The staircase in Palazzo Bartolomeo di Majo in Naples refers to a type of staircase defined in Italian treatises (from the 16^th century onwards) as ‘vacua nel mezzo’ (with open-well), that is, made up of flights of stairs arranged around an empty central space (1). In addition to this, there is an undoubtedly innovative planimetric peculiarity, represented by the development of a rhombic, mixtilinear layout that recalls the plan of the church of San Carlino alle Quattro Fontane (1638-41) by Francesco Borromini (1599-1667). A systematic study of the open-well staircase reveals that a first definition of the latter in Italian architectural treatises of the 16^th-17^th centuries appears to have been introduced by Andrea Palladio in the First Book of the Four Books of Architecture (1570), in chapter XXVIII, “Delle scale, e varie maniere di quelle, e del numero, e grandezze de’ gradi” (about staircases, various manners, and about number and sizes of the steps) (2). This example is described and represented in circular and oval forms (Figure 1, a) (3-4). Later, a more constructive description appears in Egnatio Danti’s (1536-1586) comments on Jacopo Barozzi da Vignola’s (1507-1573) treatise, Le due regole della prospettiva pratica (1583) (Figure 1, b). In dealing with the perspective construction of staircases “à lumaca doppia” (double snail), ‘fully open’, Danti illustrates their static characteristics, stating that “they stand without having no supports in the middle, the steps being stopped with the head in the wall, and placed in such a way one on top of the other, that one holds the other, and the same steps make the vault of the staircase” (5). On the other hand, Vincenzo Scamozzi (1548-1616) is responsible for a more articulate exposition of staircase design. In his treatise of 1615, L’idea della architettura universale, in Chapter XX, entitled “De’ siti, e forme convenevoli a varie maniere di Scale private ad uso de’ tempi nostri, & alcune introdotte dall’Autore” (about the sites and suitable shapes in various manners of private staircases for the use of our times, and some introduced by the author), Scamozzi defines the main types of stairs as “rette, & oblique, & anco à chiocciola” (straight, oblique, and spiral staircases) and, in distinguishing them into “primary staircases” (open to the courtyard) and “secrete staircases” (enclosed in the walls), illustrates «ten possible types, or forms»: “à rami” (on the branches), “à mandorla”, “ovali” e “rotonde à chiocciola”, (almond-shaped, oval and spiral round), specifying that they can all be “so full, as empty in the middle” (Figure 1, c) (6). Among these, the staircases of the V and VI modes are “suspended in the air [...], under which you can pass” and refer to the “hawk-winged” model later developed by Sanfelice (7).

In particular, the case study here presented reveals a strong analogy with the staircase of the VIII mode called “à mandorla” (almond-shaped) with its unusual oblong rectangular shape, smoothed at the corners. Therefore, it is possible to assume that Ferdinando Sanfelice (1675-1748), a Neapolitan architect, was inspired to conceive the unusual staircase in Palazzo di Majo in Naples (Italy) by reading these treatise indications on the staircase design and an inherent visit to Borromini’s church San Carlo alle Quattro Fontane in Rome (Figure 2) (8).

From both the compositional and constructional point of view, the typological study of these staircases could be extended to other late medieval and classicist European examples from the 16^th century, such as the double helix staircase at Chambord castle or the stairs of Pamplona Cathedral.

The spiral model also arrived in this century in Mexico, where, thanks to the master Toribio de Alcaraz, a two-sided spiral staircase was built in the tower of the unfinished Cathedral in Pátzcuaro (9). These works contribute to the appearance of these staircases in treatises and collections of drawings by authors such as Palladio, Vignola or Du Cerceau.

It is also suggestive of the similarity of technical problems when defining the shape of the intrados of the vaults, with modern stone staircases built with closed boxes. In Spain, José Antonio García Ares (10-11) and Santiago Huerta have also used the theory of equilibrium for the structural study of staircases, using graphic statics (12-13).

In the early decades of the 18^th century, Ferdinando Sanfelice renovated the palace of the nobleman Bartolomeo di Majo in Naples, along what is now Corso Sanità, and designed its majestic portal, courtyard, and staircase. In the eyes of architectural critics, the staircase of Palazzo di Majo immediately became a spatial event of exceptional formal mastery, so much so that in 1743 it was described by his contemporary biographer Bernardo De Dominici (1683-1759) as “of beautiful invention”. In his work entitled Vite de’ pittori, scultori ed architetti napoletani (lives of Neapolitan painters, sculptors, and architects), never published by any author, De Dominici states that all consider this staircase “to be the most capricious staircase in Naples, and it is marvellous how such a large staircase is situated entirely in the air, attaching the lamia only on one side” (14).

