During a makerday, I saw a someone struggling to put together a Raspberry Pi box that they'd laser cut, and were trying to hold together:

- two pieces of acrylic to make the joint,
- and the Raspberry Pi board,
- and a nut into a slot,
- while also attempting to screw in a bolt,
- and not drop everything.

So I started thinking about if it would be possible to extend the work I have been doing on making flexible areas in acrylic to make a clip mechanism that could be laser cut to make self-fixing comb-jointed parts.

A **comb joint** (also called a **finger joint**) is a carpentry joint that has come to the attention of makers with the increasing accessibility of laser cutting. The finger joint can be used to make boxes from laser cut sheet materials (there are a couple of automated tools available including BoxMaker and Box-o-Tron). It is relatively easy to implement and the fitting of the crenelations gives much better alignment of the joining parts than butting together the edges of the adjoining sheets (this is a **butt joint**). Because "complexity is included" with laser cutting, the cutting effort needed for a comb-jointed box is not much more than for a straight-edged butt-jointed box.

However, to stay as one piece, the comb joints must either be friction fitted together or glued. For both these methods, the mechanical strength of the joint is limited by the relatively small area of the mating surfaces. To give the joint greater strength, one set of combs can be closed, so it is mechanically supported in 2 directions (which then makes it a **mortice-and-tenon joint**), and if a captive T-slotted nut and bolt is also included (this effectively "closes" the other set of combs — making a **bolted mortice-and-tenon joint**) the joint is well supported against movement in all directions planar to the component plates. Variations on the t-slot and nut exist, such as the using delrin clips, and while they are very robust, they all require hardware in addition to the laser cut acrylic.

*If you're interested in seeing more possibilities for laser cut joints, MSRaynsford has shown a good selection of posibilities.*

Flexible plastic clips are a staple of contemporary product design, just look at your phone; there's almost certainly a version of a moulded plastic clip that hold the parts of the case together. If you've got an older/non-smart phone, then it's quite possible that you also have moulded button for the back panel with a living (elactically deforming) hinge in. Having already demonstrated that it is possible to make acrylic flexible with a lattice cut living hinge, I investigated how to cut acrylic into an elastic clip that could be used to secure a comb joint.

## Elastic Clip Geometry

I've been calling this an elastic clip because of it's structural properties. To operate successfully, the material must only be operating in the elastic-region of it's stress/strain capabilities. Under *elastic deformation*, once any force is removed, the material will return to it's original shape. If the yield stress of the material is exceeded, it enters *plastic deformation* where there will be a permanent change in the shape of the structure after all force has been removed; because the applied force was great enough to start permanently re-aligning the molecules that make up the material. For a brittle material, such as acrylic, the difference between the yield stress and ultimate stress (the absolute maximum it can sustain before it breaks) is very small, so for a clip to stand repeated use, the maximum stress in operation needs to stay well away from the ultimate stress of the material.

Any kind of integrated clip is going to take the form of a cantilevered beam, where the operation of the clip bends the beam along its length, until the clip is "open". Having a back stop will limit the size of the maximum deformation of the clip, and therefore the maximum stress it will experience, so the limit of motion gives a starting point for calculating maximum operating stress.

For a straight beam, the maximum bending moment at the root of a cantilevered beam with a point load at the tip is given by: \[ M_{max} = Fl \]

Material stress due to bending is also greatest at the root, at the surface of the material, where: \[ \sigma_{max} = \frac{My}{I} \]

where \(M\) is moment, \(F\) is the applied force at the tip \(l\) is the length of the beam, \(y\) is half the depth of the beam (in this case \(y=a/2\)) and \(I\) is the *second moment of area*, which describes the effect that cross-section has on the bending.

Maximum deflection occurs at the tip, and for a constant thickness, constant depth beam is given by: \[ d_{max} = \frac{Fl^3}{3EI} = \frac{4Fl^3}{Eta^3} \]

or to calculate the force required to deflect the tip by a set distance: \[ F = \frac{dEta^3}{4l^3} \]

where \(d\) is deflection, \(E\) is the *Young's modulus* of the material, \(t\) is beam thickness and \(a\) is the depth of the beam.

If the deflection required to use the clip is known, then the force required to operate the clip is a function of \(I\) and \(l\), where both of these are from the geometry of the clip, and the maximum bending moment is a function of \(F\) and \(l\). Initial calculations showed that for acrylic, the clips needed to be both long and thin to stay under the material stress limits and keep the operating force low enough to be practical.

The equations above only hold if the cantilevered clip is uniform in thickness along its length though. If the width of the clip is allowed to vary along it's length, the root can be made thicker to minimise the stress due to bending, and the tip can be thinner to minimise to operating force.

or a tapering beam, the second moment of area varies along the length, but for a linearly changing rectangular area, the tip deflection is given by: \[ d_{max} = \frac{4Fl^3}{Et(a-b)^3} \]

Therefore, operating force is given by: \[ F = \frac{dEt(a-b)^3}{4l^3} \]

And the maximum stress at the root is given by: \[ \sigma_{max} = \frac{6Fl}{ta^2} \]

where \(a\) is the root depth and \(b\) is the tip depth.

So by selecting appropriate values for \(a\) \(b\) and \(l\), it's possible to tune the geometry to the requirements of a specific material.