1 Introduction

Interaction of adjacent masonry buildings is significant in particular during earthquakes, where joints have a crucial role. Dynamic tests performed on full scale or scaled buildings provide valuable information on this topic (Benedetti et al. 1998; Mazzon et al. 2009; Lourenço et al. 2013; Mendes et al. 2014; Magenes et al. 2014; Guerrini et al. 2019; Senaldi et al. 2020). These activities are very import especially for complex structures to be modelled. Masonry buildings under dynamic actions certainly fall into this category. However, in order to perform reliable experimental programs, there is still a need for basic modelling. Hence the modelling approach must not be aimed at detailed simulation but at collecting the key aspects of a dynamic test. The present study focuses on the development of a simplified Finite Element (FE) model, based on limited information without calibration, able to provide key information to design an experimental program. The FE model cannot be developed according to refined assumptions but starting from the main information easily available or based on similar structures. Several research groups already analysed the behaviour and vulnerability of the masonry units within aggregates by means of numerical models, with different levels of refinement (Kappos et al. 2002; Senaldi et al. 2010, 2019; Betti et al. 2015; Candeias et al. 2017; Formisano 2017; Formisano and Massimilla 2018; Maio et al. 2015; Shabani et al. 2021).

The design of experimental programs of scaled buildings subjected to dynamic actions is a very critical phase (Fujii and Sakai 2018; Meoni et al. 2019). In particular, since it is not possible to perform initial calibrations, the development of simplified models able to predict the main outcomes is certainly a useful tool to plan experimental tests (Kajii et al. 2004). In this specific case, the development of a model able to provide the following outputs was the main goal:

  • The threshold at which evident damage is expected;

  • Identify the areas where the damage is located most;

  • Identify the type of damage.

These outputs are key information for a correct design of experimental programs (Tabatabaiefar and Mansoury 2016). After the experimental phase, the numerical model was not calibrated with the experimental outcomes; in fact, the data initially provided in terms of physical and mechanical properties of the materials were adopted as input data for the development of the FE model. The physical and mechanical properties have been preliminarily assigned starting from the experimental results described in Tomić et al. (2022). In this reference, similar masonries were tested, and the mechanical parameters have been chosen according to these results. In an initial phase, the level of knowledge is commonly low and the FE model must be compatible with it.

2 Finite element model

According to goals of this work (Tomić et al. 2023), a simplified approach (D’Altri et al. 2020; Rizzano 2011) was performed to analyse the scaled masonry building dynamically tested. With some further very simple assumptions, mainly due to original time constraints and initial knowledge level, the analyses were performed by means of consecutive time histories including the effects of previous signals (Aras et al. 2020). The overall masonry structure, aggregate of two masonry units, (Fig. 1) has been modelled by means of SAP2000 FE code distributed by “Computers and Structures”.

Fig. 1
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Finite Element (FE) model: main information and IDs of displacement recording locations

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3 Geometrical model

A homogenous FE model (Fig. 1) was developed to perform the numerical analysis. The masonry walls were modelled by means of nonlinear shell elements (Masciotta et al. 2020; Sharma et al. 2021). The mesh has a regular texture with average size of 0.25 × 0.25 m2, given the regularity of the scaled masonry building. The masonry simulates an ordinary masonry building, built in historical centres. The masonry material is made of natural stones with irregular joints to be representative of real cases, typically found into historical centres. The 1:2 scaled specimen is made of two main masonry building units in order to simulate the behaviour of aggregate masonry buildings. The two building units are adjacent to simulate the effects of construction that took place at different periods. Therefore, the connection between the two units was weakly guaranteed (Tomić et al. 2021a). In particular, the adjacent walls of building Unit 2 were made without an efficient connection (Shrestha and Hao 2018). Under this assumption, the behaviour of masonry at the interface is strongly influenced by the internal stress state (Valente et al. 2019). When the two masonry units are coming closer the material is compressed and the two building units collaborate. Conversely, if the two units move away, the masonry interface cracks and the two units behave independently. Obviously, this effect could be localized along vertical interfaces. For this reason, the interface between the units has been modelled by means of unidirectional springs characterized by zero stiffness in tension (TufailKhalil et al. 2019). Conversely, the stiffness in compression of the springs has been calibrated according to the stiffness of the tributary masonry material (the springs substituted a masonry strip). Without more refined knowledge level, no further nonlinear properties are defined.

