## 4 Derivation of Empirical Formulation
### Stringer-Stiffened Cylinder  
The procedures to derive the formulation to predict the local denting damage of a stringer-stiffened cylinder are similar to procedures applied to a ring-stiffened cylinder in the previous Section 4.1. First, the correlations between \(\delta_{d}\) and various parameters were checked. It is included non-dimensional geometric (\(L_{s}\)/\(R\), \(L_{s}\)/\(t\), \(h_{\text{r}m}\)/\(t_{\text{r}w}\), \(h_{\text{s}m}\)/\(t_{\text{s}w}\), and \(w_{s}\)/\(t_{\text{s}f}\)) and material property parameters (\(E\)/\(\sigma_{Y}\) and \(\lambda_{E}\)) and combinations of variable parameters (\(Z=\frac{t^{2}}{R^{4}}\)/ \(\sqrt{1-\
u}\): Batdorf slenderness; \(\sqrt{Z}\frac{E}{\sigma_{Y}}\) and \(\frac{\sqrt{I}t}{R}\frac{E}{\sigma_{Y}}\) ). The results also indicated that energy ratio \(\lambda_{E}\) is the strongest parameter affected by \(\delta_{d}\). Then, the regression analysis was performed, and Equations (27)-(30) were derived as functions of the energy ratio (\(\lambda_{E}\)) to predict the maximum permanent dent depth of the stringer-stiffened cylinder under dynamic mass impact. It is noteworthy that the maximum permanent dent depth values (\(d\)) smaller than 0.25% of the cylinder diameter (\(D\)) were not considered, which corresponded to the upper limit of tolerable imperfection for stringer-stiffened cylinders by API [38]. The Equation (27) is a mean equation while that of Equation (28) is the design equation. It is noted that when considering the purposes of the safety design under hazard and risk conditions of marine structures, the design equation, which is determined by multiplying the mean curve with (Mean + 2 \(\times\) Standard deviation), should be used. These proposed formulations could be used easily for the purposes of the initial design without any time-consuming FEM analysis.  
The accuracy of the proposed formula was quite good, with \(X_{m}\) (ratio of predicted values from Equation (27)/numerical values) of 0.987 together with a COV of 8.60%. In these equations, the effect of striker header shapes and impact locations, as well as impact angles, was also considered as the reduction factor \(C_{S}\), \(C_{L}\), and \(C_{\beta}\), respectively. The most severe case is when the load is applied through a hemispherical indenter, which resembles highly localized point loading with \(C_{S}\) = 1, then \(C_{S}\) = 0.74 and \(C_{S}\) = 0.63 for the knife-edge indenter and rectangular indenter, respectively. Therefore, the proposed formulation was derived based on the most severe case of a hemispherical indenter. The effects of impact locations and impact angles were considered as Equations (29) and (30), respectively.  
\[\delta_{d}=\frac{d}{R}=3.59\text{ CSCL }C_{\beta}(_{E})^{0.68}\text{; Mean equation} \tag{27}\]  
\[\delta_{d}=\frac{d}{R}=4.16\text{ CSCL }C_{\beta}(_{E})^{0.68}\text{; Design equation} \tag{28}\]  
where  
\(C_{S}\): indenter shape factor (\(C_{S}\) = 1: Hemisphere indenter; \(C_{S}\) = 0.74: Knife-edge indenter;  
\(C_{S}\) = 0.63: Rectangular indenter),  
\(C_{L}\): impact location factor, and  
\(C_{\beta}\): impact angle factor.  
\[C_{L}=\text{Exp}\left(-9.91\frac{x}{L}\right) \tag{29}\]  
\[C_{\beta}=0.114\beta^{2}-0.346\beta+1 \tag{30}\]  
The skewness of the proposed formulation results to the numerical analysis results (\(X_{m}\)) with various parameters is plotted in Figures 25-27. There is no apparent skewness in these figures. Therefore, these basic parameters should be not considered for inclusion in the design of the proposed formulation to predict the maximum permanent dent depth of the stringer-stiffened cylinder under dynamic mass impact.