# A Novel Deep Learning Model to Predict Ultimate Strength of Ship Plates under Compression
## 4 Prediction Result Using Deep Learning Model
### Comparison with the Empirical Formulae  
Equation (2) is an empirical formula proposed by [PERSON] et al. [19] in previous studies that can predict the ultimate strength of a curved plate under transverse compressive load. In Equation (2), \(\sigma_{U\_TC}\) is ultimate strength of the curved plate under transverse compressive load and \(\sigma_{Y}\) is yield stress of material. In Equation (3), coefficient \(C_{F}\) was assumed as a function of the double slenderness ratio, and the correction factors in Equation (4) represented the effect of curvature with the change in the thickness of the plate. \(R\) in Equation (4) is the radius of curvature.  
\[\frac{\sigma_{U\_TC}}{\sigma_{Y}}=\left(\frac{2.25}{\beta}-\frac{1.25}{\beta^ {2}}\right)\times C_{F} \tag{2}\]  
\[C_{F}=\frac{C_{a}}{\beta^{2}}+\frac{C_{b}}{\beta}+C_{c} \tag{3}\]  
\[C_{a}=2.596\left(\frac{R}{t}\right)^{2}-1.712\left(\frac{R}{t}\right)+0.415\]  
\[C_{b}=-2.095\left(\frac{R}{t}\right)^{2}+0.929\left(\frac{R}{t}\right)-0.136 \tag{4}\]  
\[C_{c}=1.009\left(\frac{R}{t}\right)^{2}-0.724\left(\frac{R}{t}\right)+0.322\]  
\[\sigma_{E}=K_{tg}\frac{\pi^{2}E}{12(1-v^{2})}\left(\frac{t}{a}\right)^{2} \tag{5}\]  
where  
\[K_{tg}=\begin{cases}\frac{a}{\sqrt{R}t}+3\frac{(Rt)^{0.175}}{a^{0.325}}&if \ \frac{a}{R}\leq 1.63\sqrt{\frac{R}{t}}\\ 0.3\frac{2^{2}}{R^{2}}+2.25\left(\frac{R^{2}}{at}\right)^{2}&if\ \frac{a}{R}>1.63\sqrt{\frac{R}{t}} \end{cases} \tag{6}\]  
Figure 10: Coefficient of determination of deep learning model (all test data).  
Equation (5) was proposed by DNVGL [20] as an empirical equation used to compare the results using the deep learning model in this paper. In the equation, \(\sigma_{E}\) is the elastic buckling stress, and \(a\) represents the length of the curved plate. The elastic modulus and Poisson ratio used in the equation were substituted with the values shown in Table 1, and the values obtained through Equation (6) were used for Buckling factor \(K_{tg}\).  
\[\sigma_{cr}=\begin{cases}\sigma_{E}&for\ \sigma_{E}\leq 0.5\sigma_{Y}\\ \sigma_{Y}\Big{[}1-\frac{\sigma_{Y}}{\omega_{E}}\Big{]}&for\ \sigma_{E}>0.5 \sigma_{Y}\end{cases} \tag{7}\]  
In many cases, elastic buckling stress needs to be corrected because the value calculated by the above method is higher than the yield stress. Therefore, critical buckling stress was calculated through the Johnson-Ostenfeld method of Equation (7), and proportional limit was assumed to be a commonly used \(0.5\sigma_{Y}\)[21].  
In this study, the presented empirical equations and values predicted using the DNN model were compared. Comparisons were made between the results of this study and those of the studies by [PERSON] et al. [19] and DNVGL [20]. Table 4 shows the error between these prediction results and FEM values. Because the geometric ranges of the curved plate applicable to each empirical equation is different, the conditions of the curved plates were selected and compared by referring to studies that proposed the empirical equation.  
The empirical formulae presented by [PERSON] et al. [19] and DNVGL [20] showed average error rates of 12.33% and 33.19%, respectively, and both showed a larger error rate as the aspect ratio, curvature, and initial deflection increased. In contrast, in the case of ultimate strength prediction using deep learning, the average error rate was approximately 1.26%, indicating a higher accuracy than the empirical formulae.