## 3 Results  
From the FEM_1--CASE A, considering the design's seismic action [33; 34] and evaluated at the base section of the chimney, it obtained an axial force, F\({}_{Z}\) = 2711.20 kN, a bending moment, M\({}_{X}\) = 3877.00 kNm, and a shear force, F\({}_{y}\) = 151.00 kN. Moreover, the top displacement is equal to 0.060 m is obtained. It is important to highlight that in this case, the maximum compression acting on the chimney base section is equal to 2.35 MPa less than the masonry compressive strength (3.55 MPa). On the contrary, more than half of the base cross-section presents tensile stresses with the neutral axis (n) located at the distance d = 1.60 m from the more compression edge of the section (Figure 14, where G is the gravity center, c is the load application point, n indicates the neutral axis and e is the eccentricity) and consequently several crack patterns can be formed. Considering that the tensile strength (f\({}_{td}\)) of the masonry, which characterizes the chimney, is equal to 0, the decompression moment (M\({}_{d}\) = 1556 kNm of the base section is calculated through the well-known Equation (8):  
\[\mathrm{f_{td}}=(\mathrm{F_{Z}}/\mathrm{A_{b}})/(\mathrm{M_{d}}/\mathrm{W})=0, \tag{8}\]  
where A\({}_{b}\) is the area of the chimney base cross-section and W is the section modulus. The obtained value of M\({}_{d}\) occurs for a seismic load equal to 41% of the design's seismic action. Considering the presence of the TMD (FEM_1--CASE B), the values of the actions acting on the chimney base section are F\({}_{Z}\) = 2777.44 kN, M\({}_{X}\) = 2847.00 kNm, and F\({}_{y}\) = 116.70 kN leading to reductions compared to the results obtained for the chimney without the TMD equal to 26.5% for the bending moment and 22.7% for the base shear. Furthermore, the maximum value of the top displacement is equal to 0.042 m (30% lower than the one of FEM_1--CASE A). In this case, the maximum value of the compression stress is equal to 1.86 MPa, significantly lower than the masonry compressive strength and the chimney base cross-section results mostly compressed. In fact, the M\({}_{d}\) value is reached for 66% of the design's seismic action. Table 4 summarizes the comparison of the results above discussed considering both the analyzed cases. The presence of the TMD modifies the dynamic behavior of the system. Figure 15 shows the trend of the equivalent damping (\(\mathrm{\xi_{e}}\)) as a function of r and calculated as indicated in Equation (9):  
\[\mathrm{\xi_{e}}=\mathrm{H_{ampl}}/2, \tag{9}\]  
Figure 14: Compressive and tensile stresses distribution on the chimney base cross-section considering the design’s seismic action. It can be noticed that the equivalent damping that characterizes the chimney with the TMD is \(\xi_{\rm e}=8.5\%\) obtained considering the average value of the \(\xi_{\rm e}\) calculated in correspondence to the two lower peaks for \(\mathbf{r}=0.95\) and \(\mathbf{r}=1.05\). To investigate the TMD functioning over the elastic behavior of the chimney, a fiber model representing the structure without the TMD (FEM_2--CASE A) was implemented to perform pushover analysis [44; 45] to obtain the capacity curve of the masonry chimney through the application of a modal load distribution proportional to the first vibration mode. To consider the nonlinear behavior of the masonry, a compression elastic-brittle constitutive law was adopted considering a tensile strength equal to 0. In particular, the chimney's fiber model is divided into beam elements having a height equal to 5 m, to which the nonlinear properties of the masonry are assigned. Figure 16 reports the fiber discretization of the base section. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Response** & **Case A** & **Case B** & \(\Delta\) \\ \hline base shear kN & 151.0.0 & 116.70 & 22.7\% \\ base moment kNm & 3877.00 & 2847.00 & 26.5\% \\ top displacement m & 0.060 & 0.042 & 30.0\% \\ \hline \hline \end{tabular}
\end{table}