Design strategies for bending-active plate structures out of multiple cross-connected layers

Abstract

This paper focuses on various analogue and numerical design strategies that were developed and implemented to generate pre-defined, free-form surfaces with parametrically differentiated bending-active components. Derived from recent research on bending-active plate structures, the authors present two case studies which both focus on the seamless interaction of material, form, and structure. Both systems illustrate a novel approach to the realisation of double curved, free-form structures by means of bending. The two systems were built in prototypical mock-ups to evaluate their geometry and structural capacity. While the resulting geometries are complex, the use of elastic bending during the assembly process allows for simple manufacturing methods. Thin plastic sheets, which are easy to cut and bend, were manipulated by simple, two-dimensional fabrication techniques and assembled into multi-layered, bent structures that are stiff and self-supporting. After a general introduction on bending-active plate structures, the authors present the key design principles used for the development of the two case studies and conclude with a reflection and comparison of these techniques.

Full text

Proceedings of the International Association for Shell and Spatial Structures (IASS)
Symposium 2015, Amsterdam
Future Visions
17 - 20 August 2015, Amsterdam, The Netherlands
Design strategies for bending-active plate structures out of
multiple cross-connected layers
Andreas SCHÖNBRUNNER*, Nicola HABERBOSCHa, Riccardo LA MAGNAa,
Simon SCHLEICHERab, Julian LIENHARDa, Jan KNIPPERSa
*Institute of Building Structures and Structural Design (ITKE), University of Stuttgart
Keplerstr. 11, 70174 Stuttgart, Germany
schoenbrunner@str-ucture.com
a Institute of Building Structures and Structural Design (ITKE), University of Stuttgart, Germany
b College of Environmental Design (CED), University of California, Berkeley, USA
Abstract
This paper focuses on various analogue and numerical design strategies that were developed and
implemented to generate pre-defined, free-form surfaces with parametrically differentiated bending-
active components.
Derived from recent research on bending-active plate structures, the authors present two case studies
which both focus on the seamless interaction of material, form, and structure. Both systems illustrate a
novel approach to the realisation of double curved, free-form structures by means of bending. The two
systems were built in prototypical mock-ups to evaluate their geometry and structural capacity. While
the resulting geometries are complex, the use of elastic bending during the assembly process allows
for simple manufacturing methods. Thin plastic sheets, which are easy to cut and bend, were
manipulated by simple, two-dimensional fabrication techniques and assembled into multi-layered,
bent structures that are stiff and self-supporting.
After a general introduction on bending-active plate structures, the authors present the key design
principles used for the development of the two case studies and conclude with a reflection and
comparison of these techniques.
Keywords: bending-active structures, plate structures, multi-layer systems, easy-assembly, flatbed
manufacturing technology, computational design, structural morphology, free-form geometry.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
1. Introduction
Using elastic bending to create curved, load-bearing structural elements has recently gained increasing
attention since this construction method opens numerous possibilities. It is particularly interesting
from an economic point of view because it allows not only to build curved shapes from initially flat
elements but also offers an advantage for their transportation, assembly, and structural adaptation
(Lienhard [4]).
Previous research has already investigated bending-active structures based on the bending of rods and
stripes (Knippers et al. [3]) (Douthe et al. [1]) (Lienhard et al. [6]). To push the topic one step further,
this paper focuses on the use of plates and sheets as starting materials. Usually, the bending of planar
sheets is limited to shapes with single curvature. In order to widen the design spectrum, the authors
explore the idea of generating more complexly curved geometries by interconnecting multiple, single-
curved elements with each other.
This approach is made possible due to the use of very thin sheets, which are additionally modified to
support specific bending deformations. While each sheet individually is rather flexible, their coupling
to a multi-layered structure provides the system with the necessary structural rigidity.
The new design possibilities resulting from this approach can best be explained through the example
of two case studies. As common for bending-active structures in general, the key challenge for both
projects lay in their design and analysis, which usually requires a profound knowledge of the sheets’
structural behaviour and a method of modelling their non-linear structural behaviour under large
deformations. For the form-finding and simulation, both projects therefore implemented the so-called
elastic cable approach developed by Lienhard et al. [5].
The found results were then fed into parametric design tools that allowed for the creation of a digital
abstracted model of the desired structure within the given range of values. In both case studies a
prototypical mock-up was realized to verify and evaluate the developed systems and tools.
2. First case study – controlled material manipulation
Figure 1: (A) Prototype (B) Abstracted digital model
The concept of the first case study is to combine several perforated sheets by locally bending and
interconnecting them to each other. Together, the deformed layers not only keep each other in their
undulated shape but also brace the combined structure significantly. The achieved stiffening effect

