Considerations on the Scaling of Bending-Active Structures

Considerations on the Scalingof Bending-Active Structures

Julian Lienhard1,* and Jan Knippers1

1Institute of Building Structures and Structural Design,University of Stuttgart, Germany

(Submitted on 16/03/2013, Reception of revised paper 01/06/2013, Accepted on 09/09/2013)

ABSTRACT: Bending-active structures are composed of curved beam or shellelements which base their geometry on the elastic deformation of an initiallystraight or planar configuration. In bending-active structures the moment ofinertia has a direct influence on the residual stress and is therefore limited bya given minimal curvature in the system and the permissible bending stresscapacities of the chosen material. These interdependencies may lead to ascaling issue that limit bending-active structures to a certain size range. Thisrange may be widened if the system’s reliance on elastic stiffness iscompensated by other stiffening factors such as coupling of structural elementsand stress stiffening effects.

This paper will analyse the scaling effects in bending-active structures.Some basic systems are studied by means of dimensional analysis and FEMparameter studies to clarify at which power each influencing factor effectsscaling. Based on these findings some more complex structures are studied fortheir scalability. This will offer the basis for some more general conclusions.

International Journal of Space Structures Vol. 28 No. 3&4 2013 137

1. INTRODUCTIONThe term “bending-active” was introduced by theauthors to describe curved beam and surfacestructures that base their geometry on the elasticdeformation of initially straight or planar elements [1].Active bending in this context is understood to be aform defining strategy based on systemised elasticdeformation i.e. bending. The main motivation foractive-bending lies in the simplicity of producingcurved elements, which per definition containconsiderable amounts of residual bending stress.Since residual stresses are dependent on the bendingradius and cross-sectional height, we are restricted bymaterial strength in the sizing of structural members.This limitation means that the size of a cross-sectionmay not be defined freely according to therequirements for strength and stiffness under externalloads. Hence scaling problems may occur whenchanging the scale of a bending-active structure.

In classical structural engineering we may consider3 ranges of scale: the physical structural model, areduced scale structure and a large scale structure, in

which dimensional analysis and the derived scalinglaws help to calibrate the proportions of test resultsbetween different scales [2]. In today’s engineeringpractice analysis is mostly based on Finite-Element-Modelling (FEM) in which dimensions are consideredby the relation between geometrical and mechanicalinput variables. Structural analysis is therefore alwaysdone on a virtual 1:1 model. With these powerfulcomputational means the necessity for structuralphysical models has been reduced to some dynamicproblems e.g. wind tunnel testing. Reduced scalemock-up structures have similarly becomedispensable.

In the development of bending-active structures thephysical structural model has regained importance as aform-finding tool in the early design stages. Anemergent amount of medium scale bending-activeresearch structures is raising the question of theirrelevance for large scale building structures. In somecases like the gridshell, visionary projects such as the‘Multihalle Mannheim’ with 64 m span have longproven their scalability. In this particular project, which

*Corresponding author e-mail: [email protected]

138 International Journal of Space Structures Vol. 28 No. 3&4 2013

was predominantly developed through physical models,scale factors for self-weight were derived to correctlysimulate dead load deformation of the scaled model [3].Other expressions of active-bending in buildingstructures are yet to be analysed for their scalingbehaviour. The research presented in this paper is basedon the experiences from various research structures inthe range of 2 to 10 m span. In all of these structures thescalar jump from a physical structural model to amedium scale structure was successfully undertaken.The question analysed in this paper concerns the scalarjump from a medium scale to a large scale structure andis thereby aiming to fathom the scaling limits of variousforms of bending-active structures.

Scaling in the most general sense is concerned withpower-law relationships between two or morevariables of a system. Investigating the scaling ofbuilding structures the variables concerned aredeformation and stability on the one side, load andmechanical properties on the other. If the relation ofthese variables is independent of the system’sdimensions we consider the system to be self-similar.

Some of the more common effects to consider forthe scaling of building structures are:- Dimension effect: cubic increase of mass with

scale- Load effect: quadratic increase of surface area

leads to quadratic increase of surface load- Size effect of material: probability of material

defects increases with size, whereas the influenceof material defects increases for small sizespecimens.

