Bending-Active Plates: Form and Structure

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170
Bending-Active Plates
Form and Structure
Riccardo La Magna,Simon Schleicher,andJan Knippers
Riccardo La Magna, Jan Knippers
Institute of Building Structures and Structural Design (ITKE), University of Stuttgart, Germany
ric.lamagna@gmail.com 
j.knippers@itke.uni-stuttgart.de
Simon Schleicher
College of Environmental Design, University of California, Berkeley, United States
simon_s@berkeley.edu
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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171
Abstract
Aiming to support the current research on bending-active plate structures, this
paper discusses the topic of form-finding and form-conversion and presents ex-
amples to illustrate the formal and structural potential of these design strategies.
Following a short introduction into the topic, the authors reflect on the specific
challenges related to the design of bending-active plate structures. While previ-
ous research has mainly focussed on a bottom-up approach whereby the plates
first were specified as basic building blocks and the global shape of the structure
resulted from their interaction, the main emphasis of this paper lies on a possi-
ble top-down approach by form-conversion. Here, the design process starts with
a given shape and uses surface tiling and selective mesh subdivision to inform
the geometrical and structural characteristics of the plates needed to assemble
the desired shape. This new concept entails some constraints, and the paper
therefore provides an overview of the basic geometries and mechanics that can
be created by following this approach. Finally, to better demonstrate the inno-
vative potential of this top-down approach to the design of bending-active plate
structures, the authors present two built case studies, each of which is a proof
of the concept that pushes the topic of form-conversion in a unique way. While
the first one takes advantage of translating a given shape into a self-support-
ing weave pattern, the second case study gains significant structural stability by
translating a given form into a multi-layered plate construction.
Keywords:
bending-active structures, elastic bending, plate structure,
form-finding, nonlinear analysis
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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172
1. Introduction
With the rise of new simulation strategies and computational tools, a new gen-
eration of architects and engineers is getting more interested in form-finding ar-
chitectural systems. The key motivation of this approach is to determine a force
equilibrium to generate and stabilise a structure just by its geometry. While the
membrane and shell structures of pioneers like Frei Otto, Heinz Isler, and Felix
Candela were often derived from model-based form-finding processes or using
pure geometrical bodies (Chilton 2000, Otto 2005, Garlock & Billington 2008), today’s structures
often arise from advanced digital simulations and the integration of material be-
haviour therein (Adriaenssens et al. 2014, Menges 2012).
A good example for the new possibilities emerging from a physics-informed
digital design process is the research done on bending-active structures. This type
of structural system uses large-scale deformations as a form-giving and self-sta-
bilising strategy (Knippers et al. 2011, Lienhard et al. 2013, 2014, Schleicher et al. 2015). Typically,
bending-active structures can be divided into two main categories, which relate
to the geometrical dimensions of their constituent elements. For instance, 1D
systems can be built with slender rods and 2D systems out of thin plates (Fig. 1).
While extensive knowledge and experience exists for 1D systems, with elastic
gridshells as the most prominent application, plate-dominant structures have not
yet received much attention and are considered difficult to design. One reason is
that plates have a limited formability since they deform mainly along the axis of
weakest inertia and thus cannot easily be forced into complicated geometries.
However, this obstructive limitation of the smallest building block can also
be understood as special advantage. Used strategically, it offers not only more
control over the global formation process, but can also be used to inform the
individual parts of the assembled structure based on the features of the overall
shape. This essentially means that form-finding in the context of bending-active
structures could evolve from a bottom-up to a top-down approach, starting with
a desired global shape first and then solving the form-force equilibrium of its
parts. Following this approach renders the ability to construct a given shape by
integrating local bending of its components while guaranteeing that stresses
remain within the permitted working range of the material.
2. Typical Design Approaches and
Resulting Challenges
Bending-active structures are often designed by following either a behaviour-based,
geometry-based, or integrated approach (Lienhard et al. 2013). While the first category
refers to traditional, intuitive use of bending during the construction process and
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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173
relies only on hands-on experience regarding the deformation behaviour of the
used building material (Fig. 2a), the latter two categories describe a more scientif-
ic take on the design of bending-active structures. Here, experimental and ana-
lytical form-finding techniques were conducted beforehand and then informed
the design process.
One example for bending-active plate structures that were built by following
a geometry-based design approach are Buckminster Fuller’s plydomes (Fuller 1959).
