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Groups of contiguous unreinforced stone masonry buildings are a common type of housing seen in old European downtowns. However, assessing their response to earthquakes poses several challenges to the analysts, especially when the housing units are laid out in compact configurations. In fact, in those circumstances a modeling technique that allows for the dynamic interaction of the units is required. The numerical study carried out in this paper makes use of a rigid block modeling approach implemented into in-house software tools to simulate the static behavior and dynamic response of an aggregate stone masonry building. Said approach is used to reproduce the results of bi-axial shake-table tests that were performed on a building prototype as part of the activities organized within the Adjacent Interacting Masonry Structures project, sponsored by the Seismology and Earthquake Engineering Research Infrastructure Alliance for Europe. The experimental mock-up consisted of two adjacent interacting units with matching layout but different height. Two rigid block models are used to investigate the seismic response of the mock-up: a 3D model allowing for the limit analysis of the building on one hand, and a 2D model allowing for the non-linear static pushover and time-history analysis on the other. The 3D model was built for the blind prediction of the test results, as part of a competition organized to test different modeling approaches that are nowadays available to the analysts. The 2D model was implemented once the experimental data were made available, to deepen the investigation by non-linear static pushover and time-history analysis. In both models, the stonework is idealized into an assemblage of rigid blocks interacting via no-tension frictional interfaces, and mathematical programming is utilized to solve the optimization problems associated to the different types of analysis. Differences between numerical and experimental failure mechanisms, base shears, peak ground accelerations, and displacement histories are discussed. Potentialities and limitations of the adopted rigid block models for limit, pushover and time-history analyses are pointed out on the basis of their comparisons with the experimental results.

In this paper, a novel numerical method for strucural analysis, called the Hybrid Discrete-Finite Element Method (HybriDFEM), is presented. In this method, a structure is modeled as an assembly of rigid blocks in contact. All the deformation is concentrated at the interfaces, which are modeled as series of distributed nonlinear multidirectional springs. The method shares similarities with the Discrete Element Methods (DEM) in its ability to account for contact interfaces and/or block deformability, and with the Applied Element Method (AEM) in the representation of interfaces as a series of normal and shear springs. However, it is close to the FEM in the way it is formulated, which offers the possibility to readily link both methods for potential hybrid applications. This paper focuses on the modeling of continuous and discontinuous frames with the HybriDFEM. It is shown how the model can do so with a nonlinear material model, and considering (or not) nonlinear geometric effects through large nodal displacements. Different nonlinear solution procedures implemented in HybriDFEM are demonstrated, such as load-control and various displacement-controlled methods. This model is able to simulate contacts between rigid or deformable units, an important feature when it comes to the modeling of, e.g., unreinforced masonry structures, with a reasonable computational cost and a formulation that is cast within the framework of the classical FEM.

TThe supports of out-of-plane loaded unreinforced masonry walls in buildings are subjected to a motion that is filtered and amplified by the building structure and, in some cases, can be significantly different from the ground motion. Moreover, because these walls span one or several storeys, their top and bottom supports are subjected to motions that differ in phase and amplitude. In state-of-the-art assessment procedures for the out-of-plane stability of masonry walls any effect of a relative support motion is neglected. The objective of this paper is to study the effect of the relative support motion on the response of out-of-plane loaded vertically-spanning unreinforced masonry walls. The acceleration capacity of the walls is investigated by means of a discrete element model representative of different wall configurations. A set of ground motions covering a wide range of peak ground acceleration and peak ground displacement is used as input to the simulations. The relative motion between the wall supports is included in the model in a systematic way: firstly, through a motion that is non-synchronous but of equal amplitude; secondly, through a motion that is synchronous but of different amplitude. The effect of the relative support motion is studied on different wall configurations where the elastic modulus of masonry, the wall height-to-thickness ratio, the wall effective thickness and the overburden at the top wall are varied. The study shows that, because of the relative support motion, the acceleration capacity of the walls can drop by 20% and, in the cases where the overburden is high, by more than 50%.

