Aftershock Effect on Seismic Behavior of 3D Steel Moment-Resisting Frames

Abstract

Aftershocks are inevitable phenomena following a mainshock, especially after a major earthquake. However, the cumulative damage caused by aftershocks and its impact on structural performance evaluation has only recently received significant attention. This study explores the effects of mainshock–aftershock (MS–AS) sequences, including multiple consecutive aftershocks, acting on 3D steel moment-resisting frame structures. Following nonlinear time history analysis, several fundamental variables such as residual interstory drift, maximum displacement, plastic hinge formation, and base shear are evaluated to examine cumulative damage. In this context, the findings depicted in terms of aftershocks play a significant role in exacerbating plastic deformations and damage accumulation in steel moment frames. Subsequently, to mitigate cumulative damage on steel moment frames, retrofitting strategies were implemented. Retrofitting strategies effectively reduce cumulative damage and improve seismic resilience under multiple earthquake events. This research highlights the limitations of single-event seismic assessments and the need to incorporate sequential earthquake effects in design and retrofit practices. Furthermore, it provides new insights into mitigating further damage by retrofitting existing structures under multiple earthquakes.

1. Introduction

Earthquakes are among the most destructive natural phenomena, which may cause major damage to structures and infrastructures, eventually leading to significant human casualties and economic loss, even though understanding of the behavior of structures under earthquakes has advanced through the years. There is a critical issue in assessing the real behavior of structures during earthquakes. Conventional seismic design analysis evaluates structures under a single earthquake; however, in real life, buildings are subjected to a mainshock and one or more large aftershocks. The aftershocks impose cumulative damage on structures, especially for those that are already damaged from the mainshock. These aftershocks sometimes cause the structures to undergo unrepairable damages or result in their total or partial collapse.

Fortunately, in recent decades, this issue has been taken into consideration; therefore, numerous studies have widely investigated aftershock effects on the seismic behavior of steel structures. The preceding studies, such as Amadio et al. [1] and Li and Ellingwood [2], indicated that repeated aftershocks significantly increase structural damage by reducing energy dissipation capacity and increasing substantial residual drift compared to a single earthquake. Their findings constituted the foundation for understanding how aftershocks affect structural performance during an earthquake. Following studies expanded on this by evaluating various influencing parameters. Hatzigeorgiou [3] and Hatzigeorgiou and Beskos [4] investigated how ground motion characteristics, such as soil types and proximity to the fault, influence displacement and ductility demands. They concluded that sequential ground motions resulted in significantly higher deformation levels. Ruiz-García and Negrete-Manriquez [5] demonstrated that near-fault aftershocks led to increased residual drift in steel moment-resisting frames compared to far-fault ground motions.

Further studies focused on more specific structural configurations. Parekar and Datta [6] examined stiffness irregularities and their effect on interstory drift distribution. It was found that irregularities, especially in lower stories, escalate damage under repeated earthquakes. Torfehejad and Sensoy [7] introduced different collapse behavior in low- to mid-rise and high-rise special moment frames under recorded sequential seismic events. With regard to retrofitted or specialized systems, Mohsenian et al. [8] evaluated the seismic stability of the eccentrically braced steel frames with vertical energy-dissipating links under mainshock–aftershock (MS–AS) sequences. They concluded that, while vertical energy-dissipating links absorb seismic energy, they help other structural elements remain elastic. It was also found that EBFs have stronger performance and reliability under design-level earthquakes. Narayan et al. [9] investigated the seismic behavior of the low-, mid-, and high-rise steel structures with moment-resisting frames under mainshock–aftershock sequential events. The results of this study showed that, while 4- and 8-story buildings withstood up to seven aftershocks, a 12-story building collapsed after the third aftershock, emphasizing the cumulative aftershock damage effect on structural integrity. Valente et al. [10] developed repairable bolted fuse devices for beam-to-column connections in composite steel frames to concentrate plastic deformations locally and protect main structural elements from damaging effects during earthquakes. Using detailed finite element and simplified nonlinear models calibrated with experiments, they showed that these fuses effectively dissipate seismic energy while allowing easy post-earthquake replacement. Frames with fuses developed larger displacements and showed longer periods but reduced base shear and improved energy dissipation compared to conventional moment frames. The fuse geometry notably influences performance, especially under hogging moments. This system offers a practical and economical approach to enhancing seismic resilience and repairability of steel structures.

