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Proceedings of the International Association for Shell and Spatial Structures (IASS)

Symposium 2015, Amsterdam

Future Visions

17 - 20 August 2015, Amsterdam, The Netherlands

Linear Double Folded Stripes

Rupert MALECZEK *, Gregoire LORIOT de ROUVRAYa, Clemens PREISINGERb

*Institute of Design | unit koge. Structure and Design

University of Innsbruck / Faculty of Architecture

Technikerstrasse 21c, 6020 Innsbruck, Austria

Rupert.maleczek@uibk.ac.at

a Bollinger Grohmann Schneider ZT-Gmbh

Franz-Josefs-Kai 31-1-4, 1010 Vienna, Austria

grouvray@bollinger-grohmann-schneider.at

b Institute of Architecture | Structural Design

University of Applied Arts Vienna

Oskar Kokoschka-Platz 2, 1010 Vienna, Austria

cp@karamba3d.com

Abstract

This paper presents the research to find a computational method for creating freeform structures

consisting of linear double folded stripes.

The author developed an algorithm that enables a freeform-surface approximation with folded

rectangular stripes. In assembled state they form either hexagonal or octagonal three-dimensional

patterns. The stripes are rectangular in unrolled condition, and get no torsion in folded state.

The denomination “double folded” has its origin in the generation of the stripe’s folding pattern. For

the assembly of several stripes to a spatial structure every second stripe’s segment is attached to

another stripe’s segment. These segments are called contact segments. Between these contact

segments so called connection segments are connecting two contact segments of each stripe. In this

particular configuration the connection segments have an additional diagonal fold to avoid torsion in

folded state, and to allow more freedom in the design phase to the orientation of the segments. This

additional diagonal fold forms a so-called double fold. This strategy is a result of earlier research on

linear folded stripes [1][2].

The paper will give a concise description of the geometrical approach for the generation of three-

dimensional stripe assemblies based on given surfaces. The limits and potentials of possible

assemblies will be discussed, regarding structural constraints between pattern and stripe assembly.

Keywords: folding, architectural geometry, structural morphology, structural pattern

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam

Future Visions

1. Introduction

Linear folded stripes can be seen as a solution to fabricate freeform structures regarding to material

efficiency, prefabrication and mobility. As the „technical human made nature“ is producing most

materials either in rectangular plates, linear strings with multiple section profiles or in very small

pieces, this approach works with rectangular stripes to be efficient in the use of plate material by

minimizing the produced offcuts. One author developed manifold variations to generate spatial

structures with this particular system [3,4]. As linear folded stripes require a particular assembly

method, some systems require a hexagonal grid, to generate the stripes.

Figure 1: members of a free stripe and an assembled configuration

The presented system differs from other stripe systems as it has an additional fold, in the connection

segment, to allow more freedom in the generation and orientation of the stripe. As the possibilities of

hexagonal grid variations are augmented by this technic, the authors became interested in the analysis

of the structural qualities of these grids.

2. Double folded stripes

During the investigation on linear folded stripes, the double fold was unveiled as a reasonable solution

to extend the range of solutions between two contact segments [2][3][5]. The strategy of the double

fold differs from the similar approach on mesh based stripes in the number of double folds. While

mesh based stripes, generate double folds based on the vertices of a mesh, the presented approach

generates a diagonal extra fold oriented on a connection segment.

Figure 2: mesh based stripe left compared to a double folded stripe

connection distance

contact distance

angle foldable edge

foldable edge

middle axis stripe

contact segment A

connection segment B

contact segment B

contact segment A'

Center of Rotation

additional foldline

a folded stripe

connection distance

contact distance

angle foldable edge

foldable edge

middle axis stripe

contact segment A

connection segment B

contact segment B

contact segment A'

Center of Rotation

additional foldline

a folded stripe

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam

Future Visions

This Vertex-based triangle can be seen as the connection segment between two contact segments that

have a common intersection in the Vertex. Double folded stripes do not have this common intersection

point of the contact segments and have therefore a connection segment that is rectangular in unrolled

condition but needs in spatial folded condition an additional fold to avoid torsion in this segment [5].

The generation of a hexagonal reticular structure is based on consecutive contact segments connected

by connection segments. The double folded stripe is therefore generated on a grid from straight lines,

that define contact and connection segments.

