Acta Geotechnica
DOI 10.1007/s11440-014-0354-8
RESEARCH PAPER
A procedure for incorporating setup into load and resistance
factor design of driven piles
Kam W. Ng • Sri Sritharan
Received: 23 February 2014 / Accepted: 9 September 2014
Ó Springer-Verlag Berlin Heidelberg 2015
Abstract In a recent study, the time-dependent increase in
axial load resistance of steel H-piles driven into cohesive
soils, due to setup, was systematically quantified using
measured field data. A method to estimate the setup based
on measurable soil properties was subsequently established.
These studies highlighted that the uncertainties of the
measurements of soil properties and thus the semi-empirical
approach to estimate setup are significantly different from
those of the methodology used for measuring the pile
resistance during retaps at any time after the end of driving.
Recognizing that the two sets of uncertainties should be
addressed concurrently, this paper presents a procedure for
determining the factored resistance of a pile with due consideration to setup in accordance with the load and resistance factor design that targets a specific reliability index.
Using the first-order second-moment method, the suggested
procedure not only provides a simplified approach to
incorporate any form of setup in design, but it also produces
comparable results to the computationally intensive firstorder reliability method. Incorporating setup in design and
construction control is further shown to reduce foundation
costs and minimize retap requirements on piles, ultimately
reducing the construction costs of pile foundations.
K. W. Ng (&)
Department of Civil and Architectural Engineering, University
of Wyoming, Dept. 3295, EN3050, 1000 E. University Ave.,
Laramie, WY 82071-2000, USA
e-mail: kng1@uwyo.edu
S. Sritharan
Department of Civil, Construction and Environmental
Engineering, Iowa State University, 376 Town Engineering
Building, Ames, IA 50011-3232, USA
e-mail: sri@iastate.edu
Keywords Foundations Load and resistance factor
design Piles Pile setup
1 Introduction
The setup in foundation design typically refers to the
increase in vertical load resistance (or capacity) of driven
piles embedded in soils over time, which results from the
healing of remolded soils surrounding the pile, dissipation
of pore water pressure induced by pile driving (i.e., consolidation), and/or increase of lateral soil stresses [22, 25].
A systematic field experimental program involving five
steel H-piles driven in cohesive soils conducted by the
authors concluded that the pile resistance can increase by
as high as 55 % within 7 days due to the dissipation of pore
water pressure induced by pile driving [18]. The presence
of setup was evident by the significant increase in pile
driving resistance measured in terms of hammer blow
counts required after the end of driving (EOD), which was
later confirmed by the subsequent static load test (SLT)
measurements. Several researchers have reported field
evidence of pile setup in literature [9, 11, 14, 15]. If pile
setup can be satisfactorily incorporated into load and
resistance factor design (LRFD) framework, it will reduce
the foundation cost by requiring: (1) shorter pile length,
less number of piles, and/or smaller dimensions for the pile
cap; and (2) smaller hammers possibly with less consumption of fuel and reduced labor during construction.
A systematic approach to routinely account for setup
during pile driving with due consideration to time has not
existed until recently. Pile setup is only accounted for in
special cases for which setup is accounted for using
dynamic strike tests and/or SLTs, which incur additional
construction costs. Given the development of a pile setup
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Acta Geotechnica
procedure using soil parameters [18, 19], this paper focuses
on incorporating setup into the LRFD framework. This
method is targeted for eventual incorporation in design
specifications, such as the American Association of State
Highway and Transportation Officials (AASHTO), which
do not have recommendations to incorporate setup into the
LRFD approach.
Ensuring simplicity in design and construction methods
as well as to allow for incorporation of setup using any
predictive methods, the paper proposes a systematic,
closed-form approach to determine separate resistance
factors for pile resistance at end of driving (REOD) and pile
setup resistance (Rsetup). These separate resistance factors
will account for varying uncertainties associated with two
different resistance components, enabling pile design using
Eq. (1) as similarly suggested by Yang and Liang [27]:
X
cQ uR ¼ uEOD REOD þ usetup Rsetup
ð1Þ
where c is load factor, Q is applied vertical load, R is the
total nominal pile resistance, u is the resistance factor for
total resistance R, uEOD is the resistance factor for REOD,
and usetup is the resistance factor for Rsetup. Each of the pile
resistance components has its own uncertainties, such as
those resulting from the in situ measurement of soil properties and the empirical approach. To satisfactorily incorporate a pile setup estimate into LRFD, the impact of the
different uncertainties should be recognized separately, yet
accounted for simultaneously to achieve the desired reliability in design. To comply LRFD framework and provide
designers with an efficient method involving little computational requirement, the proposed procedure uses the
first-order second-moment (FOSM) framework to determine two resistance factors: uEOD and usetup. The application of proposed procedure was demonstrated using a
database of steel H-piles embedded in cohesive soils. For
verification purposes, the outcomes are compared with the
resistance factors determined based on the first-order reliability method (FORM) by Yang and Liang [27].
2 Background
2.1 Pile setup resistance estimation
AASHTO suggests the use of dynamic restrike tests and/or
completing SLTs at different times to account for pile
setup, which may be appropriate for infrequent use in
special projects. Both suggestions are time-consuming and
not feasible in practice [3]. Alternatively, the pile setup can
be estimated using empirical methods available in literature
(e.g., [11, 24, 26]). Among the empirical methods, Skov
and Denver’s method [24] has been widely used to estimate
pile setup resistances [27], which is summarized in Eq. (2):
123
t
Rsetup ¼ Ro A log
to
ð2Þ
where Ro is a reference pile resistance, A is an empirical
setup factor, and to is a reference time by which Ro is
determined. This method also requires performing a
dynamic restrike at to = 1 day to determine the Ro value
and multiple restrikes over a period of time to determine
the A value. Based on 250-mm square concrete piles and
Yoldia clay collected in Germany, Skov and Denver [24]
recommended an A value of 0.6 if pile restrikes are not
feasible. Due to variability of soil and pile types, Bullock
et al. [7] and Yang and Liang [27] concluded that a more
suitable A value may be in the range between 0.1 and 1.0.
