## 2 A Review of Commonly Used Nonlinear Static Analysis Procedures
Under an assumed initial loading pattern, [30] determined the incremental loading by applying the following expression: \[\Delta F_{i}^{k+1}=V_{b}^{k}\Bigg{(}\frac{F_{i}^{k}}{V_{b}^{k}}-\frac{F_{i}^{ k-1}}{V_{b}^{k-1}}\Bigg{)}+\Delta V_{b}^{k+1}\Bigg{(}\frac{F_{i}^{k}}{V_{b}^{k}} \Bigg{)}\] (7)  
To account for higher modes, this was later modified as suggested by [31], and the following equation was applied:  
\[F_{i}=\frac{W_{i}\sqrt{\sum_{j=1}^{n}\left(\varphi_{ij}T_{j}\right)^{2}}}{\sum _{L=1}^{N}W_{L}\sqrt{\sum_{j=1}^{n}\left(\varphi_{L,j}T_{j}\right)^{2}}}* \Delta V_{b}+F_{i}^{old}, \tag{8}\]  
In Equation (1) to (8), various symbols represent key aspects of seismic load distribution in structures. \(F_{i}\) is the force distribution at each level, \(W_{i}\) denotes the weight at each level, and \(\Phi_{i}\) indicates displacement in the fundamental mode. \(h_{i}\) is the height of each level, \(V_{b}\) represents the base shear, and other symbols, such as \(\Gamma_{j}\), \(M_{i}\), \(\Phi_{j}\), \(S_{a}(\zeta_{j}\Gamma_{j})\), \(\Delta F_{i}\), and \(\Delta V_{b}\) are used for multi-modal analysis and adaptive load vectors in seismic analysis. Choosing a nonlinear static procedure (NSP) for seismic vulnerability assessment involves balancing accuracy needs with structural complexity. Simple pushover analysis is convenient but may miss multimodal effects, while multimodal analysis demands substantial computational resources, especially for complex structures. Figure 2 illustrates the evolution of the pushover analysis methods, emphasizing the ongoing drive to enhance seismic demand estimation methods and the continuous need for research in structural engineering. As computational technology advances, more sophisticated nonlinear analysis procedures, such as nonlinear response history analysis (NLRHA), have become feasible but remain computationally intense and time-consuming [32]. In response, pushover-based methodologies, such as the N2 method, the capacity spectrum method (CSM), and the displacement coefficient method (DCM), have emerged as efficient, simplified alternatives that use empirical approaches to directly estimate seismic demand and capacity, thereby reducing computational burdens [2; 33; 34]. The N2 method provides a simplified estimation of seismic demands based on a plastic hinge analysis approach, the CSM uses a capacity spectrum to assess structural response, and the displacement coefficient method estimates seismic demands based on displacement coefficients derived from experimental data [2; 3; 4]. In conclusion, nonlinear static (pushover) analysis techniques continue to evolve due to advancements in computational capabilities, demands of complex and unsymmetrical 3D geometries, the requirement for improved understanding of complex nonlinear phenomena, and the ongoing need for accurate seismic vulnerability assessment methodologies [4]. These developments are changing the way we understand and respond to seismic activities, ensuring the safety and sustainability of our buildings [1; 2; 3; 4]. These methods, developed over the past forty years, are described in the next section.