Risk, Reliability, and Uncertainty Quantification of Structural Systems Subjected to Shock and Vibration
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WindInduced Response of Inclined and Yawed IceAccreted Stay Cable Models
Abstract
During the past decades, windinduced vibrations of bridge stay cables were reported to occur under various incipient conditions. The ice formation on stay cables is one of these conditions, which causes the iceaccreted stay cables to alter their cross section geometry, thus modifying their aerodynamic characteristics. Wind tunnel tests and several CFD simulations were performed for iceaccreted inclined bridge stay cables with two iceaccretion profiles dimensions, 0.5D and 1D, where D is the diameter of the cable. Windinduced vibrations were analyzed experimentally for cable models with yaw inclination angles of 0°, 30°, and 60° and vertical inclination angles of 0° and 15°, for Reynolds numbers of up to 4 × 10^{5}. The aerodynamic drag and lift coefficients of the cable models and the pressure coefficients were determined from the CFDLES simulations. The experimental results indicated that the vertical and torsional vibrations of the iceaccreted stay cables increased with the increase of the vertical and yaw angles. Also, higher vertical and torsional vibration amplitudes were measured for the case with larger ice thickness, indicating the effect of the ice accretion profile on the cable windinduced response.
1. Introduction
Cablestayed bridges are among the most reliable and complex bridge structures, and their ability to support long spans make them an ideal solution for spanning large distances over bay or valley regions with high vehicular traffic. One of the main concerns when designing such bridges is related to the windinduced effect on the bridge decks and towers. However, during the past decades, largeamplitude vibrations of stay cables have been reported [1–7]. These vibrations can tamper with the safety and serviceability of cablestayed bridges and can cause unexpected fatigue failures at the cable anchor points. The causes for the stay cable vibrations vary in nature, such as rainwind induced vibrations [5, 8], dry inclined cable galloping [2, 9–12], and highspeed vortex excitation [8, 13, 14]. Another type vibration has been reported in recent years for bridge stay cables with ice accretion, when sudden largeamplitude wind and ice accretioninduced vibrations were noticed [15–18]. For this phenomenon, the ice accumulates on the stay cable surface under freezing rain, drizzle, and wet or dry snow conditions at low temperatures. The formed ice accretion changes the crosssectional shape of the cables, which in turn can cause aerodynamic instability [19]. Several field studies reported vibrations of stay cables in freezing conditions, which caused the ice to detach from the cables and fall on the passing vehicles. The Port Mann Bridge in Canada was temporarily closed to traffic in 2012 because of the iceaccreted cable vibrations and due to the ice falling from the stay cables causing a hazard for the traffic and people on the bridge. The Tacoma Narrows Bridge in USA was temporarily closed in 2011, due to similar hazards caused by the falling ice from the cables. Similar phenomena of ice falling from cables were reported for other bridges as well, such as the Alex Fraser Bridge in 2016, Canada; the Great Belt East Bridge, Denmark [20]; and the Severn Bridge in UK [21].
In an attempt to clarify the onset conditions for the iceaccreted cable vibrations, Koss and Matteoni [22] conducted a wind tunnel experiment for determining the effect of ice accretion on the aerodynamic forces, recorded for fullscale cables under 0° angle of attack and Re numbers of up to 3.2 × 10^{5}, using the FORCE/DTU climatic wind tunnel (CWT). Also Koss et al. [23] performed an experimental investigation for determining the shape of the ice accretion formed on cables of 0.0381 m and 0.089 m in diameter, that were oriented in horizontal and vertical directions, for different exposure times and air temperatures. They classified the regions of the ice accretion profiles into the core, center area, outer area, flowout, flowout accumulation, and runback. The effect of the ice accretion on the aerodynamics of the vertical bridge hangers was investigated by Gjelstrup et al. [20] using static and dynamic wind tunnel tests. The aerodynamic coefficients were determined for smooth and rough surfaces of the iceaccreted cables, under smooth and turbulent wind flow conditions. Furthermore, Koss and Lund [24] presented the results of a fullscale experimental study on horizontal 160 mm diameter bridge cables, under wet and dry ice accretion conditions, using an innovative spray system to determine the aerodynamic coefficients and to evaluate the galloping instability using the Den Hartog criterion. Moreover, Koss et al. [25] studied the formation of ice accretion on three fullscale horizontal bridge cables with plain, helical fillet, and indented high density polyethylene (HDPE) cover, under wet and dry climatic conditions, and they concluded that gravity plays a major role in the ice profile formation on the cables under the wet icing condition. Also, this study indicated that only the cable with plain HDPE cover was susceptible to galloping instability, as per the Den Hartog criterion. Demartino et al. [19] extended this study to fullscale HDPE vertical and inclined bridge cables, and they investigated the process of ice accretion formation, the final ice shape, and the corresponding aerodynamic drag, lift, and moment coefficients for different parameters such as temperature, wind speed, and yaw angle. Using the experimental data of Demartino et al. [19], Demartino and Ricciardelli [26] modelled different galloping vibrations for one degree of freedom (1DOF) and multiple degrees of freedom (MDOF) cables, and they analyzed the stability of bridge cables with ice accretion. Results showed that the 1DOF models usually generate conservative results, and the use of dynamic wind tunnel experiments was recommended to confirm the results of existing theoretical models, especially for the cases where no sufficient data are available.
