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This paper discusses and extends some main features of the rigid-block equilibrium (RBE) method. RBE is a numerical approach that frames the equilibrium problem of rigid-block assemblies as an optimisation problem to compute possible internal and equilibrated singular stress states. The contact between blocks is considered having a finite friction capacity and the unilateral behaviour is modelled through a penalty formulation. In particular, the penalty formulation widens the standard admissible solution space of compressive-only forces by allowing for tensile forces appearing on potentially unstable regions. The RBE objective function minimises the interface forces whereas the constraints are linear functions enforcing the static equilibrium of the whole assembly. In this paper, along with the original quadratic objective function, the authors propose a linear function to illustrate and explore the role played by both the nodal forces and the interface resultants. Moreover, the authors show how RBE can be used to explore different admissible internal stress states – for example, due to increasing, static, horizontal actions.

Masonry buildings are the main part of the Architectural heritage in the World. The need of preservation for the historical memory, together with the exigence of fruition and use, has evidenced the need of appropriate analysis tools. In general, the stresses in masonry structures are usually considerably lower than those required for their material failure, so that the stability is mainly due to their shape and self weight. These aspects allowed in the past the simple realization of construction models in a reduced scale in order to assess the stability. As the history has pointed out, the behaviour of constructions is strongly linked to the workmanship and the good realization of the construction details. Local or global collapse can occur in fact due to pathologies arising during the construction or to bad realization of construction details. This paper deals with this last aspect, examining three different masonry buildings: the first one is a brick masonry building in Bologna: San Barbaziano Church, the others are two Roman domus in two different archaelogical sites in Campania: Villa Regina (Boscoreale, Napoli) and Villa dei Misteri (Pompeii, Napoli). The defects have been divided into two classes: structural defects and non structural ones. It is shown that structural defects can cause a global damage and could involve significant interventions.

Since a few decades, the Discrete Element Modelling (DEM) method has been adopted by many authors as a reliable tool for the structural assessment of unreinforced masonry (URM) structures. In this paper, through compas_dem and using 3DEC by Itasca as a solver in the background, we investigate the mechanical behaviour of a three-dimensional URM structure combining the effects of foundation displacements and geometrical imperfections. For this purpose, we consider three different models of the above-mentioned structure. The first one is a perfect digital model, while in the other remaining two models, random geometrical imperfections are applied to the perfect model in order to investigate their influence. After post-processing the 3DEC results, the influence of the applied vertical settlement and geometrical imperfections is explored in terms of crack pattern/mechanism, internal stress states, and the thrust exerted on the supports. The aim of this paper is not to find the actual stress state of the highly indeterminate structure, but to investigate the role played by the combined effects of foundation displacement and geometrical imperfections on the internal stress state.

Unreinforced masonry is the most diffused construction material in the major part of historical centers in Europe. In a building subjected to earthquake forces the contribution of in-plane shear resistance of the masonry walls is a determinant factor for the stability of the whole structure. In particular, the masonry piers are the structural elements subjected to the combination of normal and shear forces. In general, ductile tools to model the in plane behaviour of masonry are always welcome in order to evaluate the capacity of walls subjected to vertical and horizontal actions. In this framework, two no-tension approaches to model the behaviour of masonry walls loaded with in-plane forces, involving a minimum energy procedure, are presented. Both the procedures allow the representation of the stress maps in the panel in case of monotonic increase of shear load. The results of the numerical analyses are compared and discussed., Frattura ed Integrità Strutturale, 14 (51)

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., Applied Sciences, 11 (8), ISSN:2076-3417

In this paper, we introduce an extension of the piecewise rigid displacement (PRD) method for addressing the stability of a generic two-dimensional masonry structure subjected to large displacements. So far, the PRD method has been applied to simulate cracks in the reference configuration considering small displacements. Here, we investigate both cracks and internal forces in the presence of large foundation displacements, also providing an approach to estimate their maximum allowable value. The proposed extension allows accounting for the evolution of both mechanism and corresponding internal stress state due to the increasing prescribed displacements. After benchmarking the PRD analysis of a pointed arch with a physical test and results obtained through the Discrete Element Modelling software 3DEC, its strength/use is demonstrated on a complex numerical application, based on the cross-section of a Gothic cathedral. Also in this case, the PRD results are compared with a 3DEC analysis showing good agreement in terms of cracks evolution, internal stress states and displacement capacity. This paper shows the ability of the PRD method in solving in a few seconds a critical issue in the field of masonry structures that is, the effects of large foundation displacements, both in terms of mechanisms and forces, simultaneously., Computers & Structures, 242, ISSN:0045-7949

