GENERAL SOLUTION 



In figure 3, the point of tangency of the stress circle with the strength envelope 

 is defined by the coordinates (a , t^^) . The tangent of the angle, af, of the line 

 connecting the origin of planes tOP) and the point of tangency can be expressed as 



m = tan a- = — (17) 



t a - a, 

 X b 



It can be shown that 

 - c tan ({) 



5 = ^^^^ 



^ 1 + tan 9 



and 



Oq - c tan (|) 



= f — : — 2~-^ tan ^ + c (19) 



X 1 + tan^ (f) 



Combining equations 13, 14, 18, and 19, the expression for m can be written: 

 0^ [tan <}) (1 + tan^ e) - 2 tan e (1 + tan^ (j)) ] 



+ a [tan 6 (1 + tan^ (})) (1 + tan^ 6)] 



3. 



+ c [(1 + tan2 (j)) (1 + tan^ 6) 



- tan^ (|) (1 + tan2 e) ] 

 + S [2 tan^ e (1 + tan2 (\>) 



- (1 + tan2 (()) (1 + tan2 6)] 



m = 



a„ [(1 + tan2 6) - 2 (1 + tan^ <\>)] 



+ a [(1 + tan^ <j)) (1 + tan^ 9)] 



3. 



- c [tan (]) (1 + tan^ e) ] 

 + S [2 tan 9 (1 + tan^ <i>)] (20) 



8 



