PARAMETRIC KERF BENDING MANUFACTURING GYROID SURFACE DESIGN & TECHNOLOGY
BERIN NUR KOCABAS
OWAZE ANSARI
01
CONTENT
1 ABSTRACT
ABSTRACT
Geometric understanding allows us to solve many problems in manufacturing complex surfaces and geometric rules can lead the project from theory to practice
2 THEORETICAL BACKGROUND GEOMETRY/GYROID DEVELOPABLE SURFACE
(Pottmann in Architectural geometry. Kindle edn, 2015). FEA analysis technologies are evolving and becoming more accessible to analyze surfaces and geometric rules that arise while designing. Knowledge of this makes it possible to take preempti-
3
MDF AND PLYWOOD ANALYSIS
4
HOW TO IMPROVE BENDING PROPERTY OF MDF AND PLYWOOD?
based on geometry, constructively altering the design process. As the field grows,
5
PATTERN DESIGN FOR GEOMETRY
traditional materials like plywood more innovatively. Our research explores in fabri-
6
ve design decisions and effect fabrication methods using parametric design tools,
it enhances material understanding and limitations, playing a crucial role in using
cation of non-developable surfaces by means “Kerf bending”, while keeping in mind
KARAMBA ANALYSIS
pre and post geometrical analysis. This geometrical analysis serves as a guide in translating the flat unrolled surface, from having rigid to flexible material properties,
7 PROTOTYPES
using the 3D surface data to determine the subtractive manufacturing. This type of 8 PABRICATION
geometrical analysis is quintessential in determining the manufacturing of double curvature surfaces. The aims of the research are: testing, analyzing and translating
9 CONCLUSION
different kerfing techniques on to our Gyroid prototype from the geometrical analysis; determining the scaffolding for retention of its double curve; and comparing the
10 REFERENCE LIST
transformations that emerged during fabrication.
Figure 1: Gyroid Generation
KEYWORDS: FEA analysis technologies · Non-developable surfaces · Kerf bending
02
THEORETICAL BACKGROUND
GEOMETRY/ GYROID The gyroid, illustrated above, is an infinitely connected periodic minimal surface
In this project, we wanted to explore more on the knowledge of the geometric pro-
containing no straight lines (Osserman 1986) that was discovered by Schoen (1970).
perties of surfaces. The aim in selecting the gyroid geometry was to explore the
Große-Brauckmann and Wohlgemuth (1996) proved that the gyroid is embedded.
minimal surface and experiment on the doubly curved surfaces with wood products. One of the most intrinsic properties of any surface is the developability. The “kerfing” technique deals with the transformation of a rigid material into a flexible one. In our case, we experimented on the MDF and plywood materials and applied the kerfing technique on them. The main challenge was to solve is how to cut the flat shape to obtain the design surface.
Figure 2 : Gyroid Surfaces
Figure 3: Gyroid Surfaces
“Gyroid.” from Wolfram MathWorld. Accessed November 3, 2019. http://mathworld.wolfram.
Figure 4: SURFACE IN A CUBE
MOLD PRODUCTION
In order to obtain the gyroid surfaces, first,
The grey foam cube mold is cut with CNC
the geometry needed to be unrolled. Howe-
milling. It cut into 6 pieces (as shown in
ver, since the gyroid surfaces are doubly cur-
Figure 8) in total and assebly made as 3
ved, it was not possible to unroll it to a single
by 3. Every 2 pieces are actually the mir-
surface. As the software were not allowing us
ror of themselves.
to generate the unrolled surface, we decided to start working on the physical mold of a gyroid. By using the digital model in Figure 1, the
Figure 5: COMPACT MOLD
foam mold is produced by CNC milling tech-
Figure 8: MOLD AFTER CUT
02
THEORETICAL BACKGROUND
Mirrored Pieces
nique. The gyroid surface is dividing the cube
Each set of 3 pieces are first sanded and
into 2 parts so that the mold is produced in
the surfaces and corners refined.