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 re-distribution, 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., Engineering Structures, 228, ISSN:0141-0296

In the present paper, we propose the Continuous Displacement for Fracture (CDF) method, a continuous energy-based numerical approach to find mechanisms and crack patterns exhibited by 2D masonry structures subjected to given loads and settlements. The structure is modelled through the normal, rigid, no-tension material, and the equilibrium problem is solved as the minimum of the total potential energy (TPE). With the CDF method the solution is sought in the space of continuous functions. The CDF performances are compared and illustrated against the PRD approach that finds the TPE minimum in the space of small, piecewise-rigid displacements. The CDF method is displacement-based approach, allowing for a direct control of the effects of foundation settlements. Some problems are proposed to benchmark the methodology against both PRD and analytical solutions to also clearly illustrate its peculiarities. Finally, its use and potentials are benchmarked and compared on a case study. CDF provides results in good agreement with both the PRD approach and another more sophisticated model. The main outcome is that, although more computationally cumbersome, CDF is mesh independent and perfectly captures a clear subdivision of the structural domain into macro-regions behaving as rigid or quasi-rigid bodies.

This work develops new strategies to robustly apply Thrust Network Analysis (TNA) for the assessment of unreinforced masonry (URM) structures studied within the frame of limit analysis. It formulates and solves a nonlinear optimisation problem on thrust networks considering relevant constraints for the assessment of URM structures. Geometrical and force constraints are include such as the consideration of the structural envelope and bounds on the reaction forces. The objective functions studied correspond to the minimum and maximum horizontal thrusts of the vaults. To evaluate the level of stability of an existing structure, this work develops a methodology to estimate the geometric safety factor (GSF) and the minimum thickness of masonry vaults by solving sequential optimisation problems for increasingly tightened geometrical bounds. This procedure is implemented in a Python-based, open-access tool within the COMPAS framework and illustrated here on two- and three-dimensional applications that are relevant for the structural analysis of historical constructions.

This paper proposes an extension of the Piecewise Rigid Displacement (PRD) method based on a new dual linear programming problem that minimises the complementary energy. Before, the PRD method had been applied to solve the kinematical problem for masonry-like structures composed of normal, rigid, no-tension (NRNT) material minimising the total potential energy. Specifically, the PRD method frames this minimum-energy search as a linear programming problem whose solutions are displacements and singular strain fields (cracks). Here, we show that the corresponding dual linear programming problem discretises the minimum of the complementary energy and returns, as solutions, admissible internal stress states compatible with the crack pattern obtained by solving the primal problem. Thus, these two minimum-energy criteria are dually connected, and their combined use allows coupling mechanisms and internal forces with settlements or homogeneous boundary displacements. This allows addressing different mechanical problems: equilibrium and stability of the reference configuration, effects of settlements, and mechanisms due to overloading (e.g. horizontal forces). Since the NRNT material represents the extension to continuum media of Heyman's material model, the PRD method offers an extremely fast, limit analysis-based, displacement approach that allows simultaneously finding mechanisms and compatible internal forces for any boundary condition, loads and geometry., Mechanics Research Communications, 107, ISSN:0093-6413

This paper presents a methodology to assess the stability of vaulted masonry structures using Thrust Network Analysis (TNA). It offers a new numerical strategy to compute the Geometric Safety Factor (GSF) of a given structure by directly evaluating its minimum thickness. Moreover, it provides an approach for tracing the vault's stability domain based on its extreme thrust values, which indicates the robustness of the structure. Together, these outcomes represent a proper measure of the safety level of masonry structures. Such results are obtained from constrained nonlinear optimisation problems (NLPs) with appropriate objective functions and constraints enforcing the limit analysis’ admissibility criteria. Networks with fixed horizontal projection are considered, for which the spatial geometry is a function of the independent force densities and the height of the support vertices. A faster, interactive procedure is proposed to improve the selection of such independent force densities. The range of applications of the present method includes arbitrary network topologies and different support conditions. Beyond analytically described geometries, the method can deal with geometries obtained numerically (e.g. from point clouds). The presented strategy is implemented in a Python-based package, and relevant applications illustrate the method's potential in assessing the stability of three-dimensional historic vaulted structures., Computers & Structures, 257, ISSN:0045-7949

