wind actions on cable-supported bridges - s.o. hansen · geotechnical & structural engineering...

Geotechnical & Structural Engineering Congress - Phoenix, Arizona - February 14-17, 2016

Wind Actions on Cable-Supported Bridges

S. O. Hansen1, A. D. Horn1, and R. G. Srouji2

1SOH Wind Engineering LLC, 141 Leroy Road, Williston, VT 05495; PH (802) 316-4368; FAX (802) 735-9106; email: [email protected], [email protected] 2Svend Ole Hansen ApS, Sankt Jørgens Allé 5C, Copenhagen V, Denmark; PH +4533277852; email: [email protected]

ABSTRACT

This paper addresses the wind actions on cable-supported bridges including the relevance of wind tunnel testing. Background is given on flutter and vortex-induced vibrations, from which wind tunnel testing is made necessary. References will include wind actions on long span bridges in the United States, Canada, Norway, and Denmark.

A long term experience have shown that section model wind tunnel tests may be efficient and accurate when predicting wind-induced vibrations on bridges. The combination of full-scale measurements and experimental wind tunnel data complement each other giving the most accurate wind-induced response of the structure.

For very streamlined bridges often used for European cable-supported bridges, vortex-induced vibrations measured in the dynamic test rig may depend on the Reynolds number and hence the length scale of the section model. Experimental data is taken from 1:10 and 1:50 section model tests performed in large and small boundary layer wind tunnel facilities with an h x w cross section of 3.0 x 2.5 m and 1.55 x 1.75 m, respectively. Low Reynolds numbers simulated with a 1:50 scale model give much larger vibrations than found for intermediate Reynolds numbers simulated with a 1:10 scale model. This behavior is similar to the behavior of circular cylinders. Full-scale measurements performed on Great Belt Bridge show a good correlation with wind tunnel tests on scale 1:50 section models.

INTRODUCTION

The increased spans of cable-supported bridges constructed over the last decades have resulted in several examples of wind-induced vibrations of bridge decks. More recently, cable-supported bridges have reached a structural detailing, where connections such as bearings and hangers are constructed with less friction reducing their maintenance. The focus on maintenance and a “clean” structural detailing is not beneficial for the structural damping of the bridge, and this enhances the risk of wind-induced vibrations. Wind loads and aerodynamic characteristics of these bridges need to be examined in order to ensure their safe, reliable, and comfortable performance.

Theoretical calculations of the wind load on a structure is very difficult due to the complicated equations that describe airflows and the many parameters in the


boundary conditions of the system. Despite advancements in computing, it is difficult to obtain accurate results from numerical calculations of wind loads on structures in turbulent flow. Thus, wind tunnel testing of a model is the most appropriate method to determine wind loading during bridge design. The accuracy of the methods used is found by comparing the model results with full-scale testing providing the actual wind effects.

The present paper will focus on full-scale experience from past and present bridges, and how the wind tunnel may be used as a design tool in order to determine wind-induced vibrations.

FLUTTER-INDUCED VIBRATIONS

Flutter commonly describes an oscillatory instability of a bridge deck that occurs when a certain critical wind speed is reached. Flutter is characterized by a rapid build-up of vibration amplitude reaching amplitudes that may cause a collapse of a bridge in few cycles of motion.

Flutter is normally considered a consequence of aerodynamic coupling between vertical bending and torsion, commonly referred to as heave and pitch, respectively. However, depending on the geometry of the bridge section, flutter may either be torsional dominant motions or coupled torsional and vertical motions.

The sensitivity to coupling of a vertical and torsional mode shape is dependent on the magnitude of the two natural frequencies in still air and their internal frequency ratio. The higher natural frequencies and frequency ratio, the higher critical flutter wind speeds. The sensitivity also highly depends on the product of the mode coupling coefficient �� and �� defined below by describing the similarity of the vertical and

torsional mode shapes. Similar vertical and torsional mode shapes, i.e. �� = 1,

indicate possible mode coupling. If �� = 0, the mode shapes are not likely to couple.

