318 REPORTS ON THE STATE OF SCIENCE, ETC. 
In what follows it will be convenient to use the term ‘concentration factor ’ 
to denote the ratio of the true calculated stress to the stress which would be 
calculated if the concentration of stress were to be neglected. 
A very simple example is that of a small round hole, such as an oil hole, 
drilled normally into the surface of, say, a twisted or bent circular shaft. In 
the twisted shaft the tensile stress concentration factor is 4, while the shear 
stress factor is 2. In the bent shaft the tensile and shear factors are both 3. 
These results are practically independent of the ratio of the size of the hole to 
the size of the shaft, unless the former is so large as to reduce materially the 
cross-section of the shaft. 
Similar concentration factors are found in cases where holes are drilled in 
members of other shapes. 
As another illustration we may take a small groove of semi-elliptic cross- 
section, cut in the surface of a twisted shaft. If p is the radius of curvature 
at the bottom of the groove and a is the depth, the shear stress concentration 
factor is 
tig /A a 
p 
This result is true whatever the angle between the direction of the groove and 
the axis of the shaft. The tensile stress factor, on the other hand, varies from 
ae ae 
if the groove is parallel or perpendicular to the axis, to 
if the groove runs round the shaft in the form of a 45° spiral. 
It may be remarked here that work on grooves of other shapes has shown 
that the stress concentration depends mainly on the depth and the radius of 
curvature at the bottom, provided the groove is not very shallow. For instance, 
a 60° Vee groove with a rounded corner gave factors only a few per cent. less 
than those calculated from the foregoing formule. 1t appears, therefore, that 
the latter may be used even when the groove departs considerably from the 
elliptic form. 
It will be seen from the above results that stress concentrations may 
theoretically be quite large in cases likely to occur in practice. Thus, if the 
depth of a groove is sixteen times the radius of curvature, the concentration 
factor may vary from 5 to 9, while in the extreme case of a surface crack even 
these values may be greatly exceeded. 
The theoretical stresses are lower in the case of a continuously grooved 
surface such as a screwed portion than in an isolated groove. For example, in a 
particular screwed rod the tensile stress concentration factor was estimated to 
be 3.7, while for an isolated groove of the same shape the factor was found to 
be 4.9. Possibly this fact partly explains why it is found to be of value to turn 
down the plain portion of a screwed bolt to the root diameter; if this is not 
done the stress concentration in the end groove is considerably greater than in 
the rest of the screwed part. 
As in the case of the circular holes, the above results are substantially 
independent of the size of the grooves unless these are so large that a material 
amount of metal is removed in their formation. 
_A striking consequence of this fact is that severe stress concentrations should 
arise from the scratches and other surface defects left on machine parts by the 
ordinary processes of manufacture. As will be seen later, the non-fulfilment of 
this prediction is one of the outstanding discrepancies between the theoretical 
aud experimental aspects of our subject. 
4. Practical Limitations of the Foregoing Methods. 
In practice it is not mfrequently found that the application of the foregoing 
results, on the basis of the usual criteria of failure, gives a value for the weaken- 
ing due to stress concentration which is misleading or entirely wrong. In other 
cases the agreement is quite good enough for practical requirements. It is 
