## 6 A Feasibility Analysis of the Optimization Method for the Finished State
### Optimization Results  
The objective function values for the optimization process, as plotted in Figure 10, demonstrate a significant decrease with an increasing number of iterations. Specifically, after 30 iterations, the objective function value dropped from an initial value of 592 to a final value of 21. At this point, \(\max\big{(}\big{\|}\rho\|,\|\sigma\|,\gamma_{r}\,\big{\|}\,\big{\|}\,\big{\|} \,\big{\|}\,\big{\|}\,\big{\|}\,\big{)}=4.10\times 10^{-4}<\varepsilon\), which meant that iteration had converged. Thereafter, the optimal cable force sequence, shown in Figure 11, ranged from 1883 kN to 4631 kN. The maximum force occurred in cable A22 on the side span, while the minimum was in cable J3 near the pylon. This sequence was input into a finite element model of the complete bridge (including self-weight, the secondary dead load, the optimal cable forces, and all constraints, without considering the construction stages) to validate the optimization's accuracy. If the results are reasonable (the evaluation method is referenced in the Optimization Principles section), the optimization of the finished state of the cable-stayed bridge with steel box girders is complete. Otherwise, the constraints or the objective function can be modified to reformulate the quadratic programming model for further optimization. Figure 11: Optimal sequence of cable forces. Figure 10: Evolution of the value of the objective function with the number of iterations. Figure 12 depicts the distributions of the main girder's vertical displacement and the pylon's horizontal displacement under the finished dead load condition. Optimization of the main beam results in vertical displacements ranging from \(-\)38 to +43 mm. The maximum positive deflection occurs mid-span, while the maximum negative deflection consistently occurs at the mid-span of the side span. The lateral displacement of the top of the tower is 5 mm towards the side span. The axial force distributions of the main girder and the pylons under the finished dead load state are shown in Figure 13. The optimized axial forces in the main beam and the pylons are predominantly compressive. The axial force in the main beam ranges from \(-\)93,986 to 15 kN and increases uniformly from the transition piers to the pylon. The pylon axial force ranges from \(-\)625,872 to \(-\)1810 kN and increases uniformly from the top to the bottom of the pylon. Figure 14 shows the bending moment distributions in the main girder and pylons under the finished dead load condition. Optimization resulted in main girder bending moments ranging from \(-\)28,623 to 30,868 kN\(\cdot\)m. The maximum positive bending moments occurred near the \(L/6\) point of the side spans, and the maximum negative bending moment occurred at the auxiliary pier's cap beam sections. The pylon bending moments ranged from \(-\)17,291 to 12,765 kN\(\cdot\)m, with the largest positive moment at the mid-height of the upper pylon columns and the largest negative moment at the pylon base. Figure 12: Displacement evoked by the dead load. (**a**) Vertical (main girder); (**b**) horizontal ( pylon). Figure 13: Axial force evoked by the dead load in the (**a**) main girder and (**b**) pylon. The normal stress of the main girder and pylons under the finished dead load condition is shown in Figure 15. The optimized main beam exhibited a top-surface stress ranging from \(-71.47\) to \(+4.28\) MPa. The maximum compressive stress was observed near the pylon, and the maximum tensile stress occurred mid-span. The bottom flange stresses varied from \(-73.35\) to \(17.00\) MPa, with the maximum compression near the pylon and the maximum tension near the side spans. For the pylon side-span stresses, the range was \(-6.67\) to \(-0.01\) MPa, exhibiting the maximum compression near the upper pylon connection. The main span's normal stresses fell between \(-7.00\) and \(-0.01\) MPa, with the maximum compressive stress in the mid-upper portion of the lower pylon columns.