Following on from yesterday’s post, today we’ll look at the buckling of the cylindrical tubes:
The tube in this analysis was defined to have a small (0.01 mm) asymmetric bulge at mid height, to initiate buckling. The video shows that the vertical stress is almost uniform around the tube, until it reaches a stress of 1.75 MPa, when the top of the tube starts to buckle inwards, then the buckling deformations increase rapidly, and the tube collapses. This stress will result in a force of 27.5 N in the 50 mm diameter cylinder analysed, or a total force of 110 N (about 11.2 kg force) in four tubes, or nearly 5 times the buckling load of the square tubes, which ties in well with the relative strengths seen in the Surfing Scientist video.
Finding the exact buckling load of a thin-walled cylinder is surprisingly difficult. A linear buckling analysis in Strand7 finds a minimum buckling load of 1.96 MPa:
The standard formula given by Timoshenko (and also in Roarke’s Formulas for Stress and Strain) gives a buckling stress of 4.7 MPa, but this is based on concentric buckling of the tube, rather than transverse buckling of the top.
Closer approximations to the value found in the non-linear analysis are given in Formulas for Stress, Strain, and Structural Matrices, by Walter D. Pilkey. Table 20-15-9 of this book gives a series of empirical formulas for different ratios of radius/thickness. The closest ratio is 500 (compared with 250 in the analysis), which gives a buckling stress of 1.57 MPa. The table also gives a theoretical formula, giving a buckling stress of 4.21 MPa, but the basis of this formula is not stated.
Perhaps the differences between the theoretical, analytical, and empirical results is not so surprising, because as the link below shows, actually it is rocket science: