Seismic behavior of RC moment resisting structures with concrete shear wall under mainshock–aftershock seismic sequences

A structure may be subject to several aftershocks after a mainshock. In many seismic design provisions, the effect of the seismic sequences is either not directly considered or has been underestimated. This study examined the seismic behavior of reinforced concrete (RC) moment-resisting structures with concrete shear walls under seismic sequences. Two three-dimensional structures of short and medium height were designed and analyzed under seven real mainshock–aftershock seismic sequences. The models were loaded and designed according to the Iranian seismic code (4th ed.; Standard No. 2800) and ACI-318. The structures were analyzed using the nonlinear explicit finite element method. The maximum displacement, inter-story drift ratio, residual displacement, and ratio of aftershock PGA to mainshock PGA were investigated and assessed. Because of the high lateral stiffness of the shear walls in addition to their completely elastic behavior, the aftershocks did not increase the inter-story drift ratio or relative displacement in the short structure model. The medium height model under the seismic sequences showed a significant growth in the relative displacement (roughly 25% in some cases), inter-story drift ratio, plastic strain, and residual displacement (42.22% growth on average) compared to the structure that was only subjected to the mainshock. Remarkably, in some cases, the aftershock doubled the residual displacement. Unlike the moment-resisting frames under seismic sequences, the models showed no significant growth in the drift ratio with an increase in height. Assessments indicated that the ratio of aftershock PGA to mainshock PGA is a determinative parameter for structural behavior under seismic sequences, such that high values of this ratio (> 0.6) caused significant increases in the residual and relative displacements.


Introduction
An earthquake in a seismic zone is not a single event. In most cases, a chain of aftershocks of different magnitudes will accompany it. Multiple earthquakes occur around the world, especially in regions with complex fault systems. These fault systems usually do not release all the accumulated strain energy during the first vibration or mainshock; thus, tension and strain will form in different parts of the fault system that will lead to subsequent vibrations or aftershocks. This will continue until the fault system has stabilized (Ōmori 1984).
These aftershocks will affect the dynamic behavior of structures in terms of plastic strain, residual displacement, and so on. Studies have shown that the repair of structures damaged by a mainshock and aftershocks, in most cases, leads to failure of the rehabilitation strategy, mainly due to the short time intervals between the mainshock and aftershocks. In this regard, most structural damage originates from the significant reduction in stiffness and strength of the structural members and the failure of materials under repetitive seismic loading (Dulinska and Murzyn 2016).
Because structures located in seismic zones will experience seismic sequences, a structure that has resisted the mainshock could be destroyed during the aftershocks (Amadio et al. 2003). In some cases, strength and stiffness degradation caused by cyclic seismic loading can produce up to twice the amount of lateral displacement during aftershocks compared to the mainshock (Hosseini et al. 2019;Tarigan et al. 2018). The Michoacán earthquake (Mexico City, 1985) experienced a mainshock of 8 and an aftershock of 6.6 (Richter) and the Northridge earthquake (Los Angeles, 1994) experienced a 6.7 mainshock followed by a 6 (Richter) aftershock. These are good examples of this complicated event (Amadio et al. 2003). Despite their devastative structural effects, in most analytical studies and design codes, the effect of aftershocks has been neglected, underestimated, or only indirectly considered.
The seismic design of RC moment-resisting buildings with shear wall systems have attracted attention because of their low relative displacement, increased energy absorption, and reliable seismic behavior (Astaneh-Asl 2002; Qu and Bruneau 2009). Figure 1 compares a typical RC building and an RC building with a shear wall system. Shear walls are vertical elements that resist lateral or horizontal loads. In medium-to-tall buildings, a shear wall system is used to decrease the effect of earthquake loads and handle gravitational loads.
A shear wall behaves like a deep cantilever beam which provides lateral stability and stiffness for a structure. Such a system can be used for short, specially designed structures RC frame building RC shear wall building Deformability Strength Fig. 1 Comparison between RC moment-resisting frame building and RC shear wall building moment-resisting frame structures with a concrete shear wall system under seven real mainshock-aftershock seismic sequences. In this regard, nonlinear time-history dynamic explicit FE analysis was used. Two structural models of three and six stories were seismically loaded and designed according to Standard No. 2800 of the Iranian code and ACI-318 (Standard No. 2800ACI 318-14 2014). ETABS and ABAQUS software were used to design and analyze the models (ETABS 2019; ABAQUS 2018). The seismic behavior of the models was investigated under seven real mainshock-aftershock seismic sequences from Friuli, Mammoth Lakes, Cape Mendocino, Imperial Valley, Northridge (Newhall-Fire Station), Northridge (Rinaldi Receiving Station), and Whittier Narrows. The seismic parameters of drift ratio, ratio of aftershock PGA to mainshock PGA, maximum relative displacement, residual displacement and maximum elastic-plastic strain were considered and compared under the mainshock only and under the mainshock-aftershock sequence.

