Marc Leyral
email: marc.leyral@paris-lavillette.archi.fr

Quentin Chef

Sylvain Ebode

Pierre Guerold

Zina Wided

Mélanie Cherif

Victor Bento

Marius Bardy

Albin Bauduret

Hanane Breugnot

Marie Guidou

Morgan Merle

Lu Dannenmüller

Agathe Gao

Robin Mombazet

Mathilde Pelour

Sara Armengaud

Denisa Boukili

Martin Mihaes

Léo Procureur

Nathanaël Pauvarel

Benoit Thomas

Annabelle Royannais

José Pablo Camail

Caroline Salinas

Maxence Leger

Sarah Frisson

Francisco Garcia Klein

Zahra Cruz

Victoria Garcia Asadollahi

Aurélia Flores

Marius Anastasi

Louise Bagreaux

Ronan Bouchaudon

Razan Lacroix

Anne-Louise Daham

Luis De Massary

Lizarraga Gerardo

Ana Romero

Élodie Miranda

Jeanne Roblin

Cornelia Bugnet

Alice Huller

De Peufeilhoux Villatte

The Baya Nest pavilion project: braided pattern optimization for hanging shell structures by dynamic relaxation

Keywords: Hanging, shell, braiding, weaving, dynamic relaxation, grasshopper, optimization, teaching, pedagogy

published or not. The documents may come     The Baya Nest pavilion project: braided pattern optimization for hanging shell structures by dynamic relaxation

Introduction

The art of basketry goes back to the Neolithic period and is still used today. Wicker baskets, rattan furniture, lampshades made of woven blades, peach pots, grape harvest hoods, etc. Examples of everyday objects come to mind by the hundreds, but what about architecture?

The world of construction has remained quite insensitive to basketry and when it is used, it is mostly for filling purposes. Examples of structural wickerwork architecture are extremely rare. Yet habitats with braided structures exist by the thousands... in the animal world! This is the case of the nest of the Baya weaver, among others.

The objective of this study is to design a methodology for the construction of load-bearing braided shells and to apply it to the realization of human-scale pavilions suspended from the branches of trees in the park of the Butte du Chapeau Rouge in the 19 th district of Paris. Its scientific contribution is to provide an answer to a question that has been imposed on us (and about which we have not found any studies in the literature): knowing the field of the main constraints of a loaded shell, is there an ideal way to braid it and, if so, what is it?

Nests: from reciprocal frame to braiding 2.1. Reciprocal frame structures

When we think about how a bird's nest works, reciprocal frame is the first idea that comes to mind. This structural typology forms a surface composed of inclined beams supporting each other.

It is regularly used in Japan for roof structures that do not require central pillars. The term nexorade is sometimes used to define reciprocal frames in which each element, occasionally called nexor (from the Latin node or link), is considered infinitely rigid (structure's equilibrium by reciprocity of shear forces and not by taking into account the flexural flexibility of the elements, unlike a braid for example). The end of each beam rests on the next beam that supports it, forming a circularly closed assembly since the last beam is finally placed on the first one. The failure of a single beam leads to the failure of the whole structure.

However, it can be objected that if a strand is removed from a bird's nest, it does not collapse, nothing very significant would happen. In addition, some birds, such as the Baya weaver, bend their blades so that each one flexes many other blades over its entire length. This method of construction is still analogous to a reciprocal structure: each blade, by its bending, presses on the blades that it interlaces with, generating a frictional force on them, and vice versa. The introduction of bending into the reciprocal structure, each element mobilizing the many other elements that intersect it, however, leads us to a new structural typology as part of this larger group of reciprocal structures: woven objects on which is based the art of basketry.

Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial Structures

Inspiring the Next Generation 3

Braiding: the art of basketry

Basketry is both an art and a craft whose product is the making of braided objects with plant blades. The terms weaving and braiding are often used, incorrectly, as synonyms. Weaving is the art of regularly interlacing textile threads in an orderly and often orthogonal pattern to create a fabric. The term braiding comes from the braid, the rope, which was made by assembling at least three fibers by repeating a pattern of crossings. In basketry the material evolves between the moment when it is braided (still green and flexible) and the period when the produced object is used (the blades will then be dry and rigid).

Finally, it should be noted that the result of braiding is more likely to be a rigid surface than that of weaving, which is instead a very flexible surface. It therefore seems more accurate to describe as weaving a surface that can only resist traction (no rigidity) and as braiding a surface whose rigidity allows it to resist both traction and compression. For these reasons, and although the name of the Baya weaver obviously comes from the art of weaving, we will henceforth use the verb braid in our study.

Origins: Basketry appeared even before pottery and wood and metal working. It was used to create natural barriers to protect against animals (hedgelaying). The first traces discovered date back to the Neolithic period (negatives of braided molds on coil pottery) and the oldest-found basketry work have been exhumed in Fayum (Upper Egypt) and have been dated to more than 10,000 years ago.

Material's choice: With the help of Roger Hérisset [START_REF] Hérisset | La Vannerie en Bretagne : Nouvelle approche de l'ethnologie de la vannerie et contribution à l'étude des composantes culturelles[END_REF], we have chosen the materials of our study after a multi-criteria analysis, each criterion having a weighted importance (between 1 and 5).

1. Execution's easiness M1: from a practical point of view, braiding being a skill that requires years of specific learning process. Ours being basic, this criterion has a weight of 4. 2. Availability and provenance M2: Some basketry materials are difficult to find in quantity and at a reasonable cost for a project of this size. Following the traditions of French basket makers also allows to integrate the local origin of the material or not. M2 has a weight of 3. 3. The cost M3 = fyt / Cl : as we demonstrate below, the dimensioning criterion for our suspended shell project is the tensile strength, the quantity of material to be used being inversely proportional to the tensile yield strength fyt (MPa). Thus, maximizing the strength/cost ratio means minimizing the overall cost of the structure. This criterion has a weight of 5. 4. The durability M4 = E / (1 + kdef). We are looking for a stiff material (high Young's modulus E)

to limit the initial deformation, with a creep coefficient kdef as low as possible to limit the deformation under long-term loading. It is a question, therefore, of maximizing the M4 criterion. However, the Baya project is temporary: we chose for this criterion a coefficient of 2.

This analysis, conducted on white wicker, Buff wicker, split wicker, rattan, chestnut splinters, Provence cane, bamboo blades, mulberry, hazelnut blades, rye straw and palm leaf, resulted in the selection of three materials for the project: Buff wicker, split wicker, and rattan blades.

Wicker is obtained from willow stems, which can reach up to 3 m. According to Mellgren [START_REF] Mellgren | Vannerie au fil des saisons[END_REF], they are harvested in winter from willow stumps. There are four types of wicker. The newly harvested green wicker is not used for basketry. Wicker is raw dry when it is dried naturally by keeping the bundles upright. After drying, the bark can be removed from the stems. Either by immersing their feet in water and placing them in a high temperature room, allowing the sap to "circulate again" and the bark to detach (white wicker). Or by bathing the wicker in boiling water for three to five hours (Buff wicker). The wicker is then dried again. It will be left in water for 8 to 15 days before weaving in order to soften it.

It is also possible to split larger wicker stems into splints. The end of the strand is notched in three or four parts with a sickle, it is split along its entire length with a small wooden splitter. The ribs are finally obtained after passing through a marking gauge or an éclisseusse.

Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial Structures

Inspiring the Next Generation 4 Rattan is also very commonly used in basketry, especially for furniture making. Its production comes from the climbing, sometimes thorny, palms; it is generally native to tropical regions. Rattan stems are between 3 and 10 mm in diameter. The palms are cut at the base, then the trunks are cut into sticks. These are removed from the leaves and thorns and dried in the sun. Using a splitter, two types of material are woven from them: the strips, or ribs, of the outer region, and the round marrow of the inside, which is used for fine basketry. For sizing, the ribs pass through a levelling machine and the pith through a die. There are many different braiding techniques, the ones we have used are crocane and layered. Basketry applications can be found in primitive or vernacular habitat. In Ethiopia, in the Guge mountains, members of the Dorze tribe weave ogive-shaped huts with false banana palms on frames made of bamboo and stems of local timber. The structure is lightweight and can be lifted by three or four people to be moved. They are initially built very high, about 10 to 12 meters because of the presence of termites that destroy, by devouring them, the bases of the huts whose height then gradually decreases in time! This habitat has an average lifespan of 60 years and can last up to 80 years.

The Molo tribe from Lake Turkana in Kenya, use acacia wood, grass or dried wood fibers to build their spherical-shaped huts. The hut is composed of a structure made of acacia branches on which is braided a cover of palm leaves and dried wood fibers. Some of the huts are raised by pilings to prevent pests.

Sickle and splitter

Step 2: Split the wicker along its entire length using the splitter.

Step 1: Notch the wicker stem at its end Some examples are also to be mentioned in modern architecture. However, all of them, with few exceptions, have in common that braiding is only used as a filling element, it is not structural.

Design of the "BAYA" algorithm

Our goal is to transpose the basket maker's skill to an architectural work. Our project, Baya, is a suspended pavilion, but our thinking is broader: it must be applicable to any loaded surface, such as shells, as well as simpler elements like beams.

