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Jun 30, 2025

Seismic response and collapse capacity assessment of dual RC buildings with vertical irregularities in shear walls | Scientific Reports

Scientific Reports volume 15, Article number: 9966 (2025) Cite this article 1253 Accesses 1 Citations Metrics details Buildings and other human-made structures can exhibit various irregularities due

Scientific Reports volume 15, Article number: 9966 (2025) Cite this article

1253 Accesses

1 Citations

Metrics details

Buildings and other human-made structures can exhibit various irregularities due to environmental factors or architectural considerations. These irregularities can alter the structural response to applied forces, leading to the concentration of seismic demands in specific elements, such as shear walls or braces, which may result in irreparable damage or even collapse. As a result, understanding and considering these effects is essential in the analysis and design of irregular structures. While most research on irregularities in height has focused on setback structures, this study investigates the seismic behavior of RC buildings with geometric irregularities in shear wall height. The seismic response and collapse capacity of buildings with varying levels of irregularity are evaluated in comparison to regular structures to assess the impact of these irregularities on structural capacity. The results imply that when this type of irregularity occurs at higher elevation levels, it has minimal impact on the seismic behavior and, in some cases, may even enhance it. However, when the irregularity spans multiple stories and originates from lower elevation levels, its impact becomes significant, leading to substantial increases in structural responses and a higher probability of collapse. Furthermore, it was demonstrated that present seismic standards are ineffective in providing accurate predictions of the seismic behavior of RC shear wall structures exhibiting this type of irregularity. Consequently, this study proposes a new index, ∅v, which accounts for the vertical position of wall width reduction, addressing a critical gap in current seismic design provisions for these types of structures. Finally, recommendations were provided to improve the existing seismic codes.

The primary purpose of structural and earthquake engineering is to protect structures against collapse during rare earthquakes and to achieve a life safety performance level during design-level earthquakes. An important factor influencing the seismic response is the presence of irregularities in the structural system. Various types of irregularities have been shown to adversely affect seismic performance by concentrating plastic deformations in a limited number of elements or introducing complex seismic behavior. Among these, vertical irregularities—such as reductions in shear wall length, setbacks, or the presence of soft and weak stories—are particularly common. While some studies have addressed these issues, current design provisions remain limited and lack critical parameters needed to enhance seismic design effectively.

In 1997, Valmundsson and Nau evaluated seismic code criteria for irregular buildings using time history analysis with four records. They found that mass and stiffness criteria moderately increased seismic responses, while the strength criterion led to significant increases1. In 2002, Dot and Das investigated the impact of resistance reduction on the biaxial responses of buildings designed according to regulations. The results indicated that the highest local demands become more significant when structural resistance reduction is considered2. In the same year, De Stefano and Pintochi introduced a one-floor model that accounted for the combined influence of axial forces and biaxial lateral forces in the lateral load-bearing elements of structures. Their findings revealed that earlier models of asymmetric structures, which neglected the interaction between the horizontal components of earthquake records, were associated with larger responses. Inelastic interaction, however, resulted in a 20–30% reduction in floor torsion, except in structures with low periods3.

In 2003, Das et al. investigated seismic design challenges for buildings with vertical irregularities, focusing on stiffness, resistance, mass, and the impact of non-structural masonry walls. Their results showed that the Static Equivalent Method (ELF) was overly conservative for certain vertical irregularities. Furthermore, structures with diverse vertical irregularities demonstrated favorable performance under design-level earthquakes4.

In 2005, Tremblay and Ponce investigated how mass irregularities influence the seismic behavior of a frame with steel bracing and setback configurations that lead to abrupt variations in plan dimensions and seismic mass across the building’s height. Their study revealed that, even under the condition of significant mass irregularities, the structural performance of frames designed with the use of the equivalent static analysis approach in the code was only minimally affected5.

In 2006, Michalis et al. evaluated structures with vertical irregularities using the Incremental Dynamic Analysis (IDA) method. Their study addressed mass, stiffness, strength irregularities, and combinations of stiffness and resistance irregularities6. Earlier, in 2004, Chintanapakdi and Chopra explored structures with irregularities in stiffness, resistance, and their combination. They employed nonlinear time-history analysis and modal pushover analysis to investigate how these irregularities influenced floor drifts and displacements7.