The large variability of A values exhibits no correlations
with any soil properties. Consequently, the general
application of Eq. (2) becomes limited because A cannot
be estimated easily based on soil properties. To improve
upon the pile setup estimation, Ng et al. [18] conducted an
extensive load test program to quantify the pile setup and
develop a semi-empirical method to estimate the setup for
low displacement piles in a cohesive soil profile. The
experimental results revealed that setup significantly
influenced shaft resistance but not the end bearing.
Recognizing relative small contribution of end bearing to
total resistance and to maintain simplicity in design
applications, pile setup in terms of total pile resistance
was considered. Based on extensive field evaluations and
quantifying the setup using the effect of pore water
pressure dissipation, Ng et al. [19] concluded that setup
can be satisfactorily estimated using: (1) pile resistance at
EOD obtained from either a bearing graph (ultimate pile
resistance versus hammer blow count) generated using
Wave Equation Analysis Program (WEAP) or CAse Pile
Wave Analysis Program (CAPWAP) method; (2) soil
properties; and (3) pile geometry, as described below:
!
"
#
f c ch
t
þ fr log10
ð3Þ
Rsetup ¼ REOD
tEOD
Na rp2
where REOD is the pile resistance at EOD, Rsetup is the net
increase in pile resistance immediately after EOD, fc and fr
are empirical factors (13.78 and 0.149, respectively, for
WEAP), Ch is the horizontal coefficient of consolidation
determined from CPT pore water pressure dissipation tests
and strain path method, Na is the weighted average SPT
N value accounting for different cohesive soil thicknesses
within a soil profile, rp is the equivalent pile radius based
on pile cross-sectional area, t is the time of pile setup
elapsed after the EOD, and tEOD is the time at EOD
(assumes 1 min). It is important to note that Eq. (3) was
developed based on load tests performed within 36 days as
summarized in Ng et al. [18]. Thus, its application for
quantifying long-term setup shall be used with caution.
Acta Geotechnica
computational effort, which is attributed to an iterative
procedure that simultaneously adjusts the load and resistance components until a target reliability index is
achieved. This procedure requires a purpose-built program,
which may not be readily available to establish usetup
reflecting local soil conditions.
Measured Pile Resistance, Rm (kN)
8000
6000
Ratio
Mean
R m /R EOD
R m /R t
1.643
0.996
Std.
Dev.
0.370
0.194
COV
N
0.225
0.195
39
39
4000
Huang [10]
Lukas et al. [14]
Long et al. [13]
Fellenius [8]
ISU-PILOT
ISU Field Tests
2000
3 Uncertainties of pile resistance
3.1 Evaluation based on resistance ratio
0
0
2000
4000
6000
8000
Estimated Pile Resistance Considering Setup, Rt (kN)
Fig. 1 Comparison between the total measured and the total
estimated resistances of steel H-piles using Eq. (3) and WEAP
Detailed derivation of the pile setup empirical Eq. (3) can
be found in Ng et al. [19]. The validation for steel H-piles
is shown in Fig. 1, while similar validation for other pile
types was documented in Ng et al. [19].
2.2 Incorporation of setup in pile design
Recognizing the different degree of uncertainties associated with nominal values of REOD and Rsetup, Komurka
et al. [13] proposed applying separate safety factors for
these components within the allowable strength design
(ASD) framework. Although the same concept can be
extended to the LRFD framework, establishing a suitable
resistance factor for Rsetup is not straightforward. If Rsetup is
added to REOD as a single resistance component, and is not
considered separately using the proposed Eq. (1) during a
conventional resistance factor calculation, an unrealistically high resistance factor will be yielded, such as the
resistance factor of 1.31 determined by Abu-Farsakh et al.
[1] for precast, prestressed concrete piles driven in Louisiana soil and verified using CAPWAP at the EOD condition. Using a database of 37 tests collected by Paikowsky
et al. [21] on various pile types in clay, which comprised of
five steel H-piles, five closed-end pipe piles, five timber
piles, seven prestressed concrete piles, nine concrete piles,
and six other pile types, Yang and Liang [27] selected Skov
and Denver’s [24] empirical Eq. (2) to estimate Rsetup. The
results of their statistical analysis confirmed the different
degree of uncertainties for Ro and Rsetup, indicating with
coefficients of variation (COVs) of 0.339 for Ro and 0.475
for Rsetup shown in Table 3. Yang and Liang [27] used an
invariant FORM to determine the resistance factors for
Rsetup, recommending a conservative usetup of 0.30 for a
bridge span of\60 m based on a target reliability index (b)
of 2.33. Despite its accuracy, the FORM requires intensive
The database shown in Table 1 was used to determine the
uncertainties of the two resistance components. Table 1
comprises five recently completed full-scale pile tests
within the State of Iowa, USA (data sets 1–5; see Ng et al.
[16] for more details), ten data sets from the PIle LOad
Test (PILOT) database of the Iowa Department of Transportation (data sets 6–15; refer Roling et al. [23] for more
information), and four well-documented tests (data sets
16–19) reported by others in literature [9, 11, 14, 15]. Soil
condition for data sets 1–15 was mostly glacial till, while
soil conditions for data sets 16–19 are described in Table 1,
Table 1 also summarizes the elapsed time (t) of pile
restrikes or SLT following EOD.
