# IMPA versus Cloud Analysis and IDA: Different Methods to Evaluate Structural Seismic Fragility
## 3 Numerical Application
### FE Model Description  
The nonlinear FE model of the analysed frame was developed in the OpenSEES platform [60]. To account for the nonlinearity, \"Beam With Hinges Element\", already available in the OpenSEES library, was used to model columns and beams. This element adopts a lumped plasticity formulation with plastic hinges at the end of the element connected by an elastic link. This means that all nonlinearities are concentrated at the ends of the elements and can be only propagated along the length \(L_{p}\) of the plastic hinge, unlike the distributed plasticity formulation where they may spread along the whole element. Therefore, the length of the plastic hinge \(L_{p}\) plays an important role in avoiding the concentration of strain at the element ends. In this work, in good agreement with the \(L_{p}\) evaluated by the equation proposed by [PERSON] and [PERSON] [61], it is assumed to be equal to the cross-section height. The two-point Gauss integration was used on the element interior, while two-point Gauss-Radau integration was applied over lengths of 4\(L_{p}\) at the element ends. A total of six integration points were used [62; 63]. To account for non-linearity, a four-point moment-curvature relationship was assigned to the element  
Figure 5: (a) Plan; (b) transverse section; (c) elevation; (d) modelled frame. The structural joint originally designed for thermal deformations are highlighted by orange hidden lines.  
Figure 6: Columns (C) and beams (B) cross-sections and longitudinal reinforcement details.  
ends. The four-point law accounting for crack, yield, ultimate failure and collapse state (80% of ultimate failure) was evaluated through the software Response2000 [64], which can simulate nonlinear sectional behaviour by assuming a suitable law for the material. Due to the dependence of the sectional response on the applied axial load, it is assumed to be zero for beams, while it is estimated for the column considering their area of influence. Shear failure was not considered in the model. The Newton line search method was considered as the solution algorithm for the time-history analyses, which increased the effectiveness of the Newton-Raphson algorithm by introducing line search to solve the nonlinear residual equation. The tolerance and maximum number of iterations used were OpenSees default values [62]. Newmark integrator has been used and the convergence test was the normal displacement increment. The tolerance of the test is equal to \(10^{-7}\) and the number of maximum iterations is 50. Rayleigh damping is adopted to account for energy dissipation.  
In IDA, the scale factor \(\lambda\) was chosen to scale the spectral acceleration at the fundamental period \(S_{a}(T_{1}\zeta=5\%)\), which was scaled to \(IM=a_{\lambda}\in\)[0.1 g, 0.8 g] with \(\Delta a_{\lambda}=0.1\) g. Similarly, in IMPA, the response spectra were scaled multiple times to obtain scaled spectral acceleration at the fundamental period \(S_{a}(T_{1}\zeta=5\%)\) equal to \(IM\)= \(S_{a\lambda}\in\)[0.1 g, 0.8 g] with \(\Delta S_{a\lambda}=0.1\) g. The mass damping coefficient and the stiffness damping coefficient of the Rayleigh damping are evaluated by considering the first and the second natural frequency of the case study. The percentage of critical damping is equal to 5%.