Geometrical and Static Aspects of the Cupola of Santa Maria Del Fiore, Florence (Italy)

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

A. Cecchi & I. Chiaverini Department of Civil Engineering, University of Florence, Florence, Italy A. Passerini Leonardo Società di Ingegneria S.r.l., Florence, Italy

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

1 INTRODUCTION: THE DOME OF THE CATHEDRAL OF FLORENCE

The construction of the cathedral of Firenze was begun in the year 1296, with the works related to the exten- sion of the ancient church of Santa Reparata: it was designed byArnolfo (1240,1302).The design included a great dome, based on an octagonal base, to be erected in the eastern end of the church.The dome is an unusual construction of the Middle Ages, (Wittkower 1962): Arnolfo certainly referred to the nearby octagonal bap- tistery San Giovanni, so ancient and revered that the Florentines believed it was built by the Romans, as the temple of Ares, hypothesis which was not con- firmed by excavations, that set the date of foundation Figure 1. View of the cathedral of Santa Maria del Fiore. is between the V and IX century (Rocchi 1996). In the second half of the fourteenth century the con- dimension was internally about 45 m, surpassing the struction of the octagonal base was completed; for fifty greatest known, the Pantheon in Rome (43 m about, year about the construction yard stood by,testifying the in concrete) and Hagia Sofia in Constantinople (31 m great uncertainty about the building technique of the about, in masonry); the base itself was laying on four dome, till Brunelleschi’s assignment in 1420. great high piles, so that the height of the top, 90 m We should keep in mind that in that time in Italy about, and the height of the base, 60 m. about, greater there were two other great yards open for the construc- than the Pantheon, made it practically impossible to tion of the cathedrals of Milan and Bologna, but the erect the dome by framework, as was done in Rome. construction of the dome of Florence was so excep- Besides, the Roman dome is a spherical revolution tional to enable the target to appear beyond the human surface, while the Florentine dome has a much more possibilities. complicated geometrical shape, due to its octagonal This is understandable since the dome to be built base. Notice that the Pantheon was built in concrete, a would be the largest ever known (Figure 1,2): its base technique which was probably lost in the Middle Ages

555 perchè nel murare la praticha insegnera quello ches- sara a seguire” (Brunelleschi’s specifications), nor about the mechanical apparatus he would have used later to raise the heavy weights “tirare i pesi per via di contrappesi e ruote, che un sol bue tirava quanto avrebbero appena tirato sei paia” (Vasari 1550).

2 AUTHORS WHO DOCUMENTED THE DOME. THE AIMS OF OUR RESEARCH

The silence of Filippo is the main cause of the discus- Figure 2. Plan of the cathedral of S. Maria del Fiore sions that followed on the argument. This seems to be (Ximenes). common to all the Middle Ages constructions. Fitchen speaks of “...the total lack of written documentation on both the engineering structure and the erectional procedures...” (Fitchen 1961). and was substituted in the dome of Florence by brick Many authors have related the dome and his mas- masonry. The static function of the flying buttress of ter: among many others let’s remember Giovanni di the Middle Ages cathedrals, to carry the horizontal Gherardo da Prato (1421), Manetti (1480?), Vasari forces to the foundations, (Heyman 1966) was left to (1550), Opera del Duomo (1691) (The Opera del the chapels, which surround the base, much more suit- Duomo was charged by the Duke of Tuscany to write able to the classical tradition of the town, Florentia, down an essay on the stability of the Dome, threatened founded by the Romans in 59 B.C. (Davidsohn by cracks), Guasti (1887), and recently Sanpaolesi 1956). (1962), Fondelli (2004), Rocchi (1994). Sgrilli and Vasari (Vasari 1550) is the main source of infor- Ximenes have rendered the dome accurately in the mation concerning Filippo Brunelleschi’s work. He XVIII century. wrote down an act, collected in the Museum of Nevertheless as a result of this information neither the Cathedral, where the main dimensions of the thorough study on the geometry of the dome, nor on building were specified. We should remember the its statics is carried out. following: Beyond these arguments, the topics which interest many researchers, e.g. Di Pasquale (1977), Bartoli – the dome is composed by two cupolas; (1994), are the methods of construction and erection of – the inferior one has a variable thickness from 2.35 m the dome and of the masonry, particularly the special in the bottom to 1.49 m at the top and it is vaulted one called “a spina pesce”. “a quinto acuto negli angoli”. The function of the The survey of Fondelli, based upon photogramme- superior one is: “conservalla dal umido (practical try, must be remembered for its accuracy. It started in function) e perché torni più magnifica e gonfi- 1968 and was carried on for many years. Fondelli’star- ante (esthetical function)”: its thickness varies from get was to render both the outside and the inside of the 0.72 m to 0.43 m; great cathedral and at the same time to link the survey – 24 stone ribs (pietraforte) link the two cupolas, 1 for to the Italian geodetic net. each of the 8 corners and 2 for each web; these ribs The task of this paper is to give a contribution to are tied round by 6 hoops in pietraforte cramped by the knowledge of Filippo’s work on the following four means of iron brackets; topics: – further links between the two cupolas are the “volticciole” (small vaults) chained by means of – based on Brunelleschi’s specifications, the above oak beams; papers and our observations the geometrical shape – the material used is brick masonry, even if at first, of the internal surface of the dome is proposed; stone walls were proposed. – the proposed shape is compared to the results of the survey of the dome; You can note Brunelleschi’s structural intuition, – the differential geometry of the surface is exposed; that is to say the employment of a sandwich ribbed – the statics of a Brunelleschi-like dome is defined structure, in order to lower the load of heavy vaults. and analyzed. Nothing he wrote neither about the methods of erection of the vault, even if he used a cantilever technique Further researches will discuss the masonry, the without frameworks, absolutely new for his times, nor engineering structure, the details and the erectional about the masonry “secondo sara allora consigliato procedures.

