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The paper investigates how the deviation of the centreline from the semi-circle influences the mechanical behaviour of gothic masonry vaults. The geometry of the vaults is quantified by a scalar parameter. The present work deals with the force-transmitting system, crack opening magnitude and pattern. Furthermore, the magnitude of the so-called threshold support displacement and the magnitude of the support displacement at first failure is also analysed. Masonry structures consist of separate blocks, each of which can move independently from each other, which makes the calculation process rather complicated. The Discrete Element Method (henceforth DEM) is used to solve the highly nonlinear problem. In the chosen method the equations of motion are solved by the central difference scheme. Approximate boundary conditions are applied, which assume symmetry in the analysed structures. The main findings of the work are: barrel vaults with pointed shape become statically determinate at a higher support displacement level than barrel vaults with semi-circular centreline. In the case of large deviation from the semi-circle or at large support displacement level shear failures can appear in gothic masonry barrel and cross vaults, which could decrease the magnitude of feasible horizontal support displacement.

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... This approach, which is computationally advantageous with respect to FEM strategies, has been used to study unreinforced structures and masonry vaults. Some remarkable examples of the capacities of DEM-based strategies can be found in [8,18]. ...

... Strengtheningtestnumerical model Huerta [21] Galileo was wrong: The geometrical design of masonry arches 24 Structural analysis Milani [22] Upper bound sequential linear programming mesh adaptation scheme for collapse analysis of masonry vaults 18 Strengtheningnumerical model Theodossopoulos et al. [11] Assessment of the structural response of masonry cross vaults 18 Failuretestnumerical model Theodossopoulos et al. [10] Case study of the failure of a cross vault: Church of Holyrood Abbey 16 Failuretestnumerical model Chiozzi et al. [17] Fast kinematic limit analysis of FRP-reinforced masonry vaults. II: Numerical simulations 14 Strengtheningnumerical model Coccia and Como [23] Minimum thrust of rounded cross vaults 11 Structural analysis external buttresses. ...

... One of the first attempts to use a DEM approach to model cross vaults can be found in [8,18]. Lengyel et al. [8] clarified the role played by ribs in ribbed vaults and compared their behavior with unribbed vaults. ...

Masonry cross vaults, which began to be built in Europe in the times of the Roman Empire and reached heights of extraordinary splendor during the Gothic Period, are part of an impressive architectural heritage, which stand in need of evaluation, maintenance and repairs or renovation. Nowadays, the requirements of preservation and conservation of the historical heritage call for a strategy oriented on the description of the structural behaviour from different characteristic perspectives without neglecting the functional, architectonic and structural features. The present paper is intended to discuss the different research approaches adopted by several authors during the past and to show how numerical simulations in conjunction with experimental studies can provide the tools for the quantification of the damages produced by a variety of different critical events. To this scope, this paper includes an extensive and somewhat ambitious review, dealing with a wide variety of aspects, such as: (a) a bibliometric study, (b) classification of the types of cross vaults, (c) structural behavior, (d) types of failures and damage, (e) numerical simulation, (f) experimental testing, (g) monitoring, and (h) retrofitting. The review is expected to be of great interest for architects and engineers working in the field of evaluating and retrofitting the architectural heritage, as well as to those researchers who would find it useful to have at hand in a single document the most important advances made in the field of masonry cross vaults in recent years.

... The study addresses the force-transmitting system of the masonry shell (web), the internal forces of the ribs, and the general characteristics of the motion. The characteristics of the web and the rib are defined in Lengyel (2017) and Lengyel and Bagi (2015). ...

... Further to the diagonal load-bearing direction, a highly compressed zone evolves around the crown in the x-direction. A similar phenomenon was found for static load, see Lengyel (2017) and Lengyel and Bagi (2015) For groin vault, the compression forces in the web significantly increased, while for ribbed vault, the increase was more focused in the ribs. In addition to the magnitude of the forces, a closer look at the principal directions reveals that some blocks are directly supported by the diagonal ribs (the direction of the rib and the principal direction on the masonry web are almost perpendicular to each other). ...

Historically, the first groin vaults were built without ribs, but ribs later appeared for various reasons. Some ribs have only an aesthetic role, while others have a clear structural role in stiffening the groin. The stiffening effect on the static behavior is currently of wide interest, as the preservation of architectural heritage has become important. While static behavior has various explanations, the discrete spatial structure of cross vaults allowed for only approximate solutions to dynamic problems, such as seismic load response. In this article, a 3D time-history analysis is carried out with a commercial discrete element code. Following the motion of elastic discrete blocks in the structure allows the continuation of nonlinear effects from the rocking and sliding of the blocks. The analysis only addresses the most presumably dangerous direction of ground acceleration. The time-history analysis uses a synthetic accelerogram; a real accelerogram is scaled in the frequency domain to fit to a standard response spectrum. The influence of the supporting walls is handled approximately by considering them to be an SDoF system, which affects the signal of the support acceleration.