In past years, research has been conducted on structures with setbacks (step-like structures), and this type of building frame has been widely used by engineers. In 1992, Wood studied the seismic behavior of setback RC frames. It was determined that setback frames, in comparison to uniform frames, do not exhibit increased vulnerability to damage or higher-mode effects8. In 2008, Athanasiado investigated the seismic response of RC buildings having irregularities in height. This study demonstrated that the seismic response of irregular buildings is comparable to, or even better than, regular frames9. In 2008, Karavasilis et al. presented an alternative and developed method to measure the amount of irregularity in frames as a result of the setback. In this study, two indices of irregularity were introduced for structures with setback. However, by determining only two parameters, the amount of irregularity in the same and similar structure cannot be easily determined. On the other hand, the provided parameters are solely based on the geometric features of the frame and architectural considerations10. The 2008 study by De Stefano and Pintucchi examines previous research on the response of structures exhibiting irregularities in both height and plan, and it also suggests practical approaches for using passive control systems to minimize torsional effects11. In 2010, Sarkar et al. introduced a novel approach to measure the level of irregularity in buildings with setback by considering dynamic properties of the buildings. Furthermore, they proposed a modification to the empirical formula specified by the codes and predicted the experimental period of setback building frames by introducing a new coefficient12. In 2011, Reyes and Chopra assessed the precision of existing nonlinear static methods in capturing torsional effects through a comparison of their outcomes with the accurate predictions derived from nonlinear time history analyses (NTHA). Results indicated that earthquakes from distant and near fields have similar impacts on the displacement demands when considering torsional effects13. In 2012, Sadashiva et al. studied the effects of vertical stiffness and strength irregularity using NTHA for 3- to 15-story structures. Adjusted designs achieved target interstory drift ratios, highlighting demand variations. They proposed simple equations to estimate these effects at critical locations14. In 2014, Varadharajan et al. introduced a unified parameter to evaluate irregularities in mass, stiffness, and strength, incorporating both their magnitude and distribution. They analyzed the seismic response of irregular buildings using 27 ground motions and developed regression-based equations to estimate key seismic parameters15. In 2017, the study by Mwafy and Khalifa examined the earthquake-resistant design of high-rise structures with irregularities in height. It concluded that severe irregularities result in poor seismic performance and recommended adopting more conservative design coefficients for such buildings, while suggesting adjustments for those with moderate irregularities16. In 2018, Wang et al. demonstrated that vertical irregularity has a more detrimental effect compared to mass irregularity. The probability of structural failure rises substantially with the combination of mass and vertical irregularities17. In 2020, Mohsenian et al. studied RC tunnel-form buildings with geometric irregularities. This research indicated that tunnel-form buildings have a high load-bearing capacity and strength, and the current seismic regulations need to be revised18. Ghayoumian and Emami presented a multi-directional pushover method for evaluating the performance of modern RC buildings with torsional irregularities19. Das et al. investigated the structures exhibiting irregularities in plan and height and showed how these irregularities can lead to life and financial losses and the collapse of structures. They stated that although substantial research has been carried out on asymmetric structures, there are still no credible frameworks for designing multi-story buildings that exhibit these characteristics. They have investigated this matter and highlighted the significance of conducting more research in the field of irregular structures to provide more comprehensive and general design guidelines for these types of buildings20. Amiri and Yakhchalian evaluated scalar and vector intensity measures (IMs) for collapse capacity evaluation of steel SMRFs with vertical irregularity in mass21. In 2021, Mohsenian et al. investigated how the nonuniform allocation of mass in the floor plan of concrete buildings with a tunnel system affects their seismic performance. They demonstrated that this structural system is resistant to torsional effects arising from these effects and exhibits favorable seismic performance22. In 2022, Nady et al. explored the effect of irregularity due to mass, stiffness, setback, and their combination. They introduced an index of irregularity to evaluate the extent and position of various types of vertical irregularities23. In 2024, Khosravi Larijani and Tehrani investigated the effects of irregularity due to nonparallel systems, considering the orientation of earthquake records. The results revealed that this type of irregularity increased the probability of collapse24. The study by Hussain et al. validated seismic design parameters for regular and irregular structures. It suggested raising the R-values specified in the code to promote more economical structural designs without compromising safety25.

The complex response of irregular structures under seismic loading highlights the need for advanced modeling techniques to accurately assess their performance26, using advanced methods of analysis, such as nonlinear time-history analyses27, including potential failure mechanisms of RC members in the analyses28, which ultimately contribute to improved seismic resilience and collapse prevention of structures29.

Among the studies conducted on irregular structures, there have been a few research on structures with geometric irregularities in height, and most of these studies have focused on structures with setbacks. In these structures, simultaneous with the reduction in the lateral load-bearing system length, the floor weights also decrease in the same proportion. This helps mitigate the adverse impacts of shortening the lateral load-bearing system. In many structures the floor area and weight remain constant, and the lateral load-bearing system is the shear wall. Due to the reduced shear forces in the top floors relative to the base floors, the engineers may decide to reduce the width of shear walls in the upper floors based on their engineering judgments. This creates vertical geometric irregularities in structures, and may lead to unfavorable effects on the seismic response, while in past research, this aspect has received less attention from researchers. In this study, due to the importance of this issue, attempts have been made to study the impact of reducing the width of shear walls on the seismic behavior and probability of collapse by considering various models with different configurations.

Current seismic codes define geometric irregularities in shear walls based solely on wall width reduction, ignoring the vertical position of this reduction. This simplification, driven by a lack of research data and a simplified approach to irregularity effects, assumes a uniform impact regardless of height, overlooking how different positions influence the seismic response. Ignoring this parameter can result in inaccurate seismic vulnerability assessments, as irregularity location critically affects load redistribution, plastic hinge formation, and overall structural performance. In this study, the reliability of present seismic standards to address this issue is evaluated, and recommendations are proposed to enhance these provisions.

