The 2D continuum finite element model is available in (Giardina et al., 2013b), is reproduced and enhanced in this research, incorporating advancements in modelling structures subjected to settlements since the original study. The modelling approach is schematically illustrated in Figure 1c. A sensitivity study is conducted using two masonry material models: the orthotropic Engineering Masonry Model (EMM) and the isotropic Total Strain Rotating Crack Model (TSRCM). The two chosen constitutive laws are commercially available and widely used for modelling unreinforced masonry structures (Sousamli, 2024). The material properties for both TSRCM and EMM are reported in Table 3.

The TSRCM, originally employed in (Giardina et al., 2013b), captures the non-linear cracking behaviour of masonry by evaluating failure mechanisms in the principal directions, including tensile and compressive failures. In contrast, the EMM, differentiates tensile failure in horizontal, vertical, and diagonal directions, horizontal and vertical compressive failure, and shear sliding.

For head-joint failure in the EMM, both the defined minimum tensile strength and the shearing properties, i.e., cohesion and friction angle, contribute to the overall strength. The minimum head-joint strength f_tx,min (Figure 4), is a user-defined input that accounts for the combined effects of head-joint horizontal opening and splitting cracks in the bricks, assuming a pure vertical crack running through both bricks and head-joints. Additionally, the local shear strength component provided by the bed-joints is calculated by adding cohesion to the product of the friction coefficient and the overburden level, resulting in the final head-joint strength (Figure 4). In the TSRCM, crack opening is governed by a single user-defined tensile strength value. Once the tensile stress reaches this specified threshold, a crack initiates. The crack can then propagate in the direction of the principal tensile stress, meaning that its opening is not restricted to predefined paths but can develop dynamically based on the stress distribution in the material, as the crack orientation adapts to the loading conditions.

The material properties of the TSRCM, reported in Table 3, have been retrieved from (Giardina et al., 2013b). In contrast, the additional material properties required for the EMM are derived either from material tests, such as cohesion and friction angle, described in (Giardina et al., 2012) or computed according to (Schreppers et al., 2016). These include Young’s modulus in the horizontal direction, shear modulus, and the minimum head-joint strength, based on available values for the fired-clay baked masonry used in the test.

The minimum head-joint strength employed in numerical analyses has been reported to range from 1.0 to 3.3 times the bed-joint tensile strength (Schreppers et al., 2016; Prosperi et al., 2023b; Wani et al., 2023; Korswagen, 2024; Sousamli, 2024). This variation depends on the type of masonry material or arises from calibration efforts. Consequently, while the value specified in Table 3 is set at 3.0 times the bed-joint tensile strength (Schreppers et al., 2016), a variation with a lower value of 1.5 times the bed-joint tensile strength is also considered for sensitivity analysis.

Above the doors and windows of the facade model, wooden lintels are included, with the material properties specified in Table 4. Settlements are applied to a steel profile supporting the façade model. The material properties of the steel profile are reported in Table 4. The cross-section of the steel profile is an I-section, with a height of 60 mm and a width of 50 mm. The flanges of the profile have a thickness of 5 mm (Rizzardini, 2011; Giardina et al., 2012). The façade is modelled by 8-node plane stress elements (8,251 in total) with 3 × 3 integration points, with a mesh size of 10.5 mm (Figure 5a).

The results of a simplified micro-model are compared against those of the macro-models. In the simplified micro-model, the bricks are modelled as independent linear elastic elements. Each brick is split in half, with an interface connecting the two portions to represent potential cracking at the midpoint of each brick (Figure 5b). Since no crushing is expected in the bricks, a discrete cracking behaviour is assigned to this interface, with the flexural strength of the bricks employed as interface tensile strength. Mortar joints and their contact edges with the bricks are represented as combined Cracking-Shearing-Crushing (CSC) interface elements. The CSC model employs a multi-surface plasticity approach, integrating three primary failure criteria: tension cut-off criterion, Coulomb Friction criterion and compression cap criterion. A key feature of the CSC model is its ability to account for the interaction between these failure modes. For instance, the development of tensile cracks can influence shear behaviour by reducing shear resistance, while shear sliding can affect the normal stress distribution, potentially leading to crushing. By incorporating these interactions, the model provides a more realistic simulation of interface behaviour under complex loading scenarios (Murgo et al., 2021; Chang, 2022; DIANA FEA bv, 2024). The modelling approach is schematically illustrated in Figure 1b. The material properties are listed in Table 5.