... Using buildings safely should be subject to obligatory periodic inspections of their technical condition and structure conditions. Such an approach allows for assessing their condition throughout their life cycle, planning the necessary renovation works, and identifying the negative impact of neighboring facilities, biological and chemical co-development, and weather conditions [1][2][3]. During these inspections, the method of visual construction assessment is used in the first place, which allows for diagnosing only the obvious, visible signs of damage or failure. ...

The article presents an approach to monitoring the structure’s condition with two measurement methods: the SHM-X crackmeter and the classic geodetic method of determining displacements, supplemented with additional information on the condition of the external environment obtained from thermal images. The study aimed to propose an approach combining geodetic and non-geodetic methods of assessing the condition of a structure and its effectiveness in practical application. The research facility is a public utility building of the Bydgoszcz University of Technology with a reinforced concrete structure. Objects of this type require periodic tests of their constancy. Interpreting the test results and identifying possible dangerous states that may indicate the risk of a construction failure is extremely important. The results presented in the article are an extension of the previous ones, in which several factors that could have a destructive effect on the structure were excluded. Observation of the object showed that only the reinforced construction plate is deformed. The only factor influencing the change in structure geometry is thermal changes. As part of the tests in places where cracks were noticed, the SHM-X crackmeter was used to measure the cracks’ opening. In the geodetic research, measurements of the measurement and control network displacement were carried out, in which the TDRA6000 laser station measurement technology was used. The control points were also placed in places where the width of the cracks was directly observed. The proposed approach, with the applied calculation scheme and supplementing the information with the temperature measurement with thermal images, showed the submillimeter accuracy of the determined 3D displacements of the controlled points. Additionally, the parallel application of these methods gives a complete picture of changes in the structure elements, in which signs of destruction appear under the influence of stress.

... Focusing the attention on the numerical procedures, among the others, the discrete element (DEM) and the finite element methods (FEM) have proven suitable to accurately reproduce the response of masonry curved structures, like arches, vaults, and domes. Both DEM (Dell'Endice et al., 2021;Lengyel, 2017;Sarhosis et al., 2019) and FEM (Atamturktur et al., 2012;Briccola and Bruggi, 2019) permit to model complex geometries and arrangements, and nonlinear constitutive response. At the same time, these allow to perform both static and dynamic analyses. ...

... After Heyman's model, the upper bound and the 36 lower bound methods or, alternatively, the limit equilibrium state analysis have been 37 largely used. These methods were applied with several purposes, as the definition of the 38 minimum thickness and/or the bearing capacity under vertical and lateral loads for 39 different shapes of arches [4][5][6][7][8][9][10][11], the study of arches and vaults behaviour by using the 40 thrust network analysis [12][13][14][15] or advanced numerical methods [16][17][18][19], the analysis of 41 the strengthening effects through innovative materials [20][21][22][23][24][25] and many others. 42 ...

This work is aimed at evaluating the collapse displacement of masonry arch subjected to spreading supports. This is achieved through a general application of the virtual works principle. The problem is described in a finite displacements formulation and investigated with a probabilistic approach, also considering the effects of the geometrical uncertainties. This aspect is related to the imperfections of the voussoirs, which affect the structural shape. The comparison between the numerical and experimental results, derived both by the literature and laboratory tests, confirms that the geometrical irregularities can significantly affect the results obtained on the nominal structural geometry. Moreover, the disagreement observed in the experimental tests is explained.

... Limit analysis is the best alternative to nonlinear incremental analyses for the evaluation of the ultimate load-bearing capacity of masonry vaults. In the last years different methods have been applied to the study of Gothic vaults: among others, a limit analysis based on a lower-bound approach, which is the so-called thrust network method [5,6], and techniques based on the discrete element method [42]. Other approaches based on the upper-bound theorem have been adopted in the study of seismic vulnerability of cross-vaults [31]. ...

The aim of this paper is to present an in-depth numerical investigation on the statics of historical masonry stellar vaults, a special class of masonry ribbed vaults whose three-dimensional geometry features a star-shaped projection on the horizontal plane. In particular, the mechanical behavior of the masonry stellar vault belonging to the church of Santa Maria del Monte in Cagliari (Italy) is analyzed and illustrated as an especially meaningful case study. This church, which was built during the second half of the sixteenth century, is a beautiful example of Gothic-Catalan style, and its ribbed stellar vault is one of the most representative of this type in the town of Cagliari. The geometric outline of the vault has been obtained through laser scanning techniques and a procedure of reverse engineering. Starting from a three-dimensional representation of its geometry, the ultimate load-bearing capacity of the stellar vault can be accurately estimated through a recently developed, NURBS-based upper-bound limit analysis scheme. A comparison with incremental nonlinear analyses carried out with the commercial finite element code DIANA is presented. Furthermore, the paper also includes a sensitivity study aimed at investigating the role of ribs on the ultimate load-bearing capacity of the structure.

