Graph. Variations of the critical reduced velocity to exceed 0.7D amplitude as a 

function of Scruton number for the flat-face reference model in three segments at a 

cable-wind angle of 60 degrees. .......................................................................................... 19

Figure 12. Graphs. Variations of the (a) static lift coefficient, (b) static drag coefficient, 

and (c) dynamic response of cable model IR113E-HF as a function of Reynold’s 

number for a cable-wind angle of 60 degrees and a Scruton number of 9.5. ...................... 20

Figure 13. Graph. Variations of the dynamic response of cable model IR113E-HF as a 

function of Reynolds number for a cable-wind angle of 60 degrees and three Scruton 

numbers. .............................................................................................................................. 21

Figure 14. Graph. Variations of the critical reduced velocity to exceed 0.5D amplitude as a 

function of Scruton number for the Indian River Bridge cable IR113E-HF model, 

with double helical fillets. ................................................................................................... 22

Figure 15. Graph. Variations of the critical reduced velocity to exceed 0.5D amplitude as a 

function of Scruton number for the Indian River Bridge cable IR118E model, smooth 

surface. ................................................................................................................................ 23

Figure 16. Graphs. Variations of the (a) static lift coefficient, (b) static drag coefficient, 

and (c) dynamic response of cable model IR118E as a function of Reynolds number 

for a cable-wind angle of 60 degrees and two Scruton numbers......................................... 24

Figure 17. Graphs. Variations of the (a) static drag coefficient and (b) static lift coefficient 

for the cable model with simplified ice/wet-snow accreted shape as a function of 

Reynolds number and angles of attack for a cable-wind angle of 90 degrees. ................... 25

Figure 18. Graphs. Variations of the (a) drag coefficient and (b) lift coefficient for the 

cable model with simplified ice/wet-snow accreted shape as a function of Reynolds 

numbers for selected angles of attack for a cable-wind angle of 90 degrees. ..................... 26

Figure 19. Illustration. Sketch of the simplified ice/wet-snow accreted shape marking the 

location of the angles of attack. ........................................................................................... 26

Figure 20. Graphs. Variations of the (a) static drag coefficient and (b) the static lift 

coefficient for the cable model with simplified ice/wet-snow accreted shape as a 

function of Reynolds numbers and angles of attack for a cable-wind angle of 60 

degrees. ................................................................................................................................ 27
v 

Figure 21. Graph. Variations of the critical reduced velocity to exceed 0.7D amplitude as a 

function of Scruton number for the cable model with simplified ice/wet-snow 

accreted shape. .................................................................................................................... 28

Figure 22. Photo. View of the cable model with simplified ice/wet-snow accreted shape in 

the 2- by 3-m wind tunnel with a cable-wind angle equal to 90 degrees. ........................... 28

Figure 23. Photo. View of the cable model with simplified ice/wet-snow accreted shape in 

the 2- by 3-m wind tunnel with a cable-wind angle equal to 60 degrees. ........................... 29

Figure 24. Photos. (a) A side view and (b) a close-up view of the top end of the cable 

model with simplified ice/wet-snow accreted shape in the 2- by 3-m wind tunnel. ........... 29

Figure 25. Illustration. Sketch of the three coordinate systems: global (or structural), wind, 

and cable (or body).............................................................................................................. 32

Figure 26. Illustration. Analyzed cross-section shapes: (a) IR113E-HF (exaggerated by a 

factor of 10) and (b) the flat-face symmetric cable. ............................................................ 37

Figure 27. Graphs. Across-wind aerodynamic damping predicted by equation 11 for (a) the 

total aerodynamic damping and (b) the contribution of each term. .................................... 38

Figure 28. Graph. Numerical simulations versus experimental results for Indian River 

Bridge cable IR113E-HF with Sc = 5.3 and yaw and inclination angles of 0 and 60 

degrees, respectively. .......................................................................................................... 39

Figure 29. Graphs. Comparison of numerical simulations to interpolated data for (a) drag 

and (b) lift coefficients for IR113E-HF for Φ = 60 degrees and Re = 2.4×105. .................. 40

Figure 30. Graphs. Time series of response for IR113E-HF at Φ = 60 degrees and 

Re = 2.4×105 with Sc = 5.3 where (a) is response plotted against time and (b) is yresponse plotted against x-response with the same scale on both axes. .............................. 41

Figure 31. Graphs. The (a) aerodynamic damping (right axis) plotted with angle of attack α

(left axis) for IR113E-HF at Φ = 60 degrees and Re = 2.4×105 with Sc = 5.3, 

combined with the relative probability (counts per bin/total) of the damping in (b) 

heave and (c) sway. ............................................................................................................. 41

Figure 32. Graph. Comparison of numerical predictions to experimental results for the flatface cable with Sc =17.5 with initial angle of attack set to 20 degrees, and with yaw 

and inclination angles set to 0 and 60 degrees, respectively. .............................................. 42

Figure 33. Graphs. Comparison of numerical simulations to experimental results for the 

flat-face cable of three rotated sections with initial angle of attack of (a) 80, 10, and 

10 degrees and (b) 80, 30, and 80 degrees with Sc = 20 and yaw and inclinations 

angles set to 0 and 60 degrees, respectively. ....................................................................... 43

Figure 34. Graph. Comparison of numerical simulations to experimental results of 

IR113E-HF with Sc = 5.2 with initial angle of attack set to -55 degrees, and yaw and 

inclination angles set to 60 and 60 degrees, respectively. ................................................... 44

Figure 35. Graph. Time series of response for IR113E-HF at Φ = 60 degrees and 

Re = 2.4×105 with Sc = 5.3 and Iu ≈ 5 percent where (a) is response plotted against 

time and (b) is y-response plotted against x-response with the same scale on both 

axes. ..................................................................................................................................... 45
vi 

LIST OF TABLES

Table 1. Summary of cable model properties. .............................................................................. 13

Table 2. Combinations of angle of attack investigated. ................................................................ 18

Table 3. Test log data for test number 1003. ................................................................................ 48
vii 

LIST OF ABBREVIATIONS AND SYMBOLS

Abbreviations 

3D three dimensional 

FHWA Federal Highway Administration

HDPE high-density polyethylene 

HF helical fillets

MDOF Multi-degree of freedom 

Symbols

C* generalized damping matrix 

CA, CD, CL static force coefficients: A, along; D, drag; L, lift

D cross-section depth or cable diameter 

F* generalized forces

f frequency

FX, FY, FZ structural wind loads in cable loads coordinate system (body system)

I identity matrix

Iu intensity of turbulence in longitudinal wind direction 

K* generalized stiffness matrix

m cross-section mass per unit length

  generalized response: deflections, speeds, accelerations

R radius 

Re Reynolds number (based on wind-tunnel reference speed)

Sc Scruton number 

T coordinate transformation matrix

t time

t1, tmax ice thickness 

Ul, Vl, Wl wind velocity vectors locally aligned with the cable loads 

UN wind speed normal to cable 

U(t), W(t) time-varying components of wind

u, v, w wind velocity vectors 

us, vs, ws wind velocity vectors locally aligned with structure 

  mean wind speed in longitudinal wind direction (reference speed in wind tunnel)

V/fD critical reduced velocity

V mean wind speed in longitudinal wind direction at which the cable reached an

aerodynamic stability boundary 

X, Y, Z cable loads coordinates in body system 

x, y, z structural coordinates/response deflections 

  speed of structural response

x′, y′, z′ wind coordinates 

α(t) instantaneous variation of angle of local wind attack 

α angle of local wind attack: section rotation about its longitudinal axis 

β wind direction angle (yaw) 

β0 mean wind direction 

ζ structural damping (ratio to critical)
viii

ζaero aerodynamic damping (ratio of critical)

θ cable inclination angle from the horizontal plane 

θ0 mean cable inclination angle 

μ air dynamic viscosity

ν air kinematic viscosity

ρ air density

σu standard deviation (root-mean-square) of turbulence in longitudinal wind 

direction

Φ, φ effective cable-wind angle 

ωn circular frequency of the nth cable mode
1 

EXECUTIVE SUMMARY

The Federal Highway Administration (FHWA) conducted an experimental and numerical study 

of the inclined cable-galloping phenomenon applied to the design of stay cables for cable-stayed 

bridges. This report presents a detailed description of the experimental and numerical 

approaches, the main findings of the study, and future recommendations. The study was carried 

out in two parts. 

EXPERIMENTAL CAMPAIGN 

Experiments on free-to-respond rigid bridge stay cable models were carried out in a 2- by 3-m 

wind tunnel. The models were set at an inclination of 60 degrees and were yawed to the flow 

from 0 to 90 degrees to study the influence of the cable-wind angle on the wind-induced 

excitation. The models were designed so that they could be rotated along their longitudinal axis 

to set their angle of attack as desired. The experiments were set as a logical progression of a 

concerted research effort that included: 

• The full-scale measurements of the cross-sectional shape of high-density polyethylene 

(HDPE) external sheathing of stay cables for bridges in service in the United States by 

FHWA.

(1)

 

• Static tests on replicas of the scanned shapes to characterize the variations of the 

cross-sectional aerodynamic coefficients of the stay cables as a function of Reynolds 

number, angle of attack, and cable-wind angle. The tests were carried out in 2015 in the 

2- by 3-m wind tunnel and are detailed in a 2021 report from FHWA.(2)

• The current dynamic tests on the same models. 

The final phase of experiments focused on the verification, through dynamic tests on the 

stay-cable replicas, of the range of applicability of the quasi-steady theory for the predictions of 

the damping level needed to control the vibrations of stay cables with shape distortion and/or 

with ice/wet-snow accretion. The 1.8-m-long models were supported in a 16-spring dynamic rig, 

which utilized air bearings and eddy-current dampers. The mass-damping parameter (Scruton 

number), the Reynolds number, the reduced velocity, and the surface-roughness-to-diameter 

ratio of the models were adjusted to match full-scale values. 

Five cable models were investigated, including a section with a generic ice/wet-snow accretion. 

Galloping was observed for all models for a variety of cable-wind angles, Scruton numbers, and

angles of attack. In general, based on the variations of the static force coefficients with angles of 

attack and Reynolds numbers, the onset speed of the instability could be predicted. However, for 

a cable-wind angle of 60 degrees, large wind-induced oscillations were observed in several 

instances that could not be predicted solely based on the variations of the static force coefficients 

with Reynolds number and angles of attack. For these cases, increasing structural damping (the 

typical measure used for vibration control) had only a limited influence on the amplitude of the 

vibrations. This phenomenon was not observed for the cable-wind angles greater than 60 degrees

that were investigated. 
2 

For the cable model with helical fillets (HF), increasing the Scruton number was observed to 

mitigate the vibrations as was first anticipated at the onset of the study. 

