Homogenized model for herringbone bond masonry: linear elastic and limit analysis

A kinematic procedure to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry is presented. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigidplastic interfaces with friction, representing mortar joints. A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in- plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume REV and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively. When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond pattern inclination and (2) masonry textur

Homogenized limit analysis of FRP-reinforced masonry walls out-of-plane loaded

A three-dimensional (3D) homogenized limit analysis model for the determination of collapse loads of out-of-plane loaded FRP reinforced masonry walls is presented. Homogenization is performed on unreinforced masonry, whereas strips are applied at a structural level on the already homogenized material. Unreinforced masonry strength domain is obtained by means of a compatible approach in which bricks are supposed infinitely resistant and joints are reduced to interfaces with frictional-cohesive behavior and associated flow rule. A sub-class of elementary deformation modes is a-priori chosen in the representative volume element (RVE), mimicking typical failures due to joints cracking and crushing. Masonry strength domains are obtained equating power dissipated in the heterogeneous model with power dissipated in a fictitious homogeneous macroscopic plate. Afterwards, an upper bound FE limit analysis code is implemented to study entire unreinforced and FRP reinforced walls out-of-plane loaded. For unreinforced masonry, rigid infinitely resistant wedge-shaped 3D elements are used. The utilization of 3D elements is necessary to simulate the flexural strength increase induced by the introduction of FRP strips with negligible thickness, which are modeled by means of triangular rigid elements. FRP strips contribution is taken into account assuming that masonry and FRP layers interact by means of interfacial tangential actions. Internal power dissipation is possible at the interfaces between wedge adjoining elements (masonry failure), at the interfaces between triangular FRP and wedge masonry elements (delamination) and between triangular FRP adjoining elements (FRP failure). Two different structural examples are presented to validate the numerical model, namely a FRP reinforced masonry wall in cylindrical flexion and a set of masonry walls with openings in two-way bending. Results obtained with the model proposed fit well both experimental and numerical data available for all the cases analyzed, meaning that the procedure proposed can be used in building practic

Optimal FRP reinforcement of masonry walls out-of-plane loaded: A combined homogenization-topology optimization approach complying with masonry strength domain

open2A novel approach for the rational arrangement of fiber reinforcements on masonry structures based on topology optimization is presented. Following previous experiences on the automatic achievement of strut-and-tie models in reinforced concrete structures, the minimization of the strain energy can be implemented to derive optimal layouts of reinforcement for any structural element. To cope with the brickwork limited strength, the optimal problem can be conveniently reformulated as the minimization of the amount of reinforcement that is required to keep tensile stresses in any masonry element below a prescribed threshold. The out-of-plane macroscopic elastic properties and strength domain of brickwork are derived through an original homogenization approach, which relies upon the discretization of 1/4 of any unit cell by six constant moment elements. Thanks to the limited number of variables involved, fast evaluations of masonry macroscopic strength domains can be obtained. This criterion is implemented into the multi-constrained discrete formulation of the topology optimization algorithm, to locally control the internal actions field over the design domain. Topology optimization is then applied to the investigation of the optimal reinforcement of plain and windowed panels, comparing the conventional energy-based method and the proposed stress-based approach.openBruggi M.; Milani G.Bruggi, Matteo; Milani, Gabriel

Quasi-analytical homogenization approach for the non-linear analysis of in-plane loaded masonry panels

[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.Milani, G.; Bertolesi, E. (2017). Quasi-analytical homogenization approach for the non-linear analysis of in-plane loaded masonry panels. Construction and Building Materials. 146:723-743. doi:10.1016/j.conbuildmat.2017.04.008S72374314

Simple Homogenization-Topology Optimization Approach for the Pushover Analysis of Masonry Walls

A topology optimized rigid triangular FE macro-model with non-linear homogenized interfaces for the pushover analysis of in plane loaded masonry is presented. The shape of the mesh and the position of the interfaces is evaluated through a topology optimization approach that detects the main compressive stress fluxes in the structure. Different values of the horizontal action are considered to derive an adaptive mesh or an optimal discretization that is suitable for multiple loads. Masonry properties are calibrated by means of a homogenization approach in the nonlinear range. To tackle elastic and inelastic deformations, interfaces are assumed to behave as elasto-plastic with softening in both tension and compression, with orthotropic behavior. The two-step procedure competes favorably with classic equivalent frame approaches because it does not require a-priori assumptions on the mesh and on the length of the rigid offsets. An example of technical relevance is discussed, relying into a multi-story masonry wall loaded up to failure

GURU v2.0: An interactive Graphical User interface to fit rheometer curves in Han's model for rubber vulcanization