Valente et al. [11] studied repairable fuse devices for composite steel frames, comparing bolted and welded designs that concentrate on inelastic deformations in beam-to-column connections. Experimental tests and numerical analyses demonstrate that both fuse types enhance seismic resilience by improving energy dissipation and protecting the main structural elements. Bolted fuses offer easier replacement and higher energy dissipation, while welded fuses provide greater initial stiffness. This work provides valuable insights for designing repairable, earthquake-resistant steel structures.

Despite numerous studies on the seismic performance of steel structures under a single earthquake event and mainshock–aftershock sequences, there is a notable gap in the systematic evaluation and comparison of steel moment-resisting frames before and after retrofitting under realistic multiple-aftershock scenarios. This research addresses this gap by conducting detailed nonlinear time history analyses on SAC benchmark buildings subjected to recorded mainshock–aftershock sequences, including an extended case with up to five consecutive aftershocks. Critical response parameters such as residual interstory drift, maximum displacement, plastic hinge formation, and base shear are thoroughly examined to capture the cumulative damage effects caused by sequential seismic events. Furthermore, the study evaluates the effectiveness of X-brace frame retrofitting strategies considered for mitigating aftershock-induced damage and enhancing overall seismic resilience. By integrating real seismic records, advanced nonlinear modeling, and practical retrofitting design following relevant seismic codes, this work provides valuable insights for improving seismic performance evaluation methodologies and informs retrofit strategies for steel moment-resisting frames in earthquake-prone regions.

2. Structural Finite Element (FE) Models and Seismic Inputs

2.1. Structural FE Models

In this study, 3-, 9-, and 20-story steel buildings were developed as a part of the SAC joint venture project [12] in order to investigate the aftershock effect on steel structures having moment-resisting frames. This SAC project was designed to represent typical commercial office building configurations in high seismic risk areas throughout the United States, using local seismic standards (UBC 1994 for Los Angeles and Seattle; BOCA 1993 for Boston) [13,14], and considering both gravity and lateral loading. Steel moment-resisting frame (SMRF) buildings were assumed to be located on stiff soil (Site Class D), which reflects firm soil conditions typical of urban areas. Figure 1 displays three-dimensional (3D) finite element (FE) models of SAC buildings generated using SAP2000 software.

Each building has a regular, symmetric floor plan with exterior steel moment-resisting and interior gravity frames. While perimeter moment-resisting frames function as the primary lateral force-resisting frames, interior gravity frames resist vertical loads.

The 3-story building is modeled with fixed-base supports, representing the relatively rigid foundation conditions typical for low-rise structures without basements. In contrast, the 9- and 20-story buildings incorporate pinned supports at the basement levels to simulate realistic foundation flexibility, allowing rotational movement at the base. Although the original SAC project documentation provides limited details on support modeling, this approach aligns with established seismic modeling practices. It enhances the accuracy of the dynamic response by better capturing the interaction between the superstructure and its foundations [15]. The applied loads, including gravity and seismic loading for the steel moment frames, are consistent with the original SAC report [15].

Table 1. SAC building loads.

Story	Dead Load (kN/m2)	Live Load (kN/m2)
Floor	4.6	1
Roof	4	1
Penthouse	5.6	-

displays the applied dead and live loads for the SAC buildings.

In the original SAC joint venture study [12], all the necessary information about the structural members’ cross-section and loading is given in detail. Figure 2 shows the floor plans of the steel moment-resisting frames. All three buildings are symmetric in plan, and the moment-resisting frames are shown in bold color. In Figure 3, the elevation of moment frames is illustrated. The structures have uniform bay spacing and story height for the 9- and 20-story structures incorporating basement levels.

Firstly, we aimed to show whether the dynamic behavior of all structures produced with SAP2000 v24.2.0 software was compatible with the original SAC buildings [15]. Therefore, modal analysis was performed for each building to compare the fundamental period of the structures.

Table 2. Comparison of fundamental period results obtained from FE model with those reported by Gupta and Krawinkler [15].

No. of Story	Fundamental Periods (s)
	1st Period from the Analysis	1st Period from the Report	2nd Period from the Analysis	2nd Period from the Report	3rd Period from the Analysis	3rd Period from the Report
3	1.025	1.03	0.339	0.33	0.185	0.17
9	2.46	2.34	0.929	0.88	0.548	0.50
20	4.15	3.98	1.47	1.36	0.869	0.79

and

Table 3. Comparison of modal mass participation results obtained from the FE model with those reported by Gupta and Krawinkler [15].

