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Nature-inspired generation scheme for shell structures

Authors:
Nature-inspired generation scheme for shell structures
Riccardo LA MAGNA
1
Frédéric WAIMER
1
Jan KNIPPERS
1
r.lamagna@itke.uni-stuttgart.de f.waimer@itke.uni-stuttgart.de j.knippers@itke.uni-stuttgart.de
1
Institute of Building Structures and Structural Design, University of Stuttgart, Germany
Summary
Although less researched and put into practice in the building environment, pure plate structures are
to be observed frequently in biological structures. The 3-plate principle which is common in the
morphology and growth pattern of natural systems is also found to be of a structurally optimum
content when considered from a plate point of view. This is for instance the case of the sea urchin’s
plate skeleton morphology, which served as biological inspiration for the recently built ICD/ITKE
Research Pavilion 2011 at the University of Stuttgart. The current paper will focus on the 3-plate
principle and its mechanical features, also presenting study models to analyse the structural
characteristics and advantages of the principle. Along with the theoretical background, the paper
will introduce the structural concept of the pavilion, as well as the analysis methods used for its
design and engineering.
Keywords: 3-plate principle; bio-inspiration; complex geometry; shell structure; structural design;
structural morphology.
1. Introduction
The stability of polyhedral structures is uniquely determined by the valency of their vertices, where
the valency indicates the number of structural elements converging at one node. According to the
type of elements that make up the structure we may distinguish between lattice and plate systems,
meaning that the first ones are materialised by the edges whilst the latter by the faces of the
polyhedron. Systems with triangular faces which only rely on bars and joints make up pure lattice
structures, whereas systems composed of just Y vertices (only three edges meeting at one point)
which depend on plates and hinges along the edges are called pure plate structures. Such structures
are said to comply with the 3-plate principle.
In triangulated structures, where the loads are
carried by concentrated tension and
compression forces in the edges and vertices
(Fig. 1), a minimum of three bars is needed to
fix a node in space, i.e. three mutually
independent possibilities of translations for an
unconnected node. For this reason, any
triangulated lattice system will be inherently
stable. In plate structures, the two basic
elements are plates and edges (Fig. 1), and a
minimum of three lines of support are needed
to stabilize a plate plane in space, for instance
two translations along perpendicular
directions plus one rotation in the plane of the
plate. By eliminating the nodal force
Figure 1: Axial force action in lattice systems and
shear force action in plate systems
concentration of lattice structures, the plate is stabilized by resisting internal forces which lie in the
plane of the plate itself. Each face in a pure plate structure may be reduced to carry plate forces only,
i.e. plates must not rely on carrying bending moments and torsion for their own stability. On the
other hand, the lines of support enable the transmission of normal and shear forces but no bending
moments between the joints. Three plates with three edges meeting at just one point are necessarily
to stabilize themselves, thus resulting in a bending bearing but yet deformable structure.
2. Dual principle
In geometry, each polyhedron is related to its dual [1], where the vertices of one correspond to the
faces of the other. The transformation process to produce a polyhedron’s dual is known as polar
reciprocation: in the case of a plate system, chosen a reference point (pole), the vertex of the lattice
structure is located on the line from the pole perpendicular to the plane. Finally, if the distance of
the vertex from the origin is chosen to follow the polarity relation [2]:
(1)
where p represents the distance of the plate
from the origin and P the distance of the
vertex from the origin (Fig. 2), then the two
systems will be one the dual of the other. The
same relation of polarity as described above
for the transformation of a plate structure into
its lattice dual is also valid for the opposite
transformation of a lattice structure into its
plate dual. Geometrical dualism has the
property of transforming a fully triangulated
lattice polyhedron into a pure plate system,
and vice versa. This implies that a lattice
structure in equilibrium will still be granted
the minimum stability conditions once it has
been reciprocated into its dual. The duality as
described so far acts upon the topology and
rigidity of systems [3]. However, dualism is
not only limited to morphological features,
but it may also be applied to the mechanical
properties of the structure.
If we consider all the forces F
p,n
acting along the n edges of a plate P (Fig.2), the equilibrium of the
plate imposes that the sum of the vectorial product between the force vector and the position vector
with respect to the chosen pole is zero [2]:
(2)
which must obviously result in a closed polygon of forces. Analogously, the axial forces S
p,n
in all
the n bars that converge to the vertex P dual to the plate p (Fig. 2), have to be in mechanical
equilibrium [2]:
Figure 2: Relationship between geometry and
forces in a dual system [1]
(3)
also indicating a spatial string polygon in equilibrium. Following the right-hand rule to define the
direction of the vectorial product, it is easy to see that the two vectors Fxd and S are always parallel.
which means that the two polygons of forces are necessarily identical [2]:
(4)
i.e. d = S/F for all corresponding parts of the structure.