In this paper an homogenization procedure of a Vierendeel girders with elastic joints is shown. The method is based on the Stephen transfer matrix analysis and employs as substitute continuum a polar Timoshenko beam. The polar character of the equivalent beam arises quite naturally from the analysis of the pure bending eigenvector components of the girder force transfer matrix. Through the girder unit cell, two bending moments are transmitted: one is generated by the couple of axial forces acting on each nodal section of the beam, the other is produced by the moments applied at the nodes of every ones by the adjacent cells and is modelled as the resultant of the micro-polar moments. Transfer force eigenvector analysis reveals that the unit cell bends maintaining the webs undeformed, a property that allows evaluating both the bending stiffnesses and the equivalent material micro-polar scale parameter by means of straightforward equations. The accuracy of the proposed method is verified by comparing the predictions of the homogenized model with literature data and with the results of the analysis of a series of girders carried out by the finite element method.

The archaeological site of Pompeii has been subjected to restoration interventions since the beginning of its excavation. After the second World War concrete structures replaced the old wooden roofs erected in the late XIX. A systematic study of these structures is lacking, despite the number and the significant role they play in the stress state of the ancient masonry. In particular, the structural role of atriums roofing structures in the Pompeii buildings is examined. The impact of the restoration interventions actually present in the archaeological area of Pompeii is analyzed. A dynamic analysis has been performed to assess the seismic vulnerability of the ancient buildings in their actual state. The paper shows that maintenance of the existing interventions is a fundamental topic and substitution of r.c. roofs with timber structures cannot be justified by structural reasons. As a case study the entire Domus of the Tragic Poet is analyzed. A simulation involving a wooden roof is compared with the actual structure of concrete roof building. © 2020 European Association for Structural Dynamics.

Within the frame of parametric design, in the present work we focus on a very special objective, namely parametrically generating families of purely compressed shells. A similar task can be pursued by adopting for the equilibrium analysis the so-called Thrust Network Analysis for which the shell structure is condensed into a network of bars. Here instead, we adopt a continuum approach, namely the so-called Membrane Equilibrium Analysis. With this continuum approach, a purely compressed membrane equilibrium solution is searched by solving a scalar second-order partial differential equation representing the transverse equilibrium equation of the membrane. The shell is compressed if the membrane surface is contained within the volume of the shell and if the stress potential is concave. For a given shell, the main difficulty is represented by the fulfillment of the concavity constraint for the stress potential. In the present study, this difficulty is overcome by assigning families of convenient concave stress potentials and considering the shape as the unknown. By considering stress potentials or boundary data controlled by a few parameters, such variable parameters can be manipulated in order to alter the end result. Other methods tackling the stress function with the help of a computer, exist in the literature, but the main contribution of the present paper is the shape analysis of compression-only shells with the help of finite element apparatus. A few illustrative examples are presented to demonstrate the method.

The methodical study of seismic safeguard of artistic heritage has considerably spreaded in the last years, thus increasing researchers’ interest in problems concerning monumental buildings. In particular, the seismic safety of art objects, that are the objects generally contained within Museums, is a research field of great interest, being part of research and policy in the more general field of Cultural Heritage, with contribution of several local government and European research grants. It must be noted, in fact, that buildings of historical and cultural significance may contain objects of inestimable value, for which there is no assessment of their effective vulnerability related to how they are displayed or stored. The case of statues and ceramics placed on pedestals, not equipped with any isolation system capable of mitigating the oscillations induced by possible earthquakes can be just considered as an example. This paper focuses the attention on this last problem, i.e. objects that can be considered as rigid bodies simply supported on the main structure, leaving out of account the filter’s effect due to the action of structure on the show-case or furniture. This problem is the same of a large class of non-structural components, such as mechanical and electrical hospital and laboratory equipment that can lose their functionality because of earthquake motions, and structural problem, such as isolated constructions made of superimposed rigid blocks in archaeological areas. The results of a numerical analysis on the rocking behaviour of rigid bodies are compared with experimental dynamic tests performed on simple shaped objects made of different materials and simply supported on a shaking table.

