## 2 Analytical Fragility
### Damage States for Container Crane  
For the randomness in a structural capacity, the uncertainty of the overall stiffness of the structure may mainly originate from the structure's geometric dimensions and material properties. In this study, with the assumption that the steel plates were perfectly manufactured, the elastic modulus was the only parameter to be considered as a normal distribution random variable with a mean value of 200 GPa and coefficient of variation (COV) of 0.06 [24]. A nonlinear static analysis was then performed to discover the capacity behind the elastic states. The limit states obtained by a pin-based pushover can be applied for comparison with the response obtained from a dynamic analysis because the capacity of the structure is load-independent [1]. The force-displacement curve was obtained by gradually increasing the lateral force assigned at the boom level of the crane. This approach was conducted successfully in the authors' previous study [29]; However, all input parameters were used in the previous study as best estimates (mean/median values), without considering the uncertainty of the input values to the response of the structure. To assess the uncertainty in the input value of the structural capacity, 50 input cases of the elastic modulus values were generated by Latin Hypercube Sampling (LHS). This was accomplished to reduce the computational effort in comparison with Monte Carlo Simulation (MCS) sampling because of the stratified technique for the input probability distributions of the LHS method [30]. Correspondingly, 50 different pushover analyses were conducted on finite element models with uncertain variables of the elastic modulus to identify the limit states of the damage based on predefined plastic hinges of ASCE/SEI 41 [20]. For the FE model, the structural components were modeled with one-dimensional beam-column finite elements. Thus, the concentrated (or lumped) plasticity approach was applied for the nonlinear material response. In this approach, the constitutive laws are expressed in terms of moment-rotation (M-\(\theta\)) relationships of the cross section of interest. The M-\(\theta\) relationship of each section can be defined assuming a backbone as indicated in ASCE/SEI 41. The lumped plasticity approach is more robust and reduces the computational cost. However, it restricts the inelastic deformations to prespecified regions of small length. In this study, the three predefined performance-based damage levels of ASCE/SEI 41, i.e., immediate occupancy (IO), life safety (LS), and collapse prevention (CP), are assumed as minor, moderate, and extensive damage states, respectively, in terms of the portal drift of the container crane. This is depicted in Figure 5, while Table 3 summarizes the corresponding average values. Table 3 is useful for the preliminary design of a 30.5-m-gauge container crane. For example, if an elastic state is expected for designing a 30.5-m-gauge crane with structural similarity, its portal drift, defined as the drift angle between the leg base and the portal beam (see Figure 1), should be less than the minor state listed in Table 3. The values listed in Table 3 are crucial for developing the fragility curves.