This construction principle is based on approximating the basic geometry of a
sphere with a regular polyhedron. Its edges and angles are then used to arrange
multiple plates into a spatial tiling pattern, which is fastened together by bending
the plates at their corners (Fig. 2b). The resulting structure is made out of identical
plates joined together by placing bolts at predefined positions. Even though this
technique allowed Fuller to construct a double-curved spherical shape out of
Figure 1. Classification of bending-active structures based on the member’s geometrical dimension
(from Knippers et al. 2011).
Figure 2a. Traditional Mudhif reed house.
Figure 2b. Plydome.
Figure 2c. ICD/ITKE Research Pavilion 2010.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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174
initially planar and then single-curved plates, this methodology also had several
shortcomings. First and foremost, it is limited to basic polyhedral shapes. Only
because of the repetitive angles was it possible to use identical plates. Further-
more, at his time Fuller was forced to compute the needed overlap of the plates
and the exact position of the pre-drilled holes mathematically. The only way to
calibrate this data was by producing plydomes in series and improving the de-
tails over time.
Compared to that, following an integrative design approach for bending-active
plate structures provides more flexibility and renders the opportunity for com-
putational automatisation. A prominent example is the 2010 ICD/ITKE Research
Pavilion (Fig. 2c). As characteristic for an integrative approach, this project started
with intensive laboratory testing to better understand the limiting material be-
haviour of the used plywood. The results of these physical experiments were
then integrated as constraints into parametric design tools and used to calibrate
finite-element simulations. Synchronising physical and digital studies ensured
that the form-finding techniques provided an accurate description of the actual
material behaviour while at the same time giving more feedback on the resulting
geometry of the structure. This project even went so far to re-create the actu-
al bending process by simulating the deformation of every strip into a cross-
connected and elastically pre-stressed system (Lienhard et al. 2012).
While the last project is definitely innovative, it should be pointed out that
the integrated approach here was used mainly in a bottom-up way and thus nar-
rowed the possible design space. For the future development of bending-active
plate structures, however, it may be desirable to prioritise a top-down approach,
which gives more weight to the target geometry and thus more freedom to the
designer. However, the key challenge remains and boils down to how to assess
both the global shape as well as the local features of the constituent parts for
structures in which geometrical characteristics and material properties are inev-
itably linked together and similarly affect the result.
3. Form Conversion
The principal limit to the formal potential of bending-active structures lies in
the restrictions on the material formability. The only deformations that can be
achieved within stress limits are the ones that minimise the stretching of the
material fibres. For strips and plate-like elements, these reduce to the canonical
developable surfaces: cylinders and cones. Attempting to bend a sheet of mate-
rial in two directions will either result in irreversible, plastic deformations or ul-
timately failure. Such a strict requirement severely limits the range of structural
and architectural potential for plate-based bending-active systems. To expand the
range of achievable shapes, it is therefore necessary to develop workarounds
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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175
for the induction of Gaussian curvature. To overcome such limitations, multidi-
rectional bending can be induced by strategically removing material and freeing
the strips from the stiffening constraint of the surrounding. A similar approach
is presented by Xing et al. (2011) and referred to as band decomposition. The key
principle is illustrated in Figure 3.
Here a continuous rectangular plate is reduced to two orthogonal strips. The
strips are later bent into opposite directions in a finite element simulation using
the ultra-elastic contracting cable approach based on Lienhard, La Magna and
Knippers (2014). The bending stiffness of the plate, depending proportionally on its
width b, results in a radical increase of stiffness in the connecting area between
the strips. As a result, the connecting area almost remains planar, and therefore
the perpendicular bending axis remains unaffected by the induced curvature. In
this way it becomes possible to bend the strips around multiple axes, spanning
different directions but still maintaining the material continuity of a single element.
Figure 3b depicts the resulting von Mises stresses calculated at the top fibres. The
gradient plot clearly displays an area of unstressed material at the intersection
between the two strips, as expected based on the previous arguments. A local
stress concentration appears at the junction of the strips due to the sharp con-
necting angle and inevitable geometric stiffening happening locally in that area.
Figure 3a. Multidirectional strip.
Figure 3b. von Mises stress distribution after bending.
