Multilayer elastic timber gridshells with Monge meshes

Structures and Architecture. REstructure REmaterialize REthink REuse - Rinke & Frier Hvejsel (Eds) © 2025 The Author(s), ISBN 978-1-041-11139-9 Open Access: www.taylorfrancis.com, CC BY-NC-ND 4.0 license Multilayer elastic timber gridshells with Monge meshes C. Martínez-Criado, A.J. Lara-Bocanegra & A. Roig Higher Technical School of Architecture, Universidad Politécnica de Madrid (UPM), Spain F. González-Quintial Higher Technical School of Architecture, Universidad País Vasco, Spain A. Martín-Pastor Higher Technical School of Building Engineering, Universidad de Sevilla, Spain A. Majano-Majano Higher Technical School of Architecture, Universidad Politécnica de Madrid (UPM), Spain ABSTRACT: This work presents an alternative strategy to the design and fabrication of elastic timber gridshells to bring this type of structures closer to an industrialization and prefabrication context. The research focuses on the use of Moge surfaces as a bottom-up and construction-aware strategy that naturally leads to principal meshes featuring orthogonal nodes. This geometry is used as input in the initial stages of a proposed design-to-fabrication workflow, which comprises up to five models that drive information from a very first 3D parametric design stage to the generation of 2D fabrication file. Active bending, orthogonal meshes and CNC carpentry joints are integrated to lead to a gridshell that can be easily machined using only a 3-axis CNC router on structural boards. The methodology is validated through the construction of a full-scale demonstrator. 1 INTRODUCTION The use of timber in construction, as well as the implementation of new computational design tools linked to the advantages offered by digital fabrication, have made it possible to realise impressive gridshells. These are highly efficient and expressive structures, suitable for lightweight roofs of medium and large spans, which derive much of their stiffness and load-bearing capacity from their shape. Efficiency, aesthetic appeal and the growing demand for lowcarbon construction are making these structures interesting to architects and engineers. However, the lack of an economically affordable construction technique has relegated their use to singular projects. Figure 1. (a) PEMADE gridshell, Spain; (b) Pompidou-Metz gridshell, France [Source: Jean-Marie Conraud]; (c) Wisdom Stockholm gridshell, Sweden (Source: Anna Gerdén). DOI: 10.1201/9781003658641-86 712 These structures can be classified into three main groups according to the elements used in their construction: pre-curved gridshells, which use laminated timber elements and are commonly used for complex geometries; gridshells made of large cross-section straight elements, where all the complexity is located at the nodes; and the so-called elastic gridshells, which use slender laths of small cross-section, initially straight, that are elastically bent into their final position. These bending-active structures offer great advantages from a fabrication-aware point of view, as straight, easy-to-produce timber elements are used, what means a great impact on manufacturing costs and complexity. In terms of construction, traditional elastic timber gridshells have historically required a high degree of craftmanship (Figure 1a), whereas digital fabrication and industrialization have been widely adopted in pre-curved timber gridshells (Figure 1b), where advanced technologies such as 5-axis robotic arm machining has been implemented, at the expense of notably increased economic costs. However, to the authors’ knowledge, this level of development in industrialization and prefabrication in the field of elastic gridshells has only been implemented in the design and construction of the Wisdome Stockholm gridshell (Figure 1c) (Slabbinck et al., 2023), which combines a pre-curved initial layer with on-site bent thin laths connected by dowels-like joints. An elastic timber gridshell can be understood as a grid of thin timber laths that extend over a reference surface of the structure. When these laths are curved tangentially to the surface, they generally experience a combination of stresses due to torsion and bending in the two axes of inertia of the cross-section, because of the geodesic torsion, normal curvature and geodesic curvature of the underlying network of curves over the reference surface. Depending on the surface meshing, the three curvature components can be present (triaxial compliant) within the laths, as in the Mannheim Multihalle (Figure 2a), so that the timber cross-section used in these structures tends to be square (Lara-Bocanegra, 2022). However, some patterns lack one of these components (biaxial compliant), what allows, at the expense of a certain formal restriction, to simplify the manufacturing and construction process (Oberbichler et al., 2023). This is the strategy used in the so-called ribbed shells, with only geodesic curves, where thin but wide timber laths with no geodesic curvature are successively connected, giving rise to structures such as the Uzwil Equine Therapy Centre (Figures 1b, 2) (Chilton & Tang, 2017). Another biaxial-compliant network involves asymptotic curves, in which the normal curvature