66 ROYAL SOCIETY OF CANADA 



time to time, and the one most .generally used by engineers is based upon 

 the assumption, that when one of the principal stresses reaches a limit 

 determined by the material, failure takes place. This theory has been 

 adopted in general by engineers following the lead of Poncelet and 

 Eankine. 



Another theory assumes that the material fails when a certain maxi- 

 mum strain has been reached, while a third theory adopts the view that 



yielding takes place when the shear stress 

 exceeds a certain value. Each of these theo- 

 ries satisfactorily explains the lowering of the 

 yield point in simple tension by a simulta- 

 neous bending moment, but they yield 

 different formula for purposes of calculation 

 of working strength. 



If we consider the equilibrium of a tri- 

 — Û '_j angular element of a shaft of unit thickness, 



^ ' ^ then the bending moment will produce a 



normal component of stress p on the face AB, while the torsional 

 moment produces stress of value q on the faces AB, BC. 



If on a third plane AG, the stress (p: ) is a normal (principal) 

 stress, then since the element ABC is in equilibrium, we obtain the 

 Equations of condition. 



Pn — P = q cot 6 (1) 



p„ = q tan B (2) 



and eliminating 6 we obtain 



p^ = p/-l± s/ p'A 4- 9' 

 where the — sign denotes the lesser principal stress say p\. 



According to the maximum stress theory the effect of p\ is neglig- 

 ible and failure takes place when |?„ ^ /, where / is the working strength 

 of the material. For a circular shaft writing 



p = M/\ 7t r 



and by analogy 

 we obtain 



jt)„ = ^ / ^ ;r r' 



M = (ii-f \/M^-\- T') I 2 



the formula generally used by engineers. 



The greater strain theory takes into account the effect of the lesser 

 principal stress, and if e^ is the maximum strain in tihe direction of the 

 greater principal stress then when failure takes place we have 



Pn—Pn/m = Ee^>^f 



