1 Introduction

The earthquake that occurred in Umbria and Marche (Italy) in 1997 provided an opportunity to analyse the consequences of a seismic event in historical centres, which were mainly constructed with unreinforced masonry. In most cases, structures grouped in linear formations or with complex geometries emerged as a key feature, later termed ’aggregate structures’ (Binda et al. 2006). Subsequently, the term ’aggregate’ has been used to refer to groups of adjoining buildings, regardless of their distribution shape. In these aggregate structures, damage is typically greater in the end units due to the lesser degree of constraint, while the internal units may also sustain damage if they have lower seismic capacity than the adjoining ones, even when the structures are constructed similarly (Binda et al. 2006).

The 2009 L’Aquila earthquake (Italy) caused significant damage to many historical buildings in city centres, prompting increased research on the seismic behaviour of aggregate structures (Carocci 2012). These studies have shown that aggregates are particularly vulnerable to the activation of local mechanisms which can lead to damage (Sorrentino et al. 2019; D’Ayala and Paganoni 2011). So far, research on aggregates has focused mainly on unreinforced masonry structures in European historical centres located in seismically active areas, such as Italy or Portugal. These studies have addressed aggregate structures with different shapes, either row (Battaglia et al. 2021; Angiolilli et al. 2021) or irregular (Grillanda et al. 2020), and considered dwellings separated either by a single shared wall or with perfectly joined contiguous walls, due to the layout of those historical centres (Angiolilli et al. 2021). A seismic vulnerability study of a long aggregate consisting of nine historical unreinforced masonry buildings (Greco et al. 2020) found that the ensemble featured greater longitudinal resistance than when considering each building individually, probably due to confinement effects. However, resistance in the transverse direction decreased for some of them. Furthermore, the resistance of the inner, smaller structures was significantly reduced when subjected to the relative displacements imposed by their adjacent larger structures. Hence, research on aggregates in historical centres has concluded that the earthquake effects on each structure differ depending on their position within the ensemble, mainly due to the torsion effects derived from both the local and global geometries. Likewise, previous research stated that the largest shear stresses were concentrated on the walls of end-houses in the case of row aggregates (Battaglia et al. 2020; Valente et al. 2019) and on corner units for block-type aggretates (Formisano et al. 2010). The same study proposed a modified index vulnerability method to consider the structural interactions among adjacent buildings, which was subsequently utilised in related studies (Chieffo et al. 2021; Chieffo and Formisano 2019b). These subsequent studies focused on non-regular aggregated buildings and demonstrated irregular damage distribution among the structures.

Research on aggregate structures has expanded to include various construction methods and modern building complexes. For example, a study was conducted on a row-shaped aggregate of bed-joint reinforced concrete masonry structures (Marques et al. 2012), and another study examined the vulnerability of a block-shaped aggregate consisting of four stone masonry houses (Chieffo and Formisano 2019a). The latter study found that the probability of damage was lower in the aggregate compared to the isolated structure and that the corner units were more vulnerable to seismic activity than the intermediate structures.

The seismic safety assessment of aggregates should take into account boundary conditions (Angiolilli et al. 2021), especially those related to the interaction of such structural units. This interaction involves both the internal and external behaviour of each structural unit. Identification of the quality of the connections between the walls of adjacent buildings plays an important role in the seismic assessment of the aggregate, but numerical simulation is a difficult and open task that has not been extensively studied so far (Angiolilli et al. 2021). Some recent studies propose the implementation of a simplified model that represents connections between buildings. In particular, the research carried out on historical buildings by Stavroulaki (2019) simulates the support of an adjacent structure in an aggregate, such as elastic foundations or rods. In another study, a simplified method based on the calibration of elastoplastic links for strength and rigidity (based on the main geometric characteristics) was proposed to analyse vulnerability in a series of buildings aggregated in rows, in Italy’s historical centres (Formisano and Massimilla 2018). Bernardini et al. (2019) developed a procedure to find the minimum portion of an aggregate which needs to be simulated in order to capture the influence of exterior buildings. Angiolilli et al. (2021) carried out a fragility assessment of an L-shaped aggregate of unreinforced masonry structures corresponding to different periods, making connections between buildings by placing elements with different elastic properties to emulate interactions.