NUMERICAL SIMULATIONS 

To expand the use of the experimental data, a theoretical framework via numerical simulations of

cable response was investigated. Predictions of the dynamic response of the cable models used in 

the experiments were carried out using a three-dimensional (3D), nonlinear response analysis 

simulation in the time domain. The simulations modeled with precision the dynamics of the 

mechanical systems, including mass, stiffness, frequencies, and structural damping. 

The aerodynamics were modeled through the static force coefficients and their rate of change 

with varying angles of attack, Reynolds numbers, and cable-wind angles. The coefficients were 

measured on the same models in the same wind tunnel in a test campaign conducted in 2015.(2)

By direct comparison with the dynamic tests, the simulations provided an evaluation of the range 

of applicability of the quasi-steady theory for the predictions of negative aerodynamic damping 

for stay cables with shape distortions due to sheathing deviations and/or ice/wet-snow accretions. 

Using a 3D nonlinear quasi-static approach, the predictions were generally in good agreement 

with the wind-tunnel experiment observations. Numerical tests showed that turbulence might be 

able to suppress the vibrations predicted in smooth flow. In other cases, especially for a cablewind angle of 60 degrees, the comparisons showed that the proposed aerodynamic model cannot 

explain the wind-tunnel results. The likely reasons for these discrepancies are: 

• Lack of details in the static force coefficients data. Very sharp gradient trends were seen

in the transitions from stable to unstable regions and vice versa that required detailed 

knowledge of the force coefficients over the three-parameter space: angle of attack, 

cable-wind angle, and Reynolds number. Even though very fine increments of angle of 

attack (e.g., approximately 1 degree) have been measured in some cases, it is practically 

challenging to sample the entire parameter space experimentally. 

• Different aerodynamic phenomena caused the instabilities. Some of the large-amplitude 

vibrations found during the test could not be predicted and/or did not follow the general 

expectations based on quasi-static theory. For example, physical tests with increased 

damping in the wind tunnel have shown a limited influence of the added damping on the 

oscillations, even at high levels of Scruton number. These observations indicate that 

another aerodynamic excitation mechanism (perhaps stiffness driven) may play a 

significant role in the observed responses. Identifying and modeling aerodynamic 

phenomena outside of those phenomena described by quasi-static theory was beyond the 

scope of the current research program. 
3 

CHAPTER 1. INTRODUCTION

The Federal Highway Administration (FHWA) conducted an experimental and numerical study 

of the inclined cable-galloping phenomenon applied to the design of stay cables for cable-stayed 

bridges. This study is a continuation of a multiyear multipartner experimental investigation of the 

effects of wind on inclined stay cables that FHWA initiated in 2000. 

The study reported here is a logical progression of the concerted research effort between 2015 to 

2017 that included: 

• The full-scale measurements of the cross-sectional shape of high-density polyethylene 

(HDPE) external sheathing of stay cables for bridges in service in the United Sates by 

FHWA.

(1)

• Static tests on replicas of the electronically scanned shapes to characterize the variations 

of the cross-sectional aerodynamic coefficients of the stay cables as a function of 

Reynolds number, angle of attack, and cable-wind angle.(2)

• The current dynamic tests on the same models in the same wind tunnel and numerical 

simulations of the observed wind-induced response. 

This report presents the results of the dynamic tests, including a detailed description of the 

experimental and numerical approaches, the main findings of the study, and recommendations 

for a way forward to evaluate the damping demands to mitigate inclined stay-cable galloping for 

cable-stayed bridges. 

BACKGROUND

In recent years, it has been observed that the external sheathings used to cover the steel strands of 

stay cables of cable-stayed bridges were not necessarily of regular circular shape, as many 

researchers had assumed. Today, and for the past 15–20 years, the preferred material used for the 

sheathing has been HDPE, the same material and pultrusion process used for pipes to carry water 

or natural gas in industrial applications. 

The external shape of the sheathing is generally controlled, with deviation tolerances ranging 

from ±1.5 to ±5 percent of the mean diameter, depending on the application and the quality 

standard followed. Onsite, however, depending on the shipping and storage conditions of the 

HDPE tubes (generally 12-m long), deviations larger than ±5 percent of the mean diameter can 

exist, and typically, if the deviations are larger than ±10 percent of the mean diameter, the tube is 

discarded since welding to another tube might not be possible. 

In a broad sense, imperfections of the cross sections of the stays of approximately 1–2 percent of 

the mean diameter (i.e., 2 to 4 mm for a 200-mm HDPE tube) should not be a concern on a 

cable-stayed bridge with a 350-m main span and 150-m tower heights. However, the 

combination of the diameter of the stays and the typical wind speeds of the atmospheric 

boundary layer put the stays right in the critical Reynolds number regime. In this regime, many 

fluid mechanics phenomena happen at the surface of a circular cylinder affecting the behavior of 
4 

the boundary layers, the shear layers, the pressure distribution, and the wind loading. These 

phenomena are known to be influenced by Reynolds number, surface irregularities and 

imperfections, the characteristics of the turbulence of the flow, the angle of incidence between 

the flow and the circular cylinders, and other factors. Schewe states that in the critical Reynolds 

number regime, “a very small cause can have a very great effect.”(3) For a thorough review of the 

problem specifically related to inclined stay cables, the reader is referred to the work of 

Christiansen, and for circular cylinders in general to Zdravkovich.(4,5)

In 2015, FHWA initiated a research study in three parts, following up on a series of laboratory 

experiments on a mock-up of a stay cable in a dynamic suspension rig that clearly identified that 

the eccentricity of a cable could lead to instability.

(6)

 The first part consisted of the development 

of a scanning robot that could climb up existing stay cables and provide details of the 

cross-sectional geometry and surface conditions. The robot was then put into service, and 

multiple cables of several bridges were scanned in the United States over a 2-yr period. 

Preliminary results of these scans were published by FHWA in a conference paper.

(1)

 In general, 

the stay cables were found to have a cross-sectional shape that deviated from a round circular 

cylinder. The out-of-roundness level varied significantly from cable to cable and from section to 

section on the same cable. 

The second part of the FHWA research project consisted of a series of wind-tunnel tests on 

replicas, at model scale, of the shapes of the external sheathing of cables provided by FHWA 

from direct measurements on existing bridges. The tests were carried out in a 2- by 3-m wind 

tunnel in the summer of 2015. The models were kept stationary, and the aerodynamic forces as a 

function of Reynolds number, angles of attack, and cable-wind angle were monitored with two 

internal force balances, one at each extremity of the models. These tests are referred to as the 

“static tests.” 

Among other observations, the tests confirmed that the aerodynamic characteristics of the stays 

were highly sensitive to their cross-sectional shape variation.(2,4) A small deviation from the 

mean curvature had an important influence on the aerodynamic force coefficients, especially for 

the mean across-wind force coefficients in the critical Reynolds number regime. Based on an 

analysis of the force coefficients, negative aerodynamic damping in excess of 1 percent of 

critical was derived analytically from the static tests, indicating the potential of stay cables to 

gallop at certain cable-wind angles, Reynolds numbers, and cross-sectional irregularities.(4)

The third part of the FHWA research project was to verify through dynamic tests that the 

sensitivity to wind-induced vibrations and inclined cable galloping predicted based on the static 

tests would be confirmed on free-to-respond models with the same cross-sectional shapes. This 

final part forms the basis for the experimental part of the study reported here. 

Moreover, the estimated damping demands based on the static tests carried out in 2015 indicated 

the possibility of cable dampers on bridges built according to the current designed practice not 

being fully capable of mitigating wind excitations.(2)

 Thus far, however, there is no strong 

evidence of unexplained and/or difficult-to-control vibrations with dampers on the existing 

bridges. Questions do arise, however. What are these unaccounted for conditions which work in 

favor of mitigating the destabilizing effects of the cable sheathing imperfections? Is there a 
5 

missing component in our design methodology that needs improvement? These questions are 

also covered in the current study. 

OBJECTIVES

The experiments’ main objective was to study how replicas of existing stay cables would 

respond to the potential for aerodynamic instabilities as revealed by their static force coefficients 

when suspended in a 2-degree-of-freedom suspension system with adjustable frequencies and 

structural damping. The main objective of the numerical simulations of the experiments was to 

provide an evaluation of the range of applicability of the quasi-steady theory for the predictions 

of negative aerodynamic damping for stay cables with shape distortions due to sheathing 

deviations and/or ice/wet-snow accretions.

7 

CHAPTER 2. EXPERIMENTAL APPROACH

SCOPE OF THE DYNAMIC TESTS

To ensure that the experimental conditions of the dynamic tests would be identical to the 

conditions at which the static tests were carried out, the experiments were designed to take place 

in the same wind tunnel, on the same models, in the same flow conditions and at the same 

Reynolds numbers as in the 2015 study.(2)

A total of four cross-sectional shapes were studied on five different models. The main part of the 

experiments was carried out on: 

• The cross-sectional shapes from a lower section of cables numbered 118E and 113E of 

the Indian River Inlet Bridge (officially known as the Charles W. Cullen Bridge), located 

7 miles south of the city of Rehoboth Beach, DE in Delaware Seashore State Park. For 

this study, cable 118E was modeled as a smooth surface and cable 113E was modeled as 

having double helical fillets (HF). All cables of the Indian River Inlet Bridge have double 

helical fillets. 

• A reference shape developed in the framework of the static tests identified as a “flat-face” 

cross section.

(2,4)

 

• A simplified generic ice/wet-snow accreted shape developed in the framework of the 

current study. 

All models had a constant cross-sectional shape for their entire length, except for one model that 

was built in three sections. For the model in three sections, each section had the flat-face shape 

that could be rotated around its longitudinal axis to adjust its angle of attack independently of the 

two other sections. The idea behind this model with three sections was to study the influence of 

changes of cross section on the aerodynamic instabilities, thus making a parallel with the 

fabrication and installation methods of stay cables that are generally enclosed in several HDPE 

tubes 11- to 12-m long. The field measurements and other site observations have revealed that 

each 12-m section may have a different cross-sectional shape and/or be rotated along its 

longitudinal axis thus having different aerodynamic properties for a given oncoming wind 

alignment.

The experiments were designed to respect the following similitude requirements: Reynolds 

number, mass-damping parameter, and reduced velocity. 

DESCRIPTION OF THE EXPERIMENTS

Wind-Tunnel Facility and Flow Conditions

The measurement campaign took place in a 2- by 3-m wind tunnel. The wind tunnel is a 

closed-circuit atmospheric facility with an empty test section that measures 2.7-m wide by 1.9-m 

high by 5.2-m long and has a 9:1 contraction ratio. The wind tunnel has a wind speed uniformity 

of ±0.7 percent and a turbulence level of 0.14 percent in an empty test section. The wind tunnel 
8 

is equipped with floor and roof turntables allowing precise yawing of models mounted vertically 

in the test section. The maximum speed of the wind tunnel (120 m/s) and the size of the test 

section are suitable for testing model-scale stay cables with a high aspect ratio at full-scale 

Reynolds numbers. For an elastically supported rigid sectional model, Froude number scaling did

not play a role and was not considered. In the field, gravity will influence the sag of the cables, 

affecting their frequency in one plane. This effect is simulated in the wind tunnel by a different 

frequency between heave and sway motion. The similitude parameters that were respected in the 

experiments were reduced velocity (which governed the velocity and frequency scaling), 

mass-damping parameter (Scruton number, Sc), and Reynolds number (Re). 