- Height effect: exponential growth of wind-speedwith height combined with quadratic growth ofwind-load with speed.

- Dynamic effects: Wind induced vibration etc.Since the investigations in this paper are aimed to

be of general nature we are only considering thechange in mass and load. The effects of change inmaterial properties, wind load and dynamic behaviourwith scale are very individual to each project andtherefore not taken into account in the furtherinvestigations. Their influence on scaling howeverwill play a role on the construction of some large scalebending-active structures.

2. DIMENSIONAL ANALYSIS OFELASTICAIn order to gain a principal understanding of thescalability of a simple bending-active system someinitial studies are made on the elastica arch. Theelastica describes the elastic deformation curve of a

slender single span beam in its post buckling state.After some first considerations by James Bernoulli in1691, Euler describes the shape of these curves in ‘Descurves Elastics’ in 1744, offering analytical solutionsfor post buckled curves of a pin supported beam [4](Fig. 1)

The residual stress in such an elastically deformedbeam can be determined with the Euler-Bernoulli lawwhich states that the bending moment My isproportional to the change in curvature as shown inequation (1) [5]. With the section modulus Wy and theconsideration that the width b of a cross-section has noinfluence on the maximum bending stress we can writethe residual bending stress as an expression of thecross-sectional height h, the Modulus of Elasticity Eand the Curvature 1/r (2).

(1)

(2)

(3)

In (2) we can see that both curvature (1/r) and crosssectional height h have a linear influence on theresidual stress caused by active bending. The momentof inertia Iy is therefore limited by a given minimalcurvature in the system and the permissible bendingstress of the chosen material. The radius of curvaturecan be expressed as a function of the span Li and therise fi. In (3) this relation is given with a scale factor s.Simplification of the equation shows that s can beexcluded as a linear factor, thus linear up-scaling of astructure allows for a linear up-scaling of the cross

rs L

s f

s fs

L f

fi

i

i i i

i

=⋅⋅ ⋅

+⋅

= ⋅+ ⋅⋅

( )2 2 2

8 2

4

8

σ My

y

E I

r W

E h

r=

⋅⋅

=⋅⋅2

1

r

M

E Iy

y

=⋅

Considerations on the Scaling of Bending-Active Structures

f

L

Wx

Li

EIfi

hr

Figure 1. Development elastica curves.

Julian Lienhard and Jan Knipprs


sectional height while keeping the residual stressconstant. Assuming constant material properties thisleads to the overriding question; whether the influenceof the span L on the deflection Uz can be compensatedby the moment of inertia Iy, if the scaling of the cross-sectional height is limited to be linear for keeping theresidual stress σM constant.

2.1. Deriving the dimensions using theBuckingham Pi Theorem:Dimensional analysis enables a convenientinvestigation of physical behaviour by combining thevariables of a system into dimensionless groups (Pi-terms). The Buckingham Pi-Theorem states that therelations in any physical system can be described by agroup of n–rd Pi-terms, in which n is the number ofvariables and rd the number of basic dimensionstherein (rank of the dimensional matrix) [6]. Inmechanics the basic dimensions are mass, length andtime. In the following considerations on staticstructural behaviour, force is chosen as a basicdimension without further reduction into itsconstituent components, for better comparison toknown engineering equations. Based on the Pi-terms afunctional equation can be derived which shows areduced form of the relevant variables, however doesit not give information about the nature of the solution.The exact form of the functional relationship has to beempirically obtained by a set of experiments in whichthe Pi-terms are systematically varied. Analyticalanalysis the individual Pi-terms often is sufficientenough to describe the change of system behaviourwith scale, without knowing the complete solution ofthe functional equation.

