Geotechnical Engineering

DHARM

552 GEOTECHNICAL ENGINEERING

Thus, the limitations and deficiencies of Rankine’s approach in respect of Eqs. 14.7 and

14.8 apply equally to Pauker’s equations.

Bell’s Method

Bell (1915) modified Pauker-Rankine formula to be applicable for cohesive soils; both friction

and cohesion are considered in this equation. With reference to Fig. 14.1, from, the stresses, on

element I,

σ = qult tan^2 (45° – φ/2) – 2c tan (45° – φ/2) ...(Eq. 14.24)

or σ =

q

N

c

N

ult

φ φ

−

2

...(Eq. 14.25)

with the usual notation, Nφ = tan^2 (45° + φ/2).
This is from the relationship between the principal stresses in the active Rankine state
of plastic equilibrium.
From the stresses on element II,
σ = γDf tan^2 (45° + φ/2) + 2c tan (45° + φ/2) ...(Eq. 14.26)
or σ = γDf Nφ + 2c Nφ ...(Eq. 14.27)
Equating the two values of σ for equilibrium, we have:
qult = γDf tan^4 (45° + φ/2) + 2c tan (45° + φ/2)[1 + tan^2 (45° + φ/2)] ...(Eq. 14.28)
or qult = γDf Nφ^2 + 2c Nφ(1 + Nφ) ...(Eq. 14.29)
This is Bell’s equation for the ultimate bearing capacity of a c – φ soil at a depth Df.
If c = 0, this reduces to Eq. 14.23 or 14.7. For pure clay, with φ = 0, Bell’s equation
reduces to

qult = γDf + 4 c ...(Eq. 14.30)

If Df is also zero, qult = 4c ...(Eq. 14.31)

This value of considered to be too conservative as will be shown later on.

The limitation of Bell’s equation that the size of the foundation is not considered may be

overcome as in the case of Rankine’s equation by considering soil wedges instead of elements.

Figure 14.2 may be employed for this purpose. Proceeding on exactly similar lines as in

the Rankine approach, one gets:

qult =

1

22

γγ..b NNφφ(^22 −+ 121 )DN cNNf.φ+ φ(φ+) ...(Eq. 14.32)

or qult =

1

2

γγ..bNγ++D N cNfq. .c ...(Eq. 14.33)

where Nγ =

1

2

NNφφ()^2 − (^1) ...(Eq. 14.34)
Nq = Nφ^2 ...(Eq. 14.35)
and Nc = 2 NNφ()φ+ 1 ...(Eq. 14.36)
Equations 14.34 and 14.35 are identical to Eqs. 14.13 and 14.14, already given. Nγ, Nq
and Nc are known as ‘bearing capacity factors’.
If c = 0, Eq. 14.32 reduces to Eq. 14.11 of Rankine.