The collapse factor is a significant parameter in the framework of the safety assessment and economical design of ductile structures. This fact draws attention to the necessity of a careful assessment of the limit analysis approaches. The kinematics in these structures arises in fact from the actual rotation of the plastic hinges under axial force and bending moment. It can be shown that it is possible to obtain a reliable tool capable of competing with computationally expensive methodologies. The application of the methods of limit analysis involves a simplified and idealised model of the structure and, notwithstanding the fact that hundreds of papers have been devoted to the topic, some consequences of apparently unimportant simplifications still seem to have not been properly and firmly highlighted. This paper investigates the ultimate load and collapse modes of steel frames under combined vertical and horizontal forces through limit analysis.

Most of the computational approaches based on Heyman's theory rely on the safe theorem and can only tackle kinematical problems (KPs) with non-homogeneous boundary displacements for simple geometries. So far, no computational tools have been developed to directly address the KP for generic 2D structures taking into account different mechanical scenarios and foundation settlements simultaneously. With this aim, compas_prd, a new computational Python-based tool has been developed. It is based on the piecewise rigid displacement (PRD) method, which couples the KP with the equilibrium problem (EP), allowing the search for a mechanism and its corresponding internal stress state even in large displacements. Both KP and EP are framed as linear programming problems, allowing fast computational solving. Looking at an approximate cross-section of a multi-span masonry bridge, different mechanical scenarios are addressed to illustrate the abilities of compas_prd, also taking into account an initial deformed configuration., International Journal of Masonry Research and Innovation, 6 (4), ISSN:2056-9459, ISSN:2056-9467

The recent diffusion of wind turbines has evidenced problems about their vulnerability in different site conditions, as shown by recent collapse of wind towers after severe actions. The analysis of structures subjected to variable actions can be carried out by means of several methods with different levels of accuracy, among which the nonlinear dynamics is generally recognized as the most reliable one. This paper develops a numerical procedure to obtain approximate solutions for rigid-plastic response of structures subjected to base harmonic pulse. The model is applied to a wind turbine tower subjected to inertia forces generated by an harmonic ground acceleration, in which the failure is assumed depending on the formation of shear hinges. The approach presents the advantages of rigid-plastic procedures, in particular an efficient representation of the post-elastic behaviour of the structure, low computational competence and limited number of mechanical parameters.

The equilibrium of cloister masonry vaults, treated as composed of unilateral material in the sense of Heyman, is the topic of the present work. For such a material, the safe and the kinematic theorems of limit analysis can be employed to detect equilibrium and nonequilibrium. In the spirit of the safe theorem, the structure is stable if a statically admissible stress field can be detected. On allowing for singular stresses, here we consider statically admissible stress fields concentrated on surfaces or lines lying inside the masonry vault. Such structures are unilateral membranes, whose geometry is described a la Monge, and the equilibrium of them is formulated in Pucher form, that is, in terms of the so-called projected stresses over the planform Ω. The problem, under purely parallel loads, is reduced to a single partial differential equation of the second-order, in two space variables, where the shape function f and the stress function F appear symmetrically. The unilateral restrictions require that the membrane surface S lies in between the extrados and intrados surfaces of the vault and that the stress function be concave. In the present work, by starting with a sensible choice of a concave stress function F, the transverse equilibrium equation is solved for f by imposing suitable boundary conditions. A cloister vault of Palazzo Caracciolo di Avellino, a XIV century building located along via dell'Anticaglia in Naples, is the case study. For two load conditions, membrane surfaces and geometrical safety factors are identified.

The analysis of rocking behaviour is a key aspect of safety assessment the maintenance of artifacts, like statues vases, since the rocking itself can be the cause of damage at the base of the object, due to impact with the support. By seismic excitation, a freestanding object may in fact enter a rocking state which would cause overturning, so that often the sliding is desirable, given the possibility of evaluating relative displacements to avoid collisions. In the majority of cases, museum exhibitions are not equipped with isolation devices for every single contained object. This paper presents the theoretical numerical bases of a simple low cost isolation system to protect small art objects. Smaller artifacts are in fact the more prone to overturning since the rocking response is highly influenced by the size of the object. The insertion of a further rigid body between the moving base the statue can reduce the effect of base motion due to earthquakes. The case study is a real marble statue by Donatello, placed in the Museo dell’Opera del Duomo in Florence.

