Experimental investigation of wind-induced vibrations of main cables for suspension bridges in construction phases

Journal of Fluids and Structures 93 (2020) 102846

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Experimental investigation of wind-induced vibrations of main cables for suspension bridges in construction phases ∗

Xugang Hua a , , Chaoqun Wang a , Shengli Li b , Zhengqing Chen a a

Key Laboratory for Wind and Bridge Engineering of Hunan Province, College of Civil Engineering, Hunan University, Changsha 410006, China b Zhengzhou Key Laboratory of Disaster Prevention and Control for Cable Structure, School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China

article

info

Article history: Received 4 August 2019 Received in revised form 3 November 2019 Accepted 17 December 2019 Available online xxxx Keywords: Suspension bridges Main cables Galloping Wind tunnel test Vibration mitigation

a b s t r a c t The main cables of suspension bridges show a changing cross-sectional shape with the evolution of construction phases, and they may suffer from severe wind-induced vibrations at certain conditions. The primary objective of this research was to examine the aerodynamic performance of the main cable in construction phases and to develop appropriate countermeasures to eliminate the potential wind-induced vibrations. Two cross-sections with different shapes of a main cable were chosen, and a series of wind tunnel tests were performed in a reduced wind velocity range of 32–366 using elastically mounted sectional models. Galloping occurred for the two cross-sections under certain wind incidence angles when a critical velocity was reached. No obvious hysteresis phenomenon of galloping was observed in the tests. The steady amplitude of galloping increased linearly with wind velocity and the increasing rate almost kept constant for different structural damping ratios. The aerodynamic nonlinearity, rather than the structural damping nonlinearity, is the main source leading to the limited amplitude oscillation. An empirical expression of galloping amplitudes for the two cross-sections was derived based on the test data. Meanwhile, the critical wind velocity was studied in a Scruton (Sc) number range of 108–4196 (as varied by changing the initial structural damping ratio between 0.093% and 3.62%). Results showed that the Den Hartog criterion was applicable to forecast the possibility of galloping, but not able to estimate the critical wind velocity for the main cable. Linear fitting method can be used to predict the critical velocity based on the experimental data. Finally, three vibration mitigation measures were studied, and a combination of structural and aerodynamic measures was recommended for galloping mitigation of main cables. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Erection of main cables is an essential step for construction of a long-span suspension bridge. During the construction phases, the stiffening girder has not been erected and main cables are suspended on bridge towers independently of other structures. As a result, the rigidity of a main cable is very small, and the cable will be sensitive to wind excitations. Large-amplitude wind-induced vibration of main cables has been observed during the construction phases of the 1650 m ∗ Corresponding author. E-mail address: [email protected] (X.G. Hua). https://doi.org/10.1016/j.jfluidstructs.2019.102846 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Main cable during construction.

(main span) Xihoumen Bridge (Li and Ou, 2010; An et al., 2016) and the 1480 m (main span) Second Dongting Lake Bridge in China, and the safety of structures as well as the construction workers were seriously threatened. Generally, the main cable of long-span suspension bridges consists of more than one hundred identical steel strands, and the steel strands are installed one by one (Fig. 1). Consequently, the main cable may experience a number of crosssectional shapes during construction phases, and its aerodynamic performance can be foreseen to very complex. Main cables during construction may suffer vortex-induced vibration, turbulence-induced buffeting and galloping in wind flow. A certain amount of fundamental work relating to the wind-induced vibration of main cables during construction has been published. Galloping, as the primary source of wind-induced vibrations for main cables, is a kind of aerodynamic instability, and has a strong destructive power on account of the possible divergent amplitude. When a main cable shows a special cross-sectional shape at a certain construction stage, it will have a risk of galloping, and large-amplitude oscillation will occur when the wind velocity reaches a critical value (Li et al., 2017). Both wind tunnel test and numerical simulation have been conducted to study the aerodynamic performance of main cables. Results based on the Den Hartog criterion (Den Hartog, 1932) indicate that galloping may occur on main cables not only erected by the common steepled method (An et al., 2016), but also by the flat-topped method (Li et al., 2017). Results of previous research provide an academic reference of wind resistance for main cable construction, since several typical cross sections of main cable have been investigated and contrasted, and the risky construction stages have been pointed out (An et al., 2016; Li et al., 2017). However, existing researches are all based on the classical quasi-steady theory (Den Hartog, 1932; Nigol and Buchan, 1981a,b; Desai et al., 1990). As is known, onset of galloping can be preliminary estimated by classical galloping theory (Macdonald and Larose, 2006; Guo et al., 2019), but sometimes the prediction of critical wind velocity does not coincide with the experimental results (Wen et al., 2018). On the other hand, study of super-critical galloping (Qin et al., 2019) is also necessary. It can provide a more comprehensive cognition of main cable galloping for engineers and constructors, and lay a basic foundation for further research on galloping mitigation. Super-critical galloping has been extensively studied on iced conductors (Barbieri et al., 2004), bridge hangers (Chen et al., 2018), stay cables (Gao et al., 2019), etc. In-depth investigation on onset mechanism, critical velocity and vibration amplitude of galloping were conducted by wind tunnel tests and CFD (Computational Fluid Dynamics) simulation. Selfexcited aerodynamic forces in supercritical state usually presents strong nonlinear characteristics (Chen and Kareem, 2003; Ghadami and Epureanu, 2018; Shruti and Yahya, 2015) such as bifurcation (Mannini et al., 2018) and amplitude dependency. Neglect of these nonlinear factors may lead to a prediction error of critical velocity for some structures (Gao and Zhu, 2016, 2017). Furthermore, galloping amplitude of a structure in unsteady flow is related to the nonlinearity of aerodynamic forces (Jafari and Sarkar, 2019). Response prediction of a structure at a specified wind speed for a given wind incidence angle is an important subject with great theoretical significance and practical value. From the above, it is very necessary to conduct further studies on wind-induced vibrations of main cables. With the extension of bridge span and, accordingly, the decrease of main cable stiffness, wind-induced vibration is becoming an increasingly serious threat to the construction of suspension bridges. Galloping suppression of main cables is important to ensure the construction safety. Common practice in engineering is to enhance the critical velocity of galloping, which is related to the Sc number, natural frequency and aerodynamic characteristics of structures (Sun et al., 2016; Blevins, 1990). Accordingly, there are three common methods for galloping mitigation: damping method, stiffness method and aerodynamic method (Jafari and Sarkar, 2019; Gao et al., 2019). Intensive study of galloping has been conducted on the iced conductors (Den Hartog, 1932), and a large amount of research relating to galloping mitigation has been published since then. As a feasible and economic method, spacers are widely used in electric engineering since it can enhance the stiffness of conductors effectively (Hu et al., 2012). Similar

