The staircases represent one of the most impressive architectural expressions of the building. Many authors presented a great deal of research over the years on this matter intending to understand how they are designed and laid out. This paper is concerned with a particular structural type of masonry staircase, known as stair with open well or roman staircase. It aims to demonstrate that in masonry-vaulted staircases, the close relationship between the shape and static behavior is particularly evident, and geometry and construction are essential for their stability. The authors have proved this statement by studying Palazzo Di Majo’s open-well staircase in Naples, whose main structure consists of tuff vaults. The first part of the article is substantially descriptive and presents an in-depth description of the geometric and architectural features of the stair. The second part explains all the aspects concerning the equilibrium of this kind of stairways, within Heyman’s theory of masonry.

Las escaleras representan una de las más imponentes expresiones arquitectónicas del edificio. Varios autores han presentado muchas publicaciones a lo largo de los años sobre este tema para entender cómo han sido diseñadas y cómo se sostienen. Este trabajo se trata sobre un tipo específico de escalera de albañilería, conocida como escalera “de ojo abierto” o “a la romana”. El objetivo es demostrar que en las escaleras con bóvedas de fábrica existe una estrecha relación entre la forma y su comportamiento estático. La geometría y la construcción son imprescindibles para su estabilidad. Los autores han demostrado esta tesis estudiando la escalera de ojo abierto del Palacio Di Majo en Nápoles, cuya estructura principal está constituida por bóvedas de toba. La primera parte del artículo presenta una descripción detallada de las características geométricas y arquitectónicas de la escalera. La segunda parte, explica el equilibrio de estas escaleras a partir de la teoría del equilibrio de estructuras de fábrica de Heyman.

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 (^{th}-17^{th} centuries appears to have been introduced by Andrea Palladio in the

A, Andrea Palladio,

In particular, the case study here presented reveals a strong analogy with the staircase of the

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 (

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 (

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

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

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 (

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” (

The staircase of Palazzo di Majo was surveyed by Michele Capobianco and published in 1962 in the magazine

A subsequent survey was published in 2007 by Italo Ferraro in

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 (

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 (

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 (

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:

arches and vaults work in contrast and produce thrusts;

the thrusts must be counteracted, i.e. absorbed in compression by the supporting structures;

the thrusts can be absorbed through metal chains;

arches and vaults work best when subjected to a substantial vertical load;

arches and vaults are stressed by very low compressive stresses compared to the stresses that cause crushing;

cracks in the vaults are almost always due to relative displacements of the supports (even if, in some way, related to the distribution of loads and geometry.

With particular reference to the typology analysed in this paper,

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 arches of the stairs are usually of the

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’.

(a) drawing by Roberto Pane in (

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 (

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 (

The structural analysis has been performed within the Limit Analysis framework as introduced by Heyman for masonry structures (

In this work, the static theorem (safe theorem) is applied, which ensures that the structure is stable if any statically admissible stress field can be found (

Regarding 3d structures, in the case of vaults, the surfaces representing the support of the singular stresses are unilateral membranes, whose geometry is represented

The examined stair serves a total of four levels and is built on a rhomboidal plan. The structure consists of four half-barrel vaults, which develop in sequence, interspersed with the same number of intermediate landings, supported by spheroidal nails made using a quarter of cross vaults (

The staircase equilibrium in the present study has been set through the use of several simplifications, required to reduce the difficulties connected with the complex structural system characterizing the vaults used in the construction.

The simplification adopted consists in first studying the equilibrium of a cloister vault generated by the intersection of a pavilion vault and a horizontal plane placed at a certain height from the springer. This equilibrium of the cloister vault has been then extended to the case of a central void at the mirror. In doing so, the problem has been shifted from the 3d case to the 2d case. In fact, a similar situation is precisely what is found in masonry stairs with an open well. The planform of the cloister vault has been subdivided into four trapezes (in which the stress is considered uniaxial) and a rectangle (in which the stress is assumed to be completely biaxial), as shown in _{
1
} ,_{
2
}) and a curvilinear reference system (

subdivision of the projection of the vault on the horizontal plane (left) and stress along the rays (right)

The edges of the well are assumed to be level curves of the membrane surface and, in a first step, form a structure, which can balance the stresses transmitted by the compression rays.

where (

To obtain the membrane surface carrying the transverse load, defined by the function f(_{1},_{2}) we start by prescribing an appropriate stress field. We introduce the curvilinear reference system

The covariant natural base vectors _{1} and _{2} associated with this system, are:

being

where

^{22} being the sole nonvanishing contravariant component of the projected stress in the curvilinear reference. For the two equilibrium equations [

The additional transverse equilibrium equation must be studied to verify the equilibrium in the surface

the equilibrium problem is reduced to a single scalar

The transverse equilibrium equation of the membrane, written as a function of the curvilinear coordinates

where

with

The surfaces ^{
0
} ,^{
1
} and ^{
2
} must not only be such as to ensure equilibrium but must also satisfy certain boundary conditions concerning the shape. These conditions are given below and detailed for the case under consideration:

^{
0
}

_{
red
} being the height measured at the arch springing, placed at an angle of 30° to the horizontal, considering the presence of the abutment at the sides of the arch.

^{
1
}

^{
2
}

assuming that the coordinate

The normal stress

Where _{
1
} and _{
2
} are the Cartesian components of the load

where

The simplest way to obtain the numerical solution of this system of first-order differential equations would be to provide some conditions for ^{
0
} , ^{
1
} , ^{
2
} lies inside the masonry.

This verification was carried out by superimposing in

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}

We want to thank Dr. Rosa Anna Auletta for providing us her dissertation entitled

This contribution is the result of a multidisciplinary team. The chapter 1 was written by Ornella Zerlenga; the chapter 2 by Vincenzo Cirillo; the chapters 3 and 4 by Claudia Cennamo and Concetta Cusano. The conclusions were written by all the authors.