The pile driving resistance (hammer blow count) at
EOD, needed for estimating initial pile resistance using
WEAP (Re-EOD), was available for all data sets. The force
and velocity records at EOD, measured using pile driving
analyzer (PDA) for CAPWAP signal matching, were
available only for data sets 1–5, 16, 18, and 19. Additionally, the driving resistances at the beginning of restrikes
(BOR) at different times (Re-restrike), required for estimating
pile resistance using WEAP analysis, are available for data
sets 1–5, and 18. The WEAP pile resistance at time t (Re-t)
can also be established, which is the sum of the estimated
EOD resistance (Re-EOD) and setup resistance (Re-setup)
calculated using Eq. (3). As suggested by Yang and Liang
[27], a measurement-based pile resistance at EOD
(Rm-EOD) and/or a restrike time (Rm-t) was taken as the
resistance value calculated from CAPWAP analysis using
PDA data. A measurement-based pile resistance at the time
of SLT (Rm-t) was determined based on Davisson’s criterion [8]. Consequently, it follows that the difference
between Rm-t and Rm-EOD is assumed to be the measured
pile setup resistance (Rm-setup). CAPWAP method was
chosen to determine the pile resistances at EOD (Rm-EOD),
because of the following reasons:
1.
2.
The pile resistances obtained from CAPWAP are
based on field measurements of pile force and velocity
records using PDA;
Since it is practically infeasible to measure pile
resistances at EOD using SLT, measurement-based
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Acta Geotechnica
Table 1 Five pile resistance ratios (RR) based on a database of steel H-piles
Data
set
References {Soil Condition}
(Location)
1
ISU2 {GT} (Mills, IA, USA)
2
3
4
5
6
7
8
9
10
11
12
13
ISU3 {GT} (Polk, IA, USA)
ISU4 {GT} (Jasper, IA, USA)
ISU5 {GT} (Clarke, IA, USA)
ISU6 {GT} (Buchanan, IA, USA)
PILOT {GT} (Decatur, IA, USA)
PILOT {GT} (Linn, IA, USA)
PILOT {GT} (Linn, IA, USA)
PILOT {GT} (Linn, IA, USA)
PILOT {GT} (Johnson, IA, USA)
PILOT {GT} (Hamilton, IA, USA)
PILOT {GT} (Kossuth, IA, USA)
PILOT {GT} (Jasper, IA, USA)
RR1
Restrike,
(Rm-t/Re-restrike)
RR2
Typical RR,
(Rm-t/Re-EOD)
RR3
RR at time t,
(Rm-t/Re-t)
RR4
EOD,
(Rm-EOD/Re-EOD)
RR5
Setup,
(Rm-setup/Re-setup)
9d
0.90a
1.62a
0.91a
1.05b
0.74a
d
1.14
a
1.84
a
1.00
a
1.21
b
0.76a
1.00
a
1.62
a
0.94
a
1.07
b
0.76a
0.95
a
1.70
a
1.02
a
1.24
b
0.68a
0.84
a
1.54
a
0.90
a
1.05
b
0.69a
1.63
a
1.05
a
–
1.34
a
0.85
a
0.58a
0.97
a
0.62
a
-0.05c
1.48
a
0.94
a
0.82a
1.50
a
0.97
a
0.93a
1.84
a
1.18
a
1.48a
1.33
a
0.84
a
0.56a
1.70
a
1.15
a
1.45a
1.56
a
0.96
a
0.91a
Time elapsed after
EOD, t (day)
36
d
16
9
d
d
14
3
d
5
d
5
d
5
d
3
d
4
d
5
d
1
d
d
14
PILOT {GT} (Poweshiek, IA, USA)
8
15
PILOT {GT} (Poweshiek, IA, USA)
3d
16
Huang [11] {Silty clay over silty sand}
(Shanghai, China)
2e
17
18
19
Lukas and Bushell [15] {Fill over clay}
(Illinois, USA)
Long et al. [14] {Silty clay over sandy
till} (Illinois, USA)
Fellenius [9] {Mixture of sand, silt and
clay over glacial till} (Canada)
–
–
d
31
d
10
–
d
26
7
d
d
22
e
7
0.77
a
1.11
a
–
e
13
e
15
e
16
e
18
e
21
e
28
e
32
e
44
1.14a
1.16a
0.75a
1.39b
0.87b
0.52a
2.25
a
1.23
a
1.70
a
1.05
a
1.95
a
1.16
a
1.02
a
0.61
a
2.16
a
0.87
a
1.72
b
1.07
b
1.87
b
1.13
b
0.67b
1.92
b
1.16
b
0.74b
2.11
b
1.27
b
1.02b
2.03
b
1.22
b
0.89b
1.91
b
1.14
b
0.70b
2.24
b
1.32
b
1.16b
2.32
b
1.36
b
1.26b
2.29
b
1.33
b
1.19b
0.93b
0.78b
1.61a
1.12a
–
1.38a
0.91
b
1.43
b
0.04c
0.85a
1.19b
Measured pile resistance using SLT; b measured pile resistance using CAPWAP; c pile setup was insignificant thus neglected; d time of SLT; e time of
restrike; Re = estimated pile resistance using WEAP; Rm = measurement-based pile resistance obtained using CAPWAP or SLT; and GT = glacial till
a
3.
pile setup using CAPWAP can be effectively used to
compare with the estimated value using Eq. (3); and
The relatively high-efficiency factor (u/k) for CAPWAP [21].
To investigate various sources of uncertainties for different resistance components, five different resistance
ratios (RR), ratios of measurement-based resistance values
(Rm) to analysis-based estimated pile resistances (Re), are
summarized in Table 1. In this paper, measurement-based
pile resistances were obtained from SLT and/or CAPWAP,
while estimated pile resistances were determined using
WEAP. However, it is important to note that such a comparison shall not be restricted in this arrangement. Each of
these ratios is described as follows:
123
1.
2.
3.
RR1 is a ratio of measurement-based pile resistance
using SLT at time (t) to pile resistance estimated using
WEAP during the last restrike event before SLT. RR1
is used to evaluate the uncertainty associated with
Rrestrike (i.e., RR1 = Rm-t/Re-restrike);
RR2 is a typically used ratio of measurement-based
pile resistance at time (t) using either SLT or
CAPWAP to estimate WEAP pile resistance at EOD
without any setup consideration (i.e., RR2 = Rm-t/
Re-EOD);
RR3 defines the ratio of measurement-based SLT or
CAPWAP resistances to total estimate resistances,
which is the sum of the REOD using WEAP and the
Rsetup using Eq. (3) (i.e., RR3 = Rm-t/Re-t);
Acta Geotechnica
4.