is zero. Several investigations have explored these elastic gridshells, whether made in timber or aluminium (Figure 2c) (Schling & Wan, 2022). Lastly, curves over a surface with no geodesic torsion are Principal Curvature Lines (PCL). They constitute a network of orthogonal and conjugate curves aligned with the maximum and minimum normal curvatures, having inherent advantages in the fabrication-aware design of actively bent gridshells. It allows for perpendicular torsion free nodes, which has great advantages for multilayer systems (Mesnil et al., 2015). However, the use of these patterns has been more widely adopted in pre-curved timber gridshells, such as Solemar Therme in Bad Dürrheim (Figure 2, Figure 2d). Figure 2. (a) Mannheim Multihalle. Mannheim, Germany, 1975. Frei Otto & Arup; (b) Equine Therapy Centre. Uzwil, Switzerland, 2004. Julius Natterer; (c) INSIDE/OUT pavilion. Munich, Germany, 2017. Eike Schling et al (Schling & Wan, 2022); (d) Solemar Therme. Bad Dürrheim, Germany, 1987. Geier and Geier. 713 In this work, a novel design and fabrication strategy is addressed to bring elastic timber gridshells closer to the principles of digital fabrication and industrialization. It explores the integration of orthogonal patterns generated as Monge surfaces, 3-axis CNC machining and CNC carpentry joints. The proposed methodology achieves a high degree of accuracy using an accessible 3-axis CNC router for machining structural boards, a concept that has been validated through the construction of a full-scale prototype. 2 MONGE SURFACES: A CONSTRUCTION-AWARE DESIGN STRATEGY Monge surfaces offer geometric properties of great utility for the construction of doubly curved surfaces, very suitable both in the design of structural elements and architectural envelopes. The first study of such surfaces can be found in the work of the French mathematician Gaspard Monge (Monge, 1849). In Monge’s original text, two different approaches can be identified to generate these surfaces. The first of these has a synthetic character that is absolutely contemporary and perfectly extrapolable to current digital graphic tools. It defines a Monge surface as the result of moving a planar curve (generatrix) along a directrix curve, planar or not, in such a way that the generatrix always remains in a plane normal to the directrix and moves without torsion. For this reason, some authors consider them as a type of sweeping surface, being the special case where there is no torsion between nearby generatrixes. The second approach is related to the Polar Surface of a curve, one of the three developable surfaces associated with the Frenet trihedron, by which a Monge Surface can be generated in both its continuous and discrete form (Martín-Pastor & González-Quintial, 2024). However, the design by means of a generatrix and a directrix curves (Figure 3) offers clear advantages, as the surface is defined by only two curves, which intrinsically leads to a Principal Curvature Lines (PCL) network and allows quite precise control over the resulting mesh, in contrast to PCLs of a freeform surface extracted using computational tools, where very little initial control is possible. Geometric properties of Monge surfaces were described and demonstrated for the first time in Monge (1849). The natural generation of a PCL mesh by using two curves, as stated above, implies that all the generatrix curves are planar and orthogonal to all the directrix curves, and that all the directrix curves are parallel to each other, whether they are planar curves or not. It should be pointed out that, within any surface, the PCLs of each of the two families intersect at 90º and the normal vectors to the surface along the PCLs form a developable surface. Besides, a Monge surface can be discretized into developable strips between two curves of the same family. Figure 3. Successive normal planes to a curve c intersect in lines e1, e2 (a). The axes act as rotation axes of the points contained in the normal planes (b). A Monge surface is generated by the successive movement of generatrix curves contained in the consecutive planes normal to the directrix (c). 714 In a Monge surface, the generatrix curves are both planar and geodesic curves of the surface. This has remarkable properties, such as the possibility of placing rectangular strips circumscribed to the surface over the geodesic curves. The PCL network allows a discretization of the surface into Principal Meshes, being able to form Circular Planar Quadrilateral (PQ) Meshes or Conical PQ Meshes. These two meshes are useful for generating special meshes, such as Exact Offset Meshes, which is crucial for the design of multilayer systems. Many studies on this subject and its constructive applications have been conducted (Liu et al., 2006; Pottmann et al., 2007; Pottmann et al., 2008; Mesnil et al., 2018; Jiang et al., 2022; Dellinger et al., 2023). Monge surfaces can be used as a construction-aware system for gridshell design, where the PCL mesh derives from the resulting Monge surface, offering all the geometric characteristics and constructive properties of this network of curves, as outlined below. This bottom-up process enables a formal exploration with certain formal restriction (Figure 4), but guarantees all the geometric properties of the surface, which are decisive in the subsequent construction and materialization processes (Gonzalez-Quintial & Martin-Pastor, 2024). 