## 4 Derivation of Empirical Formulation
### Ring-Stiffened Cylinder  
To find a suitable dependent basis parameter on which to derive the local denting damage \(\delta_{d}\) (\(\delta_{d}=d/R\)), the relationship between \(\delta_{d}\) and various appropriate non-dimensional geometric and material property parameters were considered such as \(L_{s}/R\), \(L_{d}/t\), \(w\
ot{v}_{f}\) and \(h_{u}\bar{u}_{w}\), \(E/\sigma_{Y}\), and \(\lambda_{E}\) (where \(L_{s}\): ring stiffener spacing; \(R\): mean radius; \(t\): shell thickness; and \(w_{f}\), \(t_{f}\), \(h_{w}\), and \(t_{w}\) are flange width, flange thickness, web height, and web thickness of ring-stiffener, respectively; E: Young's modulus; \(\sigma_{Y}\): yield strength; and \(\lambda_{E}\): energy ratio). Additionally, the combination of different parameters was also evaluated in order to identify the most reasonable method of deriving \(\delta_{d}\) formulation, which included Batdorf slenderness \(Z=\frac{L^{2}}{Rt}/\sqrt{1-v}\); and multiplying of Batdorf coefficient with material properties \(\sqrt{Z}\frac{E}{\sigma_{Y}}\) and apart from multiplying the constant of \(\frac{\sqrt{Lt}}{R}\frac{E}{\sigma_{Y}}\) (where \(v\): Poisson ratio). The degree of dependence and trends are indicated in Figures 16-20. Generally, the trend of parameter of \(L_{s}/R\), \(L_{s}/t\), \(w\
ot{v}_{f}\), and \(h_{u}\bar{u}_{w}\) has less influence on \(\delta_{d}\). Another factor \(E/\sigma_{y}\) also has less effect, but in the real case, material properties played an important effect on \(\delta_{d}\). Therefore, this parameter will be included in the combination of basic variables. It can be seen that the ratio of energy parameter \(\lambda_{E}\) is shown to have the strongest effect on \(\delta_{d}\). The energy ratio \(\lambda_{E}\) is the ratio of impact kinetic energy of the striker (\(E_{k}=\frac{1}{2}mv^{2}\)) to the energy absorption capacity of the struck structure (\(E_{a}=\frac{\sigma_{Y}+\sigma_{T}}{2}\varepsilon_{T}V_{Str}\)). This parameter was used for deriving \(\delta_{d}\). Figure 16: Tendencies of \(d/R\) with non-dimensional basic parameters \(L_{s}/R\) and \(L_{s}/t\) for the ring-stiffened cylinder. After investigating the dependent parameters, the regression analyses were performed using the best-fit curve of numerical values as shown in Figure 21. The mean of the uncertainty modelling factor (\(X_{m}\)) was quite good with 0.996, while the coefficient of variation (COV) was 6.32%. It is remarkable that the values of maximum permanent dent depth (\(d\)) in Figure 21 were less than 0.5% of the cylinder radius (\(R\)), and they were not included because these values can be allowed as the initial  
Figure 19: Tendencies of \(d\)/\(R\) with non-dimensional basic parameters \(R\)/\(\frac{\sqrt{L}}{K}\frac{E}{\sigma_{Y}}\) for the ring-stiffened cylinder. Figure 17: Tendencies of \(d\)/\(R\) with non-dimensional basic parameters \(w\)/\(t_{r}\) and \(h_{m}\)/\(t_{w}\) for the ring-stiffened cylinder. Figure 18: Tendencies of \(d\)/\(R\) with non-dimensional basic parameters \(E\)/\(\sigma_{Y}\) and \(\lambda_{E}\) for the ring-stiffened cylinder. imperfections according to the upper limit of tolerable initial imperfection of BSI [39] for ring-stiffened cylinders. The simple formulations were derived to predict the maximum of permanent dent depth of ring-stiffened cylinders as shown in Equation (15), which is corresponded to mean curve with discontinue line. When considering the safety design, the design equation is provided in Equation (16), which is determined by multiplying mean curve with (mean + 2 \(\times\)\(\sigma\)) as continue line in Figure 21. Where \(\sigma\) is the standard deviation of mean equation; \(\delta_{d}\) is non-dimensional dent depth. The proposed formulation in Equation (15) developed in this study for predicting the permanent local denting damage includes some parameters: indenter shape factor (\(C_{s}\)), impact location factor (\(C_{L}\)), and impact angle factor (\(C_{\beta}\)).