The lintels and steel profiles are modelled using the same approach employed for the macro-models selected in this study, and with the material properties summarised in Table 4. The mesh size is 9 × 10.5 mm and 20,186 elements are obtained after meshing (Figure 5b).

The models use a non-linear interface between the building base and the steel profile (Giardina et al., 2013b). A sensitivity study is conducted using two base interface models: Coulomb friction and Discrete Cracking. Coulomb friction interfaces assume the interaction is governed by a frictional behaviour, as used in the reference study, while Discrete Cracking interfaces model discontinuities between elements based on the relation between tensile tractions and relative displacements normal to the interface, keeping the shear behaviour as linear elastic. The properties of both interfaces are listed in Table 6.

Following the approach in the reference study, a negligible value for the stiffness in shear is assigned, corresponding to a value of 10⁻⁹ N/mm³. Large differences in stiffness values can cause ill-conditioning in the stiffness matrix, potentially affecting the numerical solution and the reliability of the analysis. For this reason, sensitivity analyses are carried out considering two additional negligible (compared to the normal stiffness values) values of the tangential stiffness k_t: 10⁻⁷ and 10⁻⁵ N/mm³.

In the laboratory test, first, the façade is subjected to vertical loads (Figure 3; Table 2), followed by settlement, which is induced by pulling downward the left end of the base steel profile at point A. In the numerical models, three subsequent loading phases are used to apply the loads: First, the self-weight is applied in five steps, then the vertical overburden loads are applied in five steps, and finally, the settlement is applied with a rate of 0.5 mm per step (30 steps). The displacements are reset to zero before the application of the settlement.

The features of the adopted numerical integration scheme are reported in Table 7. The Quasi-Newton method avoids computing a new stiffness matrix at each iteration. Instead, it utilises information from the previous solution vectors and the out-of-balance force vector at each increment (DIANA FEA bv, 2024). Three convergence norms for the self-weight and overburden are used to ensure a more robust and reliable evaluation of convergence because of the settlement application, thus improving the numerical stability.

A smeared cracking approach is adopted in the macro-models, treating cracking as a distributed effect. In contrast, in the simplified micro-model, the crack width can be determined from the plastic relative displacements (i.e., normal openings) of the interfaces. In both cases, the width of cracks can be retrieved for each integration point and exported as tabulated outputs at each step of the analysis. Knowing the crack width throughout the settlement load application enables a comparison of the damage severity with that observed in the experiment. Moreover, the tabulated outputs of the FEM models can be used to quantify the damage progression and accumulation using one single scalar value (Korswagen et al., 2019; Korswagen, 2024), Ψ in Equation 1: Ψ = 2   n 0.15   c w 0.3 (1)

Where “n” is the number of cracks, “c_w” is the weighted crack width (in mm) calculated with Equation 2: c w = ∑ i = 1 n c i 2   L i ∑ i = 1 n c i   L i (2) c i = c o 2 + c s 2 (3)

Where “c_i” in Equations 2, 3 is the maximum crack width of crack “i” in millimetres, and “L_i” is the length in millimetres. The crack width is composed of both the opening (c_o) and sliding (c_s) components. A summary of the relation between Ψ and the approximate crack width for the various damage levels is presented in Table 8.

Figure 6 shows the CPU time of each analysis, offering insight into their computational efficiency. Interestingly, the TSRC-based macro-model requires 1.5 times the CPU time of the EMM models. Overall, the simplified micro-model is the most CPU-intensive, with its required time ranging from 1.6 to 2.6 times that of the macro-model with EMM. The difference in CPU time between the selected macro- and micro-models is likely due to the smaller number of elements in the former and the added complexity of the material models used in the latter. No significant differences are observed based on the type of base interface or variations in its shear stiffness.

Convergence is observed for all the steps of each selected analysis. Although the iterative solution converges, differences in structural response remain. Convergence ensures numerical accuracy but does not guarantee the solution is close to the true one. The behaviour of the different models is further examined by analyzing the maximum crack width at each step of the analysis (in Figure 7), as well as the displacements at points B and C (in Figure 8). The results of the models are compared against the ones from (Giardina et al., 2012; Giardina et al., 2013b).

