1. Introduction
In 1954, the Italian engineer Sergio Musmeci (1926–1981) designed a reinforced concrete cross vault, characterized by a triangular layout and obtained by assembling three triangular webs with parabolic profiles [
1]. Although the vault geometry was a unique and pioneering result of structural optimization, it remained unbuilt and its project was only mentioned in reference [
2] as “exemplary for its modular conception of the structure based on the repetition of cylindrical shells”.
The smart geometry invented by Musmeci is revisited here to explore its potential realization as a masonry structure, as first proposed in references [
3,
4], and it is then taken for inspiration in considering the design of a more general class of triangular masonry cross vaults. As a reliable structural assessment methodology of masonry vaults, the static (or safe) theorem of limit analysis is adopted, following its formulation by Heyman under the assumptions of null tensile strength, infinite compressive strength, and no-sliding behavior [
5,
6,
7].
Departing from the classical slicing technique (for a recent application, see, e.g., reference [
8]), which models a typical masonry vault as a series of independent arch slices, thrust network analysis (TNA) has established itself as a powerful and automatic technique for the 3D funicular analysis of masonry vaults under gravitational loads [
9,
10]. It consists of finding a discrete network of compressive forces (or thrusts) in equilibrium with the vault self-weight, as reduced to nodal forces by tributary areas, and completely contained within the vault thickness. Peculiar to the method is the introduction of a form diagram, prescribing the network in horizontal projection and entailing a specific pattern of the internal force flow. Consequently, the unknown nodal heights and force densities (i.e., the thrust-to-length ratios of the branches) are determined through a suitable nonlinear optimization problem enforcing nodal equilibrium equations and static admissibility conditions [
11,
12,
13,
14,
15,
16,
17].
The TNA is mechanically interpreted by relating the network nodes to rigid masonry blocks ideally constituting the vault. Accordingly, the nodal equilibrium amounts to the equilibrium of the blocks under their self-weight and the interface forces, regarded as the thrusts in the network branches. In reference [
18], Block observed that “the equilibrium of a masonry unit, or voussoir, in the vault does not require all forces to meet at one point in 3D space”. In other words, the TNA assumption that all thrusts applied to a network node converge to that node is a sufficient, but not necessary, rotational equilibrium condition for the blocks. This observation has motivated the recent introduction of the generalized thrust network analysis (GTNA) [
19]. Among its advantages, the GTNA allows an extension of the set of equilibrated and statically admissible stress states in the vault and translates the classical minimum-thrust problem for the safe assessment of the vault into a simple linear programming problem.
Besides the intrinsically discrete formulations discussed above, continuous methods have been proposed for the static limit analysis of masonry vaults. The thrust surface analysis approach describes the statics of a masonry vault by searching for a compressed thrust membrane within the vault thickness in equilibrium with its self-weight [
20,
21,
22,
23,
24]. A generalization of that approach is offered by shell-based static limit analysis formulations, which overcome the thrust membrane concept by assuming that general shell stress states can arise within the vault to resist external actions [
25,
26,
27,
28,
29,
30,
31]. Thrust surface analysis and shell-based static limit analysis can be regarded as the counterpart continuous methods of TNA and GTNA, respectively. A general and comprehensive review of computational methods for the structural analysis of masonry structures can be found in reference [
32]. Experimental and computational investigations specifically devoted to the structural behavior of masonry cross vaults under diverse external actions have been discussed in, e.g., reference [
33,
34,
35].
In the present paper, the GTNA is adopted for investigating the structural safety of a masonry instance of the Musmeci vault. Initially, by conforming to the original geometry, the vault thickness is assumed as the only design parameter. Minimum-thrust and minimum-thickness analyses are performed, respectively enlightening the potential stress states in the vault after outward settlements of the supports and the geometric safety factor of the structure, as defined in reference [
5]. Subsequently, the Musmeci vault is regarded as representative of a class of triangular parabolic masonry cross vaults parameterized by their rise-to-span ratio and normalized thickness. Minimum-thrust and minimum-thickness analyses are thus extended through parametric analyses, informing of the structural behavior of such unconventional structures.
The paper is organized as follows. In
Section 2, the Musmeci structural model is reviewed. In
Section 3, the fundamentals of GTNA are discussed. In
Section 4, the minimum-thrust and minimum-thickness analyses of the Musmeci vault as a masonry structure are presented. Their extension to a parametric class of triangular parabolic masonry cross vaults is carried out in
Section 5. The conclusions are outlined in
Section 6.
