Variable Casting of Voussoirs for a Stereotomic Shell

Abstract

The project for an acoustic shell in the Italian city of Matera was looked as an opportunity to explore an alternative stereotomic approach. The semi-vaulted space was initially thought to be built with discrete blocks of stone following a structural system in compression bounded by tie-rods, but practical and economic sustainability issues led to a different approach to that of classic cutting and carving raw stone. The collaboration between two different research teams led to the incorporation of a reusable mold technology; with the help of robotic technology and flexible moulding, it became possible to create customized heavy blocks discarding the need for disposable one-off moulds for casting voussoirs. By surveying stereotomy as a classic discipline within the scope of this project, this paper extrapolates and reflects on the validation of a different production process facing the classic ones that have defined stereotomy in architecture and construction.

The Acoustic Shell in Matera

Context

What is stereotomy? Once a fundamental technology acclaimed by everyone involved in the construction of significant stone structures, stereotomy seems now to be relegated to a few niches of interest. But why has cut stone become so scarcely used in spanning spaces today? And is stereotomy only about cut stone? Possible answers to these questions are explored in the development of the project described in this research: the design of an acoustic shell in Matera, Italy. This project is an on-going effort by the New Fundamentals Research Group (NFRG), based in the Dipartimento di Scienze dell’Ingegneria Civile e dell’Architettura (DICAR) of the Politecnico di Bari, in which the centerpiece is a vault built of stone for the acoustic conditioning of open-air performances (Fallacara et al. 2016: 200).

The project for the Acoustic Shell was born during an architecture studio course (2015–2016) in the architecture degree program of the Politecnico di Bari. This development is to be part of a more extensive intervention including a new museum and multipurpose centre in the city of Matera. This city’s historic centre is protected by UNESCO and is famous for its millennia-old stone houses built with local material cut from quarries around the city. The presence of this material in the history of architecture and landscape of the city led the design team to consider stone as the main building material. Additionally, it has been demonstrated by the NFRG, which has a research line dedicated to the updating of stereotomic stone vaults, that stone is specially suitable as a construction material in the Mediterranean region due to its large availability and high thermal inertia, which protects interiors from harsh summer temperatures.

The location for the Acoustic Shell project is very close to one of the quarries surrounding the city. Now a sculpture park—Parco Scultura La Palomba—, this quarry is bordered on one of its sides by a big natural stone cantilevered vault, which provided inspiration for the curved profile of the shell. This shell is the focal point of the architectural intervention, as it is also the counterpoint to the seating steps distributed along the sloping terrain, from which the audience can also admire the natural display of the cliffs that characterize the territory. The steps are made of stone and placed in a concentric fashion, respecting the tradition of classic theatres. While the stone material slowly disappears amidst the green landscape in the outermost area, it rises in the centre to give shape to the acoustic shell (Fig. 1). Bringing together modernity and tradition, this architectural device gives stone, symbol of Matera, a new shape, identity and function (Barberio et al. 2016: 587).

Fig. 1

Masterplan in Matera (Fallacara et al. 2016)

The acoustic shell was designed to fulfil two main roles:

• to control the sound dispersion created by the performer;

• to provide a backdrop for the performance.

Regarding the first of these, the control of the sound is carried out by the overall geometry of the shell as well as by individually controlled acoustic reflection devices (Fallacara et al. 2016: 204). Various combinations of these two factors were analysed in collaboration with Francesco Martellota, Professor of Building Physics in the Department of Civil Engineering and Architecture of the Politecnico di Bari. The results of these experiments, carried out with digital analysis of recorded sound with technical microphones (Fig. 2), pointed to the efficiency of the shell in decreasing the time between performer action and sound reception, while the circular diffusers helped in conditioning peaks and lows, giving uniformity to the sound (Fallacara et al. 2016: 205).

Fig. 2

Experiments with microphones for acoustic characterization (Fallacara et al. 2016: 204)

Regarding the second role, the cellular structure of the stereotomic shell system defines the backdrop of the performance, which has particular aesthetic interest (Fig. 3). This cellular structure is created with the idea that a clear link between topology and stereotomy may be used to achieve the geometric configuration of the desirable structure (Fallacara 2006: 1078). In this case, the correspondence between a flat surface and a quasi-toroid surface drove the arrangement of cells in this negative mean Gaussian curvature. The tessellation of the initial flat surface takes into account the structural properties of stone. Being a low tensile-resistant material, blocks should have minimal acute angles or thin segments. The hexagon was chosen for the tessellation due to its large inner angles ideal for stone construction, and its perimeter-to-area ratio which affords optimized contact interfaces and fabrication resources.