2 pieces and when they interlock, they form a cube as in Figure 5. Figure 6 shows the 1 piece of the mold.
ces and closed up the mold. We traced all the
12 cm 12 cm
Figure 7: UNROLLED SURFACE
Figure 9: MOLD PIECES
of the one-piece) in between the 2 mold pieedged of the fabric with a marker and open
After sanding, the pieces are glued
up the mold. When we flatten the fabric, the
together and 2 piece mold is obtained
traced edges gave us the unrolled geometry
finally. In order to have a stronger mold,
as in Figure 7. However, the center part of the fabric was wrinkled inside the mold. The wrinkles showed us that the center of the surface needs to be removed in order to obtain the doubly curved geometry.
Figure 10: MOLD PIECES READY TO GLUE
Figure 6: ONE MOLD PIECE
We put a fabric (larger than the surface area
Sandpaper
the 2 pieces are joined together and tied to wait for 24 hours. After a day of drying, the mold was ready to use.
Glue
Figure 12: Plywood pieces in the
Figure 11: Tied cube mold
02
GEOMETRY BREAKDOWN
THEORETICAL BACKGROUND The mold is also used for bending the steamed plywood pieces. Some trials are recorded after we put the steamed 0.8 mm and 1.5mm plywood strips into the mold and waited for 24 hours. After the strips are places, the mold is closed and tied together in order to keep it as a cube, as Figure 11 shows.
After 24 hours of waiting, the mold is opened as in Figure 12 and the plywood pieces are taken out. After a while, the pieces did not keep their shape and tended to stretch. This is recorded in Figure 13 and found out that steam bending will not be an accurate solution for generating the doubly curved surface.
12 cm 8,5 cm
Figure 13: Plywood pieces recorded
In order to generate the gyroid geometry out of the unrolled surface, some experiments held with paper. The center of the surface is carved out with the different geometries and the optimal carving geometry has experimented as it is shown in Figure 14.
Figure 14: Paper experimenting on the unrolled surface, 5 different trials
12 cm Figure 18: Laser cut paper trial 4
Figure 15: Laser cut paper trial 1
02
DEVELOPABLE SURFACE
THEORETICAL BACKGROUND
As the experimenting on the paper continued, different patterns are cut. In figures 15-20, the surfaces are diagonally divided into 6 pieces and the inner corners of each 6 pieces are overlapped as shown in Figure 16A. The overlappings created compression on the surfaces and helped the generation of the doubly curved surface. In figures 18-20, the center of the surface is carved circular. However, these trials failed as the circle geometry did not allow the bending behavior.
Figure 19: Laser cut paper trial 5
Figure 16A : Overlapping in the inner corners
Figure 20: Laser cut paper trial 6
Figure 16: Laser cut paper trial 2
16A
Figure 17A : Compressed surface
Figure 17: Laser cut paper trial 3
17A
03
MDF AND PLYWOOD ANALYSIS
KERF TYPES
Cutting on one side The traditional kerf bending technique consists of a series of cuts on one side of the rigid panel to transform it in a flexible material that can be bent, and it looks like a continuous surface on the other side. The distance between the kerfs and their depth will determine panel flexibility and the radius of the bend. We can find the geometric spacing of the cuts so when bent, the inside edges of the cuts join to create the curve. The depth of the cuts and remaining wood thickness for the bend depends on the different kinds of wood, material behaviors, and panel thickness. To bend wood with minimum loss of strength, right kerf depth and spacing could be determined using tests. To avoid stress concentration at the end of the kerfed length caused by the use of constant kerf depth we can vary kerf depth gradually in relation to the stress distribution. Cutting on both sides Advanced methods are developed by traditional kerf bending: cutting on both sides of the board. We are analyzed uniform pattern, cut in the quad grid on each side while ensuring the material doesn’t cut through. The double-sided incision grid makes wooden panels able to be bent and twisted. The incisions cross over and span both sides meaning it can be bent into 3D shapes: it is possible manufacturing a double curvature surface. Cutting through
Figure 21: Kerf Types
One of the main goals of our research is to classify the different kerf bending techniques in relation to geometry and curvature. We have tested some of them to achieve different results, most of which were failed attempts at making a doubly Curved surface. Following a gradual study of optimizing the management and fabrication of curved Surfaces from 2D planar materials, we have chosen a case study: “the Gyroid that is a doubly ruled surface with negative curvature”. The challenge is to find a pattern that enabled bending in two directions and research an optimized solution using parametric tools. We are testing some different techniques to manufacture the same surface. We have grouped kerf bending into three, according to the type of cut: spirals, zigzags, and slit. The experimentation is part of a broader research that has involved the study of different kind of “cutting through kerf” in relation to the Gaussian curvature of the designed surfaces. Our tests show that there are some problems to manufacture double curvature using many of the analyzed patterns.
