Developments in Quantifying the Response Factors Required for Linear Analytical and Seismic Design Procedures

Abstract

Despite the recent initiatives and developments in building design provisions using performance-based design, practicing engineers frequently adopt force-based design approaches, irrespective of the structural system or building irregularity. Modern seismic building codes adopt the concept of simplifying the complex nonlinear response of a structure under seismic loading to an equivalent linear response through elastic analytical procedures using seismic design response factors. Nevertheless, code-recommended seismic design response factors may not result in a cost-effective design with a uniform margin of safety for different structural systems. Previous studies also adopted different methodologies for quantifying the seismic response factors. Hence, there is a pressing need for a comprehensive review covering the developments in quantifying these vital design factors. This paper presents a systematic review of the response factors used with linear analytical procedures recommended by modern building design provisions and the techniques employed in previous studies to evaluate these factors using SDOF and MDOF systems covering analytical, experimental and hybrid assessment approaches. Limitations and gaps identified from the previous studies indicated that most investigations focused on 2D analysis and regular low-rise buildings, while limited studies were directed to shear wall structures, considering mostly unidirectional seismic loading. This comparative review provides insights into previous studies’ methodologies and constraints and identifies the need for future research to calibrate seismic response factors to achieve more economical designs with consistent safety margins for different structural systems.

1. Introduction

Rapid urbanization in major metropolitan areas around the globe has witnessed scarcity and a high cost of land, giving rise to a remarkable increase in the number of multi-story building constructions. These multi-story buildings, if not acceptably designed, can be under significant threat from natural hazards that can cause potential damage to the structures, resulting in substantial economic losses. Lateral loads, particularly in wind- or earthquake-prone regions, usually govern the design of a multi-story building. Inelastic analysis is required to capture the seismic behavior realistically since buildings are expected to experience large deformations under the design of an earthquake. Practicing engineers adopt elastic analysis methods in the design of structures instead of nonlinear analysis, either due to economic reasons or a lack of required knowledge to utilize nonlinear analysis procedures. The inelastic response of a structure is accounted for in the elastic analysis methods by reducing seismic forces and amplifying deformations to arrive at safe designs with optimized costs. Thus, seismic design response factors play an essential part in the safety and economy of structures.

Seismic response factors prescribed in various design codes and guidelines covering different regions, structural systems, and constructional practices may not provide cost-effective designs for different structures and seismic zones with a uniform margin of safety [1,2,3,4,5,6]. Accurately calibrating these factors optimizes the seismic design forces, reducing costs for the overall structural system without compromising structural safety. This highlights the need for verifying the code-provided seismic design response factors of multi-story buildings with various structural systems using well-founded assessment methodologies. The objective of this paper is thus to systematically review the response factors recommended by contemporary building design codes and compare the methodologies adopted in previous studies to evaluate these factors, aiming to provide insights into the need for future research to achieve more cost-effective designs with consistent safety margins for different structural systems.

2. Scope and Literature Review Approach

The existing literature on the development, assessment, and verification of the seismic response factors of multi-story buildings is studied systematically in this paper, as shown in Figure 1. A brief overview of the seismic design response factors in various seismic codes is discussed. The historical perspective showing the evolution of the seismic design factors is presented, followed by an overview of the strength reduction factor defining its various components. The literature is then summarized by providing the available methodologies adopted by various researchers in evaluating the seismic design factors. Finally, the comparative review is followed by a description of the limitations and gaps in the literature, highlighting the need to consistently quantify the seismic design factors for multi-story buildings.

3. Overview of Seismic Design Response Factors

The seismic design forces of structures are derived in design codes by reducing the anticipated elastic seismic forces with a force reduction factor. The factors used to reduce seismic forces and amplify deformations to arrive at a safe design with optimized cost are termed seismic design response factors. Seismic design response factors may be based on engineering judgment and have a limited analytical basis [1]. The values of these seismic design factors adopted in seismic design codes do not provide uniform safety margins covering various structural systems, although they dictate the performance of buildings and the seismic design process. This presses the need for the proper selection of appropriate values of the seismic design factors for building structures, which has been a debatable issue in the development of seismic design provisions and highlighted in several previous studies. The shortcomings in seismic design factors are particularly evident at various performance levels and under bi-directional input ground motions [7,8,9,10]. Hence, the accurate evaluation of seismic design factors and the interrelationships between the different design parameters are essential components in the seismic design of multi-story buildings.

The reserve strength and the ductility levels in a structure are utilized to reduce the seismic forces through the response modification factor [1,2,3,4,5,6]. Lateral load-resisting systems are designed to be deflection-controlled and possess adequate inelastic deformation capacity. The ductile detailing is essential to ensure that the components of these systems achieve a desirable behavior. Some previous studies highlighted the significance of redundancy in the structure to the seismic design response factors [11,12,13,14]. Figure 2 illustrates a typical lateral force–deformation relationship defining the components of seismic response factors, including the response modification factor (R), ductility reduction factor (R_μ), deflection amplification factor (C_d), and structural overstrength factor (Ω_o), as recommended in various building codes [1,2]. The values of the R, C_d, and Ω_o factors depend on the structural system and material.

3.1. Historical Perspective of Seismic Response Factors

The evolution of seismic-resistant designed buildings can be traced several decades after the San Francisco earthquake of 1906 and classified into three phases [15,16]. The first phase adopted the application of the prescribed percentage of building weight as an applied load to the structure [17]. This was published under the first seismic code provisions of the Uniform Building Code (UBC) presented by SEAOC in 1927. The second phase used the concept of the seismic base shear (V) related to zone factor (Z), building system type (K), building period (C), and building weight (W) [18]. The response modification factor was introduced under this phase for the first time in the late 1970s to calculate the design base shear (V_s) of the structure by reducing the elastic base shear (V_e) with the reduction factor (R) using 5% damped acceleration for different systems. The present phase, also defined as the third phase, is based on applying the equivalent lateral force on the structure and employing spectral acceleration maps representing the site seismicity, importance factors, the natural building period, factors affecting the site, and the response modification factors [19].

3.2. Seismic Response Factors in Various Codes

Seismic design factors serve a similar function in all seismic design codes. These factors introduced in the seismic codes are denoted in different terms and assigned different numerical values. Brief comparisons of these factors practiced in various seismic codes are summarized in

Table 1. Comparison of seismic design coefficients.

Seismic Provisions	Applicable Region/ Country	Response Modification Factor	Deflection Amplification Factor	Deflection Amplification Factor/ Response Modification Factor
ASCE 7-22 (2022) [20]	U.S. and other countries	R	Cd	0.50–1.00
Eurocode 8 (2004) [4]	Europe	q a	q	1.00 c
NZS 1170.4 (2016) [21]	New Zealand	µ b	µ	1.00 c
NBCC (2020) [22]	Canada	Rd/Ro	Rd/Ro
MCBC (2015) [6]	Mexico	Q a	Q	1.00 c
UBC
UBC (1994) [23]	U.S. and other countries	Rw	(0.375)Rw	0.375
UBC (1997) [24]		R	0.7R	0.70

^a reduces to 1.0 at T = 0 s and is period-dependent in the short period range. ^b does not reduce to 1.0 at T = 0 s and is period-dependent in the 0.45–0.7 s range. ^c greater than 1.0 in the short period range.

.

4. Methods of Assessing Seismic Response Factors

The literature review of previous assessment studies on seismic response factors is grouped under the following two categories:

• Single degree of freedom (SDOF) systems assessing demands
• Multi-degree of freedom (MDOF) systems assessing capacities

Methodologies adopting SDOF systems for computing the reduction factor “demand” are derived based on the ratio between the elastic strength demand (V_e) and the inelastic strength demand (V_y), corresponding to a defined constant ductility μ [25]. They can also be related to the ratio between the elastic and inelastic response spectral ordinate (S_a,elastic and S_a,inelastic, respectively) related to a specific natural period T [26]. On the other hand, approaches adopting MDOF systems are focused on realistic buildings and actual structures [10,27,28]. Previous studies covering the above two methodologies by various researchers in evaluating the seismic design factors are reviewed and described briefly in the following sections.

4.1. Single Degree of Freedom (SDOF) Systems: Assessing Demands

Newmark and Hall [29,30] parameterized the reduction factor as a function of ductility. The concept proposed by Newmark focused on SDOF structures covering long period and short period ranges. The strength reduction factor was considered equal to the displacement ductility factor for elastic long period and ductile systems with similar initial stiffness and equal displacement. An equal energy approach was proposed by computing the reduction factor for the short period elastic and inelastic structures. The ductility reduction factor (R_μ) was considered equal to unity for short-period structures. Three period zones covering the short period (<0.2 s), intermediate period (0.2 to 0.5 s), and long period (>1 s) were considered in the study. Newmark and Hall established a relationship with the R_μ factor and verified it as a function of ductility (µ) that was period dependent (T) and proposed equations to evaluate the behavior factor (R_μ), as shown in

Table 2. List of equations employed for computing seismic response factors using SDOF.

Author	Equations	Remarks
Newmark and Hall (1973; 1982) [29,30]	Rμ = 1 Rμ = $\sqrt{2\mu -1}$ Rμ = μ	T < 0.2 s 0.2 s < T < 0.5 s T > 1 s
Krawinkler and Nassar (1992) [31]	Rμ = ${\left[c \left(\mu -1\right)+1\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$c$}\right.}$ c (T, α) = $\frac{T^a}{1+T^a}$ + $\frac{b}{T}$	Hardening Value α	Model Parameters
			a	b
		0%	1.00	0.42
		2%	1.01	0.37
		10%	0.80	0.29
Miranda and Bertero (1994) [32]	$\varphi$ = 1 + $\frac{1}{10T-\mu T }$ − $\frac{1}{2T }$ exp [−1.5 (ln T − 0.6)2] $\varphi$ = 1 + $\frac{1}{12T-\mu T }$ − $\frac{2}{5T }$ exp [−2.0 (ln T − 0.2)2] $\varphi$ = 1 + $\frac{T_g}{3T }$ − $\frac{3T_g}{4T }$ exp $\left[{\left(\ln \frac{T}{T_g}-\frac{1}{4}\right)}^2\right]$ Rμ = $\frac{\mu -1}{\varphi }$ + 1 ≥ 1	Rock sites Alluvium sites Soft soil sites Tg is the predominant period of ground motion corresponding to the relative velocity of the linear elastic system with 5% maximum damping for the entire period range
Vidic et al. (1994) [33]	Rμ ≈μ Rμ = c1 ${\left(\mu -1\right)}^{C_R}$ $\frac{T}{T_o}$ + 1 Rμ = c1 ${\left(\mu -1\right)}^{C_R}$ + 1 To = c2${\mu }^{c_T}$ T1	To is the period range in the two segments related to the predominant period of the ground motion T1 T ≤ To T ≥ To c1, c2, cR, and cT depend on hysteretic behavior and damping
Borzi and Elnashai (2000) [25]	q = (q1 − 1) $\frac{\mathit{T}}{{\mathit{T}}_{\mathbf{1}}}$ + 1 q = q1 + (q2 − q1) $\frac{\mathit{T}\mathit{-}{\mathit{T}}_{\mathbf{1}}}{{\mathit{T}}_{\mathbf{2}}\mathit{-}{\mathit{T}}_{\mathbf{1}}}$ q = q2	T ≤ T1 T1 < T ≤ T2 T > T2
Cuesta and Aschheim (2001) [34]	R = R(µ,T) = $\frac{V_y\left(\mu =1, T\right)}{V_y\left(\mu , T\right)}$
Genshu and Yongfeng (2007) [35]	R = Rμ Rα Rξ

.