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Fig. 2. Cross section appearance of a main cable during construction phases.

methods have been investigated in bridge engineering. Wen et al. (2018) successfully suppressed galloping of closely spaced parallel hangers using rigid spacers in the Xihoumen Bridge. He et al. (2018) proposed an elastic cross-tie for staggered stay cables to suppress the wind-induced vibration, and they also derived the scale ratio of the cross-tie stiffness for engineers. Li et al. (2014) compared the efficiency of flexible and rigid connections on the transversely connected parallel cables of cable-stayed bridges based on wind tunnel studies. More broadly, auxiliary steel wires are implemented to prevent H-section hangers in arch bridges from both flutter and galloping (Chen et al., 2012). For existing bridges, attaching mechanical dampers is another effective measure (Pacheco et al., 1993; Hoang and Fujino, 2009). Traditional oil damper, viscous damper and tuned damper have a long history for wind resistance of bridge engineering. Recent years, some new type dampers, such as eddy current damper and magnetorheological damper, are becoming more and more popular in mitigation of various wind-induced vibrations for bridge cable structures due to their multiple advantages (Niu et al., 2018). As is known, the primary cause of aerodynamic instability is the self-excited forces related to aerodynamic configuration of cross sections. In view of this, aerodynamic characteristics of slender structures with multiple cross sections have been intensively studied (Achenbach, 1971; Niu et al., 2015). Surface roughness (Zdravkovich, 1990), sharp corner (An et al., 2016) and other shape characteristics can have a strong impact on aerodynamic performance of a structure. Based on these fundamental research, various aerodynamic measures have been proposed and implemented to prevent bridge cable structures from dry inclined galloping, ‘drag crisis’, rain-wind induced vibration and vortexinduced vibration (Vo et al., 2015; Kleissl and Georgakis, 2011; Gao et al., 2019). Choice of methods for vibration control mainly depends on comprehensive consideration of efficiency, economy and feasibility. With respect to the wind-induced vibration of main cables in construction phases, appropriate vibration control methods need to be developed. In this study, two risky cross sections of a main cable in construction phases were selected based on the existing research findings. Aerodynamic performance of the main cable was studied through a series of wind tunnel tests using the elastically mounted sectional models. The structural nonlinearity of the spring-model system was evaluated from free-decay vibration responses. Critical wind velocity, vibration amplitude and hysteresis loop of galloping were studied in detail. Moreover, three kinds of measures for wind resistance of main cable in construction phases were discussed. 2. Outline of experimental procedure 2.1. Prototype main cable To study the wind-induced vibration of main cables in construction phases, the structural parameters of the Xihoumen Bridge were employed in this study. Cross section of the main cable in completion stage is of round shape with a diameter of about 90 cm. Each main cable is assembled by 169 identical steel strands, and the cross section of a steel strand is regular hexagon so that the steel strands fit closely to each other. In construction phases, steel strands are erected one by one from the bottom up. Therefore, the cross-sectional shape of a main cable gradually changes along with the construction process, and each main cable will present at least 169 different cross-sectional shapes. Fig. 2 shows seven typical cross sections of a main cable during construction phase. Natural frequency of the cable was not measured during construction phases, instead it was calculated by using the finite element (FE) method. In FE modeling, all the strands are taken as a whole and the interaction between adjacent strands (for instance, the friction between them) was ignored. Consequently, the calculated natural frequency is constant during the whole construction phase of main cable since the strands are arrayed in parallel. The natural frequency of the first vertical bending mode of the main cable is 0.091 Hz. The structural damping of the main cable during erection is not measured. It should be noted that the inherent structural damping of main cables in erection may be very different from the finished cables. As a result, a fairly wide range of structural damping is employed for wind tunnel studies, as will be shown later. 2.2. Description of section model Aerodynamic performance of the main cable is mainly related to its cross-sectional shape which varies with evolution of construction phases. An et al. (2016) studied the aerodynamic performance of main cable at various construction stages and pointed out the riskiest cross section shapes which may suffer galloping. Based on An’s research, two risky cross sections were selected for wind tunnel test: the first and third cross sections in Fig. 2. For expression convenience, the two selected cross sections were named as Cable 1 and Cable 2, respectively, and their profiles are illustrated in Fig. 3.

