Aerodynamics of Bridge Stay Cables Under Simultaneous Action of Rain and Wind – Part I: Vortex Shedding Aerodinâmica de Cabos de Pontes Estaiadas Sob a Ação

Resumo Simultaneous occurrence of cable vibrations under the combined action of wind and rain have been observed around the world in cable-stayed bridges in the last 20 years. This mechanism has caused great concern to bridge engineers and researchers due to the large vibration amplitudes. These vibrations, which are predominantly cross-wind, were observed only under conditions of light rain and low wind speed occurring simultaneously. This paper presents the main results of an experimental wind tunnel study on three sectional cable models, set to investigate the influence of rivulets on the vortex shedding mechanism. The M1 model was positioned horizontally with perpendicular wind incidence to the longitudinal axis, the M2 model was positioned horizontally with oblique wind to the longitudinal axis and the M3 model was a typical inclined cable of cable-stayed bridges. For the M3 model, the vortex shedding intensity increases for all rivulets positions. The greatest intensity occurred when the upper and lower rivulets were at 50° and 110°, respectively. For turbulent flow, the vortex shedding is stronger in the models with rivulets. For Re < 1.2 x 105 the lower rivulets has no influence on the flow. However, for > 1.2 x 105 the lower rivulets have a clear influence on the flow around the cylinder. For the inclined model (M3) the lower rivulet affected the flow for all the Re range.


Introdution
Currently, the cables of cable-stayed bridges have great flexibility, low mass and extremely low structural damping.Therefore, these cables are susceptible to vibration caused by various aerodynamic excitation mechanisms; among them a new kind of phenomenon known as wind-rain-induced vibrations.Cable vibrations of cablestayed bridges under the combined action of rain and wind have been observed around the world in the last 20 years.This paper presents results obtained by measuring instantaneous external pressures in a cross section of a sectional model of a typical cablestayed bridge with and without artificial rivulets.It was determined how the location of the rivulets influenced the position and intensity of vortex shedding in models with artificial rivulets as well as the influence of flow on the model with rivulets.Experimental investigations were carried out in the Joaquim Blessmann boundary layer wind tunnel of the Building Aerodynamics Laboratory (LAC) of the Federal University of Rio Grande do Sul (UFRGS).

Rationale
Large amplitudes reduce the service life of cables and their connections due to fatigue and therefore damage the corrosion protection duct.Moreover, the oscillations can also cause collisions between adjacent cables and cause user discomfort.It is necessary to understand the mechanisms of vibrations induced by rain and wind through field tests or wind tunnel investigation.The results presented below serve to improve the design of cable-stayed bridges.

Pressure coefficients
The pressure coefficients are dimensionless coefficients obtained from the ratio of effective pressure, p ∆ , caused by wind in a small area on the surface of a body, and the dynamic pressure, q , Equation (1).
where p c is the pressure coefficient,

Force coefficients
These coefficients are also dimensionless and depend on the shape of the cross section and the wind attack angle, β .Furthermore, they also depend on the flow regime and especially for cir-

Reynolds number, Re
The Reynolds number express a relation between the inertial forces and the viscosity forces (Equation 4).
where ν is the air cinematic viscosity.

Strouhal number, St
The frequency of vortex shedding is directly proportional to the wind velocity, being related through the Strouhal number (Equation 5), which usually depends on the geometry of the structure and the Rey-nolds number.
where s f is the frequency of a pair of vortex shedding.

Vortex excitation
In non-aerodynamic bodies, a pair of vortex shedding appears at a specific Reynolds number range.This is called Kármán vortex.The vortex shedding induces lateral forces perpendicular to the flow and consequently induces movements in this direction (Blessmann [1]).Figure 1 shows the characteristics of the flow around a stationary cylinder defining the shear layer, stagnation point, separation point, and wake.

Inclined circular cylinder aerodynamics
Additional effects appear in inclined cylinders such as axial flow, with low-frequency vortex shedding, and rain-wind-induced vibrations.
The position of the cable to the wind is defined in Figure 2. The inclination angles (the angle the cable makes with the horizontal plane) and the incidence angle (defined as the angle between the cable projection on the horizontal and a vertical plane perpendicular to the wind direction) are defined by α and β , respectvely.

Axial flow
Matsumoto et al. [4] showed the existence of a strong axial flow near the wake along the axis of a yawed cylinder ( α =0° and ≠ β 0°.).The axial flow causes disturbance in the regular Kármán vortex shedding.Figure 3 shows the flow around a cylinder with For inclined circular cylinders, it is possible to say that the axial flow near the wake has a feature similar to the separation plate submerged in the wake (Matsumoto [4]).

Low-frequency vortex
The low frequency vortex shedding is also known in the literature as axial vortex, vortex shedding of long period or vortex shedding with frequency lower than the Kármán vortex shedding.Matsumoto et al. [6] verified the existence of the axial vortex by using smoke to identify the formation of vortex in the wake of the cable as shown in Figure 4. up to 2m due to the combined effect of rain and wind.Each rivulet is formed by a filament of rain water resulting from precipitation on the cables (Figure 5).They are called lower rivulet (Figure 5a) and upper rivulet (Figure 5b).The formation of the upper fillet can take a sine wave form (Wang et al. [10]).The position of the lower and the upper rivulets is identified in this paper by 1 θ e 2 θ , respectvely in Figure 6, where V ef1 is the velocity on the cable's normal plane and γ is the attack angle perpendicular to the section.