5.
RR4 defines the ratio of measurement-based resistance
to estimated resistance at EOD using WEAP (i.e.,
RR4 = Rm-EOD/Re-EOD); and
RR5 defines the ratio of measurement-based pile setup
resistance to the setup resistance estimated using
Eq. (3) (i.e., RR5 = Rm-setup/Re-setup).
The RR4 and RR5 are two essential ratios for calibrating
uEOD and usetup, respectively.
3.2 Results of conventional FOSM analysis
Table 2 presents the resistance biases (kR) and coefficients
of variation (COVR) calculated for all five RRs (i.e., RR1 to
RR5) using the database summarized in Table 1. It shows
that pile resistances obtained from restrikes (RR1) have the
lowest COVR value of 0.14, which reveals the reliability of
pile setup quantification using restrike tests, but as noted,
they are not always feasible in routine practice. Furthermore, the significant difference in COVR values between
EOD (0.157 in RR4) and setup (0.317 in RR5) confirms the
different uncertainties with EOD and the setup, justifying
the need to develop a separate resistance factor for each
component.
In compliance with the LRFD limit state (i.e., uR C cQ)
and assuming the load (Q) and resistance (R) are mutually
independent and follow a lognormal distribution, the
resistance factors (u) for RR1 to RR4 were calibrated in
accordance with the FOSM method, using Eq. (4) as suggested by Barker et al. [5]. With the focus on the axial pile
resistance, the AASHTO [3] ‘‘Strength I’’ load combination was used. The numerical values for the different
probabilistic characteristics of dead (QD) and live (QL)
loads (c, k, and COV), as documented by Nowak [20] and
adopted by Paikowsky et al. [21], were recapitulated in
parentheses given in the definition of each parameter as
follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kR cDL COVDLR
ffio
u¼
ð4Þ
n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kDL exp bT ln 1 þ COV2R ðCOVDL Þ
cDL ¼
cD QD
þ cL
QL
kDL ¼
kD QD
þ kL
QL
COVDLR ¼
COVDL
1 þ COV2R
COVDL ¼ 1 þ COV2D þ COV2L
where kR is the resistance bias factor of the RR, COVR is
the coefficient of variation of the RR, cD is the dead load
factor (1.25), cL is the live load factor (1.75), kD is the dead
load bias (1.05), kL is the live load bias (1.15), COVD and
COVL are the coefficients of variation of dead load (0.1)
and live load (0.2) respectively, and QD/QL is the dead to
live load ratio (2.0). As recommended by Paikowsky et al.
[21] and adopted by AASHTO [3], the target reliability
indices (bT) of 2.33 (corresponding to 1 % probability of
failure) for redundant piles and 3.00 (corresponding to
0.1 % probability of failure) for non-redundant piles were
used.
Table 2 shows the respective resistance factors (u) and
efficiency factors (u/k). When the effect of setup was
considered based on RR1 using restrikes and RR3 using
Eq. (3), a realistic resistance factor of 0.69 for the bT
values of 2.33 was yielded. In contrast, for RR2, a comparison was made based on the EOD condition, yielded
unrealistically high resistance factors of 1.11 and 0.91 for
the bT values of 2.33 and 3.00, respectively. These relatively large kR values were due to indirectly and inaccurately including effect of pile setup in the calculation of the
resistance factors. Recognizing the difference in the COV
values between the EOD component in RR4 and the setup
component in RR5, a single resistance factor determined in
RR4 concluded that the conventional LRFD calibrating
procedure using Eq. (4) cannot account for the different
uncertainties associated with both components. This
implies that combining the uncertainties of REOD and Rsetup
into one resistance factor fails to satisfy the LRFD philosophy and achieve a consistent and reliable design.
Table 2 Comparison of resistance factors obtained using the conventional LRFD framework
Resistance ratio (RR)
Sample size
kR
COVR
Nominal pile resistance (R)
bT = 2.33
bT = 3.00
u
u/k
u
u/k
7
0.959
0.140
Re-restrike
0.69
0.72
0.58
0.61
RR2 = Rm-t/Re-EOD
30
1.723
0.211
Re-EOD
1.11
0.65
0.91
0.53
RR3 = Rm-t/Re-t
30
1.029
0.190
Re-t [Eq. (3)]
0.69
0.67
0.57
0.55
RR4 = Rm-EOD/Re-EOD
8
1.111
0.157
Re-EOD
0.78
0.71
0.65
0.59
RR5 = Rm-setup/Re-setup
28
0.950
0.317
Not applicable for Eq. (4)
RR1 = Rm-t/Re-restrike
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Acta Geotechnica
Lognormal Distribution - 95% CI
0.5
99
Resistance Ratio for setup (RR5)
Percent
95
90
1.0
2.0
Resistance Ratio for EOD(RR4)
Resistance
Ratio for setup
(RR5)
(RR5)
Mean
-0.09821
Std Dev
0.3109
N
28
AD
0.374
P-Value
0.392
80
70
60
50
40
30
20
Resistance
Ratio for EOD
(RR4)
(RR4)
Mean
0.09520
Std Dev 0.1532
N
8
AD
0.255
P-Value
0.620
10
5
for EOD and setup, respectively. Using a total of 17 points
given in Table 1 and having both RRs D and S, the correlation coefficient (q) was calculated as 0.071. Compared
with the q value of -0.243 computed for the Skov and
Denver [24] empirical Eq. (2), which led to the independent relationship between the pile reference resistance (Ro)
and setup resistance (Rsetup) as noted by Yang and Liang
[27], the much smaller calculated q value of 0.071 concluded that pile resistances for EOD (D) and setup (S) are
mutually independent.