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Fig. 3. Two cross sections of a main cable for wind tunnel tests: (a) Cable 1; (b) Cable 2.

Fig. 4. Section models of the main cable for wind tunnel tests. Table 1 Primary parameters of section models. Cable 1 Geometric scale λL Height H (m) Width B (m) Length L (m) Mass m (Kg/m) Initial natural frequency f (Hz) Frequency scale λf Initial damping ratio ζ Initial Sc number Velocity scale λU Wind incidence angle α

Cable 2

Section model

Prototype

Section model

Prototype

1:5 0.036 0.096 1.50 14.32 1.78 19.56:1 0.093% 108 3.91:1 0◦ , ±2◦

– 0.178 0.480 – 359.55 0.091 – – – – –

1:10 0.039 0.079 1.50 15.82 1.75 19.23:1 0.11% 117 1.92:1 0 ◦ , ±3 ◦

– 0.386 0.789 – 1582.02 0.091 – – – – –

Section models of Cable 1 and Cable 2 were made of acrylonitrile-butadiene styrene (ABS) plastic (Fig. 4). Geometrical scale ratios of the two models were 1:5 and 1:10, respectively to fit for the test section of wind tunnel. Both models had an identical length of 1.5 m, resulting in two aspect ratios of 15.6 and 19.0, respectively. Each model had two identical square plates on the two ends to weaken the end effect. The square palates were made of the same materials as section models and each had a thickness of 0.4 cm and a side length of 40 cm. Detailed parameters of the section models are summarized in Table 1. In addition, there are some simplifications of the two section models compared to the prototype main cable: (1) according to previous study of similar slender structures such as iced conductors (Lou et al., 2014) and suspension bridge hangers (Wen et al., 2018), the surface roughness of steel strands were smoothed; (2) steel strands were assembled to be close to each other, and the possible tiny gaps between them were ignored; (3) due to the large ventilation rate (more than 83%) of the wire mesh on catwalk and a certain distance between the catwalk and main cable, aerodynamic interference of the catwalk was neglected (Li et al., 2017). 2.3. Overview of wind tunnel tests Wind-induced vibration of Cable 1 and Cable 2 were investigated through a series of wind tunnel tests by employing elastically-mounted section models (Fig. 5). According to classical transverse galloping theory, galloping is a kind of

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Fig. 5. Elastically-mounted system in wind tunnel.

crosswind vibration, hence each elastically-mounted section model was a single degree-of-freedom (SDOF) system. Horizontal motion was constrained by horizontal steel wires (very slight) tied to two ends of the section model. Only the first vertical bending mode of the main cable was considered. An et al. (2016) pointed out that galloping is most likely to occur on Cable 1 and Cable 2 at wind incidence angles near +2◦ and +3◦ , respectively. Therefore, each of the two section models were tested at three angles including their risky wind incidence angles. Fig. 6 shows a sketch of the elastically-mounted system in wind tunnel. In order to look for appropriate vibration mitigation measures, the Scruton number was varied by changing the structural damping and defined as follows, Sc =

4π mζ

(1)

ρH2

where ρ = 1.225 Kg/m3 is the air density. The initial dynamic properties (i.e. natural frequency and structural damping) in Table 1 represent the corresponding values measured at small vibration amplitude of the suspension system. They are reference values used for defining reduced wind velocity and Sc number. According to the similarity criterion of wind tunnel test (Scanlan and Tomko, 1971), the reduced wind velocity between section model and prototype should keep consistent, that is Uf ff Hf

=

U fH

or λU = λf λL

(2)

where λU = U /Uf is the velocity scale, representing the ratio of model-scale velocity with respect to full-scale velocity; λf = f /ff is the frequency scale, which is the ratio of model frequency with respect to full-scale frequency. λL is the geometric scale of the cross-section. The scaling of frequency and velocity adopted in this study are also shown in Table 1. Wind tunnel tests were carried out in the high-speed test section of the wind tunnel HD-2 at Hunan University. The test section is 3.0 m wide and 2.5 m high and the turbulence intensity of wind velocity above 3 m/s is no more than 0.5%. In tests, the section model was free to oscillate in heaving direction, as already shown in Fig. 5. Vertical displacement responses of the section model and free-stream wind velocity were measured by laser displacement sensors and a cobra probe respectively. For each testing case, no less than 40 s of displacement time history of the section model was recorded at a steady state. 2.4. Dynamic characterization of elastically-mounted system For the elastically-mounted system, natural frequency mainly depends on the spring stiffness and section model mass, and structural damping was changed by the viscoelastic tape attached to springs (Fig. 7). The dynamic characteristics of the system may not keep constant as vibration amplitude varies (Gao and Zhu, 2016, 2017). The purpose of quantification of nonlinear properties of modal frequency and damping ratio is twofold. First, the initial natural frequency and structural damping identified at small amplitude of vibration are appropriate to define the Sc number and reduced critical velocity U/fH for galloping onset. Second, the study of post-critical behavior of galloping of section models needs the detailed information regarding nonlinearity in inherent structural damping of the spring-model system. In view of above, free-decay oscillation tests in still-air were performed in each test case.