However, the actual windinduced vibrations for the inclined and yawed cables in regard to the wind direction, which are very often encountered in practise, with ice accretion attached to their surface, were not specifically investigated. The current study examines the aerodynamic behavior of yawed and inclined iceaccreted stay cables. The stay cable models had vertical inclination angles of 0° and 15° and yaw angles of 0°, 15°, 30°, and 60°, while the ice accretion profile thickness were 0.5D and 1.0D. Wind tunnel wind speeds between 1.5 m/s and 15 m/s were considered to identify the critical windinduced cable vibrations.
2. Experimental SetUp and Cable Model Configuration
The experimental program was performed in the suction wind tunnel facility of the University of Ottawa, Department of Mechanical Engineering (Figure 1(a)), which has a testing section of 61 cm × 92 cm, and has three openings on each side of the testing section, used for installing the models and to coordinate the necessary vertical and yaw inclination angles for the cable models. The maximum wind speed which can be achieved in the wind tunnel is 30.0 m/s; however, for the current experiment, the iceaccreted cable was tested for wind speeds of up to 15.0 m/s, corresponding to a Reynolds number of 4.0 × 10^{5}, considering the high amplitude vibrations developed beyond these testing conditions. The wind speed varied during the tests from 1.5 m/s to 15.0 m/s, in steps of 1.5 m/s. The blockage ratio was determined as the total area of the model normal to the free stream velocity divided by the total area of the test section and was found to be 0.368 × 10^{−3} and 0.736 × 10^{−3}, for the cable models with 1.0 cm and 2.0 cm ice profiles, respectively. According to West and Apelt [27], if the blockage ratio is lower than 6%, for smooth cylinders with aspect ratios in the range of 4 to 10, the Strouhal number is not affected and no correction is needed for the aerodynamic coefficients; hence, the turbulence intensity would not be a significant concern.
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An important parameter which can significantly affect the windinduced response of the cable models during the wind tunnel tests is the relative angle of attack between the wind direction and the cable axis. Therefore, the windcable relative angle, as defined by Cheng et al. [2], was employed in the current research, that is, , where , , and represent the relative windcable angle, yaw, and vertical cable inclination angles, respectively (Figure 1(b)). To simulate the windinduced vibrations of the cables, eight supporting springs, each of elastic constant k = 1.5 N/m, were installed symmetrically at the two ends of the cables along the vertical direction, as schematically represented in Figure 1(b). Also the position of the cable models was adjusted using the three circular openings in the wind tunnel lateral wall and by moving the entire spring support system installed outside the tunnel to adjust for different vertical and yaw angles of attack, as shown in Figures 2(a) and 2(b). For flow perpendicular to the cylinder at 0° yaw, the same location openings were used for installing the model (Figure 2(c)), while for bigger yaw and vertical angles, the cable model was installed between the middle and the last opening with an elevation difference between the spring support systems (Figure 2(d)).
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The cable models were scaled to 1 : 45 from the prototype bridge cable used by Cheng et al. [2] for the fullscale wind tunnel experiments conducted for the smooth surface inclined and yawed cylinders. Ice accretion profiles with a thickness of 0.5D and 1D, where D is the diameter of the cable model, were added on one side of the cable models. These dimensions correspond, respectively, to a cable ice accretion thickness of 45 cm (0.5D) and 90 cm (1D), considering the 1 : 45 scale factor. According to Koss et al. [23] exposing the cable for 1,800 seconds to precipitations under freezing conditions, the ice accretion profile thickness can reach up to 0.5D. The ice accretion profiles with thicknesses of 0.5D and 1.0D were both tested in the current experiment, for clarifying the critical iceaccreted cable response. In order to replicate the arbitrary aspect of the ice profile, expandable foam was applied on the cable model. The ice accretion simulated by foam showed good geometrical agreement with the models obtained from the climatic wind tunnels reported in the literature [23], especially for the troughs and crests of the ice accretion. However, the foam also developed small gaps and indentations; the entire ice profile was corrected by applying aluminium foil, and thus, the cable model and the foam ice accretion could better resemble the ice surface smoothness, as it can be seen in Figures 3(a) and 3(b).