1
0.5
1.0
2.0
Fig. 2 Normality test using the Anderson–Darling method for pile
setup resistance and initial resistance at EOD
4 FOSM criteria
To consider the pile setup resistance estimated using
Eq. (3) in pile designs that conforms to the reliability
theory used in the LRFD framework, the principle of
strength limit state function (g) corresponding to a safety
margin is expanded, as in Eq. (5). The first FOSM criterion
is satisfied, and this equation is valid, but only if REOD,
Rsetup, and applied load (Q) have lognormal distributions as
described below:
g ¼ lnðREOD Þ þ ln Rsetup lnðQÞ
ð5Þ
To verify that the pile resistances given in Table 1
follow lognormal distributions, a hypothesis test, based on
the Anderson–Darling (AD) [4] normality method, was
used to assess the goodness of fit of the assumed
distributions. Unlike other goodness-of-fit tests such as
the Kolmogorov–Smirnov test [12], the AD method was
designed to better detect discrepancies in the tail regions of
a probability distribution, especially with a relatively small
sample size [10]. Figure 2 shows that the AD values of
0.255 and 0.374 are smaller than the critical P values of
0.620 and 0.392 within the 95 % confidence interval (CI)
for EOD and setup conditions, respectively. The hypothesis
test confirmed the assumed lognormal distributions for both
resistances and the limit state function (g).
In order to verify the independent relationships between
the random variables required for the second FOSM
criterion, the correlation between the RR for EOD (let
Rm-EOD/Re-EOD be D) and for the setup (let Rm-setup/Re-setup
be S) was assessed through the calculation of a correlation
coefficient (q) using
q¼
covðD; SÞ
rD rS
ð6Þ
where cov(D, S) is covariance between the RR for EOD
and setup, and rD and rS are standard deviations of the RR
123
5 Resistance factors for pile setup
Following the derivation described in ‘‘Appendix,’’ the
resistance factor for pile setup (usetup) based on the FOSM
Method can be expressed as:
cDL
ksetup
uEOD
QDL
usetup ¼
ð7Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kDL
exp bT ln½ðCOVRR ÞðjÞ
QDL
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kEOD
j
COVRR
QDL ¼ 1 þ
QD
QL
COVRR ¼ 1 þ COV2REOD þ COV2Rsetup
Q2D 2
k COV2D þ k2L COV2L
Q2L D
j¼1þ 2
QD
QD 2
kD kL þ k2L
2 kD þ 2
QL
QL
where ksetup is the resistance bias factor of the setup
resistance, COVREOD is the coefficient of variation of the
resistance at EOD, and COVRsetup is the coefficient of
variation of the setup resistance. This equation reveals that
usetup is dependent on several parameters. Considering only
the AASHTO [3] ‘‘Strength I’’ load combination, the
probabilistic characteristics (c, k, and COV) of the random
variables QD and QL are defined in Eq. (4). Considering the
database of steel H-piles summarized in Table 1, the
probabilistic characteristics (k and COV) of the random
variables REOD and Rsetup were selected from RR4 and RR5
of Table 2, respectively. Since the a value is suggested as
unity in the ‘‘Appendix,’’ the following analyses primarily
focus on the influence of the remaining parameters (i.e., bT,
uEOD, and QD/QL) on usetup.
Since the uncertainty for the pile resistance at EOD is
lower than that for setup (COVEOD = 0.157 less than
COVsetup = 0.317) and mean biases of both resistances are
close to unity (kEOD = 1.111 and ksetup = 0.950), Fig. 3
shows that the calculated uEOD values are noticeably
Acta Geotechnica
0.60
setup=0.327
0.20
0.10
0.00
1.80
2.30
2.80
Target Reliability Index,
3.30
T
0.40
T
/ =0.42 ( T=2.33)
0.40
/ =0.34 ( T=3.00)
0.20
0.10
0.30
0.20
Reduced Safety
Margin
Redundant
Safety Margin
= 2.33)
0.70
0.80
0.90
0.10
EOD
Fig. 5 Relationship between resistance factors for setup and EOD
condition
0.50
0.60
0.50
0.30
0.00
0.60
setup
= 2.33
= 3.00
0.10
Resistance Factor for EOD,
Efficiency Factor, /
setup
Resistance Factor for Setup,
Note: Based on EOD = 0.78 and 0.65
for T = 2.33 and 3.00, respectively
0.50
0.20
T
Fig. 3 Relationship between the resistance factor and the target
reliability index
0.60
T
setup=0.436
0.30
setup=0.327
setup=0.398
(
0.40
0.30
EOD=0.783
EOD=0.653
EOD
0.50
0.40
= 3.00)
( = 1.0)
T
setup
0.60
(
0.70
0.50
0.00
0.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20
= REOD /(QD+QL )
Fig. 4 Relationship between the resistance factor for setup and the
ratio of pile resistance at EOD to total load (a)
higher than the usetup values at a fixed QD/QL of 2.0.
Similar to the trend observed for uEOD, usetup decreases
with increasing bT. Based on the data tabulated in Table 1
and the probabilistic characteristics listed in Table 2, usetup
values decreased from 0.436 to 0.327 as bT values
increased from 2.00 to 3.00. This observation conforms to
the LRFD framework suggested by Paikowsky et al. [21],
whereby a smaller resistance factor is dependable for a less
redundant pile group that has a larger probability of failure,
represented with a higher bT.