Previous research focuses on the behaviour of aggregates in European urban areas with medium-to-high seismic hazard, but does not address areas with high seismic risk, such as Central America. In the urban centres of these countries, the predominant construction systems are different from the traditional systems described in the majority of studies of aggregated structures. San José, the capital of Costa Rica, is an important urban centre of Central America located in a seismic hazard zone affected by subduction between the Cocos and Caribbean plates (Alvarado et al. 2017). The single-family house construction system, the most widespread in the city, must possess appropriate behaviour against seismic loads while also ensuring affordable cost and simple on-site execution. The Costa Rican seismic code (CRSC) (Colegio Federado de Ingenieros y Arquitectos de Costa Rica (CFIA) 2016) specifies a construction system based on partially grouted concrete block masonry (PG-RCM) walls with horizontal and vertical reinforcement using steel bars, supported by a reinforced concrete beam (Hidalgo-Leiva et al. 2016; Carro and Segura 2013). However, the local construction practises for these structures differ substantially from those found in other regions (Hidalgo-Leiva et al. 2021b). PG-RCM structures are widely used as family dwellings in Costa Rica, comprising approximately 86% of the buildings in the city (Esquivel-Salas 2020). Urban centres are increasingly expanding and more condominium-type housing is being developed, so there are an increasing number of buildings that share one or more outer walls with adjacent constructions, as shown in Fig. 1.

Fig. 1
figure 1

Agregated buildings in row, San José (Costa Rica)

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Despite the increasing evidence that aggregate structures behave differently under seismic loading to isolated structures, the CRSC (Colegio Federado de Ingenieros y Arquitectos de Costa Rica (CFIA) 2016), the current Italian seismic code (Ministero delle Infrastrutture e dei Trasporti 2018) and the Eurocode (European Committee for Standardization 2004) still focus primarily on the behaviour of isolated buildings. Recent studies of vulnerability and seismic risk also assume that PG-RCM buildings behave as isolated elements, despite the fact that the type of interaction between adjacent buildings can potentially impact seismic capacity and failure mechanisms. As a result, there is a lack of regulatory clarity and research-based guidance on how to design and assess the seismic performance of aggregated buildings in Costa Rica. To bridge this knowledge gap, it is essential to investigate the behaviour of these structures when they are connected to adjacent buildings. Specifically, understanding the seismic performance of commonly used PG-RCM houses in aggregate configurations is crucial.

A computational tool for assessing the seismic behaviour of aggregate construction should take into account the characteristics of the construction materials and the actual state of the construction details. Due to the strong nonlinearity of masonry materials, including low tensile resistance, high compression resistance, and a high friction angle, elastic analysis is unrealistic (Heyman 1966; Como 2013), so nonlinear analysis or failure analysis is usually recommended. Macro-element modelling methods have been used successfully to study isolated and aggregated structures of different types of masonry, reinforced concrete, and mixed structures (Belmouden and Lestuzzi 2009; Lamego et al. 2017; Simões et al. 2015; Hube et al. 2020; Maio et al. 2017; Cattari and Lagomarsino 2013). The simplicity of these methods allows the seismic behaviour of structures to be defined through nonlinear static or dynamic analyses. Furthermore, the computational cost of these methods is very low compared to other methods such as finite or discrete elements (D’Altri et al. 2020; Parisse et al. 2022). Various macro-element modelling strategies have been proposed in the literature to idealise 3D masonry structures. The equivalent frame method strategy, in particular, divides the structure into horizontal, vertical and rigid macro-elements with nonlinear material properties, and requires only two nodes for each element. (Roca et al. 2005; Magenes 2000; Penna et al. 2014). It allows numerous analyses to be performed by varying the desired parameters, which facilitates calibration with experimental data and also permits the influence of different variables to be studied. Because it considers the behaviour of structural elements in the plane, the method attains its full potential when studying structures that ensure box behaviour (Grillanda et al. 2020), such as the PG-RCM structures.

The main objective of this study is to investigate the seismic behaviour of PG-RCM structures in aggregate configurations commonly found in San José, Costa Rica. Specifically, the study aims to analyse the different possibilities of interaction between the structures when they are joined in a shared-row aggregate and to compare the results with the seismic behaviour of the isolated structure.

The research presented in this study is novel in its approach to modelling the nonlinear static behaviour of current PG-RCM aggregate dwellings, considering a range of connection types from completely rigid to more flexible, which is more representative of modern building practises. In addition, the study examines the behaviour of individual structures within the aggregate, allowing a better understanding of the interaction effects between adjacent buildings. This investigation is particularly significant because it focuses on the four most common types of interaction between adjacent structures found in aggregates in San José, Costa Rica, providing insights into the seismic capacity and failure mechanisms of these structures.

The paper is divided into sections as follows. In Sect. 2, a model setup is performed, comparing the nonlinear static analysis response at both the wall and structure levels with reference values. After validating the isolated structure (IS) model, in Sect. 3 the aggregated structures are addressed, that is, a row alignment of single-family homes. The nonlinear static response is obtained by considering various types of connecting interfaces between buildings and comparing them. Finally, Sect. 4 highlights the relevant findings of the study.

