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Meccanica
Erschienen in: Meccanica 3/2019

19.02.2019

A non-smooth-contact-dynamics analysis of Brunelleschi’s cupola: an octagonal vault or a circular dome?

verfasst von: Valentina Beatini, Gianni Royer-Carfagni, Alessandro Tasora

Erschienen in: Meccanica | Ausgabe 3/2019

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Abstract

The cupola (dome) of Santa Maria del Fiore in Florence was ingeniously constructed by Brunelleschi using a conical bricklaying, radial-oriented toward a focus point on the central axis. Therefore, the dome is built as a surface of revolution but with parts cut away to leave the octagonal cluster vault form. This circular arrangement is compared with an octagonal horizontal corbelling in models where the dome is schematized as an assembly of rigid-blocks in frictional contact, analyzed with a Non-Smooth-Contact-Dynamics approach. The high indeterminacy of the contact reactions implies considerable difficulties in their determination, which are faced via a regularization procedure by adding a compliance at the contact points in representation of the deformability of the mortar joints. Numerical experiments, performed with a custom software, highlight the uniform flow of forces in the Brunelleschi arrangement, but evidence the disturbances induced by the herringbone spirals, mainly used for construction purposes, which are overloaded along the meridians and very weak in the direction of the parallels. This is due to the vertical narrow disposal of the blocks, which increases the stiffness in meridional direction, but diminishes the capacity of the friction-induced forces to equilibrate the hoop stress.
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Fußnoten
1
Leon Battista Alberti confirms that “Testudo camura atque item fornix armamentis substitutis inducatur, necesse est” (The Groin vault and the barrel vault have to be constructed with the support of centering), De Re Aedificatoria [3], liber III, Caput XIX.
 
2
Alberti (De Re Aedificatoria, liber III, Caput XIX), verbatim says: “Angularem quoque testudinem sphericam modo per eius istius crassitudinem rectam sphericam interstruas, poteris attollere nullis armamentis. Sed istic nexura potissimum opus est, qua huius imbecillae partes partibus illius firmioribus arctissime illigentur”. Alberti, at the begging of the same chapter, defines the various types of vaults (testudines), in particular the groin vault (spherica angularis) and the spherical vault (recta spherica).
 
3
The dual cone\({\mathcal {K}}^*\) of the cone \({\mathcal {K}}\) is a convex cone expressed as
$$\begin{aligned} {\mathcal {K}}^* = \left\{ \varvec{y} \in {\mathbb {R}}^n: \left\langle \varvec{y}, \varvec{x} \right\rangle \ge 0 \quad \forall \varvec{x} \in {\mathcal {K}} \right\} .\qquad (1) \end{aligned}$$
A Cone Complementarity Problem CCP(\({A},\varvec{b},\Upsilon\)) is the problem of finding a \(\varvec{x}\) that satisfies
$$\begin{aligned} {A}\varvec{x}-\varvec{b} \in \Upsilon ^*, \quad \varvec{x} \in \Upsilon , \quad \left\langle {A}\varvec{x}-\varvec{b}, \varvec{x} \right\rangle = 0 ,\qquad (2) \end{aligned}$$
where \(\Upsilon\) is a (convex) cone. One can also use the notation \({A}\varvec{x}-\varvec{b} \in \Upsilon ^* \bot \varvec{x} \in \Upsilon\). The CCP is equivalent to a Variational Inequality problem.
 
4
This is one of the first interpretations of the construction method used by Brunelleschi [5], not confirmed by successive studies. The particular bricks that were supposed to be used in proximity of the ribs were referred to as “mattoni a libro aperto” (open-book bricks), due to the shape that could be inferred, albeit tentatively, by observing a few spots in the interior of the dome, like that of Fig. 2b, where the brickwork is not hidden by the plaster. Later studies have confirmed that what is observed are sections of rectangular–parallelepiped shaped bricks with a dihedral angle formed by two planes intersecting on a line not parallel to any of the edges of the block.
 
5
As observed by Drucker [38], if normality fails the classical theorems of limit analysis are no longer valid for frictional sliding. Normality condition is preserved in the case of associative friction.
 