No. of Story	Mass Participation Factor (%)
	1st Mass Participation Factor from the Analysis	1st Mass Participation Factor from the Report	2nd Mass Participation Factor from the Analysis	2nd Mass Participation Factor from the Report	3rd Mass Participation Factor from the Analysis	3rd Mass Participation Factor from the Report
3	80	82.8	14	13.5	4.4	3.7
9	77	83.5	10	10.6	3.7	3.6
20	75	80.4	14.7	11.5	3.4	3.4

show the comparison of the fundamental periods and mass participation factors of these steel structures. It has been demonstrated that the modal analysis results of the developed FE models are consistent with the period values provided in the report [15]. However, some discrepancies in effective mass and first-mode period values for the 9- and 20-story models, compared to the report, are observed. These differences are primarily due to the increased complexity and size of these structures. The 3-story building, with its simpler geometry and fewer degrees of freedom, exhibits less sensitivity to modeling details such as element discretization and numerical solver parameters. Conversely, the taller buildings’ larger number of elements and more complex dynamic behavior make them more susceptible to minor variations in these factors, leading to the slight differences observed. As shown in Table 2, the differences in running times among the 3-, 9-, and 20-story models are primarily driven by the increasing number of structural elements, degrees of freedom, and numerical complexity associated with additional basement levels and varying support conditions, resulting in progressively longer analysis durations for taller and more complex structures.

2.2. Ground Motion Selection

In this study, a set of real mainshock–aftershock (MS–AS) ground motion sequences was considered to assess the seismic behavior of steel moment frame structures under multiple earthquakes. Mainshock–aftershock sequences can be constructed artificially, can be repeated, or can be recorded as real ground motions. However, Zhang et al. [16] outlined that artificial sequences often result in scenario-specific seismic responses due to their inaccurate frequency characteristics. Furthermore, as repeated ground motion approaches assume the identical spectral content for mainshock and aftershock, they oversimplify the actual structural behavior. Therefore, to avoid these limitations as well as to obtain realistic dynamic interaction, real recorded earthquakes were used in this study, following Basim et al. [17], Mohsenian et al. [8], and Hatzigeorgiou and Beskos [4]. Although each earthquake record consists of three components (two horizontal and one vertical), only the horizontal components were included in the nonlinear time history analysis to focus on the in-plane lateral response of the structures. Detailed criteria for earthquake selection are given in

Table 4. Earthquake selection criteria (a). The properties of selected ground motions (b).

(a)
Criteria				Description
Magnitude of mainshock				M > 6
Magnitude of aftershock				M > 5
Distance from fault (Rjb)				Near fault: Rjb < 15–20 km Far fault: Rjb > 15–20 km
Site class (Vs30)				(200–560)
Source				PEER NGA files (2H components)
(b)
Result ID	RSN	Earthquake Name	Station Name	Magnitude	Mechanism	Rjb (km)	Vs30 (m/s)
1	1227	“Chi-Chi_ Taiwan”	“CHY074”	7.62	Reverse Oblique	0.7	553.43
2	2490	“Chi-Chi_ Taiwan-03”	“CHY074”	6.2	Reverse	27.84	553.43
3	122	“Friuli_ Italy-01”	“Codroipo”	6.5	Reverse	33.32	249.28
4	131	“Friuli (aftershock 1)_ Italy”	“Codroipo”	5.91	Reverse	41.37	249.28
5	181	“Imperial Valley-06”	“El Centro Array #6”	6.53	strike slip	0	203.22
6	204	“Imperial Valley-07”	“El Centro Array #6”	5.01	strike slip	7.4	203.22
7	230	“Mammoth Lakes-01”	“Convict Creek”	6.06	Normal Oblique	1.1	382.12
8	248	“Mammoth Lakes-06”	“Convict Creek”	5.94	strike slip	6.44	382.12
9	959	“Northridge-01”	“Canoga Park—Topanga Can”	6.69	Reverse	0	267.49
10	3775	“Northridge-06”	“Canoga Park—Topanga Can”	5.28	Reverse	8.98	267.49
11	368	“Coalinga-01”	“Pleasant Valley P.P.—yard”	6.36	Reverse	7.69	257.38
12	383	“Coalinga-02”	“Pleasant Valley P.P.—yard”	5.09	Reverse	6.51	257.38

a. Based on the aforementioned criteria in Table 4a, Chi-Chi and Friuli earthquakes are classified as far-fault earthquakes, whereas the remaining records are characterized as near-fault earthquakes. In mainshock–aftershock sequences, a time gap of 10 s was inserted between the mainshock and aftershock to bring the structure to rest and ensure the decay of dynamic response from the mainshock before applying the aftershock.

