COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 219 
where p, = collapsing pressure of normally round tube (in lbs. per sq. im.), 
p, = collapsing pressure of distorted tube. 
M = ratio of maximum to minimum diameter. 
The utility of this formula is, however, somewhat doubtful, and 
cannot in any case be taken as indicating the effect of initial deviations 
from the true circular form, since p, is the experimentally determined 
collapsing pressure of what is described as ‘ a normally round tube,’ which 
in this case was merely a commercial tube of average quality. 
It has since been proposed by Slocum!* that the rational formula 
may be made applicable to the practical determination of the collapsing 
pressure by introducing a correction factor, C, so that 
12 QE /t \3 
Uatcle eee ary 
where ¢, is now the average thickness and d,,,, the maximum diameter. 
The assumption that the strength varies as the cube of the ratio of the 
average thickness to the maximum diameter is not entirely valid, but it 
has been found that the above formula gives a fairly close approximation, 
and that C is nearly constant for any one class of tubes. Its value has been 
found for the following cases, the tubes in each instance being of average 
quality :— 
1. Lap-welded steel tubes = -69. 
2. Solid-drawn weldless steel tubes = -76. 
3. Solid-drawn brass tubes = -78. 
The values of E and m are known for most materials, and the maxi- 
mum diameter and average thickness are the dimensional quantities 
most readily and conveniently obtainable for a given tube. It is, however, 
somewhat doubtful whether variations from true geometrical form would 
account altogether for the reduction of 25 per cent. to 30 per cent. indicated 
above. Southwell considers that too wide a range of values of ; were 
used in the comparison, and that in the thicker tubes the elastic limit of 
the material was reached before the equilibrium became unstable, thus 
producing a lower value of the collapsing pressure.* 
Physical Properties of the Material. 
It is evident, from statical considerations, that the whole of the 
material composing a tube of circular form is in a state of compression in 
a circumferential direction. The physical properties which it is natural to 
suggest as determining the strength are Young’s modulus, the elastic limit, 
and, for thick tubes, the ultimate strength, all in compression. The two 
latter quantities are difficult to determine, and the ultimate strength, in 
materials usually employed in tubes, is a somewhat indefinite quantity. 
The value of Young’s modulus is known to be approximately the same in 
compression and tension, and is easily determined. For thin, long tubes it 
appears to be the only physical property influencing the resistance to 
collapse. The custom of specifying, as in the case of boiler flues, the 
ultimate tensile strength cannot therefore have any reference to the actual 
collapsing pressure, but serves merely as a guarantee of the quality of the 
* This question is fully discussed by Mr. Southwell in a paper which is to appear in 
the Philosophical Magazine for September 1913. 
