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Bending-Active Plates: Form and Structure

Authors:
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
http://vdf.ch/advances-in-architectural-geometry-2016.html
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
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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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.
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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
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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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
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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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
© 2016 vdf Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4_13, ISBN 978-3-7281-3778-4
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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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... As seen in chapter 3, in mathematical terms a transformation that preserves lengths is also referred to as an isometry. Pure bending, inextensional and isometric deformations are all synonyms that are often used interchangeably in literature, preferring one term over another to highlight either mechanical or mathematical aspects [La Magna et al. 2016]. Strictly speaking, the only isometric transformations of the plane are into cones and cylinders, i.e. developable surfaces. ...
... The second case study investigates the design potential emerging from integrating both bending and torsion of slender strips into the design process. The results are described extensively in the author's publication [La Magna et al. 2016]. A modified Enneper surface acts as a base for the saddle-shaped design (Fig. 5.18a). ...
... In comparison to the previous case study, this project showcases a different tiling pattern and explores the possibility to significantly increase a shape's rigidity by cross-connecting distant layers with each other. As for the previous prototype, results are reported in the publication [La Magna et al. 2016]. ...
Thesis
Full-text available
Commonly referred to as bending-active, the term has come to describe a wide variety of systems that employ the large deformation of their constituent components as a primary shape-forming strategy. It is generally impossible to separate the structure from its geometry, and this is even more true for bending-active systems. Placed at the intersection between geometry, design and engineering, the principle objective of this thesis is to develop an understanding of the structural and architectural potential of bending-active systems beyond the established typologies which have been investigated so far. The main focus is set on systems that make use of surface-like elements as principle building blocks, as opposed to previous and existing projects that predominantly employed linear components such as rods and laths. This property places the analysed test cases and developed prototypes within a specific category of bending-active systems known as bending-active plate structures.
... As seen in chapter 3, in mathematical terms a transformation that preserves lengths is also referred to as an isometry. Pure bending, inextensional and isometric deformations are all synonyms that are often used interchangeably in literature, preferring one term over another to highlight either mechanical or mathematical aspects [La Magna et al. 2016]. Strictly speaking, the only isometric transformations of the plane are into cones and cylinders, i.e. developable surfaces. ...
... The second case study investigates the design potential emerging from integrating both bending and torsion of slender strips into the design process. The results are described extensively in the author's publication [La Magna et al. 2016]. A modified Enneper surface acts as a base for the saddle-shaped design (Fig. 5.18a). ...
... In comparison to the previous case study, this project showcases a different tiling pattern and explores the possibility to significantly increase a shape's rigidity by cross-connecting distant layers with each other. As for the previous prototype, results are reported in the publication [La Magna et al. 2016]. ...
Data
Full-text available
... This design approach is based on the simple idea of using soft building blocks and large elastic deformations of initially straight or planar elements for the construction of large, load-bearing structures or compliant mechanisms (Lienhard 2014;Schleicher 2015;La Magna 2017). Previous work of the authors, as seen in Figure 2, has demonstrated the innovation potential of this concept, for instance in the context of lightweight shell structures made from very thin plywood plates (Lienhard et al. 2013;La Magna et al. 2016;Schleicher and La Magna 2016). ...
... For the further development of the pavilion, the team applied a combination of bottom-up form-finding and top-down form-conversion techniques (La Schleicher et al. 2016). In a first step, a suitable shape of the pavilion was determined by using the finite element software Sofistik to deform and couple multiple flat strips to a single-layered shell that features minimal bending energy within a given set of boundary conditions ( Figure 5). ...
Conference Paper
Full-text available
The goal of this paper is to advance the research on bending-active structures by investigating the system's inherent structural characteristics and introducing an alternative approach to their design and fabrication. With this project, the authors propose the use of sandwich-structured composites to improve the load-bearing behavior of bending-active shells. By combining digital form-finding and form-conversion processes, it becomes possible to discretize a double-curved shell geometry into an assembly of single-curved sandwich strips. Due to the clever use of bending in the construction process, these strips can be made out of inexpensive and flat sheet materials. The assembly itself takes advantage of two fundamentally different structural states. When handled individually, the thin panels are characterized by their high flexibility, yet when cross-connected to a sandwich, they gain bending stiffness and increase the structure's rigidity. To explain the possible impacts of this approach, the paper will discuss the advantages and disadvantages of bending-active structures in general and outline the potential of sandwich shells in particular. Furthermore, the authors will address the fundamental question of how to build a load-bearing system from flexible parts by using the practical example of the Studio One Research Pavilion. To illustrate this project in more detail, the authors will present the digital design process involved as well as demonstrate the technical feasibility of this approach through a built prototype in full scale. Finally, the authors will conclude with a critical discussion of the design approach proposed here and point out interesting topics for future research.
... Only at the intersection between the petals a small area of stress concentration arises. The approach presented here has been termed form conversion (La Magna et al. [6]) as it establishes a one-to-one correspondence between the base geometry and the bending-active plate system that emerges from it. The previous remark demonstrates the universal character of the method. ...
