The angle, 3, shown in figure 2, can be expressed as 



6 = tan [(1 + — ) tan 6] (9) 



and the seepage force per unit area, S, is given by 



S = Y sin e (z - z ) cos e (10) 



W W 



where z is the vertical depth below ground surface to the point being considered, z^ is 

 the vertical depth below ground surface to the phreatic surface, and y is the unit 

 weight of water. 



By defining the angle, a, as shown in figure 3, Hartsog and Martin derived the 

 following general expressions for the stresses a and t on a plane oriented at any 

 angle, a, at depth z: ^ 



2 + (- 1 + tan^ ot - 2 tan a tan 9) - 2 S tan a 

 a = .... 



1 + tan^ a 



T = T, + (a - a, ) tan a (12) 



c b c b 



where 



2 Oq - 2 S tan 9 



(1 + tan2 9) ^ 



= tan G + S (14) 



^a " ^t ^w 9 + (z - z^) cos2 9 (15) 



Oq = (1 + tan^ ^) + c tan <}> j 



2 ' 



± {[ - a (1 + tan2 cj)) - c tan (J)] 



Si 



- [a 2 (1 + tan^ <^ + tan^ 6 + tan^ ()> tan^ 9) 



3. 



+ 2 a S tan 9 (1 + tan^ (})) 



3. 



+ S2 (1 + tan^ <|)) - c^]}''^ (16) 



In the above expressions, is the unit weight of the soil above ground water level 

 and is the buoyant unit weight of the soil below ground water level. The (+) and (-) 

 signs in equation 16 correspond to the passive and active stress circles, respectively. 



6 



