COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 215 
the elastic breakdown of the material in some part of the column. If the 
ratio is not too great, buckling results from irregularities thus produced, and 
the problem is not amenable to rigorous mathematical treatment owing 
to the fact that such inequalities are largely the result of initial irregularities 
in the form of the column.* When, however, $ becomes large, buckling 
does not occur, and failure takes the form of uniform lateral expansion. 
There is an analogy in the collapse of tubes to each of these cases. 
(1) When the ratio ; is very small, collapse will occur without over- 
strain in any part of the material. As in the case of long columns, this is 
the only case for which a complete mathematical solution has been found. 
The problem was first investigated by Bryan, and the theory subse- 
quently improved by Basset ! and Love,'!? and more recently by South- 
well, '7 '8 The pressure at which the equilibrium becomes unstable is 
given by 
He t 3 
p=o (7 re ex rien kde apseemte Sait | 
where c is a constant depending only on the elastic properties of the 
material, and is equal to 
2B 
l—m? 
where # is Young’s modulus, and m Poisson’s ratio. 
Very elaborate and accurate experimental work carried out during 
recent years by Carman® and Stewart!® has shown that the relation (1) 
holds very nearly in the case of tubes in which the ratio F is less than -025. 
The value of the constant c is, however, found to be considerably less than 
the theoretical value. The discrepancy has been attributed by Slocum! 
to imperfections in the geometrical form of the tube, and by Southwell to 
the fact that in the comparison of the theoretical and experimental results, 
values of 4 were included which were great enough to allow elastic break- 
down to precede instability. 
(2) When the ratio ; 
a 
relation (1) no longer holds. It is evident that no tube can withstand, 
without permanent deformation, a pressure greater than that which would 
cause any part of the material to exceed the elastic limit. By Lamé’s 
theory the maximum compressive stress occurs at the inner surface of the 
tube, and, assuming that there is no longitudinal constraint, is given by 
exceeds ‘025, it is found by experiment that the 
f=— P 
A aE 
O-8) 
d d 
* It is possible to give a mathematical explanation of the form of the curve showing 
the relation of load to . even in this case. See paper by R. V. Southwell, ‘The 
Strength of Struts,’ Engincering, Aug. 23, 1912. 