The entrance to Palazzo di Majo is through the traditional system of entrance hall, courtyard, and staircase. Over the years, however, several urban interventions have occurred, changing the state of the place, the shape of the courtyard and the access to the staircase. According to De Dominici, the courtyard was originally, “irregular in shape and (by Sanfelice) was reduced to such a magnificent form that no better could be desired”.

Cartographic sources (as Mappa topografica della città di Napoli e de’ suoi contorni by Duke of Noja in 1750-75) show the original shape of the courtyard (a large square with rounded corners) and the transformation following the construction of Corso Napoleone in 1809. The new road axis entailed the demolition of half of the courtyard and new access to the staircase from the street, which altered the visual perspective.

According to the project by Sanfelice, the access to the staircase was from the hallway located in Discesa Sanità (downhill). From here, through an opening on the left side of the hall, it was possible to turn into a small entrance tangent to the profile of the courtyard, leading to the staircase. Today the access from the ground floor is poorly lit, and the clutter of the staircase darkens the context even more. However, once up the first flight, the unusual spatial design envelope and the bold development of the vaulted system supporting the flights is revealed to the eye. The spatial layout takes shape from a rhombic cage with rounded vertices and convex towards the well (Figure 3) (15).

The four flights are arranged along the convex sides, and the profile of the shaft is concentric with that of the cage.

When viewed from below, the staircase stretches upwards like an elastic band. It is easy to understand the embarrassment of the biographer who, in commenting on the spatiality of this ‘secret staircase’, states that “its beauty cannot be described for having in such a small site made a duplicated staircase, and so comfortable, that no one could wish for a better one” (14).

The staircase of Palazzo di Majo was surveyed by Michele Capobianco and published in 1962 in the magazine L’architettura. Cronache e storie, in the first of three articles on “Eighteenth-century staircases in Naples” (16).

A subsequent survey was published in 2007 by Italo Ferraro in Napoli. Atlante della città storica. Stella, Vergini, Sanità (17). Here (in collaboration with Ornella Zerlenga, Vincenzo Laezza, Raffaele Liguori, Giuseppe Marino) carried out a more updated survey in 2017-18 (Figure 4).

The survey methodology used was direct. This methodology required a fundamental critical process of designing horizontal and vertical cross-section planes from which to extrapolate the metric information.

The data return took place in the form of planimetric and elevation representations, that allowed a graphic visualization of the various altimetric levels of which the dome is composed.

The metric differences between the various survey documents were considered negligible since the graphical analysis was oriented towards a geometric-configurative investigation (18).

From a geometric-configurative point of view, the cage and the well of the staircase are straight cylinders in which the surface is generated by a straight line that translates in space, resting on the flat line of the rhombic-shaped directrix.The intersections of the cylinders with horizontal planes located at different heights generate the landings, while those with planes of varying inclination generate the flights of stairs.

The flights of stairs are covered by the so-called ‘Roman vaults’ in southern Italy and especially in Rome, used to support the flights of stairs. In the canonical model, the intrados surface of the vault supporting the flights is cantilevered and continuously set on the corresponding perimeter wall, following its course. Vertical planes containing quarter circles, which join the quarter circles of the corner pavilion vaults (supporting the landings), limit it; it is defined towards the well by a rampant arch (19).

Under the rhombic layout, vaults respecting the described properties support the flights; this does not occur for the landings. Two quarter-vaults, spheroidal spindles sustain the landings in the staircase of Palazzo di Majo (Figure 5). Finally, the structural and typological configuration, is used to open up the large arches above the floor level, facing the courtyards as in an ‘open staircase model’. Thus, the perception of Palazzo di Majo’s staircase is ‘introverted’ (20-21).

The study of vaulted staircases cannot be separated from the analysis of masonry vaults.

The term “vaults” refers to arched or shell constructions, which cover spaces, and in which, as far as possible, tensile stresses in the materials used are minimized. The behavior of an open-well staircase is similar to that of vaults or domes with a central void (eye) and can be approached similarly. Masonry staircases consist of arches and vaults that, in terms of materials and techniques adopted, are the same as those typically employed in constructing buildings. Therefore, the only difference lies in their function: they no longer merely cover spaces but become the support for the structure forming the stairwell.

There are some peculiar aspects of masonry behavior that must be taken into account beforehand:

With particular reference to the typology analysed in this paper, stairs with open well, also known as roman staircases, are built with vaults resting exclusively on the perimeter walls. Their support is based on the mutual actions between the flights. Generally, particular cylindrical vaults are used, built only up to the geometric key, with a horizontal axis in the landings and an inclined axis in the flights (Figure 6).