Equations (1) and (4) state the correlated geometrical and
statical formulas for transformations between dual plate and
lattice structures. It has been shown that the plate-lattice
dualism is not limited to the topology and the simple
geometrical features of the system, but it may also be
extended to the mechanical characteristics of the structure.
An interesting consequence of this last statement is that a
simple way to calculate the forces and reactions in a spatial
plate structure is through its lattice dual. This is done by
translating all the data for the plate structure (including
geometry, loads and boundary conditions) into data for the
dual lattice structure which is successively solved with any
truss calculation software. Finally, by reversing the dual
transformation process, all the information concerning the
lattice structure can be brought back to the original plate
structure, where the axial forces of the lattice’s bars will
now represent the corresponding shear forces and reactions
along the line of support of the plate system.
The concept is illustrated in Fig. 3 where the FE analysis of
a simple plate structure, a cube in this case, is compared
with its lattice dual which corresponds to the regular
octahedron, having chosen the centroid of the cube as pole
for the reciprocal transformation. A test point load F has
been applied to one of the cube’s vertices having the
direction of the edge, which is transformed into a point
load of the same magnitude applied to the reciprocal node
of the truss structure and having the same direction of the
concurrent bar. To avoid any nodal effect which would
have induced stress concentrations, the vertices of the cube
have been trimmed off without affecting the general
stability of the structure. The connecting edges of the
plates have been simulated by a set of axial springs of
infinite stiffness to easily access the values of the shear
forces in the lines of support. Apart slight discrepancies
due to numerical approximation, the axial and shear forces
values of the two systems result equal, thus confirming the
dual principle as exposed so far.
Figure 3: Force distribution in plate
structure and its lattice dual
Figure 5: Detail of sea urchin’s plate structure
3. Research project
In the summer 2011 the Institute for Computational Design (ICD) and the Institute of Building
Structures and Structural Design (ITKE), together with the students at the University of Stuttgart
have realized a temporary, bionic research pavilion made of wood (Fig. 4). The project explored the
transfer to architecture of biological principles of the sea urchin’s plate skeleton morphology. This
was achieved by means of novel
computer-based design and simulation
methods, along with computer-controlled
manufacturing methods for the building
implementation. A particular innovation
consisted in the possibility of extending,
through computer-based applications, the
recognized bionic principles to different
geometries achieving high-performing
lightweight structures, which was
demonstrated by the fact that the complex
morphology of the pavilion could be built
exclusively with extremely thin sheets of
plywood (6.5 mm).
3.1 Biological model
The project aimed at integrating the performative qualities of biological structures into architectural
design and at testing these results on a spatial and structural material-system. The focus was set on
the development of a modular system which allowed a high degree of adaptability and performance
due to the differentiation of its geometric components. During the analysis of biological structures,
the morphology of the sea urchin (Echinoidea) was of particular interest, which provided the basic
principles of the bionic structure that was realized. The shell of the sea urchin (Fig. 5) consists of a
modular system of polygonal plates, which are
linked together at the edges by finger-like calcite
protrusions [4]. Shell action is very close to plate
action because a finely faceted plate polyhedron is
nothing but a slightly discontinuous shell, stabilized
only by shear forces, acting across the edges. High
load bearing capacity is thus achieved by the
particular geometric arrangement of the plates and
their joining system. Therefore, the sea urchin
serves as a perfect model for shells made of
prefabricated elements. Similarly, the traditional
finger-joints typically used in carpentry as
connection elements, may be seen as the equivalent
of the sea urchin’s calcite protrusions.
3.2 Morphology transfer
The most important feature of the sea urchin’s plate morphology is its compliance to the 3-plate
principle, meaning that it is fully trivalent and can thus be regarded as a pure plate structure [5].
This property allows the sand dollar to grow without interfering with the transfer of stabilizing
forces, as the direction of growth will be perpendicular to the shear lines, hence perpendicular to the
Figure 4: External view of the pavilion
direction of the stabilizing shear forces. Following the analysis of the sand dollar, the morphology
of its plate structure was translated into the design of a pavilion. Three plate edges always meet
together at just one point, enabling the transmission of normal and shear forces but no bending
moments between the joints, thus resulting in a bending bearing but yet deformable structure. As
the plates are mainly subjected to in-plane forces, the degree of utilization of the material’s strength
is maximized, guaranteeing an optimal structural configuration. The high lightweight potential of
this approach is evident as the pavilion, despite its considerable size, could be built out of 6.5 mm
thin sheets of birch plywood and therefore primarily needed anchoring to the ground to resist wind
suction loads.
