330 REPORTS ON THE STATE OF SCIENCE, ETC. 
Analysis. 
Although further experiments are in progress, the foregoing data—for a particular 
steel, stressed with steady and alternating loads bearing a particular ratio—may 
profitably be analysed, as a preliminary step, in order to note the application of a 
simple hypothesis regarding the beneficial action of the redistribution of stress 
attributed to ‘ primary hysteresis’ in the metal. 
According to this hypothesis, the metal on the margin of the hole, and on the 
transverse diameter, becomes subject to a modified stress that ranges from (S + 3a) 
to (S — 3a), the upper limit being that which most directly governs the development 
of slip-bands producing primary hysteresis. The value of S is less than 3s, the 
intensity that would be developed in an ideally elastic metal, and is probably very 
small in a metal endowed with sufficient capacity for developing primary hysteresis. 
In such circumstances, fatigue occurs only when the semi-range 3a exceeds 
A, = A, (1 —(S/U)?), which is then very nearly equal to Ag, the fatigue limit under a 
simple alternating stress, varying between equal intensities of pull and push. Thus, 
the nominal range of stress required to promote fatigue is 
a = A,/3, nearly. 
If 3s/U is high, this value may be considerably greater than A/3 = A,/3 (1 — (3s/U)?) 
—which would be the value if there was no redistribution—although equal to it when 
8 is zero. 
Since the two series of tests afford two alternative means of deducing the value of 
A), the validity of the hypothesis may be checked although, with such thin plates, 
the value of A, cannot be determined by direct experiment. 
_ Applying the relation indicated above, the results of the second series of tests 
give 
A, = 3a = 3x14 tons/in.? + 2-15 = 19-5 tons/in.? 
Turning to the first series of tests, on unpierced strips, the value of Ay may be 
deduced by applying Gerber’s parabolic rule, namely, A = A, (1 —(S/U)?). Thus: 
A, = 14-+(1 — (18/36-3)2) = 18-5 tons/in.2 
It may be noted that this latter value is 51 per cent. of the ultimate tensile strength, 
U = 36:3 tons/in.?; which percentage appears not unreasonable, although a some- 
what lower value, from 45 per cent. to 50 per cent., is more common in such steels 
tested in stout bars. It appears, therefore, that the approximate agreement between 
the two values (within 6 per cent.) is sufficiently close to justify the use of the simple 
working rule, namely :— 
Limit of nominal alternating component stress (a) must not exceed A,/3. 
It may be remarked, further, that the 6 per cent. discrepancy—noted above—is 
in the direction which may be attributed to either or both of two known effects : 
(1) the metal may have been hardened slightly by the ductile strain associated with 
redistribution ; (2) the application of the Gerber parabolic rule, in relation to the first 
series of tests, may tend to underestimate the value of A, in a hard-drawn metal. 
In this particular case, however, the latter effect is improbable, since it is not likely 
that A, will exceed 51 per cent. of U. 
In a more ductile metal, such as that investigated in the contribution to the Report 
for 1923, the effect of the hole is less dangerous than is indicated by mathematical 
analysis, even when the conclusions are modified by the rule explained above. The 
discrepancy may be attributed to the local hardening effect due to the slight plastic 
strain, associated with considerable primary hysteresis. 
For example, an experimental mild steel plate, having a single circular hole on 
the centre-line, showed a fatigue limit approximately at the combination s = 6-0, 
a = 5:5 tons/in.*, although the ultimate tensile strength was only 22 tons/in.*. Apply- 
ing the redistribution rule indicated, we might deduce 
A, = 3a/U = 16-5/22 = 75 per cent. ; 
but since this ratio is certainly too high, even for such a mild steel, it is inferred that 
local hardening must have occurred in an important degree, raising the fatigue limit 
by perhaps 25 per cent. 