Figure 4 shows the relationship between the usetup and a
values. The uncertainties associated with REOD have been
accounted for in terms of uEOD, and thus, there is no reason
to accommodate any additional safety margins. Any
redundant safety margins applied to REOD, such as having
an a value greater than 1.0, will reduce the usetup value
toward zero. This observation indicates that the resistance
contribution from pile setup within the LRFD framework
becomes less significant when a pile has a large resistance
Resistance Factor for Setup,
0.80
Note: Based on constant QD /QL = 2.0
EOD=0.653
Note: Based on EOD = 0.86,
0.78, 0.75 and 0.65 for T =
2.00, 2.33, 2.50 and 3.00,
respectively
EOD=0.860
setup
0.90
Resistance Factor for Setup,
Resistance Factor for EOD or
Setup, EOD or setup
1.00
0.454
0.45
(
T
= 2.33)
0.395
0.40
0.371 0.366
0.359
0.35
0.325
(
0.30
T
0.309 0.306
= 3.00)
0.25
0.20
0.15
0.10
Note: Based on EOD = 0.78
and 0.65 for T = 2.33 and
3.00, respectively
0.05
0.00
0
0.5
1
1.5
2
2.5
3
3.5
4
Ratio of Dead to Live Load, QD /QL
Fig. 6 Relationship between resistance factor for setup and dead to
live load ratio
component contributed from the uEODREOD that sufficiently overcomes the applied factored loads (RcQ). The
resistance contribution from pile setup becomes nil specifically at a values equal to 1.80 and 2.16 for the bT values
of 2.33 and 3.00, respectively. If the REOD value can be
appropriately reduced while satisfying the limit state
Eq. (1), the resistance contribution from pile setup can be
increased by having a larger usetup value. Therefore, suggesting the a value in Eq. (22) to unity, Eq. (7) is appropriately recommended to determine the usetup value.
In order to illustrate the influence of different uEOD
values that represent different regional soil conditions and
design practices, Fig. 5 illustrates the variation of usetup
values as a function of uEOD values based on a constant
QD/QL ratio of 2.0. This figure shows that higher usetup
values resulted from a higher uEOD. For a non-redundant
pile group at a bT value of 3.00, the uEOD and usetup values
were determined to be 0.653 and 0.327, respectively. For a
redundant pile group at a bT value of 2.33, the uEOD and
123
Acta Geotechnica
Table 3 Summary of recommended LRFD resistance factors
Empirical method
Pile type
Pile
resistance
FOSMa or
FORMb
Resistance bias
(k)
Coefficient of variation
(COV)
Resistance factor, u
bT = 2.33
bT = 3.00
REOD
FOSM
1.111
0.157
0.78c
0.65c
0.950
0.317
0.36
0.31
FOSM
1.158
0.339
0.58
0.45
0.65
0.50
1.141
0.475
0.27
0.20
0.30
0.00
Equation (3)
Steel
H-pile
Rsetup
Skov and Denver
[24]
All
Ro
FORM
Rsetup
FOSM
FORM
a
Based on a suggested a value of one;
b
based on Yang and Liang’s [27] results; and
usetup values were determined to be 0.783 and 0.398,
respectively.
The aforementioned observations from Figs. 3, 4, and 5
are based on a fixed QD/QL of 2.0, and it is of interest to
investigate the influence of this ratio, which is a function of
the bridge span, on the usetup values. Based on the
AASHTO [2] Specifications, QD/QL ratios of 0.52, 1.06,
1.58, 2.12, 2.64, 3.00, and 3.53 are suggested for span
lengths of 9, 18, 27, 36, 45, 50, and 60 m, respectively
[27]. Figure 6 illustrates that usetup reduces gradually with
increasing QD/QL ratios from 0.52 to about 2.12, and at an
even slower rate thereafter. This figure also indicates that
an increase in QD/QL ratio, by a factor of 6.8 (i.e., from
0.52 to 3.53), only reduces usetup by a small factor of about
1.2 (i.e., from 0.454 to 0.371). Hence, it can be generally
concluded that usetup values are almost insensitive to the
QD/QL ratios. Regardless of different QD/QL ratios, usetup
values of 0.36 and 0.31 as summarized in Table 3 can be
conservatively recommended for bT values of 2.33 and
3.00, respectively.
6 Comparison with FORM
Using dynamic and SLT data from 37 piles collected by
Paikowsky et al. [21], pile setup resistance factors (usetup)
suggested by Yang and Liang [27] based on FORM were
compared with the values calculated using the proposed
FOSM procedure. Pile resistances were determined using
CAPWAP at both EOD and BOR conditions. Skov and
Denver’s [24] empirical Eq. (2) was adopted by Yang and
Liang [27] to estimate the pile setup resistance with the
reference time to set to 1 day and an average A value of
0.50. The k and COV of Ro and Rsetup determined by Yang
and Liang [27] are given in Table 3. Accordingly, the
random variable Ro was best fitted with a lognormal distribution, while Rsetup was best fitted with a normal distribution. Using these probabilistic characteristics, usetup
values computed by Yang and Liang [27] based on FORM
were plotted in Fig. 7 with respect to QD/QL ratios ranging
123
c
based on a sample size of 8 and use with caution
from 0.52 to 4.0. Using the same probabilistic characteristics and the resistance factors for Ro determined based on
the FOSM, usetup values determined using the proposed
FOSM Eq. (7) are compared in Fig. 7. For redundant pile
groups represented by a target reliability index (bT) of 2.33,
Fig. 7 (a) shows that the usetup values determined using both
methods decrease with increasing QD/QL ratios (i.e., both
share a similar trend). The smallest usetup value based on
FORM is 0.32, while the smallest usetup value based on the
proposed FOSM method is 0.27, which is comparable to the
recommended usetup value of 0.30 suggested by Yang and
Liang [27]. The usetup values determined using FORM are
generally 20 % higher than those calculated using the
FOSM method, indicating that usetup values calculated
using the simpler and closed-form FOSM method are relatively more conservative than those determined using a
more complex and computationally intensive FORM
method. The difference usetup value is attributed to: (1) the
different reliability theory used in estimating the usetup
values; (2) the different uEOD values, as given in Table 3;
and (3) the assumed distribution for the probabilistic characteristics of the random variable Rsetup.
For non-redundant pile groups (or individual piles)
represented by a target reliability index (bT) of 3.00, Fig. 7
(b) shows a similar relationship between usetup values and
QD/QL ratios. However, usetup values determined using the
FORM are almost nil, while those calculated using the
proposed FOSM method are about 0.20. Yang and Liang
[27] concluded that a zero usetup value implies that pile
resistance will be optimized solely based on Ro, and the
benefits of incorporating Rsetup will not be realized due to
higher uncertainties associated with the estimation of
Rsetup. In fact, pile setup did occur in the 37 pile data series
reported by Yang and Liang [27]. Furthermore, extensive
pile load tests performed by Ng et al. [17] confirmed that
pile setup as high as 55 % occurred when a single pile was
embedded in cohesive soil. Hence, it is justifiable to apply
the usetup value of 0.20 determined using the proposed
FOSM method, so that the benefits of incorporating pile
setup in LRFD can be realized.
