Simulating the seismic behaviour of unreinforced masonry (URM) is challenging due to large deformations and severe damage. Capturing this highly nonlinear response requires advanced numerical modelling strategies that represent block separation, debonding, friction, and impact. Discontinuum-based modelling strategies, such as the Distinct Element Method (DEM), are well suited, as they explicitly represent bond failure and damage progression from cracking to collapse. DEM relies on the explicit time integration scheme of motion equations; hence, the choice of the damping scheme becomes critical. Typically, mass-proportional damping is used in dynamic analysis, often without complementing it with stiffness-proportional damping which requires unpractical reduction of the time steps to ensure numerical stability. Yet relying solely on mass-proportional damping can overdamp low frequencies and underdamp high frequencies. This study implements and validates an alternative damping approach, Maxwell damping, where multiple spring-dashpot elements are introduced at unit-mortar interfaces within a simplified micro-model. This work introduces an optimization algorithm to tune the Maxwell elements without heuristics, targeting near-uniform damping over a broad frequency range. Effectiveness is assessed against shake-table tests on a full-scale cross-vault URM specimen. Predicted displacements, accelerations, damage evolution, and computational efficiency is compared with mass-proportional and zero-viscous damping models. This study investigates Maxwell damping as a practical relaxation scheme for the seismic analysis of complex masonry systems using DEM, building on prior formulations in the literature and extending them to the present modelling and validation context.

This paper presents a cyclic joint constitutive model within a Distinct Element Method framework to simulate the in-plane response of unreinforced masonry structures. The model combines multi-surface failure criteria, including tensile cut-off, Coulomb friction, and an elliptical compression cap. It incorporates exponential softening, a unified damage scalar for stiffness degradation, and a hardening–softening law for compression. Shear-induced dilatancy is captured via an uplift-correction mechanism with an exponential dilatancy-decay law, while stiffness degradation governs energy dissipation. The model is validated at both material and structural scales. Material-level simulations of cyclic compression and shear tests show close agreement with experimental data. Structural-scale validation on full-height calcium-silicate walls under combined compression and cyclic lateral loading demonstrates the ability to reproduce rocking-dominated, shear-dominated, and hybrid failure mechanisms. The model successfully replicated global hysteretic force–drift loops, capturing stiffness decay and energy dissipation, as well as local failures like cracking, sliding, and toe crushing. The model also reproduced the drift-dependent transition from rocking to friction-controlled sliding, a key mechanism for earthquake assessment. By integrating these features into a single, efficient framework, the proposed constitutive model provides a robust tool for evaluating seismic performance and conserving heritage.

An existing interface material model for quasi-brittle fracture, originally developed within the Discrete Element Method framework, is implemented and enhanced for use in implicit Finite Element analyses of unreinforced masonry structures. The model captures mixed-mode fracture in tension-shear and combines cohesion with Coulomb friction in compression-shear. To address convergence issues arising when loading–unloading takes place, due to a discontinuity in the traction–separation relation, a regularization of the frictional contribution is proposed. A new model parameter is introduced and a calibration procedure to ensure numerical robustness and objectivity is presented. Furthermore, the consistent tangent stiffness matrix is derived to improve convergence in full-scale simulations. The improved model is applied within a simplified micromodelling approach to simulate the in-plane cyclic response of 2D masonry structures, including a shear wall and a spandrel subjected to a combination of horizontal and vertical actions. The results demonstrate that the model accurately reproduces key aspects of masonry behaviour, including stiffness degradation, energy dissipation, and crack patterns, while maintaining robustness and efficiency in complex cyclic loading scenarios.

The out-of-plane (OOP) dynamic behavior of unreinforced masonry (URM) gable walls was investigated in this paper using a high-fidelity block-based numerical modeling approach, building on the participation of the authors in the ERIES SUPREME blind prediction competition. In this paper, the numerical models developed for the competition were updated based on the experimental data published after the competition to further improve accuracy. The improvement was obtained by slight recalibration of mortar joint tensile strength and friction between the walls and the loading set-up. The updated models were also adopted to simulate a third wall originally excluded from the competition. The models were then used to complement the experimental campaign with additional configurations in a parametric study. Specifically, the influence of roof-wall connections and pre-existing damage on the performance of the gable walls were examined to address gaps identified in both experimental and numerical studies of the past. Stronger roof-wall connections, while improving global stability and increasing wall OOP strength in the static regime by up to 140%, led to collapse at dynamic loading intensities reduced by an average of 28% and up to a maximum of 57%. This early collapse resulted from the transfer of larger dynamic demands to the gable walls. This higher demand transfer also caused earlier damage initiation and considerable changes in collapse mechanisms, effects not captured by static analysis, highlighting the uncertainties governing dynamic behavior and the need for robust methodologies to address them. Finally, light pre-damage, modelled in this study as a crack at the base of the walls, had only a minor influence on failure mechanisms and OOP resistance.

This paper investigates settlement-induced damages in unreinforced masonry (URM) walls using a high-fidelity block-based numerical modeling approach. The research aims to address gaps in the understanding of settlement effects on URM walls with flanges, particularly with respect to their seismic out-of-plane (OOP) behavior. A parametric study is conducted on four wall specimens with varying geometries, boundary conditions, and settlement scenarios, including symmetric and asymmetric patterns. The numerical models are developed via a high-fidelity block-based finite element method that simulates masonry using expanded blocks connected by zero-thickness joints, allowing for detailed analysis of cracking patterns and damage mechanisms. Different damage states, from no visible cracks to near-collapse conditions, are identified in the response of the walls and are used as initial conditions for subsequent monotonic static pushover OOP loading. The results highlight the significant influence of settlement-induced pre-damages on the OOP response of URM walls, with varying degrees of impact observed across different specimen configurations. The findings underscore the importance of considering even “light” settlement-induced pre-damages when assessing the seismic performance of URM structures, particularly in subsidence-prone regions. Under symmetric hogging, such pre-damage level can reduce OOP stiffness and peak strength by up to 41% and 20%, respectively. This study lays the groundwork for future investigations into the seismic behavior of pre-damaged masonry structures under dynamic loading and offers valuable insights for the development of more accurate assessment and mitigation strategies for buildings subjected to settlement deformations.