This paper describes quasi-static cyclic shear-compression tests on six plastered single-leaf stone masonry walls. Of the six walls, four are tested under different axial load ratios, performing a cyclic load history with two cycles per increasing drift demand. Three are tested under the same axial load ratio but are subjected to different load histories, namely, one monotonic loading, one cyclic loading with two cycles per drift level and one cyclic loading with one hundred cycles per drift level. Stiffness, force capacity and drift measures corresponding to six different limit states of the walls are extracted from each test and the effect of the axial load ratio and of the load history on these parameters is investigated. Changing the axial load ratio on the tested walls confirms findings from previous studies, according to which, with increasing axial load the wall stiffness and the force capacity increase while the ultimate drift decreases. Varying the number of cycles in the applied load history brings new findings, showing that all the drift measures of the walls, especially those corresponding to limit states attained in the post-peak regime, are very sensitive to the number of cycles applied while the stiffness and the force capacity are not. These trends are shown in the paper and compared against code and literature provisions for in-plane loaded walls. One face of each wall is plastered and the damage of the plaster is compared to the damage of the stone and mortar. This information is required when assessing the non-structural damage of the walls.

Under seismic actions, stone masonry buildings are prone to damage. To assess the severity of damaged masonry buildings and their failure modes, engineers connect these problems to surface crack features, such as the crack width and the extent of cracking. We aim to further these assessments in this study, wherein we propose using simple machine learning models to predict: 1) three ratios encoding the degradation of stiffness, strength, and displacement capacity of damaged rubble stone masonry piers as a function of the observed crack features and the applied axial load and shear span ratio; and 2) the pre-peak vs. post-peak regime, based on the crack features. When predicting the stiffness, force, and drift ratios, the prediction error is significantly reduced when the axial load and shear span ratio are included in the feature vector. Furthermore, when predicting the pre-peak vs. post-peak regime, simple machine learning models such as the k-nearest neighbor and the logistic regression result in remarkable accuracy. The obtained results have significant implications on the automated post-earthquake assessment of masonry buildings using image data. It is shown based on documented laboratory test data, that, by selecting proper crack features and incorporating information about the kinematic and static boundary conditions, even simple machine learning models can predict accurately the damage level caused to a rubble masonry pier. The three crack features used in this study are the maximum crack width, length density, and complexity dimension. The pipeline developed in this paper is general enough and is applicable to other masonry typologies and elements upon new evaluation of crack features and image data., Automation in Construction, 140, ISSN:0926-5805

Unreinforced masonry (URM) has been the most widespread construction material for centuries and is widely used in many countries. Accurate numerical models for this material are required when designing new structures, and, even more importantly, for assessing or renovating the existing stock. However, an accurate modelling of URM structures remains a challenge due to its highly non-linear mechanical behaviour involving, among other features, large out-of-plane displacements, the variability of the material properties, and the influence of construction details. The discrete element method (DEM), for which applications to masonry structures appeared in the 1990s, has gained in popularity due to its capacity to accurately capture masonry-specific phenomena; in particular, as it represents the structure as an assembly of distinct blocks, deformations are concentrated at the interfaces with low or zero tension strength, and the effect of large displacements. Unfortunately, the majority of computer programs for structural analysis based on the DEM are proprietary software, which limits its spread among practitioners and researchers. The aim of this work is to evaluate the open-source software package LMGC90, which implements the Non-Smooth Contact Dynamics Method, as a tool for modelling masonry structures.