This paper deals with the analysis of fracture identification of a generic masonry structure subjected to loads and to kinematical actions (settlements, distorsions). The masonry is composed of normal rigid no-tension material (NRNT), that is a Heyman’s material treated as a continuous body. An energy minimum criterion is used to solve the equilibrium problem, using continuous (C°) displacement fields and adopting a classical Finite Element approximation for the geometrical description. Some examples illustrate the way in which our approximate method detects crack patterns in some simple cases. The C° method, though more cumbersome form the numerical point of view, exhibits great efficiency in the identification of the exact location and orientation of the crack system, compared to methods based on discontinuous displacement fields.

The methodical study of safeguard of artistic heritage and other devices subjected to earthquake and, in general, to time-dependent forces has considerably spreaded in the last years, thus increasing researchers’ interest in problems concerning motions of rigid objects simply supported on a base plane. The behaviour of piece of equipments, statues, storage tanks, or even tall buildings has been in fact studied as that of rigid bodies with relation to different base excitations. In some cases, the possibility of influencing the quality of motion can be a strong tool to reduce vulnerability, like in the cases in which rocking motion is to be avoided and sliding motion is welcome. This paper focuses the attention on this last problem. This is the same large class of both non-structural and structural elements that can lose their functionality because of earthquake motions. The results of numerical modelling of sliding and rocking motion in presence of both different excitations and mechanical parameters are presented and compared with experimental data performed by the authors. The results developed are in good agreement with the laboratory tests, and this assures the reliability of both the analytical procedure and the determination of the parameters involved.

The interest of the scientific community for the seismic behaviour of existing masonry buildings has largely increased in recent years, due to the fact that the seismic events have evidenced a great need of rehabilitation. Masonry structures, and especially low rise buildings, are in fact the most diffused all over the world and in particular in seismic areas in developing countries because of their ease in construction and economy. Moreover unreinforced masonry is the most diffused construction material in the major part of historical centrers in Europe. In a well engineered and constructed masonry building subjected to earthquake forces, in which out of plane mechanisms can be avoided, the contribution of in-plane shear resistance of the masonry walls is a determinant factor for the stability of the whole structure. Based on this last consideration, this paper presents an approach for the analysis of masonry walls loaded with in-plane forces, involving a minimum energy procedure. The results of the numerical analyses presented are compared with those obtained by laboratory tests on brick masonry panels.

Masonry structural modelling needs of a completely different methodology from the ones adopted for ductile structures. In fact, the concepts of strength, stiffness and elastic stability, fundamental for the latter structures, play a marginal role in masonry mechanics. In this respect, Heyman's theory, gives a modern turn to the old methods of masonry design, adopting a set of very simple and clear mechanical hypotheses. In these papers, the basic ingredients of a new method based on unilateral equilibrium and rigid block kinematics, which may allow the implementation of Heyman's model for masonry on a computer, is introduced. In particular a simple method based on energy minimisation, with the possibility of combining the effects of loads and settlements on real masonry structures, is developed.

The investigation of masonry structures behaviour, considered as Normal Rigid No-Tension material (NRNT) and subjected to loads and settlements, is the principal purpose of the present work. The equilibrated solution is a minimum of the energy and, adopting piecewise-rigid (PR) displacement, a minimization strategy procedure is proposed. Some cases will be analysed to illustrate the numerical performances of the PR approach, pointing out the subdivision into macro-blocks arises naturally in solving these minimization problems, and that the subdivisions into macro-blocks predicted by the proposed approximation procedure, are in good agreement with the ones expected and seen in real masonry structures. In particular two applications on masonry panels with irregular openings are exposed simulating the effect of an earthquake, so that variable horizontal forces are applied to the floor levels. The examples shown that the PR strategy appears efficient in reproducing both the location of cracks and the horizontal collapse load multipliers as well as the related collapse mechanisms.

The seismic protection of objects contained within museums is a topic of great interest, especially with reference to how they are displayed or stored. This problem is the same as that of a large class of non-structural components, such as mechanical and electrical hospital and laboratory equipment that could lose their functionality because of earthquakes. Statues and ceramics simply supported on the floor represent a significant set of case. In some cases, like the Bronzes of Riace, isolation systems have been developed. However, in general museum exhibits are not equipped with devices capable of mitigating the oscillations induced by possible earthquakes. The case study of a marble statue placed on a freestanding squat rigid pedestal is examined. The system of algebraic differential equations governing the problem has been derived and included in an ad-hoc numerical procedure. It is shown that the insertion of a squat rigid body with low frictional resistance at the lower interface with the floor, and high frictional resistance at the upper interface with the artifact significantly reduces the amplitude of the rocking response. As a result the artifact rocks without sliding on the rigid base that slides without rocking with respect to the floor. The numerical analysis performed can be a tool to help in the choice of the optimal friction values in the surfaces of the flat block, designed as a simple isolation system.