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Fig. 6. Sketch of elastically-mounted system for section model test.

Fig. 7. Viscoelastic tape attached to a spring of the elastically-mounted system.

Fig. 8 presents a typical vertical displacement samples from the free-decay test. Nonlinear natural frequency and structural damping of the system were identified through a ‘local analysis’ method. In this method, displacement signals of a free-decay test were divided into numerous pieces by a moving time window (Fig. 8). Vibration frequency and structural damping in each displacement piece were assumed to be constant. The frequency was easy to be obtained, and the damping was calculated using logarithmic decrement based on least square fitting method (detail in Tang et al., 2019). It should be noted that the reference amplitude Yn of each displacement piece was represented by the amplitude in the middle of the displacement piece (as shown in Fig. 8). In this way, the variation of both natural frequency and structural damping with vibration amplitude can be identified by moving the time window from the start to the end of the displacement signal. Figs. 9 and 10 show the variation of identified natural frequency and damping ratio with vibration amplitude, where f1i and ζ1i (i = 1, 2, 3, . . .) denote the frequency and damping for Cable 1 and f2 and ζ2 for Cable 2. According to Fig. 9, only slight variation of vibration frequency (less than 0.015 Hz) was observed for different vibration amplitudes for Cable 2. But for Cable 1, natural frequency at about Y /H = 0.1 increased from 1.78 Hz to 2.02 Hz when the initial damping ratio changed from ζ11 = 0.093% to ζ16 = 3.62% as caused by the viscoelastic tape. The increase in natural frequency is mainly attributed to the wrapping stiffness introduced by the viscoelastic tape. Similarly, the initial natural frequencies in Table 1 were the natural frequencies at a small vibration amplitude (about Y /H = 0.1) for each test case, and they were used to calculate the critical wind velocity in Sections 3.1 and 4.1 below. With regard to damping ratio (Fig. 10), in general, damping ratio showed an increasing trend with vibration amplitude, especially at ζ < 0.4%. For instance, the damping ratio ζ11 for Cable 1 varied from 0.093% to 0.34% in the amplitude range of 0.1 < Y /H < 2.1. This nonlinearity should be accounted for when estimating the aerodynamic performance of main cable, especially when estimating the critical velocity-damping diagram. The initial damping ratios in Table 1 referred to the damping ratios identified at a small vibration amplitude (about Y /H = 0.1) for each test case, and they are used to calculate the corresponding Sc number. As mentioned earlier, structural damping of the main cable during erection is not measured. The nonlinear damping shown in Fig. 10 has no connection with the (nonlinear) damping for the full-scale cable. For instance, friction between

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Fig. 8. Typical vertical displacement signals from free decay test.

Fig. 9. Nonlinear damping of elastically mounted system versus vibration amplitude.

Fig. 10. Nonlinear natural frequency of elastically mounted system versus vibration amplitude.

adjacent steel strands may be an important source of nonlinear damping at high amplitude, and it may differ from the nonlinear damping caused by viscoelastic tape of the elastically-mounted system.

3. Galloping performance of main cables Vibration responses of Cable 1 and Cable 2 were obtained at various wind velocities U and wind incidence angles α in wind tunnel to study the galloping performance of the two cross-sections of main cables. In this section, the experimental results are discussed only at one initial structural damping for each cable, namely ζ11 = 0.093% for Cable 1 and ζ2 = 0.11% for Cable 2, and the effect of damping on galloping will be presented in Section 4. The critical velocity and vibration amplitude of galloping as well as the hysteresis phenomenon were presented and discussed in detail.

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Fig. 11. Vibration amplitude of the main cable with wind velocity: (a) Cable 1; (b) Cable 2.

Fig. 12. Time history response of the main cable: (a) Cable 1 at α = +2◦ , U /fH = 164; (b) Cable 2 at α = 0◦ , U /fH = 164; (c) Cable 2 at α = +3◦ , U /fH = 130.

3.1. Critical wind velocity In wind tunnel tests, large amplitude vibration was observed for both Cable 1 and Cable 2 at certain conditions. Fig. 11 shows the vibration amplitude of both cables versus reduced wind velocity U/fH, where f is the initial natural frequency in Table 1. Y is the stable amplitude of main cable in tests. For Cable 1 at α = +2◦ , as well as Cable 2 at α = 0◦ and +3◦ , apparent vibration occurred once the wind velocity reached a certain value, and then the vibration amplitude progressively increased with the wind velocity. While for Cable 1 at α = 0◦ and −2◦ , as well as Cable 2 at α = −3◦ , no evident vibration was observed in the whole range of wind velocity. Fig. 12 illustrates the displacement time history of section models at three cases when large amplitude vibration occurred, all the three cases represented typical sinusoidal vibration. The frequency domain analysis shows that the predominant frequencies of displacement response of the three cases were 1.78 Hz, 1.74 Hz and 1.74 Hz, respectively. These results agreed very well with the tested natural frequencies shown in Fig. 9. Thus, it can be concluded that the large amplitude vibration occurred on Cable 1 at α = +2◦ as well as Cable 2 at α = 0◦ and +3◦ was galloping. Galloping is a kind of destructive phenomenon caused by aerodynamic instability. Once the critical velocity is reached, the vibration amplitude of main cable will continuously increase with the wind velocity (Fig. 11). Thus it is important to

