PSI - Issue 62

672 L. Zoccolini et al. / Procedia Structural Integrity 62 (2024) 669–676 L. Zoccolini, E. Bruschi, C. Pettorruso, D. Rossi and V. Quaglini / Structural Integrity Procedia 00 (2019) 000 – 000 where ( ) is the force developed by the damper, is the damping coefficient, is the damping exponent, ̇( ) is the axial velocity of the damper, and [ ∙ ] is the signum function. The values of and depend on the piston head orifices design (Castellano et al., 2004; Symans et al., 2008). The damping exponent modifies the behavior of the device. Depending on its value, it is possible to distinguish three FVDs categories: (i) when =1 the devices are called linear FVDs, and the force produced by the device is directly proportional to the axial velocity; (ii) when <1 the devices are called non-linear FVDs; (iii) when >1 the devices are called ultra-linear FVDs (Christopoulos & Filiatrault, 2006). The amount of dissipated energy increases as decreases. Indeed, non-linear FVDs are commonly used for the seismic protection of structures (Antonucci et al., 2004). Ultra-linear and linear FVDs are applied to those structures that are mainly subjected to wind load. These devices work as shock-transmitting or lock-up devices, producing large forces at high velocities (Seleemah & Constantinou, 1997). 2.1. Application of FVDs on bridges The first FVDs were introduced in the military and aerospace industry in the 1860s (Taylor, 2002b). Thanks to their great capacity to dissipate energy and structural vibration control, the FVDs were adopted to mitigate vibration starting from the last decades of the 20 th century. They were installed not only on buildings (Lago et al., 2019) but also on bridges. The first time that a FVD was used to increase the seismic resistance of a bridge was in 1997 on the Golden Gate Bridge in the USA (Aiken & Kelly, 1996; Rodriguez & Ingham, 1996). It is a bridge composed of three spans, a main span of 1280m and two side spans of 343m. The Golden Gate Bridge comprises two steel towers, cables and suspenders, stiffening trusses, concrete pylons, and anchorage blocks. After the Loma Prieta earthquake in 1989, seismic evaluation and retrofit studies were conducted to determine the vulnerabilities of the bridge. The seismic evaluation showed that the expansion joints, wind locks, and longitudinal connection between the side span and the towers were vulnerable due to the high participation of secondary longitudinal vibration modes. Moreover, secondary transverse vibration modes were responsible for an over-stress in shear and torsion of the lateral braces of the stiffening truss. To address these issues, the bridge piers, saddles that support the cables on the tops of the towers, and wind locks were strengthened. In addition to that, FVDs were installed longitudinally between the main span and the towers and transversally between the side south span and the pylons. The presence of this device allowed the longitudinal relative displacements at the expansion joints and wind lock to be reduced. In this way, the possibility of an impact between the suspension spans and the towers is eliminated. Moreover, FVDs reduced the induced stress by the longitudinal displacements into the towers and avoided coupled vibration between the south side span and the pylons by isolating them. Another representative example of the effectiveness of FVDs application can be found in the retrofit intervention on the Millennium Bridge in London. Shortly after its initial construction in 2000, this pedestrian suspension bridge experienced issues with excessive lateral sway, which not only compromised the comfort and safety of its users but also raised concerns regarding its structural integrity. To enhance the bridge performance, the Taylor Company implemented a retrofit solution incorporating 37 FVDs into its design (Klembczyk, 2017; Taylor, 2002a). These dampers are characterized by metal bellows that are seals that guarantee long life without any required maintenance. Three different types of dampers were used: 16 dampers were installed at each side of the two piers, 17 dampers were installed under the bridge deck, and 4 vertical dampers were placed in pairs under the south end of the bridge. The installation of FVDs in the Millennium Bridge addressed the issue of lateral movements induced by external forces, such as strong winds and the dynamic motions of pedestrians walking across the bridge. When the bridge experiences lateral movements due to dynamic loads, the FVDs absorb and dissipate the excess of the kinetic energy, thereby reducing the lateral vibrations and restoring the bridge's stability. The FVDs were adopted also in the design of new bridges (Infanti et al., 2017; Sartori et al., 2021). One example is the case of the Rion-Antirion Bridge, located in the Gulf of Corinth, Greece. The cable-stayed bridge has four pylons with a 2252m extended suspended deck. It is completed by a 986m long approaching viaduct on the Rion side and another 228 m long one on the Antirion side (Fig. 2a). The bridge was designed to withstand seismic excitations with a return period of 2000 years, characterized by a peak ground acceleration (PGA) of 0.48g. Therefore, fuse restraints in conjunction with FVDs were included in the design (Infanti et al., 2004). The main deck is restrained by a fuse 4