Figure 3: Relationship of stripes segments for an assembly

2.1. Grid configurations

Linear folded stripes can be seen as a relation of a reticular line structure in combination with

directions. As shown in Figure 4, for double folded stripes three main configurations of reticular nets,

are possible. An octagonal grid from closed stripes, and a hexagonal grid either generated from closed

or open stripes. Following the main stripe categorisation [3], all systems presented here are post

defined systems, as one aim of this investigation is the relation of a given surface with the stripe

system.

Figure 4: possible grid configurations with the contact and connection segments highlighted

α angle rotational axis

Dp rotational angle

f opening angle

connection segment

contact segment A contact segment A

contact segment B

middle fold

α

f

contact distance

connection distance

open Octogrid open hexgrid closed hexgrid

empty cellempty cell empty cell

empty cellempty cell

contact segments connection segments

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam

Future Visions

2.2. Geometric constraints and stripe generation

The generation of this particular system consists of two consecutive steps. After the generation of a

line network, and the definition of the stripe system, the main orientation of each contact segment

must be defined, to generate a network from continuous Baselines. In our case the main orientation of

the contact segments is oriented on the surface normal, of the surface that we are approximating. It

must be considered here that the main orientation of the contact segments can also be freely chosen.

With this combination of baselines and vectors, the stripe generation can be done.

Figure 5: stripe elements

The stripe generation starts with a contact segment. The Segment of the Baseline representing the

Main Direction of the connection segment is translated in the Main Orientation with the stripe height.

The Baseline’s Segment that belongs to the adjacent Connection segments encloses the angle f that

defines the first foldline after Berthomier’s formula [5]. This first foldline generates the first Midfold

that is the connection of the endpoint of the first foldline and the end of the connections segments

Baseline.

For the determination of the second foldline the knowledge of the already known foldlines in unrolled

condition is used. If folding is seen as a rotation of Vectors from one position to another position

using a reflection plane, we can consider the relation of the Midfold in folded position and in unrolled

position. We can determine the reflection plane’s normal vector as the relation of the Vectors

MidfoldFolded – MidfoldUnrolled.

The used method for the generation of the double fold determines one solution for each relationship of

two contact segments that are connected with one connection segment. This single solution must not

always be a potential fabricable one as the length of the contact segments, the angle of the consecutive

foldline and the stripe height are limiting the valid results, as foldline intersections are not wished in

this context.

Middle Edge

foldline 2

foldline 1

Middle Edge

unrolled stripe

oriented to

conatct segment #2 Main Orientation

segment #2

Main Orientation

segment #1

α1

α1

f1

contact segment #2

contact segment #1

f2

α2

contact segment #1 contact segment #2

connection segment

α2

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Figure 6: Foldline Intersections as non-valid solutions

For this particular investigation, the authors did not work on an algorithm that reacts on foldline

intersections, by a reconfiguration of the baseline grid. This is due to the fact that the structural

optimisation method explained more detailed in the chapter “3. Structural Behaviours” is already

intervening and manipulating this baseline grid. If the authors had implemented a redundant baseline

grid manipulation the results of the structural optimisation would have been less clear.

2.3. Stripe Orientations

In the standard configuration a hexagonal cell from linear folded stripes has a Main Direction and a

Main Orientation [4]. As shown in Figure 5 the Main Orientation is the direction normal to the

baseline, in which the height of the stripe is measured. The main Orientation is linked to the direction

of the contact segments. In other stripe assemblies the main direction of all contact segments is

identical, and therefore clearly defined. As one advantage of double folded stripes is the liberation

from these identical directions, it is also more desirable to define a Main Direction for an entire stripe

assembly. As the origin of the structures investigated for this paper are Nurbs Surfaces, we can

consider a Main Direction if the contact segments follow either the U or the V direction.

Figure 7: a surface approximated with two different Main Directions linked to the UV division

3. Structural modelling of the double folded stripe method

On the basis of the geometric principles developed so far, several structural models were derived,

analysed and structurally optimized using the parametric finite element (FE) program Karamba. The

goal was to prove the technical viability of large-scale structures based on the double folded stripe

method.

