Seismic fragility analysis of low-rise unreinforced masonry buildings subjected to near-and far-field ground motions

model according to the Eurocode 8 standard. Subsequently, incremental dynamic analysis (IDA) was performed for each case study by applying FF and NF ground motions. Finally, fragility curves were developed based on the IDA results for each damage limit state. The susceptibilities of one-and two-story URM walls subjected to FF and NF seismic events were investigated by examining the derived fragility curves.


Introduction
Recent earthquakes revealed the high vulnerability of unreinforced masonry (URM) buildings to seismic loads [1][2][3]. URM buildings constitute one of the oldest structural systems built without considering modern seismic design criteria [4]. Furthermore, the brittle material type with low-tensile strength and large mass, and the lack of connection between the walls and ceilings are other reasons for the high susceptibility of this structural typology to earthquake events [5,6]. URM is considered a prevalent structural system in high-seismicity zones and historic areas [7]. Furthermore, URM is considered to be one of the most popular and cost-effective structural systems for constructing low-rise buildings [8][9][10]. Therefore, seismic vulnerability assessment and the proposal of a strengthening strategy are required to improve the resiliency of this structural typology and sustainable reconstruction of historic areas.
Various methodologies have been proposed for the nonlinear modeling of URM structures, which can be categorized into 1) the discrete element method, 2) the continuum homogenous method, and 3) the equivalent frame method (EFM) [11,12]. The discrete element method is usually utilized for the nonlinear modeling of URM structural elements owing to its high-computational effort [13]. The continuum homogeneous method is typically utilized to model URM structures with complex architectures, such as churches, towers, or bridges [14,15]. However, various EFMs have been developed and are usually used for the nonlinear modeling of URM residential buildings with the lowest computational effort compared with the other two methods [16][17][18].
EFMs facilitate the performance of highly computationally demanding nonlinear dynamic analysis of URM buildings to obtain fragility curves [19][20][21]. Simulation-based analytical fragility curves more accurately represent damage probabilities than curves derived using simplified analytical, empirical, or hybrid methodologies [4]. The analytical fragility curves of URM buildings, which show the probability of exceeding a damage limit state for a structure subjected to an earthquake with a specific intensity measure (IM), can be obtained based on cloud analysis, multi stripe analysis, and incremental dynamic analysis (IDA) [8,[22][23][24][25]. In these approaches, various seismic records with different intensities are applied to the model to evaluate the structural response [26]. IDA has the highest computational demand among the aforementioned methods; however, simplicity of implementation and no prior assumptions for the probabilistic distribution of seismic demand for the derivation of fragility functions are the two main advantages of this method [27,28].
The seismic events that were recorded and applied to a model for dynamic analysis were categorized into 1) far-field (FF), 2) pulse-like near-field, and 3) non-pulse-like near-field based on the site-to-source distance and presence of strong velocity pulses [29]. High-amplitude long-duration acceleration or velocity pulses occurring primarily in the fault-normal direction are the main characteristics of pulse-like near-field (NF) ground motions [30]. The main cause of pulses is the forward directivity effect, which typically occurs at sites in the direction of rupture propagation when the rupture velocity is close to the shear-wave velocity [29]. Different methodologies have been developed to distinguish pulse-like ground motions [31][32][33], and various studies have emphasized the higher level of damage that can be imposed on structures compared with other ground motion types [34][35][36][37]. In one of the few studies concerning the seismic vulnerability assessment of URM buildings subjected to FF and NF excitations, a three-dimensional (3D) one-story URM building was modeled, and it was concluded that the model was more susceptible to NF seismic events than FF ground motions [38]. Most studies concerning the effect of NF ground motions on the seismic behavior of URM buildings are limited to the pushover analysis of detailed numerical models and nonlinear time history analysis of the equivalent single-degree-of-freedom model of buildings [39,40]. However, equivalent single-degree-of-freedom models do not accurately reflect the seismic behavior of URM walls compared with the aforementioned numerical modeling approaches [4]. Therefore, a comparative study of the effects of NF and FF ground motions on the seismic fragility of URM buildings is required to develop sufficiently accurate numerical models and perform nonlinear dynamic analyses.
In this study, the macroelement utilized for the nonlinear modeling of URM case studies is first introduced. Subsequently, a simplified analytical method is proposed to calculate the maximum lateral strength (V m ) of the URM piers of a perforated URM wall. The proposed simplified analytical method enhances the efficiency of the nonlinear modeling process. This study was dedicated to the seismic fragility analysis of low-rise URM buildings. Unlike previous studies, this study employs detailed nonlinear modeling with the double-modified, multiple vertical line element model (DM-MVLEM) to capture accurately the complex behavior of URM walls. Four low-rise URM-perforated walls were chosen as case studies, and nonlinear models were constructed. Pushover analyses were performed, and the damage limit states were calculated for each case study. Subsequently, IDA was performed by applying FF and NF excitations to the case studies. The selected NF ground motions were pulse-like excitations used to investigate the pulse effects on the seismic behavior of the buildings. Finally, the fragility curves of each case study subjected to three seismic events were produced to facilitate a comparison of the seismic responses of the case studies. By conducting IDA and developing fragility curves, this study aims to enhance the understanding of the seismic response and vulnerability of low-rise URM buildings, thus contributing to effective strengthening strategies.

