398 REPORTS ON THE STATE OF SCIENCE, ETC. 
that the upper five plotted points of specimen A15 lie on a straight line—corre- 
spondingly to the torsion specimens—but that the upper part of the curve of All 
is not quite a straight line. The upper three points of Al6 and the upper two points 
of Al4 lie on straight lines. It may be assumed that the range of strain had become 
approximately constant for those upper points that lie on a straight line. The bend 
at the elastic limit is not so sharp as in the case of torsion. 
The formule for, and method of, calculating the ultimate ranges of stress, and 
the range of stress at the point in the specimen where the modulus changes, are given 
in Appendix II. 
Tables VI. and VII. show the result of the calculations; fig. 22 represents these 
results graphically. It appears (Table VII. and fig. 9) that the elastic limit point R 
for all values of a is below the elastic limits observed in making the tests. Even 
for = = 0 (i.e. RS horizontal), which gives the highest value of elastic limit stress 
in the body of the specimen, this is the case. Of course the stress-strain curve will 
be rounded off in the neighbourhood of R. ‘This rounding off will have most effect 
on the calculations (7.e. in equating of resisting moment to applied moment) in the 
case of the solid specimen Al5. The curved line in fig. 23 shows a hypothetical 
rounding off for A165 for the case of H’ = 0; the effect is to raise the right-hand part 
of RS by a stress of 0-28 tons per sq. in. The effect for the other specimens would 
be less, and for other values of = the raising of the line RS will be smaller as = 
increases. It is clear, then, that the main issue, viz. the position of the line RS, is 
little affected by rounding off in the neighbourhood of R. 
It will be observed (fig. 22) that the line RS for specimen A16 (which has the 
thinner wall of the two hollow specimens tested) is always above the RS lines for the 
E 
considered as having equal probable weight with the others. The effect of rounding 
off will be to raise the RS line of this specimen less than the raising of the others, 
but this consideration does not account for more than about one-third of the 
difference in stress between the RS line of Al16 and that of the others. 
Further examination of the RS lines of fig. 22 shows that while a value of = 
somewhere between 0:2 and 0-3, makes the RS lines come nearest together, this 
ta 
other specimens for all values of So far as the author knows, this test has to be 
value is apparently about the middle of the wide range of = from 0 to 0-5, all values 
between these limits bringing the RS lines fairly close together. Considering the 
ultimate ranges of stress which produced fracture, the agreement between these 
/ 
is closest for values of 2 from 0-2 to 0-3. The assumption of a definite value of 
A (stress) 
A (strain) 
to the results for torsion. One reason for the uncertainty of this result obviously 
lies in the circumstance that the area of layers of the circular section (providing resist- 
ing moment to bending) diminishes rapidly towards the skin farthest removed from 
A (stress) 
A (strain) 
may not be an absolute constant, the general inclination of the RS stress-strain line 
through the outer part of the body of the specimen will most probably not be greater 
for bending thus leads to results that are somewhat indefinite, in contrast 
the neutral axis of bending. It seems clear, however, that although 
than 0-4. The extreme value of = = 1, i.e. the assumption of linearity of stress 
from axis to skin, which is usually adopted for calculating the skin stresses, is palpably 
wrong. 
Some idea of the probable amount of the error made by calculating skin stresses 
of fatigue tests on the assumption of linearity of stress from the axis outwards may 
be obtained from Table VIII. 
Taking 0-3 as the most probable value of 
2s the error for hollow alternately 
bent specimens would be about 15 per cent., and for alternately bent solid specimens 
a 
