1 Introduction

The Infill Masonry Walls (IMWs) have been considered as nonstructural members and were neglected in the modeling of buildings with their strength and stiffness. In addition, seismic provisions prescribed a factor to consider IMWs effects on the fundamental period of the buildings. Engineers do not consider the IMWs in their designing procedures and this may affect their models. To overcome this issue, some experimental investigations were made to find out the effects of IMWs on the structural response (e.g. Ramos et al. 2010, Preti et al. 2012, Salmanpour et al. 2015, Carpine et al. 2021). Brodsky and Yankelevsky (Brodsky and Yankelevsky 2017) focused on the effects of IMWs in the Reinforced Concrete (RC) frame by using half-scale specimens subjected to monotonic loads. Their results showed that IMWs increased the vertical loading resistance of the frame by an amount of 280% on average, and affected the failure mode. Nicoletti et al. (Nicoletti et al. 2020) proposed a procedure to determine the stiffness of IMWs in the modeling of RC frame and validated through an experimental work based on the in-situ impact load test. Uva et al. (Uva et al. 2012) investigated the effects of IMWs for an existing RC building located in Calabria, Italy, and proposed some factors to refine the structural response. Recently, scientific studies proved that the stiffness and strength provided by IMWs should be considered in the modeling process due to their effects on the global behavior of buildings (Mohamed and Romão 2020). In addition, the interaction between nonstructural components and structural members may lead to local damages in columns of buildings (Fardis et al. 1999; Perrone et al. 2017; Merino et al. 2020).

It is worth noting that the opening dimensions, boundary condition, geometry of wall (e.g. thickness and height), out-of-plane behavior, previous damage of IMWs, and double-leaf infills can highly influence the seismic behavior of buildings. Ozturkoglu et al. (Ozturkoglu et al. 2017) noted that the position and dimensions of openings could considerably influence the lateral stiffness and nonlinear behavior of RC structures, as compared to fully infilled and bare frames. Zuo et al. (Zuo et al. 2022) performed a parametric study to take the aspect ratio and opening dimensions of IMWs into account. Single and double-leaf infills were investigated under the in-plane and out-of-plane loads (Onat et al. 2018; Mazza and Donnici 2021). Misir et al. (Misir et al. 2016) proposed Z-ties as improving method that can help the double-leaf infills to withstand high lateral deformations. Furtado et al. (Furtado et al. 2016a, 2017) performed comprehensive experimental and numerical studies on lateral behavior and the out-of-plane failure of IMWs. They concluded that openings could reduce the frequency of IMWs by 20–40%, and the axial loads of lateral columns could increase the frequency of IMWs. In addition, they showed that the width of wall support and previous damages could considerably decrease the strength capacity of IMWs, even up to 60% (Furtado et al. 2020). Some investigations were carried out to evaluate the effects of material and geometry of IMWs, effects of lateral columns stiffness on the IMWs behavior, effects of the connection between IMWs and frame, as well as mechanical characteristics of IMWs on the seismic performance of buildings (Mazza and Donnici 2021; Wararuksajja et al. 2020; Huang et al. 2021). It was confirmed that previous damages of IMWs could significantly reduce the frequency of wall up to 5% and 35% corresponding to in-plane and out-of-plane loads (Furtado et al. 2017). Therefore, this effect should be considered for seismic vulnerability assessment of structures.

Many studies were performed to investigate the seismic failure probability of structures, while the IMWs were assumed as non-structural elements and their effects on the seismic responses were neglected (e.g. see Kazemi and Jankowski 2023a, Kazemi et al. 2023). It should be noted that the IMWs could provide additional stiffness and strength that may highly affect seismic parameters such as fundamental structural period and interstory drift ratio (Gara et al. 2021). Also, IMWs showed different limit-state levels, as compared to the structural elements, which affect the overall performance of the building. Moreover, previous damages of IMWs (i.e. in-plane damages) and different workmanships can affect the performance level and collapse capacity of IMWs (Furtado et al. 2022).

Nowadays, the growing need for constructing buildings in urban areas leads to having buildings on different soil types. Although many improvements made to model buildings with a high amount of accuracy, it is hard to take into account the effects of Soil-Structure Interaction (SSI) in the designing process. Therefore, designers used some simplified procedures proposed by seismic codes to consider the SSI effect. Many studies have been performed assuming the SSI effects on the buildings constructed in different soil types (Dutta et al. 2004; Pitilakis et al. 2008; Raychowdhury 2011; Petridis and Pitilakis 2020). Elwardany et al. (Elwardany et al. 2017, 2019) investigated the SSI effects on buildings with Steel Moment-Resisting Frames (SMRFs) with and without IMWs subjected to the earthquake-induced colliding. They concluded that SSI and IMWs could significantly alter the behavior and structural response of buildings. Recent studies focused on the seismic probabilities of collapse based on the fragility curves to assess the vulnerability of SMRFs and RC structures (Kazemi et al. 2020, 2021a, b). Some studies used finite element models to determine the fragility curves of RC structures and confirmed the key role of SSI effects in the seismic probabilities (Pitilakis et al. 2014; Karapetrou et al. 2015). In addition, Mitropoulou et al. (Mitropoulou et al. 2016) developed fragility curves for three systems of fixed-based, two-dimensional finite element for pile foundation and Winkler spring model. Approaches of modeling SSI is a fundamental principle and neglecting the soil nonlinearity may lead to an un-conservative prediction of the system response (Pecker et al. 2014). Rajeev and Tesfamariam (Rajeev and Tesfamariam 2012) investigated the nonlinearity effects of SSI and soil parameters such as cohesion, density, Poisson's ratio, and friction angle on the seismic vulnerability of RC structures using the Beam on Nonlinear Winkler Foundation (BNWF) model. Nonlinear behavior of shallow foundations may dissipate the seismic energy through soil yielding. Harden and Hutchinson (Harden and Hutchinson 2009) introduced a simulation methodology for modeling of nonlinearity of shallow foundations using BNWF, and the model was developed as a stand-alone module as a useful tool for designing purposes by Allotey and El Naggar (Allotey and Naggar 2008). Forcellini (Forcellini 2021) investigated the SSI effects on the fragility curves of the RC structures with and without IMWs and concluded that SSI effects increased the failure probability of both RC structures.