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
therein is quite similar to the one found in a corrugated cardboard. However, unlike its two-
dimensional counterpart, this case study uses the stiffening effect in the context of a doubly curved,
multi-layered geometry. Moreover, the aim of this case study is to showcase that the occurring
deformation of the global shape can be precisely controlled and applied to specific free-form target
surfaces. In order to do so, this case study makes use of new computational tools, which combine
findings of physical and numerical simulations with geometrical rules. As a proof of concept, the
system was applied to a challenging target surface (1.5 m x 1.1 m x 0.6 m) with different geometrical
and structural characteristics and built in prototypical scale with 1 mm thin PET-G sheets (Fig. 1A).
2.1. Structural system
The basic system consists of three layers of thin sheets, which are connected at specific points
(Fig. 2A). The slenderness and the controlled manipulation of the sheets by a perforation in specific
areas allow for an out-of-plane deformation and the creation of an actively bent structure. Pulling a
virtual triangle of three points around the holes to a slightly smaller triangle which is provided by the
other layer, an undulating deformation with three high points and three low points occurs (Fig. 2B).
Topological identical sheets, phase shifted in their relative location, and connected to each other,
result in an actively bent structure, which not only keeps its shape but is also capable of bearing loads.
A hexagonal grid can be recognized in the system, in which each vertex represents the high and low
points of one layer (Fig. 2C). To further allow for a morphological differentiation responding to
internal stresses occurring in the target surface, this grid can be adapted. High bending moments, for
example, can be addressed by grid portions with larger cells, whereas high shear forces require a
refinement of the grid.
Figure 2: Coupled layer simulation (A) Input geometry (B) Intermediate stage (C) Fully contracted
Early investigations showed that when using only two layers, the coupled system remains rather
flexible due to continuous hinge lines. To prevent these axes from reducing the structural capacity of
the entire structure, a third layer is introduced that stiffens the structure by adding additional static
height to these previously flexible locations. To explain the structural system one could therefore say
that a double layered structure (Fig. 3A) is deformed into a desired shape (Fig. 3B) and locked in this
configuration by a third layer (Fig. 3C).
Figure 3: Schematic sketch of layer principle (A) flexible (B) deformed (C) rigid

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
2.2. Design process
The geometrical complexity of the differentiated sheets and the possible application to diverse target
surfaces, asked for computational tools to automate the translation of a given surface into fabrication
data. Analysing, form-finding, and unrolling the complete structure is highly complex and would
exceed the performance of common computational processes. Therefore, the system was analysed in
smaller portions, which informed an abstracted, computational 3-D model, from which the fabrication
data is generated.
2.2.1. Abstraction
The structure can be broken down into pairs of two “Y”-shaped elements. By connecting these
elements distance L1 is pulled to a smaller distance L causing the elements to bend each other
(Fig. 4AB).
The curved shape of the elements can be simplified for the generative model into triangular and
rectangular surfaces since only the vertices of the triangle and respectively the connection points are
crucial. The ratio of pitch f to span L and the length of the arch l define the location of the triangles
and can be processed by means of the material information (Fig. 4C).
Figure 4: (A+B) Controlled deformation by contraction (L1>L) (C) Element abstraction
The smallest bending radii occur in the corners of the triangle at the location of the connecting points.
The higher the static height f becomes the larger the stresses due to small bending radii are. Therefore,
the parameter of the ratio comparing the span L to pitch f needs careful negotiation to enhance the
structural performance of the system without exceeding the stress limits.
To inform the computational tool with this crucial parameter, physical and digital experiments were
conducted. Having chosen Vivak® (PET-G) for a larger scale prototype, the material’s mechanical
properties were investigated further through physical tests to provide values for the digital analysis.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
By analysing multiple numerical simulations (Fig. 5A), an ideal range for the ratio of distance L to
arch rise f was found and incorporated in the computational tool. (Fig. 5BC)
Figure 5: (A) Simulation of “Y”-shaped element; Pitch = f, Radius = r, Span = L, Arch length = l
(B) Graphical analysis (C) Numerical analysis
2.2.2. Application to target surfaces and processing fabrication data
To generate the computational model, a hexagonal grid is mapped onto the target surface and altered
accordingly to a previously conducted FE-Analysis (Fig. 6A).
This grid serves as basis for the abstracted high and low triangles, which can be used again as a grid
for the next layer (Fig. 6BC). Thereby layers can be added until a desired structural performance is
gained (Fig. 6D). This methodology can be applied to NURBS surfaces regardless of Gaussian
Curvature (synclastic and anticlastic).
Figure 6: (A) FE-Analysis and accordingly adapted grid (B) Triangle placement on one level
(C) Triangles for adjacent levels (D) 3-D model as basis for fabrication data
In order to translate the abstracted model back to fabrication data with the desired morphology of the
built system, a few steps are necessary beginning with an unrolling of the surfaces. The edges of the
unrolled surfaces serve as system lines, indicating the position of the coupling points on the plates
(Fig. 7A). To obtain the curved appearance the system lines were offset and filleted smoothly to