Case studies
The selected case studies were limited to low-rise URM walls with a maximum of two stories. Four perforated URM walls were selected as representatives of different structural typologies. Case study A was a wall tested at the University of Pavia subjected to prescribed cyclic displacements on the first and second floors [41]. The wall was made of solid fired-clay brick, and the mortar was a mixture of hydraulic lime and sand [41]. Case study B involved a two-story URM wall with five openings. Case studies C and D involved one-story URM walls with three and four openings, respectively. The four case studies and their geometries are shown in Fig. 1. All walls have a thickness of 25 cm and are made of the same material as the Pavia door wall [41].

Development of nonlinear models
3D modeling is necessary to predict the actual seismic behavior of buildings owing to the unpredictability of the seismic loading direction. Moreover, the asymmetry of the rigidity and mass distributions of each component with respect to the loading direction could be accurately predicted by performing 3D analyses. However, for simplicity, twodimensional (2D) models were developed in this study that focused on horizontal motion in a single plane. These models suit buildings with simple plan forms that lack mass or stiffness cross-coupling between orthogonal horizontal directions [8]. Despite this simplification, it is believed that essential in-plane structural behaviors can be adequately represented. The main objective of this study was to compare intensively the analytical fragility curves obtained computationally from IDA. Hence, a 2D modeling strategy that disregards the out-of-plane behavior of walls but accurately depicts the in-plane behavior of structural elements has been adopted [18].
The geometry of the piers and spandrels was determined based on the Dolce method [42], which is defined as the most reliable method for the equivalent frame idealization of perforated URM walls [43]. Based on this method, spandrels and piers are connected with rigid elements such that the length of the spandrels equals the length of the openings. The effective height of the piers is derived from the intersection between the vertical centroidal axis of each pier and the lines forming a 30 • angle from the corners of the adjacent openings [42]. The DM-MVLEM was used to develop the nonlinear models of the case studies [18]. The DM-MVLEM is a macroelement for the nonlinear analysis of URM buildings composed of two modified MVLEM elements to simulate the flexural behavior and a nonlinear shear spring that connects the modified MVLEM elements to simulate the shear behavior of URM structural components [18,20,44]. The DM-MVLEM was verified by comparing the results of the tests to the numerical analysis of two piers, a spandrel, and a full-scale two-story URM wall subjected to cyclic loading [18]. To accelerate the modeling procedure and reduce human errors, Hyperomet was utilized for the nonlinear modeling of walls based on the DM-MVLEM [45]. Hyperomet is a graphical user interface for the OpenSees software framework [46] designed for the nonlinear modeling of URM walls [45]. Fig. 2 depicts all the models developed using the DM-MVLEM macroelement.
The stress-strain curve of the URM material is depicted in Fig. 3(a), with a diagonal tensile strength (f td ) of 0.21 MPa and shear strength of the masonry at zero compressive stress (f v0 ) of 0.345 MPa based on [41]. The strain-stress curve was assigned to the fibers of the DM-MVLEM, and a nonlinear material with a trilinear backbone curve was assigned to the nonlinear shear springs in the middle of the DM-MVLEMs [18]. Fig. 3(b) shows the typical backbone curve of a nonlinear shear spring. The lateral strength parameters of the curve are proportional to V m and the corresponding displacements are proportional to the initial in-plane shear stiffness and height of the elements, as described in Ref. [18].
V m is considered as the minimum value of the maximum lateral strength owing to the shear sliding (V S ) and diagonal cracking (V D ) failure modes calculated using Equations (1) and (2) [18].
where b is the width, h is the height, and t is the thickness of a pier, is the axial compression stress, and ζ is the shear stress distribution coefficient at the center of a pier considering the aspect ratio calculated based on  the equations in Table 1: Moreover, parameter α 0 is the zero-moment coefficient relevant to the moment distribution along the height of the pier, which is elaborated on in the next section.

