Apparently these equations contain three unknowns: the problem is then statically determinate. They are easily integrated, if r = cost, that is to say a circle, under two arbitrary functions, Fi(#>) and F2(^) to be determined from the conditions at the edges. You must also consider the Heyman solution for a cylindrical surface with two lateral frames: in this case he shows how the edges of the shell are not stress free and the shear stress N^v is not null. So two straight edge beams in tension are necessary for equilibrium.

In a Brunelleschi-like dome the lines v=cost are the ellipses (32), even if they approximate circles.

You can note also that this dome has eight symmetry axes: axes AF, BG, DH,..., and axes RC, UC, VC,... For example along RC it can be assumed that Nv^ = 0, so that according to Heyman, F1 (<p) = 0, while along AA'", because of the rib, the tangential stresses N//aa''' , although symmetric, are not null, as described in Figure 10.

Figure 11 shows the slice AA'''I'''I cut off the dome and the forces N//aa''' and N^aa'" which equilibrate its weight. Side A"T" is free of forces, but after the construction of the lantern in the real dome, weights were applied along it.

The equilibrium of momentum with reference to

AI cannot determine the two unknowns N

Niaa'" : N^aa'" is not parallel to the line AI, so that its contribution to momentum is different from zero.

The rib assumes and equilibrate the forces 2N//aa''' , according to the Figure 10, while the forces N^aa'" equilibrate themselves.

Besides, the existence of eight symmetry axes makes the structural behaviour of the dome not far

Figure 10. Equilibrium along the web ribs.
Figure 11. Slice AA'''I'"I with unit membrane forces along the ribs.

from that of a surface of revolution. For instance Timoshenko and Heyman present the solution for semi-spherical domes, in the membrane theory, also with the upper portion removed, in which, in each point, a line of curvature is the meridian, while the other one is obtained with a plane normal the meridian, containing the normal m to the surface.

The elastic solution with the upper part removed confirms the intuition of Brunelleschi on the possibility of the cantilevered erection of the dome without frameworks.

Writing down the two equations of equilibrium, not identically satisfied, the problem is statically determinate: in fact, for symmetry, shears are null.

In particular from the solution, it is well known that the forces Nv along the meridians are positive for a latitude <p > 51°50'.

The Brunelleschi-like dome has been modelled with a FEM analysis with ANSYS®.

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Figure 12. The Brunelleschi-like meshed dome.

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Figure 12. The Brunelleschi-like meshed dome.

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Figure 13. The diagram of the 1stprincipal stress in Pa. Linear solution.

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Figure 13. The diagram of the 1stprincipal stress in Pa. Linear solution.

The virtual 3D model of the dome has been meshed with the solid element Solid65 (Figure 12). A masonry density of 1800 kg/m3 was assumed.

The case without the weight of the lantern has been primarly considered, as it was during the construction.

Figure 13 shows the linear solution of the dome. In particular it shows that part of the dome has positive stresses, particularly in the lower part of the ribs and of the webs, coherently with the Heyman's both solutions of the cylindrical shell and the spherical dome.

These positive stresses are lower than 1,5 x 105 N/ m2: nevertheless this value is excessive for a Middle Ages masonry. The intuition of Filippo is really surprising. He used the right expedients for the engineering possibilities of his time, as the "lisca pesce" technique and reinforcements in strips of wood and stone (Brunelleschi's specifications). This phase, without the lantern, was run across the construction; so cracking appeared in the cupola (Opera del Duomo 1691, Fondelli 2004).

Figure 14 shows the non linear solution obtained with a crush/cracking FEM analysis, with the value of cracking near zero.

Confronting the Figure 13 and Figure 14 it can be noticed how the equilibrium is possible with minor

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