Coulomb derivation based on limit equilibrium approach for Passive Case

Figure 1 below shows the cross section of a retaining wall.

 coulomb's passive earth pressure

Weight of Wedge ABC  

From Δ ABC,

Area of Δ ABC = ½ AD × BC

BC/AB = {sin(α + β)/ sin(Ï´ – β)}

BC = AB {sin(α + β)/ sin(Ï´ – β)}

Again,        AD = AB sin[1800 – (α + Ï´)] = AB sin(α + Ï´)

columb active earth pressure formula 1  

columb active earth pressure formula 2

 

 

columb active earth pressure formula 3

 

 

Triangle of Forces for W, Pp and Rtriangle of force in passive case

From the sine rule,

columb passive earth pressure formula 2  

 

Substituting the value of W from the equation (3.34) into (3.35) we get,

columb passive earth pressure formula 3

 

In order to get the maximum value of Pp,

formula of passive pressure 2  

 

columb passive earth pressure formula 4

 

 

columb passive earth pressure formula 6

 

 

Where,

columb passive earth pressure formula 5

 

 

 

 

Note: When β = 0 (leveled backfilled), δ = 0 (no wall friction), and α = 900 (vertical interface between wall and backfill), then ka = Ka = (1 – sinɸ)/(1 + sinɸ). The point of application of Pp is at a distance of H/3 above the base of the wall.