Cross Vaults Research Articles

Modelling of the Dynamic Response of a Full-Scale Masonry Groin Vault: Unstrengthened and Strengthened with Textile-Reinforced Mortar (TRM)

ABSTRACT This article presents the results of the numerical simulations of a dynamic experimental campaign on the shaking table. The specimen is a full-scale masonry cross vault without and with the textile-reinforced mortar used as a strengthening solution. The specimen was tested in the scope of the European project SERA.ta 07 “Seismic Response of Masonry Cross Vaults: Shaking table tests and numerical validations”. The specimen replicates a generic cross-section of a central bay located in the lateral nave of a three-nave church, where the recurrent in-plane shear mechanism is induced. Finite element model (FEM) and discrete element model (DEM) were built and calibrated based on the experimental frequencies and mode shapes. Then, a unidirectional seismic action was applied to the numerical model in order to develop shear damage in the shell of the vault. A detailed description of the two simulations and their main outcomes are presented. The response was evaluated based on the displacements, damage, collapse mechanisms and hysteretic behaviour.

HBIM STRUCTURAL MODEL TO EVALUATE BUILDING EVOLUTION AND CONSTRUCTION HYPOTHESES: PRELIMINARY RESULTS

Abstract. Historic Building Information Modelling (HBIM) is a technology that has proven to be very effective for the management, preservation, and maintenance of heritage buildings. HBIM allows a digital replica of the building, in which information can be stored, designs can be made, and future actions can be planned. To do this, it is obviously necessary to have a thorough knowledge of the building and its historical evolution. The HBIM model can therefore become the ideal place in which to develop and model construction hypotheses of building portions that no longer exist, or even record its development over time using different phases of work. Based on this context, the aim of this article is to use the HBIM approach for modelling different construction hypotheses and use the model to study the behaviour of different configurations with structural analysis. To do this, the case study of the church of San Michele Maggiore in Pavia was chosen, which in the 15th century underwent major restorations due to structural failures of the vaults of the central nave, which were replaced with the current cross vaults. In the literature there are different constructive hypotheses of the ancient vaults, which have been modelled in HBIM precisely to evaluate the different structural behaviours following the method presented. This article presents the historical analyses and geometric surveys that led to the HBIM modelling and the model itself. In the future, after careful selection of the most appropriate software, structural calculations will be made to study the structural behaviour of the building.

Shaking Table Testing of an Unstrengthened and Strengthened with Textile Reinforced Mortar (TRM) Full-Scale Masonry Cross Vault

ABSTRACT This article presents the results of the LNEC-3D shaking table tests on a full-scale masonry cross vault without and with the textile reinforced mortar used as strengthening solution. The specimen has been tested in the scope of the European project SERA. TA 07 “Seismic Response of Masonry Cross Vaults: Shaking table tests and numerical validations”. The specimen replicates a generic cross-section of a central bay located in a lateral nave of a three-nave church, derived from the intersection of two semi-circular barrel vaults with low rise. It is representative of a generic monumental church in Central Italy, generated by a squared base groin vault with a net span of 3.13 m and 1.13 m rise at full scale. A unidirectional seismic action is applied to develop shear damage in the shell of the vault. A detailed description of the two tests and the conclusions are presented. The response is evaluated based on the displacements, damage, and collapse mechanisms developed as a function of an increasing intensity earthquake testing protocol, in which a pre-processed strong ground motion component of the L’Aquila (Italy) earthquake (April 6, 2009) was used.