... The form conversion approach for bending-active plate structures was tested on two full-scale prototypes ( Figure 3). Each case study presented different characteristic in terms of geometry, topology, and shape (La Magna et al. [6]). Nonetheless, the core generative strategy remained the same for both cases. ...
Conference Paper
Full-text available
By distinguishing bending-active structures based on the geometric dimension of their elements, 1-d and 2-d systems can be identified. Rods and gridshells typically belong to the first category, plates to the latter. Plate-based bending-active structures present limited formability compared to rods due to the higher out-of-plane rotational inertia of the plates’ cross-section. Nonetheless, by following simple generative rules it is possible to substantially expand the formal possibilities of bending-active plates. This approach was tested and employed for the construction of a series of full-scale prototypes in order to demonstrate the wide range of shapes that can be achieved in this way. The research conducted so far was primarily focused on the main geometric and mechanical aspects associated to the form-finding process. The current paper looks instead into the global structural behaviour of bending-active plate shells. The process of converting an arbitrary freeform surface into a buildable plate shell requires the introduction of voids in the surface to allow the bending process to take place. The effect of this operation on the global mechanical behaviour of the shell will be analysed and discussed. Considerations on the scalability and buckling of bending-active plate structures will be also presented to highlight the potential of this approach to be employed for larger structural systems.
... El trabajo aplicado a estructuras que basan su geometría en el aprovechamiento de la deformación elástica-activa de materiales laminares -inicialmente planos-abre nuevas complejidades para el proceso de diseño y construcción de envolventes arquitectónicas curvas complejas. Avances previos dan cuenta de que esta lógica proyectual brinda aportes confiables a través de la interacción de variables relacionales parametrizadas (La Magna, Schleicher and Knippers, 2016). ...
... The problem can be equivalently stated in mechanical terms, as the sought design is that which minimises the strain energy of the system, the strain energy being a measure of how much the deformed pattern deviates from the target geometry (Audoly and Pomeau 2010). In practice, we seek for patterns which will induce only bending and torsional actions in the cell walls of the honeycombs, as the strain energy under bending and torsion is much lower compared to membrane strain (La Magna et al. 2016). The problem needs to be broken down in several steps. ...
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Full-text available
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This paper investigates a novel assembly system that employs tension force to transform initially planar elements into bending-active plate structures. Tension is applied through cables connecting bending-active elements. The elements are designed through form planarization, tessellation, and stripe definition on an input geometry. The system offers the benefit of reduced joinery sequences by having minimum tension cables operating in both the joinery and assembly process. Moreover, since there is a correlation between the tessellation process and the arrangement of the cables on the input geometry in the proposed assembly system, this research followed a design-to-assembly workflow. This workflow includes a set of digital tools and various physical tests in small and large scale. Finally, the paper discusses the practicality of the system and further potentials.
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This article highlights the design potentials of a recently proposed form-driven approach for bending-active tensile structures, in which the geometry of the actively bent elements can be directly defined without the recourse to a form-finding procedure. The approach is applied to the design of a lightweight sun-shading system that can be used to protect glazed building façades, and in which actively bent beams are restrained by pre-stressing strips. Other than structural requirements, the geometry of this hybrid structure is informed by functional and environmental considerations to prevent overheating and glare inside the building.
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This work aims to provide an approach to the development of a new structural system, representing a new typology of bending-active hybrids, working with planar elements. This system can be applied as temporary or semi-permanent structures. It takes advantage of utilizing parametric design and prefabrication in order to optimize its structural and sustainable characteristics. СBA-CFS (complex bending-active continuous flexible sheet) structures are made with flexible materials and based on the interaction of bent and tensioned elements in balance, complementing one another. The main element of the structure is a bending-active surface, bent bidirectionally in a complex buckled geometry to realize double curvature. The orientation and size of the buckles take into consideration the specific rules of the form-finding and its stress distribution. The innovation behind the proposal lays in a new comprehension of the concept of a bending-active surface, avoiding tessellation or cuts. The bent shape is kept static through a set of membranes working in tension. The proposed pavilion project is made-up of a unique knitted architectural material, inspired by spacer fabrics. This specific material proposal strongly supports the concept of uniform non-tessellated bending-active sheet, adapted to a wide range of forms with a single homogeneous piece. A sample of the material is produced manually at a 1:1 scale and a model of pavilion at 1:5 scale. The work is supported by physical and digital models, using the finite element method of analysis and particle spring system for simulation and form-finding.
