NUMERICAL ANALYSIS OF NSF USING FEM
6.2 NSF ON END-BEARING SINGLE PILES
6.2.2 Interface Elements for Pile-soil Interaction
Recently, Lee et al. (2002) has demonstrated that in the numerical simulation of problems involving NSF, the incorporation and proper definition of interface elements along the pile-soil interface is a dominant factor for a correct capturing of NSF numerically. Lack of such interface elements such as those methods using elastic models generally leads to excessive overestimation of NSF. Plaxis furnishes a facility of applying interface elements along the pile-soil interface (see Fig. 6.1) to simulate the soil-structure interaction. The interface element was assigned a 'virtual thickness' which is an imaginary dimension used to separate the pile and the adjacent soil. The behaviour of the interface element is governed by the Coulomb criterion to distinguish between elastic behavior and plastic behavior when permanent slip occurs.
The ultimate negative skin friction at the pile-soil interface, fn,ult, is dictated by
fn ult, =σ' tan 'h ϕ int+c'int (6.1)
227
where σh’ is the normal effective stress on the interface element (or horizontal effective stress in the case of a vertical pile) and c’int and φ’int are the effective friction angle and effective cohesion of the interface element, respectively. For the interface to remain elastic without full mobilization of skin friction, the interface shear stress τ should satisfy
(6.2)
and for plastic behavior, τ is given by:
(6.3)
When negative skin friction at the pile-soil interface reaches the value defined by Eq. (6.1), negative skin friction will be fully mobilized and permanent slip between the soil and the pile occurs.
The effective horizontal stress σh’ can be related to the vertical effective stress σv’ by the coefficient of earth pressure upon loading, k, as follows:
(6.4) Substituting Eq. (6.4) into Eq. (6.1) leads to
(6.5)
Using the effective stress method, negative skin friction can be related to the effective vertical stress σv’ by the coefficient β as
, ' tan 'int 'int
n ult v
f =kσ φ +c
ult
fn,
τ <
'h k 'v
σ = σ
ult
fn,
τ =
228
(6.6)
Combining Eqs. (6.5) and (6.6) and ignoring the negligible term c’int, leads to the following definition of effective friction angle at the pile-soil interface:
(6.7)
The centrifuge results in Chapter 4 consistently reveal a β value of 0.24 for the soft clay. To fully define the interface friction angle φ’int, the coefficient of earth pressure upon loading, k, must be determined, which turns out to be essentially dictated by the soil constitutive model adopted. Since Mohr Coulomb soil model and Modified Cam Clay model have been adopted for the simulation of the soils in the present study, the coefficient of earth pressure inherent in these two models is further explored herein.
When using elastic-perfectly-plastic constitutive model, like Mohr Coulomb model to simulate soil in the present analyses, the behavior of the soil will dominantly elastic during the various loading stages such as the water drawdown and surcharge loading in the laterally confined model container analogous to that of a 1-D Oedometer test. As such, the coefficient of earth pressure upon loading, k, is governed by the effective Poisson ratio, ν’, as
(6.8)
, '
n ult v
f =βσ
) arctan(
'int
k φ = β
' 1
' ν ν
= − k
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For higher order of constitutive model like Modified Cam Clay model, the various loading stages in the simulation of the model test will cause the continuous plastic hardening of the soil. The energy equation for the Modified Cam Clay is as follows (see for example Wood, 1990):
(6.9)
where p’ is the effective mean normal stress; q is the deviator stress; dεsp and dεvp are the incremental plastic deviator strain and plastic volumetric strain, respectively.