For instance, the first symmetric vertical mode (VS1) can be coupled with the first symmetric torsional mode (TS1) but not with an asymmetric torsional mode (TA1), see Figure 1. The mode coupling coefficient is given by

�� = ∫ �(�)�(�)��

�� ∫ ��(�)��

�� ,

�� = ∫ �(�)�(�)��

�� ∫ ��(�)��

�� ,

where � and � are the torsional (about the bridge deck axis) and vertical mode shapes, respectively.

Flutter derivatives are additional aerodynamic load components stemming from the motion of the bridge deck. Typically, only vertical and torsional loads due to vertical and torsional motion are treated, though, the horizontal direction is sometimes included. The vertical and torsional loads are defined as

bKHKKHK

U

bKKH

UKKHbUF

defdef

defdefLm

)()()()(2/1 *

42*

32*

2*1

2

,


bKAKKAK

U

bKKA

UKKAbUF

defdef

defdefMm

)()()()(2/1 *

42*

32*

2*1

22

,

where the proportionality factors relating to vertical and torsional loads are denoted H* and A*, respectively. The reduced frequency is defined as K = b2πn/U where U is the wind speed, n is the frequency of the motion, and b is the deck width.

The principle format of the motion-induced loads given above was originally suggested by Scanlan, see e.g. (Simiu and Scanlan, 1986), and it has also been described by Dyrbye and Hansen (1997) using a slightly different normalization and sign convention.

Figure 1. Example of first symmetric vertical (VS1) and torsional (TS1) mode shapes and a first asymmetric torsional (TA1) mode shape.

VORTEX-INDUCED VIBRATIONS

Resonance wind velocity

Vortex-induced vibrations may occur when vortices are shed alternately from opposite sides of a structure. This gives rise to a fluctuating load perpendicular to the wind direction. As the vortices are shed alternately from one side and then the other, a harmonically varying cross-wind load with the same frequency as the frequency of the vortex shedding is formed. The shedding frequency ns(z) of the cross-wind load caused by vortex shedding at location z is

)(

)()(

zh

zvStzn m

s ,

in which St is the Strouhal number, vm is the mean velocity of the approaching wind, and h is the cross-wind dimension of the structure considered. The Strouhal number depends on geometry of the structure and on the Reynolds number.


Significant vibrations may occur if the dominating frequency of vortex shedding ns is the same as the natural frequency ne for the structure vibrating in a mode in the cross-wind direction. Therefore, the resonance wind velocity vr is equal to

St

hnvv e

rm ,

defined by ns = ne.

Calculation of vortex-induced vibrations

To determine the vibrations of rectangular geometries, the Scruton number is often used to judge whether vortex-induced vibrations may occur. What is worth noticing in the Scruton number is that the width of the structure is left out, which could be due to the fact that main focus has been put on circular cross sections. However, a paper recently published by Svend Ole Hansen (2013) shows a modified equation of the Scruton number under the assumption that the aerodynamic damping is proportional to the width of the structure. Therefore, the equation given in Hansen (2013) accounts for rectangular geometries by including the width b. This equation is known as the general mass-damping parameter ScG, and is given as

hb

mSc es

G

2 , (1)

in which δs is the structural damping expressed by the logarithmic decrement, me is the effective mass per unit length, and ρ is the air density. It is worth noting that the last part of the expression, me/(ρhb), is equal to the ratio between the mass of the structure and the mass of the air displaced by the structure.

The standard deviation σy of the structural deflection for a structure with a uniform mode shape may be determined by Equation (2), see (Hansen, 2013):

l

h

m

hb

haK

Sc

C

Sth e

L

yaG

G

cy

2

2

14

1 , (2)

in which l is the length of the structure and Cc, aL, and KaG are aerodynamic parameters governing the response of the structure as illustrated in Figure 2. The negative aerodynamic damping at the resonance wind velocity is proportional to KaG, see (Hansen, 2013).

The aerodynamic parameter Cc governs the standard deviation of the deflection when the lift forces are dominating. The limiting factor aL defines the normalized standard deviation of cross-wind deflection σy/h for a general mass-damping parameter of 0, see Figure 2. The proportionality damping factor KaG times 4π locates the transition range at which the smaller deflections governed by lift forces change to larger deflections governed by motion-induced forces, see Figure 2.


