The general equation for the slope of the line, GH, that passes through point 

 (au,^) , as shown in figure 14, is: 



T " T b 



tan<x = — • (31) 

 -°b 



Therefore, the equation of the line that passes through point (a^,x^), and has a slope 

 of is: 



t = t, + (d-d )tan<*. (32) 



The state of stress on a plane that is oriented at an angle <* can be obtained by 

 developing an equation for the intersection of the stress circle' and line GH. The 

 equation for the intersection is found by substituting the expression for t from equa- 

 tion 32 into equation 23: 



(o-o Q ) 2 + [x b + (a-a b ) tan-] 2 - — — = 0. (33) 



(a^tan<j)+c) 2 

 l+tan 2 <)> 



Equation 33 can be rewritten to yield a quadratic in a 



a 2 (l+tan 2 cc) + a(-2a +2x 1 tan° c -2a 1 tan 2a 

 o b b 1 



+ (a 2 +x, 2 -2x, a, tancc+a 2 tan 2 <x) (34) 

 o b b b b 



(a tand)+c) 2 



= 0. 



l + tan z <i) 



The two roots of equation 34 are determined by the "sum of roots" technique. One root 

 is the normal stress at the origin of planes, Ou, while the other root is the normal 

 stress on the plane of investigation, a : 



25 -2t, tan a: +2a,_tan 2 cc 



a = a, (35) 



c l + tan 2 - b 



Because all terms on the right-hand side of equation 35 have been previously defined, 

 the only unknown, a c , is expressed in terms of defined parameters. An alternate form 

 of equation 35 can be written by substituting the expression for from equation 30 

 into equation 35: 



2a + a, (- l+tan 2 --2tan° c tan0) -2tan a: S 



o b v (36) 



a - • — ■ — ■ • 



c l+tan 2 - 



An equation for x c can be developed by substituting o" c , into equation 32: 



x = x, + (a -a,) tan«. (37) 

 c b c b 



Equations 36 and 37 are general equations for calculating the normal and tangential 

 stresses on a plane at any inclination, when the soil within an infinite slope is at 

 the active or passive state of stress. 



16 