Krawinkler and Nassar [31] established a relationship to evaluate the force reduction factor using the nonlinear response of SDOF systems. Fifteen ground motions were selected for statistical analysis, representing alluvium soil and rock sites with an earthquake magnitude of 5.7 to 7.7 in the Western USA region. The structural system parameters were examined using the fundamental period (T), strain hardening coefficient (α), the level of yield, and type of inelastic behavior of material (bilinear versus stiffness degrading) apart from the epicentral distance in estimating the sensitivity of strength reduction factors. Estimating the strength reduction factors was proposed using the equations shown in Table 2 by giving importance to the site conditions showing a negligible influence on epicentral distance and stiffness degradation under 5% damping. The study was limited to alluvium soil and rock sites under moderate seismicity regions, and other site classes and seismicity regions were not investigated.

Miranda and Bertero [32] evaluated the strength reduction factors using the elastic strength demands under 124 recorded ground motions over different conditions of soil, representing low shear wave velocity. The study was based on different soil conditions representing rock, very soft, and alluvium sites. The average strength reduction factors were evaluated for SDOF systems on each soil group, assuming 5% critical damping under the influence of magnitude and epicentral distance. Soil conditions significantly influenced the strength reduction factors, whereas the earthquake magnitude and its epicentral distance had a negligible effect. Simplified expressions were proposed to compute the period-dependent force reduction factors (R_μ) with a 5% maximum damping based on the mean strength reduction factors covering different soil types, as shown in Table 2.

Vidic et al. [33] considered SDOF systems using a bilinear response model along with a stiffness degrading model (Q-model) to compute the ductility reduction factor (R_μ) by approximating a bilinear curve. Twenty standard records were selected from seismic events, with moderate epicentral distances and predominant periods in estimating the strength reduction factors. Simplified expressions were proposed in estimating the reduction factor for strong motions, considering two linear segments depending on the hysteretic behavior of the system and its respective damping, as shown in Table 2. The physical boundary conditions of R = μ and T = ∞ are not satisfied with the value of constants other than unity, and structures with very long periods are affected, and hence, the study is limited to short-period structures only.

Borzi and Elnashai [25] derived the response modification factors by evaluating the ratio between the elastic and the inelastic acceleration spectra using regression analysis utilizing a broad range of input ground motion data. The influence of ductility on response factors was assessed using a hysteretic model of elastic perfectly plastic (EPP) nature on the input ground motion parameters like the magnitude of an earthquake, the distance from the epicenter, and soil conditions. The study concluded that ductility played a significant role in evaluating the response modification factor under the influence of the input ground motion parameters. Analyses were conducted on all the selected SDOF systems considering various ductility levels to define functions of the period-dependent response factor for earthquake sets with different ground motion parameters, as shown in Table 2. The study was limited to earthquake magnitudes between 5.5 and 7.9, while investigations on other magnitude ranges were not considered.

The isoductile yield strengths and the corresponding strength reduction factors were studied on elastoplastic SDOF systems by Cuesta and Aschheim [34] using inelastic response subjected to waveforms containing linear, triangular, sinusoidal, and quadratic acceleration components. The SDOF systems were studied under twenty-four simple waveforms characterized by only a few cycles of repetitions, and the response characteristics were compared to assess their similarities and differences under an organized characteristic period. The strength reduction factor (R) was expressed as the ratio of the strength to isoductile yield strength (V_y) under constant ductility (µ = 1), as shown in Table 2. The inelastic response data of the systems due to gradual and shock loading pulses are compared with the expected results based on the concepts of preserved force, equal displacement, and equal energy to estimate the isoductile strengths and the strength reduction factors under constant ductility response. For all pulses, the R factor value increases with an increase in the ductility value and tends to be equal to unity for µ =1 and reaches asymptotic limits in the short- and long-period systems. Only one SDOF system (elastoplastic) with simple waveforms of not more than a few cycles was considered to assess the R factor. This study did not cover the investigation with other systems investigated under realistic ground motions representing both long- and short-period systems.

Cuesta et al. [36] presented a relationship to simplify the R factor with strong ground motions by modifying the original coefficients based on the characteristic period from the previous studies [25,33]. Models representing the bilinear load deformation and stiffness-degrading load deformation with varying amounts of viscous damping were considered with different ratios of post-yield stiffness to initial stiffness. Ground motions were selected to represent both short- and long-duration frequencies. The inelastic spectra obtained for a set of 14 ground motions were compared using several bilinear models; load-deformation relationships were studied, and the errors were compared. The bilinear and stiffness degradation systems performed well with the simplified R factor model, indicating limited stiffness degradation under the selected ground motions with a fair amount of accuracy of the strength estimates, except for the nearly harmonic motions generated at soft soil sites. However, an additional study was recommended to confirm their findings with a single bilinear R factor representing both near and far fault ground motions.

Chakraborti and Gupta [37] studied the dependence of force reduction factors on the input parameters for strength and stiffness degrading structural systems. Two models with different input parameters under a scaled spectrum representing strong ground motions were considered. The input parameters considered for the first model were based on geological site conditions, the magnitude of the earthquake and its duration, the period of the structure, ductility supply, and the ductility demand ratio. In contrast, the second model was characterized by the input based on the normalized design spectrum, the ductility demand ratio, and ductility supply. Lower values of the force reduction factors were identified in the case of high ductility demand ratios with flexible oscillators, making a slight difference to the degradation factors. The study was conducted to estimate the lateral yield strength of the structural system with known ductility demand and supply, which may not apply to existing structures due to underestimating the maximum displacement. The study did not consider structures under high-magnitude earthquakes and other seismic regions, including near source records.

Genshu and Yongfeng [35] analyzed the R factor for the modified-Clough hysteretic model. The study compared it with elastic perfectly plastic (EPP), bilinear elastic (BILE), and shear slipped (SSP) models to separate the effects of ductility, energy dissipation, and stiffness degradation with different soil types on R spectra. The above models were analyzed with varying damping and post-yield stiffness ratios by normalizing the system periods for different earthquake records to maintain the peak characteristic (T) of R spectra near the characteristic period (T_g) for better correlation. An equation for R spectra was proposed as a product of functions representing mean ductility coefficient (R_μ), mean post-yield stiffness coefficient (R_α), and mean damping coefficient (R_ξ) by maintaining the peak characteristics of R spectra to unity, as shown in Table 2. Ductility was the most prominent factor for determining R, which increased with an increase in ductility value compared with post-yield stiffness and damping, which had a minor effect on the structures with a short period range. The study was conducted with a maximum acceleration of 0.2 g by scaling all the records for easier comparison. The seismic behavior of the structures and the influence of the R factor with varying accelerations representing other ranges of seismic events need to be investigated to validate the study’s recommendation.

Jalali and Trifunac [38] analyzed the force reduction factors for structures near earthquake faults, such as near-fault, fault-parallel, and fault-normal strong motion displacements. A simple two-column structure with the strength reduction factors under different earthquake faults was considered for the study, and the results were summarized. For the structures designed close to near-fault or active-fault regions, the study recommended the use of constant strength reduction factors equal to $\sqrt{2\mu -1}$ for all elements of motion for both long and short periods. The structures designed for fault-parallel motions in the study presented smaller values of the R factor than the design R values representing non-conservatism and recommended change. However, the structures designed for fault-normal pulses had higher values of R factors than the selected R values, showing a conservative design and describing the reduction amplitudes properly near strong earthquake faults. The study used only a simplified form of ground motions using smooth pulses representing strong motions for peak amplitudes, which may not provide realistic results, as real scaled records were not employed in their analysis.

Hatzigeorgiou [39] developed a procedure to evaluate the ductility demands of structures under multiple near-fault earthquakes using the appropriate strength reduction factor (q) considering the viscous damping, period of vibration, and post-yield stiffness ratio. Empirical expressions were derived through nonlinear regression analysis to propose the estimation of mean strength reduction factors satisfying the basic principles of dynamic inelastic analysis. The results provided a good similarity between the analysis and model results. The study was focused on ideal structures where smaller earthquakes are expected to have similar behavior factors and similar ductility demands under the design earthquake. In contrast, seismic sequences provide smaller behavior factors requiring greater ductility demands, possibly resulting in more significant structural damage. The study recommended a performance-based design to overcome the above drawback for actual structures with the design earthquake.

Zhang et al. [40] investigated the strength reduction factor of SDOF systems for sequence-type records of mainshock–aftershock events classified under site classes B and C, considering the respective displacement ductility and cumulative damage. An SDOF systems series of elastic plastic type scaled to five relative intensity levels of ductility factor and the damage index were studied using nonlinear time history analysis under different site conditions to determine the strength demands and the aftershock effects of the natural period and proposed expressions as a function of the damage index, period, and the ductility factor. The study indicated that the damage-based force reduction factor (R_D) was influenced strongly by the period for short-period structures compared to the long period. Furthermore, the ductility-based force reduction factor (R_µ) was more than the damage-based force reduction factor (R_D) for long-period structures. The proposed study was limited to only two types of soil class, and investigation of other soil types was not considered.

Molazadeh and Saffari [41] considered the effects of ground motion duration on different hysteretic models to study the structural behavior and assess the effects on ductility and energy dissipation. Two sets of seismic records representing long- and short-duration events were considered in conducting nonlinear response history analysis (NLRHA) on five SDOF models. The behavior of the elastic perfectly plastic, stiffness deterioration and degradation under two sets of ground motion with different values of R_μ was evaluated. The behavior of structures with pinching-degradation models compared to EPP models was more severe under long periods than under short periods, with lower effects of R_μ on hysteretic energy dissipation and ductility. The regression equations proposed were limited to the structures with the ductility values considered in the study to ensure a satisfactory structural performance, where the required ductility capacity needs to be greater than the ductility demand of the structure obtained by the nonlinear analysis.