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Fig. 13. Aerodynamic coefficients of two cross-sections of main cable (An et al., 2016). Table 2 Comparison of tested and theoretical critical velocities of section models. Cable

Cable1

Cable 2

α

Reduced critical velocity Ucr /fH Tested (m/s)

Theorical (m/s)

−2◦

No galloping No galloping 86.64

No galloping No galloping 27.30

No galloping 75.06 63.36

No galloping 117.39 132.05

0◦ +2◦

−3◦ 0◦ +3◦

predict the critical wind velocity. According to classical galloping theory (Den Hartog, 1932), critical wind velocity Ucr can be determined as Ucr =

−2fH CL′ + CD

× Sc

(3)

where CD is the drag coefficient, CL′ refers to the derivate of lift coefficient with respect to wind incidence angle (i.e. dC L /dα ,). Aerodynamic coefficients of the main cable were obtained by An et al. (2016) through wind tunnel test, and they are illustrated in Fig. 13. To predict the critical wind velocity from Eq. (3), the initial structural damping ζ in Table 1 (0.093% for Cable 1 and 0.11% for Cable 2) was employed due to the small amplitude vibration at galloping onset. Comparisons between the tested and predicted critical wind velocity for the two models are provided in Table 2. It can be seen that the possibility of galloping instability can be successfully determined from Den Hartog criterion; however, the discrepancy in critical wind velocity is quite obvious between measurement and prediction. In other words, the Den Hartog criterion is applicable to forecast the possibility of galloping, but not able to accurately estimate the critical wind velocity. Possible reasons for the discrepancy are discussed as follows. First, the quasi-steady aerodynamic forces are employed in Den Hartog criterion, and the memory effect in flow is neglected. Moreover, critical wind velocity calculated by Eq. (3) is very sensitive to the value of CL′ , but in practical wind tunnel tests, both the gradient of α as well as the adopted fitting method of CL − α curve may cause large errors, especially at a wind incidence angles where CL changes abruptly. Therefore, the quasi-steady galloping theory can be adopted for the preliminary assessment of galloping performance for the main cable during construction phases, and further study should be conducted based on unsteady methods such as wind tunnel test for engineering application. 3.2. Hysteresis phenomenon Hysteresis is associated with discontinuity in flow around bluff body and can be found in various wind-induced vibrations such as flutter, vortex-induced vibration, and galloping. The typical feature of hysteresis is the bifurcation of unsteady aerodynamic forces which could result in transition of vibration amplitude in a certain wind velocity (Mannini et al., 2018). However, this phenomenon has not been properly considered in practice. This neglect may result in potential misjudgments for the aerodynamic performance of structures. In view of this, a special test on Cable 1 at α = +2◦ was conducted to study the galloping hysteresis of main cable. The test was performed through three steps: first, reduced wind velocity was gradually increased up to about 200, and steady responses at each wind velocity are obtained; after that an artificial excitation was exerted on the cable while keeping the reduce wind velocity; at last, the wind velocity

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Fig. 14. Vibration response of Cable 1 for increasing and decreasing wind velocity (α = +2◦ ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 3 Amplitude growth rate k for galloping of main cable.

α k

Cable 1

Cable 2

+2◦

0◦ 1.451×10−2

1.665×10−2

+3◦ 1.927×10−2

was gradually decreased until the cable went back to a stable condition. Vibration amplitude of the cable for both cases of increasing wind velocity and decreasing wind velocity was illustrated in Fig. 14. According to Fig. 14, the amplitude Y/H leaped from 2.14 to 2.24 after the artificial excitation and then decreased with the wind velocity. Generally speaking, there are slight difference in vibration responses for two branches, one being a progressive increase of wind velocity and the other a progressive decrease. Hysteresis phenomenon seems to be more obvious for higher reduced velocity range, and it gradually diminishes for lower reduced velocity range. At U/fH > 150, there was slight difference between the two branches, the relative difference of amplitude at the same wind velocity varied from 4.4% to 7.8%. And at U/fH < 150, vibration amplitude for the two branches agreed very well. According to Figs. 9 and 10, both the structural damping and natural frequency of the main cable kept almost constant in the range of 1.3 < Y /H < 2.1, thus the structural nonlinearity can be ignored. In other words, aerodynamic nonlinearity is mainly responsible for the minor discrepancies between the two branches. In summary, the vibration amplitude of main cable did not produce an evident hysteresis loop. Therefore, the weak amplitude hysteresis in galloping of a main cable can be neglected in engineering practice, and it can be considered that there is only one stable galloping amplitude at a certain wind velocity. 3.3. Vibration amplitude Amplitude of galloping were selected and shown in Fig. 15 for further study of galloping performance. Similar to iced conductors (Li et al., 2019) and rectangular cylinders (Mannini et al., 2014, 2018), the vibration amplitude of Cable 1 and Cable 2 at different wind incidence angles all increased approximate linearly with wind velocity. This character shows a significant difference to the galloping of other bridge members such as parallel hanger ropes (Wen et al., 2018), stay cables (He et al., 2018) and I-section truss members (Wardlaw, 1990). This may be on account of the aerodynamic characteristics of different cross sections. Based on this ‘linear’ character, an empirical formula for galloping amplitude was proposed as follow: Y H