These elements constitute the entire structure of the work. The loads circulate through the entire thickness of the structural surface of the shell and subject it to tensile and compressive stress. Our problem is to study these constraints fields and to question the existence of an optimal braiding of the shell in relation with them. This study consists in observing the behavior of the stems, on a 30*30 cm specimen, and understanding the relations between the geometrical characteristics of the braid and its mechanical behavior when it is subjected to a given loads field in order to feed an algorithm, called BAYA, capable of giving the optimal braiding of any shell under various loading cases. Ergonomic and open-source, it will be available on the site www.construire-l-architecture.com.

The laws of optimal braiding

Methodology: our numerical simulation is conducted by dynamic relaxation, a method that allows us to solve static equilibrium problems by a fictitious dynamic calculation. Bouhaya [START_REF] Bouhaya | Thèse de doctorat -Optimisation structurelle des gridshells[END_REF] explains that the objective is to find the equilibrium position of a structure subjected to a set of loads by a pseudo-dynamic calculation : "it is an iterative method that describes the movement of the structure from the moment of loading to its equilibrium (…)" and, "according to [Barnes], "the basis of the method is to trace stepby-step for small time increments, Δt, the motion of each node of a structure (from an initial disturbed instant) until, due to artificial damping, the structure comes to rest in static equilibrium." A more complete pedagogical presentation of dynamic relaxation applied to elastic gridshells (ELASTICA algorithm) is given by Leyral et al. [START_REF] Leyral | Elastica project: dynamic relaxation for postformed elastic gridshells[END_REF] in another paper of IASS 2021.

In a braid constrained in the plane, we observe three types of forces: the forces due to the bending of the strands that one must bend to pass above and then below the strands of the other layer, the restoring forces related to the axial deformation of the strands (Hooke's law) and the friction forces at the knots. Knowing that we can simulate the latter by looking at the normal stress (by Hooke's law) in fictitious bars connecting the two directions at each node, only Hooke's law and the bending force are thus necessary to simulate the behavior of a braid. These are therefore the same forces that are found in the dynamic relaxation of elastic gridshells (whose own weight would be neglected).

From this observation, the choice was made to take over the core of the ELASTICA algorithm, by modifying it so as to be able to study the reaction of a braided specimen subjected to a field of principal stresses in compression and tension. The objective is to understand the influence of the different braiding parameters listed below.

The material (fixed setting): the experiment is conducted on two of the materials used in the Baya project: round wicker stems and thin rectangular rotten stems. They are modeled by their geometrical and mechanical characteristics: the dimensions of the cross-section of the strands, the modulus of elasticity E= 3000 MPa and the elastic limit stresses in tension and compression fyt = fyc = 25 MPa.

Input parameters: specimen geometry and external loading. The braid is a set of strands, braided in two different directions. In each direction, the strands are parallel to each other, and pass alternately above and below the strands of the other direction.

The specimen is subjected to forces that simulate the principal stress field in the shell (it represents two directions of orthogonal stresses at the level of the considered finite element, which has been oriented to only have normal stress components).

The simulation of the different principal stress fields therefore depends only on the two following parameters:

• Traction / Compression.

• The value Fσ1 and Fσ2 of the efforts (N/ml) respectively in direction 1 and

direction 2 (| Fσ1 | ≤ | Fσ2 |).
Once the material is fixed, the geometry is defined by four parameters:

• The braiding densities n1 and n2 of each layer (in number of strands per ml)

• The orientation α of the braiding with respect to the principal stress field (angle of the layer 1 with the orthogonal forces applied on the specimen). • The angle θ of braiding: angle between the two layers of braiding. After each simulation, several outputs are collected and combined for failure criteria comparison.

BAYA algorithm for optimal braiding of architectural shells

The BAYA algorithm determines the principal stresses for any shell under a given load. Thanks to the above optimal braiding laws, it is able to give the ideal braiding for the shell. Algorithm can be applied to any loaded surface, including basketry items such as baskets, we have chosen to illustrate its presentation on an isostatic beam on two supports loaded by a uniformly distributed force.

1. Geometry of the studied shell: The shell surface, the position of the support points and the loads are the input data of the BAYA algorithm. 2. Finite element mesh of the shell: The surface is discretized into small rectangular isosceles triangles, each representing a finite element of it. 3. Obtaining efforts: in the finite element edges by dynamic relaxation. 4. Obtaining the principal stresses: deduced from the forces by rotation of stresses' coordinate system on the largest support reaction of the rightangled isosceles triangle. 5. Braiding of the shell: by using the laws defined above. The predominant mode of failure of a suspended nest is due to the breaking of the strands. We lead the verifications in the elastic domain with a safety factor on the results of the calculations taking into account various parameters: variations in modulus of elasticity E (between the different blades, due to moisture and due to creep), accuracy of shape of shell, variations in loading, accuracy of computer model and assumptions, nature and significance of collapse and consequences of failure. Our global safety factor is 6.30 (4.50 x 1.40 security coefficient on ULS combination).

Experimental verifications 4.1. Comparing BAYA to Pratt

In the making, the braiding algorithm was tested on a 2D rectangular surface since the principal stresses of a uniformly loaded isostatic beam are familiar (see Figure 19a).

Further to this test, we decide to braid the result, in order to compare our "Baya beam" to a Pratt truss made out of stacked and connected stems. For identical sizes, loads and safety factors, Baya needs half the quantity of Pratt.

This will soon be verified by loading the two specimens up to failure.

Laws of optimal braiding by studying physical specimens

The numerical model idealizes reality, and several simplifying assumptions were made for the verifications. As the results could not be compared with the bibliography, since we did not find any equivalent studies, it was necessary to verify them by a similar experimental study.

The preliminary results of this physical experiment, which is still in progress, tend to confirm, with a satisfactory margin of accuracy, those of the numerical model.

Photoelasticimetry

Light is an electromagnetic wave whose electrical component propagates orthogonally to its magnetic component. This polarization can be modified by a polarizer placed downstream of the light source, which transforms the light wave into a linearly polarized wave. If we place a second polarizer, called an analyzer, after the first one, all the light coming out of it will be transmitted if their axes are parallel, and no light will come out of it if they are perpendicular.

Some materials are birefringent when subjected to stress: they are capable of polarizing light along the axes of the principal stresses within them. This is the principle of photoelasticimetry.

Plane Polariscope: a birefringent material is placed between a polarizer and an analyzer, we obtain an experimental device called a plane polariscope. It can be shown that the light intensity at the output of the experimental device is equal to

𝐼 = 𝑎. 𝑠𝑖𝑛 2 (2𝛼). 𝑠𝑖𝑛² [ 𝜋.𝑐.𝑑 𝜆 (𝜎 1 -𝜎 2 )] (1)
α is the angle between the principal stresses and the angle of the polarizer. When sin²(2α) = 0, the light intensity is equal to zero and a black band is observed: these are the isoclines. Along these isoclines, the stresses are parallel to the angle of the polarizer. To obtain the lines of the principal stresses, called isostatics, we thus draw the isoclines for different angles of the polarizer, place the directions of the principal stresses σ1 and σ2 on them and then connect them.

Figure 23: Isochrones for one specific load (A), isoclines drawn for various polarizer angles (B), merged drawings (C), isostatics (D), comparison with the principal stresses from the algorithm (E).

By photogrammetry on a physical shell

A suspended balloon is filled with water (pretension gives the shell compression capacity) and loaded with a mass. A picture analysis algorithm recognizes marker points; then Hooke's law converts the grid deformation into strengths.

. The Baya weaver

The Baya project, described below, is a colony of suspended pavilions whose skin, the very structure of the object, is a braid of long and fine elements. It echoes many habitats of the animal world, in particular bird nests. Among them it appeared to us that the one of the Baya weaver was an object particularly close to our intention.

The Baya weaver (Ploceus philippinus) is a small bird of the Ploceidae family, which lives in Southeast Asia and builds suspended nests in the shape of weighted down stockings (close enough to a drop of water). The etymology of its name comes from the Sanskrit word vaya meaning the one that weaves. Most of the colonies (50 to 100 nests in general) are in India, on coconut trees in the northeast and trees in the southwest.

The weaving of the suspended nest begins during its reproduction period, the period of rains associated with the monsoon, caused by an unidirectional exchange of warm air loaded with humidity moving from the ocean to the land. The nest serves to protect the bird, its eggs and young from predators and the climate. The location of the tree (overhanging the water to limit the entry of rodents), its species (thorny trees are preferred), the position of the nest on it (on the side protected from the wind), and its orientation are thus judiciously chosen. The nest, composed of 1.000 to 1.500 long strands of 20 to 60 cm, is braided during eighteen days for breeding purposes.

Illustration

Knots used Description

The bird ties the first either to a palm leaf making a hole in it or to a hanging branch, forked or not. If the branch is forked (the most common case), it weaves strands around each element formed by the Y. Thus, the nest and the branch are completely interdependent and move together. If the branch is straight, it must first make a small notch or use a defect in the branch to hold the first strand. Baya's weaver then makes a bundle, a ribbon in the shape of a circle that will serve as a perch to tie the other strands without having to fly in static around his nest.

