Pile Design and Construction Practice, Fifth edition

350 Piles to resist uplift and lateral loading
6.3.8 Calculation of lateral deflections and bending
moments by elastic continuum methods
The method of preparing p–y curves described in Section 6.3.5 was based on the assumption
that the laterally loaded pile could be modelled as a beam supported by discrete springs. The
springs would be considered as possessing linear or non-linear behaviour. In the latter case the
method could be used to model pile behaviour in strain conditions beyond the elastic range.
In many cases where lateral forces are relatively low and piles are stiff the pile head
movements are within the elastic range and it may be convenient to use the elastic
continuum model to calculate deflections and bending moments.
Randolph (6.23) used finite element analyses to establish relationships between pile
deflections and bending moments with depth for lateral force and moment loading as shown
in Figure 6.36. The following notation applies to the parameters in this figure:
y0 � lateral displacement at ground surface
z � depth below ground level
H0 � lateral load applied at ground surface
M � bending moment in the pile
M0 � bending moment at ground surface
r0 � radius of the pile
4 � effective Young’s modulus of a solid circular pile of radius r0 (i.e. 4EpIp/�r0 )
Gc � characteristic modulus of the soil, that is, the average value of G * over depths
less than lc G *
where
G � shear modulus of the soil
v � Poisson’s ratio
lc � critical length of the pile
lc for homogeneous soil
�2r0(E �/m p for soil increasing linearly in stiffness with depth
* r0) 2/9
� 2r0(E �/G p * ) 2/7
� G(1 � 3
E
4v)
� p
m*
m � where G varies with depth as G � mz
G z
� c
where
3v
� m�1 � 4�
is a homogeneity factor
�c �
The use of the Randolph curves is illustrated in worked Example 8.2.
The Randolph method is useful where the shear modulus is obtained directly in the field
using the pressuremeter. If Young’s modulus values only are available the shear modulus for
an isotropic soil can be obtained from the equations:
G* at lc /4
G * at lc /2
E u � 2G(1 � v u) and E � � 2G(1 � v �)
where v u and v� are the undrained and drained Poisson’s ratios respectively.
(6.49)