## 4 Result and Analysis--A Case Study
### Numerical Model
#### 4.1.1 The Initial State  
According to the different purposes of controlling the parameters of Nanxi River Bridge under the action of dead load, five preliminary calculation ideas are divided into the following through several trial calculations. 1. The main beam of Nanxi River bridge was equivalent to a multispan continuous beam with multipoint elastic support. The vertical component of the cable force was balanced with the elastic support in the vertical direction. This goal could be realized by the simple beam method (SBM). 2. The bending energy of the main beam was minimized, which could be achieved by the minimum bending energy method (MEM). 3. The vertical displacement between the cable and the anchor point of the main beam was constrained to be between \(-20\) mm and \(20\) mm, which can be achieved by the zero-displacement method (ZDM). 4. The minimizing maximum bending moment (MMM) of the main beam is limited. The value was kept within the range of \(-180,000\)-\(200,000\) kN.m. [PERSON] [10] used to limit the sum of the maximum bending moment values of the main beam and the main tower when studying the cable force optimization of a cable-stayed bridge with two towers and three spans. Considering the symmetry of Nanxi River Bridge, this paper only sets the limit value for the maximum bending moment of the main beam. 5. Limiting the initial value of the pretension forces (PTF) of the cable was to ensure the uniformity of the cable force (limiting force method, LFM). In this paper, the initial PTF was uniformly set as \(7000\) kN, which was an integer value around the median cable force determined by several trial calculations. When determining the initial value, the cable force in working conditions 2-4 was set to the same values of \(2000\) kN, \(2250\) kN, and \(2500\) kN respectively by some scholars [22], making the consideration more abundant. Figure 9: Standard finite element model of Nanxi River Bridge. The initial state of Nanxi River Bridge obtained by calculating the above targets is shown in Figure 9(a)-c.  
Figure 9(a) illustrates that the cable force calculated by ZDM and SBM increases gradually with the increase of cable length, indicating that the cable force distribution of both is more uniform. The cable forces calculated by the ZDM method are 5431 kN, 5440 kN, and 5379 kN between S1 and S3, respectively. There is partial unevenness of the value of cable force. The SBM method also shows this phenomenon between the cable force values numbered S7 to S9, which is allowed in the local range. Overall, the cable force of ZDM is slightly smaller than that of SBM. In Figure 9(b),c, all the initial states calculated by the conventional method are at least 62.6% less than the peak vertical displacement without PTF and have at least 47.1% less negative bending moment at the tower beam consolidation (both from ZDM). To a certain extent, it confirms the critical role played by the PTF of cable in adjusting the structure's internal force distribution. According to the calculation results of the above MEM method, for variable section cable-stayed bridges such as Nanxi River Bridge, the application of the MEM method will result in sudden and uneven local variations of the cable force distribution. In particular, the cable force value closest to the tower position may be too large or even exceed the limit (for Nanxi River Bridge, this is the cable number S1). Among all the tensions of all  
Figure 10: Structural response of cable-stayed bridge in the initial state calculated by different methods. five methods, the peak force was 10,500 kN for the tension of S1 determined by the MEM method, and the minimum value was 5431 kN determined by the ZDM method. The initial state determined by the MMM method has the problem of frequent changes in the tensioned side of the main beam and poor uniformity of the cable force. However, the absolute value of the vertical displacement of the main beam is small and the positive and negative values of the bending moment are more uniform.