The general equation for the circle of stresses passing through point (a ,x ) is, 

 therefore: a 



there fore : 



(a tancj>+c) 2 



(5-5 ) 2 + x 2 = 0. (23) 



l+tan z <}> 



The origin of planes, (a^x^), is defined by the intersection of the circle of stresses 

 and the line parallel to the ground surface that passes through the point (a a ,x a ). The 

 general equation for this intersection is found by substituting the expression for x 

 from equation 19 into equation 23: 



(a Q tan4>+c) 2 



(a-a ) 2 + (atanG+S) 2 - = 0- (24) 



l+tan 2 <}) 



Equation 24 can be rewritten to yield a quadratic in a: 



a 2 (l+tan 2 6) + a(-2a +2tan6S) 



(a Q tan<})+c) 2 

 1+tan 2 - 



+ a 2 + S 2 - — = 0. (25) 



Equation 25 will lead to two solutions for a: one solution (a a ) will correspond to 

 the point of known stress (5 a ,r a ); and the other solution (oV) will correspond to the 

 origin of planes (a^,x^). Equation 25 can be solved by means of a "sum of roots" 

 solution (Rosenbach and others 1958), with the following result: 



2a -2tan6S 



o (26) 

 a + ov = • 



l+tan z 8 



By substituting o~ a for a, equation 24 can be rewritten as a quadratic in a Q , in 

 which 5 Q is the only unknown: 



a 2 + a (-2a -2a tan 2 4-2tan(l)c) 

 o o a a 



+ a 2 (l+tan 2 <J>+tan 2 6+tan 2 <j>tan 2 e) (27) 



+ 2a tan6S(l + tan 2 4>) + S 2 (l+tan 2 ^>) - c 2 = 0. 

 Solving equation 27 by means of the "quadratic equation" and rearranging, yields: 



a = a (l+tan 2 <i>) + tan<f>c ± [-a (l+tan 2 <})) -tan<j>c] 



O 3. 13. 



2 



- [ a 2 (l+tan 2 <j)+tan 2 0+tan 2 <j)tan 2 e) , 2 8) 

 + 2a tan6S(l+tan 2 <})) + S 2 ( l + tan 2 <j)) - c 2 ] 



.2^ 



The positive root defines the center of the passive stress circle and the negative root 

 defines the center of the active stress circle. 



14 



