# Investigating the Reliability of Nonlinear Static Procedures
## 2 Basics of Nonlinear Static Procedures Adopted in Standards
Table 1 summarizes the thresholds adopted for the monitoring of the NC PL in this research. Finally, the NC PL is reached in correspondence to the minimum displacement capacity of the structure, \(\mathrm{d}_{u,NC}^{*}\), between the two associated with the satisfaction of the two checks in Table 1.  
\[\mathrm{d}_{u,NC}^{*}=min\big{\{}\mathrm{d}_{u,1}^{*};\mathrm{d}_{u,2}^{*} \big{\}} \tag{2}\]  
Such an approach may be consistently applied in the case of both NLSA and NLDA. Proceeding to the step that involves the transformation of the MDOF pushover curve into the equivalent SDOF system, the criteria proposed in [55] are assumed, as they are unanimously supported in the literature. Such conversion is performed by dividing the pushover components by the participation factor \(\Gamma\) through the following expressions:  
\[V_{b}^{*}=\frac{V_{b}}{\Gamma};\qquad\qquad d^{*}=\frac{d}{\Gamma} \tag{3}\]  
with \"\(*\)\" to indicate that the quantity pertains to an equivalent SDOF system and \(\Gamma\) denoting the participation factor obtained with respect to the provided deformed shape [32]. Once the pushover is converted, it is possible to proceed to the comparison with the seismic demand to compute the maximum displacement demand (denoted as \(\mathrm{d}^{*}\mathrm{max}\)). For this issue, different possible alternatives are investigated, as depicted in Figure 2. \begin{table}
\begin{tabular}{c c} \hline
**Control** & **NC Limit State** \\ \hline Global & \(\mathrm{d}_{u,1}^{*}=d\Big{(}\frac{V_{b}}{\mathrm{max}}=0.8\Big{)}\) \\ Macroelement & \(\mathrm{d}_{u,2}^{*}=d(DL_{min}\geq\ DL4)\) \\ \hline \end{tabular}
\end{table} 
Table 1: Thresholds adopted for achievement of the NC limit state (with, at the macroelement scale, a check regarding the inter-story drift). Figure 1: Illustration of the DL\({}_{\mathrm{min}}\) criterion used to assess damage levels at macroelement level: (**a**) multilinear constitutive law adopted for URM elements and identification of DL at element scale; (**b**) the top level overall reached DL2, since all piers reached at least DL2 (four piers indeed reached DL3, but there was still a pier in DL2); (**c**) the top level overall reached DL3 since all piers reached at least DL3 (only one pier reached DL4, while all the others were in DL3). See (**a**) for the color legend. The best-known procedures, also investigated in this paper, are (1) the N2 method [55] and (2) the capacity spectrum method (CSM) [34]. Both approaches require the introduction of a reduction criterion to account for the effect of the response's nonlinearity in the response spectra evaluation. While the N2 method refers to the concept of inelastic spectra associated with ductility, the CSM method refers to the use of overdamped spectra. The N2 method is the one traditionally adopted in European countries, while the CSM is favored in the USA [9,10] and New Zealand [11] (even if it has been recently recommended in [8] as \"Method b\"). A review of the use of these two methods when applied to URM structures has already been presented in [32]. The N2 method strictly requires the pushover curve's transformation into an equivalent bilinear representation. Various criteria have been suggested for this purpose, as outlined in Table 2, which aims to summarize the three criteria investigated in this paper. According to the Italian Structural Code [8], an elastoplastic relationship is usually employed for URM structures. In this approach, the initial branch of the bilinear curve starts from the origin and extends through the pushover curve point corresponding to 70% of the maximum base shear (\(V_{max}^{*}\)), while the yield base shear \(V_{y}^{*}\) is defined by imposing the equivalence of the areas under the bilinear and the original pushover curve up to the ultimate displacement capacity (d\({}_{u,NC}^{*}\)), set in correspondence to the base shear decay equal to 20%. According to NTC2018, the bilinear curve is invariant with the examined LS.