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The present paper is devoted to the discussion of a series of unreinforced and FRP retrofitted panels analyzed adopting the Rigid Body and Spring-Mass (HRBSM) model developed by the authors. To this scope, a total of four out of plane loaded masonry walls tested up to failure are considered. At a structural level, the non-linear analyses are conducted replacing the homogenized orthotropic continuum with a rigid element and non-linear spring assemblage by means of which out of plane mechanisms are allowed. FRP retrofitting is modeled adopting two noded truss elements whose mechanical properties are selected in order to describe possible debonding phenomenon or tensile rupture of the strengthening. The outcome provided numerically are compared to the experimental results showing a satisfactory agreement in terms of global pressure-deflection curves and failure mechanisms.

The aim of the paper is to propose and assess the reliability of a modeling strategy which combines the homogenization of the masonry material and the use of zero-thickness interface elements. This strategy is specifically proposed for numerically investigating the structural response of FRP-reinforced curved masonry structures. Indeed, in order to consider the influence of the geometry curvature of the masonry substrate on the local bond behavior of the FRP-strengthening system, bond-slip laws which specifically account for the geometric curvature of the substrate are introduced at the FRP/substrate interface layer. Numerical analyses concerning masonry arches selected from the current literature are presented in the paper in order to assess the reliability of the proposed modelling approach.

The paper presents the results of a series of in-situ tests carried out on two masonry arches, one unreinforced and the other reinforced with SRG (Steel Reinforced Grout). The arches are built adopting a peculiar construction technique using common Italian bricks with dimensions 250 × 120 × 55 mm3 and 10 mm thick mortar joints. One of the two arches has been reinforced with an SRG material constituted by an inox grid embedded into a layer of lime mortar, whereas the second one is maintained unreinforced for comparison purposes. The experimental set-up is designed to apply an eccentric vertical load placed at ¼ of the span in a series of loading and unloading cycles up to the failure. The numerical analyses have been performed using a sophisticated heterogeneous micro-modeling technique, where bricks, mortar joints and the strengthening have been modeled separately. Finally, the numerical outcomes have been comparatively assessed with respect to the experimental results and the crack patterns obtained at t...

[EN] A simple holonomic compatible homogenization approach for the non-linear analysis of masonry walls in-plane loaded is presented. The elementary cell (REV) is discretized with 24 triangular elastic constant stress elements (bricks) and non-linear interfaces (mortar). A holonomic behavior with softening is assumed for mortar joints. It is shown how the mechanical problem in the unit cell is characterized by very few displacement variables and how the homogenized stress-strain behaviour can be evaluated semi-analytically. At a structural level, it is therefore not necessary to solve a FE homogenization problem at each load step in each Gauss point. Non-linear structural analyses are carried out on a windowed shear wall, for which experimental and numerical data are available in the literature, with the aim of showing how quite reliable results may be obtained with a limited computational effort.

A simple quasi analytical holonomic homogenization approach for the non-linear analysis of masonry walls in-plane loaded is presented. The elementary cell (REV) is discretized with 24 triangular elastic constant stress elements (bricks) and non-linear interfaces (mortar). A holonomic behavior with softening is assumed for mortar. It is shown how the mechanical problem in the unit cell is characterized by very few displacement variables and how homogenized stress-strain behavior can be evaluated semi-analytically.

The simple quasi analytical holonomic homogenization approach for the non-linear analysis of in-plane loaded masonry presented in Part 1 is here implemented at a structural leveland validated. For such implementation, a Rigid Body and Spring Mass model (RBSM) is adopted, relying into a numerical modelling constituted by rigid elements interconnected by homogenized inelastic normal and shear springs placed at the interfaces between adjoining elements. Such approach is also known as HRBSM. The inherit advantage is that it is not necessary to solve a homogenization problem at each load step in each Gauss point, and a direct implementation into a commercial software by means of an external user supplied subroutine is straightforward. In order to have an insight into the capabilities of the present approach to reasonably reproduce masonry behavior at a structural level, non-linear static analyses are conducted on a shear wall, for which experimental and numerical data are available in the technical literature....