To examine the impact of geometric irregularity in height on structural performance, fourteen buildings, including seven twelve-story and seven eight-story structures, were investigated. The structures featured a dual system comprising moment frames and concrete shear walls. Each of the 8-story and 12-story models include a regular structure and six irregular structures with various irregularity levels. Figure 1 illustrates the plans of these models for 12- and 8-story buildings.

Plans of (a) Twelve-story structures and (b) eight-story structures.

As evident from Fig. 1, there is no irregularity in the plans of the structures, and the buildings are entirely symmetrical. In these structures, the only irregularity is related to the reduction of the width of the shear wall, which is expressed by two indices ∅h and ∅v. The index ∅h represents the ratio of the width of the wall in the lower floors to the width of the wall in the upper floors, while ∅v denotes the fraction of the overall height of the wall relative to the height of the portion where the wall maintains its original width. These two indices can be expressed using the following formulas:

The parameters H1, H2, L1, and L2 are visible in Fig. 2.

Parameters of ∅h and ∅v indices.

To investigate the consequences of various levels of irregularity on the seismic behavior three values for ∅h (1.3, 1.7, and 2.0) and two values for ∅v (1.33 and 2.0) are considered. According to the ASCE7 code30, ∅h values greater than 1.3 lead to vertical geometric irregularity. Therefore, ∅h values of 1.3, 1.7, and 2.0 were selected to represent irregular buildings. In this research, the parameter ∅v is introduced to evaluate the impact of the elevation at which this irregularity occurs in the height, a factor currently overlooked in seismic codes.

In all models, the height of each story is 3.2 m, and the span length is 6 m. The dead load on the stories and the roof is 4.9 kN/m2, and their live load is considered to be 2.0 and 1.5 kN/m2, respectively. The load caused by the partitions was imposed as a uniform equivalent load of 1.0 kN/m2 on the floor surface. The load of external walls was assumed to be 6.4 kN/m.

To design structures using the spectral method, which is a mandatory approach used in the majority of design standards for irregular structures, the spectrum of the standard design was determined based on the characteristics of the soil and the desired site. In accordance with the ASCE7-16 code30, the base shear was matched to that resulting from the equivalent static analysis.

The buildings under investigation are located in the city of San Francisco, United States, falling within seismic design category D, as per the ASCE7-16 code30. The soil type considered for determining the target spectrum is of type C, with seismic characteristics specified in Table 1.

The structural system utilized is a combined system made up of moment-resisting frames and concrete shear walls with special ductility30. The structural modeling and nonlinear analyses were performed by OpenSees version 3.2.231. The design of the structures was done using ASCE7-1630 and ACI 318-1932 codes. After finalizing the section properties, the structures were prepared for 2D modeling in the OpenSees software.

The specifications for frame elements are given in Tables 2 and 3, respectively. Additionally, the dimensions of the shear walls are provided in Tables 4 and 5, and their reinforcement details are given in Tables 6 and 7.

For two-dimensional modeling in the OpenSees software, the approach outlined in FEMA P69533 was followed. In this manner, a shear wall and a three-span moment resisting frame was separated from the three-dimensional model. Since the structures are perfectly symmetric in plan, the behavior of the separated sections is a good representation of the behavior of the 3D structure. To account for the p-delta (P-Δ) effects in columns that bear part of the gravity load and are not modeled, a leaning column element was used, as shown in Figs. 3 and 4.

Schematic plan and elevation of 12-story frames.

Schematic plan and elevation of 8-story frames.

In OpenSees, the modeling of shear walls was accomplished using the Shear-Flexure Interaction Multiple-Vertical-Line-Element Model (SFI_MVLEM), a model derived from finite element principles that captures the nonlinear behavior of RC walls in both shear and flexure34. These elements have been experimentally calibrated and validated using the laboratory results of five relatively slender RC wall specimens35. Figure 5 presents an example of the validation of the cyclic behavior of shear walls modeled using the OpenSees SFI-MVLEM model. The experimental data from the RW-A15-P10-S78 shear wall specimen, tested by Kolozvari et al.34,35, was used to verify the accuracy of the nonlinear modeling of shear walls.

Validation of the cyclic behavior of shear walls predicted using the SFI-MVLEM model in OpenSees, compared with experimental data from the RW-A15-P10-S78 shear wall specimen34,35.

Concentrated plastic hinge method was used in the modeling of moment frames. In this method, it is assumed that the middle regions of the beam and column elements have linear behavior and yielding occurs at both ends. To model these frames, linear elastic elements (elasticBeamColumn) are used in the middle areas of beams and columns. Also, non-linear springs with zero length (zeroLength) are used in the end regions of the elements, which only have rotational-torsional behavior. The materials used for the end springs of beams and columns were uniaxialMaterial ModIMKPeakOriented31, developed using the Ibarra-Medina-Krawinkler model36. The cyclic behavior diagram used for these springs is illustrated in Fig. 6, with its parameters derived from the relationships proposed by Haselton37. These relationships, which are based on the sectional and design properties of each frame element, have been validated and calibrated through experimental results37.

Hinge Properties31.