Fig. 3

Preview of the original acoustic shell (Fallacara et al. 2016: 201)

Stereotomy as a Design Driver

The design of the acoustic shell for Matera seeks to establish a dialogue between the lightness of the general shape with the weight of the building material. A cantilevered structure is dependent on a tensile-resistant material, which is commonly used in the full extent of its mass, for instance, by means of beams or rebar in reinforced concrete. However, the main material chosen for this structure is stone. Its lack of tensile-resistance properties has led it to perform mainly compressive tasks, through the “the art of using the weight of stone against itself so as to hold it up thanks to the very weight that pulls it down”, as Perrault so wonderfully has put it (Perrault 1688: 171).

Unlike in the arch configuration, the blocks in the top of a cantilevered structure lack equilibrium, which requires compensation, through the interaction between stone and metal reinforcement. This kind of solution is found generally throughout history for reinforcing stone construction (Ariza et al. 2017: 108) and more specifically in Viollet-le-Duc’s Entretiens sur l”Architecture, where a prototype for a reinforced arch is explored by means of thrust equilibrium achieved by the interaction between stone and steel bars in continuity with the thrust vectors (Viollet-de-Duc 1863: 79). Adding these bars to a belt around the entire perimeter, this solution allows for the remainder of the stone of the structure to be tightly and accurately packed together in the form of compression resistant voussoirs (Fig. 4). The approach for the fabrication of the voussoirs is the traditional stereotomic way of cutting of stone, albeit with current technology tools. For optimization purposes, each “super-voussoir” is composed of six independent stone blocks, allowing the concavity to be cut with a saw instead of carved by milling. In the end, each of these groups are packed together within a perimeter belt (Fig. 5).

Fig. 4

Individual stones packed into a super-voussoir (Fallacara et al. 2016: 200)

Fig. 5

Groups of modules within the universe of voussoirs

New technologies have brought a new democratization (Kolarevic 2015: 52) by facilitating the access to construction methods once dependent on intensive manual labour. Although this democratization opened the possibilities to access material customization once again, it is not without considerable constraints. Various processes, in which the Industrial Revolution played a major role, have increased the cost of specialized manual labour, especially when alternatives of mass production are available. The advent of digital fabrication creates an opportunity for the usage of custom shapes once reserved for the hands of skilled masons. However, this opportunity lies more in technical accessibility than in economic feasibility; for now digital fabrication has not inaugurated a generally cheap way of production, but does instead permit a geometric freedom of creating.

In this context, it is understood that the cutting of stone into unique shapes brings an added cost to a project, both in the value of cutting time and resources, and in the amount of non-reusable stone waste in the project. These facts can amount to heavy expenditures, often playing a part in the decision-making process of a construction process. Bypassing the need for the costly dressing of stone, whether manually or digitally tooled, can lead to a cost-effective way of implementing stereotomic principles to construction.

Within this background, a research project in the Digital Fabrication Laboratory (DFL) of the Faculty of Architecture of the University of Porto (FAUP) has explored alternative paths for materializing stereotomy. One of such alternatives deals precisely with reducing time and waste material when fabricating voussoirs by using a variable geometry mould for casting unique blocks (Azambuja Varela and Sousa 2017: 198), which challenged one of the classic paradigms of stereotomy: that it is strictly related to stone cutting. After a contact and learning the characteristics of this method, the NFRG challenged the DFL to collaborate in this project, especially in developing the fabrication strategy inspired by the variable mould system. This partnership has as its main objective the search for specific solutions for the stereotomic structure proposed for Matera by joining the efforts of each team’s core interests: the stereotomic construction critical thinking of the Bari team, and the custom fabrication and design strategies developed by the Porto team.

