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Linear buckling of quadrangular and kagome gridshells: a comparative assessment

Romain Mesnila,b,∗, Cyril Douthea, Olivier Baverela, Bruno L´egerb

aLaboratoire Navier, UMR 8205, ´

Ecole des Ponts, IFSTTAR, CNRS, UPE, Champs-sur-Marne, France

bBouygues Construction SA, Guyancourt, France

Abstract

The design of gridshells is subject to strong mechanical and fabrication constraints, which remain largely unexplored for

non-regular patterns. The aim of this article is to compare the structural performance of two kind of gridshells. The

ﬁrst one is the kagome gridshell and it is derived from a non-regular pattern constituted of triangles and hexagons. The

second one results from a regular pattern of quadrangles unbraced by diagonal elements. A method is proposed to cover

kagome gridshells with planar facets, which reduces considerably the cost of fabrication of the cladding.

The sensitivity of kagome gridshells to geometrical imperfections is discussed. The linearised buckling load of kagome

gridshells is then compared to the one of quadrilateral gridshells. The most relevant design variables are considered in

the parametric study. Two building typologies are studied for symmetrical and non-symmetrical load cases: dome and

barrel vault. It reveals that the kagome gridshell outperforms quadrilateral gridshell for a very similar construction cost.

Keywords: grid shell, conceptual design, fabrication-aware design, kagome grid pattern, quadrangular grid pattern,

imperfection, buckling analysis, performance assessment

1. Introduction

Grid-shells are structures made of beam elements that

act as continuous shells structures. The choice of a

grid pattern inﬂuences both fabrication and structural be-

haviour. For example, triangulated structures are known

to be stiﬀer than quadrangular meshes. Quadrangular gr-

ishells rely on the bending stiﬀness of connections, whereas

triangulated gridshells beneﬁt from a shell-like behaviour

without the need for rigid connections. The better struc-

tural performance of triangular gridshells is however at

the cost of an increased node complexity due to higher

node valence. In quadrangular meshes, panels are however

not necessarily planar, and only speciﬁc curve networks on

surfaces or shape-generation strategies guarantee meshing

with planar quadrilaterals [1,2,3,4]. There is thus a nec-

essary trade-oﬀ between design freedom and fabrication

constraints.

This article focuses on a lesser known family of pattern,

called kagome grid pattern, composed from triangles and

hexagons and represented in Figure 1. The kagome pattern

can be found in Japanese basketry, where the members

are woven. We focus here on applications to structural

engineering and consider non-woven pattern, where all the

neutral axes of the beams are concomitant, and the beams

are rigorously straight. Like quadrilateral grids, kagome

grids present a node valence of four, which indicates a

reasonable cost of fabrication. Among other usage, kagome

∗Corresponding author

E-mail: romain.mesnil@enpc.fr

grids have been used in the architecture of Shigeru Ban and

for ornamentation purpose. Their structural possibilities

remain however largely unexplored, and little is known on

the planarity of the facets, a key element to the economy

of the envelope.

Figure 1: A kagome grid pattern covered with planar facets generated

with the method described in this paper.

Kagome grid pattern and quadrilateral grid pattern

have the same node valence, and their structural behaviour

can be compared qualitatively. Rigid connections are nec-

essary to guarantee in-plane shear stiﬀness of these pat-

terns. However, their relatively low node valence assures

the existence of a large families of torsion-free beam oﬀsets

compatible with the use of deep beams [5]. Kagome and

quadrilateral grid patterns can thus be built with very sim-

ilar technological solutions. Their relative structural per-

formances is however not quantiﬁed and will be studied in

this paper, whose main contributions are:

Preprint submitted to Engineering Structures November 15, 2016

•a strategy for the covering of kagome meshes with

planar facets, demonstrating that they could be

a viable alternative to triangular or quadrilateral

meshes;

•a parametric study comparing the linear buckling

load of kagome gridshells with quadrangular grid-

shells for shapes covered with planar facets;

•design guidelines for kagome gridshells.

The article is organised as follows: the ﬁrst section

presents the motivations for this work as well as relevant

literature in the ﬁeld of mechanics of gridshells. The sec-

ond section introduces the methodology chosen to assess

the structural behaviour of kagome gridshells. The third

section gathers the results of the conducted parametric

study. A brief discussion and conclusion sum up the con-

tributions of the present work.

1.1. Previous work on the mechanics of single-layered lat-

tice shells

The structural behaviour of gridshells is usually gov-

erned by non-linear eﬀects, most noticeably buckling [6].

Four buckling conﬁgurations can be observed in gridshells:

•Global buckling in the manner of a shell;

•Member buckling;

•Snap-through of one node;

•In-plane rotation of one node.

Some design recommendations, often emphasizing simple

shapes, like spherical cupolas have been published. Gioncu

published a state of the art on the buckling of reticulated

structures in 1995 [7]. A report produced by the Work-

ing Group of the International Association for Shells and

Spatial Structures (IASS) in 2005 completes this review

with analytical and numerical results, demonstrating the

important advances made in that ﬁeld [8]. A novel issue

is to be published in 2016. A design guide for the stability

of reticulated shells with a thorough literature review is

proposed in [9], showing a great mastery of this topic.

These guidelines identify two approaches to evaluate

the structural behaviour of a grid structure: homogeni-

sation methods and numerical experiments. This article

establishes a parametric numerical study, and uses previ-

ous work on homogenisation of grid structures to comment

the numerical results.

1.1.1. Homogenisation and equivalent shell thickness

Homogenisation techniques aim to formulate an equiv-

alent continuous behaviour of a heterogeneous structure

with a cell repeated periodically. These methods use the

superposition principle and usually work well for struc-

tures with a linear behaviour [10]. They have been suc-

cessfully used for planar grids [11], but a rigorous extension

to gridshells is diﬃcult because of the loss of periodicity,

due to the variations of curvature. A discussion on this

topic is proposed by Gioncu and Balut [12].

The advantage of equivalent thickness model is that

they provide structural engineers with simpler formulas

and can be of practical interest for conceptual structural

design. Some attempts to provide equivalent shell thick-

ness have been used in previous studies [13,14,15,16].

