Soil container boundary effect may be a concern when testing large size pile groups. The minimum boundary clearance for an axially-loaded pile group is mainly governed by the dissipation of shear stress away from the pile-soil interface. Randolph and Wroth (1978) established that it is reasonable to assume a maximum shear stress at the pile-soil interface and the shear stress decreases inverse-proportionally to the
174
increasing distance from the pile shaft. There exists a radius, rm, at which the shear stress becomes negligible, and the boundary walls of the model container should ideally be located beyond such distance from the model piles such that the boundary effects are insignificant. As presented in Chapter 3, the inner plan dimension of the model container used for the present study is 420 × 420 mm in model scale, or 33.6 × 33.6 m in prototype scale. For the largest 16-pile group, the container boundary wall is 11 m from the center of a perimeter pile in prototype scale (see Fig. 5.1). The adequacy of such clearance is explored here.
Early studies on axially loaded pile assumed that the shear strains around a pile shaft were confined to a narrow zone of about 5 to 10 cm adjacent to the pile (Burland, et al., 1966). The shear stress is a maximum at the pile-soil interface and reduces with increasing distance from the pile shaft (Cooke, 1974). Randolph and Wroth (1978) postulated that beyond an influence radius, rm, the shear stress becomes negligible.
They established that the influence radius, rm, is equal to 2.5L(1-ν) for a pile installed in a uniform soil stratum, where L is the length of the pile and ν is the poison ratio of the soil. For a socketed pile, rm is given as follows (Randolph and Wroth, 1978):
(5.1)
where
(5.2)
(5.3) {0.25 [2 (1 ) 0.25] }
rm =L + ρ − −ν ξ
) ( / ) 2 /
(L G L
Gs s
ρ =
b
s L G
G ( )/ ξ =
175
with Gs(L/2) and Gs(L) as the shear modulus of the top soil at half and base of the soil depth. Respectively; Gb as the shear modulus of the underlying stiffer soil stratum.
From Eqs. (5.1) and (5.3), it can be seen that the stiffer the underlying soil stratum, the smaller is the influence radius. For the case of end-bearing pile on rigid base, ξ assumes a negligible value and the influence radius, rm, assumes a minimum value of one quarter of the pile length L, or 4.0 m for the present study with an embedded pile length of 16 m, as deduced from Eq. (5.1). For the case of “socketed” pile, rm is normally larger and is dictated by the ratio of the shear modulus of the upper soil stratum to that of the underlying stiffer soil stratum. The dense base sand layer for the present “socketed” case has a Young’s modulus, Eb, of about 55 MPa as evaluated by careful back-analysis of the pile settlement of the single socketed pile by FEM (refer to Chapter 6). Taking an effective Poison ratio of 0.3, the calculated shear modulus Gb of the underlying dense sand layer is
kPa
On the other hand, the undrained modulus of the soft clay used for the present study, Eu, can be estimated from undrained shear strength Cu as recommended by Leung et al. (2006) for the Kaolin clay used in the present study:
Eu = 150Cu (5.4)
It is evident from Eqs. (5.1) and (5.3) that the stiffer the upper soil layer, the larger is the calculated rm. Taking the maximum undrained shear strength of about 35 kPa (refer to Fig. 4.2) at the base of the clay at the end of the consolidation after surcharge, Eu = 150 × 35 = 7500 kPa, resulting in a shear modulus of
21,150 2(1 )
b b
G E
= ν =
+
176
kPa
Likewise, the shear modulus of the clay at midway of the clay depth is calculated to be:
Gs(L/2) = 1040 kPa
Thus,
The influence radius can hence be determined as:
= 16 × {0.25 + [2 × 0.51 × (1-0.3)-0.25] × 0.1}
= 4.8 m
Comparing to the available boundary clearance of 11 m for the largest pile group in the present study, the above evaluation of rm of 4.0 m for end-bearing piles and 4.8 m for socketed piles reveals that the container boundary effect for the present test configuration is insignificant.