Shear wall design
In this study, the design of the shear walls was based on the simple tension and compression boundary element method (Simple T & C) in which it is assumed that the sizes of the boundary elements at the top and bottom (columns in this case) are fixed. Therefore, the program does not adjust the size of the boundary elements. The software calculates the area required for the steel rebars in the boundary members. The software algorithm confirms that the reinforcing ratio is less than the maximum allowable ratio and the minimum number of rebars will be used for the shear wall (Mostofinejad 2008;ETABS tutorial 2010;Shear Wall Design Manual ACI 318 2016).
The pier of the wall is designed for a factored axial force ( P f −top ) and moment ( M f−top ) based on the loading combinations. (ETABS tutorial 2010; Shear wall design manual ACI 318 2016). The applied moment and axial force are converted to tantamount force using Eqs. (1) and (2), which are also applied to the bottom of the boundary elements (ETABS tutorial 2010;Shear Wall Design Manual ACI 318 2016).
where L p is the wall length and B 1−left and B 1−right are the width of the left and right boundary members, respectively.
The values of P left−top and P right−top could be either under tension or compression (negative or positive), depending upon the loading pattern. In terms of tension, the area required for the steel rebar can be calculated using Eq. (3): where P is either P left−top or P right−top , A st is the area required for the steel reinforcements in the concrete design, s is the steel resistance factor and f y is the steel yield strength.
For the case of compression, the area required for the steel rebars ( A sc ) must satisfy either Eqs. (4) or (5) (Mostofinejad 2008;ETABS tutorial 2010;Shear Wall Design Manual ACI 318 2016). For negative values of A sc , no compressive reinforcement is necessary; however, for all cases, the reinforcing ratio should be less than the maximum allowable ratio.
where P max factor is defined by the shear wall design preferences (the default is 0.80), c is the concrete resistance factor for compression, f c is the concrete compressive strength, A g the gross area and is equal to the total area (ignoring reinforcements) and A sc is the area required for steel reinforcement in the concrete design.

Finite element analysis
Nonlinear dynamic explicit FE analysis based on the central-difference integration rule was used to study the seismic behavior of the models under mainshock-aftershock seismic sequences in a large number of small-time increments. The numerical implementation of the time integration method is explained in Eqs. (6)-(8) below (Dassault Systèmes 2017; Náprstek et al. 2011). To provide realistic analysis, the structures were modeled with distributed mass.
When considering the cumulative damage during ground-motion loading or under seismic sequences (mainshock-aftershock), both the damage initiation criterion and damage evolution have been characterized by the yield strength (plastic strain for concrete) and progressive degradation of the material stiffness (under a specific yield surface). For brittle materials such as concrete, experimental studies have shown that the compressive stiffness recovers entirely when the load changes from tension to compression; thus, the stiffness recovery factor is considered to be equal to 1. Under tensile loading, however, stiffness is not recovered because of the existence of small cracks; thus, the stiffness recovery factor is considered to be equal to 0 (Lee and Fenves 1998;Lubliner et al. 1989;Hillerborg et al. 1976;Dassault Systèmes 2017). The damage evolution in the ductile materials (steel rebars) is defined based on the relative plastic displacement and a specific exponential form developed through trial-and-error (Hillerborg et al. 1976; Dassault Systèmes 2017) as: where i is the number of increments in an explicit dynamic step, U N i is the displacement for a DOF of N (displacement or rotation component), U̇N i is the velocity for a DOF of N (displacement or rotation component), Ü N i is the acceleration for a DOF of N (displacement or rotation component), M NJ is the mass matrix and P J (i) and I J (i) are the external and the internal load vector, respectively.

