Experimental study and mathematical modeling on the unsteady galloping of a bridge deck with open cross section

Highlights

• Unsteady galloping was studied in the wind tunnel for a bridge deck with open cross section.

• Qualitatively different behaviors were observed for slightly different angles of attack.

• Aeroelastic test results for small-step increments of the Scruton number.

• Comparison with the unsteady galloping behavior of rectangular cylinders.

• For the first time, Tamura’s wake-oscillator model is applied to a bridge deck.

Abstract

Galloping instability can potentially threaten the modern launching of steel-concrete composite bridge girders, due to light weight and bluff shape of the steel box, which is normally launched first. A bridge deck with typical open cross section was selected and investigated in smooth flow through wind tunnel techniques. Aeroelastic tests showed that the classical instability arising from the interaction between vortex-induced vibration and galloping may occur for a mean flow incidence of 4°, fixing the actual galloping onset at the Kármán-vortex-resonance wind speed up to a high value of the mass-damping parameter. In contrast, a different and more complicated behavior was observed for a mean flow incidence of 0°, where the actual galloping instability occurs at a wind speed clearly higher than the Kármán-vortex-resonance wind speed even for a very low value of the mass-damping parameter. Static tests further indicated that the most evident difference between the two cases is the magnitude of the vortex shedding force, which is much lower for a null angle of attack. A rectangular cylinder with the same side ratio was also tested for the sake of comparison. Finally, Tamura’s wake-oscillator model was implemented for the bridge deck at a mean flow incidence of 4°, following a recently proposed parameter identification method. The mathematical model was found to be able to give some promising predictions even for a complex bridge deck profile.

Introduction

Across-wind galloping is an aeroelastic instability typical of slender structures with bluff noncircular cross sections, like square or D shapes. Compared to another aeroelastic phenomenon that could also occur in the across-wind direction, namely the vortex-induced vibration (VIV), the apparent difference is the unrestricted oscillation amplitude with the increase of wind speed.

The quasi-steady (QS) theory is well accepted as a classical approach to treat the across-wind galloping problem provided that a high reduced wind speed is ensured (Parkinson and Brooks, 1961; Parkinson and Smith, 1964; Novak, 1969, 1972; Novak and Davenport, 1970; Laneville and Parkinson, 1971; Novak and Tanaka, 1974; Parkinson and Sullivan, 1979). The definition of a high reduced wind speed for the applicability of QS theory is, however, dependent on the specific cross section. For a square cross section, Parkinson and Wawzonek (1981) suggested that the QS theory can be used for a reduced wind speed higher than 2.15 times the Kármán-vortex-resonance reduced wind speed V_r (defined here as V_r ​= ​1/(2πSt), being St the Strouhal number). For the same geometry, Blevins (1990) later slightly adjusted this restriction to 2.7. For a 3:2 rectangular cross section (side ratio equal to 1.5 and the short side facing the flow), Mannini et al. (2016b) suggested that the reduced wind speed should not be lower than 3.5V_r. As for the 2:1 rectangular cylinder (side ratio equal to 2), a high reduced wind speed could even mean 5–6 times V_r (Santosham, 1966). Nevertheless, a high value of mass-damping parameter, known as Scruton number, is usually needed to ensure that the galloping instability occurs in a high reduced wind speed range. If a sufficiently high reduced wind speed cannot be ensured, unsteady effects due to shed vortices and fluid memory become non-negligible. The galloping instability occurring at low reduced wind speed can therefore be named “unsteady galloping”, as opposed to the galloping instability initiating at high reduced wind speed, which can be called “high-speed galloping” or “quasi-steady galloping” due to the good performance of the QS theory.

In particular, the unsteady galloping arising from the interaction with VIV is able to promote unrestricted oscillations starting at the Kármán-vortex resonance wind speed, instead of at the critical velocity predicted by the QS theory. In some other cases, the actual galloping onset was reported to occur earlier than the quasi-steady threshold, although VIV and galloping phenomena can already be observed separately (Mannini et al., 2014, 2016b). This typical unsteady galloping was first noted by Parkinson and his co-workers during the wind tunnel tests on a square cylinder (Parkinson and Brooks, 1961; Parkinson and Smith, 1964). Later investigation indicated that this phenomenon is even more evident for a 2:1 rectangular cylinder (Santosham, 1966) and also possible for three-dimensional towers, even in turbulent flow (Parkinson and Sullivan, 1979; Novak and Davenport, 1970). Aerodynamic force measurements on forced-vibrating cylinders provided deeper insight into this phenomenon. For a square cylinder (Otsuki et al., 1974; Nakamura and Mizota, 1975; Bearman et al., 1987; Luo and Bearman, 1990) or a 2:1 rectangular cylinder (Nakamura and Mizota, 1975; Washizu et al., 1978), the phase angle of the motion-induced aerodynamic force was reported to change its sign from negative to positive for a reduced flow velocity slightly higher than V_r. This means that a positive aerodynamic damping is introduced into the dynamical system for the reduced wind speed lower than V_r, rather than a negative damping according to the QS theory. At the same time, the forced-vibration tests also suggested that the interference with vortex shedding is able to promote a higher magnitude of the motion-induced aerodynamic force compared to that predicted by the QS theory, for reduced wind speed no much higher than V_r.

