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Journal of Civil Engineering and Architecture 16 (2022) 200-226

doi: 10.17265/1934-7359/2022.04.004

Kagome Project: Physical and Numerical Modeling

Comparison for a Post-formed Elastic Gridshell

Marc Leyral1, Quentin Chef2,3, Tom Bardout2, Romain Antigny2 and Alexis Meyer3

1. AMP Research Unit & Laboratory MAP-MAACC, ENSA Paris-La Villette, 144 Avenue de Flandre, Paris 75019, France

2. ENSA Paris-La Villette, Paris 75019, France

3. École Centrale-Supélec, Gif-sur-Yvette 91190, France

Featured Application: This study focuses on the evaluation of the critical buckling load for elastic gridshells by testing scale models

or using dynamic relaxation. After discussing the differences between numerical and physical results at various scales, the study

proposes to evaluate the resistance of certain gridshell typologies that cannot be reached numerically. The results can be used to

design building covers or 1:1 pavilions made of elastic gridshells.

Abstract: An elastic gridshell is an efficient constructive typology for crossing large spans with little material. A flat elastic grid is

built before buckling the structure into shape, in active and post-formed bending. The design and structural analysis of such a

structure requires a stage of form finding that can mainly be done: (1) With a physical model: either by a suspended net method, or

an active bending model; (2) With a numerical model performed by dynamic relaxation. All these solutions have various biases and

assumptions that make them reflect more or less the reality. These three methods have been applied by Happold and Liddell [1]

during the design of the Frei Otto’s Mannheim Gridshell which has allowed us to compare the results, and to highlight the significant

differences between digital and physical models. Based on our own algorithm called ELASTICA [2], our study focuses on: (1)

Comparing the results of the ELASTICA’s numerical models to load tests on physical models; (2) The identification of the various

factors that can influence the results and explain the observed differences, some of which are then studied; (3) Applying the results to

build a full-scale interlaced lattice elastic gridshell based on the Japanese Kagome pattern.

Key words: Interlaced lattice, gridshell, timber, dynamic relaxation, numerical modeling, physical modeling, form finding, Kagome.

1. Introduction

Between 2020 and 2021 we carried out studies

which enabled us to produce the ELASTICA tool [2],

an ergonomic and open-source algorithm for the design

and form-finding of post-formed elastic gridshells, for

the verification of their structural integrity, and for

editing fabrication and assembly plans. Then, we

wanted to apply these results for the design and

fabrication of a post-formed elastic gridshell with

interlaced members.

The numerical modeling of certain types of

gridshells is very complex, due to their geometry, and

Corresponding author: LEYRAL Marc, Engineer Architect,

Lecturer at École Nationale Supérieure d’Architecture de

Paris-la Villette, research fields: post-formed elastic gridshells,

braided shells, complex structures. E-mail:

marc.leyral@paris-lavillette.archi.fr.

the reliability of the results is difficult to assess. This

is the case, for example, of non-deformable

membranes in their plane, three-dimensional patterns

and interlaced members.

We therefore decided to design this structure by

testing physical models. The aim is to calibrate the

tests using the numerical results of a “classic”

gridshell, then to access the out-of-plane inertia of the

interlaced gridshell and finally to extrapolate the

results to scale 1. During the calibration phase, we

were able to observe, analyze and study the various

biases of the numerical and physical models, trying to

explain why the results of the two types of models

could diverge. We have deduced from this a set of

recommendations and precautions to be applied to all

types of elastic gridshell projects.

D

DAVID PUBLISHING

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

201

Parts of the chapters may have been reproduced

with permission from: Marc Leyral, Sylvain Ebode,

Pierre Guerold, Clément Berthou; Elastica project:

dynamic relaxation for post-formed elastic gridshells,

In Inspiring the Next Generation-Proceedings of the

International Conference on Spatial Structures

2020/21 (IASS2020/21-Surrey), edited by: Alireza

Behnejad, Gerard Parke and Omidali Samavati,

published by University of Surrey, Guildford, UK, in

August 2021 [2]. This paper is in continuation of this

work.

2. Elastic Gridshells

2.1 Definition and Origins

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

(Fig. 1).

Labbé [3] 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”.

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 assembled 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 to significantly increase its

out-of-plane inertia (Fig. 2). The final grid is very

rigid and can cover a large span without intermediate

supports, and this with very little material.

Fig. 1 Schematic typological definition of a gridshell (WIKIARQUITECTURA/Jean-Maurice Michaud/Sofia Colabella).

Shell Town truss Elastic gridshell

CNIT, Paris, 19858 Pont de la Frontière, Potton, Quebec, 1896 Toledo Gridshell 2, Naples, 2014

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

202

Fig. 2 Schematic typological definition of elastic gridshell (Credits: [2]).

Many people around the world have built simple

shelters based on this constructive process and using

only local and natural materials: wood, plant fibers,

leaves, etc. (Fig. 3). The lightness of the structure is a

key advantage for their self-construction. Apart from a

few exceptions like the Mongolian yurt, most

vernacular gridshells are not post-formed: the stems

are bent and fixed to the ground one after the other.

(a) (b)

(c) (d) (e)

Fig. 3 Vernacular elastic gridshells: (a) Steps to build a lobembe according to Philippart de Foy [4], which is not a

post-formed gridshell, (b) Post-formed gridshell: a Mongolian yurt (Smith Archive & Alamy Stock Photo), (c) Huts of the

Haru Oms, Nama people (Exploring Africa/maison-monde.com), Huts of Xingu Indians (d) and Zulu tribes (e)

(maison-monde.com, auroraphotos.com, John Lee, Wikimedia and africa.quora.com, Atom ref ZA 0375-N-N08935).

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

203

(a) (b) (c) (d)

Fig. 4 (a) gridshell in Berkeley; (b) trial gridshell in Essen; (c) its model; (d) and its construction (from Ref. [1]).

(a) (b)

Fig. 5 The Mannheim Multihalle, exterior (a) (Image by Archive Frei Otto) and interior views (b) (Archive Frei Otto and

Gabriel Tang).

