## 3 Dynamic Analysis of Container Crane Coupled with Uplift and Derailment
### Nonlinear Time-History Analyses
For the friction isolator element, both the horizontal and vertical reactions of the legs will reach zero if the legs have uplift and derailment events [33]. In this study, the stiff spring constant (\(k_{\text{k}}\)) of the friction isolator element is assumed to be 43,782 kN/cm [24]. Based on both physical test and analytical model from previous studies, the seismic response of the container crane over time can be divided into three stages [1; 6; 34], as depicted in Figure 6: (a) the structure is shaken owing to low seismic load (so-called swaying). (b) With a higher seismic load, the landslide legs are uplifted and derailed (lateral translation) when the total gravity load is transferred to the waterside legs owing to the considerable reduction in the axial load on the landslide legs. As a result, when the load is added to the waterside legs due to the uplift of the landslide legs [1], plastic hinges can occur on the waterside legs. Finally, (c) the landslide legs land inside the rail, resulting in a residual inward displacement of each leg. In this stage, plastic hinges may also develop on the landslide legs. The first occurrence of a plastic hinge can be at either the waterside or landslide legs, which can depend on the design strength of the portal frame, ground motion profile and intensity, direction of the earthquake, or friction between the ground/rails and the crane wheels. The torsional flexibility makes a negligible contribution to the portal flexibility even with significant seismic excitation. This has been verified by many studies using push-twist analysis for the A-frame container crane [1; 29]. Therefore, in this study, the torsional effect was ignored when evaluating the seismic performance of the portal crane response. Figure 7 illustrates an example of the seismic response of a crane subjected to artificial ground motion (AGM8) with Peak Ground Acceleration(_PGA_) of 0.51 g (equivalent to a scale factor of 3.0 in comparison with the AGM8 depicted in Figure 4b), considering both the uplift and derailment behaviors. There are four occurrences of uplift at the landslide leg. The first uplift occurred at 3.24 s,  
Figure 6: Failure modes of container crane under seismic loading: (a) swaying; (b) uplifted legs with/without derailment; and (c) permanent derailment and remaining deformation. reaching its maximum value of 0.04 cm at 3.3 s. Concurrently, it was derailed by as much as 10.4 cm inward at the rail. The second, third, and fourth occurrences of the maximum uplifts were 0.5, 0.1, and 0.9 cm at 4.93, 6.94, and 8.64 s, corresponding to the total permanent derailments of 34.5, 39.2, and 52.0 cm. In this study, the uplift behavior is related to the redistribution of the load. The uplift is defined by two conditions, as mentioned in the study proposed by [PERSON] et al. [29], in which both vertical displacement and the reaction of the uplifted leg are equal to or larger than zero. In particular, Figure 7b illustrates the four occurrences of uplifting, corresponding to four occurrences of vertical reaction reduced to zero in Figure 7c. Both waterside legs and one landslide leg of the crane in this example drift over the limit state of collapse prevention, indicating a potential overturning collapse toward the sea. The analyses of this study ignore the lateral restraint provided by a small flange on each wheel. Thus, the derailment is controlled by the weight of the crane, and the friction between the crane's toes and the ground/rail. Figure 7: Example of uplift and derailment behaviors of a container crane subjected to artificial ground motion (AGM)2: (**a**) ground motion with _PGA_ of 0.51\(\times\)_g_; (**b**) uplift of landslide legs; (**c**) vertical reaction of landslide legs at time of uplifting; and (**d**) derailment of landslide legs.