Figure 3c. Gaussian curvature.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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176
This result can be compared with Figure 3c, which displays the Gaussian curvature
of the bent element. From the plot it is clear that the discrete Gaussian curva-
ture (based on Meyer et al. 2003) of the deformed mesh is everywhere zero, apart from
a small localised area at the intersection of the two branches. This confirms that,
within stress limits, flat sheets of inextensible material can only be deformed
into developable surfaces at most.
Based on the strip approach defined so far, the general procedure for an ar-
bitrary freeform surface is summarised in the following steps:
1. Mesh the target surface (Fig. 4a).
2. Perform an interior offset for each face of the mesh.
3. Connect the disjointed faces by creating a bridging element; two faces
initially sharing an edge will be connected (Fig. 4b).
4. The bridging element is modified to take into account the bending cur-
vature. Assuming that the start and end tangent plane of the bridging
element coincide with the surfaces to be connected, the element can be
defined through a simple loft (Fig. 4c).
5. Unroll the elements.
The presented method maintains general validity for any arbitrary source
mesh. In the case of an Ngon mesh, its banded dual will have strips with N arms
departing from the centre surface element. The geometry of the voids is defined
by the valence of the mesh. For the sphere example a 4-valent source mesh
produces square voids throughout the banded structure. A tri-valent hexagonal
mesh would produce triangular voids and so forth.
Figure 4a. Mesh of target surface.
Figure 4b. Offset and edge bridging.
Figure 4c. Bending of bridging elements.
Figure 4d. Plywood prototype of sphere test case.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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177
4. The Geometry and Mechanics of
Bending-Active Plate Structures
It is typical in engineering to distinguish between plate and shell structures, the
main difference being that plates are initially flat and shells already present curva-
ture in their stress-free configuration. Both structures can be identified as having
thickness significantly smaller than length and width. In this way the geometry
of a shell or plate is uniquely defined by their centre surface and local thickness
(Bischoff et al. 2004). The structural behaviour of shells and plates is characterised by
two main states of deformation, membrane and bending action. Membrane de-
formations involve strain of the centre surface, whilst bending dominated defor-
mations roughly preserve the length of the mid-surface fibres. Under bending,
only the material fibres away from the mid-surface are fully exploited, therefore
the structural elements are much more flexible. Pure bending deformations are
also called
inextensional deformations
as the neutral surface is completely free
from longitudinal extension or compression. In mathematical terms, a transfor-
mation that preserves lengths is also referred to as an isometry. Pure bending,
inextensional, and isometric deformations are all synonyms that are often used in-
terchangeably in literature, preferring one term over another to highlight either
mechanical or mathematical aspects. Strictly speaking, the only isometric trans-
formations of the plane are into cones and cylinders, i.e. developable surfaces.
In structural applications, membrane deformation states are generally pre-
ferred as the cross-section is completely utilised and the load-bearing behaviour
of the shell is significantly enhanced. On the other hand, characteristics of inex-
tensional deformations may be exploited in certain situations, for example, deploy-
able or tensile structures which might benefit from bending dominated transition
stages. In the context of bending-active structures, inextensional deformations
represent the main modality of shape shifting, as the bending elements may un-
dergo large deformations without reaching a critical stress state for the material.
Owing to the high flexibility of thin plates with respect to bending, this state of
deformation may be regarded as the dominating mechanical effect for bending-
active structures as having the strongest effect on the nonlinear behaviour of plates.
The relationship between large deformations and pure bending is well un-
derstood in light of energetic arguments explained in the following paragraph.
An important assumption in the context of bending-active structures is that of a
perfect elastic response of the material. This is the case of Hookean elasticity,
which assumes a linear elastic response of the material and therefore yields a
proportional relationship between strain and stress (Audoly & Pomeau 2010). This as-
sumption is valid for small strains in general, which is commonly the case for
bending-active structures. In the membrane approximation, the elastic energy
of a plate reads:
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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178
where the subscript
‘cs’
means that we can evaluate the density of the elastic
energy along the centre surface. The approximation
(4.1)
can be understood as
following: It is the surface integral of the squared, 2-dimensional strain along
the centre surface 𝜖αβ, multiplied by the factor Eh, which is proportional to the
thickness h and to Young’s modulus E of the material.
Conversely, the bending energy of a 2-dimensional plate assumes the fol-
lowing form:
which can be read as the surface integral of the squared curvature (dependent
on x and y) of the centre surface, multiplied by the factor Eh3, which is common-
ly called bending iffness.