This study included both near-fault and far-field ground motions to reflect the variability of expected seismic demands. Therefore, the classification of ground motions is necessary because near-fault ground motions include pulse-type velocity effects, forward directivity, and high intensity over short periods. Thus, it may impose more extreme demands on a structure. On the other hand, far-fault earthquakes have more spread-out energy content and longer durations without having the impulsive aspects that are displayed in near-fault ground motions [18,19,20].

The site class was considered stiff soil, similar to the original SAC Building Report [15]. The US Seismic Design Maps tool [21] was used to find the seismic design parameters for the chosen location.

Table 4b shows the properties of selected earthquakes. It can be seen that these records cover a range of different site classes. This variety enables us to assess the seismic behavior of structures under various ground conditions and to gain a comprehensive understanding of their structural performance.

Figure 4 presents only one horizontal component of the unscaled acceleration time histories of selected MS–AS sequences to provide a clear visual comparison. Earthquakes were scaled according to the guidelines of ASCE/SEI 41-13 [22] (see note under Table 4b).

All ground motion records have been scaled by ASCE/SEI 41-13 [22] guidelines. A 5% damping ratio was used for attenuation, and the target response spectrum corresponded to a site spectral acceleration value of S_ds = 1.321 g at the specified coordinates (Latitude 34.05°, Longitude −118.25°). The scaling was uniformly applied over the period range from 0.2 T to 1.5 T in both principal horizontal directions.

2.3. Extended Sequence with Multiple Aftershocks

In addition to mainshock–aftershock (MS–AS) sequences, a special case study was conducted to investigate the cumulative damage effects of the multiple aftershocks on the seismic behavior of the buildings. The Mammoth Lakes earthquake (1980) was selected as the mainshock earthquake. Following the mainshock, up to five aftershocks occurred within the same fault system within a short period. The selected aftershocks were combined with the mainshock. The aftershocks were chosen based on time proximity and station compatibility. In this scenario, aftershocks with a 10-s zero acceleration gap were combined with the mainshock and following aftershocks. The extended sequence scenario provided valuable real insight into structural response when the buildings were subjected to multiple aftershocks, particularly in seismically active areas. Figure 5 displays the unscaled extended sequence with five aftershocks of the Mammoth Lakes earthquake. The scaling process of the Mammoth Lakes earthquake and its successive aftershocks were the same as MS–AS sequences.

2.4. Retrofitting of Steel Structures

Retrofitting is a crucial strategy to mitigate seismic damage in existing steel moment-resisting frames. Various strengthening methods have been developed, including concentric and eccentric bracing frame systems, energy dissipation devices such as dampers and ductile fuses, base isolation, and fiber-reinforced polymer strengthening. These methods differ in their mechanisms, cost, complexity, and effectiveness, depending on the building configuration and seismic demands. In seismic retrofitting, these approaches aim to enhance lateral stiffness, energy dissipation, and overall seismic resilience, balancing structural performance improvements with practical considerations such as ease of implementation and maintenance requirements. The X-bracing frame system employed in this study represents a widely accepted and code-compliant retrofit strategy that effectively improves seismic performance while maintaining design simplicity and cost efficiency. Although alternative techniques, including advanced energy dissipation devices and base isolation, offered potential benefits under certain conditions, their complexity and applicability to existing structures could be limited. Consequently, the selected X-bracing retrofit for frames provided a pragmatic and robust solution aligned with the study’s goals of improving seismic resilience in a feasible and reliable manner. The present study focused specifically on the application and effectiveness of X-bracing; a comprehensive evaluation and comparison of other retrofit methods were beyond the scope of this work and were proposed for future research. All frames of the three structures were retrofitted with concentrically braced members. An extensive explanation is given in Section 2.5.

The retrofit aimed to enhance lateral stiffness, reduce interstory drift, and limit plastic hinge formation. X-braces for frames were placed in exterior bays, as shown in Figure 6, to maintain plan symmetry and minimize torsional irregularities.

Table 5. Properties of the X braces used for frame retrofitting.

