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This paper presents an approach to a constitutive model for anisotropic quasi-brittle materials, developed in the framework of rate independent softening plasticity, involving a yield criterion in which an anisotropic friction and cohesion tensors are involved. It turns out to be useful for materials characterized by ultimate behaviour which varies according to the direction, such as composite materials, anisotropic rocks, textiles, masonry. The intrinsic structure of a wide range of structural, geological and industrial materials, such as composites, masonry, wood, textiles and several types of rocks and clays, is the major cause of their anisotropic behaviour. The material response to the same stress state is strongly linked to the sampling orientation with respect to the principal stress axes. The position and the orientation of the clay particles, for example, define a sort of material geometry on which the stress geometry depends. Similar considerations can be applied to a Representative Volume Element in order to describe the macroscopical behaviour of an equivalent anisotropic orientation of the fibers in a composite, the disposition of the bed and head joints in a masonry wall medium resulting from an homogenization technique. Moreover, anisotropic friction plays an important role in the formation of wrinkles and folds in textiles. The intrinsic structure of a wide range of structural, geological and industrial materials, such as composites, masonry, wood, textiles and several types of rocks and clays, is the major cause of their anisotropic behaviour. The material response to the same stress state is strongly linked to the sampling orientation with respect to the principal stress axes. The position and the orientation of the clay particles, for example, define a sort of material geometry on which the stress geometry depends. Similar considerations can be applied to a Representative Volume Element in order to describe the macroscopical behaviour of an equivalent anisotropic orientation of the fibers in a composite, the disposition of the bed and head joints in a masonry wall medium resulting from an homogenization technique. Moreover, anisotropic friction plays an important role in the formation of wrinkles and folds in textiles. The theory proposed in this paper, provides a failure criterion for quasi brittle materials in which the existence of an anisotropic friction and cohesion tensors are postulated. A geometrical representation of the limit domain in the case of plane stress, together with the results of laboratory tests, is presented and discussed.

Earth is one of the most diffused building materials all over the world. Adobe constructions of different type can be found in large areas belonging to developing countries and in some cases are part of Cultural Heritage. The peculiar material characteristics with particular regard to fragility of the blocks together with the strong anisotropy have been experimentally studied, and numerical analyses have been performed in order to design optimal retrofitting techniques. In general classical constitutive models have been applied in this case. This paper proposes an approach to a constitutive model for this material, developed in the framework of rate independent softening plasticity, involving a yield criterion in which anisotropic friction and cohesion tensors are considered. It turns out to be generally useful for materials characterized by ultimate behaviour which varies according to the direction, such as the adobe masonry. A geometrical representation of the limit domain in the case of plane stress, together with the comparison with results of laboratory tests on adobe masonry is presented and discussed. It is shown that the framework of plasticity with internal variables provides a coherent description of the softening problem to represent the behaviour of adobe masonry.

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. Although 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 been only in the last years a key question. Significant is the case of statues and ceramics placed on pedestals. In particular cases, like the Bronzi 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, leaving out of account the filter’s effect due to the action of structure on the show-case or pedestal. 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. The influence of a further rigid body inserted between the moving base and the statue is examined in this paper. A preliminary sensitivity analysis is made on order to obtain the optimal friction coefficient to be created in these last surfaces inserted