A Flexible System for Variable Voussoir Casting

Principles

The system for materializing stereotomy developed at the DFL relies on a structure depending on discrete heavy voussoirs which, inside the system, support themselves. In order to reduce material waste and cutting time and resources, a casting technique is chosen. Instead of using raw stone extracted from the quarry, plain concrete or stone powder are poured into a mold and cured until solid.

Forming is typically dependent on one-off moulds, which tend to relate its application to mass production logics. Indeed, the exploration of mass customization through forming processes tends to be expensive, resulting in heavy expenditure (Clifford et al. 2014: 4). As seen in Fig. 6, the structure for the acoustic shell presents a great degree of variability—32 types in an universe of 136 items—, which would fit the latter case. By using a flexible casting system it is possible to overcome the production challenges triggered by the high degree of customization in the voussoirs of this structure.

Fig. 6

Voussoir terminology (Azambuja Varela and Sousa 2016: 35)

The structural efficiency of a stereotomic structure is bounded by various factors ruled by its geometry. The geometry must be correct under two main constraints: the general distribution of volume/mass that should follow a thrust vector system, and the contact surfaces normals which are also bound to the same vector system. The first requirement is relatively easily met; for example, arches with different types of curves are able to stand because they are thick enough to contain the thrust curve. The second requirement is less forgiving, as the contact faces are the interface between the voussoirs that make up the system. A correct orientation minimizes residual vectors, ensuring the efficient transfer of loads and avoiding slippage, which performs a key role in the success of a stereotomic structure.

Following the previous line of thought, the contact faces (Fig. 7) are the main target of the variable mould system regarding its accuracy. In order to enhance the model to reproduction accuracy, these are designed as ruled surfaces, that is, surfaces composed of straight edges. A stretched membrane between two locations is understood as composed of lines that connect every pair of two points within the minimum distance, thus giving us the Euclidean definition of a straight line. The voussoirs’ contact faces are thus designed as ruled surfaces, by connecting every two adjacent contact edges. Given that every contact edge is normal to the thrust surface, the generated double-ruled contact face is also tendentially normal to the thrust surface.

Fig. 7

Fabrication guidelines overlaying voussoir geometry (Azambuja Varela and Sousa 2016: 35)

Typical casting consists in pouring a fluid into a container of some sort, and this idea is translated to the voussoir grammar: if the sides of the container became the contact faces, the bottom of the container will be the intrados. The extrados tends to be flat, while the intrados in its simplest version is also flat. These geometrical constraints are not an obstacle to stereotomic structural efficiency (Kaczynski et al. 2011: 115), as it has been observed in previous experiments (Azambuja Varela and Merritt 2016: 772).

Description

Algorithmic Generation of Model and Fabrication Data

The principles described above call for specific design guidelines which were carefully transported to algorithm rules. Using Rhino’s Grasshopper graphical algorithm editor, a parametric model was built to create every voussoir’s geometry and inherent fabrication data. The input data of this model are the generic thrust surface of the acoustic shell, the initial grid, voussoir thickness and shaft radius (Fig. 8). With this information, the parametric model is able to automatically generate all the voussoir geometry and fabrication data through these summarized steps: (1) map the grid into the shell surface to generate individual cells; (2) flatten each cell, keeping vertex in same thrust normal vector; (3) thicken each cell along thrust normals; (4) offset contact faces inside with a distance equal to the shaft radius; (5) intersect the previous offsets to generate the axis of each shaft; (6) create helix path for the robotic drilling program.

Fig. 8

Different aspects of membrane stretching

Fabrication Physical Setup

The flexible and reusable nature of the casting system implies reusable mould items that somehow inform the nature of the finalized blocks. The perimeter geometry of the intrados is a direct consequence of the strategy for the realization of the contact faces. Given the straight nature of the contact faces intrados edge, the intrados perimeter is to be composed of straight edges. As these edge lengths and relative angles vary according to each voussoir geometry, a planar horizontal intrados perimeter is apparently unavoidable; a rigid conical or pyramidal surface for moulding the intrados would quickly become incompatible with the wide range of geometries present in this shell. Although the perimeter must be planar, its surface might be regularly or irregularly concave towards the centre of the voussoir, in any shape the design might require.