However, these models do not allow for the modelling of

localised buckling and the study of the inﬂuence of imper-

fections for shell structures remains tedious for non-trivial

shapes. Nowadays, the ever-growing computational power

makes the use of ﬁnite element modelling and non-linear

analysis ubiquitous in practice, and numerical simulations

are often preferred to homogenisation formulæ.

1.1.2. Numerical experiments

Numerical methods are used for the practical design of

gridshells, because they allow for integration of complex

issues, like material nonlinearities or geometrical imperfec-

tions. Some guidelines for the analysis of reticulated domes

have been proposed by Kato et al. [17,18]: these stud-

ies introduced geometrical imperfections and semi-rigid

nodes. Bulenda and Knippers [6] performed parametric

studies on domes and barrels vaults and evaluated the in-

ﬂuence of imperfections on the stability of gridshells. A

more complete study using ﬁnite element analysis to eval-

uate local node stiﬀness of patented connections has been

performed by Huang et al. [19]. Bruno et al. assessed

the inﬂuence of nodal imperfection and of Eigenmode Im-

perfection Method (EIM) more recently [20]. Malek et al.

[15] performed numerical investigations on the buckling

of spherical cap domes and considered geometrical values,

like grid spacing, or height over span ratio, as parameters.

This approach lead to recommendations for the design of

gridshells with triangular or quadrangular layout.

Other studies have evaluated the inﬂuence of residual

stresses in elastic gridshells [16,21]. A more complete

analysis was performed on the elastic gridshell built for the

Soliday’s festival in Paris, considering accidental ruin of

some members [22]. These studies show that high bending

stresses due to the form-ﬁnding process of elastic gridshells

have little inﬂuence on the buckling capacity of domes.

Such procedures could be extended to steel structures, in

order to assess the inﬂuence of other residual stress ﬁelds

on the stability of gridshells.

1.2. Imperfections

There are many diﬀerences between the ideal numerical

shell models and the built structures. These diﬀerences, or

imperfections can be of diﬀerent nature: loads, geometry,

material, residual stresses in the members. Thin shells are

known to be sensitive to imperfections [23]. These param-

eters are often set as a global geometrical imperfections.

Gioncu and Balut also point out that geometrical imper-

fection tend to govern over material nonlinearities for large

span structures [12].

2

Typically, the diﬀerence between the built geometry

and the computed model is of a few centimeters at most

[24]. It is therefore necessary to introduce a norm, in or-

der to asses realistic imperfections. In the following, the

norm k·k∞deﬁned as the maximal displacement is used.

Bulenda and Knippers propose a higher bound of L/500

for the imperfection with the inﬁnity norm [6]. Based on

data on the precision requirements for built project [25],

Malek et al. studied an imperfection of 3mm [15].

The choice of the shape function is discussed in Section

2.4. The ﬁrst buckling mode is recommended by design

codes, and was used for example for the design of the roof

of the British Museum and the Palacio de Comunicaciones

[26,25]. However, diﬀerent studies show that other imper-

fections shapes should be considered, as they result in a

bigger reduction of the buckling capacity of gridshells. Ex-

amples of such shapes can be found in [6] with the use of

dynamic eigenmodes, and a discussion on the choice of ap-

propriate imperfections is proposed in [20]. It has to be no-

ticed that there is no closed-form solution on the worst im-

perfection possible, some studies even demonstrated that

higher order eigenmodes can have a more critical eﬀect

on the reduction of buckling capacity [27]. The purpose

of this paper being to compare relative performance be-

tween kagome grid pattern and quadrilateral pattern, we

will consider the imperfections most commonly used in

current practice and limit the sensitivity analysis of section

2.4 to imperfection shapes following the ﬁrst eigenmode.

2. Methodology

2.1. Numerical experiment and choice of the parameters

Two typical free-form structures are barrel-vaults and

domes. Theses shapes are easily generated using transla-

tion or scale-trans surfaces surfaces, which have the advan-

tage of generating planar quadrilateral facets. A method

to convert such meshes to planar kagome meshes is de-

scribed in 3.1.

The dome is a surface of translation deﬁned with two

parabolæ.

z(x, y) = 4h

L2(y−L)·y+4H

d2x−d

2x+d

2(1)

d

L

L

h

l

Figure 2: Geometrical parameters describing the dome.

The barrel vault is a scale-trans surface. The curves on

the ground are sine curves, and the elevation is a parabola.

We write f(x) = dsin 2πx

L1, the equation of the surface

follows:

z(x, y) =

4h1−2·f(x)

L

L2(y−f(x)) (y−L+f(x)) (2)

L

h

L0

L1

L

l

d

Figure 3: Geometrical parameters describing the barrel vault.

The geometrical parameters describing the two models

are displayed in Figures 2and 3. The number of geo-

metrical parameters is quite important, we decrease their

numbers by introducing non-dimensional parameters. The

physical meaning of these ratios is explained and detailed

below. The main span Lof the structures is set to 30

meters. Three ratios Π1, Π2and Π3correspond to geo-

metrical parameters. The two ratios Π4and Π5are the

performance metrics studied in this article.

Aspect ratio

The geometry has a main span Land another charac-

teristic length d. The ﬁrst ratio is called aspect ratio and is

deﬁned by equation (3). For the domes, the ratio Π1cor-

respond to the ratio of curvatures, whereas for the barell

vault, higher values of Π1correspond to higher gaussian

curvature (the case Π1= 0 is a cylinder).

Π1=d

L(3)

Notice that diﬀerent aspect ratios could be constructed

from the barrel vault. For the simplicity of the demon-

stration, it was decided to set the ratio L0/L to 4 and

the ratio L1/L0to 2.5. These values are similar to the

conﬁguration of the gridshell roof covering the museum of

Downland [28].

Rise-over-span ratio

The name is self-explanatory: the second non-

dimensional parameter is the ratio of the characteristic

height hwith respect to the main span L. Common for-

mulas indicate that structural performance should increase

with this number.

