Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Geometrical and static aspects of the Cupola of Santa Maria del Fiore,Florence (Italy)

A. Cecchi & I. ChiaveriniDepartment of Civil Engineering, University of Florence, Florence, Italy

A. PasseriniLeonardo Società di Ingegneria S.r.l., Florence, Italy

ABSTRACT: The purpose of this research is to clarify, in the language of differential geometry, the geometryof the internal surface of Brunelleschi’s dome, in the Cathedral of Santa Maria del Fiore, Florence; the staticsof a Brunelleschi-like dome have also been taken into consideration. The masonry, and, in particular, the “liscapesce” one, together with the construction and layout technologies, have been main topics of interest for manyresearchers: they will be the subjects of further research.

1 INTRODUCTION: THE DOME OF THECATHEDRAL OF FLORENCE

The construction of the cathedral of Firenze was begunin the year 1296, with the works related to the exten-sion of the ancient church of Santa Reparata: it wasdesigned byArnolfo (1240,1302).The design includeda great dome, based on an octagonal base, to be erectedin the eastern end of the church.The dome is an unusualconstruction of the Middle Ages, (Wittkower 1962):Arnolfo certainly referred to the nearby octagonal bap-tistery San Giovanni, so ancient and revered that theFlorentines believed it was built by the Romans, asthe temple of Ares, hypothesis which was not con-firmed by excavations, that set the date of foundationis between the V and IX century (Rocchi 1996).

In the second half of the fourteenth century the con-struction of the octagonal base was completed; for fiftyyear about the construction yard stood by, testifying thegreat uncertainty about the building technique of thedome, till Brunelleschi’s assignment in 1420.

We should keep in mind that in that time in Italythere were two other great yards open for the construc-tion of the cathedrals of Milan and Bologna, but theconstruction of the dome of Florence was so excep-tional to enable the target to appear beyond the humanpossibilities.

This is understandable since the dome to be builtwould be the largest ever known (Figure 1,2): its base

Figure 1. View of the cathedral of Santa Maria del Fiore.

dimension was internally about 45 m, surpassing thegreatest known, the Pantheon in Rome (43 m about,in concrete) and Hagia Sofia in Constantinople (31 mabout, in masonry); the base itself was laying on fourgreat high piles, so that the height of the top, 90 mabout, and the height of the base, 60 m. about, greaterthan the Pantheon, made it practically impossible toerect the dome by framework, as was done in Rome.Besides, the Roman dome is a spherical revolutionsurface, while the Florentine dome has a much morecomplicated geometrical shape, due to its octagonalbase. Notice that the Pantheon was built in concrete, atechnique which was probably lost in the Middle Ages

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Figure 2. Plan of the cathedral of S. Maria del Fiore(Ximenes).

and was substituted in the dome of Florence by brickmasonry. The static function of the flying buttress ofthe Middle Ages cathedrals, to carry the horizontalforces to the foundations, (Heyman 1966) was left tothe chapels, which surround the base, much more suit-able to the classical tradition of the town, Florentia,founded by the Romans in 59 B.C. (Davidsohn1956).

Vasari (Vasari 1550) is the main source of infor-mation concerning Filippo Brunelleschi’s work. Hewrote down an act, collected in the Museum ofthe Cathedral, where the main dimensions of thebuilding were specified. We should remember thefollowing:

– the dome is composed by two cupolas;– the inferior one has a variable thickness from 2.35 m

in the bottom to 1.49 m at the top and it is vaulted“a quinto acuto negli angoli”. The function of thesuperior one is: “conservalla dal umido (practicalfunction) e perché torni più magnifica e gonfi-ante (esthetical function)”: its thickness varies from0.72 m to 0.43 m;

– 24 stone ribs (pietraforte) link the two cupolas, 1 foreach of the 8 corners and 2 for each web; these ribsare tied round by 6 hoops in pietraforte cramped bymeans of iron brackets;

– further links between the two cupolas are the“volticciole” (small vaults) chained by means ofoak beams;

– the material used is brick masonry, even if at first,stone walls were proposed.

You can note Brunelleschi’s structural intuition,that is to say the employment of a sandwich ribbedstructure, in order to lower the load of heavy vaults.