Thus,
(6.10)
The incremental volumetric and deviator shear strains compose of elastic and plastic components as follows:
(6.11) (6.12)
The elastic volumetric strain component dεve is typically negligible as compared to the plastic volumetric strain component dεvp during the soil yielding process. What is more, Modified Cam Clay model disregards the elastic deviator shear strain dεse. As such,
2 2 2 2 2
' ' ( ) ' ( )
p p p p
s v v s
qdε + p dε = p dε +M p dε
2
2 2 2
' vpp ' vpp '
s s
d d
q p p M p
d d
ε ε
ε ε
⎡ ⎤
+ = ⎢ ⎥ +
⎣ ⎦
p v e v
v d d
dε = ε + ε
p s e s
s d d
dε = ε + ε
230
(6.13)
Under the present configuration as shown in Fig. 6.1 which is analogous to a laterally confined 1-D vertical compression problem, we have
(6.14)
(6.15)
where dεa and dεr are incremental vertical and horizontal strains, respectively.
Combining Eqs. (6.13), (6.14) and (6.15) leads to
(6.16)
Substituting Eq. (6.16) into Eq. (6.10) gives
(6.17)
where η is the stress ratio defined by,
p s
p v s
v
d d d
d
ε ε ε
ε ≈
2
v a r a
dε =dε + dε ≈dε
2( ) 2
3 3
s a r a
dε = dε −dε ≈ dε
2
= 3
≈
s v p s
p v
d d d
d
ε ε ε
ε
2
4 9 2
3 = +M
η+
231
(6.18)
Combining Eq. (6.17) and Eq. (6.18) leads to
(6.19) or,
(6.20)
As such, with the proper evaluation of the coefficient of earth pressure upon loading, k, by Eq. (6.8) when using Mohr-Coulomb model, or Eq. (6.20) when using Modified Cam Clay model, the friction angle of the interface element can be properly defined as follows:
(1) When using MC model, combining Eqs. (6.7) and (6.8) defines the effective friction angle of the interface element as
(6.21)
(2) When using MCC model, combining Eqs. (6.7) and (20) gives
(6.22) k
k p
q
r a
r a
2 1
) 1 ( 3 )' 2 ' 3(
1 ' '
' +
= − +
= −
=
σ σ
σ η σ
2
4 9 2 3 2 1
) 1 (
3 M
k
k + = + +
−
2 1 4 4 9
9
2
− +
= M k
⎥⎦⎤
⎢⎣⎡ −
= '
)' 1 arctan (
' ν
ν φ int β
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎣
⎡
+
−
+
=
4 2 9
9
4 4 9
arctan '
2 2
M M
int
β φ
232
As will be demonstrated subsequently, the correct definition of the effective friction angle of the interface element at the pile-soil interface based on the above equations is crucial to capture the NSF at the pile-soil interface correctly.
It has been established above that the coefficient of lateral earth pressure upon loading, k, is uniquely dependent on the soil model selected. The validity of Eqs. (6.8) and (6.20) inherent for MC model and MCC model, respectively, was verified by running a simple FEM analysis in Plaxis using the mesh shown in Fig. 6.1. For Mohr- Coulomb model, the soil assumes the following typical properties: γ’=15.5 kN/m3, c’=0.1 kPa, φ’=24°, E’=2000 kPa, and the Poisson ratio was varied with values of 0.30, 0.34 and 0.38. Corresponding to each Poisson ratio, the theoretical coefficient of earth pressure calculated by Eq. (6.8) is 0.43, 0.52 and 0.61, respectively. The FEM analysis was conducted under self-weight loading and subsequent uniform distributed loading (UDL). The effective horizontal and vertical stress along the depth of the mesh can then be readily extracted from the FEM outputs and the ratio of the above two stresses gives the k values. As shown in Fig. 6.2(a), the k values thus obtained are noted to be constant along the soil depth and match almost exactly with those calculated using Eq. (6.8). For the Modified Cam Clay model with typical soil parameters γ’ = 15.5 kN/m3, λ = 0.24, κ = 0.05, eint = 1.71, when the M value varies from 0.6, 1.0 to 1.4, the theoretical k value as calculated by Eq. (6.20) is 0.89, 0.75 and 0.60, respectively. These also compare favorably with the derived k values from FEM outputs as shown in Fig. 6.2(b). The minor underestimation of the theoretical values by Eq. (6.20) is likely due to the omission of the relatively smaller elastic strain components when deriving Eq. (6.20).