2 Model setup

2.1 Validation of reinforced wall model

As a first stage, the calibration and validation of the equivalent frame method is proposed for PG-CRM walls with vertical and horizontal reinforcement. Therefore, the experimental results obtained for dynamic wall tests of this type, carried out at the University of Costa Rica (Hidalgo-Leiva et al. 2021a) are used. In this way, it is ensured that at the wall level, the software is capable of adequately representing the real behaviour of the structure. The verification is carried out by certifying that the two-dimensional simulations represent failure mechanisms similar to those observed experimentally and that the pushover curves are within the order of magnitude of what is observed in the envelope of the hysteretic curves of the tests. It is important to pay special attention to the stiffness of the walls, since the theoretical and experimental (real) values differ greatly for this type of masonry, especially because the stiffness values vary depending on the axial load acting on the element (Lagomarsino et al. 2013), hence it is necessary to calibrate this parameter so that the results are adjusted satisfactorily for all the dimensions of the wall studied. The experimental setup consists of four walls sharing all dimensions except width, as can be seen in Fig. 2a, which were tested by lateral cyclic loads applied to the top of the specimens as shown in Fig. 2b.

Fig. 2
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Wall geometry and experimental setup (adapted from Hidalgo-Leiva et al. 2021a)

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2.1.1 Material representation and properties

The masonry used in this study consists of walls that are \({15}\,\hbox {cm}\) thick with a compressive strength of \(f'm = 16.4\) MPa. According to Navas Carro (2011), the modulus of elasticity of this type of masonry is \(E = 932f'm\), and the CRSC (Colegio Federado de Ingenieros y Arquitectos de Costa Rica (CFIA) 2016) establishes a shear modulus of \(G = 0.4\) E. The concrete used has a compressive strength of 28.07 MPa.

Vertical and horizontal reinforcement in masonry, as well as shear reinforcement in concrete columns and beams, consists of \({10}\,\hbox {mm}\) deformed steel bars with a grade of 40 and quality ASTM A615M (ASTM International 2016a). The longitudinal reinforcement in concrete columns is made up of \({12}\,\hbox {mm}\) deformed steel bars with a grade of 60 and quality ASTM A615M, while the longitudinal reinforcement in reinforced concrete beams consists of \({20}\,\hbox {mm}\) deformed steel bars with a grade of 60 and quality ASTM A706M (ASTM International 2016b).

As described in (S.T.A. DATA 2019; Lagomarsino et al. 2013), the 3Muri software (3Muri, release 13.2.0.14) used in this study models the behaviour of masonry elements using a bi-linear elasto-perfectly plastic law. This means that after the elastic phase, the masonry elements will deform in a perfectly plastic manner until they reach maximum drift. The failure criteria adopted considers two strength domains, one for bending (ultimate moment) and one for shear (Mohr-Coulomb), both depending on the axial compression force acting on the element; also, the type of failure in an element can change according to the acting axial load. Hence, the maximum drift can be shear or bending, depending on the failure mode. The software also includes the capability to model reinforced masonry elements and reinforced concrete ring beams.

Reinforced masonry structures tend to exhibit box behaviour, which is why out-of-plane (OOP) failures are not usually observed, especially in the Costa Rican PG-RCM. Therefore, this study focuses exclusively on defining the behaviour of the walls in the plane. The reinforcement in masonry elements is subject to its own maximum drift limitations and follows the material law of steel, as well as the reinforced concrete elements, together with the behaviour of concrete and the formation of plastic hinges at the end of the elements.

2.1.2 Wall element

The software determines the maximum displacement of the reinforced masonry walls based on the prescribed drift values for shear and bending of the steel reinforcements. Typically, the values of drift range from 0.7% to 1.2%. However, in accordance with the results of the experiments, a drift value of 2% for shear and 4% for bending was assigned in the software. Furthermore, a study by Hidalgo-Leiva et al. (2021a) found that the stiffness of the masonry walls was significantly lower than theoretically predicted, particularly in the absence of vertical loading. The new CRSC Draft(Colegio Federado de Ingenieros y Arquitectos de Costa Rica (CFIA) 2020) already had provisions for a general reduction in crack stiffness of 50% for walls without axial loads, but the results of the study (Hidalgo-Leiva et al. 2021a) indicate that an even greater reduction may be necessary. Therefore, in this research, it is assumed that the real stiffness is around 10% of the theoretical value.

Figure 3 presents the results for models of walls 01, 02, 03 and 05. The results for wall 04 are omitted due to failures during the loading process, as reported in (Hidalgo-Leiva et al. 2021a). The figure displays the pushover curves of the equivalent frame models and the experimental curves, which represent the elasto-perfectly plastic bi-linear approximation of the dynamic test envelope. The solid line represents the load in the positive direction and the dashed line in the negative direction.

