Interlocking Folded Plate – Integral Mechanical Attachment for Structural Wood Panels

Abstract

Automatic joinery has become a common technique for the jointing of beams in timber framing and roofing. It has revived traditional, integral joints such as mortise and tenon connections. Similarly, but only recently, the automatic fabrication of traditional cabinetmaking joints has been introduced for the assembly of timber panel shell structures. First prototypes have used such integrated joints for the alignment and assembly of components, while additional adhesive bonding was used for the load-bearing connection. However, glued joints cannot be assembled on site, which results in several design constraints.
In this paper, we propose the use of dovetail joints without adhesive bonding, on the case study of a timber folded plate structure. Through their single-degree-of-freedom (1DOF) geometry, these joints block the relative movement of two parts in all but one direction. This presents the opportunity for an interlocking connection of plates, as well as a challenge for the assembly of folded plate shells, where multiple non-parallel edges per plate must be jointed simultaneously.

Full text

Interlocking Folded Plate - Integral Mechanical Attachment for
Structural Wood Panels
Christopher Robeller ∗1and Yves Weinand †1
1Timber Construction Laboratory IBOIS, EPFL
June 20, 2015
Abstract
Automatic joinery has become a common technique for the jointing of beams in timber framing and
roofing. It has revived traditional, integral joints such as mortise and tenon connections. Similarly,
but only recently, the automatic fabrication of traditional cabinetmaking joints has been introduced
for the assembly of timber panel shell structures. First prototypes have used such integrated joints
for the alignment and assembly of components, while additional adhesive bonding was used for the
load-bearing connection. However, glued joints cannot be assembled on site, which results in several
design constraints.
In this paper, we propose the use of dovetail joints without adhesive bonding, on the case study
of a timber folded plate structure. Through their single-degree-of-freedom (1DOF) geometry, these
joints block the relative movement of two parts in all but one direction. This presents the opportunity
for an interlocking connection of plates, as well as a challenge for the assembly of folded plate shells,
where multiple non-parallel edges per plate must be jointed simultaneously.
1 Introduction
Architectural designs have often been inspired by
folded shapes such as Origami, however the fold-
ing principle can rarely be applied to building
structures directly. Instead, many folded plates
have been cast as concrete thin-shells in the 1960s.
These constructions were labour-intensive and re-
quired elaborate formwork for the in-situ casting.
Prefabricated constructions with discrete ele-
ments made from fiber-reinforced plastics have
been researched in the 1960s. [1]
Folded plates built from laminated timber pan-
els have been presented by C. Schineis [19] (Glu-
lam) and H. Buri [2] (Cross-laminated Timber).
These designs combine the elegant and efficient
shape of folded plate shells with the advantages of
structural timber panels, such as CO2 storage and
a favorable weight-to-strength ratio. However, a
major challenge in the design of a timber folded
plate is presented by the joints: Since timber pan-
els cannot be folded, a large amount of edgewise
joints has to provide two main functions. One
of these functions is the load-bearing behaviour,
where connector features of the joints must pro-
vide a sufficient stiffness and rigidity. The second
main function of the joints is the assembly of the
parts, where locator features of the joints are es-
sential for a precise and fast positioning and align-
ment of the parts.
B. Hahn [5] examined the structural behaviour
of a first timber folded plate shell which was built
from plywood and assembled with screwed miter
joints, concluding that the load-bearing perfor-
mance could be improved significantly with more
resistant connections.
Inspiration for such improvements may be found
in integral mechanical attachment techniques, the
oldest known technique for the jointing of parts,
where the geometry of the parts themselves blocks
their relative movements [13]. Such integrated
joints have recently been re-discovered by the tim-
ber construction industry. Beginning in 1985,
mortise-and tenon joints have been repatriated in
timberframe and roof constructions [7]. Only very
recently, integrated joints have also been proposed
for the edgewise jointing of timber panels. In
the ICD/ITKE Reserach Pavilions 2011 [12] and
1