Comparing the stretching energy (4.1) with the bending energy (4.2) shows that
the small thickness h comes in the flexural energy (4.1) with a larger power than
in the stretching energy, i.e. h3 in place of h. For very thin plates, this makes the
energy of isometric deformations much lower than those involving significant
stretching of the centre surface. As a result, large deformations occur mainly
under bending, as the low energy involved in the process is generally compatible
with the strain limits of the material.
Although commonly referred to as bending-active, the term has been spe-
cifically coined to describe a wide range of systems that employ the large defor-
mation of structural components as a shape-forming strategy. Besides bending,
torsional mechanisms can also be employed to induce form, as the energy in-
volved is of similar order of magnitude compared to bending. An essential re-
quirement for bending-active structures is that the stress state arising from the
form-finding process does not exceed the yield strength of the material. Based on
the previous assumptions of perfect elastic material response and thin, slender
sections, the maximum bending curvature and maximum torsional angle twist
can be checked against the following relationships:
active structures is that of a perfect elastic response of the material. This is the case of
Hookean elasticity, which assumes a linear elastic response of the material and therefore yields a
portional relationship between strain and stress (Audoly & Pomeau 2010). This assumption is
valid for small strains in general, which is commonly the case for bending-active structures. In the
membrane approximation, the elastic energy of a plate reads:
  





where the subscript ‘cs’ means that we can evaluate the density of the elastic energy along the
centre surface. The approximation (4.1) can be understood as following: It is the surface integral of
dimensional strain along the centre surface , multiplied by the factor Eh, which is
proportional to the thickness h and to Young’s modulus E of the material.
Conversely, the bending energy of a 2-dimensional plate assumes the following form:





 




which can be read as the surface integral of the squared curvature (dependent on x and y) of the
centre surface, multiplied by the factor Eh3, which is commonly called bending stiffness.
the stretching energy (4.1) with the bending energy (4.2) shows that the small thickness
comes in the flexural energy (4.1) with a larger power than in the stretching energy, i.e. h3 in
For very thin plates, this makes the energy of isometric deformations much lower than
those involving significant stretching of the centre surface. As a result, large deformations occur
mainly under bending, as the low energy involved in the process is generally compatible with the
strain limits of the material.
Although commonly referred to as bending-active, the term has been specifically coined to describe
a wide range of systems that employ the large deformation of structural components as a shape-
ng strategy. Besides bending, torsional mechanisms can also be employed to induce form, as
the energy involved is of similar order of magnitude compared to bending. An essential requirement
(4.1)
energetic arguments explained in the following paragraph. An important assumption in the context
active structures is that of a perfect elastic response of the material. This is the case of
Hookean elasticity, which assumes a linear elastic response of the material and therefore yields a
portional relationship between strain and stress (Audoly & Pomeau 2010). This assumption is
valid for small strains in general, which is commonly the case for bending-active structures. In the
membrane approximation, the elastic energy of a plate reads:
  





where the subscript ‘cs’ means that we can evaluate the density of the elastic energy along the
centre surface. The approximation (4.1) can be understood as following: It is the surface integral of
2-dimensional strain along the centre surface , multiplied by the factor Eh, which is
proportional to the thickness h and to Young’s modulus E of the material.
Conversely, the bending energy of a 2-dimensional plate assumes the following form:





 




which can be read as the surface integral of the squared curvature (dependent on x and y) of the
centre surface, multiplied by the factor Eh3, which is commonly called bending stiffness.
the stretching energy (4.1) with the bending energy (4.2) shows that the small thickness
comes in the flexural energy (4.1) with a larger power than in the stretching energy, i.e. h3 in
For very thin plates, this makes the energy of isometric deformations much lower than
those involving significant stretching of the centre surface. As a result, large deformations occur
mainly under bending, as the low energy involved in the process is generally compatible with the
strain limits of the material.
Although commonly referred to as bending-active, the term has been specifically coined to describe
a wide range of systems that employ the large deformation of structural components as a shape-
ng strategy. Besides bending, torsional mechanisms can also be employed to induce form, as
the energy involved is of similar order of magnitude compared to bending. An essential requirement
(4.2)
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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179
R
x
y
z
l
φ
l
M
T
xy
z
ϑ
Figure 5a. Material strip subject to axial bending.