PLYWOOD/MDF CUTTING PATTERNS (4mm)
Figure 25: 1/6 th OF THE SURFACE
Figure 22: DASHED PATTERN
03
MDF AND PLYWOOD ANALYSIS
MDF, the unrolled geometry broke down to 6 pieces symmetrically. So, some pat-
23A Figure 23: ZIG-ZAG (HUMID)
For further analysis to develop the doubly curved surface from Plywood and
terns and carving techniques are tried out on the 1/6th of the surface as shown in Figure 25.
Figure 23A: Overlapping holes
4 mm Plywood is used in the Figure 22, however, the material thickness and
the pattern did not work out to increase the ability to bend. The surface remained the same, as rigid. In the Figures 23 and 24, the tests are held on the 4 mm MDF, and the overlapping geometry is carved out on the edges, as shown in Figure 23A. Since
Figure 24: ZIG-ZAG/ DASHED
the material was stiff, the overlapping technique failed and the pieces broke while trying. In Figure 24, it was the best option in terms of bending, compared to Figures 22 and 23. However, Figure 18 was only able to bend in 1 direction. So, some other carving techniques and patterns required to be explored to develop the bending capacity and ability.
04
HOW TO IMPROVE BENDING PROPERTY OF MDF AND
PATTERN SAMPLES ON MDF
PLYWOOD? 4 cm Figure 26A 4 cm Figure 26E
Figure 26B
Figure 26: MDF PATTERN SAMPLES SAMPLES / 7 TRIALS
Figure 26F
Figure 26C
Figure 26G
Figure 26D
04
HOW TO IMPROVE BENDING PROPERTY OF MDF AND
PATTERN SAMPLES ON PLYWOOD
PLYWOOD?
Figure32: FINAL PATTERN/ PLYWOOD ON 1.5mm PLYWOOD
Figure 28: 5mm hexagon edge Figure30: HEXAGONAL
Figure 31: SQUARE SPIRAL ON 1.5mm PLYWOOD
Figure 27: 10mm hexagon edge Figure 29: 3mm hexagon edge
PATTERN SAMPLES ON MDF
In Figure 26, 7 different pattern samples are produced. Most of them performed in 1 way of bending. Figure 26B was able to bend in 2 directions, however, due to the rectangular geometry, its bending capacity was limited.
After the trial of the hexagonal pattern in plywood, in Figure 31 a different pattern
In Figures 27-29 hexagonal pattern is tested. The hexagonal pattern allowed the bending in 2 directions and peformed better than the pattern 26B. So, different pattern scales are also tested in MDF. When the pattern scale gets smaller, as in the Figure29, the MDF started to be broken.
curred. As the spiral pattern allows the extraction of the surfaces during bending,
However, MDF material tends to brake in most of the bending trials, so the same pattern tried on Plywood as well as in Figure 30.
is applied. The square spiral pattern was the best option that has been explored. In the Figure32, the plywood surfaces are popping out when the bending is octhe pattern was the optimal choice for our geometry. After this pattern trial on a rectangular surface, it is also applied to our unrolled geometry.
Figure33: Plywood layering
04
HOW TO IMPROVE BENDING PROPERTY OF MDF AND PLYWOOD?
