膜構造研究論文集 第32号, 13-19, 2018年
Research Paper on Membrane Structures, No.32, 13-19, 2018

An experiment on fabrication technique to establish three-dimensional curved surface ofETFE membrane structure by stretching cutting pattern strips
Quang Hieu BUI *1
Masaya KAWABATA *2

Synopsis
In the manufacture field of plastics, high strengthening by drawing of films is general. It is easy to draw uniaxial
direction, however, there have been few cases that performed biaxial elongation, especially in three-dimensional(3D) shape. This paper presents an experiment of fabrication 3D curved surface of ETFE membrane structure bydrawing the cutting pattern strips with several prescribed stress ratios between the warp or MD direction and theweft or TD direction. The results of the experiments showed that the 3D smooth curve surfaces could beestablished with the stress ratios as same as the prescribed stresses in form finding and cutting pattern analysis. In
addition, the reduction of the rigidity of structures after the prestress construction period was seen at the pressureof around 1.3kPa.

1. Introduction
On the one hand, the final membrane structure shape is generally
doubly curved surface, while the membrane material itself ismanufactured in plane panels. Therefore, the cutting pattern is applied
to divide the 3D curved surface into plane strips. In these procedures,the difference of the prescribed prestress state and the actual stressstate in the structure can be occurred due to the material properties andnumerical errors. To overcome this discrepancy, researchers focusedon the optimization of cutting pattern with prescribed prestress [1][2]or introducing cutting pattern in form finding analysis [3][4].
On the other hand, the ETFE film is well used in membrane
structures in recent years because of its advantages such as superlightweight, excellent light transmission, superior durability, selfcleaning properties and fire resistance performance etc. ETFE
material itself had two yield points on its stress and strain curve [5],and the yield strength of ETFE film was increased by drawing the film
to its plastic region [6]. The fabrication of low-rise 3D curved surface
by drawing the plane film was confirmed by the authors’ experiments.
However, the establishment of high-rise 3D smooth curved surface bydrawing the plane film still had problem because of the occurrence of
wrinkling and insufficient prestress [7].
This paper presents the experiments to fabricate the 3D smooth
curved surface of ETFE membrane structure by drawing the cutting*1*2

pattern strips. First, the form finding analysis is carried out withseveral prescribed stress ratios between the MD and TD directions.
Second, the cutting pattern analysis based on the mathematical
approach find the plane strips without the prestress which correspondto the results from the form finding. Third, these plane strips areconnected by heat seal lines into the specimens, and the boundaries ofthese specimens are drawn to introduce the prestress of tension. The
quantity of boundary extension guarantees the absence of wrinklingand the sufficiency of prestress. Finally, the pressurization test is
carried out to confirm the strength of the ETFE structure after above
construction stage. The results of the experiments showed that the 3D
smooth curved surfaces could be established with the stress ratio
between the MD and TD directions as approximate same as theprescribed stresses in form finding and cutting pattern analysis. And
the reduction of the rigidity of structures after the prestressconstruction stage was seen at the pressure of around 1.3kPa.
2. Overview of experiments
2.1. The process of experiments
The experiments were carried out in three periods as shown in Fig.
1 as follow.

Graduate student, Graduate School of Urban Innovation. Yokohama National University, Japan
Associate Prof., Graduate School of Urban Innovation. Yokohama National University, Japan13

Fig.3 Assumptions for initial stress in form finding

Specimens
DD301
DD302

Fig.1 The process of experiments

Table 1. Types of specimens
Stress ratios
x1: y1x2: y2
3:1
1:1
3:1
3:1

3D view

The dimensions of the frame of experiments are shown in Fig.2,
and all dimensions are in millimeter.

x3: y3
3:1
3:1

3D view

Elevation view
Elevation viewa. DD301 specimenb. DD302 specimen
Fig.4 The results of form finding
2.2.2. Cutting pattern analysis
The least-squares minimization flattening approach [10] [11] wasused in this paper for cutting pattern. The difference of the length
between 3D coordinates and 2D coordinates is minimized by theequation Eq.1.
𝑚(1)

𝜙𝑖 2 (𝑥)

S x =
𝑖=1

where, m is the number of all link elements involved in plane strip, x
is the coordinate vector of plane strip to be determined.
𝜙𝑖 =𝑥𝑖,1 − 𝑥𝑖,22

+ 𝑦𝑖,1 − 𝑦𝑖,2

2

(2)