Figure 5b. Material strip subject to torsion.
for bending-active structures is that the stress state arising from the form-finding process does not
exceed the yield strength of the material. Based on the previous assumptions of perfect elastic
material response and thin, slender sections, the maximum bending curvature and maximum
torsional angle twist can be checked against the following relationships:
Figure 5a. Material strip subject to axial bending. Figure 5b. Material strip subject to torsion.
 = ?@
?A =BC
DE

?F
?A =BG
HI


"KA =BC
L "KA =BG
LG
"NO =
P
QRST
=
URSTL
DE =
/URST
DV
?F
?A =
WRSTLG
HI =
WRST
HV
where:
k curvature [1/m] θ angle of twist [rad]
MB bending moment [kNm] MT torsional moment [kNm]
E Young’s modulus [N/mm2] G shear modulus [N/mm2]
I moment of inertia [m4] J torsional constant [m4]
W = bh2/6 resistance moment [m3] WT = bh3/3 torsional resistance [m3]
σmax max. allowable stress [N/mm2] τmax max. shear stress [N/mm2]
h section height [mm] h section height [mm]
These equations refer to the classic Euler-Bernoulli model for bending and de Saint-Venant torsion
model for beams. Both models ignore higher order effects, respectively deformations, caused by
transverse shear and torsional warping. Although generally non-neglectable for large deformations,
owing to the previous assumptions of slender cross-sections and elastic behaviour, it is safe to
assume these values for a preliminary check of the master geometry.
The complexity of the structural systems and form-finding procedures still require an accurate
numerical analysis. In general, currently available simulation tools can be subdivided into two
categories. The first one, dynamic relaxation (DR), is a numerical iterative method to find the
solution of a system of nonlinear equations. It has been successfully employed in engineering
applications for the form-finding of membrane and cable net structures (Barnes 1999, Adriaenssens
& Barnes 2001) and in modified versions also for torsion related problems in surface-like shell
exceed the yield strength of the material. Based on the previous assumptions of perfect elastic
material response and thin, slender sections, the maximum bending curvature and maximum
torsional angle twist can be checked against the following relationships:
Figure 5a. Material strip subject to axial bending. Figure 5b. Material strip subject to torsion.
===
= =
=1==2ℎ ==ℎ
where:
k
curvature [1/m] θ angle of twist [rad]
M
B bending moment [kNm] MT torsional moment [kNm]
E
Young’s modulus [N/mm2] G shear modulus [N/mm2]
I
moment of inertia [m4] J torsional constant [m4]
W
= bh2/6 resistance moment [m3] WT = bh3/3 torsional resistance [m3]
σ
max max. allowable stress [N/mm2] τmax max. shear stress [N/mm2]
h section height [mm] h section height [mm]
These equations refer to the classic Euler-Bernoulli model for bending and de Saint-Venant torsion
model for beams. Both models ignore higher order effects, respectively deformations, caused by
transverse shear and torsional warping. Although generally non-neglectable for large deformations,
owing to the previous assumptions of slender cross-sections and elastic behaviour, it is safe to
assume these values for a preliminary check of the master geometry.
The complexity of the structural systems and form-finding procedures still require an accurate
numerical analysis. In general, currently available simulation tools can be subdivided into two
categories. The first one, dynamic relaxation (DR), is a numerical iterative method to find the
solution of a system of nonlinear equations. It has been successfully employed in engineering
applications for the form-finding of membrane and cable net structures (Barnes 1999, Adriaenssens
& Barnes 2001) and in modified versions also for torsion related problems in surface-like shell
elements (Nabaei et al. 2013). The second method relies on finite element simulation (FEM). Non-
linear finite element routines have advanced so much lately that it is becoming more common to
These equations refer to the classic Euler-Bernoulli model for bending and de
Saint-Venant torsion model for beams. Both models ignore higher order effects,
respectively deformations, caused by transverse shear and torsional warping. Al-
though generally non-neglectable for large deformations, owing to the previous
assumptions of slender cross-sections and elastic behaviour, it is safe to assume
these values for a preliminary check of the master geometry.