Feng Wang et al. [42] developed a spectrum of damage-based strength reduction factors for systems subject to bi-directional input ground motions more suitable for 2D analyses. The study presented an inelastic system of single-mass bi-degree of freedom (SMBDOF) for orthogonal bi-directional input ground motions, where the x-component constructs damage-based force reduction factor spectra (R_b spectra) with a constant damage index from the park-ang model. Utilizing 178 input ground motion records across site classes, the study established mean R_b spectra based on ductility capacity and damage level. Comparative analysis with SDOF system R spectra indicated increased strength demand in specific cases when coupling two-component responses in SMBDOF systems. R_b spectra trends remain consistent across site classes, reflecting site condition characteristics. The period ratio is crucial, as significant differences may worsen seismic damage. The study derives curves of R_b spectra with a period ratio of 1 through regression analysis, incorporating correction factors. Validation against actual mean spectra demonstrates a good match, confirming the proposed procedure’s effectiveness.

It is clear from the literature review of over four decades that the previous studies conducted on seismic response factors on SDOF systems were mainly based on hypothetical systems with hysteresis relationships and not actual structures. Most of the earlier work was based on ductility reduction factors without accounting for the contribution of overstrength factors. Several studies based their analytical work on seismic scenarios representing moderate to severe earthquakes and simple waveforms, while studies covering multiple seismic scenarios, including aftershock events, were limited. For the sake of brevity, the review of the SDOF systems is summarized in Table 2 and

Table 3. Summary of the literature on SDOF systems.

Author	Analysis Type and Methodology	Type of Structure and Structural System	Seismic Design Factors	Seismic Factor Dependencies	Soil Site Class	Features of Input Ground Motions
Newmark and Hall (1973; 1982) [29,30]	2D analytical	Long and short period range structures	Rµ	T, µ	Alluvium, very soft, and rock sites	Elastic response spectra for long and short periods
Krawinkler and Nassar (1992) [31]	2D analytical	Short period range structures	Rµ	T, µ, α	Alluvium soil and rock sites	15 GM with 5.7 to 7.7 EQ magnitude
Miranda and Bertero (1994) [32]	2D analytical	Short period range structures	Rµ	T, Tg, µ	Alluvium, very soft, and rock sites	124 GM with low shear velocity
Vidic et al. (1994) [33]	2D analytical	Short period range structures	Rµ	T, µ, c	Alluvium and stiff soil sites	20 strong motion EQ records
Borzi and Elnashai (2000) [25]	2D analytical	Low-rise structures	R (q)	T, µ	Soft, stiff, and rock sites	43 GM with 5.5 to 7.7 EQ magnitude
Cuesta and Aschheim (2001) [34]	2D analytical	Elastoplastic structures	R	T, µ, Tg	Alluvium and stiff soil sites	24 simple waveforms linear, triangular, sinusoidal, and quadratic
Cuesta et al. (2003) [36]	2D analytical	Bilinear and stiffness degrading models	R	T, µ, Tg	Alluvium and stiff soil sites	14 GM of short- and long-duration frequencies
Chakraborti and Gupta (2005) [37]	2D analytical	Short period range structures	R	T, µ, Tg	Alluvium, intermediate, and hard rock sites	Several accelerograms of 3 to 8 EQ magnitude
Genshu and Yongfeng (2007) [35]	2D analytical	Modified Clough hysteretic model	R	T, µ, Tg	Site classes A, B, C, and D	370 seismic records, different site classes, and 0.2 g acceleration
Jalali and Trifunac (2008) [38]	2D analytical	Short period range structures	R	T, µ	Stiff and soft soil sites	Near-field EQ with EQ magnitude ranging from 4 to 8
Hatzigeorgiou (2010) [39]	2D analytical	RC structures	R (q)	T, µ	Soft, stiff, dense, and rock sites	110 near-fault Eqn. with 5.5 to 7.8 magnitude
Zhang et al. (2017) [40]	2D analytical	Elastoplastic structures	Rµ, RD	T, µ	Site classes B and C	Mainshock > 0.10 g PGA and aftershock > 0.05 g
Molazadeh and Saffari (2018) [41]	2D analytical	Short period range structures	Rµ	T, µ	Site class D	Short and long duration records of varying magnitudes
Feng Wang et al. (2023) [42]	2D analytical	Short period range structures	Rb, R	T, µ	All site classes	178 ground motion records with bi-directional loading

. The equations employed in the respective research are also listed in Table 2, whereas the review summary of the SDOF systems is presented in Table 3.

4.2. Multi-Degree of Freedom (MDOF) Systems: Assessing the Capacities

The review of the seismic response factors under the MDOF systems is classified based on the different structural systems and presented in the following five categories in the present study:

• Steel-braced structures.
• Steel frame structures.
• Steel walls and other structures.
• RC frame structures.
• RC shear wall structures.

4.2.1. Steel-Braced Structures

Calado et al. [43] evaluated the strength reduction factor (q) of low-rise steel buildings by analyzing three structural systems using a simple steel frame, cantilever, and diagonally braced cantilever buildings. The analysis used a single-story frame to study the cantilever buildings with and without braces, whereas two and three stories were used for frame buildings. Natural and artificial accelerograms represented various seismic loading conditions with 15 to 25 s. durations. The study characterized the failure of the structures using a numerical index based on the damage accumulation. The study concluded that the q factors were very conservative based on their parametric results influenced by different natural frequencies. The study was limited to regular 2D ideal structures with limited accelerograms, and other structural systems were not investigated.

Balendra and Huang [44] explored overstrength and ductility factors in low seismic conditions for three steel frame types: moment-resisting frames (MRFs), concentrically braced frames (CBFs), and semi-rigid frames. They designed three, six, and ten-story symmetric office buildings following BS 5950 guidelines, addressing P-Δ effects. As the study region (Singapore) lacked seismic provisions, seismic performance was evaluated using inelastic pushover analysis (IPA) and the N2 method for the design earthquake. MRFs exhibited significant overstrength and ductility, with overstrength decreasing by up to 50% and ductility increasing by over 25% when rigid connections were replaced with semi-rigid ones. Response modification factor values ranged from 16 to 6 for MRFs, 8.5 to 3.5 for CBF, and 15 to 3.5 for semi-rigid frames. The obtained R factors varied widely, and the study, confined to regular structures and pushover analysis, did not assess seismic performance through nonlinear dynamic analysis.

Asgarian and Shokrgozar [45] investigated ductility, overstrength, and response modification factors for the Buckling Restrained Braced Frames (BRBFs) in four to fourteen-story buildings with split X, diagonal, chevron (V and inverted V) bracing configurations. Employing the Iranian National Building Code (2006) and seismic parameters from Iranian Earthquake Resistant Design Codes (2005), the study used two-dimensional IPA, linear dynamic, and nonlinear inelastic dynamic analysis (IDA) with three natural ground motion records. Seismic parameters for BRBF in various configurations were evaluated using the ultimate limit state and the allowable stress methods. The study found conservative seismic response values compared to design codes, indicating decreased overstrength and ductility factors with increased story height. However, the study’s reliance on a limited number of seismic records may not fully capture the variability in input ground motions.

Mahmoudi and Zaree [46] assessed response modification factors using conventional steel CBFs and BRBFs, selecting thirty conventional steel CBFs and twenty BRBFs ranging from three to twelve stories. For conventional CBFs, chevron V, X-type bracings, and chevron-inverted V were considered, while BRBFs were studied with chevron-inverted V and chevron V bracing types. The response modification factor and ductility values of CBFs and BRBFs were assessed considering the life safety structural performance levels using IPA to evaluate the factors on ductility, overstrength, and force reduction. Overstrength and R factors increased with the number of braced bays, decreasing slightly for taller BRBFs due to increased ductility. The study identified conservative response modification factors in Iranian seismic codes, recommending necessary modifications. The analysis was based on IDA for ideal structures up to 12 stories without investigating seismic response with real seismic records.

Farahi and Mofid [47] evaluated the R_μ, Ω_o, and R factors of chevron knee bracing (CKB) frames adopting the FEMA P695 approach involving inelastic static and dynamic analyses. The IPAs were conducted to evaluate the overstrength and ductility, while IDA was conducted to obtain the selected buildings’ collapse margin ratios (CMRs). The CMR values were modified to derive adjusted collapse margin ratios (ACMRs) for each frame type. These values exceeded the acceptable collapse margins of FEMA P695. The R factor values considered were conservative, and the collapse margins decreased with the increase in structure height. The ductility and stiffness were higher in the CKB frames with more spans than with fewer bays. The study was limited to a few 2D CKB frames, and further investigation with a wider range of structures was recommended.

Moni et al. [48] employed twelve regular BRBF buildings with four, six, and eight stories using split X and chevron-inverted V braces to evaluate the seismic design factors. The buildings were analyzed and designed using the NBCC [3] to match the response spectrum of Vancouver, Canada. Two-dimensional nonlinear dynamic time history analysis (THA) was performed using nine natural records and one artificial record to evaluate the seismic performance and estimate the response modification, overstrength, and ductility factors. The response modification factors, ductility, and overstrength met the code limits and were conservative, decreasing with increased building height and span length. The study was limited to regular structures with limited bracing systems.

Kheyroddin and Mashhadiali [49] proposed the R factor of steel concentrically braced frames (CBFs) forming a hexagonal pattern using the V and inverted V bracings. They quantified the R factor through FEMA P695 methodology, conducting nonlinear static and dynamic analyses on four-, ten-, and 20-story buildings. The proposed hexa-braced buildings, designed with an assumed value of R factor complying with ASCE standards, were compared with similar buildings braced by X braces as a benchmark. Two-dimensional IPA and IDA, employing two sets of near-field and far-field scaled ground motion records, were performed. The CMR based on FEMA P695 methodology obtained until collapse demonstrates adequate safety margins for both X-braced and hexa-braced frame models. The study was limited to regular buildings and recommended further investigation covering other codes with experimental verification.

Masood Yakhchalian et al. [50] evaluated the deflection amplification factor (C_d) of steel buckling restrained braced frames (BRBFs) using six regular low-to-midrise structures ranging from two to twelve stories under the design earthquake. The study compared the adequacy of the ASCE recommended C_d value of 5.0 for estimating inelastic maximum drift ratios (IMIDRs) in steel BRBFs. They conducted 2D inelastic analyses using 78 seismic records and observed that the C_d underestimates the IMIDR of lower stories while overestimating for taller structures. A more realistic approach was proposed to estimate the C_d using the particle swarm optimization (PSO) algorithm. The influence of the strain hardening ratio on C_d and C_{d Roof} relative to the response modification coefficient (R) with the ductility reduction factor (R_μ) was also investigated.

