Lionel Du Peloux

Dr Olivier Baverel

Jean-François Caron

Al Frederic Tayeb

From shape to shell: a design tool to materialize freeform shapes using gridshell structures

Keywords: gridshell, composite material, GFRP, grasshopper, compass method, freeform, double curvature

This paper introduces and explains the design process of a gridshell in composite materials built in Paris in 2011 for the festival Soliday. A brief introduction presents the structural concept and the erection methodology employed. It explains why composite materials are relevant for such applications. Following this practical case, the whole process from 3D-shape to real-shell is then detailed. Firstly, the shape is rationalized and optimized to smooth local curvature concentrations. Secondly, a specific computing tool is used to mesh the surface according to the compass method. This tool allows designers to look for optimal mesh orientations regarding the elements curvature. Finally, a full structural analysis is performed to find the relaxed shape of the grid and check its stability, strength and stiffness under loads. The authors conclude on the overall relevance of such structures.

INTRODUCTION

The emergence of gridshell structures -intensively studied by the German architect Frei Otto -is a major step in the development of complex shapes in AEC (Architecture, Engineering and Construction). Since the 70's this structural concept has led to emblematic realizations (Mannheim [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgarten-schau[END_REF], Downland [START_REF] Harris | Design and construction of the Downland Gridshell[END_REF], Savill, Hanovre [START_REF] Ban | The Japanese pavilion[END_REF]). They have shown that beyond their architectural potential, gridshells are well suitable for complex shape materialization because of their intrinsic geometric rationality.

However, the very few number of gridshells constructed up to now attests that they are quite tricky to design compared to standard buildings. Architects and engineers would face both demanding conceptual knowledge (3D geometry, form finding techniques, non-linear behaviour, large scale deformations, permanent bending stresses, etc.) and real lack of tools dedicated to their design. This paper presents a computing tool based on Rhino & Grasshopper that aims at meshing NURBS surfaces with the compass method. This tool also includes a one-way interface for GSA (a structural analysis software from Oasys) to perform the structural analysis of the resulting grid. Thereby, this tool introduces shape-driven design of gridshells. Following a case study -the construction of the first composite gridshell to host people -a methodology to design these shape-driven structures is proposed. Finally, future prospects to their development are discussed.

Gridshell: concept, erection process, materials

CONCEPT A gridshell is a structure which behaves like a shell but is made of a grid. Thus, the material is not spread continuously as shells, but it is organized in a discrete grid pattern. Like shells, gridshells derive their stiffness from their double curvature shape. These structures can cross large spans with very few materials. They offer a rich and voluble lexicon to express blob-shapes.

ERECTION PROCESS Usually, the grid morphology is not trivial and leads to design numerous costly and complex joints. To overcome this issue, an original and innovative erection process was developed that takes advantage of the flexibility inherent to slender elements.

A regular planar grid made of long continuous linear members is built on the ground (Fig. 1). The elements are pinned together so the grid has no in-plane shear stiffness. Thus, the grid can accommodate large scale deformations during erection (Fig. 2). Then the grid is bent elastically to its final shape (Fig. 3). Finally, the grid is frozen in the desired shape with a third layer of bracing members (Fig. 4). The grid becomes a shell and the structure's stiffness is multiplied by about 15. MATERIAL FLEXIBILITY FOR STRUCTURAL RIGIDITY Composite materials like glass fibre reinforced polymer (GFRP) could favourably replace wood in this case where both resistance and bending ability of the material is sought. Thus, structure's stiffness derives from its geometric curvature and not from the material's intrinsic rigidity. Moreover, using synthetic materials free us from the painful problematic of wood joining and wood durability [START_REF] Douthe | Gridshell structures in glass fibre reinforced polymers[END_REF].

HIGH TECH & LOW COST Though gridshells require high-tech design techniques, they seem to be a low-cost way to materialize non-standard morphologies (200€/m 2 ), because of their geometric rationality. The project complexity is shifted upstream.

From-finding versus Grid-finding

One can identify two different ways of designing gridshells: those with a given outline and those with a given shape.

The first approach considers the final shape a consequence of a form-finding process, driven by the supports of a grid which is thought to be an input data. The second approach consists of deriving a grid from a given shape. When erected on its supports, the grid should give back the intended morphology.

FORM-FINDING A physical or numerical grid-model is handled until a structural shape is found, in compliance with the architectural intents. This way, Frei Otto designed the Multihalle of Mannheim using hanging funicular nets and photogrammetry [Otto and Hennicke 1974] (Figs. 5-6). Nowadays, this form-finding stage would probably be done by computer, relying on numerical methods such as dynamic relaxation or force density.

GRID-FINDING

The compass method is used to develop the initial shape in a quadrangular mesh. Rebuilt on a plane, the mesh leads to a regular grid suitable to materialize the studied shape by a gridshell. An alternative method, taking into account the grid's mechanical properties, was also proposed by the Navier laboratory [Bouhay, [START_REF] Bouhaya | Mapping two-way continuous elastic grid on an imposed surface: Application to grid shells[END_REF]. This method uses explicit dynamic algorithms to pin an initially flat grid on a given shape, with a system of fictive forces.

CASE STUDY

The project

In June 2011 six students from the École des Ponts ParisTech (French engineering school), supported by the Navier laboratory, gave birth to a structure for the association "Solidarité Sida": a tent unlike any other, reminding blob architecture with its curved and rounded shape (Fig. 7).

DESCRIPTION

Design Process

The following diagram summarizes the design process from the initial architectural intent (a 2D sketch) to the final gridshell. Each step is then detailed.

Fig. 8: Step by step design process including different levels of structural control, based on curvatures and stresses checks (1,2,3).

3D Modeling

From the initial architectural intent we have modelled the shape as a wireframe (Fig. 9), including the outline and some sections. With those curves we can control the shape in-plane and drive its volumetry. The surface is then derived from the wireframe using a NURBS interpolation (Fig. 10). All along the design process adjustments were carried out on this surface until it reached the architectural and structural requirements. In this process, the initial surface was mostly deformed or sculpted by handling its control points.

Figure 9: Wireframe geometry. Figure 10: NURBS patch.

Shape Optimization

For now, we built a «space sketch» of the project. However, this NURBS surface has no reason to lead directly to a structural shape. Thus, the structural elements have to be checked to make sure they will support the stress field induced by grid shaping.

STRESS & CURVATURE Stresses in elements are mainly due to grid bending, that is to say geometric curvature imposes the grid stress state (Eq. 1). Thus, principal curvatures of the surface describing the shape are good indicators to evaluate if the structural members have the required mechanical properties. This preliminary control can be completed by an analysis of the curvature of the mesh elements. Finally, nothing but a true structural analysis considering members mechanical properties will allow us to find the exact relaxed shape and the stress field in the structure.

! = ! !"#$ + ! !"#$ ≅ ! !"#$ = ! ! ! = !. ! ! !"#$%&"#' (1)
Where σ represents the total stress (compressive plus bending) induced by the shaping. E, v and I are respectively the longitudinal young modulus, the radius and the bending inertia of the profile. R is the radius of curvature of the profile.

SURFACE OPTIMIZATION Before any attempt to mesh the shape, it is recommended to optimize the sketch shape regarding its minimal principal curvatures (Eq. 2). Using the curvature-analysis built-in function in rhino, it is easy to identify and smooth areas that are initially too curved (Fig. 11).

! !"# > !.! ! !"# (2)
Different sketches are compared according to this criterion to smooth areas where curvature is excessive.

Shape Meshing

Following a decade of research on this topic at the Navier laboratory, a specific tool has been developed on Rhino & Grasshopper for the design of such shape-driven gridshells (Fig. 12). This tool gathers several components that process basic operations (meshing with the compass method, grid-processing, structural analysis) required for the generation of a suitable grid for the materialization of a 3D shape by a gridshell structure.

Figure 12: Grasshopper canvas (compass method, grid processing, structural model generation).

COMPASS METHOD This process propagates a two way mesh of constant pitch on any NURBS surface.

Two crossing guide-curves are drawn on the surface to mesh. These curves mark the boundary of 4 quarters. Each half guide-curve is then subdivided with a compass of constant distance w (the pitch). Finally, from two consecutive half guide-curves, quadrants are meshed with the same compass distance (Fig. 13).

Figure 13: Compass method principle.

MESHED SURFACE The compass method doesn't allow to mesh the entire meshing domain. Only a smaller part could be meshed and its area varies according the chosen set of guide-curves (Figs. 1415). Thus, it is not possible to rely exclusively on the shape to be realized with the lattice. An extended surface -chosen carefully -has to be considered as the meshing domain. OVERALL PROCESS To overcome this difficulty, we propose a methodology which relies both on the creation of a meshing domain (domainSrf) from the targeted surface to materialize (gsSrf) and on the identification of a suitable set of guide-curves.

First

Step. We consider the gridshell surface (gsSrf) a part of a larger surface (domainSrf). Trimmed by a clipping plane or surface (cuttingSrf), this domain surface should give back the intended shape to build .

Figure 15: gsSrf. Figure 16: domainSrf and cuttingSrf.

Second

Step. A set of guide-curves is chosen (Fig. 17) and the mesh is propagated on the domain surface according to the compass method (Fig. 18). The guide-curves have to be chosen so that the whole gridshell surface (gsSrf) is meshed. Several trials can be necessary to get a suitable mesh.

Figure 17: Guide-curves set.

Figure 18: Resulting mesh on domainSrf.

Third

Step. The mesh is trimmed by the clipping surface (Fig. 19). The resulting mesh lays on the whole initial intended surface to mesh. The gridshell support-outline is given by the intersection of the clipping plane and the domain surface (Fig. 20).