Investigating the deflection for a given elasticacurve of span L stiffness EI and the line load qz andexcluding the influence of mass and residual axial forcewe may derive the following functional equation:

Uz = f (L, E, Iy, qz)5 Variables: Uz, L, E, Iy, qz

5-2 = 3 Pi-Terms2 Dimensions: [mm], [N]The dimensional Matrix is:

The dimensionless Pi-terms may be derived usingvarious procedures, some of which are explained anddiscussed in detail by Barr [7]. Independent of the

U L E I q

N

mm

z y z

[ ]

[ ]

0 0 1 0 1

1 1 2 4 1− −

procedure it must be noticed that there is no unique setof Pi-terms that can be derived for a given problem. Pi-terms may differ in type depending on the choice of arepeating variable that eliminates dimension,additionally transformations of Pi-terms are possible.Here the results of the system deflection under a linearload were derived with the step wise procedure using Las the repeating variable. The resultant dimensionlessPi-terms are:

(4)

An analysis of the given Pi-terms now enables toclarify at which power each variable is effected byscaling. In the common terminology of dimensionalanalysis one differentiates between the scale of themodel and that of the prototype. Investigating thescaling of Iy we can compare the Term π2 in a modelstate m to an s times larger prototype state p.

(5)

If all dimensions of the system are scaled by thesame factor s we can show that the similitude in (5) issatisfied by a linear up scaling of the cross sectionaldimensions, here shown for a flat section:

(6)

With (5) we can show that the deflection of anelastic arch is self-similar, if the effects of self-weightand residual force are excluded.

Similar to (5) the Term π3 can be used to analyse therelation of the loading condition in model andprototype state, showing that the line load qz scalelinearly for constant material properties betweenmodel and prototype state:

(7)

q

E L

q

s E Lq

s s q

z

m

z

pz p

L E

⋅

=⋅ ⋅

→

= ⋅ ⋅

) ,

zz m,

s I sb h s b s h

I b h

mm m m m

p p p

4 43 3

12 12⋅ = ⋅ = ⋅ ⋅ ⋅

= → ⋅ =

( )

ss b hm m⋅ ⋅

L

I

s L

II s I

y m y p

y p y m

4 44

=

⋅

→ = ⋅

( ), ,

π π π

φ

1 2

4

3

4

= = =⋅

→ =⋅

U

L

L

I

q

E L

U

L

L

I

q

E L

z

y

z

z

y

z

; ;

,


with and

Including self-weight with the constant for densityρ and earth acceleration g adds another Pi-term, givenin (8) where ρ and g appear as a linear factors. Thecubic growth of mass with the cross-sectional area andlength of an element however violates the similitudeshown in (5).

(8)

On the other hand, for lightweight structures such asconsidered here the cubic growth of mass with scale isper definition of minor influence, since the stiffness ofa lightweight structure is relying not solely on theelastic stiffness of their members but also on theirtopological arrangement and geometric stiffness. Theinfluence of mass on scaling may therefore benegligible for small changes in scale at a medium sizerange, however it will play a role when investigatinglarge scale structures.

In the investigations above, only linear behaviourand elastic stiffness were taken into consideration.Next to the residual bending stress, there usually alsois a considerable amount of residual axial force N in abending-active system e.g. the nonlinearly distributedaxial compression force in the post buckling state of anelastica curve. In a nonlinear investigation axial forcehas an influence on the geometric stiffness (initialstress stiffness) which, in the case of bending-active

π ρ4 =

⋅ ⋅g L

E

sE

EEp

m

=sL

LLp

m

=structures is usually also nonlinear in its distributionalong individual elements. Such behaviour cannot beconsidered by a single equation derived throughdimensional analysis.

From these considerations we can draw apreliminary conclusion: Bending-active systems areself-similar if dead load plays a minor role and axialforce is not destabilising to the system.

3. FEM ANALYSIS OF ELASTICAIn the following study, the conclusions drawn aboveare verified for several span to rise ratios of elasticacurves and then studied considering the mass and axialforce N for a set geometry at varying size ranges, usingnonlinear Finite-Element-Analysis. Each geometrywas first form-found using incremental deformationand then studied for load displacement behaviour bystep wise increasing the load up to the point of systemfailure. These studies were done using the commercialFEM Software Sofistik® with the nonlinear solverASE 3. All the studies shown in this paper were madeon multiple scales and various loading conditionsassuming constant material behaviour, of which themost representative results are shown in the graphsbelow.