Fig. 3
figure 3

Comparison between the numerical model and the experimental results obtained from Hidalgo-Leiva et al. (2021a). Continuous lines are positive direction, and dash lines are negative direction

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It can be seen that the results of the equivalent frame model agree with those of the experimental ones, in terms of base shear magnitude, plastic phase, and maximum displacement (Fig. 3). Setting a drift limit for both bending and shear earlier in the study, it is possible to determine the failure mode based on the maximum displacement observed in the curves. If the wall fails due to shear, the displacement will be smaller than in the case of a bending failure. Thus, it is evident from the figure that one wall has failed due to bending, while the other ones have failed due to shear.

2.2 Validation of single-house structural model

This case study has previously been analysed by Hidalgo-Leiva et al. (2021b). It involves a typical Costa Rican single-family home typology, which is mainly composed of \({12}\,\hbox {cm}\) thick PG-RCM walls and reinforced concrete beams. The model considers the structural elements with the dimensions shown in Fig. 4. The walls of the A and B axes are called the main shear walls of the structure.

Fig. 4
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Plan view of the structural walls (adapted from Hidalgo-Leiva et al. 2021b)

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The beams of the structure must remain within the elastic regime under seismic loading, concentrating the damage on the walls. Furthermore, since masonry of the type studied is considered to be properly reinforced, the joints with the slab and the roof ensure that no OOP failure occurs.

The walls are composed of PG-RCM vertically and horizontally reinforced, with a reinforced concrete beam at the top. The slab and the roof, due to their construction, are considered rigid diaphragms, which is what is most commonly observed in this type of housing (Esquivel-Salas 2020). The concentrated mass for the first and second levels are \({78.3}\,\hbox {kN}\hbox {s}^{2}\hbox {m}^{-1}\) and \({38.1}\,\hbox {kN}\hbox {s}^{2}\hbox {m}^{-1}\), respectively. Reinforcing steel is considered ASTM A615(ASTM International 2016a) grade 40, with a yield stress of \({280}\,\hbox {MPa}\). The bar layout is specified in the previous research (Hidalgo-Leiva et al. 2021b). The drift of the shear limit of the walls is established according to Hidalgo-Leiva et al. (2016) from the tests carried out on walls with the same characteristics. To define the bending drift of the walls, the ratio of shear and bending drift limits of the walls tested by Hidalgo-Leiva et al. (2021a) has been followed, since the materials and composition of the wall are very similar.

This study involves four pushover analyses of the structure, along with both positive and negative X and Y directions, the load is distributed following the shape of the main modes of vibration (with higher mass participation) and the registered displacement corresponds to the average displacements of the roof nodes. The results are compared with those obtained by Hidalgo et al. (2018) in a pushover analysis of a model resulting from the macro element method (Fig. 5).

Fig. 5
figure 5

Equivalent frame mesh of the isolated structure

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The periods of the first modes of vibration for the dwelling obtained by the reference study are \({0.269}\,\hbox {s}\) along the Y axis and \({0.117}\,\hbox {s}\) along the X axis, while those obtained for the proposed model are \({0.251}\,\hbox {s}\) on the Y axis and \({0.119}\,\hbox {s}\) on the X axis. Figure 6 depicts the mode shapes in both directions.

Fig. 6
figure 6

First modes of the isolated structure

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Figure 7 shows the damage to the structural walls during the last load step, just after the loss of 20% of maximum resistance. If the load is applied along the X direction, the failure of the building occurs due to shear collapse in one of the two main walls. In contrast, narrower walls fail by bending under loads acting in the Y direction.

Fig. 7
figure 7

Failure mechanisms for loads in the positive X and Y directions

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For the pushover analysis of the structure along the positive X and Y directions, Fig. 8 shows the comparison between the results obtained by Hidalgo et al. (2018) and the results of the proposed equivalent frame model’s pushover curves in both the positive and negative directions. It is important to note that Stewart’s hysteresis model (Stewart 1987) was used in the reference model. This model represents a tri-linear behaviour with reduced stiffness after yielding and negative stiffness after reaching the ultimate load (softening). In contrast, the model proposed in this research exhibits bi-linear elastic-perfectly plastic behaviour. Therefore, it is expected to have a higher elastic limit load and a similar resistance before collapse compared to the reference model proposed in a previous study (Hidalgo et al. 2018).”

Fig. 8
figure 8

Comparison between the pushover curves of the proposed model and the reference values (Hidalgo et al. 2018)

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Figure 8 shows the agreement between the proposed and the reference model. The elastic stiffness of both models aligns closely, the values for ultimate base shear being within the same order of magnitude, and the ultimate displacements before failure exhibit similarity. Given that that the computational model is based on calculated values rather than experimental measurements from an actual structure, it can be concluded that the model simulates the expected behaviour of a house with these characteristics. Therefore, the model provides a suitable tool for studying the seismic behaviour of aggregated reinforced-masonry structures, while keeping computational costs low.

