# Investigating the Reliability of Nonlinear Static Procedures
## 2 Basics of Nonlinear Static Procedures Adopted in Standards
Instead, the other two criteria in Table 2 lead to an adaptive approach, i.e., one that varies according to the position of the SL on the curve (i.e., before or after the peak base shear \(V_{max}^{*}\)). More specifically, in the proposal introduced in the version of Eurocode 8-3 [56] published on September 2019 (in the following, it is briefly recalled as \"Updating EC8-3 rel.2019\" and is currently under review also with additional modifications), an elastoplastic relationship is adopted but with the following differences from NTC2018: the initial stiffness is determined by equating the areas beneath the curve up to the peak of the pushover curve (\(V_{max}^{*}\)); the strength \(V_{y}^{*}\) is assumed equal to \(V_{max}^{*}\) until the peak, while, after, it is computed by imposing the equivalence of the areas up to the d\({}_{u,NC}^{*}\) displacement. Finally, according to [PERSON]'s proposal, introduced in [32], the following adaptive criteria are  
Figure 2: NSP research investigation framework: (**A**) equivalent bilinear according to [8] (A1), [56] (A2) and [32] (A3); (**B**) procedure for the d\({}_{\max}\) estimate according to [34] (B1), [33] (B2), [57] (B3) and [32] (B4); (**C**) computation of the maximum intensity measure compatible with the fulfillment of NC (IM\({}_{\rm MC}\)). adopted: until the shear peak, the stiffness is computed by imposing the area's equivalence up to the displacement associated with the LS under examination, which also establishes the corresponding value of the yield base shear \(V_{y}^{*}\) to be assumed; after the shear peak, \(V_{y}^{*}\) is kept constant and equal to \(V_{max}^{*}\) and the stiffness is computed according to the equivalence area's rule. The initial stiffness of such an equivalent bilinear then addresses the building's equivalent period \(T^{*}\), which plays a pivotal role in determining \(\dd_{\max}^{*}\) through the N2 method (as described in references [8,49] and evident from Table 3). The N2 method, initially introduced by [PERSON] and [PERSON] in their work [33], is explicitly recommended in the Italian Structural Code (as Method A [8]) and Eurocode 8-1 [49], besides the Updating EC8-3 rel.2019 document. Equation (4), presented in Table 3, recalls how the assessment of the inelastic displacement demand is computed according to this approach. In Table 3, \(\dd_{\text{e,max}}^{*}\) is equal to the elastic displacement spectral value \(S_{d,e}(T^{*})\) at period \(T^{*}\) (which is \(\dd_{\text{e,max}}^{*}=S_{d,e}(T^{*})\)); \(\R=\frac{m^{*}\cdot S_{d,e}(T^{*})}{V_{y}}\) is the reduction factor; \(S_{u,e}(T^{*})\) is the elastic acceleration spectral value at period \(T^{*}\); \(m^{*}\) is the mass of the equivalent SDOF; and \(T_{u}\) is the secant period at the ultimate displacement capacity of the pushover curve. Indeed, the original proposal of [33] has been demonstrated to not always be suitable for stiff structures such as URM buildings [32,57]. Thus, alternative proposals have been developed for these structures in the literature. The alternative N2 method of [PERSON] et al. [57] introduces a novel equation to calculate \(\dd_{\max}^{*}\) (refer to Equation (5) in Table 3) that aims to include the influence of the hysteretic dissipation when estimating the earthquake-induced nonlinear displacement demand. Various hysteretic responses are assumed for SDOF oscillators, as investigated in depth by executing NLDA; in particular, three distinct behavior types are classified based on the prevailing failure modes activated in piers, i.e., those primarily influenced by flexural behavior; those predominantly affected by shear behavior; and a group exhibiting a hybrid mode [57]. The parameters \(\dd_{\text{hyst}}\), \(b\), \(c\), and \(\dd_{\text{hyst}}\) in Table 3 are calibrated by these authors on the basis of the dynamic response of this first set of SDOFs. This calibration involves conducting an orthogonal regression analysis between the results of the \(\mu_{R}-R-T\) relationship and those obtained from NLTHA [57].