Building	Brace Cross-Section	Width b (mm)	Thickness t (mm)	Length L (m)	Slenderness Ratio (λ)	Compactness Ratio
3-Story	HSS 8 × 8 × 5/8	173.7	14.75	7.27	94.3	11.7
9-Story	HSS 9 × 9 × 5/8	199.1	14.75	6.4	73.1	13.5
20-Story	HSS 10 × 10 × 5/8	224.5	14.75	5.33	54.5	15.2

displays properties of the braces designed for 3-, 9-, and 20-story retrofitted structures. Their slenderness ratio and compactness ratio were checked for all the selected cross-sections.

The X-bracing members used in the retrofitting of the SAC buildings’ frames were selected as HSS (Hollow Square Section) due to their performance benefits and practical use in seismic applications. HSS members have excellent torsional and flexural stiffness. This was beneficial for use in bracing systems subjected to cyclic axial forces during earthquake loading [23]. HSS had closed cross-sectional shapes that worked well for braced systems due to their ability to uniformly distribute stresses, which can reduce the sections’ local buckling and improve their post-buckling strength in compression [24].

Furthermore, HSS braces exhibited good energy dissipation capabilities, particularly in concentric bracing systems exposed to both tension and compression loads [25]. The HSS sections were also less vulnerable to local distortions and were more resistant to out-of-plane buckling than open sections such as channels or angles [23]. From a practical perspective, HSS members are/were geometrically compact and symmetrical, making them easier to fabricate, connect, and install into existing frames. The use of HSS braces was generally supported by design codes [22,26], which specified acceptance criteria and design requirements for brace members, particularly in seismic retrofitting applications.

2.5. Applied Normative Frameworks

This section summarizes the key standards and guidelines that govern the seismic retrofit design and nonlinear modeling approaches employed in this study. The design of the seismic retrofitting system, specifically the X-bracing retrofit of frames, followed the provisions of the TBEC 2018 [26] (Turkish Building Earthquake Code), which establishes requirements for member slenderness, ductility, and material properties to ensure effective seismic performance. For the nonlinear seismic analysis, the modeling of steel moment-resisting frame elements, including plastic hinge formation, yield rotation, ultimate deformation capacity, and strength degradation, was carried out by ASCE/SEI 41-13 [22]. This standard provides detailed criteria and acceptance limits for plastic hinges, which are critical for simulating inelastic behavior under earthquake loading. By integrating these normative frameworks, the study ensured that both the retrofit design and seismic response simulation conformed to national and international seismic standards.

2.5.1. X-Bracing Retrofit of Frames: Code Requirements and Design

The cross-sections of the X-braces were designed for frames based on the Turkish Seismic Code (TBEC 2018) [26], which outlines the specific requirements for the strength, slenderness, and ductility of braces to be used in seismic retrofitting. Even though the evaluation of nonlinear behavior and plastic hinge properties of the structures was implemented according to ASCE41-13 [22], the design of the X-braces was implemented according to the requirements of TBEC 2018 [26] for two reasons: (1) code applicability, (2) practical considerations. Designing the braces in accordance with TBEC 2018 [26] and modeling any nonlinear behavior based on ASCE 41-13 [22] ensured realistic implementation while also allowing for accurate estimation of parameters for nonlinear analysis. The braces were modeled as concentric X-bracing systems. The material used for modeling the braces was A36 structural steel. The brace cross-sections were selected according to the following key checks from TBEC 2018 [26]:

• Slenderness check: To select a suitable brace cross-section for each building, the slenderness ratio λ of the brace should be checked:

  λ = KL/i ≤ λ_max
  K = buckling coefficient, L = unsupported length of the brace, i = radius of gyration of the brace cross section, λ_max is typically 200 for compression braces [26].

2.

Compactness and ductility: Compactness and ductility of the brace cross-sections should be checked to ensure that braces dissipate energy properly; therefore, the width-to-thickness ratio should be controlled [26]:

b/t ≤ (b/t)_limit

To simulate the inelastic behavior under seismic loading, axial plastic hinges are introduced at the mid-span of each brace. These hinges capture both tensile yielding and compressive buckling, enabling a realistic representation of the brace’s energy dissipation and degradation. Plastic hinge parameters such as yield rotation, ultimate rotation, and stiffness and strength degradation follow ASCE/SEI 41-13 criteria.

This numerical modeling approach allows the nonlinear time history analysis to accurately simulate brace yielding, buckling, and energy dissipation, ensuring a reliable prediction of structural performance and damage progression.

