## 3 Results & Discussion  
This section shares the results of the analysis of the bat/ball impact finite element models and the resulting data, in order to understand baseball bat durability for the given bat profiles with respect to SoG. Applying the statistical distribution assumptions outlined in Section 2.6, a probability of failure analysis was conducted to compare the profiles and their respective durability. Figure 7 compares the NF, SPF, and MPF probabilities of each bat profile and each tested SoG. Table 2 summarizes the probability of a bat profile not to crack at all (NF), to crack (SPF), or to break completely (MPF). The NF probability outcome data are shown in the green region of Figure 7. The zero-degree (0\({}^{\circ}\)) SoG was found to have the lowest NF% for Bat Profiles A, B, and D, but the highest NF% was for Bat Profile C. The highest mean values from Table 2 suggests that Bat Profile D is the most durable bat. An interesting note related to the NF probability is that there is no difference between the -3\({}^{\circ}\) and +3\({}^{\circ}\) SoG values for Bat Profiles A, B, or C, which indicates that there is no bias toward either positive or negative slopes of grain for these profiles when considering all impact locations. Bat Profile D shows signs of a bias toward the +3\({}^{\circ}\) SoG as it is 3% higher than the -3\({}^{\circ}\) SoG. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Bat Profile**} & \multicolumn{3}{c}{**Mean Values**} & \multicolumn{3}{c}{**Standard Deviation**} \\ \cline{2-7}  & **NF\%** & **SPF\%** & **MPF\%** & **NF\%** & **SPF\%** & **MPF\%** \\ \hline A & 81.5 & 14.2 & 4.3 & 3.1 & 1.3 & 2.5 \\ B & 74.1 & 21.9 & 4.0 & 3.4 & 5.7 & 2.9 \\ C & 80.0 & 13.1 & 6.9 & 3.1 & 3.5 & 4.5 \\ D & 82.9 & 10.7 & 6.4 & 2.2 & 0.2 & 2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summarized Probability Analysis Results for Centerline Impacts. The SPF results of the finite element models are shown in the yellow regions of Figure 7. Bat Profile C has the largest SPF probability and largest SPF% standard deviation, despite having the lowest NF probability. Bat Profile B also had the largest sensitivity to SoG effects due to a 7% increase in SPF probability between the \(-3^{\circ}\) and \(0^{\circ}\) SoG, followed by an 11% decrease from \(0^{\circ}\) to \(+3^{\circ}\) SoG. The large changes lead to Bat Profile B having the largest SPF% standard deviation among the tested SoGs. Bat Profiles A and D were found to be the most insensitive to SoG effects. A final interesting note related to SPF probability is that the \(+3^{\circ}\) SoG region resulted in the lowest SPF outcome likelihood across all profiles. The results in Figure 7 also provide insight related to MPF bat durability with respect to each profile and SoG. Per Table 2, Bat Profile C had the largest mean MPF probability value followed by Bat Profiles D, A, and B. The nominal mean values suggest that Bat Profiles C and D had the worst MPF durability of the bat profiles studied. Bat Profile C was also found to have the largest standard deviation of the three SoGs modeled, which indicates that it was the most sensitive to SoG of the four tested profiles. Bat Profiles A and D had their largest MPF probability values at the \(0^{\circ}\) SoG location, whereas Profiles B and C had their largest MPF probability values at the \(+3^{\circ}\) SoG location. For these profiles, a MPF outcome is over twice as likely to occur at the \(+3^{\circ}\) SoG than the other tested slopes of grain. The finding of Bat Profiles B and C having lower NF% than Bat Profile A correlates well with a proprietary and empirically observed order of relative durability; however, the NF% for Bat Profile D being higher than the NF% values for the rest of the profiles does not appear to correlate at first glance. Historically, Bat Profile D has been known to be a bat profile with poor durability. However, the data presented in Table 2 show that it has the largest probability of NF under the conditions considered here.