In Fig. 2 elastica curves with various span L to risef ratios are investigated in which L is kept constant.The system is investigated at scale factors s = 1, 2, 4and 8. In accordance to Pi-term 3 and equation (7)derived above, the line load qz is also scaled by s toaccount for the growing loading area that may bespanning between two arches. The system lines in


0

0

50

100

150

200

250

2 4 6 8 10

Scale s (−)

Uz

(mm

) (w

ith q

z =

s ×

q z )

Straightf/L: 0.05f/L: 0.15f/L: 0.20f/L: 0.30f/L: 0.50f/L: 0.60f/L: 0.75

s.f

s.q s.LUz

Maximum stiffness f/L = 15% − 35%

(a) (b)

Figure 2. (a) Study of elastica curves with constant span and varying rise at different scales.(b) deflection curve for different scales.



Fig. 2a show the various arches in the initial andcomparative deformed position, clearly showingmaximum stiffness at ratios of f/L = 0.15 to 0.35. Asthis ratio tends towards f/L = 1.0 the arches showincreasingly low stiffness and nonlinear behaviour;compare dashed line in Fig. 2a. The graphs in Fig. 2bshow the displacement curves in the linear rangewithout dead load or residual axial force. Theirlinearity prove the similitude derived above for allvariations of the elastica curve, in which the rise tospan ratio is a constant in correspondence with theincline of the graph.

In Fig.3a an elastica curve with f/L = 0.15 isinvestigated further by plotting the load deformationdiagram with corresponding compression stress for astep wise increasing scaled line load. In order toexclude findings that are limited to symmetricsystems, the line load was applied asymmetrically. Theinvestigations were compared at two different scaleswith s = 1 and s = 8 in Fig. 3b showing the nonlinear

behaviour at higher loads and final snap throughbuckling. For each scale the system was calculated inthree different scenarios.

First (indicated by the continuous lines) includingdead load and axial force, showing a difference in loadfactor ∆λ = 5 between scale s = 1 and s = 8 at the pointof snap through buckling.

In a second scenario (dashed line) the residual stressand thereby axial force N was disabled. This leads to ashift of both graphs by load factors ∆λ = 5 for s = 1 and∆λ = 6 for s = 8 higher at the point of failure. Thedifference between scales is reduced by one factorfrom ∆λ = 5 to ∆λ = 4.

Finally the dotted line shows the elastica in acalculation without dead load and residual stress. Herethe two load deflection curves almost perfectly matchand snap through buckling occurs at the same loadfactor, ∆λ = 0. This clearly supports the hypothesismade in section 2 that the bending-active elastica archis self-similar if dead load and axial force are omitted.

λ

λ

λ∆ = 5

∆ = 4

∆ = 0

Snap through

Nonlinear

Linear .s.qz 0.8.

s.L

Uz

S = 1 dl: yes N: yesS = 1 dl: yes N: noS = 1 dl: no N: noS = 8 dl: yes N: yesS = 8 dl: yes N: noS = 8 dl: no N: no

25

20

15

10

5

00 20 40 60 80 100

λ

λ .s.qzλ

Load

fact

or

(−)

Displacement Uz/s (mm)

0.80

0.90

1.00

−12.00 −10.00

−10.00

−1.79−4.01

−10.5

−6.37

−18.5

−51.8

−47.6

−39.8

−27.0

−6.55

−42.5

−65.1−75.2 −7

9.2

−7.6

7−1

6.4

−35.

1

−45.

8

−51.

2

−53.

0

−67.

4

−44.

9

−18.

9

−18.2

−35.0

−49.6

−59.6 −63.

2

−60.

0

−50.

3

−35.

8

−18.

7

−1.7

8

−8.00 −6.00 −4.00 −2.00 0.00

−8.00 −6.00 −4.00 −2.00 0.00

−10.00 −8.00 −6.00 −4.00 −2.00 0.00

0.00

−2.0

0

−12.00 −10.00 −8.00 −6.00 −4.00 −2.00 0.00

2.00

0.00

Beam line load, load factor 20

Beam elements, compression stress Mpa,nonlinear loadcase, load factor 5



(a) (b)

Figure 3. (a) Asymmetrical line load on elastica curve showing compression stress and deformation to the point of snap throughbuckling. (b) Load deflection curve of elastica curve with 15% f/L ratio at two scales; showing linear, nonlinear and snap

through failure range.