The present paper aims to investigate the effects of different placements of IMWs, and the nonlinearity effects of SSI on the seismic response of SMRFs. Due to different collapse states for IMWs and SMRFs, a Tcl programming algorithm was developed to control demand limitations during Incremental Dynamic Analysis (IDA). Therefore, it can be possible to determine the effects of IMWs on the seismic responses considering SSI effects assuming four soil types. To determine the seismic vulnerability of SMRFs, seismic limit state capacities and seismic fragility curves were plotted and the influence of assumed parameters were investigated. The results of this study can be widely used by civil engineers for design procedures or retrofitting purposes.

2 Modeling methods

2.1 Modeling of SMRFs

In this paper, buildings with three, five, seven, and nine story levels (3-Story, 5-Story, 7-Story, and 9-Story) with considering SMRFs in four outer frames were designed according to the regular plan presented in Fig. 1. In the design process, four soil types were considered to model buildings located in California with a latitude of 34.039º and longitude of − 118.324º.

Fig. 1
figure 1

Considered plan for modeling of SMRFs

Full size image

Table 1 presents values of the soil parameters and seismic design parameters for selected soil types. These quantities are available both in ASCE 7–16 (ASCE 2017) and on the USGS website (USGS 2022). SDS and SD1 are numeric seismic design values at 0.2 and 1.0 s, respectively, G0 is the initial soil shear modulus, Φ, c, γ, Vs, υ are friction angle, cohesion coefficient, unit weight, shear wave velocity, and Poisson's ratio, respectively. According to the seismic force-resisting system selected for the buildings, design parameters of R, Ω0 and Cd equal to 8, 3, and 5.5, respectively, were selected (see Table 12.2.1 ASCE 7–16 (ASCE 2017)). It is worth noting that the force-based design approach was used for designing SMRFs assuming spectral acceleration presented in Table 1 for considered soil types and provisions of ASCE 7–16 (ASCE 2017), AISC 360–16 (AISC 2016), and AISC 341–16 (see Section E3) (AISC 2016). To model three-dimensional buildings using ETABS 2016 software, the floor dead and live loads of 3.35 kN/m2 and 1.68 kN/m2 were applied, respectively. For designing the SMRFs, the parameters of Fy = 345 MPa (yield strength), E = 200 GPa, and υ = 0.3 were considered for steel, and the compressive strength of concrete for designing the foundation was considered equal to 30 MPa (Kazemi et al. 2021a).

Table 1 Soil and seismic design parameters
Full size table

Some studies used 2D structural models in Opensees (McKenna et al. 2016) for 3D buildings (e.g. see (Kazemi et al. 2018, 2020, 2021a, b; Mohebi et al. 2018; Asgarkhani et al. 2020; Yakhchalian et al. 2020, 2021)) since the regular plan of building can allow us to neglect the torsional effects. Regarding the improvements of analytical tools in Opensees (McKenna et al. 2016), it is possible to model the deterioration effects of strength and stiffness that are suitable for 2D modeling (Ibarra et al. 2005). In this study, the 2D models were verified with 3D buildings that were modeled in ETABS 2016 software. To reduce the complexity of models, 2D models were used (instead of 3D ones) and SSI effects were added to them. According to the plan, to model SMRFs in Opensees (McKenna et al. 2016), all columns, except for those in SMRF (blue colour), were assumed as leaning columns to consider the crucial role of the P-delta effects in seismic performance (Kazemi et al. 2018; Mohebi et al. 2018; Asgarkhani et al. 2020; Yakhchalian et al. 2020, 2021). All structural elements were modelled with an elastic beam-column element with two nonlinear rotational springs according to the Modified Ibarra–Krawinkler bilinear hysteretic model (IMK Hinge) (Ibarra et al. 2005; Kazemi and Jankowski 2023b; Mohebi et al. 2021; Yazdanpanah et al. 2022). Table 2 presents the design documentation for the assumed SMRFs considering four types of soil. The region of beam-to-column connection, that is known as panel zone, was modeled to capture the deformations. To model the panel zone, the procedure introduced in Opensees (McKenna et al. 2016) was used assuming eight rigid elements and one zero-length element with a trilinear backbone curve proposed by Gupta and Krawinkler (Gupta and Krawinkler 1999).