2. Musmeci Triangular Parabolic Cross Vault
The structure referred to here as the Musmeci vault consists of a reinforced concrete parabolic cross vault with an equilateral triangular bay. It was conceived by Sergio Musmeci, in collaboration with the architect Giuseppe Vaccaro, in 1954, as a module forming, through four-fold repetition, the cover of a rural market in southern Italy [
1,
2]. The cover, which accommodated the rhomboidal layout of the market space, was the result of Musmeci’s early investigations on structural design optimization. An original sketch of the planar view and a reconstructed perspective view of the whole structure are depicted in
Figure 1. In detail, each vault was formed by three triangular webs with parabolic profile, characterized by external arches with a rise
and span
. The intersection of the three webs generated the ribs of the vault as semi-parabolic arches with a rise
f and span
. In the following, the equilibrium design approach developed within the original Musmeci structural model [
1] is briefly described, as also reviewed in references [
3,
4].
For the typical triangular vault to be considered, the Cartesian reference frame shown in
Figure 2 is introduced. A slicing technique is adopted to find an equilibrated membrane stress state in the parabolic webs. By using a symmetry argument, only the web whose ridge line is parallel to the
y-axis is considered. It is discretized into a family of independent parabolic web arches of infinitesimal width
, with a span linearly increasing from 0 to
, and with mid-curve parameterized as follows:
Motivated by the web shallowness, the web self-weight is assumed to be approximately uniform in horizontal projection and with a magnitude
q per unit surface. It follows that the vertical and horizontal actions (per unit
in horizontal projection) transmitted by the web arch of span
to the rib arch are, respectively, given by:
Consequently, the rib arch, which is parameterized in terms of the abscissa
as
is subjected to the vertical and horizontal loads (per unit
in horizontal projection), respectively, given by:
Since the rib is subjected to a vertical load
p linearly varying from the crown to the abutment, Musmeci inferred that the thrust line differs from the parabolic rib mid-curve. However, he realized that the constant horizontal load
s was capable of bringing the former very close to the latter, the maximum eccentricity resulting in a few centimeters [
1]. A structural analysis of the ribs was carried out by observing that, “Since the flexural stiffness of rib is practically negligible at the abutment, where the section is theoretically reduced to a point, and at the crown, where the angle between webs becomes zero, the behavior of the rib can be considered similar to that of a three-hinged semi-arch”. On such a basis, the vertical reaction at the abutment is given by:
the thrust follows from the rotational equilibrium about the crown hinge as
and the thrust value at the crown results in
the negative sign implying that tensile stresses arise in the rib from the crown to the neutral point located at the abscissa
. In particular, the normalized thrust at the abutment is estimated to be:
The Musmeci project assumes the vault thickness to range from 12 to
(with tapering in the crown region), the ribs having a triangular section with a height of
. The webs are connected along the external sides by edge curbs recalling the
formeret arches adopted in Gothic masonry vaults. The edge curbs have an almost rectangular section, with a thickness ranging from 15 to
. The reinforcement layout of the vault is shown in
Figure 3. In detail, the whole surface of the webs is reinforced by a welded mesh with
pitch, whereas the edge curbs and the ribs are reinforced by
top and bottom longitudinal bars and
diameter stirrups with
pitch. It is observed that the arrangement of reinforcement bars along the ribs is consistent with their structural role of curved beams in the Musmeci model.
4. Structural Analysis of the Musmeci Vault Reinvented as a Masonry Structure
In this section, a masonry instance of the Musmeci vault is considered and its structural safety is investigated using GTNA.
For complying with the original geometry, the vault thickness is assumed to be the only design parameter. A uniform normalized thickness is initially considered, corresponding to a thickness . Outward horizontal settlements are considered in the rib planes at the corner supports for a minimum-thrust investigation.
To that end, the beam network topologies depicted in plan view in
Figure 6 are adopted for the discretization of the vault mid-surface. The network refinement is controlled by a refinement parameter
N, coinciding with the number of beams along the half-web (the case
is illustrated in the figure). Panels (a) and (c), respectively, refer to orthogonal and fan-like networks, obtained by adapting the form diagrams proposed for a classical TNA of cross vaults with a square layout in reference [
17]. Distinct patterns of internal force flows underlie the two network topologies. In particular, the orthogonal topology presupposes that the self-weight of the webs is first conveyed to the ribs, which are then in charge of its transmission to the corner supports. Conversely, a direct flow of the self-weight of the webs toward the corner supports is envisaged within the fan-like topology. Motivated by the potential blending of the two internal force flow patterns, the orthogonal and fan-like topologies are conveniently interpreted as the two limit cases, respectively, corresponding to
and
, of a continuous sequence of network topologies indexed by a shape parameter
. By way of example, panel (b) of
Figure 6 shows the beam network for the choice
. Thanks to the problem symmetry, in numerical simulations, one-sixth of the vault is considered and suitable symmetry conditions are accounted for on the symmetry planes. At the corner support, additional static admissibility conditions are prescribed, requiring the line of action of the resultant constraint reaction to be contained within the springing section.
