## 5 Accuracy of the Proposed Formulation
### Comparison with Other Existing Formulae  
Prior to comparing the newly derived formulation with other formulations, let us re-visit the existing equations in the open literature for predicting the permanent dent depth of stiffened cylinders. The equations by [PERSON] and [PERSON] [13], provided the smeared model based on the equivalent thickness. However, the shortcoming were that the membrane resistances of cylinder shell increased significantly when the thickness and fully plastic moment of the cylinder shell were replaced with the equivalent thickness and equivalent bending moment per length, respectively. The force-displacement relation is then expressed as Equation (31):  
\[P=16m_{0}\frac{t_{eq}}{t}\sqrt{\frac{\pi}{3}\frac{d}{R}\frac{D}{t}} \tag{31}\]  
where  
\(m_{0}\): fully plastic bending moment of the cylinder wall per unit length; \(m_{0}=\frac{c_{0}t^{2}}{4}\),  
\(d\): permanent dent depth at the impact point,  
\(\sigma_{0}\): average flow stress; \(\sigma_{0}=\frac{\alpha\gamma+\sigma_{T}}{2}\), and  
\(t_{\text{eq}}\): equivalent thickness; \(t_{eq}=\sqrt{\frac{4m_{eq}}{\sigma_{0}}}\). Equation (31) was improved by [PERSON] et al. [2]. He followed the assumption of [PERSON] and [PERSON] [13], but only \(m_{0}\) was replaced with \(m_{eq}\). The force-displacement response for the ring-stiffened cylinder is obtained as Equation (32):  
\[P=8m_{eq}\sqrt{\frac{\pi d}{8t}}+N_{0}\sqrt{\frac{\pi td}{2}} \tag{32}\]  
where  
\(m_{\text{eq}}\): equivalent bending moment per length; \(m_{eq}=\frac{M}{t}\),  
\(M\): bending moment; \(l_{r}\): ring-stiffener spacing, and  
\(N_{0}\): plastic axial resistance per unit width; \(N_{0}=\sigma_{0}t\). Unlike the case of ring-stiffeners, the stringer affects both membrane resistance and circumferential bending resistance. Thus, the stringers were smeared to obtain an equivalent shell thickness and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **SS-C-1** & **SS-C-2** & **RS-C-1** & **RS-C-2** & **RS-C-3** & **RS-C-4** \\ \hline Test result, \(d\) (mm) & 36.80 & 43.20 & 45.92 & 34.14 & 35.52 & 47.35 \\ Proposed formulation, \(d\) (mm) & 39.77 & 44.67 & 48.53 & 33.95 & 36.89 & 49.07 \\ Bias (Prop. Formulation/Test), \(X_{m}\) & 1.081 & 1.034 & 1.057 & 0.994 & 1.039 & 1.036 \\ Mean of \(X_{m}\) & & & & 1.040 & & \\ COV (\%) & & & & 2.74 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison of the proposed formulation results with the test results. then the stringers were treated as beams and adding the resistance of each stringer was added to the resistance of the cylinder shell. The force-displacement response for stringer-stiffened cylinder is obtained as Equation (33):  
\[P_{total}=P_{cyl}+\sum_{i=1}^{n_{s}}P_{str}=16n_{0}\frac{t_{eq}}{t}\sqrt{\frac{ \pi}{3}}\frac{d}{R}\frac{D}{t}+\sum_{i=1}^{n_{s}}2P_{0}\frac{w}{h_{sw}} \tag{33}\]  
where  
\(w\): beam deflection; \(h_{sw}\): stringer web height,  
\(n_{s}\): number of stringers in damaged effect zone,  
\(P_{0}\): plastic collapse resistance in bending; \(P_{0}=\frac{8M_{p}}{L}\),  
\(M_{p}\) and \(L\) are plastic bending moment and total length of the stringer-stiffener, respectively;  
\(P_{total}\): resistance of the stringer-stiffened cylinder, and  
\(P_{cyl}\): resistance of the cylinder shell. In the current study, the force-displacement relationship for both ring and stringer-stiffened cylinders can be expressed as the following equations:  
* Ring-stiffened cylinder:  
In the current study, the force-displacement response of the proposed formulation is expressed in Equations (34) and (35) for mean equation and characteristic equation, respectively. It can be obtained by the derivation of Equations (15) and (16) with respect to \(d\).