Pushover Analysis in Seismic Engineering: A Detailed Chronology and Review of Techniques for Structural Assessment

Abstract

This study analyzes the progression, utilization, and inherent challenges of traditional non-linear static procedures (NSPs) such as the capacity spectrum method, the displacement coefficient method, and the N2 method for evaluating seismic performance in structures. These methods, along with advanced versions such as multi-mode, modal, adaptive, and energy-based pushover analysis, help determine seismic demands, enriching our grasp on structural behaviors and guiding design choices. While these methods have improved accuracy by considering major vibration modes, they often fall short in addressing intricate aspects such as bidirectional responses, torsional effects, soil-structure interplay, and variations in displacement coefficients. Nevertheless, NSPs offer a more comprehensive and detailed analysis compared to rapid visual screening methods, providing a deeper understanding of potential vulnerabilities and more accurate predictions of structural performance. Their efficiency and reduced computational demands, compared to the comprehensive nonlinear response history analysis (NLRHA), make NSPs a favored tool for engineers aiming for swift seismic performance checks. Their accuracy and application become crucial when gauging seismic risks and potential damage across multiple structures. This paper underscores the ongoing refinements to these methods, reflecting the sustained attention they receive from both industry professionals and researchers.

1. Introduction

The assumption of linear elastic behavior is fundamental in seismic response evaluations, but it may not accurately estimate actual structural damage [1,2]. More precise analysis methods are needed. Dynamic time history analysis is rigorous but impractical for regular use due to its complexity and the requirement for detailed MDOF models [3]. Nonlinear static pushover analysis, a simpler alternative, is gaining traction for predicting lateral strength and inelastic deformation, although it requires detailed MDOF models [4]. Newer nonlinear algorithms using pushover analysis data have shown promise for more realistic seismic demand estimates and are being integrated into performance-based engineering guidelines [5,6,7,8,9]. Seismic analysis procedures are categorized into linear and nonlinear methods, each further divided into static and dynamic procedures. The linear static technique is widely used due to its simplicity [10,11], while linear dynamic techniques offer diverse approaches for studying time-varying structural reactions during seismic events [12,13]. Figure 1 illustrates the combination of these seismic analysis procedures.

Despite the practicality of linear procedures, their accuracy can be limited, especially for complex structures or significant seismic events, necessitating the development of nonlinear procedures [14]. Nonlinear time history (NTH) analyses are the most precise for estimating seismic demands, but their complexity and high computational requirements limit their use in routine engineering applications [15]. Nonlinear static procedures (NSPs) have been adopted as alternatives and involve monotonously increasing predefined load patterns until a target displacement is reached [16], as recommended in [5]. However, these simplified NSPs can be inaccurate in predicting inelastic seismic demands when higher mode and inelastic effects are significant [16,17]. To address these limitations, enhanced procedures, such as modal pushover analysis (MPA), modified modal pushover analysis (MMPA) [18], and upper-bound pushover analysis (UBPA) [19], have been developed. In addition, nonlinear procedures, including nonlinear modal response history analysis [20] and nonlinear response history analysis [21], are available. These methods consider the nonlinear behavior of materials and large deformations, but they require significant computational resources [22,23].

To further improve the prediction of inelastic seismic demands, adaptive pushover procedures were developed. These procedures involve progressively updating load vectors to reflect changes in system modal attributes during the inelastic phase [24]. Additional approaches, such as the adaptive modal combination (AMC) procedure, use adaptive load patterns [16,25,26]. These developments represent significant strides towards enhancing the accuracy and practicality of seismic demand estimation.

The novelty of this review lies in its comprehensive integration of various pushover methods into a single document, including their procedures, and highlighting their respective advancements. As we navigate through the following sections, we aim to provide an extensive understanding of each method, its advantages and disadvantages, and its applicability in seismic analysis. The subsequent sections are structured as follows:

• Section 2: an examination of widely adopted nonlinear static analysis techniques.
  This section represents a comprehensive review of the most utilized nonlinear static analysis procedures, tracing their development and application in seismic studies.
• Section 3: an exploration of pushover analysis methods using first modal inertial/invariant single load vectors.
  This section represents an in-depth examination of pushover analysis methods that employ first modal inertial or invariant single load vectors, detailing their methodology and significance.
• Section 4: a study of pushover analysis with invariant multi-mode vectors.
  This section investigates the intricacies of pushover analysis techniques that utilize invariant multi-mode vectors, highlighting their unique features and applications.
• Section 5: an examination of adaptive load vector techniques.
  This section explores adaptive load vector methods, examining their adaptability and how they differ from traditional techniques.
• Section 6: an overview of research efforts assessing various pushover analysis approaches.
  This section is a summary of the extensive research initiatives undertaken to evaluate the effectiveness and applicability of different pushover analysis techniques.
• Section 6.1: a comparison between nonlinear static pushover analysis and nonlinear dynamic analysis techniques.
  This section details a comparative study highlighting the differences and similarities between nonlinear static pushover analysis methods and their dynamic counterparts.
• Section 7: an analytical discussion.
  This section engages in a critical discussion, exploring the methodologies, their strengths, potential limitations, and the implications of their findings within the broader context of seismic analysis.
• Section 8: final remarks and takeaways.

This section outlines concluding thoughts on the presented methodologies, encapsulating the key findings, and offering insights into potential future directions in the field. This comprehensive examination aims to provide valuable insights into the wide array of pushover methods, aiding engineers and researchers in selecting the most suitable approach for their specific seismic analysis requirements.

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.

• 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:

$$F_i=\frac{W_i{h}_i}{\sum _{i=1}^n{W}_l{h}_l}\mathrm{\ast }{V}_b,$$

(2)

• The value of $V_b$ is given by:

$$V_b=S_d\left(T_n\right)W,$$

(3)

• Load distribution as stipulated in FEMA 356 [8]:

$$F_i=\frac{W_i{h}_i}{\sum _{i=1}^n{W}_l{h}_l^k}\mathrm{\ast }{V}_b,$$

(4)

• Uniform distribution of load pattern:

$$F_i=W_i,$$

(5)

2.

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:

$$F_i={\sum }_{j=1}^n{\alpha }_{mr}{\Gamma }_j{M}_i{\phi }_{ij}{S}_a({\zeta }_j,T_j),$$

(6)

3.

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.

Under an assumed initial loading pattern, [30] determined the incremental loading by applying the following expression:

$${∆F}_i^{k+1}=V_b^k\left(\frac{F_i^k}{V_b^k}-\frac{F_i^{k-1}}{V_b^{k-1}}\right)+{∆V}_b^{k+1}\left(\frac{F_i^k}{V_b^k}\right)$$

(7)

To account for higher modes, this was later modified as suggested by [31], and the following equation was applied:

$$F_i=\frac{W_i\sqrt{\sum _{j=1}^n{\left({\phi }_{i,j}{\Gamma }_j\right)}^2}}{\sum _{L=1}^N{W}_L\sqrt{\sum _{j=1}^n{\left({\phi }_{L,j}{\Gamma }_j\right)}^2}}\mathrm{\ast }{∆V}_b+F_i^{old},$$

(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 ϕ_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 Γ_j, M_i, ϕ_ij, S_a(ζ_j,T_j), ΔF_i, and ΔV_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.

3. The First Modal Inertial/Invariant Single Load Vectors Pushover Analysis Methods

This section provides a comprehensive exploration of modal inertial pushover analysis methods, which are fundamental techniques used to understand and predict how structures respond to seismic forces. Several of these methods have been integrated into established seismic design codes, reflecting their importance and widespread acceptance in the field of structural engineering.

3.1. The Capacity Spectrum Method (CSM) ATC 40 [5]

The capacity spectrum method, initially proposed by Freeman et al. in 1975 [35], was introduced as an alternative tool for rapid assessment of buildings. [36] It compares a structure’s ability to resist lateral loads with seismic demand (response spectrum). The capacity/pushover curve represents the structure’s capacity. The primary pushover recommended in ATC 40 (1996) [37] is based on the equivalent linearization technique [5], applying a linearization approach that uses a capacity spectrum. The following steps describe the procedure involved in this methodology [38]:

• Pushover analysis and conversion: execute a pushover analysis to chart the structure’s capacity curve, showcasing base shear, V_b, against the building’s roof displacement, D_t. Adapt this curve into acceleration-displacement (AD) metrics using the equivalent single degree of freedom (ESDOF) model. Utilize the first mode participation factor, C₀, and the modal mass for the conversion: D* = D/C₀ and A = V_b/M.
• Graphical representation: superimpose the derived capacity curve with the 5%-damped elastic response spectrum, ensuring both are in AD format.
• Deformation demand and pseudo-acceleration: opt for an initial peak deformation demand ‘dt’ and pinpoint its linked pseudo-acceleration A from the capacity curve, starting with a damping ratio ζ = 5.
• Ductility and damping calculation: Measure ductility using m = D*/u_y and compute the hysteretic damping with ζ_h = 2(m − 1)/pm. Determine the equivalent damping ratio with ζ_eq = ζ_el + kζ_h, where k is a factor adjusting for the system’s hysteretic behavior. Refine the ‘dt’ estimate using the elastic demand curve for ζ_eq.
• Convergence check: Monitor the displacement ‘dt’ for stabilization. Once consistent, the target displacement for the multi-degree-of-freedom (MDOF) system is set as ‘dt = C₀dt’.
• The equivalent linearization process necessitates an understanding of the displacement ductility ratio. The ATC-40 document outlines three iterative methods: procedures A, B, and C. While procedures A and B are clear and easily programmable, procedure C is a visual method not apt for coding. Specifics of these methods can be found in the ATC-40 [5] document and are omitted here for conciseness. The procedures are clearly defined in the [5] document. In his work [39], he also applied this methodology.

3.2. The Improved Capacity Spectrum Method

The Improved Capacity Spectrum Method (ICSM) updates the original Capacity Spectrum Method (CSM) for seismic analysis by introducing constant-ductility inelastic design spectra, as per FEMA-440 and [40,41,42]. Key advancements include new calculations for effective period and damping. ICSM’s main feature is its technique for estimating seismic demands using the equivalent Single Degree of Freedom (SDOF) system, where the capacity spectrum is mapped against the inelastic demand spectrum at various ductility levels, as shown in Figure 3 and represented by the various color lines. The intersection of one of these spectra determines the deformation demand, requiring iterative analysis. The selection is made when the ductility factor μ from the capacity diagram aligns with the ductility at the point where it intersects with the demand curve. ICSM also imposes strength constraints and iterative methods for target displacement calculation, enhancing the accuracy and robustness of seismic design frameworks.