The wind speeds considered in the experiment ranged from 10 to 80 m/s. This range allows 

testing in the Reynolds number regime of interest. A false floor and false ceiling were installed 

in the wind tunnel to shield the test rig from the flow, as shown in figure 1. The false floor and 

ceiling consisted of a 15-degree ramp upstream of the model, a flat section measuring 2.9-m 

long, and a 15-degree ramp downstream of the model. The height of the wind tunnel was reduced 

to 1.53 m with the false ceiling and floor installed. The rotating portion of the false floor and 

ceiling allowed the model to be yawed from 0 to 90 degrees. The false floor and ceiling each 

included a 0.45-m-diameter hole. The hole was necessary to allow the cable to move, 

unimpeded, with amplitudes of up to ±1.7 cable diameters. 

© 2017 RWDI.

Figure 1. Photo. View of the 2- by 3-m wind-tunnel test section with the false floor and 

ceiling installed.
9 

Dynamic Rig

A 16-spring suspension rig was developed and built to fit within the false floor and ceiling of the 

2- by 3-m wind tunnel. Six air bearings were used to suspend the models within the arrays of 

springs so that the models could move freely in the along-wind (heave mode of an inclined stay 

cable) and the across-wind (sway mode) directions. The rig was designed to allow vibration 

amplitudes of up to 1.7 cable diameters (150 mm). A combination of spring stiffness and tension 

was used to adjust the frequency of oscillation in each of the modes. 

The rig was designed to include four eddy-current dampers that provided a damping level 

constant with amplitude of motion and could be adjusted independently for each mode of 

vibration. Figure 2 presents a view of the lower rig anchored to the turntable of the wind tunnel, 

with a panel of the false floor removed. Four laser displacement transducers monitored the 

motion of the cable models at their extremities. For some of the experiments, two of the springs 

in each degree of freedom were replaced by load cells, converting the dynamic rig into a static 

rig to determine the static force coefficients of the cable model with the same end conditions as 

for the dynamic tests. Sketches of the dynamic rig are shown in figure 3 and figure 4. 

© 2017 RWDI.

Figure 2. Photo. View of the lower dynamic rig anchored to the wind-tunnel turntable.
10 

Source: FHWA.

Figure 3. Images. Sketches of the (a) ceiling and (b) floor dynamic rigs. 

Source: FHWA.

Figure 4. Image. Side-view sketch of the dynamic rig.

Model Fabrication, Geometry, and Properties

The models were designed and fabricated using the same technique developed for the static tests 

detailed in the 2021 report.(2) Selective Laser Sintering was used to grow cable model tubes that 
11 

were inserted onto a stiff carbon fiber spar. The tubes for the Indian River Bridge representative 

cables and the flat-face reference shape of the static tests were extended to 1.8 m to match the 

longer models of the dynamic tests, as opposed to 1.5 m for the static tests. These models had a 

diameter (D) of 0.089 m and an aspect ratio of 20. Two new models with a smaller diameter 

were built for the ice/wet-snow accreted shape (D = 0.070 m) and the flat-face shape 

(D = 0.070 m) in three sections. The research team anticipated that the drag crisis would occur at 

lower Reynolds numbers for the ice/wet-snow accreted shape; therefore, the decision was made 

to reduce the diameter of the model. 

The surface-roughness-to-diameter ratio of the models was adjusted to an average of 1.5×10−5

, 

similar to the surface roughness of prototype cables. A rotation of the cable tube on the 

longitudinal axis of the spar was used to adjust the angle of attack of the model. A simultaneous 

rotation of the floor and roof turntable was used to set the yaw angle. The inclination of the 

model was kept constant at 60 degrees. 

Figure 5 through figure 8 present the cross-sectional shapes of the cable models of the dynamic 

tests. A summary of the model properties, including mass per unit length (m) and frequency (f) in 

both the sway and heave directions, is presented in table 1. 

Source: FHWA.

Figure 5. Image. Cross section of the Indian River Bridge, cable 118E, smooth surface. 
12 

Source: FHWA.

Figure 6. Image. Cross section of the Indian River Bridge, cable 113E, helical fillets.

Source: FHWA.

Figure 7. Illustration. Flat-face reference shape.
13 

Source: FHWA.

Note: t1 represents average ice thickness, tmax represents maximum ice thickness.

Figure 8. Illustration. Simplified ice/wet-snow accreted shape.

Table 1. Summary of cable model properties. 

Properties IR118E IR113E HF Flat-Face Flat-Face 2

Ice/Wetsnow 

D (m) 0.089 0.089 0.089 0.065 0.070 

msway (kg/m) 7.63 7.67 7.62 5.71 5.94 

mheave (kg/m) 7.42 7.46 7.41 5.50 5.73 

fsway (Hz) 4.40 4.35 4.41 5.00 4.92 

fheave (Hz) 4.23 4.17 4.13 4.76 4.70 

Damping (percent of 

critical)

1.1–5.9 0.6–2.5 0.3–2.1 1.6–3.3 0.4–1.5 

Sc range 8.4–46.4 4.7–19.7 2.0–16.5 14.7–31.4 3.9–15.2 

Re range (×105) 0.3–3.6 0.6–4.1 0.3–3.4 0.7–3.3 0.4–4.1 

. 

A structure’s aerodynamic stability is governed by the characteristics of its main components and 

will depend on each of the following: cross-sectional geometry, mass, and structural properties 

such as stiffness and damping. For any element of the structure, the dependence on overall 

dimensions, mass and damping can be described by the mass-damping parameter known as the 

Scruton number, Sc: 

(1)
14 

Where:

m = mass per unit length. 

ζ = structural damping (ratio to critical). 

ρ = air density. 

D = section depth or cable diameter.

In general, a lower Scruton number implies that a structural component might be more 

susceptible to aerodynamic instabilities due to its low mass per unit length and/or damping 

relative to the overall dimensions exposed to wind. 

In this study, the models were kept light in relation to their diameter compared to expected 

full-scale values. To respect mass-damping similarity, the mass-damping parameter was adjusted 

by compensating with added structural damping. 

The generic ice/wet-snow accreted shape was developed based on field observations and 

experience. It represents the shape of a stay cable following a freezing rain event where a 2-mm 

layer of glaze ice would have accreted on a large part of the circumference (220 degrees), 

followed by a small drop of temperature where the freezing rain would have transformed into 

wet-snow and 8 mm of wet-snow would have accumulated on the windward face of the cable.

Accretions of glaze ice and wet-snow on cables are common for cables in moderate and 

continental climates.

Test Program 

During the static tests in 2015, critical cable-wind angles and angles of attack were identified as 

having a potential for aerodynamic instability. The dynamic tests systematically revisited these 

critical conditions for the two Indian River Bridge cable models and the flat-face reference 

shape. When vibrations were observed, the level of structural damping and Scruton number were

increased to evaluate the influence of damping on the vibrations and to compare them with 

predictions of damping demand based on a quasi-steady aerodynamic model.(7) The tests 

consisted of speed sweeps at fixed angles of attack, yaw angles, and damping levels. A view of 

the test arrangement is shown in figure 9. The list of runs for each model is presented in table 3

of Appendix A along with a summary of the test conditions. 
15 

© 2017 RWDI.

Figure 9. Photos. (a) View of cable model IR113E with helical fillets in the test section of 

the 2- by 3-m wind tunnel and (b) close-up view of the helical fillets.

A similar approach was taken for the flat-face model in three sections. The angle of attack of 

each section was deliberately selected to alternate between an angle deemed potentially unstable 

and an angle deemed stable. This approach was followed by tests with a random selection of the 

angles of attack for each section. For the ice/wet-snow accreted shape, the dynamic rig was 

converted to a static rig to determine the static force coefficients as a function of angle of attack 

before dynamic tests were conducted with this model. Inspection of the static coefficients was 

subsequently used to choose the critical angles for dynamic tests. Dynamic tests for the ice/wetsnow accreted shape were repeated for various level of damping.

17 

CHAPTER 3. RESULTS OF THE EXPERIMENTS

The main findings of the experiments presented in this chapter focus on the influence of the 

Scruton number on the wind-induced vibrations observed. The experimental data, digital photos, 

and videos for each run described in the appendix were stored on an external hard drive and 

provided to FHWA under separate cover. 

FLAT-FACE REFERENCE MODEL

Figure 10 presents the variations of the critical reduced velocity (V⁄fD) at which the vibrations 

observed exceeded 50 percent of the model diameter (written as 0.5D in the figure captions) with 

Scruton number and angle of attack for the flat-face reference model. The turntable was kept at 

0 degrees for the experiments, and the model inclination was 60 degrees, resulting in a 

cable-wind angle of 60 degrees. Ten angles of attack were investigated. Instability in the 

across-wind direction (sway mode) was observed for all angles of attack investigated for Scruton 

numbers varying from 2.2 to 16.5. In general, the amplitude of the vibrations grew gradually 

with increasing wind speed until amplitudes of 1 cable diameter were reached.

Source: FHWA.

Figure 10. Graph. Variations of the critical reduced velocity to exceed 0.5D amplitude as a 

function of Scruton number for the flat-face reference model at a cable-wind angle of 

60 degrees.

In figure 10, the variations observed in critical reduced velocity for a fixed Scruton number 

revealed that the cross-sectional shape had an important influence on the amplitude of the 

vibrations observed. However, the relativity low increase of the critical reduced velocity with 

Scruton number for all angles of attack, indicated that damping had a weak influence on the 
18 

instability process which is not in agreement with predictions that would be based on a quasisteady aerodynamics process. This finding agrees with the work of Kimura, Kato, and Kubo.(8)

 

A similar trend was observed for the flat-face reference model in three segments. Apart from 2 

randomly selected combinations of angles of attack, all combinations tested (13) were unstable 

and remained unstable up to a Scruton number above 35. The combinations of angle of attack 

investigated are shown in table 2. 

Table 2. Combinations of angle of attack investigated. 

Run Bottom Section

(Degree) 

Midsection

(Degree) 

Top Section

(Degree) 

409 20 20 20 

412 80 80 80 

417 120 120 120 

423 20 80 120 

426 20 120 80 

429 120 80 20 

434 120 20 80 

446 10 100 40 

448 60 80 30 

451 80 10 10 

455 80 30 80 

458 100 40 100 

461 120 20 0 

For example, for run 409, all three sections of the model were set at an angle of attack of 

20 degrees. For run 423, the lower section of the model was set at an angle of attack of 

20 degrees, the midsection at 80 degrees and the top section at 120 degrees. 

