1908-9.] Internal Friction in Cases of Compound Stress. 431 
expression obtained for this moment when internal friction is allowed for. 
Let M be the bending moment applied to the shaft, and T the simul- 
taneously applied twisting moment. Let M' be the bending moment which 
would have the same maximum effect upon the shaft as M and T together. 
Then, assuming that the strength of the shaft is determined by the 
maximum principal (tensile) stress, 
M' = i(M+ CM 2 + T 2 ). 
This is the rule given by Rankine, and in general use among engineers. It 
would appear to be true only for brittle materials. If, as according to 
Guest, the strength be considered as determined by the maximum shearing 
stress, the expression becomes, 
M'= 
This seems to represent, at least very closely, the conditions for ductile 
materials. Finally, if the effect of internal friction is considered, and the 
strength is supposed to be determined by the minimum resistance to 
sliding, 
M' = -=L-— • {fxM + V(1+^)(M 2 + T 2 )}. 
I* i /!■ i 
And, taking /x = 0T4, as on previous occasions, 
M' = 0T2M + 0-88 X /M 2 + T 2 . 
This last expression gives values for M' which are intermediate between 
those of Rankine and of Guest, though they cannot be said to represent 
actual conditions for ductile metals more accurately than those of Guest. 
Summary. 
The minimum resistance to deformation, and the inclination of the 
surfaces of sliding, are given for any system of stress in a body, supposing 
internal friction to be operative. 
The effect of internal friction in various cases of combined loading, and 
the value of the coefficient have been calculated from experimental data. 
The results do not allow of a definite value being assigned to p for steel. 
Certain differences in the behaviour of ductile and of brittle bodies, 
when loaded, are pointed out. 
An expression for the equivalent bending moment of a shaft submitted 
to simultaneous bending and twisting, when internal friction is allowed for, 
is given. 
{Issued separately July 9, 1909.) 
