# Investigating the Reliability of Nonlinear Static Procedures
## 1 Introduction  
The ductility and dissipation features of new and existing buildings constitute key factors in performance-based seismic engineering in terms of assuring the desired safety levels and economic feasibility. Moreover, as testified by the evidence of many real seismic events, unreinforced masonry (URM) structures are characterized by strong nonlinear behavior that may even be activated by seismic inputs of low-medium intensity [1; 2; 3]. Due to this, the ability to appropriately account for nonlinearity and estimate the seismic demand becomes crucial. Different analysis methods (i.e., linear and nonlinear, static and dynamic) are proposed in the literature [4; 5; 6] and recommended in certain standards [7; 8; 9; 10; 11]. In the case of linear approaches, the use of q-factors is one way to account for the intrinsic ability of a structure to resist energy dissipation [12; 13]. For the seismic design of new URM buildings, q-factor reference values, able to implicitly account for plastic behavior up to the near-collapse (NC) limit state, are available [14]. Their use, combined with that of capacity design structural details, aims to ensure the required seismic performance. Despite this, the application of linear approaches in the case of designing URM structures is quite problematic, particularly in areas with moderate-to-high seismic activity, as proven in somestudies [15; 16]. The use of linear methods in the case of existing URM buildings is even more challenging [17], beginning with the difficulty in defining robust q-factor reference values, considering that, often, such structures are designed according to an empirical approach rather than an engineered one, thus leading to a large variety of possible structural solutions that may affect their seismic responses [18]. These factors mean that, in the case of URM buildings, nonlinear methods (and especially static ones, which this paper focuses on) constitute the standard methods applied not only at the research level but also in engineering practice [19]. While nonlinear dynamic analyses (NLDAs) are considered highly accurate and are useful in explicitly capturing the full dynamic responses of structures, nonlinear static procedures (NSPs) based on the execution of nonlinear static analyses (NLSAs) are often more widely used for various practical reasons. In fact, NLDAs involve the use of ground motion time-history records that need to be appropriately selected [6; 20; 21; 22], as well as defining appropriate nonlinear hysteretic models. Although, nowadays, very effective tools are available to address the selection of seismic inputs [23; 24; 25; 26; 27]--and the availability of nonlinear constitutive laws has increased, proving their effectiveness in simulating the actual responses of complex buildings [28; 29; 30]--currently, NLDAs are mostly adopted at the research level. Nonlinear static analysis (NLSA), also known as pushover analysis, simulates the actual inertial forces activated within dynamic responses by statically applying horizontal conventional load patterns to a 3D model of a structure. NLSA is a very effective tool in evaluating structural seismic responses and estimating progressing nonlinear states. Meanwhile, NSPs address the use of the so-obtained pushover curve (i.e., the overall base shear versus the top displacement) to compute the expected displacement seismic demand, given a certain seismic input, expressed in terms of response spectra. To this aim, NSPs require the following additional steps [31]: (i) the definition in the pushover curve of given performance levels; (ii) the conversion of the original multi-degree-of-freedom structure's capacity, represented by the pushover curve, into an equivalent single-degree-of-freedom (SDOF) system; and (iii) the adoption of an appropriate procedure to compare such an SDOF capacity curve with the response spectrum (i.e., the method of calculating the maximum displacement demand under a seismic input of a given intensity).