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
complete the fabrication data (Fig. 7B). Since the coupling points of the unrolled layers are not
congruent, when connecting them the predicted deformations occur (Fig. 7D). This aspect effects both
the element pairs at their local scale as well as the global shape of the coupled system.
Figure 7: (A) Unrolled surfaces (thin) smoothed corners (thick) (B) Offset edges
(C) Sketch of 2D fabrication (D) Controlled deformation when cross-connected
3. Second case study – quick-action coupling
Figure 8: (A) Prototype (B) Abstracted digital model
The second case study follows the general research line but adds a new aspect to it. While the first
case study focused on the fabrication and the abstraction of thin sheet material, this case study also
takes the subsequent assembly process into account. In order to bend and connect multiple structural
elements into a desired shape, a bio-inspired snap-through effect was used. While snap-through
mechanisms are generally widespread in nature, for example to store energy up to a critical point and

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
then release it on demand (Forterre [2]), using this effect as a quick-action coupling to connect, fix,
and bend structural elements into a desired shape is a new idea. Although snap-through buckling is a
failure mode which engineers are taught to avoid, it has enormous potential for design when it is
thoughtfully incorporated in the process. Particularly for the assembly of a structure, snap-through
bucking has a high potential for application, as the following project shows. It implements this effect
in order to join individual parts together without the need for auxiliary tools or special fasteners.
Similar to the previous project, a computational design tool was developed to apply this design
strategy to free-form surfaces.
The system was eventually built in a prototypical scale (2.6 m x 1.9 m x 2.0 m) using thin sheets of
0.5 mm and 0.75 mm PET-G to evaluate the system performance (Fig. 8A).
3.1. Structural system
The structure consists of two basic elements: a series of stripes, which form the upper and the lower
surface, and pairs of truncated cones that act as spacers in-between this two-layered structure
(Fig. 9AB). The form of a truncated cone was chosen as spacer because it can resist area loads in the
surface normal direction and at the same time maintain the ability to undergo snap-through buckling
when loaded at certain points (Fig. 9CD). These cone elements are populated over the entire structure
and are used to provide both structural height and increased interconnectivity between the layers
where structurally necessary. In the resulting structure, a connection similar to mortise and tenon
joints are used to connect the individual stripes of the upper and lower layer with each other as well as
cross-connecting them with the pairs of truncated cones.
Figure 9: (A) Structural elements (B) Sandwich section (C) Stable cone under uniform stress (D) Point
loads trigger the snap-through
To additionally increase the structural performance of the system and minimize the material usage, the
height of the sandwich structure varies based on the information of a previously conducted FE-
Analysis. Areas of high bending moments are met with higher cones that provide more static height,
whereas high shear forces are coped with a higher density of cones, which locally increases the
interconnectivity of the sandwich layers. Since there is a distinct relation of cone radius and sandwich
height, circles are placed computationally onto the target surface with varying radii. These circles lead
to different cone sizes and therefore sandwich heights.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
3.1.1. Assembly process
In the assembly process, the tenons of the cones are inserted into the mortises of the stripes, which
already connects them loosely together (Fig. 10AB). By snapping the cones for a first time into their
inverted stable condition, the previously loose connection becomes fastened, which is due to a
combination of surface friction and geometric restriction (Fig. 10C). This process is then repeated for
all mortise and tenon joints at the top and the bottom layer (Fig. 10D-G). After connecting all the
cones to the top and bottom layers, the structure is still somewhat flexible. Snapping the cones a
second time, however, this time only partially and in opposite direction, finally locks the layers tightly
to each other and makes the structure finally very stable (Fig. 10HI).
Figure 10: Assembly sequence
3.2. Design process
Similar to the previous project, the computational tool implemented in the second case study uses an
abstracted geometry to enhance the computing performance and to enable an unrolling process. The
abstraction neglects details such as the mortise and tenon joints and creates the surfaces only to obtain
a developable topology which maintains the key geometrical information during the unrolling process.
3.2.1. Abstraction
The curvature of the system is created comparable to free-forming of laminated sheets: The top and
bottom layer are bent at first individually and then fixed in shape by a shear connection in this case the
truncated cones. The double curvature can be achieved by perforating the outer layers so that the
desired global curvature can be simplified to a grid of single curved stripes (Fig. 11A). The remaining
triangular areas negotiate the arriving curvatures (Fig. 11B). Little deviations of the computational
model from the built system are acceptable due to possible tolerances in the connection (Fig. 11C).
Incorporating findings gathered during the previous project allows for a controlled and still precise
abstraction process.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
Figure 11: (A) Simplifying the double curved surface to developable stripes (B) Stripe abstraction
(C) Possible tolerances
3.2.2. Application to target surfaces and processing fabrication data
To assure that the cones can be snapped by hand, parameters restricting their proportions and
dimensions were identified in physical tests and incorporated into the design tool. In order to allow for
the second snap-through buckling, the top and the bottom cones of one connection pair need to have
different angles resulting in different radii touching top and bottom surface. For changes of the
curvature direction (convex, concave) (Fig. 12A) the orientation of the cone pair needs to change
accordingly (Fig. 12B). For synclastic regions this orientation can be processed easily since both
principal curvatures have the same direction. Only anticlastic regions need further investigation which
of the principle curvatures has the larger absolute value and is therefore decisive for the orientation.
Figure 12: Orientation switch due to curvature direction
The generation of the 3-D model can be broken down to three major steps. After analysing structural
properties and curvature of the target surface it is populated by a circle packing based on the structural
needs and findings from material test. Eventually the abstracted cone surfaces are created and the gaps
in between the cone pairs are filled with developable surface stripes (Fig. 13).
In order to minimize the self-weight of the structure, a differentiation of the material thickness was
implemented. For the developed mock-up the surfaces were organized in groups of 0.5 mm and