These first experiments for this kind of strategy for fabricating voussoirs used a flat plane as a base for the casting, resulting in a flat intrados. The project for the Acoustic Shell introduced an interesting variant in the possibility for intrados design and fabrication. The original project comprised a concave super-voussoir (each of which was composed of six radially cut stones, resembling orange segments or circular sectors) with a hole in the middle which was used for the implementation of the acoustic apparatus. This concavity plays a role in housing the acoustic diffusing disc, resulting in a symbiotic relationship between aesthetics and function; the importance of this feature sparked the search for a solution adapted to the casting variant of the fabrication approach.

From a geometric point of view, the first iteration in cut stone for the fabrication strategy for this shell had the intrados concavity created by six sloping faces, each connecting an intrados perimeter edge to a hole edge. In a topologically similar fashion, and in line with the reusable mould approach, the new iteration has a membrane stretched so that it connects every pair of intrados edge and hole edge. This allows the intrados mould surface to adapt its morphology to each surrounding perimeter (Fig. 9), effectively solving the geometries needed to shape all the sides of the variable reusable casting system.

Fig. 9

Synthesis of the fabrication system

The mould system was built with a wood board, metal pins, rubber band and fabric (Fig. 10). The concave surface of the intrados is materialized with Lycra fabric draped on top of a centre hexagonal prism and tensioned on the perimeter; a second experiment had radial strings below the Lycra fabric.

Fig. 10

Steps for the fabrication of a voussoir with the variable casting system

The variable casting operation follows the following steps (Fig. 11):

Fig. 11

Detail of protrusions and shadows generated by the perimeter of the planar intrados

1) Six cylindrical holes, one for each steel tube support, are milled in a planar wood board with the help of a 6-axis industrial robot, so that each of the six cylindrical axes is parallel to its corresponding contact edge.

2.1) A previously tooled prism with the hole shape is fixed to the base board. This prism together with the board are the only constant elements of the variable geometry system.

2.2) The Lycra fabric is placed on top of the prism and stretched until the intrados perimeter.

3) The metal pins are inserted in the holes, pushing the fabric down and effectively fixing it. Additional adhesive tape is used to stretch the fabric between pins outwards. The second half of the prism is fixed into place.

4, 5) The rubber band is fixed to a designated pin with a custom hardware and is stretched around all of the pins with the help of a ratchet.

6) The casting takes place and all the volume required is filled, effectively creating a horizontally planar extrados.

7–8) After the minimal cure time, the rubber band and metal pins are removed, and the cast voussoir may be put to rest for further cure.

Fabrication Constraints

Due to the fabrication strategy, some geometric constraints apply for the design of the voussoirs and to the general design as a consequence. As defining elements of the voussoir’s geometry, the perimeters of its intrados and extrados should be polylines contained in parallel planes, coincidentally horizontal when casting. While the extrados surface is most likely fabricated as a planar surface (due to the setting of the casting fluid), the intrados surface may be modelled with the help of extra material (Fig. 9). Connecting both of these polylines, the contact surfaces must be ruled and the generatrices must be parallel to the intrados and extrados, being the simplest case a plane and the most complex a hyperbolic paraboloid surface.

The vault appearance is characterized by the cylindrical voids left by the metal pins in the casting stage. These appear in the place of the contact edges (Fig. 7) of the fabricated voussoir, which are not sharp and do not yield a watertight structure due to the voids aligned with the normal of the vault’s thrust surface, resulting in an effect similar to that of concrete tie holes. These holes are part of the material language, its own tectonic, expressing the genes that result in its phenotype (Fig. 12). On the other hand, the curvature of the shell has a direct influence in the quality of the mould. A high surface Gaussian curvature is associated with a large amplitude between the normals of adjacent contact edges. This amplitude causes one border of the rubber band to stretch much more than the other, eventually creating pleats in the shorter side. This is not desirable, as the mould should follow the principle of straight lines so that the double-ruled surface is accurately reproduced.

Fig. 12

Front elevation of acoustic shell with proposed changed voussoirs

Materialization Tests

Redefining the Component Design

The understanding of the fabrication constraints suggested modifications of the shape of the voussoirs, eventually leading to an updated conception of the Acoustic Shell surface (Figs. 12, 13). The original iteration was based on the idea of CNC cut stone blocks. This led the design team to define the contact faces as planes more easily cut with a saw. With the change to cast voussoirs, the geometry became closely related to the geometry of the mould and the effect of its stretched membrane. Regarding the geometry of the contact faces, there was a shift from a plane to a hyperbolic paraboloid. By comparing it to planar geometry, this type of contact surface is closer to a pure normal to the thrust surface, which tends to be more efficient by avoiding slippage between blocks.

