Folded assembly methods for thin-walled steel structures

Abstract

There has been significant recent interest in origami-inspired foldable structures for applications in which transportability and rapid construction are primary design drivers, for example, emergency shelters and staging structures. However, widespread application is not yet seen due to complexities in folded geometry and modelling the structural behaviour of folded sheet material. This paper proposes a fundamentally new approach whereby folded assembly methods are developed for conventional thin-walled steel construction and benchmarked in terms of their assembly effort, manufacturing accuracy, and structural performance. Manufacturing accuracy was benchmarked with 3D digital image correlation and 3D scanning and showed a folded assembly method to be accurate to within ± 50% of plate thickness with assembly by unskilled persons. Structural performance under uniaxial compressive load was assessed with experimental and numerical analyses, with consistent predictions showing that conventional thin-walled steel analysis techniques are sufficient to model folded structure behaviours. Modelling of the novel folded steel structures is therefore also shown to avoid much of the complexity normally encountered in folded structure analysis, such as characterisation of fold-line rotational stiffness or folding plasticity behaviours.

Full text

FOLDED ASSEMBLY METHODS FOR
THIN-WALLED STEEL STRUCTURES
Quan Shia, Xiaoqiang Shia, Joseph M. Gattasa,∗, Sritawat Kitipornchaia
aSchool of Civil Engineering, University of Queensland, Australia
Abstract
There has been significant recent interest in origami-inspired foldable struc-
tures for applications in which transportability and rapid construction are
primary design drivers, for example, emergency shelters and staging struc-
tures. However, widespread application is not yet seen due to complexities in
folded geometry and modelling the structural behaviour of folded sheet mate-
rial. This paper proposes a fundamentally new approach whereby folded as-
sembly methods are developed for conventional thin-walled steel construction
and benchmarked in terms of their assembly effort, manufacturing accuracy,
and structural performance. Manufacturing accuracy was benchmarked with
3D digital image correlation and 3D scanning and showed a folded assembly
method to be accurate to within ±50% of plate thickness with assembly by
unskilled persons. Structural performance under uniaxial compressive load
was assessed with experimental and numerical analyses, with consistent pre-
dictions showing that conventional thin-walled steel analysis techniques are
sufficient to model folded structure behaviours. Modelling of the novel folded
steel structures is therefore also shown to avoid much of the complexity nor-
mally encountered in folded structures analysis, such as characterisation of
fold-line rotational stiffness or folding plasticity behaviours.
Keywords: Thin-walled steel, digital fabrication, folded structures, crease
behaviour, origami-inspired engineering
∗Corresponding author
Email address: j.gattas@uq.edu.au (Joseph M. Gattas)
Preprint submitted to Journal of Constructional Steel Research July 3, 2017

1. INTRODUCTION
1.1. Origami Inspired Structures
Origami-inspired design techniques have seen much recent technological
development due to their potential for delivering a relatively large structure
in a compact package. They are thus useful when transportation capacity is
a driving design concern with famous examples including solar power arrays
[1], a space telescope [2], and biomedical devices [3]. Recent examples in civil
engineering include transportable shelters [4], bridges [5], and prefabricated
modular structures [6].
A second major benefit of origami-inspired design is its capability to
achieve a significant increase in stiffness at minimal expense of weight. This
has been utilised for example in military shelters, in which the folds can pro-
vide structural stiffness when deployed [7,8] but also facilitate packaging into
a smaller volume for transportation or storage [9,10]. High-stiffness folded
core sandwich constructions, which consist of two outer faces attached to
an origami core, have also been developed as lightweight or morphing build-
ing components [11,12] and impact-resistant or isotropic sandwich panels
[13,14].
Most of the above structural applications utilise shell forms developed
from a known origami pattern. This requires advanced geometric design
methods [15,16,17] and structural analysis methods. Characterising the
rotational stiffness of fold lines has been shown to be critical for accurately
modelling the structural behaviour of most origami-inspired or folded struc-
tures [18,19]. Despite extensive research into the creasing mechanism of com-
mon folded sheet materials, including paper [20,21], paperboard [22,23,24],
Mylar sheets[25], and steel [26], understanding of this phenomenon is still
limited. Development of engineering analysis methods and widespread ap-
plications for folded structures have therefore also been limited.
1.2. Folded Sheet Metal Fabrication
Folded sheet metal fabrication is widely applied in automotive, manufac-
turing, and building industries. The bending brake and press are typically
used to form a bend line along the sheet metal [27], however considerable force
is required. Additionally, the bending tolerance, accumulation of errors, and
geometric constraints from the bending machine mean that traditional sheet
metal bending processes are difficult to apply for origami-inspired designs,
2

particular as small eccentricities in fold-line location can significantly alter
sheet mechanics [28].
Industrial Origami has proposed a precise sheet metal folding method
based on a notional geometric fold line composed of curved slits [29], shown
in Figure 1. Adjacent edges are connected by the area between slits, termed
‘straps’, which are torqued during folding and thus achieve a much reduced
bending force [30,31]. The slits have additional structural benefits including
a reduction in stress concentrations at the curved slit ends, and ‘edge to face’
engagement during the bending process, with the twisted strap acting to pull
together adjacent surfaces [32]. Faces are therefore able to transfer shear
forces directly across folded edges. Slits can be precisely placed in a sheet
with computer numerical control (CNC) manufacturing processes for high-
accuracy part production. The combination of CNC fabrication methods
with folded geometries has also previously been shown to be a cost-efficient
means for production of modular structures [33].
a)
strap
slit
b)
strap
Figure 1: Sheet-metal bending technique with slits and straps along fold lines. a) Unfolded
and b) folded configurations.
This paper investigates manufacturing, assembly, and analysis methods
for thin-walled steel structures developed through the combination of sheet
metal bending techniques and folded structural geometry. Section 2describes
the geometric design and assembly method of the new steel prototypes, man-
ufactured with waterjet-cut fold line and edge connection details. Analysis of
prototype surface imperfections is conducted in Section 3for quantification of
the accuracy of the fabrication method. Section 4presents an experimental
study of prototype structural performance under uniaxial compressive loads,
followed by comparative numerical analyses in Section 5. A discussion of
results and effects of fold-line behaviours are given in Section 6.
3

