Solving for 5^ in equation 26, yields: 



2a Q - 2tan9S 



% = ° a • (29) 



D (l+tan 2 6) 



Where: a Q is defined by equation 28; 



S is defined by equation 18; and 

 o is defined by equation 8. 



Because a Q has two roots, it follows that 5^ will also have two roots, one for the 

 active origin of planes and the other for the passive origin of planes, as shown in 

 figure 13. An equation for can be developed, in terms of the defined quantity, 

 by substituting into equation 19: 



b tan6 + y w (Z-Z w ) sin8cos9 = o b tan9 + S. (30) 



With the expressions, equations 29 and 30, defining the origin of planes, the state of 

 stress, (o c ,t ), on any plane of investigation, can now be obtained. The conventions 

 and notation for the typical stress circle are shown in figure 14. 



Figure 14. — Typical stress circle. 



15 