The left and mid columns of
Figure 7 show the generalized thrust networks of minimum thrust for the Musmeci vault obtained assuming, from top to bottom, the values
for the network shape parameter. A refinement parameter
is considered. For the typical network beam
b, a pipe with axis on the straight line
and diameter proportional to
(
Section 3) is plotted. Green [resp., blue] dots are used to identify the beam end-sections where the generalized thrust network is tangent to the vault extrados [resp., intrados]. The orthogonal network prompts the formation of a series of web arches supported by the ribs, which thus play a fundamental static role (
Figure 7a,b). Conversely, the fan-like network is characterized by the formation of fan arches, that interact on the vault ridge line, and directly rest on the corner supports (
Figure 7g,h). An intermediate static regime is obtained for an intermediate value of the shape parameter, with the ribs that are only loaded in the regions close to the corner supports (
Figure 7d,e). The right column of
Figure 7 shows the settlement mechanisms of minimum thrust, which are kinematically dual to the obtained generalized thrust networks. A progressive transformation is recognized, from one driven by the settlement of the rib arches and triggering a settlement of the web arches for the orthogonal network, to one due to the settlement of the fan arches for the fan-like network. The obtained values of the normalized thrust are
for
, respectively. Accordingly, the orthogonal topology emerges as the most efficient pattern for the internal force flow.
The minimum thickness of the Musmeci vault reinvented as a masonry structure is then investigated. A vault is in minimum thickness configuration when the corresponding generalized thrust network is tangent to the vault extrados or intrados in a number of sections large enough that, from a kinematic viewpoint, the opening of the descending hinges implies an incipient collapse mechanism. By applying the procedure discussed in
Section 3.4, the normalized minimum thickness is
, corresponding to a minimum thickness
and a geometric safety factor
. The minimum thickness is attained using the orthogonal network topology and the relevant value of the normalized minimum thrust is
. Such a result is in good agreement with the approximate estimate given in Equation (
8) and follows from the similarity of the orthogonal internal force flow pattern with the one originally considered in the Musmeci model, where a generally compressive membrane stress state on the vault mid-surface is found. The present results are also in line with the predictions of
and
for the normalized minimum thickness and the relevant normalized minimum thrust, respectively, obtained using a slicing technique that generalizes the one adopted in the Musmeci model in order to avoid tensile stresses [
3,
4,
8].
The small value obtained for its normalized minimum thickness proves the static efficiency of the cross vault geometry ideated by Musmeci. Although its unique design makes it difficult to compare with more conventional geometries, a qualitative clue can be achieved by considering a masonry cross vault with a square layout, a circular profile, and the same rise-to-span ratio (
) of the Musmeci vault as a benchmark. Its normalized minimum thickness is 0.0155, approximately 30% larger than the 0.0105 estimate previously obtained. Indeed, such a vault is characterized by a springing angle of about
, differing from the one of about
relevant to the original geometry. Nonetheless, upon referring to the larger class of masonry cross vaults with pointed circular profile investigated in references [
17,
19] for a benchmark, the normalized minimum thickness of such a vault with the same rise-to-span ratio and springing angle of the Musmeci one is 0.0111, which is still approximately 6% larger compared to the Musmeci geometry.
In closing, it is noticed that a further thickness reduction could be achieved by dropping off the assumption of uniform thickness. For instance, under the condition that the ribs are twice as thick as the webs, the normalized minimum thickness of the latter is , corresponding to a web thickness of .
5. Parametric Analyses of Triangular Parabolic Masonry Cross Vaults
The Musmeci vault is here regarded as an instance of a class of triangular parabolic masonry cross vaults whose geometry is described by the rise-to-span ratio and the uniform normalized thickness . The structural safety of such a class of vaults is thus investigated, accounting for their parameterized geometry.
As proposed in reference [
17], the minimum and maximum thrusts are explored, resulting from outward or inward horizontal settlements in the rib planes at the corner supports, respectively. In
Figure 8, the normalized thrust values,
and
are reported versus the normalized thickness
, for the selected values
of the rise-to-span ratio. Red, yellow, and blue curves correspond to orthogonal (
), fan-like (
), and optimized (
) beam networks, respectively. As expected, an increase in
implies a general decrease in the thrust regime, with the optimized network shape parameter shifting from
to
while considering vaults that are more and more slender. Moreover, an increase in
implies a larger interval
of admissible thrusts for the vault, to be interpreted as an increase in the vault robustness.
A parametric analysis of the minimum thickness of the vault, with respect to the rise-to-span ratio
, is then performed. The relevant results are shown in
Figure 9, where the estimates obtained using the orthogonal, fan-like, and optimized networks of beams are compared. It is observed that the optimized minimum thickness versus rise-to-span ratio curve is characterized by two branches. The first [resp., second] branch, corresponding to shallow [resp., slender] vaults, is increasing [resp., decreasing] and attained with an orthogonal [resp., fan-like] network topology. The transition between the two branches takes place for vaults with intermediate values of
and requires intermediate values of the network shape parameter
. In particular, a maximum of the curve is observed within such a connection branch, implying that the geometric safety factor of triangular parabolic masonry cross vaults under self-weight improves when considering shallow or slender geometries.