A GUI software (GURU) for experimental data fitting of rheometer curves in Natural Rubber (NR) vulcanized with sulphur at different curing temperatures is presented. Experimental data are automatically loaded in GURU from an Excel spreadsheet coming from the output of the experimental machine (moving die rheometer). To fit the experimental data, the general reaction scheme proposed by Han and co-workers for NR vulcanized with sulphur is considered. From the simplified kinetic scheme adopted, a closed form solution can be found for the crosslink density, with the only limitation that the induction period is excluded from computations. Three kinetic constants must be determined in such a way to minimize the absolute error between normalized experimental data and numerical prediction. Usually, this result is achieved by means of standard least-squares data fitting. On the contrary, GURU works interactively by means of a Graphical User Interface (GUI) to minimize the error and allows an interactive calibration of the kinetic constants by means of sliders. A simple mouse click on the sliders allows the assignment of a value for each kinetic constant and a visual comparison between numerical and experimental curves. Users will thus find optimal values of the constants by means of a classic trial and error strategy. An experimental case of technical relevance is shown as benchmark

seismic vulnerability reduction of masonry churches a case study

Abstract This paper presents some advanced FE analyses conducted on a masonry church damaged by the recent Emilia earthquake, occurred on 20 th -29 th May 2012. The Nativity of the Virgin Mary parish, located in the province of Ferrara (Italy), was subjected to a post-earthquake strengthening intervention, which consisted of introducing steel profiles, rigid diaphragms and repointing technique for the damaged masonry material. The earthquake highlighted a series of pre-existing structural vulnerabilities. A probable mechanism involving the detachment of the facade was also observed due to the presence of vertical cracks. Three FE models were created in order to investigate the seismic behavior of the church in the original, retrofitted and proposed FRP-retrofitted configurations. The mechanical properties of the material were adapted from National Technical Code recommendations for existing masonry constructions by introducing a macro-scaling homogenization technique. The performed numerical analyses highlighted the seismic vulnerability of the church in the original and retrofitted configurations. A strengthening intervention performed by means of FRP strips was numerically investigated. The results obtained by using the three different FE models were compared and the use of FRPs was found to be a quite reasonable strengthening intervention, ensuring a considerable seismic upgrading

Design of the optimal fiber-reinforcement for masonry structures via topology optimization

AbstractA novel approach for the rational positioning of fiber reinforcements on masonry structures based on topology optimization is presented. Due to the brittle behavior of masonry, the minimization of the strain energy cannot be implemented to generate truss-like layouts that may be interpreted as strut-and-tie models in the discontinuity regions of reinforced concrete structures. To cope with the brittleness of brickwork, the optimal problem can be conveniently reduced to the minimization of the amount of reinforcement required to keep tensile stresses in any masonry element below a prescribed threshold. A strength criterion recently proposed for masonry is employed, based on a lower bound limit analysis homogenization model (Milani, 2011) and relying upon a discretization of ¼ of any unit cell by six CST elements. Thanks to the limited number of variables involved, closed form solutions for the masonry macroscopic strength domain can be obtained. This criterion is implemented into the multi-constrained discrete formulation of the topology optimization algorithm, to locally control the stress field over the design domain. For comparison, the phenomenological Tsai–Wu strength criterion for anisotropic solids is also implemented.The contribution discusses three sets of numerical results, addressing the fiber-reinforcement of some benchmark masonry walls. The optimal reinforcement layouts are found to be affected by the choice of the masonry strength criterion only to a limited extent, as far as failure in the masonry element is mainly due to tensile stresses. Contrary to intuition, placing the reinforcing fibers along the direction of the principal tensile stresses in masonry is also found to be not necessarily the most effective solution, for certain geometries and load conditions

seismic vulnerability mitigation of a masonry church by means of cfrp retrofitting

Abstract The paper presents some numerical results on a Romanesque masonry church located in Emilia-Romagna (Italy), a region recently stricken by a devastating seismic sequence on 20 th - 29 th May 2012. A full investigation of the damages and their comparison with advanced FE analyses, in both linear and nonlinear range are carried out. FE limit analyses are performed through non-commercial software proposed by one of the authors. A remarkable consistency is found among limit analysis results, real performance of the structure under seismic excitation and advanced nonlinear dynamic analyses. In particular, both damage patterns and active failure mechanisms found numerically are consistent with that observed on the church after the seismic event. The results put in evidence the insufficient strength of the apse for combined shear/bending actions, the columns of the central nave for bending, as well as the facade for overturning of the upper part. A seismic upgrading by means of CFRPs composite materials is proposed, designed and analysed quantitatively using FEs, finding an optimal fit between the required performance and the invasivity reduction. The interaction between CFRP strips and masonry substrate is accounted for assuming the behaviour of the reinforcement in agreement with Italian Guidelines for r.c./masonry strengthening with composite materials (CNR DT200). It is found that, with a targeted design, it is possible to prevent premature collapses of the macro-elements, strongly increasing the load carrying capacity of the structure

Fast stability analysis of masonry domes and vaults subjected to gravity-induced loads

This paper presents two different (semi)analytical methods for the limit analysis of masonry structures, i.e., a static approach known as “stability area method”, theoretically framed within the lower bound theorem of limit analysis, and a kinematic approach, based on the upper bound theorem. The analysis is conducted on case studies of masonry domes and vaults subjected to a vertical load applied at the crown. The collapse load is obtained by considering different hypotheses on the masonry tensile and compressive strengths. The results are compared with those deriving from experimental tests available in the literature