Geotechnical Engineering

DHARM

474 GEOTECHNICAL ENGINEERING

Pa

y

yad=( – )

W

R

(180° –yqf– + )

N (–)qf

f

S

R

W

Sliding

wedge

B b Ruptureplane or

sliding

surface

D

H

(–)qf

Surface of the fill

Wall

+d

H/3Pay

a

Vertical

q
A
(a) Sliding wedge (b) Force triangle
Fig. 13.19 Active earth pressure of cohesionless soil—Coulomb’s theory
The triangle of forces is shown in Fig. 13.19 (b). With the nomenclature of Fig. 13.19,
one may proceed as follows for the determination of the active thrust, Pa:

W = γ (area of wedge ABC)

∆ABC =

1

2

AC. BD, BD being the altitude on to AC.

AC = AB.

sin( )

sin( )

αβ

θβ

+

−

BD = AB. sin (α + θ)

AB =

H

sinα

Substituting and simplifying,

W = γ

α

θα αβ

θβ

H^2

2sin^2 .sin( ).

sin( )

sin( )

+ +

−

...(Eq. 13.3)

From the triangles of forces,

Pa

sin(θφ− )

=

W

sin( 180 °−ψθφ− +)

∴ Pa = W.

sin( )

sin( )

θφ

ψθφ

−

180 °− − +

Substituting for W,

Pa =

1

2 180

2

2

γ

α

θφ

ψθφ

θα αβ

θβ

H

sin

. sin( )
sin( )

.sin( ).sin( )

sin( )

−

°− − +

++

−

...(Eq. 13.32)

The maximum value of Pa is obtained by equating the first derivative of Pa with respect
to θ to zero;

or

∂

∂θ

Pa

= 0, and substituting the corresponding value of θ.