Zero-moment coefficient
The zero-moment coefficient was calculated to determine the effective uncracked section length of a pier, neglecting the tensile strength of the bed joints and assuming a simplified distribution of compression stresses [47]. The parameter was utilized to calculate the V S of the piers, as shown in Equation (1). The parameter α 0 is calculated as the maximum value of the fraction of the height of a pier (H 0 ) with a positive or negative moment value as depicted in Fig. 4 to the total height of a pier (h) [19]. For ideal typical test layouts, the α 0 is assumed to be equal to one in a cantilever scheme and 0.5 in a fixed-fixed boundary condition. No further description of this value is provided in Eurocode 8 Part 3 [48], NTC [49], FEMA 356 [50], or FEMA 273 [51]. This parameter depends on the geometry and boundary conditions of piers. Therefore, predefined ideal values cannot be assigned to α 0 of piers of a perforated URM wall due to the flexibility of spandrels on two ends of the pier.
One strategy is to apply a mode proportional load pattern based on the fundamental mode shape to the elastic model of the wall and define the moment distribution diagram of each pier and calculate α 0 [19].
However, different studies have indicated the accuracy of a pushover analysis of URM buildings with the mass-proportional load pattern compared with the mode-proportional load pattern by comparing the results with the IDA curves [18,52,53]. Therefore, in this study, the parameter α 0 was calculated by applying a mass-proportional lateral loading to the elastic model of the walls.
To investigate the variation in this parameter and define the equations that facilitate the process of attaining this parameter for the analysis of low-rise URM buildings, nine perforated URM walls were modeled in SAP 2000, considering the elastic behavior of piers and spandrels. The five added URM walls are illustrated in Fig. 5.
Mass-proportional lateral loads were applied to the case studies and the α 0 values were determined based on the moment diagram. The results show that the α 0 values of the piers of the two sides of the walls of the first story are higher than the values of the internal piers. Furthermore, if a load was applied in the right direction, the α 0 value of the leftmost pier was higher than that of the external pier. However, because the seismic events applied cyclic loads to the structures, the highest α 0 value was considered for both external piers using a more conservative approach.
To determine an equation to derive the α 0 values, the piers are subcategorized into three groups: 1) internal piers of the first story, 2) external piers of the first story, and 3) piers of the second story. The differences between the α 0 values of the interior and exterior piers of the second story were not as high as those of the first story; therefore, no distinction was considered. The piers of the one-story walls were assigned to groups 1 and 2. Moreover, b 3 h was considered a parameter proportional to the flexural stiffness of each pier, depending on the geometry of each pier utilized for the prediction of the parameter α 0 .  Table 2. The range of the parameter shows that assuming fixed-fixed boundary conditions for the piers of the first story of a two-story perforated wall which corresponds to 0.5 for the parameter α 0 will overestimate the V m of the piers since the calculated α 0 are more than 0.5, as presented in Table 2. Furthermore, assuming cantilever piers on the second floor of a two-story wall or piers of a one-story wall and assigning the value of one to the parameter α 0 will underestimate the V m values of the piers.
Finally, to investigate the accuracy of the proposed equations, the V m values of all piers of the nine walls were calculated using the α 0 values derived from the equations in Table 2 and the linear static analysis of the walls. A scatter plot of the results of the two approaches is shown in Fig. 7. The maximum R 2 value in Table 2

