# A Theoretical Study on the Resilience Evaluation Method of Operational Road Tunnel Systems
## 4 Theoretical Analysis Model of Road Tunnel Resilience  
### Introduction of the Model  
Based on the concept of resilience, it can be seen that the essence of resilience is a comprehensive ability to resist external disturbances, reflected using system function changes  
Figure 4: Road tunnel resilience triangle. over time. Therefore, the change in system function over time is used to quantitatively describe the resilience value. Referring to the conceptual model of seismic resilience proposed by [PERSON] et al. [21], a theoretical model of road tunnel resilience was proposed, combined with the characteristics of the road tunnel as shown in Figure 5. In order to quantify the resilience of the road tunnel, the following hypotheses are proposed:  
1. Before and after the road tunnel system is affected by external disturbances, the change in system function level value is not considered. 2. The functional change in the road tunnel system caused by external disturbances is linear. In the theoretical analysis model of road tunnels, the road tunnel system operates at a constant functional level before and after being affected by external disturbances. The three stages of road tunnel resilience proposed in Section 3.2 are the AE, EF, and FB segments, as shown in Figure 5. In the AE segment, \(\Delta\)F is used to describe the degree of damage to the system function or the severity of external disturbances. In the EF segment, the response time of the road tunnel system can be described using \(\mathrm{t_{3}}-\mathrm{t_{2}}\). It is used to reflect the soundness of the system's response mechanism to external disturbances. Under the sound emergency response mechanism, the road tunnel system can quickly start an effective emergency decision-making scheme and prevent secondary damage to the system's function. In the FB segment, the recovery time of the road tunnel system function is described using \(\mathrm{t_{4}}-\mathrm{t_{3}}\). In addition, resilience should be an ability of the system within a certain functional state. If the system function exceeds a certain limit, the system can be considered to enter a \"plastic stage\" called resilience failure state. Under this state, the road tunnel system should have the following three characteristics:  
1. The road tunnel system function is basically lost. Figure 5: Resilience theoretical model of a road tunnel system, where \(\mathrm{F_{before}}\) is the value of the system function level of the road tunnel before the influence of external disturbance. \(\mathrm{F_{after}}\) is the value of the system function level of the road tunnel after the influence of external disturbances. \(\mathrm{F_{recover}}\) is the value of the system function level of the road tunnel after road tunnel system restoration. \(\mathrm{F_{min}}\) is the lowest level of system function when the basic traffic capacity of the road tunnel is satisfied. \(\Delta\)F is the disaster loss value of the road tunnel system. \(\mathrm{t_{1}}\) is the moment when the road tunnel system suffers from a natural disaster or emergency event. \(\mathrm{t_{2}}\) is the moment when the function of the road tunnel system declines to the lowest value. \(\mathrm{t_{3}}\) is the moment when the function of the road tunnel system begins to recover. \(\mathrm{t_{4}}\) is the moment when the function of the road tunnel system is restored to its normal state. (2) The road tunnel system function is difficult to recover in a short time. (3) The cost of road tunnel system function restoration is high. For the road tunnel system, P\({}_{\text{min}}\) is defined as the system resilience limit, which is used to describe the system function level value when the road tunnel system maintains the most basic traffic capacity. From the perspective of resilience, the road tunnel system can predict and forewarn the external disturbance so that it can deal with the influence in advance. The threat of external disturbances to the road tunnel system function will be greatly reduced, and the road tunnel system will show better resilience.