Numerical and Theoretical Models for NFRCM-Strengthened Masonry

The shear behavior of masonry strengthened with natural fabric-reinforced cementitious matrix (NFRCM-strengthened masonry) is investigated through two different numerical models: a multi-layer model considering masonry and reinforcement as different materials and a multi-step homogenized model, where reinforced masonry is considered as a whole. The approaches are compared by performing nonlinear numerical pushover analysis with an increasing shear action applied to the panels. The parametric analysis shows the capacity and limits of both continuous diffused models – defined as a multi-or a single layer - to represent reinforced masonry in-plane behavior.


Introduction
In the field of masonry strengthening, particular attention would be devoted to the use of natural materials, such as fibers and bio-based matrices, for eco-compatible applications. The present research aims to extend the knowledge of mechanical behavior of innovative materials and the development of sustainable solutions for the reinforcement of existing structures, i.e. fabricreinforced cementitious matrix (FRCM) made with natural fibers (NFRCM), starting from interpretation, by theoretical and numerical models, of experimental tests. The modelling of FRCM-strengthened masonry represents a complex task, because both masonry and FRCM are composite materials in a different way and, being an innovative strengthening technique, there is a limited number of contributions dedicated to this topic. Some authors consider the heterogeneous micro-modelling strategy by modelling separately FRCM and masonry components, to reproduce and/or predict the detailed crack patterns and nonlinear behavior of both FRCM and masonry [1]. This behavior would include the composite failure modes, such as the crack propagation in the matrix, the reinforcement textile local failure mechanisms, the bond-slip behavior between reinforcement-matrix and composite-substrate. Due to its complexity, a few studies consider this technique and some authors adopted homogenization approaches for both masonry and FRCM, in order to model large masonry structures [2,3]. Some macro-models consider masonry components (bricks and mortar) as a homogeneous continuum, and the FRCM composites are considered as an equivalent continuum, with the textile fibers assumed as an embedded reinforcement of mortar matrix [4]. These models have also been implemented with reference to NFRCM-strengthened masonry [5]. A first FE-model is here proposed, starting from the above-mentioned hypothesis considering a multi-layer material. A second FE-model considers a whole homogenization for masonry and NFRCM [6] as a simplified alternative to the multilayered model. The two different approaches are compared and calibrated by imposing an increasing shear action on NFRCM-strengthened masonry panels. The differences and similarities of both methods are analyzed and critically compared, aiming to evaluate the applicability and reliability of each proposed model.

Homogenization
A numerical 3D homogenization procedure [6,7] is adopted to obtain the elastic properties of the materials for the multi-and single-layer models. This technique allows to define an equivalent continuum able to reproduce at the macro-scale the mechanical characteristics of the masonry panels and of the reinforcements, by considering the periodic texture of masonry and the periodicity of the NFRCM. For both materials, a representative element of volume (REV) is defined and periodic boundary conditions are applied. Masonry. This work considers a square panel (1160x1160mm 2 ) made with solid clay bricks arranged as 250 mm thick headers and 10 mm thick of mortar joints, while brick width and height are equal to 120 mm and 55 mm, respectively. The 3D homogenization for masonry panels assumes its components -brick and mortar -as isotropic materials and adopts the mechanical parameters -E, ν -reported in Table 1. The geometrical description of REV and the boundary conditions applied are reported in Fig. 1.  NFRCM. This work considers a NFRCM system constituted by a 10 mm thick mortar matrix and a Sisal plain-woven textile reinforcement (mesh 10x10mm 2 ) placed in the middle of the whole thicknessembedded in the mortar. Mechanical properties of NFRCM system are defined by means of 3D homogenization procedure. Both constituent materials -mortar matrix and sisal fiber reinforcement -are modeled as an isotropic material with mechanical properties in accordance to experimental tests [8] and reported in Table 2. The geometric description of REV and boundary conditions applied are reported in Fig.2.
Key Engineering Materials Vol. 817

Numerical models
Numerical pushover analyses of a reinforced masonry panel are performed by adopting a singlelayer model (RM) and a multi-layer model, which considers a layer of homogenized masonry and a mortar layer with an embedded Sisal reinforcement on both panel sides. Results are compared also with the unreinforced case (URM) represented by the homogenized masonry. Masonry panels are subjected to an increasing shear displacement applied at the upper edge and the boundary conditions are assumed fixed at the bottom and sliding at the top. A rigid beam is positioned on the top of the panel to distribute uniformly the lateral displacements considered (Fig.3). Multi-layer model. The modelling strategy consists in the adoption of curved layered shell elements [9] that allow the consideration of different materials arranged as layers into one single 2D

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Mechanics of Masonry Structures Strengthened with Composite Materials III finite element. The whole thickness of the element considers one half of the RM panel subdivided into 2 layers: NFRCM and masonry, with perfect bond assumed between the layers. The NFRCM layer consists in the composite mortar with the sisal textile embedded as a reinforcement element. An eccentricity is assigned to the reinforcement, to settle it in the middle of the mortar layer. Fig. 4 shows the layered shell thickness subdivision: the whole thickness (t) mortar layer (t1), masonry layer (t2) and the reinforcement grid positioned with an eccentricity (e). Most of the NFRCM mechanical properties are obtained through the experimental tests on the composite constituent materials: mortar (E, ν, ft, fc) and sisal fibers (E, ft) [8]. The fracture energy in compression (Gc) and in tension (Gt) are assumed accordingly to the existing literature [10]. All materials parameters used in the FE model are reported in Table 3.  [9]. The element thickness is equal to the whole thickness of URM and RM panels. Material elastic parameters are assigned in agreement with the third homogenization procedure described previously, and a total strain rotating crack model (TRSCM) is used for representing the nonlinear behavior, which is characterized by a parabolic response in compression and an exponential response in tension. The inelastic parameters -tensile and compressive strength, fracture energy in compression and tension -are assumed accordingly to the existing literature [10]. Elastic and inelastic parameters are collected in Table 4.

Results Discussion and Conclusion
In this work, the behavior of masonry panels reinforced with NFRCM has been studied by means of two different numerical simulation approaches: homogenized and multi-layered FE-models.

Key Engineering Materials Vol. 817
As illustrated in Fig. 5 both FE models are able to simulate the increase of resistance in the elastic phase and the peak load that may develop in masonry panels reinforced with NFRCM systems.
The multi-layer model is able to catch the stiffness increment in the plastic and post cracking phases, due to the presence of fibre stiffness input parameter. The single-layer homogenized model is not able to simulate the ductility increment for large displacements, by reference to multi-layer model results. Therefore, it would represent a not effective tool for simulating the behavior of reinforcement systems with natural fibres, where an important ductility increment of reinforced masonry is expected. Further analyses would consider different constitutive laws for NFRCM material, with the implementation of a residual tension strength value, to improve the reliability of RM model. Experimental tests also would give a more appropriate constitutive function to characterize material behavior. Despite this limitation, the homogenized model represents a simple efficient tool for quick predictions.