CIVIL ENGINEERING FORMULAS

PILES AND PILING FORMULAS 107

of the pile head on yandMcan be evaluated by substituting the value of Mt
from the preceding equation into the earlier yandMequations. Note that, for
the fixed-head case,

(4.7)

TOE CAPACITY LOAD

For piles installed in cohesive soils, the ultimate tip load may be computed from

(4.8)

whereAbend-bearing area of pile, ft^2 (m^2 )
qbearing capacity of soil, tons/ft^2 (MPa)
Ntbearing-capacity factor
cuundrained shear strength of soil within zone 1 pile diameter above
and 2 diameters below pile tip, psi (MPa)

Although theoretical conditions suggest that Ncmay vary between about 8 and
12,Ncis usually taken as 9.
For cohesionless soils, the toe resistance stress, q, is conventionally
expressed by Eq. (4.1) in terms of a bearing-capacity factor Nqand the effective
overburden pressure at the pile tip

(4.9)

Some research indicates that, for piles in sands, q, like reaches a quasi-
constant value, ql,after penetrations of the bearing stratum in the range of 10 to
20 pile diameters. Approximately

(4.10)

whereis the friction angle of the bearing soils below the critical depth. Val-
ues of Nqapplicable to piles are given in Fig. 4.1. Empirical correlations of soil
test data with qandqlhave also been applied to predict successfully end-bearing
capacity of piles in sand.

GROUPS OF PILES

A pile group may consist of a cluster of piles or several piles in a row. The
group behavior is dictated by the group geometry and the direction and location
of the load, as well as by subsurface conditions.
Ultimate-load considerations are usually expressed in terms of a group
efficiency factor, which is used to reduce the capacity of each pile in the

ql0.5Nq tan 

fs,

qNqvoql

vo

QbuAbqAbNccu

yf

PtT^3

EI 

Ay

A By

B