# A Novel Shape Finding Method for the Main Cable of
## 2 Shape Finding of the Main Cable with Eulerian Description
### Nonlinear FEM Solution for the Cable Shape
To solve this nonlinear equation related to \(y_{j}\) and \(z_{j}\), the [PERSON] iteration method [33] is introduced:  
\[y(x)=y_{i}N_{i}(x) \tag{13}\]2  
in which \(f(x)\) is the function to be solved, \(x_{n+1}\) is the approximate solution of \(f(x)=0\) at (n + 1)-step iteration, \(\omega\) is a relaxation coefficient, which is set as 1.0 in this study. \(f^{\prime}(x_{n})\) in Equations (22) and (23) is a matrix for a multivariable problem, which consists of the partial derivatives of the objective function with respect to each unknow variables:  
\[y(x)=y_{i}N_{i}(x) \tag{13}\]3  
\[y(x)=y_{i}N_{i}(x) \tag{13}\]4  
\[K_{ij}^{zy}=\frac{\partial f_{zi}}{\partial y_{j}}=\left\{ \begin{array}{cc}T_{yi }\frac{z_{i}-z_{di}}{(y_{i}-y_{di})}&,i=j\\ 0&,i\
eq j\end{array} \right. \tag{27}\]\[K_{ij}^{zz}=\frac{\partial f_{zi}}{\partial z_{j}}=\sum\
olimits_{\Omega_{e}} \left(H_{0}\frac{dN_{j}(x)}{dx}\frac{dN_{i}(x)}{dx}dL\right)+\left\{ \begin{array}[ ]{cc}T_{yi}\frac{z_{i}}{y_{i}-y_{ii}}&,i=j\\ 0&,i\
eq j\end{array} \right. \tag{28}\]  
in which \(f_{yi}\) and \(f_{zi}\) are functions described by the first members of Equations (22) and (23), which are derived from the equilibrium equations in \(y\)-direction Equation (9) and \(z\)-direction Equation (10) at \(i\)th point, respectively, \(J_{i}=\int_{0}^{L}N_{i}(x)dx=dL/2\) for the chosen one-dimensional linear shape function. To construct the Newton iteration form Equations (22) and (23) for the main cable, a matrix related to the \(f^{\prime}(x_{n})\) and a vector related to the \(f(x_{n})\) are assembled as:  
\[\mathbf{K}=\left[ \begin{array}{cc}\mathbf{K}^{yy}&\mathbf{K}^{yz}\\ \mathbf{K}^{zy}&\mathbf{K}^{zz}\end{array} \right] \tag{29}\]  
\[\mathbf{b}=\left( \begin{array}{c}f_{y}\\ f_{z}\end{array} \right) \tag{30}\]  
in which \(\mathbf{K}^{yy}\), \(\mathbf{K}^{yz}\), \(\mathbf{K}^{zy}\), \(\mathbf{K}^{zz}\), \(\mathbf{f}_{y}\) and \(\mathbf{f}_{z}\) are obtained by assembling the \(K_{ij}^{yy}\), \(K_{ij}^{yz}\), \(K_{ij}^{yz}\), \(K_{ij}^{zz}\), \(f_{yi}\) and \(f_{zi}\) for all elements, \(\mathbf{K}^{yy}\), \(\mathbf{K}^{yz}\), \(\mathbf{K}^{zy}\) and \(\mathbf{K}^{zz}\) are matrices with the size of \(n\times n\), \(\mathbf{f}_{y}\) and \(\mathbf{f}_{z}\) are matrices with the size of \(n\times 1\). To solve the deformation increments of the main cable in \(y\)- and \(z\)-directions, i.e., \(\Delta=\left(dy\;dz\right)^{\mathrm{T}}\), a linear algebra equation is then developed with the form of  
\[\mathbf{K}\Delta=-\omega\mathbf{b} \tag{31}\]  
The nodal coordinates of the main cable are readily updated using the summation of the \(\left(y_{i}\;z_{i}\right)\) and \(\left(dy_{i}\;dz_{i}\right)\). It is worth noting that he initial nodal coordinates \(\left(y_{i},z_{i}\right)\) before the iteration can be assigned as arbitrary values, the coordinates of the intersection point (IP) between the pylon and main cable are suggested for the operational convenience. The above iteration scheme is illustrated in Figure 2 with a given \(H_{0}\), which is called the inner-layer iteration in this study. However, in most cases, \(H_{0}\) is an undetermined parameter. The middle span-sag \(f\) or the coordinate of the middle point in \(y\)-direction \(y_{mid}\) of the main cable is always prescribed. An outer-layer iteration is therefore implemented, as described in Figure 2. At each outer-layer iteration, the coordinate of the middle point in \(y\)-direction \(\hat{y}_{mid}\) can be solved by a given \(H_{0}\) using the inner-layer iteration. A one-to-one mapping relationship \(\hat{y}_{mid}=g(H_{0})\) is assumed, in which \(g(\blacksquare)\) is the function symbol. As a result, the solution of \(H_{0}\) can be determined by finding the root of the equation \(G(H_{0})=g(H_{0})-y_{mid}=0\). Since \(g(H_{0})\) has no explicit expression, the Secant method is utilized to achieve a numerical solution of \(H_{0}\).