Tusked through tenons – an engineers take.

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I’m an engineer. Despite that, I’m usually fine with following the proportions which most woodworkers use to lay out a joint. They are tried and true and (as an engineer) mostly seem reasonable to me. And I don’t have the woodworking experience to (generally) stray too far from whatever more experienced woodworkers suggest.

But I’m in the midst of laying out a tusked tenon joint for a dining table, and haven’t been able to find those sort of guidelines. And it’s for a critical joint, holding the legs onto my dining table. And I subscribe to Chris Schwartz’s attitude towards joinery – I want this to last several lifetimes. So I started digging into it.

For whatever it’s worth, here’s what I found. I’m going to get to proportions at the end, so you may wish to skip to there. (I’m writing it up so I can find it again too, but I do hope it’s helpful to others.)

The problem

Tusk tenon layouts don’t seem to be discussed much. Googling turned up a handful of useful and not useful links. This joint has been used in some very demanding places, e.g. the Moravian workbench, so if laid out correctly it’ll do.

But what are the right proportions? Especially, if the tenon isn’t long enough to house the key, it’ll split. And all I can find for that length is the advice to “make it long”.

H.Fukuyama et. al. wote a paper, aimed at single ended dowels in softer woods (cedar, etc). So, not really applicable to a pin (dowel) running through the tenon, as is typical in the drawbored or pinned tenon used by furniture makers (a.k.a. double shear). But I think his work is adaptable to the tusk tenon.

The typical loading – and certainly the one I am interested in – for the tusked tenon would be a racking load, as in when someone is dragging a table about. And in that case, the wedge is effectively in single shear, as in Fukuyama’s paper. That’s precisely the loading that I did not find any other usable papers on.

Lets think about this

H. Fukuyama et. al. include a thorough discussion of failure modes – compression of one member, compression of the other, shear of the pin, etc.  I found his paper here: http://support.sbcindustry.com/Archive/2008/june/Paper_167.pdf?PHPSESSID=ju29kfh90oviu5o371pv47cgf3

(Disclaimer – I sure hope it was intended to be published to the public. I’m assuming it was and hence my use is fair use.)

Here’s an image showing the failures his group considered.

Screenshot from 2017-12-03 11-49-40

Failures where the dowel is embedded in one member or the other without deflecting (modes 1-3), and the dowel failing without damage to the members (mode 7), seem unlikely in this case. Those would occur when one member is substantially softer than the other. The in-between modes (4,5 and 6) seem likely in the current case (everything hardwood).

For a piece of furniture, I’m not interested in ultimate load. I am interested in any compression or distortion of the mortised or tenoned member, and any significant distortion of the wedge. Ultimate loads are higher, but I don’t want the joint loosening over time either.

How much load will the joint see? This is for a table with aprons under the top, not a stretcher lower on the leg. If someone lifts one end and drags the table, then the joint would be about 30 inches above the floor.  If we assume a grippy, carpeted surface, we might apply as much as 0.3*120*30=1080 inch pounds of torque to the joint. (120 pounds is about 1/2 the weight of this table). So let us aim for a joint which can handle, say, 1500 inch pounds. That ought be pretty bomb-proof. (A stretcher might be 12 inches above the floor, but would also carry more of the tables weight. So 2000 in-lbs should be OK for it, too.)

Now, I am ignoring the ability of the tenon to bear against the mortise in the leg – that’s deliberate. It means a well fit joint is actually a bit stronger than what I’m calculating – and that if the fit is a bit loose, it’s will still hold up. More importantly, it means driving the wedge into the joint is unlikely to split or dent the members.

One side of the tenoned rail is pressed against the leg, and the other tries to pull out of the leg, resisted by the wedge (in shear). The compressed shoulder of the tenon bears against the leg, so it needs a bit of area to do that – it’s not just cosmetic. Then again, the height between the compressed shoulder of the tenon and the shearing side of the wedge should be substantial too. That reduces the load by spreading it out.  Similarly, we want the wedge to have some bearing area on the leg and on the wall of it’s own mortise – so it should extend a fair ways beyond the tenon, perhaps thrice it’s width. And mortise wall  shouldn’t be less than that either.

As a first estimate, cut the wedge mortise to be, say, 1/3rd or 1/2 the thickness of the tenon. The wedge should be somewhat square-ish where is exits the tenon underside, and extend a fair ways beyond the tenon. The tenon will probably be about 3 times taller than it’s width, and the uppermost shoulder of the tenoned member would be another, say, 1/2 wedge thickness all around. And assume the leg is simply big, in all dimensions.

