Geotechnical Engineering

DHARM

558 GEOTECHNICAL ENGINEERING

Then, for φ = 0, qult = (2 + π)c = 5.14 c

(vi) Prandtl’s expression, as originally derived, does not include the size of the footing.

14.5.5 Terzaghi’s Method

Terzaghi’s method is, in fact, an extension and improved modification of Pandtl’s (Terzaghi,

1943). Terzaghi considered the base of the footing to be rough, which is nearer facts, and that

it is located at a depth Df below the ground surface (Df ≤ b, where b is the width of the footing).

The analysis for a strip footing is based on Fig. 14.8.

W

bqult

A B

f f

b

F Df A

III

G II C II

B

45° – /2f

III

E

D

qult Ca

f

Pp

C

Ca

Pp

q= Dg f

(a) Terzaghi system for ideal soil, rough base and surcharge (b) Forces on the elastic wedge

f

Fig. 14.8 Terzaghi’s method for bearing capacity of strip footing
The soil above the base of the footing is replaced by an equilvalent surcharge, q(= γDf).
This substitution simplifies the computations very considerably, the error being unimportant
and on the safe side. This, in effect, means that the shearing resistance of the soil located
above the base is neglected. (For deep foundations, where Df > b, this aspect becomes impor-
tant and cannot be ignored).
The zone of plastic equilibrium, CDEFG, can be subdivided into I a wedge-shaped zone
located beneath the loaded strip, in which the major principal stresses are vertical, II two
zones of radial shear, BCD and ACG, emanating from the outer edges of the loaded strip, with
their boundaries making angles (45° – φ/2) and φ with the horizontal, and III two passive
Rankine zones, AGF and BDE, with their boundaries making angles (45° – φ/2) with the hori-
zontal.
The soil located in zone I is in a state of elastic equilibrium and behaves as if it were a
part of the sinking footing, since its tendency to spread laterally is resisted by the friction and
adhesion between the soil and the base of the footing. This leads one to the logical conclusion
that the tangent to the surface of sliding at C will be vertical. Also AC and BC are surfaces of
sliding and hence, they must intersect at an angle of (90° – φ); therefore, the boundaries AC
and BC must rise at an angle φ to the horizontal. The footing cannot sink into the ground until
the pressure exerted onto the soil adjoining the inclined boundaries of zone I becomes equal to
the passive earth pressure. This pressure Pp, acts at an angle φ(φ = δ = wall friction angle) to
the normal on the contact face; that is, vertically, in this case.
The adhesion force Ca on the faces AC and BC is given by

Ca =

b
2cosφ. c ...(Eq. 14.55)
where c is a unit cohesion of the soil, with shearing resistance of the soil being defined by
Coulomb’s equation s = c + σ tan φ.