Continuous addition of strands extends the wad downwards while increasing the thickness and length of the ribbon that serves as a support. It then braids two horn-shaped protrusions on either side and connects them at the ends, forming a ring around himself.

It braids around the ring to widen it and add an opening downwards. The base of the nest is then made up of a first rather thick ribbon forming a temporary scaffolding during the construction. The base of the ring will then divide the chamber into two equal halves.

Stage

Description

The bird, perched in the initial ring, adds, strand by strand, material to the back edge of the upper part of the ring to make a flange that will form the roof of the egg chamber. At the same time, the front edge is also enlarged forming a porch-shaped roof that will later become the roof of the antechamber. Two openings are made in the ring: one for the entrance tunnel and the other to access the egg chamber. The shape of the nest then evokes that of a helmet. The male then calls, through whistles, the females which inspect all the nests by making them pass strength tests, pushing on the walls with their beak and pulling on fibers to check the quality of the work. The female then chooses the best nest, the builder of this one becoming her companion. At this stage, the male has designed and structured the egg chamber.

Waiting for a female, the bird pads the exterior and interior surfaces increasing the thickness of the walls and strengthen the attachment point.

As soon as a female accepts a nest at the helmet stage (whether padded or not), the male spreads the wall of the egg chamber downwards and then braids the bottom (one day's work). It still pads the inside while the female brings a sparse lining of fibers and a few feathers that make up the floor of the egg chamber. The eggs can then be laid. They add drops of mud, clay or cowpat to the walls, perhaps in order to ballast the nest against the wind. During this time, and for two days, the male adds a long vertical tubular tunnel that will protect the entrance from rodents.

Study of the nest and comparison with the BAYA algorithm's results

The objective of this part of the study is to answer the following question: does the Baya weaver follow, when it makes its nest, the laws of optimal weaving that we have defined? To answer this question, we have first carried out a survey by laser surface scanner and by photogrammetry of the nest with the support of Ph.D. Professor François Guéna, Scientific Director of the MAP-MAACC laboratory and François Goussard and Delphine Brant, researchers at the National Museum of Natural History.

Thanks to the color saturation and the bump pattern obtained, we have built an image recognition algorithm able to isolate on the nest surface most of the strands and recognize their diameter. We deduced the main directions and the density of the strands on each area of the nest. The results seem to betray the ghost of the temporary scaffolding used by Baya weaver to build its nest from the ring stage. To confirm this, we scanned the nest with X-rays thanks to the help of Dr. J.-R. Blondeau, head of Pontl'Abbé hospital's radiology dept., and accessed to the interior of the nest's walls. This study enlights the static scheme of the weaver's nest, which functions as a "swing" with the two sides perpendicular to the entrance hole being much more stressed. The braiding of the nest, which follows broadly that of its temporary perch, is not only justified by the construction's method: it is also the most efficient structural scheme for the object. This is confirmed by the striking proximity between the ideal nest braid provided by the BAYA algorithm and the one actually made by the bird. This study proved of the relevance of the optimal braiding laws and of the BAYA algorithm, and helped to determine the final static scheme of our project.

Form finding: the drop of water and the weighted down stocking

Several authors compare Baya weaver's nest to the shape of the drop of water at the time of its detachment. Others, like Newton, prefer the comparison with a suspended and weighted down stocking. These two forms have however their specificities, for some similar to our object of study, for others departing from it. The suspended stocking, like the nest and our architectural project, is punctually loaded by vertical forces directed downwards. However, its surface is a weaving, i.e., unlike our braid, it cannot bear any tensile stress, explaining why the free volume inside is greatly reduced. The water drop has a synclastic and stretched surface, coming from a field of pressure forces, that differs from that of the nest. Thus, we carried out a comparative form finding study, looking for the shape minimizing the energy involved on the two systems using the BAYA algorithm. A spherical shell held at its summit vertex, modelized by mesh in taction only and without diagonals, submitted to a hydrostatic force field converges to the shape of a water drop. The distance of this shape to the Baya weaver's nest is 91.54%. But a very flexible spherical shell, without compression forces, forced to keep a constant volume at the bottom, and subjected to a vertical force field, converges to the shape of a weighted stocking. The proximity to the nest is 97.02%, which is therefore better, ruling in favor of Newton. It is only slightly less good than the shape which best describes the nest, obtained by reactivating the shell's possibility to bear compressive forces (97.34%). This study helps us to decide on the ideal shape of our project. Before braiding the final pavilion, we decided to build a 1:2 prototype. The deformation of the prototype was critical but fitted the algorithm prediction. The observation of the loaded surface guided us to reinforce the density of the compressed area in order to avoid surface buckling.

Conclusions

The outcomes achieved so far are:

• Obtaining the optimal braiding laws • Algorithm for braiding the shells Outcomes expected during next steps:

• Construction of the project Baya.

• Going on experimental verifications.

Figure 2 :

 2 Figure 2: Examples of reciprocal frames: model by Charlotte Gambotti, Leonardo da Vinci's military bridge, Qingming rainbow bridge, Stonemason Museum (Yasufumi Kijima arch., Yoishi Kan ing.) [credits: Kelkaku-Inc.], Exhibition Hall of the Bunraku Puppet Theater in Seiwa (Kazuhiro Ishii arch.) [credits: Kazuhiro Ishii].

Figure 1 :

 1 Figure 1: Bird's nest and the project Uchronia by Arne Quinze.

Figure 3

 3 Figure 3: Anthropomorphic example of reciprocal frame by Olivier Prévost [7].

Figure 4 :

 4 Figure 4: Baya weaver weaving his nest [DURAIRAJ].

Figure 5 :

 5 Figure 5: Basket maker's tools: pruning shears, punches, pliers.

Figure 6

 6 Figure 6: hedgelaying [photo: Vincent Jones].

Figure 7 :

 7 Figure 7: Dry raw, white, and Buff wicker. Wicker splitting and an éclisseusse [Wicker Paradise / Ji-Elle].

Figure 8 :

 8 Figure 8: From left to right: climbing palm [Dinesh Valke], braided rattan blades, braided rattan pith, and splitter for splitting the rattan into blade and pith.

Figure 9 :

 9 Figure 9: From left to right: crocane, layered, loop, torch, knotted and spiral wickerwork.

Figure 10 :

 10 Figure 10: Other examples of weaving in nature: social republicans [Diego Delso, delso.photo, License CC-BY-SA], weaver ants [Frawsy Eklablog & Bernard Dupont] and caciques [Tanguy Deville].

Figure 11 :

 11 Figure 11: Dorze huts, Chencha village, under construction [Kevin Smith], and Molo huts, Turkana Lake [Eric Lafforgue, Bjorn Svensson / Alamy Banque D'Images / Richardandmarthashaw.blogspot.com].

Figure 12 :

 12 Figure 12: Ephemeral Beaubourg project built by Nikolay Polissky in the Nikola-Levinet park (Kaluga region) and bamboo tunnel by Wang Wen Chih [Cave Urban]. Weaver's Nest project by Porky Hefer of Animal Farm (South African materials and manufacturing techniques on steel frames) and another project from the same design office [Deezen & Animal Farm].

Figure 13 :

 13 Figure 13: Diagram of a braided specimen subjected to a given stress field.

Figure 15 :

 15 Figure 15: Input parameters on the geometry of a braided specimen.

  Figure 14: Fictitious vertical bar simulating friction forces at a knot.

2 ⁄

 2 Maximum axial force differential between who consecutive strands ∆𝑁 𝑚𝑎𝑥 N V3 -Knots notranslation 𝛼′. 𝑉 ≤ ∆𝑁 𝑚𝑎𝑥 with 𝛼′ the friction coefficient Maximum strength in fictitious bars 𝑉 N V4 -Knots no-rotation 𝐸𝐼.sin()/𝑙 𝑚𝑒𝑠ℎ ≤ 𝛼 ′ . 𝑉. 𝑅 with  = angle between 2 strands; R = radius of the contact surface Maximum displacement 𝑑 mm V5 -Global buckling Kinetic energy has converged + specimen is still planar The study of the specimen teaches us the following principles: • As expected, the yield ratio Tmax / fyt is a linear function of the tensile strength Fσ1 • Local buckling (Cmax compared to Euler's force) is not an issue with regular n1 and n2 densities • Global buckling (non-planar specimen) depends mainly on compressive strength Fσ2 • Friction forces raise with Fσ1 and fall with Fσ2 ; which explains why instability failure appears with lower compression values when tension is not high enough (cf. second chart) • High values of Fσ2 lead to knots translation or rotation, even negative friction (decompression) • Positive friction force possesses a physical limit (depends on the diameter of strands) • Changing the braiding angle θ has little effect if input forces are almost parallel to the strands (α close to zero); which validates the idea of braiding along the principal stresses.

Figure 16 -

 16 Figure 16 -Test's results and criteria charts examples: V1, V2, V3, V4 as Fσ1 functions; each line corresponding to a different Fσ2 value).

Figure 17 :

 17 Figure 17: Diagram of the beam to be braided.

Figure 18 :

 18 Figure 18: Principal stresses on a finite element.

8 Figure 19a :

 819a Figure 19a: Meshing, strains and principal stresses, and optimal braiding by the BAYA algorithm.