The additional fold in the connection stripe allows the designer to arbitrarily choose the direction of

the fold lines at the connection elements. A connection detail without the double fold can be seen in

Figure 3: The fold lines at the connection segments do not coincide and so the upper and lower fibres

of the cross section of the connection segments do not meet. This reduces the capacity of the

connection for transmitting bending moments and complicates the construction process. With aligned

folds conventional means like e.g. gusset plates or straps can be used in case of tension to connect the

segments.

h

1 2

l2

l1

lmin = (h / tan 1 )+ (h / tan 2)

lmin

Future Visions

3.1. Choice of material

In the past the folded stripe method has been used to construct a medium sized pavilion (Figure 8)

with a ground plan of roughly 10 x10m [5]. Wooden planks of 20 x 2cm were used as building

materials. The edges of the planks were cut and connected to mitre using wood screws and a wooden

infill (Lamello). Partly prefabricated, the pavilion could be successfully assembled on site. The

complicated flow of forces at the connections made it however difficult to reliably predict the

structural behaviour using finite element calculations. Thus in order to arrive at comparable results for

folded stripe structures all calculations assume cross sections folded from thin sheet metal of a

thickness of 3mm. This significantly reduces the complexity of the connection nodes. For all folded

stripes structures a steel of class S235 was assumed. it has a design yield strength of 23.5kN/cm2 and a

specific weight of γ = 78.5 kN/m3.

Figure 8: the “Design by Research” pavilion [5] made from wooden planks (in detail right)

3.2. Cross section properties and orientation

Depending on the geometry of the reference surface and the distribution of external loads, torsion

moments in the folded stripes structure play a significant role. Therefore the structural efficiency of

open and closed cross sections for application in folded stripe structures was assessed. In order to

reduce the complexity of digital model construction, each member was split in two with the strong

axis of the cross sections in each part tangent to the reference surfaces normal at the beams endpoints.

In real structures the geometric torsion can be accounted for by the additional fold-line of the double

folded stripe method. This would be applicable for e.g. structures made from wood planks (Figure 8).

Alternatively open cross sections built up from thin sheet like materials can be warped from a straight

initial state without inducing significant initial stresses. Their torsion stiffness can be enhanced by

closing the cross section after applying the initial warp. The torsion stiffness of open cross sections

like C, U and I-profiles is the sum of their St. Vernant and warping torsion stiffness. In all finite

element calculations the warping torsion stiffness was neglected.

In case of open cross sections made from straight segments the torsion moment of inertia for

calculating the St. Venant torsion stiffness amounts to: IT = ∑(si×ti

3/3) ,where si is the length of the

segment and ti the corresponding wall thickness. The equivalent property for closed cross sections is

given by Bredts first formula: IT=4×A×2/∑ (si/ti) ,with si and ti as before and A being the area

enclosed by the cross section. A comparison of the two above formulas shows that the torsion stiffness

of closed cross sections can be larger by several orders of magnitude than that of an open cross section

with similar height and width [6].

Future Visions

3.3. Assumed load conditions

In all calculations three typical loads were considered: dead weight, service load and wind. Dead

weight is automatically evaluated and included by the FE-program. Service and wind loads were

assumed to be of magnitude 1kN/m2 each. The application of service loads to the structure happened

in such a way as to produce the most severe stresses in the structural elements. In case of arched

structures like barrel vaults this is e.g. full load on only one half. Wind loads acting in two horizontal,

perpendicular directions were taken account of using additional, separate load-cases. Wind was

assumed to act perpendicular to the reference surface of the structure. In the finite element model the

wind pressure on the reference surface was transferred to the nodes of the beam structure.

3.4. Structural behavior of barrel vaults constructed with the double folded stripe method

As a preliminary to the investigation of more involved geometries, barrel vault structures implemented

with the double folded stripe method were analysed. A hexagonal grid structuralised from linear

folded stripes, based on regular distributed uv-coordinates on a simple vault geometry results in a

regular pattern (see Figure 9). All structure nodes along the two lower edges were fixed against

translations in all directions. Along the free edges at the front- and back-end of the vault, edge beams

of large bending stiffness were added.

In order to study the influence of different structural densities in the main directions of the vault, the

finite element model was based on a parametric set-up generated in the parametric design environment

Grasshopper (GH) [7]. On top of this the finite element program Karamba [8] was used to assess the

structural response of the system. As cross sections uniform quadratic shapes were assumed

throughout the structure.

Figure 9: Cell deformation for better structural behaviour

For the trivial case of an edge supported barrel vault it is clear, that the main load bearing direction is

the short span between the line-supports. A beam layout will thus perform better as compared to

others if it more closely adheres to the anisotropy of the force flow determined by the support

positions. Fig. 8 shows a comparison between two geometries with high structural density in

longitudinal and cross direction. The latter enables a more direct flow of forces from the loaded nodes

to the supports which results in a more efficient load bearing system.

