## 2 Analysis of Sag Effect Influence
relative to the span length \(L\) is defined as the anchor span ratio \(\beta\) (\(a/L=\beta\)), and substituted into Equation (4):  
\[k=\frac{12 EA\beta^{2}\sigma^{3}}{L\bigg{(}\Big{(}\alpha^{2}+(1-\beta)^{2}\Big{)}^ {3/2}\Big{(}12\sigma^{3}+\gamma^{2}EL^{2}(1-\beta)^{2}\Big{)}+\frac{(\alpha^{2} +\beta^{2})^{\gamma_{\beta}}(12\sigma^{3}+\gamma^{2}E\beta^{2}L^{2})}{t}\bigg{)}} \tag{5}\]  
From Equation (5), it can be seen that when the height-to-span ratio and the material properties of the crossing cable are constant, the constraint stiffness \(k\) is related to the main span length \(L\), the height-to-span ratio \(\alpha\), the ratio \(t\) of the crossing cable areas of the mid and side tower, the anchor span ratio \(\beta\), and the initial stress \(\sigma\). Based on Equation (5), the relationship between main span length \(L\), anchor span ratio \(\beta\), initial stress \(\sigma\) and crossing cable constraint stiffness \(k\) is analyzed. The ratio of the crossing cable area of the side tower to the mid tower is \(t=1\), and the relationship between the constraint stiffness of the crossing cable and the anchorage position (anchor span ratio), span length and initial stress of the crossing cable is studied, respectively. The ratio of height to span is 0.25, the area of the cable is 0.008 m\({}^{2}\), and the initial stress is 600 MPa. The cable is a high-strength steel wire, with an elastic modulus of 205 GPa and a specific weight of 78.5 kN/m\({}^{3}\). The relationship between the crossing cable constraint stiffness \(k\) and the anchor span ratio \(\beta\) under different main span lengths is shown in Figure 3.  
As can be seen from Figure 3, the constraint stiffness of the crossing cable is related to its anchorage position. When the main span lengths are 700 m, 1000 m, and 1500 m, the anchor span ratios corresponding to the maximum constraint stiffness of the crossing cable are 0.694, 0.673, and 0.633, respectively. As the length of the main span increases, the anchorage position with the maximum constraint stiffness tends to move towards the mid span. When the height-to-span ratio is 0.25 and the anchor span ratios are 0.5, 0.6, and 0.7, respectively, the relationship between the constraint stiffness \(k\) of the crossing cable and the main span length \(L\) is shown in Figure 2.0  
As can be seen from Figure 4, as the main span length increases, the constraint stiffness of the crossing cables continuously decreases. When the anchor span ratio \(\beta\) is 0.5, 0.6, and 0.7, the constraint stiffness of the crossing cables decreases by 73.04%, 74.09%, and 76.30%, respectively, as the main span length increases from 500 m to 1500 m. The main span length has a significant impact on the constraint stiffness of the crossing cables. This is mainly because as the span increases, the cable length increases and the sag effect increases, both of which reduce the axial stiffness of the cable. Figure 2.1 The relationship between \(k\) and anchor span ratio \(\beta\) under different spans. When the main span length is 1000 m and the height-to-span ratio is 0.25, the relationship between constraint stiffness \(k\) and anchor span ratio \(\beta\) under different initial stresses of cables is shown in Figure 2.2  
As can be seen from Figure 5, after increasing the initial stress of the cable, due to the reduced sag effect of the cable, the anchorage position with the maximum constraint stiffness of the crossing cable tends towards the side tower, and its constraint stiffness gradually increases. Compared to the initial stress of 500 MPa, the constraint stiffness increases by 11.46% and 18.68% when the initial stress is 600 MPa and 700 MPa, respectively. The influence of the area ratio of the crossing cable between the mid tower and the side tower on its constraint stiffness is analyzed below. The steel volume of a pair of crossing cables is defined as \(C\), which has the following relationship:  
\[\left\{ \begin{array}{l}A_{m}/A_{s}=t\\ A_{m}(h^{2}+a^{2})^{1/2}+A_{s}\Big{[}h^{2}+(L-a)^{2}\Big{]}^{1/2}=C\end{array} \right.