Figure 20 :

 20 Figure 20: Experimental device.

Figure19b: 4 . 3 .

 43 Figure19b: Braided beam according to Baya laws (in progress)

Figure 21 : 1 .

 211 Figure 21: Polarizer principle

  Figure 27: Davis' hypothesis [3] of its orientation to the wind with respect to the risk of egg fall.

Figure 26 :

 26 Figure 26: General architecture of a Baya weaver's nest according to Quader [8] at the helmet stadium and at the final stage.

Figure 30 :

 30 Figure 30: Sections of the RX scanner, reconstructed 3D model compared with predictions of BAYA algorithm.

Figure

  Figure 31: More stressed zones of the nest, as predicted by the BAYA algorithm.

Figure 29 :

 29 Figure 29: Preferred directions and density of braiding on the surface of the nest.

Figure 28 :

 28 Figure 28: Nest modeling by laser surface scanner and by photogrammetry.

Figure 33 :

 33 Figure 33: Form finding studies on a model of water drop and on a model of weighted stocking, results are compared with a 1:2 physical model.5.3. Baya Project: application to a suspended pavilion at human scale5.3.1. ContextThe park of the Butte du Chapeau Rouge, developed from 1938 by Léon and Jean Azéma, is in the 19 th district of Paris. After various tests comparing numerical and model studies, we opted, following the example of the colonies of nests of Baya weavers, for the realization of several pavilions implanted in the chemin de ronde, a peripheral walk below the esplanade, lined with lime trees and plane trees.

Figure 34 :

 34 Figure 34: BAYA algorithm results on the project, displacements are compared with a 1:2 scale model.

Figure 32 Figure 35 :

 3235 Figure 32: ideal shape from a form finding led on a Baya weaver's nest.

Table 1 :

 1 Input parameters and test steps.

	Input parameters	Symbol	Unit	Range of variation	Step
	Value of force 1	𝐹 𝜎1	N	150 -10000 N	1000 N
	Value of force 2	𝐹 𝜎2	N	100 -Min(Fσ1 ; 4000N)	500 N
	Braiding density in direction 1	𝒏 𝟏	u/ml	10	n/a
	Braiding density in direction 2	𝒏 𝟐	u/ml	10	n/a
	Braiding angle	𝛼	Degrees (°)	0 -45°	5°
	Orientation of the braid in the specimen	𝜃	Degrees (°)	0 -45°	5°

Table 2 :

 2 Outputs and criteria controls. Yield strength 𝑇 𝑚𝑎𝑥 ≤ 𝑓 𝑦𝑡 𝑎𝑛𝑑 𝐶 𝑚𝑎𝑥 ≤ 𝑓 𝑦𝑐 Max compressive strength in strands 𝐶 𝑚𝑎𝑥 N V2 -Buckling 𝐶 𝑚𝑎𝑥 ≤ 𝜋²𝐸𝐼 𝑙 𝑚𝑒𝑠ℎ

	Output	Symbol Unit	Controls	Criteria
	Max tensile stength in strands	𝑇 𝑚𝑎𝑥	N	V1 -

Table 3 :

 3 Stages of nest construction according to Crook[START_REF] Crook | Field experiments on the nest construction and repair behavior of certain weaver birds[END_REF].

	Stage

Wad stage (2 days) Advanced wad stage Ring stage (5 days)

Acknowledgements

The Paris City Hall and the 19 th district City Hall, in particular Sophie Godard, Cécile Becker, Eric de Grootte, Philippe Clayette and Sylvie Roudier, for their support, as well as the inhabitants of the neighbourhood, Mr. Mayor François Dagnaud and Mrs. First Deputy Halima Jemni.

Valéry Schollaert, ornithologist, Roger Hérisset, basketry expert and doctor in ethnology at the UBO, and Prof. Ph.D. François Guéna, Scientific Director of the MAP-MAACC laboratory (ENSAPLV). Pierre Jehel and the Paris Centrale-Supélec school. Caroline Simmenauer for proofreading.

The National Museum of Natural History, in particular Delphine Brabant and Florent Goussard, researchers in paleontology and comparative anatomy, and Dr. Jean-Robert Blondeau and the radiology department of the Pont-l'Abbé hospital for their help in modeling the nest of the Baya weaver.

The ENSA of Paris-la-Villette, Caroline Lecourtois, director, Vincentella De Comarmond, deputy director, Anne d'Orazio, President of the Board, Marc Fayolle De Mans, Jacques Bergna and Alain Raynaud (model workshop), Philippe Agricole and Philippe Bourdier for logistics.


Marc Leyral
email: marc.leyral@paris-lavillette.archi.fr

Sylvain Ebode

Clément Pierre Guerold

Clément Berthou

Geoffrey Louison

Mohamed Zitouni

Anastasia Komisarova

Jose Francisco Landa

Armand Passemard

Anabel Ginesta

Mariana Cyrino

Peralva Dias

Marta Anna Mleczkowska

Léa Lallemand

Haifa Ltaïef

Miguel Madrid

Lea Carresi

Beatriz Maldonado

Gaspard Chaine

Thibaut Morosoff

Vicente Benito Sanchis

Irene Eseverri

Marta Delgado Paez

Alejandro Mendez

Jean-Baptiste Mallard

Abderraouf Kaoula

Ana Karen Pimienta

Wided Zina

Elastica project: dynamic relaxation for post-formed elastic gridshells

Keywords: gridshell, timber, dynamic relaxation, grasshopper, form finding, teaching, pedagogy

come

Introduction

Elastic gridshells have come a long way from the primitive hut described by Eugène Viollet-le-Duc in its Histoire de l'habitation humaine and the vernacular pygmies lobembes to the post-formed gridshell of the Manheim Multihalle of Frei Otto (arch.) and Happold and Liddell (ing.). Yet this millennial typology remains anecdotal, except numerous educational pavilions and private self-building installations, in modern architecture. The Multihalle in Mannheim, built in 1975, remains almost the only large-scale architectural object meeting a defined program and need.

Why such reluctance? The scientific literature, which is becoming increasingly rich on this subject, emphasizes their capacity to overcome large spans with less material. The material used, often wood, is part of the current problem of limiting anthropogenic carbon emissions. If it is not generalized, the use of elastic gridshells should be much more prominent in the covers of large projects.

Our goal is to propose solutions on to this issue. The main problem we identifiedapart from the issues of acoustics, thermal insulation and standardization of the finishing work which would merit their own studyis the complexity for the architect to grasp the form of a post-formed elastic gridshell.

As for funicular domes, elastic gridshells take their shape in a natural way, according to the laws of physics: a rod of a given length takes one and only one natural shape when bent. This does not restrict architectural freedom: the assembly of elements of different lengths in various positions makes it possible to create free and varied shapes. However, the natural form is unknown and it must be found: the process that leads to it, the form-finding, uses complex tools such as dynamic relaxation that the architect does not possess (with a few exceptions). Some tools, such as Kangaroo, already allow an easy simulation of active bending but operate in a "black box": the detail of the performed calculation is not accessible to the designer.

To overcome this difficulty in being able to know or predict the real shape of the work, we wanted to popularize the current scientific knowledge on this typology and to gather it in a simple and ergonomic design tool: an algorithm called ELASTICA. This tool also allows anyone to access the detailed data of the calculation and to conduct a complete study of dimensioning and verification of the structure. Finally, it has been applied to the construction of a post-formed elastic gridshell at human scale called "Elastica" on the belvedere of the Parc de la Butte du Chapeau in the 19th district of Paris.

State of the art 2.1. What is an elastic gridshell?

In architecture, a shell is a continuous thin structure with a curved surface. Its rigidity is related to its curvature (shape resistance). Thus, a gridshell is a structural lattice of bars forming a curved surface. Labbé [START_REF] Labbé | Quoi de neuf / le gridshell (des jours meilleurs) ? Archistorm[END_REF] classifies gridshells into two main groups:

-"(...) those with pre-calculated members, both in their curvature and in their geometrical resolution but also in their "inactive-bending" fixings, -and those known as "active-bending" which start from an initially flat grid, which connections are not fixed until after their assembling, once the structure is established in its architectural form".

Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial Structures

Inspiring the Next Generation 3

The first category works in compression and is not the subject of this study. The elements in the second one, called elastic gridshells, are working in flexion and compression and have two main characteristics:

• They are in active bending; the shape is given by the bending of straight elements maintained fixed. This condition is necessary to qualify a gridshell as elastic. • They are post-formed which means that the grid is manufactured flat, not braced. The thin and hinged elements form a deformable unit that is then flexed during the erection. This condition is not necessary to be part of elastic gridshells, however our study will be placed in this framework.

The natural shape of an elastic gridshell depends on the initial grid and the displacements imposed on its support points. Let us take the simplest of them as an example: a simple flexible rod on the ends of which one pushes laterally. Initially the rod is in compression. Very slender, its equilibrium in compression by shortening quickly gives way to an unstable equilibrium in flexion: this is buckling. This can be generalized by describing a post-formed elastic gridshell as the post-buckling shape of a flat grid subjected to imposed displacements of its supports.

