The probability that an earthquake occurs when a train is running over a bridge in earthquake-prone regions is much higher than before, for high-speed railway lines are rapidly developed to connect major cities worldwide. This paper presents a finite element method-based framework for dynamic analysis of coupled bridge-train systems under non-uniform seismic ground motion, in which rail-wheel interactions and possible separations between wheels and rails are taken into consideration. The governing equations of motion of the coupled bridge-train system are established in an absolute coordinate system. Without considering the decomposition of seismic responses into pseudo-static and inertia-dynamic components, the equations of motion of the coupled system are formed in terms of displacement seismic ground motions. The mode superposition method is applied to the bridge structure to make the problem manageable while the Newmark-method with an iterative computation scheme is used to find the best solution for the problem concerned. Eight high-speed trains running over a multi-span steel truss-arch bridge subject to earthquakes are taken as a case study. The results from the case study demonstrate that the spatial variation of seismic ground motion affects dynamic responses of the bridge-train system. The ignorance of pseudo-static component when using acceleration seismic ground motions as input may underestimate seismic responses of the bridge-train system. The probability of separation between wheels and rails becomes higher with increasing train speed.

In cable-stayed bridges, especially asymmetric bridges, counterweights are always made to work together with cable pretension forces to get a reasonable finished state. To solve the optimization problem of the cable-stayed bridge considering the counterweights, the integrated optimization method (IOM) for estimating cable forces and counterweights is proposed. In this method, the counterweights are proposed to act on the anchor points. After that, the summary of the minimum weighted total bending energy and the summary of the counterweights are considered as two objective functions of a multiobjective problem. Finally, the dynamic weighted coefficient method is used to solve this problem and realize the Pareto solution set. IOM presents detailed procedures in a simple numerical model and is then applied to the Yong-ding special-shaped cable-stayed bridge. The results show that not only IOM can realize the priority selection of the loading position of the counterweights but also get a better reasonable finish state because of the introduction of the counterweight dimension; the dynamic weighted coefficient method can quickly find the Pareto optimal solution set and be further screened by decision-makers; counterweight is very helpful to reduce torsion and other spatial effects in cable-stayed bridges. IOM can be used as a universal optimization method for cable-stayed bridges.

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The substructure method is applied to the dynamic analysis of a train–bridge system considering the soil–structure interaction. With this method, the integrated train–bridge–foundation–soil system is divided into the train–bridge subsystem and the soil–foundation subsystem. Further, the train–bridge subsystem is divided into the train and bridge components. The frequency-dependent impedance function of the soil–foundation subsystem is transformed into time domain by rational approximation and simulated by a high-order lumped-parameter model with masses. The equations of motion of the train and bridge components are established by the rigid-body dynamics method and the modal superposition method, respectively. Finally, the dynamic responses of the two subsystems are obtained by iterative procedures, with the influence of the soil shear velocity studied. The case study reveals that it is important to consider the effect of soil–foundation interaction in the dynamic analysis of train–bridge systems, but with the increase of the shear velocity of the soil, such influence becomes weaker.

This paper presents a method for calculating the evolutionary power spectral density (EPSD) of the seismic response of bridges using the convolution summation. With zero initial values, a formula for the dynamic component of the response of bridges to spatially varying seismic ground motions is derived as the convolution summation, by assuming the seismic acceleration to vary linearly between two adjacent time stations. The convolution summation is used for calculating the convolution integral of the dynamic component response factor in the EPSD, in which the constant coefficients are independent of the harmonically modulated excitation. The constant coefficients are obtained by the time-history analysis using triangular unit impulse acceleration excitations. The computational cost of the EPSD depends mainly on the amount of degrees-of-freedom (DOFs) numbers of bridge supports in contact. The corresponding computational scheme is proposed, and its validity is indirectly verified with a single DOF system, by comparing the results obtained with those of the existing methods. Finally, a three-span continuous rigid-frame bridge is taken as a case study to illustrate the applicability and effectiveness of the proposed scheme.