The results are summarized in figure 11. However, the tests revealed that the critical reduced 

velocity at which the vibrations would be unstable was greatly influenced by the combination of 

angles of attack of the three segments. As shown in figure 11, for a Scruton number of 17, the 

critical reduced velocity increased from 150 up to 230 (240 was defined as the stability 

threshold) by a rotation of the angles of attack. The selected stability threshold corresponds to the 

maximum wind speed reached in the experiments where the cable model was still stable. It is 

assumed that the cable would still be stable beyond this wind speed. 

Note that the increase of reduced velocity in the wind tunnel was achieved by gradually 

increasing the mean wind speed of the flow. Aerodynamic stability was observed for two 

combinations of angles where two of the segments were set at the same angle of attack. This 

observation clearly indicated that the spanwise correlation of the aerodynamic forces had an 

importance on the instability observed. 
19 

Source: FHWA.

Figure 11. Graph. Variations of the critical reduced velocity to exceed 0.7D amplitude as a 

function of Scruton number for the flat-face reference model in three segments at a cablewind angle of 60 degrees. 

CABLE MODEL IR113E-HF

The results for the Indian River Bridge cable IR113E-HF are presented in figure 12 through 

figure 14. Instabilities were observed in sway for most of the angles of attack tested for a cablewind angle of 60 degrees and a Scruton number of 5. The instabilities followed the pattern that 

the test team had anticipated based on the variations of the force coefficients with Reynolds 

number. 

Typically, oscillations started after a drop of the drag coefficient and a sudden increase of the lift 

coefficient. The oscillations reached their maximum with an increase of Reynolds number, a 

reduction of the drag coefficient, and a reduction of the lift coefficient. The oscillations finally 

subsided when the Reynolds number reached a region where the lift coefficient was at a 

minimum and the drag coefficient reached a plateau or re-increased. When oscillations occurred,

they appeared to be in sync with the formation of a laminar separation bubble on one side of the 

cable (also known as the start of the TrBL1 regime based on Zdravkovich’s nomenclature) and 

continued until the disruption of the laminar separation bubble.(5)

 

For a smooth cable in the critical regime, oscillations that would start and stop in regions of 

transition between TrBL0 and TrBL1 or TrBL1 and TrBL2, have been reported on several 

occasions. Such oscillations have been well documented in previous studies initiated by 

FHWA.(6) The observations of the current study confirmed that this transition process (and 

therefore source of dynamic excitation) was also present for a cable shape directly obtained from 

a stay cable in service and with typical helical fillet of approximately 1.5- to 2.5-mm thickness at 

a 45-degree helix angle.
20 

Figure 12 illustrates the relationship between the onset of the oscillations and the variations of 

the force coefficients with Reynolds number. The lift (CL) and drag (CD) coefficients of figure 12

are from the static tests on the same model in the same wind tunnel for the same angle of attack 

and same cable-wind angle.(2)

Source: FHWA.

Figure 12. Graphs. Variations of the (a) static lift coefficient, (b) static drag coefficient, and 

(c) dynamic response of cable model IR113E-HF as a function of Reynold’s number for a 

cable-wind angle of 60 degrees and a Scruton number of 9.5. 
21 

As opposed to what was observed for the flat-face reference model, an increase of Scruton 

number had a direct influence on the amplitude of the oscillations for cable model IR113E-HF at 

a cable-wind angle of 60 degrees. This influence is illustrated in figure 13. The angle of attack of 

the cable model was kept constant, and the damping of the dynamic rig was increased 

systematically for each run. 

Source: FHWA.

Figure 13. Graph. Variations of the dynamic response of cable model IR113E-HF as a 

function of Reynolds number for a cable-wind angle of 60 degrees and three Scruton 

numbers.

As illustrated in figure 14, for all cable-wind angles that were tested above 60 degrees, no 

instabilities were observed for a Scruton number of 5. In figure 14, all points with a reduced 

velocity above 180 were stable. The influence of an increase of Scruton number on the 

oscillations is clear. Note that all cables of the Indian River Bridge have helical fillets and 

dampers.
22 

Source: FHWA.

Figure 14. Graph. Variations of the critical reduced velocity to exceed 0.5D amplitude as a 

function of Scruton number for the Indian River Bridge cable IR113E-HF model, with 

double helical fillets.

CABLE MODEL IR118E 

The results for all tests with the Indian River Bridge cable IR118E (smooth surface) are 

presented in figure 15. Instabilities were observed for all angles of attack tested for the cable 

IR118E for a cable-wind angle of 60 degrees. Increasing the Scruton number had little influence 

on the amplitude of the response, even less influence than for the flat-face reference model.

However, for cable-wind angle larger than 60 degrees, increasing damping had an influence on 

the response. The three points in figure 15 above a reduced velocity of 180 at Scruton numbers 

approaching 30 were stable for the entire range of Reynolds number studied. 
23 

Source: FHWA.

Figure 15. Graph. Variations of the critical reduced velocity to exceed 0.5D amplitude as a 

function of Scruton number for the Indian River Bridge cable IR118E model, smooth 

surface.

The tests provided a clear indication that the aerodynamic phenomenon at the source of the 

excitation for this cable shape was different than for cable model IR113E-HF, even though all 

other parameters of the test were constant. As in figure 12, figure 16 illustrates the relationship 

between the force coefficients and the dynamic response as a function of Reynolds number. The 

onset of the excitation also appeared to be associated with a reduction of drag coefficient and an 

increase of lift coefficient, indicating the presence of a laminar separation bubble and the start of 

the TrBL1 regime. However, as the Reynolds number was increased, the vibrations grew rapidly 

in amplitude up to full instability. As shown in figure 16, this rapid growth in amplitude was for 

a Scruton number of 32, which is at least three times higher than the Scruton number of the 

majority of the stay cables for bridges in service. Increasing the Scruton number from 32 to 37 

only delayed the onset of the instability by a few m/s. 

Figure 16 shows not only the variations of the force coefficients with Reynolds number from the 

static tests but also the equivalent force coefficients calculated based on the mean displacements 

of the cable model in the dynamic rig.(2)

 The stiffness of the dynamic suspension system in both 

the sway and heave modes was calibrated during the tests. The mean aerodynamic forces were 

deduced from the mean displacements and the stiffness of the rig. In general, the two sets of 

force coefficients were found to be in agreement, unless large oscillations were observed. 
24 

Source: FHWA.

Figure 16. Graphs. Variations of the (a) static lift coefficient, (b) static drag coefficient, and 

(c) dynamic response of cable model IR118E as a function of Reynolds number for a 

cable-wind angle of 60 degrees and two Scruton numbers.
25 

CABLE MODEL WITH ICE/WET-SNOW ACCRETED SHAPE

The simplified ice/wet-snow accreted shape was developed in the framework of this study. 

Before tests were performed in the dynamic rig, it was necessary to measure the static force 

coefficients as a function of Reynolds number and angle of attack. The dynamic rig was 

modified, replacing springs by rigid links connected to load cells at each extremity of the model. 

This modification allowed the static coefficients to be determined for the same testing conditions 

as for the dynamic tests, same model, same end conditions, and same flow conditions. 

Figure 17 presents the static force coefficients for all angles of attack tested for a cable-wind 

angle of 90 degrees as a function of Reynolds number. As for the coefficients for the Indian 

River Bridge cables and the flat-face reference shape, the drag coefficients reduced significantly 

with Reynolds number, and large variations of lift coefficients with Reynolds number were also 

observed. The variations were found to be more pronounced for some of the angles of attack. By 

inspection of the coefficients, dynamic excitation was expected. Figure 18 and figure 19 present 

the coefficients for five of the angles of attack, selected for their adverse variations with 

Reynolds number, and a sketch of the cross section with a marking for these angles of attack, 

respectively. Adverse conditions with regard to aerodynamic stability were defined when a drop 

of the drag coefficient with Reynolds number was combined with a sharp variation of the lift 

coefficient due to change of Reynolds number or with angle of attack. 

Source: FHWA.

Figure 17. Graphs. Variations of the (a) static drag coefficient and (b) static lift coefficient 

for the cable model with simplified ice/wet-snow accreted shape as a function of Reynolds 

number and angles of attack for a cable-wind angle of 90 degrees. 
26 

Source: FHWA.

Figure 18. Graphs. Variations of the (a) drag coefficient and (b) lift coefficient for the cable 

model with simplified ice/wet-snow accreted shape as a function of Reynolds numbers for 

selected angles of attack for a cable-wind angle of 90 degrees. 

Source: FHWA.

Figure 19. Illustration. Sketch of the simplified ice/wet-snow accreted shape marking the 

location of the angles of attack. 

Figure 20 presents the static coefficients for a cable-wind angle of 60 degrees. The variations 

follow the same trends as for the cable-wind angle of 90 degrees. 
27 

Source: FHWA.

Figure 20. Graphs. Variations of the (a) static drag coefficient and (b) the static lift 

coefficient for the cable model with simplified ice/wet-snow accreted shape as a function of 

Reynolds numbers and angles of attack for a cable-wind angle of 60 degrees. 

The results of the dynamic tests are presented in figure 21. As expected, instabilities were 

observed for cable-wind angles of both 90 and 60 degrees. For a cable-wind angle of 90 degrees, 

the angles of attack at which instabilities were observed at a Scruton number of 4.5 were 10, 230, 

240, 250 and 320 degrees. The tests at 60 degrees exhibited generally stronger excitation; all 

angles of attack investigated (10, 30, 210, 214, 240, and 250 degrees) were unstable at a Scruton 

number below 8. For the IR113E cable with helical fillets an increase of Scruton number had a 

more important influence on the amplitude of the vibrations than for the flat-face reference 

shape. For the angles of attack investigated, a Scruton number of 12 was sufficient to mitigate 

the oscillations at a cable-wind angle of 60 degrees. Views of the cable model with the simplified 

ice/snow accreted shape (represented by white material in the figures) are presented in figure 22

through figure 24. 
28 

Source: FHWA.

Figure 21. Graph. Variations of the critical reduced velocity to exceed 0.7D amplitude as a 

function of Scruton number for the cable model with simplified ice/wet-snow accreted 

shape.

© 2017 RWDI.

Figure 22. Photo. View of the cable model with simplified ice/wet-snow accreted shape in 

the 2- by 3-m wind tunnel with a cable-wind angle equal to 90 degrees.
29 

© 2017 RWDI.

Figure 23. Photo. View of the cable model with simplified ice/wet-snow accreted shape in 

the 2- by 3-m wind tunnel with a cable-wind angle equal to 60 degrees.

© 2017 RWDI.

