# Quantification of Energy-Related Parameters for Near-Fault Pulse-Like Seismic Ground Motions
## 2 Nonlinear Modeling of the Inelastic Behavior  
The reliable modeling of the inelastic response of masonry, steel or concrete buildings is a challenging task. Although the use of refined finite element models together with the implementation of sophisticated constitutive laws can simulate accurately the structural response, their effectiveness depends on a wide set of parameters, thereby jeopardizing the attempt of deriving general conclusions and practical closed-form results. On the other hand, the use of an idealized single-degree-of-freedom (SDOF) system together with a simplified, yet realistic, piece-wise linear force-deformation or moment-curvature relationship can adequately capture the main features of the global behavior and facilitate the derivation of effective design tools. On this basis, SDOF systems characterized by the constitutive laws listed in Table 1 are considered. Two non-degrading nonlinear models are taken into account, namely the elastic-perfectly plastic model (EPP) and the elastoplastic model with hardening (HYST1). A variety of hysteretic models with degradation are also considered. The nonlinear models with stiffness and strength degradation that account for the pinching effect (HYST2 and HYST3) are representative of very old structures not designed to resist seismic loads (where HYST3 is characterized by a stronger degradation than HYST2). [PERSON]'s model (HYST4) includes more refined conditions for the reloading curves and takes also into account the degradation of the stiffness due to increasing damage. It can be considered \begin{table}
\begin{tabular}{c l} \hline \hline
**Model Tag** & \multicolumn{1}{c}{**Model Description**} \\ \hline EPP & Elastic-perfectly plastic model including Bauschinger effect \\ HYST1 & Elastoplastic model with hardening \\ HYST2 & Bilinear model with stiffness and low strength degradation, and pinching effect \\ HYST3 & Bilinear model with stiffness and high strength degradation, and pinching effect \\ HYST4 & [PERSON]’s model \\ HYST5 & Bilinear model with strain softening and stiffness degradation \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Set of nonlinear constitutive laws under consideration. representative of reinforced concrete structures with stiffness degradation caused by severe shear stresses and slippage of steel bars. Finally, the bilinear model with strain softening and stiffness degradation (HYST5) is here considered because it provides a reasonable approximation of the global dynamic response of regular low-rise masonry buildings with efficient wall-to-diaphragm connections and stiff diaphragms under the assumption that the response is dictated by the in-plane behavior of the walls and dominated by a single mode of vibration. Overall, the main features of these nonlinear models can be adjusted by tuning a set of five parameters, namely \(p\), \(\alpha\), \(\beta\), \(\gamma\) and \(\rho\). The parameter \(p\) is the strain hardening ratio, and it controls the post-yielding stiffness. The parameter \(\alpha\) is related to the stiffness of the unloading branch; if it tends to infinity, then there is no stiffness degradation during the unloading phase, while \(\alpha\) = 2 can be considered as reference value. The parameter \(\beta\) controls the strength degradation as follows: for \(\beta\) = 0, there is no strength degradation due to energy dissipation, while \(\beta\) = 0.1 can be considered a limit value (for larger values, it can lead to unrealistic results or numerical stability issues). The parameter \(\gamma\) controls the pinching effect due to closing cracks during the reloading phase as follows: for \(\gamma\) = 1, there is no pinching, while \(\gamma\) = 0 corresponds to the maximum pinching effect and \(\gamma\) = 0.5 can be considered a realistic reference value. Finally, \(\rho\) controls the level of residual strength.