Pushover analysis
Pushover analysis was performed by applying the mass-proportional load pattern, and the pushover curves are illustrated in Fig. 8. Furthermore, for the Pavia door wall, a pushover curve derived by applying a load pattern similar to that in the test and the backbone of the test result were presented to demonstrate the accuracy of the DM-MVLEM method, as shown in Fig. 8(a). Although a difference can be detected between the

Table 1
Equations used to derive ζ (the shear stress distribution coefficient at the center of a pier considering the aspect ratio).

Aspect ratio
Shear stress distribution coefficient initial stiffness of the test and the numerical model curves, the maximum base shear is predicted with good accuracy, and the nonlinear behavior, evaluated in terms of ductility, was estimated accurately. Note that the vertical applied load of the Pavia door wall was based on the test, but for other case studies, a uniform load (set at 20 kN m ) was applied to each floor, which is relatively lower than the vertical load applied to the Pavia door wall [41]. It can be inferred from Fig. 8 that the inclusion of piers or extension of the wall length, as seen in the two-story walls, leads to an increase in the maximum base shear for the one-story walls. The combined flexural and shear failure mode is the dominant failure mode of the piers in the case study (A), which can be concluded from the pushover curve analysis and is consistent with the test results. However, the shear failure mode is more dominant in other case studies. Therefore, the ductility of the case study (A) is greater than that of the other case studies.

Definition of limit states
Instead of considering predefined damage limit state values, a more accurate structure-specific approach based on the pushover curve was used to derive the inter-story drifts corresponding to the limit states [54]. Qualitative and quantitative descriptions of the three limit states  selected based on Eurocode 8 Part 3 [48] are presented in Table 3.
The yielding points of the pushover curves were calculated after bilinear-elastic-perfectly plastic idealization of the pushover curves performed following the procedure recommended by Eurocode 8 part 1 [55] and using the SPO2FRAG user interface [56]. The yielding displacement was calculated by defining the area-balancing criterion between the area under the capacity curve and the idealized curve, and the corresponding maximum inter-story drift was considered as the damage limitation (DL) limit-state definition.
The ultimate displacement that corresponds to the near collapse (NC) limit state was calculated when the pushover analysis stopped converging, or the corresponding displacement was 80% of the peak base shear [56]. Furthermore, the significant damage (SD) limit state was estimated as the maximum inter-story drift corresponding to 75% of the ultimate displacement value, which approximately corresponds to the displacement of the maximum base shear [57].
The limit state values are influenced by the dominant failure modes of the piers, which are dependent on the opening configurations that affect the geometry of the piers and spandrels and the applied vertical loads, as highlighted in Refs. [57,58]. The differences between the calculated inter-story drifts corresponding to each limit state of the walls are shown in Fig. 9. Therefore, structure-specific methods for deriving the limit states of URM buildings are recommended instead of using predefined values from the literature, although for cases B, C, and D, the obtained values are similar. These results highlight the need to investigate this issue further to identify fixed values of the damage states according to specific structural features.