Nothing he wrote neither about the methods of erec-tion of the vault, even if he used a cantilever techniquewithout frameworks, absolutely new for his times, norabout the masonry “secondo sara allora consigliato

perchè nel murare la praticha insegnera quello ches-sara a seguire” (Brunelleschi’s specifications), norabout the mechanical apparatus he would have usedlater to raise the heavy weights “tirare i pesi per viadi contrappesi e ruote, che un sol bue tirava quantoavrebbero appena tirato sei paia” (Vasari 1550).

2 AUTHORS WHO DOCUMENTED THEDOME. THE AIMS OF OUR RESEARCH

The silence of Filippo is the main cause of the discus-sions that followed on the argument. This seems to becommon to all the Middle Ages constructions. Fitchenspeaks of “. . . the total lack of written documentationon both the engineering structure and the erectionalprocedures. . .” (Fitchen 1961).

Many authors have related the dome and his mas-ter: among many others let’s remember Giovanni diGherardo da Prato (1421), Manetti (1480?), Vasari(1550), Opera del Duomo (1691) (The Opera delDuomo was charged by the Duke of Tuscany to writedown an essay on the stability of the Dome, threatenedby cracks), Guasti (1887), and recently Sanpaolesi(1962), Fondelli (2004), Rocchi (1994). Sgrilli andXimenes have rendered the dome accurately in theXVIII century.

Nevertheless as a result of this information neitherthorough study on the geometry of the dome, nor onits statics is carried out.

Beyond these arguments, the topics which inter-est many researchers, e.g. Di Pasquale (1977), Bartoli(1994), are the methods of construction and erection ofthe dome and of the masonry, particularly the specialone called “a spina pesce”.

The survey of Fondelli, based upon photogramme-try, must be remembered for its accuracy. It started in1968 and was carried on for many years. Fondelli’s tar-get was to render both the outside and the inside of thegreat cathedral and at the same time to link the surveyto the Italian geodetic net.

The task of this paper is to give a contribution tothe knowledge of Filippo’s work on the following fourtopics:

– based on Brunelleschi’s specifications, the abovepapers and our observations the geometrical shapeof the internal surface of the dome is proposed;

– the proposed shape is compared to the results of thesurvey of the dome;

– the differential geometry of the surface is exposed;– the statics of a Brunelleschi-like dome is defined

and analyzed.

Further researches will discuss the masonry, theengineering structure, the details and the erectionalprocedures.

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Figure 3. Geometrical hypothesis on the dome.

3 HYPOTHESIS ON THE GEOMETRICALFEATURES OF THE INTERNAL SURFACEOF THE DOME

Figure 3 shows the geometrical hypothesis, plan andvertical section (AAIVVFIVF).

The plan of the octagon A, B, D,. . ., basis of thedome, can be observed. C is its centre. The linesA′′′A,B′′′B,D′′′D,. . ., are the projections of eight cir-cular arches whose ray is r: they are the rulings of thesurfaces. The octagon, represented by A′′′,B′′′,D′′′,. . .,is the lantern: in section it is represented by the lineAIVFIV.According to Brunelleschi’s specifications thearches’centres aren’t in the centre of the octagon: theyare in the point “quinto di sesto”. E.g. A′′′A and F′′′Fare the projections of the circular arches AAIV, FFIV,whose centres are respectively the points A′′, F′′ suchthat, for instance, A′′F = 1/5 AF. Naturally eight cen-tres form a new octagon A′′,B′′, D′′,. . .,. The pointsof the webs of the dome represented in the horizontalplan by the eight quadrilaterals AA′′′B′′′B, BB′′′D′′′D,DD′′′E′′′E,. . ., belong to eight cylinders, each havingfor rulings two circles: AA′′′, BB′′′; BB′′′, DD′′′; DD′′′,EE′′′;. . ., and straight horizontal lines as generatricesinclined of β = 67, 5◦ (Figure 3).

The main properties of these cylinders are thefollowing (Figure 3):

– a horizontal plane cuts the dome with an octagon,e.g. A′B′D′,. . . .,;

– the cylindrical web. AA′′′I′′′I has its axis in theA′′I′′ segment.This property is common to the otherwebs.

Figure 4. Surveyed points of the western web of the dome.