=k

U fH

+b

(4)

where k is a dimensionless factor representing the growth rate of galloping amplitude with wind velocity, and b is a constant. Galloping amplitude of the main cables were fitted via Eq. (4), and the results are shown in Fig. 15. The values of k are listed in Table 3. According to Fig. 15, the fitted data agree well with the test data. Therefore, the galloping amplitude at higher wind velocities could be predicted via the empirical formula. According to Table 3, the values of k for the three cases were different to each other. Results of Cable 2 show that the growth rate of galloping amplitude with wind velocity was affected by wind incidence angle. In fact, k mainly depends on aerodynamic conditions of structures such as aerodynamic configuration (Zhou et al., 2014; Niu et al., 2015) and wind incidence angle, rather than structural conditions such as Sc number, which will be further studied in Section 4.1.

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Fig. 15. Vibration amplitude of galloping of main cable versus reduced wind velocity.

Fig. 16. Vibration response of Cable 1 with different structural damping (α = +2◦ ).

4. Vibration mitigation measures As is well-known, structural damping (or Sc number), stiffness and aerodynamic configuration are the key factors affecting galloping performance. In order to find out an effective measure to mitigate galloping responses, these three factors will be studied and discussed in this section. 4.1. Effect of structural damping A set of tests on Cable 1 was performed to study the effect of initial structural damping (or Sc number) on the galloping performance of a main cable. Six damping ratios were adopted in these tests (i.e. ζ11 = 0.093%, ζ12 = 0.26%, ζ13 = 0.43%, ζ14 = 0.98%, ζ15 = 2.11% and ζ16 = 3.62%), corresponding to Sc number in the range 108–4196. Fig. 16 illustrates the wind-induced vibrations of the main cable at α = +2◦ with different initial structural damping. It should be noted that for the full-scale bridge, the actual wind velocity Uf can be calculated by Eq. (2) in accordance to similarity principle. According to Fig. 16, galloping occurred at all the test damping ratios when a critical wind velocity was reached. Again, once galloping was triggered, the steady amplitudes at different damping all increased approximate linearly versus the wind velocity. Hereupon the galloping amplitude of main cables were selected and linearly fitted via Eq. (4), and the fitted results are shown in Figs. 17 and 18. Fitted curves plotted in Fig. 17 are almost in parallel to each other, which means that the coefficient k is almost constant with an average value of 1.627×10−2 (Fig. 18). In other words, the amplitude growth rate is not sensitive to the initial structural damping. Recalled that the nonlinear variation of structural damping with vibration amplitude is different for the five cases as shown in Fig. 10, the galloping amplitude is also insensitive to the nonlinear structural damping present in suspension system. It may be concluded that the aerodynamic damping is the primary source leading to the limited amplitude oscillation of galloping. From the above, critical wind velocity at different structural damping is particularly important for galloping suppression, since the amplitude still increases rapidly at a high structural damping. Fig. 19 plots the tested critical wind velocity at various damping, and the theoretical critical velocity was calculated via Eq. (3) for comparison. In addition, tested data of critical wind velocity were linearly fitted in order to roughly predict

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Fig. 17. Measured and fitted vibration amplitude of Cable 1 at different initial structural damping (α = +2◦ ).

Fig. 18. Amplitude growth rate k of Cable 1 at different structural damping (α = +2◦ )

Fig. 19. Critical velocity of Cable 1 as function of structural damping (α = +2◦ ).

the critical wind velocity at higher damping. According to Fig. 19, fitted results are in good agreement with the tested ones. However, the theoretical and experimental results show a great difference: at small damping ratio, theoretical wind velocity is smaller than experimental wind velocity, and vice versa. This further substantiates the discussion in Section 3.1: Den Hartog criterion is applicable to forecast the possibility of galloping, but not able to accurately estimate the critical wind velocity. Although damping solution is widely adopted for galloping suppression in bridge engineering, the feasibility for a main cable in construction should be taken into account. According to Fig. 19, the tested critical wind velocity Ucr /fH of galloping is 87 at small damping ratio ζ11 = 0.093%, and it increases to 205 at a larger damping ratio of ζ16 = 3.62%. It is obvious that structural damping has an evident effect on the excitation of galloping, but this effect is still rather limited. In fact, the critical damping ratio to suppress the galloping in practice can be very large. For instance, if a critical wind velocity

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Fig. 20. Arrangement of a rigid support for galloping suppression. Table 4 Effect of rigid supports for galloping resistance. Number of supports

Natural frequency for full-scale bridge ff (Hz)

Critical wind velocity for full-scale bridge Ucr,f (m/s)

