# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 5 Hygro-Thermo-Mechanical Simulations and Discussion
### Hygro-Thermo-Mechanical Behavior of a Brick Masonry Wall  
The results of the HTM model presented in the previous section are examined in the following paragraphs. In particular, the behavior of the wall is studied in terms of minimum and maximum principal stresses, \(\sigma_{3}\) and \(\sigma_{1}\), according to the type of material and case scenario. The general sign convention is followed: thus, negative stresses indicate compression, and positive stresses denote tension. For the HTM analyses, the moment with maximum induced deformation (maximum horizontal displacement of the top of the wall) was chosen for the study. The selected deformed state is shown in Figures 10 and 11 for the LM and the CM cases, respectively, together with the history of displacements for the top of the wall and the imposed boundary conditions. Here, \(u\) and \(v\) are the displacements in the X and Y directions, respectively. It is recalled that the stress discussion is focused solely on a selected mid-height section of the wall. When considering the displacement history for the LM case (see Figure 10), it is possible to observe the initial deformed configuration of the wall before the application of the variable external boundary conditions. In particular, the wall presents a horizontal displacement to  
Figure 10: Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with natural hydraulic lime mortar: (**a**) imposed boundary conditions and displacement history measured at the top of the wall; (**b**) deformed shape selected for the study (maximum horizontal displacement). wards the interior caused by moisture shrinkage of the internal face (\(\varphi_{int}=0.50<\varphi_{0}=0.74\)), which prevails over thermal dilation (\(\vartheta_{int}=20\)\({}^{\circ}\)C \(>\vartheta_{0}=13.5\)\({}^{\circ}\)C). This horizontal displacement is approximately equal on the exterior and interior sides of the wall. On the other hand, the vertical displacement is rather small and remains always below 0.1 mm during the duration of the analysis. This vertical displacement is different on the exterior and interior sides of the wall due to rotation of the section. When the variable boundary conditions are implemented, the wall oscillates following the sinusoidal trend of the external temperature and relative humidity. In fact, the maximum horizontal displacement occurs with some delay after the external temperature peak (overall thermal expansion with greater dilation of the exterior face, i.e., \(\theta_{ext}>\vartheta_{int}>\vartheta_{0}\)) and after the minimum value of the external relative humidity (general shrinkage of the wall still more prominent on the internal face, i.e., \(\varphi_{int}<\varphi_{ext}<\varphi_{0}\)). The displacements obtained for the wall with cement mortar (see Figure 11) follow the same trends described above for the LM case. However, the CM case shows larger deformations. This was expected considering the higher thermal expansion and hygric expansion coefficients of cement with respect to the lime mortar. The stress state obtained for the LM case under the self-weight condition (body force plus roof weight) is shown in Figure 12. As expected, the distribution of minimum principal stresses (see Figure 12a,b) reveals a uniform level of compression in bricks and bed mortar joints, around \(-0.09\) MPa for the selected area (note that compression increases with wall depth). Conversely, a 2D effect can be seen in the vertical head joints, which experience a lower compression level, around \(-0.06\) MPa. A more detailed analysis of the stress  
Figure 11: Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with cement mortar: (**a**) imposed boundary conditions and displacement history measured at the top of the wall; (**b**) deformed shape selected for the study (maximum horizontal displacement).