2. FOLDED ASSEMBLY METHODS
2.1. Geometry
For investigation of folded assembly techniques, a foldable variant of a
conventional thin-walled steel structure is first developed. The proposed
structure is a triangular hollow section with patterned windows that form
integral internal bracing, shown in Figure 2a. This module is selected as it is
hypothesised to have a simple compression behaviour similar to a typical tri-
angle truss, that is with compressive loads carried by angle sections that run
continuously along the element axis and with an effective compressive length
as set by hexagonal plate bracing, shown in Figure 2b. Structural members
with openings [34,35] or high-strength fabricated square and triangular sec-
tions [36] are also well-understood, should complexities be encountered due
to hexagonal openings or fold-induced residual stress, respectively.
a)
b)
Figure 2: a) Proposed folded module and b) hypothesised load-carrying model with anal-
ogous triangle truss model.
There are two further benefits to the geometry. First, the unfolded model
forms a rectangular sheet and so can be sized to fit typical metal sheet stock
with minimal waste off-cuts. Second, in comparison with typical origami-
inspired foldable structures, this geometry can be specified with a small num-
ber of uncoupled parameters as shown in Figure 3. Wall plates are specified
with axial length Land side length W. Hexagonal plates must meet at all
three wall plates and so hexagonal plate side length is dependant and equal
to WH=W
3. Longer structures are formed by tessellation along the axial
direction, with a n-module tessellation forming a column with total length
nL.
2.2. Folding and Assembly
Two critical features are encountered in the transfer of the geometric de-
sign to a realised steel structure. First, a means by which to bend steel sheet
accurately at internal fold locations must be utilised. Second, a means by
4

W/3
nxL
W
L
W/6
W
L W/3
a)
b)
Figure 3: Parameters of the module. a) Unfolded and b) folded and tessellated geometries.
which to rigidly connect discrete edges once folded must be utilised. Both
should have minimal effect on structural performance as compared with non-
folded steel. Both should also be achievable by hand, that is without spe-
cialist equipment or training, for economic viability.
A steel folding method that possessed the necessary characteristics was
identified as the steel folding joint utilised by Industrial Origami [29]. This
connection is parametrised as shown in Figure 4a, with slit spacing as, radius
ar, and polar angle aθ. A suitable steel edge connection method could not
be identified in current literature and so two methods were developed. The
Type 1 connection is shown in Figure 4b and has male ‘tabs’ that mate with
female ‘slots’ on a matching edge. Parameters are tab spacing bs, tab width
bw, tab length bl, leading edge length be, slot width cw, slot length cl, and
flap length cf. A slit with parameters as per the folding connection is also
specified to allow tabs to fold after insertion. The Type 2 connection is a
lapped rivet connection with identical male and female edges as shown in
Figure 4c. Parameters are edge offset df, hole diameter dd, and hole spacing
ds.
a) b) c)
as
araθdd
ds
bwbs
bl
cw
be
cfcl
df
Figure 4: Parametric connection types. a) Fold line connection, b) Type 1 male (top) and
female (bottom) connection and c) Type 2 male (top) and female (bottom) connection.
5

2.3. Fabrication
Digital fabrication, that is the use of CNC manufacturing equipment, was
identified as a suitably accurate and economical method of part fabrication.
A digital fabrication method was developed for plate structures in [33] and is
capable of superimposing a parametric folded geometry with parametric edge
connection types to generate a complete part drawing. The method ‘tags’
2D folded geometry edge lines with an edge type designator. In this instance,
edge designations are one of ‘I’ folded connection, ‘M’ male connection, ‘F’
female connection, or ‘O’ solid outline connection, as shown in Figure 5a.
The corresponding parametric connection detail is then computationally su-
perimposed on the tagged edges, maintaining independence between element
and connection parameters.
I
MMF
M
F
F
I
F
M
I
IF
a)
OO OO
O
MO OO O
b) c)
I
MMF
I
F
M
I
O
O
O
O O
F
F
M
I
I
O OO
MO
F
F
F
Type 1 Connection
Type 2 Connection
Figure 5: Digital fabrication of Type 1 (top) and Type 2 (bottom) triangular sections. a)
Tagged edge connection types, b) waterjet-cut unfolded and c) folded samples. Tagged
edge connections include: ‘M’ male edge, ‘F’ female edge, ‘O’ outline, and ‘I’ folded
connection.
Specimens using of both Type 1 and Type 2 male and female connec-
tions were manufactured using a CNC-waterjet cutter from 0.9mm thick Gal-
vanised steel. Module parameters were selected as L=300mm, W=300mm,
and n=2, for a total length of 600mm. Folded connection parameters were se-
lected as as=25 mm, ar=5mm, and aθ=90◦. Type 1 male and female param-
eters were selected as bs=25mm, bw=10mm, bl=10mm, be=1mm; cw=16mm,
6