In general the graph shows how the curves arevery close in the linear range and the influence ofdead load grows with size (∆λ = 1 for s = 1 and ∆λ= 5 for s = 8).

4. SCALING OF THREE CASE STUDYSTRUCTURESWith the scaling investigations on three successfullybuilt case study structures, the above drawnconclusions are verified. A jump in scale from aprototypical structure, in the size of an exhibitionpavilion, to a large building structure is investigated.The choice of material for these bending-activestructures is limited by availability of materialsoffering high strength with low bending stiffness suchas plywood and GFRP. In addition the scalarinvestigation are made in a relatively small range of s= 0.5 to 5. Therefore material properties are assumedconstant. For all systems several wind and combinedsnow load cases were investigated; the graphs in thispaper are a selection that highlight best the scaledependent behaviour discussed here. In contrast to theinvestigations on the elastica, loads are areal andtherefore constant across scale. Height effects of windare not considered because of the relatively smallscalar range investigated.

The case study structures have been discussed in-depth in previous publications, therefore only shortintroductions are given with the references for furtherinformation.

4.1. Research pavilion ICD/ITKE 2010At the end of July 2010 the Institute of Computerbased Design (ICD) and the Institute of buildingstructures and structural design (ITKE) at theUniversity Stuttgart realised a temporary researchpavilion made of plywood. The design of thepavilion was the result of a student workshop whichfocused on the integration of physical experimentsand computational design tools to develop bending-active structures. The structure is entirely based on

the elastic bending behaviour of 6.5 mm birchplywood strips. The strips were roboticallymanufactured as planar elements, and subsequentlyconnected to coupled arch systems. Radialarrangement and interconnection of the self-equilibrating arch system lead to the final torusshaped design of the pavilion with a total diameter of10.5 m, maximum free span of 4 m and a height of3.5 m. The form-finding and structural behaviour ofthis structure was explained in [8].

The FE form finding routine developed for thisstructure was used for the scaling investigationspresented here (Fig 4). The system was studied onthree different scales in which each model wasseparately form-found. Thus all effects from residualstress are included in the investigations.

In Fig. 5a the comparison of the deformation undervertical load shows significant stiffening effects in thesystem where the residual stresses from the erectingprocess are included. This may be explained due to thefact that the curved arch sections are inducing tensilestress into the straight sections (compare normal forcesin the upper graph). This tension pre-stress increasesthe stiffness the coupled arch structure hence thegeometrical stiffness of the entire system, fromnonlinear structural analysis generally known asstress-stiffening effect.

Fig. 5b shows the load deflection curves of thesystem on three different scales. The applied load qw isa wind pressure surface load and therefore remainsconstant for all scalar investigations. The generalbehaviour is similar to that shown of the elastica curvein Fig 3; showing snap through buckling at decreasingload factors with increasing scale. With increasingscale the system also shows distinct nonlinearbehaviour. At scale s = 4 snap through already occursunder dead load which leads to a completely nonlinearbehaviour with increasing load. In contrast to theelastica curve in Fig. 3b were the residual compressionstress is destabilising to the system, the coupled arches


Structural model

Physical model

S1:50

Built structure

FE-simulation Real construction

S1:1

10.5 m

Large scale structure

FE-simulation

S1:1 S2:1 S4:1

Figure 4. Study of various models of the 2010 ICD/ITKE research pavilion.



in the pavilion store residual tension stress in theirstraight sections. The resultant increase in geometricalstiffness can be seen in the dashed line graphs of scale

s = 1 and s = 2 where the system was calculatedwithout its residual stress from the form-finding of theerection process (N: no). Here snap through buckling

qz

Residual normal force (kN/m)

Displacement Uz (mm)syst. excl.residual stress

Displacement Uz (mm)syst. incl.residual stress 5.

691.

725.

52

5.32

12.227

.0

6.46

5.80

3.87

17.3

−18.162.559.255.551.848.244.741.235.335.135.0

35.3

34.8

34.4

34.0

33.7

16.133.7

34.2

34.7

35.3

36.0

36.335.935.635.435.237.640.844.247.651.154.857.9

36.737.5

34.916.0

−21.7

−24.9−26.9−28.8

−29.6

−28.1−30.5

−32.2−3

2.9

−32.

