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My guess on the tenon 3/4 - 1"
As to length I would go as deep as the wood allows.
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I have repaired many of these, I worked in a furniture repair shop for several years. You will never be able to drill a hole in both ends that line up perfectly and allow the outer edges to align closely. Your best and strongest option is to get a piece of all-thread, pick a size based on the smaller end of the leg. I usually go with 3/8 or 1/2 inch, I have used 3/4 in large legs, trust me it will be impossible to bend a short rod that thick by leaning on the chair and it will be stronger than a wood dowel. Next drill a hole slightly larger than the rod in the lower half, you want enough room to get a good bond of epoxy all the way around the rod. I try not to drill a hole larger than 1/2 the diameter of the leg at the deepest point of the threaded rod, I like as much wood around it as I can get.
Put the broken lower half in a vise and adjust it vertically straight and then drill as straight down the center-line and deep as you can, use your best judgement. Tape off the edges of the break so epoxy doesn't stick to the leg if it overflows, painters tape is good. Epoxy a piece of all-thread in the lower half, make sure it's held in place as close to the center-line of the leg as you can until the epoxy is hard. Clean any excess that oozes out beyond the hole on the broken wood with denatured alcohol. Leave the break as clean as possible so it will line up to the other half, excess epoxy in the break will make the repair show, prevent the edges from lining up and possibly lengthen the leg.
Now use a larger drill bit that will allow you to line up the outer edges of the break while giving you room to counter act any misalignment in the all-thread from the center-line of the leg. ALWAYS do a test dry fitting, after the epoxy is mixed and on is not the time to find out the rod is too long or the halves won't align and you need a slightly larger hole in the top. Turn the chair upside down like it is in your picture and again drill the hole following the center-line of the leg. Tape off the leg and cover the chair with a tarp or old towel for protection, put epoxy in the hole and some on the protruding all-thread then push it into place aligning the edges of the break to match, make sure you fill the hole completely. Gaps/chips can be filled with epoxy now or wood filler later. clean up with denatured alcohol and keep an eye out for bleed out for about 20 minutes. I use a good quality 5 or 10 minute epoxy for this, so it is set by 20 minutes. Let it sit 24 hours and then you can touch up any chips or gaps with wood filler, stain or dye and then a light shot of spray can lacquer and you are good to go. I use dyes in the shop to hide repairs, adjust and blend colors but that is a whole other process.
I hope this helps,
Paul
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Thanks for your replies, (and the great detail, Paul) I will work on the chair this weekend and see what happens. Should be fun little project.
Carl
Carl
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What is an all thread? A threaded rod?
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Or - cut the other legs to the same length. If this was a stand alone chair you could get away with it.
I like the idea of the metal rod/screw. The big issue here is you have so little to work with on the broken off side.
John
Always use the right tool for the job.
We need to clean house.