... The parametric study on the arch curvature is performed for both rigid and deformable blocky systems, to provide a comparison of the baselines. The material and contact properties used in this study are given in Table 1, and they follow the suggestions provided in relevant literature (Lengyel 2017, Pulatsu et al. 2016). ...

The majority of historical masonry structures include arches and vaults, constructed with or without (dry-joint) any mortar. This paper focuses on dry-joint masonry, because it is common all around the world among architectural heritage. Furthermore, even if there was a mortar in the original construction, it typically suffers from deterioration over its lifetime, often causing total loss of mortar in many of the joints. Due to large horizontal thrust that can be produced, depending on their geometry, arches are typically supported by heavy buttresses. These structures tend to be difficult to model due to their nonlinear nature and inherent discontinuity, which makes it challenging to evaluate their stability. In that context, it is necessary to have realistic numerical models to better diagnose their structural behaviour in a seismic event and, ultimately, to perform only necessary and beneficial interventions. The main goal of this paper is to assess the seismic performance of various dry-joint arch forms with different masonry pier types (i.e. monolithic and regularly coursed) subjected to incrementally increasing lateral loads proportional to the mass (pushover). To achieve this goal, a parametric study is performed on arch curvature and pier morphology. Moreover, the influence of steel tie-rod reinforcement is also examined on the proposed masonry models. These complex masonry arch systems can be simulated with discrete element modeling (DEM) approach. In this research, a commercial three-dimensional discrete element code, 3DEC, is used; in which masonry units are modeled as distinct blocks with zero tensile strength at their joints. The results reveal that pointed arches provide better seismic resistance than the circular arch form. Furthermore, implemented steel tie-rods yield significant increase in stability for the arch-pier structures, which is quantified on different arch curvatures.

... Finally, block-based approaches consider the structure as an assembly of separate blocks which can interact through specific laws. This approach represents the most accurate analysis tool (Caliò, Cannizzaro, and Marletta 2010;Foti, Vacca, and Facchini 2018;Lengyel 2017;Lengyel and Bagi 2015;Lengyel and Németh 2018;McInerney and DeJong 2014;Milani et al. 2016;Pantò et al. 2017), even if the description of a historic vault block-by-block is still challenging and computationally demanding. In this framework, software tools based on limit analysis of rigid blocks have been lately developed for the 2D analysis of masonry bridges (LimitState: RING software originally developed in Gilbert and Melbourne (1994), and for the limit analysis of masonry structures modeled as assemblages of 3D rigid blocks subjected to live loads and settlements (LiABlock_3D; Cascini, Gagliardo, and Portioli 2018). ...

A principal reason of damage in historic masonry vaults consists in relative displacements of the vaults’ abutments. Excluding the case of seismic-induced damage, cracks are often produced by differential settlements generated by the lateral wall instability or soil degradation (e.g., due to stress concentrations, non-uniform soil stratigraphy, flooding phenomena etc.). When dealing with historic vaults, the effects of long-term deformation processes cannot often be linked directly to causes, which may also be unknown. In this article, the effects of differential settlements on historic masonry barrel vaults are investigated. An efficient 3D contact-based model was developed to reproduce experiments on a scaled pointed barrel vault (representative of a typology of late-medieval barrel vaults in Scotland) under non-uniform differential settlement. First, the numerical model is used to simulate the experimental campaign, achieving good agreement in terms of crack pattern (longitudinal shear) and transverse-longitudinal deformation profiles. Then, further analyses are carried out to gain insight on the effects of several plausible uniform and non-uniform settlement patterns on representative historic barrel vaults. Various settlement configurations were analysed and complex failure patterns observed. This study could help analysts in understanding the nature of on-going deformation process in historic masonry vaults and engineers in the design of strengthening strategies.

... DEM strategies are typically based on the contact penalty approach and explicit integration schemes. This formulation allowed several applications on fullscale masonry structures (Papantonopoulos et al., 2002;Lemos, 2007;Tóth et al., 2009;Sarhosis and Sheng, 2014;Ç akti et al., 2016;Simon and Bagi, 2016;Bui et al., 2017;Forgaćs et al., 2017;Lengyel, 2017;Foti et al., 2018) using elastic or rigid blocks ( Fig. 1.8). ...

Several tools for the prediction and the assessment of the structural behavior of masonry buildings have been developed in recent decades. Numerical tools have been favorably developed and preferred over analytical approaches, given the complex mechanical response of masonry and the irregular geometries of historic masonry buildings. In this chapter, a thorough review of numerical strategies for the analysis of masonry structures is presented. Additionally, classification of these strategies is also suggested to logically organize the extensive literature on this topic. Even though a wholly congruent categorization of all the numerical tools is essentially unrealistic given the specific aspects of each solution developed, the existing numerical strategies are subdivided into four classes: block-based models, continuum models, geometry-based models, and macroelement models. Each class is thoroughly reviewed and the open challenges in numerical modeling of masonry structures are critically examined.

