# Investigating the Reliability of Nonlinear Static Procedures
## 2 Basics of Nonlinear Static Procedures Adopted in Standards
Specifically, the exponents \(b\) and \(c\) are fine-tuned using data from the entire set of oscillators, while parameters \(\dd_{\text{hyst}}\) and \(\dd_{\text{hyst}}\) are subjected to separate calibration processes for the three previously reported ranges of hysteretic dissipation. In Section 4.2 of this paper, the choice among the latter is carried out on the basis of the examination of the damage simulated at the NC limit state by nonlinear analyses performed on the examined case study. Another alternative was proposed by [PERSON] et al. [32], which maintains a format similar to the equation adopted in [8,49] (see Equation (6) in Table 3); in this expression, the coefficient \(a\) defines the ductility and the demand computed using the traditional N2 method is underestimated, while \(b\) accounts for the dissipative capacity of the system. \begin{table}
\begin{tabular}{c c} \hline \hline
**Reference** & **Methods to Bilinearize Capacity Curve for the NC Limit State** \\ \hline \multirow{2}{*}{NTC2018} & The stiffness k is calculated in correspondence to the 70\% \(V_{max}^{*}\); then, the stiffness k is kept constant, and the yield strength \(V_{y}^{*}\) is calculated by the equivalence of the areas up to the displacement of collapse \(\dd_{u,NC}^{*}\). \\ \hline \multirow{2}{*}{Updating EC8-3 rel.2019} & Up to the base shear peak, the yield strength \(V_{y}^{*}\) is kept constant, equal to the peak \(V_{max}^{*}\), and the stiffness k is calculated by imposing the equivalence of the areas up to \(V_{max}^{*}\); after the base shear peak, the stiffness k is kept constant and equal to the value computed until the peak, while the yield strength \(V_{y}\) is calculated by the equivalence of the areas up to the displacement of collapse \(\dd_{u,NC}^{*}\). \\ \hline \multirow{2}{*}{[PERSON] et al. (2019)} & For LSs associated with displacements up to the base shear peak, the stiffness k is calculated by the equivalence of the areas up to the displacement associated with the SL by assuming \(V_{y}^{*}\) equal to the current value of the base shear at this point; for SLs associated with displacements after the base shear peak, the yield strength \(V_{y}\) is kept equal to \(V_{max}^{*}\) while the stiffness is computed by imposing the equivalence of the areas up the displacement of collapse \(\dd_{u,NC}^{*}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Methods to bilinearize the capacity curve and associated references. To achieve equivalence in the inelastic displacement demand, the proposal of [PERSON] and the N2 method are calibrated to match the ductility demand of 3 (or, equivalently, by setting \(a=3\)) [32]. With regard to the \(b\) coefficient, in this paper, it is assumed equal to 1.2 (or correspondingly setting \(b=1.2\)), in order to maintain the relationship with the reduction factor equal to 3 (i.e., \(a=3=R\)). It is worth noting that the coefficient \(b\) has the potential to set the limit from which the equal displacement rule starts to be valid. It is worth observing that expressions (4) and (5), presented in Table 3, reveal the strong influence of the initial period of the SDOF (T*) and its relationship with T\({}_{\text{C}}\) (i.e., the corner period of the response spectrum). In fact, the equal displacement rule is applied for T* \(>\) T\({}_{\text{C}}\), while, for T* \(<\) T\({}_{\text{C}}\), the equivalence between the elastic and nonlinear SDOF is imposed by an appropriate reduction rule. The CSM, initially formulated and pioneered by [PERSON] [34] and integrated into Standards such as [10], is employed in the analysis of the case study, using the methodology illustrated in [31]. The CSM obviates the strict requirement of converting the capacity curve into an equivalent bilinear, and it makes use of reduced spectra (see Equations (10) and (11)) based on the concept of equivalent damping \(\zeta\) (Equation (8)).