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The structural response of masonry arches is strongly dominated by the arch geometry, the stone block dimensions and the interaction with backfill material or surrounding walls. Due to their intrinsic discontinuous nature, the nonlinear structural response of these key historical structures can be efficiently modelled in the context of discrete element approaches. Smeared crack finite elements models, based on the assumption of homogenised media and spread plasticity, fail to rigorously predict the actual collapse behaviour of such structures, that are generally governed by rocking and sliding mechanisms along mortar joints between stone blocks. In this paper a new Discrete Macro-Element Method (DMEM) for predicting the nonlinear structural behaviour of masonry arches is proposed. The method is based on a macro-element discretization in which each plane element interacts with the adjacent elements through zero-thickness interfaces and whose internal deformability is related to a single degree of freedom only. Both experimental and numerical validations show the capability of the proposed approach to be applied for the prediction of the non-linear response of masonry arch structures under different loading conditions.

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The assessment of the seismic response of historical masonry buildings represents a subject of considerable importance but, at the same time, of very difficult task. Refined finite element numerical models, able to predict the non-linear dynamic mechanical behavior and the degradation of the masonry media, require sophisticated constitutive law and a huge computational cost that makes these methods nowadays not suitable for practical application. In the past many authors developed simplified or alternative methodologies that, with a reduced computational effort, should be able to provide numerical results that can be considered sufficiently accurate for engineering practice purposes. However most of these methods are based on simplified hypotheses that make these approaches inappropriate for monumental buildings. In this paper a three dimensional discrete element model, able to predict the nonlinear behaviour of masonry shell elements, is presented as an extension of a previously introduced spatial discrete-element conceived for the simulation of both the in-plane and the out-of-plane behavior of masonry plane elements. The new macro-element enriches a larger computational framework, based on macro-element approach, devoted to the numerical simulation of the seismic behaviour of historical masonry structures.

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.

This paper presents an automatic approach for the evaluation of the plastic load and failure modes of planar frames. The method is based on the generation of elementary collapse mechanisms and on their linear combination aimed at minimizing the collapse load factor. The minimization procedure is efficiently performed by means of genetic algorithms which allow to compute an approximate collapse load factor, and the correspondent failure mode, with sufficient accuracy in a very short computing time. A user-friendly original software in the agent-based programming language Netlogo, here employed for the first time with structural engineering purposes, has been developed showing its great versatility and advantages. Many applications have been performed both with reference to the classical plastic analysis approach, in which all the loads increase proportionally, and with a seismic point of view considering a system of horizontal forces whose magnitude increases while the vertical loads are assumed to be constant. In this latter case a parametric study has been performed aiming at evaluating the influence of some geometric, mechanical and load distribution parameters on the ultimate collapse load of planar frames.The problem of plastic analysis and design of frame structures has been deeply analysed by many researchers since the middle of the past century [1] by means of two different approaches. The first of these is the finite element method in which the global stiffness matrix of the system is computed and the response of the structure is obtained solving iteratively a set of non linear equations [2]. In this analysis the history of loading is applied incrementally until the failure of the structure, therefore the analysis can be very time-consuming.The second approach is directly based on the kinematic theorem of limit analysis; by analysing all the possible collapse mechanisms of a structure and the related collapse loads, the correct ultimate load is determined seeking the absolute lowest value among the considered mechanisms. This method therefore does not require the direct computation of stiffness matrix and it is not necessary to apply the complete history of loading.Many contributions can be found in the literature aiming at solving plastic collapse problems for example by means of linear programming [3][4][5] or at the optimal plastic design of beams [6] or frames considering as constraint the minimum weight of the structure [7-10] or a generic cost function of design variables [11].In the limit analysis kinematic approach, one of the most frequently used method is that first developed by Neal and Symonds [8,9] in which only the elementary mechanisms are analysed and these are combined to obtain a final collapse mechanism whose load factor is lower than all the possible

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