Further Refinement of the Corbelling Theory for the Equilibrium Analysis of Corbelled Domes

Curved and Layer. Struct. 2019; 6:30–40 Research Article Aguinaldo Fraddosio*, Nicola Lepore, and Mario Daniele Piccioni Further refinement of the Corbelling Theory for the equilibrium analysis of corbelled domes https://doi.org/10.1515/cls-2019-0003 sion of the concept of false arch: indeed, both the construc- Received Jul 07, 2018; accepted Jul 23, 2018 tions are composed by a superposition of purely horizontal stone layers slightly cantilevered toward the center un- Abstract: The aim of this paper is to present a further de- til meeting at the top [3]. Whereas at first sight, this kind of velopment of the Corbelling Theory [1] for assessing the constructions may appear to be the result of a self-building structural safety of “false domes” constructions like Trulli, process, actually they are the expression of a consider- and more generally for corbelled domes. In particular, it is able constructive knowledge, handed down over time. In well-known that a corbelled dome of a Trullo is a layered particular, the above architectural typology comes from a thick shell and that only the thin inner layer (candela) has long historical process: corbelled domes were well-known a structural role. The proposed procedure extends the ca- since the Bronze Age, and several examples can be found pabilities of the approach proposed in [2] to more general in archeological sites in Italy (Tuscany) and in Greece as load conditions, including the infill. The effectiveness of the renowned case of the Mycenaean Tholoi (XIV century the proposed approach is discussed through the analysis B.C.). of a paradigmatic case study of a Trullo located in Albero- In spite of their extensive presence, this kind of ma- bello. Since the geometry of the construction plays a fun- sonry constructions have received a scarce interest from damental role in the structural behavior, the starting point the scientific community, and consequently, their mechan- of the analysis was an accurate 3D-laser scanner survey of ics is not completely understood yet. Indeed, the structural the dome. The obtained results give some interesting in- behavior of corbelled domes was studied in only a few sights about the role of the infill load on the equilibrium of works in the scientific literature, and this issue is not cov- such kind of corbelled domes. ered neither by traditional calculation techniques [4, 5], Keywords: Corbelled vaults, Historic constructions, Equi- nor by standard limit equilibrium analysis applied to ma- librium analysis, Corbelling theory sonry structures [6–8]. Finally, also advanced structural analysis numerical approaches [9] appear to be uncapable of describing the peculiar mechanical behavior of cor- 1 Introduction belled domes. The first studies expressly dedicated to the structural analysis of corbelled constructions were mainly focused Dry-stone corbelled domes are a paradigm for the vernac- on the analysis of Mycenaean Tholoi, in the assumption ular architecture of all the Mediterranean area. Cabane in that the so-called “false dome” was behaving basically in Provence, Mitato in Greece, Girna in Malta, Trulli in Italy, the same way of a “false arch”, where the latter is a con- or the earthen-corbelled domes in the Aleppo region are struction made up of horizontal blocks simply superim- just some remarkable examples. posed and placed so that each block protrudes with re- From the architectural point of view, corbelled domes spect to the underlying one. In this vein, a corbelling dome can be considered as the natural three-dimensional exten- is considered as composed by a series of independent meridian false-arches, and it is assumed that forces can be transferred between two consecutive rings of stones only *Corresponding Author: Aguinaldo Fraddosio: Department of through vertical load paths. This approach is known as the Civil Engineering and Architecture, Polytechnic University of Bari, Corbelling Theory (CT) and it was first proposed in [10] for Italy; Email: [email protected] the case of Mycenaean Tholoi. In the latter work, the anal- Nicola Lepore: Department of Civil Engineering and Architecture, ysis is performed by imposing the overturning equilibrium Polytechnic University of Bari, Italy; Email: [email protected] Mario Daniele Piccioni: Department of Civil Engineering and Ar- for each group of blocks composing the generic infinitesi- chitecture, Polytechnic University of Bari, Italy; mal wedge of the dome. Email: mariodaniele.piccionipoliba.it Open Access. © 2019 A. Fraddosio et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Further refinement of the Corbelling Theory for the analysis of corbelled domes Ë 31 In [1] the CT was subsequently formalized in a rigorous treatment based on suitable equilibrium differential equa- tions; within this theoretical framework the study of false- arches, tholoi, and Trulli is proposed. The limitations of the CT are discussed in [11, 12] with