3.3. The N2 Method, Eurocode 8

The N2 method, initially developed by [43] and later improved by [44], is a widely used nonlinear static analysis method recommended for use by EC8 [42]. It combines pushover analysis of an MDOF model and response spectrum analysis of a corresponding SDOF model to achieve a balance between reliability and applicability for everyday design use. The N2 method differs from the capacity spectrum method in that it has adopted an $R-\mu -T$ approach rather than the over damped spectra applied in the CSM method. The detailed procedure for conducting the N2 method is outlined below,

• Pushover Analysis: perform the analysis to obtain the capacity curve represented in V_b–D terms.
• Conversion and approximation: Convert the pushover curve of the MDOF system into the capacity diagram of an ESDOF system. Approximate this curve with an idealized elastic-perfect plastic relationship to determine the period T_e of the ESDOF.
• Target displacement calculation: the target displacement is computed as follows:

$$d_{et}^{*}=S_a\left(T_e\right){\left[\frac{T_e}{2\pi }\right]}^2$$

(9)

where S_a(T_e) represents the elastic acceleration response spectrum at the period T_e. To ascertain the target displacement $d_t^{*}$, different formulas are proposed for varying period ranges.

In the short period range (T∗ ≤ T_C), if the condition F_y′/m′ ≤ S_a(T_e) holds, the response remains elastic, leading to dt′ = de′t and dt = C₀dt′. If this condition is not met, the response turns nonlinear, and the ESDOF maximum displacement is given by:

$$d_t^{\mathrm{\prime }}=\frac{d_{et}^{\mathrm{\prime }}t}{\sqrt{1+\left(qu-1\right)\mathrm{\ast }\frac{TC}{Te}\le d_{et}^{\mathrm{\prime }}t}}$$

(10)

where q_u is the acceleration ratio of the structure with unlimited elastic capacity, times the modal mass m* over its yield force, defined as:

$$qu=\frac{S_a\left(T_e\right)\mathrm{\ast }{m}^{\mathrm{\prime }}}{F_y^{\mathrm{\prime }}}$$

(11)

In the medium and long period range (T* ≥ T_C), the target displacement for the inelastic system matches that of an elastic structure, so dt′ = de′t. The MDOF system’s displacement is consistently determined as dt = C₀dt′.

3.4. The Displacement Coefficient Method FEMA 356 [8], FEMA 440 [40], FEMA 273 [45] and ASCE 41 [7]

The displacement coefficient method, initially used in FEMA 273 [45] and later incorporated in FEMA 356 [8], aims at improving the accuracy of nonlinear static procedures (NSPs) by directly estimating the values of the peak inelastic displacement using modifying factors. The displacement coefficient method, a simplified approach, is less accurate for irregular or nonlinear structures. The displacement coefficient method calculates displacement coefficients for each mode of vibration, allowing for better prediction of inelastic behavior and determination of the control-node/target displacement [38]. The DCM involves the calculation of displacement coefficients for each mode of vibration of the structure to estimate the displacement response [14].

• Carry out a pushover analysis of the multi-degree-of-freedom system (MDOF) similar to 3.1(1). At this step, the eigenvalue analysis of the system is conducted to establish the modal properties of the system.
• The following relationship is then used to determine the target displacement.

$${\delta }_t=C_0{C}_1{C}_2{C}_3{S}_a\frac{T_e^2}{{4\pi }^2}g,$$

(12)

In their work, Asıkoğlu et al. [38] clearly defined all the associated parameters.

Figure 4 depicts the displacement coefficient method in a graphical manner.

K_i is the initial or elastic lateral stiffness of a building, reflecting its stiffness under small, elastic deformations. K_eff, the effective lateral stiffness, is calculated at a point where the base shear force is 60% of the building’s effective yield strength, derived from the bilinear representation of the capacity curve, indicating the building’s behavior under larger forces.

3.5. The Improved Coefficient Method

The FEMA-440 document [40] suggests several enhancements over the FEMA-356 coefficient method [8] for target displacement estimation. Firstly, it introduces a revised equation for C₁ as follows:

$$C_1=1.0+\frac{R-1}{{\alpha T}_e^2}$$

(13)

Here, α is set to 130, 90, and 60 for site classes B, C, and D, respectively (refer to FEMA-440 [44] for a detailed breakdown of site classes).

The next enhancement pertains to the coefficient C₂, expressed as follows:

$$C_2=1.0+\frac{1}{800}{\left(\frac{R-1}{T}\right)}^2$$

(14)

Lastly, FEMA-440 suggests omitting the coefficient C₃ and introduces a strength constraint to prevent dynamic instability. This strength constraint is defined by setting a maximum value for R, as illustrated below:

$$R_{max}=\frac{{∆}_d}{{∆}_y}+\frac{{\alpha }_2^{-t}}{4};t=1.0+0.5\ln (T)$$

(15)

In this equation, Δ_d represents the deformation at peak strength, Δ_y denotes the yield deformation, and α₂ is the negative slope of the force-deformation curve’s strength-degradation segment.

3.6. The Energy-Based Pushover Analysis Method

Energy-based pushover analysis [46,47,48] is a method that enhances nonlinear static analysis by integrating inertial properties and kinetic energy into force or displacement patterns. This method is particularly relevant for seismic input as it utilizes the fundamental mode shape Φ, effective damping, and a velocity profile to represent a structure’s dynamic behavior and to scale the response spectrum. The method is insightful for understanding kinetic energy dissipation through plastic behavior but has limitations, especially for braced structures [31,49,50].

The procedure of energy-based pushover analysis can be summarized as follows:

• Model selection: choose a structural model based on the building’s characteristics and identify the dominant mode shape (ϕ) through modal analysis.
• Force application: apply lateral forces (F_k) according to the mode shape and mass (m_i) distribution.
• Incremental loading: incrementally apply loads and measure displacements (d_jk) at each step.
• Equivalent SDOF system: calculate both the equivalent mass (m* = Σm_i) and the base shear (F* = ΣF_i).
• Displacement calculation: sum incremental displacements to find total equivalent displacement (δ* = Σ(Δδ)).
• Capacity curve construction: plot the capacity curve as base shear (F*) versus total equivalent displacement (δ*).
• Energy analysis: calculate the ductility ratio (μ* = δ*_max/δ*_y) and compute the energy term (Π_x = 1/2F*_yδ*_y(2μ* − 1)).
• Assessment of limitations: review the obtained results, being aware of potential constraints, particularly when dealing with braced structures. The cumulative equivalent displacement, denoted as δ*, is determined by summing up all the incremental values of Δδ*. Figure 5 depicts the overall concept of this method

This method was applied on a three-level steel structure conceptualized for Los Angeles within the framework of the SAC initiative.

3.7. The N1 Method

The N1 pushover analysis method offers a practical alternative to the N2 method, assessing seismic response using nonlinear static analysis without a defined equivalent SDOF system.

The title ‘N1’ indicates its characteristics: ‘N’ refers to the method being nonlinear and ‘1’ denotes that it employs a single multi-degree-of-freedom (MDOF) structural model. The N1 method aims to evaluate displacement demand by adjusting the results obtained from response spectrum analysis (RSA) to reflect nonlinear structural behavior. This method borrows from FEMA 368 and FEMA 369 [51] methodologies, enhancing them by allowing for stiffness reduction and thereby lengthening the time. This is a departure from the traditional N2 method, which typically acquires this by the bilinearization of the capacity curve. The N1 method has been shown to match the accuracy of the N2 method, but only when a lateral load distribution proportional to the first mode shape is included. The following relationship illustrates the comparison between the N1 and N2 methods:

D_el = Φ_nΓ₁S_de(T₁),

(16)

In the equation, Δ_eℓ represents a displacement or deformation parameter. Φ_n is the nth mode shape of the structure, indicating how it deforms in its nth mode of vibration. Γ₁ is the modal participation factor for the first mode, quantifying the contribution of this mode to the overall response. S_de(T₁) denotes the spectral displacement for the first mode, which depends on the fundamental period T₁ of the structure. This equation is used in seismic analysis to estimate displacements under certain loading conditions.

The N1 method for pushover analysis is used in irregular buildings with vertical irregularity, primarily for comparison or seismic demands. Researchers have refined it into the “N1 corrected method,” aligning more closely with the N2 method.

3.8. The Mass Proportional Pushover Analysis

The mass proportional pushover (MPP) procedure is a pushover analysis approach introduced by Kim and Kurama [52] to estimate the peak seismic lateral displacement demands for structures. It offers a simplified method that eliminates the need for conducting a modal analysis or modal combination, even when higher mode effects are significant. The main advantage of the MPP procedure is that it uses a single pushover analysis for the structure, with the lateral force distribution being proportional to the total seismic masses assigned to the structure at different levels. This approach consolidates the effects of higher modes into a single invariant lateral force distribution, making it a more efficient and practical method for estimating displacement demands.

• Perform single nonlinear pushover analysis: carry out a single nonlinear pushover analysis of the structure with the lateral force distribution given by [m]g{1} = [w]{1}.
• Determine MDOF base shear force vs. roof displacement relationship: establish the relationship between the MDOF base shear force Vb and the roof displacement, u_r, from the pushover analysis of the structure.
• Determine equivalent SDOF pseudo-acceleration vs. displacement relationship: calculate the equivalent SDOF pseudo-acceleration A versus displacement D relationship using the following relationships:

  $$A=\frac{V_b}{M} ,$$

  (17)

  $$A=\frac{u_r}{\Gamma },$$

  (18)
• Find maximum SDOF displacement Dmax: solve the equation ${\ddot{D}}_n+2{\xi }_n{\omega }_n{\dot{D}}_n+\frac{F_{sn}}{L_n}=-{\ddot{u}}_g\left(t\right)$ with Fs/L = A to find the maximum SDOF displacement Dmax.
• Calculate maximum MDOF roof and floor lateral displacements: compute the maximum MDOF roof and floor lateral displacements of the structure using the following relationship:

  $$\left\{u_{max}\right\}=D_{max}\Gamma \left\{u_e\right\}$$

3.9. The Modified First-Mode-Based Pushover Analysis

The modified first-mode-based pushover analysis (MFPA) was proposed by Aman Mola Worku et al. [53] as an advanced assessment method. It represents the global force–displacement capacity graphically, which enables a comparison with the representative response spectrum. MFPA assumes that the maximum inelastic deformation of a nonlinear single-degree-of-freedom (SDOF) system can be estimated from the maximum deformation of a linear elastic SDOF system which has a higher period and a higher damping ratio than the initial values of a nonlinear system. In brief, the procedure involves the following steps:

• Definition of the structure’s nonlinear model.
• Conducting an eigenvalue analysis on the structural model to determine the period and normalized mode shape.
• Apply the following relationship to determine the load pattern for the MFPA:

$$S_j^{\mathrm{\ast }}={\beta }_j{m}_j{\varnothing }_{1j}{S}_a(T_1,{\xi }_1)$$

(19)

4.