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

Figure 3. Geometrical hypothesis on the dome. 4 SURVEY OF THE DOME. COMPARISON BETWEEN SURVEYED AND 3 HYPOTHESIS ON THE GEOMETRICAL GEOMETRICAL POINTS FEATURES OF THE INTERNAL SURFACE OF THE DOME The second aim of our work was to survey one web of the dome so that the hypothesis explained in Chapter Figure 3 shows the geometrical hypothesis, plan and IV IV 3 can be compared. vertical section (AA VF F). The survey was carried out only with topographical The plan of the octagon A, B, D,..., basis of the methods, using an electronic laser total station, a Leica dome, can be observed. C is its centre. The lines TCR 705, 5” angular precision and 2 mm.+2 ppm. lin- A A,B B,D D,..., are the projections of eight cir- ear precision and a digital camera Nikon D300 with a cular arches whose ray is r: they are the rulings of the 12 mega pixel CCD and 50/20 interchangeable Nikkor surfaces. The octagon, represented by A ,B ,D ,..., optics. is the lantern: in section it is represented by the line IV IV It is well known (Cecchi 2006) that the laser tech- A F .According to Brunelleschi’sspecifications the nology permits the direct survey of inaccessible points, arches’centres aren’t in the centre of the octagon: they which is the case of this dome. are in the point “quinto di sesto”. E.g. A A and F F Beyond this the quick acquisition of data in a dig- are the projections of the circular arches AAIV,FFIV, ital format in a reference frame, permits a successive whose centres are respectively the points A ,F such computer elaboration of the data. that, for instance, AF = 1/5 AF. Naturally eight cen- In June 2007 the upper balcony of the cathedral tres form a new octagon A ,B ,D,...,. The points was reached; this one was built on the bottom of the of the webs of the dome represented in the horizontal dome from which the best and nearest view of the plan by the eight quadrilaterals AA B B, BB D D, inside surface of the dome is possible. The topograph- DDEE,..., belong to eight cylinders, each having ical instrument was placed on the middle east side and for rulings two circles: AA ,BB ;BB ,DD ;DD , points of the west web were surveyed: their number is EE;..., and straight horizontal lines as generatrices = ◦ about 700 and includes points of the bordering vaults inclined of β 67, 5 (Figure 3). too . The numerical format of the surveyed points is The main properties of these cylinders are the DXF so thatAUTOCAD® immediately can place them following (Figure 3): in a three dimensional Euclidean space (Figure 4). – a horizontal plane cuts the dome with an octagon, This figure clearly shows how the points were sur- e.g. ABD,....,; veyed with zenithal constant angles: notice that the last – the cylindrical web. AAII has its axis in the series refers to points along the upper lantern sides. AI segment.This property is common to the other Then the digital coordinates can be elaborated with webs. the least squares method (Kraus 1998).

557 ◦ ◦ Figure 6. The estimated straight line at 64 degree on the Figure 5. The estimated straight line at 64 degree on the vertical plane projection. horizontal plane projection.The values in the figure represent the difference between the calculated points and the surveyed ones.