The materials used for the shear wall elements are nDMaterial FSAM34, which is a combination of steel materials (SteelMPF)38,39 for reinforcements and concrete materials (ConcreteCM)40 for concrete sections. The effect of confinement is considered in the concrete within the boundary elements of the shear walls. A compressive strength of 30 MPa was used for concrete, along with AIII and AII reinforcements. The material specifications are given in Tables 8 and 9.

For the seismic performance assessments different analysis methods were used including, pushover, NTHA and incremental dynamic (IDA). In the NTHA, the loading on the structures is applied using a series of recorded accelerograms from previous earthquakes. To achieve a more precise and detailed evaluation of the structural behavior under earthquakes, the IDA method outlined in FEMA P69533 was employed. In this method, varying intensities of each selected earthquake motion are applied to the model by scaling Sa or PGA parameters, and response curves are plotted for different intensities and for each record. These responses can provide insights into the behavior of structures in nonlinear ranges.

For NTHA and IDA, the earthquake records specified in FEMA P69533, which are related to far-field sites, were used. The earthquake records from FEMA P695 were chosen because they represent a broad range of seismic conditions and are intended to be applicable to a wide variety of locations, including those with different seismic hazard profiles. These records were specifically designed to cover a range of earthquake characteristics (e.g., magnitude, distance, and site conditions) that can occur in different regions and potentially cause substantial structural damage33. For the NTHA, records numbered 1 to 11 were used and scaled according to the ASCE 7-1630 provisions, which mandate a minimum of 11 records to determine the mean response for evaluations. The list of selected ground motions is provided in Table 10. The records were scaled to ensure that within the period range of 0.2 T to 2.0 T, the mean response spectrum of the records complies with or exceeds the target spectrum of the code (where T is the period of the main mode), as shown in Fig. 7.

Target response spectrum and scaled ground motion spectra.

For the IDA analysis, acceleration records number 1–22 were used, based on the FEMA P69533 recommendations, and the scaling of the earthquake records was performed based on the spectral acceleration parameter corresponding to the first mode (Sa(T1)) to determine the intensity of the earthquake input to the structures.

The validation of the nonlinear structural models was presented in the last section. To further verify the accuracy of structural modeling, the first-mode period extracted from the software programs ETABS and OpenSEES for 12- and 8-story structures has been compared in Tables 11 and 12, respectively. The highest difference identified in the results of these two software programs for the 12- and 8-story structures is 3.3% and 1.74%, respectively. This minor difference demonstrates the precision and reliability of the modeling performed using the OpenSEES software.

In this section, the results of nonlinear static analysis and pushover curves for the examined structures are provided. The lateral load distribution across the floors is based on the first vibration mode. As shown in Fig. 8, which illustrates the pushover diagrams for the 12-story and 8-story structures, the results reveal that reducing the wall width in a scenario where ∅v is equal to 1.33 has a minor impact on the capacity curve. In a way, the capacity curves of irregular structures almost closely match those of regular structures for this case. In the case where ∅v is equal to 2.0, it is observed that for smaller ratios of wall width reduction (∅h equal to 1.3 and 1.7), the ultimate deformation of the structure has increased. However, when ∅h reaches 2.0 (structures S12-R2-U6 and S8-R2-U4), the negative impact of reducing the wall width becomes evident, leading to a considerable reduction in resistance and ductility of the structures. The differences of various parameters between irregular and regular structures are presented in Tables 13 and 14 for 12-story and 8-story structures, respectively. It is noted that in these tables, ductility is defined as the ratio of ultimate displacement to yield displacement, while base shear refers to the total horizontal force exerted at the base of the structure (i.e., the sum of all lateral forces).

Pushover diagram of (a) twelve-story and (b) eight-story frames.

In summary, the pushover results indicate that when ∅v is 1.33, the reduction in wall width (∅h) has a minimal effect on the structural performance, and therefore, these frames may be classified as regular structures. However, for ∅v = 2.0, two distinct structural performances are observed, based on the magnitude of the ∅h index. When ∅h is equal to 1.3 and 1.7, the performance of the structure has improved, while for ∅h equal to 2.0, the performance has deteriorated.

The NTHA results, represented by mean inter-story drift values for all models, are evaluated for both DBE and MCE earthquake levels. According to FEMA P201241, story drift is a key parameter in seismic performance assessment. Figure 9 shows the drift diagram for the case where ∅v is equal to 1.33. In this scenario, the difference in drift between irregular and regular structures is negligible at both DBE and MCE earthquake levels, from the first story up to the point where the reduction in wall width begins. In the upper stories where the wall width is reduced, a rise in the value of ∅h causes a slight increase in the seismic demands of irregular structures in comparison with regular structures. In twelve-story structures, the maximum difference is observed in the twelfth story. This difference, for ∅h values of 1.3, 1.7, and 20. in DBE earthquake, is 1.5%, 2%, and 6%, respectively. In the MCE earthquake, the corresponding differences are 4%, 6%, and 13%. As evident, these differences increase with the intensity of the earthquake.

The story drifts at different earthquake levels when ∅v = 1.33.