2.2 In Modern and Contemporary Architecture

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. Later in the same year, he

built a wooden trial gridshell of 198 m² at the German

Building Exhibition in Deubau, Essen (Fig. 4).

Its height at the center point is 5 m. Two orthogonal

layers of 60 mm × 40 mm Oregon pine elements are

assembled to the floor by bolting at the knots, forming

a super elliptical—or squircle—base with a mesh size

of 48 cm. It was then lifted using a mobile crane and

fixed to an edge beam driven into the ground.

However, the first architectural project of large

scale, is undoubtedly the Multihalle in the Herzogenried

Park in Mannheim (Germany), built in 1975 for the

Bundesgatenshau (Fig. 5). 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

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

204

of the wood increases with its moisture content.

3. Design Method for Elastic Gridshells:

Form Finding and Verification

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 (RD). The work of Frei Otto, ARUP and

Happold & Liddell [1], 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.

3.1 Design with a Physical Model

3.1.1 Hanging Chain Nets Model

Hanging chain nets model is simple to realize,

although it requires sliding links to make sure that all

the cables are tight (Fig. 6). Its use can be surprising,

since the notion of an antifunicular—and therefore

pure compression—is applied 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.

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 which is a commonly

assumed hypothesis.

Let us remember that Douthe [5] studied the

differences between the funicular and the elastica

shape according to the attack angle α at the basis and

the loading rate p (Fig. 7). 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 pL3/EI

ratio below 65, confirming a posteriori the modelling of

Mannheim by Frei Otto. Like Happold and Liddell [1],

we can therefore conclude that “a funicular shape is an

advantage but is not essential.”

(a) (b) (c) (d)

Fig. 6 Hanging chain net (a) and active-bending models (b) of the trial gridshell in Essen; (c) hanging chain net model of the

Mannheim Multihalle; (d) zoom on the links (www.freiotto.com, Architekturmuseum TU München and from Ref. [1]).

Fig. 7 (a) Diagram of the problem studied by Douthe [5]; (b) evolution of the distance to the hanging chain form with the

angle α.

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

205

(a) (b) (c)

Fig. 8 Load test by adding nails to the nodes of the Essen (a) and Mannheim (b) models; (c) loading tests with water-filled

garbage cans (from Ian Liddell and from Ref. [1]).

(a) (b)

Fig. 9 Scalability tests on the section and stiffness of the material (a) and on the length of the element (b) (from Ref. [2]).

3.1.2 Active Bending Physical Model

In 1973, as Happold and Liddell [1] 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 (polymethyl methacrylate)

model at the 1/16th 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 1/60th 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 (Fig. 8). 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/aS3 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 previous experiences on

the essential question of the extrapolation of the

results from the model to the real project, we first

concluded 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 (Fig. 9).

The buckling force is transposable but subjected

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.

3.2 Design with a Numerical Model by Dynamic

Relaxation

To overcome the inaccuracies of a form finding

with a physical model, a numerical method is

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

206

generally necessary. We developed our own algorithm

on Rhino + Grasshopper. Called ELASTICA, it is a

complete, generic, open-source and ergonomic tool,

usable by all, for form-finding, dimensioning, and

optimization of elastic gridshells using dynamic

relaxation. The theory of dynamic relaxation and the

elaboration of ELASTICA algorithm are given in

Appendix A. We will focus in the following on the

analysis of the results obtained by physical and

numerical models on a gridshell project.

4. Numerical and Experimental Design

4.1 Presentation of the Studied Gridshells

The project takes place on the belvedere of the

Butte du Chapeau Rouge Park, in the 19th

arrondissement of Paris. Built from 1938 by Léon

Azéma, then by his son Jean, the park is bordered by

“Habitations à Bon Marché” (French housing at low

rent during the first half of the 20th century) built with

concrete and red bricks and offers a breathtaking view

of the Saint-Denis plain below.

It is in this context that we built in 2020 a first

post-formed elastic gridshell of “classic” design,

Elastica. In July 2022, we plan to complete the initial

project by building a second post-formed elastic

gridshell, Kagome, which will be different because it

will be made by interlaced lattices in three directions

(Figs. 10 and 11).

Fig. 10 Elevation and masterplan of the studied gridshells.

Elastica Gridshell

(already built in 2020)

Kagome gridshell (to

be built in 2022)

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

207

(b)

(a) (c)

Fig. 11 (a) Gridshell Elastica, 2020 (Credits: Salem Mostefaoui), main dimensions (b), and 3D du gridshell Kagome (c).

Fig. 12 Methodology developed for the design of the Kagome gridshell.

4.2 Issue and Methodology

Issue: Kagome is not directly accessible to

numerical calculation with the ELASTICA algorithm

(Fig. 12) because:

(1) It is already braced when built flat (because of

the three-direction pattern). There is therefore no

movement possible between the lattices in plan, and

this is true from the erection phase to the loading

phase.

Making a 1:10 physical model of Elastica

Comparing the load test results with the results of the ELASTICA

algorithm for a 1:10 version of the Elastica project

Load test

YES

Making a 1:10 physical model of Kagome

Load tests

Extrapolating the results to 1:1 scale using the

EI

xx

/

aS

3

ratio

VERIFYING

ASSUMPTIONS

NO

Additionnal load tests

Listing the possible biases and selecting those

that appear to contribute the most

Modifying the ELASTICA algorithm Load testing a 1:2 physical model of Elastica

Making a 1:2 physical model of Kagome

RECALIBRATION FINAL DESIGN

Matching results?

≈ 350 mm

500

mm

500

mm

b = 6.30 m

h = 2,60 m

d = 4.30 m

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

208

(2) The evaluation of the out-of-plane inertia is

complicated by the lathing in three directions and the

method of assembly by interlacing the lattices.

4.3 Consolidation Phase

4.3.1 Numerical Calculation of the Elastica Project

Using ELASTICA Algorithm

Load assumptions (nodal loads):

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.

1.00 G = 1.26 kg

1.00 G + 1.00 W + 0.60 S = 5.97 kg

Ultimate Limit State (ULS): a safety criterion:

loads combinations for stress and surface’s buckling

verifications.