The floors are not directly modelled because a diaphragm constrain has been considered for the three floors. The distribution of vertical loads has been assumed according to the characteristics of floors (direction of wood beams). The following section describes the masonry nonlinear stress–strain constitutive relationship adopted in the numerical model for quadrilateral shell finite elements (Ravichandran et al. 2021; Uranjek et al. 2019). Perfect fixed restraints were considered at the bases of the walls. A perfect connection has been considered to model the wall to wall connection (Kahrizi and Tahamouli Roudsari 2020; Solarino et al. 2019; Tomić et al. 2021b).

3.1 Mechanical model

The mechanical parameters have been chosen according to the information initially available before the test. The stress–strain constitutive relationship is compatible with a simplified approach (Lignola et al. 2014). The masonry has been modelled as elastic perfectly plastic in compression (Lopez et al. 1999). This assumption might seem very simple; however, it allows to verify that the structural capacity is not governed by the achievement of the compressive strength during the seismic excitation. This is common for masonry buildings characterized by low compressive stresses due to gravity loads. For these structures, the masonry is almost characterized by very low compression stresses. In fact, this assumption was confirmed a posteriori by recording the maximum value of the compressive stress in the FE model that is always lower than the compressive strength for all time histories (for this reason no further improvement of the compressive behaviour was done). The structural behaviour in tension has been modelled limiting the maximum tensile strain after linear softening of masonry at 3 × εC, where εC represents the cracking (peak) strain of masonry in tension (Segura et al. 2018; D’Ambra et al. 2016). The tensile behaviour is the most critical; it was assumed a reduced fracture energy based on preliminary information and it was confirmed after the test.

The following Fig. 2 shows the nonlinear stress–strain constitutive relationship used in the numerical model for masonry, for the quadrilateral shell finite elements (D’Ambra et al. 2016).

Fig. 2
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Constitutive model of masonry material: negative and positive values for compressive and tensile stresses/strains, respectively

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A viscous model has been used for the damping (Cecchi and Tralli 2012), fixed equal to 5% for all time history analyses (probably a higher value could also be suitable, but 5% was preferred also looking at the results of similar tests (Giamundo et al. 2016)).

3.2 Loads model

The FE model was subjected to a sequence of signals at the base. The signals (accelerograms) refer to the desired data. In reality, the experimental practice has shown that the desired signals are always affected by a modification due to many disturbing factors (Giamundo et al. 2016). As clarified later, the exact experimental accelerograms were in any case considered in the simulations, and not the desired signals. It is the only revision of the model after the blind prediction to account for the real experimental signal, compared to the desired one, since this significantly affects the numerical results (Eichmann Oehrli 2019), even in a simplified analysis of the structure.

3.3 Structural analysis

A time history analysis was performed to assess the structural behaviour of the scaled masonry building under dynamic signals. The signals sequence described in Tomić et al. (2022) has been implemented in a time history analysis. The same signal was applied in four steps of increasing intensity (25, 50, 75 and 100% of the main signal), each divided in three substeps (excitation in y-direction or longitudinal, in x-direction or transversal, and bidirectional excitation). The time histories were assigned in a single sequential analysis. In particular, in the output of sequential shaking analyses, the final (residual) displacements of the previous shakings have been subtracted (reset of baseline), so that each run starts from an imposed zero condition, even if residual displacements were found at the end of a previous shaking and this residual affects the FE modelling (Rossetto et al. 2019).

It is found that up to 50% (i.e. runs 2.1, 2.2 and 2.3) there was almost no residual displacement, so that up to runs 4.1, 4.2 and 4.3 the simulation was a continuous time history. In this way, after severe damages during run 3, it is expected that simulations with 100% scale factor (i.e. runs 4) require more refined models including geometrical nonlinearities and large displacement formulations (Gagliardo et al. 2021; Portioli and Cascini 2017; Vlachakis et al. 2019), out of the scope of this work, and not tested experimentally anymore.

The experimental results have been analysed and compared to FEM simulations, i.e. numerical results, obtained with the experimental acceleration time histories.

4 Numerical results

The numerical results are shown in this section. The main goal is the localization of the triggering of damage, hence of the internal peak stresses, and the results were shown up to run 4 where 100% of the desired signal has been applied.

