# An Improved Analytical Algorithm on Main Cable System of Suspension Bridge
## 3 Improved Numerical Analysis Method
### The Main Cable System Calculation in Side Span at Finished State
#### 3.2.2 Improved Numerical Analysis Method for Side Span and Its Iteration Steps  
The main cable system calculation at side span under the condition that saddle position is known, using the method that un-stressed cable length iterated once, is illustrated as an example, and the iteration steps are shown, as follows:  
* Set initial value of horizontal angle \(\beta_{q}\)(1), \(\beta_{q}\)(2) for tangent line of suspension cable at saddle point of support ends. * Set the initial value of \(K_{q}\) and reference value of \(K_{q1}\) of tangent slope of suspension cable at saddle point of start support: \[K_{q1}\Leftarrow\tan\left[\beta_{q1}(1)\right],\ K_{q}\Leftarrow K_{q1}\]
* Determine whether the iterative equations are solvable or not, and set initial value to sign \(IBZ\), \(IBZ\Leftarrow 1\) (Note: \(IBZ\) = 1: solvable, \(IBZ\) = 0: unsolvable). * \(J2\Leftarrow 1\) (\(J2\) is the modification time of initial slope or vertical component of initial iterated internal forces, when there is no solution for iterative equations)
* If \(IBZ\) = 0, obtain correction factor according to \(J2\) to modify the vertical component of initial iterated internal forces. \[J2\Leftarrow J2+1\] \[C3\Leftarrow 1+(-1)^{[J2-\mathrm{int}(\frac{B}{2})\times 2+1]}\times \mathrm{int}\left(\frac{J2+1}{2}\right)\times 0.05\] Assign initial value of the slope at start point: \(K_{q}\Leftarrow C3\times K_{q1}\), then go to Step 6. If \(IBZ\) = 1, go to Step 6. * Calculate the vertical component and horizontal inclination at start point: \[V(1)\Leftarrow K_{q}\cdot H(1);\ \beta_{q1}(1)=ATAN(K_{q})\] (21)where, \(H(1)\) is horizontal component at start point which is determined according to the condition that the horizontal forces at both side of the saddle are equal. * Calculate tangent point coordinate (\(X_{q2}\), \(Y_{q2}\)) between the line with horizontal angle \(\beta_{q}(2)\) and the end point of cable at saddle. * Calculate tangent point coordinate (\(X_{q1}\), \(Y_{q1}\)) between the line with horizontal angle \(\beta_{q}(1)\) and the start point of cable at saddle. * Calculate coordinates and internal forces of cable segments from left to right point (or right to left point): (1) According to the internal forces (horizontal and vertical components) at one end and the horizontal distance between two ends of a cable segment \(k\) (\(k=1,2,\ldots,n\)), calculate the unstressed cable length \(S_{0}(k)\) and the elevation difference \(\Delta h_{k}\) between two ends by Equations (1) and (2). If there is no solution, then \(IBZ\Leftarrow 0\), go to Step 5. Otherwise, \(IBZ\Leftarrow 1\), and calculates the coordinates \(y_{k}\), horizontal, vertical component \(H_{j}(k)\), \(V_{j}(k)\) of the cable segment at right point \(k\), respectively. (2) Calculate the internal force (\(H_{i}(^{k+1})\),\(V_{i}(^{k+1})\)) at left end (\(i\)) of cable segment \(k+1\) while using equilibrium condition. * Calculate the elevation \(Y_{n}\) at end point and the elevation error \(\Delta_{n}\Leftarrow Y_{n}-Y_{q2}\). * If \(|\Delta_{n}|\leq\varepsilon\) (set \(\varepsilon=10^{-4}\) m\(\cdot 10^{-6}\) m), go to Step 16; otherwise, go to Step 16. * Calculate the stiffness due to vertical deformation of main cable at side span by Equation (18), and compute \(dV\): \(dV=\frac{1}{\frac{n}{n}}\Delta Y_{n}\)
* determine \(\alpha\): If \(|dV|>\sqrt{V(1)^{2}+H(1)^{2}}\), \(\alpha\Leftarrow 0.1\times\sqrt{V(1)^{2}+H(1)^{2}}/|dV|\) Otherwise, \(\alpha\Leftarrow 1\). * modify \(V(1)\): \(V(1)\Leftarrow V(1)+\alpha\cdot dV\). * Calculate new inclination angle \(\beta_{q}(1)\) of cable at start point: \(\beta_{q1}(1)=ATAN\Big{[}\frac{V(1)}{H(1)}\Big{]}\), then go to Step 8.