3 SYSTEM PRINCIPLES The current research is based on the inherent coherence between (1) orthogonal patterns generated as Monge meshes, (2) 3-axis CNC milling on structural boards, and (3) orthogonal CNC carpentry joints, what bring elastic timber gridshells closer to an industrialization and prefabrication context. Thus, the design of how the different elements of the structure meet is crucial. In elastic timber gridshells, the connection system is a key aspect as it must resolve the encounter between the different layers (nodal joint) and create a composite section with high inertia (multilayer system) while maintaining the depth of the individual laths (which is limited by bending radii). In traditional elastic timber gridshells, the nodal connection is typically addressed by placing a bolt at the intersection of the laths to allow free rotation between them. The multilayer connection system between the timber layers is generally achieved using mechanical fasteners (shear blocks), creating semi-rigid joints with sliding in two shear planes (Lara-Bocanegra et al., 2022). In this work, considering the 3-axis CNC logic, where the milling direction is perpendicular to the board, an interlocking CNC orthogonal carpentry joint is proposed to resolve both nodal and multilayer connection. It should be pointed out that the lath depth of the proposed geometry is not constant, being higher at the nodes, which provides two significant advantages: on the one hand, it eliminates the need to drill the lath for bolts and screws, thus avoiding the section loss and the associated risk of breakage, at the time the disassembly and reuse are favoured; and on the other hand, the shear forces are transmitted through wood-to-wood contact in a single shear plane, qualitatively improving the stiffness of the joint. The assembly of the joint alternates the locking of one piece with the other, resulting in a self-stable system that also resists out-of-plane moment. Figure 5 shows a structural fragment and a possible materialization of this CNC carpentry joint. Figure 4. Formal exploration of Monge Surfaces generated from a generatrix (red) and a directrix (blue). 715 4 DESIGN-TO-FABRICATION WORKFLOW The aforementioned concepts are integrated into a design-to-fabrication workflow that consists of several interconnected models with different types of information, that can be gathered within three groups based on their focus: 3D-modelling, structural analysis or fabrication. Using the visual programming environment Grasshopper 3D and starting from two generating curves, that are the very first input of this workflow, up to five models can be generated: (1) Basic model, (2) Reference model, (3) Analysis model, (4) Detailing model, and (5) Fabrication model. As illustrated in Figure 6, this workflow follows a non-linear approach, where the structural analysis model is used to verify the proposed geometry structurally but is not input for the detailing model, which is obtained directly from the Reference Model. 4.1 Modelling approaches 4.1.1 Basic model The Basic model is an agile model in which the main geometry of the structure is defined. Using two input curves, a network of curves is generated as a Monge mesh that defines the underlying reference surface, which will be used parametrically to define constructively the structure. In this preliminary design phase, the relationship between the curvature radius (R) and the depth (t) of the laths is considered, that depends on the material used. Figure 5. (a) Structural fragment and (b) orthogonal CNC carpentry joint proposal. Figure 6. Design-to-fabrication workflow. 716 Due to the greater depth of the laths at the joints, it is not possible to bend them at these stretches. This constraint is considered in the basic model through a rationalization process applied to the directrix and generatrix curves. Maintaining the orientation of the joints, the curve between them is reconstructed using circle arcs correcting the position of the next shearblock. This is achieved through a loop script that repeats this process from the start to the end of each generating curve. Because of this rationalization process, the curvature of the laths between each of the joints increases compared to the original one, as shown in Figure 7 Fout! Verwijzingsbron niet gevonden.. 