The complexity of the structural systems and form-finding procedures still
require an accurate numerical analysis. In general, currently available simulation
tools can be subdivided into two categories. The first one, dynamic relaxation
(DR), is a numerical iterative method to find the solution of a system of nonlin-
ear equations. It has been successfully employed in engineering applications
for the form-finding of membrane and cable net structures (Barnes 1999, Adriaenssens
& Barnes 2001) and in modified versions also for torsion related problems in sur-
face-like shell elements (Nabaei et al. 2013). The second method relies on finite element
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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180
simulation (FEM). Non-linear finite element routines have advanced so much lately
that it is becoming more common to integrate them in the design process. All
the results presented here were achieved through geometrical non-linear finite
element simulations run in SOFiSTiK.
5. Case Studies
The following two case studies are both made out of the same material – 3 mm
birch plywood. This plywood consists of three layers and has different mechani-
cal behaviours along the main fibre orientation (stronger) and against it (softer).
This is due to the fact that the fibre direction of the upper and lower layers is ori-
ented in one direction and rotated by 90° in the centre layer. Based on this the
plywood also has two values for the minimal bending radius that can be achieved
as well as two values for the maximum axial twist the material can undergo. The
Young’s modulus of a 3 mm plywood along the grain is: Em
II
= 16471 N/mm²
and against the grain is: Em⊥ = 1029 N/mm².
5.1 Case Study: Berkeley Weave
The first case study investigates the design potential emerging from inte-
grating both bending and torsion of slender strips into the design process.
A modified Enneper surface acts as a base for the saddle-shaped design
(Fig.6a). This particular form was chosen because it has a challenging anticlastic
geometry with locally high curvature. The subsequent conversion process
into a bending-active plate structure followed several steps. The first was to
approximate and discretise the surface with a quad mesh (Fig. 6b). A curvature
analysis of the resulting mesh reveals that its individual quads are not planar
but spatially curved (Fig. 6c).
The planarity of the quads, however, is an important precondition in the
later assembly process. In a second step, the mesh was transformed into a
four-layered weave pattern with strips and holes. Here, each quad was turned
into a crossing of two strips in one direction intersecting with two other strips
in a 90-degree angle. The resulting interwoven mesh was then optimised for pla-
narisation. However, only the regions where strips overlapped were made planar
(blue areas), while the quads between the intersections remained curved (Fig. 6d).
A second curvature analysis illustrates the procedure and shows zero curvature
only at the intersections of the strips while the connecting arms are both bent
and twisted (Fig. 6e). In the last step, this optimised geometrical model was used
to generate a fabrication model that features all the connection details and strip
subdivisions (Fig. 6f).
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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181
A closer look at the most extremely curved region in the structure illustrates
the complexity related to this last step (Fig.7a). To allow for a proper connection, bolts
were placed only in the planar regions between intersecting strips. Since the strips
are composed out of smaller segments, it was also important to control their po-
sition in the four-layered weave and the sequence of layers. A pattern was created
which guaranteed that strip segments only ended in layers two and three and are
clamped by continuous strips in layers one and four. A positive side effect of this
weaving strategy is that the gaps between segments are never visible and the strips
appear to be made out of one piece. The drawback, however, is that each segment
has a unique length and requires specific positions of the screw holes (Fig. 7b).
Figure 6. Generation process and analysis.
Figure 7a. Analysis of Gaussian curvature.
Figure 7b. Schematic of the weaving and technical details.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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182
To demonstrate proof of concept for this design approach, this case study
was built in the dimensions of 4 m x 3.5 m x 1.8 m (Fig. 8). The structure is assem-
bled out of 480 geometrically different plywood strips fastened together with
400 similar bolts. The material used is 3.0 mm thick birch plywood with a Young’s
modulus of EmII = 16471 N/mm² and Em⊥ = 1029 N/mm². Dimensions and ma-
terial specifications were employed for a finite-element analysis using the soft-
ware SOFiSTiK. Under consideration of self weight and stored elastic energy, the
minimal bending radii are no smaller than 0.25 m and the resulting stress peaks
are still below 60% of permissible yield strength of the material.
5.2 Case Study: Bend9
The second case study is a multi-layered arch that spans over 5.2 m and has a
height of 3.5 m. This project was built to prove the technical feasibility of using
bending-active plates for larger load-bearing structures. In comparison to the pre
-
vious case study, this project showcases a different tiling pattern and explores
the possibility to significantly increase a shape’s rigidity by cross-connecting dis-
tant layers with each other.
