# 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  
A differential equation with the strong form is defined as \(f(x,u,u^{\prime},u^{\prime\prime},\cdots)=0\), in which \(f(\blacksquare)\) is a function symbol, \(x\) is the independent variable, \(u\) is the dependent variable, \(u^{\prime},u^{\prime\prime},\cdots\) are the first-, second- and high-order derivatives with respect to the \(x\). In FEM, it can be discretized using a basis function or a shape function \(N_{i}\) to produce a weak form of \(\int_{\Omega}f(x,u,u^{\prime},u^{\prime\prime},\cdots)N_{i}d\Omega=0\), in which \(\Omega\) is the computational domain of interest. To solve the cable shape, a one-dimensional linear shape function is adopted in this study with the form of  
\[N_{i}(x)=a_{i1}+a_{i2}x \tag{12}\]  
in which \(a_{i1}\) and \(a_{i2}\) are undetermined coefficients estimated by the nodal coordinates of the element. By using the Galerkin's method, the cable shape in terms of \(y(x)\) and \(z(x)\) can be formulated as:  
\[y(x)=y_{i}N_{i}(x) \tag{13}\]  
\[z(x)=z_{i}N_{i}(x) \tag{14}\]  
By introducing the \(N_{i}(x)\), the discrete algebraic form or weak form of Equations (9) and (10) can be expressed as:  
\[\int_{0}^{L}-H_{0}y^{\prime\prime}N_{i}(x)dx+\int_{0}^{L}q\sqrt{1+{y^{\prime }}^{2}+{z^{\prime}}^{2}}N_{i}(x)dx+\int_{0}^{L}T_{yi}\delta\big{(}x-x_{j} \big{)}N_{i}(x)dx=0 \tag{15}\]  
\[\int_{0}^{L}-H_{0}z^{\prime\prime}N_{i}(x)dx+\int_{0}^{L}T_{zj}\delta\big{(}x -x_{j}\big{)}N_{i}(x)dx=0 \tag{16}\]  
in which the subscript \(j\) (\(j\) = 1, 2,..., \(n\)) indicates the hanger ID, which is introduced to distinguish with the subscript \(i\) before the [PERSON] summation convention is applied. The subscripts \(k\) and \(l\) in the following sections are also utilized for the same purpose. The integration by parts is applied to Equations (15) and (16), which leads to \[\int_{0}^{L}H_{0}y^{\prime}\frac{dN_{i}(x)}{dx}dx-H_{0}y^{\prime}N_{i}(x)|_{\text{ BC}}+\int_{0}^{L}q\sqrt{1+{y^{\prime}}^{2}+{z^{\prime}}^{2}}N_{i}(x)dx+T_{yi}=0 \tag{17}\]  
\[\int_{0}^{L}H_{0}z^{\prime}\frac{dN_{i}(x)}{dx}dx-H_{0}z^{\prime}N_{i}(x)|_{\text {BC}}+T_{zi}=0 \tag{18}\]  
in which BC indicates the integral boundary condition. The substitution of Equation (8) into Equations (17) and (18) produces:  
\[\int_{0}^{L}H_{0}y_{j}\frac{dN_{j}(x)}{dx}\frac{dN_{i}(x)}{dx}dx-H_{0}y^{\prime }N_{i}(x)|_{\text{BC}}+\int_{0}^{L}q\sqrt{1+\left[y_{j}\frac{dN_{j}(x)}{dx} \right]^{2}+\left[z_{j}\frac{dN_{j}(x)}{dx}\right]^{2}}N_{i}(x)dx+T_{yi}=0 \tag{19}\]  
\[\int_{0}^{L}H_{0}z_{j}\frac{dN_{j}(x)}{dx}\frac{dN_{i}(x)}{dx}dx-H_{0}z^{ \prime}N_{i}(x)|_{\text{BC}}+T_{zi}=0 \tag{20}\]  
in which \(y_{j}\) and \(z_{j}\) are the final \(y\)-direction and \(z\)-direction coordinates of the main cable. An additional formula for describing the ratio between \(T_{yi}\) and \(T_{zi}\) is given as:  
\[\frac{T_{zi}}{T_{yi}}=\frac{z_{i}-z_{di}}{y_{i}-y_{di}} \tag{21}\]  
As formulated in Equation (12), the differential of a one-dimensional linear shape function with respect to \(x\) is a constant for an element. Meanwhile, the node of the main cable that is located at the top of the pylon is assumed to be fixed. The Dirichlet boundary condition is therefore applied to the variables of \(y\) and \(z\). The integral form of Equations (19) and (20) can be alternatively solved using a form of the summation as:  
\[y(x)=y_{i}N_{i}(x) \tag{13}\]0  
\[y(x)=y_{i}N_{i}(x) \tag{13}\]1  
in which \(dL\) is the length of element \(\Omega_{e}\) in the \(x\)-direction. It is worth mentioning that \(\left[y_{j}\frac{dN_{j}(x)}{dx}\right]^{2}\) and \(\left[z_{j}\frac{dN_{j}(x)}{dx}\right]^{2}\) are constant within each element. However, their values are not identical for different elements.