2.5.2. Plastic Hinge Criteria According to ASCE 41-13

To evaluate the nonlinear behavior of the structural elements, plastic hinges were defined according to ASCE41-13 [22]. Plastic hinges represent localized inelastic rotations that develop in steel members under seismic loading. The fundamental parameter defining plastic hinge behavior is the yield rotation angle, θ_y, which marks the onset of yielding in the member. According to ASCE 41-13 [22] (Table 9.6), θ_y was calculated differently for beams and columns as follows:

For beams:

$${\theta }_y={\displaystyle \frac{Z{F}_{ye}{I}_b}{6E{I}_b}}$$

For columns:

$${\theta }_y={\displaystyle \frac{Z{F}_{y\mathrm{e}}{I}_c}{6E{I}_c}}\left(1-{\displaystyle \frac{\mathrm{P}}{P_{y\mathrm{e}}}}\right)$$

Z is the section modulus, F_ye is the expected yield stress of the steel, l_b and l_c are the effective lengths of the beam and column, respectively, E is the Young’s modulus, I_b and I_c are the moments of inertia of the beam and column sections, P is the axial load on the column, and P_ye is the expected axial yield load.

These parameters are critical for defining the bilinear force–deformation behavior of plastic hinges, as illustrated in Figure 7. The plastic rotation capacity was further refined by slenderness-dependent factors a and b given in Table 9.6 of ASCE41-13 [22], which were used to interpolate the plastic rotation angle limits based on flange and web slenderness ratios. The residual strength ratio, c, defines the strength retained after inelastic deformation, and acceptance criteria define the plastic rotation limits for Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP) performance levels as multiples of θ_y. This detailed modeling approach enables accurate simulation of inelastic behavior, strength degradation, and performance limits of steel members under seismic loading, ensuring realistic predictions of structural response in NLTHA.

A damping ratio of 2% was adopted in the NLTHA to represent the inherent energy dissipation of steel moment-resisting frames. This value aligns with the recommendations in FEMA 355C, Section 6.5.2 [27], which identifies 2% of critical damping as the standard inherent damping ratio for steel moment-resisting frames during dynamic analyses. Thus, the use of 2% damping in this study represents a consistent and code-compliant approach to modeling structural energy dissipation in seismic performance evaluations.

3. Results

Nonlinear time history analysis (NLTHA) was performed to obtain the detailed and precise response of the considered steel moment frame structures under real ground motions and their consecutive aftershocks.

The steel moment-resisting structures were evaluated in terms of the following response parameters:

• Residual drift: displays how the structures permanently deformed after nonlinear analysis, particularly after the aftershock effect
• Maximum interstory drift ratios (IDR): show the story’s maximum displacement
• Plastic hinge formation: records the development and extent of plastic hinge in columns, beams, and braces
• Base shear: used when comparing total lateral force resistance and flexibility

Figure 8 displays the interstory drift ratio of the 3-, 9-, and 20-story steel structures, indicating that the effects of aftershocks significantly amplify the structural demands of the buildings regardless of height. In the 3-story building, mainshock-only loading results were determined in moderate drifts (below 3%), including aftershocks, particularly for Chi-Chi, Coalinga, and Mammoth Lakes records, with the upper stories being most affected. The 9-story frame began to show more distribution of larger drifts in the mid and upper stories; the Chi-Chi, Friuli, and Imperial Valley records resulted in a greater drift ratio under MS–AS. The 20-story building continued to exhibit drift ratios of nearly 2% under mainshock loading, while aftershocks yielded a more varied distribution and increased structural demands under almost all earthquake records considered herein, except for the Northridge earthquake.

The plastic hinge formations were examined to determine how the aftershock effects caused cumulative damage in the steel structures. For brevity, only the plastic hinge formations in the 9-story building are given. Figure 9 shows the plastic hinges formed under the mainshock-only and mainshock–aftershock sequence of the Imperial Valley earthquake. While the distribution of the plastic hinges in the mainshock-only case remained at the Immediate Occupancy (IO, green color) damage level, considering the aftershock effect, it was seen that the number of plastic hinges and the severity of the damage was increased and the plastic hinges were spread across more stories, indicating greater and more widespread damage. Notably, blue and red dots appear in the figure, corresponding to the Life Safety (LS) and Collapse Prevention (CP) performance levels, respectively, further highlighting the intensifying damage due to aftershock sequences.