cl=2mm, and cf=13.4mm. Type 2 male and female parameters were selected
as df=8 mm, dd=5.6 mm, and ds=46.79 mm. An unskilled person can fold
each sample extremely quickly from the unfolded specimens shown in Figure
5b. Assembly of each Type 1 sample took a single person approximately 20
min with only a hammer. Assembly of each Type 2 sample took a single
person approximately 15 min with only a rivet gun. Final samples are shown
in Figure 5c.
There are several benefits of this folded module when compared to exist-
ing origami-inspired and deployable structures. First, the flat-packed trans-
ported state of the module and resultant slenderness relationship are different
to typical deployable bar structures. These have a minimum packaged size
dictated by the structural section dimensions, for example, CHS diameter
[37], whereas the present module has a size determined by wall thickness tp.
Given the folded volume V=√3LW2
4and the unfolded volume VU= 3LW tp,
the expansion ratio λ=VU
Vfor the folded triangle truss then varies linearly
with wall slenderness ratio: λ= 4√3tp
W. This enables a much more compact
storage and transportation. Second, a complete structural element can eas-
ily be designed based on the initial module definition and during folding of
the module, complete kinematic independence is maintained between all sub-
components, i.e. the hexagonal and wall plates can all be folded separately
and in almost any order. It therefore avoids much of geometric complexity
typically encountered in origami-inspired structural designs.
3. IMPERFECTION ANALYSIS
Structural performance of thin-walled elements is potentially highly sensi-
tive to surface imperfections [38]. As this paper introduces a new fabrication
method which may generate such defects, an investigation is conducted to
quantify imperfections present in folded thin-walled steel samples.
3.1. Method
Methods of imperfection measurement frequently used for steel structures
include contact methods such as LVDTs [39] or digital touch-probes [40]; and
non-contact methods such as full-field 3D surface scanning [41]. In this study,
two non-contact measurement methods are used to evaluate manufacture
imperfections: 3D scanning and 3D digital image correlation (DIC).
3D scanning measurements were taken on a Faro Edge 1400 3D laser scan-
ner, with a single point and volumetric repeatability of 0.029 mm and 0.041
7

mm, respectively. Samples were scanned in their entirety and translated
to 3D mesh data using the accompanying commercial software Geometric
Studio 2014. A minimum surface error optimisation routine developed previ-
ously [42] was used to align 3D measured data with ‘exact’ 3D CAD meshes,
with imperfection measurements taken as the minimum distance between
measured and exact surfaces. DIC measurements were taken on a Corre-
lated Solutions two-camera DIC system, calibrated to track deformations of
a single front-camera surface of each specimen during testing. Camera im-
ages were analysed with accompanying commercial software VIC-3DTM 2010,
with the imperfection measurements taken as the out-of-plane displacement
wof each sample in the initial unloaded state.
3.2. Analysis and Discussion
Three samples, namely D, E, and F, were fabricated using Type 2 folded
assembly method described above and measured using both 3D scanning
and DIC measurement, with approximately 83,000 full-model and 30,000
single-face sampling points, respectively. Results are summarised in Table
1and Table 2in terms of average and maximum absolute surface errors,
respectively. It can be seen that there is a very good agreement between
scanner and DIC methods in terms of average error. However, discrepancies
exist with maximum error measurements. Closer inspection of 3D scan data
shows 99.3% of data values were lower than the maximum DIC errors, with
the remaining 0.7% of points with larger values concentrated on edge zones of
prototypes. The DIC, for reasons of lighting, is unable to take measurements
at these locations and so the accuracy at boundaries cannot be verified,
however the agreement at all other locations leads to the conclusion that
both 3D scanner and DIC have accurate results. Single-face comparison
of cross-sectional measurements, shown in Figure 6, and full-field contours,
shown in Figure 7a-b, also show excellent correlation and further demonstrate
the reliability of the measurement methods.
Table 1: Comparison of average absolute surface errors (mm).
Type 1 Type 2
Sample Name A B C D E F
Scan - - - 0.20 0.27 0.25
DIC 0.21 0.40 0.37 0.20 0.27 0.24
8

Table 2: Comparison of maximum absolute surface errors (mm).
Type 1 Type 2
Sample Name A B C D E F
Scan - - - 1.65 1.57 2.13
DIC 1.11 2.12 1.33 1.05 1.02 1.40
0 50 100 150 200 250 300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50100150200250300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50 100 150 200 250 300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50 100 150 200 250 300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50100150200250300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50 100 150 200 250 300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50 100 150 200 250 300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50100150200250300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
0 50 100 150 200 250 300
-0.8
-0.4
0.0
0.4
0.8 3D Scan
DIC
a) b)
c)
width of column x (mm)
Errors w (mm)
Middle Profile Curve
Surface Imperfection
Contour from 3D Scan
width of column x (mm)
Errors w (mm)
width of column x (mm)
Errors w (mm)
x (mm)
y (mm)
d)
0
300
200
100
Figure 6: Cross-sectional geometric imperfection comparison of Type 2 specimens. a)
Sample A, b) Sample B, c) Sample C and d) location of the cross section.
Three samples, namely A, B, and C, were fabricated using the Type 1
folded assembly method and measured using the DIC method only. The 3D
scanning method was not used due to reasons of expense and the accuracy
observed with DIC sample measurements for Type 2 samples. Comparing
the average surface error of Type 1 and Type 2 models, obtained from DIC as
0.33mm and 0.24mm, respectively, shows both folded assembly methods are
comfortably accurate to within ±50% plate thickness and can therefore be
considered highly accurate. Comparing single-face full-field contours, shown
in Figure 7b-c, it can be seen that larger defect regions are seen near tab
regions, and so are likely attributable to the use of a hammer at these loca-
tions during assembly. The Type 2 method is concluded to be the best of
the two methods proposed in the previous section, as it has approximately
30% smaller average surface error, more consistency in individual surface er-
ror between samples, and an approximately 25% shorter assembly time, as
9

compared with the Type 1 method.
a)
b)
d)
c)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
[mm]
Figure 7: Full-field geometric imperfection comparison. a) 3D scan contours, from left to
right: samples D, E and F, b) DIC contours, from left to right: samples D, E, and F and
c) DIC contours, from left to right: samples A, B, and C.
4. EXPERIMENTAL ANALYSIS
4.1. Material tensile tests
The prototypes were made from a 0.9 mm thickness G250 galvanised steel
sheet material. Four tensile coupons were fabricated for material testing,
with an effective width wand reduced section length Ltof 10mm and 50mm
respectively. The measured results were transformed to true stress-strain and
used in a numerical analysis given in Section 5.
4.2. Structural testing method
As a first investigation into the structural behaviour of folded thin-walled
steel sections, three samples of each of Type 1 and Type 2 assembly methods
were tested under uniaxial compressive loading. An Instron Universal Testing
machine was used to apply compression load between a rigid base plate and
10

