Why is it so hard to build freeform shapes? Mainly because most fabrication machines are meant to process flat material stock, and the complexity increases exponentially as soon as we move from 2D to 3D. Money is also a big issue. To quote a very interesting Instructable recently featured on this site:
In a small book, Digital Gehry - Material Resistance, Digital Construction (Bruce Lindsey, Bitkhäuser 2001), [the author] writes that if a flat piece costs one dollar, single curvature pieces cost two dollars, double curvature pieces cost ten dollars. This is true for all materials wood, metal, plastic, glass, concrete...
Nonetheless, double curved shells are very attractive for engineers because they maximize the stiffness of the structure without adding extra material to it. It's geometry that does all the work. Besides, modelling software are making it increasingly easy to create complicated shapes, and new ways to build efficiently are required to meet the more demanding designs.
In this Instructable I'll be going through the process of designing and building a full-scale bending-active pavilion. Bending-active structures exploit the large deformation of materials to achieve shape and stiffness, often in counterintuitive but fascinating ways. Along with the fabrication steps I'll also provide some rules of thumb to better understand the behaviour of your materials and be able to quickly estimate the possibilities and the limits of the design. There is no better tool than your own judgement and experience to make the correct decisions when building large-scale installations (well, a Finite Element model can also be very useful, but this is for another Instructable).
Step 1: Creating the Global Design
The global design can be developed in many different ways and on many different platforms. For this project I used Fusion 360 and specifically the sculpt tools, which proved to be very intuitive and easy to use. I first started with a lofted surface generated with four rail curves. This provided me a first draft surface which I could later modify to accommodate specific requirements like the height and openings. The T-splines integration in Fusion makes it really easy to modify freeform surfaces, and with a little effort I could get a good result by using the modification handles that you see in the picture.
I chose this particular shape because I wanted to display the full potential of this system. Double-curved surfaces are notoriously hard to build, and transitioning between areas of different curvature (from the right dome-shaped support to the saddle of the left support) proves to be even more complex. Nonetheless I was up for the challenge.
Step 2: Model Validation
As this approach has the potential to be applied to any freeform shape, the design constraints are dictated only by the mechanical properties of the material you are employing to build the piece. During the design process it is important to keep in mind that materials can break if overstressed, which is something we definitely want to avoid. A good rule of thumb to check if the bending of the pieces exceeds their structural capacity is condensed in the following formulas:
M = E*I*kappa, where M is the bending moment in the strip, E is the material's elastic modulus, I the second moment of inertia of the strip and kappa the curvature.
If we know the moment we can easily calculate the stress through the following relationship:
sigma = M/W, where sigma is the maximum stress in the top or bottom fibres of the strip and W is bh^2/6 for a rectangular section like in our case, where b is the length and h the height of the section.
These formulas relate the stress arising from the bending action to the radius of curvature. You need to make sure that the stress sigma in the areas with the tightest curvature does not exceed the ultimate strength of the material. In my case I used a 3 mm (1/8") birch plywood to be able to achieve a very tight radius of curvature, as plywood is a very flexible material and it can withstand large deformations before failing. From the model it is clear that the area of highest curvature is where the transition between positive and negative curvature is happening, which corresponds to a radius of curvature of 0.17 m. Breaking down the calculation:
E = 6800 N/mm^2
I = (400 mm x (3 mm)^3) / 12 = 900 mm^4
kappa = 5.56 1/m
M = 6800 N/mm^2 * 900 mm^4 * 0.00556 1/mm = 34000 N*mm
We can now calculate the stress from the bending action occurring in the critical area:
sigma = 34000 N*mm / 600 mm^3 = 57 N/mm^2
Considering that the bending strength of 3mm plywood sheets is roughly 60 N/mm^2 (source), we are under the maximum utilization of the material. That is a relief! Having validated our initial design, we can now proceed to the next step.
Step 3: Mesh Generation
This step condenses all the essence of the project. By placing voids in the right spots we allow the material to bend in different directions, thus giving us the opportunity to cover all the design surface with our flat components. This would not be possible if we were using full sheets, as trying to bend the material in two directions would automatically lead to very high stress concentrations and ultimately failure of the pieces.
From the surface model we need to first generate a discrete mesh version of it. For this step I've exported my Fusion 360 model to IGES and imported it in Rhino where it was also meshed.