Figure 10 demonstrates the interstory drift ratio for the steel structures without and with bracing systems under MS–AS sequences. The results showed that after retrofitting, the overall pattern of the responses remained similar to the original structures without braces, and aftershocks continued to impose greater interstory drift demands than in the mainshock-only cases. However, the magnitude and distribution of these increases had reduced substantially. The retrofitted frames with braces exhibited a similar distribution (i.e., upper stories had larger demands for low-rise cases, while tall frames showed a more uniform distribution of drift); however, the overall response was more moderate.

Although there continues to be evidence of an aftershock sequence effect, the retrofitting with steel braces improved the ability of the structures to absorb and distribute seismic energy, and reduced the overall severity of drift amplification due to aftershocks. This ultimately suggested that, while the trend of MS–AS behavior did not fundamentally change, retrofitting successfully manages the severity of this behavior.

Figure 11 shows that the maximum interstory drift ratios produced by MS–AS sequences increased for all buildings when compared to the mainshock-only (MS) cases. The impact of this increased drift was greatest for the 3- and 9-story buildings, and particularly under the Coalinga and Chi-Chi records. After retrofitting (RT MS–AS), the interstory drift demands were significantly reduced for each case, but especially for the low- and mid-rise buildings. In the 20-story building, the obtained results verified that retrofitting had produced a significant reduction in drifts resulting from aftershocks and had improved overall seismic performance. Moreover, in some earthquakes, such as Friuli, Imperial Valley, and Northridge, drift values were relatively small. It could be stated that this was due to the frequency content of the aftershock as well as the structure’s capacity to dissipate seismic energy during the aftershock.

Figure 12 displays the base shear forces of the 3-, 9-, and 20-story buildings under MS, MS–AS, and the retrofitted structures with braces under mainshock–aftershock (RT MS–AS) sequences. As can be seen in Figure 12, it was observed that the base shear forces for all buildings were increased from the MS-only case to the MS–AS case; thus, the aftershocks caused an increase in demand. When comparing the MS–AS cases with the RT MS–AS cases, it was clear that the base shear force increased even more due to additional stiffness provided by the braces. While the retrofitting caused the base shear force to increase, it also provided improved drift control and seismic performance. The 20-story building exhibited lower base shear compared to the 3- and 9-story buildings due to its longer fundamental period, which shifted its response to the lower-frequency range of ground motions where spectral acceleration was typically lower. Additionally, its response was distributed across multiple modes, reducing the concentration of force at the base. As a result, despite its larger mass, the taller structure attracted less seismic force, which was consistent with the dynamic response characteristics of flexible, high-rise buildings.

Additionally, the seismic performance of the steel moment-resisting frames was evaluated under multiple aftershocks to examine how the number of aftershocks could affect the behavior of the building. The Mammoth Lakes earthquake and its five successive aftershocks were selected to investigate the multiple aftershock case. Figure 13 shows the interstory drift ratios for 3-, 9-, and 20-story buildings under the mainshock and multiple aftershock sequences (up to 5). In all cases, aftershocks increased drift demands, especially between the mainshock and the first few aftershocks. For the 3-story building, drifts increased steadily with each aftershock. In the 9-story frame, the increase slowed after the third aftershock, indicating a limit in damage progression. In the 20-story structure, it was observed that the interstory drift ratios suddenly increased in the lower floors as a result of the fifth aftershock. Overall, the interstory drift ratio highlights the cumulative impact of aftershocks on structural response.

Figure 14 shows the maximum interstory drift ratios for the 3-, 9-, and 20-story buildings under the multiple aftershocks. The results obtained highlighted that for all buildings, the drift ratios increased significantly from the MS to the MS–AS, demonstrating the immediate impact of the first aftershock. With multiple aftershocks (MS-2AS to MS-5AS), the drift ratios continued to increase, especially for the 20-story building, which showed a steady increase across all cases. In contrast, the 3- and 9-story buildings experienced smaller variations after applying the second aftershock, suggesting that their drift demands tended to stabilize beyond the second or third aftershock.

