# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 5 Hygro-Thermo-Mechanical Simulations and Discussion
### Influence of the Initial Conditions on the Hygro-Thermo-Mechanical Response  
It is recalled that the thermal- and moisture-induced strains are dependent on the specific initial temperature and relative humidity values for which the free-strain or zero-strain condition is assumed. As a result, the initial conditions play a major role in the HTM analysis. For instance, the results presented in the previous section were calculated assuming that the wall had initial conditions equal to the average temperature and relative humidity of the exterior environment, i.e., 13.5 \({}^{\circ}\)C and 74% RH. However, if the initial conditions of the wall were assumed equal to the interior space, i.e., 20 \({}^{\circ}\)C and 50% RH, the obtained results would change drastically. Note that the steady-state hygro-thermal equilibrium reached by any of these two cases would be the same. However, the deformed configuration attained with the steady-state equilibrium does depend on the chosen hygro-thermal conditions used as a reference, see Equations (23) and (24). For the sake of clarity,Figure 16 shows the maximum horizontal displacement obtained for the HTM analysis discussed in the previous section performed on the same LM and CM cases but assuming different initial conditions. Figure 14: Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with natural hydraulic lime mortar. Principal stress distribution obtained for the hygro-thermo-mechanical case: minimum principal stress, \(\sigma_{3}\) [MPa], in (**a**) bricks and (**b**) mortar joints; maximum principal stress, \(\sigma_{1}\) [MPa], in (**c**) bricks and (**d**) mortar joints. Figure 15: Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with cement mortar. Principal stress distribution obtained for the hygro-thermo-mechanical case: minimum-principal stress, \(\sigma_{3}\) [MPa], in (**a**) bricks and (**b**) mortar joints; maximum principal stress, \(\sigma_{1}\) [MPa], in (**c**) bricks and (**d**) mortar joints. It is worth recalling that the coefficient of thermal expansion, \(\alpha_{T}\), has been assumed constant. Therefore, the thermally induced strains are directly proportional to the imposed temperature gradient, \(\Delta T\). Consequently, assuming \(\Delta T=T_{high}-T_{low}\), where \(T_{high}>T_{low}\), the positive gradient leads to a positive strain and, thus, dilation or swelling. Conversely, the same gradient with a negative sign, i.e., \(\Delta T=T_{low}-T_{high}\), would produce the same absolute value for the strain but, in this case, associated with shrinkage. As a result, the initial conditions defined for a thermo-mechanical analysis determine whether the material undergoes dilation or contraction, but the absolute value of the resulting strain would be the same. On the contrary, it must be noted that the moisture-induced strains greatly depend on the initial conditions for which the zero-strain configuration is assumed. This behavior is due to the non-linear nature of the coefficient of hygric expansion, \(\alpha_{h}\). In order to illustrate this, the previously introduced brick wall with natural hydraulic lime mortar can be used as an example. Assuming \(\varphi_{50}=0.50\), the coefficient of hygric expansion for brick is \(\alpha_{h}(\varphi_{50})=11.8\times 10^{-6}\) m\({}^{3}\)/kg. Moreover, for \(\varphi_{74}=0.74\), the corresponding coefficient for brick is \(\alpha_{h}(\varphi_{74})=5.6\times 10^{-6}\) m\({}^{3}\)/kg. Time mortar presents a similar trend: namely, \(\alpha_{h}(\varphi_{50})=7.9\times 10^{-6}\) m\({}^{3}\)/kg and \(\alpha_{h}(\varphi_{74})=4.2\times 10^{-6}\) m\({}^{3}\)/kg. Therefore, the coefficients of hygric expansion of both materials double for \(\varphi_{50}\) in comparison with \(\varphi_{74}\). This implies that if the initial conditions are considered as the free-strain equilibrium, a drying wall (\(\Delta\varphi=\varphi_{74}-\varphi_{50}\)) will experience greater deformations than the same material becoming wet (\(\Delta\varphi=\varphi_{50}-\varphi_{74}\)).