Shen Li et al. [51] assessed the seismic response factors of regular Y-shaped eccentrically braced high-strength steel frames ranging from four to sixteen stories. The study employed ordinary steel in links and high-strength steel in beams and columns. The high-strength steel in beams and columns was used to reduce the cross-section, maintaining good ductility with excellent plastic deformation ability during seismic events. The study assessed R and C_d by conducting 2D performance-based design, static elastic plastic, and incremental dynamic analyses utilizing ten ground motion records. The R and C_d were calculated considering the number of stories and the link lengths and established linear relationships between the seismic design response factors.

Mahdi Mokhtari and Ali Imanpour [52] verified the overstrength and the ductility-related force modification and deflection amplification factors for 14 prototype steel-resisting braced frames (MKFs) in Vancouver, Canada. Regular buildings ranging from six to fifteen stories were designed, and 2D numerical models were developed to assess the seismic design factors. The IDAs were conducted using three sets of ten seismic records, each representing subduction intraslab, shallow crustal, and subduction interface events. The study suggested designing the proposed MKF system as a moderately ductile system in high seismicity regions, with a maximum height of 40 m using 1.60 as the overstrength factor and 3.0 as the ductility factor.

The presented review of the seismic response factors of MDOF systems for steel-braced structures in the present study covered investigations published since 1995. Most of the earlier work was based on 2D analytical work with unidirectional seismic loading. Regular low-rise- to medium-rise-braced frames were analyzed using IPA and IDA with limited research on relatively tall buildings. In earlier studies, irregular structures under the effect of bi-directional loading representing real buildings with varying heights employing 3D inelastic analysis were not considered. The review of the MDOF systems representing steel-braced structures is summarized and presented in

Table 4. Summary of the literature on MDOF systems—steel-braced structures.

Author	Analysis Type and Methodology	Type of Structure and Structural System	No. of Stories	Seismic Design Factors	Remarks
Calado et al. (1995) [43]	2D analytical	Regular MRFs, cantilever, and braced	1, 2, and 3	R(q)	Adequate margin of safety
Balendra and Huang (2003) [44]	2D analytical and IPA	Regular MRFs, CBFs, and semi-rigid	3, 6, and 10	Rμ and Ωo	Adequate margin of safety
Asgarian and Shokrgozar (2009) [45]	2D analytical, IPA, and IDA	BRBFs with diagonal, split X, chevron V, and inverted V bracings	4, 6, 8, 10, 12, and 14	Rμ, Ωo, and R	Adequate margin of safety
Mahmoudi and Zaree (2010) [46]	2D analytical and IPA	Regular CBFs and BRBFs with X, chevron V, and inverted V bracings	3, 5, 7, 10, and 12	R	Adequate margin of safety
Farahi and Mofid (2013) [47]	2D analytical, IPA, and IDA	Regular chevron knee bracings	2, 4, 6, 10, and 14	Rμ, Ωo, and R	Adequate margin of safety
Moni et al. (2016) [48]	2D analytical, IPA, and THA	Regular BRBFs with chevron-inverted V and split X bracings	4, 6, and 8	Rμ, Ωo, and R	Adequate margin of safety
Kheyroddin and Mashhadiali (2018) [49]	2D analytical, IPA, and IDA	Regular CBFs with hexagonal patterns and X-frames	4, 10, and 20	R	Adequate margin of safety
Masood Yakhchalian et al. (2020) [50]	2D analytical, IPA, and IDA	Regular BRBFs	2, 4, 6, 8, 10, and 12	R and Cd	Adequate margin of safety
Shen Li et al. (2021) [51]	2D analytical, IPA, and IDA	Y-shaped eccentrically braced high strength	4 to 16	R and Cd	Adequate margin of safety
Mahdi Mokhtari and Ali Imanpour (2023) [52]	2D analytical, IPA, and IDA	Moment-resisting knee brace (MKF)	6 to 15	Rd and Ro	Adequate margin of safety

for a better understanding.

4.2.2. Steel-Framed Structures

Mohammadi [53] considered elastoplastic MDOF models with various dynamic characteristics to evaluate the effect of the deflection amplification factor of steel frame buildings. Buildings with five to fifteen stories were selected and designed, as per AISC design procedures. A two-dimensional nonlinear dynamic analysis assessed the seismic performance under twenty-one accelerograms from ten different earthquakes. The study proposed an approximate approach to evaluate maximum inelastic deformation in a structure using given strengths and deflection amplification factors. The study concluded that the values of the deflection amplification factor for the MDOF systems obtained were more significant than the theoretical values. The study was limited to regular structures with a specific structural system, while other structural systems were not investigated.

Foutch and Wilcoski [54] proposed an approach for determining R factors of steel MRFs using both analytical and previous experimental work. Steel regular buildings were designed and detailed per the International Building Code (IBC). The IDA was performed on each building under 20 near-fault input ground motions to suit the Los Angeles site. The R factors evaluated were conservative, indicating that the values can be reduced considerably. The demand and capacity for the proposed study were based on story drift, which may not be effective on stiff systems, like steel-braced frames and shear walls. The study was limited to the moment-resisting frames, while other structural systems were not investigated.

Stefano et al. [55] studied the impact of overstrength in plan-irregular multi-story steel frame buildings, comparing the inelastic seismic response of a six-story asymmetric steel building with a single-story building with negligible overstrength. Single-story and six-story asymmetric buildings were designed in two different directions based on Eurocode 8. Two-dimensional IPAs and THAs using 30 artificial accelerograms were employed to assess the ductility demand and the torsional behavior. The study emphasized the significant role of overstrength, especially in buildings under torsion, and highlighted greater ductility demands in the upper floors of multi-story asymmetric buildings with higher overstrength, contrary to results from corresponding single-story buildings. Furthermore, the study was limited to only two ideal buildings investigated with 2D IPA and THA, while 3D nonlinear analysis was not considered to estimate the torsional effects of irregular structures.

Karavasilis et al. [56] analyzed a set of 36 regular-plane steel MRFs to evaluate their response in estimating the strength reduction factors (q). The Eurocode was used for frame design, and ground motions were based on the EC8 response spectrum using a PGA of 0.35 g for soil class B. Employing two-dimensional IDA with thirty-four natural records from the PEER (2005) database, the study compared results with experimental findings from the nine-story SAC steel frame (SAC steel project) by Gupta and Krawinkler [57]. The proposed strength reduction factor was independent of sectional dimensions but significantly dependent on the story height and the capacity factor. The study observed that the conventional strength reduction factor recommended by the seismic design code may or may not be conservative. The study was limited to buildings of 20 stories and did not account for long duration–high amplitude seismic motions.

Kang and Choi [58] proposed the evaluation of the response modification factor (R) of steel MRFs as a product of the ductility factor (R_μ) for SDOF systems, the modification factor of MDOF systems (R_M), and the strength factor (R_S). Regular frames of four to sixteen stories were designed with varying soil profiles and seismic zone factors, as per UBC standards and the provisions of the AISC code. A two-dimensional IPA was conducted on selected frames to estimate the R_µ, R_M, and R_S factors, and the failure mechanisms were assessed. The R factor was evaluated for frames with strong column–weak beam (SCWB) models and weak column–strong beam (WCSB) models. The mean value of R factors for the SCWB models exceeded the code-recommended values, while the values for the WCSB models were below the code recommendations. The proposed rule was verified on regular steel MRFs using IPAs only. The study recommended evaluating the seismic performance with other structural systems to ensure a uniform margin of safety.

Izadinia et al. [59] derived and compared the seismic response factors using capacity curves from different pushover analysis methods. Three regular steel frames from the SAC project of three to twenty stories were considered in the study to evaluate the R_μ, Ω_o, and R factors. Conventional pushover analysis (CPA) and adaptive pushover analysis (APA) were conducted on each frame, and the results were compared. The study adopted force- and displacement-based adaptive pushover analyses (FAPA and DAPA) using various constants and load patterns. The study identified a relative difference of 16% and 17% of R factor and ductility ratios (µ) between CPA and APA (FAPA or DAPA) due to larger results in APA using different seismic records. The study was limited to regular 2D MRFs with static pushover analysis, while dynamic analysis was not considered.

Ferraioli et al. [60] assessed redundancy, overstrength, ductility response modification, and strength reduction (q) factors in six regular and six vertical irregular steel MRFs of varying heights (three to nine stories) and bay configurations. Designed for soil class A as per the Italian seismic code, IPAs and IDAs with 12 ground motions were conducted. The redundancy factor for both regular and irregular frames was conservative compared to that recommended by Eurocode 8 and the Italian seismic code values. The strength reduction factor evaluated was below the code-recommended values for higher story frames due to the limited rotation capacity of the first-story columns. The code recommended values of the strength reduction factor were conservative in the case of a fixed connection at the base and recommended a full-strength connection at the column base-to-foundation. The study was limited to 2D frames and recommended further investigation of other types of connections.

Reyes-Salazar et al. [61] investigated ductility (µ), ductility reduction factor (R_µ), and energy dissipation (E) in low-rise and medium-rise steel buildings of 3- to 10-story, designed with strong column-weak beam principle according to IBC. Two-dimensional nonlinear THA was conducted using 20 strong ground motions to assess the µ, R_µ and E, at local, story, and global structural levels. The values of the local ductility capacity of beams were larger in medium-rise buildings than in low-rise buildings. The study proposed the findings based on the ratios of global ductility to local ductility (Q), local ductility reduction factor to local ductility (Q_L), story ductility reduction factor to story ductility (Q_S), global reduction ductility factor to global ductility (Q_G), story ductility reduction factor to story energy dissipation (P_S) and global ductility reduction factor to global energy dissipation (P_G). The values were evaluated as 0.33 for the Q ratio, 0.32 to 0.6 for the Q_L ratio, 0.60 to 1.10 for the Q_S ratio, and 0.73 to 0.86 for the Q_G ratio. The study was limited to medium-rise regular steel buildings with only one structural system, and other structural systems with high-rise buildings were not investigated.

The study by Macedo et al. [62] aimed to evaluate the impact of a more logical selection of the behavior factor (q) of steel moment-resisting frames (MRFs) designed according to Eurocode 8. The seismic design of steel MRFs was conducted on 30 buildings with five frame heights ranging from two to eight stories, using both code-prescribed q and an improved force-based design (IFBD) approach. The seismic behavior was assessed using nonlinear static and response history analyses utilizing 40 input ground motions. The IFBD-designed steel frames exhibited inelastic demands evenly distributed along the building height and ductility demands that align better with design assumptions.

Fayaz Rofooei and Ali Seyedkazemi [63] determined the seismic performance factors (SPF) for steel diagrid structures spanning six to twenty-four stories. The study considered the post-buckling behavior of diagonal compression members, along with variations in span length and diagonal angles. SPF calculations were based on the ATC-19 coefficient method, while FEMA P695 was employed to evaluate accuracy. Seismic collapse capacity was assessed through IDAs with 22 far-field records. Their results indicated that as diagrid structures increased in height, there was a decrease in the response modification factor, overstrength, and median collapse intensity, associated with an increase in the ductility factor. Additionally, the collapse margin ratios of the diagrid structures exceeded FEMA P695 recommended limits.

