# Research on Effective Design Methods of Core Beam of Full Bridge Aeroelastic Model
## 3 Numerical Solution of Torsion Constant
### Five-Point Difference Method
 \tag{25}\]  
Note that the first and last elements of the vector on the right side of Equation (25) take into account the boundary conditions at \(y=c\) and \(y=d\). Difference equations of all points form a closed system of linear algebraic equations, of which the coefficient matrix is symmetric. After solving the equations, stress results can be obtained. A typical solution is shown in Figure 6 and the torsion constant is the double volume enclosed by the stress surface and the \(XY\) plane, according to Equation (3). Taking a U-shaped cross-section as an example, the target values of three moments of inertia, Target_Iz, Target_Iy and Target_Ix, are listed in Table 1. In fact, the target values listed in Table 1 are based on an actual suspension bridge, which will be mentioned later. In the process of ANSYS optimizing, the initial geometric sizes (_h, d, th_ and _td_) of the section is first assigned, and then _Iz, Iy_ and _Ix_ are calculated. The errors of _Iz, Iy_ and _Ix_ are defined separately as  
\[ \begin{array}{l}Err\_Iz=100\%\cdot|(\text{Target\_Iz}-\textit{Iz})/\text{ Target\_Iz}|\\ Err\_Iy=100\%\cdot|(\text{Target\_Iy}-\textit{Iy})/\text{Target\_Iy}|\\ Err\_Ix=100\%\cdot|(\text{Target\_Ix}-\textit{Ix})/\text{Target\_Ix}|\\ \end{array}  \tag{29}\]  
The three errors are all limited to 0-5%. The sum of the three errors is used as an objective function, as  
\[ \begin{array}{l}\textit{TErr}=\textit{Err\_Iz}+\textit{Err\_Iy}+\textit{ Err\_Ix}\\ \end{array}  \tag{30}\]  
where _Err_Iz_ is the sum of the errors of _Iz, Iy_ and _Ix_. The purpose of optimization is to minimize the total error. In order to investigate the influence of the initial parameters, two different sets of them and the searching domain are given, as listed in Table 2. It is seen that the initial value of d in the first set is not included in the searching domain. Figure 8 shows the optimization process of the zero-order and the first-order method, where the ordinate is _TErr_ and the abscissa is the number of iterations. It can be seen from Figure 8 that, based on the first set of initial values, the parameters are firstly optimized four times by zero-order method, resulting in a total error of 31.318%. Then the first-order method is performed three times, resulting in a total error of 3.738%. However, a total error as small as 0.31593% (see Table 3) is obtain \begin{table}
\begin{tabular}{c c c c} \hline _Target_Iz (mm\({}^{\textbf{4}}\))_ & _Target_Iy (mm\({}^{\textbf{4}}\))_ & _Target_Ix (mm\({}^{\textbf{4}}\))_ \\ \hline
79.04 & 4518.74 & 241.44 \\ \hline \end{tabular}
\end{table} 
Table 1: Target section properties. \begin{table}
\begin{tabular}{c c c c c} \hline  & _h (mm)_ & _d (mm)_ & _th (mm)_ & _td (mm)_ \\ \hline Initial set 1 & 5.038 & 3.612 & 3.0579 & 19.585 \\ Initial set 2 & 4.138 & 2.712 & 3.06 & 19.6 \\ Searching & 4.0–7.0 & 0.1–3.0 & 1.0–3.99 & 10.0–30.0 \\ domain & 5.038 & 3.612 & 3.0579 & 19.585 \\ \hline \end{tabular}
\end{table} 
Table 2: Initial values and searching domains. Figure 8: Optimization process. initial values is selected and optimized with the first-order method. Comparing the two sets of initial values, it is easy to find that the number of iterations of the second set of initial values is significantly reduced and the accuracy is obviously improved, which indicates that the initial values dominate the total error. In order to investigate the influence of an individual variable, \(h\), \(d\), \(th\) or \(td\) on the target values in Table 1, the second set of initial values are taken as a reference. The optimization is performed with three variables fixed and the fourth one varies in the given range. The results are plotted in Figures 9-12, where it is noted that \(Err\_lx\), \(Err\_ly\) and \(Err\_Ix\) vary simultaneously. As h increases, \(Err\_lz\) increases exponentially, while \(Err\_ly\) and \(Err\_lx\) are only slightly affected (see Figure 9). The other three cases are similar in the way they change, with the error decreasing first and then increasing to some extent as Figures 10-12 show.