In the 8-story structures of this group (∅v equal to 1.33), in the upper floors where the wall width is reduced, the story drift of irregular structures has not exceeded that of regular structures for ∅h values of 1.3 and 1.7. Only when ∅h equals 2.0 does the irregular structure exhibits greater drift in comparison with the regular structure, with a maximum increase of 3.6% during DBE earthquakes and 5.3% during MCE earthquakes, occurring on the eighth story. Comparison of the results for 12- and 8-story structures reveals that the impact of this irregularity on story drift increases with the number of stories.

The story drifts of twelve and eight-story structures under different earthquake levels for the case where ∅v equals 2.0 are presented in Fig. 10. It is observed that irregularity has a more noticeable effect in this case. In 12-story structures, the difference in story drift between irregular structures with wall width reduction ratios (∅h) of 1.3, 1.7, and 2.0, and regular structures in DBE earthquake is 13%, 23%, and 34%, respectively. In the MCE earthquake, the corresponding differences are 15%, 23%, and 47%. In 8-story structures, the maximum difference in story drift between irregular and regular structures is observed on the eighth story. For wall width reduction ratios (∅h) of 1.3, 1.7, and 2.0 in DBE earthquake, the increases in drift compared to regular structures are 8%, 15%, and 42%, respectively. In the MCE earthquake, the corresponding increases are 11%, 20%, and 53%.

The story drifts at DBE and MCE earthquake levels when ∅v = 2.0.

As shown in Fig. 10, for ∅v = 2.0, an increase in the ∅h index leads to higher inter-story drifts in the upper floors while reducing them in the lower floors. This outcome occurs because a higher ∅h index corresponds to a decrease in the strength and stiffness of the shear wall at the upper floors, resulting in greater drift values. Additionally, a more significant reduction in the shear wall width at the upper floors (i.e., higher ∅h values) increases the likelihood of developing a secondary plastic hinge at the location of the width reduction. This, in turn, limits the transfer of shear forces from the upper floors to the lower floors, thereby reducing drift values in the lower levels.

A comparison of Figs. 9 and 10 highlights the significant influence of the ∅v index, as higher values of this index amplify the effects of irregularity. The findings suggest that a combination of the ∅v and ∅h indices is essential to accurately assess the adverse impacts of irregularities on structural response. Since current code provisions overlook the ∅v index employed in this study, it appears imperative to incorporate this factor into future code updates.

In this section, bar charts illustrating the ratio of maximum drift in MCE earthquakes to DBE earthquakes are provided for 12- and 8-story structures to assess the impact of earthquake intensity. Figures 11 and 12 show the effects of increasing earthquake intensity on the maximum drift in the models.

The ratio of maximum drift in MCE earthquake to DBE earthquake in 12-story buildings.

The ratio of maximum drift in MCE earthquake to DBE earthquake in 8-story buildings.

The intensity of the MCE earthquake is typically assumed to be 1.5 times that of the DBE earthquake. Given the inelastic characteristics of the models, an increase in earthquake intensity results in the formation of additional plastic hinges, causing responses to increase by more than 50% with a 50% increase in earthquake intensity. Figure 11 illustrates that in 12-story structures, the impact of increased earthquake intensity is more pronounced in the lower and upper stories than in the middle stories. In 8-story structures, as shown in Fig. 12, the lowest impact is observed in the lower stories, while the impact increases for the upper stories. In the upper stories of structures with ∅v = 2.0 (i.e., S12-R2.0-U6 and S8-R2.0-U4), the increase in earthquake intensity has a more significant effect on drift compared to other structures. This can be attributed to the initiation of yielding in the longitudinal reinforcements of the wall at the level where the wall width decreases, leading to the creation of a secondary plastic hinge at this level.

The maximum plastic rotations of beams in 8 and 12-story structures are presented in Figs. 13 and 14, respectively, for ∅v equal to 1.33 and 2.0. By comparing Figs. 13 and 14, it can be observed that when the reduction in the width of shear walls starts from lower stories (∅v equals 2.0), the rotations of plastic hinges in beams are greater compared to the scenario where the reduction in wall width starts from upper stories (∅v equals 1.33). This matter again indicates the influence of the level at which the irregularity initiates on the structural responses.

Rotation of beam plastic hinges when ∅v is 1.33 for: (a) 12-story and (b) 8-story structures.

Rotation of beam plastic hinges when ∅v is 2.0 for: (a) 12-story and (b) 8-story structures.

In 12-story buildings where the width reduction starts from the sixth story and above, a comparison of various cases reveals that, as expected, the maximum beam rotations are observed sequentially in structures with width reduction ratios of 2.0, 1.7, and 1.3, with the structure without geometric irregularities exhibiting the lowest rotations. Therefore, the impact of the wall width reduction on the increase in beam rotations is evident. The largest difference in beam rotations between irregular and regular structures occurs at the twelfth story, with values of 27%, 33%, and 71% for structures with ∅h equal to 1.3, 1.7, and 2.0, respectively. It is important to note that, although an increase in ∅h from 1.3 to 1.7 results in a modest rise of 6% in the observed differences (from 27 to 33%), a further increase in ∅h from 1.7 to 2.0 leads to a substantial jump of 38% (from 33 to 71%). This underscores the need to impose a limit on the maximum ∅h value to mitigate the adverse effects of irregularity.