Acta Geotechnica
0.60
0.60
(a)
Proposed FOSM
Note: Based on
FORM by Yang and Liang [26]
value of 1.00
0.50
setup
value of 1.00
0.40
Resistance Factor for Setup,
Resistance Factor for Setup,
Note: Based on
setup
0.50
(b)
Proposed FOSM
FORM by Yang and Liang [26]
0.40
0.30
(Recommended setup = 0.30 by Yang &
Liang [26] using FORM for T = 2.33)
0.20
0.10
0.30
0.20
0.10
0.00
(Recommended setup = 0.00 by Yang &
Liang [26] using FORM for T = 3.00 )
0.00
-0.10
0
1
2
3
4
Ratio of Dead to Live Load, Q D /QL
0
1
2
3
4
Ratio of Dead to Live Load, Q D /QL
Fig. 7 Comparison of usetup based on FOSM and FORM for (a) bT = 2.33, and (b) bT = 3.00
Using the proposed FOSM procedure, Table 3 summarizes the recommended resistance factors for both the Skov
and Denver’s [24] method and the setup Eq. (3). Comparing these resistance factors calculated for Skov and
Denver’s [24] method and all pile types (i.e., 0.58 and 0.45
for Ro; 0.27 and 0.20 for Rsetup), higher resistance factors
were obtained for the setup Eq. (3) and steel H-piles. The
higher resistance factors are attributed to (1) the application
of a more accurate empirical pile setup Eq. (3) as demonstrated in Ng et al. [19], and (2) the use of the regional
database given in Table 1, which contained only steel
H-piles, hence reducing the uncertainties caused by various
pile types used by Yang and Liang [27]. However, it is
important to remind that Eq. (3) was developed based on
load tests performed within 36 days, the recommended
resistance factors shall be used with caution for a pile setup
estimated longer than 36 days. Furthermore, the recommended uEOD for Eq. (3), 0.78 and 0.65 for bT of 2.33 and
3.00, respectively, shall be used with caution as they were
determined based on a limited sample size of 8. If more
pile load test data are available in future, recalibration of
the resistance factors using the proposed procedure is
encouraged.
while the pile performance in terms of achieving the
desired Rtarget is verified using either a dynamic analysis
method such as WEAP or SLT. The target nominal
pile resistance estimated using a static analysis method
(Rtarget,static) is determined by:
P
cQ
Rtarget;static
ð8Þ
/static
where RcQ is the total factored applied load, and ustatic is
the resistance factor of any static analysis method listed in
Table 4 recommended by AASHTO [3]. If pile setup
would be considered during design using Eq. (3), the limit
state Eq. (1) can be written as Eq. (9) by replacing Rsetup
with Eq. (3), such that:
X
cQ uEOD REOD
!
"
#
fc Ch
t
þ usetup REOD
þ fr log10
ð9Þ
tEOD
Na rp2
7 Application framework
Rearranging this equation, REOD becomes the target
nominal pile driving resistance at EOD (Rtarget-EOD), which
may be verified using WEAP during construction, via
P
cQ
!
"
Rtarget;EOD
#
fc Ch
t
þ fr log10
uEOD þ usetup
tEOD
Na rp2
In current practice, driven production piles are designed
using a static analysis method to determine a suitable pile
length for a given target nominal pile resistance (Rtarget),
Note that the denominator is always higher than ustatic,
since the uEOD of 0.50 for WEAP (or a higher uEOD of 0.78
based on the data given in Table 1) is larger than any of the
ð10Þ
123
Acta Geotechnica
Table 4 Summary of AASHTO [3] recommended LRFD resistance
factors
Method
Soil type
Resistance factor (u)
bT = 2.33
bT = 3.00
WEAP
All
0.50
0.40
a-Method
Clay
0.35
0.28
b-Method
k-Method
Clay
Clay
0.25
0.40
0.20
0.32
CPT-method
Clay
0.50
0.40
Based on 19 data sets of steel H-piles embedded in
cohesive soil with pile resistance at EOD estimated using
WEAP and pile setup estimations determined using the
proposed empirical Eq. (3), conservative resistance factors
of 0.36 and 0.31 were recommended for pile setup for
redundant and non-redundant pile groups, respectively. The
suggested application framework of incorporating pile
setup in LRFD facilitates economic advantages in pile
designs.
Acknowledgments The authors would like to thank the Iowa
Highway Research Board for sponsoring the research presented in this
paper.
static analysis methods (i.e., a-method, b-method,
k-method, and CPT-method) listed in Table 4. Having a
larger denominator term and the same RcQ, the Rtarget,EOD
determined using Eq. (10) considering pile setup will
certainly be smaller than the Rtarget,static determined using
Eq. (8). If the pile setup is considered and incorporated in
design using the proposed LRFD procedure: (1) the target
pile driving resistance determined using Eq. (10) will be
smaller than that determined using the conventional static
analysis method; (2) a shorter pile embedment length will
be required to achieve the target pile resistance; (3) the
retapping of piles after EOD, for which the assumed target
pile driving resistance at EOD based on a static analysis
method has not been met, can be reduced since a smaller
target pile driving resistance will be required; and (4) the
economic advantages of pile setup can be realized while
complying with the LRFD framework and ensuring a target
reliability level. The aforementioned advantages have been
confirmed in a recent study that investigated the impact of
incorporating setup on 604 production steel H-piles driven
in cohesive soils [17]. This study found that the target
driving resistance on average was reduced by 17 % and the
number of production piles requiring retapping reduced
from 37 % to 15 %.