− 𝑑𝑖

where, (xi,1, yi,1) and (xi,2, yi,2) are the unknown coordinates in the plane
strip for the ith link element, andFig.2 Dimensions of frame of experiments
𝑑𝑖 =2.2. Period 1: Fabrication of specimens
2.2.1. Form-finding analysis
In this paper, the MD direction is named as X direction, and the TD
direction is named as Y direction. The isotropic stress surface method[8][9] was used for form finding in this paper. It is assumed that themembrane has a very small Young’s modulus, and the initial stressratios between the MD and TD directions are indicated in Fig.3. There
were two types of specimens which are described in Table 1.
The results of form finding analysis are shown in Fig.4 in 3D view
as well as in elevation view. The cutting pattern analysis is carried out
to determine the dimensions of plane strips of the film whichcorrespond to the shapes of form finding as shown in section 2.2.2.

𝑋𝑖,1 − 𝑋𝑖,2

2

+ 𝑌𝑖,1 − 𝑌𝑖,22

+ 𝑍𝑖,1 − 𝑍𝑖,2

2

(3)

is the actual length of the ith link element with 3D node coordinates(Xi,1, Yi,1, Zi,1) and (Xi,2, Yi,2, Zi,2) determined in the form finding
procedure.
The results of these analyses are shown in Fig. 5.

a. DD301a/b
b. DD302a/b
Fig.5 The cutting pattern shapes

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In Fig.5, the numbers show the lengths of cutting lines, while thepercentages indicate the expansions of the lengths of drape strips andthe corresponding lengths of the form finding shapes. In thisexperiment, each type of cutting pattern shapes had two specimens
named as DD301a/b and DD302a/b in the following parts.

measured by four laser displacement meters. The setup of laser
displacement meters is shown in Fig. 8. In addition, those heights
were also measured by the ruler at the first and final steps ofexperiments. The results of those heights are shown in Fig.9 regarding
the target height of minimum surface.

2.3. Period 2: Stretch fabrication by drawing the boundary
After the specimens were spread out across the experiment frame,
the boundaries of specimens were drawn by special technique whichcannot be described in this paper. Fig.6a shows the values of drawing
boundary in Z direction at four special positions in a quarter part ofspecimens, while the Fig.6b indicates that the values of drawing
boundary are interpolated linearly at other positions of specimens.

Initial shape, DD301a

Final shape, DD301a

Initial shape, DD301b

Final shape, DD301b

Initial shape, DD302a

Final shape, DD302a

Arch

Frame

a

c

b

Frame

d

b

a. 3D viewb. Plane viewFig.6 Process of drawing boundary
The experiments were carried out in five steps, and the values ofdrawing boundary are shown in Table 2.
Table 2. The process of experimentsa (mm)b (mm)c (mm)d (mm)
Step 10
0
0
0
Step 21010
10
10Step 320
15
20
20/25*
Step 430
20
30
30/40*
Step 54030
4040/50** The first value was used for DD301a/b, while the second value wasused for DD302a/b

Initial shape, DD302b
Final shape, DD302b
Photo 1. Smooth curved surfaces

90

h2

h1
300

2.4. Period 3: Pressurization test
The pressurization test was carried out after around ten hours ofstretch fabrication period. The process of pressure is described asfollow: 0kPa → 0.5kPa → 0kPa → 1kPa → 0kPa→ 1.5kPa→ 0kPa
→ 2kPa → 0kPa.

1000

500

Frame
1000

55

85

④
③
500

1000

a. 3D view

20

②
①
75

Arch

Frame

500

Frame
1000

1000

b. Elevation viewFig.8 Setup of laser
Fig.7 Saddle heights
displacement meters
All dimensions are in millimeter
In Fig.9, while the dash lines show the results of hand measurement
by ruler, the marked solid lines present the results of laser
displacements meters. Moreover, solid lines without marker indicatesthe results of form finding in section 2.2.1 with stress ratio was 1:1 allover specimens. The saddle height h1 of specimens DD301a/b at the
final shape was higher than the results of minimum surface, while thisheight of specimens DD302a/b moved forward to the minimum
surface results. Therefore, the prescribed stress ratios in form finding
analysis should be chosen depended on the view points of architecture.