1.35 G = 1.70 kg

1.35 G + 1.50 W + 1.05 S = 9.09 kg

A safety factor on the results of the calculations

considering various uncertainties (variations in modulus

of elasticity E—due to natural inhomogeneity, moisture

and creep—accuracy of shape of shell, variations

in loading, accuracy of computer model and

assumptions, nature and significance of buckling

collapse, consequences of failure) of 3.46 has been

applied on the ULS.

Buckling limit load of the surface: predominant ruin

mode for gridshells. To determine the critical buckling

load, we proceeded by dichotomy on loads in the

ELASTICA algorithm. We tested all the main design

parameters: lathing type (single layer or double layer),

the use or not of bracing, the use or not of shear blocks

(cf. Table 1 and Fig. 14). To interpret these results, we

compared them with those obtained by model and

numerical modelling by Happold and Liddell [1] for

the Mannheim Multihall (Fig. 13 and Table 2).

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 (26bh3/12) and without

(2bh3/12) 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 2.00 to 2.02 by designing a double

layer grid.

• By a factor of 1.81 to 1.83 adding bracing.

• By a factor of 11.09 to 12.29 adding shear blocks.

Table 1 Parameters of the ELASTICA algorithm for the Elastica project.

Input data Symbol Value Unit Mechanical parameters Symbol Formula Unit

Lath width b 0.045 m Surface Simple layer grid A bh m

2

Lath height h 0.012 m Double layer grid 2bh

Initial mesh length L

0

0.5 m Nodal mass m Vρ+ma kg

Mass of the connecting

element m

a

0.4 kg

Inertia

Simple layer

I

bh

3

/12

m

4

Timber density ρ 500 kg/m

3

Double layer without shear

b

locks 2bh

3

/12

Timber modulus of elasticity E 11,500 MPa Double layer with shear blocks 26bh

3

/12

Axial stiffness R

a

EA/L

0

MN/m

Bending stiffness R

f

2EI/L

03

MN/m

Fig. 13 Kinematics of gridshell’s global buckling when nodal loads exceed critical load (from Ref. [2]).

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

209

Table 2 Critical buckling loads for Elastica and Mannheim gridshells.

Layer Bracing Shear blocks Critical nodal load Elastica

(

k

g

at each node

)

***

Critical nodal load Mannheim

(

k

g

/m2

)

***

Simple no n/a 1.05 3.8**

yes n/a 1.90 Non-evaluated

Double

no no 2.10 63***/100**

no yes 25.80 -

yes no 3.85 100*/160**/280***

yes yes 42.70 -

* Results of the numerical model.

** Extrapolated predictions based on the Essen model.

*** Results extrapolated on the basis of the Multihall model.

Fig. 14 Principle of increasing inertia by connecting the layers with shear blocks (from Ref. [2]).

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 [5] 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/R3” (I is here the

inertia per unit of length). The proximity to the results

on the three designs tested (cf. Table 3) – the radius of

curvature, 2.4 m, is measured at the median

curvilinear position – shows a correlation between the

buckling of a gridshell and that of a cylindrical shell,

giving an a priori validation of the hypothesis.

We concluded from this studie that Elastica should

be a double-layered gridshell with bracing and shear

blocks (Fig. 15).

4.3.2 Loading Tests on a 1:10 Physical Model of

Elastica

Following the studies described so far, the Elastica

gridshell was built in September 2020. The initial

objective of our research in 2022 was to theorize the

extrapolation of these results for the design of

gridshells with different lathing such as the Kagome

gridshell.

However, during the development of the

ELASTICA tool, the Covid-19 prevented physical

meetings of the team and studies on real models could

not be carried out. In 2022, we therefore decided to

check the validity of our digital tools upstream by

comparing them to load tests on 1:10 scale models

(Fig. 16).

Let us recall that during the studies of the

Mannheim Multihalle by Happold and Liddell [1], the

buckling collapse load of a physical model was

measured at 2.8 kg/m² without bracing and at 12.5

kg/m² with it. An extrapolation of the critical load

being possible from a model to a real project by

multiplying it by the ratio of the EIxx/aS3 of the project

and the model (Ixx being the out-of-plane inertia, a the

spacing of the lattices and S the span of the gridshell),

the structure’s collapse load is thus assessed at 63 kg/m²

for double lathing without bracing and at 280 kg/m²

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

210

Table 3 Critical buckling loads by the ELASTICA algorithm and the formula of cylindrical shells.

Layer Bracing Shear blocks Critical nodal load

by

modellin

g

(

k

g)

Critical nodal load calculated

with

p

cr

= 3EI/R

3

(

k

g)

Simple no n/a 1.05 1.02

Double no no 2.10 2.05

no yes 25.80 26.71

Fig. 15 Synthesis of the project loads and critical buckling loads according to the different possible designs, and details of

the Elastica gridshell: our study shows that the expected loads on the Elastica gridshell require a double lath design with

bracing and shear blocks.

Table 4 Comparison between the numerical and experimental results on the post-formed elastic gridshell Elastica in its

version with simple lathing and without bracing.

G01: Elastic gridshell with simple lathing and without bracing

Parameters Model 1:10 Project

Radius of curvature at the center R 0.24 m 2.40 m

Lattice section b×h 10 mm × 1 mm 50 mm ×12 mm

Number of nodes by m² 400 4

Gridshell span S/distance between lattices a 0.356 m/50 mm 3.56 m/500 mm

Timber elastic modulus 10,200 MPa 11,500 MPa

Scale model 1:10 Real project 1:1

Load test on the

physical model

Numerical

simulation (RD)

Comparaison to

cylinder shells

Extrapolation by

upscaling 1:10

result

Comparaison to

cylinder shells

Numerical

simulation (RD)

Buckling critical load 0.027 kg (0) 0.011 kg (0) 0.011 kg (1) 2.80 kg (2) 0.92 (1) 1.05 kg (0)

Lattice inertia I

xx

2.358 mm

4

(1) 0.83 mm

4

21,468 mm

4

7,200 mm

4

Inertia by length unit I 0.047 mm

4

/mm (1) 0.017 mm

4

/mm 42.9 mm

4

/mm 14.4 mm

4

/mm

(0) By direct measure (physical model) or by dichotomy (using ELASTICA algorithm).