Furthermore, for the obtained model, the positive and negative curves in the X direction show a similar response, whereas a notable difference is observed for loads applied along the Y-axis. This discrepancy arises due to the torsional effects present in the structure, which result in distinct behaviour when loaded in one direction compared to the other. Being able to observe and account for these effects implies a significant advantage of the proposed model over other simplified models.

3 Buildings in aggregate

A compound dwelling model has been used, adopting a row arrangement that resembles the configuration studied by Battaglia et al. (2020). This row arrangement, as illustrated in Fig. 1, represents a common configuration found in the city of San José. To aid in the analysis of the aggregate, each dwelling is assigned a distinct number, as shown in Fig. 9a, which displays the central node for each house. Furthermore, Fig. 9b exhibits the three-dimensional mesh of the aggregated structures.

Fig. 9
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Distribution of aggregated structures, and its equivalent frame mesh

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3.1 Interaction between adjoining walls representation

Defining the type of interaction between adjacent dwellings is a complex task due to various factors such as construction quality, modifications made over time, and a lack of detailed documentation. In the context of the studied model, connection elements are used when necessary. These elements are modelled by placing bi-articulated connecting beams at the end nodes of the adjoining walls (Fig. 10a), aligned with the reinforced concrete height. If one wall has smaller dimensions than the adjoining wall, its ends are joined to intermediate nodes of the larger wall (Fig. 10b). The spacing between adjacent structures, when connection elements are present, is determined by the sum of half the thickness of the adjoining walls, assuming that the elements are located on their central axis \({12}\,\hbox {cm}\).

Inspired by a recent study by Angiolilli et al. (2021), this study explores four types of connection, adapting them to the specific context. The models of the aggregates will be named and modelled according to the type of connection:

  • Case ’Shared wall’ (SW). This represents dwellings that share a common wall between them, and despite being independent houses, they function structurally as a single unit. The model considers structures with a single wall between them, prioritising the wall with the greatest extent.

  • Case ’Rigid connection’ (RC). This represents structures that have walls in contact and behave as if they were perfectly joined due to their construction. The model considers the previously defined distance and beams, including the presence of a fictitious floor between buildings. Beams and slabs are infinitely rigid, ensuring complete load transmission between structures.

  • Case ’Compression link’ (CL). This represents structures with walls that are not connected but have touching surfaces capable of separate movement and of sliding parallel to each other. The model uses the beams and spacing defined earlier, with stiffness and compression resistance values equal to the crown beams of the structure, but lacking any tensile strength. As a result, this model only allows for the transmission of compression loads between dwellings.

  • Case ’Flexible link’ (FL). This represents a fictitious case, considering connection elements with no resistance between the dwellings (no compression nor tension resistance) but having the same stiffness as the CL model. This model is needed for an in-depth comparison of the transmission of forces in the CL case and its effects on load redistribution, especially in a modal pushover analysis. Although it has the same stiffness matrix as the CL model, there is no strength interaction between buildings, so the structures are loaded as isolated entities

Fig. 10
figure 10

Strategy employed for connecting structures

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3.2 Modal analysis

Figure 11 shows the shapes of the fundamental mode of vibration in the Y direction for the different aggregate configurations and the isolated structure, while Fig. 12 shows those that correspond to the X direction.

Fig. 11
figure 11

Shape of the main modes of vibration along the Y axis

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Fig. 12
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Shape of the main modes of vibration along the X axis

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In the modes of the Y axis shown in Fig. 11, there are no observable differences in the general shape for the different cases. This is because the torsion effects of the IS are negligible in the Y direction and, therefore, the aggregated structures also have low torsional effects.

However, in Fig. 12, it can be perceived that for the modes of the X-axis, the behaviour of the different aggregates tends to concentrate displacements in some of the dwellings located at the ends of the row. This is due to the accumulated torsion effects produced by the asymmetry of each dwelling as a unit, which can be observed in the main mode of vibration in the IS case.

For the cases analysed, it was noticed that the RC case behaves like a fully rigid large structure due to its connections. Bearing in mind that the dwellings are orientated in the same direction, the aggregate will present a similar torsion to that observed in the IS, with greater displacements at the end of the shear wall with the greatest extension.

On the other hand, the CL and FL cases show the displacements concentrated at one end of the row. However, different displacements and torsion effects can be seen in each individual structure in a manner similar to that of the IS case. This difference in the shape of the principal mode of vibration, with more localised displacement concentrations, is a result of considering more flexible connections between buildings, as opposed to the fully rigid connections in the RC case.

The SW case presents the smallest torsion effects among all cases due to the uniformity of the shear walls along the X direction, resulting in less torsion for each dwelling. However, in the SW case, the largest deformations are found at the opposite end of the row compared to the RC case, despite both having rigid connections between dwellings.