a spherical seat top plate, as shown in Figure 8. Quasi-static controlled dis-
placement loading was applied with a crosshead speed of 0.25mm/min, with
reaction force and displacement recorded at 0.1s intervals at the crosshead
location. The aforementioned DIC system was adopted to record full-field
displacements on one front-camera surface with measurements taken at 0.5s
increments. It should also be noted that additional care was taken to elimi-
nate load eccentricities in the Type 2 tests, due to the observed scattered in
the Type 1 tests. The influence of eccentricity on the structural behaviour
will be further discussed with testing results in Section 6.
Extra lighting
DIC cameras
Bottom plate
Table
Top plate
Spherical seat Crosshead
Column
Ground
Speckle pattern surface
Extra lighting
Figure 8: Experimental configuration.
4.3. Results of axial compression tests
Prototype typical failure modes are shown in Figure 9with maximum
loads listed in Table 3. Type 2 has a 15% higher mean peak load than
Type 1, and ±3% variation between samples, as compared with ±17% for
Type 1. The lower variation of Type 2 correlates with the lower fabrication
11

defects present in this assembly type and suggests the fabrication quality
arising from the folded assembly process is suitable for consistent structural
behaviour. A secondary reason is the aforementioned uncontrolled loading
eccentricities in Type 1 tests. Because of the scale of eccentricities, two types
of failure modes are exhibited, unsynchronised (failure mode 1) and synchro-
nised (failure mode 2). The majority of specimens exhibited mode 1, with
elastic buckling failure occurring at one corner in the region with the small-
est cross-sectional area, see Figure 9a. Only the sample C showed a mode
2 elastic buckling failure whereby all corners failed near-simultaneously, see
Figure 9b. Although this sample had the highest compressive capacity, the
failure is evidently atypical and cannot represent Type 1 samples. There-
fore, the better fabrication method and lower loading eccentricities in Type
2 samples, contribute to better structural performance and lower variation
of maximum load. A numerical study is undertaken in the following section
to further investigate these behaviours.
Table 3: Experimental maximum compressive loads and failure modes.
Type 1 Type 2
Sample Name A B C D E F
Maximum Compression Load (kN) 16.7 20.9 22.9 22.0 21.3 22.7
Failure mode 1 1 2 1 1 1
a)
Sudden Collapse
Snap back
Snap back
b)
Location of
the point of interest
Location of
the point of interest
Figure 9: Experimental samples after failure. a) Failure mode 1, unsynchronised failure
and b) failure mode 2, synchronised failure.
12

5. NUMERICAL ANALYSIS
5.1. Material model
Material elasto-plastic properties were obtained from material tensile
tests. Material coupons 1, 2, 3 and 4 had measured Young’s modulus of
elasticity values of 208.2, 214.7, 170.4, 210.6 GPa, respectively, and so a typ-
ical steel value of E =210 GPa was used. Instrumentation was not sufficient
to record Poisson’s ratio and so a typical steel value of ν=0.3 was used.
Material properties were obtained from an averaged engineering stress-strain
curve with the following formulas [43].
σtrue =σeng(1 + εeng) (1)
εln = ln(1 + εeng) (2)
where σeng and εeng are engineering stress and strain respectively, σtrue
stands for ‘true’ strain, and the εln means the logarithmic strain. The point of
0.02% proof stress 243MPa is considered to be the last point with zero plastic
strain. The corrected plastic stress-strain data for the numerical model is
listed in Table 4.
Table 4: Logarithmic plastic stress-strain data of Galvanised steel sheet.
Stress(MPa) 243 289 319 364 391 397
Plastic Strain(%) 0 0.02 0.05 0.09 0.13 0.14
5.2. Finite element model
A numerical model was developed using finite element analysis package
Abaqus, with the compression test simulated with a nonlinear Riks analysis
method [44]. The model consisted of a triangular column and a deformable
top plate. The column mesh was composed of 3-node S3R and the 4-node S4R
shell elements, with an element size of approximately 5mm, found following
a mesh convergence study. The top plate mesh was composed of 8-node
brick elements C3D8R with approximately 5mm mesh size. The inclusion of
steel fold-line geometry was found to be necessary for modelling of column
behaviour and these were modelled with a local mesh detail as shown in
Figure 10a.
13

Model 1 geometry Model 1 failure
(unsynchronised)
a)
Move down Move down
Move up
Move up
Location of
the point of interest
Model 2 geometry Model 2 failure
(synchronised)
Sudden Collapse
Snap back
Snap back
b)
Location of
the point of interest
Von Mises
Stress (MPa)
397
364
330
298
265
232
199
165
132
99
66
33
0
Figure 10: Geometry and failure modes of numerical models. a) Model 1 with unsynchro-
nised failure generated with free-rotation top plate boundary condition. b) Model 2 with
synchronised failure generated with fixed-rotation top plate boundary condition.
The boundary condition at the base of the numerical column was rigid,
which corresponded to the steel base plate bearing condition used in the ex-
periment. The boundary condition at the top of the column was restrained
against lateral translation but free against axial translation and rotation.
As shown in Figure 10, a displacement-controlled deformable top plate was
used to simulate the load application plate, with deformability found to be
necessary as top plate flexural deflection between the crosshead and proto-
type boundary was small but non-negligible. For the compression test, the
top plate had a spherical seat at the crosshead connection point and so the
numerical top plate was only restrained against lateral translation and axial
torsion. A frictionless edge-to-surface contact was used between the top plate
and the column edges.
5.3. Assumptions for model simplification
As specimens from Type 1 to 2 are very similar in structure, the geometry
was simplified to represent both assembly methods. Three major simplifying
assumptions were used and are summarised as follows. a) The specimen
was rotationally symmetric, with all three corners having the same fold-line
properties. b) Four lateral braces are regular hexagons and connected to
the outer wall with a rigid edge connection that is without folding slits or
14

