## 4 Result and Analysis--A Case Study
### Numerical Model  
The whole Nanxi River Bridge is discredited into 220 nodes and 180 elements, and the standard finite element model is established in Midas/Civil. The so-called standard model is a complete bridge model with acting dead weight and two-phase dead load. The spatial beam unit is used to simulate the main girders, piers, and towers. The truss unit is used to simulate the diagonal cables. Nanxi River Bridge is a single-tower cable-stayed bridge with tower and beam consolidation. The properties of the structural members are shown in Table 1 and the boundary conditions are shown in the following Table 2. The parameters in Table 1 are from the Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts (JTG 3382-2018). The connections between the tower and the main beam, between the main beam and the stay cable nodes, and between the tower and the stay cable nodes simulate coupling through rigid connections with \"master-slave constraints\".  
Initializing the population and establishing the search space may have a certain impact on the results, then five finite element models are firstly established in the structure calculator. The zero-displacement method, the rigid supported continuous beam method,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Position** & **Material** & **Material Type** & **Element Type** & \
\begin{tabular}{c} **Elasticity Modulus (MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Density** \\ **(kg/m\({}^{5}\))** \\ \end{tabular} & **Poisson’s Ratio** \\ \hline Main girder & Concrete & C60 & beam & \(3.60\times 10^{4}\) & 2500 & 0.2 \\ Tower & Concrete & C50 & beam & \(3.45\times 10^{4}\) & 2500 & 0.2 \\ (above bridge deck) & Concrete & C60 & beam & \(3.60\times 10^{4}\) & 2500 & 0.2 \\ Tower & Concrete & C50 & beam & \(3.45\times 10^{4}\) & 2500 & 0.2 \\ (under bridge deck) & Concrete & C50 & beam & \(3.45\times 10^{4}\) & 2500 & 0.2 \\ Main pier & Steel stand & Strand 1860 & Truss & \(1.95\times 10^{5}\) & 7850 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of the structural members.  
Figure 8: General layout of Nanxi River cable stayed T-frame bridge (unit: cm).  
and minimum bending energy method are used to determine a initial state without cable force optimization. The above three methods establish different finite element models based on the standard model for calculating the initial bridge state. The standard calculation model is shown in Figure 9.  
Considering the span of Narxi River Bridge is \((120+120)\) m, the sag effect of the cable is considered when modeling the cable by the equivalent truss element whose elastic modulus is modified by [PERSON]'s formula (1965). [PERSON]'s correction formula is as follows:  
\[E_{eq}=\frac{E_{0}}{1+\frac{(\gamma S\cos\alpha)^{2}E_{0}}{12\sigma^{3}}} \tag{15}\]  
where \(E_{0}\) is the original elastic modulus of the cable (kPa). \(\gamma=\frac{\text{Heavy material per cable and protective structure}}{\text{Area of cable section}}\) is the gravity force per unit length of the cable (kN/m), \(S\) is the cable length, \(\alpha\) is the angle between the cable and the horizontal line (\({}^{\circ}\)), and \(\sigma\) is the cable stress (kPa).