... DEM approaches are based on contact penalty formulations and explicit integration schemes. In this context, several applications have been conducted on real masonry structures [84,85,86,87,88,89,90,91,92,93] using rigid or linear elastic blocks ( Figure 9). ...

Masonry structures, although classically suitable to withstand gravitational loads, are sensibly vulnerable if subjected to extraordinary actions such as earthquakes, exhibiting cracks even for events of moderate intensity compared to other structural typologies like as reinforced concrete or steel buildings. In the last half-century, the scientific community devoted a consistent effort to the computational analysis of masonry structures in order to develop tools for the prediction (and the assessment) of their structural behavior. Given the complexity of the mechanics of masonry, different approaches and scales of representation of the mechanical behavior of masonry, as well as different strategies of analysis, have been proposed. In this paper, a comprehensive review of the existing modeling strategies for masonry structures, as well as a novel classification of these strategies are presented. Although a fully coherent collocation of all the modeling approaches is substantially impossible due to the peculiar features of each solution proposed, this classification attempts to make some order on the wide scientific production on this field. The modeling strategies are herein classified into four main categories: block-based models, continuum models, geometry-based models, and macroelement models. Each category is comprehensively reviewed. The future challenges of computational analysis of masonry structures are also discussed.

... The last approach concerns the use of discrete element methods to assess the bearing capacity of masonry vaults, which in the recent years has been successfully applied to many typologies of masonry structures (see e.g. [27,28]). ...

The ultimate limit state behavior of masonry domes under axisymmetric gravity loads is nowadays well known and it has been proved how a generalization of the thrush line method used successfully for arches is quite effective also in this case. However, the behavior of a dome under horizontal loads, which is important in case of seismic action, becomes incredibly hard to tackle and still remains an open issue.
The present paper aims at presenting a fast and reliable automatized kinematic limit analysis approach able to accurately predict the actual behavior of masonry domes subjected to horizontal static loads. The model uses a rough discretization of the dome obtained by means of few rigid-infinitely resistant NURBS generated elements, adapting step by step the initial mesh in order to progressively overlap the element edges (where all dissipation is lumped) with the hinges forming the failure mechanism. The adoption of a rough mesh makes the code extremely fast, much more competitive than a standard FE model, allowing at the same time to approximate the actual geometry and load distributions in an extremely accurate way. The utilization of geometries obtained with laser scanner acquisitions is straightforward and the presence of pre-existing cracks can be accounted for as well. Three complex case studies are analyzed in detail to benchmark the approach proposed, relying into existing domes belonging to the Italian cultural heritage. The first example has the geometrical parameters of a typical late Renaissance dome, the Cathedral of Montepulciano, the second is the dome of Anime Sante church (collapsed during the L’Aquila 2009 earthquake with a paradigmatic failure mechanism) and the last is the dome of Caracalla baths, whose causes of collapse remain still unknown. In all cases inspected, the approach proposed quickly provides collapse accelerations and active failure mechanisms at a fraction of the time needed by non-linear FE analyses, providing interesting hints into the actual behavior of such kind of structures under horizontal loads.

... Moreover, as stated in [21], by using specific values for the mechanical parameters, DEM is suitable for approximating Heyman's assumptions [1]. Several works have been conducted using DEM to investigate the static and dynamic behaviour of masonry structures, [22][23][24][25][26][27][28][29]. The application of this method shows good results in the description of collapse mechanisms and displacement capacity. ...

The structural assessment of doubly-curved vaulted masonry structures, such as pavilion vaults, poses challenges specific to their high degree of indeterminacy. Two-dimensional equilibrium analysis methods may provide a lower bound of load or displacement capacity, but they do not accurately describe the three-dimensional (3D) behaviour of these structures, particularly when shear deformation (e.g. sliding) is important. Therefore, discrete element modelling (DEM) methods, which can effectively simulate 3D load redistribution, have been used to investigate support displacement capacity and corresponding 3D collapse mechanisms. DEM analyses are usually conducted on perfect digital geometries. Meanwhile, both real structures and small-scale physical models have implicit assembly and fabrication imperfections, which may significantly alter their response. The present paper aims to investigate the influence of geometrical and mechanical imperfections by comparing DEM analyses with the results obtained from tests on a scale model. In particular, a new method to simulate imperfections within the DEM framework is proposed, and a DEM parametric analysis is compared to the measured behaviour of a 3D-printed scale model of a pavilion (or cloister) vault on spreading supports. The influence of both mechanical imperfections and geometrical imperfections, due to element geometry deviations or imprecision of the assembly process, have been investigated. Based on these analyses, the three-dimensional behaviour of a pavilion vault subjected to horizontal displacement of the supports is described, and the variability of results due to imperfections is demonstrated. A good agreement between DEM analyses and 3D-printed scale model tests is achieved, in terms of crack patterns and mechanisms. Geometrical imperfections did change the load paths within the vault, as expected, and also influenced the displacement capacity.