Maysam Samadi and Norouz Jahan [64] examined the impact on seismic design parameters such as the response modification factor (R), overstrength factor (Ω), deflection amplification factor (C_d), and the damping ratios for tall steel buildings. The study examined regular steel buildings with 28 and 56 stories, featuring steel-braced and reinforced concrete (RC) shear wall cores with outriggers placed at every quarter of the building height, resulting in 44 building models. Seismic parameters were assessed using the modal response spectrum (MRS), pushover, and nonlinear time history (NLTH) 3D analyses. Including the outriggers increased the response modification factor, overstrength, stiffness, and damping ratios, particularly in the buildings with RC core walls, while reducing ductility in both systems. Their study also identified inadequacy in the code-recommended C_d values.

The evaluation of the seismic response factors of MDOF systems for steel MRF structures in the past two decades was mainly based on 2D analytical works. Regular low-rise- to medium-rise-braced frames were analyzed using IPAs and IDAs, with little work on relatively tall buildings. In earlier studies, irregular structures under the effect of bi-directional loading representing real buildings with varying heights employing 3D inelastic analysis were not considered. Limited experimental validation was performed to validate the analytical work. For the sake of brevity, the review of the MDOF systems representing steel frame structures is summarized and presented in

Table 5. Summary of the literature on MDOF models—steel frame structures.

Author	Analysis Type and Methodology	Type of Structure and Structural System	No. of Stories	Seismic Design Factors	Remarks
Mohammadi (2002) [53]	2D analytical and IDA	Regular MRF buildings	5, 10, and 15	Cd	Adequate margin of safety
Foutch and Wilcoski (2005) [54]	2D analytical, experimental, and IDA	Regular MRF buildings	3, 9, and 20	R	Adequate margin of safety
Stefano et al. (2006) [55]	2D analytical, IPA, and THA	Plan irregular MRF buildings	1 and 6	Ωo	Proposed improvement
Karavasilis et al. (2007) [56]	2D analytical, experimental, and THA	Regular MRF buildings	3, 6, 9, 12, 15, and 20	R (q)	Not valid for long duration–high amplitude GMs
Kang and Choi (2011) [58]	2D analytical and IPA	Regular MRF buildings	4, 8, and 16	R	Adequate margin of safety
Izadinia et al. (2012) [59]	2D analytical and IPA	Regular MRF buildings	3, 9, and 20	Rμ, Ωo, and R	Conservative with CPA and APA
Ferraioli et al. (2014) [60]	2D analytical, IPA, and IDA	Six regular and six irregular MRF	3, 5, 7, and 9	Rμ, Ωo, and R (q)	Adequate margin of safety
Reyes-Salazar et al. (2018) [61]	2D analytical and THA	Regular MRF buildings	3 and 10	µ, Rμ, and E	More conservative in medium-rise buildings
Macedo et al. (2019) [62]	2D analytical, IPA, and IDA	Five regular MRF frames	2 to 8	R (q)	Adequate margin of safety
Fayaz Rofooei and Ali Seyedkazemi (2020) [63]	2D analytical, IPA, and IDA	Regular steel diagrid structures	6 to 24	R and Ωo	Adequate margin of safety
Maysam Samadi and Norouz Jahan (2021) [64]	3D analytical, MRS, IPA, and NLTH	Two regular steel-braced buildings with outriggers	28 and 56	R, Cd, and Ωo	Adequate margin of safety, R, and Ωo. Inadequate Cd

.

4.2.3. Steel Walls and Other Structures

Elnashai and Broderick [65] evaluated the strength reduction factors (q) of moment-resisting composite frames. Twenty frames from two, three, six, and ten stories grouped under two sets of ten frames each were analyzed and designed to the member capacity, as per the Eurocodes. The first set represented frames with bare steel columns, while the second set represented partially encased composite columns. Six sets of ground motions with various intensities were employed by performing dynamic analysis to determine the seismic response of the selected frames. The strength reduction factor evaluated for steel moment-resisting and composite frames was exceptionally higher than the code recommendation. The study was conducted on the selected class of structures and recommended further studies on various structures, including braced frames.

Moroni et al. [66] evaluated the seismic response factors for confined masonry buildings by comparing the linear and nonlinear analyses of eight Chilean buildings with three and four stories. Three-dimensional THA was conducted under severe earthquakes representing Chilean and Mexican records to assess the seismic behavior and evaluate the force reduction factor, ductility reduction factors, displacement amplification factors, and ductility ratios. The wall density of the buildings reflecting the building period and the nature of earthquake intensity influenced the strength reduction factor. Lower damage levels were witnessed in the buildings with higher wall density, resulting in lower values of the strength reduction factor. Furthermore, Chilean seismic records had higher damage levels than Mexican records, which had low to moderate damage. The study was limited to confined masonry buildings with a short to medium epicentral distance from the source of the earthquake; hence, the R values were not compared for other structural systems for different ductility requirements.

Mitchell et al. [67] proposed changes in the force modification factors for the 2005 edition of the NBCC by describing the overstrength-related modification factor (R_o) and the ductility-related modification factor (R_d). The proposed values for seismic force-resisting systems (SFRS) were derived from the methodology in the 2005 NBCC, covering steel, concrete, timber, and masonry structures. The R_d was referred to as the force reduction factor (R) used in the 1995 NBCC. Recommendations were based on sub-assemblage tests, nonlinear analyses across structural systems, post-earthquake evaluations, and structural behavior assessments. The study proposed an increase in the R_d values in the 2005 NBCC, ranging from one for brittle systems (unreinforced masonry) to five for the most ductile systems (ductile steel moment-resisting frames). R_o was introduced by eliminating the calibration factor (U) and proposing values of R_o between 1.00 and 1.70, causing further reduction in the designed base shear considering the reserved strength in the SFRS.

Kurban and Topkaya [68] assessed the seismic design factors of shear plate shear wall (SPSW) systems with different geometrical characteristics designed as per AISC seismic provisions. They analyzed forty-four SPSW systems from two, four, six, eight, and ten-story buildings, considering the story mass, plate thickness, and plate aspect ratio as the prime variables. Twenty near-field and far-field earthquake records were employed in the 3D finite element analysis to compute the overstrength (Ω_o), ductility reduction factor (R_µ), displacement amplification (C_d), and response modification factors (R). Equations were developed for mean values and lower and upper bound ranges, proposing a relationship between the C_d and R factors. C_d increased with an increase in R, which increased with the ISDR values. The study was limited to regular ideal SPSW frames, and further research was recommended on realistic SPSW systems that represent a part of moment-resisting systems.

Fiorino et al. [69] performed a numerical study on cold-formed steel (CFS) strap-braced stud walls based on the IDA procedures of FEMA P695 to evaluate the strength reduction factor (q). Fourteen regular ordinary buildings classified under residential and office building categories with varying heights of one, two, three, and four stories were designed with a PGA of 0.15 g, 0.25 g, and 0.35 g and soil class C, as per the Eurocodes, assuming a trial value of q as 2.5. Three-dimensional IDA was performed in both directions using a set of 44 earthquake records scaled until global collapse. Collapse fragility was evaluated, validating the assumed q value of 2.5 for CFS strap-braced stud walls with acceptable CMR values. Since the study was limited to low-rise regular buildings based on numerical works, buildings with irregular configurations and verification of shake table tests were recommended.

Reza Salimbahrami and Majid Gholhaki [70] focused on the ductility reduction factor (R_μ) and inelastic displacement ratio (C_R) in SDOF structures. Their study extended to examine the impact of R_μ and C_R on the lateral strength adjustment and their role in determining the response modification factor (R) for the MDOF structures. Regular six, twelve, and twenty-story RC moment-resisting frames with steel plate shear walls were investigated. Nonlinear time history analysis, utilizing seven far- and near-field ground motions on the 2D models was employed to assess C_R, which was then compared with FEMA 440’s C₁ factor. The ratio of C_d to the ultimate response factor (R_u) of the MDOF system was evaluated at various ductility levels. They observed variations in R_μ factors for near- and far-field input ground motions and recommended modifications in MDOF structures, resulting in an increased modification factor with higher periods and ductility demands.

The evaluation of the seismic response factors of MDOF models for steel walls and other structures since 1996 included work on steel MRF composite buildings, confined masonry buildings, shear walls, and stud walls. Ideal buildings representing low-rise to mid-rise frames were analyzed using IDA with limited work on relatively tall buildings. In earlier studies, the investigation of irregular structures under the effect of bi-directional loading representing real buildings of varying heights was not considered. For the sake of clarity, the review of the MDOF systems representing steel walls and other structures is summarized and presented in

Table 6. Summary of the literature on MDOF models—steel walls and other structures.

Author	Analysis Type and Methodology	Type of Structure and Structural System	No. of Stories	Seismic Design Factors	Remarks
Elnashai and Broderick (1996) [65]	2D analytical and IDA	Steel MRFs and composite buildings	2, 3, 6, and 10	R (q)	Adequate margin of safety
Moroni et al. (1996) [66]	3D analytical and THA	Regular confined masonry buildings	3 and 4	Rμ, Cd, and R	Reasonable margin of safety
Mitchell et al. (2005) [67]	2D analytical and experimental	Steel, concrete, timber, and masonry building	Buildings up to 12 stories	Rd and Ro	Proposed changes by enhancing R values
Kurban and Topkaya (2009) [68]	3D analytical and THA	SPSW systems	2, 4, 6, 8, and 10	Rμ, Ωo, Cd, and R	R increased proportionally with height
Fiorino et al. (2017) [69]	3D analytical and IDA	CFS strap-braced stud wall	1, 2, 3, and 4	R (q)	Adequate margin of safety
Reza Salimbahrami and Majid Gholhaki (2019) [70]	2D analytical and IDA	Steel MRFs with steel plate shear walls	6, 12, and 20	Rμ, CR, Rυ, and Cd,	Adequate margin of safety

.

4.2.4. RC Frame Structures

The strength reduction factors (q) of low-rise- to medium-rise reinforced concrete (RC) buildings were evaluated based on the ductility and overstrength by Kappos [71] for earthquake motions in Southern Europe. Five RC buildings with one to five stories were analyzed using six sets of ground motions representing rock and alluvium sites by scaling the design spectrum to 35%. Two-dimensional IPAs and THAs were employed to evaluate the strength reduction factor dependent on overstrength. They observed that the calculated ductility-dependent part of the strength reduction factor exceeded the code values. The combined reduction factor was reasonably conservative compared to the recommended EC8 values for short and intermediate periods. The study only focused on regular RC buildings, 2D nonlinear analysis, and seismic records representing southern Europe.