Conduct a static pushover analysis to get the V_b–U_r, which is later transferred to the idealized capacity curve.

5.

Determine the target displacement value from the idealized pushover curve.

6.

Establish the building’s performance estimate at the Maximum Considered Earthquake.

4. Invariant Multi-Mode Vectors Pushover Analysis

The procedure involves combining different modes, as cited in [29]. Relevant modes are identified and incorporated into the analysis. A particular expression is used to determine the distribution of the loads in this process.

4.1. The Modal Pushover Analysis

The modal pushover analysis (MPA), introduced by Chopra et al. [54], is a seismic performance evaluation method designed to enhance the accuracy of predicting how structures respond to earthquakes. Unlike traditional pushover analyses, MPA uses a modal-based approach that distributes inertia forces for each mode, which are then combined to approximate the total seismic demand on inelastic structures. For elastic structures, MPA is equivalent to response spectrum analysis (RSA). This method aims to provide a more accurate and efficient way to assess the seismic performance of structures, offering comparable accuracy to RSA for inelastic systems [54]. A step-by-step summary of the MPA procedure to estimate the seismic demands for building is presented as a sequence of steps:

• Compute the natural frequencies and modes of the building for linear elastic vibration.
• Perform nonlinear static analysis to develop the base shear-roof displacement pushover curve for each mode.
• Idealize the pushover curve as a bilinear curve, considering any negative post-yielding stiffness.
• Convert the pushover curve to the force-displacement relationship for the n-th mode inelastic SDF system.
• Calculate the peak deformation of the n-th mode SDF system using the force–deformation relationship and damping ratio, using the following second order differential equation.

$${\ddot{D}}_n+2{\zeta }_n{\omega }_n{\dot{D}}_n+\frac{F_{sn}(D_n,{\dot{D}}_n)}{L_n}=-{\ddot{u}}_g\left(t\right),$$

(20)

6.

Determine the peak roof displacement associated with the n-th mode SDF system. The following relationship applies:

$$u_{rno}={\Gamma }_n{\varphi }_{rn}{D}_n,$$

(21)

7.

Extract the desired responses from the pushover database at the roof displacement equal to the peak displacement.

8.

Repeat Steps 3–7 for all modes.

9.

Compute the dynamic response due to each mode by subtracting the contribution of gravity loads alone.

10.

Combine the gravity response and peak modal responses using the complete quadratic combination (CQC) rule to obtain the total response. The following relationship applies:

$$r_{MPA}=\sqrt{{\sum }_{n=1}^j{r}_{no}^2},$$

(22)

4.2. Multi-Mode Pushover for Deformation Demand Estimates of Steel Moment-Resisting Frames

The procedure proposed by [55] is a method for estimating the deformation demands of steel moment-resisting frames during seismic events [56]. The multi-mode pushover procedure accurately predicts seismic demands of steel moment-resisting frames by calculating the target interstory drift and using modal combinations. It considers frequency content, higher mode effects, and the interaction between modes. It offers superior response predictions with minimal computational effort. This approach allows for more accurate prediction of the deformation demands of the structure. The multi-mode pushover procedure involves the following steps:

• Modal analysis: Perform a modal analysis to determine the natural frequencies, mode shapes, and modal participation factors of the structure. This can be done using a general-purpose nonlinear static analysis tool.
• Response spectrum: Obtain the response spectrum for the design earthquake. This spectrum represents the maximum response of the structure at different periods.
• Mode combination: Combine the modal responses to obtain the total response of the structure. The modal responses are combined using a modal combination rule, such as the complete quadratic combination (CQC) rule or the square root of the sum of squares (SRSS) rule.
• Target interstory drift: Estimate the target interstory drift, which is the maximum interstory drift that the structure is expected to experience during the earthquake. This can be done using one of three procedures: the modal pushover procedure (MPP), the adaptive capacity spectrum method (ACSM), or the multi-mode pushover procedure (MP3).
• Generalized force vectors: Calculate a set of generalized force vectors using the modal analysis results. These force vectors represent the instantaneous force distribution acting on the structure when the interstory drift at one story reaches its maximum value during dynamic response to the seismic excitation.
• Maximum response: determine the maximum value of each response parameter by taking the envelope of the results obtained from the generalized force vectors.

4.3. Improved Modal Pushover Analysis

In contrast to the modal pushover analysis (MPA) procedure which utilizes invariant multi-mode lateral load pattern vectors for obtaining responses, the improved modal pushover analysis (IMPA) [57] incorporates the redistribution of inertia forces after the yielding of the structure. The key enhancement in IMPA is its use of post-yielding structural deflection shape as an invariant lateral load pattern. However, to circumvent extensive computation, it suggests a two-phase lateral load distribution for the first mode, while the force patterns for the higher modes follow the MPA approach. The steps encompassed in the IMPA procedure are as follows:

• Initiate with first mode MPA: Begin with the initial steps (1–3) of the MPA method for the primary ‘mode’. The initial load pattern is determined by the lateral force distribution, represented as S₁* = mϕ₁.
• Determine displacement at yield: from the pushover analysis in the first step, ascertain the structure’s displacement vector, ψ1y, at its yielding point.
• Progress with an updated load pattern: From the point of structural yielding, extend the pushover analysis using a new load distribution S1y∗ = mϕ1y. This is the second-phase lateral load pattern. Utilize this updated curve and the subsequent Steps 4–7 of the MPA method to assess the structure’s behavior.
• Compute overall response (IMPA): Merge the primary ‘mode’ response from Step 3 with the responses of higher modes derived from the conventional MPA method. Combine using methods like the square root of the sum of squares (SRSS) or complete quadratic combination (CQC).

4.4. The Modified Modal Pushover Analysis

Chopra et al. [18] developed the modified modal pushover analysis (MMPA) to estimate seismic demands in frame buildings, with a key simplification that assumes higher vibration modes are elastic. This reduces computational effort compared to the traditional modal pushover analysis (MPA). The study found that MMPA generally yields larger and potentially more accurate demand estimates, especially in cases where MPA underestimates demands. However, for cases where MPA already overestimates demand, MMPA may increase conservatism. MMPA’s main limitation is in lightly damped systems (with damping less than 5%), where its linear elastic assumption for higher modes may not accurately represent inelastic behavior, potentially leading to an overestimation of demand. The accuracy of the MMPA procedure is also found to be sensitive to specific building and ground motion characteristics. The following phases in the modified modal pushover analysis (MMPA) approach to calculate the peak inelastic response of a multistory structure with a plan symmetric about two orthogonal axes subjected to earthquake ground motion along an axis of symmetry are summarized below:

• Modal pushover analysis application: implement Steps 1–4 like those outlined in the standard modal pushover analysis.
• Determine peak deformation D1: For the primary mode’s inelastic single degree of freedom (SDF) system, compute D1 using parameters T1 and ζ1 [20]. This can be achieved via nonlinear response history analysis (RHA), an inelastic design spectrum, or specific empirical formulas.
• Ascertain peak roof displacement ur1: using the equation ur1 = Γ1ϕr1D1, calculate ur1 linked to the primary mode’s inelastic SDF system.
• Retrieve relevant responses: from the pushover data (from Step 2), extract response values r1 + g for the combined effects of gravity and lateral forces at a displacement of ur1 + urg.
• Calculate primary mode dynamic response r1: determine r1 using the relation r1 = r1 + g − rg, where rg represents the contribution from gravity loads.
• Evaluate higher-mode dynamic responses: For modes beyond the first, assume the system retains its elasticity and compute the dynamic responses. This mirrors the traditional modal analysis for a linear multi-degree-of-freedom (MDF) system, eliminating the need for extra pushover analyses and thus saving computational time. The number of modes to consider for accurate results varies based on the building’s height.
• Compute overall response (demand): integrate the gravity response with the peak ‘modal’ responses using the square root of the sum of squares (SRSS) rule: r≈max[rg ± Σnrn2], where r1 is derived from Step 5 and rn(n > 1) from Step 6.

4.5. The Generalized Pushover Analysis (GPA)

Sucuoglu and Gunay [58] introduced a new method called generalized pushover analysis (GPA). The GPA method uses generalized force vectors to predict building sway during earthquakes. Pushover analyses create an envelope of results, allowing for better understanding of building reactions under different conditions.

The GPA algorithm is composed of the following five basic steps:

• Eigenvalue analysis: calculate natural frequencies ω_n, modal vectors u_n, and modal participation factors Γ_n from eigenvalue analysis.
• Response spectrum analysis: obtain modal spectral amplitudes A_n, D_n from the corresponding linear elastic spectra.
• Calculate modal interstory drift ratios at the j-th story, Δ_j,max, and the maximum interstory drift ratio at the j-th story, Δ_j,max, from RSA.
• Generalized force vectors. Compute generalized force vectors fj that produce the maximum response Δ_j, as calculated from the expression below:

$$f_j\left(t_{max}\right)={\sum }_n\left({\Gamma }_n{m\phi }_n{A}_n\frac{{∆}_{j,n}}{{∆}_{j,max}}\right),$$

(23)

Target interstory drift demands:

calculate the target interstory drift, Δ_jt, at the j-th story using Equation (24):

$${∆}_{jt}={\sum }_n{\Gamma }_n\frac{{∆}_{j,n}}{{∆}_{j,max}}{D}_n({\phi }_{n,j}-{\phi }_{n,j-1}),$$

(24)

Note: If the first mode spectral displacement demand D₁ in Equation (24) is replaced with its inelastic counterpart, $D_1^{*}$, then a first mode pushover analysis is conducted. $D_1^{*}$ is obtained from either NRHA or the inelastic response spectrum of the bi-linear SDOF system. For higher modes n = 2 − N, D_n values are obtained from the linear elastic response spectrum.

5.

Generalized pushover analysis (GPA): Conduct N GPAs. In the j-th GPA (j = 1 − N), incrementally push the structure in the lateral direction with a force distribution proportional to f_j.

At the end of each loading increment, i, compare the interstory drift Δ_ji obtained at the j-th story with the target interstory drift Δ_jt calculated from Equation (24).

Continue displacement-controlled incremental loading (i = 1, 2, …) at the j-th GPA until Δ_ji reaches Δ_jt.

4.6. The Consecutive Modal Pushover Analysis (CMP)

The consecutive modal pushover (CMP) procedure, introduced by [59], is an innovative pushover method designed to address the limitations of the conventional nonlinear static procedure (NSP) when analyzing higher-mode responses in tall buildings. Unlike the traditional NSP, the CMP procedure integrates both multi-stage and single-stage pushover analyses, with final structural responses determined by combining the results of these separate analyses. The research demonstrated that the CMP procedure significantly improves the accuracy of predicting seismic demands of tall buildings, especially in estimating hinge plastic rotations at mid and upper floor levels, when compared to results from nonlinear response history analysis (NL-RHA) and modal pushover analysis (MPA). Thus, the CMP procedure represents a substantial advancement in seismic demand prediction for tall buildings, effectively incorporating higher-mode effects [59]. The steps explained below illustrate the details of this procedure.