1−32.

4−33.

9

−34.

4

−34.

4−34.4−34.2

−33.7

−32.8

−31.5−29.4−29.5

−30.9

−31.4−31.6−31.5

−28.5−27.8−26.6−24.8−22.7−20.4

−17.8

−15.0

−11.4

−31.430.2

−32.3 −34.

5

−31.

9−3

3.6

−30.

431

.043

.459

.0

89.1

94.1

104.

2

105.

4 108.

4

96.2

82.3

65.7

49.6

35.6

19.8

6.62

2.77

5.84

36.1

45.9

59.887

.580

.069

.036

.5

13.9

33.6

22.0

4.77

11.0

5.25

1.01

5.27

82.5

89.1

72.190

.5

16.1

s.L

8

6

4

2

0 100 200 300 400 500

Load

fact

or λ

(−)

Displacement Uz/s (mm)

Uz

.qw

∆λ = 4

∆λ = 5

= .0.1 kN/m2

s = 1 N: yes

s = 2 N: yes

s = 1 N: no

s = 2 N: nos = 4 N: yes

(b)

(a)

λλ

Figure 5. (a) Displacement Uz of a coupled arch section from the pavilion, showing 17% increase in stiffness for the systemincluding residual stress induced by active bending. (b) Load deflection curve of the ICD/ITKE 2010 research pavilion at 3

different scales for a given wind pressure load case.


occurs at an earlier point if the geometry only is takeninto consideration (∆λ = 0.5 for s = 1 and ∆λ = 1 for s = 8). However, the advantageous influence ofresidual tension stress to the system is reduced overtime due to enhanced relaxation of timber.

These investigations show that the reduced sizestructure is already close to the limit size of thissystem. In this case increasing the scale wouldnecessitate a change in material and topology.

4.2. Marrakech umbrellaThe project is based on a student workshop thatdeveloped shading solutions for an outdoor plazaspace at an architecture school in Marrakech Morocco.The design proposal of a funnel shaped membraneroof was further developed by the authors with the aimof minimising anchoring forces to the surroundingbuildings. The introduction of a bending-activesupporting structure for the free edges of themembrane proved to be a very efficient solution. Afterthe successful test setup in Stuttgart June 2011, thestructure was mounted by students from Stuttgart andMorocco in spring 2012 [9]. The structure features 6elastically bent glass fibre rods with a length ofapprox. 7.5 m. The rods are pushing out threeadditional corner points on both free edges of thestructure. The funnel shaped membrane has a span ofapprox. 11 m ∞ 11 m and an eaves height of 5.5 mresulting in a membrane surface of approx. 110 m2.This structure highlights the possibility of using activebending in hybrid form-and bending-active structureswhere the system is stabilised by the elastic beamswhich, in turn, are restrained by the membrane surface.

For the scaling of this system, special attention hadto be paid to the change of membrane pre-stress. Sincethe pre-stress of the membrane has a direct influenceon the curvature of the bending-active beam elements,it was altered in such a way that the geometry resultingfrom combined form-finding of bending-active beams

and pre-stressed membrane was similar at allinvestigated scales. This resulted in under-proportionalscaling of membrane pre-stress.

In this system the bending-active beams areshortcutting tension forces between the edge cablesand the low point. Independent of the load directionthe bending-active beam elements are therefore alwaysunder compression. These compression forcesincrease over proportionally with scale and generallydestabilise the system. In this combination of bending,shear and axial load the linear beam elements cannotcompensate the quadratic growth in membrane surfaceas seen in Fig. 7. Here the arm-end deformations Uz

are shown for both wind suction (left) and windpressure (right), showing highly nonlinear loaddeflection curves and significant differences in scale.

Scaling of a membrane structure with bending-active beam elements therefore necessitates a changein topology i.e. increasing the number of beamelements when scaling up in overall size. However,beyond spans of approx. 30 m other effects maybecome decisive, as the membrane loses its stabilisingeffect on the beam elements due to flexibilityincreasing with size. Finally, even the availability ofthe necessary large size GFRP cross sections maybecome limiting factors to the scale of membranestructures with integrated bending-active supportsystems. Similar conclusions were drawn byAdriaenssens [10].