Chryssanthopoulos et al. [72] proposed a probabilistic assessment methodology of the strength reduction factors (q) of RC frames designed as per Eurocode 8. The study analyzed a three-bay, ten-story regular RC frame for the medium ductility class, considering various spatial distribution scenarios, failure criteria, random member capacity, and inter-story drift. Adequate safety margins were estimated for the ultimate limit state compared to the service limit state, which depended mainly on the adopted structural criterion. The strength reduction factors were calibrated based on actual behavior factors, considering hazard and ultimate limit state vulnerability curves. The study focused on a 2D regular frame with limited input ground motions.

Elnashai and Mwafy [73] calibrated the overstrength (Ω_o) to investigate the relationship with force reduction factors (R) by undertaking a comprehensive study on RC buildings designed with modern seismic codes. The R factors were evaluated using refined expressions, and the “supply” values were verified with “design” and “demand” on the selected medium-rise RC buildings. Twelve RC buildings with eight and twelve stories were classified into three groups based on ductility and PGA, as per EC8. Irregular frame buildings and regular frame wall buildings with eight stories were classified under Group 1 and Group 3, respectively. Twelve-story regular frame buildings were classified under Group 2. IPAs and IDAs were performed using eight natural and artificial records to evaluate the Ω_o and R factors under different components of ground motions. The calculated R factors were over-conservative compared to the design code and recommended an increase, particularly for regular frame structures with high ductility levels at lower PGA values. The study focused on medium-rise buildings designed to Eurocode standards.

Maheri and Akbari [74] investigated the seismic behavior factor (R) on a dual system with RC frames and steel bracings, braced with steel X and knee-braced systems. Three regular RC buildings with four, eight, and twelve stories were considered to assess the effect of story height, load sharing of the bracing system, and the type of bracing on the R factor. The design base shear was obtained using a PGA of 0.3 g for the dual system. The elements of the R factor, including the ductility reduction factor and overstrength factor, were evaluated using the 2D IPA based on a study by Mwafy and Elnashai [75]. The results generated from the numerical IPAs were verified with three similar model results obtained from the experimental pushover results [76]. The results were found to be conservative with the code-recommended values. The study was limited to regular RC-braced buildings investigated with 2D pushover analysis without inelastic dynamic analysis.

Husain and Tsopelas [11] provided two measuring indices to quantify the structural redundancy in RC buildings using a 2D IPA. The deterministic redundancy effects were quantified using the redundancy strength index, while probabilistic redundancy effects were quantified using the redundancy variation index. RC frames with three to nine stories with one to six bays were analyzed using the IPA. Simplified expressions were developed based on the number of structural members, ductility capacity, and strength and stiffness distribution in the members. The results indicated that the structural redundancy increased as the ductility capacity of the member increased. However, the study had a limitation since the time-dependent characteristics of the seismic response, such as frequency content, excitation amplitude, and duration, were not accounted for, which emphasized the need to quantify the impact of structural redundancy using the inelastic time history analysis.

FEMA P695 [1] proposed a rational methodology to quantify the seismic response factors of force-resisting systems. Six regular RC frame buildings with one to twenty stories were developed and designed according to ASCE 7-05 criteria for the B, C, and D seismic design category (SDC) to evaluate the Ω_o, C_d, and R factors. Two-dimensional nonlinear IPA was performed to assess the overstrength factor and ductility capacity, while nonlinear dynamic response history analysis was employed to evaluate collapse capacities and CMR. Twenty-two sets of far-field and twenty-eight sets of near-field seismic records were employed by deriving the seismic response factors of the buildings, which were designed with an assumed trail value of R. The CMR values were adjusted depending on the uncertainty of the system and the acceptable limits of collapse probabilities. The trail R value was considered acceptable when the adjusted collapse margin ratio was more significant than the probability of collapse at the maximum considered earthquake (MCE). It was recommended to limit the methodology to 20–30 stories for MRF structures and 30–40 stories for shear wall and braced frame structures due to variations in response characteristics of very tall buildings. Further, the study was based on unidirectional seismic loading and the plan, and elevation irregularities were not addressed.

Alhamaydeh et al. [77] investigated three RC-framed buildings with four, sixteen, and thirty-two stories to evaluate the seismic design factors. The buildings were modeled and designed as per the IBC (2009). The study considered two sets of seismic records for a 2% probability of exceedance and a 10% probability of exceedance in 50 years. The seismic design parameters, such as the overstrength factor (Ω_o), response modification factor (R), and deflection amplification factor (C_d), were evaluated using 2D THA with inelastic inter-story drift as a criterion to assess the seismic response. The seismic design factors evaluated depended significantly on the input ground motion. The study indicated that the code values were conservative compared to both exceedance probabilities. The study was limited to regular structures with few case study buildings, and further investigation was recommended on a more extensive range of buildings with different heights.

Alam et al. [78] performed a study on three, six, and eight-story RC framed concrete buildings strengthened with a super elastic shape memory alloy (SMA) rebar to assess the seismic overstrength (R_o) and ductility (µ). The beams of each RC building were reinforced with three different types: (a) regular steel reinforcement (steel), (b) SMA rebar in the plastic hinge region and normal steel in other areas of beams (steel SMA), and (c) SMA rebar as the main reinforcement in beams (SMA RC). The columns representing the frame system of all the buildings were reinforced with normal steel only. The buildings were designed and detailed to suit the high seismic zone in Canada (Vancouver), as per the NBCC. The R_o and µ were evaluated using 2D IPA and THA under the effect of 10 earthquake records. The R_o of the SMA RC frame and normal steel RC frame were similar and in the acceptable range, with a lower ductility in the SMA RC frame due to a lower modulus of elasticity. The study focused on regular frame structures with and without SMA and recommended experimental research and further studies with structures, including slabs and transverse beams.

Thuat [79] investigated the strength reduction factor demands of RC buildings designed under different ground motion intensity levels with varying soil types and strength reduction factors (q). Regular RC MRFs with five and nine stories were designed for earthquake intensities of 0.25 g and 0.50 g, covering soil types from A to D, with strength reduction factors ranging from 1.5 to 5.0. Nonlinear dynamic response analysis was conducted with several scaled records to assess the strength reduction factor demands. The results indicated that the code-recommended strength reduction factor demands were conservative. The study expressed the strength reduction factor for the maximum ground acceleration by a linear relationship. The study was limited to 2D regular frame structures, and further research covering different seismic zones was recommended.

Seismic design response factors of the regular RC frames braced with steel chevron braces were evaluated by Akbari and Maheri [80] using the IPA. Overstrength, the response modification factor, and the ductility of four- to twelve-story RC-braced frames were investigated under different ductility demands. The parameters affecting the strength reduction factor under the applied lateral load, including the bracing system and height of the frame, were investigated. The braced RC frame had a larger ductility capacity than the moment-resisting RC frame, indicating a significant effect on the R factor. Buildings with braced frames of shorter height had a higher ductility than those with taller heights with higher R factors. The study indicated conservative R factor values under different ductility demands for the investigated buildings and recommended tentative R factors for future works. However, the study was limited to pushover analysis on regular 2D frames only, and nonlinear dynamic analysis was not considered.

Nine-story plan-symmetric RC MRF structures were assessed under the effect of higher modes, and their seismic response was investigated by Maniatakis et al. [81]. Using 34 seismic records obtained from the PEER strong motion database, 2D nonlinear response history analyses were conducted on a building designed with the Greek seismic design code EAK 2000. The study concluded that the flexural overstrength and the ground motion characteristics influenced the story inertial and shear forces, which were attributed to higher modes. The modal reduction factors decreased with the increasing mode order compared to behavior and ductility factors. The study was limited to only one building and one structural system, and the flexural overstrength was developed based on the deformation of the frame subjected to second and third modes and cannot be generalized to higher modes.

Mondal et al. [9] focused on evaluating the actual values of the response reduction factor for realistic RC MRF buildings. The buildings were designed based on Indian standards for seismic requirements, and RC designs were detailed to meet the requirements of ductile detailing. The study was conducted on a ductile (special) RC MRF to verify and validate the code-recommended response reduction factor (R) value. Four typical plan-symmetric RC frame regular office building structures with two, four, eight, and twelve-story configurations were considered in the seismic zone IV study region, as per Indian standards. The results indicated that the Indian standard recommended a higher R value than the actual value of R. However, the study was limited to one single plan-symmetric configuration and one seismic zone, and nonlinear response time history analysis was not considered.

Hossain et al. [82] considered a set of RC buildings with three and eight stories with varying reinforcement detailing for the horizontal elements of the building systems to estimate the seismic force reduction factors. Vertical elements of the 13 RC frames used in the study were reinforced with normal steel. The beams were reinforced with normal steel in two frames along the height of the buildings, while eleven frames were reinforced at the plastic hinges with super elastic shape memory alloy (SMA) bars as a gradual replacement to the steel from level 1 to the building height. The NBCC was used to perform linear analysis to design the buildings. Twenty records were employed in the nonlinear IDAs to investigate the seismic force reduction factors. The result demonstrated satisfactory performance with sufficient margins for the seismic factors but was limited only to regular frame structures.

Vona and Mastroberti [83] investigated the existing RC frame buildings to estimate the strength reduction factor (q) based on the force-based assessment procedure. Existing RC buildings with two, four, and eight stories were considered, with three sets representing buildings with frames without effective infill, frames retrofitted with masonry infill, and frames with masonry infill at the ground floor level. Structural modeling was performed using IDA with 50 seismic records on each building type. The study proposed the strength reduction factors for the investigated buildings for the existing code, including retrofitting measures, which improved and extended the existing seismic performance. The study recommended further studies on different structural types covering a more comprehensive range of buildings.

Seismic response coefficients of tunnel-form RC buildings with horizontal irregularity were investigated on three, five, seven, and-ten-story buildings by Vahid Mohsenian et al. [84]. IDAs were performed on the 3D models using 10 far-field earthquake records. Response modification factors were calculated at both demand and supply values. The demand response was estimated depending on site seismicity and physical and geometrical specifications, while the supply response was based on the capacity of the building to resist the inelastic deformations to meet the desired performance levels. Their study indicated that an increase in the building height witnessed an increase in ductility-related modification factors and a decrease in those related to overstrength and response modification factors. The seismic response exhibited a good safety margin compared to the code-recommended value of five.

