Efficient and Expressive Thin-tile Vaulting using Cardboard Formwork Lara DAVIS Research Assistant ETH Zurich Zurich, Switzerland davis@arch.ethz.ch

Matthias RIPPMANN Research Assistant ETH Zurich Zurich, Switzerland rippmann@arch.ethz.ch

Tom PAWLOFSKY Research Assistant ETH Zurich Zurich, Switzerland pawlofsky@arch.ethz.ch

Philippe BLOCK Assistant Professor ETH Zurich Zurich, Switzerland block@arch.ethz.ch

Summary This paper presents research in the construction of freeform, thin-tile vaulting, demonstrated through a full scale prototype and application of an expandable cardboard guidework system. This form of complex structural design is enabled through Thrust Network Analysis (TNA) – a novel three-dimensional design tool for exploring funicular form. These free form shells present new challenges to the tiling patterns, sequence of building and especially the structural stability and guidework during construction. A continuous formwork system was developed using 2-D CAD-CAM cutting and gluing processes. The combined implementation of traditional thin-tile vaulting, TNA and the efficient cardboard formwork allows the construction of increasingly complex forms with the same (or even improved) benchmarks of sustainability and economy – radical form with efficient construction. Keywords: Catalan/ Thin-tile vaulting; freeform vault construction; compression-only forms; form finding; cardboard guidework; CAD-CAM fabrication; tiling geometry; construction sequence.

1. Introduction Thin-tile (or Catalan) vaulting, in its traditional form, is a materially efficient, sustainable construction technique [1,2]. It employs the use of good structural form to achieve a minimal shell thickness. A rapid-setting plaster-of-paris mortar can be used to add thin masonry units to the vault to construct the first layer of the masonry surface in space, essentially without formwork when proper edge conditions are provided. The bond of the plaster mortar allows masonry units to be carried with small amounts of bending until closed-course arches or a vaulted surface is formed carrying the load of the structure in compression. A minimal, lightweight guidework must be employed to visually define the masonry surface for builders, so that the structural form may be accurately constructed. Such guidework may consist of light wood constructions or mason’s line drawn in space, between which the mason must loft the surface of masonry by eye. Critical constructional logics limit the forms of vaults achievable with such systems. On account of the geometric constraints of these minimal guidework systems, the ability of the mason to interpret the structural surface, and the requirement for simple structural behavior to stabilize a vault during construction, it is only possible to build relatively predefined, regular geometric shapes with this technique. Thrust Network Analysis (TNA), a novel design tool for exploring three-dimensional funicular form [3,4], enables the development of fully 3-dimensional equilibrium shells. TNA gives rise to a whole new range of complex shapes and makes possible a new type of vault: the freeform, compression-only Catalan vault. This class of freeform, thin-tile vaulting presents new challenges in the realization of the Catalan vaulting process. For instance, the three-dimensional force flow in freeform shells is too complex to be simplified or

abstracted into e.g. arch action or hoop forces. Therefore, in order to sequence masonry courses in stable sections during construction, to establish in-situ structural stability, it is necessary to have a comprehensive formwork system to carry the load of the shell before it is completed. Additionally, the simple geometric guides of traditional thin-tile vaulting no longer suffice to aid the builder in the execution of such double-curved surfaces. A comprehensive surface guide must be employed to describe the complex doubly curved masonry surface to the mason for the purpose of accurate construction. Thus, both on the grounds of surface description and stability during construction, a continuous formwork system is more pragmatic for these new forms possible. While this conclusion seems to negate the inherent material and labor efficiency of thin-tile vaulting, this research introduces a formwork system, which is capable of describing the freeform Catalan shell, whilst still possessing the material economy of the traditional Catalan shell. The cardboard guidework implemented in this project is fabricated with 2-D CAD-CAM cutting and gluing processes and is assembled on site. The system’s rapid fabrication, lightweight transportation, and speed of erection and de-centering dramatically reduce the material and labour-based costs of construction. An inexpensive and potentially reusable/recyclable material, this lightweight cardboard guidework extends the viability of thin-tile vaulting to freeform construction. The combined implementation of traditional thin-tile vaulting, TNA and the efficient cardboard formwork allows the construction of increasingly complex forms with the same (or even improved) benchmarks of economy – radical form with efficient construction.