2013 [11], fingerjoints have been applied to ply-
wood panels and an application of dovetail joints
for cross-laminated timber panels (CLT) was pre-
sented in the IBOIS Curved Folded Wood Pavil-
ion 2013 [18]. In these prototype structures, the
integrated joints have played an important role
for the assembly of the components. They have
also participated in the load-bearing connection
of the parts, but additional adhesive bonding was
needed. With few exceptions [6], such glued joints
cannot be assembled on site, because they require
a curing period with a specific constant tempera-
ture and humidity [15]. Therefore, their applica-
tion is limited to off-site assembly of larger compo-
nents, which complicates both transport and han-
dling while still requiring additional connectors for
the final assembly.
In this paper, we propose the use of dovetail
joints without additional adhesive bonding, on the
case study of a timber folded plate shell. (Figure
1).
Through their single-degree-of-freedom (1DOF)
geometry, these joints block the relative movement
of two parts in all but one direction. This presents
the opportunity for an interlocking connection of
plates, as well as a challenge for the assembly
of folded plate shells, where multiple non-parallel
edges per plate must be jointed simultaneously.
1.1 Dovetail joint geometry and me-
chanical performance
Using polygon mesh processing, we describe an
edgewise joint based on its edge E. From the
mesh connectivity, we obtain the edge vertices p
and qand the adjacent faces F0and F1with their
face normals n0, n1. We use the polygon mesh
to represent the mid-layer of timber panels with
a thickness tand offset F1and F2at ±t
2to ob-
tain the lines L(Figure 2a). From a division of
E, we obtain the points Xjfor a set of refer-
ence frames {u1, u2, u3}, where u1k~pq and u2kn0
(Figure 2b). A finger joint geometry is obtained
from an intersection of planes located at Xj, nor-
mal to u1, with the four lines L.
Without additional connectors, finger joints are
a kinematic pair with three degrees of freedom
(3DOF), also called planar joints. They can re-
sist shear forces parallel to the edge and in-plane
compressive forces. However depending on the
plate geometry, thickness and most of all rota-
tional stiffness of the connection detail, bending
moments are also transferred between the plates.
Also, due to the rotation of the plate edge caused
Figure 3: FEM analysis (top view) of a 3x3m,
21mm Kerto-Q folded plate thin shell assuming
fully stiff joints. Distribution of traction (red)
and compression (blue) stresses in the y direction.
Top: gravity load case. Bottom: asymmetric snow
load.
by bending, in-plane traction forces perpendicu-
lar to the edge line appear and their magnitude
increases under asymmetrical loads. Such forces,
which occur as a result of out-of-plane loading,
cannot be supported only by shear and in-plane
compression resistant joints.
On a dovetail joint (Figure 2d,e), the intersec-
tion planes on the points Xjare normal to a ro-
tated vector w1. It is obtained from a rotation
of the reference frame {u1, u2, u3}about u3at an
alternating angle ±θ3. The resulting rotated side
faces reduce the dovetail joints degrees of freedom
to one translation ~w3(1DOF). Simek and Sebera
[20] have suggested θ3= 15◦for spruce plywood
panels. Such prismatic joints can only be assem-
bled or disassembled along one assembly direction
~v =~w3. In addition to the finger joints resistance
to shear and compressive forces, dovetail joints
can, without adhesive bonding, also resist bend-
ing moments and traction forces which are not
2