The broad topic of the present work is Statics and Kinematics of masonry structures, made of monolithic blocks, modelled a la Heyman, that is rigid bodies loaded by external forces, submitted to unilateral constraints, and undergoing small displacements, under the simplifying assumption that sliding on rough interfaces is prevented. Specifically, in this work, we study the effect, in terms of internal forces, of specified loads, by using given settlements/eigenstrains to trigger special regimes of the internal forces. Although our scope here is the analysis of masonry structures composed by monolithic pieces, and whose blocks are not likely to break at their inside, the method we propose can also be applied to generic masonry structures, such as those made of bricks or small stones. Heyman’s assumptions translate, for unilateral continua, into a normality assumption which allows to employ the two theorems of Limit Analysis. These continuous structures may actually fracture everywhere at their inside, forming rigid blocks in relative displacement among each other. Such piecewise rigid-body displacements in masonry are physiological, and rather than the result of over-loading, are most likely the direct product of small changes of the displacement type boundary conditions. However, when in a part of the structure a specific piecewise rigid-body displacement nucleates, that part of the structure exhibits a one degree of freedom mechanism, and becomes statically determined. Therefore, the internal forces can be computed, despite the original uncracked structure being abundantly overdetermined, and then admitting infinite many statically admissible stress regimes. With these assumptions, in the present paper we study the equilibrium and the effect of settlements in a masonry structure made of monolithic blocks. In particular, the triple helical stair of the convent of San Domingos de Bonaval, located in the Bonaval district of Santiago de Compostela, is considered as case study.

The structural restoration interventions in the archaeological site of Pompeii have been performed since the beginning of excavation in the area. In the last century the old wooden roofs erected in the late 1800s were replaced with concrete structures similar to the old ones, realized after the Second World War. A systematic study of these structures is lacking, despite the number and the significant role they play in the stress state of the ancient masonry. The structural role of these roofing structures in the Pompeii buildings and atriums and in the suburban Villas is examined. Starting from a detailed geometrical and historical analysis of the roof, the impact of the restoration interventions actually present in the archaeological area of Pompeii is analyzed in order to assess the seismic vulnerability of the buildings and the key aspects in case of restoration or maintenance. The paper shows that maintenance of the existing interventions is a fundamental topic and substitution of r.c. roofs with timber structures cannot be justified with structural reasons. As a case study the entire structure of the Domus of the Tragic Poet (masonry walls and concrete roof) is investigated.

The present work is concerned with the prediction of the crack pattern produced by given kinematical data (settlements/distortions) in masonry constructions. By adopting the simplified model of Heyman, extending it to masonry structures treated as continuous bodies, we restrict the analysis to the search of displacement fields which are piecewise rigid. Restricting to small strains and displacements we look for the solution of the kinematical problem by minimizing the potential energy of the structure. A variational approximation of the minimum problem is obtained by considering a fixed finite element subdivision of the structure into rigid blocks. Two case studies are presented to illustrate the way in which a particular fracture pattern can be identified as the one associated to the minimum of the energy in this restricted class of piecewise rigid displacements.

The study of the rocking response of statues and in general the objects contained within Museums, is a research topic of great interest, being part of research and policy in the more general field of Cultural Heritage. The seismic protection of buildings of historical and cultural significance has been developed in all the last century. The protection of the contained objects, with reference to how they are displayed or stored, has become only in the last years a key question. Significant is the case of statues and ceramics placed on pedestals. In particular cases, like the Bronzes of Riace, on purpose isolation systems have been developed, while in general museum exhibitions are not equipped with devices capable of mitigating the oscillations induced by possible earthquakes. This paper focuses the attention on this last problem, i.e. objects that can be considered as rigid bodies simply supported on the main structure. This problem is the same of a large class of non-structural components, such as mechanical and electrical hospital and laboratory equipments that can lose their functionality because of earthquake motions. This paper presents the theoretical and numerical bases of a simple and low cost isolation system to protect small art objects, as the first step toward the choice of the optimal friction coefficient to be created in the pedestal. In fact the insertion of a further rigid body between the moving base and the statue can reduce the effect of base motion due to earthquakes. The results of an analysis made by means of a numerical procedure on purpose developed are presented The problem studied concerns the motion of two superimposed blocks of different aspect ratio: a top slender block placed on a flat one, both on a moving base. The hypotheses are made in order to represent the real situation of a marble statue placed on a flat rigid pedestal freestanding on a moving floor.