Figure 19: Trimmed mesh. Figure 20: Final mesh and support outline.

MESH OPTIMIZATION As the form is known and the procedure to mesh the surface is now known, an optimization of the mesh can be performed (Fig. 21). The aim is to find a mesh that can mesh the entire surface and that creates acceptable stresses in the beams. To this end the curvature of the elements is checked using the following equation (Eq. 3) :

! !"#! > ! !"# !.! (

GRID-PROCESSING

The generated mesh can be shaped in a matrix. This allows both its transformation in a planar grid of regular pitch (Fig. 22) and automatic definition of triangulation elements for bracing. GENERATION OF ANALYSIS MODEL A procedure gathers and processes all the geometric information. It creates an import file for the automatic generation of an analysis model in GSA, a third-party software dedicated to structural analysis. Additional components assist the designer in the definition of complex load cases, such as non-uniform wind and snow loads, directly in Rhino & Grasshopper.

Structural Analysis

Once the structural model is built by our tools, it can be loaded in the structural analysis software to perform:

• Computation of the permanent flexural stress and the relaxed shape using a dynamic relaxation algorithm (Fig. 23).

• Loading analysis according to the Eurocode (self-weight, snow, wind, ...).

Final mesh Permanent bending stress

CONCLUSION

This paper has presented the different steps for the design of a gridshell in composites materials built for the Solidays festival in 2011 in Paris. The first step was the optimization of the shape in order to avoid concentrations of curvature locally. The second step showed a tool to automatically mesh a surface using the compass method. With this tool, the optimum orientation of the mesh is studied. The last step showed the details of the structural analysis of the gridshell. This construction demonstrated the technical feasibility and also the economical feasibility of the gridshell in composites materials.

Figure 1 :

 1 Figure 1: Regular grid on the ground.Figure2: Grid erection.

Figure 2 :

 2 Figure 1: Regular grid on the ground.Figure2: Grid erection.

Figure 3 :

 3 Figure 3: Erected grid.Figure4: Grid triangulation.

Figure 4 :

 4 Figure 3: Erected grid.Figure4: Grid triangulation.

Figure 5 :

 5 Figure 5: Hanging net.

Figure 6 :

 6 Figure 6: Resulting structure.

  This structure of 300m² was the first gridshell in composite materials (GFRP) to receive an audience. It had to get a certificate of approval that involved administrative requirements and EUROCODE justifications, a first for such a structure. Beyond the technical performance, this large scale project designed to house up to 500 people at a time has shown the economic relevance of this concept. It became reality thanks to key partnerships including T/E/S/S and Viry.SOLIDAYS FESTIVAL Each year in June, Solidarité Sida organizes the Solidays festival. It is a music festival that has attracted around 160 000 people during 4 days and has raised about 1.7 million Euros in 2011. It is also a forum. Its purpose is to raise awareness about AIDS, and raise funds for medical research and outreach initiatives. The gridshell structure was designed to house the forum during the festival.

Figure 7 :

 7 Figure 7: Photo of the final structure (http://vimeo.com/31341461).

Figure 11 :

 11 Figure 11: Geometric curvature minimization in three steps. Rmin ∈ [blue = 3,00m ; red = 10,00m].

Figure 14 :

 14 Figure 14: Two different meshes are obtained from two distinct sets of guide-curves. The meshed area never takes on the whole surface. Convergence phenomena could be observed (right picture).

Figure 21 :

 21 Figure 21: Mesh testing.

Figure 22 :

 22 Figure 22: Developed surface and derived grid.

Figure 23 :

 23 Figure 23: Final compass mesh and corresponding relaxed mesh (stress diagram).

ACKNOWLEDGEMENTS

The authors would like to thank the students (E. Roux, E. Blache, J.-R. Nguyen, A. Grandi, G. Frambourt, T. Perarnaun), their university supervisor (R. Mège) and the association Solidarité Sida, for their initiative and confidence. Special thanks also to T/E/S/S and Viry for their technical and financial support which permitted this project to become real. Thanks also to all our partners who provided significant material assistance: Serge Ferrari, Top Glass & Solutions Composites, Owens Corning Reinforcement, DSM Resins, ENSG, Esmery Caron, Axmann, Chastagner and ¨Paris Voile.


L Du Peloux

F Tayeb

O Baverel
email: baverel@enpc.fr

J-F Caron
email: caron@enpc.fr

Faith Can Also Move Composite Gridshells

Keywords: gridshell, composite, glass fibre reinforced polymer, cathedral, freeform, double curvature, large span, church !

This paper presents the overall design and construction process of a gridshell in composite materials at Créteil, in Paris suburb. This religious edifice of 350m² is a temporary cathedral meant to gather the parishioners during the two-years renovation of their permanent cathedral. It can accommodate up to 500 people and complies with all the required performances for such a building: structural stiffness, fire safety, waterproofness, lightning, thermal comfort, etc. This project arises thanks to a long-term collaboration between T/E/S/S and NAVIER and marks the accomplishment of a ten-years research project in the area.

INTRODUCTION

The structure is a gridshell in composite material (Fig. 1a), made up of two kilometers of glass fiber reinforced polymer tubes (GFRP), covered by a PVC coated fabric. An initially flat grid is bent elastically to give the intended shape (Fig. 1b). It is then triangulated with a third layer of pipes so the structure becomes a shell. It weights only 5kg/m 2 and covers an area of 350m 2 without any internal support. After an architectural and technical presentation of this project, the paper discusses what gridshells are following on from Frei Otto's work (the association of a structural typology, a flexible material and a specific erection process). It presents recent developments in the area, focusing on the contribution of composite materials.

The paper then investigates the overall design process of the structure, from the architect's sketch to the final gridshell. Each step of this sequential and iterative process is described: 3D modeling, meshing, formfinding, structural analysis and post processing of the geometry.

Afterwards, the paper highlights the issue of existing standards, which do not account specifically for composite materials. It explains how this difficulty was overcome.

Finally, complexities brought by connections are highlighted. Solutions are identified and validated by tests.

PROJECT OVERVIEW

Context and challenges

Creteil is a city of 90.000 inhabitants in the southeast suburb of Paris. It started to urbanize itself in the late 50's, impelled by the French architect Charles-Gustave Stoskopf. In 1976 he designed Notre Dame of Créteil, a modest catholic church made of concrete, which became a cathedral 10 years later.

Recently, the diocese of Créteil has undertaken a major architectural redevelopment project of its cathedral, including a timber shell covering the religious area and the creation of a new cultural area. Once transformed, the edifice shall be more visible, more hospitable and livelier for citizens. Inevitably, such a molt takes time and a temporary place of worship was required to ensure liturgical services during the two-years work.

In November 2011, T/E/S/S, the structural design office in charge of the cathedral renovation project, made an ambitious proposal to the diocese. Based on a previous successful experience -the construction of a composite gridshell for the festival Solidays (see video [START_REF]Video of the Forum Solidays project[END_REF]) -T/E/S/S suggested that rather installing a basic tent, the parishioners should construct themselves a temporary cathedral (see video [START_REF]Video of the Temporary Cathedral of Créteil project[END_REF]).

The overall cost of the project is 400k€, about 30% more than a standard tent of the same size that would have been rented for 2 years.

Architectural form of the building

The generation of the form was driven by two objectives that of providing a variety of appropriate internal spaces within which the community could assemble, and externally to be a welcoming and visually interesting form.

Today, the internal organization of a Roman Catholic Church is in large part driven by the post Vatican II vision of a religious celebration being a collective gathering of the community around the Eucharist, center of spiritual life. A circular seating arrangement is often considered the most convivial form to create a sense of belonging while minimizing a sense of hierarchy (Fig. 2).

Fig. 2. In space, a circular form leads naturally to a blob shape.

!

However the community is not only using the building for religious celebration but also for encounters on a more informal manner, for example spontaneous gatherings after religious ceremonies. In the early Roman church, such gathering of the community was facilitated by the presence of an anti-space to the main space called a narthex, through which one passed on entering the church. It was therefore felt appropriate that the formal freedom which the gridshell system offered would be used to explore forms composed of an agglomeration of major and a minor volumes which contain the two functions: formal and informal gatherings (Fig. 3).

Fig. 3. Major and minor volumes are agglomerated into one volume.

Formal explorations were undertaken using modeling clay. The final form is based loosely on two adjacent semi spherical volumes of different size, which are merged into one complex form. Externally the fear of the design team was that the totally convex blob form could look intimidating. It was therefore decided that the two spherical virtual forms, which would be joined to make the final form, would be arranged not in a symmetrical axial manner, but in an asymmetrical curved composition. The resulting form seen in plan is convex on one side and concave on the other. The concave form in plan allows for double curvature to be introduced into what would be otherwise a simpler blob and gives sensuality and visual interest to the building.

Placing of the building on the site

The temporary cathedral is located on a land owned by the municipality, which is used for sporting and other communal gatherings. The curve in the building defines an external area where the church community could meet in the open air and this is where the entrance to the church is situated. The building was positioned on the site so that the entrance addresses a grass planted area forming a garden forecourt or "parvis" (Fig. 3). A service building housing plant, toilets and vestry are housed in a portacabin positioned to the rear of the building.

Entrance

It is formally quite difficult to integrate doors, which must be verticals, into a complex geometry. Either the gridshell could be deformed to accommodate the geometrical requirements of doors, or the doors could be integrated into an independent form. The latter approach was chosen.