A correction to this paper has been published: https://doi.org/10.1617/s11527-021-01653-6

Displacement-based methods contained in recent standards for seismic safety assessment require the determination of the full nonlinear pushover curve for local failure mechanisms in historic masonry structures. This curve should reflect both the initial elastic behavior and the rigid body behavior after the activation of rocking. In this work, a rigid block model is proposed for the displacement-based seismic assessment of local collapse mechanisms of these structures. Masonry is modeled as an assemblage of two-dimensional rigid blocks in contact through frictional interfaces. Two types of contact models are formulated to capture, respectively, the pre and postpeak branches of the pushover curve: a unilateral elastic contact model, capturing the initial nonlinear behavior up to the force capacity of the structure, corresponding to the activation of the collapse mechanism, and a rigid contact model with finite friction and compressive strength, which describes the rigid-body rocking behavior up to the attainment of the displacement capacity of the structure. Tension-only elements are also implemented to model strengthening interventions with tie-rods. The contact problems associated with the elastic and rigid contact models are formulated using mathematical programming. For both models, a sequential solution procedure is implemented to capture the variation of the load multiplier with the increasing deformation of the structure (P–Δ effect). The accuracy of the modeling approach in reproducing the pushover curve of masonry panels subjected to horizontal seismic loads is evaluated on selected case studies. The solution is first tested against hand calculations, existing analytical models, and distinct element simulations. Then, comparisons against experimental tests follow. As a final application, the failure mechanism and pushover curve of a triumphal masonry arch are predicted by the model and its seismic assessment is performed according to codified force- and displacement-based methods, demonstrating the adequacy of the proposed tool for practice.

Cracks are the most important source of information about the damage that occurs to unreinforced masonry piers under seismic actions. To predict the structural state of unreinforced masonry piers after an earthquake, research models have been developed to quantify important features of crack patterns. One of the most used crack features is the width, but this can be influenced by several parameters such as the axial load ratio, the shear span ratio, and the loading protocol, which have not been fully studied in previous research studies. In this study, we use experimental data to investigate the evolution of cracking in stone masonry piers during the application of cyclic shear–compression loading. The data consists of gray-scale images taken during quasi-static shear–compression tests performed on six plastered rubble-stone masonry walls subjected to constant axial force and cycles of increasing drift demand. Through the combined use of digital image correlation and a pre-trained deep learning model, crack pixels are identified, post-processed, and quantified based on their width. The dependency of the crack width on the axial load ratio, the shear span ratio, and the loading protocol at the peak force and ultimate drift limit states of the piers is clarified by a displacement vector field analysis, histogram of the crack width, and the concentration of deformation in the cracks. We show that, as opposed to flexural cracks, diagonal shear cracks do not fully close when moving from the applied drift demand to the residual drift measured upon removal of the lateral load. Furthermore, we provide the maximum residual crack width at peak force and ultimate drift limit states. This study will improve the decision making abilities of future models used to quantify earthquake-induced damage to stone masonry buildings.

Masonry walls are bulky and heavy and have therefore the potential to act naturally as a protective system to blasts. Yet, they are known to have a limited flexural and torsional capacity, particularly when unreinforced. When exposed to shockwaves, they experience out-of-plane failure mechanisms which may affect the overall stability of the building and engender flying debris inside the building. The out-of-plane response of unreinforced masonry walls to blasts depends on many factors characterizing both the wall and blast action, making any sort of prediction difficult. In this context, experimental tests and numerical models become key tools that can be used to study the wall’s response on a case-by-case basis. This review covers the major experimental and numerical approaches to assess the out-of-plane response of unreinforced masonry walls subjected to blasts. A methodological appraisal is used for the test methods, focusing on the preparation of the test items and test setup, the boundary conditions and failure mechanisms investigated, as well as the commonly employed measurement techniques. The survey on the modelling approaches includes key topics such as level of detail and cost, and reports strategies to model the wall and blast scenario. The review provides a thematic analysis of the available literature, aimed to assist the analyst in selecting a suitable tool for the investigation of masonry in the field of blast engineering. Furthermore, the findings presented herein can support amendments of existing codes and guidelines pertaining to the design of protective masonry structures.

