## 6 Conclusions  
In the preliminary design of cable-stayed bridges, it is strategically significant to properly determine the design parameters. However, it is challenging to obtain the optimum design of a cable-stayed bridge because of numerous variables, multiple load cases, and \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & **Verification Item** & \multicolumn{6}{c}{**Load Cases**} \\ \hline \multirow{4}{*}{Strength} & \multirow{4}{*}{\begin{tabular}{c} Tensile stress of the main girder (MPa) \\ Compressive stress of the main girder (MPa) \\ Tensile stress of the main girder (MPa) \\ Compressive stress of the tower (MPa) \\ Compressive stress of the towers (MPa) \\ Maximum stress of the cables (MPa) \\ \end{tabular} } & \begin{tabular}{c} LCB1 \\ 32.26 \\ \(-126.73\) \\ \(-126.73\) \\ \(-126.66\) \\ \end{tabular} & \begin{tabular}{c} LCB2 \\ 25.21 \\ \(-144.51\) \\ \(-118.39\) \\ \(-18.66\) \\ \(-17.76\) \\ \(-17.76\) \\ \(-1566.61\) \\ \end{tabular} & \begin{tabular}{c} LCB1 \\ 36.45 \\ \(-118.39\) \\ \(-18.66\) \\ \(-17.76\) \\ \(-17.66\) \\ \(-1566.61\) \\ \end{tabular} & \begin{tabular}{c} LCB3 \\ 17.11 \\ \(-118.39\) \\ \(-17.76\) \\ \(-17.76\) \\ \(-1566.61\) \\ \end{tabular} & \begin{tabular}{c} LCB5 \\ 17.11 \\ \(-118.39\) \\ \(-18.66\) \\ \(-17.76\) \\ \(-17.66\) \\ \(-1566.61\) \\ \end{tabular} & \begin{tabular}{c} LCB5 \\ 17.11 \\ \(-118.39\) \\ \(-18.66\) \\ \(-17.76\) \\ \(-1566.61\) \\ \end{tabular} \\ \hline Stiffness & \multirow{4}{*}{\begin{tabular}{c} Maximum deflection of the main girder (m) \\ \end{tabular} } & \begin{tabular}{c} VL1 \\ 0.611 \\ \(-0.611\) \\ \end{tabular} & \begin{tabular}{c} VL2 \\ 0.952 \\ \(-0.651\) \\ \end{tabular} & \begin{tabular}{c} VL3 \\ 0.551 \\ \end{tabular} & \
\begin{tabular}{c} VL4 \\ 0.551 \\ \end{tabular} \\ \hline Stability & \multirow{4}{*}{
\begin{tabular}{c} Safe factor \\ \end{tabular} } & LCB6 & LCB7 & LCB8 & LCB9 & LCB10 & LCB11 \\ \cline{1-1} \cline{3-6}  & Safe factor & 8.13 & 7.63 & 7.44 & 7.79 & 7.22 & 8.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Check of constraints under each load combination. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Constraints** & **Boundary** & **Load Cases** & **Optimum Design** \\ \hline Concrete Design Strength & \multirow{2}{*}{\(-22.4\) MPa\(\sim\)1.83 MPa} & LCB1\(\sim\)LCB5 & \(-18.6\) MPa\(\sim\)1.83 MPa \\ of the towers & & \multirow{2}{*}{\(-270\) MPa\(\sim\)270 MPa} & LCB1\(\sim\)LCB5 & \(-144.51\) MPa\(\sim\)52.1 MPa \\ Steel Design Strength of & & & & \\ the main girder & & & & \\ Safety Stress Factor of & & & & \\ the cables & & & & \\ Vertical Deflection of the & & & & \\ main girder & & & & \\ Elastic Buckling & & & & \\ Stability Factor & \(\geq\)4 & LCB6\(\sim\)LCB11 & 7.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Check of strength, stiffness, and stability constraints. diverse constraints. Integrated methods are adopted in this paper to enable efficient design optimization of cable-stayed bridges:  
* to simplify the complexity of the problem, an improved two-layer framework is presented for cable-stayed bridge optimum design problems. The formula consists of two mutually iterative sub-problems: optimizing the internal force distribution by adjusting the cable prestressing forces and optimizing the sizing and geometrical parameters. The sub-problems exhibit fewer variable coupling features, making them easier to solve. * to decrease the dimension of the design variables, B-spline interpolation curve is adopted to condense the variables, instead of setting all cable forces as variables. B-spline curve stands out when confronting large-span cable-stayed bridges with dense cables, because it can fit cable force distribution in cable-stayed bridges with only a few controlling points or fitting points. * to improve optimization efficiency, a surrogate model-assisted predictor for optimum cable forces is constructed.