In looking for forms to house the doors, reference was made to the conical monumental doorways with rings of concentric decoration, which welcome the faithful to Romanesque and Gothic churches in France. The conical forms were found to by coherent to the overall geometry of the building. The entrance doors were therefore inserted into a conical hooded form made of rolled steel plates and stiffened by concentric steel tubes, which not only make reference to historic precedence but also refer to the gridshell to be discovered inside.

The cone of the entrance doors was positioned in the concave side of the building giving access directly to the narthex part of the internal volume (Fig. 4). To the rear of the church is situated a service door. The steel hood, which houses this door, is curved tightly around the door and takes up an ovoid form.

Daylight

The gridshell is covered in a PVC membrane, which is opaque. How to introduce daylight into the interior was a major subject of reflection. The simplest way found was to use transparent membrane placed occasionally on the membrane. A small amount of light was required in the interior to create a contemplative atmosphere. The lights would in consequence glow and would be seen as luminous insertions in the vault, like stars in the celestial vault or the apse of some Romanesque churches. The stars were patterned on the joints of the PVC membrane.

The almond shape came from simplification of the cutting into the panels either side of the joints and to avoid stress concentrations around cuts in the membrane. This shape, known as Mandela, is frequently used in Marian religious imagery. The distribution of the transparent insertions is quite uniform but gets denser above the pinnacle.

Technical description

The gridshell structure is made of long glass-fiber pipes (∅ext 42mm) pinned together with scaffold swivel couplers. The structural members of the grid, all of different length, are built from one, two or three composite pipes connected with steel sleeves ( §6.2). The length of the pipes is restrained to 12m so standard trucks can deliver them. The pipes are organized in three layers. The first two layers are placed perpendicularly on the ground. They form the "quadrangular primary grid". The distance between the pipes of these two layers is constant so the grid is regular. This primary grid is elastically deformed to obtain the final shape. The third layer of pipes acts as bracing. It gives the structure a shell behavior. Those pipes are fixed to the primary grid once it has been shaped.

The structure is anchored to a concrete strip footing with a special steel system, which ensures the transit of loads from the composite structure to the ground. A similar system enables to fix the structure to the doors.

A PVC coated fabric, tailor patterned, covers the structure. It includes transparent strips that bring daylight inside. The fabric is stretched on a devoted peripheral edge beam with a double lacing system (halyard and strap). At the ground, the lacing edge beam is made of a bent composite rod nailed regularly to the concrete slab. At the doors, it is made of a steel arch welded to the doorframes.

The membrane is waterproof and as a continuous membrane has no joints except at the perimeter. At the perimeter a continuous strip of membrane is prefixed to the internal face of the membrane and fixed to the ground slab being clamped between the concrete and an aluminum flat bar. At the doors, the flexible strip is riveted to the doorframes.

Brief history and definition

The inventiveness of the gridshell concept is commonly attributed to Frei Otto, a German architect and engineer who devoted much time to gridshells. In 1975, he achieved the famous Mannheim Multihalle [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgarten-schau[END_REF] in collaboration with the engineer Edmund Happold (Arup). Thanks to this achievement, the gridshells appeared then among structural typologies allowing various rounded shapes.

Literally, the word « gridshell » refers to grids behaving like shells: from a mechanical point of view, stresses acting on the structure are mainly transmitted through compression and traction.

Nonetheless, according to the historic evolution of the concept, characterizing a gridshell as the combination of a structural concept -a grid behaving like a shell -and of a specific construction processusing the bending flexibility of the material -seems to be more accurate.

The Mannheim project -in which a wooden regular and planar grid, lacking shear stiffness, is elastically deformed up to a selected shape with the help of stays, and then braced and covered -is the launching of this new concept.

25 years later, with the development of the numerical tools, two new projects saw the light: the wooden gridshell of Downland in 2002 [START_REF] Harris | Design and construction of the Downland Gridshell[END_REF] and the Savill one in 2006 [START_REF] Harris | The Savill Garden gridshell: design and construction[END_REF]. The principles of these projects are similar to Mannheim one but new constructive methods are used. In 2000 Shigeru Ban innovates shifting from wood materials to cardboard for the Japan Pavilion at EXPO 2000, in Hannover [START_REF] Ban | The Japanese pavilion[END_REF].

Recent developments

Based on this groundwork, the Navier laboratory has developed a research program on gridshells for the last ten years, focused on both the use of new materials and the development of more efficient numerical methods [START_REF] Douthe | Study of slender prestressed structures in composite materials: application to the conception of gridshells[END_REF]. These developments have been validated by the construction of two prototypes whose areas were about 150 m² (Fig. 5a and Fig. 5b). In 2011, the laboratory used its expert knowledge for a large-scale project, the forum of the Solidays festival [START_REF] Baverel | Gridshells in Composite Materials: Construction of a 300m 2 Forum for the Solidays[END_REF]. This achievement, built by voluntary workers, has been the first composite material gridshell to house public (Fig. 5c). Supported by T/E/S/S engineering company and Viry firm, it was an important step for the spreading of the laboratory knowledge.

In 2012, the context is favourable for a new achievement named « Temporary Cathedral of Créteil ». Although this gridshell has an area very similar to the Solidays one, the project raised new challenges, in particular the challenge of reliability. Indeed, its period of use is at least two years. Additionally, the skills coming from T/E/S/S company made possible important developments such as for doors, lacing edge beam, anchorages and sleeves.

Unlike the two first prototypes, the gridshells built for Solidays and at Créteil are based on a new approach regarding shape-structure relationship. Indeed, thanks to a numerical tool performing the compass method, the geometry of the object is no longer defined as the reversal of a hanging net -in this case, only the flat geometry could be mastered [START_REF] Addis | Toys that save millions" -a history of using physical models in structural design[END_REF] -but now the flat geometry is straight deducted from the one proposed by the architect.

This new approach opens up new architectural horizons, making possible the exploration of new shapes and new meshes for gridshells.

Gridshell in composite material

The gridshells built in composite material, being at the heart of this paper, are consistent with the framework defined in the section §3.1.

Structural Typology Their mechanical behaviour is very similar to the one of real shells even if the material is discrete and located in a grid more or less open. In spite of that, gridshells benefit from the same advantages as the ones showed by an eggshell: they can cross large span using a low amount of material. Their stiffness is mainly linked to their double-curved shape.

Material Flexibility for Structural Rigidity

In this field of application, composite materials like glass fibre reinforced polymer (GFRP) could favourably replace wood, where both resistance and bending ability of the material is sought [START_REF] Douthe | Gridshell structures in glass fibre reinforced polymers[END_REF]. The stiffness of the structure does not derive from the intrinsic material rigidity but principally from its geometric curvature. Ideally, the composite profiles are produced by pultrusion, an economic continuous moulded process. The standardization of the process guaranties very stable material and mechanical properties. It frees designers from the painful problematic of wood joining and wood durability. The characterization of this material is presented at §5.3.

Erection Process Usually, the grid morphology is not trivial and leads to design numerous costly and complex joints. To overcome this issue, an original and innovative erection process was developed that takes advantage of the flexibility inherent to slender elements. A regular planar grid made of long continuous linear members is built on the ground (Fig. 6a). The elements are pinned together so the grid has no inplane shear stiffness and can accommodate large-scale deformations during erection (Fig. 6b). Then, the grid is bent elastically to its final shape (Fig. 6c). Finally, the grid is frozen in the desired shape with a third layer of bracing members (Fig. 6d) and the structure becomes a shell.

Nota: Other ways to triangulate the grid could be investigated, like cables. Moreover, it could be interesting to freeze the grid in a nonrelaxed position, so the exact intended form is achieved.

Overall design process

The goal of the design process is to identify a gridshell structure that works and which respect as faithfully as possible the architectural project -a shape and a program. It represents "the path from shape to structure". Its progress, sequential and iterative, revolves around three major stages: shape, mesh and structure (Fig. 7).

It is not trivial to go through this complex process. Indeed, for each step, the method, the tool, the criteria, that offer both a sufficient explorative richness to find out enough candidate solution, and the means to evaluate and compare the suitability of those solution, have to be found. Fig. 7. Design process from shape to shell.

In the next part of this paper the authors present, for each previously mentioned stage, the studied options and the selected evaluation criteria.

From sketch to shape

The first step of the process consists in building a precise geometric model from the sketch of the architect and to evaluate its mechanical potential (Fig. 8). At this stage, the goal is to estimate the probability a given shape would lead to the generation of a structurally feasible gridshell.

Fig. 8. Selection of 3 slightly different 3D shapes, derived from the targeted shape, based on an analysis of their principal curvatures.

Stresses in the grid are mainly due to the bending of the profiles. They derive directly from their geometric curvature. Thus, the principal curvatures of the surface -because they give a qualitative measurement of the local curvature of any curve drawn on a surface -are relevant indicators to evaluate the stress rate of a grid laying on it. Particularly, one should ensure the following condition is satisfied everywhere, where r is the pipe's outer radius, Rmin is the minimum principal radius, E is the flexural modulus, σk,flex the characteristic flexural strength ( §5.3) and γlt the long-term partial coefficient of material resistance ( §5.4):

! ! !"#$ ! < ! ! !,!"#$ !!" ! (1)
Ideally, the shape is controlled by few key parameters. Thus, it can be adapted and optimized through an iterative process, towards this criterion (1).

From shape to mesh

During the second step, the candidate surface is meshed and the mechanical potential of the resulting grid is evaluated. At this stage, we try to estimate the probability a given mesh could lead to the generation of a viable gridshell structure (Fig. 9). Simultaneously, meshes are compared according to their architectural relevance.

