# What's the difference between critical load and yield stress?

So far, I have learned of three quantities that are related to the failure of a beam (axial and longitudinal loads). The first illustrates the stress under which balsa wood will undergo plastic buckling:

$$\sigma_{pb} = m \sigma_{ys}\left(\frac{\rho}{\rho_s}\right)^{\frac{5}{3}}$$

Where $$m$$ is a constant (estimate = 2 for balsa, $$\sigma_{ys}$$ is yield strength, and $$\rho_s$$ is $$1500$$ kg m$$^{-3}$$. $$\rho$$ is obviously density.

The second is a graph of compressive failure strength of balsa wood at varying densities:

And finally Euler's Critical Load equation: "The critical load is the maximum load which a column can bear while staying straight." $$P_{cr} = \frac{\pi^2 E I}{(KL)^2}$$

Here, $$E$$ = elastic modulus, $$I$$ = area moment of inertia, $$K$$ is a constant varying from 1-2 (based on whether the member is pinned or not on ends, estimate = 1), $$L$$ is the length, and $$P_{cr}$$ is the critical load in Newtons (I believe.)

The first two are from this research paper, and the last is from Wikipedia. My question is, if there is a length at which the load will not bend the beam, then why is there a plastic buckling stress, which only depends on area? Shouldn't the beam not buckle under any load as long as the length is small enough? I believe there is something I am missing with the fundamental definitions of these. If stress is $$F/A$$, then the first equation could be changed to $$F = A$$ *... which does not include length, as Euler's equation does. And how does failure, as shown in the graph, play into this?

• It would be helpful if you would define the variables in the equations you present. Also, check out this video youtube.com/watch?v=wrdO8hPJGyg the answer to the question in the title of your post is contained therein-- when something is loaded critically it has not undergone plastic deformation while if it passes its yield stress it has. Nov 5, 2017 at 5:35
• Sorry, I wasn't too familiar with these but I thought they were somewhat fundamental to the more experienced so did not include them. They have been put in. Thanks. Nov 5, 2017 at 6:00