## 3 Dynamic Analysis of Container Crane Coupled with Uplift and Derailment  
### Nonlinear Time-History Analyses  
In this study, a direct-integration nonlinear time-history analysis was used in which the Hilber-Hughes-Taylor (HHT) method was applied to improve the numerical dissipation of the time integration algorithm. The parameters of the HHT method were \(\gamma\) = 0.5, \(\beta\) = 0.25, and \(\alpha\) = 0 (assuming that the formulation will be identical to the average acceleration method) [31,32]. The 3D finite element model of the Korean container crane is composed of 9916 elements, whereas four friction isolator elements are used to simulate the behavior between wharf and crane's toes. Most of the structural components of the crane were made of stiffened hollow box sections. The properties of materials comply with the Japanese Industrial Standards (JIS) including JIS-SM490Y (for hollow box sections and wide-flange shapes) and JIS-STK490 (for steel tubes of diagonal braces), as summarized in Table 4.  
To develop the fragility curves, a total of 20 recorded and artificial horizontal ground motions matched for the design spectrum as mentioned in Section 2.2 were used under 23 _IM_ levels ranging from 0.17 to 0.75\(\times\)_g_. Thus, 460 ground motions were applied as input data for the seismic demand. For container cranes, the impact of the vertical ground motion on the response of the crane was approximately 0.1% of the portal drift for the level of excitation, \(S_{a}\) (0.3 s) \(<\) 0.5\(\times\)_g_[1]. The effect of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties of Materials** & **Unit** & **Value** \\ \hline Yield strength of steel for JIS-SM490Y & & \\ Thickness & & \\ t \(\leq\) 16 mm & MPa & 365 \\
16 mm \(<\) t \(\leq\) 40 mm & MPa & 355 \\ t \(>\) 40 mm & MPa & 335 \\ Yield strength of steel for JIS-STK490 & MPa & 313 \\ Young’s modulus of both steel types & GPa & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Properties of materials. Figure 5: Pushover curves for the 50 random cases generated by Latin Hypercube sampling. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Limit States** & **Minor** & **Moderate** & **Extensive** \\ \hline Portal drift (\%) & 1.61 & 1.82 & 2.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Limit states in terms of portal drift. the vertical ground motion should be analyzed on an entire complex soil-structure interaction and quay wall system [24]. In this study, a simpler model was used to reduce the cost and time required. Thus, the vertical ground motions were neglected. The _IM_ of this research is the spectral acceleration at the fundamental natural period of the container crane, \(S_{a}\) (\(T=1.68\) s). The fundamental period of this study (applying a friction isolator element at the base) was found to be longer than those obtained from our previous studies using the pin-support and gap element [24; 29]. Zero-length link elements of SAP2000 (or a friction isolator element as above-mentioned) were assigned to the model's toes to simulate the contact friction, enabling both uplift and derailment, because the behavior of these elements was similar to the real interaction between the crane wheel and the rail during the operation of the crane. The uplift and derailment behaviors changed the horizontal displacement of the entire structure; thus, the corresponding fragility curves were different from those obtained from other boundary conditions such as pin or gap elements [29; 33]. The friction isolator element exhibited a vertical behavior similar to that of the gap element (see [24]); However, the gap element did not remove itself from the horizontal direction during uplift. The friction isolator element could solve the horizontal force capacity throughout the compressive vertical force and frictional coefficient (\(\mu\)). Here, the frictional coefficient (\(\mu\) is assumed to be 0.75) ignored the difference between static and kinetic friction.