Non-linear dynamics is generally recognized as the most reliable method to carry out analyses of structures subjected to earthquake actions. In general non-linear dynamics requires a great level of expertise, as well as cost and time necessary for calculation. The lack of suitable computational techniques has induced seismic analyses involving two classes of simplified procedures: modal approximations and constitutive and structural models with sufficient accuracy and low numerical complexities. The dynamic response of building structures subjected to seismic loads has been often examined using the single-degree-of-freedom model, that provides a good estimation of the fundamental response mode, which is normally responsible for overall structural failure. A SDOF analysis can give a preliminary assessment for a protective structure, even in cases in which the constitutive models are somewhat more complex. The rigid-plastic cantilever beam can be in fact a simple structural scheme to clarify the behaviour of more complex structures and to verify the accuracy of the numerical methods in a non-linear dynamic analysis. In general, significant differences exist between a complete building and uniform beams, nevertheless the continuum model provide useful results. This paper presents a general treatment to develop approximate solutions for rigid-plastic response of structures subjected to base harmonic pulse, that has been shown in literature as an appropriate approach to the seismic analysis. A numerical procedure has been on purpose developed, taking into account two different approaches: a step-by-step solution of the general non linear dynamic problem and the evaluation due to a modal approximate response, satisfying both kinematical admissibility requirements and boundary conditions. An estimation of the error due to the second approach is given. In order to assess the reliability of the approximate procedure it is shown that the approximation does not depend on the forcing accelerogram.

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.

The intrinsic structure of a wide range of structural, geological and industrial materials, such as composites, masonry, wood, textiles and several types of rocks and clays, is the major cause of their anisotropic behavior. The material response to the same stress state is strongly linked to the sampling orientation with respect to the principal stress axes. The position and the orientation of microfibers, for example, define a sort of material geometry on which the stress geometry depends. Some of the research efforts made in recent years to achieve a better understanding of constitutive behaviour of quasi-brittle materials have been developed in the framework of plasticity. The theoretical and numerical problems about the variational formulation of non-associated plastic flow rule has been developed although the complexity of the problem welcomes more studies in the future. Most contributions towards establishing a rational failure theory of quasi brittle materials exhibiting a strong anisotropic behaviour are phenomenological: experimental results are interpreted on the basis of a criterion which is assumed to be suitable for the observed mode of failure. The yield condition, based on the generalization of the Mohr-Coulomb criterion, involves six material parameters. A geometrical representation of the failure domain is provided 3 in the case of plane stress. Experimental tests arranged for the identification of the criterion and performed on composite samples are considered and discussed in this paper. It is shown that the agreement between the theory and experiments is reasonably good.

In this paper the unilateral model for masonry is applied to explain the equilibrium of vaults. In particular, the present study is concerned with the application of the safe theorem of limit analysis to spiral vaults, that is, curved constructions modeled as continuous unilateral bodies. On allowing for singular stresses in the form of line or surface Dirac deltas, statically admissible stress fields concentrated on surfaces (and on their folds) lying inside the masonry, are considered. The unilateral restrictions require that the membrane surface lies in between the extrados and intrados surfaces of the vault and that the stress function, representing the stress, be concave. Such a constraint is, in general, not satisfied on a given shape for given loads: in such a case, the shape has to be modified to fit the constraint. A particular application, namely the double spiral stair of Sanfelice’ palace in Naples, is considered.

Conflicts between conservation and safety assume in the archaeological areas a determinant role, since the constructions to be evaluated are ruins. In these areas safeguard of heritage, safeguard of tourists and exigencies of fruition cannot coincide. An emblematic case in the Pompeii site is the safeguard of isolated constructions made of superimposed rigid blocks, such as isolated columns, groups of columns connected with a trabeation, jambs made of grey tuff cutted blocks without connections with the walls behind. The original state of the elements is often compromised by cracks, out of plane configurations, material detach. Often the archaeological exigency is the preservation of the defects in order to record historical facts. Usually repairs are made by sealing the cracks with grout, reattaching of stone piece with mortar only or pins. The choice of the structural model to assess the seismic vulnerability of elements strongly influences the types of interventions, and this is a significant point in the case of Pompeii. In this paper the out of plane behaviour of superimposed masonry blocks is examined. A significant role is played by the material parameters, often difficult to evaluate. The safety assessment and the comparison with the conservation need can lead to different approach to the cultural heritage with respect to ordinary constructions.