Simplified Analysis of a Masonry Cross Vault Under Dynamic Actions

ABSTRACTDynamic tests performed on masonry cross vaults provide important information to improve the knowledge under seismic actions of these particular structural elements. Under seismic actions, curved masonry elements show a behaviour strongly influenced by several parameters (both physical and mechanical). Their structural behaviour is hard to predict due to the significant variability of the input parameters and the strong heterogeneity of these elements. Refined nonlinear models often require specific numerical calibrations of nonlinear parameters in order to assess the structural capacity of tested element. Conversely, simplified Finite Element models represent useful tools to develop a basic knowledge on the expected structural behaviour. Initial knowledge level in SERA TA blind competition favoured simple assumptions. Therefore, the main goal was to evaluate key information of a masonry vault under dynamic action: triggering and the type of damage, the most stressed areas and the threshold at which evident damage is expected. The results obtained by these preliminary analyses have been compared with the experimental tests.

Numerical Simulation of the Response of an Unreinforced Brick-Masonry Cross Vault Subjected to Seismic Loading

ABSTRACTThis paper presents the numerical evaluation of the seismic response of a masonry cross vault using the Discrete Macro-Element Method (DMEM). The case study corresponded to a full-scale unstrengthened cross vault that was experimentally investigated within the scope of the SERA Project — Seismic Response of Masonry Cross Vaults: Shaking table tests and numerical validations. The cross vault was subjected to repeated shaking table and dynamic identification tests until reaching significant damage. The numerical simulations involved the calibration of the Young’s modulus of the masonry material aiming at reproducing the cross vault’s experimental natural frequencies and mode shapes. The comparison of frequencies was carried out by estimating the difference between experimental and numerical results, whereas the correspondence between mode shapes was studied using the Modal Assurance Criterion. Subsequently, a sensitivity analysis was performed to identify the influence of nonlinear properties on the seismic response of the cross vault (displacement and acceleration time histories and failure mechanism). The accuracy of the numerical time histories was evaluated by estimating magnitude and phase discrepancies. The results aimed at demonstrating the applicability of the DMEM for assessing the seismic response of masonry cross vaults with an acceptable degree of accuracy and low computational cost.

3D Rigid Block Limit Analysis of a Masonry Cross Vault Subjected to Shaking Table Tests

ABSTRACT This paper presents the results of a numerical study on a brick masonry cross vault that was tested on a shaking table, on a scale of 1:1. The study was part of the blind prediction competition organized within the activities of the SERA-AIMS project “Seismic Response of Masonry Cross Vaults: Shaking table tests and numerical validations”. The main objective of the blind prediction competition was to compare various numerical techniques. Among others, the one presented in this paper is based on 3D rigid block limit analysis. In the numerical model, the masonry units of the cross vault are modeled with rigid blocks interacting at no tension-frictional interfaces. A simple iterative procedure is used to take into account the non-associative behavior in sliding failure. Preliminary analyses carried out at the blind prediction competition stage as well as post-diction studies, after the experimental tests, are presented. The paper emphasizes the differences between the pre-diction and post-diction phases to highlight the improvements made during the numerical investigation. The study shows the influence of support and action modeling as well as of friction coefficient on the failure mechanisms and force capacity.

Finite-Element Seismic Assessment of a Masonry Cross Vault Through Blind Predictions and Numerical Simulations

ABSTRACT In this paper, a continuum 3D finite-element model is presented to determine the seismic behaviour of masonry cross vaults under seismic motions. The work regards the investigation of a regular cross vault made of clay bricks and hydraulic mortar, accounting for two distinct configurations: an un-strengthened design and one strengthened through the application of a fibre-based reinforcement. The research adopts parts of the blind prediction simulation developed during the SERA project. For both configurations, the manuscript follows two main steps: a prediction simulation, based on laboratory tests executed on specimens made with the same materials of the cross vault, and a post-diction calibration, based on the outcomes of the experimental tests on the full-scale model. The final FE models, realized through the commercial software ABAQUS CAE, have allowed a satisfactory simulation of the experiment, providing insights for the modelling of these structural systems through continuum approaches.

