## 4 Solutions to the Influence Line Equations of the Main Beam
### Characteristics of Influence Lines  
When a unit load 1 kN is applied at an arbitrary location \(\xi\in[-1218,\ 1218]\) m, the corresponding vertical deflection, rotation, moment, and shear of the main beam at a point \(x\in[-1218,\ 1218]\) m is computed using the proposed model. Then, the influence lines of vertical deflection, rotation, moment, and shear of the main beam at all \(x\in[-1218,\ 1218]\) m are computed and the results are shown in Figure 9a, 9b, 9c, and 9d, respectively. As seen in Figure 9a, the larger values of the vertical deflection influence line are symmetrical about the vertical axis at \(\xi=x\). When the unit load is applied at \(\xi=0\) m, the maximum value of the vertical deflection influence line is \(8.01\times 10^{-5}\) m, which occurs for the influence line of the middle-span \(x=0\). Meanwhile, the values of the vertical deflection influence lines within the pure suspension section are obviously larger than the hybrid and cable-stayed sections. Similarly, in Figure 9b, the larger values of the beam rotation influence the line distributed at the two sides of \(\xi=x\), and the distribution range is larger than that of the vertical deflection. When the unit load is applied at \(\xi=-84\) m, the maximum value of the beam rotation influence line is \(4.98\times 10^{-7}\)\(rad\), which occurs for the influence line at \(x=-161\) m.  
It is seen in Figure 9c that the beam moment influence lines are all approximately symmetric about \(\xi=x\), and its distribution range is larger than that of the vertical deflection. When the unit load is applied at \(\xi=35\) m, the maximum value of the beam rotation influence line is \(25.35\) kN\(\cdot\)m which occurs at the concerned location \(x=35\) m.  
As for the beam shear in Figure 9d, the influence lines are symmetric about \(\xi=x\). When the load is applied around the vertical constraint, the value of the beam shear  
Figure 9: Distribution of the influence lines of the main beam. influence line is relatively larger. When the unit load is applied at the auxiliary pier \(\xi=-1106\) m, the maximum value of the beam shear influence line is 0.98 kN, which occurs for the influence line at \(x=-1106\) m.  
Based on Figure 9, the influence line at an arbitrary location \(x\) can be easily obtained by cutting a cross profile at \(x\). For a comparison purpose, the influence lines at six locations are obtained and plotted in Figure 10a, 10b, 10c, and 10d for the vertical deflection, rotation, moment, and shear of the main beam, respectively, which are at the middle-span \(x=0\), \(1/4\) of the pure suspension section \(x=91\) m, the intersection between the suspension and hybrid sections \(x=182\) m, the middle of the hybrid section \(x=266\) m, the intersection between the hybrid and cable-stayed sections \(x=350\) m, and the middle of the cable-stayed section \(x=525\) m.  
It is seen in Figure 10a that the vertical deflection influence line of the beam at any location is approximately symmetrical about the concerned location \(x\), with the peak values occurring at the concerned location. In addition, affected by the deflection of the main cables, the vertical deflection influence line of the beam at any location has negative values at the two sides of the concerned point. As seen in Figure 10b, the beam rotation influence line at any location is approximately anti-symmetrical with the concerned location. In addition, the peak values of the rotation influence lines at the pure suspension section are larger than those at the hybrid and cable-stayed sections. For the beam moment in Figure 10c, the influence line at any concerned location, the line is approximately symmetrical about the concerned location, and the maximum value occurs at the concerned location. Moreover, the maximum values of the beam moment influence lines at the pure suspension section are larger than those at the hybrid and cable-stayed sections.