• Frequency and mode shape analysis: Conduct an eigen analysis on the linear elastic structure, focusing on the initial three modes. Ensure mode shapes are normalized such that the roof component, denoted as ϕn, is set to one.
• Determine lateral force distribution: calculate Sn∗ = mϕn, where Sn∗ represents the distribution of incremental lateral forces throughout the structure’s height for the nth stage.
• Total roof displacement calculation: ascertain the structure’s total target roof displacement, represented as δt.
• CMP analysis steps:
  • For medium-rise structures, execute a single-stage pushover analysis using an inverted triangular load pattern. For taller structures, use a uniform force distribution. Continue until the roof’s control node reaches the predefined δt.
  • Implement a two-stage pushover analysis, where the second stage starts from the end state of the first stage.
  • For structures with a fundamental period of 2.2 s or more, conduct a three-stage pushover analysis. Each stage begins from the end state of the preceding stage.
• Determine the peak responses: From the earlier pushover analyses, calculate the peak values for targeted responses, such as displacements, drifts, and hinge rotations. Represent these peak values from the one-, two-, and three-stage analyses as r2, and r3 respectively.
• Calculate the overall peak responses:
  • If T < 2.2 s, then r = max{r1,r2}.
  • If T ≥ 2.2 s, then r = max{r1,r2,r3}.
• Sequential modal analysis note: The structure’s nonlinear behavior is intertwined with its response and is influenced by the loading path. In the CMP approach, modal pushover analyses should be sequentially executed, starting with the primary mode, and advancing to the subsequent ones, as detailed in [60].

4.7. Modified Consecutive Modal Pushover Analysis

The study by [61] aimed to refine seismic demand estimation in asymmetric-plan tall buildings, accounting for higher modes and torsion. It introduced the modified consecutive modal pushover (MCMP) procedure, which was applied to 10, 15, and 20-story structures. The results, compared with modal pushover analysis (MPA) [54] and nonlinear time history analysis (NLTHA [42], showed that the MCMP procedure provided accurate seismic demand estimates, as validated by NLTHA. Notably, the MCMP excelled in predicting specific structural responses, such as plastic hinge rotations, and outperformed both FEMA load distribution patterns and MPA, especially regarding displacements and story drifts. These benefits were most evident in mid- and upper-story buildings, marking the MCMP as a valuable tool for assessing seismic demands in taller structures.

• Carry out the initial steps, 1 and 2, as per the procedure explained in Section 4.6 above.
• Determine the total target displacement and increment. Calculate the values for the total target displacement and target displacement increment using the following relationships:

$$U_{ri}={\beta }_i{\delta }_i$$

(25)

$${\beta }_i=1-\sum _{j=1}^{N_s-1}{\alpha }_j,$$

(26)

3.

Apply gravity loads. Consider gravity loads as an initial condition for each pushover analysis.

4.

Conduct a single-stage pushover analysis. Perform a pushover study employing a uniform lateral load distribution until the target displacement at the roof is reached.

5.

Conduct a multi-stage pushover analysis. Determine the number of analysis stages, ns, using the following relationship:

$${\sum }_{i=1}^{N_s}{M}_i^{*}\ge 0.9{\sum }_{i=0}^N{m}_j,$$

(27)

• For the first stage, apply the initial distribution φ1s1* = mφ1 to the building to reach the displacement increment ur1 = β1δt.
• Continue the excitation with the incremental forces $S_2^{\ast }=m$φ₂, until the displacement increments at the roof equals u_r2 = β₂δ₂ in the second stage.
• Perform the pushover analysis until the number of stages equals Ns and the displacement at the roof reaches the total target displacement δt.
• At each stage of the multi-stage analysis, the initial condition is the same as the state at the end of the previous stage.

6.

Ascertain the peak values of seismic reactions. For both single-stage and multi-stage pushover analyses, determine the peak values of seismic reactions, such as displacements, story drifts, rotations of plastic hinges, and axial force of the braces.

7.

Determine peak responses’ envelope. Calculate the envelope of peak responses, r, as follows:

$$r=max\left\{r_s,r_M\right\},$$

(28)

4.8. Upper-Bound (UBPA) Pushover Analysis

Engineers often use simplified nonlinear static analytical procedures, also known as pushover analyses, to assess the seismic demands of tall buildings, as these methods are simpler than nonlinear response history analysis. However, this conventional approach has limitations in accurately predicting the inelastic seismic demands of high-rise buildings. To address this, a study [19] introduced an improved simplified pushover analysis method, which incorporates higher-mode effects into the analysis. This method consists of three key components: (1) response spectrum-based higher-mode displacement contribution ratios; (2) an alternative formula for determining the lateral load pattern; and (3) an upper-bound modal combination rule for establishing the target roof displacement. This proposed method was validated using five buildings of varying heights and compared with three other types of analysis [19]. Results show that the proposed method is more accurate in estimating important response attributes of tall buildings, including roof displacements, story drift ratios, and plastic hinge rotations.

The principal steps of the proposed pushover analysis procedure are as follows:

• Build the selection and elastic properties. Select a building and determine its elastic structural properties, including natural frequencies and mode shapes. Normalize mode shape Φ_n such that the roof degree of freedom ϕ_rn = 1 for all modes.
• Calculate the upper-bound contribution of the second mode. Use the elastic response spectrum of the selected earthquake to determine the upper-bound of the contribution of the second mode (q₂/q₁)UB, as given by:

$${\left(\begin{array}{c}{q}_2\\ q_1\end{array}\right)}_{UB}=\left|\frac{{\Gamma }_2{D}_2}{{\Gamma }_1{D}_1}\right|,$$

(29)

3.

Determine the lateral load distribution vector. Compute the distribution vector of the lateral loads over the height of the building using:

$$f_s=F_1+F_2={\omega }_1^2{m\phi }_1+{\omega }_2^2{m\phi }_2{\left(\frac{q_2}{q_1}\right)}_{UB},$$

(30)

4.

Compute the target roof displacement Ur. Determine the target roof displacement Ur as per the expression:

$$u_{r,UB}=u_{r,TLP}\left[1+{\left(\frac{q_2}{q_1}\right)}_{UB}\right],$$

(31)

5.

Perform a pushover analysis. Conduct a pushover analysis using the lateral load calculated in Step 3 until the target displacement computed in Step 4 is reached (single-run analysis).

6.

Extract the seismic responses. Determine the desired seismic responses (e.g., story drifts, element internal forces, etc.) at the step corresponding to the target displacement obtained from the single run in the previous step.

4.9. Modified and Extended Upper-Bound (MUB) Pushover Analysis

The upper-bound (UB) pushover analysis method, initially developed for 2D tall building frames, tends to underestimate seismic demands at lower levels of the structures [19]. To address this issue, a study [62] suggested combining the seismic demands from the UB pushover method with those from a conventional pushover analysis, aiming to better regulate seismic demands at lower building levels. Additionally, the study proposed an extension of the UB pushover method to accommodate tall buildings with unsymmetrical plans. This extension, termed the extended upper-bound (EUB) method, incorporates upper-bound lateral forces with patterns tailored to various unsymmetrical-plan tall buildings and considers higher-mode effects in calculating these forces. The results indicate that the EUB method can accurately estimate inelastic seismic demands of unsymmetrical-plan tall buildings, considering both higher modes and torsional effects.

• Initial calculations: perform calculations as outlined in Steps 1 to 4 of Section 4.8 above.
• Pushover analyses: After applying gravity loads, conduct both the upper-bound pushover and conventional pushover analyses. Continue these analyses until the control node, located at the roof’s center of mass, reaches the target displacement determined in the previous step.
  • For medium-rise unsymmetric-plan buildings, perform a conventional pushover analysis using an inverted triangular load pattern, and use a uniform force distribution for high-rise buildings until the target displacement is reached. For torsionally flexible systems, perform the conventional pushover analysis using the lateral force fs1, which is derived from Equation (30).
  • Perform the upper-bound pushover analysis using the upper-bound lateral forces fsi derived from Equation (30) until the target displacement at the roof is reached.
  • Determine the peak seismic demands. Calculate the peak values of the seismic demands individually for the upper-bound pushover and conventional pushover analyses. Denote the peak values obtained from these analyses by r_fsi and r₁, respectively.
• Determine the envelope of results. Calculate the envelope (r) of the results computed in the previous step as follows:

$$r=\left\{\begin{array}{c}max\left\{r_1,r_{fs2}\right\} for the TS and TF systems\\ max\left\{r_1,r_{fs2},r_{fs3},r_{fs4}\right\} for the RSS Systems\end{array}\right.,$$

(32)

where

r₁= r_TLP for the TS and TSS medium-rise systems

r₁= r_ULP for the TS and TSS high-rise systems

r₁= r_fs1 for the TF systems

4.10. The Improved Upper-Bound (IUB) Pushover Analysis

To enhance the accuracy of seismic behavior predictions in high-rise buildings, an advanced and improved upper-bound pushover procedure was proposed. This method, suggested by [63], refines the lateral load pattern and has undergone testing on nine, twelve, fifteen, and twenty-story steel buildings. The findings demonstrate that this new approach is generally more accurate and aligns more closely with nonlinear time history analysis (NTHA), showing its effectiveness and potential for improved seismic behavior prediction. The improved upper-bound (IUB) pushover procedure involves the following steps:

• Initial calculations: perform Steps 1 to 3 in a similar way as outlined in Section 4.8 above.
• Define the lateral load distribution vector. Use the following relationship to define the distribution vector of the lateral forces over the height of the structural system:

$$f_s={\omega }_1^2{m\phi }_1+\left({\omega }_2^2{m\phi }_2{\left(\frac{q_2}{q_1}\right)}_{UB}\right)\ast 0.5,$$

(33)

3.

Evaluate the target lateral displacement at roof. Evaluate the roof’s target lateral displacement, U_r, by applying the following relationship:

$$U_r=U_{rM1}\left(1+{\left(\frac{q_2}{q_1}\right)}_{UB}\ast 0.5\right),$$

(34)

4.

Perform pushover analysis. Use the lateral load determined in Step 2 to perform a pushover analysis of the structure until the target displacement, determined in Step 3, is reached (single-run analysis).

5.

Identify seismic responses. Determine the relevant seismic responses at the step corresponding to the target displacement from the single run in the preceding phase. This includes parameters such as story drifts, element internal forces, etc.