Conflicts between conservation and safety assume in the archaeological areas a determinant role, since the constructions to be evaluated are ruins. In these areas safeguard of heritage, safeguard of tourists and exigencies of fruition cannot coincide. An emblematic case in the Pompeii site is the safeguard of isolated constructions made of superimposed rigid blocks, such as isolated columns, groups of columns connected with a trabeation, jambs made of grey tuff cutted blocks without connections with the walls behind. The original state of the elements is often compromised by cracks, out of plane, material detach. Often the archaeological exigency is the preservation of the defects in order to record historical facts. Usually repairs are made by sealing the cracks with grout, reattaching of stone piece with mortar only or pins. The choice of the structural model to assess the seismic vulnerability of elements strongly influences the types of interventions, and this is a significant point in the case of Pompeii. In this paper the role of the mortar joint is examined with reference to the out of plane behaviour of superimposed masonry blocks. A significant role is played by the material parameters, often difficult to evaluate. The safety assessment and the comparison with the conservation need can lead to different approach to the cultural heritage with respect to ordinary constructions. I conflitti tra conservazione e sicurezza nelle aree archeologiche assumono un ruolo determinante, dato che spesso le costruzioni oggetto di valutazione sono ruderi. In queste aree la sicurezza intesa sia come tutela del bene che come tutela del visitatore e possibilità di fruizione del bene stesso sono aspetti che non sempre coincidono. Un caso particolarmente emblematico nell’area archeologica di Pompei è la conservazione di elementi strutturali isolati di notevole sviluppo verticale come colonne, gruppi di colonne con trabeazione, torri dell’acquedotto in opera laterizia, stipiti di ingresso costituiti da grossi blocchi tufacei messi in opera a secco e senza ammorsatura alle pareti retrostanti. Lo stato originale degli elementi è spesso compromesso da numerose fessurazioni, fuori piombo, distacchi e lacune, e spesso l'istanza archeologica è quella di conservare questi dissesti perché documentano dei fatti storici. Vengono di prassi effettuati dei consolidamenti con sarcitura delle lesioni, riadesione dei pezzi distaccati solo con malta o anche con perni di vario materiale. La scelta del modello strutturale per valutare la vulnerabilità sismica degli elementi, vista l’unicità del bene Pompei nel suo complesso, influenza significativamente le successive scelte di intervento. In questo lavoro verrà mostrato come tenere conto del ruolo del giunto di malta e quindi valutare il comportamento fuori piano della muratura oppure valutare soltanto l’attrito tra le due superfici dei blocchi comporta modellazioni, risultati e valutazioni di tipo diverso. Un ruolo essenziale viene giocato dalla conoscenza accurata dei parametri materiali, che spesso sono totalmente o parzialmente sconosciuti. Le successive valutazioni di sicurezza, confrontate con l’esigenza di conservazione, inducono comportamenti nei confronti del bene culturale diversi rispetto alle ordinarie costruzioni.

The paper deals with the collapse behavior of a historical masonry arch bridge subject to combined seismic loads, by means of the limit analysis and shakedown theorems. The assessment of the collapse loads and shakedown multiplier have been analyzed using lower bound theorems of the limit analysis and shakedown theory. The retrofitting consists in the application of CFRP strips on the extrados of the arch with a significant increase of the collapses and shakedown multipliers. The results of a FEM analysis have been achieved with ANSYS code involving the non-linear material behavior and the structural role of the spandrel walls and filling.

In many seismically active regions of the world there are large numbers of masonry buildings. Most of these buildings have not been designed for seismic loads. Recent earthquakes have shown that many of these buildings are seismically vulnerable and should be considered for retrofitting. Different conventional and unconventional retrofitting techniques are available to increase the strength and/or ductility of unreinforced masonry (URM) walls. This paper reviews and discusses seismic retrofitting of masonry walls with emphasis on the conventional techniques. Retrofitting procedures are discussed with regard to a case study: a stone masonry building in Irpinia region, damaged by the 1980 earthquake. The interventions are evaluated by means of finite elements with a macroelement obtained with an homogenization technique. Linear and nonlinear procedures are compared, and peculiarities of each procedure are shown.

