## 3 Results and Discussion
### Damage Assessment
Figure 18e illustrates the damage pattern formation at the tunnel's right arch foot. The predicted PPVs at which minor damage occurred are shown in Figure 17.  
Figure 16: Parametric analysis results for different amplitudes. Figure 17: Predicted PPV for Minor and No damage at different loading frequencies. the allowable PPVs in empirical models were found to be more conservative in contrast to the PPVs for No Damage, which were numerically predicted. Equations (5) and (6) provide the best fitted curves in Figure 17 for Minor and No Damage states across a frequency range of 10 to 100 Hz. \[\text{PPV}=4.06\text{ln}(f)+7.16\ \ \ (10\text{ Hz}\leq f\leq 100\text{ Hz}) \tag{5}\]  
where PPV is peak particle velocity, cm/s; and \(f\) is the frequency of the blast wave, Hz. The Equation (6) proposed for the No Damage state represents the ultimate Peak Particle Velocity (PPV) based on the site characteristics and structural configurations that have been taken into account. \[\text{PPV}=4.11\text{ln}(f)+5.96\ \ \ (10\text{ Hz}\leq f\leq 100\text{ Hz}) \tag{6}\]  
On the other hand, this Equation is enhanced for practical usage by adding a safety factor of 1.2 and 1.5, according to the safety requirement and the significance of the tunnel construction. Figure 19 depicts the best fitted curves for factors of safety of 1.2 and 1.5, which may be represented by Equations (7) and (8), respectively. \[\text{PPV}=3.42\text{ln}(f)+4.96\ \ \ (10\text{ Hz}\leq f\leq 100\text{ Hz}) \tag{7}\]  
\[\text{PPV}=2.74\text{ln}(f)+3.96\ \ \ (10\text{ Hz}\leq f\leq 100\text{ Hz}) \tag{8}\]  
This study conducted numerical modeling to investigate the effect of blast load frequency and predict the minimum PPV at which the cracking of concrete tunnels starts. Previous studies have ignored the effect of frequency content and predicted underground tunnel damage based on intensity level, with varying distances between the tunnel and blast source and the tunnel shape effect. This study encompasses the use of different loading amplitudes with varying frequency content to examine the structural damage. The results show that higher loading intensities are associated with more tunnel damage, as shown by a rise in the number of cracked elements and peak particle velocity (PPV). Likewise, it has been shown that lower blast wave frequencies result in more damage than higher frequencies, with different areas of damage occurring inside the tunnel structure depending on the blast wave frequency. Figure 15:0 Comparison of No damage state with Proposed factor of safety. The research categorized the structural damage of the tunnels into two distinct categories, namely No Damage and Minor Damage. The study included the prediction of No Damage (PPVs) and the identification of certain loading frequencies at which the cracks initiated. The findings indicate that the degree and placement of cracks inside the tunnel varied depending on the frequency of the blast wave, with distinct frequencies resulting in damage to certain sections of the tunnel's framework. This study compared the predicted PPVs with empirical models derived from China and South Korea, therefore emphasizing disparities in the threshold PPVs for the No Damage state. The Korean code exhibited a more cautious approach in its forecasts when compared to the numerical outcomes. Two equations were proposed for a safety factor of 1.2 and 1.5 based on the No Damage state. These equations will provide a revised threshold for acceptable PPV that takes the selected safety level into account. The scope of this study is limited to a large horseshoe-shaped tunnel cross-section that is surrounded by hard rock formations and features a tunnel lining of 0.3 m and with specific lining material properties. Determining damage based on PPVs and the frequency characteristics of blast waves is contingent upon several factors, such as site conditions, tunnel geometry, and the material properties of structure and rock.