214. REPORTS ON THE STATE OF SCIENCE.—1913. 
practical importance of the subject, definite and exact knowledge is 
lacking. A universal formula has not yet been found by which the 
strength of a tube of given dimensions and material may be estimated. 
It is perhaps safe to say that it is impossible to devise such a formula 
which will be sufficiently simple to be of any practical value. This will 
at once be evident when the number of factors which enter into the 
problem, and the lack of knowledge with regard to each, irrespective of 
their mutual relations, are considered. These factors may be divided into 
two main classes: (a) those relating to dimensions and geometrical 
form ; (6) those relating to the physical properties of the material. The 
first of these may be subdivided as follows :— 
(1) Lateral dimensions; 7.c., diameter and thickness. 
(2) Length. 
(3) The boundary conditions at the end of the tube. 
The statical condition of the tube at the moment of collapse being one 
of unstable equilibrium, the influence of slight variations from the circular 
form or uniform thickness which are invariably found in practice must 
also be considered. 
It is proposed in the course of this report to consider separately the 
influence of the above factors. 
Lateral Dimensions. 
Although the influence of length will be dealt with later, it may be 
stated here that it is found both from experiment and theory that, as the 
length increases, the strength of a tube of given lateral dimensions tends to 
a minimum constant value, which appears to be attained, for practical 
purposes, when the length is greater than six times the diameter.® It 
is therefore proposed to consider here only the case of a tube of infinite 
length. The strength of such a tube is dependent upon its diameter and 
thickness, and it appears to be established, both from theoretical con- 
siderations and from the experimental data available, that the collapsing 
pressure is some function of the ratio of the thickness to the diameter é ) : 
A complication is at once introduced by the fact that the form of that 
function deyends upon the value of the ratio iM 
The problem is, in many respects, analogous to that of a column under 
a direct compressive load, in which the conditions determiming failure 
depend on the ratio of k, the least radius of gyration of the cross-section, to /, 
the length. In the failure of a column, two ranges of values of k may be 
distinguished. 
(1) When : is very small, failure occurs by pure buckling, without 
any departure from perfect elasticity, and it can be deduced mathemati- 
cally that the stress at failure is 
we k\2 
p=a(7) 
where a depends only on Young’s modulus and the end conditions. 
(2) When = exceeds a certain fairly definite value, failure is caused by — 
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