However, some researchers argued that the above influence radius is only applicable for the case of piles subjected to external axial loads applied at pile head, and may not be applicable to the case of piles subjected to NSF (Lim, 1994). For
( ) 2020
2(1 )
u s
u
G L E
= ν =
+
51 . 0 ) ( / ) 2 /
( =
=Gs L Gs L ρ
1 . 0 /
)
( =
=Gs L Gb ξ
} ] 25 . 0 ) 1 ( 2 [ 25 . 0
{ + ρ −ν − ξ
= s
m L
r
177
example, for the case of an end-bearing pile subjected to NSF, the relative pile-soil settlement will be the largest because the pile base is essentially prevented from moving downwards. Consequently, the induced negative skin friction along the pile shaft will be the largest, which could result in a larger influence radius, rm. On the other hand, for an end-bearing pile subjected to external axial load at the pile head, the mobilized skin friction at the pile-soil interface may be the least as compared to other pile toe conditions, resulting in the least rm. This is just the opposite to the case of pile subjected to NSF.
Based on the matching of the results from a discrete element method with those from rigorous elastic theories, Lim (1994) proposed an empirical linear distribution of rm along the pile shaft in a consolidating soil. For the case of end-bearing piles subjected to NSF, the maximum rm at the top of the clay surface can be expressed as follows (Lim, 1994):
(5.5)
where h is the depth of the upper consolidating soil and d is the pile diameter.
The minimum rm located at the pile toe is expressed as:
(5.6)
For the case of socketed piles subjected to NSF, the maximum rm at the top of the clay can be expressed as (Lim, 1994):
,
70 125
[1.26 1.32 ( 5)]
m top 9500 r h h
d ν − ν
= × − + −
, , (1 )
m bottom m top
r =r − −ν h
178
(5.7)
where h, d and ν are the same as those in Eq. (5.5); Es and Eb are the average Young's modulus of the upper settling soil and the underlying stiff soil, respectively. Parameter a is defined as follows:
(5.8)
where e is the socketed length. The corresponding rm,bottom of the socketed pile can be calculated using Eq. (5.6).
For the present test configuration on end-bearing piles, Eqs. (5.5) and (5.6) can be used to readily calculate rm,top = 16.6 m; rm,bottom = 3.6 m; with an average rm of about 10.1 m. For the socketed pile case, Eb = 55,000 kPa, average Yong’s modulus of the clay layer at the end of surcharge Es = 2700 kPa, socketed length e = 0.64 m. The calculated rm,top = 14.8 m; rm,bottom = 3.1 m; with an average rm of about 8.9 m, which is slightly smaller than that of the end-bearing case. Thus, even if the empirical formulae proposed by Lim (1994), which give an average rm of 10.l m for end-bearing piles and 8.9 m for socketed piles, are adopted, the container boundary effect for the present study is still deemed to be insignificant in view of the available boundary clearance of 11 m for the largest pile group in the present study.
,
log( )
70 125
1.26 1.32 ( 5) 1.1 9.53 15 3 ( 5)
log(200) 9500
b s m top
E
E h h
r h a
d d
ν ν ν ν
⎧ − ⎡ ⎤⎫
= × ⎨⎩ − + − − ⎢⎣ − − − − ⎥⎦⎬⎭
1.1 9.53 1.5 10 3 (h 5)
a⎡ ν −ν d ⎤
+ ⎢⎣ − − × − ⎥⎦
)2
( 44 . 0 66 . 0
1 h
e h
a= + e −
179
The residual shear stress at the container boundary, τ, can be approximately evaluated by the formula established by Randolph and Wroth (1978):
(5.9)
where τ0 is the shear stress at the pile shaft surface; r0 is the pile radius and r is the distance from the pile centre. For the present case with r0 = 0.64 m and r = 11 m, the shear stress at the container wall induced by the shear stress at the pile shaft surface, τ
= r0/r × τ0 = 0.64/11× τ0 = 5.6% τ0. In other word, the shear stress at the location of the boundary wall is only 5.6% of the maximum shear stress at the pile-soil interface, which is essentially insubstantial. This once again illustrates that the container boundary at 11 m from the centre of the perimeter piles of the largest pile group in the present study is sufficiently far enough without significant boundary effects.