tabs. c) Residual stress from the fabrication process was concentrated in the
fold-line ‘straps’ and had minimal effect on overall structural behaviour; it
can therefore be neglected in the model.
A final simplifying assumption was made with respect to imperfections in
the column geometry and the loading condition. A perfect column under cen-
treline axial compression is ideal and non-existent in the real world. Imper-
fections have been quantified for the columns and, while varied from specimen
to specimen, are overall within the range of a conventional thin-walled steel
structure and so a common imperfection simplification [45,46,47,40] was
chosen for use in the model. A linear bifurcation analysis is firstly carried out
to obtain the first eigenmode. The buckled geometry from this eigenmode is
scaled and used to generate a geometric imperfection for the subsequent non-
linear analysis model. An eccentric loading condition was introduced due to
observed variation in alignment of load plate and prototypes during testing.
This was achieved in the model by moving the top plate in model approxi-
mately 4mm away from the centroid of column top. The introduction of an
eccentricity was not found to dictate whether the numerical model exhibit-
ing a synchronised or unsynchronised failure. Therefore, a central loading
condition was accepted in the models, discussed further below.
6. DISCUSSION
6.1. Comparison of experimental and numerical failures
The recorded numerical and experimental maximum failure loads are
listed in Table 5and show a good agreement. The numerical prediction
is within 5% for failure 1 experimental average and within 4% for failure
2 experimental average. Type 1 samples are not involved in failure 1 av-
erage because of their higher level of manufacturing defects and resultant
experimental scatter.
Table 5: Maximum axial compression forces: experimental averages and numerical results.
Experiments FE model Difference
Failure mode 1 22.1 23.2 5.0 %
Failure mode 2 22.9 23.6 3.1 %
The numerical and experimental failure mode shapes are shown in Figure
10 and Figure 9, respectively, and good agreement is again seen between
experiments and numerical models for both modes.
15

The generation of unsynchronised and synchronised failure modes was as-
sumed to be simulated in numerical models with and without an applied load
eccentricity, respectively. However, the numerical study indicated that both
centrally and eccentrically loaded models failed unsynchronised, see Figure
10a. The reason may be that the introduction of imperfection changed the
centroid slightly and resulted in an unobservable eccentricity. An equivalent
method solved the simulation of the failure mode 2. Models with a rigid or
rotationally-restrained top plate exhibit synchronised failure, even with an
eccentric loading condition, see Figure 10b. This occurs because both model
variations have uniform load transfer from the top plate into the column,
similar to the non-eccentric case, and so buckling occurs simultaneously in
all three corners as a synchronised failure. The single instance of the syn-
chronised failure mode is also explained with this result, as perfect centreline
axial loading conditions are necessary for the synchronised mode to manifest
and this is difficult to achieve for a thin-walled tube specimen with a large
cross-section.
The numerical and experimental models also have similar pre and post-
buckling failure characteristics. A comparison between the out-of-plane dis-
placement wof experimental sample C and the numerical model are shown in
Figure 11. DIC displacement data are used for the experimental model and
measurements are taken through a cross-sectional slice located at a collapsed
corner along the long axis, shown in Figure 11a. The first three buckling
profiles, shown in Figure 11b, are taken at axial displacements Uof 0.09mm,
0.2mm and 0.4mm, which occur in the elastic pre-failure region. They show
the formation of an elastic buckling profile with a mode exhibiting an upper
and lower quasi-sinusoidal peaks. The second three profiles, shown in Figure
11c, are taken at axial displacements 0.5mm, 1.0mm and 2.5mm, which occur
in the plastic post-failure region. Elastic buckling has now propagated into
plastic buckling at the lower location, with the upper region showing ‘snap
back’ behaviour where it elastically returns to an undeformed state. The nu-
merical and experimental profiles are seen to have excellent correspondence
over the entire failure, again demonstrating the accuracy of the numerical
analysis.
6.2. Comparison of experimental and numerical load paths
The prototype with synchronised failure is first analysed. In this case,
the numerical model shows relatively low axial shortening of approximately
0.5mm before failure occurs. However, the raw displacement records from
16

-1 0 1 2
w (mm)
0
100
200
300
400
500
600
Height of specimen y (mm)
U=0.09mm U=0.2mm U=0.4mm
-3 2 7 12
w (mm)
0
100
200
300
400
500
600
Height of specimen y (mm)
U=0.5mm
-3 2 7 12
w (mm)
U=1.0mm
-3 2 7 12
w (mm)
U=2.5mm
Exp
FE
Exp.
FE
Exp.
FE
Exp.
FE
b)
c)
Exp
FE
Exp
FE
-1 0 1 2
w (mm)
-1 0 1 2
w (mm)
x (mm)
300
0
100
200
0
100
200
300
400
500
600
y (mm)
Line of measurement taken
a)
Figure 11: Comparison of experimental and numerical buckling profiles. a) Location of
profiles, b) elastic stage and c) plastic stage.
the Instron crosshead show failure occurring at 1.2mm. Subsequent testing
of the Instron crosshead on rigid samples showed a discrepancy between dis-
placement measurements taken centrally and taken at the top plate, and so
Instron displacement data was concluded to be inaccurate. Analysis of pro-
totype imperfections above showed DIC displacement measurements to be
17