Both structures in Figure 10 feature the same dead weight and external loads. Due to the more

efficient load bearing mechanism of the structure on the left hand side its maximum deflection is only

10% of that calculated for the right hand side alternative.

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Figure 10: Displacement patterns of linearly supported barrel vaults with different structural densities

in longitudinal (left) and cross direction (right)

4. Optimisation of the folded stripes panelisation on arbitrary reference surfaces

4.1. Parameterization of the hexagon pattern for arbitrary reference surfaces

a) b) c) d)

e) f)

Figure 11: Procedure for generating a hexagonal panelisation for folded stripes structures on arbitrary

surfaces

The parameterisation of the geometry in the previous vaulted barrel example depended on two

parameters only – the number of cells in each of the main directions of the reference surface. In order

to allow for more complex solutions the following parameterization for the hexagon grid was

implemented. Starting from the normalized uv-representation of an arbitrary surface (see Figure 11a)

the boundaries of the uv-surface were populated with points (Figure 11b). Parameters control the

position of the points on the boundary which serve as endpoints of lines that connect opposite edges of

the uv-rectangle. The nodes of the hexagon patter for the folded stripes structure result from line-line

intersection. Figure 11e and Figure 11f show the application of the procedure to an arbitrary reference

surface.

The advantage of this parameterization approach is its simplicity and the fact that only valid hexagons

patterns result. In the context of this investigation convex hexagons are considered valid. Concave

Future Visions

cells were a priori excluded based on the assumption that these are structurally less effective than

convex ones and that they incur higher production efforts and costs.

4.2. Structural optimization of folded stripes structures

There exist three different levels of complexity for tackling structural optimization tasks: sizing of the

member cross sections, shape- and topology optimization of the overall structure. For the following

investigations the cell topology was considered to be fixed. The shape optimization of the hexagon

pattern was done on the basis of the parameterization described above. The whole geometric setup

was implemented as a Grasshopper (GH) definition.

GH comes with a range of optimization engines which can be easily integrated into any definition.

The plug-in ‘Galapagos’ [9] is one of these. It implements an evolutionary optimization algorithm

which works well for complex problems like structural shape optimization. Galapagos takes control of

the parameters which define the folded stripes geometry and tries to find a set of values for them that

minimize or maximize a dependent value – the so called objective function of the optimization

problem. Fig. 10 shows a symbolic representation of the applied optimization script.

Figure 12: Symbolic representation of the optimization procedure.

The finite element program Karamba features a module for optimizing the cross sections of individual

members. This was used as an additional optimization cycle inside the geometric optimization routine.

Karamba’s cross section optimization routine lets the user set an allowable maximum displacement

and chooses the members cross sections in such a way that the maximum stress in the structure is

equal or less than the material strength. The relative structural efficiency of two geometric variants of

a folded stripes structure can be assessed by comparing the resulting mass after cross section

optimization.

5. Optimization of folded stripes structures on three characteristic reference

surfaces

In order to assess the structural viability of double folded stripe structures made from sheet metal,

three reference surfaces were investigated (see Figure 13). In ground plan they measured

approximately 14 x 14m. The first one is single curved only, whereas the other two feature double

curvature. In structural respect surface a) acts like an arch, the double curved geometries show the

typical load bearing behaviour of shells.

In Figure13 red lines and red points symbolize line supports and point supports respectively. The main

difference between surfaces b) and c) is the support condition.

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Figure 13: Three reference surfaces with different support conditions for generating optimized folded

stripes structures. a) single curved surface with line support; b) double curved surface with point

support; c) double curved surface with line support.

5.1. Cross section optimization using fixed geometries

In order to assess the influence of the cross section shapes on the structural performance of folded

stripes structures the reference surfaces of Figure 13 were populated with fixed, regular hexagon

patterns. On each surface the same number of hexagonal cells was created.

Figure 14 shows a rendering of the resulting cross sections after cross section optimization under dead

weight, wind- and service loads using I-profiles only. It can be seen that in some regions huge I-

profiles result. This is due to the fact, that I-profiles have only a small torsion resistance as compared

to their strength in bending. The presence of torsion in the structure thus necessitates large I-sections,

which are utilized in an inefficient manner.

Figure 14: Rendering of the resulting cross sections after cross section optimization using I-profiles

only.