In general, seismic studies of aggregate structures are usually carried out by considering adjoining walls which are completely joined or only have a simple wall in between, which is more similar to the SW case. However, in modern aggregates, buildings with near frictionless contact between them are more commonly found, the CL case being intended to emulate these.

Thus, as shown in the different cases, assuming only one type of connection can result in oversimplification, leading to results that differ greatly from reality.

Table 1 Main modes of vibration of the different aggregates
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Table 1 shows the vibration periods obtained and the participating mass of every model case. In general, very small variations in the vibration periods are observed, but the modes differ in shape, which implies important differences in the domain of large deformations. The periods of vibration for the SW case differ from the rest and yield a more flexible pattern along the X direction because of the lack of shear walls. The masses for each mode shown in Table 1 are high enough to consider them as the main modes of vibration. However, there is still a large mass corresponding to other shapes, mainly in the X direction for the CL, FL and RC cases. This must be taken into account when these main modal shapes are used for loading the structure in a pushover analysis, since the rest of the modes are being ignored. This simplification has been adopted to avoid modal combinations that lead to unrealistic deformation shapes.

3.3 Pushover analysis of the aggregate

In the selected pushover analysis implemented in the 3Muri software, horizontal forces are applied at all nodes of each storey, following the shape of the main mode of vibration of the structure (Figs. 11 and 12). Analysis control is carried out using displacements of a control node, which are related to the loads applied on the rest of the nodes. This type of control allows displacements of the structure to be obtained while maintaining the load distribution, even when some elements may have failed. This is particularly useful for analysing aggregate structures, where gradual failure situations of some houses are expected to be observed. To achieve this, the control node may be different from the one plotted and should be chosen carefully, as it is possible to better represent the collapse if it belongs to a weak part of the structure (Galasco et al. 2006).

Figures 13, 14, 15, and 16 show the pushover curves for the positive and negative X and Y directions of the four aggregate building cases studied. The graphs indicate the point of failure for each structure in the aggregate. To analyse them correctly, the displacement of the central node in the roof of the third dwelling (central) has been considered as the reference to be plotted. The base shear corresponds to the sum of what was recorded for all nodes at the aggregate base. To capture the failure of a variety of structures, pushover analyses were performed to reach an 80% reduction of the maximum shear capacity of the aggregated structures as a whole. Due to the reference node chosen, these curves show their greatest displacement extension when the deformations are concentrated in dwelling 3 and the maximums are only considered up until the collapse of this structure, given that after the collapse the displacements measured for this node will not be real.

Fig. 13
figure 13

Displacement of the roof of dwelling 3 and total base shear, loads in +X direction

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Fig. 14
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Displacement of the roof of dwelling 3 and total base shear, loads in \(-\)X direction

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Fig. 15
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Displacement of the roof of dwelling 3 and total base shear, loads in +Y direction

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Fig. 16
figure 16

Displacement of the roof of dwelling 3 and total base shear, loads in \(-\)Y direction

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Figures 13, 14, 15, and 16 show that although the behaviour in the positive and negative directions of each axis can be similar, the results differ as the houses are not symmetric and therefore the walls that initially support the loads in both cases are different. Thus, higher maximum resistance values can be observed for all models on the +X and \(-\)Y axes with regard to their opposite signs. In addition, the main behavioural differences between the cases of aggregated structures correspond to the X axis response. The different points on each curve when the structures fail are shown in Table 2 for the positive direction and Table 3 for the negative direction.

Table 2 Failure point of the first three structures in the +X direction
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Table 3 Failure point of the first three structures in the -X direction
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Figure 13 shows that the curves corresponding to the FL and CL connections have similar behaviour before the second significant drop in load (corresponding to the failure of the second dwelling) and these have a very different shape from that observed for cases of more generalised behaviour, such as the RC and SW aggregate, the differences between which are associated with the initial stiffness and resistance related to the absence of walls in the SW case. However, as shown in Table 2, both have considerably grater deformation capacity in the cases of the first and second dwellings.

In the +X direction pushover curves, the FL and CL cases begin to differ from each other after the second drop in load, this being the moment corresponding to the failure of the second dwelling. From this point on, the CL connection begins to dissipate significantly more energy through plastic deformations, showing greater strength and deformation capacity in this direction. This suggests that in certain specific cases, the support between the walls of adjoining houses could be beneficial for the seismic behaviour of the structure.

Figures 17a–c show the failure mechanisms observed before the failure of the first, second and third structures (20% drop in base shear), respectively, when applying the nonlinear static analysis in the positive direction of the X axis. The failures that can be observed correspond to the points indicated for each case in Table 2.