(1) Buckling critical load of a cylindric shell under hydrostatic load: p

cr

= 3EI/R

3

.

(2) Extrapolated using the EI

xx

/aS

3

between the model and the project.

Double layer only 2.10 kg

Simple layer onl

y

1.05 kg Double layer + bracing 3.85 kg

Double layer

+ shear blocks

25.8 k

g

Double layer + bracing

+ shear blocks

42.7 k

g

Simple layer + bracing 1.90 kg

Self-weight

1.26 kg SLS Combination

5.97 kg

Supported structure: 0 kg

ULS Combination

9.09 kg ULS Combination with

security factor

31.45 k

g

Nodal load (kg)

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

211

(a)

(b)

Fig. 16 Buckling kinematics predicted by ELASTICA (a) and on a 1:10 model of the Elastica project (b) at approximately

35%, 80%, 90% and 99% of their respective critical buckling load.

with it. However, numerical calculations predicted are

100 kg/m². The study attributes the difference, a factor

of 2.8, to the non-variability of scale of the stiffness in

shear or to a greater stiffness of the nodes of the scale

model.

4.3.3 Conclusion of the Consolidation Phase

The results of the “calibration” test show that the

critical buckling load of the scale model is 2.48 to

2.53 times greater than anticipated by the calculation

via the dynamic relaxation simulation (Table 4). This

non-negligible ratio is quite close to that observed by

Happold and Liddell on Mannheim (2.80).

To try to understand it, let us come back to the

reflections of Happold and Liddell [1] on the

extrapolation of the results on a scale model. The

authors identified several properties of the structure

“which define and control its behaviour. [They] are

listed as follows:

S = Span. If the model is geometrically scaled then

its size can be represented by a typical dimension, say

the span

EIxx/a = The out of plane bending stiffness of the

surface a per unit length (a = spacing of laths)

EIyy/a3 = [The in-plane bending stiffness] is

proportional to the contribution of the timber members

to diagonal stiffness, if the joints between timber

members are rigid

EA/a is the axial stiffness along the timber members

per unit length

E’A’/ka = is proportional to the contribution of the

ties to the diagonal stiffness ([…] ka being the tie

spacing)”.

List to which is added a contribution related to the

slip per unit force of each node.

We then undertake the following reasoning in order

to determine the parameters that seem to us to be the

most significant in their contribution to the differences

observed:

As “the deformation of the grid shell is mainly due

to out-of-plane bending and diagonal distortions of the

grid squares [and] if the diagonal stiffness is much

less than the axial stiffness” [1], we will consider as

negligible the contribution of the axial stiffness of the

lattices with respect to the out-of-plane bending

stiffness.

As the tested gridshell is designed without bracing,

the contribution of the bracing elements’ axial

stiffness E’A’ has no place in the reflection.

In agreement with Happold and Liddell, we will say

that if the bracings have a very high axial stiffness, the

contribution of the stiffness in plan is negligible.

However, the tested gridshell had no bracing.

We therefore identify at this stage the two

parameters on which we will focus our evaluation:

(1) Semi-rigid nodes. The contribution of the nodes

which are modeled in dynamic relaxation as perfectly

articulated in the plane whereas they in fact have a

certain stiffness due to the tightening of the assembly

and the friction induced by the curvature of the

lattices.

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

212

To evaluate this parameter, we will:

Carry out loading tests on braced models,

because the blocking of the deformation of the mesh

induced by the axial stiffness of the braces, if it is

large enough, should make the numerical and physical

results converge.

Modify the ELASTICA algorithm to take into

account a spring torque in rotational friction at each

node, compare the results obtained with loading tests

on models at 1:10 and 1:2 scales. The contribution of

friction, which tends to fictitiously increase the

extrapolated critical load, should decrease with the

increase of the scale of the physical model.

(2) In-plane bending stiffness. Moreover,

considering this nodes’ rotational stiffness implies a

possible mobilization of the in-plane bending stiffness

of the lattices, since it allows the deformation by

bending of the elements inside the local plane of the

surface, which is only possible if the nodes are

semi-rigid (Fig. 17). To analyze this contribution, we

will modify ELASTICA to take into account a biaxial

bending of the timber elements.

In addition, each model, numerical or physical,

includes a set of biases, the nature of which should be

listed and, if possible, the deviations they may cause in

the result should be assessed. These are mainly geometric

imperfections and model loading imperfections, which

are all the more important as the model is reduced. We

will devote part of our analysis to them.

4.4 Recalibration Phase

4.4.1 Loading Tests on a 1:10 Scale Model with

Loose Connections

In order to reduce as much as possible the

contribution of the stiffness of the nodes

and—possibly—of the bending in the plane of the

lattices, we carried out a new test on the 1:10 model,

loosing the nodes as much as possible. The results of

this test give a critical buckling load of the model at

0.0177 kg per node, i.e. a deviation of 1.61 to 1.65

with the numerical predictions. A preliminary result

motivates us to study more precisely the contribution

of the rotational stiffness of the connections and that

of the out-of-plane bending stiffness of the lattices in

the evaluation of the critical load of the tests.

4.4.2 Loading Tests on a 1:10 Scale Model with

Bracings

In order to go further in this reflection, we carried

out a loading test on a 1:10 braced model (Fig. 18). If

our assumption is correct, and since the axial stiffness

per unit length of the braces E’A’/ka is large enough,

the possible contributions of the rotational stiffness of

the nodes and the out-of-plane bending stiffness of the

chords should be negligible.

The results of this experiment show a difference of

only 1.5% between the physical model (0.0182 kg

per node) and the numerical model (0.0185 kg per

node).

(a) (b)

Fig. 17 (a) Hinged nodes; (b) semi-rigid nodes: bending in the plane of the lattices is only possible if the nodes are

semi-rigid.

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

213

Fig. 18 Loading tests on a 1:10 scale model with bracings (single layer).