4.11. The Extension of the Improved Upper-Bound Pushover Analysis Procedure (EIUB)

This approach was developed by [64] to accurately predict the behavior of both ordinary and tall structures. The extended IUB examines the seismic response of irregular structures with setbacks. This procedure aims at expanding the improved upper-bound (IUB) analysis to account for the influence of setbacks on the seismic response of mid-rise structures. In addition to the first two modes of vibration utilized in the IUB approach [63], a third mode is used to construct the applied load vector, which may portray the irregularity characteristic. The proposed expanded IUB version was applied to fifteen 10-story structures with varying setback configurations, as well as a reference structure. This new version’s performance was assessed in terms of target displacement, story displacements, story drift, and plastic hinge rotations, which were compared to those produced from prior pushover processes using nonlinear time history analysis (NTHA) as a benchmark.

The extension of the improved upper-bound pushover analysis procedure involves the following steps, summarized:

• Conduct an eigenvalue analysis. Conduct an eigenvalue analysis to compute the natural frequencies of the structure, ωn, and the mode shapes, ϕn, such that the lateral component of ϕn at the roof equals unity.
• Determine the upper-bound of mode contributions. Using the elastic response spectrum of the specified earthquake data, determine the upper-bound of the contribution of the 2nd and 3rd modes q_i/q₁, as described by the following relationship:

$$f^{\prime }={0.48q}_1\left[\left({\omega }_1^2{M\phi }_1\right)\pm \frac{0.26}{0.48}\left({\omega }_1^2{M\phi }_2\frac{q_2}{q_1}\right)\pm \frac{0.26}{0.48}\left({\omega }_3^2{M\phi }_3\frac{q_3}{q_1}\right)\right],$$

(35)

3.

Calculate the lateral load distribution vector. Calculate the lateral load distribution vector throughout the building’s height using the following relationship:

$$f_{i,EIUB}=\mathrm{m}\mathrm{a}\mathrm{x}(f_{i,UB}^{\mathrm{″}},f_{i,Unif}),$$

(36)

4.

Calculate the target roof displacement. Calculate the desired roof displacement Ur by applying either one of the following relationships:

$$U_r=U_{rM1}(1+\frac{q_2}{q_1}{C}_r),$$

(37)

$$U_r=U_{rM1}(1+\frac{q_2}{q_1}{C}_r+\frac{q_3}{q_1}{C}_r),$$

(38)

5.

Perform a pushover analysis. Perform a pushover analysis using the lateral load estimated in Step 3 until the target displacement calculated in Step 4 is reached (single-run analysis).

6.

Determine the seismic responses. Determine the maximum values of the seismic responses from the single-run analysis conducted in step 5.

4.12. The Extended N2 Method

The extended N2 method was developed by Maja Kreslin and Peter Fajfar at the University of Ljubljana [65]. The extended N2 method improves the estimation of seismic demand compared to the basic N2 method by considering higher-mode effects both in plan and in elevation [65]. The extended N2 method provides a more accurate distribution of seismic demand throughout the structure, particularly in the upper part of the building and at the flexible edges. This improvement is achieved by combining the results of basic pushover analysis and those of standard elastic modal analysis. The extended N2 method yields fair, conservative estimates of response and greatly improves the prediction of seismic demand in the specified areas. The extended N2 method involves the following detailed procedure:

• Perform the basic N2 analysis [42,65], which includes a pushover analysis to determine the target displacement at the roof level. This analysis is done separately for each horizontal direction of the building.
• Perform an elastic modal analysis of the building using a 3D mathematical model [66]. This analysis considers all relevant modes of vibration in each horizontal direction. The results of the modal analysis are normalized so that the roof displacement at the center of mass (CM) is equal to the target displacement obtained from the pushover analysis.
• Determine the seismic demand by applying correction factors to the results of the pushover analysis. Two sets of correction factors are used: one for displacements in plan and the other for story drifts along the elevation. The correction factors are determined separately for each horizontal direction and depend on the location in plan.

4.13. The Spectrum-Based Pushover Analysis (SPA) Method

The proposed spectrum-based pushover analysis (SPA) method was developed by Zhang et al. [67] to establish a quick and effective seismic assessment method for tall buildings. The spectrum-based pushover analysis (SPA) procedure consists of several steps, as follows:

• Modelling of the building and applying the seismic gravity loads.
• Eigenvalue analysis of the structure to compute the (ω_n), T_i, (αi, βi, λi), values.
• Compute the force vector of different modes: calculate s_n∗ = mϕ_n, where s_n∗ represents the distribution of incremental lateral forces over the height of the structure for the nth stage of the multi-stage pushover analysis.
• Obtain the design spectrum and compute the target displacements. Obtain the design spectrum for the sites of interest or the mean spectrum of a series of ground motion data. Compute the target roof displacements of the structure for different modes u_ir0, the total roof displacement u_r0, and contributions using Equations (26) and (27):

$$u_{ir}={\alpha }_i\ast u_{r0},$$

(39)

$$u_{ir0}\approx {\left({\sum }_{i=1}^N{\left(u_{ir0}\right)}^2\right)}^{0.5},$$

(40)

5.

The spectrum-based pushover analysis procedure (SPA):

6.

In the first stage, perform a nonlinear static pushover analysis using incremental lateral forces S₁* = mϕ1 until the roof displacement reaches u_1r = α₁u_r0. Obtain the peak response for this stage, noted by r₁. In the second stage, apply incremental lateral forces S₂* = mϕ₂ until the roof displacement equals u_2r = α₂u_r0. Note that the initial condition in this stage is the same as the state at the last step of analysis in the last stage. Obtain the peak response r2 for this stage. Repeat Step 5.2 for the remaining modes considered. For each ith step, the target displacement is uir = αiu_r0, and the initial condition remains consistent with the state at the end of the previous stage.

7.

Envelop the peak response for each step: determine the maximum response from all stages using:

$$r=max\left\{r_1,r_2,\dots ..,r_n\right\},$$

(41)

The value of r represents the seismic responses estimated by the SPA procedure.

5. Adaptive Load Vectors

This method involves a dynamic load pattern that updates in real-time during the analysis, reflecting the ongoing response of the structure.

5.1. The Adaptive Pushover Analysis Method

The adaptive pushover method represents the most advanced nonlinear static method [38,68], as it updates the load pattern uniformly in the inelastic range through each analysis step. The method consolidates instantaneous mode shapes, enables spectral amplifications, and determines modal shapes, using complex calculations and flexible options for modal forces scaling. The basic steps of this method can thus be summarized as follows:

• Eigenvalue analysis: perform an eigenvalue analysis using the stiffness matrix at the previous step.
• Displacement calculation: calculate the displacement profile for the current step based on the modal response. The following relationship is applied:

$$D_{ij}={\Gamma }_J{\Phi }_{ij}{S}_D\left(j\right),$$

(42)

3.

Displacement normalization: normalize the displacements obtained in the previous step to keep the top displacement proportional to the load factor by applying the following formula:

$$\overline{D_i}=\frac{D_i}{{maxD}_i},$$

(43)

4.

Load factor (λ) update: update the load factor and calculate the displacement vector based on the updated load factor.

5.

Displacement application: apply the updated displacement to the structure and solve the system of equations.

6.

Stiffness matrix update: calculate the updated stiffness matrix after the loading is applied.

7.

Loop: return to the first step of the loop to proceed with the next step.

8.

The flow chart in Figure 6 illustrates this procedure.

5.2. The Adaptive Modal Pushover Analysis (AMPA)

The Adaptive modal pushover analysis (AMPA) method was introduced by [70] in their paper titled “Adaptive Spectra-Based Pushover Procedure for Seismic Evaluation of Structures”. Gupta and Kunnath’s adaptive modal pushover analysis addresses the limitations of the traditional modal pushover analysis by iteratively updating the modal load pattern based on the deformed shape of the structure at each step of the analysis. This adaptive process allows for better consideration of higher-mode effects, nonlinear behavior, and changes in structural characteristics under seismic loads, leading to a more accurate assessment of the structure’s performance.

The adaptive modal pushover analysis (AMPA) procedure can be summarized as follows:

• Create a mathematical model: develop a comprehensive mathematical model of the structure.
• Define nonlinear relationships: establish the nonlinear force-deformation relationships of the structural components. This includes the initial stiffness, yield moment, and post-yield stiffness, or an encompassing force-deformation envelope.
• Compute the damped elastic spectrum: calculate the damped elastic response spectrum for site-specific ground motions, considering appropriate damping constants, such as 5% for reinforced concrete structures.
• Perform an eigenvalue analysis: Conduct an eigenvalue analysis to determine periods and eigenvalues. Compute the modal participation factors using:

$${\Gamma }_j=\frac{1}{g}{\sum }_{i=1}^{i=N}{W}_i{\varnothing }_{ij},$$

(44)

5.

Calculate the story forces for modes: compute the forces at each story level for each mode using:

$${\Gamma }_j=\frac{1}{g}{\sum }_{i=1}^{i=N}{W}_i{\varnothing }_{ij},$$

(45)

6.

Compute and combine the base shears: calculate the modal base shears V_j and combine them using the SRSS method to compute the building base shear V:

$$V_j={\sum }_{i=1}^{i=N}{F}_{ij},$$

(46)

$$V=\sqrt{{\sum }_{j=1}^N{V}_j^2},$$

(47)

7.

Scale story forces: uniformly scale the story forces by a scale factor S_n, expressed as:

$$V_j=S_n{V}_j,$$

(48)

where

$$S_n=\frac{V_B}{N_{S}V},$$

(49)

8.

Perform a static analysis for each mode: Conduct a static analysis using scaled incremental story forces for each mode independently. Ensure a bidirectional analysis for modes other than the fundamental mode.

9.

Compute element responses: calculate the element forces, displacements, story drifts, and member rotations using the SRSS combination, updating the results of the previous step accordingly.

10.

Check and update for yielding: At the end of each step, compare the member forces. If yielding is detected, recompute the member and global stiffness matrices and return to Step 4.

11.

Iterative analysis: repeat the process until the maximum base shear or global drift exceeds a predetermined limit.