highly accurate and so these were used instead of Instron data. Comparison
between experimental DIC and numerical measurements is shown in Figure
12b with a better agreement seen, including good prediction of the linear elas-
tic region, axial stiffness, peak force, and the post-buckling non-linear region.
As the remaining five samples have unsynchronised failures and similar load-
displacement curves, the typical experimental DIC data from sample D and
the numerical model 1 are compared directly in Figure 12a. In this model,
one corner collapses first and the two other corners fail at a later stage. Exact
comparison of load paths is thus complex because the column has different
displacement histories at different locations around the top edges. However,
due to rotational symmetry of the sample, the 3D DIC system was able to
consistently collect data from second or third collapsed corners and so numer-
ical and experimental comparisons are made at a location marked in Figure
9and Figure 10. At these corners, a dramatic snap back phenomenon can be
seen clearly on their load paths shown in Figure 12a and deformation-scaled
numerical shapes in Figure 12c, which is similar in nature to the elastic snap
back discussed in the previous section. Again, numerical and experimental
results show good agreement in the elastic stage, including axial stiffness and
peak force. Some small discrepancies between model paths exist in the post-
buckling region however this is to be expected due to the instability of the
phenomenon being modelled and instability in the boundary condition. The
corresponding DIC records indicate some base points from snap-back corners
are lifted at times which leads to a dynamic and unsymmetrical boundary
conditions which were not captured in the numerical model.
6.3. Influence of fold lines
As discussed in Section 1, accurate modelling of folded structures require
the rotational stiffness of fold lines. In this paper, the adoption of a rel-
atively simple folded geometry and a precise, parametrically-characterised
fold-line detail have together led to an unexpected outcome; the ability to
accurately model complex folded structure behaviours with a simplified nu-
merical model that considers only fold-line slit geometry. The simplified
model avoids the need to separately characterise rotational stiffness or to
consider elastic-plastic material deformations and residual stresses that arise
during the folding process. To confirm that modelling of the slit geometry
is critical in determining behaviour, various models were established with
‘dashes’ consisting of mesh cut-outs that were straight, rather than rounded.
Even with this minor change, dashed models showed a large (20%) increase
18

a) b)
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8
Force F (kN)
Displacement U (mm)
Model-1
Sample D
0
5
10
15
20
25
0 0.2 0.4 0.6 0.8
Force F (kN)
Displacement U (mm)
Model-2
Sample C
Von Mises
Stress (MPa)
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
326
299
272
245
218
190
163
136
109
82
54
27
0
Von Mises
Stress (MPa)
255
234
213
191
170
149
128
106
85
64
43
21
0
Von Mises
Stress (MPa)
Sudden Collapse
Snap back
Snap back
Snap back distance
(scale factor 150)
Location of
the point of interest
Stage 1 (initial stage)
U = 0.0005mm,
F = 0.06kN.
Snap back distance
1
2
3
1 2 3Stage 2 (max. load)
U = 0.41mm,
F = 23.2kN.
Stage 3 (snap back)
U = 0.29mm,
F = 17.0kN.
c)
Figure 12: a) Load versus axial displacement of Model 1 and an unsynchronised failed
specimen (sample D), b) load versus axial displacement of Model 2 and a synchronised
failed specimen (sample C) and c) development of snap-back behaviour in Model 1 (axial
displacements are scaled by 150 times for illustration).
in compressive load and there is thus an implication that selection of dif-
ferent creasing schemes, be slitted, dashed or other, has a strong effect on
structural behaviour. Similarly, [48] found that fold-line depths, a geometry
factor, has great influence on the overall behaviour of origami arches.
The accuracy of the fabrication techniques and numerical modelling meth-
ods developed in this paper therefore suggest that folded thin-walled steel
structures could be the key to resolving large and ongoing research questions
19

in the field of origami-inspired engineering, namely the consistent character-
isation of fold-line axial and rotational stiffnesses from geometric fold-line
parameters; and the subsequent validation of simplified analysis models for
folded structures, of which several have been developed but require accurate
fold-line stiffness characterisation. However, the deformations developed in
this study were mainly axial displacements. They were not directly from
the rotation around fold lines. The simplified simulation method may not be
able to capture deformations accurately for structures in which displacements
come from fold-line rotation, for example a Miura-ori core or origami tube.
Thus, it is necessary to conduct a further study to examine this conjecture.
7. CONCLUSION
This paper parametrised and applied a method for assembling thin-walled
steel structures through the combination of sheet metal bending techniques
and a folded geometry. Different assembly methods were assessed with 3D
scanning and DIC measurement imperfection analyses, which showed the
fabricated prototypes were highly accurate. Experimental axial compression
tests showed the proposed folded geometry replicates structural behaviours of
a typical triangular truss, with a comparative numerical study showing excel-
lent predictions of elastic buckling and nonlinear post-buckling behaviours.
Use of a parametric fold-line geometry enabled the accuracy in both fabri-
cation and analysis, and in the latter case, the inclusion of a fold-line slit
geometry was found to be critical in modelling structural behaviours and
avoiding complexities typically encountered in the analysis of folded struc-
tures.
The work completed in the present paper suggests many avenues for fu-
ture work. In terms of the mechanics, work is needed to quantify the crease
behaviours required for employed fold-line hinge simulation. Specifically, it
is hypothesised that the parametrised fold line would enable control of the
rotational stiffness. Understanding this would enable not only engineering
application of folded steel structures, but also a means by which to validate
current origami analysis theories that utilise rotational stiffness relationships.
In terms of the engineering application, the proposed assembly method is
simple, accurate, reliable, and suitable for the development of potential ap-
plications in conventional structures and temporary buildings. Applying the
same method to fold other structural sheet materials is also worthy of further
investigation.
20

8. ACKNOWLEDGEMENTS
The corresponding author is grateful for the financial support provided by
Australian Research Council Grant DE160100289. The authors would also
like to give thanks to Mr Christopher Russ for helping set up experiments
and Dr Zhiming Guo and Ms Ya Ou for constructive suggestions.
21