Figure 1: Folded thin shell prototype built from 21mm LVL panels, assembled with single-degree-of-
freedom dovetail joints without adhesive bonding. Components interlock with one another
p
q
p
q
p
q
n0n1
w3
F0
F0
F+
F0
F-F1
F1
F+
F1
F-
a. b. c. d. e.
u2u3
u1
Ei
L1
L
L3
LL2
L
L0
L
Xj
j+1
j+2
j+3
j+1
j+2
j+3
j+1
j+2
j+3
v
w1
w2
Figure 2: Joint geometry. a: Basic parameters, b: Intersection planes (grey) normal to ~pq, c: 3DOF
joint, d: Rotated intersection planes (grey) normal to ~wj, e: 1DOF joint
parallel to ~v. Due to the inclination of the side
faces of the joint, resistance to these forces can be
improved significantly. In that way the inclined
faces take over the role that the glue would have
in a finger joint. (Figure4)
1.2 Fabrication Constraints
One of the main reasons for the resurgence of fin-
ger and dovetail joints is the possibility of auto-
matic fabrication. However, the mechanical per-
formance of the joints depends on fabrication pre-
cision. At the same time, fast machine feed rates
are important for a time-efficient production. We
have fabricated such joints with a robot router and
a gantry router, achieving higher precision with
the gantry machine, which is more stiff and pro-
vides a higher repeat accuracy.
The variability of the machine-fabricated joints
is enabled by the 5-axis capability of modern
routers: Although traditional edgewise joints in
cabinetmaking were used for orthogonal assem-
blies, both the finger and dovetail joint can also
be applied for non-orthogonal fold angles, which
was essential for the reference projects mentioned
before. However, there are certain fabrication-
related constraints for machine-fabricated dovetail
joints. In order to integrate the joint fabrication
directly with the panel formatting, we use a side-
cutting technique [8], which is limited to a tool
inclination βmax. We obtain this limit from the
specific geometry of the tool, tool-holder and spin-
dle used for the joint fabrication. (Figure 5)
The parts can be assembled in two ways, as
shown in figure 5, which allows to address a larger
range of dihedral angles ϕ. From this we ob-
tain the fabrication-constrained most acute fold
ϕmin = 90◦−βmax and most obtuse fold ϕmax =
90◦+βmax. With standard cutting tools, this tech-
nique allows for the jointing of acute folds up to
3

Figure 4: FEM simulation of bending on a dovetail joint connecting two Kerto-Q 21mm LVL panels.
The bending moment applied is transformed into compression, normal and shear forces parallel to the
inclined contact faces.
βmax
φmax
φmin
βmax
βmax
TCP
TCP
TCP
TCP
TCP
TCP
TCP
TCP
TCP
TCP
TCP
TCP
TCP
Figure 5: Fabrication Constraints. Side-cutting technique used for the automated fabrication of 1DOF
edgewise joints with common 5-axis CNC routers. The maximum tool inclination βmax results from
the tool and the tool holder geometry. From this we obtain the range of possible dihedral angles ±ϕ
between panels.
ϕ= 50◦, which is ideal for folded plate structures.
Very obtuse fold angles ϕ≥140◦, which might be
required for smooth segmented plate shells, can-
not be fabricated with this method.
1.3 Simultaneous Assembly of Multiple
Edges
The assembly of doubly-corrugated folded plates
requires the simultaneous joining of multiple edges
per component (Figure 1), which has implications
on both the shell and the joint geometry.
For multiple 1DOF-jointed edges, simultaneous
assembly is only possible if the individual assem-
bly directions ~v are parallel. With a normal dove-
tail joint geometry (Figure 7a), this is not the case:
A simultaneous assembly is only possible for par-
allel edges, which allows only for rectangular as-
semblies, such as drawers or a cabinets.
In order to simultaneously join non-parallel
edges, we must rotate the assembly direction vof
the joints to make them parallel. This possibility
is known from Japanese cabinetmaking [9], where
certain joints, like the Nejiri Arigata Joint (Figure
7b), are assembled diagonally, along a vector that
does not lie on either one of the two planes. While
European dovetail form a prism with a single tab,
e1
e2
e1
e0
e2e1
e0
x3
x2
Figure 6: The assembly of a folded plate from dis-
crete elements (left side) requires the simultaneous
assembly of non-parallel edges. (right side) We
rotate the insertion direction of our 1DOF joints,
to make the insertion vectors of simultaneously
jointed edges parallel. We chose a hexagon re-
verse fold pattern which requires only moderate
rotations.
4