Π2=h

L(4)

3

Structural density ratio

We consider here the grids to have a mean member

length of l. The comparison of this number to the main

span, as done in equation (5) gives indications on the grid

coarseness.

Π3=l

L(5)

Buckling ratio

The last parameter compares the buckling pressure pcr

found by linear buckling analysis to the member bending

stiﬀness EI/L4. The number described by equation (6) is

the value that is compared between kagome and quadri-

lateral meshes. The ratio I /L4being kept constant in this

study, the buckling ratio will be a measure of the stiﬀness

due to the form and mesh topology independently of the

section properties.

Π4=pcrL4

EI (6)

Notice that only the quadratic moment of inertia Iis

considered. A comparable non-dimensional number could

be constructed with the span L, the axial stiﬀness EA and

the critical pressure pcr. However, it is a well-known fact

that member shortening has more impact on very shallow

structures which won’t be considered in our study.

Structural eﬃciency

We introduce ﬁnally a variable, later called structural

eﬃciency, in order to compare the performance of kagome

and quadrilateral grid pattern. The parameter is deﬁned

as:

Π5=pcr.A

m·g(7)

where Ais the horizontal surface covered, mthe mass of

the structure and gthe acceleration due to gravitational

forces on Earth. The number deﬁned by equation (7) com-

pares the total resultant of vertical forces to the resul-

tant of gravity forces. It must be noticed that for a same

structural density, i.e. individual member length, the total

length of members diﬀers between the kagome and quadri-

lateral grid. For a square grid with edge length l, the total

beam length per unit area is 2

l. For a kagome grid made

of regular hexagons and triangles and edge length l, the

total beam length per unit area is √3

l. From this simple

case, an estimation of the ratio of the masses is given by:

mKagome

mQuad ∼√3

2'86% (8)

In other terms, for a same structural density, the kagome

grid is slightly lighter than the quadrilateral grid. This dif-

ference justiﬁes the fact to look more closely at the struc-

tural eﬃciency, and not only at the buckling load.

Table 1sums up the range of variations of each pa-

rameter. It is chosen to ﬁt existing designs: for exam-

ple the rise-over-span ratio remain in general superior to

0.1 to avoid high bending stresses or snap-through. The

structural density are chosen so that the minimal mem-

ber length is 1.253, a reasonable value compared to built

projects. Each set of geometrical parameters generates a

geometry for a quadrilateral and a kagome grid. Two load

cases are considered, as discussed in Section 2.2. The para-

metric study proposed in this paper consists thus of 500

linear buckling analysis and several fully nonlinear analysis

for the study on imperfections sensitivity.

2.2. Material, loads and boundary conditions

The material used is steel, and we restrict our study to

a linear elastic material law. Detailed studies with plas-

ticity have been made previously and are reviewed in [8].

These studies are necessary to evaluate with high ﬁdelity

the post-buckling behaviour of gridshells, at the cost of

high computational eﬀort. In the ﬁrst steps of the design

process, engineers need to perform many analyses, often

with simpliﬁed assumptions and a linearised buckling load

is already a good indicator of the structural performance.

It was already chosen as design criterion in [15] and [21].

The modelling hypothesis follow:

•the supports are pin joints with full translational re-

straint;

•the joints are assumed to be fully rigid;

•in the barrel vault, the arches are simply-supported;

•distributed loads are replaced by concentrated loads

at connections.

The members are made of circular hollow section, with

a wall thickness of 10mm and a diameter of 200mm for the

dome and the barrel vault. With these geometries, there

is no diﬀerence between Iy,Iz, and torsional buckling of

members is not possible, which simpliﬁes the parametric

study. In the followings, we use beam elements with three

elements per member. This subdivision allows to capture

eventual localised buckling modes, which can arise in grid-

shells.

Two load cases are considered: a uniform projected

vertical load of 1kPa and a non-symmetrical load of 1kPa

applied following the normal of the surfaces with the pat-

tern of Figure 4.

+

--

+

Figure 4: Areas of positive and negative pressure for the non-

symmetrical load case, top view of Figure 2.

Current literature focuses more on uniform symmetri-

cal load cases [15]. Koiter has shown than spherical caps

are subject to geometrical imperfections for such load case,

but not for concentrated load. Therefore, it is meaningful

4

Π1Π2Π3

Barrel Vault [0,0.025,0.05,0.075,0.1,0.125,0.15] [0.1,0.2,0.3,0.4,0.5] [ 1

24,1

16,1

12]

Dome [1,1.33,1.67,2] [0.1,0.2,0.3,0.4,0.5] [ 1

32,1

24,1

16]

Table 1: Variations of the parameters in the present study.

to consider this kind of load case in our sensitivity analysis.

Furthermore, non-symmetrical load cases are known to be

more critical than symmetrical ones for buckling and of-

ten govern the sizing of gridshells. The asymmetrical load

case is thus also considered in order to provide guidance on

situations closer to the engineering practice. The chosen

asymmetrical load represents here a wind load, which usu-

ally features areas of positive and negative pressure. Wind

loads computed from the Eurocode can usually be decom-

posed between a symmetrical and asymmetrical compo-

nent. Since we already study a symmetrical load case, we

focus only on the non-symmetrical component of this load.

2.3. Buckling analysis

This study mainly adopts linear buckling analysis of

perfect gridshells. In addition, geometric non-linear anal-

ysis on structures with imperfections are carried out to

preliminary evaluate the eﬀects of imperfections. In linear

buckling analysis, it is often considered that the stiﬀness

matrix of a structure can be written as the sum of KEthe

elastic stiﬀness (independent of the applied load P) and

of KGthe geometric stiﬀness (which decreases here with

P). The linear buckling analysis makes the assumption

that the coeﬃcients of KGvary linearly with the ampli-

tude of Pand ﬁnds thus couples of buckling factor and

displacement vector (λ, Φ) so that:

(KE+λKG)·Φ=0(9)

The non-linear buckling problem becomes therefore

the eigenvalue problem shown in equation (9), the lowest

eigenvalue λ1giving an estimate of the buckling capacity

of the structures. The linearisation hypothesis is in fact

a Taylor development, and it is valid if the displacements

before buckling are small. In structures subject to large

deformations, like gridshells, the linear buckling analysis

can overestimate largely the real buckling capacity. In

detailed design, fully non-linear analysis is thus required

to assess the bearing capacity of gridshells, but the lin-

ear buckling analysis can be quickly estimated and can be

helpful in conceptual design stage [15].