Incremental dynamic analysis
The Rayleigh damping model was implemented with an equivalent damping ratio of 2% at the first and second modal frequencies of each model [18,59,60]. Considering the convergence issues and prolonged analysis associated with using the current stiffness matrix, it may be advisable to replace it with a committed stiffness matrix [61]. This alternative results in significant computational time savings. Therefore, Rayleigh damping was incorporated based on the mass and last committed stiffness matrix to perform IDA.
IDA was performed by applying 22 FF and 14 NF pairs of seismic records from FEMA P695 [62]. A boundary of 10 km was considered to categorize the records into FF and NF [62]. The average of the Campbell and Joyner-Boore fault distances was used as the source-to-site distance. NF records were selected based on the three criteria mentioned in Ref. [31]. Wavelet analysis was used to extract the largest velocity pulse from a given ground motion. The presence of a pulse that occurs early in the time history is satisfied by calculating an indicator and checking the   pulse occurrence time [31]. Another criterion is that the ground motion should have a PGV greater than 30 cm / s . The occurrence of an early pulse in the ground motion was independent of the scaling process of the IDA. However, a PGV greater than 30 cm / s may not be satisfied for ground motions scaled to intensities lower than the original ground motion. Therefore, the PGV criterion was neglected in this study because of the downscaling process of the IDA. A summary of the ground motions selected from FEMA P695 [62] and their characteristics, including the record sequence numbers (RSN) in the PEER database [63] and the maximum peak ground acceleration (PGA) of the two components of each earthquake, are presented in Table 4. The FF and NF ground motions' spectra are shown in Fig. 10. An IDA curve is a diagram of the ground motion IM against an engineering demand parameter (EDP) [28,65]. In this study, the IM is the spectral acceleration corresponding to the first mode of the structure considering 5% of damping (Sa (T 1 , 5%)), and EDP is the maximum inter-story drift. Note that the fundamental period of the two-story walls (case studies (A) and (B)) was 0.21 s. The fundamental periods of one-story walls were 0.12 s and 0.13 s for case studies (C) and (D), respectively. The IDA curves for the case studies are shown in Fig. 11.

Seismic fragility analysis
A log-normal cumulative distribution function was used to define the fragility function based on Equation (3).
where P[C|IM = im] is the probability that a ground motion with IM = im causes the structure to collapse. Φ() is the standard normal cumulative distribution function, ln (Z) is the natural logarithm function, η and β are the mean and standard deviation of the ln(im) values, respectively [66,67]. The fragility curves were produced based on the results from the IDA, considering together all 72 time-history analyses. The analyses outcomes related to the FF and those related to the NF seismic events for all three mentioned limit states are presented, as shown in Fig. 12. Unlike the fragility curves for the DL limit state, the fragility curves for the NC and SD limit states are close to each other. This is in part due to the brittle behavior of URM material that causes a lower difference between the seismic demand when it is significantly or heavily damaged, especially for earthquakes characterized by a medium-high magnitude (Magnitude = 6.5-7.5 M), as in the cases if the selected ones (see Table 4). It can be concluded that case study (B) is the most vulnerable case study, with the lowest IM corresponding to a 50% probability of reaching a limit state (median IM), whereas case study (D) yielded the opposite results.
The maximum difference between the median IMs of FF and NF ground motions is 6.2%. This corresponds to the case study (A), considering the Sa (T 1 , 5%) as the IM. Therefore, it can be inferred that the seismic fragility analysis of low-rise URM buildings by applying FF

Table 3
Quantitative and qualitative descriptions of defined three performance levels [48].  and ignoring NF ground motions was sufficiently accurate for the selected case studies subject to strong ground motions. However, further investigations are required for seismic events characterized by medium intensities (Magnitude = 5.5-6.5 M). Furthermore, the results showed that the median IMs of the NF ground motions applied to the two-story walls are lower than those of the FF seismic events. This difference is negligible in the case study (B). Nevertheless, for one-story buildings, the obtained results are the opposite. In one-story walls, the differences between the fragility curves of the FF are indistinguishable from those of the NF ground motions. It can be concluded that for two-story walls, the NF ground motions are more damaging than those for the FF motions; however, for one-story walls, the FF ground motions are more destructive. This difference shows that for the two-story walls with a fundamental period of 0.21 s, the NF records can impose more damages in terms of inter-story drifts. However, for one-story walls with a fundamental period of around 0.12 s, the FF seismic events are more damaging, which is consistent with the results of nonlinear analysis of stiff and low-period structures, such as masonry mosques [68], masonry bridges [69,70], and nuclear structures [71]. The NF ground motions caused more damage to the models with higher fundamental period values, which can be observed in the two-story walls.
A general rule that can exhibit the vulnerability of low-rise URM buildings cannot be determined because the seismic demand depends on the dynamic characteristics and failure modes of the structural elements. The same results were derived from a similar study on the seismic damage assessment of low-rise moderate reinforced concrete frames subjected to NF and FF ground motions [72]. Furthermore, it is mentioned that the derived fragility curves from both groups of ground motions are close to each other [72]. Applying NF ground motions is not required for the seismic fragility analysis of one-story URM buildings subjected to medium-high seismic events. Nonetheless, applying NF ground motions for the seismic fragility analysis of the two-story URM building mode is suggested for accurate analysis.