The behaviour of unreinforced masonry walls has been widely investigated in order to assess the seismic vulnerability of masonry structures. Even if partial analyses are not sufficient for a full structural analysis under seismic actions, they can be used to have quick estimation of both collapse load and failure mechanisms. Simplified out-of-plane collapse mechanisms, taking into account connections with floor and transversal walls, are considered to evaluate the behaviour of masonry walls subjected to out of plane forces. In general cantilever walls and walls supported only at the top and bottom edges can be represented as equivalent SDOF system, in order to evaluate by simple static analysis the limit horizontal force at the threshold of rocking, assuming that the wall has cracked and hence the tensile strength of the mortar can be neglected. Similarly, the displacement at the threshold of overturning can be obtained from that simple kinematic model. The value of the collapse load and the consequent load-displacement behaviour is strongly related to the way of computing the displacements and to the way of cracking. This paper presents some remarks on the evaluation of the effective secant stiffness of an unreinforced masonry wall in out of plane loading. Comparisons with the Priestley model are shown.

Con il termine “edifici in c.a.” si intendono usualmente edifici in cui la morfologia strutturale è caratterizzata da telai spaziali in c.a. ordinario. Gli edifici esistenti, che sono stati costruiti nell’arco di tempo che va dallo spazio fra le due guerre mondiali ai nostri giorni, presentano invece una incredibile variabilità di soluzioni strutturali e di materiali costituenti. Queste caratteristiche vengono riscontrate soprattutto negli edifici che oggi vengono inseriti tra quelli strategici o rilevanti. Nelle aree a medio-alta sismicità, o nelle aree che a seguito della recente riclassificazione sismica del territorio nazionale lo sono diventate, la valutazione della vulnerabilità sismica di tali edifici, pur non presentando le peculiarità e le variabilità tipiche delle costruzioni in muratura, costituisce comunque un cimento significativo proprio a causa di questa moltitudine di strutture particolari. Non va inoltre dimenticato che proprio negli edifici in muratura costruiti negli anni ’60 la presenza di strutture o di parti di strutture in c.a. è significativa. In questo lavoro vengono presentati tre casi di studio in cui sono presenti gli aspetti or ora descritti: un edificio a nucleo con finestre a nastro, un edificio costruito negli anni ’30 con calcestruzzo con macroinerti, costruzioni in c.a. di grande luce inserite nel più ampio contesto di un edificio in muratura.