4.4.3 Preliminary Outcomes

These results tend to confirm our hypotheses on the

contribution on the critical load of buckling of the

rotational stiffness of the nodes and—possibly—of the

in-plane bending stiffness of the timber lattices in the

case of an unbraced gridshell. However, these results

should be considered with great caution at this stage

because:

The number of trials is very low.

At this scale, the biases linked to geometric

imperfections and to the loading protocol—we only

loaded 1 node out of approximately 15 on the

model—are probably significant and their

contributions to the results are still poorly controlled.

We could not scale the bolts and therefore the

washers. They are proportionally larger and

mechanically increase the friction at the nodes and

therefore their stiffness. It is a fact that the difference

in the physical tests between the model with the loose

nodes (0.0177 kg per node) and the braced gridshell

model (0.0182 kg per node) suggests that once the

shell has been shaped, the friction between the bolts of

the nodes and the lattices plays a non-negligible role

in blocking by friction, especially as the scale of the

model is reduced and the scale of the bolts is

proportionally larger compared to that lattices.

However, this bias does not contradict—on the

contrary—our previous interpretations.

These conclusions would imply that the consistency

between the critical buckling load of a gridshell

(braced or not) and that of a cylinder shell subjected to

hydrostatic pressure is valid only on a sufficiently

large scale. We note that a cylindrical shell has no

possible movements in the plane of its surface.

We did not test a simple and symmetrical

cylindrical vault but the project of architecture

students from ENSA-Paris la Villette, which presents

a strong asymmetry. Subsequent tests showed that

this asymmetry reduced the critical buckling load

by 33% according to the physical tests on a 1:10

scale model and by 46% according to the extrapolated

numerical model (Fig. 19). We also noticed that

this asymmetry very substantially increased the

sensitivity of the loading imperfection (depending on

the position of the loads, the result could vary by

56%).

We conclude that additional tests on a 1:2 scale

model are necessary in order to assess the impact of

imperfections in the result. Indeed, on a larger scale,

the loading and geometry imperfections and also

the nodes friction will be reduced. A modification of

the ELASTICA algorithm taking into account the

rotational stiffness of the nodes and the in-plane

bending stiffness of the lattices will also make it

possible to compare the new physical results to

numerical ones.

4.4.4 Methodology for the Analysis of the

Contribution of Rotational Stiffness of Nodes

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

214

Fig. 19 Effect of gridshell symmetry: adding only the red parts increased the critical buckling load of Elastica by 86%

according to the numerical simulation. A similar modification on Kagome led to an increase of 50% according to a loading

test on a 1:10 physical model.

Taking into account the rotational stiffness of the nodes: it is modeled at each node by a spring force Ff

proportional to the rotation angle θ between the two directions of the lattices (Fig. 20):

Fig. 20 Spring force due to the friction opposing the relative rotation of the lattices.

It combines the contribution of:

(1) a tightening force Nser equal to 90% of the

elastic limit of the wood perpendicular to the fibers

σy,w,90 multiplied by the washer area Ar:

𝑁 =0,9∗𝜎

,, ∗

𝐴

(1)

(2) a force Ncon linked to the contact between the lattices

of the 2 opposite directions when the curvature of one

exceeds a limit defined by an average slack (Fig. 21):

𝑵𝒄𝒐𝒏 =𝑭

𝒊𝟏,𝒊 𝑭

𝒊,𝒊𝟏∗𝐜𝐨𝐬𝜶

𝒊𝟒𝑬𝑰 𝐬𝐢𝐧𝜶𝒊𝐜𝐨𝐬𝜶𝒊

𝑳𝟎𝟐 (2)

𝑵𝒔𝒆𝒓 =𝑭

𝒊𝟏,𝒊 𝑭

𝒊,𝒊𝟏∗𝐜𝐨𝐬𝜶

𝒊

𝟒𝑬𝑰 𝐬𝐢𝐧𝜶𝒊𝐜𝐨𝐬𝜶𝒊

𝑳𝟎𝟐

The bearing moment Mr equals then:

𝑴𝒓=𝜶

𝒇𝒓 ∗𝑵𝒔𝒆𝒓 𝑵

𝒄𝒐𝒏∗𝒃

𝟐 (3)

where αfr is the static friction coefficient of

wood-on-wood.

Hence the fictitious force Ff:

𝑭𝒇=𝑴𝒓

𝑳𝟎=𝜶𝒇𝒓 ∗𝑵𝒔𝒆𝒓 𝑵

𝒄𝒐𝒏∗𝒃

𝟐∗𝑳

𝟎

(4)

We carry out numerical tests with and without

tightening of the assemblies. The contact force Ncon is

applied only if the curvature is sufficient to create the

contact. With one-millimeter tolerance in assemblage,

contact appears in about one-third to one-half of the

gridshell nodes, according to our testing. The results

are presented in Tables 7-9.

F

f

dθ/2

F

f

F

f

F

f

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

215

𝐹, =2𝐸𝐼 sin𝛼

𝐿, ∗𝐿

,

𝐹, =2𝐸𝐼 sin𝛼

𝐿, ∗𝐿

,

Fig. 21 Forces due to bending (see Appendix A).

(a) (b) (c)

Fig. 22 (a) “Arch” gridshell; (b) “critical” gridshell; (c) “collapsed” gridshell.

Table 5 Model data for the 1:2 scale model of Elastica (single layer version, no bracings).

Parameters Model 1:2 Project

Radius of curvature at the center R 1.20 m 2.40 m

Lattice section b×h 24 mm × 7 mm 50 mm × 12 mm

Number of nodes by m² 16 4

Gridshell span S/Distance between lattices a 1.78 m/250 mm 3.56 m/500 mm

Timber elastic modulus 11,500 MPa 11,500 MPa

Table 6 Comparison and calibration of the physical tests for the 1:2 scale model of Elastica.

Models Buckling start (RD) Rupture (physical) End of buckling (RD)

- Load: 57.15 kg Load: 65.00 kg Load 74.14 kg

During these more precise tests, we realized that

there were in fact three distinct states on our model

which prompted us to define more stages of global

buckling load on the structure instead of one. In order

to correctly characterize the observed results, we

define them in Fig. 22.