The analysis software used is Karamba, a plug-in in-

tegrated with parametric CAD tools RhinocerosTM and

GrasshopperTM[29,30]. This software enables to perform

structural analysis within a 3D-modelling environment,

which considerably eases the design process for structural

engineers.

2.4. Inﬂuence of imperfections

This section focuses on the inﬂuence of imperfection

on kagome gridshells. The tested geometry is a dome sup-

ported on a circular plan (Π1= 1, Π2= 0.2, Π3= 1/32),

and subject to a uniform vertical load. Figure 5shows dif-

ferent plots of linear buckling load pcr normalised by the

linear buckling load of the structure without imperfection

pcr,0computed with diﬀerent imperfection amplitudes for

the inﬁnity norm. The kagome pattern is more sensitive to

imperfections than the quadrangular pattern. For the am-

plitude of L/500, the reduction of the linearised buckling

load is approximately 10%.

0.6

0.8

1

−L/200

−L/500

0

L/500

L/200

Imperfection a

L

Normalised load factor pcr

pcr,0

Kagome gridshell, Mode 1

Quadrilateral gridshell, Mode 1

Figure 5: Inﬂuence of imperfection scale on the linear buckling load.

A second order analysis is thus performed on both per-

fect and imperfect geometry to evaluate more precisely

the inﬂuence of imperfections. The load/displacement dia-

grams for the kagome and quadrilateral gridshells obtained

are displayed in Figure 6and 7respectively. The four hor-

izontal lines represent the linearised buckling loads. Three

imperfection amplitudes are considered: the ﬁrst one is a

small imperfection (L/1500) and can be compared to the

one used by Malek et al. [15], the second corresponds to

the (L/500), as proposed by Bulenda and Knippers [6],

the third one is of (L/200) as recommended in EC3.

The structures with imperfections do not reach their

linearised buckling loads, contrary to perfect structures.

The load/displacement graphs are less curved than the

5

ones of the gridshells without imperfection. Consider for

example the imperfect geometry with a norm of 15cm

(L/200): the structure behaves in a fully non-linear man-

ner and it is hard to distinguish a linear domain. This

indicates that the linearised buckling load is not suited for

structures with high imperfection norm, as high stresses

are at stake before buckling. Considering the imperfection

amplitude of L/500 proposed by Bulenda and Knippers

[6], we notice that the bearing capacity of the imperfect

structure decreases by approximately 15%. With the norm

proposed by Malek et al. [15], the loss of bearing capacity

with imperfections is negligible for the considered geome-

try.

0 50 100 150 200

100

200

300

Displacement (cm)

Load factor

Perfect geometry

Mode 1, e∞= 2cm (L/1500)

Mode 1, e∞= 6cm (L/500)

Mode 1, e∞= 15cm (L/200)

Figure 6: Load/displacement diagram for a dome covered with a

kagome grid pattern.

0 50 100 150 200

50

100

150

200

Displacement (cm)

Load factor

Perfect geometry

Mode 1, e∞= 1cm (L/1500)

Mode 1, e∞= 6cm (L/500)

Mode 1, e∞= 15cm (L/200)

Figure 7: Load/displacement diagram for a dome covered with a

quadrilateral grid pattern.

The qualitative behaviour of kagome gridshells with re-

spect to imperfections is similar to what has been analysed

in previous research on quad pattern [6,15], and Figure

6and 7are indeed similar. In the treated example, the

decrease of bearing capacity with respect to imperfection

amplitude is however less important than in previous lit-

erature, for example in [6] who considered cable braced

quadrilateral gridshells. It can however be compared to

the decrease observed in a previous study for quadrilat-

eral gridshells [15]. The explanation given by Malek et

al. is that quadrilateral (and kagome) gridshells rely on

in-plane bending stiﬀness of the members to withstand

loads, whereas triangulated gridshells can transfer out-of-

plane loads with axial forces in the members. Small im-

perfections can therefore introduce bending moments in

triangular gridshells and change their load transfer mech-

anism, from axial forces to axial and bending combined.

To go further, we propose to study the load where

the displacement reaches the service limit state. This dis-

placement is found with a second order analysis. We set

δSLS =L

200 and compare the inﬂuence of imperfection for

quad and kagome gridshells. Figure 8shows the critical

non-dimensional service load deﬁned as pSLS L4

EI . Kagome

grids remain stiﬀer regardless of the imperfection. This

additional criterion shows also that the imperfection does

not change the relative performance of kagome and quadri-

lateral grids for simple performance metrics used in pre-

liminary design.

0246

·10−3

0

2

4

·104

Imperfection a

L

pSLS L4

EI

Quad

Kagome

Figure 8: Non-dimensional SLS load for Kagome and Quad grid-

shells

In summary, this section discussed the inﬂuence of im-

perfection on the structural performance of gridshells. The

study of sensitivity to imperfection suggests that kagome

and quadrilateral gridshells without cable-bracing have

qualitatively similar behaviours for linear buckling anal-

ysis. Even classical approaches, like Eigenmode imper-

fection, illustrate the limitations of linear buckling anal-

ysis for detailed stages of design. In the following, we

propose nonetheless to compare the bearing capacity of

kagome and quadrilateral gridshells by studying the lin-

earised buckling load without imperfections, because this

performance indicator is commonly used in conceptual de-

sign stages. In detailed design, geometrical and material

non-linear analysis will be required to assess the structural

response of gridshells with full accuracy.

3. Generation of planar kagome grid pattern

3.1. An algorithm for kagome pattern with planar facets

We present here an original method that converts pla-

nar quadrilateral (PQ) meshes to planar kagome (PK)

meshes. This method guarantees a control of the cost of

6

the cladding, which is an important issue in free-form ar-

chitectural design.

