COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 317 
So far as stress concentrations are concerned, the most important numerical 
results which have been obtained by method I. are those relating to the stresses 
in a plate containing an elliptic hole of any eccentricity (1) (and, approximately, 
those in a plate from whose edge springs a small semi-elliptic crack or groove) ; 
and in some cases the stresses in bent and twisted cylindrical shafts, the 
boundaries of whose cross-sections include re-entrant portions. These results 
may further be extended to find approximately the stress concentrations due to 
small semi-elliptic holes or cracks, and also scratches or grooves, formed in the 
surface of members which are not cylindrical, but in which the general stress 
distribution has been found by other methods. ee 
In a further application, the general two-dimensional solution in elliptic co- 
ordinates (1) may be employed to find the stresses in a flat plate having the 
form of a hyperbolic cylinder of any eccentricity, when it is subjected to a 
tensile load directed along the imaginary axis of the hyperbola. 
Method II. (2) may be used to find directly the stresses in a flat plate of any 
shape, to which given edge tractions are applied. As with the previous method, 
concentrations due to small cylindrical holes or cracks, cut normally into the 
surface of a member, may be found, but in this case there is no restriction 
cn the shape of the hole. Further, the effect of cutting any scratch or groove, 
in a surface along which the principal stresses are parallel and perpendicular 
to the direction of the scratch or groove, may be determined. 
There is another possible application of this method which, so far as the 
author is aware, has not yet been used. If a thin plate, initially flat, be bent 
by couples applied at its edge, the equations satisfied by the principal curvatures 
of its surface are mathematically identical with those satisfied by the principal 
stresses in a plate, of the same shape, subjected to appropriate edge tractions. 
Hence, if a photo-elastic experiment be performed in which the applied edge- 
tractions represent, on some convenient scale, the prescribed conditions: of 
curvature at the boundary of the bent plate, then the measured stresses at any 
point in the stretched plate will represent, on the same scale, the curvatures at 
the corresponding point of the bent plate. Hence the stresses in the latter may 
be found. 
Turning now to III. (3), it is possible by this method to find the shearing 
stresses in a bent or twisted cylindrical beam or shaft, having a cross-section of 
any given shape. The weakening effect of any small groove, formed in the 
surface of a member which is not cylindrical, may be found in those cases where 
the stress in the neighbourhood is a shearing stress acting on planes respectively 
parallel and perpendicular to the groove. 
The soap-film solutions for small surface grooves are complementary to those 
obtained by the photo-elastic method, so that a combined method may be used 
to find the effect of such a groove in any case whatever. 
The thermal method depends on the fact that there is a sudden generation 
of heat in soft metals such as mild steel when the stress reaches the yield point. 
The resulting rise of temperature may be detected by means of a thermo-couple 
applied to the surface of the material. It follows that advantage may be taken 
of this phenomenon to determine the magnitude of stress concentrations, and 
that, unlike the other available methods, there is no restriction to two-dimen- 
sional cases. So far, however, very few attempts have been made to develop 
this method. 
In this branch of the subject progress is most urgently needed in the esti- 
mation of stresses in solids of revolution. On the purely mathematical side 
the most promising line of attack seems to be the investigation of solutions 
applicable to ellipsoids and hyperboloids of revolution, and it may be regarded 
as probable that an important advance in this direction will shortly be made. 
As regards solids of revolution in general, what is required is some method 
of the type of the soap-film and photo-elastic methods. 
8. Examples of Stress Concentrations. 
3. Examples of Stress Concentrations.—lt is not proposed to attempt here 
anything like a comprehensive summary of the detailed work which has been 
performed by the foregoing methods. Nevertheless, it is desirable for the 
purposes of the present paper to illustrate by means of examples the general 
nature of the results obtained. 
