The video here
shows a rough and ready experiment demonstrating the superior strength under axial load of cylindrical tubes, compared with square tubes (skip to 1:30 if you are not interested in the chat).
The conclusion about the greater strength of the cylindrical tubes is fair enough, but I wasn’t convinced by the explanation that the early collapse of the square tubes was due to the stress concentrations at the corners, so I did a finite element analysis of an axially loaded paper tube:
The contours show axial stress in the vertical direction. It can be seen that the stress is uniform for the first few stages, until it reaches about 0.02 MPa, when the sides start to buckle inwards and outwards in a series of waves. As these waves increase in amplitude there is indeed a transfer of stress from the centre of the plates to the corners, until the maximum stress reaches about 3 MPa, when the corners start to buckle, but the initial change from a uniform stress distribution was the result of the deflections of the plates, rather than the corners causing a stress concentration.
The total reaction force at the base at the final initiation of buckling was 5.8 N (an average stress of 0.29 MPa), which would give a total failure load of 23.2 N (about 2.3 kg force) for four tubes, tying in pretty well with the failure load of something less than 2 heavy books in the video.
The next post will look at the failure mechanisms of cylindrical tubes.
Update 7th July 2011:
Stills from the Surfing Scientist video, showing the square columns at the point of buckling, provided by Georg (see comment below):
Strand7 data files and avi files of the animations may be downloaded here:
The data files need a licenced copy of Strand7 to run, but may be viewed with the free Strand7 Viewer. The avi files can be viewed with a standard video viewer, and are much higher resolution than the Youtube videos.
Hi Doug, I managed to extract some frames from the Surfing Scientist’s video using the VLC video player (slowing down playback velocity to 0.06 or so allows for accessing single video frames). On these, you can see (I’ve sent you 3 snapshots via email) that two different mode numbers (I’d say 2 in the right and 5 in the left column, resp.) of the instability are established in the two quadratic front columns (with respect to the camera). And this is where linear theory of elasticity comes to an end. Then, caused by inhomogeneities of the material unknown to us, the amplitude of a single full period of the instability grows larger than that in all other parts of the column. I think a stability analysis would show that indeed there is an instability, which means the part with the largest amplitude will show the greatest gain in amplitude under an increasing load. Just before breakdown, one edge of the columns shows a single full period of the bending instability the rest of the same edge remaining straight. Collapse then begins at that very single bended part of the column. So this is a highly non-linear phenomenon. And imho, it is very difficult to judge whether of-the-shelf software allows for such non-linear phenomena or not. That’s why earthquake engineering is an art…
Georg – thanks a lot for the pictures; I have added them to the end of the post.
I agree that modelling the exact behaviour during buckling is very difficult, if not impossible, but if we have a reasonable estimate of the scale of the imperfections in the column, and the end conditions, I think we can do a reasonably accurate analysis up to the point of buckling, which is what we need in practice.
In the case of the paper colums of course I don’t even know the right dimensions, let alone the size of imperfections, so I’m quite surprised the analyses came out with a result of the same order of magnitude as the video.
I could not down load the FE results
I have added links to the avi files of the animations, and the Strand7 data files. The results files are about 50 MB each, so I haven’t uploaded them, but if you have Strand7 you can do the analysis yourself. If you don’t have Strand7 you can download the free viewer to look at the input files.