Table 2 Design documentation of SMRFs for soil types
Full size table

2.2 SSI modeling

The BNWF model, which can be widely used for modeling shallow foundations, retaining walls, and piles, have the degrading hysteretic behavior with the ability of defining rules. This model employs a series of nonlinear springs and dashpots in parallel to model soil beneath the foundation. Figure 2 presents the BNWF model and implementation of the springs and dashpots. The kind of springs used in this model plays a crucial role in soil responses. Therefore, three types of springs known as the q–z spring, p–x spring, and t–x spring were used to capture the vertical and rotational resistance, and the sliding and passive resistance of the foundation, respectively. Normalized load-normalized displacement curves of these springs were plotted in Fig. 3. The BNWF model was modeled in Opensees (McKenna et al. 2016) using material models of QzSimple for q–z spring, PxSimple for p–x spring, and TxSimple for t–x spring (Raychowdhury 2011; Rajeev and Tesfamariam 2012; Harden and Hutchinson 2009; Raychowdhury and Hutchinson 2009; Kazemi and Jankowski 2023b)

Fig. 2
figure 2

BNWF model and implementation of the springs and dashpots

Full size image
Fig. 3
figure 3

Normalized load-normalized displacement curves of the nonlinear springs

Full size image

The BNWF consists of structural beam elements between foundation and soil nodes, and soil springs defined with zero-length element (Fig. 2). To define the BNWF model parameters, the foundation of the 3-Story, 5-Story, 7-Story, and 9-Story SMRFs were designed according to the soil parameters presented in Table 1 using SAFE 2016 software. It should be added that the dimensions of foundations were controlled based on the assumed provisions and load combinations. Table 3 presents the dimensions of the foundation considered for SMRFs.

Table 3 Foundations' dimensions considered for SMRFs
Full size table

To define the materials used in the BNWF model, the equations presented by Boulanger et al. (Boulanger et al. 1999) were used. According to the definition, in the elastic region of the QzSimple material model, load qe was assumed to have linear behavior regarding deformation, z.

$$q_{{\text{e}}} = k_{{{\text{ini}}}} z$$
(1)

where kini is the initial elastic stiffness. It is experimentally achieved that the elastic maximum load, q0, is a factor Cr of the ultimate load, qult, then:

$$q_{0} = C_{{\text{r}}} q_{{{\text{ult}}}}$$
(2)

Then, the nonlinear part of the curve can be defined by:

$$q = q_{{{\text{ult}}}} - (q_{{{\text{ult}}}} - q_{0} )\left[ {\frac{{Cz_{50} }}{{Cz_{50} + \left| {z - z_{0} } \right|}}} \right]^{n}$$
(3)

where z50 is the displacement in the half of ultimate load, z0 is displacement in the yield point, C and n are constitutive values to control the post-yield portion and curve shape. The calculation for PxSimple and TxSimple material models is similar to the QzSimple material model with assuming various values of Cr, n, C. Gazetas (Gazetas 1991) proposed the following equations for calculating the vertical stiffness and lateral stiffness of springs, kv, and kh, respectively, based on the width and length of foundation, B and L, respectively.

$$\begin{gathered} k_{{\text{v}}} = \frac{{GL}}{{1 - \upsilon }}\left[ {0.73 + 1.54\left( {\frac{B}{L}} \right)^{{0.75}} } \right] \hfill \\ k_{{\text{h}}} = \frac{{GL}}{{2 - \upsilon }}\left[ {2 + 2.5\left( {\frac{B}{L}} \right)^{{0.85}} } \right] \hfill \\ \end{gathered}$$
(4)

2.3 Numerical modeling approach to represent infill masonry wall

For decades, researchers experimentally studied the IMWs effects on the lateral response of RC and SMRFs (see (Buonopane and White 1999; Tasnimi and Mohebkhah 2011)), and proposed different types of numerical models (see Koutromanos et al. 2011, Trapani et al. 2018, Mazza 2019). For instance, the less complicated and simple model is a double strut element developed by Smyrou et al. (Smyrou et al. 2011), three nonparallel strut elements that consider the interaction between IMWs and frames were developed by El-Dakhakhni et al. (El-Dakhakhni et al. 2003). Crisafulli and Carr (Crisafulli and Carr 2007) developed four nodes panel connected to the beam and column joints to include both shear and compressive behavior of IMW. In this research, the model proposed by Rodrigues et al. (2010) that is commonly used for simulation of nonlinear behavior of IMWs was used. Figure 4 presents the schematic illustration of the IMW modeling approach in Opensees (McKenna et al. 2016). Some experimental tests assuming cyclic loads validated this model. Therefore, to validate the modeling process, the experimental test used by Kakaletsis and Karayannis (Kakaletsis and Karayannis 2008) was assumed. Figure 5 presents the shear-displacement curve of the IMW model validated with the result of Test S-full infill used by Kakaletsis and Karayannis (Kakaletsis and Karayannis 2008). The results show good agreement with the numerical modeling of IMW.

Fig. 4
figure 4

Schematic illustration of IMW modeling approach in Opensees (McKenna et al. 2016)

Full size image
Fig. 5
figure 5

Verification of shear-displacement curve of IMW model in Opensees with the result of Test S-full infill used by Kakaletsis and Karayannis (Kakaletsis and Karayannis 2008)

Full size image

To define the IMW model, the nonlinear uniaxial element (see Fig. 4) should characterize by a multi-linear hysteresis curve defined as uniaxial Material Pinching4 in Opensees (McKenna et al. 2016). Furtado et al. (Furtado et al. 2015, 2016b, 2018, 2021) numerically investigated the IMWs and their influences on the seismic structural behavior. They showed that the first cracks of IMW could appear between 0.075% and 0.12% drift (dc). Therefore, in this study, dc was selected equal to 0.1%, and the corresponding cracking force, Fc, was determined equal to 570 kPa, based on the relationship between elasticity modulus of IMW, Em, and dc. According to experimental tests and recommendations of seismic codes, the ratio of cracking strength and maximum strength (Fc/Fcr) for the IMW considering the brick and mortar properties can be adopted equal to 0.55 (Furtado et al. 2015). In addition, Fy and dy are the yielding force and the corresponding displacement that can be assumed between 65 and 75% of Fcr and 0.35–0.50% of dcr, while Fcr can occur for drift value of 0.5% (Furtado et al. 2021). Moreover, by estimating the diagonal compressive strength of IMW from experimental specimens, fIMW, the maximum strength can be calculated from the following equations:

