## 4 Dynamic Time-History Analysis of Wharf Structures Subjected to Impact Loads
### Analysis of Structural Dynamic Response under Impact Loading  
The ship impact load is applied at the midpoint of the rubber fender in the structure of the high-piled wharf. To exemplify the dynamic properties of the high-piled wharf structure under the impact load of the ship, the maximum displacement of the structure is calculated for a dynamic load of 0.2 m/s from a 20,000 t ship and the stable displacement of the structure is calculated under the equivalent level of a static load. A comparison of the displacement characteristics between these two scenarios is conducted to ascertain the dynamic amplification effect of dynamic loads on the high-piled wharf structure. Using the finite difference method, the structural displacement under a 409.2 kN static load was computed and is presented in Table 5. This table displays the displacement values of three nodes on the horizontal bent structure of the high-piled wharf, signifying the displacement at the top of the inclined pile, the top of the straight #5 pile, and the inclined #6 pile at the mud surface. Employing the nonlinear dynamic response analysis method, the structural displacement characteristics under dynamic loads were calculated. Nodes were selected based on achieving equivalent output displacement values as observed under static loads. The results, including the displacement duration curve and the maximum displacement, are illustrated in Figure 9 and detailed in Table 5. Observations from Figure 9 reveal that the displacement trends at these three nodes are approximately similar over time, consistently increasing from time zero and peaking at around 0.75 s, followed by varied degrees of reduction. The top displacement of the fifth straight pile is marginally greater than that of the inclined pile, while the displacement at the mud surface of the inclined pile is less than that at the pile's top. These variations suggest that soil constraint significantly influences pile displacement. Post ship impact (at ls), the structure retains a certain level of displacement due to damping effects, inducing free decay vibrations until returning to the initial state. As this study mainly focuses on attaining the maximum response under dynamic loads, subsequent displacement responses were not explicitly monitored. The ratio of displacement under dynamic load to displacement under an equivalent static load is defined as the dynamic amplification factor. Based on the data in Table 5, it can be inferred that, for the transverse bent structure of the high-piled wharf in this example, the dynamic amplification coefficient is approximately 1.5 when subjected to a half-wave sinusoidal load with a duration of 1.0 s. This result confirms that the prevalent use of static methods for applying ship impact loads, without adequately considering the influence of  
Figure 8: The distribution of maximum principal stress in soil under a normal operational load. dynamic amplification effects, is inherently limited. In the context of structural design for high-piled whaves or theoretical and simulation research related to ship collisions with high-piled whaves, ship impact loads should be determined with due consideration of dynamic amplification effects. Figure 10 displays the envelope diagram depicting the peak bending moment of each pile body under the action of the service load alone and when subjected to the superimposed ship impact load in Scenarios 1, 2, and 3. Broadly, under the ship collision load, noticeable bending moments are observed primarily on straight #1 pile and the inclined #6 pile. The highest bending moments are concentrated at the tops of the piles, followed by a segment just below the mud surface and within the load-bearing layer of the pile body. A sudden change in bending moment occurs at the mud surface. A comparison between the bending moments of each pile solely under the service load and those under the combined ship impact load reveals the considerable contribution of ship impact load to the overall bending moment of the pile body.