Figure 2. Physical interpretation of the aerodynamic parameters in Equation (2).

FULL-SCALE EXPERIENCE

Flutter and vortex-induced vibrations are two phenomena that may occur on full-scale bridges. While flutter throughout the history has only been observed very few times in full scale, vortex-induced vibrations are more common on long span bridges. Well-known occurrences of these phenomena are on the Tacoma Narrows Bridge, Golden Gate Bridge, and several Norwegian and Danish streamlined suspension bridges.

A few months after the Tacoma Narrows Bridge opened to the public in 1940, the bridge collapsed due to wind-induced vibrations. The now famous collapse of the bridge occurred in wind speeds of about 19 m/s. Whether the collapse was caused by vortex shedding, flutter, or a combination of the two phenomena is still being debated.

One of the advocates for the collapse being driven by vortex shedding was Theodore von Karman (Kármán and Edson, 1967). Von Karman showed that blunt bodies such as bridge decks could, like circular cylinders, shed periodic vortices in their wakes. However, in Billah and Scanlan (1991), it is noted that the frequency of vortex shedding determined from the Strouhal relation would have been close to 1 Hz at 19 m/s, significantly different from 0.2 Hz, which was the frequency of the torsional vibration mode that was observed. However, it cannot be ruled out that the initial vibrations of the bridge at lower wind speeds may have been caused by vortex shedding easing the risk of self-exciting flutter motions at higher wind speeds.

Theories for flutter suggest that the collapse occurred when torsional vibrations due to flutter grew until failure. Flutter caused energy to be imparted into the system due to negative aerodynamic damping, and this energy then caused the amplitude of vibration to grow until the bridge failed.

After the collapse, a vast amount of experimentation was performed on section models of the bridge in order to determine the cause. Through experimentation and calculations, the critical flutter wind speed of the bridge was determined to occur at roughly 7 m/s, considerably lower than the wind speed during the collapse. The rebuilt version of the bridge featured multiple design changes including a wider deck, open truss girders, and steel grating between lanes in order to increase the critical flutter velocity of the bridge to about 56 m/s (University of Washington Libraries).

The Golden Gate Bridge overtook the George Washington Bridge as the longest spanning bridge in the world when it was opened in 1937. The bridge has historically experienced wind-excited vertical oscillations that were not catastrophic. In 1951, the


bridge was closed to traffic for three hours during a gale where the bridge had a vertical oscillation amplitude of 3.3 m. Shortly after this event of poor aerodynamic stability, a lower lateral bracing system between the bottom chords of the vertical trusses was added to the bridge to correct for this motion (Gimsing and Georgakis, 2012).

Norwegian and Danish suspension bridges have since 1970 been constructed with a streamlined steel box girder, see Table 1. The locations of the Norwegian and Danish suspension bridges may be seen in Figure 3. These are typical one span suspension bridges, where the span length varies from 331 m to 1,624 m.

In the 1960s, wind tunnel tests of streamlined steel box girder section models discovered that these box girders were susceptible to vortex-induced vibrations, see for instance the results obtained for the Little Belt Suspension Bridge (Ostenfeld et al., 1970). Remedies like guide vanes to suppress vortex-induced vibrations were developed, and the bridges were constructed on the basis of installment of guide vanes if necessary. Four bridges in Table 1 needed urgent support of guide vanes, namely Great Belt, Osterøy, Bømla, and Storda. On several occasions few months before opening, Great Belt Bridge experienced relatively large harmonic oscillations in vertical modes caused by vortex shedding for wind perpendicular to the bridge and in the range of 4 m/s to 12 m/s. To suppress the vortex-induced vibrations, guide vanes were installed beneath the bridge section along the main span. Hardanger and Dalsfjord are exceptions because guide vanes and a vortex spoiler were included in their design. Guide vanes were also included in the design of Little Belt and Fedafjord. After 20 years in service, guide vanes were installed at Gjemnessund. Then, two of the bridges included in Table 1 remain without remedies for suppression of vortex shedding vibrations: Askøy and Lysefjord.

Table 1. Main dimensions and properties for Norwegian and Danish suspension bridges (Hansen et al., 2015).

