COMPLEX STRESS DISTRIBUTIONS IN ENGINEEBRING MATERIALS. 315 
and is essentially negative. Thus the greatest algebraic value of » occurs at theinner 
radius r=a when $= 2(b* —a*)/a and the least algebraic value at the outer :adius 
7 =b where »=(a*—b*)/b. We thus see that @ changes sign between the two and 
that the greatest. numerical value is at 7 =a, leading to: 
an Y, (#-@ 
AbOmax. =(1— No); % s .) ’ 
27a \b* +a" 
from which it appears at once that this corrective stress becomes small if }/a is near 
unity, that is, if the thickness of the ring is small in comparison with its radius. 
’ = 
If we take 7, =0.2, n=0.3 and d/a as large as 1.25, we get A®Onax, = (0.022) Y,/2ma, ie. 
about 2% of a stress equal to the total load distributed uniformly over the internal 
circumference. This last stress will in general itself be small compared with the 
bigger existing stresses, so that, in this case, the correction, al; any rate near critical 
or dangerous points, will be entirely negligible. 
Y oa Fae \ 
Coming now to Arr and A796, we have to make ¢ = (b? — 77) (7? —a@*)/7* a maximum. 
Here we have clearly no change of sign inside the ring and a positive maximum 
occurs between 7r=a and r=)b., 
The value of 7 corresponding to this maximum is readily found from : 
V (a +b)? + 1247? a? 1? 
= 5 
This, however, leads to a somewhat awkward algebraic expression for the 
maximum stress corrections, involving lengthy radicals, and since we are really 
chiefly concerned with finding an upper limit for the stress corrections, we notice 
that we necessarily increase ¢, if in the denominator we replace r* by its least 
possible value a*. The greatest value of (b?—7*) (7*—a*) is then well known to 
occur when 7?=(a’+0?)/2, so that the maximum value of ¢ is certainly less than 
2 @2\2 
4H" = 2 , leading to 
Y, (=a?) 
e "8, 
A7?' max, = 47 <(n- 3 
max, max. < (7 1) oan Sa? (a? + b%) 
Since this contains the square of J?—«a? it leads in general to an even smaller 
“o™ 
correction than for the stress 0@. Taking the same values of 7, 7, and b/a as before, 
we find 7 
Ar?max. = A79max, < (0'00154)Y,/2ma 
which will usually be entirely negligible. 
The investigation of the case of the circular ring therefore indicates that the 
correction due to variation in the ratio of the elastic constants is usually very small. 
The more general theorems previously obtained show how, even when this correction 
cannot be neglected, or computed mathematically, it can nevertheless be allowed 
for if a suitable experimental exploration is undertaken. This justifies the use of 
xylonite models for the exploration of stress, even when such models are multiply 
connected. 
TABLE OF REFERENCES. 
(1) J. H. Micwexu. —On the direct determination of stress in an Elastic Solid, with 
application to the Theory of Plates. (Lond. Math. Sve. Proc. 1899, vol. 30.) 
(2) A. TrMPE.—Probleme der Spannungsverteilung in ebenen Systemen einfach 
gelcst mit Hilfe der Airyschen Funktion. (Doctoral Dissertation, Géttingen: 
Leipzig, 1905, B. G. Teubner.) 
(8) Viro Vo.TERRA.—Sur l’€quilibre des corps élastiques multiplement connexes. 
Paris, Annales Scientifiques de VEcole Normale Supérieure, 1907, Series 3, 
vol. 24.) 
(4) A. E. H. Love.—The Mathematical Theory of Elasticity, 3rd edition, 1920, 
pp. 219 et seq. 
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