... In the present work, the static analysis of the vaults was performed using Discrete Element Method (DEM) because it is particularly suitable to model the bricks and the interaction between them. The DEM method was already used in the literature for the analysis of vaults: several studies were performed by means of the commercial software 3DEC [42], [43], [44], [12], [11]. Usually, in DEM the motion equations of the blocks are solved in time by explicit integration algorithms that, requiring small time-steps, are computationally demanding. ...

The design of brick masonry vaults has frequently been one of the main topics of historical construction handbooks, where also the orientation of the blocks was an examined issue. However, the influence of brick-texture has not yet been thoroughly investigated computationally. In the present paper the static behavior of masonry vaults of different shapes (barrel, pavilion, and cross vaults), was studied by analyzing and integrating two points of view: one historical and the other computational. First, the instructions provided by ancient documents were described. Then, a Non-Smooth Contact Dynamics Software was used to model vaults with different brick patterns suggested in the past. The vaults were studied with self-weight to observe load descent. Then, a concentrated force was added up to collapse. The analyses show how the load descent and load-bearing capacity are influenced by the brick pattern. Finally, the results of the various analyses were compared with the insights of the past.

... Determination of deformation and deformation of utility objects and objects of practical historical value allows for assessing their technical condition. In the event of alarming changes in the facilities, it also provides for the appropriate planning of protection, renovation, and modernization works and points to the negative impact of the neighborhood and weather conditions [2][3][4][5]. The geodetic monitoring method primarily depends on the object's structure, the speed of changes, and the measurement technology [6]. ...

The article aims to present a hybrid approach combining information entropy and an evolutionary algorithm to optimize a geodetic network's measurement structure to determine an engineering object's horizontal displacements. The objective function was defined, which in the case under consideration was the information entropy of the geodetic observation system in terms of the parameter vector's entropy with the true values. The optimal number of observations in the geodetic network depended on the observation system's increase in information. During the research, it was noticed that the application of the hybrid approach allowed the selection of only those observations with the highest information content. It shortened the measurement time without reducing the accuracy of the displacements obtained. The obtained results of numerical analyses showed the proposed solution's effectiveness for optimizing the geodetic network structure.

... In the framework of numerical methods, discrete (or distinct) element models (DEM) were widely used to study the collapse of masonry arches and vaults due to support displacements (e.g., D'Altri et al., 2020;Foti et al., 2018;Lengyel, 2017;McInerney and DeJong, 2015;Van Mele et al., 2012). DE models offer the possibility of using rigid or quasi-rigid blocks and simulating large relative movements between the units (Lemos, 2007). ...

... Discrete Element Method (DEM) and Finite Element Method (FEM) models are among the most widely adopted block-based numerical approaches and several research papers aim at discussing pros and cons given by their application to specific case-studies (Giamundo et al. 2014;Sarhosis et al. 2018). In particular, DEM models are developed by the discretization of the vault into rigid or elastic blocks connected by means of non-linear interfaces, commonly modelled with elasto-plastic behaviour with softening branch, which represents the degradation of contact (McInerney and DeJong 2015;Lengyel 2017;Lengyel and Németh 2019;Foti et al. 2018). Conversely, FEM models generally assume elastic and cracked behaviour of masonry by means of several approaches such as macro-or micro-modelling and isotropic or anisotropic constitutive laws (Gaetani et al. 2017(Gaetani et al. , 2021Milani et al. 2016Milani et al. , 2017Milani et al. , 2019. ...

Cross masonry vaults are common structural elements in historical buildings. They are largely diffused in all European countries, including those characterized by higher levels of seismicity. Although they have been constructed for centuries, they represent some of the most vulnerable elements of traditional architecture, especially with reference to horizontal loads. The understanding of their structural behaviour under seismic loading is a crucial aspect for the accurate assessment of the safety of historical buildings. In the present work, the seismic response of cross masonry vaults is analysed through the Finite Element Method (FEM) and static non-linear analyses considering the effect of different brick patterns and boundary conditions. A simplified micro-modelling approach is adopted for the generation of the FEM models and two different brick arrangements are considered, i.e., radial bricks and diagonal bricks, which are the most widespread in European cross vaults. Two different boundary conditions are assumed in order to simulate a vault with and without lateral confinement. Static non-linear analyses are performed by monotonically incrementing a lateral acceleration until collapse. Results are analysed in terms of maximum load factor, stiffness, ductility, crack pattern and damage mechanisms. The analysis of the results shows that not only boundary conditions, but also the brick pattern strongly influences the vault seismic response both in terms of stiffness and ductility as well as in terms of global capacity.

... By decreasing the lateral thrust larger windows can be built and more light could enter through these windows. In the work of the author [19] it was shown that large deviation from the semicircular generator curve could lower the admissible horizontal support displacement in the case of barrel and groin vaults due to shear failure in the masonry shell. In the study of Rosa and Galazia [7] it was revealed by LSA that the equilateral pointed arch is the most resistant against live loads. ...