Once the ends of the bars are in their final position, the bent gridshell, which is by nature very deformable, must be stabilized and rigidified by adding bracing to limit the deformation of the mesh and possibly by adding shear blocks. The final grid is very rigid and can cover a large span without intermediate supports, and this with very little material.

The design of an elastic gridshell must consider many constraints: material properties and section, types of connections between elements, initial grid layout, support conditions, deployment, use and context, etc. It is therefore not the result of a simple artistic reflection; on the contrary, it is the result, through modeling and/or calculation, of a long process of research into the natural shape of the desired shell.

A typology very present in vernacular and primitive architecture

According to the anthropologist Reclus [START_REF] Reclus | Les ancêtres. -Histoire ancienne[END_REF], who studied different ethnic groups, vernacular architectures, by using local materials, are a singular continuity of the environment in which they are established. The flexible and resistant natural materials that constitute the main resources of certain communities explain why the use of elastic gridshells took place really early in history: they could date back to the Neolithic period.

Vitruvius, in his treatise De architectura, already foreshadowed the active bending potential of certain materials: "The elm tree and the ash contain much water and but little air and fire, with a moderate portion of earth. They are therefore pliant, and being so full of water, and from want of stiffness, soon bend under a superincumbent weight."

The way of life of the pygmies of the great equatorial forest (Congo and Central African Republic) has remained traditional. Being a nomadic civilization, their huts, the lobembes, built by single women or men, are temporary habitats. The mongulu is a more solid variant for a life period of two weeks to a month. They are domes made of latticework of thin branches bent and anchored in the ground. Leaves are used to form a primitive form of waterproofing barrier. Elastic gridshells are therefore used in primitive and vernacular architecture. One could deduce that it is therefore a very accessible typology. However, its use in contemporary architecture is rare and the studies that allowed the understanding of this typology only began in the 1970s.

A typology rarely used in contemporary architecture

In 1897 the Russian engineer Vladimir Shukhov built a production workshop for a steel company in Vyksa. It is the first double-curved grid structure working in compression. Contrary to this example, an elastic gridshell is built flat, before being set up by bending it.

It was not until 1962 that this typology was highlighted by the work of Frei Otto who, using a study he had been carrying out since the late 1950s on lightweight shells from suspended net models, built a first trial model of an elastic gridshell during a visit to the University of Berkeley (California). This structure is designed from a square grid of 52 m² round steel rods anchored to the ground at its four corners.

Later in the same year, he built a wooden trial gridshell at the German Building Exhibition in Deubau, Essen. Developed with Bernd Friedrich Romberg using suspended models, this 198 m² structure has a super-elliptical plan (squircle), sort of intersection between a 15 m × 15 m square and a circle in plan.

Figure 6: From left to right: gridshell in Berkeley, trial gridshell in Essen and its model [from [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF] and [START_REF] Roland | The Work of Frei Otto[END_REF]].

Its maximum diagonal span is 16.82 m, with a center height of 4.85 m. Two orthogonal layers (lattice is composed with doubled laths in one direction and single laths in the other) of 60 mm x 13 mm pine elements are assembled to the floor by bolting at the knots, forming a square grid with a mesh size of 482 mm per side. The grid, composed of 1,140 linear metres of laths, was then lifted using a mobile crane and wooden stilts on the perimeter before being fixed to a rim beam driven into the ground. However, the first architectural project answering to a program, and one of the few of this scale, is undoubtedly the Multihalle in the Herzogenried Park in Mannheim (Germany), built in 1975 for the Bundesgatenshau. The winning architects of the competition, Carlfried Mutschler, Winfried Langner, and Heinz Eckebrecht, encountered difficulties in developing their idea of a free-form, airy and light structure: their proposal for large parasols suspended by helium balloons was rejected by the authorities. They then asked Frei Otto to help them, who became their engineering consultant.

The project was designed using the suspended net method, to which we will come back later, then numerically calculated and tested. The grid built on the ground is composed of two interlaced orthogonal networks, each composed by a double layer of laths 55 mm wide, forming a square mesh of 500 mm side. The knots are held by initially loose bolts to allow their rotation during erection, which was carried out using height-adjustable scaffold towers. The curved grid, still flexible at the time, is then blocked at the ends and braced to stiffen it. The western hemlock timber was shaped green and not dried because the flexibility of the wood increases with its humidity level.

This project will make the concept of elastic gridshells popular among architects and engineers around the world. However, the complexity of its constructive approach, highlighted by the work published by Happold and Liddell [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF], probably explains why similar projects are exceptionally rare to this day.

ELASTICA algorithm design method

Study of bending behavior

Designing an elastic gridshell requires a good understanding of bending. We know from the work of Galileo, later corrected by Huygens, Leibniz and Bernoulli, that the natural shape of a chain suspended in pure traction is a hyperbolic cosine. Similarly, to understand an elastic gridshell, it is fundamental to question the natural shape of the flexible rod in pure flexion.

In the framework of classical beam theory, which is based on the Euler-Bernoulli hypotheses (plane sections remain plane and normal to the axis when it deforms), bending moments induce deflection and not the opposite which implies that the supports, including the sliding supports, remain immobile on the drawing. The deformation thus shows an elongation of the neutral axis, whereas it is supposed to remain constant in length: this theory is therefore valid only for small deflections and rotations, where this inconsistency is considered insignificant. This theoretical framework, which is the one in which engineers and architects usually evolve, is sufficient for most structures.

In fact, when the initially flat grid is shaped by the bending resulting from its buckling, the hypothesis of small displacements and fixed supports is fundamentally questioned since it is precisely the displacement of the mesh supports that will give the shape to the gridshell.

Let us take the case of flexible rod, with quadratic moment I and modulus of elasticity E, on two sliding supports, subjected to a following force (a load on the support that continues to act at a point even if it moves).

Where ω² = F/EI. A and B can be determined by the boundary conditions (no deflection at the ends):

𝑦(0) = 0 → 𝐴 = 0 𝑎𝑛𝑑 𝑦(𝐿) = 0 → 𝐵 * 𝑠𝑖𝑛(𝜔𝐿) = 0 (2)
As long as ωL < π, the only solution is A = 0 and B = 0: this is the stable compression equilibrium (shortening of the rod). When ωL = π, the equation is verified for every value of B and the system becomes unstable: the rod buckles as soon as the force exceeds the Euler's critical load Fcr = π²EI/L. Once the rod's buckling is initiated, the hypothesis of small deflections is no longer valid. The curvature can no longer be written d²y/dx² but is still by definition the inverse of the radius of curvature (1/R):

𝐸𝐼 𝑅 + 𝐹 * 𝑦(𝑥) = 0 (3)
The EI/R moment at any point, and so the curvature 1/R, are thus proportional to the distance from the point to the axis y = 0, the guiding line: this is the geometrical definition of singular curve called elastica.

Watzky and Bueno [START_REF] Watzky | L'elastica revisitée sous l'angle des efforts[END_REF] explained that Jacques Bernoulli questioned in 1691 the form taken by a vertical elastic lamina embedded at the bottom end and pulled downwards on the other end. The resolution of this simple problem is so complicated that Huygens and Leibniz gave up but Bernoulli "obtains the equations of the rectangle elastica [in 1692], then those of the general case in 1694." Fifty years later, Euler [START_REF] Euler | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (Additamenta 1 De curvis elasticis & 2)[END_REF] characterizes its various forms. More than a century later, the values, tabulated, of a closed form of the solutions are approached and Born measures them precisely from photographs of bent blades in 1906. He wrote in his memoirs on this subject: "I felt for the first time the pleasure of finding a theory in agreement with the measurementsone of the most pleasant experiences I know." In line with this reasoning, we filmed the buckling of a flexible rod to digitally obtain the kinematics of the curve with an image recognition algorithm, measured the radius of curvature at any point and deduced the bending moment. The experimental results, which converge towards elastica, have been confronted with numerical calculations made with the ELASTICA algorithm which confirmed them.

Form finding studies

Form finding can be carried out either experimentally, by means of hanging chain nets or active bending models, or numerically, for example through dynamic relaxation. The work of Frei Otto, ARUP and Happold & Liddell, particularly on the gridshells of Essen and Mannheim, has been tested by all three methods and provides valuable data for assessing the relevance of each method. [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF] and [START_REF] Roland | The Work of Frei Otto[END_REF]].

The shape of a hanging chain (hyperbolic cosine) is determined only by its axial stiffness and a flexible rod (elastica) is determined both by its axial stiffness and by its bending stiffness. To claim that one is close to the other is therefore equivalent to saying that the bending stiffness of the flexible rod is negligible compared to its axial stiffness. This is what Douthe [START_REF] Douthe | Thesis -Étude des structures élancées précontraintes en matériaux composites : application à la conception des gridshells[END_REF] wanted to verify, arguing that this hypothesis was based on the fact that "the stiffness of the shell comes from its shape and that this shape has been specially studied to take up the dead weight loads by membrane stress, i.e. only by axial forces in the gridshell elements. However, (...) the material resists well to axial forces (much better at least than to bending forces), the dimensions of the gridshell elements can therefore be reduced and, consequently, the total weight of the structure as well. This reduction in cross-sections is accompanied by a decrease in inertia so that the bending stiffness of the elements is low, which legitimizes retrospectively the initial hypothesis and the form finding using a net model."

