A - - (tan 9 + tan (j)) 



^ " (1 + tan2 9) (28a) 



B = cos^ 6 (1 + tan^ 4))^ ■ (28b) 



a 



(tan (fi - tan 9) + c (1 + tan^ 9) (29a) 



b = (tan (f) - tan 9) - tan 9 (29b) 



a' 



= z y (tan 9 + tan ((>) + c (1 + tan^ 9) (29c) 



W X. 



b' = (tan 9 + tan <t>) + tan 9 (29d) 



u = a + bn (29e) 



V = a' + b' n (29f) 



k = a b' - a' b (29g) 



Likewise, from equations 24 and 26, for z <^ ^^^(o^ ^^^^ the case of no seepage) , 



X - Xq = A (Z - Zg) ^ 



- rn v^uv" Bk ^ V- bb ' uv , , 



+ B , tan ( ) (30) 



b b/- bb' bv 



^0 



where Xq and Zq correspond to ground surface (i.e., Xq = Zq = 0) and the (-) and (+) 

 signs correspond to the passive and active cases, respectively. Here A and B are as 

 defined in equation 28, and the remaining parameters and variables are defined as 

 fol lows : 



a 



= c (1 + tan^ 9) 





(31a) 



b 



= (tan (p - tan 



6) 



(31b) 



a' 



= a = c (1 + tan^ 



e) 



(31c) 



b' 



= Y^ (tan 9 + tan 



*) 



(31d) 



u 







(31e) 



V 







(31f) 



k 







(31g) 



Equation 30 can be shown to agree with the solutions obtained using Frontard's 

 approach . 



Note that the above solutions are limited to situations where c > and 



e > tan''^[ — — tan <{)] , where y ^ is the saturated unit weight of soil below the water 

 ^sat 



table (y = y + y 1 For the case of no seepage, the above solutions are limited to 

 'sat 'b 'w 



situations where Q > ^. For conditions other than these, special (and generally sim- 

 pler) solutions can be found. 



11 