Finally, a half-scale model was produced for the

purpose of loading tests (Table 5). During this

experiment, only the 73 most critical to buckling

nodes were loaded (out of 203 nodes in total). This

induces a loading bias that must therefore be corrected

using a numerical test where the own weight of the

α

i

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

216

gridshell is distributed over all the nodes, but the

additional load is applied over the 73 selected nodes

only (Table 6).

It should also be noted that the condition of the

collapse occurred in the model at an earlier load

compared to the dynamic relaxation modelisation.

This is due to the fact that in dynamic relaxation, the

bars are considered to be infinitely resistant, allowing

a complete overturning of the gridshell. In reality, this

geometric deformation generates a localized drop in

the radii of curvature and therefore an increase in the

stress which will exceed the breaking stress limit of

the wood (Fig.24). This observation invites to

introduce a safety coefficient on the result of the

dynamic relaxation that our present study places

around 1.15.

4.4.5 Methodology for the Analysis of the

Contribution of the In-Plane Bending Stiffness

As explained in Appendix A, on a given lattice, for

a series of 3 consecutive points A, B, C, the forces due

to the bending moment belong to the plane (ABC).

These forces are then proportional to the sine of angle

α between 𝐀𝐁

and 𝐁𝐂

and to the inertia of the

section. Until now, ELASTICA tool has considered a

bending of the laths in the “weak” direction

𝑰𝒙𝒙 =𝒃∗𝒉

𝟑𝟏𝟐

⁄, thus as if the lath’s bending was

entirely in the plane perpendicular to the gridshell’s

surface plane. This assumes that the generative curves

of the “arch” are single curvature, which is not the

case. In reality, the laths are in bi-axial bending: in the

weak direction as previously described (out-of-plane

bending) and in the shell surface (in-plane bending).

The latter therefore involves the strong sense of inertia

𝑰𝒚𝒚 =𝒉∗𝒃

𝟑𝟏𝟐

⁄.

To take this into account, we construct two planes

(Fig. 23): a plane TB which is the tangent to the shell

at point B (a smooth surface is defined by

interpolation of the nodes of the gridshell at each time

step of the dynamic relaxation), and a plane NB which

contains the normal to the surface at point B and line

(AC). Taking in-plane stiffness into account means

that the planes (ABC) and NB are not identical. We

project point B on plane NB (obtaining B’) and points

A and C on plane TB (obtaining A’’ and C’’), then we

calculate the forces due to bending in each plane.

4.4.6 Results of Physical and Numerical Loading

Tests

The compared results of the tests relating to the

rotational stiffness of the nodes and in-plane bending

stiffness on physical models and on numerical models

are given in Tables 7-9.

It is noted that the difference between scale model

and numerical simulation is greatly reduced when the

rotational stiffness of the nodes is taken into account

in the calculation: it is for example of the order of 4%

on the tests on the 1:10 model. However, taking these

parameters into account requires a “guesswork”

evaluation of the actual tolerance in the connections

and of the tightening force of the nuts. We also

observe that the contribution of in-plane bending

stiffness has a very little effect on the results.

The second observation is that the contribution of

the rotational stiffness of the nodes decreases with the

scale of the model, thus—on this precise parameter—we

could conclude that the numerical simulation should

give results close to reality at scale 1 and that this is

especially true if the gridshell is braced.

Fig. 23 Definition of the planes used to evaluate bi-axial

bending.

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

217

Table 7 Impact of rotational stiffness of nodes on numerical and physical tests at a 1:10 scale.

Type of test 0.010 kg 0.011 kg 0.013 kg 0.015 kg 0.016 kg 0.017 kg 0.018 kg 0.021 kg 0.026 kg 0.027 kg

Dynamic

relaxation with

articulated nodes

Dynamic

relaxation with

friction and loose

connections

Physical test,

loose connections

Dynamic

relaxation with

friction and tight

connections

Dynamic

relaxation with

friction and tight

connections

considering

in-

p

lane stiffness

Physical test, tight

connections

Table 8 Impact of rotational stiffness of nodes on numerical and physical tests at a 1:2 scale.

Type of test 0.40 kg 0.45 kg 0.50 kg 0.55 kg 0.60 kg 0.65 kg

Dynamic relaxation with

articulated nodes

Dynamic relaxation with friction

and loose connections

Dynamic relaxation with friction

and tight connections

Physical test, tight connections

(recalibrated according to loaded

nodes)

?

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

218

Table 9 Impact of rotational stiffness of nodes on numerical and physical tests at a 1:1 scale.

Type of test 0.9 kg 1.0 kg 1.1 kg 1.2 kg 1.3 kg 1.4 kg 1.5 kg 1.6 kg

Dynamic

relaxation with

articulated nodes

Dynamic

relaxation with

friction and loose

connections

Dynamic

relaxation with

friction and tight

connections

(a) (b)

Fig. 24 By running simulations, a stress concentration (95-105 MPa) is observed in the region where the curvature is

maximum (a). At 65 kg, the stress values are higher than stresses in other regions and compared to stresses with lower loads,

hence the failure happens before the instability as demonstrated with the physical loading test (b).

Fig. 25 First proposal for a safety factor to be applied to the result of a test on an unbraced model to take into account the

non-linearity of friction and nodal stiffness, depending on the scale.

5. Conclusions and Applications

From these studies, it can be concluded that:

For unbraced gridshell, the discrepancies

between the critical buckling load obtained by the

numerical method (with articulated nodes) and by the

load tests on physical scale models seem to come

mainly from the rotational stiffness of the nodes.

The contribution of the in-plane bending stiffness

seems much less significant.

These results still need to be confirmed, given the

possible biases in the tests and the observed sensitivity

M

o

d

e

l

w

i

t

h

t

i

g

h

t no

d

es

M

o

d

e

l

w

i

t

h

l

oose no

d

es

1:10

1:5 1:2 1:1

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

≥ 113 MPa

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

219

of the gridshell to them. We carried out about ten

loading tests, which is still too small a number to

correctly analyze the effect of these biases.

