Geotechnical Engineering

DHARM

LATERAL EARTH PRESSURE AND STABILITY OF RETAINING WALLS 479

B b

W

Sliding wedge

D

R

N

f

S

(+)qf

H

Ppyyd(= + )

W

(180° –yqf– – )

(+)qf

H/3

Pp

+d

y

a

A

(a) Sliding wedge (b) Force triangle

C

Fig. 13.24 Passive earth pressure of cohesionless soil—Coulomb’s theory

∴ Pp = W.

sin( )

sin( )

θφ

ψθφ

+

180 °− − −

Substituting for W,

Pp =

1

2 180

2

.sin 2 .sin( ).

sin( )

sin( )

. sin( )
sin( )

γ

α

θα

αβ

θβ

θα

ψθφ

H

+

+

−

+

°− − −

...(Eq. 13.38)

The minimum value of Pp is obtained by differentiating Eq. 13.38 with respect to θ

equating

∂

∂θ

Pp

to zero, and substituting the corresponding value of θ.

The value of Pp so obtained may be written as

Pp =

1

2

γHK^2

. p

where Kp =

sin ( )

sin .sin( )

sin( ).sin( )

sin( ).sin( )

2

2

2

1

αφ

ααδ

θδ φβ

αδ αβ

−

+−

++

++

L

N

M

M

O

Q

P

P

...(Eq. 13.39)

KP being the coefficient of passive earth resistance.

For a vertical wall retaining a horizontal backfill and for which the friction is equal to φ,

α = 90°, β = 0°, and δ = φ, and Kp reduces to

Kp =

cos

cos sin cos .sin

cos

2

2

1 2

φ

φ

φφ φ

φ

−

L

N

M

M

O

Q

P

P

or Kp =

cos

(sin)

φ

12 − φ^2

...(Eq. 13.40)