REFERENCES
[1] K. Miura, The science of Miura-Ori: A review, in: R. J. Lang (Ed.),
Origami 4: Fourth International Meeting of Origami Science, Math-
ematics, and Education, A K Peters, Natick, MA, 2009, pp. 87–100.
doi:10.1201/b10653-12.
[2] R. J. Lang, Origami: Complexity in creases (again), Engineering and
Science 67 (1) (2004) 5–19.
URL http://calteches.library.caltech.edu/689/2/Origami.pdf
[3] K. Kuribayashi, K. Tsuchiya, Z. You, D. Tomus, M. Umemoto, T. Ito,
M. Sasaki, Self-deployable origami stent grafts as a biomedical applica-
tion of Ni-rich TiNi shape memory alloy foil, Materials Science and Engi-
neering: A 419 (1) (2006) 131–137. doi:10.1016/j.msea.2005.12.016.
[4] N. De Temmerman, K. Roovers, L. Alegria Mira, A. Vergauwen,
A. Koumar, S. Brancart, L. De Laet, M. Mollaert, Lightweight trans-
formable structures: Materialising the synergy between architectural
and structural engineering, Mobile and Rapidly Assembled Structures
IV 136 (2014) 1. doi:10.2495/MAR140011.
[5] E. T. Filipov, T. Tachi, G. H. Paulino, Origami tubes assembled into
stiff, yet reconfigurable structures and metamaterials, Proceedings of
the National Academy of Sciences 112 (40) (2015) 12321–12326. doi:
10.1073/pnas.1509465112.
[6] M. D. Tumbeva, Y. Wang, M. M. Sowar, A. J. Dascanio, A. P. Thrall,
Quilt pattern inspired engineering: Efficient manufacturing of shelter
topologies, Automation in Construction 63 (2016) 57–65. doi:10.1016/
j.autcon.2015.12.005.
[7] A. P. Thrall, C. P. Quaglia, Accordion shelters: A historical review of
origami-like deployable shelters developed by the US military, Engineer-
ing Structures 59 (2014) 686–692. doi:10.1016/j.engstruct.2013.
11.009.
[8] F. J. Mart´ınez-Mart´ın, A. P. Thrall, Honeycomb core sandwich panels
for origami-inspired deployable shelters: Multi-objective optimization
for minimum weight and maximum energy efficiency, Engineering Struc-
tures 69 (2014) 158–167. doi:10.1016/j.engstruct.2014.03.012.
22

[9] T. U. Lee, J. M. Gattas, Geometric design and construction of
structurally stabilized accordion shelters, Journal of Mechanisms and
Robotics 8 (3) (2016) 031009. doi:10.1115/1.4032441.
[10] X. Liu, J. M. Gattas, Y. Chen, One-DOF superimposed rigid origami
with multiple states, Scientific Reports 6 (2016) 36883. doi:10.1038/
srep36883.
[11] R. Sturm, P. Schatrow, Y. Klett, Multiscale modeling methods for anal-
ysis of failure modes in foldcore sandwich panels, Applied Composite
Materials 22 (6) (2015) 857–868. doi:10.1007/s10443-015-9440-9.
[12] M. Schenk, S. D. Guest, Geometry of Miura-folded metamaterials, Pro-
ceedings of the National Academy of Sciences 110 (9) (2013) 3276–3281.
doi:10.1073/pnas.1217998110.
[13] S. Heimbs, Foldcore sandwich structures and their impact behaviour:
An overview, in: S. Abrate, B. Castani´e, Y. D. S. Rajapakse (Eds.),
Dynamic failure of composite and sandwich structures, Springer, 2013,
pp. 491–544. doi:10.1007/978-94-007-5329-7_11.
[14] K. Miura, Zeta-core sandwich-its concept and realization, Tech. Rep.
480, Institute of Space and Aeronautical Science, University of Tokyo
(1972).
[15] T. Tachi, Geometric considerations for the design of rigid origami struc-
tures, in: Proceedings of the International Association for Shell and
Spatial Structures (IASS) Symposium, Vol. 12, Shanghai, China, 2010,
pp. 458–460.
[16] J. M. Gattas, W. Wu, Z. You, Miura-base rigid origami: parameteriza-
tions of first-level derivative and piecewise geometries, Journal of Me-
chanical Design 135 (11) (2013) 11011. doi:10.1115/1.4025380.
[17] A. Qattawi, A. Mayyas, H. Thiruvengadam, V. Kumar, S. Don-
gri, M. Omar, Design considerations of flat patterns analysis tech-
niques when applied for folding 3-D sheet metal geometries, Journal
of Intelligent Manufacturing 25 (1) (2014) 109–128. doi:10.1007/
s10845-012-0679-9.
23

[18] E. T. Filipov, T. Tachi, G. H. Paulino, Toward optimization of stiffness
and flexibility of rigid, flat-foldable origami structures, in: K. Miura,
T. Kawasaki, T. Tachi, R. Uehara, R. J. Lang, P. Wang-Iverson (Eds.),
Origami 6: Proceedings of the Sixth International Meeting on Origami in
Science, Mathematics and Education, American Mathematical Society,
Providence, RI, 2015, pp. 409–419.
[19] E. T. Filipov, K. Liu, T. Tachi, M. Schenk, G. H. Paulino, Bar and
hinge models for scalable analysis of origami, International Journal of
Solids and Structuresdoi:10.1016/j.ijsolstr.2017.05.028.
[20] H. Yasuda, T. Yein, T. Tachi, K. Miura, M. Taya, Folding behaviour
of Tachi-Miura polyhedron bellows, Proceedings of the Royal Society
A: Mathematical Physical and Engineering Sciences 469 (2159) (2013)
20130351. doi:10.1098/rspa.2013.0351.
[21] C. Pradier, J. Cavoret, D. Dureisseix, C. Jean-Mistral, F. Ville, An
experimental study and model determination of the mechanical stiffness
of paper folds, Journal of Mechanical Design 138 (4) (2016) 041401.
doi:10.1115/1.4032629.
[22] S. Nagasawa, R. Endo, Y. Fukuzawa, S. Uchino, I. Katayama, Creasing
characteristic of aluminum foil coated paperboard, Journal of Materi-
als Processing Technology 201 (1) (2008) 401–407. doi:10.1016/j.
jmatprotec.2007.11.253.
[23] L. A. A. Beex, R. H. J. Peerlings, An experimental and computa-
tional study of laminated paperboard creasing and folding, Interna-
tional Journal of Solids and Structures 46 (24) (2009) 4192–4207.
doi:10.1016/j.ijsolstr.2009.08.012.
[24] L. Mentrasti, F. Cannella, M. Pupilli, J. S. Dai, Large bending be-
havior of creased paperboard. I. Experimental investigations, Inter-
national Journal of Solids and Structures 50 (20) (2013) 3089–3096.
doi:10.1016/j.ijsolstr.2013.05.018.
[25] F. Lechenault, B. Thiria, M. Adda Bedia, Mechanical response of a
creased sheet, Physical Review Letters 112 (24) (2014) 244301. doi:
10.1103/PhysRevLett.112.244301.
24

