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

Abstract

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
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, 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.

Full text

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
𝒊𝟏,𝒊 𝑭
𝒊,𝒊𝟏∗𝐜𝐨𝐬󰇛𝜶
𝒊󰇜𝟒𝑬𝑰 𝐬𝐢𝐧󰇛𝜶𝒊󰇜𝐜𝐨𝐬󰇛𝜶𝒊󰇜
𝑳𝟎𝟐 (2)
𝑵𝒔𝒆𝒓 =𝑭
𝒊𝟏,𝒊 𝑭
𝒊,𝒊𝟏∗𝐜𝐨𝐬󰇛𝜶
𝒊󰇜
𝟒𝑬𝑰 𝐬𝐢𝐧󰇛𝜶𝒊󰇜𝐜𝐨𝐬󰇛𝜶𝒊󰇜
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 sin󰇛𝛼󰇜
𝐿, ∗𝐿
,
𝐹, =2𝐸𝐼 sin󰇛𝛼󰇜
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.
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edited by de la Villette, C. Paris: Savoir Faire de
L’Architecture. (in French)
[12] Rombouts, J. 2019. “Optimal Design of Gridshells.” Ph.D.
thesis, Faculty of Engineering Science of Bruxelles.

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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
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(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
𝒊𝟏,𝒊𝟏
𝟐𝐬𝐢𝐧󰇛𝜶
𝒊
󰇜
, we obtain
𝑴=
𝟐𝑬𝑰 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.
(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
𝒕𝒅𝒕
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍

at the same instant,
we deduce nodal velocities at the following instant t + dt:
𝐷=𝑣



󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍
󰇍

∗d𝑡=𝑎

󰇍
󰇍
󰇍

∗𝑑𝑡

2𝑣

󰇍
󰇍
󰇍

∗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
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(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.