This time, the geometric curvature of the polylines drawn on the surface is the criterion to characterize the mechanical potential of the grid. In particular, one should ensure the following condition is satisfied everywhere, where Rspline is the spline's local curvature radius:

! ! ! !"#$%& ! < ! ! !,!"#$ !!" ! ( 2
)
The mesh is obtained by the compass method, described in [START_REF] Otto | Gitterschalen Gridshells[END_REF], which develops a regularly spaced grid on a surface from two secant directrix. For a given shape there are an infinite number of meshes. The aim is to identify at least one grid, suitable towards architectural and structural criteria (Fig. 9). The laboratory tried various numerical methods to generate such grids [START_REF] Bouhaya | Mapping two-way continuous elastic grid on an imposed surface: Application to grid shells[END_REF]. Here, a specific software [START_REF] Du Peloux | From shape to shell: a design tool to materialize freeform shapes using gridshell structures[END_REF], developed for rhino & grasshopper, allows generating this kind of mesh on any nurbs surface. It performs the following elementary operations: surface meshing with the compass method, trimming, control of geometry's integrity and flattening of the grid. The tool also generates automatically a text file, which can be imported in structural analysis software, containing all the required information to perform the formfinding of the structure. An add-on facilitates loads application of various complexities (snow, wind, etc.), which is tricky for free forms.

Formfinding: from mesh to structure

From now on, the initial form has been optimized and promising meshes for the materialization of the future gridshell have been identified. However, they do not take account of any true mechanical reality, because only geometric rules have led to their generation. The formfinding step consists precisely in finding the mechanical equilibrium geometry of a grid, and the corresponding permanent bending stresses. The calculations, performed numerically thanks to a dynamic relaxation algorithm with kinetic damping [START_REF] Barnes | Applications of dynamic relaxation to the topological design and analysis of cable, membrane and pneumatic structures[END_REF], is composed of the following steps:

Table 2: Formfinding steps

Step Description

1
The grid is bent, by a set of applied displacements, from its rest position to the compass position.

2

The grid is then relaxed through its mechanical equilibrium.

3 Bending stresses of the triangulation are calculated relative to the equilibrium's geometry.

4

Geometry and bending stresses of the triangulation are reinjected in the model of step 2.

Two analysis models are built during this process to study the structure with or without bracing pipes.

Note that the algorithm has been improved to take account for the eccentricity due to connections [START_REF] Douthe | Form-finding of a grid shell in composite materials[END_REF].

Structural analysis

A full structural analysis is performed on the using the two mechanical models created previously during the formfinding stage.

The non-braced model is used to check the grid's behavior during the construction stages. In particular, it must be verified that the primary grid -the one with no triangulation pipes -has no risk of buckling, both for obvious safety reasons and to ensure the accuracy of the final geometry. Indeed, the more the form is likely to buckle, the more it can be triangulated in a buckled geometry different to the targeted geometry.

The model with the triangulated grid is used to confirm the gridshell complies with all the structural requirements during its lifetime. Its behavior under standard loadings is evaluated.

From the analysis model to the as-built geometry

Previously, the geometric and structural models employed during this project never took account for the eccentricity between the three layers of pipes, inserted by the connectors. Although the mechanical consequences of this eccentricity could be neglected to a first-order approximation, this is not the case for the geometric consequences it induces. Indeed, the eccentricity remains small compared to the span of the shell but is not negligible compared to the mesh size (e = 8cm / l = 17m / w = 1.00m). Thus, pipes length and anchorages position could not be determined with sufficient accuracy without giving back to the gridshell its "thickness". The employed method, purely geometric, assesses the neutral fibre of the shell is equidistant from the first two layers of pipes. Remind that the formfinding is performed only with those two layers. Connexion's axes are supposed to be parallel to the local normal of the shell's surface. This assumption is not exact but, in this case, it gives a sufficient accuracy. The red pipe is offset -e/2, the green +e/2 and the blue +3e/2 along the normal (Fig. 11).

STANDARDS: ISSUES WITH COMPOSITE MATERIALS

Beyond the technical difficulties related to both design and structural analysis of the shell, the regulatory framework was a vital issue for the project's success. Because it was the first time a structure of this kind would host regularly a large number of people in a long-term period, the question of its reliability over time was a major issue. To be built the gridshell had to comply with existing standards, which do not take into account such an innovative edifice, all in composite material. The strategy adopted to bypass this obstacle is presented later in the publication.

First level: administrative classification of the building

The first level, administrative, consisted in getting from the French authorities an appropriate classification for the building, regarding the project's reality: a lightweight structure with a short lifespan. As expected, the edifice was classified as a "building open to the public" form the category "big tops & tents" [START_REF]Règlement ERP type CTS (big tops & tents[END_REF]. In this classification, construction procedures and regulations are adapted to the short lifespan of buildings.

Second level: compliance with existing standards

The second level, normative, consisted in making the most of the existing regulatory framework to justify the compliance of a structure that would not, at first sight, be taken into account by standards that does not include composite materials.

As far as possible, the design was led in compliance with the Eurocodes, where the structural design is made according to limit states under normalized loadings (self-weight, snow, wind, etc.). Despite the Eurocodes do not directly take into account composite materials, they propose some probabilistic methods to introduce new materials (Annexe D). As far as possible, the mechanical properties of the GFRP pipe were determined by tests in conformance with these methods. Otherwise, values where taken according to the Eurocomp [START_REF] Clarke | Structural Design of Polymer Composites -Eurocomp[END_REF]. In some cases, as the sleeve, the design of the construction details has also benefited from this approach.

Nota: The Eurocomp is a kind of pre-standard intended for the structural design of buildings and civil engineering works in GFRP composites, consistent with the Eurocode approach. It is considered has the referent design code in the matter.

Characteristic flexural strength of the pipes

Naturally, the characteristic flexural strength (σk,flex) of the GFRP pipe is needed to verify if the structure complies with Eurocodes. This parameter has a critical impact on the structure's reliability because in this particular application stresses in pipes are mainly due to bending. Thus, it was important to confirm the manufacturer's value by assays.

Three-point flexural tests were led with and without connexions (tightening torque set to 20Nm) to determine the characteristic strength according to the protocol of the Eurocode (Annex D):

! !,!"#$ = ! 1 -! ! ! ! (3)

!

For five tests, the factor kn,5% is 1.80 , assuming a normal distribution. One can note that the connections scatter the results more. Finally, the manufacturer value of 400MPa (ASTM D790) was confirmed and retained for further calculations.

Partial coefficient of composite material resistance

The partial coefficients of material resistance used in the project are calculated according to the Eurocomp. The short-term coefficient proposed in the Eurocomp (γst = 1.3) was raised to consider the critical stage of erection, where the deformations are not controlled accurately (Table 4).

Note that designers should take care about long-term effects in permanently loaded pultruded composite materials subjected to creeping and relaxation [START_REF] Kotelnikova | Mechanical and thermal optimization of fiberreinforced plastic building envelope[END_REF] and [START_REF] Bank | Composites for Construction: Structural Design with FRP Materials[END_REF]. Here, it is reflected in a high partial coefficient for long-term effects (Table 4).

CONSTRUCTION DETAILS

Mies Von der Rohe liked to say: "God is in the details", thus expressing that the quality of a building highly relies on the care taken in designing project's details. In this part, after a brief review of the project construction details, the authors focus on the design of a sleeve for connecting composite pipes.

Overview on construction details

In this project, one can identify 4 major structural details: the swivel coupler for connecting composite pipes to assemble the grid (Fig. 12a); the steel sleeve for connecting several composite pipes to make long members from initially short piece of tubes (Fig. 12b); ground anchorages for fixing the structure to the concrete slab (Fig. 12c) and the lacing edge beam of the fabric (Fig. 12d). Note that the tricky issue of connecting steel and composite parts is solved in a similar way through sleeve and anchorage details.

Sleeve

Sleeves are major components in the structural system. The presented component is a great innovation compared to the composite gridshells built previously, where members were simply interrupted or overlapped.

By establishing mechanical and architectural continuities between pipes, this sleeve brings closer the real behavior and the theoretical behavior of the shell.

System description

The sleeve is a steel system that sets up mechanical continuity between two adjacent composite pipes for both tension and bending. It is made of three parts: two connectors linked by a threaded rod (Fig. 12b).

Each connector is a 48.3x2.9mm steel pipe, slightly larger than the composite pipes on which it is put on, with a welded M20 nut at one of his end. The connector is pinned to the composite pipe with three 10mm bolts. Some structural adhesive is also employed to fill gaps and to guaranty a good rigidity of the assembly. However, the sleeve is designed ignoring the adhesive contribution to the mechanical strength of the system.

A M20 threaded rod links the two connectors. It allows tension forces and bending moments to pass form one pipe to the other. It cannot transfer any twisting moment.

Mechanical behavior: tension

Tension forces are transferred from the composite pipe to the connector through shear in the pins. Thanks to a lower bearing resistance in the composite than in the steel, each of the three pins can be gradually loaded. When loading the system, at first, only one of the three pins is really in contact with both the steel pipe and the composite pipe, because of inevitable small manufacturing gaps. When increasing the axial load, this pin starts to "eat" into the composite pipe until the second pin comes also in contact (Fig. 13). This scheme is reproduced until another failure mode happened. For this mode of composite failure, which prevails in this case, the total bearing capacity of the assembly is thus three times the capacity of a single pin. It could be calculated from the compressive strength, the composite thickness and the pin diameter as:

F!=!3× f u,c .d.t !=!3× 300MPa ×10mm×3.5mm !=!3150 daN (4)
In §6.2.4, tests led at the laboratory Navier to confirm the predicted value are presented.