Generalized Thrust Network Analysis of Triangular Masonry Cross Vaults Inspired by Musmeci

A triangular parabolic cross vault, that was designed by Musmeci in the 1950s as a reinforced concrete structure but remained unbuilt, is revisited from the original perspective of its reinvention as a masonry structure. In the framework of static limit analysis under classical Heyman’s assumptions, a generalized thrust network analysis is adopted for a structural safety assessment. The performances of the vault, subject to its self-weight, are investigated through minimum-thrust and minimum-thickness analyses by conforming to the original geometry and assuming the vault thickness as the only design parameter. Further insight is achieved by exploring a more general class of triangular parabolic masonry cross vaults, whose rise-to-span ratio is an additional design parameter. The static efficiency of the smart and unconventional geometry proposed by Musmeci is thus proven, motivating the possibility of bringing it to new life in the form of a masonry structure.

A Cross-Vault System’s Relative Periodisation Based on Geometric Analysis •

The cross-vault system of the Inner City Parish Church of Budapest was the subject of numerous scientific works so far. In our present article, we aimed to supplement the information known about the vaults by providing exact geometric data gathered via laser scanning. Based on this data, we attempted to filter out common characteristics of the individual vaults, regarding the geometry of their ribs, their webs and the positions of their keystones. As a result, we proved that the cross vaults composing the system cannot be hallmarked as the remains of a given building period, and even within a given cross vault, more building periods can appear. Combining the results of the different geometric analyses, we concluded six distinct scenarios concerning the relative periodisation of the elements composing the cross-vault system, complemented by the evaluation of these. Thus, in the present article, we give an example of how to use point cloud-based building survey techniques to approach the questions of a given building’s relative periodisations.

Modelling of the dynamic response of a reduced scale dry joints groin vault

The relevance of the safeguard of masonry cross vaults is evident in churches, palaces, and cloisters of most European countries. Nevertheless, their seismic behaviour is still an aspect to be deepened. In this regard, the purpose of this paper is to study the dynamic three-dimensional behaviour of a reduced scaled dry joint groin vault. The work involved simplified micro-modelling, based on the finite and discrete element methods, for reproducing the experimental campaigns of the vault subjected to seismic loads.The research can be divided into four phases: the first one corresponds to an overview of numerical modelling approaches, specifically focused on cross vaults; the second one consists of the calibration of the numerical models, based on the experimental outputs; the third phase highlights advantages and disadvantages of the two modelling strategies adopted through qualitative and quantitative comparison, such as ultimate strength capacity, failure mechanisms and time histories of displacements. Lastly, a sensitivity analysis on the influence of several recorded inputs on the same mock-up is shown.In general, both numerical models simulate correctly the experimental responses, representing an improvement with respect to previous models available in the literature, in fact, the ability of the discrete element method to reproduce the collapse mechanism of the vault was rarely investigated. Moreover, in this contribution the importance of the modal properties for the numerical calibrations, the quantitative differences between the discrete and finite element methods and finally the role of the seismic parameters at the level of the collapse are investigated and commented.

Generalized thrust network analysis for the safety assessment of vaulted masonry structures

A generalized thrust network analysis is proposed for the safety assessment of vaulted masonry structures under their self-weight. The method requires introducing a 3D network of beams as a discretization of the vault mid-surface. The stress state in each beam is given by an internal thrust acting in the vertical plane containing the beam. By using three suitable statical descriptors per beam to represent those thrusts, the equilibrium equations of the network nodes and the static admissibility conditions of the stress state in the network beams are formulated consistently with the classical Heyman’s assumptions on masonry. As crucial merit of the formulation, those conditions result to be linear equalities and inequalities, respectively, in the unknown statical descriptors. It is then proven that the proposed method generalizes the classical thrust network analysis, which is recovered by imposing the further nonlinear constraint that the thrusts associated with the beams attached to a node converge into a single point. The minimum thrust problem for masonry vaults is formulated within the proposed framework, taking the form of a straightforward linear programming problem. On such a basis, the minimum thickness problem is also addressed. An extensive campaign of numerical results is presented, dealing with the safety assessment of masonry spherical domes, ogival domes and cross vaults. Compared to the classical thrust network analysis, the proposed method (i) overcomes computational issues related to the solution of nonlinear/non-convex optimization problems; (ii) accounts for more general stress states; and (iii) explicitly provides the settlement mechanism associated with the minimum thrust state.

