Collapse of Short Thin Tubes. 561 



end constraints, I, d, and t are the length, diameter, and wall 

 thickness, and k represents the number of lobes into which 

 the tube collapses. As is known, long tubes collapse into 

 two lobes, shorter tubes into three lobes, and still shorter 

 tubes into four lobes (see Phys. Rev. vol. xxi. p. 396, figs. 3, 4, 

 1905). This formula is represented by a family of curves, 

 corresponding to the values of 2, 3, 4, etc., for k. Southwell 

 points out that the envelope of this family of curves is very 

 nearly a rectangular hyperbola. For longer tubes for which 



d* 



k = 2, and -^ is very small, Southwell's formula reduces 



directly to Bryan's formula. Southwell at first used the 

 term " critical lengths " to designate the lengths at which 

 the tube may collapse into either 2 or 3 lobes, or into either 

 3 or 4 lobes, that is, the points of intersection of the branches 

 of this curve. He has, however, also deduced theoretically 

 an expression for the critical length in the sense in which 

 Love and Carman used the term *. In this expression 

 Jj = k s/tfijt, k is a constant. Mr. Cook, in the Phil. Mag. 

 for July, 1914, gets a value of 1*74 for k from the discussion 

 of a series of careful experiments made by him on short 

 steel tubes of three inches mean diameter and different 

 thicknesses. Cook concluded that the critical length is 

 apparently " from about 13 to 18 times the diameter " 

 instead of about 6 diameters as originally suggested by 

 Carman. Unfortunately, Cook's apparatus limited him to 

 tubes less than 13 inches in length; that is, to lengths of 

 about 4 diameters, and so his curves do not reach the 

 important bends or "critical" points. Mr. Cook himself 

 says : " the tests cannot be regarded as sufficient in number 

 or covering enough range of dimensions to confirm definitely 

 the equation L = k hjd*jt" 



The following experiments have been made to obtain more 

 data on the collapsing pressures of tubes, particularly near 

 and inside of the " critical ** bend in the curve for the 

 relation between pressure and length. The work to be 

 described is part of work which is being done under the 

 auspices of the Engineering Experiment Station of the 

 University of Illinois, and with the help of Mr. S. Tanabe, 

 Research Fellow. The apparatus was mostly that which 

 was used by the author a number of years ago, and the 

 methods of the experiments used were similar |. The tube 



* Southwell, Phil. Mag. vol. xxix. p. 69. 



t Bulletin No. 5, Engineering Experiment Station, University of 

 Illinois. 



Phil. Mag. S. 6. Vol. 32. No. 192. Dec. 1916. 2 Q 



