# Investigating the Reliability of Nonlinear Static Procedures
## 2 Basics of Nonlinear Static Procedures Adopted in Standards  
The performance-based evaluation of buildings' seismic performance relies on comparing the seismic forces acting on the structure and its ability to withstand these forces in terms of displacement. This approach involves making specific assumptions to minimize the computational and analytical efforts. When conducting a seismic performance-based assessment, it is crucial to establish the expected structural response at various damage levels under specific seismic hazard conditions. The structural response can be quantified using displacement levels associated with the attainment of corresponding limit states (LSs) or performance levels (PLs) (for example, fully operational, operational, life safety, and near collapse). These limit states--even if they are strictly correlated to the structural damage--encompass non-structural aspects such as reusability, immediate occupancy, operational functionality, and also economic considerations. However, different performance levels may also differ in terms of probability. Accordingly, the obtained results could be checked and evaluated for different performance states (for example, the life safety limit state) [48]. In this paper, attention is focused on the near collapse (NC) limit state. As briefly recalled in the Introduction, NSPs presuppose the following:  
* The execution of an NLSA to obtain the pushover curve (representative of the original multi-degree-of-freedom--MDOF--system) aimed at describing the evolution of the structural response in a nonlinear range;
* The adoption of criteria to define the attainment of given limit states on the pushover curve;
* The conversion of the MDOF into an equivalent single-degree-of-freedom (SDOF) representation of the system;
* The adoption of a procedure to assess the displacement demand expected given a seismic input, which is expressed in NSPs in terms of response spectra. In the following, for the specific alternatives proposed in the Standards to address these issues, reference is mainly made to the European Code [49] and the Italian Structural Code [8]. The execution of the NLSA in turn implies various choices, such as that of the control node, of which the displacement is incrementally increased, and that of the load pattern (LP), adopted as an approximate representation of the inertial forces that the building may experience during an actual earthquake. The control node is typically defined as a roof level's point; in this paper, the control node is assumed in proximity to the center of mass of the top floor. This choice is reasonable considering that the selected case study is characterized by rigid diaphragms. Regarding the choice of the LP, the European Code [49] prescribes the application of a minimum of two distinct LPs, which are kept invariant during the pushover analysis: (a) a \"uniform\" pattern, in which the lateral forces vary in direct proportion to the masses involved (associated with a uniform deformed displacement profile); (b) a \"modal\" pattern, where the lateral forces are in line with a deformed shape assumed to be equal to the one consistent with the first mode (producing an inverse triangular displacement profile). The Structural Italian Code [8] defines two groups of LPs: the first one contains the LP (b) and other possible alternatives that conceptually recall the dynamic behavior of the structure in the pseudo-elastic phase; the second one contains the LP (a) together with other possibilities, such as the adaptive LP (see [50; 51]). In the first group, it is also possible to approximate the first mode shape with an LP proportional to the one adopted in the case of the linear analysis, i.e., based on the assumption of a linear mode shape (producing an inverse triangular profile). For the purpose of the safety assessment, the worst result (i.e., the one associated with the higher displacement demand) between two LPs is assumed as a reference.