4.3. Flectofin®, an elastic kinetic shellsystemThe focus of the Flectofin® project is the optimisationof deployable systems in architecture using bio-inspired solutions. The Flectofin® is based on asophisticated pollination mechanism, which performsa reversible deformation when an external mechanicalforce is applied. The system is comprised of a stiffbeam element with attached flexible shell elements.


Structural model

Physical model

S1:50

Built structure


S1:1

11 m


FE-simulation

S1:1 S2.5:1 S5:1

Figure 6. Study of various models of the 2011 Marrakech umbrella.



Uniaxial bending of the beam causes the shellelements to deviate into out-of-plane bending due totheir low lateral stiffness, see Fig. 8 and for furtherinformation [11]. The prototype lamellas are 2.2 mlong and 0.25 m deep, they are made out of GFRP andproduced in a vacuum bagging process.

In Fig. 9 the load deflection curves of a Flectofin®lamella are plotted in three different scales for bothwind suction and wind pressure. Besides minornonlinearities the system shows almost perfectsimilitude. This behaviour may be explained by acombination of reasons:

In its deformed state the Flectofin® lamellaspredominantly hold residual tensile stress, leading toconsiderable stress stiffening effects. Destabilisingeffects due to residual compression forces aretherefore eluded. Positioned vertically on a façade thedead load direction acts in its upright position whichoffers maximal elastic stiffness, dimension effectsconsidering dead load may therefore be of minor

influence to scaling. Additionally, the bending-activesurface element makes up the entire load bearingsurface, the quadratic growth in surface is thereforecompensated by the quadratic growth of the loadbearing shell element itself.

This system suggests that the above madeassumptions of dead load and stability only, being thelimiting factors for scalability of bending activestructures may the generalized. This is additionallysupported by a recently built large scale adaptivefaçade: For the theme pavilion of EXPO Yeosu, Korea2012, an elastic kinetic façade shading system inspiredby the Flectofin® was built. Some early feasibilitystudies using the Flectofin® already suggested thatscalability to a size range of 14 m high lamellas waspossible. Covering a total length of 140 m and varyingin height between 3 and 14 m it was designed towithstand the very high wind loads at the Korean coast[12], proving the applicability of this system for largescale structures.

Load

fact

or

(−)

−1000 −800 −600 −400 −200 0

2

4

6

8

200 400

Displacement Uz / s (mm)Displacement Uz / s (mm)

600 800 1000

s.L s.L

Uz Uz

s = 1

s = 5

s = 2.5s = 1

s = 5

s = 2.5

.qw = .0.1 kN/m2λ λ .qw = .0.1 kN/m2

λ λλ

λFigure 7. Load deflection curves of the Marrakech umbrella at 3 different scales.

Structural model

Physical model

S1:50

Built structure


S1:1

2.2m

90°0°


FE-simulationS0.5:1 S1:1 S5:1

Figure 8. Study of various models of the Flectofin® lamella


5. CONCLUSIONBy dimensional analysis, general FEM analysis and theinvestigations on three case studies it could be shownthat the scaling of bending-active structures isdependent on the significance of dead load andstability. Since all bending-active structures arecomposed of slender profiles and comparatively elasticmaterials it can be concluded that the governing failuremode is that of snap through buckling as seen in theresults of the investigated systems above. It cantherefore be generalised with some certainty thatstability plays a decisive role in the structural integrityof all bending-active structures. In contrast to othertypes of building structures these effects cannot simplybe compensated by adaptation of elastic stiffness, sinceelement thickness is limited by material strength andbending curvature. As an important influence on thestability, it could be shown that residual compressionstresses are destabilizing (see example of elastica curveand Marrakech Umbrella) and tension stresses arestabilizing due to nonlinear stress-stiffening effects (seeexample of ICD/ITKE pavilion and Flectofin®).

The consideration of axial forces in the geometricalstiffness is therefore of particular importance and maybe used advantageously for the scaling of bending-active structures.

ACKNOWLEDGEMENTSThe research on bending-active structures is supportedwithin the funding directive BIONA by the GermanFederal Ministry of Education and Research.

