18 R. T. CHAMBERLIN AND W. Z. MILLER 
since the resistance due to the normal stress is reduced more 
rapidly than the intensity of tangential stress, as the angle is 
lowered from 45°, fracturing so far as determined solely by these 
two factors is made easier with the reduction of angle. But on the 
other hand, the intensity of tangential stress must exceed the 
shearing strength of the material in order to produce fracturing, 
and since this intensity diminishes with a lowering of the angle 
from 45° downward, the angle must not be too low to allow the 
Fic. 6.—Diagrams to show how the relative intensity of the normal (Pz) and 
tangential (Pt) stresses on an oblique section is a function of the inclination of the 
section to the direction of application of the force. In the left-hand figure the direct 
stress exceeds the tangential in intensity; in the right-hand figure, with a lessening of 
the angle between the direction of the force and the plane of the section, the tangential 
stress is the more intense. 
requisite intensity. This is the limiting factor which prevents 
breaking at too low an angle. The breaking plane is thus deter- 
mined by a balance of these three factors. This helps to explain 
why homogeneous material, when subjected to compression, 
so persistently fractures at angles lower than 45°, although the 
shearing stresses reach their greatest intensity in the 45° planes." 
1A mathematical analysis leads to the conclusion that the inclination of the 
surface along which there is the greatest tendency to rupture is lowered from 45° by : : 
where ¢ is the angle of friction, or angle of repose, of the grains of the material. For 
