Geotechnical Engineering

DHARM

SHALLOW FOUNDATIONS 637

Equation 15.25 is established from experiments and Eq. 15.26 from the pressure bulb
concept.
Effect of shape
For footings with the same width b under the same pressure q and supported on the same soil,
k decreases with increasing length L of the footing.

k =

kbLs(/)

.

1

15

• ...(Eq. 15.27)
  where k = coefficient of subgrade reaction for a rectangular footing, size b × L,
  and ks = coefficient of subgrade reaction for square footing, b × b.
  This indicates that k value for an infinitely long footing is equal to two-thirds of that for
  a square footing.

Effect of detph

The elastic modulus, E, of sand increases with depth and it may be expressed by :
E = c. γ. z ...(Eq. 15.28)

where c = constant, depending on the properties of sand,
γ = unit weight of sand, and
z = depth.

E =

Average stress

Average strain Depth of pressure bulb

=

1

2 q

S/

= cγ (Df + b/2) ...(Eq. 15.29)

∴ k′ = q/S = cγ (1 + 2Df /b) ...(Eq. 15.30)
where k′ = coefficient of subgrade reaction at depth Df.
If Df = 0, k′ = c
∴ k′ = (1 + 2Df /b)(k′ < 2k) ...(Eq. 15.31)
This indicates that the settlement of a footing is reduced to one-half, if it is lowered from
the ground surface to a depth equal to one-half of the width of the footing.

A general equation may now be written to include the effect of size and depth for square
footings:

On granular soils

k = k 1

b

b

F +

HG

I

KJ

03.^2

(1 + 2Df /b) ...(Eq. 15.32)

with k |> 2 k 1 b
b

F +

HG

I

KJ

03.^2

On cohesive soils

The modulus of elasticity for a purely cohesive soil is practically constant throughout the depth.

Therefore, the depth has no effect on the value of modulus of subgrade reaction.

On c – φ soils:

k = ka

b

b

F +

HG

I

KJ

03.^2

(1 + 2Df /b) + kb/b ...(Eq. 15.33)