vv
a. b.
faces across
edge only faces
across edge
faces along
edge +
Figure 7: a. Dovetail Joint, b. Nejiri Arigata Joint
using faces both acrioss and along the edge, the
Nejiri Arigata joints form a prism using multiple,
differently shaped tabs.
We extend this Japanese technique to a vector
subset of possible assembly directions. Figure 8
shows that the rotation about the edge line is con-
strained to 180◦−ϕi. The vector subset is large for
acute and small for obtuse fold angles. This is par-
ticularly important when joining multiple edges
simultaneously, because an intersection must be
found between multiple vector subsets (Fig. 9).
If there is an intersection, the parts can be joined
simultaneously along any direction within the in-
tersection of the subsets.
Finally we extend this concept to a 3-
dimensional rotation (Fig. 10). This is possible
through a second rotation θ2, which is constrained
to a maximum value of ±θ2,max. The limitation
results from multiple other corelated parameters,
such as θ1and βmax. We call the resulting 3D
vector subset rotation window.
With this method, we can search for a joining
solution for the prototype in figure 6. We compute
rotation windows S1,S2,S3for the edges E1,E2,
E3and overlay them at their center point. Fig-
ure 11 shows that there is a common vector sub-
set S1TS2TS3between these three edges, we can
choose our assembly direction within it.
As a result of these limited rotations, the an-
gle between neighbouring, simultaneously joined
edges cannot be very acute. Folded plate pat-
terns like the Herringbone, the Diamond, or the
Hexagon pattern, which we chose for our proto-
types, (Figure 6) work well for our joining tech-
90° 130°50°
"
#
!
!
E34 68344
a.
Figure 8: 2D vector subset
3"3$
3"
3$
E1E2
E1
E2
b.
Figure 9: 2D simultaneous assembly
2, max
1
Figure 10: 3D vector subset
5

S3
S1
E1
E2
S2
v
S1ŀS2ŀS3
E3
3x
Figure 11: 3D simultaneous assembly
nique. Another essential feature provided by these
reverse-folds are the acute fold angles, which eas-
ily satisfies the fabrication-constrained range of
ϕmin = 50◦to ϕmax = 140◦.
2 Interlocking Arch Prototype
In an assembly of multiple components (Figure
12), a step-by-step sequence must be planned for
the assembly of the parts. The completed struc-
ture can only be disassembled piecewise in the re-
verse order of assembly. In this way, the elements
interlock with one another like a Burr Puzzle [22].
Each joint consists of two parts, which must be
parallel during assembly. We therefore chose a
folded plate geometry with relatively short edges.
The manual assembly of long edges may be more
difficult but can be simplified with a modified joint
geometry. It is important to know the approxi-
mate direction of insertion for each part, as this is
not easily visible through the joint geometry. De-
formations of the arch during the assembly should
be minimised. We have assembled this first pro-
totype lying on the side. However larger assembly
may require temporary punctual supports. Al-
though the in-plane dimensional stability of the
Kerto-Q panels is very high, panels may be slightly
warped and some force may be necessary during
assembly. While we have simply used a rubber
hammer, more advanced techniques could be ap-
plied.
To understand the mechanical behaviour of the
built prototype, we have applied a vertical load at
mid-span of the arch and measured the vertical de-
0 1 2 3 4 5 6 7
0
200
400
600
800
1000
Load cycle #1
Load cycle #2
Force [N]
Displacement [mm]
Figure 13: Series of 3-point flexural tests on the
small scale interlocking arch prototype built from
Metsawood 12mm birch plywood panels.
flection at the same point. The total load of 821N
was applied in two identical load cycles consist-
ing of four loading/unloading sub-cycles. First, a
vertical load of 117N was applied in seven steps,
after which the load of the last four steps was re-
moved. The loading and unloading of the last four
steps was repeated three more times, after which
the complete load was removed and the residual
deflection was measured. (Figure 13)
Under a vertical load equal to the archs dead
weight of 9.8kg (98N), the deflection measured at
mid-span was 2mm. From this we obtain a span-
to-deflection-ratio of L/750 and the arch’s struc-
tural efficiency which reaches 8.6 when loaded
with 821N (ratio of the maximal load over the
dead weight of the arch).
3 Interlocking Shell Prototype
3.1 Automatic Geometry Processing
Using the RhinoPython application programming
interface, we have developed a computational tool
which lets us instantly generate both the geom-
etry of the individual components and the ma-
chine G-Code required for fabrication. The tool
processes arbitrary polygon meshes, and gener-
ates 1DOF joints for all non-naked edges where
the fold angle ϕis larger than ϕmin and smaller
than ϕmax shown in figure 5(non-smooth meshes).
It also requires an input of edge identifier tuples
identifying those edges which must be jointed si-
multaneously, as well as the thickness of the LVL
panels. Exploiting this geometrical freedom, we
have tested our computational tool on the design
of a folded plate shell prototype with an alter-
6