Conclusion
Because of the widespread use of low-rise (URM) buildings in highseismic zones, this study aimed to evaluate the vulnerability in terms of the fragility curves of this structural system subjected to far-field (FF) and near-field (NF) ground motions. Four low-rise perforated URM walls were chosen, and nonlinear models were developed based on the double-modified multiple vertical line element model (DM-MVLEM). The maximum lateral strength (V m ) is an effective parameter on the trilinear backbone curve of the nonlinear shear spring of the DM-MVLEM macroelement, which is calculated based on the minimum value of the maximum lateral strength due to the shear sliding (V S ) and diagonal cracking (V D ) failure modes. Moreover, for the calculation of the V S , the zero-moment coefficient (α 0 ) was required to be determined.
The value of α 0 can be defined based on the moment diagram of each pier by applying lateral loads to a wall and performing linear static analysis. The main findings of this study can be synthesized as follows: the results showed that using the α 0 values proposed for piers with ideal fixed or fixed-free boundary conditions was not sufficiently accurate for piers of a perforated wall with flexible boundary conditions owing to the presence of spandrels. Note that the proposed equations can only be used to define the parameter α 0 of the piers of a low-rise URM wall (maximum two-story) with regular opening configurations. • A nonlinear pushover analysis was performed, and structure-specific damage limit states were calculated based on the pushover curves.
The results showed that using predefined fixed inter-story drift ratios for the limit states of URM walls was not sufficiently robust. These differences were due to the different displacement capacities of the piers with shear or flexural failure modes. • IDA was performed by applying 44 FF and 28 NF ground motions to the models, and the fragility curves for the three performance levels were obtained for each case study. The fragility curves for the NC and SD limit states were relatively close to each other, unlike the fragility curve for the DL limit state. The fragility curves revealed that the difference between the seismic demands of the buildings subjected to FF and NF ground motions was not significant for the low-rise URM buildings analyzed in this study. The maximum difference between the median IMs of FF and NF ground motions required to reach a limit state was 6.2%. This was achieved in case study (A), considering the Sa (T 1 , 5%) as the IM. This behavior can be attributed to the brittle nature of the URM material, which results in a reduced disparity in seismic demand when the material is significantly or heavily damaged, especially for buildings subject to strong ground motions, such as those considered in this study, characterized by a magnitude in the range of 6.5-7.5 M. • Thus, the seismic fragility analysis of low-rise URM buildings following the application of FF ground motions and the ignorance of NF ground motions is sufficiently accurate for medium-highintensity seismic events. The study found that for two-story walls with a fundamental period of 0.21 s, the NF ground motions were more damaging than FF motions, while for one-story walls with a fundamental period of around 0.12 s, the FF ground motions were more destructive. Additionally, the fragility curves of the FF for onestory buildings were indistinguishable from those of the NF ground motions. Therefore, the use of NF ground motions was not necessary for the seismic fragility analysis of one-story URM buildings; however, for an accurate analysis of two-story URM buildings, NF ground motions should be included.
While efforts were made to encompass various opening configurations and geometries, additional case studies can be conducted to reduce further the uncertainty associated with the prediction of α 0 . The comparative study procedure outlined in this study can also be extended to 3D models, considering the impact of irregular mass and stiffness in the plan and local collapse mechanisms. Additionally, investigating the influence of different types of ground motion IM on seismic fragility curves could serve as a potential subject for future research.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
No data was used for the research described in the article.