Rain-wind-induced vibrations
The rivulets' position varies almost linearly with wind velocity.Hikami and Shiraishi [3] showed that the occurrence of vibrations induced by rain and wind is limited to cables geometrically inclined against the wind.The largest amplitudes of vibration induced by wind and rain occur for cable inclinations between 25º and 45º and incidence angles between 30º e 45º.The vibrations that reach large amplitudes occur only in the presence of light to moderate rain (rain intensity of about 8 mm / h (Ni et al. [7])).Water rivulets oscillate over the surface of cables in the circumferential direction.This movement is mainly caused by the temporary acceleration of the cross section.The rivulets' movement is influenced by the fluidstructure interaction and the forces acting on them.The change in rivulets' position causes a continuous variation of the cross section cable shape, which leads, in turn, the variation of force coefficients (Verwiebe [9]).The rivulets oscillate with the same frequency of the cable movement (Hikami and Shiraishi [3]).

Experimental investigation
Three stationary sectional models of a typical cable-stayed bridge were covered with artificial rivulets having geometric scale of 1:1 identified as M1, M2 and M3.The models represent the corrosion protection duct of steel tendons that make up a typical system of cables in a cable-stayed bridge.Figure 7 is a cross section scheme of the corresponding model.Figure 10 shows a schematic view of the positioning of the three models tested in the wind tunnel.
The water rivulets have a semi-elliptical cross section (rivulet width b fil =13 mm and rivulet height h fil =5 mm) and were made using a 1 A cable under wind effect with velocity V, will have a inclination α with horizontal plane and incidence angle β.For geometric definition the effective velocity -function of V, α e β -acts perpendicular to the cable at a wind attack angle γ to normal cable section.For calculation details of V ef and γ see Phelan et al. [8].

Aerodynamic of Bridge Stay Cables Under Simultaneous Action of Rain and Wind -Part I: Vortex Shedding
plastic tube cut in the longitudinal (Figure 8).The rivulets' length are the same as the models length.Figure 9 shows the pressure taps distribution on the sectional cable model.Instantaneous pressures around of the cable's cross section were measure in the mid-span with and without the presence of rivulets, under different circumstances.Measurements were made using electronic pressure transducers.The M1 model was positioned horizontally (α = 0 o ) and with wind incidence angle ( β = 0 o ).The M2 model was positioned horizontally (α = 0 o ), however with wind incidence angle ( β = 30 o ).The M3 model was positioned inclined (α = 45 o ) and with wind incidence angle ( β = 30 o ). Figure 10 shows the models in the wind tunnel test chamber.For the generation of turbulent wind a grid capable to generating turbulence intensity of 10.5% was used.Table 1 shows the variable test parameters.Power Spectral Density (PSD) functions of the instantaneous force coefficients obtained from the external pressure measurements were determined.Pressure taps were used to measure the external pressure.The instantaneous drag and lift coefficients were obtained from the integration of the instantaneous external pressure measurements.The spectra are presented by S(f) x 1000 [1/Hz] as a function of the reduced frequency fD/V for clarity in the analysis.For a normal circular cylinder (without rivulets), the characteristic spectra for drag and lift will present a frequency ratio of two, that is, the alternating transversal forces will occur in the shedding frequency of a pair of vortices, while in the along-wind direction the alternating forces will occur in the frequency of an individual vortex [1].This can be clearly seen in figures 11a and 12a.When the rivulets are present, note that a second, lower frequency, peak appears in the spectra.This is probably due to the shedding of the vortices from the upper rivulets which are registered in their individual frequencies.Figure 13 shows the PSD at pressure taps which present representative energy peaks for model M1 without rivulets in smooth flow, Re ≅ 7.7 x 10 4 .
Figure 13 shows that the highest peaks occur at pressure taps 8 and 30 (see figure 10), which are symmetrical to the horizontal sections axis.Due to the higher energy achieved, these positions represent the vortex detachment location in the model without fillets.