2. Method/ Approach

2.1 3D Form Finding Method Thrust Network Analysis (TNA) is a recently developed form finding method especially suited for the design (and analysis) of compression-only structures with complex three-dimensional form [3]. The design criteria of the prototype vault described in this paper were chosen in order to present new challenges to the method by including a diverse and complex set of boundary conditions, which address free “edge arches”, a high degree of curvature, formwork constraints, and suitability for possible tiling patterns. Furthermore, other specific design criteria include the following: a shallow arch with high horizontal thrust, a steep arch with low thrust, and an arch which curves in plan; a double-domed condition and a structural fold; a curved boundary and a point springing. Figure 1 shows the form finding process with TNA: a) the form diagram defines the force pattern; b) the compression-only thrust network for the chosen force pattern, in equilibrium with a given loading; and c) its corresponding reciprocal force diagram visualizing the distribution of thrust which result in the freeform vault.

2.2 Cardboard Formwork system

2.2.1 Description

The proposed formwork system for freeform thin-tile construction is an expandable cardboard system, essentially composed of simple boxes. Post-consumer recycled cardboard was used, which is a sustainable improvement upon prevalent formwork materials, since corrugated cardboard uses between 60 and 100% recycled paper fibers, and the fibers can be used up to seven times before they become too short for reuse. [5]

Fig. 1: The TNA form finding for freeform vault: a) the form diagram; b) the resulting compression-only thrust network in equilibrium with a given loading; and c) its corresponding reciprocal force diagram.

As shown in Fig. 2, the boxes are CNC cut and then assembled together to form a continuous, primary formwork surface. The cardboard boxes are supported by a system of stacked shipping palettes. Using palettes for the first rough approximation of the final vault shape offers several advantages: it reduces the volume of cardboard to be used; it facilitates easy access during construction as the palettes can be arranged in step-like configuration; and it decreases the size of the boxes, ensuring that the unrolled cutting pattern of the boxes fit to the limited machine-size of the CNC cutting machine. Note that using smaller boxes also increases the resolution of the surface and therefore refines the visual guidework for masons, allowing them to build the surface curvature more accurately with respect to the generated compression surface. Since large patches of the tiles may be constructed to follow the complex geometry (unlike in the traditional method), the cardboard formwork must have the capacity to support sections of the structure until the masonry surface can stand on its own. As the self-weight of the Catalan vault is quite low, the cardboard framework provides sufficient support to unfinished, and hence cantilevering shell segments.

To evaluate the stability of the cardboard boxes used for the formwork, the following approximation has been used: the crushing strength of a box of 600x400x400mm, made out of 6.5mm thick double wave corrugated board, 0.7 kg per square meter, with a ECT value of around 6kN/m is 4.4kN [6]. Since the cardboard selected for use is much stronger than this (7.3mm, ECT 14 kN/m), the compression strength of the box should be at least 5.5 kN. Translated, this renders a formwork capacity of at least 6 kN per square meter. These numbers demonstrate that the static aspects are not decisive for the box dimension. The final dimension has been mainly selected based upon the constraints of palette-size, the surface resolution in relation to the brick size, and the ability for bricklayers to stand inside the open boxes, i.e. the base dimension wanted to be bigger than the size of a worker’s foot.

Fig. 3: Cardboard box fabrication: a) CNC fabrication process; b) Diagram of cardboard box assembly; c) standard box cutting pattern; b) customized CNC cutting patterns for freeform vault surface description.

Fig. 2: Cardboard formwork system, showing a) masonry surface, b) cardboard boxes, c) palettes, d) de-centering tubes with cardboard spacers, e) edge registration.

2.2.2 Box Fabrication Figure 3 shows the design and production process for the boxes, which is established by a digital workflow, implementing custom RhinoScripts [7] to translate the compression surface into the final machine code needed to produce the 200 individual cardboard boxes. The steps in the scripts are for the purpose of defining the positions of the palettes, arranging the cardboard boxes, establishing naming and labeling, organizing the CAD document, cropping the boxes with the vault surface, unrolling the 3D geometry of each box, identifying the curves for cutting, creasing and labeling, identifying unexpected exceptions, monitoring the position of the 2D labeling, and querying the user for its optimization. Ultimately, the final tool paths for a maximum of two boxes are nested on single cardboard sheets, and the translation of the tool paths to the final machine code is completed with a custom CAD-CAM interface.