4.1 Research of the curves connecting points defined by constant zenithal angles The constant zenith points were projected on the horizontal plane and the line which best approximates these points was researched. Then with the least square method the approximat- ing lines can be found. Figure 4 shows these constant zenith points are not straight lines but they differ a little from it especially in the middle area of the web. These deviations have been computed through projections of these points on the horizontal and vertical planes. The results have been shown (Figures 5,6), for the ◦ sake of brevity, only for the angle 64 , where the Figure 7. The estimated round circle. greatest deviations are noticed. Out of Figure 5 the maximum deviation in the hori- in Figure 4, but by now it can be possible to formulate zontal plan projection is 0.083 m, while in the vertical some hypothesis. plane projection is 0.089 m (Figure 6).As the length of The regularity of these deviations and their increase the chord is 10.666 m, the deviation is at about 0.8%. in correspondence of the middle web allow us to sur- So it can be observed that this deviation is very modest mise that Filippo created it for precise reasons, perhaps and it can be disregarded in the global geometry of the these regarding statics. dome. A second hypothesis results from the assumption The following step was to consider the intersection that these deviations occurred during and after the con- of two such neighboring lines determines a point of struction of the dome because of the movements which the curve intersection of two border web (Figure 4). were due to cracks (Opera del Duomo 1691, Fondelli The next one was to discover the surface on which 2004), elastic deformations, thermal expansion. these points are placed. This surface is a plane and then, by projecting these points on this plane, the curve which best approximates these projections was 4.2 Further research on the dimensions of the dome studied: these curves are circles (Figure 7). The two The following angles were considered: surveyed arches radii are respectively 36,11 m and 36,23 m. – the angle made by the planes containing the two The authors leave to the proposed future research of circles. This angle is 44,92◦. Notice the analogous Chapter 2 a study on the surveyed differences pictured geometrical angle of the octagon is 45◦.

558 Table 1. Surveyed dimensions. Notice: curves φ=cost are parallel horizontal lines, whose projections on i1, i2 are inclined β with respect Dimensions m. to line AC; curves v=cost are translations of the ruling circle in direction v. AB 17,26 Deriving the vector x with respect to each Gaussian AA 36,11 coordinate (φ , v): BB 36,23 AF 44,94 BG 44,96 AA/AF 0,803 BB/BG 0,805 Zenithal angle AIV (lantern) 36.21◦ Through the cross vector product the unit vector normal to the cylinder is readily obtained:

– the angles made by the projections of these two planes with the side of the surveyed octagon. These angles are respectively 67,30◦ and 67,79◦. Notice: for the geometrical octagon this angle is 67,5◦. From the survey more dimensions have been retrieved and for clarity’s sake there can be identi- fied through the analogous geometrical dimensions of and the Gaussian quantities E,G,F,that define the First Figure 3: Fundamental Form, can be calculated: It can be observed that 0,80 is the geometrical coefficient for the arch “quinto di sesto”.

4.3 Conclusions of the comparison The surveyed dimensions and, especially, the dimensionless measures confirm the geometrical hypothesis of Figure 3 and, besides, show how carefully the dome To define the Second Fundamental Form let us form was laid out. the derivatives: Chapter 4.1 shows that lines with constant zenithal angle approximate well straight lines: so that the surface 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 intersect each other in a circular cross section. Each circle is the ruling of the two bordering cylinders (Figure 3). so that the quantities:

5 DIFFERENTIAL GEOMETRY OF THE INTERNAL SURFACE OF THE DOME

Now the third goal of our research was to represent the dome with the differential geometry methods, e.g. Do Carmo (1976), Sokolnikoff (1951). With reference to the Cartesian frame (O,i1, i2, i3) Figure 3, let x be a point of the cylinder AAII, r define the Second Fundamental Form. the radius of the circle, φ the zenithal angle, v the unit The Gaussian quantities: LN − M2 = 0 and L2 + M2 vector of direction IA, v a scalar. Then: + 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:

and the unit vector v:

559 Then:

κ1 and κ2 represent the main curvatures at any point of the surface. The first one is null: it refers to the straight lines of the cylinder, φ = cost. The other one depends only on φ, as the lines v = cost are translations in the direction v. This curvature is the one of the normal section ellipse, normal to the vector v. The lines of curvatures, tangent to the principal directions, are then 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 us consider a plane through C parallel to the unit vectors Figure 8. The estimated ellipse. In blue the surveyed points, in red the geometrical ones. s and i3 (Figure 3):

which is the equation of the ellipse, line of curvature. For instance for k = 0, the point R is obtained, for k = 1 the point T (Figure 3). = In order to show an application of the equations let With the position CA a, its equation is obtained compare points of (32) with the surveyed points of (notice that s and k are parameters): Chapter 4. At first let us fold the vertical plane of trace TR to TR (Figure 3) so that the ellipse con be drawn as the dotted line RAIVVS. The surveyed ellipse has its minor axis 32,26 m, and Let us consider the intersection of y with x: its major is 36,12 (Figure 8). Using of the letters of Figure 3, RT = 32,26 m., TS = 36,12 m. Out of (32) the geometrical sizes are respectively RT = 32,44 m., TS = 36,11. It is evident that the difference between geometric and surveyed dimensions is only 0.18 m, that means the error is less than 0.01. Let us eliminate three of the four parameters; for instance resolving for k, there can be obtained: 6 STATICS OF A BRUNELLESCHI-LIKE DOME

Let us define a Brunelleschi-like dome as an octagonal dome, formed by ribbed cylindrical webs, intersecting and the equation of the intersecting curve in the ◦ along circles, inclined 67,5 with respect to the cylin- reference (O,i , i , i ): 1 2 3 der axis. The surface of this dome is represented by equation (32). Let us consider the statics of this dome.Timoshenko (1959) and Heyman (1977) consider the statics of cylindrical shells according to the membrane theory. According to Filippo’s specifications, Chapter 1, this assumption could be hazardous for a practical application: this is only one way of understanding the If the reference system is changed (C, i1, i2, i3) and main static aspects of this dome. In the FEM analysis rotated in (C, s, v, i3): the correct thickness has been used. With reference to Figure 9 and to the well known symbols of shell structures, the equilibrium equations are the following:

560 Figure 9. Equilibrium of a cylinder surface with the lines of curvature. q is the vertical force for unit area. Figure 10. Equilibrium along the web ribs.

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, F1(ϕ) and F2(ϕ) to be deter- mined 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 Nφv 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 Figure 11. Slice AAII with unit membrane forces along ellipses (32), even if they approximate circles. the ribs. 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 from that of a surface of revolution. For instance that Nvφ = 0, so that according to Heyman, F1(ϕ) = 0, Timoshenko and Heyman present the solution for while along AA, because of the rib, the tangential semi-spherical domes, in the membrane theory, also stresses N//AA , although symmetric, are not null, as with the upper portion removed, in which, in each described in Figure 10. point, a line of curvature is the meridian, while the Figure 11 shows the slice AAII cut off the dome other one is obtained with a plane normal the meridian, and the forces N//AA and N⊥AA which equilibrate containing the normal m to the surface. its weight. Side AI is free of forces, but after the The elastic solution with the upper part removed construction of the lantern in the real dome, weights confirms the intuition of Brunelleschi on the possibil- were applied along it. ity of the cantilevered erection of the dome without The equilibrium of momentum with reference to frameworks. AI cannot determine the two unknowns N//AA and Writing down the two equations of equilibrium, N⊥AA: N⊥AA is not parallel to the line AI, so that its not identically satisfied, the problem is statically contribution to momentum is different from zero. determinate: in fact, for symmetry, shears are null. The rib assumes and equilibrate the forces 2N//AA , In particular from the solution, it is well known that according to the Figure 10, while the forces N⊥AA the forces Nv along the meridians are positive for a equilibrate themselves. latitude ϕ>51◦50. Besides, the existence of eight symmetry axes The Brunelleschi-like dome has been modelled with makes the structural behaviour of the dome not far a FEM analysis with ANSYS®.

561 Figure 12. The Brunelleschi-like meshed dome. Figure 14. The diagram of the 1STprincipal stress in Pa. Nonlinear solution.