$$\begin{gathered} F_{{{\text{cr}}}} = 0.818\frac{{L_{\inf } \times t_{\inf } \times f_{{{\text{IMW}}}} }}{{C_{{\text{t}}} }}(1 + \sqrt {C_{{\text{t}}}^{2} + 1} ) \hfill \\ C_{{\text{t}}} = 1.925\frac{{L_{\inf } }}{{h_{\inf } }} \hfill \\ \end{gathered}$$
(5)

where Linf, hinf, tinf are the length, height, and thickness of the assumed IMW. Table 4 illustrates modeling properties used for IMWs. Figure 6 presents the uniaxial material model adopted for numerical modeling of IMW according to Table 4.

Table 4 Numerical properties adopted for modeling IMWs (Furtado et al. 2021)
Full size table
Fig. 6
figure 6

The uniaxial material model was adopted for numerical modeling of IMW according to Table 4

Full size image

2.4 Numerical models

To investigate the effects of assuming IMWs and SSI in the 3-Story, 5-Story, 7-Story, and 9-Story SMRFs, four placements of IMWs were considered. Figure 7 presents different placement of IMWs considered for the 3-Story SMRF. Similarly, all these placements were considered for other SMRFs with and without SSI effects.

Fig. 7
figure 7

Illustration of different placement of IMWs considered for the 3-Story SMRF model

Full size image

Table 5 presents fundamental periods of SMRFs with and without IMWs assuming different cases and soil types. In each soil type, Case D had the lowest fundamental period and Case A had the highest one. It is worth noting that the vibration mode of structures were influenced by adding IMWs. For instance, in Case B, the values of the first vibration mode was increased by 2.33%, as compared to Case A. Moreover, in Case C and D, the values of the first vibration mode were increased by 4.67 and 6.07%, respectively. Regarding different placement of IMWs, structure founded on soil type E had the lowest fundamental period and structure founded on soil type B had the highest one. Therefore, the fundamental period of structures were affected by changing the soil types.

Table 5 Fundamental periods of SMRFs with and without IMWs
Full size table

3 Incremental dynamic analysis

It is important to evaluate the current state of a building after a severe earthquake. The state of a building can be defined based on damage states of the non-structural and structural members. Seismic codes propose useful tools to determine the seismic performance levels of the structures. To investigate the seismic performance levels of SMRFs, IDA was employed to determine the seismic limit-states of SMRFs regarding the increasing amount of Intensity Measure (IM). In this study, the spectral acceleration in the fundamental period of the SMRF, Sa(T1), was selected as IM, and the Maximum Interstory Drift Ratio (M-IDR) as controlling parameter. Then, the seismic limit-state capacity assessment of SMRFs was conducted by performing IDAs assuming the Far-Fault (FF) records introduced by FEMA-P695 (FEMA P695 2009). In addition, FEMA 356 (FEMA-356. 2000) propose three performance levels of Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP) for corresponding M-IDR of 0.7%, 2.5%, and 5.0%, respectively, which can be used for limit-state assessment. Moreover, the Total Collapse (TC) limit of the structure can be achieved by considering the flat part of the IDA curve.

According to a full-scale experimental test of RC structure done by Pujol and Fick (Pujol and Fick 2010), IMWs that were assumed to fill all bays from bottom to top witnessed initial cracks with the width of 4 mm in M-IDR of 0.25%. Their results confirmed that IMWs were in the collapse state at an M-IDR of 1.75%. Therefore, in this M-IDR, IMWs were collapsed and could not be considered in the numerical model. To assume the collapse effects of IMWs, a Tcl programming algorithm was developed in OpenSees (McKenna et al. 2016) to control the collapse of IMWs in each story assuming M-IDR of the collapse limit of IMWs equal to 1.75%. Figure 8 presents IDAs of the 3-story SMRF neglecting and including SSI effects in the soil D subjected to FF record subset. Figure 9 presents IDAs of the 3-story SMRF with IMWs (Case D) and without IMWs (Case A) in soil C subjected to FF record subset.

Fig. 8
figure 8

IDA curves of the 3-story SMRF, a without SSI, and, b with SSI, in the soil D subjected to FF record subset

Full size image
Fig. 9
figure 9

IDA curves of the 3-story SMRF, a with IMWs (Case D), and, b without IMWs (Case A), in the soil C subjected to FF record subset

Full size image

3.1 Seismic performance level

To evaluate the influence of considering IMWs in modeling, the seismic performance levels of each aforementioned case based on the median IDA curves (M-IDAs) were compared. Figure 10a–d present M-IDA curves of the 3-Story SMRF considering four cases of IMWs placement and four soil types of B, C, D, and E, respectively, subjected to FF record subset. According to Fig. 10a–b, Case D had higher values of M-IDAs and showed that the placement of IMWs in all floor levels increased the lateral resisting strength of the 3-Story SMRF. While, the placement of IMWs according to Case C had lower values of M-IDAs compared to Case D, and higher values than other cases. Therefore, increasing the placement of IMWs in the 3-Story SMRF with soil types of B and C showed an increase in the values of Sa(T1) in M-IDA curves. For cases in the soil types of D and E presented in Fig. 10c, d, respectively, the placement of IMWs in the 3-Story SMRF increased the values of Sa(T1) in M-IDA curves. In all soil types, the placement of IMWs according to Case B had lower values of Sa(T1) in M-IDA curves that show the negative effects of using this case for the 3-Story SMRF. It can be concluded that having an opening in the 3-Story SMRF (i.e. Case B) can significantly decrease the values of Sa(T1). Similar results were obtained from M-IDA curves of the 3-Story SMRF considering four cases of IMWs with SSI effect presented in Fig. 11a–d.

