## 3 Optimization on Reasonable Constructing State for Cable-Stayed Bridge with Steel Box Girder
### Analysis of Optimization Results During Construction Process  
The ratio of the 2-norm value to the initial value of the four objective functions was used to represent their relative magnitudes. The changes in these ratios with respect to the number of iterations are shown in Figure 16. After eight iterations, the convergence criterion for the objective functions of Function III was met, with values of \(\sum\limits_{i=1}^{n}\left|\Delta N_{i}^{\left(k\right)}/nT_{i}\right|=0.004< \varepsilon_{1}\), and \(\max\left|\Delta N_{i}^{\left(k\right)}/T_{i}\right|=0.018<\varepsilon_{2}\), indicating that the iteration process had converged. For the other three optimization functions, convergence was achieved after 14, 8, and 9 iterations, respectively. The optimized construction cable tensions for the four objective functions are shown in Figure 17. All distributions exhibit a similar trend, with cable tension increasing from the area near the pylon to the transition pier on both sides. The maximum cable tension for each objective function occurred at cable J22 on the mid-span side, while the minimum cable tension was found at cable J5 near the pylon. The optimized cable tensions with Function III the as the objective function ranged from 1481 kN to 3237 kN. The maximum cable tension was 1.70% lower than the result from Function I as the objective function but 0.69% and 3.26% higher than the results from the other two objective functions, respectively. The minimum cable tension was 3.46% lower than that the result with Function I as the objective function, and 0.54% and 27.12% higher than the results from the other two objective functions, respectively. The envelopes of vertical displacement of the girder and horizontal displacement of the pylon during the construction process, optimized by the four objective functions, are shown in Figures 18 and 19. The vertical displacement of the girder optimized with Function III as the objective function was in the range of \(-49\) mm to 1198 mm. The maximum upward deflection was 46 mm smaller than the result  
Figure 16: Change of objective function value along with number of iterations. Figure 17: Optimized construction tension force of stay cables. I as the objective function, and 6 mm and 54 mm larger than the results from the other two objective functions. The maximum downward deflection was the same as the results obtained from the other three objective functions. The horizontal displacement of the top of the pylon optimized with Function III as the objective function was in the range of \(-\)307 mm to 15 mm. The maximum horizontal displacement inclined to the side span was 21 mm larger than the results optimized with Function IV as the objective function and was the same as the results from the other two objective functions. The maximum horizontal displacement inclined to the main span was 4 mm smaller than the result optimized with Function I as the objective function, and 3 mm and 8 mm larger than the results optimized by the other two objective functions, respectively. The envelopes of axial force in the girder and pylon during the construction process, optimized by the four objective functions, are shown in Figures 20 and 21. The maximum axial compressive force in the girder was located near the pylon, while the maximum axial compressive force in the pylon was located at the pylon bottom. Using Function III as the objective function for optimization, the axial force of the main girder during the construction phase ranged from \(-\)94,061 kN to 191 kN, while the axial force of the bridge towers ranged from \(-\)619,899 kN to \(-\)1810 kN. The results showed minimal differences compared to the optimization outcomes of the other three objective functions, which can be considered negligible. Figure 19: Horizontal displacement envelope of pylon during construction. Figure 18: Vertical displacement envelope of girder during construction.