reference to the well-known tholos “Treasury of Atreus” (IV century BC). By considering the masses involved in the equilibrium of the construction, i.e., the self-weight of each block applied in its centroid, the stabilizing and overturning moments and the ratio between them are evalu- ated. This calculation shows a meridian wedge wouldn’t be able to sustain itself autonomously. This apparent paradox is due to the inadequacy of the standard CT in describing the structural behavior of corbelling domes. Indeed, for corbelling domes, the spatial shape based on horizontal rings of stones with closed geometry gives a crucial contribution to the stability of the single meridian wedge. In particular, the accurate brickwork of the horizontal rings of stones, thanks to the high technical exper- tise in inserting by the force small pieces of stones between the vertical stone joints, makes them capable to bear com- pression forces along parallels. In this way, the overturning of a part of the meridian wedge is prevented by the blocks of the adjacent meridian segments: this actually de- termine a fully 3D shell behavior of the structures. Notice that the capacity of transmitting compressive forces along parallels is yet available also when the construction of the rings is not completed, thanks to the fric- Figure 1: tion between the lateral surfaces of adjacent courses. A Example of the partial collapse of a Trullo vault [14]. practical evidence of this, and thus of the effectiveness of the described constructive technique, is provided by sev- the corbelling domes with respect to previous approaches eral examples of partially collapsed domes, showing that in literature. Moreover, in [2] also a computational proce- a finite-dimension wedge, even with a partial circumferen- dure based on the Thrust Network Analysis (TNA) [15, 16] tial development, is still able to stand without the collabo- is applied for understanding the structural behavior of cor- ration of the collapsed surrounded structure (Figure 1). Of belling domes. course, the above described three-dimensional structural For the sake of the completeness of the literature re- behavior is out of the capability of the standard CT. view we notice that also studies based on the application In order to overcome this limitation, Rovero and Toni- of the Finite Element Method have been proposed [17–19]. etti in [13] proposed a Modified Corbelling Theory (MCT) In this paper we start in Section 2 from the analysis based on the balance of the stabilizing and the overturning of the building techniques and of the materials used for moments imposed on a finite-dimension meridian wedge. the construction of Trulli located in Alberobello, the well- This allows for taking in account the three-dimensional known city of Apulia (Italy), UNESCO site since 1996. Then, behavior of corbelling domes, based on the possibility of in Section 3 we underline the fundamental mechanical is- the transmission of forces along horizontal rings. sues arising in the structural analysis of such kind of con- Recently, a further development of the approach in [13] structions, and we review the literature on this subject has been proposed in [2], where a new integral formula- starting from the standard CT introduced in [1], passing tion of the equilibrium problem for corbelling domes has from the MCT introduced in [13] and finally arriving to our been developed. We will call this approach from now on improvement of the MCT, the NFMCT proposed in [2]. New Formulation of the MCT (NFMCT). The obtained re- On this basis, we propose in Section 4 a further devel- sults show the capability of the proposed theory of ob- opment of the analytical approach in [2], by taking into ac- taining more accurate predictions of the actual behavior of 32 Ë A. Fraddosio et al. Figure 2: Constructive section and detail of an Italian Trullo [21]. count both the peculiar three-dimensional structural be- is called “candela” and it is built by overlapping layers of havior of corbelled domes and the role of the infill load in horizontal rings of stones. Each ring is made up of care- the equilibrium of these structures, not yet considered in fully chosen stone blocks, similar in form and size for each the literature. The aim of this refinement is that of obtain- course, and arranged in concentric circles with decreasing ing a more accurate procedure for the static assessment of diameters upwards until the last ring topped by a large corbelled domes. stone called “chianca” or “serraglia” [21]. Special care is The effectiveness of the proposed approach is dis- dedicated to achieving the continuity of the rings brick- cussed in Section 5 through the analysis of a case study work, by filling by force the gaps between blocks through represented by a Trullo in Alberobello. Since the geom- little pieces of stones; this way, practically a rigid annular etry of the construction plays a fundamental role in the system is formed.

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