Summary An analytical model describing the flexural response of vertically spanning out-of-plane loaded unreinforced masonry walls is presented in this paper. The model is based on the second-order Euler-Bernoulli beam theory and captures important characteristics of the out-of-plane response of masonry walls that have been observed in experimental tests and from numerical studies but for which an analytical solution was still lacking: the onset and the evolution of cracking, the peak strength of the out-of-plane loaded walls, and the softening of the response due to P−Δ effects. The model is validated against experimental results, and the comparison shows that the model captures both the prepeak and postpeak response of the walls. From the analytical model of the force-displacement curve, a formula for the maximum out-of-plane strength of the walls is derived, which can be directly applied in engineering practice.

Out-of-plane failure of masonry walls is often responsible for the partial collapse of unreinforced masonry structures. Modeling the out-of-plane response of these walls is therefore key in the assessment of existing buildings. The paper presents a new trilinear model describing the force-displacement response of vertically spanning unreinforced masonry walls subjected to out-of-plane loading. Different factors that affect the response of the walls are captured by the model: the support conditions, the level of applied axial load, the slenderness ratio, and the deformability of the wall. The model is validated against experimental results from shake table tests. The force and displacement parameters of the model are described by analytical expressions that are derived from a mechanical model previously developed for unreinforced masonry. They offer an alternative to existing trilinear models in which corner displacements are mainly defined by empirical relationships.

When designing unreinforced masonry buildings, the wall stiffness and, consequently, the masonry elastic and shear modulusEandGare essential parameters. Current codes provide empirical estimates of the masonry elastic modulus and a ratio between the shear and elastic modulus,G/E. This ratio, commonly taken as 0.4, is not based on scientific evidence and there appears to be no consensus concerning its value and influencing parameters, meaning that current code standards might not accurately portray the shear deformations of masonry elements. To give the choice of theG/Eratio a theoretical foundation, this paper presents closed-form expressions for theG/Eratio of running-bond masonry that capture the effects of finite joint thickness, finite wall thickness and orthotropic block properties. Based on the geometry of blocks and joints as well as their elastic parameters, a validation of the developed expression using 3D finite element analyses shows good performance. For modern masonry typologies with hollow clay bricks, aG/Eratio of 0.20–0.25 is obtained. For historical masonry typologies, such as dry stacked or mortared stone masonry, as well as solid clay brick masonry, ratios between 0.30 and 0.40 are computed.

Summary The macroscopic behavior of materials is affected by their inner micro-structure. Elementary considerations based on the arrangement, and the physical and mechanical features of the micro-structure may lead to the formulation of elastoplastic constitutive laws, involving hardening/softening mechanisms and non-associative properties. In order to model the non-linear behavior of micro-structured materials, the classical theory of time-independent multisurface plasticity is herein extended to Cosserat continua. The account for plastic relative strains and curvatures is made by means of a robust quadratic-convergent projection algorithm, specifically formulated for non-associative and hardening/softening plasticity. Some important limitations of the classical implementation of the algorithm for multisurface plasticity prevent its application for any plastic surfaces and loading conditions. These limitations are addressed in this paper, and a robust solution strategy based on the singular value decomposition technique is proposed. The projection algorithm is then implemented into a finite element formulation for Cosserat continua. A specific finite element is considered, developed for micropolar plates. The element is validated through illustrative examples and applications, showing able performance. Copyright © 2016 John Wiley & Sons, Ltd.

There is limited available research on rubble stone masonry walls, which are vulnerable under seismic loading. This paper presents an experimental campaign of cyclic shear compression tests on six large-scale walls of this topology. The effect of the axial load and shear span ratio on the wall behaviour, notably on the wall stiffness, strength, and drift capacity, was investigated. It was found that the drift at crack onset is only half of that in previous campaigns on stone masonry walls, likely because one face of each wall was plastered, making the damage more visible. Additionally, splitting cracks opening between the wall leaves appear to play a key role in the collapse mechanism. Finally, testing the walls up to the loss of their axial-load-bearing capacity provides new input for the collapse risk analysis of stone masonry buildings.