Fig. 17
figure 17

Progression of damage throughout the failure of the first 3 structures

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In Fig. 17a it can be seen that the failure mechanisms at this point are very similar for CL and FL. In addition, both cases show that two dwellings are still in their elastic phase, in contrast to the RC and SW cases, due to their rigid inter-dwelling behaviour. This is the reason why lower displacements are observed in the central dwelling in the CL and FL cases, since these concentrate higher loads and displacements in the exterior dwellings. In Fig. 17b, c, the way in which the mechanisms evolve along the aggregates can be observed. The latter shows important differences for all cases. For this reason, it is important to consider more than one case of interaction between dwellings, especially for structures with significant torsion effects, which can lead to very different stress distributions when considering different inter-dwelling connections.

To verify our previous findings, in Fig. 18 the behaviour of base shear and displacement of dwelling 5 (the first to fail) are compared in two types of aggregates, that corresponding to a connection that allows compressive forces between dwellings (CL) and that which is joined by links with elastic behaviour without any type of restriction (FL). The analysis of this house, in particular, has been chosen since it is the first to fail and, as can be seen in Fig. 13, the CL and FL models present the greatest similarities in behaviour before the failure of this structure. Therefore, if there are differences in the case of this structure, it is assumed that they will be even greater for the others. Since the stiffness of the joining elements is the same and, as shown in Table 1, the periods are very similar, the distribution of loads applied in both cases should be almost the same. This means that the differences that can be observed between the behaviour of the models are mainly linked to the effect of interaction between dwellings. Although there is a ”free” torsion (not stopped by adjoining houses) in the aggregate that considers houses connected by elements without resistance (FL), the wall elements in the numerical model do not have a real thickness. For this reason, the 12 free centimetres between the wall axes provide enough free space to displace the houses without causing penetration in the adjoining ones.

Fig. 18
figure 18

+X load response of dwelling 5, comparison between the FL and CL models

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Figure 18 plots the curves for CL, FL and IS. These models have the same elastic branch. For the first change in slope due to damage, CL shows slightly stiffer behaviour than in the FL case. It can be observed that the drop in strength of the CL aggregate is more abrupt, so the failure has more brittle characteristics than its FL counterpart, which fails with smaller displacements (\(8.4\%\) less) but in a more ductile manner. This difference is due to its resistance to compression, the CL connection contributes to reducing the torsion effects of dwelling 5, since it is supported by dwelling 4, which causes the forces to be concentrated mainly in the shear walls of the X axis. Furthermore, the FL connection allows for the torsion of the house, which translates into the appearance of failures both in the shear walls of the X axis and in other resistant elements of the structure, resulting in a smaller displacement of the reference node prior to the first drop in resistance and a more ductile final collapse.

Figure 18 shows that the resistance in both cases is approximately 9.2 percent higher than that of the isolated structure (IS). This suggests that the distribution of loads applied on these aggregates (from the main mode of vibration) allows the structure to withstand higher horizontal stresses. In the case where the interaction between adjacent dwellings is neglected (FL), the drop in resistance occurs earlier than in the CL case, both corresponding to the failure of one of the two main shear walls of the structure. However, upon extending the curve, it becomes apparent that the failure of the second main shear wall occurs with considerably larger displacements (\(50.0\% larger\)). In the elastic phase, the stiffness and stresses of the two types of connections were comparable, indicating that the differences observed in the plastic phase were caused by the redistribution of loads. This suggests that the interaction between adjacent dwellings in a direction perpendicular to the applied loads is significant, due to the torsional effects of the aggregated structures. Based on the analysis, it can be inferred that adjacent dwellings experience an interaction effect due to torsion, in which they are essentially applying a push force against each other. This effect is captured and represented by the chosen modelling strategy.

Furthermore, to compare the results obtained for the aggregates and contrast them with the behaviour of the isolated structure, the capacity curves are represented in their spectral coordinates, according to ATC-40 (Applied Technology Council 1996), in such a way that the mass of each model is ignored in the comparison. To focus on the differences between each dwelling and different aggregates, only one analysis per axis is shown, in Fig. 19 the study for loads in +X and in Fig. 20 for loads in +Y.

Fig. 19
figure 19

Spectral displacement and spectral accelerations of aggregated structures compared with the isolated structure, loads in +X direction

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Fig. 20
figure 20

Spectral displacement and spectral accelerations of aggregated structures compared with the isolated structure, loads in +Y direction

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It is important to carefully analyse the results shown in Figs. 19 and 20, as the capacity curves of the aggregated dwellings consider the complete aggregate as a single structure. Therefore, significant decreases in spectral acceleration imply the failure of one house, but not necessarily that of the rest. It should be noted that, for the isolated structure, a significant load reduction results in collapse. Additionally, the displacements considered in the aggregate correspond only to the central node of the aggregate (house 3), and thus do not provide information on the displacement related to the other central nodes of the associated dwellings. However, these curves clearly indicate that aggregate structures and isolated dwelling are different structural systems with markedly different behaviours. Overall, it can be observed that an isolated structure has a significantly higher seismic capacity compared to the aggregated buildings studied when considered as a complete structure.

