Geotechnical Engineering

DHARM

624 GEOTECHNICAL ENGINEERING

qmax = V

bL

e

b

F 1 +^6

HG

I

KJ

...(Eq. 15.4)

qmin =

V

bL

e

b

F 1 − 6

HG

I

KJ

...(Eq. 15.5)

Equation 15.3 is merely a special form of the basic formula for the resultant stress on a
section subjected to a direct load P and a moment M, expressed in strength of materials, in the
form:

f =

P

A

Mc

I

±

The maximum eccentricity for no tension to occur in the base is obtained by equating
qmin to zero, and solving for e:
emax = b/6 ...(Eq. 15.6)
Since the eccentricity can occur to either side of the middle depending upon the direc-
tion of H, the resultant force should fall within the middle-third of the base in order that no
tensile stresses occur anywhere in the base.

If the eccentricity occurs with respect to the axis which bisects the other dimension L of
the footing.

q =

V

bL

e

L

F 1 ± 6

HG

I

KJ ...(Eq. 15.7)

emax = L/6 ...(Eq. 15.8)

This leads to the concept of ‘kern’ or ‘core’ of a section, which is the zone within which

the resultant should fall for the entire base to be subjected to compression.

For a rectangular section, the kern is a centrally located rhombus with the diagonals

equal to one-third of the breadth and length; for a circular section it is a concentric circle with

diameter one-fourth of that of the circle.

Most footings are designed so that the resultant of the loads falls within the kern and

the soil reaction everywhere is compressive. However, in certain cases such as the design of

the base slab of a cantilever retaining wall, the resultant may fall outside the kern, and the

distribution of pressure shown in Fig. 15.15 must be used for the structural design of the

footing.

Resultant force outside the middle-third of the base

If the horizontal component of the total load increases beyond a certain limit in relation to the

vertical component, the resultant force falls outside the middle-third of the base, the eccentric-

ity being more than the limiting value of one-sixth the size of the base. It must be remembered

that soil cannot provide tensile reaction; it just loses contact with the footing in the zone of

tension. This situation is shown in Fig. 15.16.

From the laws of statics, the total upward force must be equal to V and also collinear

with V. That is to say:

V =

qxLmax

2

...(Eq. 15.9)

and xbe
32

=−FHG IKJ ...(Eq. 15.10)