## 2 Mechanical Model of Shifting-Girder System Undergoing Girder Loads
### Global Mechanical Model  
The global mechanical model of the RCSG system undergoing girder loads is established as shown in Figure 3, where the subscripts \(z\), \(g\), and \(d\) in the variables, which respectively represent the main-cable, rail-cable, and sling, and \(o\)-\(XY\) are the global Cartesian coordinates. Here, the rail-cable is divided into \(m\) segments at the connected points with \(m\)-1 slings; and the main-cable is divided into \(n\) segments according to actual condition, whose ends includes the connected points with \(m\)-1 slings. \(A_{zi}\) (\(i\) = 0, 1, \(\ldots\), \(n\)) are the node locations of the main-cable segments; \(H_{zi}\), \(V_{zi}\) (\(i\) = 0, 1, \(\ldots\), \(n\)) are the horizontal and vertical force at the left section of node \(A_{zi}\) for the main-cable, and \(H_{z0}\), \(V_{z0}\), \(H_{zm}\), and \(V_{zm}\) are the horizontal and vertical pretension forces of the main-cable at the left and right supports; \(L_{zi}\), \(S_{zi}\) (\(i\) = 1, 2, \(\ldots\), \(n\)) are, respectively, the horizontal length and arc length of \(i\)th segment of main-cable after the installation of slings, and \(h_{zi}\) (\(i\) = 1, 2, \(\ldots\), \(n\)) are the vertical height differences between the two ends of the \(i\)th segment of main-cable; \(L_{z}\) and \(h_{z}\) are respectively the total horizontal span and vertical height difference between two end points of the main-cable, i.e., the left and right towers of suspension bridge. \(B_{zj}\) (\(j\) = 0, 1, \(\ldots\), \(m\)) are the node locations of rail-cable segments; \(H_{gj}\), \(V_{gj}\) (\(j\) = 0, 1, \(\ldots\), \(m\)) are the horizontal and vertical force at the left section of node \(B_{zj}\) for the rail-cable, and \(H_{g0}\), \(V_{g0}\), \(H_{gm}\), \(V_{gm}\) are the horizontal and vertical pretension forces of the rail-cable at the left and right supports; \(L_{zi}\), \(S_{gj}\) (\(j\) = 1, 2, \(\ldots\), \(m\)) are the horizontal length and arc length of \(j\)th segment of rail-cable after the installation of slings, and \(h_{gj}\) (\(j\) = 1, 2, \(\ldots\), \(m\)) are the vertical height differences between the two ends of the \(j\)th segment of rail-cable; \(L_{g}\) and \(h_{g}\) are, respectively, the total horizontal span and vertical height difference between the anchor points at two ends of the rail-cable. \(S_{dj}\) (\(j\) = 1, 2, \(\ldots\), \(m\)-1) are the vertical or arc lengths of the \(j\)th sling after the installation of all slings; \(q_{1}\), \(q_{2}\), and \(q_{3}\) are, respectively, the self-weight per unit length of the main-cable, rail-cable, and slip. The weights of girder segment and shifting-girder trolley are equivalent to concentrated forces on the two nodes of the rail-cable. Due to the standard girders adopting open-type structure, the distributed loads on two nodes are unequal. The two node-loads are respectively expressed by \(W_{1}\) and \(W_{2}\), the load ratio on lifting points is 1.7:1 [19], i.e., \(W_{1}\) = 1.7 \(\times\)\(W_{2}\). The weights of girder and shifting-girder trolley are known, therefore, \(W_{1}\) and \(W_{2}\) are determined. To describe the shifting-girder process, an inter-node model is built shown in Figure 4. Under loading conditions, each segment of main-cable between two slings is further discretized into two main-cable segments \(A_{zi}\)-\(A_{zi+1}\) and \(A_{zi+1}\)-\(A_{zi+2}\). The cable equations [19] for these two  
Figure 3: Mechanical model of the shifting-girder system undergoing girder loads. segments are established by using the segment catenary element.