The paper investigates how the shape affects the Couplet–Heyman minimum thickness of the masonry pointed arch. The minimum thickness is such a structural thickness, at which a vault made of rigid voussoirs is stable for self-weight. It is expressed as a function of the pointed generator curve’s deviation from the semicircle. The arch is analysed in its undisplaced, geometrically perfect state. In the present study, perfect symmetry is assumed, and any disturbance in the symmetry is not considered. The joints between the voussoirs are placed in the radial direction. Two approaches are applied to derive the minimum thickness of the pointed arch, like Limit State Analysis (henceforth LSA) and Discrete Element Modelling (henceforth DEM). The application of the LSA leads to a nonlinear optimisation problem, which is solved by the so-called active set method. DEM technique is also applied, in which the model consists of discrete blocks each of which can move independently from each other. In DEM, sliding failure can freely develop during the loading process, which is neglected in the LSA. The results of the analyses show great correspondence, if sliding failure does not appear.

This study presents three advanced non-linear modeling strategies for the evaluation of the non-linear behavior under horizontal loads of a historical groin vault of major importance, located in Jerusalem. The vault exhibits the typical features of the Crusaders architecture, with the masonry of the bearing structure, such as the piers and the main arches, made of big stocky stones with a high quality surface finish, and the vaults made of smaller irregular cobblestones. The first model is based on a FE Concrete Damage Plasticity (FE-CDP) macroscopic approach implemented in Abaqus, where masonry is assumed elasto-plastic with damage in both tension and compression and scarcely resistant to tensile stresses. The second model is a non-commercial Rigid Body and Spring (RBS) model, where the structure is discretized into rigid elements and elasto-plastic with softening interfaces. The arches are modeled with a heterogeneous approach, whereas the vault itself through an isotropic material with quasi zero tensile strength and softening behavior. The last model is based on Kinematic Limit Analysis with adaptive mesh and consists of few rigid infinitely resistant NURBS elements (NURBS-KLA). For validation purposes, at the University of Florence a 1:5 scale model of the groin vault was tested under horizontal loads up to collapse, also in presence of FRP reinforcement glued at the extrados after the activation of the collapse mechanism in the unreinforced case. Excellent agreement with experimental results is obtained for all the models, both in terms of load carrying capacity prediction and active failure mechanism, also in presence of FRP reinforcement. The reliability of the procedures presented is therefore demonstrated.

This paper presents a parametric stability study of groin, or cross vaults, a structural element widely used in old masonry construction, particularly in Gothic architecture. The vaults’ stability is measured using the geometric safety factor (GSF), computed by evaluating the structure’s minimum thickness through a thrust network analysis (TNA). This minimum thickness is obtained by formulating and solving a specific constrained nonlinear optimisation problem. The constraints of this optimisation enforce the limit analysis’s admissibility criteria, and the equilibrium is calculated using independent force densities on a fixed horizontal projection of the thrust network. The parametric description of the vault’s geometry is defined with respect to the radius of curvature of the vault and its springing angle. This detailed parametric study allows identifying optimal parameters which improve the vaults’ stability, and a comprehensive comparison of these results was performed with known results available for two-dimensional pointed arches. Moreover, an investigation of different force flows represented by different form diagrams was performed, providing a better understanding of the vaults’ structural behaviour, and possible collapse mechanisms were studied by observing the points where the thrust network touches the structural envelope in the limit states. Beyond evaluating the GSF, the groin vault’s stability domain was described to give additional insights into the structural robustness. Finally, this paper shows how advances in equilibrium methods can be useful to understand and assess masonry groin vaults.

Determining the internal forces acting within a masonry dome or vault loaded by an assigned distribution of external actions is a problem that is still open, as evidenced by even recent contributions to the scientific literature on the topic. The present work intends to address this issue by proposing a method for determining admissible distributions of stresses in vaults and masonry domes. The problem is solved analytically with the aim of obtaining, when possible, explicit expressions for the internal forces. Although applicable in principle to arbitrarily distributed loads, the procedure adopted herein for searching for statically admissible internal forces is described in detail for the case of distributed and concentrated vertical loads. The analysis is performed by building suitable analytical solutions to the so-called “direct” and “inverse” problems of a thin shell in which bending forces are nil and only membrane forces are present. The solutions thusly obtained are applied to vaults and domes by making use of the so-called “thrust surface” concept, which represents a natural generalization of the thrust line for masonry arches. According to Heyman’s hypothesis for masonry, when the thrust surface is entirely contained within the vault thickness, a corresponding statically admissible stress field can be found. Thrust surfaces corresponding to admissible stress fields are determined by means of an expressly developed iterative procedure that begins by assigning an initial shape to the thrust surface. Then, by suitably using the solutions of the inverse problem, the shape of the thrust surface is modified so that the corresponding stresses become, when possible, statically admissible. By using the well-known theorems of limit analysis, both the mechanical and geometric safety coefficients are assessed for vaults and existing domes. As an example, the proposed procedure is applied to three practical case studies: a hemispherical dome of constant thickness, the dome of the Rome Pantheon, and the dome of Bernini’s Santa Maria Assunta Church in Ariccia (Rome).