At a guess, the tenon should extend perhaps three times the wedge thickness beyond the wedge mortise. This dimension will be key to ensuring the joint holds up. (And it’s the one which I haven’t found much advice about.)

So if the wedge width is T, then our guesses lead us to something like :

  • The wedge is TxT wide and thick at the exit through the tenon. It extends about 3T beyond the tenon.
  • The tenon is 3T x 9T, extending about 3 T beyond the wedge.
  • The shoulder of the tenoned member is more like 4T x 10T or 6T x 10 T.

Time to solve for T.

Mr. Fukuyama’s equations

Equation 2 looks a bit forbidding. Time to fire up  Google Sheets.

Some values for the strength of the wood are needed – walnut, in my case. Sources :

Between the two we can find the properties needed.

Here is the spreadsheet : Single Shear Wedge

So, answers, for the first pass geometry, using walnut, and T=.375 inches :

Failure Load Lb N
Bearing failure of Leg Eq 2.1 426.1 1895.4
Bearing Failure of Tenon Eq 2.2 2982.7 13267.5
Mode 3 Eq 2.3 636.6 2831.5
Mode 4 Eq 2.4 299.1 1330.3
Mode 5 Eq 2.5 650.0 2891.2
Mode 6 Eq 2.6 414.2 1842.4
Shear of Wedge Eq 2.7 398.0 1770.3
Shear of long grain in tenon (blowout) 578.0 2570.9
Transverse bearing limit of leg 805.7 4738.4

Well that’s gratifying. It seems the tenon is more than long enough to avoid blowout at 578 vs. 299 (though I will take that with a grain of salt). It’s not at all surprising that the tenon (end grain) is stronger than the leg (transverse grain). As expected, combined shear/bending/partial bearing failure (Mode 4) is limiting (in english : the wedge will bend a bit, and dent the leg below the tenon on the wedge side).

299 pounds gives some 1065 inch pounds of load before failure.  A little more is desired,  but the guesses were close.

Adjust all the proportions a bit, and account for other design constraints. Here’s a final  result:

Failure Load Lb N
Bearing failure of Leg Eq 2.1 568.1 2527.1
Bearing Failure of Tenon Eq 2.2 3976.9 17690.0
Mode 3 Eq 2.3 848.7 3775.4
Mode 4 Eq 2.4 438.9 1952.4
Mode 5 Eq 2.5 884.8 3935.8
Mode 6 Eq 2.6 621.3 2763.6
Shear of Wedge Eq 2.7 597.0 2655.4
Shear of long grain in tenon (blowout) 578.0 2570.9
Transverse bearing limit of leg 751.7 4316.2

Which gives a maximum torque of 1399 lb-inches, and an adequate tenon (578 vs 438 pounds). Proportions chosen :

Proportion Inch Metric
1 Wedge width 0.375 9.53
1.5 Wedge thickness 0.563 14.29
3.333 Width across shoulder 1.250 31.75
9 Height across shoulder 3.375 85.73
2.25 Tenon width 0.844 21.43
8 Tenon height 3.000 76.20
2 Tenon Length beyond wedge mortise 0.750 19.05
4 Wedge length (protrusion) 1.500 38.1

Solving it again, same geometry but using silver maple :

Failure Load Lb N
Bearing failure of Leg Eq 2.1 416.3 1851.6
Bearing Failure of Tenon Eq 2.2 2913.8 12961.0
Mode 3 Eq 2.3 595.2 2647.6
Mode 4 Eq 2.4 321.6 1430.5
Mode 5 Eq 2.5 648.3 2883.7
Mode 6 Eq 2.6 413.6 1840.0
Shear of Wedge Eq 2.7 1336.3 5944.1
Shear of long grain in tenon (blowout) 624.4 2777.4
Transverse bearing limit of leg 710.9 3162.4

Maple is a bit weaker – 321 vs. 438 pounds – which is expected. That gives about 1000 inch-lbs of torque before denting. The proportions and resulting failure are the same, and especially, the tenon is still long enough to not blow out (624 vs 321). In fact, maple is better than walnut for that – 1480 psi vs. 1370 psi, so the tenon could be a touch shorter.

Grain and orientation

Ok, I’ve dug into grain orientation a little more, based on Brian’s thought provoking comments.

The tests are conducted such that the compressive load is applied “to a radial plane”, and the bending load is applied perpendicular to that. That’s buried deep into the ASTM D143 spec covering the tests.