Kader Newaj Siddiquee et al. [85] assessed seismic performance using the collapse margin ratio (CMR) for regular concrete-reinforced frames with shape memory alloy (SMA) rebars. Their study, involving 2D nonlinear static pushover and IDA using 20 input ground motion records on three, six, and eight-story buildings, aimed to evaluate the collapse safety of frames with SMA rebars compared to frames reinforced with conventional steel rebars. Four reinforcement scenarios were considered: steel reinforcement only, SMA rebar in the plastic hinge region in the columns on ground floor and steel rebars in other areas (SMA-CM), SMA rebar in the plastic hinge region in the beams on the ground floor and steel rebars in other areas (SMA-BM), and SMA rebars in the plastic hinges in both beams and columns on the ground floor (SMA-BM-CM). The study concluded that three and eight-story frames exhibited increased collapse capacity with SMAs, and seismic response modification factors were deemed adequate, as per the FEMA P695 guidelines.

Farrokh Fazileh et al. [86] recommended a methodology to evaluate the seismic strength reduction factors for different seismic force-resisting systems (SFRSs) using a performance-based unified (PBU) procedure. This procedure was designed as a performance-based approach used in systematically quantifying the ductility-related modification factor (R_d) and the overstrength-related modification factor (R_o) for diverse SFRSs in the NBCC. Twenty-one concrete moment frame archetypes were utilized with a two-level screening process to optimize the number of archetypes efficiently in assessing the seismic force modification factors. Nonlinear static pushover analysis was used to calculate the Ro as a preliminary screening process before conducting IDA. The proposed methodology enhanced the performance level intensity by 90% in the SFR systems.

The evaluation of the seismic response factors of MDOF systems for RC frame structures for three decades was mainly based on 2D analytical works with unidirectional seismic loading. Regular low-rise to medium-rise frame buildings were analyzed using IPA and IDA, with limited work on relatively tall buildings. Earlier studies did not consider irregular buildings under the effect of bi-directional loading representing real buildings employing 3D inelastic analysis. For the sake of clarity, the review of the MDOF systems representing RC frame structures is summarized and presented in

Table 7. Summary of the literature on MDOF models—RC frame structures.

Author	Analysis Type and Methodology	Type of Structure and Structural System	No. of Stories	Seismic Design Factors	Remarks
Kappos (1999) [71]	2D analytical, IPA, and THA	Regular MRF buildings	1 to 5	R (q)	Adequate safety margins
Chryssanthopoulos et al. (2000) [72]	2D analytical and THA	Regular MRF buildings	10	R (q)	Adequate safety margins
Elnashai and Mwafy; Mwafy and Elnashai (2002) [73,75]	2D analytical, IPA, and IDA	Regular and irregular MRF buildings	8 and 12 regular and eight irregular	Ωo and R	Adequate safety margins
Maheri and Akbari (2003) [74,76]	2D analytical, experimental, and IPA	Regular MRF with steel bracing	4, 8, and 12	R	Adequate safety margins
Husain and Tsopelas (2004) [11]	2D analytical and IPA	Regular MRF	3, 5, 7, and 9	RR	RR increased with an increase in ductility capacity
FEMA P695 (2009) [1]	2D analytical, IPA, and THA	Regular MRF	1, 2, 4, 8, 12, and 20	Ωo, Cd, and R	R acceptable for ACMR > collapse probability at the MCE
Alhamaydeh et al. (2011) [77]	2D analytical, IPA, and THA	Regular MRF buildings	4, 16, and 32	Ωo, Cd, and R	Adequate margin of safety
Alam et al. (2012) [78]	2D analytical, IPA, and THA	Regular MRF buildings with SMAs	3, 6, and 8	µ and Ro	Adequate margin of safety
Thuat (2012) [79]	2D analytical and THA	Regular MRF	5 and 9	R (q)	Adequate safety margins
Akbari and Maheri (2013) [80]	2D analytical and IPA	Dual system (frame with steel bracings)	4, 8, and 12	Ωo, Cd, and R	Adequate safety margins
Maniatakis et al. (2013) [81]	2D analytical, IPA, and THA	Regular MRF buildings	9	R	Adequate safety margins
Mondal et al. (2013) [9]	2D analytical and IPA	Regular MRF buildings	2, 4, 8, and 12	R	Adequate margin of safety
Hossain et al. (2015) [82]	2D analytical, IPA, and IDA	Regular MRF buildings with SMAs	3 and 8	R	Adequate margin of safety
Vona and Mastroberti (2018) [83]	2D analytical and IDA	Regular MRF buildings retrofit with masonry infill	2, 4, and 8	R (q)	Proposed R values for a retrofit building
Vahid Mohsenian et al. (2019) [84]	3D analytical and IDA	Horizontal irregular MRFs	3, 5, 7, and 10	R, Ωo, and Cd,	Adequate margin of safety
Kader Newaj Siddiquee et al. (2021) [85]	2D analytical, IPA, and IDA	Regular MRFs with normal rebars and SMA bars	3, 6, and 8	R	Adequate margin as per FEMA P695
Farrokh Fazileh et al. (2023) [86]	2D analytical, IPA, and IDA	Regular SFRS using the PBU procedure	21 SFRS archetypes	Rd and Ro	Proposed work enhanced performance level

.

4.2.5. RC Shear Wall Structures

Challal and Gauthier [87] evaluated the seismic response of RC-coupled shear walls (CSWs) through nonlinear deformation and ductility response, designed as per the NBCC [3] and Canadian Concrete Standards [88]. Five buildings with six, ten, fifteen, twenty, and thirty stories were considered in the design using three Canadian seismic zones. Nonlinear dynamic analysis under five seismic records verified inter-story drift and assessed plastic hinges, displacement, and rotational ductility in walls and coupling beams. The code-specified drift limit was conservative, with lower drifts for taller CSWs. Maximum displacement and ductility demand factors were conservative in comparison with the NBCC limit, which decreased with an increase in the story height. The study was limited to regular structures using 2D analysis with few seismic records and recommended further investigation with different irregularities under a more extensive range of ground motions.

Elnashai and Mwafy [73,75] evaluated Ω_o and R on RC wall buildings designed with modern seismic codes. Regular frame-wall buildings with eight stories were designed according to EC8. The seismic design factors were evaluated using IPAs and IDAs with eight natural and artificial records. The calculated R factors were over-conservative compared with the design code, prompting a recommendation to increase R values, especially for structures with high ductility levels at lower PGA values. The study focused on medium-rise buildings designed to Eurocode standards.

Seismic design response factors of RC wall buildings were investigated by Mwafy [7] in densely populated areas in the United Arab Emirates (UAE). The study included factors such as the overstrength (Ω_o) factor, the force reduction (R) factor, and the deflection amplification (C_d). Five buildings from 20 to 60 stories, representing various regular concrete wall structures, were designed according to the modern seismic design codes. Nonlinear analyses were conducted under twenty natural and artificial seismic records using IPAs and IDAs to evaluate the seismic design response factors. Under moderate earthquakes, it was observed that the medium-rise buildings had higher seismic risk due to a lower contribution of the building fundamental mode than high-rise buildings. An adequate margin of safety was witnessed in the C_d and R factors. The study confirmed that the evaluated R factors were higher than the code-specified R values and recommended an increase in seismic design response factors for a more cost-efficient design of the buildings. The study focused on regular RC wall buildings with 2D analysis.

Mwafy [8] proposed a simple theoretical approach to arrive at a cost-efficient and safe design using actual ductility and first yield overstrength of RC structures. RC wall buildings with ten to sixty stories and RC frames with eight to twelve stories were investigated to assess their inelastic structural response. Force demands obtained from the elastic procedures were compared with the inelastic 2D IDA under realistic input ground motions. The study proposed suitable ranges based on the actual ductility and the first yield overstrength to identify the design efficiency. The results indicated an over-conservative design concerning the recommended seismic design factors provided in the seismic design code and recommended using IDA to verify the design for structures with seismic responses outside the acceptable range. The study was focused on regular buildings using 2D nonlinear response history analysis.

Mwafy et al. [10] studied five sixty-story regular RC shear wall buildings with varying concrete strengths designed with similar periods in evaluating the seismic design coefficients. The impact of the varying concrete strengths on Ω_o, R, and C_d was investigated using 2D IPAs and IDAs with 20 input ground motions. The increase in concrete strength in high-strength buildings reduced a significant amount of steel reinforcement with a considerable reduction in vertical elements’ cross-section, resulting in reduced lateral stiffness. Higher design overstrength (Ω_o) was observed in all the buildings than the code-recommended values with slightly reduced levels of overstrength factors (Ω_o and Ω_y) for buildings with increased building material strengths. The adequate safety margins of C_d were reflected with a constant increase in the material strength. The median R values were sufficiently higher than the design values of ASCE 7. The study recommended the possibility of increasing the R values to achieve cost-effective building designs and was limited to investigation with 2D inelastic analysis on regular-wall buildings.

Mwafy and Khalifa [89] assessed the safety margin of seismic coefficients of realistic-reinforced concrete high-rise buildings with vertical irregularities in medium-seismicity regions. Five 50-story buildings representing typical vertical irregularities were designed employing 3D models, as per international codes and seismic standards. Forty seismic records were chosen to conduct the 2D nonlinear IPAs and IDAs covering far-field and near-source seismic events. The seismic design coefficients evaluated by IPA and IDA with vertical irregularities were found to be conservative with adequate safety margins compared to the seismic code provisions, which can be revised to achieve economical designs. The study focused on unidirectional seismic loading using 2D IDA for irregular shear wall buildings.

Zerbin et al. [90] recommended an alternative formulation to evaluate the ductility reduction factor of RC wall and frame structures. Two sets of frame and wall buildings ranging from three to twelve stories were designed with medium ductility class requirements, as per Eurocode 8, to investigate three levels of sectional ductility. Simplified analytical models of frame and wall structures were proposed to resemble an equivalent SDOF system to evaluate the ductility reduction factor. Pushover and THA were performed with 34 natural ground motions to estimate the ductility reduction and modification factors. The ductility reduction factors of the frame and wall models derived from the proposed formulation provided good results with the numerical analysis from the Eurocode 8 design approach. The ductility reduction and modification factors of the frame and wall systems for MDOF models decreased with an increase in the number of stories. The code recommended force reduction factors were more conservative than those evaluated with the proposed formulation method and depended mainly on structural ductility and overstrength. The study was limited to ideal structures, and further studies on overstrength factors and real structures were recommended.

Matteo Zerbin et al. [91] proposed a numerical formulation to evaluate the ductility force reduction factors for regular RC wall and RC frame structures compared to Eurocode 8. Simple four, eight, and twelve stories representing both wall and frame structures were investigated using 2D pushover and inelastic time history analysis models, employing thirty-four natural ground motions. Their study revealed that in both wall and frame systems, the ductility reduction factors for the MDOF systems decreased with the increase in the number of stories, indicating a reduced system capability to utilize base sectional ductility and an increased importance and impact of the higher mode effects.