Hollow box cross sections exhibit a much larger ratio of torsion to bending capacity than I-profiles. In

order to assess the influence of cross section type, the above structural optimization was redone using

hollow box cross sections only. Figure 15 shows that the resulting distribution of cross sections is

more uniform than in Figure 14.

The structures in Figure14 and 15 fulfil all structural boundary conditions. The resulting mass

therefore constitutes a useful measure of relative structural efficiency. For surface a) the resulting

mass using I-profiles is 25kg/m2 compared to 7.9kg/m2 for the hollow box variant. For surfaces b) and

c) the corresponding comparative weights are 24kg/m2 to 7.7kg/m2 and 22kg/m2 to 7.8kg/m2. These

numbers were calculated by dividing the total weight by the area in ground plan.

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Figure 15: Rendering of the resulting cross sections after cross section optimization using hollow box-

profiles only

5.1. Shape optimization combined with cross section optimization

The combination of shape- and cross section optimization leads to the results shown in Figure 16. The

shape optimization was done using Galapagos and the uv-parameterization approach depicted above.

The result of surface a) shows that the beams of the folded stripes structure get aligned with the

principal load bearing direction. In an equivalent shell this would coincide with the lines of principal

bending moment. The shape optimization of the hexagon layout thus positions the beams in such a

way as to minimize torsion moments in the members. This also applies to the grid layout of the double

curved surfaces b) and c).

Figure 16: Rendering of the resulting cross sections after shape and cross section optimization

As expected the geometric optimization step further decreased the structural weight as compared to

the above analysed variants. The specific structural weights of surfaces a), b) and c) were calculated as

5.8kg/m2, 6.7 kg/m2 and 7.4 kg/m² respectively.

6. Conclusions

The structural assessment of three characteristic shell geometries shows that the double folded stripes

method can be efficiently used for large scale structures. The calculations show that torsion moments

play an important role in hexagonal grid shells. It is therefore advantageous to use closed cross

sections in regions of the structure were torsion moments prevail. This can be easily done using

corresponding folding patterns for the stripes.

The application of shape optimization tends to position the structural members in such a way that the

overall torsion moments in the structure are minimized. The calculations presented above show that

irregular hexagonal grids therefore perform better than their regular counterparts.

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Outlook

The interaction between stripe structure and its underlying line-network with the structural ability is

an interesting filed, that asks for a more intense investigation. This first investigation is therefore a

first step towards this more intense investigation. Especially the relation of the form that is chosen to

generate the stripes and its force flow requires a more profound investigation. The authors assume that

these structural dependencies will lead to the application of linear folded stripes by the building

industry in large-scale structures.

Figure 17: a potential pavilion from linear double folded stripes

References

[1] Maleczek, R., Joyce, C., & Geneveaux, C., Linear folded V-shaped stripes. Design Modelling

Symposium 2013;321-334.

[2] Maleczek, R., Genevaux, G., & Ladinig, J., Linear folded mesh based stripes. From Spatial

Structures to Space Strucctures. Seoul: IASS.2012

[3] Maleczek, R., & Geneveaux, C., Open and closed linear Folded Stripes”. Taller,Longer,Lighter;.

London: IASS 2011.

[4] Maleczek, R., „Linear Folded (parallel) stripe(s).“In Computational Design Modeling, von

Christoph Gengnagel, Axel Kilian, Norbert Palz und Fabian Scheurer, 153-160. Berlin:

Springer, 2011.

[5] Preisinger, C., Karamba - Wooden-folded-parallel-stripes www.karamba3d.com/ Wooden-

folded-parallel-stripes / 29 Sep. 2014. Web. 27 Mai 2015.

[6] Petersen, C.: Stahlbau – Grundlagen der Berechnung und bauliche Ausbildung von Stahlbauten,

3. Auflage

[7] Wikipedia contributors, Grasshopper 3D, Wikipedia, The Free Encyclopedia. Wikipedia, The

Free Encyclopedia, 24 Sep. 2014. Web. 27 Mai 2015.

[8] Preisinger, C., Karamba – parametric structural modeling, user manual for version 1.1.0,

http://karamba3d.com, 2015.

[9] Rutten, D., Galapagos: On the Logic and Limitations of Generic Solvers, Architectural Design,

Special Issue: Computation Works: The Building of Algorithmic Thought, Volume 83, Issue 2,

pages 132-135, March/April 2013.