5.3. The Adaptive Modal Combination Method

The adaptive modal combination (AMC) approach, proposed by [71], is an innovative method for evaluating the earthquake resilience of building structures. It integrates elements of the ATC-40’s capacity spectrum approach, Gupta and Kunnath’s adaptive pushover technique, and Chopra and Goel’s modal pushover analysis [54]. The AMC method uniquely employs lateral forces to mimic the instantaneous inertia force distributions during seismic events and dynamically updates target displacements using energy-based modal capacity curves and constant-ductility capacity spectra. This feature eliminates the need for pre-estimating target displacement before conducting the pushover analysis. The method has been rigorously tested on steel moment-frame buildings, demonstrating accurate estimations of critical seismic performance parameters, such as roof displacement and interstory drift [71]. The procedure can be summarized as follows:

• Modal property assessment: analyze the structure to determine its natural frequencies, mode shapes, and the modal participation factors based on its current state.
• Adaptive lateral load pattern construction: for each mode, devise an adaptive lateral load pattern using:

$$S_n^{\left(i\right)}=m{\Phi }_n^{\left(i\right)},$$

(50)

where $S_n^{\left(i\right)}$ is the load distribution, m is the mass matrix, and ${\Phi }_n^{\left(i\right)}$ is the mode shape at step i. This pattern can be recalculated at specific intervals, balancing computational demands with accuracy.

3.

Energy-based incremental evaluation: determine the next step of the capacity curve for each ESDOF system using:

$${∆D}_n^{\left(i\right)}={∆E}_n^{\left(i\right)}/V_{b,n}^{\left(i\right)},$$

(51)

where ${∆E}_n^{\left(i\right)}$ is the increment of work done by the lateral force pattern and $V_{b,n}^{\left(i\right)}$ is the base shear at step i.

4.

Inelastic response assessment: If the system exhibits inelastic behavior, approximate the global system ductility and post-yield stiffness ratio. A preliminary pushover analysis might be required to establish these parameters.

5.

Generate capacity spectra: for the specific ground motion under consideration, produce capacity spectra in an acceleration-displacement response spectrum (ADRS) format.

6.

Overlay the modal capacity curve with demand spectra: Plot the modal capacity curve alongside inelastic demand spectra. The intersection point represents the dynamic target for the pushover analysis, given by:

$$S_{a,n}^{\left(i\right)}=V_{b,n}/\left({\alpha }_n^{\left(i\right)}W\right),$$

(52)

7.

Extract the response parameters: at the step where the dynamic target point is achieved, retrieve the desired response parameters, such as displacements or member rotations.

8.

Repeat the above steps for the necessary number of modes, typically the first few, for low to medium-rise buildings. Finally, combine the peak modal responses using an appropriate scheme, such as the square root of the sum of squares (SRSS), to determine the total response:

$$r\left({\left(\sum _n{\left(r_n^{\left(ip\right)}\right)}^2\right)}^{0.5}\right)$$

(53)

5.4. Adaptive Upper-Bound (AUB) Pushover Analysis

The adaptive upper-bound (AUB) pushover analysis procedure was developed by A.Y. Rahmani et al. [72] to assess the performance of tall buildings. It considers the effects of higher modes and the dynamic changes in structural characteristics during inelastic responses. The accuracy of the proposed procedure is tested using four steel moment-resisting frame buildings. The procedure can be summarized as follows:

• Initialization: Begin with the model. Set the step to k = 0. Initialize the load factor Δλ and conduct an eigenvalue analysis. The upper-bound (UB) load vector is determined using the given Equation (54):

$$f_{UB}={\omega }_1^2{M\phi }_1+{\omega }_2^2{M\phi }_2{\left(\frac{q_2}{q_1}\right)}_{UB},$$

(54)

2.

Target displacement calculation: using Equation (55), compute the predefined target displacement:

$$u_r=u_{rTLP}\left[1+{\left(\frac{q_2}{q_1}\right)}_{UB}\right],$$

(55)

3.

Load vector application: implement the newly determined load vector.

4.

Displacement check: Examine if the roof displacement has reached the predefined target. If it has, the procedure ends; if not, proceed to the next step.

5.

Frequency update: Check if there is a need to update the frequency due to yielding. If required, apply the UB lateral load pattern as per Equation (54).

6.

Eigenvalue analysis: conduct an eigenvalue analysis using the updated stiffness (K) and mass (M) matrices.

7.

Inelastic response spectrum construction: construct the inelastic response spectrum considering the current ductility ratio using Equation (56):

$${\mu }_k=\frac{d_k}{d_y},$$

(56)

8.

Determine values: Identify the values of S_d1k and S_d2k. Then, calculate the (q₂/q₁) (k) ratio using Equation (57):

$${\left(\frac{q_2}{q_1}\right)}_{UB}=\left|\frac{{\Gamma }_2{S}_{d2}}{{\Gamma }_1{S}_{d1}}\right|,$$

(57)

9.

Lateral load pattern calculation: determine the new lateral load pattern using Equation (58):

$$f_i^{\left(k\right)}=\mathrm{m}\mathrm{a}\mathrm{x}(f_{i,UB}^{\left(k\right)},F_{I,Unif}^{\left(k\right)}),$$

(58)

10.

AUB lateral load pattern application: apply the AUB lateral load pattern as per Equation (59):

$$P^{\left(k\right)}=P^{(k-1)}+∆P^{\left(k\right)}=P^{(k-1)}+∆{\mathsf{\lambda }}^{\left(k\right)}{\overline{f}}^{\left(k\right)}°P_0,$$

(59)

11.

Iteration: return to Step 3 and repeat the process until the roof displacement meets the predefined target.

5.5. A Multi-Mode Adaptive Displacement-Based Pushover Analysis Procedure for Estimating the Seismic Demands of RC Moment-Resisting Frames (MADP)

The multi-mode adaptive displacement-based pushover (MADP) method, cited in [56], combines ‘multi-modal’ and ‘adaptive’ pushover approaches. It starts with a load pattern based on the structure’s elastic mode shape and updates it as the analysis progresses. The method uses the structure’s ‘first-mode’ inelastic mode shape to estimate target displacement accurately and is employed to assess the seismic performance of RC moment-resisting frames. Key advantages include overcoming traditional NSP limitations by considering higher-mode effects and structural changes [73], enabling adaptive and detailed seismic demand evaluation, and utilizing a simple, computationally efficient target-displacement calculation method [44].

The procedure is summarized in the following steps:

• Create a structural mathematical model: develop a mathematical model of the structure that integrates the monotonic nonlinear behavior and degradation features of the structural components.
• Eigenvalue analysis and mode shapes: Conduct an eigenvalue analysis of the elastic building model to compute the natural frequencies ω_n and the associated mode shapes ϕn. Normalize the mode shapes so that the roof component of ϕn equals unity (ϕnr = 1).
• Compute the lateral force distribution: calculate Sn = mϕn, where Sn represents the distribution of lateral forces across the structure’s height for the first stage of the multistage pushover analysis.
• Perform a nonlinear static analysis: Conduct a standard nonlinear static analysis using the lateral load pattern Sn = mϕn. From this analysis, determine the base shear-roof displacement (Sn − mϕU_n) or pushover curve of the structure for the n-th mode. (See Figure 7a.)
• Idealize the pushover curve as bilinear: Draw a straight line (‘Line 1’ in Figure 7b) through the origin to match the elastic segment of the pushover curve. Draw a flat line (‘Line 2’ in Figure 7b) through the pushover curve at the maximum base-shear capacity, V_{bn, max}. The ultimate roof displacement, u_rnu, is assumed as a point of the pushover curve where a loss of 20% in base-shear capacity is observed (see Figure 7b).
• Determine the force–deformation relationship: calculate the force–deformation relationship (see Figure 7d) of the ‘nth-mode’ inelastic SDOF system based on the bilinear idealized pushover curve using the following relationships:

$$\frac{F_{sn}}{L_n}=\frac{V_{bn}}{M_n},$$

(60)

$$D_n=\frac{u_{bn}}{{\Gamma }_n{\varphi }_{nr}},$$

(61)

$$L_n=\frac{m_n}{{\Gamma }_n},$$

(62)

7.

Construct the inelastic SDOF system: Create the ‘nth-mode’ inelastic SDOF system (see Figure 7c) with unit mass and a force-deformation relationship defined in Step 2. Assign a damping ratio equal to the value for the nth mode of the original MDOF structure (ζn).

8.

Compute the peak deformation of the SDOF model:

9.

Calculate the peak deformation of the SDOF model, D_n, using Equation (62), where D_i is the absolute peak deformation for the ith ground motion record.

10.

Calculate the target displacement for nth mode: determine the target displacement of the structure at the roof level for the nth mode, u_rnt, using Equation (63):

$$u_{rnt}={\varphi }_{nr}{\Gamma }_n{\widehat{D}}_n$$

(63)

11.

Perform adaptive multi-stage pushover analyses (the MADP procedure): Apply gravity loads and perform adaptive displacement–control pushover analyses. This involves multiple sub-steps, including calculating new lateral load patterns based on the formation of plastic hinges and updating the target displacement values (see Steps 10.1–10.5.4) [64].

12.

Repeat for the essential modes: Repeat Steps 3–11 for as many modes as deemed necessary. Typically the first three modes of vibration are adequate for most common buildings.

13.

Determine the final structural responses: combine the modal responses obtained from Step 12 using an appropriate modal combination rule, such as square root of the sum of squares (SRSS) or complete quadratic combination (CQC), as guided by references [37].

Figure 7. A schematic representation of (a) the structure’s ‘nth-mode’ pushover curve, (b) the proposed bilinear idealization technique, (c) the ‘nth-mode’ inelastic SDOF model with unit mass (m = 1 kg), n damping ratio, n natural frequency, and (d) the force–deformation relationship as defined by a bilinear idealized pushover curve and Equations (60) and (61).

Figure 7. A schematic representation of (a) the structure’s ‘nth-mode’ pushover curve, (b) the proposed bilinear idealization technique, (c) the ‘nth-mode’ inelastic SDOF model with unit mass (m = 1 kg), n damping ratio, n natural frequency, and (d) the force–deformation relationship as defined by a bilinear idealized pushover curve and Equations (60) and (61).

5.6. Optimal Mode Pushover Analysis

The optimal multi-mode pushover analysis method, developed by Attard and Fafitis in 2005 [74], is a dynamic seismic analysis approach that adjusts continually based on a building’s evolving vibration properties during a simulation. In each step of the analysis, an “optimal” vibration shape for the building is calculated, forming the basis for a simplified vibration model. Lateral forces, based on the building’s previous vibration characteristics, are then applied until the building yields. After each yielding event, the building’s stiffness and vibration properties are recalculated and the simplified model is updated, informing the next step of the analysis. This iterative method is designed to provide valuable insights for creating buildings that are more resistant to earthquakes.