Nevertheless, as the inertia of the grid is not zero, Douthe studies the differences between the funicular and the elastica shape according to the attack angle α at the base and the loading rate p. He carries out this study on a simple beam, a rectangular grid and a free-form grid. He concludes that the shape of the gridshell is almost funicular if the angle of attack α is less than 65° (optimum at 57.5°), which corresponds to a pL 3 /EI ratio below 65, confirming a posteriori the modelling of Mannheim by Frei Otto. Like Happold and Liddel [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF], we can therefore conclude that "a funicular shape is an advantage but is not essential."

About the active bending model. In 1973, as Happold [START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF] reminded us, "there was no previous engineering experience in this field." To overcome this, his team first studied a simpler example by loading a PMMA model at the 1/16 th scale of the trial gridshell in Essen and comparing the results with the data collected by the Warmbronn Workshop on the actual project. The tests were conducted with pinned or rigidly glued nodes and with or without bracing. The team found, and retained for the Mannheim project, that the addition of bracing on the diagonals of the lattice reduced deflection and increased the maximum nodal load causing buckling of the shell, but that the collapse was more sudden. An active bending model of the Multihalle was then fabricated in PMMA at 1/60 th scale and tested in the same way. The buckling collapse load of the model was measured at 2.8 kg/m² without bracing and at 12.5 kg/m² with bracing. Happold and Liddell demonstrate that an extrapolation of the critical load is possible from a model to a real project by multiplying it by the ratio of EIxx / aS 3 of the project and the model (EIxx is the out-of-plane bending stiffness, a the spacing of laths and S the gridshell span).

From these studies and our experiences on the essential question of the extrapolation of the results from the model to the real project, we conclude that:

• The shape of a funicular and the shape of an elastic gridshell can be transposed from the model to the real project, regardless of the stiffness, the section and length of the material used. • The buckling force is transposable but subject to several measurement biases.

• The shear and node stiffness are difficult to transpose, and this may reduce the relevance of the results of the previous point.

Figure 14: Scalability tests on the section and stiffness of the material, and on the length of the element.

The numerical method: discretization of the Chebyshev lattice surface

To overcome the inaccuracies of a form finding with a model, a numerical method is generally necessary. The shape resulting from the initial phase of intention, which we will now call "architect's shape", is not the real shape of the project, which must respect the rules of physics (especially bending). The form finding consists in determining, from the architect's shape, what the real shape is going to be.

The first step consists in dividing the architect's shape into a network of two layers of bars forming equilateral meshes (a necessary condition for flat fabrication). The division of any surface into equilateral parallelograms is called a Chebyshev lattice, named after the mathematician who, in 1878, having a rather modest salary, accepted a contract to optimize the cutting of military uniforms. The human body is indeed a complex shape that it is complicated to cover with a flat fabric without folds. Until the end of the 19 th century, clothes were made by eye and therefore not fitted. Custom tailoring was a time-consuming and expensive art. Chebyshev [START_REF] Chebyshev | Sur la coupe des vêtements[END_REF] therefore devised a method to create a piece of clothing adapted to the human anatomy, in large quantities, quickly and at low cost.

The problem formulated by Ghys [START_REF] Ghys | Sur la coupe des vêtements. Variation autour d'un thème de Tchebychev[END_REF] highlights the link with gridshells: "A flattened fabric is formed by two networks of interwoven straight threads (...) which form small squares. (…) The initial small squares can become deformed: their sides do not change in length but the angle between the threads is no longer necessarily straight." Thus, the change of angle between the threads allows them to envelop

Lattice on a Euclidean plan

Lattice on an architect's shape

Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial Structures

Inspiring the Next Generation 9 a double-curved surface without any fold. To achieve a Chebyshev lattice on any surface, one can go for a dynamic relaxation method, or by the geometric method, called "the compass", used by Frei Otto.

Figure 16: Compass method and application by the ELASTICA algorithm.

Dynamic relaxation

This form finding method, even though simple since it is based on the laws of Newtonian physics, is iterative and heavy by its quantity of calculations: its development had to wait until the end of the 20 th century and computer-assisted numerical modeling. According to Bouhaya [START_REF] Bouhaya | Thesis -Optimisation structurelle des gridshells[END_REF], "Southwell has developed a relaxation method for the solution of partial differential equations by finite difference approximations. The concept of the method itself was much earlier by Rayleigh (…). The development of this method is attributed to Otter in 1964 for the study of pre-stressed concrete pressure vessels. One year later, the dynamic relaxation method was developed by Day for the same type of problems". It was then further developed and applied, among others, to tensioned structures in the 1970s and to inflatable structures in 1977 (Barnes), as well as to shells (Otter 1964), prestressed cables (Barnes 1975), folds and creases (Zhang 1989), and tensioned and inflatable structures (Barnes 1977 and1999 [2]).

Dynamic relaxation allows to solve static equilibrium problems by a fictitious dynamic calculation. It is valid for large deflections. Bouhaya [START_REF] Bouhaya | Thesis -Optimisation structurelle des gridshells[END_REF] explains "it is an iterative method that describes the movement of the structure from the moment of loading to its equilibrium (…)"

Theory from an analytical point of view: According to Barnes [START_REF] Barnes | Form finding and analysis of tension structures by dynamic relaxation[END_REF], "the basis of the method is to trace step-by-step for small time increments, Δt, the motion of each node of a structure (from an initial disturbed instant) until, due to artificial damping, the structure comes to rest in static equilibrium."

Indeed, the architect's shape is not the natural form of the project, so it is not at rest. It wants to move to its natural position: it needs to relax. Thus, the fictitious motion of a structure modeled by a discrete mesh of bars (for a gridshell, this comes from a Chebyshev lattice, it therefore represents the real physical elements of the structure), at the intersection of which are located the nodes subjected to forces, must be calculated. Indeed, according to Newton's second law ∑ 𝐹 ⃗ = 𝑚 * 𝑎 ⃗, if the forces at each node do not balance, then the nodes (to which we attribute a mass, real or fictitious) experience a fictitious acceleration 𝑎 ⃗ = ∑ 𝐹 ⃗ 𝑚 ⁄ , and therefore move at a velocity that varies with time. This lets us calculate at each iteration, the position of each node at the next instant.

Steps 1 and 2: Choice of two guidelines prefiguring the two main directions of the gridshell, subdivision of them with a compass.

Step 3: First intersection.

Step 5: Next Generations and obtaining the Chebyshev lattice.

Step 4: First generation of points.

L ΔL F L L L 1) Initial state: flat grid.
2) Two nodes at opposite angles are suspended, causing a fictitious imbalance.

3) As the fictitious forces do not balance at each node, they move according to Newton's Second Law, leading to new nodes' position and a different distribution of forces. Figure 18: trial gridshell by Rombouts [START_REF] Rombouts | Thesis -Optimal design of Gridshells[END_REF]: mesh of the architect's form, form finding after RD and built project.

Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial Structures

Inspiring the Next Generation 10

In the case of a gridshell, there are (at least) three forces acting at the nodes: the nodal dead weight (𝐹 ⃗ = 𝑚 * 𝑔 ⃗), the force induced by the bending of the elements as described by Barnes, and the Hooke force in each element, proportional to the stiffness and deformation (𝐹 ⃗ = 𝐸𝑆 * ∆𝐿 ⃗⃗⃗⃗⃗ ), which ensures the equilibrium of each node.

To calculate the value of the bending forces, let us start from the bending moment M which causes the bending of the elements. Since the moment is the product of the force by its lever arm at the point considered, We then obtain the positions of each node at time t + dt. The operation is repeated until an equilibrium of forces is reached at each node: the structure is then at its natural position. This can only be done by adding a damping in the system, which is explained below.

General theory from an energetic point of view: at the initial instant, the deviation between the initial architect's shape and the equilibrium shape being maximum, the potential energy (which we define by simplification as being related to the forces resulting from a deviation between the position at instant t and the equilibrium position) of the system is maximum, the initial nodal velocities being zero, the kinetic energy ∑ 1 2 * 𝑚 * 𝑣² 𝑛𝑜𝑑𝑒𝑠 of the system is as well. The relaxation then causes the nodes to move. At convergence, equilibrium position is obtained when the potential energy is zero (the forces are in equilibrium) and the kinetic energy is zero (the nodes no longer move: their velocities is zero).

In the intermediate stages, an energy transfer takes place between potential and kinetic energies. When the system passes through its equilibrium position, the forces balance and the potential energy becomes zero. But nodal velocities, which is then maximum (and therefore the kinetic energy as well), causes a continuation of the movement in the opposite direction: the system oscillates. On the other hand, when the deformation reaches a maximum, the velocities and thus the kinetic energy are zero and the potential energy regains a local maximum. This fictitious oscillation around the equilibrium position would be endless without the addition of damping, i.e. a way to fully dissipate the total energy of the system.

Viscous damping: it is the most intuitive method, because it is based on physical notions. Indeed, in the real world, a ball rolling in a valley is slowed down by the friction of the air and ends up stopping. In the same way, the roller coaster summits must be descending. In dynamic relaxation, viscous damping is modeled by adding to each node a force proportional to a damping coefficient c and the opposite of the nodal velocities: 𝐹 𝑎 ⃗⃗⃗⃗ = -𝑐 * 𝑣 ⃗.

