COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 217 
length of the tube ; the shorter the tube, the greater is the number of lobes. 
Unwin ”? was the first to appreciate the importance of this fact, and he 
used it in an attempt to set on a more rational basis the results of the 
classical experiments of Fairbairn. Southwell has shown, however, that 
the collapsing pressure of any tube in which the ratio ; is very small may 
be expressed by 
t Z fs stael et oe aaa 
p=28 | BeSS E° 3 (=m) a 
where « is the number of lobes in the collapsed cross-section and Z is a 
constant depending on the type of the end constraints. If, for a tube of 
given thickness and diameter, the value of p be plotted against / for values 
of xequal to 2, 3,4... . aseries of curves is obtained ; and Southwell has 
pointed out that, from an inspection of these curves, it will be seen that long 
tubes will always tend to collapse into the two-lobed form, since the curve 
for x = 2 then gives the least value for the collapsing pressure, but that at 
a length corresponding to the point at which this curve intersects the curve 
for x = 3, the three-lobed form becomes natural to the tube, and for 
shorter lengths still, for which the point of intersection of the curves for 
« = 3 and x = 4 gives the upper limit, the four-lobed form requires least 
pressure for its maintenance. Thus the true curve connecting pressure and 
length is a discontinuous one, and therefore the collapsing pressure is not a 
continuous function of the length. 
It may be suggested therefore that the term ‘ critical length ’ be applied 
to the points of discontinuity ; that is, the points of intersection of the curves 
for x = nand x = n +1, these points being rightly described as ‘ critical ’ 
in the sense that the tube may collapse into either n or n + 1 lobes at these 
points. With this meaning a tube will have a number of critical lengths 
corresponding to the configuration of the collapsed cross-section, and it may 
be shown that for different tubes the length corresponding to the critical 
points is proportional to 
a? 
ie Saree meremy patie (i) 
Southwell has pointed out that the above expression is also the factor 
determining the value of the critical length in the sense in which that term 
has been generally used, e.g., by Love and Carman. Prof. Love has 
accepted the above result as superseding the expression (2) given above. 
It would appear from (3) that a thin tube may collapse into a greater 
number of lobes than a thicker tube of the same length and diameter. 
This has been verified in experiments carried out by the writer, but beyond 
this no definite experimental confirmation of the above results has yet been 
made, although work in this direction is in progress. It has usually been 
assumed—and the assumption appears to be sufficiently justified by the 
experimental work of Carman and Stewart—that, for practical purposes, 
the influence of the length vanishes when it exceeds about six diameters, 
and that below this value the strength of the tube may roughly be taken 
as proportional to the reciprocal of the length, although the experimental 
evidence in regard to the latter cannot be regarded as conclusive. 