Seismic behaviour of cross vaults with different brick pattern

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.

АНАЛИЗ РАБОТОСПОСОБНОСТИ ОПОРНЫХ УЧАСТКОВ КАМЕННЫХ СВОДОВ

The article substantiates that in cylindrical vaults with stripping and cross vaults of historical buildings, the most stressed areas are the areas where the vaults rest on the supporting structures (walls and pillars) and the zones of interface between strippings and vaults. In particular, in cross vaults, the most loaded are the diagonal ribs, in which the compressive forces act at an angle of 45° (it is in this direction that the masonry has the least resistance to compression), and the flow of compressive forces within the diagonals of the vault accumulates, reaching maximum values in its supporting heels. From which it is concluded that it is the supporting consoles that most often determine the bearing capacity of the cross vault as a whole. The article gives approximate calculated dependences for assessing the bearing capacity of the supporting consoles of the cross vaults and concludes that the most accurate picture of the stress state of the supporting consoles can be established by numerical modeling, and to assess the bearing capacity, it is necessary to use particular strength criteria.

Numerical Investigation of the Influence of Constructive Aspects on the Structural Behaviour of Masonry Cross Vaults

ABSTRACT In this paper, the structural performance of masonry cross vaults is investigated with regard to the most common differential settlements occurring in historical constructions. Numerical parametric models are developed for vaults with different brick patterns and lateral constraints, in the presence/absence of infill. In particular, radial, diagonal and pitched brick patterns are taken into account and opening, shear and vertical settlements are analyzed. Moreover, the behaviour of laterally unconfined vaults is compared with the response of vaults restrained with rigid perimeter walls or with lateral masonry arches. The analyses highlight the strengths and weaknesses of each configuration, providing a scientific validation to some speculations found in historical technical literature on the most widespread use of some brick patterns. The outcomes of this research provide a deep knowledge of masonry vaults behaviour in view of more effective interventions on historical buildings, supplying the practitioners with a crack-pattern chart useful for better interpreting the damage pattern in historical vaults with different brick courses.

FROM TLS POINT CLOUD DATA TO GEOMETRICAL GENESIS DETERMINATION OF RIBBED MASONRY VAULTS

Abstract. The contribution aims to explore the possibility of tracing the geometry of ribbed vaults from two different Mediterranean regions to a single matrix, verifying the presence of possible local variations of the same rules. In particular, the analyses are being carried out in parallel on some case studies of the regions of Sardinia in Italy and Aragon in Spain. The two case studies include the Iglesia Parroquial del Salvador la Seo in Zaragoza and the Church of Santa Lucia in Cagliari. Both constructions can be traced back to the style known as Late Mediterranean Gothic, which characterised the architecture of the countries bordering the Mediterranean basin between the 14th and 17th centuries. The two case studies chosen were almost at the extreme ends of the Late Gothic period, to determine whether some invariants sought could persist even in relatively distant periods. The analysis focused on cross vaults covering the two naves, which included a laser scanner survey in order to obtain a cloud of points of sufficient precision to carry out studies on the geometry of the vaulted systems, the identification of the intrados profiles of the ribs and therefore the definition of the curvatures and centres of all the arches making up the vaults. Finally, the results are presented by means of summary diagrams and comparison tables.