In this paper a novel method for the design of complex affine vaults is proposed. By merging the thrust line theory with Rankine’s concept of the transformation of structures, the system of complex groin vaults whose constituent parts are mutually affine cylindrical vaults is treated. The method enables the omission of the analysis of the actual structural action within a vault without jeopardising the vault’s stability. That provides simplicity of the overall computation as well as the fast evaluation of the vault’s thrust. The proposed parametric model for the equilibrium analysis of affine vaults is based on the geometrical properties of a suitably chosen initial vault’s part exposed to transformations, where the disposition of supports in a vault’s plan becomes the main input for the computation. The method provides the design of the variety of complex affine vaults, as well as the automated calculation of the resultant loading exerted to supports with their graphical representation.

In general, masonry structures are divided into two categories in terms of the connection between components: mortar-free masonry, which is called dry joint, and structures that use mortar to connect the components. Due to environmental conditions, when the mortar loses its adhesion over time, the components of the wall can be considered as discrete elements. The poor performance of masonry structures leads to the failure of a significant number of these structures even in moderate earthquakes. In particular, in-plane failure in the walls of masonry structures is very common. The results of laboratory research show that shear failure is very likely in lateral loading. One of the characteristics of this failure mode is its ability to resist high shear force and low ductility, which leads to brittle failure in the wall. Previous numerical studies on masonry structures have been mainly based on the finite element method in which the wall components are considered as continuous elements and the local behavior of the constituent parts of the walls is ignored and causes their real function to be misunderstood. In this paper, first, the specifications of unreinforced masonry wall specimens were determined based on the results obtained from previous experiments, and then numerical models were calibrated and analyzed using two software ABAQUS (finite element software) and 3DEC (discrete element software). Then, behavioral diagrams of walls with different characteristics were extracted under different loading conditions. Finally, a comprehensive comparison between the results obtained from the discrete element and finite element methods was presented. Finally, it was concluded that the discrete element approach provides a more accurate prediction of unreinforced masonry walls than finite element one.

The history of cross vaults began almost two thousand years ago with a widespread use during the Middle Ages and Renaissance, becoming nowadays one of the most diffused and fascinating structural typologies of the European building cultural heritage. However, conversely to the undeniable excellence achieved by the ancient masons, the structural behaviour of these elements is still at the centre of the scientific debate. In this regard, with the aim of reviewing the knowledge on this subject as a concise and valuable support for researchers involved in conservation of historical buildings, with a focus on design rules and structural analysis, the present study firstly introduces the cross vaults from a historical perspective, by describing the evolution of the main geometrical shapes together with basic practical rules used to size them. Then, the paper deals with the subsequent advancements in structural analysis methods of vaults, until the development of modern limit analysis.

This study focuses on domes the ground plan of which, instead of the more common circular shape, is an oval, and aims at finding the minimally necessary uniform wall thickness for domes of different geometries loaded by their selfweight. The discrete element code 3DEC was applied because of its capability of simulating the collapse mechanisms of masonry structures. Results on the minimal wall thickness, corresponding masonry volume and failure mechanisms for different dome geometries are presented. Three ranges of the friction coefficient were found. For very low frictional resistance collapse happens with pure frictional sliding, for any arbitrarily large wall thickness. In the range of relatively high (i.e. realistic) friction coefficients the structure collapses without any sliding if the wall is not sufficiently thick, and in the observed range of the friction coefficient the necessary wall thickness is nearly insensitive to its value (collapse initiates with hinging cracks only). Between the two domains an intermediate behaviour was found: combined cracking and sliding collapse modes occur for insufficient wall thickness, and the minimal thickness strongly depends on the friction coefficient. The critical and transitional friction coefficients separating the failure modes were determined for different eccentricities of the groundplan.

This paper aims to evaluate the limit equilibrium condition and the minimum thickness of masonry arches in presence of horizontal loads. The analysis fits into the frame of limit analysis referring to Heyman's theory. Two types of arches are analysed, the circular and pointed one. The loading system consists of vertical and horizontal loads, which refer respectively to the self-weight of the voussoirs and to the seismic actions. The collapse mechanism and the corresponding horizontal load multiplier are determined, in the condition of rigid abutments, as functions of the geometrical features of the structure. The results are supported by some simple experimental tests and a sensitivity analysis, which considers the effect of geometrical irregularities on the load multiplier.

The role of ribs in the mechanical behavior of masonry cross vaults has been the subject of intense debates since the 19th century. Literature on the subject diverges from considering the ribs as the main load-bearing units which carry the weight of the masonry web, to the opinion that the ribs are merely decorations. This research focused on the simplest type of cross vaults, i.e. groined vaults formed by the intersection of two semicircular cylindrical mid-surfaces. Instead of the widely used Limit State Analysis which is reliable only if specific conditions are satisfied, discrete element modelling (the commercial code 3DEC, based on an explicit time integration scheme), and a classical finite element code (ANSYS) was applied in the investigations. In the applied DEM code (3DEC) the elements (corresponding to the voussoirs) may slide along each other, and can be separated from their neighbors in any directions; and new contacts may be formed between them, in a computationally efficient automatized manner.

