Image Courtesy 25Seven
Here's an example of buckling in a thin walled steel frame (think beer can). Shown above is a 20 year old Breezer MTB designed and built by the renowned Joe Breeze. I'm not sure if this was a stock frame at the time, but it appears that the tubing wasn't engineered to provide safety against buckling forces. This website reports that if the tube's outer diameter-to-wall thickness ratio gets above 60 or 70 to one, the tube is more likely to suffer failure from buckling. Hence, tubes this thin are not to be used on frames that could be subjected to substantial abuse off road. They will dent and buckle.
It is likely that the rider in the picture above had a head-on collision (or even braked really hard as in this picture), and the compressive forces as a result were more than the tubes could handle. The way the tubes handled this load without breaking was to shorten in length and deform. The tube buckled just behind the head tube on both the top tube and down tube. In such a scenario, the front wheel is likely to move backwards, thereby striking the frame.
In thin walled shell structures (bicycle frame, airplane fuselage etc), the buckling phenomenon is a very important mode of failure. It is often among the controlling design criteria and hence, a buckling analysis is always done.
An important thing to realize is that buckling failures do not depend on the strength of the material, but rather is a function of :
1) Part dimensions and geometry : Length, and shape of cross section dictated by their area moment of inertia.
2) Modulus of elasticity E of the material, also called Young's modulus - a measure of stiffness of an elastic material.
Books on statics, structures or solid mechanics would have the formulas you need to calculate the buckling loads in a member. For instance, the critical negative tension force in a two force member that will make it buckle can be given by the following Euler's relation (assuming the action of the load coincides with the axis of the column) :
where Lo = Effective Length
and I = Area Moment Of Inertia or Second Moment of Area.
For a symmetric tubular shell structure, I is calculated by :
The buckling load is negative because it is compressive. The designer needs to ensure that this critical force is lesser than the maximum the structure can take to ensure stability.
The interesting fact however, is that nowadays, frame builders and many bicycle companies don't have to grapple with this issue since they buy stock tubes from tube manufacturers such as Easton, Reynolds, True Temper etc.
Therefore, it is the responsibility of the tubing makers to ensure that the tubes are designed to be stable. I have no clue how they size these tubes and what resources they use to do so. It will be interesting to learn. If anyone has any thoughts or comments to enlighten that process for us, comment away!
Additional Resources :
Redesign Of Scott Bicycle Frame