In 8-story buildings, where the width reduction starts from the sixth story and above (∅v = 1.33), the irregularity has minimal impact on beam rotations. However, when it initiates from the fourth story and above (∅v equal to 2.0), the effect of irregularity becomes more pronounced. In this scenario, the rotations of the beams for width reduction ratios (∅h) of 1.3, 1.7, and 2.0 have increased by 17%, 21%, and 59%, respectively, relative to the regular structure.

It is clear that the impact of reducing the width of shear walls is greater for structures with ∅v = 2.0, including S12-R2.0-U6 and S8-R2.0-U4, compared to other structures. This highlights the distinct behavior of these structures under high-intensity earthquakes compared to others. As earthquake intensity increases, the stress in the wall rises. If the wall width reduction is substantial enough to lower its resistance, the reinforcements may yield, leading to the formation of plastic hinges. The forces are redistributed, and the contribution of the frame to the incoming forces increases, leading to greater deformation and rotation.

Another issue that is very important in evaluating the performance of structures is the creation of plastic hinges in structural members. In this section, the results related to strain in critical sections of shear walls are presented to investigate the development of plastic hinges in shear walls.

Figures 15, 16, 17 and 18 show the results of compressive and tensile strains in all critical sections of twelve-story structures at different earthquake levels. As can be seen for the DBE earthquake level, the tensile strain has not reached the yield point of the steel (i.e., εy = 0.002 for longitudinal bars with a yield stress of 400 MPa) in any of the critical sections, which indicates the absence of plastic hinges in the critical sections of the shear walls. In this case, the compressive strain remains below the ultimate strain of the concrete (i.e., εcu = 0.003 based on the ACI 318-1932 code), and the concrete does not experience crushing.

(a) tensile strain and (b) compressive strain of shear wall under DBE earthquake in 12-story structures when ∅v is 1.33.

(a) tensile strain and (b) compressive strain of shear wall under MCE earthquake in 12-story structures when ∅v is 1.33.

(a) tensile strain and (b) compressive strain of shear wall under DBE earthquake in 12-story structures when ∅v is 2.0

(a) tensile strain and (b) compressive strain of shear wall under MCE earthquake in 12-story structures when ∅v is 2.0

With the increase in seismic intensity level (i.e., MCE level), the tensile strain near the bottom of the walls increases, reaching the yield point and resulting in the development of plastic hinges at the bottom of all walls. Compressive strain in the concrete at the base of the walls has increased beyond the ultimate strain of unconfined concrete in all structures, except for structure S12-R2.0-U6, resulting in concrete crushing in the unconfined areas. Since the concrete has reached its ultimate strain, it can be inferred that the walls in these structures have utilized their maximum capacity. The equal strength of these structures in the pushover diagrams can be explained by these results.

In the S12-R2.0-U6 structure, despite the fact that tensile strain has increased to the point of yielding in the steel and the reinforcements have yielded, compressive strain has not reached the ultimate strain of concrete. This suggests that the wall has not utilized its full capacity, and it can withstand more forces. The reason why additional forces are not applied to the base of the wall in this structure to fully utilize its maximum capacity is the development of a secondary plastic hinge at the section of the wall with a reduced width. As illustrated in Fig. 18, the tensile strain at the level of the wall where the width reduction occurs exceeds the yield strain of the steel. This suggests that some of the reinforcements in this section have yielded. With the increase in applied forces, more bars yield, resulting in significant deformations at this point. Consequently, this floor behaves as a soft story, inhibiting the transfer of additional forces to the lower floors. As a result, the wall sections in the lower floors do not reach their ultimate capacity. The reduction in resistance observed in the pushover curve for this structure is a direct outcome of this issue.

The IDA is utilized to predict the collapse capacity of various structures under earthquake records with different characteristics and intensities42. The results obtained can be used, employing the methods specified in FEMA P69533 and FEMA P-201241, to calculate Collapse Margin Ratios (CMR) and the probability of structural failure. By conducting analyses at different intensities for each earthquake record, IDA curves are obtained. These curves for 12-story and 8-story structures are depicted in Figs. 19 and 20, respectively.

IDA curve of 12-story structures.

IDA curve of 8-story structures.

After plotting the IDA curves, the corresponding spectral accelerations for the collapse level were determined for different records. Using IDA data, fragility curves can be developed to represent the probability of a structure reaching or exceeding specific damage states under varying levels of seismic intensity. These curves are essential for assessing the vulnerability of structures to earthquakes. Subsequently, a Cumulative Distribution Function (CDF) was applied to the dataset to determine the collapse probabilities at different spectral acceleration levels., and fragility curves for the structures were plotted. The cumulative distribution function follows the form of Eq. (3), based on the assumption of a lognormal distribution33.

where in Eq. (3), the parameters ST, ŜCT, and βtot represent the collapse spectral acceleration for each record, the median collapse acceleration, and total collapse uncertainty, respectively. Based on the guidelines provided by FEMA P69532, a total uncertainty (βtot) of 0.55 was determined, assuming that the quality of structural modeling, design requirements, and test data are classified as Good, Superior, and Superior, respectively.