WOOD LAYERING Plywood is a material manufactured from thin layers or “plies” of wood veneer that are glued together with adjacent layers having their wood grain rotated up to 90 degrees to one another. All plywoods bind resin and wood fiber sheets (cellulose cells are long, strong and thin) to form a composite material. A typical plywood panel has face veneers of a higher grade than the core veneers. The principal function of the core layers is to increase the separation between the outer layers where the bending stresses are highest, thus increasing the panel’s resistance to bending. As a result, thicker panels can span greater distances under the same loads. In bending, the maximum stress occurs in the outermost layers, one in tension, the other in compression. Bending stress decreases from the maximum at the face layers to nearly zero at the central layer. Shear stress, by contrast, is higher in the center of the panel, and at the outer fibers. Kerfing helps in reducing the panels resistance to bending based on the kerfing pattern applied, thickness of the ply and material removed during subtractive manufacturing. As the majority of the tension occurs in the outermost layers, the method explored by us makes the material bendable by increasing the number of edges. Since the material is not homogeneous in one direction it is less susceptible to shear. The goal of this research is to explore the possibility of applying these principles to achieve a double-curved wooden panel.
Figure34: Efficiency Graph
Based on our tests the graph illustrates that of the thicknesses we selected 0.8 mm, 1.5 mm, 3 mm and 5 mm. Only the 0.8mm and 1.5 mm could easily be bent with kerfing.
05
PATTERN DESIGN FOR GEOMETRY
Figure35: System Logic Step 1
Figure36: System Logic Step 2
SYSTEM LOGIC
The system logic reflects upon the method involved in subdividing our unrolled ge-
The intersection between the between middle offset and the extended curve serve as the
ometry, which eventually translates into a cut logic.
starting point for the next extension. This point is extended on the long path to the outermost offset to create another intersection and process is repeated to create the final ex-
First, the surface is translated into a radial quad grid and then separated into two
tension. The same is repeated in the even grid but the beginning sequence is in the vertical
lists of odd and even in an alternating manner. For the first case, the odd sequence,
axis rather than a horizontal one.
grids are offset thrice and the innermost quad serves as the starting point of the curve lines to be created. The horizontal edges from this are extended to the next
The system logic ensures the surface remains connected throughout the unrolled
offset in one direction, on opposite sides.
surface and the cut pattern allows tolerance when the surface is propagated into a double curved geometry.
05
PATTERN DESIGN FOR GEOMETRY
Figure37: Pseudo Code
PSEUDO CODE
Pseudo code 1 The pseudo code begins with the unroll of the input local geometry. Since the unrolled surface presented itself to be radially symmetrical, it made it easier to isolate one part. The isolated section can then be subdivided by a radial quad grid. The grid is further offset thrice and then separated into two lists of odd and even in an alternating manner, for the system logic to be applied. Finally, based on the data collected from physical tests and digital models, regions that may prevent the surface to develop into its complex geometry are removed.
05
PATTERN DESIGN FOR GEOMETRY
Figure 38: Surface Pattern 1
Figure 39: Surface Pattern 2
Figure 40: Surface Pattern 3
Figure 41: Surface Pattern 4
As the pattern is selected in Figure 32, it is applied to the 1.5 mm plywood as a next step. In Figure 38, the pattern is cut in the whole surface, without any extraction except the central circle. As it is seen in the figure, some central patterns which are in smaller pattern scale burned while laser cutting, and also some diagonal parts burned as well.
Figure 40 was the carved out version of Figue 39 to test the bending capacity with solid diagonals, however, it did not work properly.
In Figure 39, in order to avoid the burning, diagonal solid surfaces are added and tested. The solid parts were limiting the bending capacity of the surface in terms of creating 6 different curvatures.
In Figure 41, we tried out without the solid diagonal surfaces and it worked well in terms of creating the doubly curved surfaces that are desired. In Prototype 1, Figure 41 will be tested.
06
KARAMBA ANALYSIS
SUPPORT ANALYSIS As the pattern geometry and strategy is determined in Figure 41, we also had to
Figure 42: SUPPORTED FROM GROUND
analyze the support locations. In the Karamba analysis, the surface is tested with different support options.
In the first trial, supports are only located in the ground and the surface is supported from 5 locations in Figure 42. The color analysis showed that the upper parts also should be supported as they turned out in purple color.
Figure 43: GROUND & TOP SUPPORTS
In Figure 43, the supports also added to the upper corners and the analysis only showed that the central parts of the surface are weak.