The coefficient c is at the choice of the user because it does not change the nature of the equilibrium, only the number of oscillations required before it. However, a too high damping coefficient may cause the system to diverge. In the general case, the damping of the system is done in a pseudo periodic regime (sufficiently low damping, therefore the convergence is slow).

Kinetic damping: More abstract, this method does not require setting new parameters and often allows for faster convergence. Barnes [START_REF] Barnes | Form finding and analysis of tension structures by dynamic relaxation[END_REF] explains that kinetic damping "is an artificial damping (...). In this procedure the undamped motion of the structure is traced and when a local peak in the total kinetic energy of the system is detected, all velocity components are set to zero. The process is then restarted from the current geometry and repeated through further (generally decreasing) peaks until the energy of all modes of vibration has been dissipated and static equilibrium is achieved."

The ELASTICA algorithm

This algorithm is the concrete application of the above, usable for any type of elastic gridshell, and available in open source on the website www.construire-l-architecture.com. We have chosen a discretization by the compass method and kinetic damping.

Figure 24: Chronology of form finding stages and stability control using dynamic relaxation by Rombouts [START_REF] Rombouts | Thesis -Optimal design of Gridshells[END_REF], and extract of the ELASTICA algorithm.

Choice of the gridshell's material

Based on the Douthe method [START_REF] Douthe | Thesis -Étude des structures élancées précontraintes en matériaux composites : application à la conception des gridshells[END_REF], we have chosen the material after a weighted multi-criteria analysis.

1) The performance criterion M1 = σy / E is to maximize in order to have the material with the best ratio strength (high elastic limit σy)flexibility (low E) in order to achieve relatively small bending radii.

2) The criterion M2 = E. Counter-intuitively, according to the above, it should be maximized because too much flexibility is harmful for a gridshell in the service phase. Indeed, stress variations under loads are not decisive for the structural analysis of the gridshell (higher inertia in service phase and load moment generally opposite to the bending moment) contrary to the phenomenon of surface buckling. The more flexible a structure is, the more likely it is to buckle. However, it is easy to stiffen the gridshell by adding bracing and shear blocks.

3) The criteria M3 = KIC/E. KIC [MPa.m 1/2 ] is the tenacity of the material (its ability to resist crack propagation). During erection, stress concentrations can locally occur at the pseudo-punctual contact between the bolt and the wood at the holes, the risk of cracking is important (it is common to have to change laths during the installation). 𝑬 . Thus, maximizing the stability/cost ratio means maximizing M4.

Other criteria proposed by Douthe related to environmental impact are not considered in our study because of our pavilion is small, and the materials' environmental data is not easily accessible. This shows that ash is the most suitable material for our project. We also decided to increase the maximum curvature allowed by the wood by maintaining it moisture content at the fibre saturation point, soaking it before erection in order to make it more flexible.

Verification and adjustment of details

Nodal load assumptions according to the Eurocodes:

• Dead loads: G = 1.26 kg • Live loads due to wind W = 3.42 kg • Live loads due to snow: S = 2.15 kg maximum • Accidental loads: people with a mass of 100 kg hanging from various gridshell's nodes.

Load combinations:

• Service Limit State: for deflections' calculation. However, as the pavilion is temporary, no limit is imposed on this serviceability criterion and creep in timber is neglected. for the part of the stress received in the service phase. We have therefore simulated and combined three stress cartographies to obtain the resulting stresses in timber elements in service stage. 38% of the elements remain in the elastic domain during the erection stage and 39% in the service phase, all remaining below the breaking point (respectively 49% and 50% with timber soaking, reducing the elastic modulus E of about 27% at fibre saturation point during the erection). The local plastification of laths is allowed by the load transfer on the parts remaining in the elastic domain. Horizontal reactions: their average value at each lath end is 37.5 kg under self-weight and 261 kg at the SLU. The gridshell's base is anchored by steel rods driven 25 cm into the ground. Benches and plant containers act as buttresses. The whole system can resist a 270 kg horizontal reaction.

Conclusions

The outcomes achieved so far are:

• Popularization work of elastic gridshells • Calculation and verification algorithm • User's manual Outcomes expected during next steps:

• User-friendly interface • Algorithm's adaptation to other typologies like braiding (BAYA study [START_REF] Leyral | The Baya Nest pavilion project: braided pattern optimization for hanging shell structures by dynamic relaxation[END_REF]).

  de la Frontière, Potton, Quebec, 1896 Toledo Gridshell 2, Naples, 2014

Figure 1 :

 1 Figure 1: Schematic typological definition of the gridshell [Jean-Maurice Michaud / Sofia Colabella]

Figure 3 :

 3 Figure 3: Primitive hut in active bending according to Viollet-le-Duc.

Figure 2 :

 2 Figure 2: Schematic typological definition of elastic gridshells.elements

Figure 4 :Figure 5 :

 45 Figure 4: Steps to build a lobembe according to Philippart de Foy [Les Pygmées d'Afrique centrale. Éditions Parenthèse, Paris, 1984].

Figure 7 :

 7 Figure 7: Construction of the Essen Gridshell [ibid].

EIFigure 9 :

 9 Figure 9: buckling of the flexible rod and elastica graph at various stages.

Figure 8 :𝑑𝑥 2 +

 82 Figure 8: The Mannheim Multihalle, exterior and interior views [Verein Multihalle Mannheim, Daniel Lukac].

Figure 10 :

 10 Figure 10: From top to bottom: extracts from the video of the experiment; deflection and bending moment of the flexible rod by image recognition processing; post-buckling shape obtained by the experiment and by the numerical dynamic relaxation algorithm ELASTICA.

  R) (m -1 )Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial StructuresInspiring the Next Generation 73.2.1. Form finding with a modelHanging chain nets model: it is simple to realize, although it requires sliding links to make sure that all the cables are tight. Its use can be surprising, since the notion of an antifunicularand therefore pure compressionis used to model an object in flexion and compression. To confirm its relevance, in 1973, Linkwitz digitally modeled the Mannheim model using photogrammetry. The calculations conducted by Happold took into account the bending and led to results similar to those of the hanging chain nets.

Figure 11 :

 11 Figure 11: Hanging chain net and active-bending models of the trial gridshell in Essen; and hanging chain net model of the Mannheim Multihalle and zoom on the links [from[START_REF] Happold | Timber lattice roof for the Mannheim Bundesgartenschau[END_REF] and[START_REF] Roland | The Work of Frei Otto[END_REF]].

Figure 12 :Figure 13 :

 1213 Figure 12: Diagram of the problem studied by Douthe, and evolution of the distance to the hanging chain form with the angle α.

Figure 15 :

 15 Figure 15: Discretization of a fabric and Chebyshev pattern for dressing a halfsphere; Chebyshev lattice: principle, by model and by dynamic relaxation [BOISSE, P. and BOUHAYA, L.].

4 )

 4 Steps 2 and 3 are repeated iteratively until the new set of forces at each node is balanced: the stretched canvas has then reached its natural position.

Figure 17 :

 17 Figure 17: General principle of dynamic relaxation applied to an example of the form finding of a stretched canvas [10].

  the force field applied to the nodes is deduced from the relation 𝑀 ⃗⃗⃗ = 𝑂𝐴 ⃗⃗⃗⃗⃗⃗ ^ 𝐹 ⃗ . The algebraic value of the moment being 𝑀 = 𝐸𝐼 𝑅 and, by definition, 𝑅 = 𝐿 𝑖-1,𝑖+1 2 sin(𝛼 𝑖 ) , we obtain 𝑀 = 2 EI sin(𝛼 𝑖 ) 𝐿 𝑖-1,𝑖+1 . We deduce that in a system composed of curved beams, each trio of consecutive nodes admits on the ends of each of the two segments formed two opposite forces of the same values, 𝐹 𝑖-1,𝑖 = 2𝐸𝐼 sin(𝛼 𝑖 ) 𝐿 𝑖-1,𝑖 * 𝐿 𝑖-1,𝑖+1 for the first segment and 𝐹 𝑖,𝑖+1 = 2𝐸𝐼 sin(𝛼 𝑖 ) 𝐿 𝑖,𝑖+1 * 𝐿 𝑖-1,𝑖+1 for the second. Once the acceleration 𝑎 𝑡 ⃗⃗⃗⃗ at a fictitious instant t has been calculated and knowing the initial velocities 𝑣 𝑡+𝑑𝑡 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ at the same instant, we deduce nodal velocities at the following instant t + dt: 𝑣 𝑡+𝑑𝑡/2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑎 𝑡 ⃗⃗⃗⃗ * d𝑡/2 + 𝑣 𝑡 ⃗⃗⃗⃗. In the same way, it is easily demonstrated [10] that the body moves on average at the velocity it has at average time t + dt/2. The displacement is then 𝐷 = 𝑣 𝑡+𝑑𝑡/2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ * d𝑡 = 𝑎 𝑡 ⃗⃗⃗⃗

Figure 21 :

 21 Figure 21: Energy transfer during the dynamic relaxation of a gridshell.

Figure 20 :

 20 Figure 20: Spring forces from Hooke's law and forces due to bending in the case of a gridshell.

Figure 22 :

 22 Figure 22: Kinetic energy of dynamic relaxation with viscous damping.

Figure 23 :)

 23 Figure 23: Kinetic energy of dynamic relaxation with kinetic damping.

4. Elastica project: application to a pavilion at human scale 4 . 1 .

 41 Site and architectural designThe park of the Butte du Chapeau Rouge, developed from 1938 by Léon Azéma and then by his son Jean, is located between the Mouzaïa and America quarters, in the 19 th district of Paris. Perched on a hill and overlooking the plains of Saint-Denis, it is representative of the 1930s architecture. The project has been designed after various tests comparing numerical and model studies.