This paper covers the results of some tests, and shows clearly that the stiffness is slightly higher at 90 degrees than at 0, for larch.  This paper is for chestnut, and shows the same behavior – but included strength limit data as well. And this shear test paper includes shear test data. These first three clearly show the compressive strength and stiffness is higher if the forces is against faces which are tangential to the radial grain (and not against a radial face). That said, Bruce Hoadley wrote a book and on page 82 he suggests exactly the opposite, at least for ash and ring porous hardwoods. This blog post would seem to agree. Hankinson’s formula does not consider radial vs. tangential grain orientation.

The US Forest Service addresses this (page 40-30 and 4-31). They suggest that, depending on species, either 0 and 90 degrees are about equal, or that 90 degrees (radial) is a bit stronger.

I’m afraid I haven’t yet found a clear guideline, aside from staying near either 0 or 90 degrees, and avoiding the intermediate orientations. Which is to say, either rings running from shoulder to wedge, which matches my calculation, or perpendicular to that. In either case, the calculation I did should be close.

Proportions and Conclusions

That all leads to some rules of thumb.

  • The wedge is T wide and 1.5 T thick at the exit through the tenon. It extends about 4T beyond the tenon.
  • The tenon is 2T or 3T by at least 8T, and extends at least 2 T beyond the wedge.
  • The shoulder of the tenoned member is more like 3.33T x 9T
  • And for an application like a dining table, T is at least 3/8ths of an inch [9.5mm].

Now, aesthetics may lead to a somewhat different solution, and of course some applications may need less strength. As long as you follow the proportions here, then the wedge should still fail first.

If the wedge is thicker, say 1T x 2T, then the leg will dent a bit more. If it is thinner, as in the first pass, then the wedge is a bit flimsy. A wider wedge would be worth considering, but I don’t have a wider mortising chisel, and the resulting tenon width does not suit the available stock.

I haven’t tested this in practice, so some caution is warranted. It’s all theory. But I think I did it right. Comments, feedback, and someone checking my work is all welcome !

I hope that was helpful for some of you, feel free to ask questions.

That’s all folks!

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After far too long – we finished!

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A few shots of the Mr. Rogers’ bench. It took rather a while but we did finally finish the knit cover.

Here’s the first post regarding this bench : After far too long…

My lovely wife used a brioche stitch, using several colors of Brown Sheep Lambs Pride worsted weight wool.

As for the benchtop, I ended up adding a slight back-bevel to the edges (visible in the first image) and some of the pigmented white oil finish from Woca (http://www.wocadirect.com/master-oil/), which I highly recommend (on ash, anyway). The porous ash picks up the pigment nicely, it doesn’t stink, and it went on nicely. Really nice look, in my opinion.

Coat Rack

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Just a couple quick shots of a new coat rack. Along with one showing why said rack was desperately needed.

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Just a plainsawn walnut plank, with walnut 3/4″ dowels. Nothing fancy. Hidden french cleats to hold it up. Oil finish. 10 degree tilt on the pegs, 5 degree back-bevel on the sides of the plank.

Winding sticks – finished

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I cut a shallow rabbet for the inlay.

I happened to find reasonably priced stripe inlay at my local Woodcraft store.

Trimmed the ends on the shooting board

Chiseled shallow mortises for center markers

I chose to make the markers one chisel width wide, spaced at center, 10 and 20 inches. I’m hoping this spacing will work well for stock widths up to the full width of the sticks.
Once the markers were glued in, I planed it all nicely flush, then trued the upper and lower faces.

Et finis

An example – planing stops

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I’ve discovered that the wooden fences I bought from Veritas could use a little flattening.

It happens to be a good example of why I’ve gone with my somewhat unorthodox arrangement of planing stops.

The fence is quite small, and thin, but nestles nicely into the corner. And the stops are a bit thinner.

The plane didn’t strike the stops, the work sat still. Flattened and reassembled.

Winding sticks – first steps

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I’ve started turning a piece of walnut stock I had on hand into a pair of winding sticks.

I’ll inlay the sticks, a bit like http://www.walkemooretools.com/winding-sticks/, for better visibility.

The first step is simply to plane the stock square, rip it in two, and then re-true it.

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This 2nd image is of one end, with about 80% of the rip cut made. The other end is still joined, and you can see that the two pieces have twisted a bit as I rip them. The half further from the camera has lifted up, by perhaps 1/32nd or so.

Fortunately the two pieces are still reasonably flat after ripping – they didn’t twist too badly. Not as flat as they were, but not bad. I’ll let them sit a while, then take the plane to them again.

Planing Beam – Finished

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(This is the third and final post on this subject. The earlier posts about this planing beam are here :  A bench of sorts, a planing beam of sorts and Planing Beam – First Steps . A related post is here An example – planing stops )

So I’ve finished putting it all together. It seems more than sturdy enough, and is light enough to be movable, in pieces, by me working alone. While I admit to not yet having done much chopping of mortises, it doesn’t shift around easily. And the trestles can be used by themselves, if needed.