Literatur
1.
King R (2000) Brunelleschi’s dome. How a renaissance genius reinvented architecture. Chatto & Windu, London
2.
Di Pasquale S (2002) Brunelleschi. La costruzione della cupola di Santa Maria del Fiore. Marsilio, Venice
3.
Alberti LB (1485) Leonis Baptistae Alberti florentini viri clarissimi De re aedificatoria. N. Laurentus, Florence
4.
Godard A (1965) The art of Iran. Praeger, Westport
5.
Sanpaolesi P (1977) La Cupola di Santa Maria del Fiore. Il Progetto la Costruzione. Edam, Istanbul
6.
Antonio M (1985) Filippo Brunelleschi: sa vie, son oeuvre. Ecole nationale superieure des Beaux-Arts, Paris
7.
Bartoli L (1977) Filippo Brunelleschi. Nardini, Florence
8.
Pizzigoni A (2015) Brunelleschi’s bricks. J Int Assoc Shell Spat Struct 56(2):137–148
9.
Corazzi R (2012) The loxodromic curve and the herringbone. Disegnarecon 5(9):85–92
10.
Moreau J-J (1988) Unilateral contact and dry friction in finite freedom dynamics. In: Moreau JJ, Panagiotopoulos PD (eds) Nonsmooth Mech Appl. Springer, Vienna, pp 1–82CrossRef
11.
12.
Beatini V, Royer-Carfagni G, Tasora A (2017) A regularized non-smooth contact dynamics approach for architectural masonry structures. Comput Struct 187:88–100CrossRef
13.
Chetouane B, Dubois F, Vinches M, Bohatier C (2005) Nscd discrete element method for modelling masonry structures. Int J Numer Methods Eng 64(1):65–94MathSciNetCrossRefMATH
14.
Lancioni G, Gentilucci D, Quagliarini E, Lenci S (2016) Seismic vulnerability of ancient stone arches by using a numerical model based on the non-smooth contact dynamics method. Eng Struct 119(Supplement C):110–121CrossRef
15.
Rafiee A, Vinches M, Bohatier C (2008) Application of the nscd method to analyse the dynamic behaviour of stone arched structures. Int J Solids Struct 45(25):6269–6283CrossRefMATH
16.
Dubois F, Jean M (2006) The non-smooth contact dynamic method: recent lmgc90 software developments and application. In: Wriggers P, Nackenhorst U (eds) Analysis and simulation of contact problems. Springer, Berlin, pp 375–378CrossRef
17.
Beatini V, Royer-Carfagni G, Tasora A (2018) The role of frictional contact of constituent blocks on the stability of masonry domes. Proc R Soc A Math Phys Eng Sci 474(2209), 20170740ADSCrossRefMATH
18.
Beatini V, Royer-Carfagni G, Tasora A (2019) Modeling the shear failure of segmental arches. Int J Solids Struct 158:21–39CrossRefMATH
19.
Casapulla C, Maione A (2018) Modelling the dry-contact interface of rigid blocks under torsion and combined loadings: concavity vs. convexity formulation. Int J Non Linear Mech 99:86–96CrossRef
20.
Casapulla C, Portioli F (2015) Experimental and analytical investigation on the frictional contact behavior of 3D masonry block assemblages. Constr Build Mater 78:126–143CrossRef
21.
van Zijl GPAG (2004) Modeling masonry shear-compression: role of dilatancy highlighted. J Eng Mech 130(11):1289–1296CrossRef
22.
De Saxcé G, Feng Z-Q (1998) The bipotential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms. Math Comput Model 28(4):225–245MathSciNetCrossRefMATH
23.
Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Oxford University Press, OxfordMATH
24.
Anitescu M, Hart GD (2004) A constraint-stabilized time-stepping approach for rigid multibody dynamics with joints, contact and friction. Int J Numer Methods Eng 60(14):2335–2371MathSciNetCrossRefMATH
25.
26.
Anitescu M, Tasora A (2010) An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput Optim Appl 47(2):207–235MathSciNetCrossRefMATH
27.
Birgin EG, Martnez JM, Raydan M (2000) Nonmonotone spectral projected gradient methods on convex sets. SIAM J Optim 10(4):1196–1211MathSciNetCrossRefMATH
28.
Heyn Toby, Anitescu Mihai, Tasora Alessandro, Negrut Dan (2013) Using Krylov subspace and spectral methods for solving complementarity problems in many-body contact dynamics simulation. Int J Numer Methods Eng 95(7):541–561MathSciNetCrossRefMATH
29.
30.
Tasora A, Anitescu M, Negrini S, Negrut D (2013) A compliant visco-plastic particle contact model based on differential variational inequalities. Int J Non Linear Mech 53(0):2–12 Multibody system dynamics: a selective perspectiveCrossRef
31.
Michel J (2009) Numerical Simulation of granular materials. In: Cambou B, Jean M, Radjaï F (eds) Micromechanics of granular materials. ISTE-Wiley, Hoboken, pp 149–315
32.
Koziara T, Bićanić N (2011) A distributed memory parallel multibody contact dynamics code. Int J Numer Methods Eng 87(1–5):437–456CrossRefMATH
33.
34.
Keerthi SS, Gilbert EG, Johnson DW (1988) A fast procedure for computing the distance between complex objects in three-dimensional space. Robot Autom 4(2):193–203CrossRef
35.
Xiuzhi Q, Stucker B (2003) A 3D surface offset method for STL-format models. Rapid Prototyp J 9(3):133–141CrossRef
36.
Coli M, Haines M, Bianchini P (2013) Le pietre della cupola del brunelleschi. In: D’Andrea M., Rossi R. (eds), 2015 –Quinto Congresso Nazionale Geologia e Turismo, Bologna, 6-7 giugno 2013, Atti, ISPRA, Roma, pp 226–236. ISBN  978-88-448-0721-4
37.
Heyman J (2010) Equilibrium of masonry arches. Proc Inst Civ Eng Eng Comput Mech 163:129–133
38.
Drucker DC (1954) Coulomb friction, plasticity and limit loads. J Appl Mech 21(1):71–74
Metadaten
Titel
A non-smooth-contact-dynamics analysis of Brunelleschi’s cupola: an octagonal vault or a circular dome?
verfasst von
Valentina Beatini
Gianni Royer-Carfagni
Alessandro Tasora
Publikationsdatum
19.02.2019
Verlag
Springer Netherlands
Erschienen in
Meccanica / Ausgabe 3/2019
Print ISSN: 0025-6455
Elektronische ISSN: 1572-9648
DOI
https://doi.org/10.1007/s11012-018-00934-9

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