After a first run (1.1 with 25% of desired signal) the interface model (masonry has no tensile strength at the interface between the two units) confirms that the cracking between the two units occurs already at 25% scale factor (vertical red lines in Fig. 3, where FE model predicts clear opening of the joints).

Fig. 3
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Cracking between two units along the vertical connection already at 25% scale factor (vertical red lines, joints location)

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The FE model provides information on the in-plane behaviour only and the opening of the vertical joints between the two units, however, according to the failure modes typically found in masonry buildings, kinematic (macroblocks) analyses (Belliazzi et al. 2021) have been performed to assess also the out of plane (local) behaviour. It is worth noting that failure (out of plane) mechanisms were also related to the main cracks detected with the in-plane FE model outcomes; longitudinal cracks can be the cylindrical hinges for the development of kinematic analysis. No out of plane mechanisms were detected by kinematic analysis, nor additional damages have been triggered by the FE model for all signals with scale factor equal to 25%, in fact, there is no exceedance of the expected tensile strength of masonry in the shells, assumed as 0.17 MPa. However, due to potential scatter for tensile strength, some portions of the structure where low damage is expected (because masonry is already in tension, close but lower than tensile strength) were highlighted, too, with blue colour in addition to black in the contour plots (tensile stresses are positive, values are in MPa in following figures).

The 50% shake table runs provide clear damages to several portions of the structure (Figs. 4, 5, 6, 7). In those figures on the left-hand side there is the schematic significant damage (in red the one simulated for Y signal and in blue that for X signal), while on the right hand side there is principal tensile stress contour plot from the FE analysis at one of the most significant time steps, however it is remarked that not all the damage is achieved at the same time, so it is not visible in the same screenshot. It is worth noting that internal stress exceedance of the tensile strength (black coloured areas in the contour plots) corresponds to triggering of damage. The characteristics of the FE code do not allow to visualize the complete development of the crack patterns. Therefore, the most significant screenshots were reported on the right-hand side of the figures when the damage was triggered (in one direction, while the cyclic nature of signal provides also damages in the opposite direction, i.e. mirrored). The highlighted areas (with black overimpressed lines) represent the location where the expected damage is triggered and cracks are traced in the direction of the steepest change in the stress contour map. This information (i.e. development of longitudinal cracks potentially representing cylindrical hinges for the development of mechanisms) is associated to the deformed shapes provided by the FE animation and the local behaviour analysis (i.e. kinematic/macroblocks analyses). Engineering judgement (coupling stress contours, deformed shape animations and local kinematic analyses) allows to represent a schematic damage on the left hand side. For instance, in Figs. 4 and 5, there is a tensile stress concentration at the edges of the windows, crossing diagonally the joint panel (right side of figure), hence it was depicted on the left side of the same figure a diagonal failure sign, accounting also for the sign reversal during time (i.e. mirroring). There is also a tensile stress concentration (with a peak not at the same time step, so not directly visible in the screenshot) along the diagonal of the panel below the windows (right side) and a deformed shape yielding to a diagonal failure sign in the panel (left side of the figure). Those panels below the windows have significant stresses due to their reduced thickness.

Fig. 4
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Simulated damage during the seismic action along X direction for the façade 1 (scale factor 50%)

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Fig. 5
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Simulated damage during the seismic action along X direction for the façade 4 (scale factor 50%)

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Fig. 6
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Simulated damage during the seismic action along Y direction for the façade 3 (scale factor 50%)

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Fig. 7
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Simulated damage during the seismic action along Y direction for the façade 2 (scale factor 50%)

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The signal along the Y direction (i.e. run 2.1) provides the cracking of joint masonry panels (red lines) and of the joint panels and the panels below the windows. The signal applied along the X direction (i.e. run 2.2) provides a masonry cracking highlighted with blue lines, while the third bidirectional signal (i.e. run 2.3) increases the damages occurred during the previous signals.

It is worth noting that, after this sequence, the structural behaviour of the specimen could change significantly because the cracking of masonry portions could promote rocking motions of the cracked portions. Therefore, the numerical results at 75% and 100% shake table runs could be jeopardized due to rocking effects, geometric nonlinearities and out of plane mechanisms hard to predict without a refined numerical calibration and more sophisticated approaches.

Figures 4, 5, 6, 7 show the location (triggering of the expected damage) of the predicted damages identified after a dynamic action with an amplification factor equal to 50%. Obviously, the masonry walls (namely façade 1 and façade 4 in X direction) show the greatest damage when they are subjected to the action along X (main stress plane), but the same repeats for the panels in the Y direction.