Fig. 10
figure 10

M-IDAs of the 3-Story SMRF considering four cases of IMWs, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image
Fig. 11
figure 11

M-IDAs of the 3-Story SMRF considering four cases of IMWs with SSI effect on the, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image

To compare between results of all cases and soil types, the seismic limit-state capacities of the 3-Story SMRF in four soil types according to the aforementioned seismic performance level are presented in Table 6. It is possible to compare the values of Sa(T1) in four performance levels of IO, LS, CP, and TC, and four soil types, assuming different placement of IMWs. For instance, assuming Case D in soil types of B, C, D, and E, the limit-state capacities of the 3-Story SMRF experienced an increase by 16.45% (from 1.428 to 1.663), 17.22% (from 1.428 to 1.663), − 3.9% (from 1.428 to 1.663), and 6.95% (from 1.428 to 1.663), respectively.

Table 6 Seismic limit-state capacities of the 3-Story SMRF in four cases assuming FF record subset
Full size table

Table 7 presents the seismic limit-state capacities of the 3-Story SMRF in four soil types with SSI effects subjected to FF record subset. This table prepares the effects of SSI and can be easily used for comparing assumed cases. For example, in Case A with and without assuming SSI effects, the performance level of TC in soil B decreased from 1.428 (Table 6) to 1.133 (Table 7). Moreover, the values of Sa(T1) in four performance levels of IO, LS, CP, and TC, similarly decreased due to SSI effects. Therefore, the SSI effects can significantly affect the limit-state capacities.

Table 7 Seismic limit-state capacities of the 3-Story SMRF with SSI effects assuming FF record subset
Full size table

Figure 12a–d present M-IDA curves of the 5-Story SMRF considering four cases of IMWs placement and four soil types of B, C, D, and E, respectively, subjected to FF record subset. According to Fig. 12a, b, Case D had higher values of M-IDAs and showed that the placement of IMWs in all floor levels significantly increased the lateral resisting strength of the 5-Story SMRF. In addition, the placement of IMWs according to Case C had lower values of M-IDAs compared to Case D, and higher values than other cases. It can be noted that increasing the IMWs can significantly affect the Sa(T1) values of M-IDA of the 5-Story SMRF with soil types of B and C. Similar results were observed for soil D and soil E in Fig. 12c, d, respectively. Moreover, it can be show that the effects of the placement of IMWs in the 5-Story SMRF in soil types of B and C is more than those in soil types of D and E. The main reason for this may be lie in the type of structural elements. The 5-Story SMRF had higher structural strength in soil type E comparing with soil type B and for this reason, fundamental period of the 5-Story SMRF in soil E is less than in soil B (e.g. 1.56 and 1.81, respectively, according to Case A in Table 5). Therefore, the structural elements are important factor in the collapse of structure and can increase the values of Sa(T1) in M-IDA curves.

Fig. 12
figure 12

M-IDAs of the 5-Story SMRF considering four cases of IMWs, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image

Figure 13a–d present M-IDA curves of the 5-Story SMRF considering four cases of IMWs and four soil types with SSI effect subjected to FF record subset. The SSI effects can significantly reduce the values of Sa(T1) in M-IDA curves of all cases. Therefore, it is suggested to consider the SSI effects in the evaluation process due to its high influence.

Fig. 13
figure 13

M-IDAs of the 5-Story SMRF considering four cases of IMWs with SSI effect on the, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image

Table 8 presents seismic limit-state capacities of the 5-Story SMRF in four cases of IMWs and four soil types assuming FF record subset. It can be seen that the placement of IMWs in all cases can significantly affect the values of Sa(T1) in M-IDA curves in all seismic limit-state capacities. For instance, in soil type B, assuming IMWs placement of Case B, Case C, and Case D increased the performance level of TC of the 5-Story SMRF by 41.42%, 29.28%, and 20.18%, respectively. In addition, for soil type C, the performance level of TC of the 5-Story SMRF increased by 43.56%, 30.02%, and 25.06%, respectively. Table 9 presents seismic limit-state capacities of the 5-Story SMRF with SSI effects subjected to FF record subset. The SSI effects can significantly reduce the Sa(T1) values in M-IDA curves. For instance, in soil type E, the Sa(T1) values in the performance level of CP for Cases of A to D decreased by 37.41% (from 0.751 to 0.470), 29.55% (from 0.758 to 0.534), 24.27% (from 0.828 to 0.627), and 17.44% (from 0.854 to 0.705), respectively. Therefore, it is suggested to the designers for assuming the effects of IMWs and SSI in their evaluations, and Tables 8 and 9 can be easily used for the prediction of IMWs and SSI effects.