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Three cable models were tested for vertical and yaw inclinations between 60° and 15° as follows (Table 1): Cable Model 1 (CM1) was the cable perpendicular to the flow (0°) and had an aspect ratio of 46, a natural frequency of 0.395 Hz, and a Scruton number of 2.1. This model was also used for 15° yaw angle tests; Cable Model 2 (CM2) was used for vertical angles of 30° and 60° and for yaw angles of 0° and 15°; this had an aspect ratio of 53.5, a natural frequency of 0.33 Hz, and a Sc = 5.1. For Cable Model 3 (CM3), the aspect ratio was 92, and the natural frequency was 0.29 Hz while the Sc number was 10.5. This cable model was used for tests with 0°, 30°, and 60° vertical angles and 0° and 15° yaw angles. The yaw and vertical inclination angles were varied by changing the location of the model, from the middle opening to the last or first opening, as represented in Figure 3. Table 1 summarizes the experiments performed for different yaw and vertical inclination angles. The ice accretion profile of 0.5D was tested for all three cable models, at yaw inclination angles () of 0°, 30°, and 60° and vertical inclination angles () of 0° and 15°. Since 1.0D is considered as an extreme case of the ice accretion thickness, more tests were performed for the 0.5D ice accretion profile, which is more often encountered.

3. Vertical and Torsional WindInduced Vibrations of IceAccreted Cables
3.1. Effect of Ice Accretion Thickness
The torsional and vertical vibrations for the ice accreted cables were recorded for different wind speeds, from 1.5 m/s to 15 m/s at intervals of 1.5 m/s. Figure 4 shows the response time histories for the vertical and rotational vibrations, for the CM1 cable model at 0° relative angle, with an ice accretion thickness of 0.5D. The maximum vertical and torsional responses were measured as 23.92 mm and 2.63°, respectively. For the model CM1 at 0° relative angle, with 1.0D ice thickness, Figure 5 reports the vertical and torsional vibrations at 15 m/s, for which the maximum vertical displacement was 21.95 mm, while the maximum torsional displacement was 5.8°. It is interesting to note that despite the slight decrease in the maximum vertical displacements recorded for the CM1 at 0°, with 1.0D ice thickness, the maximum torsional response increased by a factor of 2.1, when compared with the CM1 model with 0.5D ice thickness.
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Figure 6 presents the time histories of vertical and torsional response of the cable for model CM2, inclined at relative angle = 30°, for the wind speed of 15 m/s. In this case, the maximum vertical displacements due to windinduced vibration were recorded as 21.94 mm at 14.8 s; the maximum vibration amplitude for torsional displacement was 3.13° at 41 s. The torsional vibration was increasing steadily, but no strong fluctuations were noticed for this case. For higher ice accretion thickness, of 1.0D, the cable model CM2 registered vertical displacement with the maximum value of 27.3 mm, while the mean value for this case was 20.66 mm (Figure 7(a)). The maximum amplitude of the torsional vibration was 7.3°, as it can be noticed in Figure 7(b), and the average value for this case was 3.3°.
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For clarifying the effect of the ice accretion effect on the mean vertical displacement of the tested cables for different wind speeds, the response of the same inclination models, but with 0.5D and 1.0D ice accretion thickness, was compared in Figures 8(a)–8(c). As expected, the vertical displacements of all models increased with the increase of wind speed; for lower wind speeds of up to 3.0 m/s for CM1 at = 0° and CM2 at = 30° and up to 4.5 m/s for CM1 at = 15°, the vertical response for the 0.5D cable models was consistent with the response of the 1.0D cable models; however, a sudden decrease in amplitude was noticed for the models with 0.5D ice accretion at 4.5 m/s for CM1 at = 0° and at 6 m/s for at CM1 at = 15° and CM2 at = 30°, respectively. The cable models with 1.0D encounter a small decay in amplitudes at low wind speeds of 4.5 m/s for CM2 at = 30° and at 7.5 m/s for CM1 at = 15°; however, for the cable model CM1 at = 0°, a sudden increase of amplitudes was noticed for 7.5 m/s. The vertical vibration response for the cables with 1.0D ice thickness was higher than that of the cable models with 0.5D ice thickness, especially for wind speeds higher than 9.0 m/s.