Aerodynamic of Bridge Stay Cables Under Simultaneous Action of Rain and Wind -Part I: Vortex Shedding
Figure 14 shows the PSD on some pressure taps which present representative energy peaks for model M1 at setting F40 with a smooth flow, Re ≅ 7.7 x 10 4 .Observe in Figure 14 that the PSD peak intensity on pressure tap 5 (upper rivulet) is highest than on tap 4; in pressure tap 7 PSD peak intensity decreases and after increases in 11.A vortex shedding occurs on the upper rivulet (stronger intensity) with flow reattachment on pressure tap 7 (weaker intensity) and new detachment at 11 (stronger intensity).At the bottom of the model the vortex shedding occurs at pressure tap 29. Figure 15 shows the external pressure PSD on the pressure taps 5, 6 and 30 which present representative energy peaks for model M1 at setting F50 with a smooth flow, Re ≅ 7.7 x 10 4 .Observe the vortex shedding on upper rivulet (tap 6).At the bottom of the cable the vortex shedding occurs on pressure tap 30. Figure 16 shows the external pressure coefficients PSD on pressure taps 6, 7 and 30 which present representative energy peaks for model For Reynolds numbers higher than Re ≈ 7.7 x 104, it is possible to identify the lower rivulet influence on vortex shedding.M2 model: In this model it was only possible to identify peaks for Re between 3.9 x 10 4 and 1.2 x 10 5 , for both smooth and turbulent flow.At high values of Re, the PSD of drag and lift coefficients present a broad band spectrum with less defined peaks.At settings with rivulets the spectral peaks were more defined, but with lower intensity than in the model without rivulets.In the S(f) of C D , it was not possible to identifie pronounced peaks.In the model under turbulent flow and the presence of rivulets, the peaks in S(f) of C L were higher than in the model under smooth flow.For the model without rivulets under turbulent flow, the peaks were lower than under smooth flow.
m²] is the wind dynamic pressure corresponding to the average velocity measured at the middle of the sectional cable model, V the av- erage wind velocity at the same position [m/s] and a r ρ is the air density [kg/m³].

F
cular sections they vary significantly with Reynolds number, Re.These force coefficients are: a) Drag coefficient: coefficient related to the drag force that is the component of the global force in the flow direction (Equation 2); b) Lift coefficient: coefficient related to the lift force that is the component of global force perpendicular to the flow (Equation 3); where D is the average drag force per unit length [N/m], L F is the average lift force per unit length [N/m] and D is the cable diameter [m].
Under Simultaneous Action of Rain and Wind -Part I: Vortex Shedding These vibrations are a new kind of instability phenomena (Matsumoto et al. [5]).These vibrations, which are predominantly cross-wind, were observed only under conditions of light rain and low wind velocity, occurring simultaneously.The oscillations are caused by the formation of two rivulets along the surface of the cables in the axial direction, which, due to its motion, constantly change the distribu-tion of pressures around the cables.Matsumoto et al. [5] observed double amplitude (peak to peak) on the Bridge Islands (Denmark) The following observations were made from the results obtained.M1 model: Figures 11 and 12 present the PSD of C L (PSD of instantaneous lift coefficients) and the PSD of C D (PSD of instantaneous drag coefficients) under a smooth flow (Re ≅ 7.7x10 4 ) applied to the configurations SF (cable without rivulets), F40 (cable with rivulets at 1 θ = 40 o and 2 θ = 110 o ), F50 (cable with rivulets at 1 θ = 50 o and 2 θ = 110 o ) and F60 (cable with rivulets at 1 θ = 60 o and 2 θ = 110 o ).Other researchers also observed maximum amplitudes at Re ≅ 7.7x10 4 [2].

1 θ
M1 at setting F60 with a smooth flow, Re ≅ 7.7 x 10 4 .Observe in Figure 16 the vortex shedding occurs on upper rivulet (tap 7).At the cable bottom the vortex shedding occur on pressure tap 30.According to Figures 11 to 16, it was observed that: for setting F40 the Strouhal number or reduced frequency (fD/V) is larger than for setting SF, while for setting F50 and F60 the fD/V value is less than for the SF (no rivulet) case.At setting F40, the peaks have lower intensity compared to the other configurations.At setting F50, note an increase in the intensity of the vortex shedding and reduction of the reduced frequency compared to the settings SF and F40.Observe in the S(f) of C D (Figure 12c) the second harmonic almost disappears.The upper rivulet appears to suppress the vortices on one side of the model making the S(f) of C L (Figure 11c) coincide with the S(f) of C D .The along-wind response has the same frequency of the crosswind response, but with lower intensity.At setting F60, both S(f) D. S. MACHADO | A. M. LOREDO-SOUZA of C D and S(f) of C L , peaks had higher intensity than the model without rivulets.When the upper rivulet is in position = 60° an increase of vortex shedding intensity on the rivulet may occur.The peaks with dif-ferent reduced frequencies S(f) of C D (Figure 12d) indicate different intensities of vortex shedding on each side of the model.Figure 17 shows a scheme of vortex shedding on M1 model at settings, SF, F40, F50 e F60.At Figure 17, the vortices show where the highest Aerodynamic of Bridge Stay Cables Under Simultaneous Action of Rain and Wind -Part I: Vortex Shedding intensity of PSD peaks of C D occurred, with its respective value of S(f) x 1000.
M3 model: The same observations made for model M2 in the previous paragraph are valid for model M3.General considerations: Figures 18 to 20 show the PSD (with Aerodynamic of Bridge Stay Cables Under Simultaneous Action of Rain and Wind -Part I: Vortex Shedding n regardless of the rivulets' presence, the inclined model (M3) had frequencies of vortex shedding lower than those obtained by conventional Kármán vortex.It is a possible effect of the axial vortex; n for the horizontal models, the lower rivulet has no influence on the flow for Re below 1.2 x 10 4 .At Re above 1.2 x 10 4 , the lower rivulet has a clear influence on the flow.For the inclined models, the lower rivulet influenced the flow for all Re values.