[EN] The final results of advanced FE analyses performed on a Roman arch bridge, namely the Augustus Bridge (Ponte di Augusto) in Narni, center Italy are presented. The bridge, one of the most impressive Roman artworks, has been injured by several traumatic events during the millennia, the result of which is its present ruined condition. The aims are manifold, starting from a better understanding of the causes at the base of the partial collapse occurred on the central pier, passing through a seismic assessment of the ruined still standing part and ending with a discussion on the role played by Roman concrete on the stability against horizontal actions. An advanced material model exhibiting damage, plastic deformation, and softening in both tension and compression is adopted for Roman concrete. Both the case of a foundation settlement of the central pier and the application of a seismic excitation are investigated, by means of nonlinear static and nonlinear dynamic analyses. Numerical simulations are carried out within the FE code ABAQUS by means of detailed 3D models, using historical documentation and previous results of the latest research carried out on materials, assuming realistic models on both the uniaxial stress-strain relationships under nonlinear load-unload conditions by using independent damage parameters in tension and compression, and the multiaxial behavior ruled by a regularized Drucker-Prager strength criterion. The methodological approach turns out to be potentially valid for all existing Roman bridges. Results highlight the vulnerability of the ruins, that the collapse of the central part was probably due to settlement of the central pier and that Roman concrete plays a crucial role in increasing the stability against earthquake actions.

A two-step homogenisation model, formulated by the authors for the in-plane case, is herein extended for the nonlinear out-of-plane analysis of masonry structures. A rectangular running bond elementary cell is discretised by 24 elastic CST elements and inelastic zero-thickness interfaces. The mechanical meso-scale problem is briefly recalled, whereas the out-of-plane homogenised behaviour is evaluated by means of a simple on-thickness integration of the in-plane homogenised curves. At a macro-scale, the rigid body and spring model is slightly modified to allow both flexural and torsional failure mechanisms. The validation of the numerical approach is achieved comparing with some full-scale masonry panels tested in two-way bending up to failure. A series of nonlinear structural analyses are conducted considering different parameters, which have been varied during the experimental campaign. The numerical results are promising and demonstrate the capability to deal with different failure mechanisms as result of a combination of various experimental aspects.

The 50 m high masonry chimney located in the old industrial facility that houses the School of Engineering of the University of Ferrara, Italy, suffered severe damages during the 2012 Emilia seismic sequence. Afterward, for security reasons, the upper damaged 12.40 m were disassembled. Both before and after shortening, the ratio between the effective mass of the fundamental mode and the total mass is approximately 20%, leading standard pushover analysis methods not to be appropriate for estimating the seismic demand. Using a single, consistent 3D FE formulation, the results of a Modal Pushover Analysis (MPA) and four nonlinear Response History Analyses (RHA) for the shortened and the original chimney were presented in the paper. The ground motions considered in the simulations are accelerograms recorded during recent and less recent devastating seismic events in Italy, New Zealand, and Japan. For both chimneys, a very good agreement between MPA and RHA was observed in terms of lateral displacements. Moreover, for the 50 m high chimney, a strong similarity was observed between the damage maps deriving from the MPA and those obtained with the RHA. All analyses confirmed a significant contribution of the higher modes. For the shortened chimney, the MPA revealed damages in the lower part of the stack (8–21 m), because of a prevailing influence of the fundamental mode. In the RHA, a more evident contribution of the higher modes was observed, probably because of the effect of the vertical component of the ground motion, not accounted for in the MPA.

A simple homogenized rigid body and spring model (HRBSM) is presented and applied for the non-linear dynamic analysis of 3D masonry structures. The approach, previously developed by the authors for the modeling of in-plane loaded walls is herein extended to real 3D buildings subjected to in- and out-of-plane deformation modes. The elementary cell is discretized by means of three-noded plane stress elements and non-linear interfaces. At a structural level, the non-linear analyses are performed replacing the homogenized orthotropic continuum with a rigid element and non-linear spring assemblage (RBSM) by means of which both in and out of plane mechanisms are allowed. All the simulations here presented are performed using the commercial software Abaqus. In order to validate the proposed model for the analyses of full scale structures subjected to seismic actions, two different examples are critically discussed, namely a church facade and an in-scale masonry building, both subjected to dynamic excitation. The...

