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I am considering using a sliding dovetail joint (without glue) between two timber pieces: a vertical wall with a slot and a horizontal shelf with a tail. The thickness of the shelf is $h_1 = 18mm$, and the thickness of the wall is $h_2 = 25mm$ or $h_2 = 18mm$. The material is a scots pine glued panel (density $510\ kg/m^3$). The width of the shelf is $w = 600mm$. The shelf grain is directed towards the wall (horizontally, parallel to the force $F_2$). The wall grain is directed vertically (parallel to the force $F_1$). The depth of the tail penetration is $d$, and its bottleneck is $b$.

What is the optimal angle of the dovetail, and what forces can it theoretically withstand? For simplicity, we may consider the case when $F_2$ and $M_1$ are negligibly small and $F_1$ is the subject, and the case when $F_1$ and $M_1$ are negligibly small and $F_2$ is the subject. Also, I would like to know the dependency between the maximum momentum $M_1$ and depth $d$ and bottleneck width $b$.

My current understanding is as follows:

  • $F_2$ will fight the shear strength of the shelf parallel to the grain in the case of small $b$ and the shear strength of the wall perpendicular to the grain in the case of small $d$
  • Analogously, $F_1$ will fight the shear strength of the shelf perpendicular to the grain if $b$ is small and the shear strength of the wall parallel to the grain if $b$ is significant.
  • $M_1$ will compete with the tensile strength of the wall parallel to the grain if $d \approx h_2$, with the bending strength of the shelf if $b$ is small, or with the bending strength of the wall if $d$ is small (using the tail end as a lever).

However, I am unsure whether it is right and how to apply the strength values to the dovetail's geometry to calculate ultimate and design stresses.

Sliding dovetail drawing

Charlie
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  • Probably better asked on a carpentry or woodwork stack. – Solar Mike Sep 11 '24 at 08:11
  • @SolarMike, as far as I understand, those forums are dedicated to practical questions about woodworking, like how to make a dovetail or what angle somebody uses/suggests. However, I am interested in the mathematical insight on the question, as I see an optimisation problem here, and the dovetail shape is not so complicated for integration — the problem for me is to know what to integrate – Charlie Sep 11 '24 at 08:48
  • So how old is the timber, aka what is the grain spacing as that affects the strength, looking at old pine it appears to be more dense than the new stuff on sale now. That will affect the timber strength and therefore the angles possible. – Solar Mike Sep 11 '24 at 08:50
  • @SolarMike, the supplier did not provide me with such information. However, once I order the panels, I can conduct my experiments to obtain the values required for the calculations. I thought that the glue quality, grain spacing, and other parameters are already incorporated into strength and thus can be implicitly considered during calculations, so I do not need to assess them separately. – Charlie Sep 11 '24 at 10:14
  • You'll just have to assume a shear strength between grains, ideal grain direction, failure mode being somewhere on the dovetail, and see what the math gives you. Consider only the dovetail joint, make the two members sufficiently large, and one of the members fully fixed at one or both ends that are sufficiently far from the joint. – Abel Sep 11 '24 at 23:29
  • @Abel, yes, you rephrased my question correctly. I would not have asked it if I had known for sure how to apply math to the sliding dovetail situation. Also, I believe the $M_1$ momentum will compete with bending and tensile strengths, not with shear --- could you explain why you think I should assume only shear strength? – Charlie Sep 12 '24 at 09:39
  • Failure tends to occur in shear, specifically a piece of dovetail shears off between grains, unless you made b small, in which case, that would fail first. – Abel Sep 13 '24 at 00:06

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