4.1.2 Reference model From the Basic model geometry (Figure 8a), nodes are computed at the intersection of the two families of curves. To generate the multilayer wireframe structure (Reference model), those nodes are moved along the normal axes of the reference surface by a distance depending on the joint geometry. Based on these nodes, it is possible to construct the curves in the space within the corresponding reference surfaces of each layer that are calculated as an offset of the original one. Reference planes are then established at each node using an origin point and two perpendicular vectors, defined by the following parameters: the node as the origin, the surface’s normal vector, and the tangent vector of the curve at each node. This Reference model (Figure 8b) will be the initial input for the upcoming models, along with cross-section dimensions and material property data. 4.1.3 Detailing model From the Reference model, a Detailing model (Figure 8c) is generated, containing all the geometry necessary for the subsequent visualization and for the design of additional details such as supports and structure cladding. The geometry of the laths is obtained by offsetting the lath’s axis according to the cross-section size along its axes (section’s y-axis and z-axis), which coincide with the normal and geodesic vectors of the laths evaluated with respect to the reference surface. Furthermore, the geometry of the joints in space can be obtained through a reorientation procedure from global XY plane (where the joints are modelled) to local reference planes of the Reference model Figure 7. Schematic curve’s curvature before (a) and after (b) rationalization process. Figure 8. (a) Basic model; (b) Reference model; (c) Detailing model. 717 4.2 Structural analysis model Once the reference geometry is established, structural analysis can be performed in two stages: one focusing on curvature stress calculation and the other on external loads (the latter not considered in this work). The first step can be approached geometrically (Roig et al., 2023) so, for a given network, the curvature vector (k) along the curves can be extracted discretely within its entire length and projected according to the vectors N and G of the surface, obtaining the normal kN ð Þ and geodesic (kGÞcurvatures (Figure 9). Regarding elastic bending of the laths, it must be checked that the bending stresses do not exceed the board’s bending strength and that a sufficient strength reserve ratio is maintained to support external loads. Denoting the y-axis and the z-axis as the principal axes of inertia of the lath’s cross-section, tangent and perpendicular to the surface of the gridshell respectively, and b and h as the cross-sectional dimensions associated with each axis, the corresponding maximum initial bending stress due to normal and geodesic curvature, σ m,y,0 and σ m,z,0 respectively, can be calculated following Equations (1 and (2, being E edge y E flat the corresponding Young modulus of the board tested edgewise and flatwise, respectively. Furthermore, the strength reserve depends mainly on the material properties and can be considered using the utilisation ratio (U) that can be calculated according to Equation 3. In this expression, and given that wood-based panels are generally considered orthotropic, each bending stress σ m,y and σ m,z is compared to the edgewise and flatwise design bending strengths f m,0,Edge,d and f m,0,flat,d , respectively. σm;y¼kNEedge h 2ð1Þ σm;z¼kGEflat b 2ð2Þ U¼UNþUG¼σm;y fm;0;edge;dþσm;z fm;0;flat;d51ð3Þ 4.3 Fabrication model For the 3D to 2D translation, from the Reference model, a deployment strategy is proposed based on length measuring according to local parameters. Each curve of the network is divided according to the points where it is decided to segment the laths for manufacturing, resulting in several segments per curve subdivided simultaneously by the nodes. The segmentation process is carried out using a segmentation pattern that avoids alignment of splice joints along the structure. Figure 10 shows the deployment process for the upper layer of a grid, where each curve is firstly segmented in red nodes. This defines two segments per curve (rendered in Figure 9. (a) Relationship between curvature vectors in space; (b) Curvature around G; (c) Curvature around N; (d) Torsion around t. 718 blue (A) and orange (B)). Then, for each segment, lengths between nodes are evaluated, being this the parameter used for 2D deployment as straight lines and nodes. To ease the assembly process, each node is assigned a unique code. By incorporating the lath’s depth and the joint geometry according to local planes along the curve (defined by tangent and normal vectors), the CNC milling geometry is obtained. 5 DESIGN IMPLEMENTATION AND CONSTRUCTION To validate the system on a constructive and assembly level, the proposed workflow has been successfully used for the design and fabrication of a full-scale demonstrator in poplar plywood (Figure 11). Covering a maximum span of 7.5 m, the structure is based on an anticlastic surface underlying a grid of PCLs with a grid step of about 0.50 m between nodes in both directions and 25×15 mm 2 cross-section. Elastically bent 3 mm thick boards were used for cladding. The remaining components were machined on 2440×1220×25 mm 3 boards using a 3-axis CNC router. Figure 10. 3D Reference model and 2D deployment with labelling of members. Figure 11. Full scale demonstrator and associated workflow models. 719 6 CONCLUSIONS This work aims to offer an alternative approach to traditional design and fabrication of elastic timber gridshells by using Monge surfaces, which provide a bottom-up and construction-aware strategy that leads intrinsically to PCL networks (naturally orthogonal). The need of a rightangled connections at the joints is addressed through the proposed interlocking orthogonal CNC carpentry joint. 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