Benefitting from economic development and technological progress, long-span suspension bridges, with their superior span capacity and good economy, have been built in large numbers in recent decades. However, the excessive cumulative longitudinal displacement at the girder ends in the process of bridge operation leads to the degradation of, and fatigue damage to, the connecting components. This study aims to solve the problem with an effective parallel damping scheme of friction dampers and viscous dampers. Firstly, the phenomenon that quasi-static longitudinal displacement accounts for the majority of cumulative displacement is confirmed by the decomposition of measured displacement data at the girder end, which is caused by the asymmetric geometric deformation of the main cable induced by the moving vertical loads of a long-span suspension bridge. An efficient control performance analysis method is proposed based on the formation mechanism of the quasi-static longitudinal displacement. Secondly, the friction damper with a continuous damping model is employed to achieve an effective control performance with respect to the quasi-static longitudinal displacement. Thirdly, in order to realize the target of operational and seismic dual control, a parallel scheme of friction dampers and viscous dampers is proposed, aiming to reduce the cumulative value in the operational state, and maximum value in the seismic state, for longitudinal displacement at the girder ends of a long-span suspension bridge.

The accurate description of plasticity and fracture behavior is essential in numerically investigating the mechanical responses of high-strength bolts under tension, shear and coupling loads. However, based on the von Mises criterion, inputting the constitutive relation and damage model from the tensile coupon test into the finite element method cannot properly predict the shear behavior of high-strength bolts. Cylindrical tensile coupons and shear specimens of common and weathering high-strength bolts are tested to obtain the complete tensile and shear responses. The combined S-V model and the modified shear constitutive model are collaboratively used to calibrate and describe the tensile and shear constitutive relations of high-strength bolts, and then the Bao–Wierzbicki model is used to predict the tensile and shear fracture behaviors. Furthermore, the collaborating method is used to discuss the applicable range of tensile and shear constitutive models for high-strength bolts under a tensile–shear coupling load, based on numerical analysis against available experimental data in the literature. The loading angle of 30° along the bolt rod is defined as the cut-off to differentiate high-strength bolts under a tensile- or shear-dominated state, and the corresponding mechanical responses of high-strength bolts can be predicted well based on the tensile and shear constitutive models, respectively.

The eddy current damper (ECD), as a high-performance seismic protection device, has been increasingly applied in the field of civil engineering. However, most existing finite element software lacks a nonlinear constitutive model for this type of damper, and an equivalent simplified model capable of accurately capturing its real damping characteristics under complex seismic excitations has not yet been established. To address this gap, this paper proposes a stochastic optimization framework for calibrating a bilinear ECD model, using the Wouterse model as a reference. Specifically, the core of this framework employs a Monte Carlo Simulation (MCS) approach, in which a suite of spectrum-compatible seismic excitations is generated and analyzed to evaluate the corresponding stochastic structural responses. The Grey Wolf Optimizer (GWO) is then utilized to minimize the discrepancy between the dynamic responses predicted by the proposed bilinear model and those of the reference model. A key outcome of this study is a practical bilinear model in which these parameters can be treated as constants, enabling designers to focus exclusively on the physical parameters $f_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $v_{\mathrm{c}\mathrm{r}}$. This study bridges the gap between high-fidelity simulations and practical engineering design, providing a feasible pathway for integrating ECDs into existing finite element software for dynamic analysis.