The algorithm takes a PQ-mesh as an input, like il-

lustrated in Figure 9. Not all PQ-meshes are acceptable,

but only those which can be coloured as a chequerboard.

In the algorithm, the dark faces become hexagons and the

white one become triangles. Starting from a quad mesh

(left), one must determine intermediary points (middle)

which deﬁne new vertices of the kagome grid (right).

Figure 9: Conversion of a quadrilateral mesh to a kagome mesh

The choice of the intermediary point is restricted by

the fact that the two adjacent hexagons have to be pla-

nar. Consider three consecutive planar quads Qi−1,Qi

and Qi+1. The algorithm determining the new vertices

can be written as follows, and is detailed in Figure 10:

1. Compute the barycentre Giof the quadrangle Qi;

2. Compute the intersection of the planes (Qi−1),

(Qi+1);

•If the intersection is a plane, create the node

Ni=Gi;

•If it is a line (L), create the node Nias the

orthogonal projection of Gion (L). Niis the

closest point to Gion (L).

3. Repeat steps 1 and 2 in a chequerboard pattern.

Other points on (L) could be chosen, but the choice pro-

posed in this algorithm yields satisfactory and regular re-

sults, as illustrated in Figure 1.

Finally, we noticed in our formal explorations that the

algorithm can encounter some diﬃculties if the curvature

of the surface is very low. With numerical imprecisions,

the binary choice of the second step of the proposed al-

gorithm can lead to instabilities. Therefore, we introduce

a number εcorresponding to the fabrication tolerance for

planarity. If the distance between Gand each of the two

planes is inferior to ε, we set the point Gas a vertex of

the kagome mesh.

3.2. Generality of the method

The previous algorithm gives a systematic method to

generate PK-meshes. Using the work by Liu et al. [31], we

can transcribe this result into notions of smooth diﬀeren-

tial geometry. They prove indeed that planar quadrilateral

meshes correspond to parametrisations (u, v) of surfaces

satisfying a simple equation:

det ∂u, ∂v, ∂2

uv= 0 (10)

The curves networks satisfying this equation are called

conjugate-curve networks. Notice that equation (10) is

satisﬁed by lines of curvatures as ∂2

uv =0. Therefore,

any surface admits conjugate curve networks. This means

that the method proposed in this article can be applied

on any shape. Practically, the post-rationalisation tech-

niques used in [32] or bottom-up techniques like [33] can be

used to ﬁnd conjugate curve networks on free-form shapes.

Kagome meshes laid along these networks will therefore be

close to PK-meshes.

3.3. Applications to gridshells

Several strategies for shape generation of gridshells

with planar facets have been employed in the past. Among

them, surfaces of revolution, scale-trans surfaces [1] or

moulding surface, which generalise the notion of surface

of revolution [2]. They can be combined with our con-

version algorithm to generate kagome meshes with planar

facets.

It must be noticed here that the kagome mesh obtained

from a square grid in Figure 9is irregular: the hexagons

seem a bit stretched. Simple trigonometric considerations

show that the regular kagome pattern in the plane is ob-

tained from a rectangular grid with an aspect ratio of √3.

The grids generated in this paper use the same rule, as we

aim for visually regular patterns, like the ones displayed

in Figure 1and 18. Some simple geometrical properties of

these grids are discussed in Appendix A.

4. Stability of kagome gridshells

4.1. Buckling of barrel vaults

Linear buckling analyses were performed on barrel

vaults with diﬀerent geometrical conﬁgurations under sym-

metrical loading, and the results are shown in Figure 12

in a non-dimensional form. In Figure 12a, Π1= 0 and

there is no corrugation, while in Figure 12b Π1= 0.15 and

the shape is ondulating like the one shown in Figure 11.

We notice that the corrugation is signiﬁcantly improving

the structural behaviour. The case Π1= 0.15 has a buck-

ling load almost four times higher than the cylinder (case

Π1= 0). The best design is shown in Figure 11.

It appears that, in general, kagome grids have a higher

buckling load. In order to quantify this assertion, we intro-

duce the number r, later called ratio of eﬃciency, deﬁned

by equation (11).The same parameters values are chosen

identical for both grids. A ratio superior to 1 indicates that

the kagome gridshell is more eﬃcient than the quadrilat-

eral gridshell.

r(Π1,Π2,Π3) = Π5,Kag ome (Π1,Π2,Π3)

Π5,Quad (Π1,Π2,Π3)(11)

In the following, Π1, Π2and Π3have been varied and

results are shown in Figure 13. We have chosen to repre-

sent the buckling load in terms of Π1and to compare the

best design of both structures deﬁned by equation (11).

7

G

Qi−1

Qi

Qi+1

(a) Step 1

G

N

Qi−1

Qi

Qi+1

(b) Step 2

N

Qi−1

Qi

Qi+1

(c) Creation of the new edges

Figure 10: Details of the conversion to a Planar Kagome mesh.

Figure 11: Optimal barrel vault: Π1= 0.15, Π2= 0.3, Π3=1

24 .

This ratio remains above 1.5, with a peak at 2.6. The

most eﬃcient designs correspond to moderate rise-over-

span ratio (Π2= 0.3) and a dense grid.

rmin (Π1) = min

Π2,Π3

r(Π1,Π2,Π3)

rmax (Π1) = max

Π2,Π3

r(Π1,Π2,Π3)

rbest (Π1) =

max

Π2,Π3

Π5,Kagome (Π1,Π2,Π3)

max

Π2,Π3

Π5,Quad (Π1,Π2,Π3)

(12)

4.2. Barrel vaults under non-symmetrical loads

Non linear analysis with non-symmetrical loads were

then considered with the distribution shown in Figure 4.

The behaviour of the structure is then dominated by bend-

ing and becomes very diﬀerent for both structures. Con-

sider Figure 14: the quadrilateral grid has a higher buck-

ling load, but the buckling occurs for a high level of dis-

placements, superior to 20% of the span. Of course, the

ruin of members will occurs before the structure buckles

and the results on linear buckling analysis would be sub-

ject to caution for the quadrilateral grid in this case.