Bridge Main span

[m]

In-wind width b [m]

Cross-wind height h [m]

Mass me [t/m]

Damping δs [%LD]

ScG (eq. (1))

[-]

Askøy 850 15.5 3.0 9.5 2.0* 6.5

Gjemnessund 623 13.2 2.6 6.4 2.0* 6.0

Lysefjord 446 12.3 2.7 6.0 2.0* 5.8

Osterøy 595 13.6 2.6 7.4 2.0* 6.7

Storda 677 13.5 2.7 7.0 2.0* 6.1

Bømla 577 13.0 2.6 6.5 2.0* 6.2

Fedafjord 331 13.6 2.6 7.3 2.0* 6.6

Dalsfjord 523 12.9 2.5 7.5 2.0* 7.4

Hardanger 1,310 18.3 3.3 12.5 2.0* 6.6

Little Belt (DK) 600 26.6 3.0 11.7 2.0* 4.7

Great Belt (DK) 1,624 31.0 4.0 19.0 1.0 - 2.0** 2.5 - 4.9

*The logarithmic damping decrement is assumed to be 2% based on full-scale measurements on Askøy Suspension Bridge (Hansen et al., 2015). **The logarithmic damping decrement is based on full-scale measurements on Great Belt Suspension Bridge (Larsen et al., 1999).


FULL-SCALE MEASUREMENTS

Nowadays, the cable-supported bridges have reached a structural detailing, where connections such as bearings and hangers are constructed with less friction reducing their maintenance. The focus on maintenance and a “clean” structural detailing is not beneficial for the structural damping of the bridge, and this enhances the risk of in particular vortex-induced vibrations. As it can be seen from Figure 2, the damping of the bridge is an important parameter in order to avoid vortex-induced vibrations. The damping may be determined from full-scale measurements.

Full-scale measurements on the existing suspension bridges Great Belt (Larsen et al., 1999) and Askøy (Hansen et al., 2015) have been conducted in order to determine their damping and natural frequencies. On the Askøy Suspension Bridge, the vibrations were recorded using mobile phone accelerometers mounted in boxes made of foam board to insulate the accelerometers from wind and sun in order to avoid output shifts in the frequency band of interest (0.1 Hz) due to sudden temperature changes. The boxes with the accelerometers were mounted to the base of the outermost position of the bridge, where the largest torsional accelerations were experienced. In order to differentiate between symmetric and asymmetric vibration modes, measurements were made at mid span and at the quarter span points. To identify heave and torsional motions, accelerometers were placed on both sides of the bridge at these locations, see Figure 4 for the Askøy Suspension Bridge.

Continuous 10-minute measurements have been conducted at Askøy Suspension Bridge, providing more than 4000 minutes of vibration data just a few months after installation on March 27, 2015. The damping and frequencies were found by analyzing the measured power spectral densities (PSD). Fitting curves were implemented to determine the frequency and damping, see Figure 5.

The aerodynamic damping during the measuring campaign was evaluated in order to determine the structural damping using the measurements carried out. When wind is perpendicular to the bridge and the velocity is close to the resonance wind

Figure 3. Location of Norwegian (left) and Danish (right) suspension bridges with a streamlined steel box girder.


velocity for vortex-induced vibrations, the aerodynamic damping based on estimates of KaG, see Equation (2), was used in this evaluation.

Figure 4. Measuring locations on Askøy Suspension Bridge. The italic numbers above the bridge refer to the box girder number (counting from Bergen). The circles indicate the location of the accelerometers. The triangle show the position of where the wind was measured. The pylons are located at box girders 1 and 213. The main span of the bridge is 850 m.

Figure 5. Spectrum measured with an accelerometer at mid span with appropriate fitting to determine natural frequencies and damping (Hansen et al., 2015).

The results of the investigations on the Askøy and Great Belt suspension

bridges have shown a damping of approximately 2% and 1-2% logarithmic decrement, respectively, for modes with risk of vortex-induced vibrations.

WIND TUNNEL AS A DESIGN TOOL

One of the most useful and economic tools for determining the aerodynamic behavior of cable-supported bridges is through wind tunnel testing on a representative section of the full-scale bridge deck. Section model tests have been widely used to evaluate wind-induced response of bridges, and are typically carried out at the preliminary design stage in order to access the aerodynamic stability of the bridge deck.