To fully exploit the large deformations that plywood allows for, the thick-
ness of the sheets had to be reduced to the minimum, leading once again to the
radical choice of employing 3.0 mm birch plywood. Since the resulting sheets
are very flexible, additional stiffness needed to be gained by giving the global
shell a peculiar geometry which seamlessly transitions from an area of positive
curvature (sphere-like) to one of negative curvature (saddle-like) (Fig. 9a). This pro-
nounced double-curvature provided additional stiffness and avoided undesirable
deformation modes of the structure. Despite the considerable stiffness achieved
through shape, the choice of using extremely thin sheets of plywood required
additional reinforcement to provide further load resistance. These needs were
met by designing a double-layered structure with two cross-connected shells.
As in the previous example, the first step of the process was to convert the
base geometry into a mesh pattern (Fig. 9b). In the next step a preliminary analysis
of the structure was conducted, and a second layer was created by offsetting
the mesh. As the distance between the two layers varies to reflect the bend-
ing moment calculated from the preliminary analysis, the offset of the surfaces
changes along the span of the arch (Fig. 9c). The offset reflects the stress state in
the individual layers, and the distance between them grows in the critical areas
to increase the global stiffness of the system. The following tiling logic that was
used for both layers guarantees that each component can be bent into the specific
shape required to construct the whole surface. This is achieved by strategically
placing the voids into target positions of the master geometry, as described in
Section 3, and thereby ensuring that the bending process can take place with-
out prejudice for the individual components (Fig. 9d). Although initially flat, each
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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183
Figure 8. View of the plywood installation Berkeley Weave.
Figure 9a. Base geometry.
Figure 9b. Mesh approximation.
Figure 9c. Double layer offset.
Figure 9d. Conversion to bent plates.
Figure 9e. Finite-element analysis.
Figure 9f. Fabrication model.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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184
element undergoes multi-directional bending and gets locked into position once
the neighbours are added to the system and joined together. The supple 3.0 mm
plywood elements achieve consistent stiffness once assembled together, as the
pavilion, although a discrete version of the initial shape, still retains substantial
shell stiffness. This was validated in a second finite-element analysis that con-
sidered both self-weight as well as undesirable loading scenarios (Fig. 9e). Finally,
a fabrication model was generated and the structure fabricated (Fig. 10).
The built structure employs 196 elements unique in shape and geometry (Fig.
11a). 76 square wood profiles of 4 cm x 4 cm were used to connect the two ply-
wood skins (Fig. 11b). Due to the varying distance between the layers, the connec-
tors had a total of 152 exclusive compound mitres. The whole structure weighs
only 160 kg, a characteristic which also highlights the efficiency of the system
and its potential for lightweight construction. The smooth curvature transition
and the overall complexity of the shape clearly emphasise the potential of the
construction logic to be applied to any kind of double-curved freeform surface.
6. Conclusions
The two built case studies clearly illustrate the feasibility of a construction logic
that integrates bending deformations strategically into the design and assem-
bly process. Both structures presented are directly informed by the mechanical
properties of the thin plywood sheets employed for the project. Their overall ge-
ometry is therefore the result of an accurate negotiation between the mechan-
ical limits of the material and its deformation capabilities.
The assembly strategy devised for both prototypes drastically reduces the
fabrication complexity by resorting to exclusively planar components which make
up the entire double-curved surfaces. Despite the large amount of individual
geometries, the whole fabrication process was optimised by tightly nesting all
the components to minimise material waste, flat cut the elements, and finally
assemble the piece on-site.
The very nature of the projects required a tight integration of design, simu-
lation, and assessment of the fabrication and assembly constraints. Overall, the
Bend9 pavilion and Berkeley Weave installation exemplify the capacity of bend-
ing-active surface structures to be employed as a shape-generating process.
For on-going research, the buildings serve as first prototypes for the exploration
of surface-like shell structures that derive their shape through elastic bending.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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185
Figure 10. View of the Bend9 structure.
Figure 11a. Detail of the elements.
Figure 11b. Detail of the connecting elements.
S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges, M. Pauly (eds.): Advances in Architectural Geometry 2016
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186
Acknowledgements
For the Weave installation the authors would like to thank Sean Ostro, Andrei Nejur, and Rex Crabb for their support. The
Bend9 pavilion would not have been possible without the kind support of Autodesk’s Pier 9 and its entire staff.
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