This is a general situation: under non-symmetrical

loads, quad gridshells are considerably softer than kagome

gridshells. Considering the large displacements of the

quadrilateral gridshells under non-symmetrical loads, it

did not seem relevant to display the results on linear buck-

ling analysis for this load case. For the studied exam-

ple, the kagome grid is indeed 5 times stiﬀer. Using the

same SLS criterion than previously ( L

200 ), the kagome grid

clearly outperforms the quadrilateral grid.

4.3. Buckling of domes

The same parametric study is then reproduced for the

dome geometry. Figure 15 shows the non-dimensional

buckling loads computed for the symmetrical load case.

Kagome and quad grids have a similar behaviour: the

buckling load is a decreasing function of Π3. For slender

domes (smaller values of Π2), increasing the height also

increases the buckling load, but a maximum is reached

when Π2is approximately 0.3 (this optimal value of Π2

depends on the cross-section used). The buckling becomes

then more localised, and a change of the shape has lower

impact on the buckling.

Figure 16 shows then rmin,rmax and rbest for diﬀerent

values of Π1. It is noticed that the kagome gridshell is

more eﬃcient than the quadrilateral gridshell in the sense

of equation 11 and this for all the conﬁgurations consid-

ered in the present study. The gain in structural eﬃciency

is very important, especially for domes with a moderate

aspect ratio, when the shell is the most eﬃcient. It can be

concluded that kagome gridshells are more eﬃcient than

quadrilateral gridshells when considering linear buckling

analysis. The most interesting geometrical conﬁgurations

(moderate rise-over-span, and small aspect-ratio) are also

the ones where the relative performances of the two meshes

typologies diﬀer the most. The eﬃciency can be doubled

in those cases.

Finally, Figure 17 compares the performance of kagome

and quadrilateral gridshells for nonsymmetrical load cases

with linear buckling analysis. The kagome gridshell re-

mains more eﬃcient in all cases, with a minimum gain

in structural eﬃciency of 23%. The tendency is inverted

compared to the symmetrical load cases: kagome grids are

more eﬃcient when the ratio Π2is higher.

8

0.1 0.2 0.3 0.4 0.5

0

1,000

2,000

Rise over span ratio Π2

Π4

(a) Cylinder: Π1=d

L= 0

0.1 0.2 0.3 0.4 0.5

2,000

4,000

6,000

8,000

Rise over span ratio Π2

Π4

Quad,Π3= 1/24

Kagome, Π3= 1/24

Quad Π3= 1/16

Kagome, Π3= 1/16

Quad Π3= 1/12

Kagome, Π3= 1/12

(b) Corrugated barrel vault: Π1=d

L= 0.15

Figure 12: Comparison of the buckling capacity of kagome and quadrilateral grids for the barrel vault geometry.

05·10−20.1 0.15

0

1

2

3

Aspect ratio Π1

rKagome

rQuad

rbest

rmin

rmax

Figure 13: Comparison of the best designs for diﬀerent values of Π1.

0.1 0.2

0

1,000

2,000

3,000

4,000

Displacement δ

L

p·L4

EI

Quad

Kagome

Figure 14: Load-displacement for a non-symmetrical load

Π1= 0.075,Π2= 0.3,Π3=1

24

5. Discussion

5.1. Shape of buckled domes

A more detailed look at the parametric study shows

that the nature of buckling modes diﬀers between kagome

and quadrilateral gridshells. Figure 18 illustrates the ﬁrst

buckling modes for two domes with the same member

length, both for quadrilateral and kagome meshes. On

these images, darker colours indicate larger displacements.

Each dark spot corresponds to an ’anti-node’ on the buck-

led shape. Counting these spots, it can be noticed that

the number of anti-nodes is higher in the kagome grid,

and that the wavelength is shorter. This diﬀerence has

been observed for coarse and ﬁne grids.

This diﬀerence illustrates the fact that kagome grid-

shells have a higher in-plane shear stiﬀness than quad-

rangular meshes. Their higher buckling capacity can be

explained by the fact that they activate buckling modes

with shorter wavelength. This remark also holds for barrel

vaults. Figure 19 shows the same kind of phenomenon for

the buckling modes of the most eﬃcient designs of barrel

vaults of our study. There is the same number of anti-

nodes in Figure 19a and 19b, but the anti-nodes are more

concentrated in the kagome grid.

5.2. Inﬂuence of mesh reﬁnement

The results of the previous section indicate that reﬁn-

ing of meshes (diminishing Π3) improves the critical buck-

ling load of gridshells. We show a more detailed analysis

of this statement by studying a dome with Π1= 1.33. The

convergence of the structural eﬃciency with respect to the

number of cells is interpreted with homogenisation princi-

ples and analytical formulæ from [11] and [34] and detailed

in Appendix B

Consider a unit cell with characteristic length l(deﬁned

in Figure 2and 3). If one builds an equivalent shell, it is

meaningful to consider that the bending and axial stiﬀness

Dand Adepend linearly on 1/l (doubling the number of

beams would double the bending stiﬀness). This is found

9

0.1 0.2 0.3 0.4 0.5

0

2

4

6

·104

Π2(rise-over span ratio)

Π4

(a) Π1=d

L= 1

0.1 0.2 0.3 0.4 0.5

0

0.5

1

1.5

2·104

Π2(rise-over span ratio)

Π4

Quad,Π3= 1/32

Kagome, Π3= 1/32

Quad Π3= 1/24

Kagome, Π3= 1/24

Quad Π3= 1/16

Kagome, Π3= 1/16

(b) Π1=d

L= 2

Figure 15: Comparison of the buckling capacity of kagome and quadrangular gridshells for the dome geometry.

1 1.2 1.4 1.6 1.8 2

0

1

2

Aspect ratio Π1

rKagome

rQuad

rbest

rmin

rmax

Figure 16: Comparison of the structural performance of domes under

symmetrical load.

1 1.2 1.4 1.6 1.8 2

0

1

2

Aspect ratio Π1

rKagome

rQuad

rbest

rmin

rmax

Figure 17: Comparison of the structural performance of domes under

non-symmetrical load.

in homogenised models by Leb´ee and Sab for ﬂat thick

quadrangular beams layouts [11].

