Design of single piles - Piling Engineering - page 159

G L c /2

G L c /4

Modified shear
modulus, G *

L c /4

L c

L c /2

G c = G L c /2
r c = G L c /4 / G c

L

Depth

Figure 4.45 Definition of ρ c and G c .

Referring to Figure 4.45, a characteristic modulus, G c , is defined as the average
value of G ∗ over the active length of the pile. In addition a parameter, ρ c , is introduced

reflecting the degree of homogeneity in the soil stiffness. For the linear variation of
modulus shown in Figure 4.45, ρ c may conveniently be defined by

G L c / 4
G L c / 2 =

G L c / 4
G c

ρ c = (4.73)

It is helpful to cast the solution in terms of an equivalent solid pile of the same cross-
sectional area and the same bending rigidity as the real pile. For an equivalent pile of
diameter d , the appropriate Young's modulus may be calculated as

( EI ) p
π

E p = (4.74)

d 4 / 64

where ( EI ) p is the bending rigidity of the actual pile.
The critical pile length is now defined by

G c ) 2 / 7 L c = d ( E p / (4.75)

The form of this equation is similar to equation (4.68), with an exponent of 2/7 instead
of 1/4. Also, since E p for an equivalent pile is used instead of the bending rigidity ( EI ) p ,
it is possible to think in terms of a critical slenderness ratio, L c /

d , for the pile which

is deduced directly from the stiffness ratio, E p / G c .

It may seen from Figure 4.45 that the definition of G c requires the knowledge of
the critical length, L c , which is in turn defined in terms of G c . Thus some iteration is
required except for the extreme cases of homogeneous soil (

ρ c = 1) and soil where G is

m ∗ d ) 2 / 9 , proportional to depth ( ρ c = 0 . 5 and the critical length reduces to L c = d (2 E p /