Investigating Coupled Train-Bridge-Bearing System Under Earthquake- and Train-Induced Excitations

Abstract

The dynamic interaction between a bridge and a moving train has been widely studied. However, there is a significant gap in our understanding of how the presence of isolation bearings influences the dynamic response, especially when an earthquake occurs. Here, we formulate a coupled model of a train-bridge-bearing system to examine the bearings’ dynamic effects on the system responses. In the analysis, the train is modeled as a moving oscillator, the bridge is a one span simply supported beam and one isolation bearing is installed under each support of the bridge. A mathematical model using fractional derivatives is used to capture the viscoelastic properties of the bearings. The vertical response is the focus of this investigation. Dynamic substructuring is used in modeling to efficiently capture the coupled dynamics of the entire system. Illustrative numerical simulations are carried out to examine the effects of the bearings. The results demonstrate that although the presence of bearings typically decreases the bridge seismic responses, there is a potential to increase the bridge response induced by the moving train.

Seismic analysis of 3D train–wheel–bridge–bearing systems: Modeling

“…Our previous work [ 46 ] proposed a strategy to build an efficient state‐space model of a simply supported beam based on the modal decomposition method. In the following sections, this strategy is adopted and extended to establish the SSEs for the three motions of the bridge section.…”

“…It has been demonstrated that using the first five modes ($$N=5$$) in the simulation would achieve sufficient accuracy. [ 46 ] Since the bridge's horizontal and vertical modal shape functions are identical, both are represented by $$\phi _{\imath ,n}(x_\imath )$$.…”

“…The horizontal EOM of the ıth bridge, which ignores the external damping related to friction and air resistance, is [ 46,47 ] $$$$\begin{equation} E_bI_{b\imath z}\dfrac{\partial ^4y_{b\imath }(x_\imath ,t)}{\partial x_\imath ^4}+c_iI_{b\imath z} \dfrac{\partial ^5y_{b\imath }(x_\imath ,t)}{\partial x_\imath ^4\partial t}+m_{b\imath }\dfrac{\partial ^2y_{b\imath }(x_\imath ,t)}{\partial t^2}=\sum _{k=1}^{4N_T}{\Delta _{k\imath }(t)F_{ry}^{wk}(t)\delta [x_\imath -x_{k\imath }(t)]}, \end{equation}$$$$where $$E_b$$ is the Young's elastic modulus of the bridge material, $$I_{b\imath z}$$ is the moment of inertia of the bridge section around z axis passing through the centroid, $$c_i$$ is the bridge internal damping, $$m_{b\imath }$$ is the bridge mass per unit length, …”

“…If no extra action is taken, this process is completed by treating the bridge section shear forces, for example, $$V_{y\imath }(t)$$, as the bearing forces to establish a force‐input bearing subsystem , which sends bearing motions back to the bridge subsystem. [ 46 ] However, as we will discuss in the bearing subsystem section, the displacement‐input bearing subsystem , which takes bearing motions as inputs, needs to be established for the multidimensional bearing designed in this paper. The adopted strategy to create the displacement‐input bearing subsystem is to virtually set an extra mass block between the bridge and bearing to reverse the order of input and output signals in the bearing subsystem.…”

Simplification of earthquake induced vehicle bridge interaction contact simulation for urban viaducts

“…In order to eliminate the visual interference caused by different orders of magnitude such as Figure 5A,B, the error is further explained by Figure 6. Different from NRMSD, the Error(t) is used to judge the error value between the contact force calculated by the simplified model and by the Hertz model at each time t, which is determined by Equation (20).…”