Geotechnical Engineering

DHARM

CAISSONS AND WELL FOUNDATIONS 797

From equations 19.40 and 19.43,

P =

()

()

HWmKI.

D

+ vv

+′

βμ =

βμμ

θ

1

2

Using Eq. 19.46,

P =

2 mI

D

v(/) /MI Mr= ...(Eq. 19.48)

where r = (/)D.

I

mIv

2

Also H + βμW =

M

r

()1+′βμμ

Simplifying,

H + βμW =

M

r

M

r

+

βμμ′

or βμW μμ

M

r

FHG − ′I

KJ

= M

r

−H

or β =

(/)

()

Mr H

W M

r

−

L −′

NM

O

QP

μμμ

...(Eq. 19.49)

As – 1 < β < 1, we have

M

r

WHM

r

F ()+′− ()11W

HG

I

KJ

<<F −′+

HG

I

KJ

μμ μ μμ μ

The vertical soil reaction is given by

σz = Kvθ(x + xc)

Also W – μ′P = σθzv cdA=+K zz ()x x dA

or W – μ′P = KvθxcA

or Kvθxc =

()WP

A

−′μ

∴σz = Kx

WP

v A

θ +()−′μ ...(Eq. 19.50)

The stresses at the toe and the heel are given by

pt =

()WP

A

−′ +KvFB

HG

I

KJ

μ θ

2

...(Eq. 19.51 (a))

ph =

()WP

A

−′ −Kv FB

HG

I

KJ

μ θ

2

...(Eq. 19.51 (b))

Substituting the value of Kvθ from Eq. 19.45,

pt =

()WP

A

MB

I

−′μ +

2

...(Eq. 19.52 (a))

ph =

()WP

A

MB

I

−′

−

μ

2

...(Eq. 19.52 (b))
For the soil to remain in the elastic state, the maximum pressure at any depth should
not exceed the passive resistance.