## 2 Numerical Model  
### Finite Element Model  
The model established in this paper (Figure 2) is based on a domestic railway viaduct (including the bridge of lines I, II, IV) located in a right-angle turning section of the river (Figure 3). The distance between the I and II lines of the railway viaduct is 5 m, while the spacing between line IV and line I is 7.5 m. The bridges of the lines I and II are simple-supported girder bridges of length (\(8+10+20+2\times 10+2\times 8\)) m, while the length of the line IV bridge is (\(3\times 10+20+2\times 10+2\times 8\)) m.  
The finite element model is mainly composed of the superstructure (box girder, ballast bed, rail, sleepers, etc.), temporary steel bracket system, etc. The temporary steel bracket system structure is composed of supporting longitudinal beams, supporting cross beams, steel columns, diagonal braces, cross-linkages, temporary steel bracket bearings, and steel pipe piles from top to bottom. The supporting longitudinal beams and supporting cross beams are made of H500 \(\times\) 500 special steel with a thickness of 4 cm. The steel column is divided into two types: side-span steel column; and middle-span common steel column. The side-span steel column has a 40 cm diameter and a 10 mm wall thickness, while the diameter of the middle-span common steel column is 60 mm, and the wall thickness is 10 mm. The diagonal braces, cross-linkages, and longitudinal linkages are made of H300 \(\times\) 200 sealed edge steel diagonal braces; the temporary steel bracket bearing platforms are 1.5 m in height, with a width that corresponds to the original bearing platform spacing of the bridge piers. The steel pipe piles are solid piles with a diameter of 30 cm  
Figure 1: Bridge pier and steel bracket system.  
and a length of 20 m, and the center distance between each pile is 0.7 m. Relevant material parameters are shown in Table 1[23; 24].  
Figure 3: The bridge to be reconstructed.  
Figure 2: Finite element model: (**a**) temporary steel support system; (**b**) superstructure.  
In the establishment of the ballast track finite element model, the interlocking force between ballast particles is ignored, which is considered a continuous medium and simulated by solid elements [25; 26; 27]. The type of rail is CHN60, coupled to the sleepers by fasteners simulated by spring-damper element [28; 29]. The type-III fastener is adopted, whose stiffness is 120 MN/m in the vertical direction, 40 MN/m in the lateral direction, and 20 MN/m in the longitudinal direction [30]. The damping is 2 \(\times\) 104 N\(\cdot\)s/m in all three directions. In the process of establishing the finite element model, the shape of the sleeper is simplified to a regular rectangular body. The eight-node element-C3D8R is adopted to simulate the ballast, rail, and sleeper.  
The bottom of the steel pipe pile is fixedly restrained, and the two ends of the bridge girder are fixedly restrained for simulating the support of the bridge piers. The ballast bed is composed of a bulk structure, which keeps close to the bridge deck under the trainload; thus, the ballast bed and the bridge deck are bonded together by the \"tie constraint\". The contact between the rail sleepers and the ballast bed is simulated as a \"surface-to-surface contact\", the normal contact is set as \"hard contact\", while the Coulomb friction model with a friction coefficient of 0.3 is adopted as the tangential contact. The supporting longitudinal beams, supporting crossbeams, steel columns, diagonal braces, and other bars are connected by continuous fillet welds, thus setting the contact between them as a \"tie restraint\".