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The mean torsional response for the cable model CM2 at = 30° with 0.5D ice accretion thickness was more consistent with the mean torsional response of the same inclination cable model, CM2 at = 30°, but with 1.0D ice accretion, as it can be noticed in Figure 9(C); however, discrepancies were noticed for the other investigated cable models. For CM1 at = 0° and CM1 at = 15°, the mean torsional response was higher for 1.0D ice accretion thickness, when compared with the 0.5D ice accretion models, for all the tested wind speeds (Figures 8(a) and 8(b)). Sudden decays in amplitude were still noticed for models with 0.5D ice thickness at lower wind speeds of 4.5 m/s and 6 m/s for the CM1 and CM2 models, respectively, while for 1.0D ice thickness models, the torsional response decay occurred at 7.5 m/s and 4.5 m/s for models CM1 and CM2, respectively.
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3.2. Effect of Relative Angle
The average amplitudes for vertical and torsional vibrations were investigated for different relative angles of attack, , and it was noticed that the highest responses corresponded to the highest relative angles. For the cases with 0.5D ice accretion, the cable models CM3 at = 61° and = 60° showed the highest vertical and torsional responses (Figures 10(a) and 11(a)), which is similar to the critical cases reported by Cheng et al. [2] for vertically and horizontally inclined stay cables, without ice accretion. Also for relative angles of 60° and 61°, the sudden decay of amplitude at lower wind speeds was not noticed. For the CM2 cable model, both vertical and torsional responses were smaller for wind speeds up to 3.0 m/s; however, from 4.5 m/s and up to 10.5 m/s, the responses for the model inclined at relative angle 33° were higher than the one registered for the model inclined at relative angle 30° (Figures 10(b) and 11(b)).
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In general, for CM3 and CM2 models, the torsional and vertical mean responses were higher for higher relative angles; however, by comparing the magnitude of the recorded vibrations, it can be concluded that the vibrations were consistent with each other, for different wind speeds. For the CM1 model, the mean vertical response was higher for = 0° at higher wind speeds between 9.0 m/s and 13.5 m/s and at 6 m/s (Figure 10(c)); the mean torsional response however was much higher for the CM1 model at = 15° between wind speeds of 7.5 m/s and 15 m/s and at 6 m/s (Figure 11(c)).
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3.3. WindInduced Response Frequency Analysis
In order to observe the variation of the response frequency under different wind speeds, Fast Fourier transform (FFT) was applied for the measured vertical vibrations and the dominant frequency for each response time history was identified. The spectral distribution obtained through the FFT analysis showed very small frequencies without a dominant peak for wind speeds lower than 3.0 m/s for CM1 at 0°, with 0.5D ice accretion, and for CM2 at 30°, with 1.0D ice accretion; frequencies difficult to identify were noticed for wind speeds lower than 4.5 m/s for models CM1 at 0° with 1.0D ice accretion and CM2 at 30° with 0.5D ice accretion, as represented in Figures 12(a) and 12(b).
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The frequencies of the windinduced response were higher for the models CM1 at 0° and CM2 at 30° models with higher ice accretion (1.0D), having a similar trend of slightly higher frequencies at 10.5 m/s and at 15 m/s. For 10.5 m/s wind speed, other peaks, of smaller intensity, were identified in the FFT spectra, around frequencies of 0.025 Hz and 0.21 Hz for the model CM1 at 0° and 0.025 Hz for the model CM2 at 30°, both with 1.0D ice accretion (Figures 13(a) and 13(b)). For the 0.5D ice accretion, the two models, CM1 at 0° and CM2 at 30°, showed trends similar to each other, for the vertical response frequencies obtained at the wind speeds between 4.5 m/s and 15 m/s were (Figures 12(b)), with a slight increase at 6.0 m/s and a sudden decrease at 10.5 m/s, followed by an ascending frequency at 15 m/s, of up to 0.34 Hz for the model CM1 at 0° and up to 0.4 Hz for the model CM2 at 30°, both with 0.5D ice accretion. A second peak at 0.18 Hz was noticed only for the model CM2 at 30°, at 10.5 m/s (Figure 14(b)), while a single dominant frequency at 0.3303 Hz was signaled for the CM1 at 0° model.
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Any changes of the frequency can indicate the change of the dynamic response of the cable model, under the effect of the increasing wind speed. As shown in Figures 8 and 10, a sudden decrease in the frequency response is observed at 4.5 m/s for the models CM1 at = 0°, 0.5D, CM2 at = 30°, 1.0D, which corresponds to a low frequency point in the vertical vibration FFT shown in Figures 12(a) and 12(b), for the same wind speed. Similarly, for other models, such as CM1 at = 15°, 0.5D and 1.0D, and CM2 at = 30°, 0.5D, the sudden decrease of the vertical response occurred at 6.0 m/s (Figures 8 and 10), which correspond to a low frequency point as well (Figure 12).
