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Prediction of pile set-up in clays and sands
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2010 IOP Conf. Ser.: Mater. Sci. Eng. 10 012104
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Prediction of pile set-up in clays and sands
W M Yan1
and K V Yuen2
1Department of Civil Engineering, The University of Hong Kong, Pokfulam Road,
Hong Kong2
Department of Civil and Environmental Engineering, University of Macau, Taipa,
Macau
Email:[email protected],
Abstract. Increase in pile capacity after initial driving has been well observed in clays and
sands over decades. The phenomenon is referred to as pile set-up by geotechnical engineers.
More economical pile design may benefit from this time-dependent increase subject to a
reliable prediction. Simple empirical relations of the current capacity with the initial capacity
and elapse time after driving are available in the literature with different model parameters
being suggested for clays and sands, respectively. Nevertheless, appropriateness of the
relations and confidence interval of the model parameters are rarely investigated and thishinders the application of these formulae. In this study, a revised single-parameter empirical
relation is proposed based on the existing formulae. A comprehensive database from pile fieldtests data in clayey and sandy ground in the literature is compiled and Bayesian analysis is
conducted on both these empirical formulae independently for clays and sands. Bayesian
inference allows not only the estimation of the uncertain parameter but also the quantification
of the associated uncertainty in the form of probability distribution. This study sheds lights on
the confidence interval of the model parameter and it provides designers more reliable
prediction of the additional capacity due to pile set-up.
1. Introduction
Case histories have reported a significant time-dependent increase in the axial load capacity of driven
piles that installed in both fine- and coarse-grained soils ([1-3] and many others). The phenomenon is
referred to as pile set-up by geotechnical engineers and such effect is definitely favorable toengineering designs. Over the years, two major mechanisms have been postulated to explain the set-up
effect: (1) dissipation of excess pore water pressure; and (2) soil aging. The first mechanism originates
from the dissipation of excess pore water pressure which is generated due to soil
remolding/disturbance during pile driving. The associated increase in lateral effective stress with
increasing time gives rise to an increase in shear strength and thus the axial capacity of the pile.
Obviously the duration of this reconsolidation depends on the permeability of the soil. It ranges from
days in coarse-grained soils to months or years in fine-grained soils. The second mechanism is a
collective term used by Axelsson [4] to describe particle rearrangement around the pile shaft. This
rearrangement may accompany by the collapsing of temporary arches formed around the shaft which
increases the lateral stress on the pile and thus the soil’s shear strength and pile’s axial capacity.
1Corresponding author
1
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While numerous research projects are still being carried out to study the underlying mechanisms
causing the pile set-up, simple empirical relations are available in the literature to predict this increase
in pile capacity with time [5-6]. The relations forecast the pile capacity from the initial capacity(described as end of driving, EOD) and elapse time after driving in which two sets of model constants
were suggested for clayey and sandy ground, respectively, based on limited data set. Firstly, reliability
of adopting these model constants is doubtful. Secondly, the form of existing empirical relations has
certain weaknesses. The current investigation first reviews existing relations and proposes a revised
one. A set of field pile load tests on sandy and clayey soils is then compiled. A Bayesian probabilistic
approach is employed to evaluate the reliability and confidence interval of the model constant
independently for sands and clays. Bayesian inference allows not only for the optimal estimate of the
uncertain parameter but also the quantification of the associated uncertainty in the form of probability
distribution.
2. Empirical relations
Skov and Denver [5] proposed the following empirical relation to describe the time-dependentincrease in pile load capacity:
0 100
1 logt
tQ Q A
t
⎛ ⎞= ⋅ +⎜ ⎟
⎝ ⎠(1)
where tQ is the pile capacity at time t , 0Q is the pile capacity at initial driving (EOD), t is the elapse
time after initial driving, A and 0t are two empirical parameters which depend on the soil type. Based
on two case histories of 13 driven prefabricated square concrete piles and 21 data records (both static
and dynamic load tests), Skov and Denver [5] suggested two sets of model constants for clayey and
sandy ground respectively, as shown in table 1. It is worth noting that in their calibration procedure t 0
was first assigned by experience and the value of A was then obtained by curve-fitting for thepredefined value of 0t .
Table 1. Model parameters suggested for equation (1) by Skov and Denver [5].