Excluding experimental biases, the numerical

model with articulated nodes always gives the safest

value. The results extrapolated on physical models tend

to overestimate the actual buckling load of the project.

This difference decreases with the scale of the

physical model (Fig. 25), the experimental biases also

decrease.

The results observed in the case of a braced

gridshell are more reliable, even with small scale

models. We recommend applying to the numerical

results a safety coefficient of at least 1.10 in this type

of design.

In reality, the rupture occurs earlier than

numerically envisaged because of the stress generated

by the deformation. We recommend applying a safety

factor of at least 1.15 to the result of the dynamic

relaxation to take it into account.

Thanks to this study we were also able to

extrapolate the resistance of the Kagome interlaced

gridshell (Fig. 26). The results, including the safety

factors stated in this paper, are as follows:

Fig. 26 Load tests on the 1:10 and 1:2 physical models of the Kagome gridshell (single and double layer), and view of the

scale 1 pavilion during construction.

Single layer (asymmetric plan): 2 tests; average:

8.82 kg/m²; s = 0.42;

Double layer (asymmetric plan): 2 tests; average:

12.86 kg/m²; s = 0.33;

Double layer (symmetric plan): 1 test; 23.50

kg/m² according to scale 1:10 extrapolation and 28.13

kg/m² according to scale 1:2 extrapolation.

On the basis of these results, we decided to carry

out the last design (symmetrical plan and double lathing).

Supplementary Materials

ELASTICA algorithm is freely available online at:

https://www.construire-l-architecture.com/07-elastica.

Acknowledgments

The Paris City Hall and the 19th district City Hall, in

particular Sophie Godard, Cécile Becker, Eric de

Grootte, Céline Belgrand, 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.

Würth France and its engineering department

(Fig. 27), our main entrepreneurial partner, in

particular Stephen Conord, National technical referent;

and B. M. Montage of the Busson family which

manufactures and gives us every year the metal pieces,

and in particular Arnaud Busson.

The ENSA of Paris-la-Villette, Caroline Lecourtois,

director, Vincentella De Comarmond, deputy director,

Christian Brossard and Frédéric Sallet general

secretary, Anne d'Orazio, President of the Board,

Jérôme Candevan, accounting officer, Anaïs

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

220

Campanaud, jurist, 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). Yann

Nussaume, Rosa de Marco and Olivier Jeudy from

AMP research unit. François Guéna from

MAP-MAACC laboratory.

The Paris Centrale-Supélec school, in particular

Pierre Jehel and Arnaud Lafont.

Dr. Jef Rombouts, engineer and architect, and

Ludovic Regnault, architect, from Bollinger +

Grohmann and Dr. Bernardino D’Amico whom we

contacted in 2017, and Dr. Almudena Majano Majano

from the Technical University of Madrid for sharing

their knowledge.

Caroline Simmenauer for the review and corrections.

Funding

This research received no external funding except

Würth’s furniture participation to the models at scale

1:2.

Author Contributions

Studies conducted with the participation of:

ENSA-Paris-la-Villette students: 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, Romain Antigny*, Tom

Bardout*, Nina Bargoin, Sofia Casero, Flora

Cassettari Chiaratto, Marisa De Guzman Reyes,

Adrien Echeveste, Noémie Girardi*, Linus

Grimminger, Ines Hitmi, Amalia Iglesias Crespo,

Bryan Katekondji*, Jongmin Kim, Yeji Kim*,

Antoine Maisonneuve, Hortense Majou, Ana Martin

de Castro, Asnathé Mouanda, Martina Paolino, Pablo

Pouget*, Alicia Tresgots*, Farah Ladhari, Mariem

Ben Ali, Clotilde Burdet, Marie Gaskowiak, Élise

Romagnan, Rafik Khendek, Farrah Mistoihi, Fanny de

Blas*, Davy Kima, Jean Siau, Alexis Soria, Julia

Tornel-Medina Matas, Julia Hladun, Alice Eising,

Oriana Rey.

* Goup référent

CentraleSupélec students: Alexis Meyer, Bastien

Frobert, Marc Pouyanne, Louise Delahaye, Jérôme

Garcia.

Conflicts of Interest

The authors declare no conflict of interest.

Fig. 27 Acknowledgments to our partners.

References

[1] Happold, E., and Liddell, W. 1975. “Timber Lattice Roof

for the Mannheim Bundesgartenschau.” The Structural

Engineer 53 (3): 99-135.

[2] Leyral, M., Ebode, S., Guerold, P., and Berthou, C. 2021.

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

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“Elastica Project: Dynamic Relaxation for Post-formed

Elastic Gridshells.” In Proceedings of the IASS

Annual Symposium, 23-27 August 2021, Surrey, U.K., pp.

1-16.

[3] Labbé, C. 2018. “Quoi de neuf/le gridshell (des jours

meilleurs)?” Archistorm. (in French)

[4] Philippart de Foy, G. 1984. Les Pygmées d'Afrique

centrale. Paris: Éditions Parenthèse. (in French)

[5] Douthe, C. 2007. “Étude des structures élancées

précontraintes en matériaux composites: application à la

conception des gridshells.” Ph.D., thesis, École Nationale

des Ponts et Chaussées. (in French)

[6] Chebyshev, P. L. 1878. Sur la coupe des vêtements.

Congrès de Paris: Association française pour

l’avancement des sciences. (in French)

[7] Boisse, P. 1994. “Modèles mécaniques et numériques

pour l’analyse non-linéaire des structures minces.” Ph.D.,

thesis, Université de Besançon. (in French)

[8] Bouhaya, L. 2010. “Optimisation Structurelle des

Gridshells.” Ph.D. thesis, École Doctorale Science

Ingénierie et Environnement. (in French)

[9] Ghys, E. 2011. “Sur la coupe des vêtements. Variation

autour d'un thème de Tchebychev.” L'Enseignement

Mathématique 57: 165-208. (in French)

[10] Barnes, M. R. 1999. “Form Finding and Analysis of Tension

Structures by Dynamic Relaxation.” International

Journal of Space Structures 14 (2): 89-104.