In order to compare the frequencies for the windinduced response recorded at different wind speeds, for cable models with 0.5D and 1.0D ice accretion profiles, the variation of the Strouhal number for the aforementioned cases was investigated. The Strouhal number was determined as St = fD_{eq}/U, where f, D_{eq}, and U are the frequency of the vertical response, and the equivalent diameter of each cable model was exposed to the wind direction and the mean wind speed, respectively. It should be noted that the thickness of the ice accretion on the cable and the relative cablewind direction angle were considered in estimating the equivalent cable diameter, D_{eq}, for the Strouhal number calculation, as shown in Equation (1). Also, D_{eq}, in Equation (1), is the equivalent cable diameter considering the ice thickness and relative cablewind direction angle; D_{c} and h_{i} are the cable diameter and mean thickness of the ice profile, respectively, while is the relative windcable direction angle:
Figure 15 shows that despite the frequency variations indicated in Figure 10, the normalized frequencies (Strouhal numbers) for all the performed cases decreased with the increase of wind speed, as expected. Also Figure 15 shows that, for different relative windcable angles, the normalized frequencies for the cases with the same ice thickness were almost identical. According to Hao [28], the galloping divergent vibration can occur for Strouhal numbers lower than 0.05, the value corresponding to the horizontal dashed line in Figure 15, showing the incipient conditions from which the galloping divergent vibration could occur, for both cable models, with 0.5D ice accretion and with 1.0D ice accretion, from wind tunnel wind speeds as low as 3.0 m/s.
The critical wind speed, after which galloping instability can be expected, for all cable models tested can be determined using Equation (2) [16, 29]. In Equation (2), , , , and are critical wind speed, natural frequency of the fundamental mode of vibration, cable diameter, and the Scruton number, respectively:
Using Equation (2), the critical wind speeds were determined spanned between 4.5 m/s and 10.5 m/s for the models CM1 at = 0° with 0.5D and CM3 at = 61° with 1.0D, respectively. These wind speeds coincide with the sudden changes in the vertical response frequencies presented in Figures 8 and 10, showing that the higher windinduced response occurred at different wind speeds, depending on the relative angle of attack and the thickness of the ice profile tested.
4. Conclusions
Cablestayed bridges stability rely on all the structural members composing these massive structures, and the stay cables, which are the most flexible elements of the bridge, and have a significant role in the overall bridge design. The wind tunnel experiment performed for cables with ice accretion reported herewith clarifies some aspects related to the windinduced response for the iceaccreted bridge yawed and inclined stay cables. Different parameters such as the vertical inclination angle (0° and 15°), yaw angle (0°, 15°, 30°, and 60°), ice accretion profile thickness (0.5D and 1.0D), and wind tunnel wind speed (1.5 to 15 m/s) were considered. The increase of ice accretion thickness was shown to increase the windinduced response, especially for wind speeds higher than 4.5 m/s. Both vertical and torsional displacements increased with the increase of the relative angles of attack; however, the investigated angles did not determine a significant increase of the windinduced response for the 0.5D and 1.0D iceaccreted stay cables. Also, at certain wind speeds, the vibration for the cables with higher inclination angles was smaller than the cases with lower inclinations; however, for wind speeds beyond 7.5 m/s, the response of the cables with higher inclination angles surpassed the case with lower inclination angles. A sudden decrease in the vertical vibration occurred for models CM1 at = 0°, 0.5D, CM2 at = 30°, 1.0D, and CM2 at = 33°, 1.0D, for wind tunnel wind speeds of 4.5 m/s, for which the frequency analysis showed lower frequency points. A similar decrease in response was noticed at wind speeds of 6.0 m/s and above, for models CM1 at = 15°, 0.5D and 1.0D, and CM2 at = 30°, 0.5D. The frequency analysis showed multiple vibration values for the vertical windinduced response between wind speeds 4.5 m/s and 9.0 m/s for models with 0.5D ice accretion and between wind speeds of 7.5 m/s and 15 m/s for models with 1.0D ice accretion, which can be an indication of an aerodynamic instability.
Data Availability
The data supporting the current research project can be found at CVG Department, University of Ottawa, and can be made available if necessary by the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Discovery Grant 06776, 2015.
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Copyright © 2018 Songyu Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.