4 SURVEY OF THE DOME. COMPARISONBETWEEN SURVEYED ANDGEOMETRICAL POINTS

The second aim of our work was to survey one web ofthe dome so that the hypothesis explained in Chapter3 can be compared.

The survey was carried out only with topographicalmethods, using an electronic laser total station, a LeicaTCR 705, 5” angular precision and 2 mm.+2 ppm. lin-ear precision and a digital camera Nikon D300 with a12 mega pixel CCD and 50/20 interchangeable Nikkoroptics.

It is well known (Cecchi 2006) that the laser tech-nology permits the direct survey of inaccessible points,which is the case of this dome.

Beyond this the quick acquisition of data in a dig-ital format in a reference frame, permits a successivecomputer elaboration of the data.

In June 2007 the upper balcony of the cathedralwas reached; this one was built on the bottom of thedome from which the best and nearest view of theinside surface of the dome is possible. The topograph-ical instrument was placed on the middle east side andpoints of the west web were surveyed: their number isabout 700 and includes points of the bordering vaultstoo . The numerical format of the surveyed points isDXF so thatAUTOCAD® immediately can place themin a three dimensional Euclidean space (Figure 4).

This figure clearly shows how the points were sur-veyed with zenithal constant angles: notice that the lastseries refers to points along the upper lantern sides.

Then the digital coordinates can be elaborated withthe least squares method (Kraus 1998).

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Figure 5. The estimated straight line at 64◦ degree on thehorizontal plane projection.The values in the figure representthe difference between the calculated points and the surveyedones.

4.1 Research of the curves connecting pointsdefined by constant zenithal angles

The constant zenith points were projected on the hor-izontal plane and the line which best approximatesthese points was researched.

Then with the least square method the approximat-ing lines can be found.

Figure 4 shows these constant zenith points are notstraight lines but they differ a little from it especiallyin the middle area of the web. These deviations havebeen computed through projections of these points onthe horizontal and vertical planes.

The results have been shown (Figures 5,6), for thesake of brevity, only for the angle 64◦, where thegreatest deviations are noticed.

Out of Figure 5 the maximum deviation in the hori-zontal plan projection is 0.083 m, while in the verticalplane projection is 0.089 m (Figure 6). As the length ofthe chord is 10.666 m, the deviation is at about 0.8%.So it can be observed that this deviation is very modestand it can be disregarded in the global geometry of thedome.

The following step was to consider the intersectionof two such neighboring lines determines a point ofthe curve intersection of two border web (Figure 4).

The next one was to discover the surface on whichthese points are placed. This surface is a plane andthen, by projecting these points on this plane, thecurve which best approximates these projections wasstudied: these curves are circles (Figure 7). The twosurveyed arches radii are respectively 36,11 m and36,23 m.

The authors leave to the proposed future research ofChapter 2 a study on the surveyed differences pictured

Figure 6. The estimated straight line at 64◦ degree on thevertical plane projection.

Figure 7. The estimated round circle.

in Figure 4, but by now it can be possible to formulatesome hypothesis.

The regularity of these deviations and their increasein correspondence of the middle web allow us to sur-mise that Filippo created it for precise reasons, perhapsthese regarding statics.

A second hypothesis results from the assumptionthat these deviations occurred during and after the con-struction of the dome because of the movements whichwere due to cracks (Opera del Duomo 1691, Fondelli2004), elastic deformations, thermal expansion.

4.2 Further research on the dimensions of the dome

The following angles were considered:

– the angle made by the planes containing the twocircles. This angle is 44,92◦. Notice the analogousgeometrical angle of the octagon is 45◦.

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Table 1. Surveyed dimensions.

Dimensions m.

AB 17,26AA′′ 36,11BB′′ 36,23AF 44,94BG 44,96AA′′/AF 0,803BB′′/BG 0,805Zenithal angle AIV (lantern) 36.21◦

– the angles made by the projections of these twoplanes with the side of the surveyed octagon. Theseangles are respectively 67,30◦ and 67,79◦. Notice:for the geometrical octagon this angle is 67,5◦.

From the survey more dimensions have beenretrieved and for clarity’s sake there can be identi-fied through the analogous geometrical dimensions ofFigure 3:

It can be observed that 0,80 is the geometricalcoefficient for the arch “quinto di sesto”.