The section model is mounted in the wind tunnel in such a way as to measure the static and dynamic lift, drag, and moment produced by the wind. The section model is modelled stiff with a low weight, allowing additional mass to be added when the mass scaling, mass moment of inertia scaling, and natural frequencies are set for the

1 54 86 107 160 213


dynamic measurements. The model is typically located between the parallel walls of the wind tunnel, giving a two-dimensional flow over the model. However, the behavior of the model cannot entirely represent the full-scale bridge and considerable interpretation of the results are required.

Nominally static set-up

In the nominally static set-up, the drag, lift, and moment are measured on the section model. The rigs are connected to the force transducers by stiff wires, giving a nominally non-vibrating model, see Figure 6.

The bridge deck is rotated with respect to the airflow, simulating wind inclination to measure the variation in the forces with a range of angles of attack, usually between +10° and -10°. The drag, lift, and moment coefficients may be determined by

mm

DD

hlq

FC ,

mm

LL

blq

FC ,

mm

MM

lbq

FC

2 ,

with h being the cross-wind dimension of the bridge deck, b being the along-wind dimension of the bridge deck, lm being the wind tunnel length of the bridge deck, and qm being the mean velocity pressure.

Dynamic set-up

In the dynamic set-up, the rigs are connected to force transducers by springs, see Figure 6. Thus, the model is responding in a mode shape close to uniform. The springs are connected to two force transducers at each side of the wind tunnel measuring lift forces and moments. The position of the force transducers and springs may be adjusted in order to achieve the desired natural frequencies and ratio between the vertical and torsional frequencies. Additional mass may be mounted on the rigs to achieve a correctly scaled mass per unit length, mass moment of inertia per unit length, structural damping, and natural frequencies. When required, the dampers may be activated to simulate structural damping on the model. Differences in the damping of individual modes are handled by adjusting the spacing between the dampers.

In the dynamic set-up, the section model may be used to investigate the dynamic response to vortex shedding and to ensure that the section is aerodynamically stable to buffeting and flutter. The buffeting and flutter tests are conservative when the full-scale �� is below 1 since �� is 1 in the wind tunnel.

The response may be determined in either low turbulent or turbulent flow. Turbulent conditions will usually reduce the response of the bridge deck to vortex shedding and modify the flutter behavior of the section. The dynamic tests in the matter of vortex-induced vibrations are typically conducted in low turbulent flow, and this turbulence is mainly of small scale nature making the results obtained conservative. Vortex-induced vibrations usually occur at full-scale wind speeds below 15 m/s, which sometimes contain a low amount of turbulence due to stable stratified flow. For buffeting and flutter tests, the full-scale wind speeds are typically significantly above 15 m/s, thus giving a turbulent flow.


Figure 6. Test set-up of the section model. Four force transducers fv1, fv2, fv3, and fv4 measure forces/translations in the vertical direction. Two force transducers, fh1 and fh2, measure forces/translations in the horizontal direction. The transducers are connected to the section model by springs in the dynamic set-up and by stiff wires in the static set-up.

Example: Section model tests on proposal for Champlain Cable-Stayed Bridge

Section model tests were performed on a proposal for the Champlain Cable-Stayed Bridge planned to span the St. Lawrence River in Montreal. Tests covered the in-service stage as well as the construction stage, which represented how the bridge would appear during the launching of the main span. Figure 7 shows that the decks of the construction cross section are connected by cross beams. The cross beams are spaced 12 m apart on center giving a porosity ratio of approximately 67% in the space between decks. An example of the cross beam spacing can be seen in Figure 8. Selected results of the construction stage in the dynamic rig are presented below.

Results from testing the construction stage of the Champlain Bridge are shown in Figure 9 and Figure 10 for the vortex shedding and buffeting/flutter tests, respectively.

During vortex shedding tests, the mass-damping parameter ScG was 3.4 with a structural damping of approximately 1% logarithmic damping decrement. The test results show a very evident critical velocity for vortex-induced vibrations corresponding to a Strouhal number St of 0.13. The vibrations observed are for a very low structural damping. A higher and more likely structural damping will increase the mass-damping parameter and thus decrease the vibrations significantly.