Figure 24. Photos. (a) A side view and (b) a close-up view of the top end of the cable model 

with simplified ice/wet-snow accreted shape in the 2- by 3-m wind tunnel. 
30 

CONCLUSIONS OF THE EXPERIMENTS

Dynamic tests on replicas of cables for bridges in service revealed aerodynamic instabilities for 

all models for a variety of conditions. In general, inspection of the variations of the static 

coefficients with Reynolds number provided a good indication of the onset of the instability.

However, for an inclination of 60 degrees at 0-degree yaw angle, large wind-induced oscillations 

were observed for most of the models and angles of attack investigated. These large oscillations

could not be predicted solely based on the variations of the static force coefficients with 

Reynolds number and angles of attack. For these cases, increasing structural damping had only a 

small influence on the amplitude of the vibrations. However, for the cable model with helical 

fillets, increasing the Scruton number mitigated the vibrations as was first anticipated at the onset 

of the study.
31 

CHAPTER 4. NUMERICAL SIMULATIONS

MOTIVATION

The full-scale measurements of HDPE cable casings on stay cables have shown deviations from 

the intended circular cross section. Analysis of wind-tunnel tests suggested that this anomaly 

may cause instabilities over a range of wind directions and angles of attack based on a linearized 

quasi-static model.(2,7) Moreover, the estimated damping demands indicated the possibility that 

cable dampers may not be fully capable of mitigating wind-induced excitations on bridges built 

according to current design practice. Thus far, however, there is no strong evidence of 

unexplained and/or difficult-to-control cable vibrations on the existing bridges. 

For reasons of practicality, numerical methods were used to replicate actual full-scale conditions. 

At a certain point, expanding the wind-tunnel tests by adding and varying additional conditions 

was not feasible. The researchers hoped that the quasi-static theory might be able to predict 

reasonably well the observed responses and extrapolate closer to full-scale conditions. 

The linearized quasi-static aerodynamic damping model, in particular, allows for screening of 

potential quasi-static instabilities. However, the model does not allow for assessment of the 

self-limited features occasionally observed in classical galloping-like responses. Due to the rapid 

changes observed in the force coefficients with angle of attack, the cable may be expected to 

transition quickly from an unstable to a stable regime on the force coefficient curve once it 

begins to move. Therefore, extending the current linearized formulation to a nonlinear numerical 

method where these effects can be reproduced is necessary. 

In this study, a comprehensive effort, including static and dynamic wind-tunnel tests, has been 

carried out to measure loads and simulate cable vibrations due to the shape variations. The 

Methodology section presents the ongoing research to evaluate, via testing and numerical 

response analyses, the conditions at full scale that may mitigate the vibrations, such as the 

influence of level of correlation of the cross-sectional shapes of the stay cables as a function of 

their length and wind turbulence. 

METHODOLOGY

Natural Wind Effects

Previous research has shown that turbulence may have mitigative effects on stay-cable 

vibrations.(9)

 Depending on the intensity and spatial properties, natural turbulence may have a 

significant impact on the cable response to wind. The turbulent fluctuations may cause 

transitions from subcritical to supercritical flow regimes (and vice versa). As the cable vibrates, 

its motions will superimpose with those caused by the turbulence fluctuations, further modifying 

the direct effects on the flow about the cable. These effects can be attributed to small length scale 

turbulent fluctuations (fluctuations at high frequencies that affect the flow in the boundary layer 

at the surface of the cable). In addition, the larger length scale variations will also affect the 

overall correlation of the fluctuating loads acting on long cables. The problem becomes 

extremely complex, given the cross sections are generally found nonsymmetric where even slight 

out-of-roundness (about 0.5–1 percent) was demonstrated to cause significant changes in the 
32 

aerodynamic properties. A sensitivity analysis of the wind-tunnel measurements showed that 

small changes in the mean angle of attack may cause a cross section to exhibit aerodynamic 

instabilities or to become stable. Therefore, variations of wind speed and direction over a few 

minutes until oscillations build up may also have a significant impact on the vibrating cables. 

Coordinate Framework

Considering the 3D stay alignment, wind directionality, and structural response shown in 

figure 25, the following coordinate frameworks were selected to represent: 

• The wind with coordinates x', y', z' and velocity vectors u, v, w. 

• Structural coordinates x, y, z, and the locally aligned wind velocity vectors us, vs, ws. 

• Cable loads with X, Y, Z (body system) and locally aligned wind velocity vectors Ul, Vl, 

Wl. 

Source: FHWA.

Note: Shown are the definitions of the yaw angle β, the inclination angle θ, the cable-wind angle Φ, and the angle of 

attack α. The drag force lies in the cable-wind plane, and the lift force is normal to this plane.

Figure 25. Illustration. Sketch of the three coordinate systems: global (or structural), wind, 

and cable (or body). 
33 

The cable inclination angle from the horizontal plane is θ with wind direction angle β (yaw) 

yielding the effective cable-wind angle (Φ): 

(2)

Due to wind turbulence and structural motions, these angles will vary in time where: 

(3)

Where are speed of structural response and θ0 is the static cable inclination angle. For

equation 3, wind is aligned in its mean direction, β0. The general transformation from structural 

to wind coordinates is: 

(4)

The normalized transformation matrix [Tʹ] from structural to body coordinates is: 

(5)

The transformations between structural and body coordinates performed as: 

(6)

and for wind from global to local coordinates via: 

(7)
34 

The quasi-static forces are computed from: 

(8)

Where CA is the along-cable static force coefficient and  , where  is th

fluctuating component. Given the cross section is not symmetric about its longitudinal axis Y, the 

forces will generally depend on the angle of local wind attack, as well on the instantaneous 

velocity (comprising structural motions) and effective wind angle reflecting the variations of 

both β and θ. 

Similar to previous studies the angle of local wind attack is calculated from:

(9,10) 

(9)

Where α(t) is the instantaneous variation of angle of wind attack. This angle of attack is 

measured off the drag axis X. The Reynolds number is calculated as: 

(10)

Where v is air kinematic viscosity. Due to its insignificant effect (actual wind speed fluctuations 

much higher compared to structural speeds) the structural motions term xẋ is omitted. All force 

coefficients are derived based on the effective wind direction Φ. Likewise, the reference wind

velocity uū is the speed used for force coefficient normalization, thus, the mean speed in the wind 

coordinate system (or wind tunnel). 

Linear Quasi-Static Formulation 

The potential for dry cable galloping has been assessed using a linearization of the quasi-static 

forces by Macdonald and Larose.(7,11,12) In this linearized quasi-static formulation, the 

aerodynamic damping is expected to be a function of angle of attack α, the cable-wind angle Φ, 

and the Reynolds number Re: 
35 

(11)

Where: 

μ = air dynamic viscosity. 

ζaero = aerodynamic damping (ratio of critical). 

ωn = circular frequency of the nth cable mode. 

It should be noted that the angle of attack used in equation 11 has an inherent implication about 

the direction of body motion. For along-wind motions the value for the angle of attack is 

α = 0 degrees, and for across-wind motions it should be α = 90 degrees. However, as indicated 

by the “0” subscript, the values for the change in drag and lift with angle of attack should be 

calculated about the 0-degree α position (i.e., the along-wind axis in figure 25). Therefore, for a 

yaw angle of 0 degrees, the predicted damping in the structural y-direction reduces to a form 

similar to the familiar Den Hartog expression for galloping:(13) 

(12)

Nonlinear Quasi-Static Formulation 

Equations 2, 3, 9, and 10 are all functions of time and can be computed at each time step of a 

numerical simulation. Based on the instantaneous values of angle of attack α, the cable-wind 

angle Φ, and the Reynolds number Re, the instantaneous force coefficients are computed at each 

time step based on interpolations from the measured values. The computed CD(t) = CD(α, Re, Φ) 

and CL(t) = CL(α, Re, Φ) are substituted into equation 8 to obtain the instantaneous force per unit 

length in the cable coordinate system. These cable-oriented forces are rotated back to the 

structural coordinate system following the transformation of equation 6, and the system of 

ordinary differential equations is solved at each time step using the generalized multidegree of 

freedom (MDOF) state-space representation, e.g.(14) 

(13)

Where: 
36 

  = generalized responses of the cable.

F* = the generalized response-dependent forces. 

I = the identity matrix. 

C* = the generalized (diagonal) damping matrix. 

K* = the generalized (diagonal) stiffness matrix. 

No modifications to the damping or stiffness matrices are performed to account for the selfexcited forces as these forces are implicitly calculated by using the instantaneous values of angle 

of attack α, the cable-wind angle Φ, and the Reynolds number Re. The time step in the 

simulation was set to 0.005 s, and the simulation was computed until a steady-state motion was 

achieved. Given the nonlinear nature of the loading, at every time step an iteration procedure 

based on response convergence has been applied where the solution above is carried out in a 

MDOF framework, which allows for the flexibility to numerically model more closely the 

assembly process of actual stay cables. 

As previously discussed, it has been observed that cable cross sections are not perfectly circular, 

and the aerodynamic forces have a strong dependence on the particular cross-sectional shape. 

Stay cables are sheathed with HDPE in a segmented fashion. Thus, even if each cross section 

was the same from one segment to the next, it is unlikely that the initial angle of attack αα is the 

same for each segment. In addition to modeling the assembly process of stay cables, the 

proposed numerical methodology can also incorporate the effect of turbulence. To simulate the 

effects of turbulence, one can either use available site measurement data or synthesize turbulent 

flows through autoregression or other available numerical techniques.1

EXPERIMENTS

There are two sets of experimental data used in the current study. The first set of wind tunnel 

experiments were performed to measure the drag and lift coefficients over a wide range of values 

of angle of attack α, cable-wind angles Φ, and Reynolds numbers Re.

(2) These data are 

interpolated so that the instantaneous drag and lift coefficients can be obtained based on the 

instantaneous values of each variable. This 3D interpolation “volume” forms the input required 

for the numerical simulations.

The second experiment set, described in chapter 3 of this report, provided free-vibration response 

data to compare with the performance of the proposed numerical simulation methodology. 

Figure 26 shows the cases considered in the current study. 

1 RWDI Reference BR01-2007, “Numerical Simulation of Wind Turbulence”, March 7, 2007.
37 

Source: FHWA.

Figure 26. Illustration. Analyzed cross-section shapes: (a) IR113E-HF (exaggerated by a 

factor of 10) and (b) the flat-face symmetric cable.

RESULTS

Aerodynamic Damping

Each term in equation 11 has been computed for the case of IR113E-HF at a yaw angle of 

0 degrees, a cable inclination of 60 degrees, and Re = 2.4×105. With most terms, identically zero 

for this case, it is clear from figure 27 that the dominant term providing negative aerodynamic 

damping is the rate-of-change of lift with the angle of attack, where the drag term provides 

positive aerodynamic damping. It is observed that the variation in predicted damping with angle 

of attack is not smooth but changes abruptly over small changes in the angle of attack. Based on 

the observations from figure 27, the initial angle of attack producing the most negative 

aerodynamic damping in this case was near −51 degrees. 
38 

Source: FHWA.