Mechanical behavior: bending

Bending moments are transferred through the threaded rod of the sleeve. This part is designed to reach simultaneously the two following qualitative criterions:

Firstly, the rod bending stiffness should be roughly equivalent to the composite bending stiffness itself to preserve curvature's continuity along the system. Note that this continuity is of prime importance from an architectural point of view.

!" !"#$ !" !"!# ≈ 1 (5)

Secondly, the steel quality of the rod should be adjusted so it starts its plastification when the composite pipe tends to approach its maximum design stress (a third of the yield stress). Thus, the rod acts as a "fuse" which concentrates rotational deformations avoiding failures in the critical elements of the structure (the composite pipes).

! !"#$%&'!!"# !"#$ ! !!!""!"#

!"!# ≈ 1 (6)
In this project, the numerical values for ratios ( 5) and ( 6) are respectively 0.79 and 0.96. The graph below shows the tensile test of a 3 pins connection between a connector and the corresponding composite pipe (Fig. 14). Then the compressive stress applied by each pin to the composite pipe exceeds its compressive strength. Progressively, the pins are pulled of through the pipe under a residual force that tends to stabilize at around 20kN.

CONCLUSIONS

This paper has presented the different steps for the design of a gridshell in composite material: a Temporary Cathedral at Créteil, in Paris suburb, in 2013. The first step was the optimization of the shape in order to avoid concentrations of curvature, locally. The second step showed a tool to automatically mesh a surface using the compass method. With this tool, the orientation of the mesh is studied according to structural and architectural criterions. The last steps showed the structural analysis of the gridshell and how to get the as-built geometry from the analysis model.

Architecturally, the structure offers a very interesting space where the textual richness of the pipes against the membrane accentuates the reading of the complex curved surfaces.

This project demonstrates that gridshells in composite material are suitable for constructing freeform buildings. However, the long-term behavior of these materials needs to be better characterized to extend their lifespan. This is one of the laboratory's research projects.

At the moment, further developments are also conducted by the laboratory to take account for torsional effects and non axisymmetric sections in such structures, as it is studied in [START_REF] Barnes | A novel torsion/bending element for dynamic relaxation modeling[END_REF]. The interaction between the structure and the fabric is also a vast research field to be explored.

Fig. 1 .

 1 Fig. 1. Interior view (a) | Grid erection (b) | Exterior view (c).

Fig. 4 .

 4 Fig. 4. Main door's exterior (a) and interior (b) views.

Fig. 5 .

 5 Fig. 5. Prototypes (a) & (b) | Solidays 2011 (c).

Fig. 6 .

 6 Fig. 6. Assembly (1) | Erection (2) | Erected grid (3) | Braced grid (4)

Fig. 9 .

 9 Fig. 9. Resulting meshes and flat grids, depending the directrix.

Fig. 10 .

 10 Fig. 10. Geometry of the relaxed mesh and its bending stresses diagram.

Fig. 11 .

 11 Fig. 11. Restoring the geometric eccentricity between layers.

Fig. 12 .

 12 Fig. 12. Major construction details.

Fig. 13 .

 13 Fig. 13. Local bearing failure due to pin pression.

! 6 . 2 . 4 .

 624 Test: load bearing capacity of the pinned connection

Fig. 14 .

 14 Fig. 14. Tensile test of the pinned connection.Till 35kN the graph reflects the elastic behavior of the composite pipe, with slight deviations corresponding to pins rearrangement ( §4.2.2). Then the compressive stress applied by each pin to the composite pipe exceeds its compressive strength. Progressively, the pins are pulled of through the pipe under a residual force that tends to stabilize at around 20kN.

Table 1 :

 1 Key figures

	1. celebration area
	2. narthex
	3. parvis
	Item	Value(s)
	Public	360 seats / 500 people standing
	Dimensions	S=350m 2 / V=1600m 3 / h=7m / l=17m / L=29m
		Pipes: 1775m / Connections: 1130u.
	Gridshell
	Weight	5kg/m 2

Sleeves: 125u. / Anchorages: 123u.

Fabric

Opaque: 530m 2 / Transparent: 25x0.5 = 12.5m 2

Table 3 :

 3 Three-point flexural tests of the GFRP pipes

	Connections	σ1	σ2	σ3	σ4	σ5	σk,flex
	Without	456	441	445	460	477	430 MPa
	With	444	478	434	479	427	408 MPa

Table 4 :

 4 Short-term and long-term values for material resistance

	Impact	Partial coefficient of material resistance	Design flexural strength
	Short-term	γst = 2.0	200 MPa
	Long-term	γlt = 3.0	133 MPa

ACKNOWLEDGEMENTS

First of all, the authors would like to thank the Catholic Church of Créteil for their trust and their courage, which permitted to get into this ambitious and successful project.

Secondly, the authors would greet the engineers from T/E/S/S, which have developed this challenging project during 18 th months. They made a valuable work and permitted research works to become a reality throughout this amazing edifice. They also thank the firm Viry for the supervision of construction works, including the delicate erection stage.

Thirdly, the authors would like to thank warmly all the people involved in the construction process: the numerous parish's volunteers, the technicians and researchers from the Laboratory Navier and the engineers from T/E/S/S and Viry firms. Beyond the technic, their enthusiasm made this project a powerful human experience.

Fourthly, the authors would like to thank the local firms for their work: BATEM (concrete), Eloi (steel), Esmery Caron (fabric), Solutions Composites (composite material), Axmann (connections), ENSG.

Finally, the authors would thank especially Tom Gray, project's architect, for his heavy contribution to this article.


L Du Peloux

J-F Caron
email: caron@enpc.fr

F Tayeb

O Baverel

2 faite d'une résille de tubes composites en GFRP The ephemeral cathedral of Créteil : a 350m 2 lightweight structure made of a GFRP composite gridshell

Keywords: composite GFRP, gridshell élastique, structure légère, mise en forme GFRP composite, elastic gridshell, lightweight structure, active bending

La cathédrale éphémère

The inventiveness of the gridshell concept is commonly attributed to Frei Otto, a German architect who devoted much time to gridshells. In 1975 he achieved the famous Mannheim Multihalle [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF], a wooden shell of 7500m 2 , in collaboration with the engineer Edmund Happold (Arup).

Literally, the word "gridshell" refers to grids behaving like shells: from a mechanical point of view that means stresses acting on the structure are mainly transmitted through compression and traction. These structures can cross large-span with very little material. Here, the whole structures spans 17x29 meters and weights only 5kg/m 2 .

However, according to the historic evolution of the concept, characterizing a gridshell as the combination of a structural concept (a grid behaving like a shell) and a specific construction process (using the bending flexibility of the material) seems to be more accurate. The Mannheim project (in which a wooden regular and planar grid, lacking shear stiffness, is elastically deformed up to a targeted shape with the help of stays, and then braced and covered) is regarded as the starting point of this new concept.

This type of gridshell, known as elastic gridshell, offers a very elegant manner to materialize freeform shapes from an initially flat and regular grid, which obviously has many practical benefits: planar geometry, standard connexion nodes, standard profiles, etc.

After a brief description of the structure the paper gives an insight on the field of elastic gridshells and discusses the benefits of composite materials in such structures. The overall design process employed to go from the 3D shape designed by the architect to a fully structural shell is then explained. Finally, complexities brought by connections are highlighted and solutions are presented.

Building Overview

Architectural considerations

The generation of the form was driven by two objectives that of providing a variety of appropriate internal spaces within which the community could assemble, and externally to be a welcoming and visually interesting form. The form gathers under the same roof a religious space with a circular arrangement and an area for more informal gatherings after ceremonies (Figure 3).

The building can accommodate up to 500 people and complies with all the required performances for such a building: structural stiffness, fire safety, waterproofness, lightning, thermal comfort, etc.

Technical description

This gridshell structure is made of long glass-fiber tubes (Ø42mm, Figure 2a) pinned together with scaffold swivel couplers. The structural members of the grid, all of different length, are built from one, two or three composite tubes connected with steel sleeves. Tubes length is restrained to 12m so standard trucks can deliver them. The tubes are organized in three layers. The first two layers are placed perpendicularly on the ground. They form the "quadrangular primary grid". The distance between the tubes of these two layers is constant so the grid is regular. This primary grid is elastically deformed to obtain the final shape. The third layer of tubes acts as bracing. It gives the structure a shell behavior. Those tubes are fixed to the primary grid once it has been shaped. The structure is anchored to a concrete strip footing with a special steel system, which ensures the transit of loads from the composite structure to the ground. A similar system enables to fix the structure to the doors.

A PVC coated fabric (Figure 2d), tailor patterned, covers the structure. It includes transparent strips that bring daylight inside. The fabric is stretched on a devoted peripheral edge beam with a double lacing system (halyard and strap, Figure 2b). At the ground, the lacing edge beam is made of a bent composite rod nailed regularly to the concrete slab. At the doors, it is made of a steel arch welded to the doorframes (Figure 2c). The membrane is waterproof and as a continuous membrane has no joints except at the perimeter. At the perimeter a continuous strip of membrane is prefixed to the internal face of the membrane and fixed to the ground slab. At the doors, the flexible strip is riveted to the doorframes.

Cost

The overall cost of the project is 500k$, about 30% more than a standard tent of the same size that would have been rented for 2 years.

