COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 351 
Step-by-step integration of the stress-equations 
Oxx , Ory _, Oxy , Oyy _ 
BPN RIG Tay FN Ie or onivg Bo weith gutbaP) 
along straight lines parallel to the axes then gives xa and yy at every point if their 
values (as is usually the case) are known at the boundary. 
The weakness, however, of this last method in practice is that the observations 
—~ 
are never really accurate enough for the differential coefficients oy So be deter- 
mined with sufficient exactness from the values of xy given by (1), using the observed 
values of @ and P — Q. 
For these reasons neither of the methods given in the report referred to has been 
employed in practice to obtain the stress-system. 
The method at present generally used is one which has been described fully by 
Prof. Coker in various places, in particular in his Presidential Address to Section G 
of the British Association in 1914 (Report, p. 495), and which was originally suggested 
by Mesnager. It consists in obtaining the sum P + Q of the average principal 
stresses in the plane of the plate by measuring the lateral contraction of the plate 
at the point in question. 
This method has given good results, but it is not, from the theoretical standpoint, 
entirely unexceptionable, for in the calculation it is assumed that the material is 
perfectly elastic, with a definite Poisson’s ratio, which has to be measured by a separate 
experiment. But, as a matter of fact, the experiments of Filon and Jessop (Phil. 
Trans., A., vol. 223, pp. 89-125) have shown that celluloid exhibits considerable strain 
ereep for limits of stress well below those which occur in the large majority of photo- 
elastic experiments, and this introduces an undesirable element of uncertainty into 
the results, unless, as in Coker’s investigations, the material employed is selected 
with the greatest care. 
Moreover, it should theoretically be possible to deduce the stresses completely 
from observation of the isoclinic and isochromatic lines alone, and the introduction 
of a strain measurement really brings in superfluous data, with a possibility of in- 
consistencies which it may be difficult to trace to their source. 
Finally, the measurement of such minute lateral contractions is one of extreme 
delicacy, and very few investigators have at their disposal the necessary apparatus 
for carrying it out. 
For the above reasons it appears of importance to describe in some detail a practical 
method of deriving the stresses in a transparent plate directly from the isoclinic and 
isochromatic lines, and to give an actual.example of its application. 
It is believed that this method is free from the defects inherent in the other method 
of step-by-step integration, described in the 1914 B.A. Report, p. 206, and referred 
to previously. : 
2. The method is based upon equations given by A. Mesnager (Annales des Ponts 
et Chaussées (partie technique) ; Sér. 9, tome 16, vol. 4, pp. 135-186) for the space- 
rates of change of the principal stresses, taken along the lines of principal stress. 
If we denote by s,, sz, arcs taken along the lines of principal stress corresponding 
to stresses P, Q respectively (fig. 3), ds, being obtained from ds, by a counter-clock- 
wise rotation of 90°; and if p,, p, are the radii of curvature of the two lines of principal 
stress, being measured positive when the tangents to the curves rotate counter- 
clockwise as the arcs s1, s increase, we have the equations 
oP P—Q 
OF =i : P : : ele 
pit : (3) 
0Q Bae es 4 
a, t i =0 : : : . . (4) 
These equations are very readily obtained by considering the equilibrium of a 
‘ curved elementary rectangle’ bounded by four near lines of principal stress, and 
expressing the conditions that the total force resolutes parallel to the tangents to the 
lines of principal stress at one corner A are zero, it being assumed that the plate 
is under no ‘ body-force.’ 
1923 BB 
