## 3 Probabilistic Seismic Demand Model and Definition of Limit States
### Definition of Limit States  
After developing the PSDM, intensity measures (IMs) and demand measures (DMs) need to be defined for the PSDM of the gantry crane. Though there was no previous research investigating the fragility curves of the gantry crane, researchers in other fields have made great effort to select the optimal IMs and DMs. The normally used IMs are spectral velocity (S\({}_{v}\)), peak ground acceleration (PGA), spectral acceleration (S\({}_{a}\)) at the fundamental period [38; 39], etc. In this paper, the direct, and convenient IM, PGA, was selected as the IM of the gantry crane for a preliminary analysis. Different from the conventional steel frame, the gantry crane needs to be operated under variable working conditions. To simplify the calculation of the limit states, three major working conditions were considered: fully-loaded trolley located in the midspan of the girder, fully-loaded trolley located in the cantilever beam close to the flexible legs, and fully-loaded trolley located in the cantilever beam close to the rigid legs, respectively. For a more convenient description, the three conditions are called Case 1, Case 2, and Case 3 below. As for DMs, many parameters have been used to create the fragility curves of structures. These are peak inter-story drift [40; 41; 42], the maximum base shear [43], peak roof drift [44], etc. Due to the special structure shape of the gantry crane, all of the DMs mentioned above could not be directly utilized in the study of the gantry crane. Considering that the inter-story drift of the structure showed a better correlation with the level of structural damage, a new DM called the local displacement ratio (LDR) of the gantry crane was proposed for the first time based on the inter-story drift as follows:\[LDR=\left\{ \begin{array}{l}\frac{\Delta x}{h},\\ \frac{\Delta y_{g}}{L_{g}},\\ \frac{\Delta y_{c}}{L_{c}}\end{array} \right. \tag{5}\]  
in which,  
\(\Delta x=\) displacement of the joint of the leg and the girder in the X-direction. \(h=\) height of the leg. \(\Delta y_{g}=\) displacement of the girder in the Y-direction. \(L_{g}=\) length of the girder. \(\Delta y_{c}=\) displacement at the valid cantilever end in the Y-direction. \(L_{c}=\) length of the valid cantilever. According to the operating conditions of the gantry crane, four groups of thresholds should be determined. Besides, the deflection of the trolley girder is a significant parameter to define the state of the gantry crane, which is a judgment parameter to evaluate the working ability of the gantry crane [45]. The very deflection only considers the deformation caused by the payload and ignores that caused by the dead load. If the deflection exceeds the limit value, the gantry crane cannot work even though the structure is still in an elastic state. Given that, a new DM the called deflection ratio (DR) was proposed to evaluate the level of the working condition of the gantry crane. DR is not a parameter related to the seismic capacity of the gantry crane, but an independent parameter defined by the Chinese code for cranes. They are 1/1000, 1/800, and 1/700, respectively. A significant step to develop fragility curves is to define limit states for different structures since their structural features will influence the capacity of the structures. In previous studies, three limit states [29], four limit states [46; 47], and five limit states [48] have been recommended to describe the structural performance levels. In this research, three limit states were adopted for the gantry crane according to [29], and they were serviceability (SA), damage control (DC), and collapse prevention (CP), correspondingly. To date, there are two widely used methods to determine the thresholds for different limit states, i.e., pushover analysis [49; 41] and incremental dynamic analysis [50; 51]. It was proven that the pushover analysis was accurate enough for a structure whose first mode period was less than 2.0 s [52].