• Define the objective function. Define the objective function using specified constraints and variables.
• Compute a variant load pattern. Calculate a variant load pattern and optimally determine a representative single-DOF system (R-SDOF).
• Determine the building’s capacity curve. Establish the building’s capacity curve and calculate a story force distribution based on the building’s vibration properties. Use this distribution to push the building until a plastic hinge develops [5].
• Capacity curve degradation: As the capacity curve degrades, new vibration properties of the degraded building are determined. Each time a yield stage is reached, this is treated as a ‘new’ building [5].
• Analysis at different stages of degradation. Perform building analysis at various stages of degradation. This considers modes of vibration of the yielded building, which may have multiple yield mechanisms, in the final target displacement prediction.
• Implement an optimization algorithm. Develop an optimization algorithm to minimize the target displacement error between the pushover and nonlinear time history analyses of a few buildings for a specific ground record [8].
• Determine optimal parameters. Determine optimal capacity and demand parameters. Use these parameters to combine the individual mode shapes from each stage of yielding into a single overall response spectrum for the structure. This integrated response spectrum can then be used to assess the performance and safety of the structure under various loading conditions.

6. A Summary of Research Initiatives on Evaluating Different Pushover Analysis Techniques

This section evaluates the efficacy of specific pushover analysis methods by comparing them to popular techniques, such as nonlinear time history analysis and incremental dynamic analysis. Their reliability and relevance are emphasized using post-earthquake data. Moreover, their adequacy is ascertained by contrasting them with outcomes from comprehensive seismic risk assessment methodologies.

6.1. A Comparison of Nonlinear Static Pushover Analysis Methods and Nonlinear Dynamic Analysis Methods

In seismic engineering, the comparison between nonlinear static pushover analysis and dynamic methods, including nonlinear time history and incremental dynamic analysis, has been a significant area of research. Rahmani et al. [72] introduced an adaptive upper-bound pushover analysis for tall buildings, showing its results align closely with nonlinear time history analysis. Concurrently, Rossetto et al. highlighted the efficiency of combining earthquake nonlinear response history analysis with other methods as an alternative to full dynamic analyses. Numerous studies have explored the use of pushover analysis methods alongside incremental dynamic and nonlinear time history analysis. The following sections present insights from research that employed various pushover techniques to assess seismic structural behavior.

The capacity spectrum method (CSM), a traditional tool in seismic engineering, has been found to sometimes underestimate the response of structures due to incorrect assumptions about the equivalent damping ratio. However, an improved version of the CSM [75] was introduced, which considers damage effects. This improved method has shown efficacy when compared with nonlinear dynamic procedures. Furthermore, the results from the study by Gentile and Galasso [76] indicate that nonlinear time history analysis of equivalent SDoF systems is not substantially superior to a nonlinear static analysis coupled with the capacity spectrum method.

The N2 method has been applied to assess the seismic behavior of asymmetric buildings. Research by Bosco et al. [77] found that improved nonlinear static methods provide more accurate estimates of maximum dynamic response than the original N2 method.

The extended N2 method has been applied to structures like an 8-story reinforced concrete building [67]. The results obtained showed that this method reproduces the real torsional behavior of the analyzed buildings effectively.

In the study by Yang Liu et al., the reinforced concrete (RC) shear wall structure’s significance in medium- and high-rise buildings for seismic resistance was discussed. They introduced the spectrum-based pushover analysis (SPA) as an effective method for predicting seismic demand and compared its accuracy with the nonlinear response time history analysis (NLRHA) and other methods [67]. K. Shakeri and Samaneh Ghorbani further explored a pushover procedure tailored for buildings with bi-axial eccentricity under bi-directional seismic excitation [78]. Additionally, Phaiboon Panyakapo [79] validated the cyclic pushover analysis’s efficacy in gauging seismic demands.

The seismic performance assessment of structures has been a focal point of research, leading to the development of various methodologies over the years. Among these, the modal pushover analysis (MPA) and its advanced variants have emerged as pivotal tools to address the inherent limitations of traditional pushover analysis, especially when dealing with structures that exhibit intricate modal interactions. K. Shakeri and Samaneh Ghorbani introduced a modal pushover procedure tailored for asymmetric-plan buildings subjected to bi-directional ground motions, demonstrating its accuracy in predicting peak seismic responses [78,79]. Jing-Zhou Zhang and colleagues developed an improved consecutive modal pushover (ICMP) procedure, which factored in the inelastic properties of structures and the interplay between vibration modes. Their findings suggested that this method surpassed conventional procedures in estimating seismic demands for tall edifices [80]. Additionally, F. Khoshnoudian and M. Kiani proposed a modified consecutive modal pushover (MCMP) procedure designed for one-way asymmetric-plan tall structures with dual systems. This method was found to outperform the FEMA load distribution and the standard MPA procedure in terms of accuracy [61].

When juxtaposed with the nonlinear response time history analysis (NLRHA), Yang Liu and J. Kuang’s research indicated that the spectrum-based pushover analysis (SPA) method was adept at predicting seismic demands, especially in structures where the seismic demand and subsequent damage were pronounced [67]. H. R. Khoshnoud and Kadir Marsono further underscored that the MPA procedure was more precise in estimating the seismic demands of buildings compared to the FEMA356 procedure [81]. Notably, R. Goel and A. Chopra [54] have been instrumental in evaluating the efficacy of pushover analysis methodologies, emphasizing the importance of MPA in the seismic performance-based design of structures [54].

Exploring the realm of upper-bound pushover analysis, A. Y. Rahmani and colleagues introduced an adaptive upper-bound pushover analysis, tailored for the seismic performance assessment of tall buildings. This method considers both the effects of higher modes and the progressive changes in the dynamic characteristics of structures during inelastic response. The results of this approach were found to be closer to those obtained by NTHA [74]. An enhanced version of the upper-bound pushover procedure was proposed for the seismic assessment of high-rise moment-resisting steel frames, showcasing better results and alignment with outcomes from NTHA [63].

To conclude, the realm of seismic engineering has witnessed rigorous exploration and comparison between nonlinear static pushover analysis with dynamic methodologies. Pioneering works by researchers such as Rahmani et al. [64], Rossetto et al., Yang Liu et al. [82], and the notable contributions of Goel and Chopra [54] have underscored the significance of advanced pushover techniques. These methodologies, especially the modal pushover analysis and its variants, have emerged as robust tools, often aligning closely with the results from nonlinear time history analysis, thereby enhancing our understanding and prediction of seismic demands in structures.

7. Discussion

The study looks at the numerous nonlinear static analysis methods applied to evaluate the seismic vulnerabilities and dangers of various structures. Numerous studies have undoubtedly been conducted to address these concerns because of the shortcomings of traditional pushover methods. The pushover analysis method, a performance-based design approach, was developed to identify and evaluate the degree of deformation induced by lateral loads. The research methods, their uniqueness, and their limitations/benefits are clearly defined in the discussion below.

7.1. The First Modal Inertial Pushover Analysis Methods

• The capacity spectrum method (CSM). It is a graphical tool in seismic analysis that visually represents the supply-demand equation, accommodating multiple limit states on a structure’s load-displacement curve. It is simpler and faster than other methods, making it efficient for evaluating a structure’s earthquake resilience. While it can enhance result accuracy by providing larger seismic demand estimates, it has drawbacks: it uses highly damped elastic spectra, which may not accurately reflect inelastic systems’ responses, and it may overestimate seismic demands in irregular, multi-story concrete buildings.
• The improved capacity spectrum method (ICSM) represents an advancement over the traditional capacity spectrum method by introducing new formulas for effective period and damping, as well as iterative procedures for determining target displacement and setting strength limitations. A notable feature of ICSM is its integration of constant-ductility inelastic design spectra, which aligns a structure’s capacity more closely with its inelastic spectrum. This method enhances the accuracy and realism in seismic performance evaluations but also adds complexity due to its iterative nature and the need for precise input parameters. While it offers a more accurate representation of seismic behavior, the increased complexity and reliance on accurate data may pose challenges in practical applications.
• The N2 method uses time-independent lateral displacement shapes and an inelastic response spectrum approximating structural and ground motion characteristics. While simple and useful for a range of buildings, including those with irregularities, it faces limitations in predicting seismic demands for structures influenced by higher modes and is less effective for torsionally flexible or high-rise buildings. The method’s reliance on idealized spectra can also lead to inaccuracies in representing real structural behavior.
• The displacement coefficient method (DCM), introduced in FEMA 273 and later incorporated into FEMA 356, is unique in nonlinear static procedures (NSPs) for its direct estimation of peak inelastic displacements using modifying factors and its approach of calculating displacement coefficients for each mode of vibration. This method enhances the accuracy in predicting inelastic behavior and determining control-node/target displacements in NSPs. However, DCM faces challenges with less accuracy when applied to irregular or highly nonlinear structures and the complexity of calculating displacement coefficients for each vibration mode, which can limit its applicability in certain seismic scenarios or structural types.
• The improved coefficient method, as an advanced version of the coefficient method, uniquely estimates maximum forces and displacements in nonlinear static analysis, focusing on roof displacement to assess structural response. ICM offers more precise estimates of frame roof displacement and base shear than other methods, with a user-friendly approach that enhances design and evaluation processes and enables easy parametric studies. However, its accuracy declines for local response measures like story drifts, and it may introduce errors due to partial plastic mechanisms. Additionally, its reliance on highly damped elastic spectra for seismic demand estimation might not accurately represent the dynamic response of inelastic systems.
• The energy-based pushover analysis enhances nonlinear static analysis by integrating inertial properties and kinetic energy, making it particularly apt for seismic scenarios. It utilizes the fundamental mode shape, effective damping, and a velocity profile to represent a structure’s dynamic behavior, offering deep insights into kinetic energy dissipation through plastic behavior. However, its effectiveness is limited for certain structural types, particularly braced structures, indicating a constraint in its applicability across different architectural designs. Despite this, the method stands out for its ability to merge dynamic aspects into static analysis, providing a more nuanced understanding of seismic impact on structures.
• The N1 method stands out for not requiring a well-defined equivalent single-degree-of-freedom (SDOF) system, facilitating its use in practical structures. It evaluates displacement demand using response spectrum analysis and adjusts for nonlinear behavior, offering accuracy comparable to the N2 method for certain load distributions. While effective for irregular or high-rise buildings by considering higher modes, the N1 method is primarily suitable for low-rise buildings and less effective for high-rise or torsionally flexible plan-asymmetric structures. A significant limitation is its inability to accurately predict seismic demands in structures heavily influenced by higher modes, owing to its limited capture of these effects.
• The mass proportional pushover (MPP) procedure, introduced by Kim and Kurama, simplifies pushover analysis for estimating peak seismic lateral displacement demands. This method stands out by removing the need for complex modal analysis or combination, even when higher-mode effects are significant. Its main advantage lies in conducting a single pushover analysis with lateral forces distributed in proportion to the seismic masses at different levels of the structure, effectively consolidating higher-mode effects into one invariant distribution. This unique approach not only streamlines the estimation of displacement demands but also enhances practicality and efficiency in seismic analysis, particularly for structures where higher-mode influences are crucial.

