In 1922, Frontard (1922) succeeded in solving equation 4 for the coordinates of 

 the passive failure surface londer the conditions of the Rankine-Resal stress assump- 

 tions. His solution was presented as follows: 



c cos (}) tan (Aq cos e + cos (}> sin Ag 



- A cos 6 - cos <t> sin A) 



x- = f^^) 



Y sin (0 - (t>) /sin (0 - (j)) sin (6 + (j)) 



, _ c cos (j) tan .- sin (j) 



^ ~ y sin (9 - (})) sin (9 + (t>) '■tan 9 . 



- /sin (0 - (j)) sin (9 + cf)) sin A + cos cos A] (5b) 



where x' and y' are the coordinates in the directions parallel to the ground slope and 

 vertically below ground slope, respectively. 



The parameter A is defined by the expression 



A /sin + sin <J) ^ . , 9 - (1)^ . 



tan = /— : 5 -. r tan (a' ::r-^) (6a) 



2 V sm - sm ()) ^ 2 ^ ^ 



Corresponding to the coordinates x' = y' = 0, the angle a' = tt/4 - (J)/2 in the passive 

 case and accordingly, the parameter Aq is defined by the expression 



Ag /sin + sin (j) ^ ,tt 9 , , , 



tan = /— : r : ^ tan (t - t) C^b) 



2 V sm 9 - sm (|) 4 2 ■ 



By using Frontard 's approach, similar expressions can be derived for the active 

 failure surface. They are as follows: 



c cos <t> tan (Aq cos + cos (j) sin Aq 



- A cos - cos (j) sin A) , . 

 x' = (/a) 



Y sin (0 - (j)) /sin (0 - i>) sin (0 + i>) 



, _ c cos (}) tan 9 r sin <l> 



^ " Y sin (9 - 4)) sin (0 + <l>) '■tan 



+ /sin (0 - i>) sin (0 + (}>) sin A + cos cos A] (7b) 



A /sm 9 + sm * ^ ^ , 9 - *, .„ . 



tan ^ = —. -. f tan (a' + — ^-^) (8a) 



2 Vsm - sm (}> 2 ^ 



'^0 /sin + sin d) ,Tr 9, , 



tan ^ = /— : : f tan (t + ^) (8b) 



2 v sm 9 - sm (}) M 2"^ 



4 



