# 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
When using this FC element for simulating the boundary of the container crane, both the horizontal and vertical reaction of legs will reach zero if  
Figure 2: Boundary condition in SAP2000: (**a**) FC element, (**b**) GAP element. the legs have uplift and derailment events. In addition, the horizontal and vertical movement of legs could be captured in uplift and derailment events. These special behaviors of the link could be modelled by commercial software, such as ABAQUS, OpenSees, ANSYS, and SAP2000, and are simulated by an elastomeric bearing as a two-node discrete element with stiffness in each of the six principal directions (three translations and three rotations), represented by linear or nonlinear springs between two nodes. Analytical expressions for force and stiffness can be used to define a spring in any direction. In SAP2000, the friction isolator, as shown in Figure 2a, allows compression-only property in the axial direction, similar to a GAP element as mentioned above, and has coupled frictional behavior in the two horizontal shear directions. The model using the base shearing behavior was introduced by [PERSON] et al. [16], and was then studied for base-isolated structures by [PERSON] et al. [17]. This element could also be used to model gap and friction behavior between contacting surfaces by taking into account efficient numbers for the stiff spring constant and damping coefficient, so the uplift and derailment events of wheels could be obtained in the simulated model. This FC boundary element was used by [PERSON] [18] in the OpenSees platform, and also compared to an experimental model. As a result, the author recommended using the FC boundary element to capture the critical uplift and derailment responses in a realistic but simplified manner. [PERSON] et al. [19] used this type of element to simulate the interaction of crane-wharf in nonlinear time history analysis by SAP2000. They recommended the stiffness for sliding and uplift to be eight times the gravity reaction, and assumed the coefficient of friction to be 0.8, and the damping 5%. In our study, the properties of friction isolators in SAP2000 were chosen to exactly capture the uplift and derailment events of the container crane as follows: the stiff spring constant (\(k_{k}\)) is assumed to be 43,781 kN/cm (25,000 kips/in), and the damping coefficient (\(C_{k}\)) is based on the coefficient of restitution (e) by an equation of loss energy during impact, as proposed by [PERSON] [13] and shown in Equations (3) and (4). These parameters have also been shown to not be very sensitive to changes of magnitude for simulating collisions [14; 15]. The coefficient of friction is suggested to be 0.6 to 0.9 by experimental model in the research of [PERSON] and [PERSON]. If the parameter is large enough, the simulated model of the leg base acts as essentially pinned until it uplifts; if a suitable parameter is used to capture the uplift and derailment event, at that point the simulated support behaves freely. In this study, the coefficient of friction was assigned to be 0.75. The damping coefficient is calculated by the following equation:  
\[\zeta=-\frac{ln\;e}{\sqrt{\pi^{2}+(lne)^{2}}} \tag{3}\]  
\[C_{k}=2\zeta\sqrt{k_{k}m} \tag{4}\]  
where \(m\) is the structural mass contributing to dynamic vibration of the stiff spring, it was considered for each leg of the crane (\(m=3470\) kN); \(\zeta\) is the damping ratio; and \(e\) is the coefficient of restitution, which contributes as a dashpot constant of impact elements to determine the amount of energy dissipated during impact. If two masses collide each other with arbitrary velocities, it is easy to express the dashpot constant of the impact element in terms of the \(e\) by equating the energy losses during impact. The elastic impact (\(e=1.0\)) corresponds to a damping ratio \(\zeta=0\), and the completely plastic impact (\(e=0\)) corresponds to \(\zeta=1.0\). In this study, \(e=0.6\) was assumed, as recommended by [PERSON] [13], corresponding to a damping ratio of \(\zeta=0.16\).