The cracks highlighted at the base of the masonry pier panels and in the joint panels (blue lines in Figs. 4 and 5) remark possible rocking phenomena around those cracks, in fact, it is probable that after 50% signal, the behaviour is governed by macroblocks rather than by a continuous behaviour of the masonry.

Along the Y direction the rocking problems seem to affect only the top joint masonry panels of the second building unit (Figs. 6 and 7). This is justified by the higher axial load acting in the panels at the ground floor.

Following Figs. 8 and 9 show the portions where the damage is expected (black areas) at runs with an amplification factor equal to 75%. Macroblocks mechanisms have been checked and overturnings are expected, as reported in the following section. The behaviour of the structure under a seismic action with 75% scale factor seems to be governed by out of plane wall mechanisms. The following figures show a schematic of the overturning mechanisms of the main walls. The tensile stress concentration at the bases of the wall suggests longitudinal cracks formation (i.e. potential cylindrical hinges) and the deformed shape animation of the FE model confirm the outcomes of the kinematic analysis. This can be repeated at mirrored walls due to the symmetry.

Fig. 8
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Simulated damage during the seismic action along X direction for the façade 1 (scale factor 75%)

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Fig. 9
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Simulated damage during the seismic action along Y direction for the façade 2 (scale factor 75%)

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Numerical analyses were performed in the blind prediction phase (even if the test was not conducted anymore) and it has been observed that the accelerations during the 100% runs could induce collapse due to local mechanisms. The dynamic action along the Y directions causes an out of plane mechanism (red lines). This condition has been confirmed also by the numerical FEM prediction where a stress concentration is clear. A second probable mechanism could be due to the dynamic action along the X direction (blue lines) as confirmed by the numerical stress concentrations shown in the following figures.

Obviously, the damage already occurred during the run 3.1 (i.e. 75% scale factor) increases due to the increasing dynamic action, up to 100% scale factor as shown in the following Figs. 10, where the areas with high tensile stresses enlarge.

Fig. 10
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Simulated damage during the seismic action along X and Y directions for the scaled building (scale factor 100%)

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5 Experimental–numerical comparison

The experimental–numerical comparison was performed in terms of expected damage and displacement field; the comparison was performed up to run 2.1 due to the clear damage occurred on the specimen.

The experimental program had some variations during its execution compared to preliminar theoretical program for blind predictions. In particular, the experimental tests were prematurely stopped at run 2.1 (50% along the Y direction). This is probably due to the evident damage achieved at the end of that test, hence round 2.1 was the last considered for the unreinforced structure. The building was subsequently strengthened and retested again with a similar set of signals; however, the retrofitted case was not analysed according to the main objectives of this work.

5.1 Damage evaluation

Starting from the lowest signal (25%) along the Y direction (i.e. run 1.1), the structure showed the separation between the two structural units. This was confirmed by the numerical predictions, in fact, as expected, without an efficient connection between the adjacent walls, small dynamic excitations could be enough to facilitate the detachment of the two structural units, with significant relative displacements. The signal with the same amplification factor given along the X direction (i.e. run 1.2) did not generate additional damage. Finally, also the simultaneous signal along the X and Y directions (i.e. run 1.3) did not cause additional damage. From this point of view, the numerical predictions are satisfactory when compared to the experimental results.

The 50% signal gave the first evident damage to the structure. For a careful analysis of the experimentally detected damage, see reference Tomić et al. (2022). In this phase, the reliability of the numerical predictions is discussed with respect to experimental outcomes and the comparison of the numerical predictions with the experimental results was satisfactory also in terms of damage identification (Fig. 11). In particular, the model satisfactorily identified the areas experimentally affected by the greatest damage, large displacements of the upper storey of Unit 1, with out of plane mechanism of façades 1 and 4, and in-plane failure for façades 2 and 3. Mainly diagonal failure in the joint panels and in the panels below the windows occurred experimentally. The damage was already visible starting from the signal along the Y direction (i.e. run 2.1), and, as predicted, some damage then developed along the other direction. The subsequent bidirectional signal essentially emphasized the state of damage reached at the end of the previous signals.

Fig. 11
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Experimental damage after the seismic action along X and Y with 50% scale factor

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The test was prematurely stopped before 75% scale factor runs due to the strong damage detected. The comparison between numerical and experimental results with the signal at 75% and 100% amplification factor was not carried out due to this modification of the experimental program.