A t 0 (days)
Clays 0.6 1.0
Sands 0.2 0.5
Svinkin and Skov [6] then proposed a revised formula as shown below:
[ ]0 101 (1 log )tQ Q B t= + + (2)where tQ , 0Q and t are defined as the same as equation (1), and B is an unknown parameter. For
simplicity, they suggested B to be taken as the corresponding values of A in table 1. A careful
comparison between equations (1) and (2) revealed that equation (2) actually adopts 0 0.1t =
regardless the soil types. One common problem with equations (1) and (2) which contain 0t (either
explicitly or implicitly) in the formulation is that the formulae predict a decrease in pile capacity when
0t t< . Equation (2) is slightly modified herein to resolve this problem and the proposed empirical
formula reads:
[ ]0 101 log (1 )tQ Q C t= + + (3)
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where C is an uncertain parameter describing the rate of increase in the pile load capacity with time
and it will be determined with the database in this study. It can be seen that at 0t = (i.e., EOD),
0tQ Q= .
3. Bayesian analysis
Consider a database D with N records. The goal is to estimate the model parameter θ (θ = B or C in
equations (2) and (3) respectively) and to quantify the associated uncertainty. This can be done by
Bayesian analysis. By assuming that the model output error (measurement noise and modeling error)
is Gaussian with zero mean and variance 2η σ , the updated probability density function (PDF) of the
model parameter B given database D can be expressed as [7-8]:
20 2
( | ) ( )(2 ) exp ( | )
2
NN
g
Np B D c p B J B Dη
η
π σ σ
−−
⎡ ⎤= −⎢ ⎥
⎢ ⎥⎣ ⎦
(4)
where c0 = normalizing constant so that the volume under the posterior PDF is unity;2
η σ = variance of
the fitting error; ( )p B = prior PDF of the model parameter B expressing the user’s judgment about
the relative plausibility of the values of the model parameters before the data is used. The goodness-
of-fit function ( | )g J B D is given by
( ) ( ) ( )( ) ( )
2
01
1| , ;
N
g t n n nn
J B D Q Q t B QN =
⎡ ⎤= −⎢ ⎥⎣ ⎦∑ (5)
where ( ) ( )( )0 , ;t n nQ Q t B = model output of the n-th record with the parameter B and ( )n
Q is the
corresponding observed value. This function represents how well the model fits the data points. The
most probable model parameter B can be obtained by maximizing the updated PDF ( | )p B D in
equation (4). Nevertheless, by using a non-informative prior distribution for the model parameter, it is
equivalent to minimize the goodness-of-fit function ( | )g J B D over the entire parameter space. In
other words, the optimal parameter can be obtained by solving 0g J B∂ ∂ = . The minimum value of
the goodness-of-fit function is given by minˆ( | ) ( | )g gB D J B D= . In this one-dimensional problem,
the optimal estimation of the parameter B that gives the minimum value of g J can be obtained easily
by different approaches. Finally, the estimated value of the prediction-error variance2
η σ can be found
by solving ( )| 0p B D η σ ∂ ∂ = and it turns out to be the minimum value of the goodness-of-fitfunction:
( ) ( )2
minˆˆ | |g gB D J B Dη σ = = (6)
Moreover, the mean B and variance2Bσ of model parameter B can be obtained by direct integration
with the posterior PDF ( | )p B D in equation (4). The same Bayesian analysis procedure is repeated
for parameter C in equation (3).
4. Database and results
A database is compiled from a total of 143 pile load tests (60 for clays and 83 for sands) available in
literature. It consists of driven concrete and steel with various shapes and timber piles in sandy andclayey ground. As a preliminary study, only piles driven in a single layer of soil (either sands or clays)
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are included in this analysis. The pile capacities were derived from dynamic as well as static load tests.
Table 2 provides a summary of the data.
Table 2. Summary of pile set-up database
Pile capacity (kN)Soil
Type†
Pile
Type‡
Numbe
r of
records Initial Set-up
Wait
period
(days)
Determination of pile
capacity£
Reference
Cl C 20 220-3420 770-4840 1-184 CAPWAP and Case [5,9-10]
Cl CPC 8 534-2581 2113-4449 21-45 SLT [11]
Cl PC 15 200-1300 890-3545 0.667-32 CAPWAP; Case; SLT [6, 10]
Cl H 9 489-2983 1090-7250 1.33-132 SLT and CAPWAP [6,12-13]
Cl SP;
SPC;
SPO
8 850-3910 1500-7120 0.625-
360
CAPWAP and Case [9-10,14]
S C 20 529-1290 678-2540 0.028-
216
CAPWAP; SLT and some
not mentioned
[5,15-16]
S CM 14 935-2250 1504-3730 15-49 SLT; WEAP [14,17]
S CPC;
CT
2 2360-
2670
3203-3820 11-37 SLT [11,14]
S PC 13 913-2949 1145-4900 2-23 CAPWAP; SLT [18]
S H 11 335-649 445-2002 1-51 CAPWAP; SLT [1,13]
S SM;
SP;
SPC;
SPO;
ST
22 694-
13100
1380-21900 0.083-51 CAPWAP; Case; SLT;
WEAP; some cases not
mentioned
[14,16-17,19-20]
S T 1 498 868 224 WEAP [17]
†: S: sandy ground; Cl: clayey ground‡: C: precasted concrete; CPC: concrete pipe (closed end); PC: prestressed concrete; CM: concrete monotube;
CT: concreted-steel taper tube; SP: steel pipe; SPO: steel pipe (open-end); SPC: steel pipe (closed-end); ST:steel taper tube; H: steel H-pile; SM: steel monotube; T: timber
*: SLT: static load test; Case: Case method; CAPWAP: CAse Pile Wave Analysis Program; WEAP: Wave
Equation Analysis of Pile driving.