Rig

fv1

fv2

fv3

fv4

Flow direction

z

x y

fh1

fh2


Figure 7. Cross section of the construction stage of the Champlain Bridge. The three decks are connected by cross beams. All dimensions are in full scale [m].

Figure 8. 1:50 model of a construction span on the Champlain Cable-Stayed Bridge. Photo of the model tested in the wind tunnel of SOH Wind Engineering LLC, Burlington, Vermont, USA in March 2015

Figure 9. Vortex-induced heaving vibrations as a function of reduced wind speed for Champlain Bridge in construction stage. The mass-damping parameter ScG was 3.4 with a logarithmic damping decrement of 1.0%. The tests were performed in low turbulent flow. r/h is the maximum amplitude normalized with the cross-wind height of the bridge deck.

The flutter tests were conducted with a frequency ratio between the torsional

and heaving modes of nT/nH = 1.12, which is very close to the ratio where critical flutter velocities are found to be the lowest for flat plates (Dyrbye and Hansen, 1997). The results showed a critical flutter velocity of 82 m/s.

The difference between vortex- and flutter-induced vibrations is clearly illustrated in Figure 9 and Figure 10, respectively. Vibrations caused by vortex shedding may “lock in”, giving vibrations slightly below and above the resonance frequency. This gives a bell-shaped curve with the maximum response at the wind speed in which the vortex shedding frequency is the same as a natural frequency of the bridge. Generally, these vibrations are not severe and thus not important for ultimate limit state designs, but the serviceability requirements should specify maximum vibration amplitudes for typical wind velocities relevant for vortex-induced vibrations.

However, for buffeting-induced vibrations shown in Figure 10, the response does not decrease at higher wind velocities. Once the critical flutter wind speed has

Max

Mean+stdMean

Mean-stdMin

Normalised wind speed U/(nh) [-]1513.12511.259.3757.55.6253.751.8750

Ver

tica

l d

isp

l., r/

h [

-] 0.12

0.06

0.0

-0.06

-0.12


been reached, the vibration amplitude will generally rapidly build-up making the bridge self-exciting. This may result in vibration amplitudes that can collapse the bridge. It is therefore essential to have critical flutter wind speeds well above wind speeds expected at site.

Figure 10. Buffeting response as function of the reduced wind speed, based on the heaving frequency in still air and the bridge width. The damping (log. decrement) is 4.1% for torsion and 4.0% for heave. The frequency ratio nT/nH was 1.12. The turbulence in the in wind direction is approximately 10%.

Example: Streamlined single box girder

When investigating section models of bridge decks, the length scales are typically within 1:30 to 1:80 depending on the size of the wind tunnel and the dimensions of the full-scale bridge. One may think that a larger scale would resemble the full-scale bridge deck more correctly. However, recent studies performed on scale 1:50 and 1:10 streamlined single box girder bridge decks, see Figure 11 through Figure 13, by Hansen et al., (2015) have shown that a larger scaled section model might not necessarily resemble the full-scale bridge more correctly than a smaller scaled section model. The investigations conducted on the two different scales showed a Reynolds number dependency similar to what is experienced on circular cylinders even though the section models were sharp-edged, see Figure 14. To demonstrate the aerodynamics of the cross section itself, the studies were performed on clean sections without any fences or railings.

Full-scale experience from the bridges mentioned in Table 1 shows that streamlined bridge decks with generalized mass-damping parameters ScG of less than approximately 6 may be susceptible to vortex-induced vibrations if no aerodynamic remedies are added on the cross section. This is also predicted by the scale 1:50 section model tested at low Reynolds numbers of 5-8,000, but not by the 1:10 section model tested at intermediate Reynolds number 20-70,000, see Figure 14.

Full-scale measurements on the existing Great Belt Suspension Bridge during construction (Larsen et al., 1999) do show a good correlation with wind tunnel tests on scale 1:50 section models, indicating that the vibration amplitudes may drop at an intermediate Reynolds number range. This could mean that scale 1:10 section model tests at intermediate Reynolds numbers might underestimate vibrations caused by vortex shedding.