While the macro-models with EMM and the micro-models exhibit minimal sensitivity to the adopted value of the interface tangential stiffness or the material model used for the interface, the reproduced macro-model with TSRCM demonstrates a higher sensitivity to these parameters (Figure 7). The variability of each set of curves is herein quantified by the Coefficient of Variation (CoV). Accordingly, the curves in Figure 7 are used to compute CoV for each type of model distinguished by the value of tangential interface stiffness and by the type of interface model. The values of the CoV vary depending on the considered step, for this reason, only the minimum and the maximum values are considered and reported in Table 9.

The macro-model with TSRCM has the highest variability, ranging from a minimum of 2%–69%, which aligns with the visual observations shown in Figure 7. The model with the EMM and f_tx,min of 0.30 MPa demonstrates the least variability. The model using TSRCM with a k_t equal to 10⁻⁵ N/mm³ is the one that best replicates the numerical results published in the reference study, for both the selected interface models. Therefore, this value is considered the reference for the analyses presented in the following sections.

For applied deflection ratios below 2.5 × 10⁻³, both the micro-model and the model with EMM and an f_tx,min of 0.30 MPa tend to underestimate the maximum crack width compared to the experimental results. In contrast, the calibrated macro-model with EMM and f_tx,min of 0.15 MPa shows good agreement, while the TSRCM-based model overestimates the damage severity. However, for deflection ratios above 2.5 × 10⁻³, the experimental trend reveals a rapid increase in the maximum crack width, which the numerical models do not capture. Additional considerations on crack patterns are discussed in the following sections.

Figure 8 compares the displacements at points B and C with the displacement applied at point A across different models. All models show good agreement in vertical displacement at point B. However, horizontal displacements vary, with only the macro-model using TSRCM capturing the experimental behaviour. The EMM and simplified micro-models exhibit a stiffer response, resulting in smaller horizontal displacements. For example, when the applied vertical displacement at point A is 10 mm, the horizontal displacements at point C from the EMM and simplified micro-model range from 8 to 9 mm, approximately 20%–30% lower than the experimental value of around 11.5 mm.

With the availability of photogrammetric measurements over the grid points of the experimental masonry façade, displacement fields from the experiment can be plotted at various stages and compared with the results from the numerical model.

Figure 9 compares the displacement fields to discuss the models’ performance in representing the overall displacements. For an applied displacement of 2.5 mm (Figures 9a1–e1), the displacement fields show good agreement. However, the experimental façade exhibits a rapid increase in displacement at the top, which the model does not capture accurately. The entire left portion of the façade shows higher displacements than those observed in the models, which appear more homogeneous. The TSRCM model most closely matches the experimental results, although all models underestimate the displacement magnitude in the central part of the façade.

While the relationship between the applied settlement and the crack width is detailed in Figures 7, 10 illustrates the crack patterns throughout the analyses, aiming to distinguish the overall failure mechanism of the façade. The smallest crack width observed during the experiment corresponds to 0.044 mm (Giardina et al., 2012). To ensure a consistent comparison, cracks narrower than 0.04 mm are not shown in the crack patterns of the models.

The crack pattern of the TSRCM-based macro-model reproduces with good accuracy the damage mechanism of the model reported (Giardina et al., 2013b; Giardina et al., 2015) in terms of maximum principal strains.

Cracks form exclusively on the right side of the façade during the experiment. While most cracks in the models also appear on the right side, cracks are also detected on the left side. However, some cracks observed in the experiments do not always appear in the models, and vice versa. In the TSRCM model, cracks are localized and follow a diagonal pattern around the openings. Conversely, the EMM models show more smeared cracks, typically following vertical or horizontal directions. However, the micro-model results in less damage than the experiment or macro-models, with cracks primarily following a stair-step pattern. Overall, the macro-model with TSRCM exhibits the most severe cracking, with the highest number and widest cracks compared to the other models. The EMM-based macro-models and the simplified micro-model follow it.