Verification
The nonlinear FE analysis results were compared with the results of an experimental study conducted by Thomson and Wallace (2004) in three parts: 1. As shown in Table 1, top displacement (Δ) of the shear-wall RW1 model ( Fig. 2) was calculated under a lateral load (F max ) of 141 kN and then was compared with the results of the experimental model under the same load. 2. The nonlinear cyclic behavior of the FE model was studied by analyzing the intended shear wall model under cyclic displacement, as illustrated in Fig. 3a. Then, the maximum reaction force (base shear) due to cyclic loading (F n c ) was compared with the same parameter for the experimental study of Thomson and Wallace (2004) (Table 2). Figure 3b shows the number of base shear cycles for the FE model under the cyclic displacement protocol. 3. Backbone curves and hysteresis loops provide valuable information about the nonlinear behavior of the models under cyclic loads (Londoño et al. 2015). Backbone curves, calculated from hysteresis curves, are used predominantly in seismic provisions (Londoño et al. 2015;Epackachi et al. 2019;Luo and Paal 2018). In order to verify the cyclic behavior of the RW1 model, the backbone curves derived from the hysteresis curves and the hysteresis loops of the experimental study of Thomson and Wallace (2004) were compared to the FE model of the current study Fig. 4a, b. There was good agreement on both the compressive and tensile sides and there was less than a 5% difference between the areas under the backbone curves.    Comparison of the results of the experimental and the FE models in the verification cases shows good agreement between the results (under 5% error in all cases).

Design of models
To study the seismic behavior of the models under seismic sequences, two models of different heights (9 and 18 m) intended for residential use (importance factor = 1) in an area with a high seismic hazard level (Ag = 0.35 g) were designed in ETABS (ETABS 2019). Figure 5 shows the plan view and the specification of the models.
The seismic loading and design of the models were in accordance with Standard No. 2800 and ACI 318 (Standard No. 2800ACI 318-14 2014). In the X-direction, the system was an intermediate moment-resisting frame (response modification = 5). In the Y-direction, the system was an intermediate moment-resisting frame with a shear wall (response modification = 6). The shear walls were located between axes 2 and 3. The sitesoil had the following characteristics: S = 1.5 , T s = 0.5 , T 0 = 0.1 , 375 m/s < V s < 750 m/s. These were matched in the selected soil types of the mainshock-aftershock records (Sect. 3.5).
The types of rebar and concrete were AIII and M30, respectively, with a yield strength ( f y ) of 400 MPa and compressive strength (f � c ) of 30 MPa. Figure 6 presents the 3D view of the structural models and Tables 3 and 4 list the details of the shear walls. The transverse reinforcement of the shear walls was 7.16, 5.00 for story one and 5.00 cm2/m for stories two and three in the 3-story model. In the 6-story model, the shear reinforcement was 12.25 cm2/m (for stories one and two), 9.06 cm2/m (for stories three and four), and 6.96 cm2/m (for stories five and six).