5.2 Displacement evaluation

A comparison was made in terms of positive and negative peak displacements between the numerical predictions and experimental results. The displacement field is on average underestimated by the model with respect to experimental outcomes, as expected because the geometrical nonlinearities and large displacement effects were not simulated. However, it is satisfactory at low amplification scale factors. Once the damage has been reached, numerical displacements were underestimated, and this is mainly due to the rocking effects not modelled within the FE model. In fact, as long as the material remains in the elastic range, the underestimations are negligible. The rocking effects start to become important at the run with 50% amplification scale factor. Tables 1 and 2 show the numerical and experimental comparison in terms of peak displacements. For each control point (Tomić et al. 2022) the percentage variation has been evaluated. Note that Rd1 to Rd6 are the sensors to measure the absolute displacements of top masonry (Tomić et al. 2022). The IDs of monitored joints are labelled according to Fig. 1. As expected, predictions are lower than experimental outcomes and the difference increases at higher signal amplification factors.

Table 1 Experimental and numerical displacements comparison
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Table 2 Experimental and numerical displacements comparison
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The same comparison was performed with reference to the relative displacement between the two building units, see Tables 3 and 4). Note that Id1 to Id4 are the sensors to measure the relative displacement at the joints location between the two masonry units (Tomić et al. 2022).

Table 3 Experimental and numerical relative displacements between the two units
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Table 4 Experimental and numerical relative displacements between the two units
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It is interesting to observe how the percentage differences are reasonable up to the first signal with 50% scale factor. Once that intensity is reached, the rocking effects become important, and the model fails to predict them as it is based on a FE homogeneous continuum modelling of masonry in small displacements. A macroblock modelling would be able to better simulate the behaviour of the heavily damaged building. However, such simplified modelling was used for damage triggering, but not for nonlinear kinematic analysis of large displacements.

The following Fig. 12 shows the comparison in terms of displacements detected during the time history; the comparison was performed for the most significant results (Rd2 and Rd5 for signal along Y direction and Rd3 and Rd6 along X direction), also in the case of time history there is a satisfactory agreement between numerical and experimental results.

Fig. 12
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Experimental and numerical displacements comparison: a, b (scale factor 25%, run 1.1), c, d (scale factor 25%, run 1.2), e, f (scale factor 50%, run 2.1)

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6 Conclusions

The aim of this work was to develop a sufficiently simplified numerical model, based on limited information without calibration, compatible with preliminary analysis at a blind prediction stage of experimental tests. Many experimental programs are planned on the basis of assumptions and choices which are subsequently amended. This is a very serious issue, especially for masonry structures subjected to dynamic actions. A model able of capturing the main information on structural behaviour is a useful tool, especially in the planning phase of experimental programs. The numerical model was in fact developed on the basis of mechanical properties initially provided and to be updated only afterward. Therefore, the model was developed to test the reliability of a simplified approach (pre diction) and not to accurately model the behaviour of the tested building (post diction). A reliable model is necessarily calibrated on the basis of deeper data from experimental tests (e.g. constraints and restraints by means of dynamic identification, among others). However, such refined calibration would make the simulations highly sensitive to the peculiarities of the case study, while this simplified modelling and approach would provide a more general outcome. In this sense it was explicitly decided not to carry out a re-calibration of the model after specific tests on materials or to refine the model by using a dynamic identification of the structure.

The numerical-experimental comparison highlighted the limits of the modelling developed in terms of displacements. In particular, numerical modelling provides satisfactory displacements until a clear damage occurs. Once a clear damage is achieved, displacements are underestimated. However, this is due to homogeneous shell elements perfectly fixed at the bases in the models, while rocking effects and geometrical nonlinearities occurred experimentally. The simplified FE model was unable to accurately predict internal redistribution once evident damage has been achieved. In order to perform this, it is obviously necessary to develop more refined models after specific numerical calibrations or other kinematic approaches should be used, while the goal of this study was mainly to detect damage activations.

From the point of view of damage, the adopted modelling was satisfactory under several profiles. In particular, the threshold at which the damage was detected has been identified. Furthermore, most of the damage was satisfactorily identified (as area of tensile strength exceedance or after local macroblock analyses, or combination of the two) in accordance with the location and typology experimentally found.