# Brunelleschi's Dome: A New Estimate of the Thrust and Stresses in the Underlying Piers
## 12 Stresses at the Pillar Base  
The analysis concerns the two abscalal piers even if analogous results can be obtained for the other two piers flanking the nave. At the pillar head (\(h\) = 28.0 m),  
\[N_{28}=12,652\ t\]  
The load \(N_{28}\) is directed along the vertical axis passing through the center of the trapeze section of the single side segment of the octagonal crown. This section, part of the whole cross section of the pillar and sustaining a side of the drum, directly carries the load \(N_{28}\). The area of this section equals about \(\%\) (20.67 + 17.21)0.5 = 79.2 m\({}^{2}\) and the corresponding compression stress is 12,652/79.2 = 16 kg/cm\({}^{2}\). 1. Section of the pillar  
The pillar cross section is hollow in the center, up to a height of 20.0 m, while it is solid from a height of 20.00 m up to 28.00 m.  
The masonry structure of the pillar is made up of about 2.0 m thick stone facings, tightly bound to a rubble and mortar core. All the various dimensions of the section have been taken from documents and drawings and processed via a computer. The hollow section is shown in Figure 13.  
The area A of the base section (hollow) and the distances of the center from its external sides (Figure 13) are  
A = 2,264,890 cm\({}^{2}\); \(Y_{gs}\) = 609.5 cm; \(Y_{gi}\) = 957.3 cm  
The moment of inertia \(J_{x}\) of the section with respect to the central axis \(x\), orthogonal to the symmetry axis \(y\), is:  
\[J_{x}=3.98\cdot 10^{11}\text{ cm}^{4}.\]  
The distances of the core points from the section center C are  
\[d_{\text{sup}}=\frac{W_{i}}{A}=\frac{J_{x}}{A\cdot Y_{gi}}=188.6\text{ cm. }d_{\text{inf}}=\frac{W_{s}}{A}=\frac{J_{x}}{A\cdot Y_{gs}}=290.5\text{ cm}\]  
2. Pillar weight  
The weight \(W_{P\text{ 0-20}}\) of the segment of the pillar from the height 0.00 m up to 20.00 m: Area of the hollow section: \(A\) = 226.5 m\({}^{2}\). \(W_{P\text{ 0-20}}\) = 226.5 \(\times\) 20.00 \(\times\) 2.2 = 9966 \(t\), by assuming the unit weight of the masonry of the pier equal to 2.2 t/mc. The weight \(W_{P\text{ 20-28}}\) of the segment of the pillar from the height 20.00 m up to 28.00 m:  
Figure 13: Position of the pillar hollow section center. The cross section \(A^{\prime}\) of this segment of the pier is solid. A\({}^{\prime}\) = 226.5 m\({}^{2}\) + 90 m\({}^{2}\) = 316 m\({}^{2}\). \[W_{P\ 20\text{-}28}=:316\times 8.00\times 2.2=5562\text{ t}\]  
Total pillar weight: \(W_{pill}\) = 9966 t + 5562 t = 15,528 t.
3. Stresses at the pillar base section Axial load: The axial load \(N_{oo}\) at the pillar base is obtained by summing the various vertical forces transmitted with the weight of the pillar: \[N_{oo}=V_{28}+W_{pill}=12,652+15,528=28,180\text{ }t\]  
Eccentricities of the various vertical loads conveyed to the pillar with respect to the center C of its base section: A force of \(V_{28}\) = 12,652 \(t\) is conveyed through the side segment of the octagonal drum section, at a distance of \(\text{d}_{28}=\text{\char 37}4.18\text{ m}=2.09\text{ m}\) from the internal boundary of the crown. With reference to Figure 13, the corresponding eccentricity \(e_{28}\) of the force \(V_{28}\) = 12,652 \(t\) is:  
\[e_{28}=\text{Y}_{\text{CS}}-2.09=6.09-2.09=4.00\text{ m}.\]  
The weight of the pillar \(W_{pill}\) = 15,528 \(t\) passes through the center of the hollow section. Thus, the following equation gives the eccentricity \(e_{oo}\) of the resultant force \(N_{oo}\):  
12,652-4.00 = 28,180-\(e_{oo}\) and \(e_{oo}\) = 1.8 m. The eccentric load \(N_{oo}\) = 28,306 \(t\) acts outwards at the distance \(e_{oo}\) = 1.8 m from the center C of the section (Figure 13). Bending moments: Vertical cracks cross all of the body of the drum, as shown in Figure 7 on page 413 of [5]. The drum has lost its hooping function. Brunelleschi concentrically inserted in the dome, at the level of first, second and third galleries, three girdling belts of \"pietra di macigno\", a strong sandstone, cramped by iron brackets. A wood chain, made of wood linlets, connected by iron clamps, was also inserted over the first stone chain.