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The basics of construction with synthetic materials From transparent to translucent – new construction options with a versatile material Whether as translucent tiling, wide-spanning membranes, air-filled foil cushions or in organically curved form: Plastics are used in architecture in the widest variety of forms and application areas. Innovative technical developments constantly improve their material properties. Plastics today are an alternative to be taken seriously in the building trade, whether they are used in the supporting structure, roof, facade or interior furnishings. The 'Construction Manual for Polymers + Membranes' returns to the basics of the series by addressing an individual building material. From the material properties to the requirements for drafting and construction, it encapsulates well-founded and comprehensive expertise in familiar DETAIL quality. Select project examples complete the reference work and make it indispensable for day-to-day planning. Historical development of plastics and membranes in architecture Comprehensive information regarding the basics of manufacturing, processing and application Precise descriptions of materials and semi-finished products Physical-structural properties and environmental effects Form finding and calculation of plastic supporting structures and membranes For the first time a complete overview of the most important details compiled according to the most recent state of the research
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In this paper structures that actively use bending as a self-forming process are reviewed. By bringing together important material developments and various historical as well as recently built samples of such structures, the aim is to show coherences in their design approach, structural systems and behaviour. Different approaches to bending-active structures are defined and described. By making this work accessible and categorising it, this paper aims to contribute to an emerging development. A differentiation of such structures is suggested based on their design approaches. Three such approaches are differentiated: the behaviour based approach, the geometry based approach and current research that seeks to integrate the two. In this paper the nature of these approaches and some important project samples are discussed
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This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.
Thesis
This thesis aims to provide general insight into form-finding and structural analysis of bending-active structures. The work is based on a case study approach, in which findings from prototypes and commercial building structures become the basis for generalised theoretical investigations. Information is continuously fed back from these case study structures into theoretical research, which creates the basis for overall working methods. The behaviour of five investigated structures is found to be independent of clearly predictable load bearing categories. Their load bearing mechanisms are largely dependent on the boundless variety of topologies and geometrical expressions that may be generated. The work therefore understands active bending as an approach to generating new structural forms, in which common load bearing behaviour is found due to the structures inherently large elasticity and inner stress state. Based on engineering and historical background, methodological, mechanical and material fundamentals of active-bending are discussed in Chapter B and C. The case study structures introduced in Chapter D open a wide field of active-bending applications, in lightweight building structures. Whether the conclusions drawn from case studies, are generally viable for bending-active structures is then discussed in the core of the work presented in two chapters on Form-Finding (Chapter E) and Structural Behaviour (Chapter F). The chapter on form-finding introduces the working methods and modelling environments developed for the present work. The chapter on structural behaviour is concerned with the influence of residual bending stress on the stiffness, scaling and stability of bending-active structures. Based on these findings, generalised design rules for bendingactive structures are highlighted in a concluding chapter.
Book
Bringing together experts from research and practice, Shell Structures for Architecture: Form Finding and Optimization presents contemporary design methods for shell and gridshell structures, covering form-finding and structural optimization techniques. It introduces architecture and engineering practitioners and students to structural shells and provides computational techniques to develop complex curved structural surfaces, in the form of mathematics, computer algorithms, and design case studies.
Article
Timber Fabric structures (TFS) initiate from a correspondence between textile principles and recent industrial developments in producing cross laminated timber panels. Several individual timber strips are interlaced according to a pattern and result in an innovative space structure. The obtained three-dimensional geometry can be regarded as the relaxed configuration of deformed panels under the imposed boundary conditions. We herein propose a form-finding procedure, which reproduces this deformed configuration as the steady state of a pseudo transient constrained dynamic problem. The corresponding nonlinear problem involves finite rotation regime and contact handling through the cross section and on both panel faces. To effectively deal with nonlinear constraints, a new modified dynamic relaxation method is herein used which combines elastic material behavior with a fictitious stiffness proportional damping into an equivalent fictitious viscous material model. The procedure is implemented as an ABAQUS/Explicit user subroutine VUMAT and the overall accuracy of the numerical results has been studied for a number of geometrically nonlinear shell benchmark problems. This numerical approach is then employed to simulate the assembly process for a Timber Fabric Module (TFM), an interlaced assembly of two timber strips. The simulated geometry for the deformed surfaces is then extracted and is compared with a 3D processed surface mesh obtained from scanning a built-in prototype with noncontact Laser scanner arm to validate the simulation procedure.
Article
The production of architecture, both intellectually and physically, is on the brink of a fundamental change. Computational design enables architects to integrate ever more multifaceted and complex design information, while the industrial logics of conventional building construction are eroding rapidly in a context of increasingly ubiquitous computer-controlled manufacturing and fabrication. A novel convergence of computation and materialisation is about to emerge, bringing the virtual process of design and the physical realisation of architecture much closer together, more so than ever before. Computation provides a powerful agency for both informing the design process through specific material behaviour and characteristics, and in turn informing the organisation of matter and material across multiple scales based on feedback from the environment. Computational design and integrated materialisation processes allow for uncovering the inherent morphogenetic potential of materials and thus are opening up a largely uncharted field of possibilities for the way the built environment in the 21st century is conceived and produced. In order to effectively introduce and outline the enabling power of computational design along with its inherent relationship to a biological paradigm, this publication looks at formation and materialisation in nature, integrative computational design, and engineering and manufacturing integration. * Architectural contributors include: Cristiano Cecatto, Neri Oxman, Skylar Tibbits and Michael Weinstock. * A scientific perspective by Philip Ball and J Scott Turner. * Features: Buro Happold's SMART group, DiniTech, Foster + Partners' Specialist Modelling Group, the Freeform Construction group and Stuttgart University's Institute for Computational Design.