DHARM

468 GEOTECHNICAL ENGINEERING

The total active thrust per unit length of the wall is obtained by considering a net area
of the pressure distribution diagram or, by integrating the general expression for pressure
over the entire height of the wall.

Thus, Pa = 0 σ (^3) c
H
z .dz

γ
φ φ
z
N
c
N
dz
H
−
F
H
GG
I
K
z JJ
2
0

γ
φ φ
z
N
c
N
z
2 H
0
2
2
−
L
N
M
M
O
Q
P
P
.
or Pa =
1
2
γ^22
φ φ
H
N
c
N
− .H ...(Eq. 13.25)
In practice, cracks occur over the entire depth, zc, of the tensile zone, making the backfill
soil lose contact with the wall in that zone.
zc may be got by equating σ (^3) c to zero.
σ (^3) c = γ
φ φ
z
N
c
N
c −=^20
zc =
222 c 45 2
N
N c
N c
φ
φ
.. .tan(/)γγ== °+φ γ φ ...(Eq. 13.26)
If the total active thrust per unit length of the wall is to be obtained ignoring the tensile
stresses, one has to proceed as follows:
Pa =
1
2
2
()Hz
H
N
c
c N
−−
L
N
M
M
O
Q
P
P
γ
φ φ

1
2
H^22 c N H
N
c
N
−
F
HG
I
KJ
−
L
N
M
M
O
Q
P
γ P
γ
φ
φ φ
.
or Pa =
1
2
γ^2222
φ φ γ
H
N
cH
N
−+c
...(Eq. 13.27)
This may also be got by integrating the general expression for active pressure between
the depths zc and H:
Pa =
γ
φ φ
z
N
c
N
dz
z
H
c
−
F
H
GG
I
K
z JJ
2

γ
φ φ
z
N
c
N
z
z
H
c
2
2
2
−
L
N
M
M
O
Q
P
P
.

γ
(^2) φ φ
222
N
Hz
c
N
()()−−ccHz−