None 1 3 5 7 9 11

0.09 0.22 0.35 0.55 0.75 0.95 1.14

1.41 3.39 5.39 8.48 11.56 14.64 17.63

Ucr ,f = 10 m/s for full-scale bridges (corresponding to a reduced wind velocity U /fH = 616, calculated by Eq. (3)) is required, a damping ratio about ζ = 15.82% will be necessary (calculated by the linear equation in Fig. 19), which is quite impractical by external mechanical dampers. More than that, it would be uneconomical to use mechanical dampers in construction phases. From the above, it is not recommended to suppress galloping of main cables by damping measures only. As mentioned earlier, the Sc numbers discussed in this study are only indicative of the wind tunnel tests, yet possibly inconsistent with the actual full-scale cable. This may result in discrepancies between predictions and full-scale behavior. Even so, tested results covering a wide range of structural damping may still provide a reference for galloping suppression. 4.2. Structural measure Stiffness is an essential factor for structural stability, particularly the wind-induced vibration of structures in wind flow. According to existing theories (Den Hartog, 1932; Nigol and Buchan, 1981a,b), critical wind velocity of galloping can be enhanced by increasing the stiffness of structures. In practice, rigid connections of parallel cables in cable-stayed bridges (Giaccu and Caracoglia, 2012), spacers of hangers in suspension bridges (Wen et al., 2018) and auxiliary steel wires of H-section hangers in arch bridges (Chen et al., 2012) are widely implemented to prevent cable structures from galloping and other aerodynamic instabilities. Catwalks is the temporary platform for erection of main cables of suspension bridges. For a large span suspension bridge, two catwalks are generally connected by several transverse gangways, and multifarious wind-resistance ropes are usually utilized for the security of catwalks. Hence the catwalks and wind-resistance ropes form a stable structural system. Measures to enhance the stiffness of catwalks have been widely studied by researchers (Li and Ou, 2010). Conversely, the main cables are freely hanging to the bridge towers and are independent of other structures and facilities in construction phases. As a result, main cables would be relatively flexible compared to catwalks (Li and Ou, 2010). From the above, main cables and catwalks can be connected together using some temporary members at certain intervals along the bridge span direction to enhance the main cable stiffness, and the critical wind velocity of galloping can be increased to some extent. Fig. 20 shows the arrangement of a rigid support which can be used for galloping suppression. When fixed to the catwalk using the proposed rigid support, the main cable is divided into several segments, and its natural frequencies mainly depend on the spacing of the adjacent rigid supports. Therefore, finite element analysis was implemented on Cable 1 to study the appropriate number of rigid supports. Table 4 summarized the numerical results of the fundamental vertical frequency and galloping critical velocity for different number of supports. In the analysis, rigid supports were arranged at equal spacing and the damping ratio of ζ11 = 0.093% was adopted. According to Table 4, the fundamental natural frequency of the main cable for the vertical vibration mode increases with the number of rigid supports. Therefore, the corresponding critical wind velocity for full-scale bridge Ucr ,f increases as well. For instance, if a critical wind velocity Ucr ,f = 10 m/s is required, the appropriate number of the rigid supports will

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Fig. 21. Foamed plastic material used for aerodynamic measure.

Fig. 22. Section model wrapped by foamed plastic materials: (a) half-wrapped; (b) totally-wrapped.

be 7. Unfortunately, the tests for the effect of stiffness on aerodynamic performance of main cables were not performed on account of the test condition, and the exact effect of stiffness on the critical wind velocity of galloping was not experimentally validated. Furthermore, as is pointed out by Li and Ou (2010), common natural frequency for the connected main cable and catwalk mainly depends on the stiffness of the reinforced catwalk, and the enhance of catwalk stiffness is limited. When the bridge suffers a strong wind with a velocity of 10 m/s, galloping may still occur: the main cable drives the catwalk and they could vibrate together. In spite of this, the proposed rigid support provides an alternative effective method for galloping suppression of main cables in construction phases. 4.3. Aerodynamic measure Central to aerodynamic instability of a main cable is its aerodynamic configuration. Thus, galloping can be effectively suppressed by aerodynamic modification of cross-sectional shapes (Christiansen et al., 2018a,b; Carassale et al., 2013). An et al. (2016) analyzed the flow field characteristic around main cable at various construction stages and wind incidence angles, and pointed out that the aerodynamic instability mainly depends on the flow separation and reattachment around a main cable. Considering the construction condition of main cable, a temporary aerodynamic measure may be helpful for galloping suppression. For instance, changing the aerodynamic configuration by wrapping the main cable using foamed plastic materials (Fig. 21) may have a beneficial effect on the aerodynamic performance. A sequence of tests was conducted to study the efficiency of this method. Firstly, the appropriate wrapping rate of the main cable was studied through a contrast experiment on Cable 1. There were two aerodynamic measures taken in the contrast test: half-wrapped and totally-wrapped (Fig. 22). The vibration responses of Cable 1 under two aerodynamic measures were tested and shown in Fig. 23. As expected, the proposed aerodynamic measure is beneficial to the vibration control for Cable 1: the reduced critical wind velocity was significantly enhanced from 87 to 164 when the cable was half wrapped by foamed plastic materials,

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Fig. 23. Effect of aerodynamic measure on galloping of cable 1 at α = +2◦ .