Gridshells

Recent projects

More recently, with the development of numerical tools, two new projects saw the light : the wooden gridshells of Downland in 2002 [START_REF] Harris | Design and construction of the Downland Gridshell[END_REF] and the Savill one in 2006 [START_REF] Harris | The Savill Garden gridshell : Design and construction[END_REF]. The principles of these projects are similar to Mannheim one but new constructive methods are used. In 2000 Shigeru Ban innovates shifting from wood materials to cardboard for the Japan Pavilion, in Hannover [START_REF] Mcquaid | The Japanese pavilion[END_REF].

Recent developments

Based on this groundwork, the Navier laboratory has developed a research program on gridshells for the last decade, focusing on both the use of new materials and the development of more efficient numerical methods [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites : application à la conception des gridshells[END_REF]. These developments have been validated by the construction of two prototypes In 2011, the laboratory used its expert knowledge for a large-scale project [START_REF] Baverel | Gridshells in composite materials : construction of a 300 m2 forum for the solidays' festival in Paris[END_REF], the forum of the Solidays festival (Figure 4c). This achievement, built by voluntary workers, has been the first composite material gridshell to house public.

In 2012, the context is favorable for a new achievement named "Temporary Cathedral of Créteil". Although this gridshell has an area very similar to the Solidays one, the project raised new challenges, in particular the challenge of reliability. Indeed, its period of use is at least two years. Additionally, the skills coming from T/E/S/S company made possible important developments such as for doors, lacing edge beam, anchorages and sleeves.

Unlike the two first prototypes, the gridshells built for Solidays and at Créteil are based on a new approach regarding shape-structure relationship. Indeed, thanks to a numerical tool performing the compass method, the geometry of the object is no longer defined as the reversal of a hanging net -in this case, only the flat geometry could be mastered [START_REF] Addis | Toys that save millions' -A history of using physical models in structural design[END_REF] -but now the flat geometry is straight deducted from the one proposed by the architect. This new approach opens up new architectural horizons, making possible the exploration of new shapes for gridshells.

Gridshell in composite material

The gridshells built in composite material, being at the heart of this paper, are consistent with the framework defined previously, that is to say :

Structural Typology

Their mechanical behaviour is very similar to the one of real shells even if the material is discrete and located in a grid more or less open. In spite of that, gridshells benefit from the same advantages as the ones showed by an eggshell : they can cross large span using a low amount of material. Their stiffness is mainly linked to their double-curved shape.

Material Flexibility for Structural Rigidity

In this field of application, composite materials like glass fibre reinforced polymer (GFRP) could favourably replace wood, where both resistance and bending ability of the material is sought [START_REF] Douthe | Gridshell structures in glass fibre reinforced polymers[END_REF]. The stiffness of the structure does not derive from the intrinsic material rigidity but principally from its geometric curvature. Ideally, the composite profiles are produced by pultrusion, an economic continuous moulded process. The standardization of the process guaranties very stable material and mechanical properties. It frees designers from the painful problematic of wood joining and wood durability. The characterization of this material is presented further in the paper.

Erection Process

Usually, the grid morphology is not trivial and leads to design numerous costly and complex joints. To overcome this issue, an original and innovative erection process was developed that takes advantage of the flexibility inherent to slender elements. A regular planar grid made of long continuous linear members is built on the ground (Figure 6a). The elements are pinned together so the grid has no inplane shear stiffness and can accommodate large-scale deformations during erection. Then, the grid is bent elastically to its final shape (Figure 5). Finally, the grid is frozen in the desired shape with a third layer of bracing members (Figure 6b) and the structure becomes a shell.

Design Process

Overall design process

The goal of the design process is to identify a gridshell structure that works and respects as faithfully as possible the architectural project -a shape and a program. It represents "the path from shape to structure". Its progress, sequential and iterative, revolves around three major stages: shape, mesh and structure. It is not trivial to go through this complex process. Indeed, for each step, the method, the tool, the criteria, that offers both a sufficient explorative richness to find out enough candidate solutions, and the means to evaluate and compare the suitability of those solutions, have to be found.

3D modelling of the intended shape

The first step of the process consists in building a precise geometric model from the sketch of the architect and to evaluate its mechanical potential (Figure 7). At this stage, the goal is to estimate the probability a given shape would lead to the generation of a structurally feasible gridshell.

Stresses in the grid are mainly due to the bending of the profiles. They derive directly from their geometric curvature. Thus, the principal curvatures of the surface -because they give a qualitative measurement of the local curvature of any curve drawn on a surface -are relevant indicators to evaluate the stress rate of a grid laying on it. Particularly, one should ensure the following condition is satisfied everywhere, where r is the pipe's outer radius, R min is the minimum principal radius, E is the flexural modulus, σ k, f lex the characteristic flexural strength and γ lt the long-term partial coefficient of material resistance :

E = r R min < σ k, f lex γ lt (1)
Ideally, the shape is controlled by few key parameters. Thus, it can be adapted and optimized through an iterative process, towards this criterion.

Mesh from the compass method

During the second step, the candidate surface is meshed and the mechanical potential of the resulting grid is evaluated. At this stage, we try to estimate the probability a given mesh could lead to the generation of a viable gridshell structure. Simultaneously, meshes are compared according to their architectural relevance. This time, the geometric curvature of the polylines drawn on the surface is the criterion to characterize the mechanical potential of the grid. In particular, one should ensure the following condition is satisfied everywhere, where R spline is the spline's local curvature radius :

E = r R spline < σ k, f lex γ lt ( 2
)
The mesh is obtained by the compass method, described in [START_REF] Otto | IL10 Grid Shells[END_REF], which develops a regularly spaced grid on a surface from two secant directrix. For a given shape there are an infinite number of meshes. The aim is to identify at least one grid, suitable towards architectural and structural criteria. The Fig. 8 Swivel coupler, steel sleeve, ground anchorage and lacing edge beam laboratory tried various numerical methods to generate such grids [START_REF] Bouhaya | Mapping two-way continuous elastic grid on an imposed surface : application to grid shells[END_REF]. Here, a specific software [START_REF] Du Peloux | From shape to shell : a design tool to materialize freeform shapes using gridshell structures[END_REF], developed for rhino & grasshopper, allows generating this kind of mesh on any NURBS surface.

Formfinding and bending prestress

From now on, the initial form has been optimized and promising meshes for the materialization of the future gridshell have been identified. However, they do not take account of any true mechanical reality, because only geometric rules have led to their generation. The formfinding step consists precisely in finding the geometry of the grid at mechanical equilibrium, and the corresponding permanent bending stresses. The calculations, performed numerically thanks to a dynamic relaxation algorithm with kinetic damping [START_REF] Barnes | Applications of dynamic relaxation to the design and analysis of cable, membrane and pneumatic structures[END_REF], is composed of the following steps:

1. The grid is bent by a set of applied displacements from rest to compass position 2. The grid is then relaxed through its mechanical equilibrium 3. Bending stresses of the triangulation are calculated relative to the relaxed geometry 4. Geometry and bending stresses of the triangulation are reinjected in the model Two analysis models are built during this process to study the structure with or without bracing tubes. Note that the algorithm has been improved to take account for the eccentricity due to connections [START_REF] Douthe | Formfinding of a grid shell in composite materials[END_REF].

Structural analysis

A full structural analysis is performed on the gridshell, using the two mechanical models created previously during the formfinding stage. The non-braced model is used to check the grid's behavior during the construction stages. In particular, it must be verified that the primary grid -the one with no triangulation tubes -has no risk of buckling, both for obvious safety reasons and to ensure the accuracy of the final geometry. Indeed, the more the form is likely to buckle, the more it can be triangulated in a buckled geometry different to the targeted geometry. The model with the triangulated grid is used to confirm the gridshell complies with all the structural requirements during its lifetime. Its behavior under standard loadings is evaluated.

Construction details

In this project, one can identify 4 major structural details : the swivel coupler for connecting composite tubes to assemble the grid (Figure 8a); the steel sleeve for connecting several composite tubes to make long members from initially short piece of tubes (Figure 8b); ground anchorages for fixing the structure to the concrete slab (Figure 8c) and the lacing edge beam of the fabric (Figure 8d). Note that the tricky issue of connecting steel and composite parts is solved in a similar way through sleeve and anchorage details. Beyond the technical difficulties related to both design and structural analysis of the shell, the regulatory framework was a vital issue for the project's success. Because it was the first time a structure of this kind would host regularly a large number of people in a long-term period, the question of its reliability over time was a major issue. To be built the gridshell had to comply with existing standards, which do not take into account such an innovative edifice, all in composite material. The strategy adopted to bypass this obstacle consisted in making the most of the existing regulatory framework to justify the compliance of a structure that would not, at first sight, be taken into account by standards that does not include composite materials. As far as possible, the design was led in compliance with the Eurocodes, where the structural design is made according to limit states under normalized loadings (self-weight, snow, wind, etc.). Despite the Eurocodes do not directly take into account composite materials, they propose some probabilistic methods to introduce new materials (Annexe D). As far as possible, the mechanical properties of the GFRP pipe were determined by tests in conformance with these methods. Otherwise, values where taken according to the Eurocomp [START_REF] Clarke | structural design of polymer composites -eurocomp design code and handbook[END_REF]. In some cases, as the sleeve, the design of the construction details has also benefited from this approach.

Flexural strength of the tubes

The characteristic flexural strength (σ k, f lex ) of the GFRP tube has a critical impact on the structure's reliability because in this particular application stresses in the tubes are mainly due to bending. Thus, it was important to confirm the manufacturer's value by assays. Three-point flexural tests were led with and without connexions (tightening torque set to 20Nm) to determine the characteristic strength according to the protocol of the Eurocode (Annex D) :

σ k, f lex = σ (1 -k n σ x ) (3)
For five tests, the factor k n,5% is 1.80 , assuming a normal distribution. One can note that the connections scatter the results more (Table 1). Finally, the manufacturer value of 400MPa (ASTM D790) was confirmed and retained for further calculations.