Note: The yaw and inclination angles are set to 0 and 60 degrees, respectively, with Re = 2.4×105

. The legend 

entries keep only the force coefficient portion of each term.

Figure 27. Graphs. Across-wind aerodynamic damping predicted by equation 11 for (a) the 

total aerodynamic damping and (b) the contribution of each term.

Indian River Bridge IR113E-HF at 0-Degree Yaw

Numerical simulations have been performed based on the approach described in the chapter 4

Methodology section. The worst-case angle of attack α = −51 degrees for IR113E-HF as 

estimated from equation 11 has been simulated numerically as tested in the wind tunnel. The 

results are shown in figure 28 for a Scruton number equal to 5.3. In general, the numerical 

simulations did not predict any steady-state motion of the cable, whereas significant responses 

were clearly observed in the wind-tunnel tests.
39 

Source: FHWA.

Note: The x-direction is along-wind and the y-direction is across-wind. 

Figure 28. Graph. Numerical simulations versus experimental results for Indian River 

Bridge cable IR113E-HF with Sc = 5.3 and yaw and inclination angles of 0 and 60 degrees, 

respectively. 

However, good agreement is found for a Reynolds number of 2.4×105

. It should be noted that the 

numerical simulations were performed with and without initial excitations of the cable, and the 

same steady-state results were achieved regardless of the initial excitation.

As shown in the 2021 FHWA report, the drag and lift coefficients change rapidly with both 

Reynolds number and angle of attack for this case.(2) The drag and lift coefficients as a function 

of angle of attack are presented in figure 29 for Re = 2.4×105. It should be noted that the “data 

points” in figure 29 themselves are a result of Reynolds number interpolations. The force 

coefficient data were obtained by fixing the angle of attack and increasing the wind speed to 

cover a wide range of Reynolds number. Therefore, to extract data at a particular Reynolds 

number, an interpolation from these runs was required. 
40 

Source: FHWA.

Figure 29. Graphs. Comparison of numerical simulations to interpolated data for (a) drag 

and (b) lift coefficients for IR113E-HF for Φ = 60 degrees and Re = 2.4×105. 

As observed from figure 29, the aerodynamic excitation grows over the portion of the lift curve 

where there is a significant negative slope. However, as soon as the instantaneous angle of attack 

(equation 9) exceeds this negative slope region, the aerodynamic forces return to a stabilizing 

regime. Thus, the shape of the force coefficient curves leads to a steady-state response, as shown 

in figure 30. As a further illustration, the linear estimate of aerodynamic damping, in equation 11 

has been calculated at each instant of time in figure 31. 
41 

Source: FHWA.

Figure 30. Graphs. Time series of response for IR113E-HF at Φ = 60 degrees and 

Re = 2.4×105 with Sc = 5.3 where (a) is response plotted against time and (b) is y-response 

plotted against x-response with the same scale on both axes.

Source: FHWA.

Note: Relative probability histograms are calculated based on the linear approximation shown in equation 11 at 

each instant of time corresponding to the instantaneous parameters α, Φ, and Re. 

Figure 31. Graphs. The (a) aerodynamic damping (right axis) plotted with angle of attack α

(left axis) for IR113E-HF at Φ = 60 degrees and Re = 2.4×105 with Sc = 5.3, combined with 

the relative probability (counts per bin/total) of the damping in (b) heave and (c) sway. 

This estimate is based on the instantaneous drag and lift coefficients and their slopes determined 

from the angle of attack α, cable-wind angle Φ, and Re. 
42 

The structural damping in this case was approximately 0.6 percent of critical damping. Figure 31

shows that the negative aerodynamic damping exceeds this basic structural damping at some 

angles of attack but returns to positive values at sufficiently negative angles of attack (as 

measured from the initial angle of attack).

Wind-tunnel data are also available for a case with increased Scruton number (Sc = 9.7). For this 

case, motions up to approximately 0.5 times the cable diameter were observed in the wind tunnel 

but could not be recreated for any Re in the numerical simulations. Therefore, it appears these 

motions might be a different phenomenon causing the instability, which cannot be simulated by 

the quasi-static numerical model. 

Flat-Face Reference Shape Cable at 0-Degree Yaw 

The free vibration wind-tunnel data indicate that the flat-face cable model will be unstable for 

Re > 2×105 with initial angles of attack of ±20 degrees (due to symmetry). Numerical 

simulations have been carried out for this case and comparisons with the wind-tunnel data are 

shown in figure 32. There is reasonable agreement observed between the numerical predictions 

and the wind-tunnel results for this case acknowledging that the wind-tunnel test had to be 

stopped due to large cable motions. 

Source: FHWA.

Note: The x-direction is along-wind and the y-direction is across-wind.

Figure 32. Graph. Comparison of numerical predictions to experimental results for the 

flat-face cable with Sc =17.5 with initial angle of attack set to 20 degrees, and with yaw and 

inclination angles set to 0 and 60 degrees, respectively. 

Flat-Face Cable Model with Spatial Variation

Following the initial tests and simulations of the flat face cable model with a constant angle of 

attack, wind-tunnel tests were performed with spatial variation of the initial angle of attack. The 

cable was divided into three segments of the same cross section such that each segment could 

have a different initial angle of attack. Spatial variation was modeled numerically using multiple 

nodes along the cable length. Simulations have been performed for two of these cases as shown 

in figure 33. In figure 33(a) the lowest portion of the cable was set to 80 degrees with the middle 
43 

and top parts both set to an initial angle of attack of 10 degrees. In the 2015 static tests, it was 

observed that the flat face cable model had adverse force coefficients at an angle of attack of 

80 degrees. For figure 33(b), the bottom and top of the section were set to an initial angle of 

attack of 80 degrees with the middle portion set to 30 degrees. 

Source: FHWA.

Note: The x-direction is along-wind and the y-direction is across-wind.

Figure 33. Graphs. Comparison of numerical simulations to experimental results for the 

flat-face cable of three rotated sections with initial angle of attack of (a) 80, 10, and 10 

degrees and (b) 80, 30, and 80 degrees with Sc = 20 and yaw and inclinations angles set to 0 

and 60 degrees, respectively. 

In the first case (figure 33(a)), no motion was observed in either the wind-tunnel experiments or 

the numerical simulations. In the second case (figure 33(b)), large amplitude motions still 

persisted in the wind tunnel at Re > 2.7×105

; however, no responses were predicted numerically. 

Indian River Bridge IR113E-HF at 60-Degree Yaw 

The numerical framework here combined with the extensive database of force coefficients 

permitted simulations for a wide range of yaw angles in addition to variations in the initial angle 

of attack. Wind-tunnel tests were also performed over a range of yaw angles; however, no 

significant responses indicative of an aerodynamic instability were observed. 
44 

This observation is consistent with the numerical simulations where no quasi-static instability 

was observed. An example of these simulations and wind-tunnel measurements is shown in 

figure 34. 

Source: FHWA.

Note: The x-direction is along-wind, and the y-direction is across-wind.

Figure 34. Graph. Comparison of numerical simulations to experimental results of 

IR113E-HF with Sc = 5.2 with initial angle of attack set to -55 degrees, and yaw and 

inclination angles set to 60 and 60 degrees, respectively. 

Simulations for IR113E-HF at 0-Degree Yaw with Turbulence

The case considered earlier for IR113-HF at 0-degree yaw and Re = 2.4×105 (figure 28) where 

good agreement was observed with the wind-tunnel tests has been subjected to numerically

generated turbulence with turbulence intensity Iu = σu/uū of approximately 5 percent. The 

turbulence generally follows the shape of the von Kármán spectra for each wind direction. 

As observed in figure 35, the previously predicted instability in smooth flow was suppressed by 

turbulence. The rapid variation of instantaneous angle of attack, Re, and cable-wind angle due to 

the approaching turbulent flow implies that little time is spent at the unstable portion of the force 

coefficient curve to allow an instability growth; the typical buffeting response is predicted 

instead.
45 

Source: FHWA.

Figure 35. Graph. Time series of response for IR113E-HF at Φ = 60 degrees and 

Re = 2.4×105 with Sc = 5.3 and Iu ≈ 5 percent where (a) is response plotted against time and 

(b) is y-response plotted against x-response with the same scale on both axes.

DISCUSSIONS AND CONCLUSIONS

The 3D nonlinear quasi-static modeling approach showed good agreement to what was observed 

in the wind-tunnel experiments for cables inclined in the wind direction. In other cases, 

especially for a cable-wind angle of 60 degrees, the comparisons showed that the proposed 

aerodynamic model cannot explain the wind-tunnel results. The likely reasons for these 

discrepancies are: 

• Lack of details in the static force coefficients data: There were trends of very sharp 

gradients in the transitions from stable to unstable regions and vice versa requiring 

detailed knowledge of the force coefficients over the three-parameter space: angle of 

attack, cable-wind angle, and Reynolds number. Even though very fine increments of 

angle of attack (e.g., approximately 1 degree) had been measured in some cases, it was a 

practical challenge to sample the entire parameter space experimentally. 

• Different aerodynamic phenomena caused the instabilities: Some of the large amplitude 

vibrations found during the test could not be predicted and/or did not follow the general 

expectations based on quasi-static theory. For example, physical tests with increased 

damping in the wind tunnel had shown a limited influence of the added damping on the 

oscillations, even at high levels of Scruton number. These observations indicated that 

another aerodynamic excitation mechanism (perhaps stiffness driven) might play a 

significant role in the observed responses. Identifying and modeling aerodynamic 

phenomena outside of those described by quasi-static theory was beyond the scope of the 

current research program.

Numerical tests showed that turbulence might be able to suppress the vibrations predicted in 

smooth flow. 

47 

APPENDIX. TEST LOGS

Test logs of the dynamic tests conducted for this study are provided in table 3.
48 

Table 3. Test log data for test number 1003. 

