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Abstract:A reliable numerical evaluation of the nonlinear behaviour of historical masonry structures, before and after a seismic retrofitting, is a fundamental issue in the design of the structural retrofitting. Many strengthening techniques have been introduced aimed at improving the structural performance of existing structures that, if properly designed and applied, provide an effective contribution to the preservation of their cultural value. Among these strategies, the use of fabric-reinforced polymeric (FRP) materials on masonry surface is being widely adopted for practical engineering purposes. The application of strips or 2D grid composite layers is a low invasive and easy to apply retrofitting strategy, that is able to improve both the in-plane and the out of plane behaviour of masonry elements also in the presence of complex geometries thanks to their flexibility. For this reason, these techniques are frequently employed for reinforcing masonry curved elements, such as arches and vaults. In this paper, taking advantage of an existing general framework based on a discrete element approach previously introduced by the authors, a discrete element conceived for modelling the interaction between masonry and FRP reinforcement is applied to different curved masonry vaults typologies. This model, already used for evaluating the nonlinear behaviour of masonry arches, is here employed for the first time to evaluate the effectiveness of FRP reinforcements on double curvature elements. After a theoretical description of the proposed strategy, two applications relative to an arch and a dome, subjected to seismic loads, with different reinforced conditions, are presented. The benefit provided by the application of FRP strips is also compared with that associated to traditional retrofitting techniques. A sensitivity study is performed with respect to the structure scale factor.

Summary The seismic performance of unreinforced masonry structures is strongly associated with the interaction between in‐plane and out‐of‐plane mechanisms. The seismic response of these structures has been thoroughly investigated by means of experimental testing, analytical procedures, and computational approaches. Within the framework of the numerical simulations, models based on the finite element method provide a good prediction of the seismic performance of unreinforced masonry structures. However, they usually require a high computational cost and advanced user expertise to define appropriate mechanical properties and to interpret the numerical results. Because of these limitations, simplified models for practical applications have been developed during the last decades. Despite this, a great number of these models focus mostly on the evaluation of the in‐plane response, assuming box (or integral) behavior of the structure. In this paper, a simplified macroelement modeling approach is used to simulate the seismic response of 2 masonry prototypes taking into consideration the combined in‐plane and out‐of‐plane action. The numerical investigations were performed in the static and dynamic fields by using pushover analyses and nonlinear dynamic analyses respectively. The latter is a novel implementation of a model previously developed for static analysis. The results obtained from this study are in good agreement with those provided by a detailed nonlinear continuum FE approach, demonstrating the applicability of this macroelement model with a significant reduction of the computational cost.

the ones obtained using expert-based approaches. 50 51 Keywords: Brick masonry structure, Multi-directional pushover analysis, 52 Nonlinear dynamic analysis, Displacement capacity, Analytical fragility curves, 53 HiStrA software.54 3 109formulation for assessing the seismic vulnerability of masonry structures is based 110 on the interstory drift capacity. As reported in the EC8-Part3 [9], the definition of 111 this displacement-based formulation is associated with the type of mechanism 112 governing the collapse of the structure. For instance, a lateral drift of 0.4% is 113 proposed for a Significant Damage LS when the structure experiences a shear 114 failure, and 0.8% (H0/L) when the collapse is ruled by a flexural mechanism, being 115 H0 and L the distance between the contra-flexure point and the point in which the 116 flexural capacity is attained, and the in-plane length of the wall, respectively. It is 117 worth to note that similar failure mechanism-based procedures have been adopted 118

In the city of Cusco and in other parts of Peruvian Andes, adobe masonry is the primary construction material. Adobe continues to be used for the construction of housing because of its low cost, its thermal properties, the use of unskilled labor, and the local traditions of the Peruvian highlands. The Peruvian National Statistics Office estimates that 67% of rural housing in Cusco is made of adobe masonry. Besides, previous seismic events and laboratory tests demonstrated that adobe dwellings without reinforcement are prone to collapse during an earthquake. Therefore, seismic vulnerability assessment of this type of dwellings is necessary aiming at developing proper contingency and mitigation risk policies. Then, fragility curves constitute a key tool when conducting seismic loss assessment because they provide information regarding the probability of exceeding a certain damage limit state as a function of a given engineering demand parameter. This work aims at developing fragility curves, combining in-plane and outof-plane loading conditions, for typical adobe buildings located in the city of Cusco. Initially, a set of one-and two-story adobe houses were studied to determine the geometrical characteristics of representative local building typologies. Subsequently, 1,000 artificial buildings were generated by means of Monte Carlo simulation based on the information gathered. The structural capacity of each artificial building was represented by simplified bilinear and trilinear capacity curves for in-plane and out-ofplane mechanisms, respectively. In order to represent the characteristics associated with subduction processes, a set of ground motion records was established. The damage state of each building was assessed for each seismic record, and this information was collected into a Probability Damage Matrix (DPM). Finally, fragility curves were fitted for each damage state of the cumulative DPM. Preliminary results show that one-and twostory adobe dwellings have a probability of collapse of 30 and 60%, respectively, when considering a peak ground acceleration of 0.30 g, which corresponds to the expected acceleration related to a return period of 475 years over a soil type 2 according to the Peruvian Standards.

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In the Peruvian Andes, a major portion of heritage buildings are made of adobe masonry. Due to the low mechanical properties of this material, historical constructions are mostly characterized by a weak and brittle behavior especially when subjected to seismic loading. In addition, heritage adobe buildings are characterized by timber roofs which do not guarantee a box-type behavior and enable the occurrence of local mechanisms in the out-of-plane direction. During the last decades, there has been an increasing interest in understanding the complex seismic response of this type of buildings. This task is usually conducted by means of computational models together numerical simulations. The current approaches such as Finite Element or macro-element methods present some important limitations such a high computational burden or oversimplified mechanical schemes that consider only the in-plane mechanisms of masonry. In this paper, an innovative numerical tool, which involves the combined interaction between the in-plane and out-of-plane mechanisms with a reduced computational demand, is used for the seismic assessment of an adobe church located in Cusco, Peru. This discrete macro-element modeling approach is validated considering the results obtained with a sophisticated computational tool in terms of capacity curves and collapse mechanisms. These results demonstrate that the proposed modelling approach is capable of properly evaluating the seismic behavior of earthen buildings, and will allow further investigations in a nonlinear dynamic context.

The prediction of the dynamic response of Unreinforced Masonry Structures (URMS) is a very complex task, since it is governed by material degradation and cyclic hysteric behaviour. Procedures based on nonlinear static analyses have been proposed for the seismic assessment of URMS, without properly considering hysteretic energy dissipation during the dynamic response. Even though dynamic nonlinear analyses provide satisfactory simulations of the seismic response, its application requires considerable computational effort and high user expertise for the accurate definition of the material properties, making it unsuitable for practical applications. However, simplified macro-element strategies, capable of simulating in-plane and outof-plane nonlinear responses, could represent a satisfactory engineering solution in the dynamic context. In this study the nonlinear static and dynamic in-plane behaviour of URMS was assessed by means of plane discrete models. The preliminary numerical investigation evidenced the need to define suitable hysteric constitutive laws for reliable nonlinear dynamic analyses of URMS.