Three distinct criteria were employed to characterize the collapse threshold for structures: a drift exceeding 5%30, numerical divergence of models due to dynamic instability, and reaching a point where the slope of the IDA curve is 20% of its initial value43. Some researchers have considered reaching a zero slope in the IDA curve as structural collapse44.

Fragility curves for 12-story and 8-story structures are presented in Figs. 21 and 22, respectively. In these figures, the fragility curve for the regular structure is plotted alongside the fragility curves for irregular structures for better comparison. Figure 21 illustrates the fragility curves for 12-story structures. For buildings with ∅v = 2.0 (i.e., figures b, c, and d), the fragility curves of the irregular structures are positioned above that of the regular structure, signifying an elevated probability of collapse across varying seismic intensities. Among these, structure S12-R2.0-U6, characterized by the highest ∅h index of 2.0, shows the most pronounced difference when compared to the regular structure. Conversely, for structures with ∅v = 1.33, depicted in figures e, f, and g, the fragility curves of the irregular buildings display minimal differences or even a lower probability of collapse relative to the regular structure. This trend is also evident in the 8-story structures, as demonstrated in Fig. 22.

Fragility curve of 12-stdory structures.

Fragility curve of 8-story structures.

Collapse Margin Ratio (CMR) is calculated as the ratio of the median spectral acceleration that induces structural collapse (\({\widehat{S}}_{\text{CT}}\)) to the spectral response acceleration at MCE level (SMT) corresponding to the fundamental period of the structure, defined by Eq. (4). The CMR is used to assess a structure’s seismic safety by comparing the median seismic demand to its collapse capacity. A higher CMR indicates better earthquake resilience and lower risk of failure.

Research conducted by Baker and Cornell45 demonstrated that the level of collapse capacity is significantly influenced by the characteristics of the input records, notably in the case of severe earthquakes or, in other words, rare earthquakes. To account for these effects, CMR is adjusted using the spectral shape factor (SSF), resulting in the adjusted collapse margin ratio (ACMR).

To facilitate a clearer comparison of the fragility curve results presented in Figs. 21 and 22, Tables 15 and 16 summarize the data for the 12- and 8-story structures, respectively. These tables highlight the impact of irregularity on the median collapse capacity,\({\widehat{S}}_{\text{CT}}\). In the most irregular case (i.e., ∅h and ∅v = 2.0), a reduction of approximately 15% in collapse capacity is observed. Additionally, a comparison of all cases reveals that the maximum variation in collapse capacity among structures of the same height reaches approximately 30%.

The values of the ACMR have been calculated for 12-story and 8-story structures, and the results are presented in the bar chart in Fig. 23. In this figure, the impact of irregularity on the ACMR values is observable.

ACMR of (a) 12-story structures and (b) 8-story structures.

The FEMA P-201241 mentions that structures designed for Dmax seismic zones have a high collapse safety margin. For this reason, in addition to the ACMR value, it is recommended to use the ACMR ratio of the irregular model to the regular model to investigate the influence of irregularity on the seismic response of the structures. To accomplish this, and to ensure a clearer basis for comparisons, Fig. 24 demonstrates the ratio of ACMR for irregular structures relative to regular structures for all buildings.

ACMR ratio of (a) irregular to regular 12-story structures and (b) irregular to regular 8-story structures.

The results indicate that when irregularity occurs from the sixth story and above in 12-story structures and from the fourth story and above in 8-story structures (i.e., ∅v = 2), the value of the ACMR decreases compared to the regular structure. As irregularity increases, the reduction in ACMR becomes more pronounced. In 12-story structures, for ∅h values of 1.3, 1.7, and 2.0, the ACMR decreases by 4%, 7%, and 12%, respectively. Similarly, in 8-story structures, the reduction for ∅h values of 1.3, 1.7, and 2.0 is 4%, 6%, and 7%, respectively.

When the reduction in wall width begins at the ninth story in 12-story structures and the sixth story in 8-story structures (with ∅v equal to 1.33), the adverse effect of irregularity on structural performance becomes less significant. In this scenario, a decrease in ACMR is observed only when ∅h is equal to 2.0. This reduction is 6% and 2% for structures S12-R2-U9 and 8S-R2-U6, respectively. For ∅h equal to 1.3 and 1.7, the collapse margin increased in both 12-story and 8-story structures, indicating that the reduction in wall width has improved the performance of the structures. In 12-story structures, this increase in ACMR ratio for ∅h values of 1.3 and 1.7 is 10% and 4%, respectively. Similarly, in 8-story structures, the increase is 8% and 3% for ∅h values of 1.3 and 1.7, respectively.

The influence of various levels of vertical irregularity at different locations of shear walls on the seismic response of dual structures comprising RC moment frames and shear walls was investigated. The evaluations were conducted using various methods of analysis. The major conclusions from the structures under study are summarized below:

Although current seismic codes define geometric irregularity solely through the ∅h index, which represents the reduction in wall width, another index, ∅v, was also introduced to account for the vertical location of this irregularity within the wall height—a critical factor excluded from existing codes. The study demonstrated that evaluating both indices is essential for accurately predicting the detrimental effects of irregularities for these types of structures.