Besides these constructional and organizational principles, hierarchy is another fundamental
property of biological structures which has been applied in the project. The pavilion is organized on
a two-level hierarchical structure (Fig. 6). On the first level, the finger joints of the plywood sheets
are glued together to form a cell. On the second hierarchical level, a simple screw connection joins
the cells together, allowing the assembling and disassembling of the pavilion. Within each
hierarchical level only three plates - respectively three edges – meet exclusively at one point,
therefore assuring bendable edges for both levels.
3.3 Digital design and robotic production
A requirement for the design,
development and realization of the
complex morphology of the pavilion
was a closed, digital information chain
linking the project’s model, finite
element simulations and machine
control. Form finding and structural
design were closely intertwined. An
optimized data exchange scheme
made it possible to repeatedly read the
complex geometry into a Finite
Element program to analyze and modify the critical points of the model. Shear load tests of the
joints (Fig. 7) were performed in order to obtain experimental data for the Finite Element analysis
of the overall structure. The FE setup (Fig. 8) was modeled with plane surfaces and lines of axial
Figure 6: Cell hierarchy and hierarchical distribution of the 3-plate principle
Figure 7: Shear testing of the finger joints
Figure 8: Finite Element model of the pavilion
Figure 9: Night view of the pavilion
springs acting in the direction of the
edges of connection between the plates.
The stiffness values of the linear
springs were obtained from the
laboratory tests on the shear strength of
the finger joints.
The plates of each cell were finally
produced with the university's robotic
fabrication system. Using custom
scripting routines the digital building
model provided the basis for the
automatic generation of the machine
code (NC-Code) for the control of the
industrial robot. This enabled the
economical production of more than
850 geometrically different components,
as well as more than 100,000 finger
joints freely arranged in space.
Following the robotic production, the
plywood panels were joined together to
form the cells, which were successively
primed and stained. The prefabricated
modules were finally joined with a
bolted connection and assembled at the
downtown campus of the University of
Stuttgart.
4. Conclusions
Plate action in today’s buildings is
more or less limited to the resisting of
horizontal forces. The research pavilion
showed how this principle can be
successfully implemented in a complex
and expressive spatial system and how
it is possible to create an extremely
light but at the same time very stiff
supporting structure only by the
optimal arrangement of its constitutive
elements. Through the plate-lattice
dualism it has been shown that any
structure which complies with the 3-
plate principle is inherently stable. This
fundamental principle, along with the
reciprocal transformation process,
offers a powerful tool to investigate a
wide range of structural morphologies
by simple equilibrium assumptions. Finally, the research pavilion offered the opportunity to
investigate methods of modular bionic construction using freeform surfaces with different
geometric characteristics.
5. Acknowledgments
The Research Pavilion was a collaborative project of the Institute for Computational Design (ICD –
Prof. Achim Menges) and the Institute of Building Structures and Structural Design (ITKE – Prof.
Jan Knippers) at Stuttgart University, made possible by the generous support of a number of
sponsors including: KUKA Roboter GmbH, OCHS GmbH, KST2 Systemtechnik GmbH,
Landesbetrieb Forst Baden-Württemberg (ForstBW), Stiftungen LBBW, Leitz GmbH & Co. KG,
MüllerBlaustein Holzbau GmbH, Hermann Rothfuss Bauunternehmung GmbH & Co., Ullrich &
Schön GmbH, Holzhandlung Wider GmbH & Co. KG.
Responsible for the project concept and development were Oliver David Krieg and Boyan
Mihaylov. The project team also included: Peter Brachat, Benjamin Busch, Solmaz Fahimian,
Christin Gegenheimer, Nicola Haberbosch, Elias Kästle, Yong Sung Kwon, Hongmei Zhai.
The scientific development of the project was carried out by: Markus Gabler (project management),
Riccardo La Magna (structural design), Steffen Reichert (detailing), Tobias Schwinn (project
management), Frédéric Waimer (structural design).
6. References
[1] Cundy HM, and Rollet AP, Mathematical Models, 1981, Tarquin Publications, Norfolk, UK
p. 78
[2] Wester T, “The Plate-Lattice Dualism”, Proceedings of the ICSB-IASS Colloquium, Beijing,
1987, Elsevier, London, pp. 321-328
[3] Wester T, “The Structural Morphology of Basic Polyhedra”, Beyond the Cube: The
Architecture of Space Frames and Polyhedra, 1997, Wiley & Sons, pp. 301-342
[4] Wester T, “Nature Teaching Structures”, International Journal of Space Structures, Vol. 17
Nos. 2 & 3, 2002, pp. 135-147
[5] Nachtigall W, Bau-Bionik, 2002, Springer, pp. 6-10
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The Plate-Lattice Dualism
  • T Wester
Wester T, "The Plate-Lattice Dualism", Proceedings of the ICSB-IASS Colloquium, Beijing, 1987, Elsevier, London, pp. 321-328