This article presents a dataset from an experimental campaign investigating the out-of-plane (OOP) seismic response of unreinforced masonry (URM) gables in existing buildings. Addressing a critical gap in published research, the dataset provides novel experimental data on the incremental dynamic OOP behavior of three URM gables tested under seismic loading until full collapse. All three gables were nominally identical but differed in their interaction with the supporting roof structure. This interaction was experimentally reproduced by imposing differential motions at the top of the gables, which were either linearly amplified or both amplified and phase-shifted relative to the motion at the base. This approach ensured idealized and numerically replicable boundary conditions, making the dataset an ideal benchmark for refining existing and developing new modeling approaches for URM structures. The dataset includes measured and calculated acceleration, displacement, and force time histories. Beyond supporting the validation and development of numerical models, it can also contribute to improving guidelines for the out-of-plane seismic assessment of URM gables and is openly available for further research and engineering applications.

This paper presents a comprehensive review of the effects of interactions between in-plane (IP) and out-of-plane (OOP) behaviors, referred to as IP-OOP interactions, on the seismic behavior of framed unreinforced masonry structures, consolidating findings from experimental, numerical, and analytical studies available in the literature. While masonry structures are highly vulnerable to seismic loading and undergo multi-directional seismic actions, most existing research focuses on their response to unidirectional forces, overlooking the complex interaction effects observed during real earthquakes. Moreover, although design and assessment standards acknowledge these interactions, they offer limited prescriptive guidance. The literature predominantly addresses the impact of IP pre-damage on OOP strength and stability (IP/OOP interaction), with comparatively fewer studies examining the reverse scenario, i.e., OOP pre-loading affecting IP resistance (OOP/IP interaction). Experimental data remains scarce, particularly for multi-bay frames and walls with openings, limiting the generalizability of current findings. Numerical simulations have significantly advanced the understanding of these interactions, yet their reliability relies on proper calibration against benchmark experiments, which are still limited in number. Among the most influential parameters affecting the effect of IP pre-damage on the OOP response, the height-to-thickness slenderness ratio plays a dominant role. Slender walls are especially prone to severe OOP strength degradation due to reduced arching action and increased instability. The length-to-height aspect ratio also influences failure modes under IP/OOP interaction, particularly in short walls where horizontal arching action reduces. Other critical factors, such as masonry material properties, boundary conditions, and frame stiffness, have been identified, but their effects remain less systematically studied. Analytical approaches have primarily focused on IP/OOP interaction effects. However, existing equations are often derived from limited datasets, restricting their predictive capabilities. The equation widely adopted in seismic guidelines has been shown to overestimate OOP strength reduction, underscoring the need for more refined models that incorporate broader experimental and numerical data. Future research should address these gaps by expanding experimental campaigns, enhancing numerical methodologies, and refining analytical frameworks to better represent real-world conditions.

This study deals with the high-fidelity block-based finite element simulation of dynamic out-of-plane (OOP) responses of unreinforced masonry (URM) walls, explicitly focusing on two-way bending behaviors under seismic loads, which is a common critical failure mode in real-world masonry structures. While experimental shake-table tests provide valuable insights into these behaviors, their high costs, complexity, and limited scalability highlight the need for advanced numerical modeling approaches. A state-of-the-art block-based finite element modeling strategy that conceives masonry as an assemblage of 3D damaging blocks interacting via contact-based cohesive-frictional zero-thickness interfaces, previously proposed for simulating cyclic quasi-static and dynamic one-way bending tests, is here extended for the first time to the simulation of incremental dynamic shake-table tests on OOP two-way spanning URM full-scale walls, subjected to a sequence of dynamic loads. The numerical models track the reference experimental behaviors with high accuracy in terms of collapse onset, failure mechanism, experienced acceleration and displacements, and hysteretic response. The effects of variations in mechanical properties, boundary conditions, and damping on the dynamic response are explored in a sensitivity study. The results indicate that slight changes in these parameters can lead to considerable differences in outcomes. This highlights the chaotic nature of the dynamic response of masonry walls, especially in near-collapse conditions, which makes probabilistic approaches more suitable for predicting masonry OOP dynamics. The proposed numerical methodology appears compatible with statistical frameworks, given the limited costs with respect to experimental tests, and it extends knowledge beyond physical experiments.

Low-rise masonry buildings worldwide frequently feature unreinforced masonry (URM) walls coupled with various pitched roof configurations supported by masonry gables. Past earthquakes have highlighted the vulnerability of these components to out-of-plane seismic loads due to their high slenderness, insufficient roof connections, and exposure to amplified accelerations while being subjected to minimal overburden due to their location at the upper part of buildings. This study presents key insights from the experimental campaign of the ERIES-SUPREME project, aimed at enhancing the understanding of the out-of-plane seismic behavior of masonry gables. Incremental dynamic tests were performed on three full-scale URM gables, simulating both induced and tectonic earthquake scenarios until collapse, using two shake tables. Differential motions at the top and bottom tables reproduced the interaction of the gables with three different roof diaphragm configurations, each introducing a unique filtering effect on the seismic input. The outcomes of the experiments can be used for refining existing numerical modelling strategies as well as contribute to developing improved tools for the seismic assessment of URM gables.