The pushover analysis results concerning the impact of irregularities in structural performance showed that, depending on the values of ∅v and ∅h, this type of irregularity can have either an adverse or minimal impact on seismic behavior. Adverse effects were particularly evident when both ∅v and ∅h were equal to 2.0, indicating a significant reduction in shear wall width at the structure’s mid-height. In contrast, when ∅v was 1.33, the irregularity had minimal impact on the seismic behavior.

The analysis results from the nonlinear time history method showed that the negative effect of irregularity on the drift of structures and the rotation of beams is negligible when ∅v is equal to 1.33, and can be almost ignored. Therefore, these structures can be considered as part of the regular structures. When irregularity is applied at lower height levels (∅v equals 2.0), its adverse effects become significantly greater and with increasing amounts of ∅h, the structural responses increase further. By examining the responses between the structures studied, it is observed that when ∅h changes from 1.7 to 2.0, the responses increase with greater intensity. This indicates that when the ratio of the reduction in the width of the walls exceeds a certain value, the detrimental effects of this irregularity increase significantly. By examining the strain in the critical sections of the walls, it was determined that the reason for this issue could be the creation of a plastic hinge at the level of reducing the width of the wall in the S12-R2.0-U6 structure.

The IDA results demonstrate that the ACMR values associated with the structures S12-R1.7-U9, S12-R1.3-U9, S8-R1.7-U6, and S8-R1.3-U6 (∅v equal to 1.33 and ∅h equal to 1.3 and 1.7) are higher than those of regular structures, indicating that irregularity in these structures not only did not have a negative impact but also improved the performance of the structures. In structures with a ∅v of 2.0, irregularities noticeably affect seismic performance in all cases. The ACMR value decreases compared to regular structures, indicating an increased probability of collapse. While the findings of this study are based on specific cases, further research is essential to validate the proposed parameters across a broader range of structures. Future studies should focus on multi-story irregularity modeling, different irregularity types, real-world case studies, and the influence of varying seismic conditions. Additionally, experimental validation and advanced numerical simulations could enhance the reliability and applicability of these findings in diverse structural scenarios.

The presented results indicate that the criterion specified by ASCE7-1630 for determining geometric irregularity alone cannot be an accurate measure for assessing the irregularity of a structure. In the results of pushover analyses, nonlinear time history analyses, and IDA, it was clear that when ∅v is equal to 1.33, there is negligible difference in responses between irregular and regular structures. In some cases, the reduction in width not only had no adverse effect but also improved the structural performance. The results indicate that, despite the adverse effects of increasing the width reduction ratio, ∅h, from 1.3 to 1.7 and then to 2.0 on the structural performance, a more crucial parameter, not explicitly addressed in the code, is the story at which the width reduction initiates. This parameter is shown in this research with ∅v and should be defined to determine the geometric irregularity at height. For example, the results show that a substantial wall width reduction at mid-height or lower levels (high ∅v values) causes premature plastic hinge formation, load path alterations, and stress concentrations, significantly reducing collapse resistance, while a reduction at higher levels may delay collapse but increase upper-story drift demands. Another deficiency in the code related to geometric irregularity at height is the lack of specified ranges for this irregularity. For this type of irregularity, specific ranges can be defined to categorize structures into different levels of irregularity, such as regular, highly irregular, and extremely irregular, based on various degrees of the irregularity parameter. In the boundary regions of each of these ranges, specific values for ∅v and ∅h can be defined so that structures with varying degrees of irregularity can be distinguished from one another. Moreover, the code lacks a clear penalty for such irregularities and is limited to imposing constraints on the methods of analysis. By defining different ranges of irregularity, appropriate constraints, and penalties can be established for structures with various degrees of irregularity, similar to other types of irregularities, including torsional irregularity and soft-story formations. The inclusion of these design constraints in the codes can enhance safety and prevent the design of structures with a high probability of collapse.

The introduction of the ∅v index represents a significant advancement in evaluating geometric irregularities in shear walls by incorporating the vertical position of irregularities—an aspect currently overlooked in seismic codes. To effectively integrate this index into seismic design standards, its practical implications must be further investigated. Its adoption could refine irregularity classifications, enhance design methodologies, adjust response modification factors, and improve accuracy of structural analysis. For practical application, seismic codes should provide simplified design equations or look-up tables correlating ∅v with necessary modifications in structural detailing. Key challenges include defining acceptable ∅v thresholds and ensuring experimental validation. Integrating ∅v into seismic codes would enhance structural performance and improve the prediction of irregularity effects.

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

Mehran Aslani & Payam Tehrani

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M.A. contributed data and analysis tools; writing the paper. P.T. conceived and designed the analysis; supervised the work. All authors reviewed the manuscript.

Correspondence to Payam Tehrani.

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Aslani, M., Tehrani, P. Seismic response and collapse capacity assessment of dual RC buildings with vertical irregularities in shear walls. Sci Rep 15, 9966 (2025). https://doi.org/10.1038/s41598-025-94328-z

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Received: 29 December 2024

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DOI: https://doi.org/10.1038/s41598-025-94328-z

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