# Effects of Boundary Condition Models on the Seismic Responses of a Container Crane
## 2 Description of a Numerical Model of the Container Crane
### Modeling of Boundary Conditions  
As mentioned above, the dynamic response of container cranes with uplift and derailment events is a complicated event. In this study, three simulated models of the base boundary conditions are investigated by SAP2000 to evaluate their effects on the response of the container crane. First, the normal boundary condition idealization is to use a simple pin (PIN), so that the container crane's legs are in bonded contact with the ground (rail). In this case, neither uplift or derailment event is permitted to develop, so it is appropriate for dynamic analyses, in which the level of load does not create an uplift and derailment event for the container crane. In fact, there are cases of analysis when the uplift response is not essential, i.e., for static analysis (gravity, service, etc.) where the uplift event rarely happens, for dynamic analysis of cranes located in regions of low seismicity, and for cranes that tie down in harbor because of large wind loads, and where therefore, PIN is sufficient. When a  
Figure 1: 3D numerical model of the Korean container crane. PIN support is applied, a simple check should be done to ensure that all vertical reactions of supports throughout the analysis remain compressive. Second, the elastic-no-tension (ENT) element simulated by a link between the structure and the ground was recommended by some previous studies, such as [PERSON] et al. [11; 12], who when analyzing a 3D model of a container crane by the time history method, and used a link element in SAP2000 for simulating a connection between container crane and crane rail. By using this element, the compressive vertical reactions of legs may be developed under general load. However, when the dynamic load makes a vertical reaction to reduce to zero, no tensile reaction developed. This means that the ENT element allows the legs to move freely in a vertical direction (allowing an uplift event). The weakness of this link is that it does not allow a horizontal movement of legs during uplift events. Therefore, the derailment event is impossible to develop as a result of the leg base being continuously fixed in a horizontal direction despite uplifting, so the leg base moves perfectly in a vertical direction. Consequently, during an uplift event the horizontal reaction is developed, while the vertical reaction reduces to zero. This means that the reaction will redistribute the internal forces and the deformation of the structure, so that the uplifted leg can carry some portion of base shear. In this study, the elastic-no-tension element is defined by the GAP element in SAP2000. Equation (1) gives the nonlinear force and deformation relationship of the support:  
\[F= \begin{cases}k(d+open)&if\ d+open<0\\ 0&otherwise\end{cases}  \tag{1}\]  
where \(k\) is the spring constant, and \"open\" is an initial gap opening that must be zero or positive (open = 0 for the link of legs of container crane and ground). All internal deformations are independent, which means that the opening of a gap for one deformation does not affect the behavior of the other deformations. Equation (1) shows that this approach is relatively straightforward, so the energy loss during a collision cannot be modeled. In this study, the stiff spring constant (\(k\)) was recommended to be 43,781 kN/cm (25,000 kips/in) [13], which was also shown to be insensitive to changes of magnitudes [14; 15]. Figure 2b shows the model of GAP. Third, the frictional contact (FC) boundary element relates the horizontal force capacity to the compressive vertical force, as can be seen in the Mohr-Coulomb Equation (2):  
\[F_{H,max}=\mu.F_{v} \tag{2}\]  
In the above equation, the largest horizontal force that can resist a slip of legs is equal to the frictional coefficient (\(\mu\)) multiplied by the vertical reaction. Here, the frictional coefficient ignores the difference between static and kinetic friction.