Analytical model
The structures were designed in ETABS and analyzed in ABAQUS for nonlinear FE analysis (ETABS 2019; ABAQUS 2018). To simulate the tensile behavior of the concrete, a  Hsu and Hsu (2015) and Carreira and Chu (1986).
The stress-strain behavior of the concrete in compression was calculated using Eqs. (9)-(12) and was presented and verified experimentally by Hsu and Hsu (2015). The former compressive concrete stress-strain behavior can be used for concrete with a compressive strength of up to 62 MPa. Figure 8 shows the compressive stress-strain diagram of the concrete. Parameters c , cu and E 0 are in kip/in 2 (1 MPa = 0.145 kip/in 2 ). Fig. 6 Designed structural models: a three-story, b six-story Table 3 Shear walls and boundary members details of the three-story structural model 1 3 Table 4 Shear walls and boundary members details of the six-story structural model where c is the calculated compressive stress, cu is the ultimate compressive stress, E 0 is the Young's modulus and c is the concrete compressive strain. The stress-strain behavior of the steel rebars was calculated based on the Ramberg and Osgood (1943) equations Eqs. (13)-(15). This approach is especially useful for metals that harden with plastic deformation, such as steel.
where is the stress (maximum = ultimate tensile stress), is the calculated steel material strain, F tu is the ultimate strength of the steel, F ty is the yield strength of the steel, r is the plastic strain at F tu and K is a constant that depends on the material being considered (0.002 in this study).
The concrete damage plasticity model was used to simulate the behavior of concrete elements under arbitrary loading. The model is a continuum, plasticity-based, damage model for concrete or brittle materials. It can model the plastic behavior of the concrete in both tensile and compression by considering the strength and stiffness degradation under cyclic loading (such as seismic loading). This model showed a good ability to consider the interaction between the concrete and rebars (Dassault Systèmes 2017; Wahalathantri et al. 2011;Lee and Fenves 1998). The transverse and shear rebars were considered according to the design calculations to provide concrete confinement. The FE analytical models are shown in Fig. 9a, b. Additionally, it has been assumed that models are placed on a rigid surface to prevent the remarkable effect of the soil-foundation interaction on the dynamic behavior of the models (Far 2019).

Mesh generation
The 4-node linear tetrahedron and 2-node linear 3D truss mesh elements were selected according to the analytical and geometric conditions for the concrete and steel rebar materials, respectively (Johnson 2006). To achieve the best mesh size, the six-story model was analyzed with different mesh element sizes using frequency-AMS analysis. The optimal element size was obtained by comparing the former results with the fundamental period calculated using Standard No. 2800 (0.67 s) (Standard No. 2800. Table 5 shows the calculated fundamental period of the model for different mesh element sizes. The results shown in Table 5 indicate that the calculated fundamental period diverged from the other cases (1.00, 0.50, and 0.25 m) when the element size was increased to 2.00 m. An optimum element size of 1.00 m was chosen for this study.

Mainshock-aftershock records
Seven real mainshock-aftershock acceleration records were chosen from the PEER database to investigate the seismic behavior of the models under seismic sequences (Pacific Earthquake Engineering Research Center 2011). The reason for the use of for finding aftershocks was to employ seismic declustering methods (independent-dependent) which were empirically determined by the data sequences from previous earthquakes and by measurement of the space-time history (Hainzl et al. 2010).
The shear wave velocity of the ground motion records agreed with the seismic design assumptions (375 < V s (m/s) < 750). A five-second zero acceleration interval was considered between the mainshock and aftershock (Pirooz et al. 2021). To provide a comprehensive evaluation of the behavior of structures under seismic sequences, an effort was made to select records having different aftershock PGA to mainshock PGA ratios, frequency contents, and magnitudes. The specifications, acceleration time-histories, and response acceleration spectra of the mainshock-aftershock ground motions are shown in Table 6 and Figs. 10,11,12,13,14,15 and 16.

Relative displacement
Studies have shown that, under seismic loading, shear wall systems can reduce the relative displacement up to 70% (Astaneh-Asl 2002; Mostofinejad 2008). Different design

Three-story model
For the mainshock-aftershocks of Friuli, Cape Mendocino, Mammoth Lakes, Imperial Valley, Northridge-1, Northridge-2, and Whittier Narrows, the maximum relative displacements of the model were 0.72 cm, 1.48 cm, 1.68 cm, 1.54 cm, 2.02 cm, 1.71 cm, and 1.50 cm, respectively. The relative displacement time-history of the model of the roof is shown in Fig. 17a-g. Because of the high lateral stiffness of the shear wall system (for lowrise structures), the model showed completely elastic behavior (plastic strain = 0) under three seismic sequences. The aftershocks caused no growth in displacement; thus, there was no need to consider seismic sequences in the design of short, low-rise structures with shear wall systems. Table 7 shows some of the results of three-story model analysis.

