Equation 41 is exactly the same (except for notation) as Martin's equation for the 

 normal stress on a vertical plane. Therefore, the general equations given in this 

 analytical development agree with equations given by other investigators for their 

 more restrictive conditions. 



Critical Depth 



The soil properties, slope of ground surface, depth of the water table, and depth 

 to the plane of investigation will all affect the stress condition (o a ,x a ). Active and 

 passive stress circles can be drawn through point (a a ,x a ) if, and only if, this stress 

 condition remains within the Mohr envelope. The Mohr diagram, with the stress condi- 

 tions (o a ,r a ) superimposed, shows conditions where failure is impending. The critical 

 depth can be defined as the depth at which the stress condition on plane A-A' is equal 

 to the strength of the soil. When the stress condition (c a) T a ) lies on the Mohr enve- 

 lope, only one stress circle may be drawn. For this condition, the active and passive 

 stress circles coincide, and failure is impending. 



The stress on plane A-A', (a , x ), at the critical depth, in a cohesive material, 

 is shown in figure 15. If: 



1. $>ji>_6 , no critical depth exists; 



2. 3><|>>_9, a critical depth exists; and 



3. 8_>8><}>, a critical depth exists. 



The stress on plane A-A', (a„,x a ) , at the critical depth, in a cohesionless mate- 

 rial, is shown in figure 16. If: 



1. <}>>g>_9, no critical depth exists; 



2. <f>=8>8, no critical depth exists; 



3. c}>=B = 0, a failure condition exists at all depths; 



4. g>cf)>0, a critical depth exists; and 



5. 8>9>(}>, a failure condition exists at all depths. 



Figure 15. — Stress condition at the critical depth when the unit cohesion 



is greater than zero. 



18 



