## 1 Introduction  
Long-span bridges play a crucial role in civil engineering and are considered major engineering feats. Since the construction of the Brooklyn Bridge in 1883, the design of modern suspension bridges has greatly advanced. The span length of suspension bridges rapidly increased from 486 m for the Brooklyn Bridge to 1067 m for the George Washington Bridge in 1931 and up to 1991 m for the Akashi Kaikyo Bridge in 1998 [1; 2]. The design of these bridges is constantly advancing along with the materials of construction [3], allowing engineers to construct longer, slender, and more flexible structures [4]. However, the flexible nature of these structures requires precise assessments of their aerodynamic properties for stability and safety [5]. For example, pipeline suspension bridges, used for water, oil, or gas transportation, can be highly sensitive to wind due to their narrowness, lightness, and flexibility [6]. In addition, non-engineering defects can also contribute to the failure of reinforced structures [7]. Wind-induced vibrations on long-span bridges are a complex phenomenon that can cause significant structural damage if not properly addressed. During construction, these bridges are especially vulnerable as the main cables are hung from the bridge towers and are not supported by a stiffening girder, leading to low rigidity and sensitivity to wind excitations [8; 9]. This has been demonstrated by several bridges in China that experienced large-amplitude, wind-induced vibrations during construction, posing a risk to both structures and workers. Substructure failures, such as in piles, also cause numerous bridge failures [10], and it is crucial to understand the importance of predicting appropriate pile displacement responses in linear elastic soil systems [11; 12]. Historically, wind loads have been treated as static, but they can cause structural resonance [13]. Vibrations caused by the wind's interaction with the bridge's structure can excite resonant frequencies, leading to excessive motion. This can result in fatigue damage, cracking, and even collapse if left unmitigated. Engineers employ various techniques, such as aerodynamic shaping, dampers, and tuned mass dampers, to reduce wind-induced vibrations on long-span bridges. Aerodynamic measures are commonly used to enhance the wind-induced vibration performance of bridges. Some nondestructive testing methodologies, such as magnetic flux leakage, can also be employed to detect early damage in long-span bridges [14]. Wind tunnel tests are frequently conducted to study the performance of these measures. Sectional model tests with large-scale ratios are often used to measure aerodynamic coefficients, analyze vortex-induced vibrations (IVVs), and investigate two-degree-of-freedom flutter [15]. The effects of handrails and wind-fairing guide vanes on bridge aerodynamics can also be studied through sectional model tests. Full-bridge aeroelastic model tests can accurately simulate the multi-order modal responses and wind-induced vibrations of the bridge in actual atmospheric boundary conditions [16]. Tests have shown that suitable wind fairing can improve flutter stability and reduce VIV response [17; 18]. The collapse of the Tacoma Narrows Bridge in 1940 is a famous example of wind-induced vibrations on a long-span bridge [19]. These vibrations can manifest as flutter, galloping, VIVs, or random buffeting. Flutter and VIVs are the primary forms of wind-induced instability vibrations that pose the greatest threat to the safety of long-span bridges [20]. The bridge's collapse was due to aeroelastic flutter, which was caused by the wind's interaction with the bridge's structure. Flutter can lead to structural collapse after amplitude divergence, while VIVs cause bending or torsional single-mode vibrations that affect driver and pedestrian comfort. During the construction of the Old Tacoma Narrows Bridge, VIVs were present and transformed into flutter just prior to the bridge's collapse. Ref.