Figure 16 .--Stress condition at the critical depth when the unit cohesion equals zero. 



The equations for critical depth are found by equating stress conditions to 

 strength conditions. Equating the right-hand sides of equations 19 and 13, yields 



atan0 + S = atan<{> + c . 



(42) 



Substituting the expressions for a a and S from equation 8 and 18 into equation 42 and 

 solving for the critical depth, Z > yields: 



cr 



c - y Z cos 2 8 (tan9-tancj>) 

 , t w 



I + = r : 



w Y, cos 6 (tan 9- tan <(>) + y sm9cos( 

 b w 



(43) 



For the special case of no ground water (Z-Z ) is zero. Solving equations 8 and 18 for 



Z : W 

 cr 



cr Y t cos 2 9 (tan8-tan<j>) 



(44) 



which is the critical depth for the special case of no ground water. Note that in all 

 the equations presented, a slope will be stable as long as the depth of the soil mate- 

 rial is less than the critical depth. 



Figure 17 shows the stress condition at the critical depth for a typical slope, 

 This figure is also helpful for visualizing the expressions presented in previous 

 chapters for the stresses acting on plane A-A'. 



19 



