392 REPORTS ON THE STATE OF SCIENCE, ETC. 
at their respective elastic limits (under torsional cycles), and afterwards proceed in 
sensibly straight lines, whose inclination is nearly the same. ‘The average inclination 
of these lines corresponds to a value of Asie) 2-50 x 108 Ib. per sg. in. This 
A (strain) 
stress is not a ‘ modulus of elasticity,’ but it will be convenient to denote it by the 
, 
symbol C’, while the elastic modulus 12-1 x 10® is denoted by C. The ratio a 
for these four specimens is thus approximately 0-225. The feature of these curves, 
then, is that for increments of range of stress beyond the elastic limit—the cycles at a 
range of stress being continued until the strain attains a constant value—the incre- 
ment of range of strain is proportional to the increment of range of actual stress. 
This proportionality, it will be observed, is for the middle of the tube-wall. 
Distribution of Stress throughout Wall of a Tube subjected to Alternating Torsion. 
The distribution of stress throughout the tube-wall has not entered into the pre- 
ceding calculation, the stress at the middle of the wall only having been calculated 
and plotted in fig. 174. 
Let RS (fig. 18) represent a curve like those of fig. 174. At an epoch 1, after 
cycles of an actual range of stress -- B,S, (at the middle of tube-wall), numerous 
enough to attain constancy of strain at this range of stress, let the constant semi- 
range of strain attained be OB, at the middle of wall. 
BB, _tm—"1 
Set off a length B,B, such that Te sre where 7, is the internal radius 
¥ 2 m 
and 7, the mean radius ; 
or, what amounts to the same thing, lay off OB, so that oa = Now, suppose 
1 1 
OB, and B,S, to be the strain and corresponding stress respectively, at the middle 
of wall, after a sufficient number of cycles at the larger range of stress B,S,. Then, 
by the previous section, S, will lie on RS. Now with linearity of strain, OB, must be 
equal to the strain at the internal radius.r; at epoch 2. The assumption will now be 
made provisionally that the stress at the inner skin at epoch 2 will be equal to B,8). 
This is a large assumption to make, and it can only be considered to have validity 
if the deductions from it have experimental confirmation. Stated in general terms, 
and including in its scope solid as well as hollow specimens, this assumption amounts 
to the following : At any epoch after a number of cycles sufficient to produce constant 
range of strain, the relation between the ranges of stress and strain at any point of 
the specimen is expressed thus : 
q = Cee + C'(e— e), 
C being the elastic modules ; 
A (stress) 
Cie ee waluesoL AiGtrain) corresponding to the line RS (fig. 18) ; 
Cs) s », semi-range of strain corresponding to the intersection of OP and RS 
(fig. 18) ; 
e semi-range of strain corresponding to the range of stress ++ q. 
9° 39 
Illustrating this assumption a little further by the aid of fig. 18, lay off BB, 
such that 
Led 
OBS Hom, 
Let OB; and B,S, be the semi-ranges of strain and stress at middle of wall at an 
epoch 3. Then linearity of strain makes the range of strain at inner skin at epoch 3 
equal to OB,, and the above assumption makes the range of stress at inner skin equal 
to B.S,. Thus the stresses and strains at the inner skin of the tube at various epochs 
ccd be represented by the same graph as the stresses and strains at the middle of the 
wall. 
In the same way, following the above assumptions, the graph RS represents the 
relation between the ranges of strain and stress not merely at the middle of the tube- 