8 Conclusions
A successful incorporation of pile setup in LRFD offers
cost-effective foundation solutions in cohesive soils.
However, the pile setup is not routinely included in foundation design due to lack of an easily usable setup method
in compliance with the LRFD framework. In a recent
study, it has been shown that the pile setup can be estimated using soil parameters. To incorporate the setup
estimated based on this or other similar procedures into the
LRFD framework, a closed-form FOSM method has been
presented, so that the uncertainties associated with the pile
resistance at EOD and the setup can be accounted for
separately.
123
Appendix: Derivation of the resistance factor for setup
Satisfying the lognormal distributions and independent
relationships of loads and resistances, the reliability index
(b) is expressed as a ratio of mean to standard deviation of
the limit state function (g), which can be expanded as
follows:
EðgÞ EðlnðREOD ÞÞ þ E ln Rsetup EðlnðQÞÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
q
b¼
¼
rg
r2lnðREOD Þ þ r2ln R
þ r2lnðQÞ
ð setup Þ
ð11Þ
where REOD is the pile resistance at EOD, Rsetup is the pile
setup resistance, E(g) is the expected value of the limit
state function g, rg is the standard deviation of the limit
state function g, E(ln(REOD)) is the expected value of the
natural logarithm of the pile resistance at EOD, and
rln(REOD) is the standard deviation of the natural logarithm
of the pile resistance at EOD, which can be similarly
defined for other random variables. To express Eq. (11) in
terms of simple means (i.e., R and Q) and coefficients of
variation (COV) for resistances and loads of the normal
distributions, the mean and standard deviation of a
lognormal distribution for any resistance or load can be
transformed using the following general expressions:
EðlnðRÞÞ ¼ ln R 0:5 ln 1 þ COV2R
ð12Þ
r2lnðRÞ ¼ ln 1 þ COV2R
ð13Þ
Using these expressions for the three random variables
(REOD, Rsetup, and Q) and substituting them into Eq. (11),
the reliability index can be expressed as follows:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
0v
u
u
1 þ COV2Q
REOD þ Rsetup
A
ln
þ ln@t
COVRR þ 2COV2REOD COV2Rsetup
Q
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b¼
h
i
ln COVRR þ 2COV2REOD COV2Rsetup 1 þ COV2Q
ð14Þ
Acta Geotechnica
Replacing the simple mean values with their respective
bias factors (k), a ratio between average measured and
predicted values (i.e., Rm/R or Qm/Q), and neglecting the
terms that involve multiplying two squared coefficients of
variation (i.e., COV2COV2) since their contribution would
be insignificantly small, the expression for b is simplified as:
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
1 þ COV2Q
kREOD REOD þ kRsetup Rsetup
A
ln
þ ln@
COVRR
kQ Q
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b¼
h
i
ln ðCOVRR Þ 1 þ COV2Q
ð15Þ
using the LRFD strength limit state equation (cQ = uR)
and replacing uR with uEODREOD ? usetupRsetup, the
equation can be rearranged for Rsetup as:
Rsetup ¼
cQ uEOD REOD
usetup
ð16Þ
substituting Eq. (16) into Eq. (15) and isolating the usetup as
the variable of interest by rearranging, a preliminary
equation for usetup can be established as follows:
usetup ¼
ksetup ½cQ uEOD REOD
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
iffi
ðkQ QÞ exp b ln ðCOVRR Þ 1 þ COV2Q
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kEOD REOD
u
u 1 þ COV2
Q
t
ðCOVRR Þ
ð17Þ
considering only the dead (QD) and live (QL) loads, as per
the AASHTO [3] ‘‘Strength I’’ load combination, the
factored load (cQ) and bias load (kQ Q) are expanded to:
cQ ¼ cD QD þ cL QL
ð18Þ
kQ Q ¼ kD QD þ kL QL
ð19Þ
As defined in AASHTO [3], cD is the dead load factor of
1.25, cL is the live load factor of 1.75, kD is the dead load
bias of 1.05, and kL is the live load bias of 1.15.
Furthermore, the coefficient of variation of the load
(COVQ), derived by Bloomquist et al. [6], in terms of
dead and live load components is given as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u 2
u QD 2
u kD COV2D þ k2L COV2L
u Q2
L
ð20Þ
COVQ ¼ u
uQ 2
QD
t D 2
2
k
þ
2
k
k
þ
k
D L
L
QL
Q2L D
substituting Eqs. [18], [19], and [20] into Eq. (17), the
usetup can be revised as:
usetup ¼
ksetup ½cD QD þ cL QL uEOD REOD
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðkD QD þ kL QL Þ exp b ln½ðCOVRR ÞðjÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kEOD REOD
ðjÞ
ðCOVRR Þ
ð21Þ
Normalizing the above expression with respect to the
total load (QD ? QL), and further rearrangement of Eq.
(21) in terms of the dead load to live load ratio (i.e., QD/
QL) and representing a as the ratio of pile resistance at
EOD to the total load (i.e., a = REOD/[QD ? QL]), the
resistance factor of pile setup at a target reliability index
(bT) can be expressed as:
c
ksetup DL uEOD
QDL
usetup ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kDL
exp bT ln½ðCOVRR ÞðjÞ
QDL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
kEOD a
j
COVRR
ð22Þ
The parameter a, a ratio of pile resistance at EOD to
total load, noted above is analogous to a safety factor
applied to the REOD if the traditional allowable stress
design (ASD) approach would have been considered. The
uncertainties associated with REOD have been accounted
for in terms of uEOD in Eq. (22) to comply with the LRFD
approach. Since the uncertainties were addressed, the
parameter a is suggested as unity in order to eliminate the
redundancy in the safety margin applied to REOD, and the
resistance factor for pile setup based on the FOSM method
can be expressed as:
c
ksetup DL uEOD
QDL
usetup ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kDL
exp bT ln½ðCOVRR ÞðjÞ
QDL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
kEOD
j
COVRR
ð23Þ
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123
Acta Geotechnica
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