Geotechnical Engineering

DHARM

LATERAL EARTH PRESSURE AND STABILITY OF RETAINING WALLS 475

The value of Pa so obtained is written as

Pa =

1

2

1

2

2

2

γ 2

αφ

ααδ

φδ φβ

αδ αβ

.. sin ( )

sin sin( )

sin( ) sin( )

sin( ) sin( )

H +

−+

+−

−+

L

N

M

M

O

Q

P

P

...(Eq. 13.33)

This is usually written as

Pa =

1

2

γHK^2

. a,

where Ka =

sin ( )

sin .sin( ) sin( ).sin( )

sin( ) sin( )

2

2

2

1

αφ

ααδ φδ φβ

αδ αβ

+

−+ +−

−+

L

N

M

M

O

Q

P

P

...(Eq. 13.34)

Ka being the coefficient of active earth pressure.

For a vertical wall retaining a horizontal backfill for which the angle of wall friction is
equal to φ, Ka reduces to

Ka =

cos

(sin)

φ

12 + φ^2

...(Eq. 13.35)

by substituting α = 90°, β = 0°, and δ = φ.

For a smooth vertical wall retaining a backfill with horizontal surface,

α = 90°, δ = 0, and β = 0;

Ka =

1

1

−

+

sin

sin

φ

φ

= tan^2 (45° – φ/2) = 1/Nφ,

which is the same as the Rankine value.

In fact, for this simple case, one may proceed from fundamentals as follows:

H

Pa

q

f

R

W

Pa

R

W

(–)qf

(a) Sliding wedge (b) Triangle of forces

Fig. 13.20 Active earth pressure of cohesionless soil special case: α = 90°, δ = β = 0°

With reference of Fig. 13.20 (b),

Pa = W tan (θ – φ),

W =

1

2

γφH^2 .cot

∴ Pa =

1

2

γθθφH^2 cot tan( − ) ...(Eq. 13.36)