Displacements experienced by many historic masonry structures concentrate at masonry joints and can be large before collapse is a concern, making modelling of stability using discrete element modelling (DEM) particularly suitable. In this study, masonry groin vault and arch models with several geometries were subjected to horizontal and vertical support displacements using DEM. Support movements were applied in a quasi-static manner to simulate the support settlement process. Displacements at collapse and at the point when the first block fell from the vault were recorded. Block separation and mechanisms were also noted during the simulations. A 2D analytical model using thrust line analysis was developed to help evaluate the DEM results. In general, the displacements at first block fall were relatively large but significantly less than those at collapse. The groin vaults and arches exhibited significantly higher capacity to sustain vertical support displacement compared to horizontal displacements. For many geometries, the DEM collapse displacements of the groin vaults compared reasonably well to similar arches, indicating that the displacement capacity of groin vaults can be reasonably estimated using two dimensional simplifications. However, for certain geometries, three-dimensional effects were found to significantly affect displacement capacity.

Aim of the present paper is the analysis of a series of existing masonry cross vaults exhibiting meaningful structural deterioration and diffused crack patterns, by means of an advanced non-linear and limit analysis software.
The approach utilized is a non-standard and non-commercial one and bases both for the non-linear and limit analysis procedure on a FE discretization of the domain by means of rigid infinitely resistant wedges, where all the non-linearity is concentrated on interfaces between adjoining elements.
When dealing with the non-linear code, a sequential quadratic programming scheme is used at each iteration in order to deal with the deterioration of mechanical properties of interfaces, provided that the actual non-linear behavior is approximated by means of a linear piecewise constant function.
Several numerical simulations are performed varying constraint conditions, material properties, infill modeling and presence of FRP strips as reinforcement devices, comparing and discussing in detail the results obtained.
From simulations results, it is found that the approach commonly used in practice to study cross vaults by means of the assemblage of single arches is not always reliable, providing failure loads and mechanism quite different from the real ones. Furthermore, similarly to what occurs for masonry arch bridges, it is found that the role played by the infill is crucial and that, depending on the actual mechanical properties of the infill, both the failure mechanisms and the collapse load may vary significantly.

This paper addresses the long-standing problem of the equilibrium of the circular, pointed, and elliptical arches commonly found in historical masonry buildings and bridges that are subjected to their own weight and the weight of superimposed masonry walls. The equilibrium problem is studied by applying two different complementary methods: the first is a simple extension and analytical re-reading of the Durand-Claye stability area method; the second is based on the application of a nonlinear elastic one-dimensional model, already used by the authors in previous studies. It is assumed that the arch’s constituent material has limited compressive strength and null tensile strength. In addition, the load transferred to the arch by the wall is determined under the common assumption that each vertical strip of wall bears directly down on the underlying arch element. The study focuses on the maximum height that the superimposed wall can reach under equilibrium conditions while maintaining acceptable values of arch residual stiffness. One noteworthy finding is confirmation of the decidedly better behavior of pointed and elliptical flat arches compared with that of circular arches.

In this paper, an analytical and numerical analysis on the collapse mode of circular masonry arches is presented. Specific reference is made to the so-called Couplet-Heyman problem of finding the minimum thickness necessary for equilibrium of a masonry arch subjected to its own weight (Heyman 1977). The note reports the results of an on-going research project at the University of Bergamo. First, analytical solutions are derived in the spirit of limit analysis, according to the classical three Heyman hypotheses and explicitly obtained in terms of the unknown angular position of the intrados hinge at the haunch, the minimum thickness to radius ratio and the non-dimensional horizontal thrust (Colasante 2007, Cocchetti et al. 2010). Results are then compared to Heyman solution. Though only the first of these three characteristics is perceptibly influenced in engineering terms, especially at increasing opening angle of the arch, the treatment settles an important conceptual difference on the use of the true line of thrust, along the line of Milankovitch work. Second, numerical simulations by the Discrete Element Method (DEM) in a Discontinuous Deformation Analysis (DDA) computational environment are provided, to further support the validity of the obtained solutions, with good overall matching of the obtained results (Rusconi 2008, Rizzi et al. 2010).

a b s t r a c t The equilibrium problem of unreinforced masonry vaults is analyzed via a constrained thrust network approach. The masonry structure is modeled as no-tension membrane (thrust surface) carrying a discrete network of compressive singular stresses, through a non-conforming variational approximation of the continuous problem. The geometry of the thrust surface and the associated stress field are determined by means of a predictor–corrector procedure based on polyhedral approximations of the thrust surface and membrane stress potential. The proposed procedure estimates the regions exposed to fracture dam-age according to the no-tension model of the masonry. Some numerical results on the thrust network and crack pattern of representative vault schemes are given.