3. Observation methods, results and discussions of experiments
3.1. Three-dimensional smooth curved surfaces
The 3D curved surfaces without wrinkling could be established by
the suggested values of drawing boundary as shown in Table 2. Photo
1 presents the initial and final shapes throughout four specimens.
3.2. Saddle heights
The saddle heights of h1 and h2, which are represented in Fig.7, are

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The saddle height h2 of four specimens moved forward to the heightof minimum surface. The stress ratio between MD and TD directions
as 3:1 for the part between arches and frame were accepted highly.

cutting pattern analysis.

y a

y m

h1 (mm)h1 (mm)d/1000c/1000

1

2

3

5 Step

4

1

2

3

x c

x m

5 Step4

a. The results of h1h2 (mm)h2 (mm)a/1000a/1000

Fig.11 Observed engineering strains
5 Step1

2

3

The average ratio between xc and xm at the final step was 1.93 incase of DD301. This ratio was larger than the results of DD302 as
1.52. The reason can be explained by the effect of friction between
arches and ETFE film. The saddle height h1 of DD301 was higher
than DD302, so the contact area between ETFE and arches in case ofDD301 was larger than DD302. As a result, the contact angle in caseof DD301 was larger than DD302. Summary, the above results canbe understood clearly based on the belt friction theory [12].

5 Step4

b. The results of h2
Fig.9 The results of saddle heights
3.3. Observed engineering strains
Four lengths shown in Fig. 10 were measured by tape measure with
precision of 0.5mm during five steps of experiments. The engineeringstrains at step k are calculated by Eq. 4.

Offset line

3.4. Green-Lagrange strains at some parts on specimens
The lengths of four edges and two diagonals of part A and part B in
Fig.12 were measured by paper tape and caliper during theexperiments. The Green-Lagrange strains at those parts could be
calculated based on the assumption that X direction is unchanged
during the experiments. Fig. 13 indicates the assumption for thiscalculation.

50

Frame

l3

Arch

l4
50

l1

Offset line

l2

Frame

a. 3D viewb. Plane viewFig.10 Observed four lengths

1𝑙𝑖

X2

(4)

80

1𝑙𝑖 − 𝑘 𝑙𝑖

L2

𝑘𝜀 =

Y2

Part A

1000

Part B
1000

100

L1

Y1

Y2

X1

X1

Part I

Part II

X2
L2

Frame

X1
L2

where, k is the engineering strain at step k, 1li and kli are the lengths ofsegment i at step 1 and step k, respectively. These strains are shown in
the relation with the ratios between the boundary extension a, b, c ord and the span of structures in Fig.11.
In Fig.11, the solid lines with circle, square, triangle and diamondmarkers show the results of specimens DD301a, DD301b, DD302a
and DD302b, respectively. In the part between two arches, theaverage ratio between xm and ym at the final step was 0.78 in case ofDD301 and 2.87 in case of DD302. The suggested values of boundary
extension kept the stress ratio between MD and TD directions as
approximate same as the prescribed stress ratio in form finding and

Y1

L1

1000

4

Frame

3

100

2

Arch

1

X2
Y1

Y2
L1

Part IIIPart IV

a. Position of part A & B
b. Observed lengths
Fig.12 Observed lengths at part A&B
The Green-Lagrange strains can be obtained by Eq.5
𝜀𝑥𝑦 − 𝑦𝑖
1 𝑘
𝜀𝑦 =
0
2𝑆
𝜀𝑥𝑦
𝑥𝑖 − 𝑥𝑘16𝑦𝑖 − 𝑦𝑗0𝑥𝑖 − 𝑥𝑗0
𝑥𝑗 − 𝑥𝑖𝑦𝑖 − 𝑦𝑗

𝑢𝑗
𝑢𝑘
𝑣𝑘

(5)

The 2nd Piola-Kirchhoff stresses of part A and part B can beobtained by Eq.6. Here, part A and part B were assumed in plane stresscondition.

where, xi = 0, yi = 0, xj = 1Lk, yj = 0, xk = 1Lj.cos(1), yk = 1Lj.sin(1), uj= nLk − 1Lk, uk = nLj .cos(n)− 1Lj.cos(1), vk = nLj .sin(n)− 1Lj.sin(1) ,

𝑆𝑥
𝑆𝑦
𝑆𝑥𝑦

and 1Lk, 1Lj, 1Lk, nLk, 1, n are described in Fig. 13.

n

uk n

Lj
1

Li
Lj
1
1 n Lkn
Lk

Shape at step 1
Shape at step n

vk

𝑆𝑒𝑞 =

j
x

uj

(7)

𝑆𝑥 2 + 𝑆𝑦 2 − 𝑆𝑥 𝑆𝑦 + 3𝑆𝑥𝑦 2

The average values of equivalent stress are shown in Fig.15. The
equivalent stress of part A and part B was almost the same in bothDD301 and DD302 specimens at the final step. Therefore, thedescribed stress ratio in form finding and cutting pattern analysis canbe chose either 3:1 or 1:1 for the middle part depended on the demandof saddle height h1.