Run 

Num Points 

Tare 

Run 

Inclination

Angle

(Degrees) 

Cable Axis

Rotation 

(Degrees) 

Turntable

Yaw 

(Degrees) 

Phi

(Degrees) 

Sway

(Hz) 

Heave

(Hz) 

Sway

(Percent) 

Heave

(Percent) 

Cable 

Name Case Type 

31 1 30 60 0 0 60 4.39 4354 0.27 0.27 Flat-face Dynamic 

33 14 32 60 0 0 60 4.39 4354 0.27 0.27 Flat-face Dynamic 

56 7 54 60 10 0 60 4.41 4.25 0.32 0.30 Flat-face Dynamic 

57 8 54 60 10 0 60 4.41 4.25 0.32 0.30 Flat-face Dynamic 

60 12 58 60 20 0 60 4.41 4.24 0.33 0.30 Flat-face Dynamic 

63 13 61 60 30 0 60 4.42 4.24 0.35 0.30 Flat-face Dynamic 

66 14 64 60 40 0 60 4.41 4.24 0.32 0.30 Flat-face Dynamic 

69 14 67 60 60 0 60 4.42 4.24 0.35 0.30 Flat-face Dynamic 

72 15 70 60 80 0 60 4.42 4.24 0.32 0.30 Flat-face Dynamic 

75 16 73 60 90 0 60 4.41 4.24 0.32 0.29 Flat-face Dynamic 

78 12 76 60 100 0 60 4.41 4.24 0.35 0.30 Flat-face Dynamic 

81 18 79 60 120 0 60 4.41 4.25 0.32 0.30 Flat-face Dynamic 

87 14 84 60 0 0 60 4.41 4.25 0.42 0.35 Flat-face Dynamic 

92 15 90 60 0 0 60 4.41 4.24 0.72 0.70 Flat-face Dynamic 

96 15 94 60 0 0 60 4.41 4.24 0.72 0.68 Flat-face Dynamic 

99 16 97 60 20 0 60 4.41 4.24 0.74 0.69 Flat-face Dynamic 

102 14 100 60 30 0 60 4.41 4.24 0.74 0.70 Flat-face Dynamic 

105 15 103 60 40 0 60 4.41 4.24 0.74 0.70 Flat-face Dynamic 

108 16 106 60 60 0 60 4.41 4.24 0.80 0.70 Flat-face Dynamic 

111 16 109 60 80 0 60 4.41 4.24 0.75 0.69 Flat-face Dynamic 

114 12 112 60 100 0 60 4.41 4.24 0.78 0.68 Flat-face Dynamic 

117 18 115 60 120 0 60 4.41 4.24 0.78 0.70 Flat-face Dynamic 

120 19 118 60 120 0 60 4.40 4.23 1.45 1.32 Flat-face Dynamic 

123 17 121 60 100 0 60 4.40 4.23 1.45 1.27 Flat-face Dynamic 

126 18 124 60 80 0 60 4.40 4.23 1.40 1.30 Flat-face Dynamic 

132 16 130 60 60 0 60 4.40 4.23 1.35 1.27 Flat-face Dynamic 

135 10 133 60 40 0 60 4.40 4.23 1.35 1.30 Flat-face Dynamic 

138 9 136 60 30 0 60 4.40 4.23 1.35 1.30 Flat-face Dynamic 

141 12 139 60 20 0 60 4.40 4.23 1.20 1.35 Flat-face Dynamic 
49 

Run 

Num Points 

Tare 

Run 

Inclination

Angle

(Degrees) 

Cable Axis

Rotation 

(Degrees) 

Turntable

Yaw 

(Degrees) 

Phi

(Degrees) 

Sway

(Hz) 

Heave

(Hz) 

Sway

(Percent) 

Heave

(Percent) 

Cable 

Name Case Type 

144 9 142 60 0 0 60 4.40 4.23 1.13 1.22 Flat-face Dynamic 

152 8 150 60 0 0 60 4.40 4.23 2.07 2.01 Flat-face Dynamic 

155 13 153 60 20 0 60 4.40 4.23 2.06 2.09 Flat-face Dynamic 

159 8 157 60 30 0 60 4.40 4.23 2.06 2.09 Flat-face Dynamic 

162 10 160 60 40 0 60 4.40 4.23 2.00 1.97 Flat-face Dynamic 

165 9 163 60 60 0 60 4.40 4.23 1.98 1.95 Flat-face Dynamic 

168 9 166 60 80 0 60 4.40 4.23 1.98 1.88 Flat-face Dynamic 

171 7 169 60 100 0 60 4.40 4.23 2.10 2.00 Flat-face Dynamic 

174 7 172 60 120 0 60 4.40 4.23 1.90 1.88 Flat-face Dynamic 

177 8 175 60 0 90 90 4.40 4.23 1.87 1.85 Flat-face Dynamic 

181 15 179 60 −155 0 60 4.40 4.23 1.97 1.88 IR118E Dynamic 

184 15 182 60 −140 0 60 4.40 4.23 1.98 1.84 IR118E Dynamic 

188 15 185 60 −126 0 60 4.40 4.23 1.97 1.91 IR118E Dynamic 

191 14 189 60 180 0 60 4.40 4.23 1.89 1.88 IR118E Dynamic 

194 14 192 60 −54 0 60 4.40 4.23 1.92 1.87 IR118E Dynamic 

197 15 195 60 0 0 60 4.40 4.23 1.85 1.86 IR118E Dynamic 

201 15 199 60 90 0 60 4.40 4.23 1.87 1.86 IR118E Dynamic 

204 14 202 60 110 0 60 4.40 4.23 1.88 1.84 IR118E Dynamic 

207 15 206 60 130 0 60 4.40 4.23 1.96 1.86 IR118E Dynamic 

211 14 208 60 150 0 60 4.40 4.23 1.87 1.85 IR118E Dynamic 

214 15 212 60 −100 0 60 4.40 4.23 1.89 1.90 IR118E Dynamic 

217 9 215 60 70 0 60 4.40 4.23 1.89 1.85 IR118E Dynamic 

221 10 219 60 134 0 60 4.40 4.23 1.88 1.89 IR118E Dynamic 

224 10 222 60 −105 0 60 4.40 4.23 1.88 1.88 IR118E Dynamic 

227 11 225 60 −105 0 60 4.40 4.23 3.43 3.72 IR118E Dynamic 

230 16 228 60 −126 0 60 4.40 4.23 3.92 3.78 IR118E Dynamic 

233 12 231 60 134 0 60 4.40 4.23 3.90 3.80 IR118E Dynamic 

236 13 234 60 180 0 60 4.40 4.23 3.90 3.70 IR118E Dynamic 

239 26 237 60 180 0 60 4.40 4.23 4.70 4.80 IR118E Dynamic 

242 12 240 60 134 0 60 4.40 4.23 4.80 4.50 IR118E Dynamic 
50 

Run 

Num Points 

Tare 

Run 

Inclination

Angle

(Degrees) 

Cable Axis

Rotation 

(Degrees) 

Turntable

Yaw 

(Degrees) 

Phi

(Degrees) 

Sway

(Hz) 

Heave

(Hz) 

Sway

(Percent) 

Heave

(Percent) 

Cable 

Name Case Type 

245 14 243 60 −126 0 60 4.40 4.23 4.50 4.70 IR118E Dynamic 

248 12 246 60 −105 0 60 4.40 4.23 4.80 4.70 IR118E Dynamic 

251 14 249 60 −105 0 60 4.40 4.23 5.90 5.90 IR118E Dynamic 

259 14 257 60 134 0 60 4.40 4.23 5.90 5.90 IR118E Dynamic 

262 11 260 60 −100 0 60 4.40 4.23 3.60 3.70 IR118E Dynamic 

265 24 263 60 −136 32 65 4.40 4.23 3.50 3.60 IR118E Dynamic 

268 16 266 60 −151 47 70 4.40 4.23 3.80 3.70 IR118E Dynamic 

273 12 271 60 −162 59 75 4.40 4.23 3.70 3.60 IR118E Dynamic 

277 22 275 60 −172 70 80 4.40 4.23 3.70 3.60 IR118E Dynamic 

283 20 281 60 170 90 90 4.40 4.23 3.80 3.60 IR118E Dynamic 

287 16 285 60 −172 70 80 4.40 4.23 1.30 1.20 IR118E Dynamic 

290 15 288 60 −136 32 65 4.40 4.23 1.21 1.11 IR118E Dynamic 

296 11 294 60 −136 32 65 4.40 4.23 2.40 2.10 IR118E Dynamic 

299 11 297 60 −106 5 60 4.40 4.23 2.20 2.20 IR118E Dynamic 

302 13 300 60 358 0 60 5.20 4.90 2.40 2.50 LargeCF Dynamic 

306 25 305 60 −80 0 60 4.35 4.17 2.50 2.50 IR113HF Dynamic 

309 16 307 60 −80 0 60 4.35 4.17 1.17 1.22 IR113HF Dynamic 

312 18 310 60 −48 0 60 4.35 4.17 1.22 1.25 IR113HF Dynamic 

315 15 313 60 −28 0 60 4.35 4.17 1.18 1.24 IR113HF Dynamic 

318 16 316 60 10 0 60 4.35 4.17 1.23 1.16 IR113HF Dynamic 

321 17 319 60 35 0 60 4.35 4.17 1.20 1.20 IR113HF Dynamic 

324 18 322 60 45 0 60 4.35 4.17 1.21 1.14 IR113HF Dynamic 

327 16 325 60 50 0 60 4.35 4.17 1.19 1.18 IR113HF Dynamic 

331 16 329 60 55 0 60 4.35 4.17 1.20 1.22 IR113HF Dynamic 

334 17 332 60 100 0 60 4.35 4.17 1.25 1.26 IR113HF Dynamic 

337 21 335 60 63 0 60 4.35 4.17 1.19 1.25 IR113HF Dynamic 

340 14 338 60 63 0 60 4.35 4.17 2.27 2.18 IR113HF Dynamic 

344 14 341 60 100 0 60 4.35 4.17 2.27 2.09 IR113HF Dynamic 

347 14 345 60 50 0 60 4.35 4.17 2.10 2.10 IR113HF Dynamic 

350 14 348 60 49 0 60 4.35 4.17 2.10 2.00 IR113HF Dynamic 
51 

Run 

Num Points 

Tare 

Run 

Inclination

Angle

(Degrees) 

Cable Axis

Rotation 

(Degrees) 

Turntable

Yaw 

(Degrees) 

Phi

(Degrees) 

Sway

(Hz) 

Heave

(Hz) 

Sway

(Percent) 

Heave

(Percent) 