Six-story model
The six-story model (mid-rise) was subjected to seismic analysis under mainshock-aftershock records and showed different seismic behavior from the three-story model. During the Friuli and Northridge-1 mainshocks, the first floors underwent elastic to plastic behavior (Fig. 18), although the plastic strain did not increase or grow under the aftershock. This indicates that, under the intended mainshock-aftershock records, the seismic behavior of the model was not susceptible to the seismic sequence. In other words, the aftershock did not increase residual displacement.
Under the Cape Mendocino mainshock-aftershock, the model showed plastic behavior or residual displacement on the first floor; however, the aftershock caused a significant growth in the relative displacement as well as the residual displacement. The seismic sequence increased the maximum relative displacement up to 21.31% compared to to increase by about 27.97% and 55.61%, respectively, in comparison with considering only the mainshock. The mid-rise model showed unique seismic behavior under the Whittier Narrows seismic sequence. Both the mainshock and aftershock caused increases in the plastic strain and residual displacement, mainly in the first story. However, the relative displacement of the top story did not growth under the aftershock. Figure 19a-g shows the relative roof displacement of the six-story model and Table 8 lists the analysis results.

Inter-story drift ratio
The inter-story drift ratio is a valuable index for evaluating the seismic behavior of all types and heights of structural systems. The inter-story drift ratio was considered in the current study to be a standard for judging the seismic behavior of structures under seismic sequences (Zhou et al. 2012).

Inter-story drift ratio of three-story model
Section 4.1.1 describes how aftershocks did not cause increases in the relative displacement of the three-story model compared to the mainshocks. Therefore, in this case, there was no difference between the mainshock and the mainshock-aftershock drift ratio diagrams in Fig. 20a-g.

Inter-story drift ratio of six-story model
Investigation of the six-story model drift ratio under earthquake loading with a slight aftershock, such as Friuli or Northridge-1 (Fig. 21a, e), shows that there was no significant increase in the drift ratio due to the seismic sequence (aftershock), but this model showed plastic behavior on the first floor under the Friuli and Northridge-1 mainshocks. Indeed, there were no differences between the drift ratio diagrams for the mainshock and mainshock-aftershock.
The behavior of this model under the aftershocks of Cape Mendocino, Mammoth Lakes, Imperial Valley, Northridge-1, and Whittier Narrows showed significant growths in the drift ratio in Fig. 21b-g. Studies have shown that the seismic sequences (mainshock-aftershock) in low-to mid-rise structures with moment-resisting frame systems will cause growth in the drift ratio as the height increases (Hosseini et al. 2019;Ruiz-García et al. 2012;Hatzivassiliou and Hatzigeorgiou 2015). The present study showed that the seismic behavior of structures with shear wall systems of the same height behaved differently Fig. 21a-g. The drift ratio showed no remarkable growth, with an increase in the height, which could be considered to be better seismic behavior for structures with a shear wall system under a seismic sequence. In the low to mid-rise shear wall structures, the maximum increase/growth in drift occurred in the first floor.

Residual displacement
The comparison between the residual displacement of two models under mainshock and mainshock-aftershock shows that it increased sharply, up to almost 100%, in the mediumrise structures. On average, the six-story model showed a roughly 42.22% growth in residual displacement. The residual displacement of the roofs of the models under seismic sequences are shown in Tables 9 and 10.