The variations in crack patterns highlight that evaluating damage severity solely based on the maximum crack width is inadequate for accurately representing the damage severity. Therefore, the damage parameter Ψ, briefly described in Section 3.5, is computed for all the steps of the numerical analyses and the experiment, and compared in Figure 11. Crack width measurements from the experimental façade are available for applied displacements ranging from 3.5 mm to 11.5 mm, so the comparison between the models and the experiment is limited to this range. While the damage parameter Ψ computed for the models accounts for the number of cracks, their width, and length, the parameter derived from the experimental results is limited to the number of cracks and their width, as crack length measurements are not available. Since the analytical formulation of Ψ incorporates weighted crack widths by their length, error bands are considered in Figure 11, to account for the simplification applied to the experimental results.

All the models overestimate the damage for applied displacements below 4.5 mm, whereas better agreement with the experimental results is observed for higher applied displacements. Figure 11 confirms the trends observed in Figure 7: in terms of damage severity, the EMM-based macro-models exhibit the best agreement with the experimental results, while the simplified micro-models underestimate the damage and the TSRCM-based macro-model overestimates it. Overall, the damage severity assessed with the EMM-based macro-models falls within the 10% error band of the experimental results. For applied displacements greater than 4.5 mm, the damage severity estimated with the TSRCM is 17%–56% higher than the experimental results.

The results of the sensitivity analysis revealed that the type of interface model (Coulomb vs Discrete cracking) has a limited impact on the outcomes of the macro-model with EMM or the simplified micro-model. In contrast, the macro-model with TSRCM exhibits greater variability. The trends of the applied deflection ratio against the maximum crack width support this observation. When considering both the maximum and minimum CoV values, the interface type does not significantly influence the magnitude of the CoV for each model; instead, the variability of the results is influenced by the interface tangential stiffness. Specifically, the TSRCM shows a maximum CoV of 69%, followed by the EMM models with a calibrated value of f_tx,min (38%), the simplified micro-model (24%), and the EMM (13%).

As briefly mentioned in Section 3.1, the TSRCM-based macro-model replicates the model from (Giardina et al., 2013b). The results of the reproduced TSRCM-based model show minor differences compared to the reference study, as summarized in Table 10 and discussed in the following.

Since the reference study, advancements in DIANA FEA software (from version 9.4–10.9) have introduced changes that influence the results. Notably, the crack bandwidth algorithm differs, and while (Giardina et al., 2013b) employed a custom 10.5 × 13.3 mm mesh, a uniform 10.5 mm mesh is used here for simplicity, reducing additional complexity. Although the impact of these factors has not been specifically assessed in this study, other features are expected to have a greater influence. For instance, the type of constitutive model used for the base interface has been observed to influence the results. The reference publication employed a no-tension interface with a Coulomb friction criterion (Giardina et al., 2013b). This interface has been reproduced and compared with a Discrete-Cracking model, as detailed in Section 3. The Coulomb friction interface better replicates the results from the reference study, though the Discrete-Cracking model is a viable alternative. With the interface tangential stiffness value (k_t) of 10⁻⁹ N/mm³ used in (Giardina et al., 2013b), a possible concern is related to the large differences in the stiffness matrix, potentially leading to unstable solutions. Variations in the k_t can influence façade damage, as shown in sensitivity analyses in (Giardina et al., 2015), which considered the combined effect of shear stiffness and horizontal settlement components. In contrast, the analysis here focuses on the specific impact of k_t on the reproducibility of the experimental test.

A different approach to applying gravity loads is also used. In this work, self-weight and additional vertical load are applied in 10 steps with three simultaneous convergence criteria, ensuring high numerical accuracy and stability before settlement is applied. In the previous model (Giardina et al., 2013b), it is not specified how many gravity load steps have been applied. A different application of gravity loads influences the initial stress distribution and, consequently, the response to settlement.

Overall, the reproduced model aligns with the one in (Giardina et al., 2013b). Nevertheless, the challenges encountered in its reproduction highlight how variations can emerge even when modelling the same case, underscoring the importance of thoroughly documenting all aspects of the research process for transparency, reproducibility, and understanding of potential discrepancies.

Both the TSRCM and EMM are available in the selected FE software and have been used in prior research to reproduce or predict the response of masonry structures and are well-established compared to more recently developed non-linear models. However, the current literature lacks comparisons of these material models’ performance in capturing the response to settlements, which is the primary focus of this research. In general, the EMM has been observed to have good, strong numerical stability and reliable convergence (Sousamli, 2024). Although prior research has used it to accurately reproduce test results (Korswagen, 2024), the model has limitations in damage localization when subjected prevalently to shear loading, as cracks tend to be diffused rather than localized, as observed in the TSRCM (Schreppers et al., 2016; Sousamli, 2024).