2.3 Considerations for Thin-Tile Vaulting

2.3.1 Tiling geometry, Construction phases & In-situ structural stability Tiling geometry must always be based upon patterns, which confer structural stability during construction. This means, to be stable in-situ, the sequence and pattern of masonry must rely upon closed-course arch action, hoop forces, or stiffness from increased structural depth. With the typical simplified geometries of traditional Catalan vaulting, this is achieved by building edge arches, cantilevering short distance from wall boundaries, or closing in a double-curved vault surface with hoop forces. With irregular geometries, such as those produced by TNA for the prototype vault, one may arguably abstract certain simple structural behaviors and build with a pattern and sequence conducive to in-situ stability (i.e. first building “edge arches” to establish arching action, closing in adjacent masonry surfaces, and then breaking the pattern with a groin to begin another arch section.) However, to test the loading capacity of the cardboard system to carry load in-situ, two continuous patterns – representing the topology of the form – were chosen. These patterns did not represent the most obvious closed-course strategies for in-situ structural stability. As shown in Figure 4, masonry sequencing followed the following logic: (Ref. Fig 6) 1. Masonry was first laid registering the profile of Footing 1. This section relied on high double-curvature and the curved structural depth of the footing to establish a stiff shell without undue bending strain; 2. Footing 2 was then begun, taking care not to extend further than necessary, since this section of the vault is very flat and thus more predisposed to bending stresses at the footing; 3. The edge arch and subsequently the interstitial region between Footing 1 and 2 was closed as rapidly as possible to establish arch action; 4: Arch B and E were then built up separately in several courses to provide stability; 5: The established coursing pattern was then continued, spanning between Arch B and E at middle-span, to avoid asymmetrical loading of Arch B and to leave working space for the installation of the second layer; 6: The median range was completed, relying on arching action of Arches B and E, which established the structure in a stable state.

Fig. 4: Tile pattern and phasing diagram of a) first tile layer and b) second tile layer.

2.3.2 Curvature and Cutting logic: For a flat, rectilinear masonry unit to achieve a high degree of surface curvature, cutting logics must be employed. Cutting logics are dependent upon the relation of the scale of the masonry unit to the scale and degree of curvature of the masonry surface it describes. The most ideal cutting logic for the high double-curvature of the prototype vault would thus be a two-cut system, employing a combination of oblique and bevel cuts to “bend” a surface in space [2]. Due to time and tool constraints, a simplified version of this system had to be used, allowing for high degrees of curvature in one axis of the masonry unit, while relying on hinging and the compensation of the mortar joint in the other. The degree of double curvature of this prototype is at the limit of what is achievable by a single-cut cutting logic.

2.3.3 Shell Thickness Traditional thin-tile vaulting often uses increased withes to establish thickness where it is structurally necessary. For example, in a regular dome structure, more layers of tiles are used at the base than at the crown. Note that this is easily realized, because the tiles are laid out in simple radial coursing with an increasing number of tile layers – the tiling geometry and geometry of required structural depth coincide. However, locally increasing thickness of a freeform shell is not that straightforward, since the tile pattern required to realize the complex geometry of the shape does not necessarily align with the pattern of forces as it does, for example, in the dome. In the case of the structure described in this paper, the forces require varied thickness of masonry, (between 1 and 3 layers). The third layer is only necessary in several isolated sections, where higher forces are concentrated. A satisfactory compromise was to surface the entire vault with two layers, with the third layer sandwiched between the intrados and extrados only where necessary. This also provides a safety factor for the experimental de-centering process (see 2.4). This solution allows for a continuous shell on the intrados and extrados, while still providing the necessary thickness where greater thrust is accrued in the form.

2.4 De-Centering Strategy De-centering is the process of removing the formwork from the surface of the shell. This is a sensitive process, because the entire formwork should be removed equally and simultaneously to avoid dangerous asymmetric loading cases from below. Such asymmetrical loading would induce bending in a compression-only structure and potentially cause cracking and failure. Whereas the lightweight guidework of the traditional Catalan construction method is simple to de-center, the relatively heavy, unitized box and palette formwork system is relatively challenging to lower systematically. This demanded the development of a de-centering mechanism, which would slowly and uniformly drop the formwork system away from the masonry surface. To facilitate de-centering, therefore, the entire formwork sits on top of a series of sealed plastic tubes containing cardboard spacers, as shown in Figure 5. Each spacer, which consists of a folded stack of cardboard sheets, taped together, supports the corners of typically 4 palettes. After the vault construction is completed, the tubes are filled with