The ICD/ITKE Research Pavilion 2010 was acollaborative project of the Institute forComputational Design (Prof. Achim Menges) and theInstitute of Building Structures and Structural Design(Prof. Jan Knippers). The work on the Umbrellastructure for Marrakech was part of a cooperationalstudent Project with the HFT Stuttgart (Prof. Fritz U.Buchmann). The Flectofin® was developed in acollaborative research Project on ‘pliable surfacestructures’ together with the PBG Freiburg, ITVDenkendorf and clauss markisen.

REFERENCES[1] KNIPPERS, J.; CREMERS, J.; GABLER, M.;

LIENHARD, J.: Construction Manual for Polymers +Membranes, Institut für internationale Architektur-Dokumentation. München: Edition Detail, 2011pp. 134.

[2] HARRIS, H.G., SABNIS, G.M. (1999): Structuralmodelling and experimental techniques, CRC Press, NewYork

[3] HAPPOLD, E.; LIDDELL, W.I.: Timber lattice roof forthe Mannheim Bundesgartensschau, The StructuralEngineer, No. 3, Vol 53, 1975, p. 131

[4] EULER, L. (1744): Methodus Inveniendi LineasCurvas.

[5] FERTIS G. (2006): Nonlinear Structural Engineering.With Unique Theories and Methods to Solve EffectivelyComplex Nonlinear Problems. Springer Pp. 9

[6] BUCKINGHAM, E. (1914): On physically similarsystems; illustrations of the use of dimensionalequations.” Phys. Review, 4.

[7] BARR, D.I.H. (1983): A survey of procedures fordimensional analysis, Int’l J. of Mechanical Eng.Education, Vol. 11, pp 147–159.


λ.qw dl

λ.qw = λ.0.1 kN/m2λ.qw = λ.0.1 kN/m2

s.L

s.LUy Uy8

6

4

2

0Displacement Uy / s (mm)

Load

fact

or λ

(−)

Displacement Uy / s (mm)

20−20−40−60−80−100 40 60 80 100

λ.qw dl

s = 0.5s = 1s = 5

s = 0.5s = 1s = 5

Figure 9. Load deflection curve of the Flectofin® Lamella at 3 different scales, for a given wind- suction and pressure load case.



[8] LIENHARD, J., SCHLEICHER, S., KNIPPERS,J.(2011) Bending-active Structures – Research PavilionICD/ITKE, in Nethercot, D.; Pellegrino, S. et al. (eds)Proceedings of the International Symposium of theIABSE-IASS Symposium, Taller Longer Lighter,London, UK

[9] LIENHARD, J., KNIPPERS, J. (2012): Permanent andconvertible membrane structures with intricate bending-active support systems, Proceedings of the InternationalIASS Symposium, Seoul, Korea

[10] ADRIAENSSENS, S. (2008): Feasibility Study ofMedium Span Spliced Spline Stressed Membranes.International Journal of Space Structures Vol. 23 No. 42008, pp. 243–251

[11] LIENHARD, J., SCHLEICHER, S., POPPINGA, S.,MASSELTER, T., MILWICH, M., SPECK, T.,KNIPPERS, J.. (2011) Flectofin: a nature based hinge-lessflapping mechanism. Bioinspiration & Biomimetics 62011, special issue on Biomimetics of Movement, 045001.

[12] KNIPPERS et al (2012): Bio-inspired Kinetic GFRP-façade for the Thematic Pavilion of the EXPO 2012 inYeosu, Proceedings of the International IASSSymposium, Seoul, Korea, 2012

NOMENCLATURE s scale factor [-]λ load factor [-]Uz Deformation in global Z direction [mm]L Span [mm]f rise of an arch [mm]r radius [mm]h height of a cross section [mm]b width of a cross section [mm]E Modulus of Elasticity [N/mm2]ρ Density [kg/mm3]g earth acceleration [m/s2]Iy Moment of Inertia [mm4]Wy Section modulus [mm3]σM Bending stress [N/mm]qz Line load [N/mm]qw Windload [kN/m2]N Residual axial force [N]dl Dead load [N/mm3]

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Considerations on the Scaling of Bending-Active Structures