218 REPORTS ON THE STATE OF SCIENCE.—1913. 
End Conditions. 
In dealing with the question of length no assertion was made in regard 
to the statical conditions at the ends of the tube. In the experimental 
work the ends were rigidly fixed to a true circular form of invariable 
diameter, and were also held rigidly in a longitudinal direction. Southwell 
has shown ?” that any variation in these conditions in the case of a short tube 
will considerably affect the resistance to collapse. It is reasonable to sup- 
pose that where the ends are more or less flexible in any direction, the 
strength will not be as great as when the tube is rigidly held parallel and 
circular. Itis, however, hardly to be expected, when the knowledge of the 
general effect of length is so vague, that the effect of the end conditions 
could be expressed in more exact terms. 
Variations from True Geometrical Form. 
The ideal conditions assumed in the derivation of the rational formula 
are never realised in practice. The phenomenon of collapse is, however, 
due to imperfections in form and material, and, in the mathematical 
analysis of the ideal case, the collapsing pressure is that which would 
produce a state of neutral equilibrium in the shell, although actual collapse 
would only occur if some shght unsymmetrical deformation were caused. 
It is evident that slight initial deviations from the perfect form must affect 
the value of the collapsing pressure to a considerable extent. The earlier 
experimental work in this subject by Fairbairn ® was carried out on tubes 
which were lap-riveted and in which therefore the variation from the true 
circular form was at least equal to the thickness of the plate. Although 
serving the purpose for which they were intended at the time, the results 
have little application to modern tubes, which are either solid drawn or 
welded, and are much more perfect in form. The slight variations, how- 
ever, which still occur are probably responsible in a large measure for 
a considerable departure from the theoretical strength. Experiments 
carried out within recent years on modern tubes have shown that the 
relation 
P d,)’ 
where ¢, and d, are the average thickness and diameter, holds when the 
average of a large number of tests is taken, but the constant is smaller than 
the theoretical constant by about 30 per cent. Further, wide differences 
in the collapsing pressure of tubes of the same average dimensions have 
frequently been found. A theoretical formula for the calculation of the 
effect of inequalities in diameter and thickness is hardly to be looked for, 
even if it were possible or convenient to measure the latter, for any given 
tube. Such a problem is, moreover, complicated by the fact that the 
positions of greatest and least thickness may have any relation to those of 
oreatest and least diameters, with a corresponding variation in the com- 
bined influence. 
A series of tests was carried out by Stewart ?° some six years ago on 
steel tubes 10 inches in diameter in order to determine the effect of distor- 
tion on the collapsing pressure. It was found that the results could be 
expressed by the formula 
p> = 0:0926 pi — 47°55 
it oer) + AT-B5 