Il ruolo della meccanica strutturale nella rappresentazione del Mediterraneo è fondamentale, data la molteplicità di campi di applicazione e di possibilità che lo sviluppo delle tecnologie offre. Un’opera d’arte o di architettura è sottoposta alle leggi della fisica, della meccanica ed agli eventi che la storia e la natura dispongono. Non è quindi sorprendente che si possa ottenere una “rappresentazione” delle emergenze del Mediterraneo in termini di riconoscimento delle leggi che ne regolano il comportamento meccanico. Il lavoro presenta alcune applicazioni di modelli derivati dalla meccanica strutturale per l’analisi del comportamento di opere che costituiscono Beni Culturali. Vengono in particolare, in questo lavoro, esaminati il comportamento delle colonne isolate considerate come blocchi rigidi e il collasso degli archi in muratura con il metodo del meccanismo. Il comportamento delle colonne dei templi, spesso costituite da rocchi sovrapposti semplicemente appoggiati, come nel caso delle rovine di Cartagine, costituisce un classico problema di meccanica dei blocchi rigidi, che a partire dal primo lavoro di Housner sulla dinamica del pendolo inverso ha trovato il suo sviluppo in una serie di lavori di modellazione teorica Allo scopo di ottenere una prima serie di informazioni, è stata effettuate una serie di prove sperimentali, su oggetti di sagoma semplice disposti su una tavola vibrante con diverse tipologie di supporto. Le prove dinamiche presentate nel lavoro costituiscono una prima analisi qualitativa della tipologia di moto che si innesca in un blocco rigido semplicemente appoggiato, con attrito, su una base in movimento, con moto armonico per accelerazione di picco variabile. Nel lavoro si mostra anche come la tipologia di moto dipendente dalla sola accelerazione di picco, è fortemente influenzata e dal valore del coefficiente di attrito tra blocco e supporto e dalla forma dell’oggetto. L’osservazione dei resti dei crolli di strutture a blocchi lapidei, come i ponti ad arco, ha mostrato che la causa principale del collasso è la perdita dell’equilibrio piuttosto che la crisi del materiale. In molti casi si osserva infatti che i blocchi lapidei crollati si presentano in buone condizioni ed il restauro può essere effettuato riassemblando gli stessi blocchi, come è avvenuto nel caso del ponte di Mostar. Per questo sono stati sviluppati dei codici per la valutazione della sicurezza dei ponti ad arco basati sull’analisi limite e che vanno sotto il nome di “metodo del meccanismo”. Il ponte ad arco viene, nella formulazione più classica del metodo, rappresentato come un assemblaggio di conci rigidi che collassa perché diventa una catena cinematica, a causa della formazione di quattro “cerniere”. Il metodo trascura l’influenza della presenza del riempimento, dei rinfianchi collaboranti con l’arco e della resistenza passiva del terreno in corrispondenza dei vincoli, influenza che gli autori hanno valutato mediante analisi parametrica. Il lavoro inserisce i risultati di una serie di analisi volte ad esaminare l’influenza dei parametri che il metodo classico del meccanismo trascura in un contesto più ampio della meccanica strutturale.

Recent advances in masonry arch bridges, substantiated by extensive testing programs in Europe and United States, provide bridge engineers with increasing confidence that reasonable estimates can be made of the capacity of these structures. Observation of ultimate strength testing indicate that spandrel walls and fill contribute greatly to the strength and stiffness of these structures, although the classical mechanism method do not take into account these effects. Nevertheless the computer programs based on the mechanism method are largely diffused in practice, because of their simple structure. On the other hand, the numerical analyses based on the finite elements involve complication and uncertainties in parameter estimation, so that are often unfit for structural engineering practice. This paper presents the results of a mechanism program developed to take into account the different real effects due to the presence of fill, lateral earth pressure, strong “backing”. The case study is the Roman arch bridge in Pont Saint Martin, Aosta, Italy, which is today in perfect condition. The arch ring was chosen for the study because its geometry (single span, circular shape) and material (cut stones) are among the most widely diffused.

Il problema della protezione sismica dei piccoli oggetti, nell’ultimo decennio, è stato esaminato con riguardo, prevalentemente, alla salvaguardia sismica degli oggetti d’arte custoditi nei musei. Non va trascurato che il problema degli oggetti assimilabili a blocchi rigidi non ancorati e semplicemente appoggiati su una base in movimento è di importanza fondamentale per la tutela delle piccole attrezzature ospedaliere e nei trasporti terrestri e marittimi. I primi studi generali risalgono alla metà degli anni ottanta, poi sono stati ripresi organicamente da vari gruppi di ricerca, anche in Italia. In questi lavori la modellazione teorica del problema, che affonda le sue radici in una classica trattazione di dinamica dei grandi blocchi, viene affrontata diffusamente. Per i parametri determinanti la caratterizzazione del problema (fra i quali la tipologia dell’oggetto, la qualità del supporto, l’interazione con i dispositivi di appoggio) vengono assunte quantità “ragionevoli” ma non riconducibili ad un dato sperimentale. Poche sono, infatti, le campagne sperimentali al riguardo. In questo si presentano i risultati di una sperimentazione di laboratorio atta alla determinazione del coefficiente di attrito tra l’oggetto d’arte ed il supporto ( a questo proposito viene descritta una attrezzatura realizzata appositamente) nonché una prima serie di risultati di prove di laboratorio realizzate su oggetti di sagoma semplice, disposti su una tavola vibrante con diverse tipologie di supporto.

