-10- 



/.O a 14 Li 1.8 20 12 2A 2.6 



Fig. 3. The least critical pree- 

 svire for various wall thicknesses 

 2h and tube lengths 1. 



This is identical with the above-mentioned equation (a) with o" = 0.3, 

 If a// and, therefore, ? have values other than zero, equation (C) 

 represents a particular straight line which will cut the axis of ordinates in 

 some point for each value of n. The ordinates of the point of intersection - 



equal to the term which is free 

 from I in eq. (C) - decrease with 

 increasing n and constant a/1, 

 while simultaneously the slopes of 

 the lines - represented hy the co- 

 efficient of I - increase. It fol- 

 lows, therefore, that the straight 

 lines represent a portion of a pol- 

 ygon that bends downward. 



The point 1= 0, y= 0, is at- 

 tained ,only for the polygon, n =0° . 

 Now n, however, signifies the num- 

 ber of lobes that appear in the cir- 

 cumference at the time of collapse, the figures 4 to 6 show the shape of the de- 

 formation of the circle through w = C sin n 3> when n =» 2, 3, 4, (Translator's 

 note: w is the displacement in the radial direction.). We have, therefore, the 

 result that the smaller the wall thickness In proportion to the radius , the great 

 er the number of incipient waves . (Translator's note: This can be seen directly 

 from Fig. 3). Of course, in the resulting deformation, not all the waves will ap- 

 pear completely formed, but each wave will have a length equal to the circumfer- 

 ence divided by n. 



In Fig. 3, the polygons 



are shown corresponding to 

 the values of a// = 0.5i 0.4 

 0.3, 0.2, .0.1, and the 

 straight line for / = c» , 

 Fig. 7 shows the lines to 

 four times the scale for the 

 particular region in question 

 to about h/a = O.OI4. In both figures, the lengths along the z-azis (on which i 

 ■ \ (^) is given simultaneously) are given in values of lOOH . 



Figs. 4 to 6. Deformations of the 

 circular cross section for 

 n equals 2 to 4« 