Max

Mean+std

Mean

Mean-std

Min

Normalised wind velocity U/(nb) [-]80

Ver

tica

l d

ispl.

, r/h

[-]

0.4

0.2

0.0

-0.2

-0.4

Max

Mean+std

Mean

Mean-std

Min

Normalised wind velocity U/(nb) [-]80

Tor

sion

al d

isp

l. [

°]

10.0

5.0

0.0

-5.0

-10.0


Figure 11. 1:50 model with a wind-exposed length of 1.70 m. The tunnel cross section has a height of 1.55 m and a width of 1.75 m. Photo of the model tested in the wind tunnel of Svend Ole Hansen ApS, Copenhagen, Denmark.

Figure 12. 1:10 model with a wind-exposed length of 2.40 m. The tunnel cross section has a height of 3.00 m and a width of 2.50 m. Photo of the model tested in the wind tunnel of SOH Wind Engineering LLC, Burlington, Vermont, USA.

Figure 13. Streamlined single box girder. Dimensions are in full scale [m]. The slope turning angle α is 15.8°.

Figure 14. Measurements at several ScG and predictions based on Equation (2) with St=0.25, aL=0.017, KaG=0.55, and Cc=0.00083 for the 1:50 model and St=0.32, aL=0.0065, KaG=0.14, and Cc=0.00010 for the 1:10 model (Hansen et al., 2015).

CONCLUSION

Long-span cable-supported bridges have reached a structural detailing that often reduce the structural damping of the bridge to a minimum. Typically, the structural damping is not crucial for flutter-induced vibrations more governed by wind-induced stiffness reductions. However, structural damping normally governs the risk of vortex-induced vibrations, and the tendencies towards a “clean” structural detailing giving minimum structural damping emphasizes the need for a good aerodynamic behavior of the cross sections considered for long-span cable-supported bridges.

1b


The paper clearly illustrates the need for better understanding of wind loading on long span bridges. Full-scale measurements and section model wind tunnel tests offer insight that may be used to improve bridge design for more safe and reliable performance.

REFERENCES

Billah, K. Y. and R. H. Scanlan, (1991), Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks, American Association of Physics Teachers, pp. 118-24.

Dyrbye, C. and S. O. Hansen, (1997), Wind Loads on Structures., John Wiley & Sons. Gimsing, J. N. and C. T. Georgakis, (2012), Cable Supported Bridges: Concept and

Design., Third edition. John Wiley & Sons. Hansen, S. O., (2013), Vortex-induced vibrations – the Scruton number revisited.,

Proceedings of the Institution of Civil Engineers – Structures and Buildings, 166, pp. 560-71.

Hansen, S. O., T. Schnipper, S. Merlung, (2015), Structural damping of the Askøy

Bridge, in-situ measurements of natural frequencies and damping., Tech. rep., Svend Ole Hansen ApS.

Hansen, S. O., R. G. Srouji, B. Isaksen, and K. Berntsen, (2015), Vortex-induced

vibrations of streamlined single box girder bridge decks., in: Proceedings of the fourteenth International Conference on Wind Engineering, Porto Alegre.

Kármán, T. v., L. Edson, (1967), The Wind and Beyond: Theodore von Kármán,

Pioneer in Aviation and Pathfinder in Space., Little, Brown and Company. Larsen, A., S. Esdahl, J. E. Andersen, T. Vejrum, H. H. Jacobsen, (1999), Vortex

shedding excitation of the Great Belt suspension bridge., in: Proceedings of the tenth International Conference on Wind Engineering, Balkema, pp. 947-54.

Ostenfeld, C., A. G. Frandsen, and G. Haas, (1970), Motorway Bridge across Lillebælt

– Aerodynamic Investigations for the Superstructure., Bygningsstatiske meddelelser.

Simiu, E., R. H. Scanlan, (1986), Wind effects on structures: an introduction to wind

engineering., John Wiley & Sons. University of Washington Libraries. “History of the Tacoma Narrows Bridge.”

http://www.lib.washington.edu/specialcollections/collections/exhibits/tnb/reconstruction (Oct. 6, 2015).

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