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
0.75 mm Vivak® sheets based on material limitations. The top and bottom surfaces are joined in
developable groups of stripes. To prevent the structure from being weakened by continuous structural
cuts, the stripes are split in different directions in the top and bottom layer. After unrolling, the
abstract surfaces are supplemented by constructive details.
Figure 13: 3-D model generation
4. Conclusion
Although the presented case studies and their design intents are similar and utilise active bending to
create loadbearing structures by means of a multi-layered system, topology and appearance differ
strongly. Whereas the first project consist of topologically and typological identical layers the second
one uses different structural typologies. This indicates the numerous design potentials of bending-
active plate constructions.
Both of the systems showcase a design alternative to the form-finding of bending-active structures.
The case studies share the approach of translating complex geometries to simple manufacturing data
without computationally intensive methods. Informing geometrical abstractions by principle form-
finding methods allows for a fast yet precise computing. The design approaches illustrate that once the
principles of a system are identified and analysed, even complex processes can be broken down to
simple procedures, which can be applied to innumerable target surfaces. Future research could further
investigate the different limitations of the target surfaces and the material possibilities.
Acknowledgement
Besides from the involved Institutes of the Architecture and Urban Planning Faculty (ITKE and ICD)
of the University of Stuttgart the authors want to express their gratitude to the Institute of Aircraft
Design at the University of Stuttgart namely Dr.-Ing. Yves Klett for sharing fabrication tools and
expert knowledge of sandwich structures.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions
References
[1] Douthe C., Baverel O. and Caron J.-F., Form-finding of a grid shell in composites materials, in
IASS 2006. Proceedings of the International Association for Shell and Spatial Structures, 2006;
47; 53-62.
[2] Forterre Y., Slow fast and furious: understanding the physics of plant movements. Journal of
Experimental Botany, 2013; 64; 4745-4760.
[3] Knippers J., Cremers J., Gabler M., and Lienhard J., Construction Manual for Polymers +
Membranes. Edition Detail. (1st ed.), Institut für internationale Architektur-Dokumentation,
2011.
[4] Lienhard J., Bending-active structures: form-finding strategies using elastic deformation in
static and kinetic systems and the structural potentials. Dissertation, Universität Stuttgart,
Institut für Tragkonstruktionen und Konstruktives Entwerfen, 2014.
[5] Lienhard J., La Magna R., and Knippers J., Formfinding bending-active structures with
temporary ultra-elastic contraction elements, in MARAS 2014. 4th International Conference on
Mobile, Adaptable and Rapidly Assembled Structures, 2014; 136.
[6] Lienhard J., Schleicher S. and Knippers J., Bending-active Structures – Research Pavilion
ICD/ITKE, in Proceedings of the International Symposium of the IABSE-IASS, Symposium,
2011.