Maysam Samadi and Norouz Jahan [64] examined the impact on seismic design parameters such as (a) the response modification factor (R), (b) the deflection amplification factor (C_d), (c) the overstrength factor (Ω), and (d) the damping ratios for tall steel buildings. The study examined regular steel buildings with 28 and 56 stories, featuring steel-braced and RC shear wall cores with outriggers placed at every quarter of the building height, resulting in forty-four building models. Seismic parameters were assessed using the modal response spectrum (MRS), pushover, and nonlinear time history (NLTH) 3D analyses. Including the outriggers increased the response modification factor, overstrength, stiffness, and damping ratios, particularly in the buildings with RC core walls, while reducing ductility in both systems. Their study also identified inadequacy in the code-recommended C_d values.

The previous studies conducted since 2001 on assessing the seismic response factors of MDOF systems for RC shear wall structures were based on 2D analytical works using IPAs and IDAs. Earlier studies were based on regular shear wall buildings, and the evaluations of seismic response factors were based on unidirectional seismic loading. In earlier studies, irregular shear wall buildings under the effect of bi-directional loading employing 3D inelastic analysis were not considered. For the sake of brevity, the review of the MDOF systems representing RC shear wall structures is summarized and presented in

Table 8. Summary of the literature on MDOF models—RC wall structures.

Author	Analysis Type and Methodology	Type of Structure and Structural System	No. of Stories	Seismic Design Factors	Remarks
Challal and Gauthier (2001) [87]	2D analytical and IDA	Regular coupled shear wall (CSW) system	6, 10, 15, 20, and 30	µd	Adequate margin of safety
Elnashai and Mwafy; Mwafy and Elnashai (2002) [73,75]	2D analytical, IPA, and IDA	Regular shear wall building	8	Ωo and R	Adequate safety margins
Mwafy (2011) [7]	2D analytical, IPA, and IDA	Regular shear wall buildings	20 to 60	Ωo, Cd, and R	Adequate margin of safety
Mwafy (2013) [8]	2D analytical, IPA, and IDA	Regular shear wall buildings	10 to 60	Ωo, Cd, and R	Adequate margin of safety
Mwafy et al. (2015) [10]	2D analytical, IPA, and IDA	Regular shear wall buildings with varying material strengths	60	Ωo, Cd, and R	Adequate margin of safety
Mwafy and Khalifa (2017) [89]	2D analytical, IPA, and IDA	Regular shear wall buildings with varying vertical irregularities	50	Ωo, Cd, and R	Adequate margin of safety
Zerbin et al. (2018) [90]	2D analytical, IPA, and THA	Regular frame and wall buildings	3 to 12	Cd, and R	Adequate margin of safety
Matteo Zerbin et al. (2020) [91]	2D analytical, IPA, and THA	Regular RC walls and frames	4, 8, and 12	Rμ,	Rμ decrease with an increase in story height with an adequate safety margin
Maysam Samadi and Norouz Jahan (2021) [64]	3D analytical, MRS, IPA, and NLTH	Two regular steel-braced buildings with outriggers	28 and 56	R, Cd, and Ωo	Adequate margin of safety, R, and Ωo. Inadequate Cd

.

5. Limitations, Gaps, and Future Research

Based on the detailed review conducted since the early 1970s, it was noticed that the R factor is influenced by ductility, the period of the structure, structural system, construction materials, and the shape of the hysteresis loops used to simplify the response of buildings using SDOF systems. The comprehensive state-of-the-art review of the seismic response factors provided specific findings, which are more fully expressed under the limitations and gaps and the possible areas for future research, as provided in the following sections.

5.1. Limitations and Gaps

Seismic response factors were reviewed from previous studies considering the assessment of SDOF and MDOF systems. The MDOF systems contribute 75% of the review compared to 25% of the studies from SDOF systems, as shown in Figure 3a. Regular and irregular buildings covering RC, steel, masonry, and composite construction with various structural systems were reviewed, and the limitations were identified and described as follows.

Type of systems considered: The presented review of seismic response factors is studied under the SDOF and MDOF systems and summarized in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8. The SDOF systems were based on hypothetical models with hysteresis relationships, while real structures were not investigated. Most of the earlier work on MDOF systems was limited to regular low-rise to medium-rise buildings. This shows the need for future exploration and further research on real buildings with different irregularities and high-rise structures.

Type of modeling: The previous studies primarily used two-dimensional analysis to evaluate seismic response factors. About 94% of the previous research was based on 2D nonlinear analyses, with only 6% of work on 3D analysis, as shown in Figure 3b. Since the 2D analysis approach is based on unidirectional seismic loading, it will not capture the realistic 3D behavior of the structures. The impact of bi-directional input ground motions is not accounted for in this approach, particularly with the irregular buildings. Since only 6% of previous research was carried out using 3D analysis, there is a gap, emphasizing the need to investigate realistic 3D numerical models of buildings employing bi-directional seismic loading.

Type of analysis: Several studies employed only IPA in evaluating the R factors, while IDA was not utilized in assessing the seismic performance, as shown in Table 4, Table 5, Table 6, Table 7 and Table 8. It is shown that about 30% of the previous studies on steel and RC-framed structures employed 2D IPA in evaluating R factors, while 70% of the research was based on IDAs. The former analysis procedure may not yield reliable results, particularly for complex structures, such as high-rise buildings, which were seldom covered in previous investigations. Moreover, IDAs were mainly considered using unidirectional seismic loading, while bi-directional loading was limited. This emphasizes the need for future investigations on 3D high-rise buildings with bi-directional loading using IDAs to realistically capture the seismic performance of these complex structures and assess their seismic design response factors.

Components of seismic design factors: In the previous studies, the seismic response factors investigated using SDOF systems were based on ductility reduction factors. The contribution of the system overstrength in evaluating the strength reduction factors of SDOF models was not considered. Similarly, structural overstrength was not evaluated in most of the previous studies conducted on MDOF models, as shown in Table 4, Table 5, Table 6, Table 7 and Table 8. This brings in the need to evaluate structural overstrength in future studies, which is vital in evaluating the strength reduction factors.

Type of buildings: The earlier studies primarily covered simple regular buildings in evaluating seismic response factors. About 92% of the previous research was based on regular buildings, compared with 8% on irregular buildings, as illustrated in Figure 3c. Most of the previous work was limited to ideal buildings of regular configuration with limited heights, while buildings with varying heights and irregular configurations were not investigated. Since modern buildings may exhibit several sources of irregularity, and with only 8% of previous studies focused on structural irregularities, there is a pressing need to assess the seismic design response factors for irregular buildings.

Structural systems: Seismic response factors were primarily evaluated on frame systems comprising RC, steel, and steel-braced frame buildings. It is noteworthy to state that in this review, one-third of the research was made on RC frame buildings, 20.5% on steel frame buildings, 17.9% on steel-braced buildings, 15.4% on shear wall buildings, and 12.87% on steel walls and other structural systems, as shown in Figure 3d. As modern structures involve several irregular and high-rise buildings, investigating the impact of bi-directional seismic loading on shear walls and other structural systems are needed in future research.

Input ground motions and seismic loading: Previous studies on the seismic assessment of SDOF systems were based on analytical work considering seismic scenarios representing near-field earthquakes suitable for medium seismic zones. Some SDOF studies employed simple waveforms to evaluate the seismic design factors without employing real earthquake records. The seismic behavior of structures with aftershock events was hardly considered in the SDOF systems. Several studies were limited to very few seismic records, which do not adequately account for the variability in the input ground motions [43,70], while other studies were limited to near-field records with little investigation on far-field earthquakes [54]. Most structures were employed using unidirectional seismic loading on 2D regular frames without considering the effect of bi-directional loading on real structures using 3D inelastic analysis. This emphasizes the need for future research on 3D buildings with bi-directional seismic loading to predict the seismic design response factors, particularly for irregular structures influenced by torsion.

5.2. Areas for Future Research

The limitations and gaps identified in the previous studies while evaluating the seismic response factors emphasize the pressing need for further research in the following areas:

• Limited research (only 8% of previous studies) was conducted to evaluate the seismic response of irregular buildings. This emphasizes the need to investigate practical buildings covering different types of irregularities with various heights;
• Most of the earlier work was based on 2D inelastic analysis with unidirectional seismic loading. Only 6% of research was conducted using 3D inelastic analysis. This presses the need to evaluate the seismic response factors using 3D inelastic analysis under the effect of bi-directional seismic loading;
• Many previous studies did not account for structural overstrength. This shows the need to consider structural overstrength in assessing the seismic response of buildings;
• Studies on regular RC shear wall buildings were limited to 15.4% in the previous investigations. This emphasizes a need to evaluate the seismic behavior of RC shear wall buildings with varying heights;
• Previous studies focused on buildings in seismic zones representing moderate to severe earthquakes. Furthermore, the effects of aftershock events need to be investigated to better evaluate the seismic performance of buildings with post-earthquake events.

6. Conclusions

This paper presented a systematic and comparative literature review on seismic response factors over four decades. It provided a brief overview of the use of seismic response factors in various international codes, historical perspectives, and their respective definitions. Seismic response factors were reviewed from the previous studies considering the SDOF and MDOF system assessment. The MDOF systems contributed 3/4 of the review, while the SDOF system contributed 1/4 of the previous research. Regular and irregular buildings covering RC, steel, masonry, and composite construction with various structural systems were reviewed and summarized.

The review also identified that the ductility, structural system, natural period, construction materials, and shape of the hysteresis loop influenced the R factor. Limitations and gaps in the previous studies were presented considering the type of systems, type of analysis, components of seismic design factors, irregularity, structural systems, input ground motions, and the seismic loading applied to the structures. Most previous research in the SDOF system was based on hypothetical models, while MDOF systems were limited to regular low-rise to medium-rise buildings. A significant research contribution was based on 2D inelastic analysis by employing unidirectional seismic loading, while very little work was conducted using 3D inelastic analysis. Only 6% of the previous research was based on 3D analysis, and 8% of the studies were conducted on irregular buildings. Regular RC, steel frame, and braced buildings were investigated at 72%, whereas shear walls had only 28% of the research in the previous studies. About 30% of the research on steel and RC-framed buildings was conducted using inelastic static analysis, while 70% of the previous studies focused on dynamic analysis, mainly employing uni-directional seismic loading. Several studies recommended improving the code-recommended R values for regular buildings due to observed adequate safety margins.

Based on the review of seismic response factors and the identified limitations and gaps, the present study provided possible future research covering structural irregularity, 3D inelastic analysis, and the investigation of different structural systems related to multi-story and high-rise buildings under bi-directional seismic loading. Further investigations to evaluate the seismic response of RC shear wall buildings with varying heights and irregularities are emphasized using 3D inelastic analysis covering regions influenced by various seismic scenarios. This would calibrate the seismic design factors recommended in design standards, resulting in more cost-effective designs with consistent safety margins for different structural systems.