## 2 Numerical Analysis
### Finite Element Modelling  
Nonlinear finite element analyses were performed by using the explicit solution of the ABAQUS software. The full geometry of the test model was modelled by applying four-node doubly curved shell elements S4R in the ABAQUS library (version 6.14, ABAQUS Inc., Providence, Rhode Island). The Simpson rule was applied with five points of the thickness integration. For the striking mass, it was assumed as a rigid body with a 3-D bilinear rigid quadrilateral element (R3D4). To control the inertial properties and initial impact velocity of the striker, a reference node was attached at the area centroid of the striker's faces. It is noted that the rotations of the striking mass during the impact were also considered in the numerical model. The optimum mesh size was selected after performing the convergence studies. The mesh size of the contact zone was \(5\times 5\) mm, while that of the out of the contact zone was \(10\times 10\) mm. This mesh size is sufficiently fine for recording the local denting response precisely. For the boundary conditions, the ends of both thick support structures of the model were restrained in all degrees. The support structures were fully clamped at four bottom bolt holes as the clamped boundary conditions in the experiment. The general contact algorithm with a penalty method was applied for the contact between the striking mass and the cylinder shell surface. The contact friction coefficient is designed by 0.3 to account for the slipping between the surfaces of the indenter and cylindrical shell. The full geometry and boundary conditions of each model are provided in the finite element modelling, as shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **RS-C-1** & **RS-C-2** & **RS-C-3** & **RS-C-4** & **SS-C-1** & **SS-C-2** \\ \hline Radius (mm), \(R\) & 400 & 400 & 550 & 550 & 550 & 550 \\ Thickness (mm), \(t\) & 3.96 & 3.95 & 4.97 & 4.94 & 2.98 & 2.97 \\ Total length (mm), \(L\) & 1060 & 1060 & 1060 & 1060 & 1060 & 1060 \\ Ring-stiffener spacing (mm), \(l\) & 200 & 200 & 150 & 150 & 600 & 600 \\ Number of rings, \(n_{r}\) & 6 & 6 & 8 & 8 & 2 & 2 \\ Number of stringers, \(n_{r}\) & - & - & - & - & 20 & 20 \\ Ring-stiffener web height (mm), \(h_{trw}\) & 35.0 & 35.0 & 40.0 & 40.0 & 200 & 200 \\ Ring-stiffener web thickness (mm), \(t_{trw}\) & 3.94 & 4.0 & 5.81 & 5.81 & 4.90 & 4.92 \\ Stringer-stiffener web height, \(h_{sw}\) & - & - & - & - & 65.0 & 65.0 \\ Stringer-stiffener web thickness (mm), \(t_{sw}\) & - & - & - & - & 4.98 & 4.87 \\ Ring-stiffener flange width (mm), \(b_{rf}\) & - & - & - & - & 50.0 & 50.0 \\ Ring-stiffener flange thickness (mm), \(t_{rf}\) & - & - & - & - & 4.89 & 4.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured dimensions of test models. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **RS-C-1** & **RS-C-2** & **RS-C-3** & **RS-C-4** & **SS-C-1** & **SS-C-2** \\ \hline Yield strength (MPa), \(\sigma_{Y}\) & 302.2 & 309 & 274.9 & 274.9 & 335.9 & 335.9 \\ Ultimate strength (MPa), \(\sigma_{T}\) & 396.2 & 397.3 & 370.5 & 370.5 & 417.0 & 417.0 \\ Elasticity modulus (GPa), \(E\) & 206.0 & 200.0 & 202.4 & 202.4 & 210.0 & 210.0 \\ Hardening start strain, \(\epsilon_{HS}\) & 0.0227 & 0.0224 & 0.0145 & 0.0145 & 0.0223 & 0.0223 \\ Ultimate tensile strain, \(\epsilon_{T}\) & 0.196 & 0.194 & 0.234 & 0.234 & 0.1727 & 0.1727 \\ Fracture strain, \(\epsilon_{F}\) & 0.376 & 0.366 & 0.427 & 0.427 & 0.3811 & 0.3811 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of test models. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **RS-C-1** & **RS-C-2** & **RS-C-3** & **RS-C-4** & **SS-C-1** & **SS-C-2** \\ \hline Drop height (mm) & 1600 & 1200 & 1200 & 1600 & 1600 & 1580 \\ Collision velocity (m/s) & 5.602 & 4.852 & 4.852 & 5.602 & 5.600 & 5.570 \\ Striker mass (kg) & 295 & 295 & 500 & 500 & 500 & 600 \\ Kinetic energy (J) & 4629 & 3472 & 5249 & 7846 & 7848 & 9300 \\ \hline \hline \end{tabular}
\end{table}