7.2. The Invariant Multi-Mode Vectors Pushover Analysis

• The modal pushover analysis (MPA) is distinguished by its inclusion of higher modes in estimating seismic demands, offering more accuracy than traditional pushover methods and a precision akin to nonlinear response history analysis for inelastic range responses. Particularly effective for irregular structures, MPA outperforms adaptive pushover analysis in drift profiles and capacity determination, and it accurately captures phenomena in structures with strength irregularities. While simpler than adaptive pushover methods, it is especially adept at estimating demands for unsymmetrical-plan buildings. However, MPA may fall short in accuracy compared to dynamic analysis for buildings with a strong first story or lower half and might inaccurately estimate seismic demands in vertically irregular frames. It also requires incorporating enough modes to accurately estimate story drifts in such structures, highlighting a balance between its advanced capabilities and inherent limitations.
• The multi-mode pushover procedure, designed for steel moment-resisting frames, excels in estimating deformation demands during seismic events. This method distinguishes itself by accurately predicting seismic demands through the calculation of the target interstory drift and the use of modal combinations, considering the frequency content, higher-mode effects, and the interactions between modes. Its ability to provide superior response predictions with minimal computational effort marks a significant advantage, especially in practical engineering applications where efficiency is key. The focus on a comprehensive range of seismic influences ensures more precise deformation demand predictions, enhancing the structural resilience against seismic events. While the specific limitations of this method are not detailed, general challenges could include accurately capturing complex seismic interactions in various structural forms and the need for precise parameter considerations to maintain its effectiveness in diverse seismic scenarios.
• The improved modal pushover analysis (IMPA) method advances beyond the standard modal pushover analysis (MPA) by incorporating the redistribution of inertia forces following the yielding of structures, a significant deviation from MPA’s invariant multi-mode lateral load patterns. Its primary enhancement lies in using the post-yielding structural deflection shape as an invariant lateral load pattern, offering a more accurate reflection of dynamic behavior during seismic events. To balance computational efficiency with accuracy, IMPA employs a two-phase lateral load distribution for the first mode and retains the MPA approach for higher modes. This method represents a sophisticated approach in seismic analysis, aiming to achieve a practical yet precise evaluation of structural responses under seismic loading. However, it might entail complexities in execution due to its advanced analytical requirements.
• The modified modal pushover analysis (MMPA), developed by simplifies seismic demand estimation in frame buildings, assumes that higher vibration modes remain elastic, thereby reducing computational efforts compared to traditional modal pushover analysis (MPA). MMPA often yields larger, more accurate demand estimates, particularly where MPA underestimates, but it can lead to increased conservatism if MPA already overestimates demand. Its primary limitation lies in lightly damped systems (less than 5% damping), where its linear elastic assumption for higher modes might not accurately represent inelastic behavior, potentially resulting in demand overestimation. The method’s accuracy is also notably sensitive to the specific characteristics of buildings and ground motions, making its applicability variable depending on structural details and seismic conditions.
• Generalized pushover analysis (GPA) offers a balance between simplicity and accuracy. GPA utilizes generalized force vectors target deformation demands and generates an envelope of results. It is suitable for preliminary design and design review but limited to elastic behavior and higher-mode consideration.
• The upper-bound pushover analysis method, integrating aspects of the capacity spectrum method, energy balance, and modified modal pushover analysis, uniquely considers higher-mode effects in the nonlinear seismic evaluation of planar building frames, using a single load vector combining the first and a factored second mode shape. This method excels in predicting seismic demands in high-rise buildings, where higher-mode effects are significant, and offers a simplified procedure that yields more accurate nonlinear response estimates, applicable to both new and existing medium high-rise buildings. However, it tends to underestimate nonlinear responses at lower levels while overestimating them at higher levels in medium high-rise structures. Its accuracy may falter, particularly in irregular structures, and it is unsuitable for torsional flexible plan-asymmetric irregular structures. Additionally, the method overlooks variations in inertial load patterns and dynamic responses due to degrading strength and stiffness and does not account for other energy dissipation sources such as duration effects, viscous damping, and kinetic energy.
• The modified and extended upper-bound (MUB) pushover analysis method emerges as an advanced and versatile approach to estimating seismic demands for tall buildings. It seamlessly combines the advantages of the UB pushover method (higher-mode consideration) and the conventional pushover method (accurate lower-level demands) to provide a more balanced and accurate assessment. Its adaptability to various building configurations, including symmetric and unsymmetrical plans, makes it a valuable tool for a broad spectrum of structures. However, it is important to be mindful of its simplifying assumptions and ensure adequate data availability for a comprehensive analysis.
• The improved upper-bound (IUB) pushover analysis refines the lateral load pattern, resulting in more accurate predictions of roof displacements, story drift ratios, and plastic hinge rotations for tall buildings. It is more complex to implement than conventional pushover analysis but provides more reliable results. Its effectiveness and potential for improving seismic behavior prediction have been demonstrated through testing on various steel buildings.
• The extension of the improved upper-bound pushover analysis procedure (EIUB) extends the capabilities of conventional IUB to accurately predict the seismic response of irregular structures with setbacks. It incorporates a third mode of vibration to account for the irregularity characteristic, resulting in improved predictions of roof displacements, story drift ratios, and plastic hinge rotations for irregular buildings. However, its complexity and data requirements make it more suitable for advanced engineering analyses.
• The extended N2 method stands out as a refined version of the conventional N2 method, leveraging its strengths to enhance the accuracy of seismic demand estimation. Its unique feature lies in the integration of results from both basic pushover analysis and standard elastic modal analysis, providing a more reliable assessment of seismic demand distribution throughout the structure. This approach particularly excels in predicting seismic demand at the upper part of the building and at flexible edges, where the conventional N2 method may fall short. However, its complexity and data requirements make it more suitable for advanced engineering analyses compared to the conventional N2 method.
• Spectrum-based pushover analysis (SPA) utilizes response spectra to estimate seismic demand in tall buildings. It offers a balance between simplicity and accuracy, making it suitable for preliminary design and design review purposes. SPA’s key features include response spectra integration, simplified pushover analysis, and the generation of an envelope of results. However, its simplifying assumptions, limited consideration of higher modes, and overestimation of lower-level demands limit its applicability.

7.3. Adaptive Load Vectors

• Adaptive pushover analysis allows for the application of exact force profiles calculated by modal analysis at every step, resulting in a more realistic response that is consistent with the expected system behavior under seismic loading. Adaptive pushover analysis is often more accurate than conventional methods in determining the drift profiles and capacity of irregular structures. However, it may require greater computational efforts for calculating the response at discrete times.
• Adaptive modal pushover analysis (AMPA) is an advanced pushover analysis method that enhances seismic demand estimation through iterative modal load pattern updates based on the structure’s deformed shape. This approach improves accuracy, particularly for taller buildings with complex geometries, but requires higher computational efforts and detailed structural information.
• The adaptive modal combination method (AMC) enhances the precision of seismic demand estimation by dynamically updating the target displacements using energy-based modal capacity curves and constant-ductility capacity spectra. AMC eliminates the need for pre-estimating the target displacement, a limitation of conventional pushover analysis. It accounts for instantaneous inertia force distributions during seismic events, making it more accurate for tall buildings with complex geometries and irregularities. However, its complexity and data requirements make it more suitable for advanced engineering analyses compared to conventional pushover analysis.
• Adaptive upper-bound (AUB) pushover analysis refines the accuracy of seismic demand estimation for tall buildings by considering higher-mode effects and dynamically updating the load pattern based on the structure’s deformed shape. This approach improves the accuracy of predictions, particularly for taller buildings with complex geometries, but requires more detailed structural information and computational resources compared to conventional pushover analysis.
• Multi-mode adaptive displacement-based pushover (MADP) combines the strengths of both multi-modal and adaptive pushover approaches. It considers higher-mode effects and structural changes to provide a more comprehensive and realistic assessment of seismic demand. MADP utilizes the structure’s first-mode inelastic mode shape to accurately estimate the target displacement, making it particularly useful for assessing the seismic performance of RC moment-resisting frames. However, its complexity and data requirements make it more suitable for advanced engineering analyses.
• Optimal mode pushover analysis (OMPA) is a dynamic seismic analysis method that continuously adjusts based on the building’s evolving vibration properties during a simulation. It utilizes an “optimal” vibration shape for the building in each step, forming the basis for a simplified vibration model. Lateral forces are applied until yielding events occur, and the building’s stiffness and vibration properties are recalculated after each yielding event. This iterative method provides more comprehensive and accurate seismic demand estimates, particularly for structures with complex geometries. However, OMPC’s complexity and data requirements make it more suitable for advanced engineering analyses.

8. Conclusions

This paper primarily focuses on outlining methodologies and the evaluation steps for assessing the seismic performance of structures. Emphasizing the critical importance of this performance, especially in earthquake-sensitive areas, the paper underscores the direct impact on inhabitant safety and the economic viability of infrastructure. It presents a comprehensive review of various pushover analysis methods, detailing the unique characteristics and historical development of each technique. The paper also includes in-depth evaluations and comparisons of these methods, anchored in the analysis of previous research, with a specific focus on approaches such as the adaptive modal combination (AMC) method.

The main goal of this research was to offer a comprehensive overview of pushover analysis methods, targeting a varied audience that includes engineers, architects, and urban planners. The study acknowledges the challenge of comparing different pushover analysis techniques due to their complex nature. It focuses on assessing various methodologies regarding their computational efficiency and practical application, aiming to elucidate their effectiveness and suitability in specific contexts.

One of the salient advantages of the methods discussed is their adaptability across different geographical regions, with necessary modifications. These techniques can be instrumental in formulating regional seismic risk scenarios and strategies.

However, it is imperative to note that different pushover analysis methods might yield varying results when gauging seismic performance. These variations arise from the distinct parameters each method considers, with the influence of each parameter differing across methodologies. As a result, there is a pressing need to calibrate existing pushover analysis techniques or to develop new methods. Understanding the current pushover analysis methods in-depth is crucial to make informed modifications or developments, addressing challenges like site-specific characteristics and data uncertainties.

To enhance the existing pushover analysis methods:

• A diverse range of buildings could be examined using refined seismic assessment techniques.
• The outcomes of pre-seismic evaluations, possibly using digital tools, can be juxtaposed with post-seismic data to ascertain the accuracy of the pushover analysis methods.
• Advanced computational techniques, such as machine learning algorithms, can be integrated with post-seismic building evaluation data to rectify the limitations of traditional pushover analysis methods.

By employing these strategies, refinements can be introduced to the parameters used in pushover analysis, leading to significant advancements in the methodology. A profound understanding of each parameter’s impact on structural performance is essential, and their relative significance should be determined to make the pushover analysis more robust and reliable.