# Reliability Assessment of Steel-Lined and Prestressed FRC Slabs against Projectile Impact
## 6 Discussion of Results  
The last two columns of Table 2 show the probability of failure (\(P_{f}\)) and reliability indices (\(\beta\)) of PFRC slab specimens determined by the MCS technique and the output of the analysis are given in Table 3. The nominal strike velocities (\(V_{0}\)) shown in this table are the same as those used in the experiment. The PFRC slab specimens of 800 \(\times\) 800 \(\times\) 100 mm were tested for impact resistance against steel projectiles with hemispherical nose shapes. Table 3 shows the \(P_{f}\) and \(\beta\) for PFRC slab specimens. There is a clear correlation between the projectile strike velocity and the \(P_{f}\) of the PFRC specimens. The \(P_{f}\) of the PFRC specimens increases with increasing projectile impact velocity. This is a trend that is expected. In Table 3, specimens with \(\beta\) values above 1.0 have a sufficiently low failure probability. Due to their combined uncertainty, the parameters affecting the ballistic limit velocity of the PFRC specimens do not bring it below the strike velocity. A \(P_{f}\) of 0.04 for RE-NL-F1-N-2, for example, indicates that among 100 specimens subjected to the same impact velocity, only four would fall below the ballistic limit, i.e., only four specimens out of 100 would be perforated, and 96 would not be perforated. In other words, there is a very small probability that this specimen would be perforated when impacted by a projectile with a nominal strike velocity of 143.8 m/s. It is imperative to note that when the striking velocity exceeds the ballistic limit, failure occurs deterministically. However, when viewed probabilistically, it means that failure is more likely. In Table 3, all specimens with a ratio greater than one support this claim. In the case of specimens RE-NL-F0-N-2, RE-NL-F0-N-3, RE-NL-F0-P-1, RE-SL-F0-P-1, RE-NL-F1-P-2, RE-NL-F2-P-1, and RE-NL-F2-P-3, failure probabilities are substantially high (\(\beta\) are less than 3.0). Generally, the reliability index of a structure or structural component should exceed 3.0 to consider it to be safe enough [13; 14; 15; 16; 21]. The probabilistic analysis of the specimens shows that only six specimens (RE-SL-F1-N-1, RE-SL-F2-N-3, RE-SL-F0-P-3, RE-SL-F1-P-3, RE-SL-F2-P-1, and RE-SL-F2-P-3) are as reliable as desired for the considered strike velocities. All other specimens have reliability indices less than 3.0. Figure 5 plots the reliability index against the \(V_{0}/V_{P}\) ratio in order to determine the impact velocity at which slab specimens can be considered reliable. All specimens were plotted on the graph. Figure 5 shows that reliability decreases sharply with increasing impact velocity. The reliability index is around 4 when the projectile strikes PFRC slabs at half of their ballistic limit (\(\sim\)0.5\(V_{P}\)). However, if the impact velocity approaches 1.2 times  
Figure 4: Variation of \(COV\left(P_{f}\right)\) with the number of simulations. the slab's ballistic limit (\(\sim\)1.2\(V_{P}\)), the reliability index reduces to around 0.75, indicating less reliability (i.e., high failure probability). When the ratio \(V_{0}/V_{P}\) is 0.7 or less, the reliability is three or above. For PFRC slabs to achieve at least a 3.0 reliability index, their ballistic limit must be designed for \(V_{0}/0.7\)-\(1.4V_{0}\). Figure 6 illustrates the variation in the reliability of PFRC slabs with the proportion of steel fibers subjected to the projectile strike velocity equal to the ballistic limit of that slab which is without fibers, prestress, or steel lining (118.7 m/s), i.e., RE-NL-F0-N. Without fiber, the slab's reliability was substantially less, as shown in the figure. The reliability of slabs improved considerably after fibers were added. For specimens with the highest amounts of steel fibers (1.2%), the maximum increase in reliability was observed. The second-highest reliability belongs to the specimen with the second-highest percentage of steel fibers.