# Do I understand column buckling?

I'm going to make such a stool:

Originally it's made of thick plywood, and I want to switch to a thinnest possible OSB to make it as cheap and light as possible.

It's a bar stool of 70 cm height, 33 cm width. Front plate is 16° inclined from vertical.

My gut tells me that the weakest elements are regions on the sides of the opening(The left one is encircled on the image). I have to keep the opening, it's bottom edge acts as a footrest.

At the narrowest point the width of this area is $$b=5$$ cm, opening height is $$L=28$$ cm.

I calculate that $$h=12~\text{mm}$$ OSB would suit me.

My first question: Can I consider it as centrally loaded column?

1. I calculate slenderness ratio: $$\lambda = \frac{L_{eff}}{r}=\frac{L_{eff}}{\sqrt{I/A}}$$ where $$I=\dfrac{bh^3}{12}=7200~\text{mm}^4$$. So $$\lambda={28~\text{cm}}/{\sqrt{7200~\text{mm}^4/(50~\text{mm}\cdot12~\mathrm{mm})}}=81$$. It tells me that it's a long column. So the load would be limited by buckling.

2. Critical load $$F=\dfrac{E_{OSB}I\pi^2}{L^2}=\dfrac{3~\mathrm{GPa}\cdot7200~\mathrm{mm}^4\cdot\pi^2}{(28~\mathrm{cm})^2}=270~\mathrm{kgf}$$

Second question: Am I right that the stool made of 12 mm OSB would be able to hold over a ton of static load?

Third question: You can notice screwheads on the picture. How can I estimate the load they need to withstand?

• I'd remove the question regarding the screws, since it has absolutely nothing to do with the rest of the question. Ask that in a separate question.
– Wasabi
May 30 '16 at 14:49
• In my model the screws are playing a role in pinning the end of the column.
– Gleb
May 30 '16 at 16:36
• Yes, the way the parts are connected plays a significant role in how the stool will behave. But that is basically a question of whether the connection should be fixed or hinged. A connection's "fixity" is more a matter of its layout than the size or strength of its components. All of the connections on the stool, for instance, are probably best modeled as pinned (or at most as partially-fixed). Therefore, the matter of estimating the forces in the screws is entirely different and mostly unrelated to the primary question asked here.
– Wasabi
May 30 '16 at 17:37
• If you asked "will a 100mm wide (2 x 5 cm) strut of 12mm OSB hold over a ton of direct compressive load in an ideal situation" I would guess the answer is "yes". But in real life there are bending and torsional loads, and the compressive load won't be evenly distributed over the whole end surface of the strut. "Will a ton of force applied to the top of the stool make the 6 nails holding the leg split the top of the leg apart" is a different (and more relevant) question. Finding some "rules of thumb" based on practical woodworking experience is the best way to answer that question. May 30 '16 at 23:40
• Like alephzero pointed out in his comment above, the most likely failure in a stool, (or any other furniture) isn't a pure compression failure under an ideal load. A certain amount of robustness is what makes furniture last as people will load them unevenly, move around, tilt them, jump on them, etc. Jun 1 '16 at 15:39

If you want to make sure though you can calculate two different scenarios, one where you have a central load applied and other where you have a bending load applied with value $B = P\times l$ where $l$ is the distance from your point load to the center of the cross-section. That would cause and increase of tension with a magnitude of $$\sigma = \frac{B\times\frac{h}{2}}{I}$$ it being traction on the outside and compression on the inside. Putting together both tensions (normal and bending) you would have a total tension of $$\sigma = \frac{P}{A} \pm \frac{P\times l\times\frac{h}{2}}{I}$$ you should evaluate in which extreme of your cross-section which signal of the second factor is appropriate.

With this expression, two things point to the fact that the second term will be irrelevant:

• $I$ is much greater than $A$
• $l$ and $\frac{h}{2}$ are small compared to the other values

This is enough to convince me that modelling as a centered load is appropriate.

Of course you must still be aware of the critical load to avoid buckling, but looking at your calculations it shouldn't be a problem (if one section needs $270 kgf$ to buckle I would expect it take more than a ton to cause buckling in one section.

2nd Question: You should know (maybe get this info with the manufacturer) what is the admissible traction and compression tensions on your OSB board and calculate how much load that is. Always think of tension rather than load because if you have a smaller cross-sectional area the same load will cause greater tensions.

However, it you based yourself on your previous calculations to make this affirmation, you are not correct. You would be correct saying that the stool made of 12 mm OSB is able to withstand more than 1 ton without suffering buckling, but it would fail by mechanisms other than buckling.

3rd Question: The function of the nails in this piece is simply to keep the piece together i would say, nails and bolts are solicited when the piece needs to resist shearing stress. In your case, the top nails will not be solicited by the person sitting, because the load is transmitted by contact to the lower parts of your structure. The lateral nails will only need to resist a component of the person's weight that tries to "open" the legs of your stool, if you want to calculate that you should calculate how much of the person's weight becomes horizontal force that the nails must resist. If you have a value of that force, divide it by the number of nails and you should know to how much effort each nail will be subject. You would then need to know that this effort is normal to the nail as in trying to rip it of the board. Instinctively i wouldn't expect you to have problems with this (since the angle of the legs is close to vertical) and I would disregard this analysis. Not blindly though, but considering that the effort in the nails would be small.

While the calculations you've done are correct for an ideal structure, we are not dealing with an ideal structure here.

For one, you don't take into consideration the fact that the column is inclined. This will generate a static bending moment along the entire column, meaning you aren't dealing with buckling under compression, but under flexo-compression, which reduces the buckling load.

Also, your $L_{eff}$ assumes that the column is pinned on both ends. This is a conservative estimate, since there will be some rotational stiffness on both ends, but it's fine.

However, the biggest flaw in your calculation is that you assume that the buckling length of your "column" is equal to the height of the opening, but that is incorrect. The correct buckling length is probably more like the distance from the midpoint of the lid (where you sit) to the midpoint of the bracing beam near the ground. You then need to perform a buckling analysis for a column with a changing cross-section (from a solid section to a section with an opening to a solid section, with a changing width throughout). It this can be done analytically (no such guarantee), it will be a huge pain. This would be best done by letting a FEM program deal with it.

When modelling the column this way, the boundary conditions will probably be fixed at the base (due to the high stiffness caused by the tall bracing) and pinned at the top (the simple screw connection will offer little resistance to small rotations), which reduces your $L_{eff}$.

Also, as @Mr.P already mentioned in a comment beneath the OP, this calculation you've done is for Euler buckling, which is the theoretical limit for a perfect specimen. We are not dealing with perfect specimens in the real world, so designing anything by the Euler load is non-conservative (your actual structure will resist less than this load). In structures there are a multitude of codes which describe how to take imperfections into consideration, but I'm not sure if they are valid or practical for a small stool.

The best solution is probably as suggested by @alephzero in a comment beneath the OP: woodworking rules of thumb. I'm not a DIYer, so I don't know what these are, but I'm sure they exist and will give you a good notion of what's reasonable and what isn't. Also, there's no solution that's better than a prototype. Go ahead and build the stool. Sit (and stand) on it every which way, put some ridiculous weight on it (stacks and stacks of books and then sit on top of them), see how sturdy it is. Also make sure to test how "flexible" it is. A wobbly stool might not actually collapse, but the person sitting on it won't feel very safe on it.

• +1 for prototype suggestion, i.e. just do it already! Jun 27 '16 at 22:36