So far, so good.

Assembly was pretty simple.
-Chisel out the corners on the through mortises.
-Fit the tenon cheeks to the mortises using a plane.
-Chisel off the corners of the stub tenons.
-Drill, dry fit, mark the hole centers using a transfer punch, then drill the drawbores.
-Assembly was done dry, no glue at all, and the joints all drew up tight and rigid.
-Final trimming and smoothing (much smoothing and chamfering was done as I went – depending on sequence and what was accessible when.)

The “tricky” bit was trimming the upper stub tenons to the top rail – the lower cross rail sets the width between uprights, so the upper stub tenons must match both position and size. The lower stub tenons simply be match the size of their mortises. Not so tricky, just required some care and forethought – clamp the lower cross rail to the top rail, centered, and then use it to position the uprights for trimming.

I didn’t apply any finish, and even left the beam top rough, as delivered – it’s pretty flat. I’m hoping the friction will keep the workpieces from sliding about. I may try something different down the road. I’ll eventually have to flatten the top, and my jointer will leave me with a smoother surface.

The blue painters tape did seem to help avoid chipping. My brad pointed bits did well, but did leave a slightly ragged face without the tape.

img_20161227_163021_sm

At Mark’s suggestion, I ended up using doubled pins for the drawbored mortise and tenons. 3/8ths [9.5mm] oak dowels, with about 1/16th offset [1.5mm]. I used a hand drill with a Big Gator drilling guide – worked well enough. (I did firmly clamp the drilling guide in place.)

I left the through tenons proud – I’d planned on flush, but this looked nice and should be just as functional.

img_20170104_101641_sm

The pins mate to holes in the underside of the beam. But honestly, it’s pretty solid without them.

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The planing stops, the holdfasts, and the dogs should provide most workholding. I can also use a clamp, a Veritas Wonder-dog, or a does-foot batten, if needed.

img_20170104_101318_sm

Planing Beam – First Steps

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My planing beam kit has arrived ! Looks like Mark at Plate 11 has done a great job, and he’s given me good instruction on how to build it. Very happy so far.

The first step is to chamfer the edges a bit, while doing so I took a few photos.

You can see I’m using two of Veritas’ planing stops at 90 degrees to position the work (in this case, one of the feet). That’s working well so far and I hope that will be my primary setup for planing almost any shorter workpiece. The dog holes should allow me to use the long planing stop at several different places along the beam, depending on workpiece length – or I could use a pair of dogs if the piece is very long.

A bench of sorts, a planing beam of sorts

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I’ve decided to purchase a workbench, or more accurately a kit.

I didn’t have space for a Roubo – a Japanese style planing beam sounded more interesting. Especially the flexibility of moving it about, and using the trestles alone, struck me as interesting.

But the western styles of workholding (dogs, planing stops, and holdfasts) seemed to me to be the best approach, especially given that I am using western planes (Stanley & Veritas).

So I’ve combined the two approaches – and Mark Hicks at Plate 11 (www.plate11.com)  has kindly agreed to make a kit for me to my plan. It’s currently in-work, I’m looking forward to receiving it in a few weeks. (Like many, I simply didn’t have much time to build my own from scratch.)

Here is the plan : planing_beam

It can also be set up to use a kanna – simply spin the beam about. Mark contributed a few ideas – notably the pins – and some suggested modifications regarding the choice of through vs. stopped mortises. (Note – after using it for a while – some form of pin or stop to connect the beam to the trestle has proved necessary – as I’d expected. But if I did it over, I might use a cleat, rather than pins – the pins are a bit hard to dis-engage. Refer to the Schwarz link below.)

I didn’t include any form of vise – for now. I’m intending to work using various planing stops, dogs, and holdfasts. To work the edge of a piece, I will use a clamp to hold the piece. We will see if practice matches theory – and if not I may drill a few more dog holes, or add a vise.

Some reference information / inspirations:

Toshio Odate discusses a planing beam in his book “Japanese Woodworking Tools”

http://www.popularwoodworking.com/workbenches/schwarz-workbenches/a-japanese-workbench

http://digitalwoodworker.com/2013/05/31/a-slab-top-for-the-japanese-workbench/

Another example of the trestle; Jay van Arsdale wrote some articles years ago in American Woodworker. You may be able to find it on Google Books (I did).

As for my choice of dog holes and their placement, Schwarz has convinced me : http://www.popularwoodworking.com/workbenches/schwarz-workbenches/i-have-a-dog-in-this-fight