International audience; In the frame of Cosserat continuum theory, an upscaling procedure for the assessment of the in-plane strength domain of discrete media is developed. The procedure is the extension to the Cosserat continuum of a procedure initially formulated for the Cauchy continuum, based on the kinematic approach of limit analysis and the classical homogenisation theory. The extension to the Cosserat continuum is made in order to take into account the effect of particles' rotation on the strength of the discrete medium. The procedure is illustrated with regard to periodic assemblies of blocks in contact and is then generalised to the whole class of discrete periodic media with particles of the same type. The case of masonry is considered as an application. Strength criteria of columns and walls are formulated in terms of non-symmetric stresses and in-plane couples. The procedure allows to show how the in-plane strength of the medium is reduced as a result of particles' rotation.

A displacement and rotation-based dynamic finite element formulation for Cosserat plates is discussed in detail in this paper. Special attention is given to the validation of the element through adequate benchmarks and patch tests. The choice of the interpolation functions and the order of integration of the stiffness and the mass matrices are extensively argued. The possibility of local system deficiencies is explored, and a shear locking investigation specifically elaborated for Cosserat plates is carried out. It is shown how the present formulation has interesting computational properties as compared to a classical continuum-based formulation and how it can provide suitable results despite the use of reduced integration. An example of application of the finite element is given, in which the natural frequencies of a masonry panel modelled by means of discrete elements are computed and compared with the finite elements solution.

International audience; A Finite Elements formulation previously developed for Cosserat elastic plates, has been extended herein to the elastoplastic framework. Material non-linearities are taken into account through the implementation of a backward-Euler closest-point-projection algorithm, for which the definition of non-smooth yield loci and non-associated plastic potentials and evolution laws is made possible. An existing homogenized elastic constitutive model and a set of yield criteria for the out-of-plane behaviour of block-masonry are implemented in the code and their validity is discussed based on the comparison with Discrete Elements simulations. The comparison is carried out in both the static and the dynamic regime.

Reliable methods for detecting pixels that represent cracks from laboratory images taken for digital image correlation (DIC) are required for two main reasons. Firstly, the segmented crack maps are used as an input for some DIC methods that are based on discontinuous fields. Secondly, detected crack patterns can serve as inputs for predictive empirical models to obtain the level of damage to a body. The aim of this paper is to compare the performance of two approaches for crack segmentation on grayscale images acquired from two experimental campaigns on stone masonry walls. In the first approach, a threshold is applied to the maximum principal strain map calculated using post-processed DIC results. In the second approach, a deep convolutional neural network is used. The two methods are compared in terms of standard segmentation criteria, namely precision, dice coefficient and sensitivity. It is shown that the precision and dice coefficient obtained from the deep learning approach are much higher than those obtained from the threshold method (by almost 47% and 34%, respectively). However, the sensitivity computed from the deep learning method is slightly (~4%) lower than the threshold method. These results show that the deep learning method can better preserve the geometry of detected crack patterns, and the prediction in terms of pixels belonging to a crack is finally more accurate than the threshold method.

International audience; The effect of the dilatancy of masonry interfaces and of the size of the building blocks on the strength of masonry structures is quantified herein. The study focuses mainly on out-of-plane loadings, which can appear due to various factors such as wind, earthquakes or explosions. The analysis is performed using the Discrete Element Method (DEM), which allows to access directly various micro-mechanical parameters, such as the joints dilatancy angle and the size of the building blocks. Detailed DEM numerical models of existing experimental configurations are presented. The numerical results are first compared and validated towards the experimental observations and then they are used to derive qualitative and quantitative conclusions regarding the effects of joints dilatancy and blocks size. It is shown that dilatancy plays an important role on the overall strength of masonry even under low confinement. The size of the blocks is also an important parameter that needs to be considered in the modeling of masonry structures.