Table 8 Seismic limit-state capacities of the 5-Story SMRF in four cases assuming FF record subset
Full size table
Table 9 Seismic limit-state capacities of the 5-Story SMRF with SSI effects assuming FF record subset
Full size table

Figure 14a–d present M-IDA curves of the 7-Story SMRF considering four cases of IMWs placement and four soil types of B, C, D, and E, respectively, subjected to FF record subset. Results presented by Fig 14a and b showed that Case D had higher values of M-IDAs compared to other cases. While Case B decreased the values of M-IDAs of the 7-Story SMRF founded on soil type B, other cases had a positive influence on the values of M-IDAs of the 7-Story SMRF. Therefore, the placement of IMWs can significantly increase the values of Sa(T1) in M-IDA curves of the 7-Story SMRF founded on soil types of B and C. Figure 14c and d show that the placement of IMWs can affect the values of Sa(T1) in M-IDA curves in a positive way, while the number of values between cases has small differences. It should be noted that the 7-Story SMRF founded on soil types of D and E had higher structural strength compared to corresponding to the 7-Story SMRF founded on soil types of B and C (see Table 2). It can be concluded that the placement of IMWs effects can increase the structural strength of the SMRF. Figure 15a–d present M-IDA curves of the 7-Story SMRF considering four cases of IMWs and four soil types with SSI effect subjected to FF record subset. Although the SSI effect decreased the Sa(T1) values of bare frames in all soil types (e.g. Case A), assuming SSI significantly affects M-IDA curves of the SMRFs with IMWs (e.g. Case B, Case C, and Case D). In other words, the Sa(T1) values of M-IDAs for the 7-Story SMRF considering SSI effects were lower than those for the 7-Story SMRF neglecting SSI effects. Therefore, SSI effects play a key role in seismic capacity evaluations and should be taken into account.

Fig. 14
figure 14

M-IDAs of the 7-Story SMRF considering four cases of IMWs, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image
Fig. 15
figure 15

M-IDAs of the 7-Story SMRF considering four cases of IMWs with SSI effect on the, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image

Table 10 presents seismic limit-state capacities of the 7-Story SMRF in four cases of IMWs and four soil types assuming FF record subset. Results showed that the placement of IMWs in all the cases increased the values of Sa(T1) in M-IDA curves of the 7-Story SMRF founded on soil type B. For instance, assuming IMWs placement of Case C and Case D, the performance level of TC of the 7-Story SMRF founded on soil type B increased by 5.67% and 11.57%, respectively. Table 11 presents seismic limit-state capacities of the 7-Story SMRF with SSI effects subjected to FF record subset. Compared to Table 10, SSI effects significantly decrease the values of Sa(T1) in M-IDA curves. In IMWs placement of Case B, Case C, and Case D, the performance level of TC of the 7-Story SMRF founded on soil type B decreased by 7.94%, 12.32%, and 15.06%, respectively, assuming SSI effects. Similar trends were observed for other soil types.

Table 10 Seismic limit-state capacities of the 7-Story SMRF in four cases assuming FF record subset
Full size table
Table 11 Seismic limit-state capacities of the 7-Story SMRF with SSI effects assuming FF record subset
Full size table

Figure 16a–d present M-IDA curves of the 9-Story SMRF considering four cases of IMWs placement and four soil types of B, C, D, and E, respectively, subjected to FF record subset. Case D had higher values of M-IDAs compared to other cases in the four soil types, and Case A had the lowest values among other cases. Similar to other SMRFs, it can be seen that the placement of IMWs can significantly increase the values of Sa(T1) in M-IDA curves of the 9-Story SMRF. Figure 17a–d present M-IDA curves of the 9-Story SMRF considering four cases of IMWs and four soil types with SSI effect subjected to FF record subset. Compared to Fig. 16a–d, differences between Case D and Case A increased. In other words, the SSI effects had a lower influence on the cases with IMWs placement than the bare frame (e.g. Case A).

Fig. 16
figure 16

M-IDAs of the 9-Story SMRF considering four cases of IMWs, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image
Fig. 17
figure 17

M-IDAs of the 9-Story SMRF considering four cases of IMWs with SSI effect on the, a soil B, b soil C, c soil D, and d soil E, subjected to FF record subset

Full size image

Tables 12 and 13 present seismic limit-state capacities of the 7-Story SMRF in four cases of IMWs and four soil types assuming FF record subset without and with SSI effects, respectively. Results confirm the increasing trends of the values of Sa(T1) in M-IDA curves of the 9-Story SMRF with the IMWs placement of Case B, Case C, and Case D. For instance, assuming IMWs placement of Case D, the performance level of TC of the 9-Story SMRF founded on soil type B increased by 15.2%, and with assuming the SSI effects, the performance level of TC increased by 15.48%. Results of tables can be used for estimating the effects of IMWs and SSI on the performance levels of the 9-Story SMRF.

Table 12 Seismic limit-state capacities of the 9-Story SMRF in four cases assuming FF record subset
Full size table
Table 13 Seismic limit-state capacities of the 9-Story SMRF with SSI effects assuming FF record subset
Full size table

4 Seismic fragility curves

In this section, the fragility curves of the aforementioned SMRFs were compared based on the analytical procedures and probability of reaching or exceeding the aforementioned damages states. It is important to evaluate the influence of IMWs placement and their effects on the seismic performance. Therefore, the performance levels of IO and LS, which present nonstructural damages without death or serious structural failure, were considered. According to the aforementioned performance levels, the probability of exceedance of a limit state (i.e. IO or LS) was determined and different cases were compared. It is worth noting that the difference between seismic fragility curves of IO and LS performance levels for the 9-story SMRF can be justified due to difference between their corresponding M-IDR.