Li

1

i

(6)

where, E=800MPa is Young modulus, and =0.44 is Poisson’s ratioof ETFE material. The equivalent stress is calculated by Eq.7.

y
k

1 𝜈
0
𝜀𝑥
𝐸
𝜈 1
0
𝜀𝑦
=
1−𝜈
1 − 𝜈2
𝜀𝑥𝑦0 0
2

Fig.13 The assumptions for calculation of Green-Lagrange strains
Fig. 14 shows the results of Green-Lagrange strains of part A and
B in the final step. The bar and error bars type of this graph indicatesthe average, maximum and minimum values of the strains at thoseparts.
x

y

xy

x

y

xy

a. Part A
b. Part B
Fig.15 Equivalent stress at part A&B
3.5. Observed sliding values
During the experiments, the motions of marked points on
specimens versus fixed points on frames and arches were measured
by ruler with precision of 0.5mm. Fig.16 shows the positions of four
special points on the offset lines of specimens and five points on
specimens at the position of arch.

a. Part A
b. Part B
Fig.14 Green-Lagrange strains at part A&B
In Fig.14, the solid bar shows the strain in MD direction, while thediagonal pattern and horizontal pattern columns presents the strain in
TD direction and the shear strain, respectively. At part A, strain in TD
direction y is around five times larger than strain in MD direction x
in case of DD301, while the MD and TD strains of DD302 are almost
the same. The length of middle cutting pattern line was 2043mm for
DD301 as shown in Fig.5. This length was larger than 2009mm of
DD302. Therefore, the strain in MD direction x of part A of DD302
was larger than DD301 with the same values of drawing boundary.
The same phenomenon was observed in TD direction.
At part B, the drawing process must guarantee two criteria: (1) the
ETFE will not be teared, (2) the wrinkling will not occur. As it can beseen in Fig.14b, although the shear strain at this part was large, theminimum principle strain was controlled to be larger than zeros at the
final step. As a result, the smooth curved surface could be establishedwith the suggested values of drawing boundary as shown in Table 2.

50

X
I
II
III
IV
3*
V

50

4*
1000

1*
1000

Y

2*
1000

a. 3D viewb. Plane viewFig.16 Observed motion positions
The results of motions of four points (1* to 4*) versus boundary
drawing values are shown in Fig. 17. In Fig.17, the solid lines withcircle, square, triangle and diamond markers show the results ofDD301a, DD301b, DD302a and DD302b, respectively. As we cansee from Fig.17, the motion of the points on the offset lines areapproximate a half of boundary drawing quantity.

17

a. Initial shape at 0 kPa
b. Final shape at 2 kPa
Photo 2. Pressurization test, DD302b
In Fig. 19, the solid and dotted lines show the results of DD301,
DD302 respectively, while the dot-and-dash lines present the resultsof PD15. It should be noted that PD15 were specimens which usedflat type of cutting pattern in the part between two arches in previous
experiments [7]. The residual displacements were observed at thepressure of 1.5kPa in both middle and corner parts over four
specimens of DD301 and DD302. In addition, the reduction of therigidity of structures can be seen at the pressure of 1.3kPa from thoserelation curve. In F19.a, the stiffness of the structures at the middle
parts was larger than the previous experiments. The reason can be
explained by the enough sufficient pre-stress of tension and the final
smooth curved surface. In contrast, the stiffness of the structures at thecorner parts was the same in both experiments because the same
cutting pattern types were used in this part.

Fig.17 Motion of offset points
Fig.18 presents the motions of marked points on ETFE specimens
versus the fixed points on arches in X and Y directions at the final step.
It should be noted that the values of observed motions in Fig. 18 were
measured on the curved of films in X direction and the curved ofarches in Y direction by ruler with precision of 0.5mm. The motion in
Y direction was depended mainly on the boundary extension quantity
at c position. The motion of the points near boundary were larger thanthe points in the middle of arches. In contrast, the motion in X
direction was depended not only on the boundary extension quantity
at a and d positions but also on the experiment process. Although thedrawing process of boundary was carried out in order of c, a, d to b
positions, this process could not take place similarly over four
specimens. As a result, the motion in X direction at arches wasdifferent over four specimens.