Two important mathematical models aiming at capturing the interaction mechanics of VIV and galloping were later proposed by Corless and Parkinson (1988) and Tamura and Shimada (1987), by coupling an equation modeling the dynamics of the vortex-shedding force to the classical nonlinear equation describing the transverse motion of the cylinder according to the quasi-steady theory. In describing the self-excited and self-limited characteristics of the vortex shedding force, the former adopted the lift-oscillator model according to Hartlen and Currie (1970), which is a Rayleigh-type oscillator, while the latter adopted the wake-oscillator according to Tamura and Matsui (1979), which is a Van der Pol-type oscillator. In particular, the model proposed by Tamura and Shimada (1987) was deduced based on physical considerations rather than simply with an empirical approach. For a square cylinder, both models were found to be able to give at least qualitative agreement with the experimental results, except for the amplitude close to the vortex-resonance wind speed, which was apparently overestimated (Corless and Parkinson, 1988; Tamura and Shimada, 1987). Some recent contributions to the understanding of the mechanism of the interaction of VIV and galloping were provided by Mannini et al. (2014, 2016b, 2018a, 2018b) for a 3:2 rectangular cylinder, Gao and Zhu (2016, 2017) for a 2:1 rectangular cylinder, and Ma et al. (2018) for a square cylinder. In all cases, the Scruton number is recognized as a key parameter to distinguish the complicated unsteady galloping behaviors observed during wind tunnel tests.

The mean flow incidence also plays an important role in the unsteady galloping problem. For a square cylinder, Luo et al. (1994) found that the strength of the vortex-shedding force on the stationary body significantly decreases with the angle of attack up to about 12°, thus implying that the interference between VIV and galloping could be different for a non-null flow incidence. Through forced-vibration tests on a square cylinder, Carassale et al. (2015) showed that the evolution of the phase angle of the motion-induced aerodynamic lift with reduced wind speed is strongly dependent on the mean flow incidence. In particular, they showed that the change of the phase angle from negative (stable) to positive (unstable) occurs at a reduced wind speed significantly higher than V_r for mean flow incidences of 6° and 9°. In addition, no quasi-steady limit was found for high reduced flow speeds when the mean flow incidence is set to 12°. This is quite different from the previously mentioned case of null mean flow incidence. The dependence of the phase angle evolution on the mean flow incidence was also highlighted by Luo et al. (1998) for several trapezoidal and a triangular cross sections.

The unsteady galloping occurring at low reduced wind speeds is not only physically interesting, but also of great importance for modern structures. Indeed, either for the development of construction materials and techniques or just for aesthetic requirements, some structures are nowadays very slender and light. In fact, evidences of unsteady galloping instability have already appeared in the recent engineering practice. Examples are two aesthetic arches in Milan, Italy (Mannini et al., 2016a), a footbridge deck with solid parapets (Cammelli et al., 2017) and a steel beam bridge during the construction phase (Salenko et al., 2017). Nevertheless, unsteady galloping seems particularly relevant for the modern launching of steel-concrete composite bridges, since the steel box girder, which is normally launched first, could feature light weight, low damping and bluff cross section. It is also frequently noticed in the literature reports, especially for European countries, that a single steel box girder without any auxiliary support can be launched to cross spans longer than 120 ​m, at heights above the ground of more than 100 ​m (Wagner, 2013; Hanswille, 2014; Mansperger et al., 2017). In particular, the launching of the Aftetal bridge, Germany, should be mentioned, since its 90-m steel box girder was totally modified during the construction phase with temporary wind fairings to eliminate the risk of transverse galloping (Hanswille, 2014). Fig. 1 shows the complete cross-section geometry of the Aftetal bridge and the aerodynamic shape modification of the steel box during launching phase.

In spite of the fact that an open cross section like the one shown in Fig. 1(b) represents a typical state of the steel-concrete composite bridge during the construction phase, only few case studies can be found in the literature. Besides the Aftetal bridge, wind tunnel tests on this type of bridge deck are also discussed in Consolazio et al. (2013), where a negative slope of the lift coefficient around the null wind angle of attack is shown, suggesting the possibility of across-wind galloping instability. Also in Cammelli et al. (2017), though referring to a footbridge, the aerodynamic and aeroelastic behavior of a cross-section geometry somehow similar to the one studied here is investigated. The experimental tests revealed a strong tendency of VIV and galloping to interact for low values of the Scruton number.

Motivated by this engineering background and by the lack of studies in the literature on this bridge deck typology, a bridge deck model with open cross section, inspired by the prototype of the Aftetal Bridge, was realized to conduct a wind tunnel investigation in smooth flow. A rectangular cylinder model with the same side ratio of 2 was also tested for the sake of comparison. In addition, the unsteady galloping behavior of the bridge deck section was modelled with Tamura’s wake-oscillator model (Tamura and Shimada, 1987; Mannini et al., 2018b). Since this model is built over physical concepts, such a study is expected to help a better understanding of the complex mechanism of unsteady galloping. Finally, it is to note that this is the first time that this model is applied to a sectional geometry different from the circular and rectangular ones, and that it is put to the test of two angles of attack able to induce markedly different aeroelastic behaviors.