## 2 Engineering Background and Numerical Model  
Located in Shanxi Province, China, Renyihe Bridge is a concrete rigid-frame continuous highway bridge with the spans 80 m + 4 \(\times\) 145 m + 80 m (Figure 1). As shown in Figure 1, the superstructure is a single-cell single-box concrete continuous girder with variable cross-sections. The prestresses are applied to the girder in longitudinal and vertical directions. The bridge was built in 2003 and has been in service since then. The finite-element (FE) model of Renyihe Bridge was established on the MIDAS platform (v6.71) using 241 beam elements with 248 nodes with the help of the modeling assistant module provided by the platform (Figure 2). The boundary conditions were assumed to be fixed. The axial stiffness of piers depends on the length of the piers and the tensile/compressive stiffness of the piers. The shorter the length of the piers and the greater the tensile/compressive stiffness of the piers, the higher the axial stiffness of the piers. For the case of Renyihe Bridge, although the length of the piers is comparatively long, the tensile/compressive stiffness of the piers is sufficiently large. Therefore, the axial stiffness of the piers of Renyihe Bridge is still very large, and they can be simply considered rigid. We admit that the simplified beam FE models cannot satisfactorily simulate all the geometric details of the structures and therefore inevitably contain model structure errors. According to our experiences, these errors can hardly be completely eliminated through traditional  
Figure 1: Side view and elevation sketch of Renyihe Bridge. (**a**) Actual side view. (**b**) Elevation sketch (unit: cm). parameter identification. However, MIDAS is a specialized FE platform designed for bridge engineering, and the beam elements provided by the MIDAS platform are different from those provided by other general FE software (e.g., ANSYS), being able to accurately reflect the realistic sectional properties of the structure. Therefore, the inherent uncertainties associated with the beam model established are greatly reduced on the MIDAS platform. On account of the reduced modeling workload and for the sake of computing efficiency, we finally chose to use the beam model established on the MIDAS platform. The numerical simulation procedure, in order, consisted of basic information (the elevation geometry, the material properties, the division of the structure for construction, and the construction duration) input, cross-section geometry input, and steel strand information input. The geometry of the structure was modeled in detail in accordance with Figure 1b, and the concrete material models adopted were those of uniaxial compression and the tension linear elastic constitutive relation. To take into account the influences of steel strand pre-stressing, the shrinkage and the creep of the concrete, and other effects during the construction stages on the mechanical state of the constructed structure, the model of Renyihe Bridge was established progressively, reflecting each constructional state in temporal order, including the concrete pouring of the piers, the concrete pouring of the 0 blocks above the piers, carriage traveling, the concrete pouring and steel strand tensioning for each block, the closure of the T-shaped structures, and the structural system transformation. Therefore, the simulation process accurately considered the cumulative influences of the multiple constructional effects. After the FE model of Renyihe Bridge was established, an initial in-plane modal analysis was performed. The results are shown in Table 1 and Figure 3. As can be seen in Table 1, the 1 st-18 th order natural frequencies calculated using the initial FE model of Renyihe Bridge are distributed uniformly in the frequency interval 0.7-7 Hz, and most of them are greater than 1 Hz, indicating that the vehicle excitations with the energies roughly in the range 2.85-11.8 Hz [17] usually cannot excite the structure's low-order (1 st-9 th order) resonances.