## 4 Application Discussion
### Discussion Application  
(1) Engineering background  
Project 1: Mingzhou Bridge shown in Figure 5 is located on the waterway of the Yongjiang River and its main bridge is a \(100+450+100\) m double-limb center-beam steel box arch bridge. There are 38 pairs of mid-span cables with 9 m spacing in the same plane with the arch ribs. A total of four groups of side cables are made of parallel steel wire cables of 19945, and the remaining 34 groups of mid-span cables are made of parallel steel wire cables of 9145; both ends of the cables are anchored by cold casting, and the cables are tensioned at the top of the arch. The stiffeners are supported on the arch ribs by suspenders or columns, and the two ends of the mid-span stiffeners are supported on the crossbeam at the intersection of the mid-span arch beams, with expansion joints at both ends of the stiffeners and longitudinal sliding bearings at the end supports, and damping limit devices set horizontally and longitudinally. Project 2: Sanshan West Bridge shown in Figure 6 is located at the intersection of the Pingzhou waterway and the Pearl Zhujiang waterway with a main span of 200 m. The main arch rib is composed of two 40750 steel pipe concrete, connected by plate embellishment strips to form a steel pipe concrete lattice column. In the section from the first suspender to the foot of the arch, there are nine parallel cross braces, each of which is a truss beam made of empty steel pipes. The whole structure uses the self-weight of the side span to  
Figure 5: Mingzhou Bridge. balance the huge thrust of the main arch through the floating flexible structure, so that the side span, the main span, and the lower foundation can be deformed and stressed together. (2) Application verification  
In order to apply the above engineering discussion to the actual bridge project, two representative cables, D1b and D19b, on the Mingzhou Bridge shown in Figure 5 and the C33 suspender on the Sanshan West Bridge shown in Figure 6 were selected as the verification objects for the discussion. The basic parameters of cables and suspenders are shown in Table 4. The error between the corresponding frequency values calculated by the analysis of various boundary conditions and the actually measured frequency values was analyzed, and the results are shown in Table 5. The discussion in this paper is preliminarily applied to the actual bridge engineering, and the accuracy of the boundary conditions at both ends of the cable defined by the cable length is tested to a certain extent. As can be seen from Table 5, for the cables with \(l\geq 45\sqrt{\frac{EI}{H}}\), the boundary conditions have little effect on their frequencies, but for the cables with \(l\leq 15\sqrt{\frac{EI}{H}}\), the frequencies calculated by applying the fixed support boundary conditions are closer to the frequency values in actual engineering, while for the cables with a length between the two, it is recommended to treat both ends as elastic boundaries. By calculating the frequency values of the cables under the three different boundary conditions and comparing them with their \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Cable Number** & \(\mathbf{E(\text{N}/\text{m}^{2})}\) & \(\mathbf{I(\text{m}^{4})}\) & \(\mathbf{m(\text{kg}/\text{m})}\) & \(\mathbf{I(\text{m})}\) & \(\mathbf{H(\text{N})}\) & \(15\sqrt{\frac{EI}{H}}\) & \(45\sqrt{\frac{EI}{H}}\) \\ \hline D1b & \(1.975\times 10^{11}\) & \(1.215\times 10^{-6}\) & 125 & 15.007 & \(1.135\times 10^{6}\) & 6.897 & 20.691 \\ D19b & \(1.975\times 10^{11}\) & \(2.541\times 10^{-7}\) & 55 & 61.731 & \(7.213\times 10^{5}\) & 3.957 & 11.870 \\ C33 & \(1.95\times 10^{11}\) & \(7.26\times 10^{-7}\) & 24.556 & 2.560 & \(1\times 10^{6}\) & 5.644 & 16.932 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Basic parameters of each cable. Figure 6: Sanshan West Bridge.