D=EI

l

A=EA

l

(13)

The buckling load of an isotropic spherical shell un-

der uniform pressure pcr is given by following the formula,

found for example in [35] or [34]:

pcr =2√AD

R2(14)

By combining equations (13) and (14), the critical buck-

ling load of an equivalent isotropic shell depends thus lin-

early on the number of cells. It is well-known that ho-

mogenised models describe accurately the actual model

when the number of cells is large enough. Having these

considerations in mind, we should expect the ratio Π5to

be constant for suﬃciently small values of Π2, because the

mass mvaries linearly with 1/l and so does the homoge-

nized buckling load. Figure 20 shows the variations of the

structural eﬃciency with respect to Π3. It appears that

Π5tends to a constant when 1

Π3increases. The conver-

gence is reached for 1

Π3'25. This value is linked with the

chosen cross-section, here 200mm pipes. The slenderness

of the members at 1

Π3'25 is of 5, which is higher than

what is found in built projects.

The convergence of the structural eﬃciency to a con-

stant indicates that kagome grids tend to behave like

isotropic shells. On the contrary, the eﬃciency of the

quadrilateral grid does not converge to a constant when

the grid is reﬁned. To better understand this, consider

that the buckling capacity of orthotropic shells depends

on the in-plane shear stiﬀness. An example of analytical

formula is given for shells of revolution in [34]. Still re-

ferring to the results of Leb´ee and Sab [11], we give an

estimate of the in-plane shear stiﬀness Gxy for quadran-

10

(a) First mode, Π4= 889 (b) First mode, Π4= 6885

Figure 18: Comparison of the ﬁrst buckling modes for quadrilateral and kagome meshes.

(a) First mode, Π4= 6145 (b) First mode, Π4= 9361

Figure 19: Comparison of buckling modes on the most performant barrel vaults in our study.

20 30 40 50 60

500

1,000

1,500

1

Π3(number of cells)

Π5

Π2= 0.1

Π2= 0.2

Π2= 0.3

Π2= 0.4

Π2= 0.5

Figure 20: Structural eﬃciency for diﬀerent reﬁnements of a kagome

grid (Π1= 1.33).

gular grids (we make the assumption that the beams are

Euler-Bernoulli beams):

Gxy =EI

l3(15)

This term clearly increases faster than the bending stiﬀ-

ness when the unit-cell becomes smaller. This non-

linearity explains why the graph of Figure 21 increases

with the number of cells 1

Π2without reaching a plateau.

The convergence study is thus a good indicator of the fact

that kagome gridshells are isotropic, whereas quadrilateral

grids are orthotropic.

5.3. Design guidelines for kagome gridshells

Structural engineers can improve the eﬃciency of their

designs by using diﬀerent strategies. We discuss here some

of these. In our study, the shape is an important fac-

tor of performance: changing a rise-over-span ratio from

10% to 20% doubles the structural eﬃciency. The opti-

mal rise-over-span ratio is around 30%. For larger values,

the gridshells become subject to localised buckling, and

overall curvature of the shape does not provide any help.

The change of geometry does not bring signiﬁcant changes

in the cost of fabrication of the elements, as our method

guarantees meshing with ﬂat panels.

Increasing the structural density also increases the

structural performance of quadrilateral and kagome grids.

For kagome grids, this strategy has a limit, as the struc-

tural eﬃciency tends to a constant when the number of

11

20 30 40 50 60

200

400

600

800

1,000

1

Π2(number of cells)

Π5

Π3= 0.1

Π3= 0.2

Π3= 0.3

Π3= 0.4

Π3= 0.5

Figure 21: Structural eﬃciency for diﬀerent reﬁnements of a quadri-

lateral grid (Π1= 1.33).

cells tends to inﬁnity. Even for very high structural den-

sity, this phenomenon does not occur for the quadrilateral

grids studied in this paper. This strategy has however

a practical limitation, as the number of connections in-

creases when increasing the density of the grid. Connec-

tions are very expensive and often govern the cost of the

structure in gridshells. Denser grids are also costly, and

the beneﬁt in structural performance might be tempered

by an increased construction cost.

We note here that the conversion rule chosen implies

that, for a same value of Π2, kagome grids have less vertices

(and thus connections) than quadrilateral grids. Writing

NKagome and NQuad the number of nodes, we have follow-

ing simple relation proven in Appendix A:

NKagome

NQuad

=√3

2'86% (16)

Therefore, the kagome grids of the present study are struc-

turally more eﬃcient than quadrilateral grids for gridshells

designed with a linear buckling criterion, and their cost of

connections is also signiﬁcantly lower.

6. Conclusion

This article has introduced a method to cover kagome

meshes with planar facets. This simpliﬁes the fabrication

of the cladding, a major concern in free-form architec-

tural design. The bearing capacity of kagome gridshells

was then studied and compared to the one of quadrilat-

eral gridshells. The results seem promising, as the kagome

gridshells has a signiﬁcantly higher performance in our

case studies, both for symmetrical and non-symmetrical

load cases. The better performance of kagome grid pat-

tern seems to come from its higher in-plane shear stiﬀness.

The gain in structural eﬃciency compared to quadrilateral

gridshells is higher when biaxial stresses are at stake in the

structure.

The gridshells were not compared to other common so-

lutions, like cable-braced or triangular gridshells. While

these solutions are probably structurally more eﬃcient

than kagome grid patterns (a comparison between kagome

and triangular pattern for a GFRP gridshell has been done

in [36], and shown that triangular pattern can be two or

three times stiﬀer than kagome grid pattern), they are also

more complex to build, due to high node valence or tun-

ing of cable tension. Steel contractors might prefer to build

moment connections rather than installing cables [4].

The study proposed in this paper considered mainly

linear buckling analysis, a tool suited for the exploration

of the design space in preliminary phases of design. The in-

ﬂuence of imperfections was however discussed and showed

that, like triangular grids, kagome grids are sensitive to ge-

ometrical imperfections. Some parameters were not con-

sidered in our studies: introducing realistic node stiﬀness

and material non-linearities in the parametric study could

greatly improve the estimation of real collapse loads and

would provide in further work reliable values for detailed

design.