The unique element of the masonry staircase typology is the flight of stairs, realised employing an arch or vault; the flying buttress, for example, allows the support of several vaulted structures set on staggered levels. In this particular type of construction, the rampant half-barrel vault constitutes the main component of the stair. They are characterised by a system that can be thought of as generated by a lame half-barrel with its impost on the two resting landings and, as if it were a real cantilevered structure, relies on the continuous support on the perimeter walls (Figure 7, a).

The arches of the stairs are usually of the crippled arch and type and are generally found in open-well staircases where the central spinal wall is absent. In fact, on the one hand, the structural function of the central spinal wall may be delegated to supporting pillars. On the other hand, as in the analyzed case, the supports entirely disappear. In the latter situation, it is necessary to sustain the vaults with arched structures on which they can rest, given the elimination of the spinal wall. The flying buttress arch (Figure 7, b) is a particular type of arch in which the intrados chord must not be tangent to the extrados chord, and no point is higher than the top impost. It is generally used as the free edge of a lame barrel vault, cut in the keystone.

The stair system of the Bartolomeo Di Majo noble palace is characterised by an intense internal spatiality, which is not revealed on the outside and combines the model of the open staircase with that of the ‘Roman staircase’.

The flights of stairs develop in subsequent convex curves, giving rise to triangular-shaped intermediate landings. The steps give way to a central well; in this case, the steps are supported by a system of vaults and flying buttresses and the staircase is statically self-supporting. The vaults rest exclusively on the perimeter walls and their support is based on the mutual contrast between the flights; the flying buttresses allow the stairwell to be left completely free (Figure 8). The lack of central supports for the well and the arches that mark the façades of the perimeter walls at the landings allow light reaching the floors below. Precisely because these stairs are built on walls that are not rectilinear, the vaults that make up the flights exert a strong thrust on the landing from which they start; this thrust, however, is balanced by the analogous thrust exerted by the incoming flight. This complex equilibrium system is based on the reciprocal contrast between the vaults and constitutes the basic principle of the structure’s stability. The masonry composing the staircase is covered with a layer of traditional plaster, decorated with grey frames on a white background, highlighting the vaults themselves. The plaster appears to be standard because of the lack of cracks due to technological incompatibility. In fact, the overlapping of a cement-based plaster on a traditional masonry would result in a crack pattern, which is absent in this case.

The plaster, however, leaves no space for the characterization of the underlying masonry. There are no gaps from which to detect the primary size of the ashlars and the type of material used, thus completing the analysis of the scale object of study.

However, due to its great availability in the subsoil of Naples, masonry is usually made of grey tuff, Neapolitan yellow tuff, and stratified yellow tuff (26).

The research reflects on the identifying dimension of the drawing and survey as a data collection tool, directed towards the knowledge of architecture through a dialectic relationship between material (the architecture of staircase) and immaterial sources (the drawing of the staircase in the treatises). The illustration of the case study was carried out employing a geometric-configurative graphic analysis. At the same time, the comprehension of spatial models and structural behavior integrated the results of the architectural survey and geometry with the disciplines of structural mechanics and construction history ¹ . A few years after his death, Sanfelice’s legacy would lead Mario Gioffredo (1718-1785) to replicate, albeit in a straight line, the rhombic staircase model in the palazzo De Sinno and in Via Tommaso Caravita. As in di Majo’s case, it was no longer a question of designing noble residences but according to the future culture of living, of building profitable properties where the staircase, looking at the sources, continued to be the representative space. From a more strictly structural point of view, the paper has dealt with the equilibrium of the type of open-well staircase. In the case study presented, the stairs are composed of no-tension Heyman material, for which the theorems of limit analysis can be applied. In the framework of Limit Analysis, the authors have already tackled the study of masonry staircases by using graphic statics (39-40). The method applied in this paper is the so-called membrane equilibrium analysis (MEA), originated in the paper on vaults (27) and further developed in (28). MEA is a general tool for computing the stress field on curved membrane surfaces (41), and it has also been adopted recently by the authors to assess the equilibrium of masonry domes (32-46). In the present work, we essentially used the ideas put forward in (33, 47), by applying the method to open-well staircases, treated as cloister masonry vaults. Above these considerations, it is possible to conclude that, for the case study analyzed, the structure can safely support the permanent and accidental actions to which it is subjected in the absence of significant settlements of the supporting walls.