## 4 Analysis of Structural Uncertainty
### Sensitivity Analysis of Tornado Diagram  
The sensitivity analysis of structural uncertainty to the SFDP of an extradosed cable-stayed bridge is an effective method to explore the laws of seismic response for high-speed railway bridges. Therefore, the \"Tornado graphic method\" is adopted for uncertainty analysis. This method is mainly used in the field of decision analysis in the early stage and is later introduced into the structural seismic damage assessment [43]. The basic idea behind this method for random parameter sensitivity analysis is to calculate the variation of SFDP relative to the input parameters, that is, to change a single parameter while other parameters remain unchanged, and to then conduct dynamic nonlinear time-history analysis to obtain a variation of the SFDP. The variation corresponding to each parameter is a horizontal bar graph, and n random variables correspond to n horizontal bar graphs. The wider the bar graph is, the more significant the influence of the random parameter is. Finally, the horizontal bar graphs are arranged from wide to narrow and from top to bottom. The basic idea of the method is shown in Figure 12.  
The \"Tornado graphic method\" is used to analyze the sensitivity of structural uncertainty parameters. The specific steps can be described as follows:  
1. For high-speed railway bridge damage, the SFDP of an extradosed cable-stayed bridge are selected, such as angles of girder, support displacement, pier displacement, pier curvature, tower curvature and cable force;  
Figure 12: Principle of sensitivity analysis of the Tornado diagram.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Correlation** & \(f_{c,core}\) & \(f_{c,core}\) & \(e_{c,core}\) & \(e_{c,core}\) & \(f_{c,cover}\) & \(f_{c,cover}\) & \(e_{c,cover}\) & \(e_{c,cover}\) & \(e_{c,cover}\) \\ \hline \(f_{c,core}\) & 1 & 0.8 & & & 0.8 & 0.64 & & \\ \(f_{c,core}\) & 0.8 & 1 & & & 0.64 & 0.8 & & \\ \(e_{c,core}\) & & & 1 & 0.8 & & & 0.8 & 0.64 \\ \(e_{c,core}\) & & & 0.8 & 1 & & & 0.64 & 0.8 \\ \(f_{c,cover}\) & 0.8 & 0.64 & & & 1 & 0.8 & & \\ \(f_{c,low}\) & 0.64 & 0.8 & & & 0.8 & 1 & & \\ \(e_{c,low}\) & & & 0.8 & 0.64 & & & 1 & 0.8 \\ \(e_{c,low}\) & & & 0.64 & 0.8 & & & 0.8 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Correlation coefficient matrix of concrete material.  
2. According to the probability distribution characteristics of each random parameter in Table 4 and considering the correlation, the upper limit and lower limit values can be determined using Latin hypercube sampling. Then, the single random variable x is changed, and its upper limit and lower limit values are brought into the dynamic analysis model, while the other parameters remain unchanged. Subsequently, the difference between the upper limit and lower limit value of the SFDP can be calculated. Finally, the difference is divided by the maximum value of the SFDP of a benchmark model, and its ratio is defined as the sensitivity of the SFDP;
3. The sensitivity of the demand parameters to random variables can be analyzed after repeating step (2), and then the sensitivity as a \"horizontal graph\" of the random variable X can be drawn in the graph;
4. Finally, a \"horizontal graph\" of each parameter can be obtained by repeating step (3); these can be arranged in descending order to analyze the parameter sensitivity of dynamic damage of the extradosed cable-stayed bridge.