Acknowledgment

We want to thank the anonymous reviewers who con-

tributed to the improvement of this paper with construc-

tive remarks. The authors also appreciate preliminary cal-

culations by Fernando Vianna Brasil Medeiros and Lean-

dro Dos Reis Lope, both graduate students at l’´

Ecole Na-

tional des Ponts et Chauss´ees. Finally, the authors thank

Charis Gantes (NTUA) for fruitful discussions that led to

the introduction of the simpliﬁed SLS criterion used in this

paper.

This work was made during Mr. Mesnil doctorate

within the framework of an industrial agreement for train-

ing through research (CIFRE number 2013/1266) jointly

ﬁnanced by the company Bouygues Construction SA, and

the National Association for Research and Technology

(ANRT) of France.

Appendix A. Basic properties of kagome grid pat-

tern

The kagome grids generated in our study tend to have

a uniform member length. We propose simple calculations

to estimate the number of connections or member length

per unit area for a planar kagome grid made out of regular

hexagons and triangles.

Appendix A.1. Description of the pattern

The regular kagome pattern is made out of regular

hexagons and triangles. The pattern is periodic, and can

thus be described by the study of a unit-cell shown in Fig-

ure A.22. In this image, all the edges have the same length

l, the dimensions of the unit cell are easily found based on

properties of equilateral triangles.

12

2√3l

2l

Figure A.22: A basic cell of a kagome grid.

The pattern is compared to a square pattern, where

the unit cell is obviously a square with edge length l.

Appendix A.2. Structural density

We compute now the edge length per unit area. In the

unit cell, we count 10 edges and 4 half edges. The edge

length per unit area LAis thus:

LA=10 + 4 ·1

2·l

2√3l·2l=√3

l(A.1)

We can compare this value with the edge length per unit

area for the square pattern, where LA=2

l. For a same

edge length, the ratio of member lengths is thus equals

to √3

2. This gives the estimation for the mass ratio of

equation (8).

Appendix A.3. Number of connections

The number of nodes per unit area is an important

question, as the cost of connections highly impacts the

cost of gridshells. For the unit cell depicted in Figure A.22,

there are 5 nodes that belong only to the cell (in white),

whereas 4 nodes belong to 4 adjacent cells (in black). The

number of connections per unit area is thus:

Nnodes =5+4·1

4

2√3l·2l=√3

2l2(A.2)

For a square grid, the number of nodes per unit area is

simply 1

l2. The ratio of these two values is thus equals to

√3

2, which gives an estimate for the ratio used in equation

(16).

Appendix B. Homogenisation approach and

equivalent buckling loads

In this Section, we adapt the formula of an anistropic

spherical cap of radius Runder uniform pressure found

by Crawford [34] to a quadrangular gridshell with equiva-

lent properties derived from [11]. The problem treated by

Crawford considers that the shell is isotropic with prin-

cipal axis along parallel and meridians. The geometry

is diﬀerent from the domes studied in this paper, but it

one of the only analytical formulæ available in the litera-

ture for orthotropic shells. We take count of the fact that

our problem deals with circular hollow sections to simplify

Iy=Iz=I.

Appendix B.1. Equivalent shell stiﬀness of a quadrangular

gridshell

Let us construct the equivalent axial and bending stiﬀ-

ness tensors from the homogenisation of a quadrilateral

grid. From [11], we have:

Axx =Ayy =A=ES

l

Gxy =l

GS+l3

12EI −1

νx= 0

νy= 0

(B.1)

and

Dxx =Dyy =D=EI

l

Dxy =GJ

l

(B.2)

The grid relies only on bending of elements for the in-

plane shear stiﬀness, and on beam torsion for the torsional

stiﬀness of the equivalent shell. All terms depend linearly

on 1

l(equivalently the number of cells) except the in-plane

shear stiﬀness, which depends on 1

l3.

Appendix B.2. Buckling of orthotropic spherical cap under

uniform pressure

We derive now the theoretical buckling load of an

anistropic shell from the work of Crawford [34]. The equa-

tions simplify greatly when Dxx =Dyy and Axx =Ayy .

Crawford introduces the quantities D3,G3and Ψ deﬁned

by:

D3=νx·Dy+Dxy

G3=2Gxy

1−νxνy−2νy

Gxy

Ayy

Ψ = A1−ν2

x

DR

(B.3)

With these notations, the buckling load of the isotropic

spherical shell is:

pcr =4D√Ψ

R(B.4)

13

Crawford computes then the ultimate buckling load of the

anistropic shell pcr given by:

pcr =

pcr if D3

D≥A

G3

pcr 1 + D3

D

1 + A

G3!

1

2

if D3

D<A

G3

(B.5)

In the case of gridshells, the second inequality is veriﬁed,

and using equations (B.1) and (B.2), we obtain:

pcr =4√AD

R2v

u

u

t

1 + Dxy

D

1 + A

2Gxy

(B.6)

So that ﬁnally:

pcr =4E√SI

lR2v

u

u

u

t

1 + GJ

EI

1 + ES 1

2GS+l2

24EI (B.7)

The ﬁrst term corresponds to the buckling capacity of an

istropic shell, it is proportional to 1

l. The second term

(under the square root) varies nonlinearly with 1

lbecause

of the term in l2, which comes from the equivalent in-

plane shear stiﬀness. The limit of structural eﬃciency for

a high number of cells is given by equation (B.8), as ltends

towards 0. We write ρthe volumic mass of steel (the mass

per unit of a quad grid being 2ρ

l) and get:

Π∗

5= lim

N→∞ Π5=2E√SI

ρR2v

u

u

u

t

1 + GJ

EI

1 + ES

2GS

(B.8)

Figure B.23 shows the application of equation (B.7)

with the cross-section used in our parametric study and

Π1= 1, which is the closest conﬁguration to a spherical

cap. The limit of the structural eﬃciency Π∗

5is also shown.

It is noticed that the orthotropic shell converges slowly to-

wards the limit, which explains why our convergence study

does not show a plateau for the quadrilateral gridshell.

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