Figure 25 :

 25 Figure 25: Masterplan, model, renderings, and photos of the Elastica project [photo Salem Mostefaoui].

m 4 )

 4 Proceedings of the IASS Annual Symposium 2020/21 and the 7th International Conference on Spatial StructuresInspiring the Next Generation[START_REF] Rombouts | Thesis -Optimal design of Gridshells[END_REF] The economic criterion M4 = E 1/2 / Cρ. The cost of the structure is C = Cρ*S*Ltot where Cρ is the volume cost of the material, S = b*h the lath section and Ltot their total length. The critical buckling force being proportional to EI/R3 and I, in first approach, proportional to S², we can write 𝑺 ∝ √ 𝑭 𝒄𝒓 𝑹3

1 .*Figure 26 :

 126 Figure 26: ELASTICA's kinematics of gridshell's global buckling when the nodal load exceeds the critical load, and critical buckling loads for Elastica and Mannheim gridshells.

  Accidental loads: According to our tests, the critical position of punctual loads is in the middle of the vault on the lateral plane (top), and very close to an edge (entrance arch). 24 people, each weighing 100 kg, can hang simultaneously in this critical area.Stress in timber elements: they are due to its bending during the assembly phase and then, in the service phase, to the loading (self-weight and variable loads). Stress is equal to 𝜎 = 𝑀 * 𝑦 𝐼 where y is the relative position (distance to the neutral axis). However, shear blocks are added after erection, increasing the inertia of the members from 2bh 3 /12 in the erection stage to 26bh 3 /12 in service phase. In addition, during erection stage the laths work independently (maximum stress at ymax = h/2), while in the service phase, double lathing works as a single Vierendeel truss (ymax = 3h/2). As M = EI/R, the maximal stress is equal to 𝜎 𝑚𝑎𝑥 = 𝐸 * ℎ 2 * 𝑅 for the part of the stress received in the erection stage and 𝜎 𝑚𝑎𝑥 = 3𝐸 * ℎ 2 * 𝑅

Figure 30 :

 30 Figure 30: Stress cartography in timber elements, and forces in bracing, under SLU combination.

  Bracing verification. Flat sections in steel S235 (width b = 30 mm and height h = 3 mm) in traction are dimensioned by the intrinsic resistance of their section Nmax = fy.b.h = 2156 kg and those in compression by their critical buckling force 𝑁 𝑐𝑟 = 𝜋²𝐸𝐼 𝑙 𝑓 ² ⁄ = 89 𝑘𝑔 for a 40 cm long element.

Figure 29 :

 29 Figure 29: Gridshell deflection under the action of its self weight and of 7 persons of 100 kg suspended at 7 nodes in the critical zone.

Table 1 :

 1 Parameters of the ELASTICA algorithm for the Elastica project.

	Input data	Symbol Value Unit		Mechanical parameters	Symbol	Formula	Unit
	Lath width Lath height	b h	0.045 0.012	m m	Surface	Simple layer grid Double layer grid	S	bh 2bh	m 2
	Initial mesh length	L0	0.5	m		Nodal mass	m	Vρ + ma	kg
	Mass of the connecting element	ma	0.4	kg		Simple layer		bh 3 /12
	Timber density	ρ	500 kg/m 3	Inertia	Double layer without shear blocks	I	2bh 3 /12	m 4
	Modulus of elasticity of wood	E	11 500 MPa		Double layer with shear blocks		b(3h) 3 /12 -bh 3 /12 = 26 bh 3 /12
						Axial stiffness	Ra	ES/L0	MN/m
						Inflectional stiffness	Rf	2EI/L0 3	MN/m

Table 2 :

 2 Mechanical and economic material parameters and multi-criteria evaluation.

	Material	Density [kg/m 3 ]	Elastic modulus E [Gpa]	Yield strength σy [MPa]	Flexural resistanc e [MPa]	Tenacity KIC [MPa.m 1/2 ]	Volume cost Cρ [€/m 3 ]	M1 Coef. 4	M2 [MPa] Coef. 1	M3 [MPa/m 3/2 ] Coef. 3	M4 [MN 0,5 *m 0,5 /€] Coef. 5	Global Score
	Oak	380 -480	13 300	40	105	0.6	1 150	3.01 x 10 -3 13 300 4.51 x 10 -5	100.28 x 10 -3	3.96/5
	Pine	300 -400	10 200	24	97	0.5	850	2.35 x 10 -3 10 200 4.90 x 10 -5	118.82 x 10 -3	3.60/5
	Ash	600 -750	12 900	40	113	0.75	1 200	3.10 x 10 -3 12 900 5.81 x 10 -5	94.65 x 10 -3	4.08/5
	Black locust	670 -770	16 900	40	126	0.9	1 500	2.37 x 10 -3 16 900 5.33 x 10 -5	86.67 x 10 -3	3.60/5
	Niangon	620 -780	14 430	42	103	1	2 000	2.91 x 10 -3 14 430 6.93 x 10 -5	60.06 x 10 -3	3.62/5
	Steel	7 850	210 000	275	600	85	23 550	1.31 x 10 -3 210 000 40.48 x 10 -5	19.46 x 10 -3	2.58/4
	Carbon Fiber Reinforced Plastic	1 800	90 000	2500	1200	45	175 000	27.78 x 10 -3 90 000 50.00 x 10 -5	1.71 x 10 -3	2.85/4
	Glass Fiber Reinforced Plastic	2 600	26 000	3200	1000	65	14 500	123.08 x 10 -3 26 000 250.00 x 10 -5	11.12 x 10 -3	3.77/4

Acknowledgements

Dr. Jef Rombouts, engineer and architect, and Ludovic Regnault, architect, from Bollinger + Grohmann and Dr. Bernardino D'Amico whom we contacted in 2017, for sharing their knowledge. Quentin Chef et Caroline Simmenauer for the review and corrections. The Paris City Hall and the 19 th district City Hall, in particular Sophie Godard, Cécile Becker, Eric de Grootte, Philippe Clayette and Sylvie Roudier, for their support, as well as the inhabitants of the neighbourhood, Mr. Mayor François Dagnaud and Mrs. First Deputy Halima Jemni. BM Montage of the Busson family which manufactures and gives us every year the metal pieces. The ENSA of Paris-la-Villette, Caroline Lecourtois, director, Vincentella De Comarmond, deputy director, Christian Brossard general secretary, Anne d'Orazio, President of the Board, Bruno Petit and Philippe Agricole (maintenance shop), Philippe Bourdier, in charge of student life and Marc Fayolle De Mans, Jacques Bergna and Alain Raynaud (model workshop).

According to Happold and Liddell, all other things being equal, the use of bracing on double lathing increases the critical load by a factor of between 1.60 and 4.44. Furthermore, we can predict that the addition of shear blocks will increase the critical buckling load by a factor of about 13, determined by the ratio of inertias with (26bh 3 /12) and without (2bh 3 /12) these blocks. As for the results of our modeling of the Elastica project, we can conclude that, all other things being equal, buckling resistance is increasing:

• By a factor of 1.97 to 2 by designing a double layer grid.

• By a factor of 1.67 to 1.86 adding bracing.

• By a factor of 11.03 to 12.14 adding shear blocks.

These results correlate with our theoretical predictions and with Happold and Liddell's analyses.

We also wished to compare them with the formula proposed by Douthe [START_REF] Douthe | Thesis -Étude des structures élancées précontraintes en matériaux composites : application à la conception des gridshells[END_REF] who believes that "in order to obtain an expression of the critical pressure pcr that will cause the shell to collapse, it is assumed that this load is close to that which causes the instability of an equivalent cylindrical shell subjected to hydrostatic loading, i.e. of the type: pcr = 3EI/R 3 " (I is here the inertia per unit of length). The proximity to the results on the three designs tested confirm this theory.

Before concluding, let us recall that any structure must be dimensioned with a safety factor on the results of the calculations. Happold and Liddell evaluate it by considering various parameters: variations in modulus of elasticity E (between the different laths, due to moisture and due to creep), accuracy of shape of shell, variations in loading, accuracy of computer model and assumptions, nature and significance of buckling collapse and consequences of failure. Their global safety coefficient was 2.85, and ours is 4.85. Since the variability in loading is already taken into account in the ULS combination (9.09 kg), we apply an overall safety factor of 4.85/1.40 = 3.46 (31.45 kg) to this load. The results of our study show that the expected loads on the Elastica gridshell require a double lath design with bracing and shear blocks.