4.3 Conclusions of the comparison

The surveyed dimensions and, especially, the dimen-sionless measures confirm the geometrical hypothesisof Figure 3 and, besides, show how carefully the domewas laid out.

Chapter 4.1 shows that lines with constant zenithalangle approximate well straight lines: so that thesurface is a ruled one and the ruling curves are circles.

In other words the internal surface of the dome con-sists of eight cylindrical surfaces; two of them intersecteach other in a circular cross section. Each circle is theruling of the two bordering cylinders (Figure 3).

5 DIFFERENTIAL GEOMETRY OF THEINTERNAL SURFACE OF THE DOME

Now the third goal of our research was to represent thedome with the differential geometry methods, e.g. DoCarmo (1976), Sokolnikoff (1951).

With reference to the Cartesian frame (O,i1, i2, i3)Figure 3, let x be a point of the cylinder AA′′′I′′′I, rthe radius of the circle, φ the zenithal angle, v the unitvector of direction IA, v a scalar. Then:

and the unit vector v:

Notice: curves φ=cost are parallel horizontal lines,whose projections on i1, i2 are inclined β with respectto line AC; curves v=cost are translations of the rulingcircle in direction v.

Deriving the vector x with respect to each Gaussiancoordinate (φ , v):

Through the cross vector product the unit vectornormal to the cylinder is readily obtained:

and the Gaussian quantities E,G,F, that define the FirstFundamental Form, can be calculated:

To define the Second Fundamental Form let us formthe derivatives:

so that the quantities:

define the Second Fundamental Form.The Gaussian quantities: LN − M2 = 0 and L2 + M2

+ N2 �= 0. The points of the surface are then parabolic.It is well known that the main curvatures κ are

obtained from the solution of the following equation:

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Then:

κ1 and κ2 represent the main curvatures at any pointof the surface. The first one is null: it refers to thestraight lines of the cylinder, φ = cost. The other onedepends only on φ, as the lines v = cost are translationsin the direction v. This curvature is the one of the nor-mal section ellipse, normal to the vector v. The linesof curvatures, tangent to the principal directions, arethen the lines v=cost and the normal section ellipses,normal to the unit vector v.

In order to find the equations of these ellipses let usconsider a plane through C parallel to the unit vectorss and i3 (Figure 3):

With the position CA′′ = a, its equation is obtained(notice that s and k are parameters):

Let us consider the intersection of y with x:

Let us eliminate three of the four parameters; forinstance resolving for k, there can be obtained:

and the equation of the intersecting curve in thereference (O,i1, i2, i3):

If the reference system is changed (C, i1, i2, i3) androtated in (C, s, v, i3):

Figure 8. The estimated ellipse. In blue the surveyed points,in red the geometrical ones.

which is the equation of the ellipse, line of curvature.For instance for k = 0, the point R is obtained, for k = 1the point T (Figure 3).

In order to show an application of the equations letcompare points of (32) with the surveyed points ofChapter 4.

At first let us fold the vertical plane of trace TR toT′R′ (Figure 3) so that the ellipse con be drawn as thedotted line R′AIVVS.

The surveyed ellipse has its minor axis 32,26 m, andits major is 36,12 (Figure 8). Using of the letters ofFigure 3, R′T′ = 32,26 m., T′S = 36,12 m. Out of (32)the geometrical sizes are respectively R′T′ = 32,44 m.,T′S = 36,11.

It is evident that the difference between geometricand surveyed dimensions is only 0.18 m, that meansthe error is less than 0.01.

6 STATICS OF A BRUNELLESCHI-LIKE DOME

Let us define a Brunelleschi-like dome as an octagonaldome, formed by ribbed cylindrical webs, intersectingalong circles, inclined 67,5◦ with respect to the cylin-der axis. The surface of this dome is represented byequation (32).

Let us consider the statics of this dome.Timoshenko(1959) and Heyman (1977) consider the statics ofcylindrical shells according to the membrane theory.

According to Filippo’s specifications, Chapter 1,this assumption could be hazardous for a practicalapplication: this is only one way of understanding themain static aspects of this dome. In the FEM analysisthe correct thickness has been used.

With reference to Figure 9 and to the well knownsymbols of shell structures, the equilibrium equationsare the following:

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Figure 9. Equilibrium of a cylinder surface with the linesof curvature. q is the vertical force for unit area.

