# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 3 Hygro-Thermal Simulations and Discussion
### Description of the Numerical Model  
A common type of load-bearing masonry wall was selected as a case study to perform the hygro-thermal simulations. In particular, the chosen wall is made up of bricks with dimensions 205 mm \(\times\) 95 mm \(\times\) 50 mm, and mortar joints 15 mm thick. The resulting structure is a 315 mm thick (three-wythe brick) masonry wall with a total height of 2650 mm. The geometrical configuration of the wall is depicted in Figure 2. The described geometry was used to prepare a two-dimensional (2D) model. The material properties used for the hygro-thermal simulations are presented in Table 1. Two types of mortar were considered for the analyses: namely, natural hydraulic lime mortar (LM) and cement mortar (CM). Moreover, the brick-mortar interfaces were assumed to have perfect contact for heat transfer, whereas imperfect contact was considered for moisture transport. Thus, a hydraulic resistance \(R_{IF}=2.0\)\(\times\)\(10^{9}\) m/s was assigned to the brick-mortar interfaces. The wall is assumed to be located in Guimaraes (Portugal), which has an annual average temperature \(\theta=13.5\)\({}^{\circ}\)C and an annual average relative humidity RH = 74% [60]. Additionally, the wall is considered to have initial conditions equal to these yearly average values; thus, \(\vartheta=13.5\)\({}^{\circ}\)C and \(\varphi_{0}=0.74\). One face of the wall is exposed to the exterior, whereas the opposite face is in contact with a conditioned interior space with constant temperature and relative humidity: namely, 20 \({}^{\circ}\)C and 50% RH. It must be noted that the presented configuration is rather unusual for a building envelope in contact with a modern habitable interior space since it lacks any type of dedicated moisture barrier or thermal insulation. Nonetheless, this configuration is reasonable for certain historical walls and expedient for the analyses at hand, in which thermal and moisture gradients are of interest. In the case of a masonry wall with good insulation on the interior side, adiabatic boundary conditions could be assumed for the internal face. The hygro-thermal simulations were performed through both steady-state (SS) and time-dependent (TD) analyses. The SS analysis was used to evaluate the final hygro-thermal equilibrium attained by the wall considering constant environmental conditions as explained in the previous paragraph. Moreover, two sets of time-dependent analyses were defined. In particular, TD1 was performed to evaluate the evolution of temperature and relative humidity profiles across the wall from the initial state to hygro-thermal equilibrium with the environment. Therefore, the final stage of TD1 is analogous to the results obtained with the stationary analysis SS. Additionally, the transient study TD2 considered the wall already in equilibrium with the environment (the initial conditions were taken from the results of SS or the final stage of TD1) and then imposed variable external conditions. In particular, the external conditions for TD2 were assumed to vary with time following a sinusoidal law, given by:  
\[\vartheta(t)=13.5+10\cdot\cos(2\pi t-\pi/2), \tag{20}\]  
\[\varphi(t)=0.74+0.12\cdot\cos(2\pi t+\pi/2), \tag{21}\]  
for the external temperature and relative humidity, respectively, with \(t\) expressed in days. The proposed thermal variation corresponds to the average maximum temperature excursion that can be experienced in a single day during mid-season periods, i.e., \(\Delta\vartheta\approx 20\)\({}^{\circ}\)C. Additionally, average values of the seasonal relative humidity were selected to determine  
Figure 2: Schematic geometrical configuration of the brick masonry wall: **(a)** transversal cross-section; **(b)** vertical cross-section; **(c)** bond arrangement and geometry of the brick units. Dimensions in mm. the range of daily moisture variation, and the corresponding sinusoidal function was set to follow an opposite trend to the one defined for the temperature.