Figicre 2. --An elemental volume of soil, AA'BB', within an infinite slope. 



The slope profile with an elemental volume of soil (AA'BB') is shown in figure 2. 

 If a unit dimension is assumed normal to the plane of figure 2, and if AA' is assumed 

 to be of unit length, the resulting plane (which shall be referred to as plane A-A') 

 will be of unit area. The force, W, acting on plane A-A', is equal to the weight of 

 the vertical column of soil above plane A-A'. 



The equation for the vertical force, W, is found by adding the products of 

 respective unit weights and volumes within the vertical column of soil: 



W = y Z cos9 + y .(Z-Z )cos9. (1) 

 t w sat w 



Equation 1 is valid for the general case; that is, plane A-A' is below the ground 



water table. When plane A-A' is above the ground water table, the quantity (Z-Z ) is 



zero and Z is replaced with Z. 

 w r 



The vertical force, W, can be resolved into components normal (N) and tangential 

 (T) to plane A-A', as shown in figure 3. The expressions for the normal force and the 

 tangential force are, therefore: 



N = y Z cos 2 9 + y .(Z-Z )cos 2 9, (2) 

 t w 'sat w 



T = yZ sinGcosO + y .(Z-Z )sin6cos6. (3) 

 t w sat w 



The normal stress, a & , shown in figure 4, is equal in magnitude to the force N, 

 because plane A-A' is of unit area. It follows, from equation 2, that: 



a = yZ cos 2 9 + y .(Z-Z )cos 2 9. (4) 

 a 't w sat v 



