# Investigating the Reliability of Nonlinear Static Procedures
## 2 Basics of Nonlinear Static Procedures Adopted in Standards
As aforementioned, appropriate criteria must be introduced to define the attainment of PLs on the pushover curve. Current approaches adopted in the Standards are based on (a) global criteria (or by assessing the strength degradation of the pushover curve) [49]; (b) local criteria associated with the attainment of a given damage level in single elements or groups of them (e.g., a wall composed of a set of structural elements, usually defined as a macroelement). In some cases [8], a combination of such criteria (c) is adopted. Such an approach is adopted also in other Codes, such as the Turkish Building Earthquake Code (TBEC), which uses both member-based strain definitions and building performance definitions separately [48]. In this research, approach (c) is adopted as a reference for the assessment of the NC LS by adopting the criteria proposed in [32], which are conceptually consistent with the Italian Structural Code [8]. Thus, a combination of two criteria is used: a criterion at the macroelement scale (i.e., the wall scale), to monitor the progression of damage in structural components (specifically, piers and spandrels in URM buildings), and a global criterion based on checking on the pushover curve when the decay rate of the maximum base shear (\(V_{max}\)) of the pushover curve is reached. Concerning the macroelement criterion, the goal is to monitor the possible concentration of damage in specific portions of the URM building that could be incompatible with the safety requirements (e.g., to identify the activation of a soft story mechanism in local portions); this is particularly relevant in the case of structures with irregular configurations or in the presence of flexible diaphragms. The first proposal to detect such conditions was introduced in [31], using the inter-story drift (\(\theta_{w,l}\)) within a particular story or level (\(l\)) of a wall (\(w\)) as an engineer demand parameter to be checked against the exceedance of the predetermined threshold. In [31], the computation of the inter-story drift is performed according to Equation (1):  
\[\theta_{w,l}=\frac{D_{w,l}-D_{w,l-1}}{h_{i}}\pm\frac{\varphi_{w,l}-\varphi_{w,l-1}}{2} \tag{1}\]  
where \(h_{i}\) represents the height of a story at level l, while \(D_{w,l}\) and \(\varphi_{w,l}\) denote the average horizontal displacement and the nodes' rotation, situated at either level l or level l-1 within wall w. The use of \(\pm\) depends on the direction of rotation, with the positive sign indicating clockwise rotation [32]. The contribution of rotations becomes relevant, in particular, in the absence of an RC ring beam or in the case of flexible diaphragms, i.e., when the shear-type idealization is far from satisfied. The adoption of the inter-story drift as a proxy for the attainment of damage conditions was also extensively used in the RINTC project [52; 53], which was aimed at the evaluation of the risk level implicit in Italian code-conforming buildings, and it is common also in the case of reinforced concrete (RC) buildings [54]. Despite the effectiveness of the inter-story drift, in the case of URM buildings, the definition of robust thresholds may be quite conventional since the drift limits adopted to check the damage level in piers vary with the prevailing damage mode (whether flexural or shear-dominant), and the activation of the latter depends on several factors (e.g., the slenderness of the panel and the acting axial load, in addition to material strength parameters). To overcome such a limitation, in this research, the criterion proposed in [32] is used, namely, the variable \"DLmin\". According to this approach, the achievement of a damage level (DL) (then associated with a corresponding PL) occurs when all the piers on a wall's story reach or exceed the monitored DL (see Figure 1). Instead, for the global check, the pushover analysis point at which the building's base shear (\(V_{b}\)) experiences a 20% reduction is assumed, as also proposed in [8,49].