COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 389 
the mean radius of the tube will be very little different from the stress calculated on 
the assumption of uniform distribution of stress throughout the tube-wall. 
Let q, = shear stress on assumption of uniformity of stress, 
ry = outside radius of tube ; 
r, = internal radius of tube ; 
then the resisting moment of the section 
9 
oT (75? — 13). 
3 
Let gm = real stress at middle of tube-wall, then g, cannot differ appreciably 
from qu. 
Qarqin 
And resisting moment = (ro25—13). 
« 
Now, since it has been convenient to express the resisting moment in terms of the 
stress at the skin (= q,, say) calculated on the assumption of linearity of stress, 
Aa Tannai 
Resisting moment = /¢ . = (To — 11’). 
ee 
Hence 
9. 
aT. 9m 1.3 3) Ie ni 
peepee Ti ee cee (fg 1 3*)s 
3 ry 2 
and 
Qu 3 1—14/to* 
qe 4 1—r,3/r0° 
This formula has been used to calculate the values of q,, for four mild-steel tubular 
specimens—viz. B22, B27, Al0, and A2. Full data concerning these will be found 
in the former paper referred to by the author.t Table I. gives certain data taken 
from that paper, as well as other items of which use is now to be made. These speci- 
mens were not made to the same dimensions, though all were of the same general 
form.’ It has been convenient to reduce the observed scale readings—in cms. (which 
are proportional to the angle of twist)—to values of the shear strain. This has been 
done by assuming that the elastic modulus of rigidity was the same for each specimen. 
All the four specimens were from the same batch of steel; B22 and B27 were cut 
from one long bar, and Al0 and A2 from another bar. The value of the elastic 
modulus has been assumed to be 12-1 x 10° lb. per sq. in., a value which is consistent 
with other measurements made while each specimen was below the elastic limit. 
Thus, for specimen B22, a range of 5-86 cm. of scale was observed under a range of 
stress of + 5-00 tons per sq.in. If Fis a factor for converting cm. on scale to strains 
at the skin, then— 
5:00 x 22 
5:00 x 2240 = 12-1 x 10%, 
2 
5:86 x F 
and 
5-00 x 2240 
ee | ats eocaal | Um 
RBA Ole 10s, 
F 
Factors F, given in Table I., have been calculated for each of the four specimens 
using the same value of modulus, 12-1 x 10% In this table, strains at the skin for 
each specimen are given for a number of epochs when the range of strain had become 
constant. The number of cycles is not in all cases sufiicient to make the range of 
strain quite constant; but constancy of range had been approximately attained, 
at least, in each case. The curves of fig. 174 have been plotted from the values of 
the strain and stress (at middle of tube-wall) given in Table I. 
These curves have, by reason of the method of plotting just described, a common 
elastic line, whose inclination is 12-1 x 10° lb. per sq.in. ‘The curves then turn away 
1 Proc. Inst. Mech. Eng., Feb. 1917. 
2 Tbid. See fig. 1, p. 122. 
