Geotechnical Engineering

DHARM

796 GEOTECHNICAL ENGINEERING

The moment of P about the base, Mp, is given by

Mp = LDzdzx

D

z .(σ−)

0

= LmK z D D zv dz

D

z .. (/ )(− )..^2

0

θ

or Mp = m.Kv.θ. Iv ...(Eq. 19.41)

Pressure Distribution at the Base

The vertical deflection at a distance (x + xc) from the centre of rotation is given by

ρ = (x + xc). θ

∴ Vertical soil reaction, σz = Kv(x + xc)θ

Moment about the base, MB = Kx x xdAvc

B

B

()..

/

/

+

z−

θ

2

2

= KxdAKxxdAvvc

B

B

B

B

θθ^2

2

2

2

2

+

− zz −

.

/

/

/

/

Since the reference axis passes through the centroid of the base second term vanishes.

∴ MB = Kv.θ. IB ...(Eq. 19.42)

where IB = Second moment of area of the base about an axis passing through the centroid and

perpendicular to the horizontal force, H.

Applying

Σ Horizontal forces = 0 for static equilibrium,

H + βμ.W – βμμ′P = P

or P(1 + βμμ′) = H + βμW

or P =

()

()

HW+

+′

βμ

1 βμμ ...(Eq. 19.43)

Taking moments of all the forces about the base,

MB + HD = MB + Mp + μ′P(αD)

where αD is the distance from the axis passing through the centroid of the base to the point at

which the resultant vertical frictional force on the side acts normal to the direction of the

horizontal force (= B/2 for rectangular wells and 0.318 B for circular wells).

The above equation may be written as follows:

Mo + HD = KvθIB + mKvθ. Iv + μ′αD

(.)2mK I

D

vvθ

or Mo + HD = Kvθ[IB + mIv(1 + 2μ′α)] ...(Eq. 19.44)

or Kvθ =

MHD

ImI

o

Bv

+

[()]++′ 12 μα

or Kvθ =

μ

[()]ImIBv++′ 12 μα

...(Eq. 19.45)

or Kvθ = M/I ...(Eq. 19.46)

where M = Mo + HD

and I = IB + mIv (1 + 2μ′α) ...(Eq. 19.47)