COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS, 333 
With phase angle 90°, the stress » due to bending and the shear stress gy due 
to twisting may at any instant ¢ be represented thus— 
p =p cos am, q= cos (5+2"5) 
where p, 1s the maximum direct stress due to the cycle of bending, 
BG + 55 nh shear 5 a: By twisting, 
and T is the period of either cycle. 
The principal stresses at the time ¢ will be— 
4 {» cos 2m + / (2 cos? 2m tag? sin? 2m) } 
Now if p,=2q,=2a (say), as in the three tests just mentioned, the principal 
stresses are 
t 
acos 27_+a 
po 
and the maximum shear stress due to the continued action is thus always equal 
to ‘a.’ 
The regions at which the maximum shear stress ‘a’ always exists are, of 
course, at the diametrically opposite skins of the specimen in the plane of the 
bending moment; the planes of thé shear ‘a’ are at right angles -to the skin 
and rotate uniformly, half a revolution made in the time T of one cycle. On 
other planes at 45° to a principal stress (the principal planes also make one half 
revolution in time T), the maximum shear stresses of the cycle vary between 
‘a’ and sg according to the position of the 45° plane with respect to the plane 
of the principal stresses. Comparing this state of stress with the stress-system 
induced by an alternating bending or an alternating twisting separately applied, 
the number of planes exposed to the maximum shear stress ‘a’ is indefinitely 
increased, and correspondingly the gliding planes of a much larger number of 
crystals are subjected to alternating shear stress of maximum value ‘a.’ 
In the tests of Nos. A9 and B17 the phase angle was zero. The range ot 
twisting moment on A9 was constant throughout, while the bending moment: 
was increased by stages. B17 was subjected to constant ranges of both bending 
_ and twisting until fracture occurred. Both tests were continuous, the machine 
_ being run throughout without stopping. 
Discussion of Tables II. and III. 
Tables II. and II1. have been prepared in order that a comparison may be 
_ made conveniently between the stresses and strains (i) in the 90° phase com- 
bined stress tests, and in (ii) tests wherein the bending and twisting were 
_ applied separately. Hollow and solid specimens are put in separate Tables in 
_ view of a previous result of one of the authors, viz. that a solid specimen under 
alternating 
bending or twisting is apparently considerably stronger than a 
hollow one.? The result of the comparison of stresses may be summarised 
—_ 
thus— 
If P be the maximum direct stress of bending cycles separately applied, 
_ if Q be the maximum shear stress of torsion cycles separately applied, 
and if p,, q,, a, have the meaning previously assigned, 
then for the ranges of stress which produced fracture, it was found that 
P>p, and Q>q:. ; 
The maximum shear stress due to bending separately applied being }P, the 
relations between the maximum shear stresses are therefore 3P>aand Q>a. 
It should be observed that these results are got by comparing only two 
‘Specimens tested with combined alternating stress at phase angle 90°. A con- 
Sideration of B31 also supports these conclusions if the smaller number of cycles 
endured by it is considered. It may be remarked, also, in view of the small 
2 * Alternating Stress Experiments,’ Proc. Inst. Mech. Eng., January-May, 
1917, p. 151. The stresses are calculated (see Table I.) from formule which 
assume that stresses are proportional to strains, which gives'a calculated 
maximum stress greater than the actual, more especially for solid specimens. 