While it is widely recognized that material models require calibration of material properties to reproduce the response of experimental tests accurately, it is important to note that altering the features of the incremental-iterative solution procedure and convergence criteria also influences the results. Different material models can be more or less sensitive to such variations. Prior to the analysis presented in this study, a preliminary sensitivity study was conducted to identify the features of the incremental-iterative solution procedure that best reproduce the results of the reference study. This sensitivity study was then extended to the macro-model using the EMM (with the f_tx,min = 0.30 N/mm³), and the results are presented in Figure 12.

The results indicate that the TSRCM model is significantly more sensitive to variations in the features of the incremental-iterative procedure compared to the EMM. For the EMM-based macro-model, most results are in good agreement, except for the model incorporating geometric non-linearities, which can severely impact numerical stability, and models utilizing the Regular Newton-Raphson method for iteration, where some non-convergent points are observed. In contrast, the TSRCM-based macro-model achieves convergence in all analyses, except for a single step in the model employing energy, force, and displacement norms during the settlement application. However, the results for TSRCM-based macro exhibit high sensitivity, resulting in irregular discrepancies.

While it is widely recognized that convergence in finite element analyses is a necessary condition for reliability, it is not sufficient on its own to guarantee the accuracy or appropriateness of the analyses. The sensitivity analysis results confirm this observation and further underscore the need to develop more accurate material models that are less sensitive to such variations. This advancement is crucial for improving the accuracy of structural response predictions. Although these aspects are not the primary focus of this study, further research is recommended to investigate their effects in greater detail.

While this study highlights the strengths and limitations of the modelling approaches, with particular emphasis on the response of structures subjected to differential settlements, the discussion that follows also considers the broader context of state-of-the-art comparisons between micro- and macro-modelling strategies. These comparisons span both quasi-static and dynamic actions and involve a range of constitutive laws and software packages.

In this study, the performance of the various models is initially evaluated based on computational time. As all the analyses do not show non-convergent steps, the CPU time provides a good indication of the computational costs of the model. The most computationally expensive model, as expected, is the simplified micro-models, due to the highest number of elements, followed by the macro-models with TSRCM and the ones with EMM. One argument supporting the use of macro-models is their computational efficiency compared to micro-models (Truong-Hong and Laefer, 2008; Murano et al., 2023; Sousamli, 2024). Although the EMM involves more parameters and thus a more complex definition of a constitutive model compared to the TSRCM, it is distinguished by a direct definition of the relationship between total stresses and strains along the material directions that align with the global axes. In contrast, the TSRCM requires a transformation from the principal to the global direction (Sousamli, 2024). As a result, the EMM outperforms the TSRCM in terms of computational efficiency.

In this study, the Total Strain Based macro-model uses a rotating crack approach to accurately reproduce the model in (Giardina et al., 2013b). Compared to the models with TSCM-fixed, models with TSCM-rotating have been observed to have a more consistent response with EMM-based models (Truong-Hong and Laefer, 2008), allowing a more equitable comparison.

While all the selected models can adequately reproduce the vertical displacements of the experimental 1/10th scaled masonry façade, the TSRCM-based macro-model better represents the horizontal displacements measured in the control point C, compared to the model with EMM and the simplified micro-model. The additional comparison between the experiment’s overall deformation and the model reveals that all the models underestimated the overall horizontal deformation of the façade, resulting in stiffer behaviour in the horizontal direction, particularly in the central part of the wall.

Prior research has observed how the selection of the constitutive law for masonry strongly influences the activation of different failure mechanisms and, hence, the response of the structure (Fusco et al., 2021; Fusco et al., 2022). In particular, EMM has been observed to provide a more accurate response of the damage response compared to the TSCM (Schreppers et al., 2016; Shabani and Kioumarsi, 2022; Ademović et al., 2024). In this study, the EMM models and the simplified micro-model correctly localize the damage mainly in the right part of the façade, whereas additional cracks open also in the left portion in the macro-model with TSRCM. The cracks in the TSRCM model are more localized, whereas in the EMM model, the cracks are more diffuse, making them less realistic compared to the experiment. The best agreement in terms of crack patterns with respect to the experiment is achieved with the simplified micro-model. Regarding the damage severity, the EMM models provide the most accurate prediction, while the TSRCM overestimates the damage, and the simplified micro-model underestimates it.

