386 REPORTS ON THE STATE OF SCIENCE, ETC. 
V. 
The Distribution of Stress in Round Mild Steel Bars under Alternating 
Torsion or Bending. 
By Professor W. Mason, D.Sc 
Abstract. 
The distribution of stress in round bars overstrained in alternating torsion or 
bending is studied with the object of finding a relation between stresses in hollow 
and solid specimens. 
The probability of linearity of strain distribution is discussed. 
Alternating torsion or bending of round bars leads to a condition in which the 
range of strain, at a given range of torque or bending, increases to a stationary value, 
provided that such given range is less than the fatigue range and greater than the 
primitive elastic limit range. 
For mild steel in this stationary condition of strain the following assumptions are 
made concerning the distribution of strain and stress throughout the body of the 
specimen at instants of maximum strain of the cycle: 
(1) A distribution of strain linear from axis to skin. 
(2) (a) A distribution of stress conforming to Hooke’s law (7.c. of perfect elasticity) 
from the axis for a portion of the radius. (b) For the remainder of the radius, a 
relation between stress and strain similar to the linear relation found experimentally 
between the ranges of strain and stress in thin-walled tubular specimens in alternating 
torsion. 
Applying these assumptions, it is found that the results of certain alternating tests 
of solid specimens in torsion become practically identical with those of hollow 
specimens, and that the results of the author’s alternating tests of hollow and solid 
specimens in bending (bending always about the same axis) are in fair agreement. 
The author concludes that the difference between (a) fatigue stresses calculated 
from the range of torque or bending moment on the usual assumption of linearity of 
stress, and (b) the actual fatigue stresses, must be greater than is commonly supposed 
for solid specimens, and also for hollow specimens in alternating bending. 
Aim of the Work. 
When the elastic limit of a round bar has been exceeded in bending or torsion, 
the stress in it is unknown, except in the case of a hollow bar with thin walls under 
torque. If the stresses are cyclic, asin a torsion or bending fatigue test, the range 
of stress throughout the body of the specimen follows a straight line distribution only 
so long as the elasticity is unimpaired. If the ranges of stress at the skin of the 
specimen are calculated, as is usual, by a formula which is founded upon linearity 
of stress, considerable error results, even when the ranges of stress are of the magnitude 
that produce fracture by fatigue. One object of this paper is to form an estimate 
of the amount of this error in accordance with certain experimental data. 
Linearity of Strain Distribution. 
Imagine a wedge bounded by a pair of planes passing through the axis of a round 
bar and by a pair of planes perpendicular to the axis of the bar. Suppose that the bar 
is overstrained in torsion. Linearity of strain entails that the latter pair of planes 
AA 
Fig. 14 
remain plane after the overstraining, and that the radial lines of intersection of the 
two pairs of planes remain straight lines. Suppose these radial lines do not remain 
straight, but take any deviation from the radial direction such as shown by the dotted 
lines in fig. 14. Now at each section normal to the axis the relative position of lines 