Figure 18a–d illustrate seismic fragility curves in the performance of IO and LS of the 3-Story SMRF considering four cases of IMWs founded on the four soil types subjected to FF record subset. The result shows that in the performance of IO for all soil types, the Sa(T1) values of Case D corresponding to the seismic failure probability of 50% are higher than those for Case C, Case B, Case A, respectively. According to Fig. 18a, in soil type B, the Sa(T1) values of Case A, Case B, Case C, and Case D, corresponding to the seismic collapse probability of 50% in the performance level of IO are 0.15, 0.17, 0.21, and 0.24, respectively, and in the performance level of LS are 0.56, 0.64, 0.74, and 0.79, respectively. Therefore, Case D with the full placement of IMWs can increase the seismic failure probability of the 3-Story SMRF founded on soil type B by 60% and 41.07%, in the performance of IO and LS, respectively. According to Fig. 18b, in soil type C, the Sa(T1) values of Case A, Case B, Case C, and Case D, corresponding to the seismic collapse probability of 50% in the performance level of IO are 0.19, 0.21, 0.25, and 0.27, respectively, and in the performance of LS are 0.70, 0.70, 0.87, and 0.94, respectively. Therefore, Case D with the full placement of IMWs can increase the seismic failure probability of the 3-Story SMRF founded on soil type C by 42.10% and 34.28%, in the performance of IO and LS, respectively. According to Fig. 18c, in soil type D, the Sa(T1) values of Case A, Case B, Case C, and Case D, corresponding to the seismic collapse probability of 50% in the performance level of IO are 0.21, 0.22, 0.28, and 0.30, respectively, and in the performance of LS are 0.75, 0.75, 0.93, and 1.01, respectively. Therefore, Case D with the full placement of IMWs can increase the seismic failure probability of the 3-Story SMRF founded on soil type D by 42.85% and 34.66%, in the performance of IO and LS, respectively. According to Fig. 18d, in soil type E, the Sa(T1) values of Case A, Case B, Case C, and Case D, corresponding to the seismic collapse probability of 50% in the performance level of IO are 0.21, 0.23, 0.28, and 0.30, respectively, and in the performance level of LS are 0.76, 0.77, 0.93, and 1.02, respectively. Therefore, Case D with the full placement of IMWs can increase the seismic failure probability of the 3-Story SMRF founded on soil type E by 33.33% and 34.21%, in the performance of IO and LS, respectively.

Fig. 18
figure 18

Seismic fragility curves of the 3-Story SMRF considering four cases of IMWs on the, a soil type B, b soil type C, c soil type D, and d soil type E, subjected to FF record subset

Full size image

Figures 19a–d, 20a–d, and 21a–d illustrate seismic fragility curves in the performance levels of IO and LS of the 5-Story, 7-Story, and 9-Story SMRFs considering four cases of IMWs founded on the four soil types subjected to FF record subset, respectively. According to Fig. 19a–d, comparing the Case A and Case D in the performance level of IO show that the seismic failure probability of the 5-Story SMRF founded on soil type B, C, D, and E increased by 48.75%, 39.36%, 31.19%, and 31.19%, respectively. Moreover, the performance level of LS experienced an increase by a factor of 37.70%, 23.93%, 20.94%, and 16.54%, respectively. According to Fig. 20a–d, comparing the Case A and Case D in the performance level of IO show that the seismic failure probability of the 7-Story SMRF founded on soil type B, C, D, and E increased by 52.83%, 36.23%, 31.94%, and 33.80%, respectively. Moreover, the performance level of LS experienced an increase by a factor of 24.88%, 19.76%, 22.56%, and 16.91%, respectively. According to Fig. 21a–d, comparing the Case A and Case D in the performance level of IO show that the seismic failure probability of the 9-Story SMRF founded on soil type B, C, D, and E increased by 48.93%, 49.05%, 45%, and 45%, respectively. Moreover, the performance level of LS experienced an increase by a factor of 31.39%, 22.68%, 17.57%, and 14.41%, respectively.

Fig. 19
figure 19

Seismic fragility curves of the 5-Story SMRF considering four cases of IMWs on the, a soil type B, b soil type C, c soil type D, and d soil type E, subjected to FF record subset

Full size image
Fig. 20
figure 20

Seismic fragility curves of the 7-Story SMRF considering four cases of IMWs on the, a soil type B, b soil type C, c soil type D, and d soil type E, subjected to FF record subset

Full size image
Fig. 21
figure 21

Seismic fragility curves of the 9-Story SMRF considering four cases of IMWs on the, a soil type B, b soil type C, c soil type D, and d soil type E, subjected to FF record subset

Full size image

In all soil types and SMRFs, the Sa(T1) values in the seismic failure probability of 50% corresponding to the performance levels of IO and LS in Case D were higher than those for Case C, Case B, and Case A. Moreover, it can be seen that assuming Case D for SMRFs founded on soil type B (e.g. 5-Story SMRF) resulted in the highest increase in the Sa(T1) values of seismic failure probability of 50% (e.g. 48.75%), as compared to the SMRFs founded in soil type E (e.g. 31.19%).

Therefore, it can be concluded that the IMWs can significantly decrease the seismic failure probability of the SMRFs and it should be assumed in the modeling process. The IMWs effects on the SMRFs founded on different soil types cannot be neglected and civil engineers should consider this effect to improve the seismic performance levels.