Сalculation methodology and determination of efforts in vault

Abstract. Problem. In this article we offer the meth-odology of calculation, based on the idea of dismem-berment of vault on the separate archs of single width. It is recommended for the estimation of efforts in the designed and damaged vaults. Also offered a methods of calculation effort in serried vaults. As spatial inflexibility of cross vault below than inflexi-bility of serried vault, then it is necessary to execute the calculation of the cheek arch abstracted from the tray of cross vault also. If the outline of tray is exe-cuted on a circular curve, then it is possible to take methodology of calculation of archs with the outline of axis a curve as catenoid. Goal. The proposed methodology for calculating closed and cross vaults is also based on the idea of dividing the arch into separate arches of unit width. The results of studies of efforts and rational forms of arches of V.A. Kiselev [1]. Methodology. Calculation based on the idea of dismemberment of vault on the separate archs of single width. Recommended for the estimation of efforts in the designed and damaged vaults. A method over of calculation of dome is brought for determina-tion of effort in serried vaults. Results. At the choice of outline of axis of arch it is necessary aspires to that an axis, on possibility, coincided with the curve of pressure. Such outline of axis is rational, as there are only central forces of N in an arch. Flexion mo-ments M and transversal forces of Q are equal to the zero. The coincidence of axis of arch with the curve of pressure can be got for the three-hinged arch. For the statically undetermined arch of complete coinci-dence of axis with the curve of pressure attaining is impossible, so that the origin of flexion moments at any her outline is inevitable. Originality. The tech-nique based on the separation of flat elements from the spatial structure does not show the reserves of strength of the structure. Experience of long-term operation of vaulted structures designed or verified by calculation using similar methods [2], [9] shows caution and reliability. Practical value. The tech-nique is also suitable for assessing the strength of damaged vaults with collapsed corners or walms.

From Stress to Shape: Equilibrium of Cloister and Cross Vaults

The assessment of the equilibrium and the safety of masonry vaults is of high relevance for the conservation and restoration of historical heritage. In the literature many approaches have been proposed for this tasks, starting from the 17th century. In this work we focus on the Membrane Equilibrium Analysis, developed under the Heyman’s theory of Limit Analysis. Within this theory, the equilibrium of a vault is assessed if it is possible to find at least one membrane surface, between the volume of the vaults, being in equilibrium under the given loads through a purely compressive stress field. The equilibrium of membranes is described by a second order partial differential equation, which is definitely elliptic only when a negative semidefinite stress is assigned, and the shape is the unknown of the problem. The proposed algorithm aims at finding membrane shapes, entirely comprised between the geometry of the vault, in equilibrium with admissible stress fields, through the minimization of an error function with respect to shape parameters of the stress potential, and then, with respect to the boundary values of the membrane shape. The application to two test cases shows the viability of this tool for the assessment of the equilibrium of existing masonry vaults.

Parametric Stability Analysis of Groin Vaults

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.

A combination of terrestrial laser-scanning point clouds and the thrust network analysis approach for structural modelling of masonry vaults

Terrestrial laser-scanning (TLS) is well suited to surveying the geometry of monumental complexes, often realised with highly irregular materials and forms. This paper addresses various issues related to the acquisition of point clouds via TLS and their elaboration aimed at developing structural models of masonry vaults. This structural system, which exists in numerous artifacts and historical buildings, has the advantages of good static and functional behaviour, reduced weight, good requisites of insulation, and aesthetic quality. Specifically, using TLS, we create a geometric model of the ancient masonry church, S. Maria della Libera, in Aquino, largely characterised by naves featuring cross vaults and previously used as a case study in the paper entitled ‘Terrestrial laser-scanning point-clouds for modeling masonry vaults’, presented at the 2019 IMEKO TC-4 International Conference on Metrology for Archaeology and Cultural Heritage. The results of the TLS survey are used as input for a structural analysis based on the thrust network analysis. This recent methodology is used for modelling masonry vaults as a discrete network of forces in equilibrium with gravitational loads. It is demonstrated that the proposed approach is both effective and robust in terms of assessing not only the safety conditions of existing masonry vaults, the actual geometry of which significantly influences the safety level, but also to design new ones.

Effects of Brick Pattern on the Static Behavior of Masonry Vaults

ABSTRACT 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.