Figure 13. The diagram of the 1ST principal stress in Pa. Linear solution. Figure 15. The diagram of the 1ST principal stress in Pa with the lantern. Linear solution. The virtual 3D model of the dome has been meshed with the solid element Solid65 (Figure 12). A masonry positive stresses, but with a cracking especially diffuse density of 1800 kg/m3 was assumed. on the lower part of the dome. The case without the weight of the lantern has been Now the effect of the construction of the lantern primarly considered, as it was during the construction. (5.000 kN about) on the dome can be shown. Figure 13 shows the linear solution of the dome. In Out of Figure 15, in the linear solution, the posi- particular it shows that part of the dome has positive tive effect of the weight of the lantern on the dome stresses, particularly in the lower part of the ribs and of is significant: it strongly reduces the positive stresses, the webs, coherently with the Heyman’sboth solutions especially in the lower areas of the dome. of the cylindrical shell and the spherical dome. Figure 16 represents the non linear solution , with These positive stresses are lower than 1,5 × 105 N/ the value of cracking near zero, probably the actual m2: nevertheless this value is excessive for a Middle state of the dome. Ages masonry. The intuition of Filippo is really sur- So an other intuition of Brunelleschi can be noted: prising. He used the right expedients for the engi- the heavy lantern placed on the top of dome decreases neering possibilities of his time, as the “lisca pesce” positive stresses and then increases the masonry sta- technique and reinforcements in strips of wood and bility. stone (Brunelleschi’sspecifications).This phase, without the lantern, was run across the construction; so cracking appeared in the cupola (Opera del Duomo 7 CONCLUSIONS 1691, Fondelli 2004). Figure 14 shows the non linear solution obtained The geometrical shape pictured in Figure 3 is con- with a crush/cracking FEM analysis, with the value of gruent in all its parts; the surveyed dimensions and, cracking near zero. especially, the dimensionless measures confirm the Confronting the Figure 13 and Figure 14 it can be geometrical hypothesis and besides show how care- noticed how the equilibrium is possible with minor fully the dome was laid out by its builders.

562 DAVIDSOHN, R. 1956. Storie di Firenze. Sansoni DI PASQUALE, S. 1977. Primo rapporto sulla Cupola di Santa Maria del Fiore, CLUSF, Firenze DO CARMO, M. P. 1976. Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey FITCHEN, J. 1961. The Construction of Gothic Cathedrals, Oxford FONDELLI, M. 2004. La Cupola di Santa Maria del Fiore, in Giuseppe Rocchi Coopmans de Yoldi, S. Maria del Fiore, ALINEA, Firenze GIOVANNI DI GHERARDO DA PRATO, 1421. Document conserved in the Museum of Opera del Duomo. Firenze GUASTI, C. 1887. Santa Maria del Fiore: la costruzione della chiesa e del campanile secondo i documenti tratti dall’archivio dell’Opera Secolare e da quello di Stato,Tip. M. Ricci, Firenze Figure 16. The diagram of the 1ST principal stress in Pa with HEYMAN, J. 1977. Equilibrium of Shell Structures (Oxford the lantern. Nonlinear solution Engineering Science), Oxford University Press HEYMAN, J. 1966. The Stone Skeleton, Int. Journ. Solids Then, the equations of the differential geometry of and Struct., 2 the dome are useful to understand and quantify the KRAUS, K. 1998. Fotogrammetria, Levrotto & Bella, Torino geometrical properties of the dome. MANETTI, A. 1480?. Vita di Filippo Brunelleschi Finally, the linear solution of the finite element NAGHDI, P.M. 1972. The Theory of Shells and Plates. analysis fits the considerations of Timoshenko and Handbuch der Physik VI, Springer-Verlag Heyman, but shows positive stresses, unsuitable for OPERA DEL DUOMO, 1691. Archivio di Stato. Filza 366. the brick masonry. Firenze The equilibrium of the dome is possible with a ROCCHI COOPMANS DE YOLDI, G. 1996. S. Maria del diffuse cracking, especially in the lower part. Fiore, Università degli Studi di Firenze, Dipartimento di Storia dell’Architettura e del Restauro delle strutture architettoniche, Firenze SANPAOLESI, P. 1962. Brunelleschi, G. Barbera, Firenze REFERENCES SOKOLNIKOFF, I. S. 1951. Tensor Analysis – Theory and applications, John Wiley and Sons ANSYS® Inc. Southpointe 275 Technology Drive Canons- TIMOSHENKO, S. P. 1959. Theory of Plates and Shells, Mc burg, PA Graw Hill BARTOLI, L. 1994. Il disegno della cupola del Brunelleschi VASARI, G. 1550. Le vite de’più eccellenti pittori scultori e Firenze Leo Olschki Editore architetti Firenze CECCHI, A. & PASSERINI, A. 2006. Survey, digital recon- WITTKOWER, R. 1962. Architectural Principles in the Age struction, finite element model of the Augustus Bridge in of Umanism, Alec Tiranti Ltd., London Narni (Italy). 5th International Conference on Structural Analysis of Historical Constructions, New Delhi 2006

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