To study the individual resistance of each dwelling, a pushover analysis of the complete aggregate is proposed. However, nodal results are utilised to create capacity curves for each of the five individual structures. The displacement of the central node of each roof and the total base shear recorded in each structure are plotted, as shown in Fig. 21. This study is carried out for the CL case with +X loads and allows the resistance of each house to be analysed separately. The capacity curves obtained for each dwelling correspond to different load patterns and boundary conditions based on their position in the aggregate. In contrast to the isolated dwelling, Table 4 displays peak base shear, ultimate displacement, and elastic limit displacements for each structure.

Fig. 21
figure 21

Response of each dwelling of the CL aggregate for +X load

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Table 4 Elastic limit, ultimate displacements, and peak base shear values for each structure under +X load
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When analysing the pushover curve of the aggregate in +X direction (Fig. 13), it was observed that dwellings 4 and 5 failed first, despite having higher resistance and deformation capacity in the +X direction. This is attributed to the significant torsion effects suffered by the entire aggregate, which concentrated displacements in the dwellings at one end and rotations at the other. Interestingly, the houses at the end with the highest displacements (4 and 5) experienced fewer torsion effects at individual level than the isolated house. Consequently, they concentrated loads on the shear walls, which had greater resistance and achieved greater displacements, as shown in Fig. 21. In contrast, houses 1 and 2 suffered significant torsion effects due to the load pattern and the absence of adjacent structures to restrict their rotations, which reduced their resistance and displacement capacities considerably. It is noteworthy that, despite having the least displacement capacity and resistance, houses 1 and 2 were the last to fail in the aggregate

4 Conclusions

In this study an equivalent frame model has been developed and validated to analyse the nonlinear static behaviour of Costa Rican PG-RCM dwellings. The model was used to investigate the seismic performance of isolated and aggregate dwellings, revealing significant differences in their behaviour under seismic loads that act perpendicular to the aggregate shape of a row. Specifically, the analyses demonstrated that the isolated structure and the aggregate structure typology studied exhibit distinct behaviours, highlighting the need for a more nuanced approach to understanding and addressing the seismic performance of aggregate structures.

The research represents an important step towards identifying and addressing the effects of inter-building interactions in seismic design and retrofitting strategies for aggregates of PG-RCM dwellings. The study revealed that differences in the rigidity and resistance of the modelled connections between adjacent buildings can significantly alter the response of the entire aggregate, both in the linear and nonlinear range. Specifically, in the X direction, a comparison between two commonly observed cases, CL and RC, for the central structure in the aggregate showed mean differences of \(21.8\%\) in ultimate base shear resistance and \(11.5\%\) in ultimate displacements. These findings highlight the crucial role that variations in connection characteristics play in the seismic performance of aggregates. Hence, appropriate treatment of the interaction between buildings is essential in the analysis of seismic performance. This finding underscores the importance of taking into account the construction practises of the country, as it is assumed that rigid connections are not usually typical of modern building construction.

This research has shown that the aggregate configuration leads to variable resistance capacities among individual dwellings. Particularly in the case of the CL aggregate, which is the most commonly observed in practice, in the +X direction, structures located at one end of the aggregate exhibit greater resistance, with an increase of \(9.1\%\) compared to the isolated structure. Additionally, these structures demonstrate a higher deformation capacity, with an increase of \(56.6\%\) compared to the isolated structure. On the other hand, structures at the opposite end of the aggregate exhibit lower resistance (a decrease of \(38.5\%\)) and displacement capacities (a decrease of \(30.2\%\)) compared to the isolated structure. However, because of the concentration of stresses at the end with higher resistance, these structures are the first to fail, resulting in lower overall resistance for the aggregate structure. These findings highlight the importance of considering the interaction effects among individual structures in an aggregate when assessing their overall performance under seismic loads. The results of this study could lay the foundations for the country’s standards to determine design strategies for improving the overall seismic performance of the aggregates. Based on the observed results, it may be beneficial to consider the total number of dwellings that are joined for the seismic analysis of this particular type of building and aggregate. However, it is important to note that these findings are based on a specific case and further research is needed to explore this approach in more detail.

Future research should aim to investigate the seismic behaviour of aggregate structures with different materials, shapes, and ground levels, and larger areas of interconnected buildings. Additionally, more research is needed to understand the effect of different types of connection on the seismic behaviour of aggregate structures and to develop more comprehensive models that can accurately capture this behaviour. Furthermore, the effect of soil conditions on inter-structure interaction may be an interesting point to consider in future research.