Once the mesh topology and size of the triangular elements is satisfactory (remember that if the elements are too small they will be very difficult to bend!), I generated the components by offsetting each triangle to its interior and reconnecting these with quad elements. This way we end up having a mesh with mixed elements made of triangles (the centre pieces) and quads spanning in between. Once assembled the triangles will remain flat, whilst the quads will take up all the bending.
Step 4: Planning the Double Layer and Component Logic
To be able to bend the material, the sheets have to be thin enough. To withstand loads, the material needs a certain thickness to it. What a contradiction! To overcome this problem I had to plan a second layer to increase the stiffness of the structure once installed. It's the same principal an I-beam or composite structures employ to better resist external loading. The two external layers withstand tension and compression, and the connection between the layers ensures that these work together. The transition area between synclastic and anticlastic curvature is the weakest spot of the whole construction. I therefore decided to adopt a variable offset which increases in this area and gradually tapers off when reaching the footpoints. In this way we have a long lever arm in the weak spot that ensure enough structural stability.
Each component is designed to overlap with its neighbour to be able to set the screws that keep them together. In the image it can be seen how this logic works: red is top, green is middle, blue is bottom.
Step 5: Lasercut Components
Once the components have been unrolled they can be nested to minimize material waste. For this operation I used Aspire, a CNC related software available at Pier 9 which integrates a very good nesting algorithm.
All the information about the numbering, orientation and layer position of the components have to be etched on the plate. This minimises the chances of committing errors during the assembly process. The more information you can embed on the component itself, the easier it will be when putting everything together. The components were cut on the Coherent Metabeam lasercutter available at Pier 9, as the cutting envelope reaches 4 ft. x 4 ft. This allowed me to use larger sheets, cutting down fabrication time and material waste. 1/8" plywood sheets are extremely flexible, and they never come perfectly flat. In their natural state these sheets are very wavy, and this might cause potential trouble when lasercutting. To avoid this I've added lateral plywood tabs hold in place by socket cap screws that bolt directly into the Metabeam's metal edge profile.
Step 6: Miters and Bevels
The way I decided to solve the connection between the two layers was to cut these pieces from a 1 3/4" wood square profile and screws them directly through the plates with wood screws. Having decided to vary the layer distance for structural reasons, I've ended up having non-parallel faces on the connectors. This made for an interesting fabrication case, as compound miters can be tricky, especially if you have to cut more than 200 of them.
Having screwed up a full bunch of connectors on the chop-saw because all I had was a badly hand-written spreadsheet, I decided to tackle this problem more methodically. On the Metabeam I etched all the angle information I needed for the miter and bevel cuts of the pieces. In this way I minimized the risk of error (just like with the information on the plates), and managed to get 200 cuts in a record time of 3 hours. Once you have everything written under you nose, it becomes much more difficult to screw up!
Step 7: Assembly
For the assembly I had some friends helping me out. The size of the piece required a couple of extra hands in order to align all the panels and get the pieces bolted together. Where the curvature becomes very high, aligning the screw holes can require a substantial amount of force.
The whole piece was divided in three chunks, the two footpoints and the central keystone. After securing the left footpoint on the ground we lifted the keystone with the forklift and secured the hanging components together. We finally proceeded to install the right support, and by the time all the pieces were installed the pavilion was standing on its own feet :) Assembling the structure on site took us about 8 hours, from bringing the pieces on site, securing them to the ground, forklifting and attaching all the pieces together.
For those of you interested in the details of the project I'm also attaching here all the nested components in .dxf format as exported from Aspire, as well as the .dxf files for the connectors with all the information embedded on them.
Step 8: The Pavilion
The pavilion measures 5.0 m x 2.8 m (16.4 ft x 9.2 ft) and is 3.0 m (10.0 ft) high at its highest point. The whole structure was entirely lasercut out of fifty 1.2 m x 1.2 m (4 ft x 4 ft) sheets of 3 mm (1/8") birch plywood and weighs only 160 kg (360 lbs). Production time was reduced to ten hours of lasercutting for the panels and three hours of cutting the spacers on the chopsaw + one hour of data etching for a total fabrication time of 14 hours. For each section of the pavilion it took one day and three people to build, so three days + one day of on-site assembly = four days for assembly.
The smooth transition between positive and negative Gaussian curvature of the pavilion showcases the potential of this system to perfectly adapt to complex geometries. By carefully analysing the initial shape in the early stages of design, the pavilion could be successfully built by exploiting the deformation properties of the material, and still retain enough structural capacity to safely load the structure. The overall weight and speed of execution clearly display the versatility and potential for lightweight construction of this system, opening up new possibilities for large scale installations.
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