Fig. 13

Fabrication sequence of one voussoir

On one hand, there is an effort to maximize surface orientation accuracy, on the other hand there is not a total usage of the contact surface area in what concerns force flow. This is due to the negative mean Gaussian curvature of the surface used in this project, which does not allow for a continuous tessellation of convex hexagons. The fabrication process creates a planar intrados perimeter which is responsible for the subtle protrusions and shadows in the corners of each block (Fig. 11), contributing to an updated aesthetic of the project surface.

As we have noted above, the intrados surface design does not follow structural requirements, although it might cover other type of concerns. To embed sound diffusers in each voussoir, an inverted pyramid cavity was considered in the original design, conceived for stone cutting. By using a casting process to fabricate the voussoir, one can include that concavity geometry within the mould itself. Further, with regard to the production of the contact faces, the elasticity of a fabric was explored to create the intrados shape as an emergent minimal surface. With the weight of the cast material, the fabric is pushed downwards, becoming flat in contact with the perimeter of the intrados.

The Fabrication Process

A first prototype was developed (Fig. 14) at 1:2 scale, choosing plaster as the material because of its fast setting properties. The material to be used in the actual construction has not yet been decided, but the possibilities range from concrete to artificial stone, mixed together with other materials. The most important factor in the material decision for the experiment is the similarity in fluidity to those possible future materials.

Fig. 14

Fabricated prototype with smooth transition from intrados perimeter to center void

Following the principles of the casting system described in “Principles”, the first part of the fabrication process is the creation of the holes for the pins in the base board. These holes are drilled with a 20 mm flat mill mounted on a spindle driven by a 6-axis robot arm. The program for the robot’s drilling movements is directly derived from the shell-generating algorithm, which also created the voussoirs’ geometry. The contact edge vectors are then oriented from the three-dimensional design space to the robotic fabrication space. A tangential movement (for drilling a 30 mm deep hole) and a circular movement (for the widening of the hole so that the steel tube can fit) are programmed for each contact edge and subsequently milled, creating six oriented holes.

A prefabricated hexagonal wooden prism (40 mm high × 25 mm side) is attached to the base board in order to inform the location of the central hole of the voussoir where the sound diffuser will be attached. Subsequently, Lycra fabric is stretched over the prism and across the board so that it covers all the holes. Over the stretched fabric, the second half of the prism is fixed, so that it will create the void in the center of the cast voussoir.

The support tubes are then inserted with the help of a hammer, pushing the Lycra fabric down and securing it in place. The rubber band is then fixed to of one of the tubes and stretched around the remainder tubes. Its tension is maximized by means of a ratchet strap, maximizing its tension.

The pouring process was carried out in two stages. The first stage uses a thicker mixture as to solve two issues dealing with the rubber band. The first issue is the leaks created by the gaps between the rubber band and the base plate. The second issue is an unwanted outward bulging effect on the membrane caused by the weight of cast material, which is solved by manually spreading the mixture in the membrane so that is adheres and results in a shape crystallization after it dries out. The second stage uses a much more liquid material, producing a clean finish of the extrados.

The removal of the mould takes place after 20 min of cure. When the plaster is already solid, both the rubber band and the steel tubes are carefully removed, unveiling the finished block. In the end, some small adjustments were made with the help of a file. The first fabricated prototype of the block is shown in Fig. 15.

Fig. 15

Finished voussoir

Alternative Component Design

When comparing the first prototype with the geometry featured in the original design, a clear difference can be observed. The concavity in the latter loses some of its depth and polygonal expression in the former. In the attempt to produce those formal qualities with the casting system, a second experiment was carried out. By attaching a system of tensioned strings connecting the corners of the intrados to the centre of the support block for the void, it was possible to control the deformation of the fabric due to the weight of the cast material (Fig. 16). This strategy resulted in the creation of six radially triangular faces in the intrados. Although this geometry more closely resembles the shapes featured in the original design, a clear difference is noted between the flatness of the faces of the original polyhedral intrados and the second prototype’s bulged surfaces (Fig. 17). The pleats observed are mainly due to a change in the type of fabric from Lycra to serge.

