How about a 3/4" stool leg?

I ran across this thought-provoking post by Chris Wong last week, which was about understanding material and joint strength.  It pointed out the downside of overbuilding things in the way of lost design opportunities and less than ideal appearance.  As it happens, I had just done that very thing.

I am designing and building saddle seat stools for our kitchen from white oak.  Making sure every component was more than strong enough was dominating my thinking.  Motivated by the post, I looked up the properties of white oak.  This is a very technical subject, and the reported values for various aspects of the strength of a wood species are nearly impossible for most of us to interpret.  A more pragmatic approach is required.  I went out into the shop, cut up some 24" pieces, propped up the ends on 2x4s and stood on them.  I'm a big guy and I can stand on the middle of a straight-grained 1x1 without breaking it.  It bent but it didn't break.  My sense of how thick my stool legs needed to be was way way off.

 Think about a four-legged stool with riven legs 3/4" square and 1/2" tenons.  Suppose they are mortised into the seat at an angle and that there are no stretchers at all.  What would fail and how much force would it take to make it fail?  If the mortises were too close to the edge of the seat, they would probably break out due to the side loading from the angled leg, but let's say that's not the case.  Then, I'd say that the tenon would rupture at the mortise, but I think it might take a whole lot of force to make that happen.  My sense is that, for straight grained wood, it's the joint and not the member that will fail.  The broken furniture I can remember seeing tends to bear this out.  When I have seen a broken member, it's because it wasn't straight-grained, so it split along the grain.

What does a stretcher really do?  It protects other joints on the legs by sharing their load, often with mechanical advantage or in a way that takes advantage of the properties of wood.  Maybe not always, but generally.  I guess this is obvious, but I hadn't thought about it in those terms.

Do an image search for shaker stools.  The legs on some of them are so darn thin and have only one stretcher per side; they seem like they would never hold up, but we know that they do.  Those thin stretchers are enough to keep the joint at the seat from failing.  I'll bet those pieces are riven.  Looking at these pictures and thinking about the physics tells me that one set of stretchers adds a lot to the strength of a piece, more than one not so much.

Look at this old, mass-produced chair I bought at a garage sale:

Most of the pieces of this chair are very thin and you'd think it would be rickety by now, but it isn't.  The ankles of the legs are 3/4".  Yet the chair is solid.  It's strong where it needs to be, at the joints.  This is just a mass-produced knock off of a cabriole leg.  Real cabriole legs are very strong though extremely delicate because they are so beefy at the joint.  Look at this cabriole leg joint I made:

Another thing I realized during these ruminations is that a stool with no back doesn't experience the same kinds of stresses as a chair.  Just think about what happens when someone rocks back in a chair, their weight on the seat and the front legs off the ground.

There is nothing here that you didn't already know, but, in my case at least,  I don't really think about any of this explicitly.  I should.   

I think that a major reason we don't test the limits of our material and joints is that it takes a lot of work to build something, so building test prototypes isn't very appealing.  Since we don't want the finished piece to fail, our solution is to overbuild.  Of course, you can always use a proven design, but what's the fun in that?

I went back to rethink my design, keeping these thoughts in mind.  It changed a lot, like it went on a diet.