## 4 Derivation of Empirical Formulation  
After investigating the effects of various parameters on the impact response of stiffened cylinders in the previous section, the series of parametric studies were performed on actual design scantlings  
Figure 14: Displacement-time response of model RS-3.  
Figure 15: Typical deformation progress under collision load of model RS-3 (units: mm).  
of ring- and/or stringer-stiffened cylinders, such as an actual submarine design concept and tension leg platforms in the ABS [37] and API [38] rules. The numerical analysis strategy method developed in the previous section was used. The details of dimension and material properties were provided in Tables 7 and 8 for stringer-stiffened cylinders and ring-stiffened cylinders, respectively. The range of \(R\)/\(t\) for reference ring-stiffened cylinder was determined from 97 to 454, while that of reference stringer-stiffened cylinder was from 111 to 475.  
In collision analysis, the rigid hemisphere indenter was applied for generating the damage on the model. For each model, the series of numerical analysis were performed with different impact velocities from 1.0 m/s, 2.5 m/s, 5.0 m/s, 7.5 m/s, and 10 m/s. These selected velocities have represented the collisions of offshore installations with dropped objects from platform decks or floating objects, which were large enough to achieve the plastic deformation of the dented stiffened cylinder without any fracture occur. The striker masses were 10 tons, 20 tons, 50 tons, and 100 tons with each velocity, respectively. Additionally, the collision of offshore installations with supply vessels is also considered. The impact velocities were 1 m/s, 2 m/s, and 3 m/s. For each velocity, the striker masses were 1000 tons, 3000 tons, 5000 tons, and 7500 tons for the stringer-stiffened cylinder. It should be noted that the added mass was assumed to be 10% to 40% of striker mass. Thus, there were a total of 200 and 256 numerical analysis cases for ring-stiffened and striker-stiffened cylinders, respectively. Subsequently,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **Unit** & **RS-1** & **RS-2** & **RS-3** & **RS-4** & **RS-5** & **RS-6** & **RS-7** & **RS-8** & **RS-9** & **RS-10** \\ \hline \(R\) & mm & 3100 & 3023 & 3175 & 3100 & 2550 & 5150 & 2500 & 3500 & 550 & 3180 \\ \  
these parametric study results were used to derive empirical equations for predicting the local denting damage of stiffened cylinders.  
The procedure to derive empirical formulations for assessment of permanent local denting damage as following steps:  
* _Step 1_: Selecting the scantling of actual design example ring- and/or stringer-stiffened cylinders.
* _Step 2_: Performing finite element analysis to assess the collision response of each model.
* _Step 3_: Preparation of non-dimensional form of geometric properties and collision loadings for applying in the derived formulations.
* _Step 4_: Checking the degree of dependence of explanatory variables.
* _Step 5_: Performing the regression analysis to find the best-fit curve by using the strongly-influenced parameters in the previous step.
* _Step 6_: Checking the accuracy of the proposed formulations.
* _Step 7_: Checking the skewness of \(X_{m}\) (ratio of the numerical results to the proposed formulation results) against the basic parameters.