Partial safety factors

The partial coefficients of material resistance used in the project are calculated according to the Eurocomp (Table 2). The short-term coefficient proposed in the Eurocomp (γ st = 1.3) was raised to consider the critical stage of erection, where the deformations are not controlled accurately. Note that designers should take care about long-term effects in permanently loaded pultruded composite materials subjected to creeping and relaxation [START_REF] Kotelnikova-Weiler | Optimisation mécanique et énergétique d'enveloppes en matériaux composites pour les bâtiments[END_REF]. Here, it is reflected in a high partial coefficient for long-term effects.

Conclusions

This paper has presented the different steps for the design of a gridshell in composite material : a Temporary Cathedral at Créteil, in Paris suburb, in 2013. The first step was the optimization of the shape in order to avoid local concentrations of curvature. The second step showed a tool to automatically mesh a surface using the compass method. With this tool, the orientation of the mesh is studied according to structural and architectural criterions. The last steps showed the structural analysis of the gridshell and how to get the as-built geometry from the analysis model. Architecturally, the structure offers a very interesting space where the textual richness of the tubes against the membrane accentuates the reading of the complex curved surfaces. This project demonstrates that gridshells in composite material are suitable for constructing freeform buildings. However, the long-term behavior of these materials needs to be better characterized to extend their lifespan.

At the moment, further developments are conducted by the laboratory to take account for torsional effects and non axisymmetric sections in such structures, as it is studied in [START_REF] Barnes | A novel torsion/bending element for dynamic relaxation modeling[END_REF].

Fig. 1

 1 Fig. 1 Interior view of the Ephemeral Cathedral in Paris

Fig. 2 Fig. 3

 23 Fig. 2 GFRP tubes, steel door, grid, PVC coated fabric

Fig. 4 Fig. 5

 45 Fig. 4 Prototypes 1 & 2, Solidays 2011

Fig. 6

 6 Fig. 6 Erection plan and construction stages

Fig. 7

 7 Fig. 7 Evaluating the mechanical potential of shapes regarding principal curvatures

Fig. 9

 9 Fig. 9 Section on the footer

2

 2 Short-term and long-term values for material resistance.

Acknowledgements

The authors would like to thank the Catholic Church of Créteil for their trust and their courage, which permitted to get into this ambitious and successful project. They also want to greet the engineers from T/E/S/S, which have developed this challenging project during 18th months. They also thank the firm Viry for the supervision of construction works, including the delicate erection stage.


Lionel Du Peloux

Frédéric Tayeb

Jean-François Caron

Olivier Baverel

Lionel Du Peloux
email: lionel.dupeloux@tess.fr@www.thinkshell.fr

The Ephemeral Cathedral of Créteil : a 350m2 lightweight gridshell structure made of 2 kilometers of GFRP tubes

Keywords: GFRP, glass fibre, gridshell, connection, active bending

Introduction

The inventiveness of the gridshell concept is commonly attributed to Frei Otto, a German architect who devoted much time to gridshells. In 1975 he achieved the famous Mannheim Multihalle [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF], a wooden shell of 7500m 2 , in collaboration with the engineer Edmund Happold (Arup). Literally, the word "gridshell" refers to grids behaving like shells: from a mechanical point of view that means stresses acting on the structure are mainly transmitted through compression and traction. These structures can cross large-span with very little material. Here, the whole structures spans 17x29 meters and weights only 5kg/m 2 .

However, according to the historic evolution of the concept, characterizing a gridshell as the combination of a structural concept -a grid behaving like a shell -and a specific construction process -using the bending flexibility of the material -seems to be more accurate. The Mannheim project -in which a wooden regular and planar grid, lacking shear stiffness, is elastically deformed up to a targeted shape with the help of stays, and then braced and covered -is regarded as the starting point of this new concept. This type of gridshell, known as "elastic gridshell", offers a very elegant manner to materialize freeform shapes from an initially flat and regular grid, which obviously has many practical benefits: planar geometry, standard connexion nodes, standard profiles, etc.

After a brief description of the structure the paper gives an insight on the field of elastic gridshells and discusses the benefits of composite materials in such structures. The overall design process employed to go from the 3D shape designed by the architect to a fully structural shell is then explained. Finally, complexities brought by connections are highlighted and solutions are presented.

Building Overview

Architectural considerations

The generation of the form was driven by two objectives that of providing a variety of appropriate internal spaces within which the community could assemble, and externally to be a welcoming and visually interesting form. The form gathers under the same roof a religious space with a circular arrangement and an area for more informal gatherings after ceremonies (Figure 3). The building can accommodate up to 500 people and complies with all the required performances for such a building: structural stiffness, fire safety, waterproofness, lightning, thermal comfort, etc.

Technical description

This gridshell structure is made of long glass-fiber tubes (Ø42mm, Figure 2a) pinned together with scaffold swivel couplers. The structural members of the grid, all of different length, are built from one, two or three composite tubes connected with steel sleeves. Tubes length is restrained to 12m so standard trucks can deliver them. The tubes are organized in three layers. The first two layers are placed perpendicularly on the ground. They form the "quadrangular primary grid". The distance between the tubes of these two layers is constant so the grid is regular. This primary grid is elastically deformed to obtain the final shape. The third layer of tubes acts as bracing. It gives the structure a shell behavior. Those tubes are fixed to the primary grid once it has been shaped. The structure is anchored to a concrete strip footing with a special steel system, which ensures the transit of loads from the composite structure to the ground. A similar system enables to fix the structure to the doors. A PVC coated fabric (Figure 2d), tailor patterned, covers the structure. It includes transparent strips that bring daylight inside. The fabric is stretched on a devoted peripheral edge beam with a double lacing system (halyard and strap, Figure 2b). At the ground, the lacing edge beam is made of a bent composite rod nailed regularly to the concrete slab. At the doors, it is made of a steel arch welded to the doorframes (Figure 2c). The membrane is waterproof and as a continuous membrane has no joints except at the perimeter. At the perimeter a continuous strip of membrane is prefixed to the internal face of the membrane and fixed to the ground slab. At the doors, the flexible strip is riveted to the doorframes.

Cost

The overall cost of the project is 500k$, about 30% more than a standard tent of the same size that would have been rented for 2 years.

Gridshells

Recent projects

More recently, with the development of numerical tools, two new projects saw the light: the wooden gridshells of Downland in 2002 [START_REF] Harris | Design and construction of the Downland Gridshell[END_REF] and the Savill one in 2006 [START_REF] Harris | The Savill Garden gridshell : Design and construction[END_REF]. The principles of these projects are similar to Mannheim one but new constructive methods are used. In 2000 Shigeru Ban innovates shifting from wood materials to cardboard for the Japan Pavilion, in Hannover [START_REF] Mcquaid | The Japanese pavilion[END_REF].

Recent developments

Based on this groundwork, the Navier laboratory has developed a research program on gridshells for the last decade, focusing on both the use of new materials and the development of more efficient numerical methods [START_REF] Douthe | Etude de structures élancées précontraintes en matériaux composites : application à la conception des gridshells[END_REF]. These developments have been validated by the construction of two prototypes (Figures 4a and4b) whose areas were about 150 m 2 .

In 2011, the laboratory used its expert knowledge for a large-scale project [START_REF] Baverel | Gridshells in composite materials : construction of a 300 m2 forum for the solidays' festival in Paris[END_REF], the forum of the Solidays festival (Figure 4c). This achievement, built by voluntary workers, has been the first composite material gridshell to house public.

In 2012, the context is favorable for a new achievement named "Temporary Cathedral of Créteil". Although this gridshell has an area very similar to the Solidays one, the project raised new challenges, in particular the challenge of reliability. Indeed, its period of use is at least two years. Additionally, the skills coming from T/E/S/S company made possible important developments such as for doors, lacing edge beam, anchorages and sleeves.

Unlike the two first prototypes, the gridshells built for Solidays and at Créteil are based on a new approach regarding shape-structure relationship. Indeed, thanks to a numerical tool performing the compass method, the geometry of the object is no longer defined as the reversal of a hanging net -in this case, only the flat geometry could be mastered [START_REF] Addis | Toys that save millions' -A history of using physical models in structural design[END_REF] -but now the flat geometry is straight deducted from the one proposed by the architect. This new approach opens up new architectural horizons, making possible the exploration of new shapes for gridshells.

Gridshell in composite material

The gridshells built in composite material, being at the heart of this paper, are consistent with the framework defined previously, that is to say :

• Structural Typologytheir mechanical behaviour is very similar to the one of real shells even if the material is discrete and located in a grid more or less open. In spite of that, gridshells benefit from the same advantages as the ones showed by an eggshell : they can cross large span using a low amount of material. Their stiffness is mainly linked to their double-curved shape.

• Material Flexibility for Structural Rigidityin this field of application, composite materials like glass fibre reinforced polymer (GFRP) could favourably replace wood, where both resistance and bending ability of the material is sought [START_REF] Douthe | Gridshell structures in glass fibre reinforced polymers[END_REF]. The stiffness of the structure does not derive from the intrinsic material rigidity but principally from its geometric curvature. Ideally, the composite profiles are produced by pultrusion, an economic continuous moulded process. The standardization of the process guaranties very stable material and mechanical properties. It frees designers from the painful problematic of wood joining and wood durability. The characterization of this material is presented further in the paper.

