# Brunelleschi's Dome: A New Estimate of the Thrust and Stresses in the Underlying Piers
## 10 Thrust Estimate of Brunelleschi's Dome  
The thrust of Brunelleschi's Dome has been evaluated with regard to a Dome slice equal to \(1/8\) of the entire Dome, referring to the single web supported directly by the underlying pier. The outer and the inner shells forming the Dome are strictly connected by ribs and the horizontal arches, so that the vault, despite its complexity, can be accurately described as behaving like a single solid structure. Making reference to Figure 8 that shows the octagonal horizontal section of the dome, we notice that the meridian sections of the Dome change with the position of the vertical sectioning planes so that the thrust, corresponding to the various slices, will change along the drum. The thrust, on the unit length of the average circle at Dome springing, will thus exhibit the undulating pattern sketched out in Figure 8, in which the middle sections of the webs correspond to the dotted lines. The computation of the thrust of the dome is made by firstly dividing the slice into 31 voussoirs and determining their weights and the positions of their centers with respect to a definite coordinate system, as shown in Table 1. The unit weight of the masonry voussoirs has been assumed to be equal to 1.85 t/mc. The first voussoir defined by the number 1 corresponds to the voussoir at the crown. The corresponding share of the weight of the lantern is reported in the last row of Table 1.  
According to the foregoing calculations, the total weight \(G_{s}\) of the slice, evaluated up to the height of the extrados of the drum and taken as \(1/8\) of the whole dome, including the share of the lantern, equals 3575 t.  
The maximum kinematic thrust \(S_{kin}(\textbf{v})\) is found by trials, varying the position of the internal hinge O along the intrados curve of the slice. Four positions, denoted by 1, 2, 3 and 4 (Figure 9), have been chosen to place the internal hinge at the intratrados of the dome. In Figure 9, the pressure curve, corresponding to the calculation of the minimum thrust performed in [16] (pp. 260-261) by means of the statical approach, is also sketched. We can recognize the approximation of this evaluation due to graphical imperfection in the construction of the sequence of funicular polygons. Figure 8: Variation of the thrust along the sections: dotted lines indicate the web centers. The maximum kinematic thrust \(S_{kin}(v)\) is found by varying the position of the internal hinge O along the intrados curve of the slice. In the following section, four different mechanisms, numbered from (1) to (4), are considered and the corresponding kinematic thrust \(S_{kin}(v)\) is evaluated. Quantities \(D^{*}\)\({}_{i}\) and \(H^{*}\)\({}_{i}\) (\(i\) = 1, 2, 3, 4) are, respectively, the distances of the internal hinges (1, 2, 3, 4) of the assumed mechanisms from the vertical and the horizontal alignments passing through the top A of the slice (Figure 9). Table 2 develops the calculation of the kinematical thrusts according to the various settlement mechanisms corresponding to the chosen four positions of the internal hinge O. The four kinematical thrusts corresponding to the assumed positions of the internal hinge, respectively, are  
\[ \begin{array}{l}\sum G_{i}(D*_{1}-x_{i})/H*_{1}=425.48t\\ \\ \sum G_{i}(D*_{2}-x_{i})/H*_{2}=460.64t\\ \\ \sum G_{i}(D*_{3}-x_{i})/H*_{3}=470.81t\\ \\ \sum G_{i}(D*_{4}-x_{i})/H*_{4}=452.70t\end{array}  \tag{32}\]  
Other choices of the positions of the internal hinges give lower values. The maximum of the kinematical thrust in the slice reaches a value of 470.8 t, corresponding to position 3 of the hinge O. With a previous calculation of the thrust performed by means of the statical approach, i.e., with the construction of a sequence of funicular polygons, a value of 400 t was obtained in [16] (pp. 260-261). This last value, because of the approximations inherent the funicular polygon layout, can be considered less accurate than the value of 470.8 t obtained by the kinematical approach.