# In-Plane Structural Analysis of Coated Masonry Walls via a Homogenized Model
## 4 Static Structural Analysis and Model Validation
### Results
#### 4.2.1 Sample System 1  
Results relevant to 3 LM and 1 LM-3D of the Sample System 1 are reported first.  
Figure 4: FE models of Sample System 1: (a) 3 LM; (b) 1 LM-3D; (c) 1 LM-2D.  
Figure 5: Load case schemes and qualitative contour plots of the displacement magnitude on deformed configuration, at the middle plane (\(z=0\)), of: (a) Sample System 1; (b) Sample System 2. The displacements range from a maximum of about 0.004 cm (red color) to a minimum of 0 cm (blue color).  
The curves in Figure 6a,b show a comparison in terms of displacements, evaluated at the sections \(x=\dfrac{L}{2}\) and \(y=\dfrac{H}{2}\), respectively, of the middle plane of the wall (\(z=0\)); a very strong agreement is observed as the curves overlap.  
In Figure 7, normal and shear stresses in the 3 LM (gray lines) and 1 LM-3D (black lines), along given selected lines, are shown: the curves (-a, -b, -c) refer at the vertical section \(x=\dfrac{L}{2}\) of the middle plane of each element, while the curves (-d, -e, -f) refer to variations along the thickness \(z\) at the section \(x=\dfrac{L}{2}\) of the base wall. It is seen that the results obtained by the homogeneous model fit _on average_ those of the non-homogeneous model, since the true stress distribution fast changes in passing through internal and external layers.  
Figure 6: Displacement curves at the middle plane of the panel: (a) \(u\left(\frac{L}{2},y\right)\); (b) \(v\left(x,\frac{H}{2}\right)\). 3 LM (gray line) and 1 LM-3D (black line).  
Then, the stresses soliciting the (homogenized) masonry and the reinforcement, as furnished by the analytical model (Equations (17) and (18)) starting from 1 LM-3D, are shown in Figure 8 (red lines). Remarkably, the analytical predictions approximate the stress distributions detected from the 3 LM (gray lines) to a very satisfactory extent, despite the fact that the homogeneous model is aimed at describing global, and not local, behavior. However, upon closer inspection of Figure 8b, a slight deviation is observed between red and gray curves near the free end (\(y=H\)), where the stresses of the 3 LM (gray lines) cancel out, while those resulting from the analytical distribution (red lines) are nonzero, although with zero average. Such a phenomenon is related to the existence of boundary layers (e.g., [32]), where it is known that the homogenization is satisfactory far away from the free edges. In order to further investigate this issue, normal and shear stresses acting along the horizontal section \(y=\dfrac{H}{2}\) are also determined in Figure 9. It is seen that the 3 LM exhibits a boundary layer near the loaded edge (\(x=0\)), which the homogeneous model (1 LM) fails to capture.  
In addition, the effect of the self-stresses is highlighted in Figure 10, where the laws \(\sigma_{x}\left(\frac{L}{2},y\right)\) and \(\sigma_{y}\left(\frac{L}{2},y\right)\), excluding (dashed lines) and including (solid lines) the self-stresses, are plotted in a small \(y\)-range. It appears that self-stress is negligible at the homogenized masonry (the curves are quasi superimposed), while it is significant at the reinforcement layers.  
Finally, a comparison between 1 LM-3D and 1 LM-2D is also made in terms of displacement and global stress in Figure 11. The coarse shell model (i.e., 1 LM-2D), is observed to approximate the results of 1 LM-3D very effectively, despite having a reduced number of degrees of freedom.