The safeguard of museum exhibits like vases and statues, laboratory equipments, storage tanks, or even tall buildings and other structural elements subjected to earthquake and in general to time-dependent forces has been in the last years a topic focus. The classical mechanical problem of the motions of rigid objects simply supported on a base plane, developed in the first year of sixties, thanks to the seminal paper by Housner on the inverted pendulum, can be now take advance due to the development of calculation tools. In some cases the quality of motion can be determined varying the material parameters involved. This is a strong tool to reduce vulnerability, especially when rocking motion should be avoided and sliding motion is welcome, as in the case of artistic heritage. This paper focuses the attention on this last problem, common to a large class of both non-structural and structural elements that can lose their functionality because of earthquake motions. The results of a numerical modelling of sliding and rocking motion in presence of both different excitations and mechanical parameters are presented and compared with experimental data performed by the authors. The results developed are in good agreement with the laboratory tests and this assures the reliability of both the analytical procedure and the determination of the parameters involved. A model for a safeguard proposal is provided.

Structural engineering generally deals with structures subjected to dynamic loads with the hypothesis of deformable behaviour. However, in some cases such as art objects, isolated columns, statues, tanks of oil storage, water towers, nuclear reactors, cement blocks used for radiation shielding and special artifacts the simplified model of a rigid block simply supported on a plane can be useful. The case of free oscillations for one simply supported rigid block and two superimposed blocks, together with forced excitations of the base of one block are analyzed in this paper.

The lower bound theorem of limit analysis and masonry vaults made with no-tension materials is the starting point of this paper. Statically admissible stress field, concentrated on surfaces and their folds contained into the masonry vault are taken into account. These ideal surfaces have the form of Dirac delta lines and surfaces for the stress field and can be considered as membranes/arches, in which the singular stress field represents the gen-eralized stress surface. The unilateral assumptions give as a result a membrane contained in the thickness of the vault (the surface lies between extrados and intrados of the vault) and generalized compressive stress on the surface, i.e. a concave stress function. The equilibrium of the membrane can be written in the Pucher form, so that both the scalar stress function and shape function can be determined by the transverse equilibrium and unilateral conditions. The model can be generally applied to no-tension materials, such as masonry and concrete. The paper examines the classical combination of two vaults present in several architectural examples since the Roman age: a domical vault and a barrel vault. Particular attention is given to the fold between the two vaults. The case study is a Roman concrete combination in the plaster room in the Villa dei Papiri at Ercolano.

The present paper deals with the collapse behavior of an historical stone arch bridge. Ancient stone or masonry arch bridges are still a useful and a not rare engineering road in Italy, as well as in other parts of Europe. Some of the most ancient arch bridges are concentrated in Italy and constitute an integral part of the Italian transportation system. They also represent a valuable engineering achievement of the people that have lived in Italy for centuries and a cultural heritage of the world. A finite element modal analysis and seismic load evaluation has been performed building a suitable program code. The assessment of the collapse and shakedown states, related to a nonlinear constrained optimization problems, has been analyzed using lower bound and Bleich-Melan theorems of limit analysis. Whereas the retrofitting of this stone structure cannot be executed in the intrados surface of the stone vault, so that an application of CFRP strips on the arch extrados is a more useful constructive technique, especially if a significant increase of the limit and shakedown load multipliers occurs.

The analysis of structures subjected to earthquake actions can be carried out by means of several methods with different levels of accuracy, among which the nonlinear dynamics is generally recognized as the most reliable one. This paper presents a general treatment to develop approximate solutions for rigid-plastic response of structures subjected to base harmonic pulse, by means of a numerical procedure on purpose developed. The case study is a vertical cantilever beam, with base support, constant mass and inertia distribution, subjected to only ground acceleration. The failure is assumed depending on the formation of a single shear hinge and the results are expressed in general terms. As a result, the motion proceeds through successive spreading and contractions of the plastic front and related elastic returns. Possible applications of the method to real structures can be envisaged. © 2020 European Association for Structural Dynamics.