Based on the analyses presented in this study, it can be concluded that the choice of a modelling approach for structures undergoing settlements, including the constitutive relationship, depends on the specific purpose of the analysis: for instance, in this research, while the simplified micro-model more accurately represents the overall crack pattern, the severity of the damage is better captured by the (EMM-based) macro-models.

The non-linear finite element (FE) analysis of masonry structures is known to be influenced by the specific constitutive model adopted, modelling assumptions, and incremental-iterative settings. This study examines variations in incremental-iterative solutions and soil-structure interfaces while excluding factors such as mesh characteristics, load step size, and material parameter sensitivity, as these have been extensively addressed in previous research (Al-Chaar and Mehrabi, 2008; Mukherjee et al., 2011; Giardina et al., 2015; Bejarano-Urrego et al., 2018; Fusco et al., 2021; Fusco et al., 2022; Prosperi et al., 2023b; Korswagen, 2024; Sousamli, 2024). Overall, macro-modelling represents the most appropriate approach for practice-oriented engineering applications, offering a balanced trade-off between accuracy and computational efficiency when compared to micro-modelling (Murano et al., 2023; Sousamli, 2024).

The results indicate that none of the selected modelling approaches fully capture the behaviour of the experimental façade, underscoring the need for advancements in the modelling of masonry structures undergoing settlements. This may involve developing new constitutive relationships or exploring alternatives to traditional incremental-iterative non-linear analyses, such as the Sequentially Linear Analysis (SLA) (Rots et al., 2007). Future research should aim to advance constitutive models for masonry to enhance the accuracy of damage prediction and enable the integration of damage mechanisms with time-dependent behaviour such as creep and relaxation, as explored in studies like (Van Zijl et al., 2001; Roca et al., 2012). These effects are particularly relevant for capturing the long-term response of structures subjected to settlement over long terms, i.e., decades (Netzel, 2009; Giardina et al., 2013a). In addition, discrete element modelling offers a valuable approach for studying the response of URM façades to settlement (Ehresman et al., 2021), although it is generally associated with increased computational demands.

Additionally, evaluating different software packages can help identify potential discrepancies and advantages, contributing to improved modelling accuracy. In this study, the Engineering Masonry Model (EMM) and the Total Strain Rotating Crack Model (TSRCM) are selected as constitutive laws for the macro-models due to their commercial availability within the chosen software package and their widespread use in modelling unreinforced masonry (URM) structures (Sousamli, 2024). While this study focuses on a specific case, broader insights into damage prediction, displacement accuracy, and computational demand could be gained by exploring alternatives, including existing open-source or non-commercially available constitutive models.

The response of the FE models is compared against the experimental results of a 1/10th scaled masonry façade. In one-gravity (1 g) masonry testing, as adopted in the experiment considered here, appropriate scaling requirements must be addressed to minimise unintended scale effects (Laefer et al., 2011). In the considered experimental test, scale factors and additional loads have been applied to reproduce the real stress gradient, providing a practical alternative to more complex methods such as centrifuge testing (Giardina et al., 2012). Furthermore, (Giardina et al., 2015), explored the influence of the scale effect on the response of the TSRCM-based macro-model, comparing it against a true-scaled numerical model. The results of the two analyses showed substantial agreement in terms of displacement in the control points and maximum principal strains. However, further analyses are needed to fully understand the impact of scale, especially regarding damage severity, since crack width and length may be affected by the façade’s dimensions. Prior research (Liu et al., 2025) using scaled models has also applied the same scaling factor used for the structure to the crack widths associated with the various damage categories presented in Table 8. While this aspect does not influence the comparison between the models and the experimental benchmark, which represents the focus of this study, it may have implications for the interpretation of absolute damage levels and their correspondence to real-scale structural behaviour.

Finally, while this study focuses on one specific façade geometry and masonry material, future research should apply the proposed methodology to other scaled experiments, incorporating varied geometries, materials, and boundary conditions, to evaluate its general applicability and reliability across diverse structural scenarios.