Cable 

Name Case Type 

353 10 351 60 49 0 60 4.35 4.17 1.28 1.05 IR113HF Dynamic 

357 14 354 60 49 0 60 4.35 4.17 0.68 0.62 IR113HF Dynamic 

359 14 354 60 50 0 60 4.35 4.17 0.67 0.64 IR113HF Dynamic 

362 16 360 60 63 0 60 4.35 4.17 0.66 0.64 IR113HF Dynamic 

365 15 363 60 100 0 60 4.35 4.17 0.66 0.64 IR113HF Dynamic 

371 14 369 60 −80 0 60 4.35 4.17 0.66 0.64 IR113HF Dynamic 

374 19 372 60 19 32 65 4.35 4.17 0.64 0.65 IR113HF Dynamic 

377 19 375 60 −84 32 65 4.35 4.17 0.64 0.64 IR113HF Dynamic 

380 20 378 60 4 47 70 4.35 4.17 0.66 0.64 IR113HF Dynamic 

383 19 381 60 −7 59 75 4.35 4.17 0.65 0.63 IR113HF Dynamic 

386 19 384 60 −17 70 80 4.35 4.17 0.66 0.63 IR113HF Dynamic 

389 14 384 60 −26 80 85 4.35 4.17 0.65 0.64 IR113HF Dynamic 

392 14 390 60 −35 90 90 4.35 4.17 0.64 0.63 IR113HF Dynamic 

395 14 393 60 −138 90 90 4.35 4.17 0.62 0.63 IR113HF Dynamic 

399 12 396 60 180 0 60 3.77 3.64 4.40 4.40 IR118E Dynamic 

402 13 400 60 180 0 60 4.30 4.17 6.60 6.40 IR118E Dynamic 

410 11 408 60 20 0 60 5.00 4.76 1.60 1.56 Flat-face2 Dynamic 

413 17 411 60 80 0 60 5.00 4.76 1.59 1.54 Flat-face2 Dynamic 

415 12 414 60 80 0 60 5.00 4.76 1.59 1.54 Flat-face2 Dynamic 

418 13 416 60 120 0 60 5.00 4.76 1.62 1.57 Flat-face2 Dynamic 

421 9 419 60 20 0 60 5.00 4.76 1.63 1.53 Flat-face2 Dynamic 

424 19 422 60 20 0 60 5.00 4.76 1.61 1.56 Flat-face2 Dynamic 

427 15 425 60 20 0 60 5.00 4.76 1.61 1.56 Flat-face2 Dynamic 

430 23 428 60 20 0 60 5.00 4.76 1.61 1.58 Flat-face2 Dynamic 

435 13 433 60 20 0 60 5.00 4.76 1.64 1.52 Flat-face2 Dynamic 

440 12 438 60 20 0 60 4.90 4.70 3.28 3.10 Flat-face2 Dynamic 

443 17 441 60 20 0 60 4.90 4.70 3.20 3.14 Flat-face2 Dynamic 

446 18 445 60 80 0 60 4.90 4.70 3.25 3.00 Flat-face2 Dynamic 

449 11 447 60 30 0 60 4.90 4.70 3.26 3.37 Flat-face2 Dynamic 

452 16 450 60 10 0 60 4.92 4.72 3.40 3.10 Flat-face2 Dynamic 
52 

Run 

Num Points 

Tare 

Run 

Inclination

Angle

(Degrees) 

Cable Axis

Rotation 

(Degrees) 

Turntable

Yaw 

(Degrees) 

Phi

(Degrees) 

Sway

(Hz) 

Heave

(Hz) 

Sway

(Percent) 

Heave

(Percent) 

Cable 

Name Case Type 

456 12 454 60 80 0 60 4.92 4.72 3.27 3.14 Flat-face2 Dynamic 

459 15 457 60 100 0 60 4.92 4.72 3.32 3.08 Flat-face2 Dynamic 

462 10 460 60 120 0 60 4.92 4.72 3.32 3.08 Flat-face2 Dynamic 

671 21 668 60 250 90 90 4.92 4.70 0.57 0.41 SIS1 Dynamic 

674 19 672 60 240 90 90 4.92 4.70 0.55 0.42 SIS1 Dynamic 

677 21 675 60 230 90 90 4.92 4.70 0.56 0.44 SIS1 Dynamic 

680 18 678 60 220 90 90 4.92 4.70 0.59 0.42 SIS1 Dynamic 

683 17 681 60 210 90 90 4.92 4.70 0.56 0.43 SIS1 Dynamic 

686 12 684 60 200 90 90 4.92 4.70 0.59 0.44 SIS1 Dynamic 

689 16 687 60 180 90 90 4.92 4.70 0.60 0.42 SIS1 Dynamic 

693 19 690 60 80 90 90 4.92 4.70 0.53 0.42 SIS1 Dynamic 

696 17 694 60 85 90 90 4.92 4.70 0.54 0.45 SIS1 Dynamic 

699 18 697 60 70 90 90 4.92 4.70 0.56 0.44 SIS1 Dynamic 

702 19 700 60 30 90 90 4.92 4.70 0.62 0.44 SIS1 Dynamic 

708 26 706 60 10 90 90 4.92 4.70 0.64 0.42 SIS1 Dynamic 

711 15 709 60 320 90 90 4.92 4.70 0.68 0.43 SIS1 Dynamic 

714 9 712 60 210 0 60 4.92 4.70 0.58 0.41 SIS1 Dynamic 

720 14 718 60 250 0 60 4.92 4.70 0.76 0.44 SIS1 Dynamic 

723 18 721 60 30 0 60 4.92 4.70 0.58 0.40 SIS1 Dynamic 

726 17 724 60 214 32 65 4.92 4.70 0.51 0.42 SIS1 Dynamic 

729 16 727 60 240 90 90 4.92 4.70 1.22 1.23 SIS1 Dynamic 

734 18 735 60 10 90 90 4.92 4.70 1.54 1.24 SIS1 Dynamic 

476 18 475 60 20 0 60 0.00 0.00 0.00 0.00 Flat-face Static

478 21 477 60 10 90 90 0.00 0.00 0.00 0.00 SIS1 Static

480 23 479 60 30 90 90 0.00 0.00 0.00 0.00 SIS1 Static

484 15 483 60 50 90 90 0.00 0.00 0.00 0.00 SIS1 Static

486 22 485 60 50 90 90 0.00 0.00 0.00 0.00 SIS1 Static

488 22 487 60 70 90 90 0.00 0.00 0.00 0.00 SIS1 Static

491 22 490 60 90 90 90 0.00 0.00 0.00 0.00 SIS1 Static

493 22 492 60 170 90 90 0.00 0.00 0.00 0.00 SIS1 Static
53 

Run 

Num Points 

Tare 

Run 

Inclination

Angle

(Degrees) 

Cable Axis

Rotation 

(Degrees) 

Turntable

Yaw 

(Degrees) 

Phi

(Degrees) 

Sway

(Hz) 

Heave

(Hz) 

Sway

(Percent) 

Heave

(Percent) 

Cable 

Name Case Type 

495 22 494 60 190 90 90 0.00 0.00 0.00 0.00 SIS1 Static

497 22 496 60 210 90 90 0.00 0.00 0.00 0.00 SIS1 Static

499 22 498 60 230 90 90 0.00 0.00 0.00 0.00 SIS1 Static

501 22 500 60 250 90 90 0.00 0.00 0.00 0.00 SIS1 Static

503 22 502 60 10 0 60 0.00 0.00 0.00 0.00 SIS1 Static

506 22 505 60 30 0 60 0.00 0.00 0.00 0.00 SIS1 Static

508 22 507 60 70 0 60 0.00 0.00 0.00 0.00 SIS1 Static

510 22 509 60 90 0 60 0.00 0.00 0.00 0.00 SIS1 Static

512 22 511 60 250 0 60 0.00 0.00 0.00 0.00 SIS1 Static

514 22 513 60 230 0 60 0.00 0.00 0.00 0.00 SIS1 Static

516 22 515 60 210 0 60 0.00 0.00 0.00 0.00 SIS1 Static

518 22 517 60 190 0 60 0.00 0.00 0.00 0.00 SIS1 Static

519 24 517 60 170 0 60 0.00 0.00 0.00 0.00 SIS1 Static
54

REFERENCES

1. Bosch, H., and J. Pagenkopf. 2017. “Development of a Robot to Measure Stay Cable

Roundness.” Presented at the 2017 International Symposium on the Dynamics and

Aerodynamics of Cables. Oct. 30–31. Porto, Portugal.

2. Christiansen H., and G. L. Larose. 2021. Wind-Tunnel Investigations of the Aerodynamics of

Bridge Stay-Cable Cross-sectional Shapes. Report No. FHWA-HRT-21-074. Washington,

DC: Federal Highway Administration.

3. Schewe, G. 1986. “Sensitivity of transition phenomena to small perturbations in flow around

a circular cylinder.” Journal of Fluid Mechanics 172, Nov: 33–46.

4. Christiansen, H. 2016. “Aerodynamics of bridge stay cables—Wind tunnel studies.” Ph.D.

dissertation UiS no. 294. Department of Mechanical and Structural Engineering and

Materials Science, University of Stavanger, Norway.

5. Zdravkovich, M. M. 1997. Flow Around Circular Cylinders Volume 1: Fundamentals.

Oxford, UK: Oxford University Press.

6. Larose, G. L., and A. D’Auteuil. 2014. Wind Tunnel Investigations of an Inclined Stay Cable

with a Helical Fillet. Report No. FHWA-HRT-14-070. Washington, DC: Federal Highway

Administration.

7. Macdonald, J. H. G., and G. L. Larose. 2006. “A Unified Approach to Aerodynamic

Damping and Drag/Lift Instabilities and its Application to Dry Inclined Cable Galloping.”

Journal of Fluids and Structures 22, Feb: 229–252.

8. Kimura, K., T. Kato, and Y. Kubo. 2012. “Aeroelastic Response of an Inclined Circular

Cylinder in Smooth and Turbulent Flow.” Presented at the 7th International Colloquium on

Bluff Body Aerodynamics and Applications. Shanghai, China.

9. Symes, J., J. Macdonald, and S. Stoyanoff. 2007. “Combined Buffeting and Dry Galloping

Analysis of Inclined Stay-Cables.” Presented at the 7th International Symposium on Cable

Dynamics, Vienna, Austria.

10. Jones, K. 1992. “Coupled Vertical and Horizontal Galloping.” Journal of Engineering

Mechanics 118, Jan: 92–107.

11. Macdonald, J., and G. L. Larose. 2008. “Two-Degree-of-Freedom Inclined Cable

Galloping—Part 1: General Formulation and Solution for Perfectly Tuned System.” Journal

of Wind Engineering and Industrial Aerodynamics 96, March: 291–307.

12. Macdonald, J., and G. L. Larose. 2008. “Two-Degree-of-Freedom Inclined Cable

Galloping—Part 2: Analysis and Prevention for Arbitrary Frequency Ratio.” Journal of Wind

Engineering and Industrial Aerodynamics 96, March: 308–326.
55

13. Den Hartog, J. P. 1932. “Transmission Line Vibration Due to Sleet.” Transactions of the

American Institute of Electrical Engineers 51, Dec: 1074–1076.

14. Stoyanoff, S. 2001. “A Unified Approach for 3D Stability and Time Domain Response

Analysis with Application of Quasi-Steady Theory.” Journal of Wind Engineering and

Industrial Aerodynamics 89, Dec: 1591–1606.
HRDI-40/08-22(WEB)E Recycled

Recyclable

Recommended citation: Federal Highway Administration,

Aerodynamic Stability of Bridge Stay Cables—Dynamic Tests and Simulations

(Washington, DC: 2022) https://doi.org/10.21949/1521926.