[11] Leyral, M. 2021. Faire Tenir—Structure et architecture,

edited by de la Villette, C. Paris: Savoir Faire de

L’Architecture. (in French)

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Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

222

Appendix A: Numerical Modelisation by Dynamic Relaxation and ELASTICA Algoritm

Reproduced and adapted with permission from: Marc Leyral, Sylvain Ebode, Pierre Guerold, Clément Berthou; Elastica project:

Dynamic Relaxation for Post-formed Elastic Gridshells, in Inspiring the Next Generation: Proceedings of the International

Conference on Spatial Structures 2020/21 (IASS2020/21-Surrey), edited by: Alireza Behnejad, Gerard Parke and Omidali Samavati,

published by University of Surrey, Guildford, UK, in August 2021 [2].

(1) Discretization of the Chebyshev Lattice Surface

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, the division of any surface into equilateral parallelograms (a necessary condition for flat fabrication) 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. Chebyshev therefore devised a method to create a piece of clothing adapted to the human anatomy, in

large quantities, quickly and at low cost (Fig. 1) [6].

(a) (b)

Fig. 1 (a) Discretization of a fabric and Chebyshev pattern for dressing a half-sphere [2]; (b) Chebyshev lattice: principle, by model and by

dynamic relaxation (Boisse [7] and Bouhaya [8]).

The problem formulated by Ghys [

9

] 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 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 a geometric approach, called “compass method”, used by Frei Otto (Fig. 2).

Fig. 2 Compass method and application by the ELASTICA algorithm [2].

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

223

(2) 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 (Figs. 3 and 4): its development had to wait until the end of the 20th century and computer-assisted numerical

modeling. It was then applied, among others, to shells (Otter, 1964), tensioned and inflatable structures [10].

Fig. 3 General principle of dynamic relaxation applied to an example of the form finding of a stretched canvas [11].

Dynamic relaxation allows solving static equilibrium problems by a fictitious and iterative dynamic calculation. It is valid for large

deflections. According to Barnes [10], “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.”

(

a

)

(b)

(c)

Fig. 4 Trial gridshell by Rombouts [12]: mesh of the architect’s form (a), form finding after dynamic relaxation (b), and

built project (c).

(3) Theory from an Analytical Point of View

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

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

224

at a velocity that varies with time. This lets us calculate at each iteration, the position of each node at the next instant.

In the case of a gridshell, there are (at least) three forces acting at the nodes (Fig. 5): 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 their axial

stiffness and deformation (𝑭

=𝑬𝑨∗∆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, 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

𝒊𝟏,𝒊𝟏

𝟐𝐬𝐢𝐧𝜶

𝒊

, we obtain

𝑴=

𝟐𝑬𝑰 𝒔𝒊𝒏𝜶

𝒊

𝑳

𝒊𝟏,𝒊𝟏

. 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

𝒊𝟏,𝒊

=

𝟐𝑬𝑰 𝐬𝐢𝐧𝜶

𝒊

𝑳

𝒊𝟏,𝒊

∗𝑳

𝒊𝟏,𝒊𝟏

for the first segment and 𝑭

𝒊,𝒊𝟏

=

𝟐𝑬𝑰 𝐬𝐢𝐧𝜶

𝒊

𝑳

𝒊,𝒊𝟏

∗𝑳

𝒊𝟏,𝒊𝟏

for the second.

(a) (b)

Fig. 5 Equilibrium of a node by the spring forces from Hooke’s law on the example of a loaded cable (a), and forces due to

bending in the case of a gridshell (b).

Once the acceleration 𝒂

𝒕

at a fictitious instant t has been calculated and knowing the initial velocities 𝒗

𝒕𝒅𝒕

at the same instant,

we deduce nodal velocities at the following instant t + dt:

𝐷=𝑣

∗d𝑡=𝑎

∗𝑑𝑡

2𝑣

∗𝑑𝑡 (5)

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.

(4) 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 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: the structure is going closer to its equilibrium, the potential energy is reduced by

conversion into kinetic energy. When the system passes through its equilibrium position, the forces balance and the potential energy

becomes zero. But nodal velocities, which are then maximum (and therefore the kinetic energy as well), cause a continuation of the

movement in the opposite direction: the system oscillates.

At convergence, which can only be reached by adding damping to dissipate the total energy of the system, 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), see Fig. 6.

α

i

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

225

Fig. 6 Energy transfer during the dynamic relaxation of a gridshell [2].

Damping can be either viscous damping by the addition of an additional force 𝑭

𝒂

= 𝒄 ∗ 𝒗

or kinetic damping. Our ELASTICA

algorithm uses this last method because it does not require setting new parameters and often allows for faster convergence. Barnes [10]

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.”

(5) Calculation of the Time Interval dt of the Iterations

A time interval too short or a nodal mass too high can lead to a divergence. Commonly, we choose a time interval dt and deduce

from it the fictitious nodal masses—different from the real nodal masses of the project—able to ensure the convergence of the

algorithm by the Barnes-Han-Lee formula: 𝒎=

𝐝𝒕²

𝟐

∗

∑𝑹

𝒂𝝁

∑𝑹

𝒇𝝁

where μ is the number of bars connected to each node (4

without bracing and 6 with). For convenience, we have set in ELASTICA tool the nodal mass m equal to the real mass and deducted

dt. This simple formula does not always ensure convergence: it is advisable to divide it by a safety factor (1.2 has been chosen in our

case after several tests).

(6) 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 (Fig. 7).

Kagome Project: Physical and Numerical Modeling Comparison for a Post-formed Elastic Gridshell

226

(a) (b)

Fig. 7 Chronology of form finding stages and stability control using dynamic relaxation by Rombouts [12], and extract of

the ELASTICA algorithm [2].

The ELASTICA algorithm is able to:

Divide any surface into a Chebyshev network using the compass method.

Solve form finding according to different parameters (geometry, single or double layering, with or without bracing, with or

without shear blocks, etc.).

Find the critical load that will cause the buckling of the structure.

Give stress in timber lattices.

Give stress in metallic or timber bracing elements.

Verify the local non-buckling condition of every bar in the project.

Calculate horizontal reactions.

Automatically draw up the fabrication and assembly plans of the structure.