The problem of reducing the seismic risk for art objects, that is, the objects generally contained within Museums, is of great interest. The first studies were performed in Japan and were successively organized in a general framework by a research program performed at Southern California University and sponsored by the Getty Museum at Malibu, California. In these papers and in the following Italian studies, the theoretical models for the problem are based on the dynamic behaviour of rigid blocks and have been deeply developed. Unfortunately, because of the great lack of experimental data, determinant parameters for the characterization of the problem (like the friction between two superimposed blocks or between the art object and the support plane) are often assumed without reference to laboratory tests. This paper presents the results of a research program containing the experimental determination of the friction coefficient between the art object and the support (by means of a testing apparatus ad hoc realized) together with dynamic tests performed on simple shaped objects made of different materials. The dynamic tests were performed using an unidirectional shaking table and different supporting surfaces, so that the influence of different friction coefficients has been analyzed.

The results of a simple computer program for the assessment of masonry arch bridges, based on the mechanism method developed by Heyman at the end of sixties, are discussed in this paper. It is shown, by means of a parametrical analysis, that the position of hinges and the safety factor of the bridge are strongly dependent on the load function shape, so that the collapse mechanism in the case of symmetric load needs the presence of six hinges. The study is applied to circular arches, with different parabolic load functions. It is shown that in some cases the safety degree can be improved or reduced by means of simple interventions related to the load distribution.On present, par cette communication, les résultats d’un simple programme d’élaboration pour l’évaluation de fiabilité de ponts à arches en maçonnerie, par le ‘mechanism method’ developpé par Heiman à la fin des années soixante. On va demontrer, par une analyse paramétrique, que la position des charnières et le facteur de sécurité sont fortement dépendants de la forme de la fonction du charge, de façon que le mécanisme de rupture en cas de charge symétrique nécessite de la présence de six charnières. L’étude a été appliqué sur des arches circulaires, avec de différentes fonctions de charge. On éprouve que en quelques cas le degré de sécurité peut être affiné par des interventions sur la distribution du charge. On present, par cette communication, les résultats d’un simple programme d’élaboration pour l’évaluation de fiabilité de ponts à arches en maçonnerie, par le ‘mechanism method’ developpé par Heiman à la fin des années soixante. On va demontrer, par une analyse paramétrique, que la position des charnières et le facteur de sécurité sont fortement dépendants de la forme de la fonction du charge, de façon que le mécanisme de rupture en cas de charge symétrique nécessite de la présence de six charnières. L’étude a été appliqué sur des arches circulaires, avec de différentes fonctions de charge. On éprouve que en quelques cas le degré de sécurité peut être affiné par des interventions sur la distribution du charge

The rehabilitation and conservation has shown in recent years the need of reliable methods for assessing masonry arch bridges: it is important not only to maintain ancient structures in good conditions, but also, when necessary, to be able to estimate their safety factor as accurately as possible. Starting from a real case, this paper presents the results of a 3D FEM analysis of a stone masonry arch bridge, performed involving non-linear material behaviour, in which the structural role of the spandrel walls and filling is involved.

The case of biaxially stressed masonry occurs in a large number of walls subjected to complex systems of in-plane loads. Masonry is a material which exibits distinct directional properties, so a failure criterion, because of the material anisotropy, cannot be postulated in terms of principal stresses, like the failure of an isotropic material. In the case of biaxial stress it is necessary to take into account the reference to the effect of a third variable, the orientation of the bed joints relative to the direction of principal stresses. Hence, to define a failure criterion for masonry in a plane stress state, a three-dimensional surface is required in terms of the principal stresses and their respective directions relative to the bed joints. The main objective of this paper is to present and discuss a surface of this type, in which a Mohr-Coulomb behaviour is considered, involving a friction tensor, to take into account the material anisotropy. Experimental tests on masonry specimens, performed by means of the Casagrande box for the identification of the model, are explained in detail.