In Figure 44, when the support is added to the center and also the remaining edges, the color analysis was not giving any purple surface. So that the doubly curved surface needs to be supported from as many corners possible.
So, in Protoype 1, the surface of Figure 41 will be supported according to the analy-
Figure 44: CORNER & CENTER SUPPORTS
sis we get from Caramba. However, since the central part of the surface is carved out, there will be no need for support in the center.
Central support
07
PROTOTYPES
PROTOTYPE 1 In Prototype 1, the pattern is cut on the 1.5mm Plywood surface and joined the edges of MDF frames with zip ties. In terms of bending, the surface performed well and it was able to bend towards the 6 edges of the frame. However, the inner patterns cracked after the bending as shown in Figures 46 and 47. The reason why is, first the scale of the patterns was too small and during the laser cut, the plywood surfaces burned in some places, thus made the surface weaker. The second reason is, as the surface took its final shape after joining to the frame, the inner surfaces twisted and overlapped. So, due to these 2 reasons, the Prototype 1 was decided not to be the
Figure 47: CRACKS ON CORNERS
Figure 46: CRACKS ON CORNERS
Figure 45: Prototype 1
optimal choice.
07
PROTOTYPES
PROTOTYPE 2
Prototype 2, was also is cut on the 1.5mm Plywood surface and joined the edges of MDF frames with zip ties. As the inner raw of patterns failed in Prototype 2, we decided to extract the 3 raws of patterns from the center and test it. After we joined the surface to the frame, we observed that there occur no cracks. So, the final model decided to be produced with this surface of the prototype, as the
Figure 49: FINAL PATTERN
Figure 48: Prototype 2
pattern drawing shown in Figure 49.
Figure 50: FINAL MODEL
Figure 52: ARCH FRAMES ASEMBLY
Figure 51: CORNER JOINT DETAIL
Figure 53: MATERIAL REDUCTION LIMIT
08 FABRICATION FINAL PHYSICAL MODEL
08
FABRICATION
The flexible kerfed plywood needs to be assembled into a global geometry, thereby shaping the local geometry by means of a mold. The global geometry is subdivided into smaller local geometries, where the division curves, serve as the supports for the kerfed plywood to achieve its doubly curved surface. The beams that are translated from the division curves are held in place by 3D printed joinery. The combination of subtractive and additive manufacturing, with zip ties holding the kerfed member in tension, results in the formation of a Gyroid.
PSEUDO CODE
08
FABRICATION
FABRICATION ELEMENTS
FINAL MODEL 8 SURFACES/ 1.5mm PLYWOOD 36 ARCH FRAMES/ 6mm PLYWOOD 252 ZIP TIES/ 7 EACH FRAME 24 CORNER JOINTS/ PLA 1 CENTER JOINT/ PLA 1 BOLT
Figure 56: Digital model elevation
Figure 56: Digital model isometric
Figure 57: Digital model perspective
Figure 55: FINAL MODEL For the fabrication, 8 surfaces of 1.5mm plywood are joined with a central grey PLA joinery and 6mm plywood arch frames. On the corners, we used differently typed of joints with white PLA (that will be evaluated in the conclusion). In order to attach the surfaces into the frames, white plastic zip ties are used.
08
JOINERY
FABRICATION
Center joint is designed to hold together the 6 central arch frames that are all joining in different angles. The joined is produced in 3d printer with gray PLA and
Figure 61: CENTER JOINT
Figure 59: CENTER JOINT
stuck together with 2mm bolts.
Corner joints are designed to join 2, 3 or 4 arch frames depend on the location. They work in the same logic, as they are joining the frames with a centralized sphere. The
Figure 62: CORNER JOINTS
Figure 60: CORNER JOINTS
joints are produced in 3d printer with white PLA and attached to the frames.
08
MOULD / 8 SURFACES
FABRICATION
In the fabrication process, first, the plywood arch frames are joined together with
FIGURE 64: PERSPECTIVE
FIGURE 63: ISOMETRIC
PLA joints. 36 arch frames are joined with 24 corners and 1 central joint.
