## 2 Prototype and Finite Element Model
### Finite Element Models  
The OpenSees software is an object-oriented, earthquake engineering software framework developed to offer researchers an analysis tool to analyze advanced structural and geotechnical systems [31; 32]. Because of the versatile material and element libraries and the ability to control the solution procedures and algorithms [31; 33], OpenSees was selected as the computational platform for the study. Based on the above information, the finite element (FE) model of the gantry crane was established in the OpenSees software, which is used to analyze advanced structural and geotechnical systems [31; 32]. The schematic 3D model of the gantry crane is shown in Figure 4. The blue points in this model play an important role in the definition of the variable section parameters of beams and columns as reference points. Since the section parameters of the columns varied with length, the gap between two reference points on columns was relatively small to construct the columns more accurately. The black lines between two points are the nonlinear columns and beams used to simulate the structure of the gantry crane. Moreover, a centerline approximation was applied, and the effect of the panel zone was neglected since the extra flexibility introduced to the system due to centerline dimensions counteracted the influence of the omission of the panel zone [11]. Besides, the second-order (P\(-\Delta\)) effect was considered in this case to take geometric non-linearity into account, the influence of which was too remarkable to ignore since the nonlinear deformation or permanent deformation of the structure would increase dramatically [34]. Furthermore, the elastic modulus, mass density, the yield stress of Q235 steel, and the damping ratio were respectively equal to \(2.01\times 10^{11}\) Pa, 7800 kg/m\({}^{3}\), 235 MPa, and 3%. It is worth noting that the trolley-travel direction, vertical direction, and the gantry-travel direction was correspondingly described as the X, Y, and Z coordinates, respectively (see Figure 4).  
In addition, the junctions where trolley beams and legs were connected were modeled as rotational springs to simulate plastic hinges, which was an efficient way to define the nonlinear behavior of the joints. A schematic of the spring model and constitutive model of the spring i shown in Figures 5 and 6. In previous studies [10], the joints between columns and rows were simulated with plastic hinges as well, but the rows were considered elastic given the relatively small span of rows and less computational effort. However, this strategy could not be applied in this research since the span of the girder (35 m) was too long to ignore its nonlinear behavior during an earthquake. Hence, all the columns and rows were considered as nonlinear elements, which were achieved by the \(dispBeamColumn\) element along the length of the element and the \(Aggregator\) section in OpenSees.  
According to FEMA356 [35], the moment versus rotation curve model is established in Figure 6. Linear response is shown between Point A and Yield Point B. The slope from B to C of 1% was chosen in this research as an appropriate percentage of the plastic slope. C represented the beginning of a considerable strength degradation. Beyond Point D, the component responded with significantly reduced strength to Point E. Finally, the strength was basically zero beyond Point E. [35]. Table 1 shows the material and section properties of the gantry crane's components. They were used to determine the values of \(a,b\), and \(c\) as characteristic parameters. The section properties were variable for the flexible and rigid legs, which were chosen near the joints connected with girders. Based on FEMA356 and Table 1, \(a=4\beta_{y},b=6\beta_{y}\), and \(c=0.2M_{y}\), respectively.  
Figure 3: Schematic of the sample gantry crane.