## 2 A Review of Commonly Used Nonlinear Static Analysis Procedures  
Nonlinear static (pushover) analysis is an essential simplified static nonlinear procedure used for estimating structural deformations. It is employed to achieve various set objectives, including assessing and devising seismic retrofit solutions for existing buildings, as well as designing novel structures that incorporate non-conventional structural materials, systems, or features not compliant with current building code requirements. Additionally, evaluating the performance of buildings is necessary to tailor them to the specific demands and preferences of owners and stakeholders. These endeavors play a crucial role in enhancing the resilience and safety of buildings, both old and new, while ensuring they align with the unique needs and objectives of the stakeholders involved [27]. A host of techniques have been developed over time, and several, such as the N2 method, the capacity spectrum method (CSM), and the displacement coefficient method (DCM) [14], are widely adopted in the field [2; 4]. Nonlinear static procedures (NSPs), also called pushover analyses, are widely used in seismic engineering to assess structures' seismic vulnerability. These procedures are more computationally efficient than dynamic analyses while still providing valuable insights into a structure's behavior under lateral loads. NSPs categorize the lateral load patterns applied to the structural model during the analysis into three primary groups: first-modal inertia, multi-mode, and adaptive mode vectors. An explanation of each category follows. 1. First modal inertia/invariant single load vectors (FEMA 356) pushover analysis: In a first modal inertia pushover analysis, the lateral load pattern is based on a single fundamental mode of vibration of the structure. The applied lateral load is proportional to the mass participating in the first mode of vibration. This approach considers primarily the inertia forces of the first mode and neglects higher modes' contributions. The analysis is often more straightforward than multimodal pushover, but it may not capture the full response behavior of complex structures that significantly engage higher modes during seismic events [8]. This method proposed three lateral load patterns, as shown below [16; 28]: * Distribution according to the fundamental mode shape: \[F_{i}=W_{i}\varnothing_{i},\] (1) * Inverted triangle distribution: \[F_{i}=\frac{W_{i}h_{i}}{\sum_{i=1}^{n}W_{i}h_{l}}*V_{b},\] (2) * The value of \(V_{b}\) is given by: \[V_{b}=S_{d}(T_{n})W,\] (3)* Load distribution as stipulated in FEMA 356 [8]: \[F_{i}=\frac{W_{i}h_{i}}{\sum_{i=1}^{n}W_{i}h_{i}^{k}}*V_{b},\] (4)
* Uniform distribution of load pattern: \[F_{i}=W_{i},\] (5)
* Invariant multi-mode vectors pushover analysis: In this procedure, different modes are combined [29]. Appropriate modes are identified and included in the analysis in a manner that aligns with the analysis undertaken. The following expression is generally applied in determining the distribution of the loads: \[F_{i}=\sum_{j=1}^{n}\alpha_{mr}T_{j}M_{i}\varphi_{ij}S_{a}(\zeta_{j},T_{j}),\] (6)
* Adaptive load vectors: This is a method where the load pattern is not constant but changes and updates instantaneously during the analysis, based on the evolving response of the structure. The following expression is generally applied in determining the distribution of the loads.