Apparently these equations contain three unknowns:the problem is then statically determinate. They areeasily integrated, if r = cost, that is to say a circle, undertwo arbitrary functions, F1(ϕ) and F2(ϕ) to be deter-mined from the conditions at the edges. You must alsoconsider the Heyman solution for a cylindrical surfacewith two lateral frames: in this case he shows how theedges of the shell are not stress free and the shear stressNφv is not null. So two straight edge beams in tensionare necessary for equilibrium.

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

You can note also that this dome has eight sym-metry axes: axes AF, BG, DH,. . ., and axes RC, UC,VC,. . . For example along RC it can be assumedthat Nvφ = 0, so that according to Heyman, F1(ϕ) = 0,while along AA′′′, because of the rib, the tangentialstresses N//AA′′′ , although symmetric, are not null, asdescribed in Figure 10.

Figure 11 shows the slice AA′′′I′′′I cut off the domeand the forces N//AA′′′ and N⊥AA′′′ which equilibrateits weight. Side A′′′I′′′ is free of forces, but after theconstruction of the lantern in the real dome, weightswere applied along it.

The equilibrium of momentum with reference toAI cannot determine the two unknowns N//AA′′′ andN⊥AA′′′: N⊥AA′′′ is not parallel to the line AI, so that itscontribution 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 axesmakes 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 alongthe ribs.

from that of a surface of revolution. For instanceTimoshenko and Heyman present the solution forsemi-spherical domes, in the membrane theory, alsowith the upper portion removed, in which, in eachpoint, a line of curvature is the meridian, while theother one is obtained with a plane normal the meridian,containing the normal m to the surface.

The elastic solution with the upper part removedconfirms the intuition of Brunelleschi on the possibil-ity of the cantilevered erection of the dome withoutframeworks.

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

In particular from the solution, it is well known thatthe forces Nv along the meridians are positive for alatitude ϕ > 51◦50′.

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

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Figure 12. The Brunelleschi-like meshed dome.

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

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

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

Figure 13 shows the linear solution of the dome. Inparticular it shows that part of the dome has positivestresses, particularly in the lower part of the ribs and ofthe webs, coherently with the Heyman’s both solutionsof the cylindrical shell and the spherical dome.

These positive stresses are lower than 1,5 × 105 N/m2: nevertheless this value is excessive for a MiddleAges masonry. The intuition of Filippo is really sur-prising. He used the right expedients for the engi-neering possibilities of his time, as the “lisca pesce”technique and reinforcements in strips of wood andstone (Brunelleschi’s specifications).This phase, with-out the lantern, was run across the construction; socracking appeared in the cupola (Opera del Duomo1691, Fondelli 2004).

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

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

Figure 14. The diagram of the 1STprincipal stress in Pa.Nonlinear solution.

Figure 15. The diagram of the 1ST principal stress in Pa withthe lantern. Linear solution.

positive stresses, but with a cracking especially diffuseon the lower part of the dome.

Now the effect of the construction of the lantern(5.000 kN about) on the dome can be shown.

Out of Figure 15, in the linear solution, the posi-tive effect of the weight of the lantern on the domeis significant: it strongly reduces the positive stresses,especially in the lower areas of the dome.

Figure 16 represents the non linear solution , withthe value of cracking near zero, probably the actualstate of the dome.

So an other intuition of Brunelleschi can be noted:the heavy lantern placed on the top of dome decreasespositive stresses and then increases the masonry sta-bility.

7 CONCLUSIONS

The geometrical shape pictured in Figure 3 is con-gruent in all its parts; the surveyed dimensions and,especially, the dimensionless measures confirm thegeometrical hypothesis and besides show how care-fully the dome was laid out by its builders.

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Figure 16. The diagram of the 1ST principal stress in Pa withthe lantern. Nonlinear solution

Then, the equations of the differential geometry ofthe dome are useful to understand and quantify thegeometrical properties of the dome.

Finally, the linear solution of the finite elementanalysis fits the considerations of Timoshenko andHeyman, but shows positive stresses, unsuitable forthe brick masonry.

The equilibrium of the dome is possible with adiffuse cracking, especially in the lower part.

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