DHARM

LATERAL EARTH PRESSURE AND STABILITY OF RETAINING WALLS 469

Substituting for zc from Eq. 13.26,

Pa =

γ

(^2) φ γ φ φ γ φ
2 4222
N H^2
c N c
N
− H c N
F
HG
I
KJ
−−
F
HG
I
KJ
.

1
2
2
2
γ^2242
φ γγφ
H
N
ccH
N
c
−− +
or Pa =
γ
φ φ γ
H
N
cH
N
(^22) c
2
−+^22 ,
as obtained earlier.
For pure clay, φ = 0
∴ Pa =
1
2
2
2 22
γ
γ
HcH
c
−+ ...(Eq. 13.28)
This acts (H – zc)/3 above the base.
The net pressure over depth of 2zc is obviously zero. This indicates that a cohesive soil
mass should be able to stand unsupported up to this depth which is known as the critical
depth.
The critical depth Hc, is given by
Hc = 2zc =
4 c
N
γ φ

. ...(Eq. 13.29)

If φ = 0, Hc =

4 c

γ

...(Eq. 13.30)

Equations 13.24 and 13.26 may be derived from the geometry of the Mohr’s circles IV

and III respectively, instead of from Eq. 13.23. The proofs are let to the reader.

13.6.9 Passive Earth Pressure of Cohesive Soil
Cohesion is known to increase the passive earth resistance of a soil. This fact can be math-
ematically demonstrated from the relationship between the principal stresses that may be
derived from the geometry of the Mohr’s circle relating to the passive case for a c – φ soil,
taking cognizance of the fact σ 3 = γ.z and σ 1 = σh (Fig. 13.16).

σ 1 = σ 3 NcNφφ+^2

σ 3 = γz and σ 1 = σhc

∴ σσγ (^1) cc== +h zNφφ^2 c N ...(Eq. 13.31)
(Here, Kp = Nφ in the usual notation).
The pressure distribution with depth is shown in Fig. 13.17.
The total passive resistance per unit length of wall is PP = PP' + PP" = 21 γH^2 Nφ +^2 cH Nφ.
PP′ acts at H/3 and PP" acts at H/2 above the base. The location of PP may be found be
moments about the base.