Discussion
After comparing the maximum relative displacement of models under mainshock-aftershock seismic sequences, it could be concluded that this parameter did not always relate to the PGA and magnitude. For example, the relative displacement of the three-story model under the Mammoth Lakes earthquake was 13.51% greater than under the Cape Mendocino ground motion, although the PGA of the Mendocino record was about 58% less than for the Cape Mendocino record. In the case of the six-story model, the maximum relative displacement under the Cape Mendocino mainshock was 29.80% greater than under the Mammoth Lakes mainshock. This suggests that factors such as the frequency content, response acceleration spectrum (Figs. 10,11,12,13,14,15,16), fundamental period of the structure (this case study), earthquake energy, and energy release procedure could affect the behavior of structures under seismic sequences. The cumulative energy diagrams of the ground motions are shown in Fig. 22a-g. The ratio of aftershock PGA to mainshock PGA (α) is a determinative factor in the behavior of structures under seismic sequences. The Friuli and Northridge-1 aftershocks did not increase the drift ratio or the residual or relative displacements; thus, the reason for this behavior could be the low aftershock PGA to mainshock PGA ratio (0.3 ≤ α ≤ 0.35). With the high aftershock PGA to mainshock PGA ratios for the Cape Mendocino, Mammoth Lakes, Imperial Valley, Northridge-2, and Whittier Narrows seismic sequences (0.6 ≤ α ≤ 0.88), the aftershocks caused a remarkable increase in the relative and residual displacements. For example, the relative displacement grown 17.91% and 28.10% under the Cape Mendocino and Mammoth Lakes aftershocks, respectively.
Shear wall systems are mainly used in 4-to 35-story buildings (Megson 2005). As stated in Sect. 4.1.1 for the three-story model, the relative displacement under the ground motions was very low (insignificant) and the seismic sequences did not increase the relative and residual displacements. Somehow this model behaved as if it was overdesigned under seismic loads and showed only elastic behavior under all three mainshock-aftershocks.
As stated in Sect. 4.1, in the design codes, the allowable relative displacement at the top of structures with shear wall systems was H∕50 to H∕2000 (Khouri 2011). Even considering the effect of the seismic sequences (mainshock-aftershock), the relative displacement at the tops of both models satisfied the H/50 range (on average H∕160 for the six-story model).

Conclusions
This research studied the behavior of structural models having six and three stories along with a shear wall system under seven real mainshock-aftershock seismic sequences. The specifications of the selected ground motions agreed with the design assumptions. Nonlinear dynamic explicit time-history analysis was used to evaluate the seismic behavior of the models. The results produced a number of conclusions.
Nonlinear analysis of the three-story model showed that the shear wall system in the short structure significantly increased the lateral stiffness and the structural behavior was completely elastic during the seismic sequences. This implies that the effect of the seismic sequence is negligible in the seismic design of structures of this size.
The ratio of aftershock PGA to mainshock PGA (α) was a very important index for the study of the seismic behavior of the structures under seismic sequences. A structure under a mainshock may have exhibited residual displacement, but this displacement did not increase in response to a weak aftershock. For example, under the Friuli and Northridge-1 seismic sequences (0.3 ≤ α ≤ 0.35), there were no increases in the residual displacement or drift ratio. On the other hand, the other studied sesimic sequences (0.6 ≤ α ≤ 0.88) caused significant grows in the residual displacement and drift ratio.
Under a mainshock-aftershock sequence, the total height of a structure with a shear wall system had a significant effect on the seismic behavior of that structure. The 3and 6-story models showed completely different seismic behaviors under the seismic sequences. The 3-story model behaved linearly under the seismic sequences and the 6-story model, along with the nonlinear behavior, was subjected to increases in the residual displacement and drift ratio due to the aftershocks.
Unlike the medium-height moment-resisting structures in which the drift ratio increased with an increase in the height of the structure, under seismic sequences, the structures with shear wall systems behaved differently. The drift ratio did not increase as the height of the structures increased. This could be considered to be better seismic behavior for a structure with a shear wall system compared to a structure with a moment-resisting system under seismic sequences.
The greatest increases in the residual displacement and drift ratio occurred on the first story. This indicates that this story requires special attention during seismic design for seismic sequences.
The seismic sequences increased the residual displacement of structures up to 100% (42.22% increase on average) and the maximum relative displacement about 30%. For example, the residual displacement and maximum relative displacement of the6-story model increased 55.61% and 28.1%, respectively, under the Mammoth Lakes mainshock-aftershock record. This significant increase should be considered in the seismic design codes.
Aside from the effect of the aftershocks on the seismic behavior of the 6-story model, the relative displacement of the top story was an average of H/160, which satisfies the design code limitations for structural systems with shear walls.
Funding This research was not funded by any funding bodies.