## Structural Responses of the Deck and Breakwater
### Structural Response Analysis  
#### 5.3.1 Structure Responses of the Deck and Breakwater  
This paper focuses on the structural strength assessment under slamming loads, and the evaluation process mainly involves verifying the yield strength of the structure. For plate elements, the Mises stress \(\sigma_{vm}\) is:  
\[\sigma_{vm}=\sqrt{\sigma_{x}^{2}-\sigma_{x}\sigma_{y}+\sigma_{y}^{2}+3\tau_{xy }^{2}} \tag{17}\]  
where \(\sigma_{x}\) and \(\sigma_{y}\) are the normal stresses in the \(x\) and \(y\) directions of the element, and \(\tau_{xy}\) is the shear stress of the element. The yield factor \(\lambda_{y}\) is used to determine whether there is a yielding phenomenon in the ship's structure, and the specific judgment formula is as follows [29]:  
\[\text{Deck}:\ \lambda_{y}=\frac{k\sigma_{vm}}{225} \tag{18}\]  
\[\text{Breakwater}:\ \lambda_{y}=\frac{k\sigma_{axial}}{235} \tag{19}\]  
where \(k\) is the material coefficient, which for the high tensile steel used in this model is 0.92 [62]. Using the dynamic analysis method, structural responses of the deck and breakwater structures were obtained. Figure 18 shows the contour plots of total displacement at different moments of the ship bow. During 14.0 to 15.5 s, the green water gradually flows from the deck front to the breakwater, causing the increase of the displacement. The area of the deck affected by green-water loads becomes larger over time. At 15.5 s, most of the  
Figure 17: Sketch of external load in finite element model of ship bow: (**a**) t = 14.5 s, (**b**) t = 15 s, (**c**) t = 15.7 s.  
deck is subjected to the impact of the green-water loads, but the water has not yet struck the breakwater. At 15.7 s, the impact load is applied to the breakwater, which results in a significant displacement of the breakwater. The maximum displacement occurs above the centerline near B105, and the value of the maximum displacement reaches as high as 1 m. This indicates that the area is likely to have been damaged or even fractured due to the impact. Over time, the green water gradually flows out of the bow, and the deck displacement caused by the impact of green-water loads gradually decreases to zero. It can be preliminarily inferred that the deck remains undamaged during the green water event. However, residual displacement of different degrees still exists in the breakwater area, indicating that the breakwater is likely to be damaged under the green-water loads. Figure 19 shows the contour plots of the Von Mises stress at different moments of the ship bow. To facilitate the visualization of the stress distribution beneath the deck, half of the deck plate was hidden in the Mises stress contour plot. Similar to the displacement contour plots, during 14.0 to 15.5 s, the green water event happens as time increases, resulting in larger impact forces and affected deck areas over time. At 15.7 s, the green water impacts the breakwater with a sudden and enormous force. Further investigation reveals that the point of maximum stress response occurs at the stiffeners of the breakwater B101. The stress response value exceeds the material's yield limit, suggesting that this area may have been damaged. Subsequently, the impact force gradually decreases, but residual stresses still exist in the breakwater area. It is noteworthy that the stress response of the deck near the breakwater is greater than that of the forward deck, and the stress response near the longitudinal centerline of the deck and breakwater is greater than that on both sides. Therefore, particular attention should be paid to the above regions when evaluating the safety of the ship's bow. Based on the previous description, the stress response increases as the deck approaches the position of the breakwater. Figure 20 presents the time histories of the stress response and total displacement at locations D108, D110, and D111 on the deck (the positions of D108, D110, and D111 are shown in Figure 8).