Two simple homogenization models in the inelastic range for running bond masonry.Subdivision of the RVE into few triangular elastic elements for bricks and non-linear joint interfaces.Closed form or semi-analytical solution of the discretized non linear homogenization model.Utilization at a structural level of a homogenized rigid body and spring model (HRBSM).Comprehensive validation at a structural level on shear walls and a deep beam. Two simple homogenization models suitable for the non-linear analysis of masonry walls in-plane loaded are presented. A rectangular running bond elementary cell is discretized by means of twenty-four constant stress three-noded plane-stress triangular elements and linear two-noded interfaces. Non-linearity is concentrated on mortar reduced to interface, exhibiting a holonomic behavior with softening. The paper shows how the mechanical problem in the unit cell can be characterized by very few displacement/stress variables and how homogenized stress-strain behavior can be evaluated by means of a small-scale system of non-linear equations. At a structural level, it is therefore not necessary to solve a homogenization problem at each load step in each Gauss point and a direct implementation into commercial software as an external user supplied subroutine is straightforward. Non-linear structural analyses are conducted on a variety of different problems, for which experimental and numerical data are available in the literature, in order to show that accurate results can be obtained with a limited computational effort.

Two FE modeling techniques are presented and critically discussed for the non-linear analysis of tuff masonry panels reinforced with FRCM and subjected to standard diagonal compression tests. The specimens, tested at the University of Naples (Italy), are unreinforced and FRCM retrofitted walls. The extensive characterization of the constituent materials allowed adopting here very sophisticated numerical modeling techniques. In particular, here the results obtained by means of a micro-modeling strategy and homogenization approach are compared. The first modeling technique is a tridimensional heterogeneous micro-modeling where constituent materials (bricks, joints, reinforcing mortar and reinforcing grid) are modeled separately. The second approach is based on a two-step homogenization procedure, previously developed by the authors, where the elementary cell is discretized by means of three-noded plane stress elements and non-linear interfaces. The non-linear structural analyses are performed replacing the homogenized orthotropic continuum with a rigid element and non-linear spring assemblage (RBSM). All the simulations here presented are performed using the commercial software Abaqus. Pros and cons of the two approaches are herein discussed with reference to their reliability in reproducing global force-displacement curves and crack patterns, as well as to the rather different computational effort required by the two strategies.

A fast and reliable non-linear heterogeneous FE approach specifically conceived for the analysis of FRP-reinforced masonry arches is presented. The approach proposed relies into the reduction of mortar joints to interfaces exhibiting a non-linear holonomic behavior, with a discretization of bricks by means of four-noded elastic elements. The FRP reinforcement is modeled by means of truss elements with elastic-brittle behavior, where the peak tensile strength is estimated by means of a consolidated approach provided by the Italian guidelines CNR-DT200 on masonry strengthening with fiber materials, where the delamination of the strip from the support is taken into account. The model is validated against some recent experimental results relying into circular masonry arches reinforced at both the intrados and the extrados. Some sensitivity analyses are conducted varying the peak tensile strength of the trusses representing the FRP reinforcement.

[EN] In the present paper, a simple homogenisation two-step procedure is proposed for the analyses of both in and out of plane loaded masonry walls and 3D structures. The ¿unit cell¿ is discretized with 24 triangular plane constant stress (CST) elements and interfaces. Bricks are specified to behave elastically, whereas mortar joints are set as zero thickness non-linear interfaces. The mechanical response of joints is modelled with different holonomic relationships including two dominant failure modes, namely cracking (mode I) and shear (mode II), or a combination of both (mixed mode). In particular, either a piecewise linear or an improved version of the Xu-Needleman exponential law are used, both exhibiting post peak softening. At the structural level, the homogenisation model is implemented into general purpose commercial FE software, modelling the homogenized orthotropic continuum as a discrete assemblage of Rigid Bodies and Homogenized softening Springs (HRBSM). In such a rigid element model, a variety of springs are introduced, to properly characterize both the in plane homogenized shear and normal behaviour, as well as bending and torque (out of plane behaviour). Momentcurvature relationships are evaluated simply by on thickness integration from the knowledge of in plane homogenized stress-strain relationships. Mechanical properties of homogenized springs are identified via classic energetic identification. Three case studies of technical relevance are finally presented for benchmarking purposes: (i) a windowed shear panel, (ii) a church façade modelled with a portion of the perpendicular walls and (ii) a 3D half scale two story masonry building.