Figure 22a–d present the seismic fragility curves of the 3-Story, 5-Story, 7-Story, and 9-Story SMRFs in Case A and Case D on the soil type B with SSI effects, respectively, subjected to FF record subset. According to the results presented by Fig. 22a–d, SSI effects decreased the seismic failure probability in both performance levels of IO and LS. In the performance level of IO, the seismic failure probability of the 3-Story SMRF in the Case A and Case D decreased by 32.89% (from 0.152 to 0.102) and 60.92% (from 0.238 to 0.145), respectively, and in the performance level of LS, decreased by 22.34% (from 0.564 to 0.438) and 28.03% (from 0.792 to 0.570), respectively. In the performance level of IO, the seismic probability of collapse of the 5-Story SMRF in the Case A and Case D decreased by 22.22% (from 0.09 to 0.07) and 10.92% (from 0.119 to 0.106), respectively, and in the performance level of LS, decreased by 10.49% (from 0.305 to 0.273) and 14.52% (from 0.420 to 0.359), respectively. In the performance level of IO, the seismic probability of collapse of the 7-Story SMRF in the Case A and Case D decreased by 30.55% (from 0.072 to 0.050) and 15.78% (from 0.095 to 0.080), respectively, and in the performance level of LS, decreased by 30.35% (from 0.257 to 0.179) and 27.30% (from 0.315 to 0.229), respectively. In the performance level of IO, the seismic failure probability of the 9-Story SMRF in the Case A and Case D decreased by 14.89% (from 0.047 to 0.040) and 111% (from 0.07 to 0.06), respectively, and in the performance level of LS, decreased by (from 0.172 to 0.141) and 14.28% (from 0.226 to 0.179), respectively. For brevity, the results of SSI effects for soil type B were presented, while similar trends were observed for other soil types and SMRFs.

Fig. 22
figure 22

Comparison of the seismic fragility curves of Case A and Case D on the soil type B with SSI effects, a the 3-Story, b the 5-Story, c the 7-Story, and d the 9-Story SMRFs, subjected to FF record subset

Full size image

According to Fig. 22a–d, it can be observed that the Sa(T1) values in the seismic failure probability of 50% corresponding to the performance levels of IO and LS in Case D were higher than those for Case A considering the SSI effects. Therefore, the results of analysis can be used for SMRFs considering the SSI effects. In addition, in both situation of Case A and Case D, it is shown that the SSI effects increased the seismic failure probability. Therefore, the negative effects of considering SSI should be assumed in modeling process.

5 Conclusions

This research evaluates the seismic performance levels of 3-Story, 5-Story, 7-Story, and 9-Story SMRFs founded on the four soil types assuming four cases of IMWs placement prone to FF subset used by FEMA-P695 (FEMA P695 2009; Yazdanpanah et al. 2022). The results have been focused on the effects of considering IMWs and their significant influence on the fundamental period and seismic responses. To accurately model the IMWs and their effects on the structural response, a Tcl programming algorithm was developed in OpenSees (McKenna et al. 2016) to intelligently control the collapse state of IMWs in each floor level assuming M-IDR of the collapse limit of IMWs equal to 1.75%. It worth noting that, due to considering 2D models, the results of analysis can be used for 2D structures within the context of assumptions. The main remarks can be summarized as follows:

  • According to the results of the fundamental period of SMRFs (Table 5), the placement of IMWs can add more stiffness to SMRFs and this can significantly decrease the fundamental periods. For instance, in soil type B, assuming Case D in the 3-Story SMRF can cause the fundamental period to decrease by 18.80% (from 1.17 to 0.95). Therefore, in each soil type, Case D with the full placement of IMWs has a lower fundamental period.

  • The Sa(T1) values in M-IDA curves for the aforementioned SMRFs increased by adding IMWs (i.e. Case B, Case C, and Case D) into the bare frame (i.e. Case A). Therefore, the results confirm that the placement of IMWs in all floor levels increased the lateral resisting strength and this can cause an increase in the seismic performance levels. For soil type B, the Sa(T1) values in the performance level of TC for the 5-Story SMRF assuming IMWs placement of Case B, Case C, and Case D increased by 41.42%, 29.28%, and 20.18%, respectively.

  • The SSI effects on the seismic limit-state capacities of the aforementioned SMRFs have been investigated. The results show that considering the SSI effects significantly decreased the Sa(T1) values in all performance levels compared to neglecting the SSI effects. For example, assuming SSI effects in the soil type E, the Sa(T1) values in the performance level of CP for the 5-Story SMRF with IMWs placement of Case B, Case C, and Case D decreased by 37.41% (from 0.751 to 0.470), 29.55% (from 0.758 to 0.534), 24.27% (from 0.828 to 0.627), and 17.44% (from 0.854 to 0.705), respectively. Therefore, it is suggested to consider the SSI effects on the modeling process and seismic vulnerability evaluation.

  • Comparing the seismic probability of collapse for the performance levels of IO and LS showed that the Sa(T1) values corresponding to the seismic collapse probability of 50% for the placement of IMWs in all floor levels (i.e. Case D) are higher than those for the bare frame (i.e. Case A). For brevity, the results of 3-Story SMRF founded on soil type B showed that assuming Case D could increase the seismic collapse probability of the performance levels of IO and LS by 60% and 41.07%, respectively.

  • Assuming SSI effects had a reduction influence on the seismic failure probability. Thus, this is important to model SSI and assume its significant influence on the seismic responses. In soil type B and the seismic performance level of LS, assuming SSI effects can decrease the collapse probability of the 5-Story SMRF by 10.49% (from 0.305 to 0.273) and 14.52% (from 0.420 to 0.359), for the Case A and Case D, respectively.

  • According to results of analysis presented for SMRFs, IMWs have considerable influence on the fundamental period of SMRFs, which can change the seismic behavior of structures. In addition, the significant effects of IMWs on the seismic performance levels and seismic failure probability cannot be ignored. For future work, three-dimensional modeling of buildings with possibility of out-of-plane failure of IMWs is recommended. In addition, it is suggested to investigate previous damages of IMWs (i.e. in-plane damages), different workmanships and the effects of near-fault seismic records.