Figure 12: Folded-plate arch prototype built from 12mm birch plywood (9-layer, I-I-I-I-I). Assembled
without adhesive bonding or metal fasteners. Span 1.65m, self-weight 9.8kg.
nating convex-concave transversal curvature. The
shell spans over 3m at a thickness of 21mm, us-
ing Kerto-Q structural grade LVL panels (7-layer,
I-III-I). (Figure 14)
Comparing this doubly-curved folded plate with
a straight extrusion (as tested by H. Buri [2]), it
can be concluded that the slight double-curvature
proves to be very beneficial when it comes to
global deflections, for example those caused by
wind loads. Deflections for the doubly-curved
shell geometry in the vertical direction are up to
39% smaller and up to 13% smaller in the lateral
direction than the ones for the straight extrusion
one.
3.2 Assembly
Due to the different assembly directions of its 239
joints, the 107 components components in our pro-
totype interlock with one another, similar to a
Burr Puzzle [22]. Figure 15 shows a part of a
so called non-directional blocking graph (NDBG),
which was inroduced by Wilson and Latombe [21].
In such a graph, single arrows indicate that
parts can be removed from the assembly. Two
opposite arrows between parts indicate that the
connection is blocked. In order to remove blocked
parts, the blocking parts must be removed first.
Our graph illustrates a left-to-right assembly. On
the right side, part number 86 is being inserted. It
connects to three other plates and blocks all other
parts in the graph. In such a configuration, the
final part remains removable, it is called the key.
Figure 16 shows the parts from figure 15 in 3D,
demonstrating how the component based on mesh
face F86 is being inserted. Its three edgewise joints
E41,E68 and E89 must be assembled simultane-
ously. The three assembly vectors of the edges
~v41,~v68 and ~v89 have been rotated to be parallel.
The same applies for the adjacent edges on the
left side of the faces F67,F69 ,F88,F103 and F105
(see figure 15). Within the rotation window of the
edge, we can freely rotate ~v for these edges (the
greater the angle between ~v and the main direction
of traction e1, the better).
3.3 Completed Shell Prototype and
Load Test
Figure 17 shows the completed folded plate pro-
totype, with a span of 3m and a shell thickness
of 21mm. Boundary conditions that restrain dis-
placements of the supports in every direction, but
allow rotations, were applied on both sides. A lon-
gitudinal line load was introduced along the top of
the shell and vertical displacement was measured
at center point. (Figure 18)
The prototype structure was also modelled in
7

8889
6869
71
107
87
104
86
70 Blocked
Free
65
98
35
68
89
41
105
106 7
107
90
44
100 91
59
66
36
99
18
Figure 15: Partial connectivity, assembly and blocking graph of the folded plate shell prototype.
(Left-to-right assembly) Large numbers represent mesh faces, small numbers represent mesh edges.
Pi+1
Pi+2
Pi+3 Pi+4
Pi+5
Pi+6
Pi
ni+1
ni+2
ni+3 ni+4
ni+5
CV
1
CC1
CV
2
CV3
CC2
CC3
CC: Concave
CV: Convex
A
B
A
A
B
B
B
A: min length
B: max length
Figure 14: Doubly-curved folded-plate: The ra-
dius (R= 17m) of the transversal curvature is
determined by the folded plates maximum ampli-
tude h[2], which is inversely proportional to the
number of segments mof the cross-section poly-
line (grey). We obtain this polyline from a circular
arc divided into segments of equal length. The in-
terior angle γ= ((m−2)∗180)−mof this polyline
is proportional to all fold angles ϕ. The geometry
of our prototype was fabrication-constrained to a
maximum component length B≤2.5m
FE analysis software (Abaqus) and loaded in the
same way. The plates were modelled using shell el-
ements, where the mid-surface is used to represent
the 3D plate and transverse shearing strains are
neglected. Connections between the plates were
considered as completely rigid in order to obtain
minimal displacements of the structure. By com-
paring the displacements of the structure with in-
finitely stiff joints with the ones measured on the
prototype, we obtained information about the ac-
tual semi-rigidity of the joints. The results ob-
tained from the testing of the large scale prototype
showed that the load of 25kN, that corresponds
to the proportional limit of the load-displacement
curve, causes a vertical displacement of 23mm. In
the FE model, the load applied in the same man-
ner caused a vertical displacement of 2.6mm.
Figure 18: Load-displacement curve of the shell
prototype. A longitudinal line load was intro-
duced along the top of the shell. Vertical displace-
ment was measured at the center point.
8

