GENERAL STRESS SYSTEM. 



41 



the twelve tangential forces must consist of six couples, each tending to 

 produce rotation. 



In this paper consideration is confined to those cases in which there is 

 a tendency to rotation only about the line o t, and this limitation elimi- 

 nates four of the couples. Thus the case to be considered here consists 

 of three pairs of normal forces and two unequal couples tending to pro- 

 duce rotation in opposite directions. This force system is shown in the 

 following diagram : 



& 



->c, 



-*T 



Figure i. — System of Forces. 



It has already been shown that any normal force, whether finite or 

 infinitesimal, is resoluble into a dilation and two shears, exactly one- 

 third of the force producing dilation, and the remainder producing two 

 equal shears at right angles to one another. Analyzing each of the nor- 

 mal forces P, Q, R separately, it will appear that the action of all of them 

 may be tabulated as two shearing stresses and a dilation— thus : 



Axes of .'■ >i : 



Dilation 

 Shear . . 

 Shear . . 



Sum . . . 



':!/'+ Q+ R) 



KQ + R-2P) 







'; IP+Q + R) 



}(Q+ R — 2P) 

 I (Q+ P — 2R) 



i(P+ Q+ R) 





 HQ + -P-2 R) 



Q 



i; 



Turning now to the coiiples <\ and C 2 , and supposing Q x > C!,, their 

 combination is equivalent to two equal and opposite couples, each equal 