Figure 15 illustrates the progression of plastic hinge formation in the 9-story building subjected to a mainshock followed by up to five aftershocks. As seen in the figure under the mainshock case, the structure remained largely elastic with no significant hinge formation. As aftershocks were introduced, plastic hinges began to form and gradually increase in number, reflecting the cumulative damage effect of sequential seismic loading. In the figure, green dots represent Immediate Occupancy (IO)-level hinges, while red dots indicate Collapse Prevention (CP)-level hinges, signifying more critical damage. The appearance of more critical hinges in later cases showed that repeated aftershocks could push certain structural elements closer to their capacity limits. Overall, the seismic response results differed among the 3-, 9-, and 20-story structures, primarily due to their varying dynamic characteristics, including fundamental periods, modal shapes, and mass distributions. Taller buildings typically had longer fundamental periods, which caused them to respond differently to the frequency content of ground motions compared to shorter buildings. Specifically, low-rise buildings with shorter periods were more sensitive to high-frequency ground motion components, often resulting in larger interstory drifts under near-fault pulses. Conversely, mid- and high-rise buildings tended to be more influenced by the low-frequency content of the seismic records, exhibiting distinct patterns of displacement and plastic hinge formation. Additionally, variations in the floor plan configurations and structural framing layouts of the three buildings were affecting lateral stiffness and mass distribution, which could influence torsional response and damage localization. These geometric and structural differences, combined with the characteristics of the input ground motions such as near-fault earthquakes or far-field motions, contributed to the observed variation in structural responses. Therefore, the interplay between building properties and ground motion characteristics had to be carefully considered to accurately assess seismic performance across different building heights and seismic scenarios.

4. Discussion

The nonlinear time history analyses of the 3-, 9-, and 20-story SAC steel moment-resisting frames under specific mainshock–aftershock (MS–AS) sequences reveal that aftershocks significantly modify the seismic response compared to single-event scenarios. Among the ground motions considered, the Imperial Valley record imposed the most severe demands, producing notably higher interstory drifts, base shear forces, and plastic hinge formations due to its intense velocity pulses and forward directivity effects, which contribute to substantial cumulative damage when followed by multiple aftershocks.

The damage patterns vary distinctly with building height and the nature of the seismic input. For instance, the Imperial Valley and Northridge records tend to induce concentrated plastic hinges and peak drift demands in low- and mid-rise buildings, as their shorter fundamental periods resonate with the high-frequency content of these motions. In contrast, longer-duration records like Chi-Chi produce more distributed damage, particularly in high-rise buildings, whose longer fundamental periods respond differently to such inputs.

The X-bracing retrofit demonstrated considerable effectiveness in mitigating damage across all accelerograms by increasing lateral stiffness and limiting interstory drifts. However, the retrofit also caused an expected increase in base shear forces, highlighting the balance between deformation control and force demands under varying seismic inputs. Plastic hinge distributions in retrofitted models showed delayed formation and reduced severity, especially under the more demanding Imperial Valley sequence.

These results highlight the importance of using diverse and representative seismic records, such as those included in this study, for performance assessments. Ignoring the distinct characteristics of sequential mainshock–aftershock records risks underestimating structural vulnerabilities, residual drifts, and post-earthquake safety needs.

5. Conclusions

This study advances seismic performance assessment by evaluating the cumulative impact of mainshock–aftershock sequences on steel moment-resisting frames using nonlinear time history analyses of SAC benchmark structures. The key findings can be summarized as follows:

• Aftershocks substantially exacerbate structural demands, increasing interstory drifts, base shear forces, and plastic hinge formation compared to single-event seismic responses. This cumulative damage phenomenon is critical to consider for realistic seismic risk assessments.
• Building height and dynamic characteristics strongly influence damage distribution patterns. Low- and mid-rise buildings experience concentrated plastic hinge formation and peak drifts, whereas high-rise buildings show more distributed but sustained damage, underscoring the importance of tailored assessment strategies.
• The X-bracing retrofit system effectively reduces cumulative damage by enhancing lateral stiffness and limiting drift demands, although it increases base shear forces due to the added stiffness. This trade-off highlights the need for balanced retrofit designs that optimize both deformation control and force demands.
• Neglecting aftershock sequences in performance-based seismic assessments risks underestimating residual drifts and structural vulnerabilities, which can compromise post-earthquake safety and recovery decisions.
• The methodology and findings underscore the necessity of integrating multiple earthquake effects into the seismic design codes and assessment protocols, particularly in regions prone to clustered seismic activity.

Future research should focus on exploring alternative retrofit technologies, incorporating fragility analyses and residual drift metrics, and considering soil–structure interaction effects to further refine seismic resilience evaluations under complex earthquake sequences. Overall, these results advocate for the explicit incorporation of MS–AS sequences in seismic performance evaluations and retrofit considerations, particularly in seismically active regions where aftershocks are frequent and significant.