# An Improved Analytical Algorithm on Main Cable System of Suspension Bridge
## 3 Improved Numerical Analysis Method
### The Main Cable System Calculation in Main Span at Finished State
#### Improved Numerical Analysis Method and Its Iteration Steps  
The main cable system calculation in main span under the condition that theoretical vertex position is known using step by step iteration method is illustrated as an example, and the iteration steps are shown in the following:  
1. All vertical loads in main span were simplified as uniform distributed load along the span, and the internal forces at both support ends \(H_{1}\)(1:2), \(V_{1}\)(1:2) were calculated using traditional parabola theory (actually only the internal force at start point is needed). 2. The start end forces \(H_{1}\)(1:2), \(V_{1}\)(1:2) were regarded as the reference value \(H\)(1:2), \(V\)(1:2) of initial iterated internal forces. 3. Input \(H\)(1:2), \(V\)(1:2) into iterative equations, and determine whether the iterative equations were solvable or not, and set initial value to sign \(IBZ\), \(IBZ\Leftarrow 1\) (Note: \(IBZ\) = 1, solvable; \(IBZ\) = 0, unsolvable). 4. \(J2\Leftarrow 1\) (\(J2\) is the modification times of iterated initial internal force when there is no solution for iterative equations)
5. If \(IBZ\) = 0, obtain correction factor according to \(J2\), modify the overall level of initial iterated internal forces (this algorithm called search algorithm, which searching a suitable internal force at start point by changing \(J2\) to make the iterative equation solvable), namely: \[J2\Leftarrow J2+1;\] \[C3\Leftarrow 1+(-1)^{\left[J2-\mathrm{int}(\frac{J2}{2})\times 2+1\right]} \times\mathrm{int}\left(\frac{J2+1}{2}\right)\times 0.05;\] \[H(1:2)\Leftarrow C3\times H1(1:2);\] \[V(1:2)\Leftarrow C3\times V1(1:2);\] If \(IBZ\) = 1, go to Step 6. 6. On the basis of Step 5 or Step 2, we determine the initial iteration horizontal force multiplier (\(KK\)) at start point, and get the elevation error at different points. Then, find suitable initial horizontal force iteration multiplier, and obtain the internal force and deformation in the main span by secant method, go to Step 7. If there is no solution, then \(IBZ\) = 0, go to Step 5 (i.e., correcting overall level of initial iterated internal forces). The details of step 6 are shown in the following:  
1. Set initial iteration horizontal force, vertical force at start point: \[H0(1)\Leftarrow KK\times H(1),V\ 0(1)\Leftarrow V\ (1)\ \mathrm{or}\ H0(2) \Leftarrow KK\times H(2),\ V\ 0(2)\Leftarrow V\ (2)\]
2. \(J3\Leftarrow 1\) (set initial value of iterative times \(J3\) based on Step 6.1). 3. Calculate internal forces from left to right point (or right to left point) in the follows:According to the internal forces (horizontal and vertical components) at one end and the horizontal distance between two ends of a cable segment \(k\) (\(k=1,\,2,\,\ldots,\,n\)), the unstressed cable length \(S_{0}(k)\) and the elevation difference \(\Delta h_{k}\) between two ends were calculated by Equations (1) and (2). If there is no solution, then \(IBZ\Leftarrow 0\), go to Step 5. Otherwise, \(IBZ\Leftarrow 1\), calculate the coordinates \(y_{k}\), horizontal and 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 point (\(i\)) of cable segment \(k+1\) using equilibrium condition. _Step 6.4_: \(J3\Leftarrow J3+1\). _Step 6.5_: Based on the elevation \(Y_{1}\) at start point and the elevation difference \(\Delta h_{k}\) of each cable segment (\(k=1,\,2,\,\ldots,\,n-1\)), the elevation \(Y_{n}\) at end point and the elevation error \(\Delta_{n}\Leftarrow Y_{n}\,-\,Y_{R}(Y_{R}\) is the actual elevation at end point) were determined. _Step 6.6_: If \(\mid\Delta_{n}\mid\leq\varepsilon\) (\(\varepsilon=10^{-4}\) m\(\cdot 10^{-6}\) m or \(10^{-7}\)\(\sim\)\(10^{-9}\) times of the main span), go to Step 6.10; if \(\mid\Delta_{n}\mid>\varepsilon\), go to Step 6.7. _Step 6.7_: If the iteration time \(J3>60\) (this value can be taken as 100, etc. ), it was considered non-convergence, \(IBZ\Leftarrow 0\), go to Step 5; if \(J3\leq 60\), then go to Step 6.8.