Euler's buckling formula applicable for impact calculations?

$$F = \frac{\pi^2 EI}{(KL)^2}$$

Is Euler's buckling formula applicable for impact calculations, considering speeds relevant for a car or aircraft crash?

If there is a level where the formula becomes inapplicable or inappropriate in impact calculations, what determines this, and what behavior (and hence other formula) will then be relevant?

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1 Answer

The bucking formula comes from a stability analysis of the restoring moment inside the beam. Actually as compressive forces are applied to a beam, its natural frequency drops, and when you reach the Euler's limit the natural frequency of the beam becomes zero. By definition this is the point the beam will not behave at all in a static fashion and will move even when the applied force is static.

So in an impact scenario the beam will move at lower force levels lower than Euler predicts due the variability of the applied loads. When impact begins there is compressive wave going through the beam (like an earthquake) possibly reflecting of the supports and setting up a standing wave back and forth. For steel these wave have speed of about 5000 m/s, or $c=\sqrt{E/\rho}$. This motion imparts inertial loading on the beam which may initial buckling even though the external loads is below the Euler levels.

This is very complex scenario that requires careful consideration. Also getting impact forces as opposed to impulses is also very difficult.

Reference paper from NASA: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760006440_1976006440.pdf

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Thanks! I found this ref also interesting: math.psu.edu/belmonte/spaghetti.html –  bretddog Jun 19 '12 at 19:46