• Erection Processusually, the grid morphology is not trivial and leads to design numerous costly and complex joints. To overcome this issue, an original and innovative erection process was developed that takes advantage of the flexibility inherent to slender elements. A regular planar grid made of long continuous linear members is built on the ground (Figure 6a). The elements are pinned together so the grid has no in-plane shear stiffness and can accommodate large-scale deformations during erection. Then, the grid is bent elastically to its final shape (Figure 5). Finally, the grid is frozen in the desired shape with a third layer of bracing members (Figure 6b) and the structure becomes a shell.

Overall design process

The goal of the design process is to identify a gridshell structure that works and respects as faithfully as possible the architectural project -a shape and a program. It represents "the path from shape to structure". Its progress, sequential and iterative, revolves around three major stages: shape, mesh and structure. It is not trivial to go through this complex process. Indeed, for each step, the method, the tool, the criteria, that offers both a sufficient explorative richness to find out enough candidate solutions, and the means to evaluate and compare the suitability of those solutions, have to be found.

3D modelling of the intended shape

The first step of the process consists in building a precise geometric model from the sketch of the architect and to evaluate its mechanical potential (Figure 7). At this stage, the goal is to estimate the probability a given shape would lead to the generation of a structurally feasible gridshell.

Stresses in the grid are mainly due to the bending of the profiles. They derive directly from their geometric curvature. Thus, the principal curvatures of the surface -because they give a qualitative measurement of the local curvature of any curve drawn on a surface -are relevant indicators to evaluate the stress rate of a grid laying on it. Particularly, one should ensure the following condition is satisfied everywhere, where r is the pipe's outer radius, R min is the minimum principal radius, E is the flexural modulus, σ k,f lex the characteristic flexural strength and γ lt the long-term partial coefficient of material resistance : Ideally, the shape is controlled by few key parameters. Thus, it can be adapted and optimized through an iterative process, towards this criterion.

E = r R min < σ k,f lex γ lt (1)

Mesh from the compass method

During the second step, the candidate surface is meshed and the mechanical potential of the resulting grid is evaluated. At this stage, we try to estimate the probability a given mesh could lead to the generation of a viable gridshell structure. Simultaneously, meshes are compared according to their architectural relevance.

This time, the geometric curvature of the polylines drawn on the surface is the criterion to characterize the mechanical potential of the grid. In particular, one should ensure the following condition is satisfied everywhere, where R spline is the spline's local curvature radius :

E = r R spline < σ k,f lex γ lt ( 2
)
The mesh is obtained by the compass method, described in [START_REF] Otto | IL10 Grid Shells[END_REF], which develops a regularly spaced grid on a surface from two secant directrix. For a given shape there are an infinite number of meshes. The aim is to identify at least one grid, suitable towards architectural and structural criteria. The laboratory tried various numerical methods to generate such grids [START_REF] Bouhaya | Mapping two-way continuous elastic grid on an imposed surface : application to grid shells[END_REF]. Here, a specific software [START_REF] Du Peloux | From shape to shell : a design tool to materialize freeform shapes using gridshell structures[END_REF], developed for rhino & grasshopper, allows generating this kind of mesh on any NURBS surface.

Formfinding and bending prestress

From now on, the initial form has been optimized and promising meshes for the materialization of the future gridshell have been identified. However, they do not take account of any true mechanical reality, because only geometric rules have led to their generation. The formfinding step consists precisely in finding the geometry of the grid at mechanical equilibrium, and the corresponding permanent bending stresses. The calculations, performed numerically thanks to a dynamic relaxation algorithm with kinetic damping [START_REF] Barnes | Applications of dynamic relaxation to the design and analysis of cable, membrane and pneumatic structures[END_REF], is composed of the following steps:

1. The grid is bent by a set of applied displacements from rest to compass position 2. The grid is then relaxed through its mechanical equilibrium 3. Bending stresses of the triangulation are calculated relative to the relaxed geometry 4. Geometry and bending stresses of the triangulation are reinjected in the model Two analysis models are built during this process to study the structure with or without bracing tubes. Note that the algorithm has been improved to take account for the eccentricity due to connections [START_REF] Douthe | Formfinding of a grid shell in composite materials[END_REF].

Structural analysis

A full structural analysis is performed on the gridshell, using the two mechanical models created previously during the formfinding stage. The non-braced model is used to check the grid's behavior during the construction stages. In particular, it must be verified that the primary grid -the one with no triangulation tubes -has no risk of buckling, both for obvious safety reasons and to ensure the accuracy of the final geometry. Indeed, the more the form is likely to buckle, the more it can be triangulated in a buckled geometry different to the targeted geometry. The model with the triangulated grid is used to confirm the gridshell complies with all the structural requirements during its lifetime. Its behavior under standard loadings is evaluated.

Construction details

In this project, one can identify 4 major structural details : the swivel coupler for connecting composite tubes to assemble the grid (Figure 8a); the steel sleeve for connecting several composite tubes to make long members from initially short piece of tubes (Figure 8b); ground anchorages for fixing the structure to the concrete slab (Figure 8c) and the lacing edge beam of the fabric (Figure 8d). Note that the tricky issue of connecting steel and composite parts is solved in a similar way through sleeve and anchorage details.

Codes for composite materials

Beyond the technical difficulties related to both design and structural analysis of the shell, the regulatory framework was a vital issue for the project's success. Because it was the first time a structure of this kind would host regularly a large number of people in a long-term period, the question of its reliability over time was a major issue. To be built the gridshell had to comply with existing standards, which do not take into account such an innovative edifice, all in composite material. The strategy adopted to bypass this obstacle consisted in making the most of the existing regulatory framework to justify the compliance of a structure that would not, at first sight, be taken into account by standards that does not include composite materials. As far as possible, the design was led in compliance with the Eurocodes, where the structural design is made according to limit states under normalized loadings (self-weight, snow, wind, etc.). Despite the Eurocodes do not directly take into account composite materials, they propose some probabilistic methods to introduce new materials (Annexe D). As far as possible, the mechanical properties of the GFRP pipe were determined by tests in conformance with these methods. Otherwise, values where taken according to the Eurocomp [START_REF] Clarke | structural design of polymer composites -eurocomp design code and handbook[END_REF]. In some cases, as the sleeve, the design of the construction details has also benefited from this approach.

Flexural strength of the tubes

The characteristic flexural strength (σ k,f lex ) of the GFRP tube has a critical impact on the structure's reliability because in this particular application stresses in the tubes are mainly due to bending. Thus, it was important to confirm the manufacturer's value by assays. Three-point flexural tests were led with and without connexions (tightening torque set to 20Nm) to determine the characteristic strength according to the protocol of the Eurocode (Annex D) :

σ k,f lex = σ(1 -k n σ x ) (3)
For five tests, the factor k n,5% is 1.80 , assuming a normal distribution. One can note that the connections scatter the results more (Table 1). Finally, the manufacturer value of 400MPa (ASTM D790) was confirmed and retained for further calculations.

Partial safety factors

The partial coefficients of material resistance used in the project are calculated according to the Eurocomp (Table 2). The short-term coefficient proposed in the Eurocomp (γ st = 1.3) was raised to consider the critical stage of erection, where the deformations are not controlled accurately. Note that designers should take care about long-term effects in permanently loaded pultruded composite materials subjected to creeping and relaxation [START_REF] Kotelnikova-Weiler | Optimisation mécanique et énergétique d'enveloppes en matériaux composites pour les bâtiments[END_REF]. Here, it is reflected in a high partial coefficient for long-term effects.

Impact

Conclusions

This paper has presented the different steps for the design of a gridshell in composite material : a Temporary Cathedral at Créteil, in Paris suburb, in 2013. The first step was the optimization of the shape in order to avoid local concentrations of curvature. The second step showed a tool to automatically mesh a surface using the compass method. With this tool, the orientation of the mesh is studied according to structural and architectural criterions. The last steps showed the structural analysis of the gridshell and how to get the as-built geometry from the analysis model. Architecturally, the structure offers a very interesting space where the textual richness of the tubes against the membrane accentuates the reading of the complex curved surfaces. This project demonstrates that gridshells in composite material are suitable for constructing freeform buildings. However, the longterm behavior of these materials needs to be better characterized to extend their lifespan. At the moment, further developments are conducted by the laboratory to take account for torsional effects and non axisymmetric sections in such structures, as it is studied in [START_REF] Barnes | A novel torsion/bending element for dynamic relaxation modeling[END_REF].

Figure. 1 .

 1 Figure. 1. Interior view of the Ephemeral Cathedral in Paris

Figure. 2 .

 2 Figure. 2. GFRP tubes, steel door, grid, PVC coated fabric

Figure. 3 .

 3 Figure. 3. Interior view of the Ephemeral Cathedral in Paris

Figure. 4 .

 4 Figure. 4. Prototypes 1 & 2, Solidays 2011

Figure. 5 .

 5 Figure. 5. Erection of the primary grid by two cranes

Figure. 6 .

 6 Figure. 6. Erection plan and construction stages

Figure. 7 .

 7 Figure. 7. Evaluating the mechanical potential of shapes regarding principal curvatures

Figure. 8 .

 8 Figure. 8. Swivel coupler, steel sleeve, ground anchorage and lacing edge beam

  Table. 2. Short-term and long-term values for material resistance.

Acknowledgements

The authors would like to thank the Catholic Church of Créteil for their trust and their courage, which permitted to get into this ambitious and successful project. They also want to greet the engineers from T/E/S/S, which have developed this challenging project during 18th months. They also thank the firm Viry for the supervision of construction works, including the delicate erection stage.