Fig. 24. Displacement time history of half wrapped cable 1 at α = +2◦ .

and no evident vibration occurred when the cable was totally wrapped. Nevertheless, galloping was not completely inhibited for the half-wrapped cable. Displacement time history of the half-wrapped model was very similar to those of the uncovered model once obvious vibration occurred at high wind velocity: both of the vibration frequency and amplitude were stable (shown in Fig. 24). When galloping was excited, the amplitude was not constant but it dramatically changed with the increase of wind velocity, reaching a maximum Y /H = 2.1 at U/fH = 248. Based on the above comparison, it can be concluded that the foamed plastic materials improved the aerodynamic performance of the wrapped portion of the segment model. And the galloping of half wrapped cable was mainly on account of the unwrapped portion of the segmental model. Therefore, the totally wrapping scheme is recommended for practical application. Same tests were performed on Cable 2 for further validity check of this aerodynamic measure. Fig. 25 illustrates the effect of aerodynamic measure on vibration response of Cable 2. No less than Cable 1, galloping was effectively suppressed in the test wind velocity while the cable was totally wrapped by foamed plastic materials. Although it is hard to acquire the form of flow field around a wrapped main cable, it can be speculated that the original flow field was disturbed by the foamed plastic materials since the overall outline of the main cable was changed. The cable’s cross section is no longer continuous along the axis once it is wrapped by foamed material. And regular flow separation along the cable axis may be disrupted by the bubbles on foamed material. Therefore, the aerodynamic force is no longer capable to excite cable galloping. Similar measurements have been adopted for galloping mitigation of other structures (Benidir et al., 2015; Wang et al., 2019). It is worth noting that, if the foamed plastic materials are used for galloping suppression, the scale ratio should be taken into account. In other words, size and distance of the bubbles of the foamed plastic material should be scaled using the same scale ratio of main cable (In fact, the foamed plastic material is very common, and its size is multiple). On the other hand, the proposed efficient aerodynamic measure is not applicable to the whole construction period. It is apparent that the construction of main cable would be interrupted by such a temporary measure. Thus, it is recommended to be implemented only when the construction suffers a strong wind and is forced to be interrupted. Or in other words, it can be adopted as a remedy countermeasure for galloping suppression. Based on the above discussion in Section 4, appropriate structure and aerodynamic measures can make an alliance to suppress galloping of main cables which is possible to occur in the construction phases of a suspension bridge, especially in areas prone to typhoons.

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Fig. 25. Effect of aerodynamic measure on galloping of cable 2: (a) α = 0◦ ; (b) α = +3◦ .

5. Conclusions The aerodynamic performance of the main cable in construction phases was studied through a series of wind tunnel tests, and appropriate countermeasures to eliminate the potential wind-induced vibrations were developed. Based on the existing research findings, two risk cross-sections with different shapes of a main cable were chosen for the wind tunnel tests. Elastically mounted sectional models were adopted and tested at various conditions, and the results were discussed in consideration of the structural nonlinearity of the elastically mounted system. In the tests, limited amplitude oscillation occurred at the two typical construction stages under certain wind incidence angles. This further proved the possibility of galloping occurrence, which was generally neglected by engineers. No obvious hysteresis phenomenon of galloping was observed in the tests, or in other words, approximately, there would be only one steady amplitude for a certain wind velocity. Once galloping is triggered, the steady amplitude will increase linearly versus the wind velocity and the increase rate will almost keep constant with a changing structural damping. The nonlinear aerodynamic damping, rather than the nonlinear structural damping, is the primary source leading to the limited amplitude oscillation. Empirical formulas for a main cable in different conditions were obtained based on the test data, which can be used to predict the galloping amplitudes. Den Hartog criterion was applicable to forecast the possibility of galloping, but not able to estimate the critical wind velocity. A linear fitting method can be used to predict the critical wind velocity based on the tested data. Damping has a significant effect on the critical wind velocity of galloping, but the critical damping ratio to suppress the galloping in practice can be very large, which is generally hard to achieve for a main cable by attaching external dampers. What is more, it is uneconomical to use mechanical dampers during construction. But it can be helpful to implement appropriate structural measures, for instance, by connecting the main cable and the catwalk using rigid supports. Besides, galloping could be efficiently eliminated by wrapping the whole main cable using foamed plastic materials. And the use of obsoleted foamed plastic materials in construction phases will be quite convenient and economic. In general, a combination of structural and aerodynamic measures was recommended for galloping mitigation of main cables in construction phases. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This study is supported by the National Science Foundation of China under grant No. 51422806, and by the Hunan Provincial Innovation Foundation for Postgraduate under grant No. CX20190288, which are greatly acknowledged. References Achenbach, E., 1971. Influence of surface roughness on the cross-flow around a circular cylinder. J. Fluid Mech. 46 (2), 321–335. An, Y.H., Wang, C.Q., Li, S.L., Wang, D.W., 2016. Galloping of steepled main cables in long-span suspension bridges during construction. Wind Struct. 23 (6), 595–613. Barbieri, N., Souza Jr, de O.H., Barbieri, R., 2004. Dynamical analysis of transmission line cables. Part 2-Damping estimation. Mech. Syst. Signal Process. 18, 671–681.

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