Fig. 5: a) Plan of cardboard box/ palette formwork assembly with de-centering spacers, b) details of cardboard de-centering elements during compression testing in load-bearing and saturated/ compressed states

water, saturating the cardboard, causing it to compress under the load of the palettes and effectively to lower the formwork. The dry compressive strength of the spacers was calculated to carry the load of the palette/box system, the in-situ loading of the masonry shell and the live-loading of the masons. Tests were conducted to insure that the robust, non-water-soluble cardboard spacers could carry this loading in a dry state, yet which would compress under the load of the palettes in a saturated state. The de-centering elements have a bursting strength (a standard evaluation of cardboard) of at least 1700 kPA [5].

3. Discussion One of the greatest challenges in the construction process was controlling in-situ structural stability with a masonry pattern based on topology. The goal of supporting the masonry unit by the formwork system was, in effect, not achievable, because the masonry unit was too coarse to effectively touch the formwork in all places. Therefore, it was still necessary to carefully sequence arches to prevent cracking (see Figure 4). This required some modification of the tiling pattern on the first layer to insure that the load would be efficiently transferred to the supports as quickly as possible.

Since the construction occurred in open-air, another obstacle arose from the fine line between the desired load-bearing capacity and compressibility of the cardboard de-centering spacers. This was evident in the initial installation of the palette assembly, in which a lower quality of cardboard was first selected. In this case, a small amount of condensed water on the inside of the tubes was sufficient to compress the spacers in a 12 hour time period. For such a formwork system, both spacers and boxes, it is absolutely necessary to control moisture and water saturation.

The final and most significant challenge is the experimental de-centering method, the final analysis of which is pending one week.

4. Future work While the traditional methods of thin-tile construction tremendously limit the forms for vault construction, this scale prototype construction has demonstrated that the combined application of Thrust Network Analysis and an expandable cardboard formwork system suggest exciting new possibilities for the freeform, compression-only Catalan vault. Future work is proposed to streamline the TNA form finding process and several aspects of

Fig. 6: Rendering of TNA-generated form

Fig. 7: Documentation of prototype construction: a-c) General construction sequence

constructability, perhaps identifying as design criteria such factors as the maximum achievable curvature possible for a given masonry unit size. Also, this project has suggested a beneficial coordination between the force diagram and clear strategies for masonry pattern, sequencing and structural stability during construction.

Fig. 8: Documentation of finished prototype

5. Acknowledgements The authors would like to thank Marcel Aubert for his critical efforts in materials acquisition and industry support; vault construction assistants Oscar Mazuera Sanmiguel, Tom Van Mele, Lindsay Howe, Luka Piskorec, Raphael Fitz, and Lex Schaul; Dominik Werne, Thomas Jaggi and Patrick Morf for their logistical support; and industry supporters Daniel Beeler at Rigips AG and Kay Blechschmidt and Josef Ronner at ZZ Wancor AG.

References [1] OCHSENDORF, J., “Guastavino Vaulting: The Art of Structural Tile”, New York,

Princeton Architectural Press, 2009. [2] DAVIS, L., “The 4-Dimensional Masonry Construction”, MArch thesis, Massachusetts

Institute of Technology, Cambridge, USA, 2010. [3] BLOCK P., “Thrust Network Analysis: Exploring Three-dimensional Equilibrium”, PhD

dissertation, Massachusetts Institute of Technology, Cambridge, USA, 2009. [4] LACHAUER L., RIPPMANN M., and BLOCK P. “Form Finding to Fabrication: A digital

design process for masonry vaults”, Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2010, Shanghai, China.

[5] Bundesministerium für Umwelt, Naturschutz und Reaktorsicherheit, http://www.bmu.de/abfallwirtschaft/abfallarten_abfallstroeme/altpapier/doc/3146.php, Oktober 2010, "Kurzinfo Altpapier"

[6] Information of Rondo Ganahl AG, Rotfarbweg 5, 6820 Frastanz Austria [7] RUTTEN D., RhinoScriptTM 101 for Rhinoceros 4.0. Retrieved on April 7, 2008 from

http://en.wiki.mcneel.com/default.aspx/McNeel/RhinoScript101.html