a. Middle partb. Corner partFig.19 Relations between displacements and pressure
4. Conclusions(1) In tensile membrane structure using ETFE film, the form finding
analysis can be carried out with several prescribed stress ratios
between the MD and TD directions. Those ratios will be determinedby the dimensions and the shapes of structures. The cutting patternanalysis based on the mathematical approach finds the cutting strips
without the prestress consideration which correspond to the form
finding results. The drape strips are connected by heat seal lines, andthe boundary will be drawn to introduce the prestress. The 3D smooth
curved surface with sufficient prestress could be established by those
construction process.
(2) The suggested values of boundary drawing can keep the stressratio between MD and TD directions as approximate same as theprescribed stress ratio in form finding and cutting pattern analysis.

Fig.18 Motion at the position of arch
3.6. Pressurization test results
Photo 2 shows the shapes at the initial step of 0 kPa and final stepof 2 kPa. During the pressurization test, the vertical displacements atfour positions in Fig. 8 were observed by laser displacement meters.
The relations between the displacements at ① and ④ positions
and pressure are presented in Fig. 19.

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(3) The sufficient prestress could be confirmed not only by theengineering strains, the equivalent stress but also by the preservationof the rigidity of structures up to around 1.3 kPa of pressure.

[5] M. Kawabata, E. Jeong and K. Nishikawa: Curved surface
fabrication and strengthening of film structures by stretching plane
film, Proceedings of IASS Symposium, 2008[6] E. Jeong and M. Kawabata: An Experimental and Analytical
Study on Stretching Effect of ETFE film, Research Report on
Membrane Structures, 2014[7] E. Ishida, Q.H. Bui and M. Kawabata: Study on fabrication of film
panel of parallel arch model by 3D composite cutting pattern, AIJ
2017 conference, pp.739-740 (in Japanese).
[8] B. Tabarrok, Z. Qin: Nonlinear analysis of tension structures,
Computers and Structures Vol.45, No.5/6, pp. 973-984, 1992
[9] Kazuo Ishi: State of the art report on form finding problem ofmembrane structures, Research Report on Membrane Structures,
1990[10] B. Tabarrok, Z. Qin: Form finding and cutting pattern generation
for fabric tension structures, Microcomputers in Civil Engineering 8,
1993, pp. 377-384[11] Jae-Yeol Kim, Jang-Bog Lee: A new technique for optimum
cutting pattern generation of membrane structures, Engineering
structures 24, 2002, pp. 745-756[12] Stephen W. Attaway: The mechanics of friction in rope rescue,
International Technical Rescue Symposium, 1999

Acknowledgements-The support for the experiments from Taiyo
Kogyo Corporation is gratefully acknowledged.
References[1] M. Ohsaki, K. Uetani: Shape-stress trade-off design of membrane
structures for specified sequence of boundary shapes, Comput.
Methods Appl. Mech. Engrg. 182(2000), pp. 73-88[2] Jae-Yeol Kim, Jang-Blog Lee: A new technique for optimum
cutting pattern generation of membrane structure, Engineering
Structures 24(2002), pp. 745-756[3] J. Linhard, R. Wuchner and K. Bletzinger: Introducing cutting
pattern in form finding and structural analysis, Textile Composites andInflatable Structure II, 2008, pp. 69-84[4] Kazuo Ishii: Form finding analysis in consideration of cutting
patterns of membrane structures, International Journal of Space
Structures Vol.14, No.2, 1999, pp.105-119

裁断図を延伸して ETFE 膜構造の 3 次元曲面を成形するための作製技術に関する実験的研究
BUI Quang Hieu *1)
河端 昌也 *2)梗 概塑性範囲の製造分野では、フィルムの延伸しによる高強度化が一般的である。一軸方向を引張することは容易である
が、特に三次元（3D）形状で二軸引張を行うケースはほとんどない。本研究では、縦糸方向または MD 方向と横糸方向または TD 方向の様々な定める応力比で複合立体裁断を延伸することにより、ETFE 膜構造の 3 次元曲面加工を実験した。 実験の結果は、形状解析および裁断解析における規定された応力と同じ応力比で 3D 滑らかな曲面を確立できることを示した。
加えて、延伸形成実験の後は構造の剛性の低下は、約 1.3kPa の圧力で見られた。

*1）
*2）

横浜国立大学大学院 都市イノベーション学府 大学院生
横浜国立大学大学院 都市イノベーション研究院 准教授

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