[26] T. G. Nelson, J. T. Bruton, N. E. Rieske, M. P. Walton, D. T.
Fullwood, L. L. Howell, Material selection shape factors for compli-
ant arrays in bending, Materials & Design 110 (2016) 865–877. doi:
10.1016/j.matdes.2016.08.056.
[27] S. K. Gupta, D. A. Bourne, K. H. Kim, S. S. Krishnan, Automated pro-
cess planning for sheet metal bending operations, Journal of Manufac-
turing Systems 17 (5) (1998) 338–360. doi:10.1016/S0278-6125(98)
80002-2.
[28] A. A. Evans, J. L. Silverberg, C. D. Santangelo, Lattice mechanics of
origami tessellations, Physical Review E 92 (1) (2015) 013205. doi:
10.1103/PhysRevE.92.013205.
[29] M. W. Durney, A. D. Pendley, Method for precision bending of sheet of
materials, slit sheets fabrication process, Patent US6877349 (Apr. 2005).
[30] M. W. Durney, A. D. Pendley, Fatigue-resistance sheet slitting method
and resulting sheet, Patent US20060021413 (Feb. 2006).
[31] M. A. Ablat, A. Qattawi, Finite element analysis of origami-based sheet
metal folding process, in: ASME 2016 International Mechanical Engi-
neering Congress and Exposition, American Society of Mechanical En-
gineers, 2016, p. V002T02A058. doi:10.1115/IMECE2016-67324.
[32] M. W. Durney, A. D. Pendley, Sheet material with bend controlling dis-
placements and method for forming the same, Patent US7350390 (Apr.
2008).
[33] J. M. Gattas, Z. You, Design and digital fabrication of folded sandwich
structures, Automation in Construction 63 (2016) 79–87. doi:10.1016/
j.autcon.2015.12.002.
[34] J. Song, Y. Chen, G. Lu, Light-weight thin-walled structures with pat-
terned windows under axial crushing, International Journal of Mechan-
ical Sciences 66 (2013) 239–248. doi:10.1016/j.ijmecsci.2012.11.
014.
[35] D. Sonck, J. Belis, Weak-axis flexural buckling of cellular and castellated
columns, Journal of Constructional Steel Research 124 (2016) 91–100.
doi:10.1016/j.jcsr.2016.05.002.
25

[36] X. L. Zhao, D. Van Binh, R. Al-Mahaidi, Z. Tao, Stub column tests
of fabricated square and triangular sections utilizing very high strength
steel tubes, Journal of Constructional Steel Research 60 (11) (2004)
1637–1661. doi:10.1016/j.jcsr.2004.04.003.
[37] Y. Chen, Z. You, Mobile assemblies based on the Bennett linkage, Pro-
ceedings of the Royal Society of London A: Mathematical, Physical
and Engineering Sciences (2056) (2005) 1229–1245. doi:10.1098/rspa.
2004.1383.
[38] A. Jansseune, W. De Corte, J. Belis, Imperfection sensitivity of lo-
cally supported cylindrical silos subjected to uniform axial compres-
sion, International Journal of Solids and Structures 96 (2016) 92–109.
doi:10.1016/j.ijsolstr.2016.06.019.
[39] A. Wheeler, M. Pircher, Measured imperfections in six thin-walled steel
tubes, Journal of Constructional Steel Research 59 (11) (2003) 1385–
1395. doi:10.1016/S0143-974X(03)00089-0.
[40] M. Nassirnia, A. Heidarpour, X. L. Zhao, J. Minkkinen, Innovative hol-
low corrugated columns: A fundamental study, Engineering Structures
94 (2015) 43–53. doi:10.1016/j.engstruct.2015.03.028.
[41] A. T. Tran, M. Veljkovic, C. Rebelo, L. S. Da Silva, Resistance of cold-
formed high strength steel circular and polygonal sections-Part 1: Ex-
perimental investigations, Journal of Constructional Steel Research 120
(2016) 245–257. doi:10.1016/j.jcsr.2015.10.014.
[42] T. U. Lee, J. M. Gattas, Folded fabrication of composite curved-crease
components, in: The 8th International Conference on Fibre-Reinforced
Polymer (FRP) Composites in Civil Engineering(CICE 2016), The Hong
Kong Polytechnic University, 2016.
[43] Abaqus 6.14 Analysis Users Guide, Providence, RI (2014).
[44] E. Riks, An incremental approach to the solution of snapping and buck-
ling problems, International Journal of Solids and Structures 15 (7)
(1979) 529–551. doi:10.1016/0020-7683(79)90081-7.
26

[45] W. T. Koiter, The stability of elastic equilibrium (in dutch), Ph.D.
thesis, Delft University of Technology, Translated in NASA TT F-10
833 (1967) and AFFDL Report TR 70-25 (1970) (1945).
[46] J. Hutchinson, Axial buckling of pressurized imperfect cylindrical shells,
AIAA Journal 3 (8) (1965) 1461–1466.
[47] A. C. Orifici, C. Bisagni, Perturbation-based imperfection analysis for
composite cylindrical shells buckling in compression, Composite Struc-
tures 106 (2013) 520–528. doi:10.1016/j.compstruct.2013.06.028.
[48] J. M. Gattas, W. Lv, Y. Chen, Rigid-foldable tubular arches, Engineer-
ing Structures 145 (2017) 246–253. doi:10.1016/j.engstruct.2017.
04.037.