Based on the proposed Bayesian analysis, the optimal parameter θ , its mean θ and standard
deviation θ σ , and prediction-error standard deviation η σ for clayey and sandy ground are computed
and they are summarized in table 3. Moreover, the posterior PDFs ( )|p Dθ for clays and sands,
respectively, are shown in figure 1. In both empirical formulae, the optimal values for clays ( ˆclaysB and
ˆclaysC ) are larger than that for sands and this implies that the increase in capacity for piles installed in
clays is more significant than that in sands at the same elapse time after initial driving. In both cases,
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the posterior PDF ( )|p Dθ is highly symmetric about the optimal value θ (see figure 1) which can
also be confirmed from the insignificant difference between θ and θ . One can see that the
uncertainty associated with sandsB and sandsC is much smaller than that for clays for this database.
Furthermore, the prediction error for sands is also much smaller and this can be confirmed by figure 2
which shows the predicted pile load capacity against the measurement. It can be clearly seen that the
fitting of sands samples exhibits much higher correlation. On the other hand, the optimal model
significantly under-predicted the set-up for low-capacity piles driven in clays. Besides, as one can read
graphically, the discrepancy between the predicted and measured capacity in clays is in general much
larger than that in sands. Comparing the two different empirical models (i.e., equation (2) and (3)), it is
noted that their prediction errors at the optimal parameter are in very similar magnitudes.
Table 3. Summary of results.
[ ]0 101 (1 log )tQ Q B t= + +
[ ]0 101 log (1 )tQ Q C t= + +
B B Bσ η σ (kN) C C Cσ η σ (kN)
Clays 0.355 0.3546 0.0351 1006 0.524 0.5236 0.0581 1077
Sands 0.232 0.2323 0.0115 719 0.418 0.4179 0.0196 688
Figure 1. Updated PDF for clays and sands data using the two models.
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5. Further remarks
As shown in table 3, model parameter B in equation (2) shows smaller standard deviations than
parameter C in equation (3), for both clays and sands data, despite the fact that both formulae exhibit avery similar form. This can be demonstrated graphically in figure 1 that C spreads out slightly wider,
which is more noticeable for the clays data.
To further reduce the uncertainty of the model constant, independent Bayesian analysis should be
performed not only on different ground soils but also on different pile types. It is suggested that driven
piles (i.e., displacement piles) with different displacement volumes would affect the amount of excess
pore water pressure generated around the pile and subsequently influences the set-up effect originated
from pore water pressure dissipation with time. As one may expect, driven steel H-piles displace
comparatively small volume of soils during installation and should give a smaller set-up increase in
the load capacity owing to the pore water pressure dissipation effect. Concrete piles, on the other hand,
should result in the contrary. Therefore, a larger value of the model parameter B or C is expected for
concrete or closed-end pipe piles than H-piles. However, a database with more records on individual
pile type is necessary to offer a further investigation.
102 103 104
Measurement (kN)
102
103
104
102 103 104 105
Measurement (kN)
102
103
104
105
Clays Sands
11
11
Qt=Q0 [1+B (1+log10(t))]^
102 103 104
Measure ent(kN)
102
103
104
102 103 104 105
Measurement (kN)
102
103
104
105
Clays Sands
11
11
Qt=Q0 [1+C log10(1+t)]^
Figure 2. Comparison between the model output and measurement.
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6. Concluding summary
Pile set-up is used by geotechnical engineers to describe the increase in pile axial capacity with time
since initial driving. This paper first reviews two existing empirical formulae which predict pile set-upwith time. Based on limited data on two case histories, different sets of parameters were suggested for
clayey and sandy ground in these formulae. A revised single-parameter empirical relation is then
proposed to predict the pile set-up effect. A database of 143 pile load records on clayey and sandy
ground is compiled from literature. Based on Bayesian analysis, optimal parameter for each soil type
is suggested with the associated uncertainty. Prediction error of the empirical relation is also estimated
and it provides designers additional information for these empirical formulae.
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