# In-Plane Structural Analysis of Coated Masonry Walls via a Homogenized Model
## 1 Introduction  
Many masonry buildings are located in seismic areas around the world. However, the intrinsic low tensile strength of masonry makes existing structures vulnerable to seismic forces [1], and, for this reason, reinforcing interventions are often required. Some reviews about the retrofitting methods for unreinforced masonry structures can be found in [2; 3]; in particular, they mention certain traditional techniques, such as methods involving surface treatments (e.g., reinforced mortar coating [4]), or to more advanced materials, such as Textile Reinforced Mortar (TRM) [5], Fiber Reinforced Polymers (FRPs) [6], and Fiber Reinforced Cemertitious Matrix (FRCM) [7]. Here, the attention is mainly addressed to an alternative technique that combines the benefits of traditional methods with the utilization of advanced materials, namely the Steel Fiber Reinforced Mortar (SFRM) coating [8]. It involves the application of thin mortar coatings, reinforced with randomly distributed short fibers in the mortar matrix, either on one or both sides of masonry walls. Such a technology entails several advantages, e.g., the time-saving related to constructing thin coatings (since recommendations for minimum rebar cover are no longer necessary) and an enhanced crack control under service loading conditions (due to the improved tensile strength offered by steel fibers in the mortar). Above all, experimental tests [8; 9; 10; 11] have proved the effectiveness of this reinforcing technique for masonry panels, especially improved performances, including increased strength and stiffness, are observed, with outcomes influenced by the steel fiber content. In the literature, a number of researchers resort to homogenized models to investigate both unreinforced and reinforced masonry. Indeed, regarding the former, even today, macro-modeling approaches are employed, where the overall material is modeled as an equivalent homogeneous Cauchy continuum (e.g., [12; 13; 14; 15; 16]). The latter obeys to macroscopic constitutive laws, often derived using homogenization techniques both in elastic and non-elastic regime, e.g., [17; 18; 19; 20; 21; 22]. Among these, ref. [17] is a pioneering work, which calculatesthe equivalent elastic constants of unreinforced brick masonry using a two-step (horizontal and vertical) homogenization process; it is based on a straightforward mechanical system consisting of elastic springs. A similar more recent paper is [22], where a homogenization method is formulated to characterize the behavior of unreinforced masonry walls featuring a regular brick pattern. This involves employing appropriately arranged in-series and in-parallel springs to model an elementary cell and subsequently determining the \"equivalent\" elastic constants of a homogeneous and orthotropic material. Similarly, concerning reinforced masonry, numerous homogenization models have been developed in the literature, according to the reinforcement technology (e.g., ref. [23] relative to CFRP reinforced masonry, ref. [24] referred to FRP grids, ref. [25] for FRCM composites). However, when addressing masonry walls reinforced by SFRM coatings, there are few modeling attempts documented in the literature [8; 26; 27; 28; 29]; moreover, the only homogenization approach is proposed in the recent paper [30]. In the latter, the methods and philosophy developed in [22] for the unreinforced masonry are extended to solve a different problem, namely that of reinforced masonry. A two-step homogenization procedure (in- and out-of-wall) is employed to determine the (membrane) macroscopic elastic constants of coated masonry subjected to stress in their plane. The main result entails deriving simple closed-form expressions for the linear elastic constants of an \"equivalent\" homogeneous and orthotropic material.