e3
v41 = v68 = v89
v35 = v65 = v98
v7 = v36
v13 = v42
e2
e1
v7v13
v59
v44
v18 v35 v36
v41 v42
v70
v43
v68
v66
v65
v84
v89
v90
v100
v99 v98
v107
v94
e1
e2
F86
F86
Figure 16: Left-to-right assembly of the Interlocking folded plate shell prototype. Built from Kerto-Q
structural grade LVL panels (7-layer, I-III-I))
9

Figure 17: Folded-plate shell prototype, built from 21mm LVL panels. With a self-weight of 192kg,
the prototype with a span of 3m was tested with a line-load up to 45kN.
4 Conclusion
A timber folded plate shell combines the struc-
tural advantages of timber panels with the effi-
ciency of folded plates. However, in such discrete
element assemblies, a large amount of semi-rigid
joints must provide sufficient support for the ad-
jacent plates in oder to ensure an efficient load-
bearing system. This remains a challenge with
much potential for improvements [5].
Integrated edgewise joints present an interest-
ing addition and an alternative to state-of-the-art
connectors: Compared to adhesive bonding, such
joints can be assembled rapidly on site. Also, com-
pared to costly metal plates and fasteners, which
are typically required in large quantities [14], the
fabrication of integrated joints is not more ex-
pensive. The replacement or reduction of metal
fasteners with an integrated mono-material con-
nection includes advantages such as improved aes-
thetics, ease-of recycling or a homogenous thermal
conductivity of the parts, which can reduce con-
densation and decay. [4] Another particular ad-
vantage is the possibility to join thin panels: The
current technical approval for the Kerto-Q panels
does not permit screwed joints on panels with a
thickness of less than 60mm. [3]
Recent experimental projects, which we intro-
duced in chapter 1, have already demonstrated
first applications of integrated edgewise joints for
timber panels. This paper followed up on these
projects, examining the particular advantages, po-
tential and challenges of 1DOF joints for timber
folded plate shells. We have demonstrated how
this joint geometry helps resisting the forces which
occur in such structures. In addition to the load-
bearing connector features, the joints provide lo-
cator features, which allow for precise positioning
and alignment of the parts through the joint ge-
ometry. This improves both accuracy and ease of
assembly. Furthermore, we have presented a so-
lution for the simultaneous assembly of multiple
edges per panel, which is essential for the applica-
tion of 1DOF joints in a folded plate shell struc-
ture. The per edge ”rotation window” introduced
in section 1.3 integrates the joint constraints re-
lated to assembly and fabrication. It can be pro-
cessed algorithmically and gives instant feedback
on whether or not a set of non-parallel edges can
be jointed simultaneously. This provides a tool for
the exploration of a variety of alternative folded
plate shell geometries.
10

The prototypes presented in this paper already
suggest possible patterns and demonstrate the re-
ciprocal relationship between the geometry of the
plates and the joints. Two built structures allowed
us to test and verify the proposed methods for fab-
rication and assembly while providing valuable in-
formation about the load-bearing capacity of the
integrated joints.
For the application in a large-scale building
structure, further research is required to deter-
mine if the integrated joints can replace additional
connectors entirely or reduce their amount. A pos-
sible combination of integrated joints with addi-
tional metal fasteners has been demonstrated re-
cently in the LaGa Exhibition Hall [10]. Another
possibility would be a combination of the 1DOF
joints with integrated elastic interlocks. [17]
Acknowledgments
We would like to thank Andrea Stitic and Paul
Mayencourt for their support with the finite ele-
ment models and load testing of the prototypes,
as well as Gabriel Tschanz and Francois Perrin
for assisting with the fabrication and assembly of
prototypes. We also thank Jouni Hakkarainen and
the Metsa Group for the supply of information and
materials.
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