For the 8 plywood surfaces, they are all put in the water 48 hours before the assembly. Every 12 hours, the surfaces are checked if they were losing their stiffness or if they are still malleable and not getting rotten. So, after 48 hours, they are taken out of the water and dried out for 4 hours. After the drying process, the surfaces were ready to assemble.
CORNER HOLE
Each surface attached to the frames one by one, with 2 people assistance, as the surfaces were still humid, they tended to broke easily. For a surface, first, the middle points are ties to the mold one by one for 6 edges of the surface. After the middle points, the corners are zipped to the frames followed by the remaining points. The
FIGURE 65: SURFACE GEOMETRY & PATTERN
MIDDLE HOLE
zip ties are attached loosely at first stage, as soon as all the edges attached, the ties are tightened and the assembly completed.
09 09A
CONCLUSION
Figure 66: Alternative geometries generated from gyroid
At first, the project started as exporting the gyroid geometry and the
whole focus was to generate the doubly curved gyroid surface. As we progressed in the project, we focused to the idea of the gyroid geometry and only explored this option. We could have expanded our exploration range for geometry and implied what we learned from the gyroid and tried different options as well. The options different scales or to stretch the geometry as shown in Figure 66.
09B
Existing corner joint
As we assembled the physical model, we realized that the corner joints
Existing center joint
were not suitable for the overall geometry. Although they were functioning well in terms of joining the arch frames properly, we could have designed a better corner joint, as we did for the central joint. The central joint could have been copied to the corners as well, also for emphasizing that the further modules can be attached to our module.
New corner joints
Figure 67: Alternative corner joints
could have been generated by scaling the gyroid geometry and joining them in
Another advantage of the central joint was the slits in the middle. As the plywood surfaces are attached to the frames, the corners were inserting the slits, so that the assembly will fit properly.
Slits to insert surfaces
Figure 68: Center joint
Bolt
09 09C
CONCLUSION
Figure 69: Inner surface
As the surface geometry is designed in Figure 41 and the assembled
state is shown in Figure 53, in the central parts there occurs some foldings and these
Parts to be controlled
are uncontrolled geometries. Even though we can call them “emergent geometries� in the center and keep them as they are, as a further development, they can also be controlled in terms of introducing some tension cables or fixing them with additional frames.
09D
The Karamba Analysis is applied on the surface without the cutting pat-
terns. So, as the surface in the analysis did not have the carved out patterns, it was not so accurate for the final model. Additionally, the arch frames could have been structurally tested as well. As a further analysis, the frame and the tensile analysis can be done as follows:
Frame
Tensile
The global geometry is subdivided by curves resulting in one subdivided surface
The local geometry as a result of our kerfing method applied appears to have negli-
that is repeated throughout. Creating Beam Elements from subdividing curves utili-
gible compressive and flexural strength. The behavior of the material is like the case
zing a rectangular cross-section, would then be taken forward for the location of its
of membrane structures, hence we removed regions of kinks in the center – to avoid
supports and material assignment. A series of points will be defined in the yz-plane
wrinkles. To further recreate this behavior on or plywood we could try to reduce the
corresponding to the module, as endpoints of the beam elements and will be sup-
shell thickness so that bending effects get neglectable and define the supports, to
ported by joints.
get negligible compression forces. If we set G=0.5E the material results with a lateral expansion factor of zero, this would correspond to our desired behavior.
09 09E
CONCLUSION
As a further development for the pattern, we realized that in the square
spiral pattern, there occur some shear forces through the majority (%77 overall) of the cuts within the subdivision as shown in Figure 70. So, we would take a pattern that prevents the shear propagating through the cut surface better compared to the square pattern. In a circular pattern (Figure 71), as the sheer will propagate to the center directly, this could be a better option. However, a circular pattern may have some potential problems in applying it to our geometry. Furthermore, hexagonal spiral pattern (Figure 72) can also have a potential to experiment and it may be easier to apply in our surface, as the surface also has 6 edges and the whole surface performs as 6 different surfaces.
Figure 71: Circular spiral grid
Figure 70: Square spiral grid
Figure 72: Hexagonal spiral grid
10
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