## Structural Responses of the Deck and Breakwater
### Structural Response Analysis
#### 5.3.3 Comparison of Linear and Nonlinear Analysis Methods  
The finite element analysis method used in this paper was a nonlinear dynamic analysis; however, there is another commonly used finite element analysis method, namely, linear dynamic analysis ([PERSON] et al. [63]). Linear analysis has a broader range of applications and can solve various dynamic-response problems. Additionally, it has a simpler calculation process and faster computational speed. However, linear analysis involves a series of assumptions, such as assuming that materials remain in an elastic state, stress-strain relationships are always proportional, loads are proportional to structural responses, and structures undergo only small displacements and deformations. Nonlinear dynamic analysis, on the other hand, can consider factors such as large deformations, large displacements, and material nonlinearity under load, resulting in the capability to simulate more complex structural responses, more accurately reflect real-world conditions, and provide more reliable design results. To compare the differences between linear and nonlinear \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Part**} & \multicolumn{2}{c}{**Von Mises Stress (MPa)**} \\ \cline{2-3}  & **Equivalent Static Analysis Method** & **Dynamic Analysis Method** \\ \hline D111 of the deck & 232.9 & 95.7 \\ B101 of the stiffener & 735.0 & 315.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of structural response between equivalent static method and dynamic analysis method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Part**} & **Von Mises** & **Allowable** & \multirow{2}{*}{**\(\lambda y\)**} & **Safety** \\  & **Stress (MPa)** & & & **Results** \\ \hline Deck & 95.7 & 244.5 & 39.1\% & qualified \\ Deck stiffener & 155.4 & 244.5 & 63.6\% & qualified \\ Breakwater & 264.3 & 255.4 & 103.5\% & unqualified \\ Breakwater & 293.0 & 255.4 & 114.7\% & unqualified \\ stiffener & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Safety evaluation results of the ship’s bow. dynamic analysis, this paper compared the structural response results obtained by the two methods. Figure 22 presents a comparison of the stress response and total displacement results obtained at deck location D111 and breakwater stiffener location B101 using nonlinear dynamic response analysis and linear dynamic analysis methods. Under the same mesh and time step, for places where the stress response does not exceed the material yield limit, such as the deck, the results of nonlinear dynamic analysis method and linear dynamic analysis method were basically consistent. Figure 22a,b show the analytical results at deck D111. However, for places where the stress response exceeds the material yield limit, such as the breakwater stiffener, there was a significant difference between the results of the nonlinear dynamic analysis method and the linear dynamic analysis method. As shown in Figure 22c,d, in the nonlinear dynamic analysis method, when the impact load causes the failure of the breakwater stiffener and enters the plastic stage, the structure undergoes significant displacement, and the rate of increase in the stress response of the breakwater stiffener begins to slow down. After the impact load decreases, the stress response of the breakwater stiffener does not immediately decrease but decreases slowly. However, the results of the linear dynamic analysis method show a significant stress response and small displacement. This indicates that the nonlinear dynamic analysis method can accurately simulate plastic behavior or instantaneous damage that cannot be described by the linear dynamic analysis method. Therefore, in the assessment of the slamming strength of the ship hull, if the slamming load is small and does not cause the structure of the ship to exceed its yield limit, the linear dynamic analysis method can be used to simplify the calculation process.