Fig. 16

Fabrication sequence of alternative design featuring strings to support the base fabric

Fig. 17

Fabricated prototype with radial grooves created by strings in the casting process

Conclusions

Through its various iterations, the project for an Acoustic Shell in Matera provided an opportunity for stereotomic research. This project was born within a broad masterplan studied within an academic context, but it has since been target of various developments which has brought it successively closer to a possible construction reality. Despite being profoundly inspired by stereotomic construction methodology, this project evolved by moving away from the use of stone cutting processes to the fabrication of its components. As that type of production is economically challenging, our research proposes an alternative fabrication strategy consisting in a casting system based on variable mould.

The adaptation of this novel technology to the Matera project brought specific design and material challenges which enriched the acoustic shell project and the system itself. The simplicity and flexibility of the system based on the orientation of the support pins and the elastic rubber band proved successful in the materialization of different shapes. Facing the production 32 unique pieces, which are then repeated four times on average, this method showed a real and feasible possibility to embrace a high degree of customization. Besides the direct adaptation of the system, new features, such as interior voids or a flexible intrados, were considered, developed and prototyped. The materialization experiments described in this paper thus suggest the capacity of the system to fabricate the acoustic shell voussoirs. Nonetheless, some specific issues can still be subjects for improvements before a definitive real application. Future research avenues would address the subtle deformation of the rubber band during the casting process, possibly by adding rigid elements to increase the accuracy of the geometry of the contact faces. In that investigation, the simplicity of the means will continue to be a key concern.

In the beginning, this paper started by questioning the validity of stereotomy today, by verifying its dependence on a close relationship with stone. By not involving any physical cutting or sectioning of the building material, the stereotomic blocks for the Matera project (Fig. 18) could arguably be considered as alien to the classic ideal of stereotomy. However, through a synthetic analysis over the key treatises on the subject, one can build an argument for the importance of structural and material properties of the construction components versus the geometry and qualities of the generated architectonic space.

Fig. 18

Both prototypes intrados view

The term “stereotomy” was born out of an argumentation between a professional of the stone construction and a mathematician interested in the relation between shapes in space. Desargues, a mathematician, used the term coupe des pierres in his theoretical work and sparked a discussion with Curabelle, who coined the word stereotomy, probably by making a reference to the divine firmament (i.e., stereoma) (Fallacara and Minenna 2014: 19). By using this extreme comparison, Curabelle effectively diverted attention away from the idea of architectonic space present in treatises titles such as Le Premier Tome d’Architecture (1567) or L’Architecture des voûtes (1643) and towards the technical art of stone cutting present in subsequent works such as Traité de la Coupe des Pierres (1690). Frézier’s influential work La théorie et la pratique de la coupe de pierres (1738) seems to have quieted the discussion, until the emergence of another famous dispute. In the second half of the nineteenth century, Gournerie fiercely disputed Monge’s ideas on stereotomy (Rabasa 2011: 718). Once again abstract theory is put against hands on practice, calling into question the view of stereotomy purely as a stone-cutting discipline, as the issues of mathematics and space modelling emerge to become equally important.

The real value of stereotomy to architecture today appears to be that of providing a way of conceiving and building space. The architect fluent in stereotomy has the tools to conceive a covered space where form, material and structural performance are integrated in the design act. Furthermore, he knows how the form should be divided in significant parts and how to materialize those parts to fulfil his original vision. The stereotomic blocks needed for building the Acoustic Shell in Matera do not need to be carved from raw stone extracted from the quarry. The main issue in this architecture approach is acknowledging those blocks as part of a coherent grammar in which its constituents relate to each other and are readily accessible. The cast voussoirs are part of a proposed update of the language of stereotomy today, one that is as concerned with the economic and ecologic sustainability as with the architecture expression of vaulted spaces. This research work reinforced the notion that stereotomy today might be achieved in different materials and production techniques, paving new avenues of research both for the NFRG and the DFL, and contributing to the development of techniques for the re-enactment of stereotomic construction.