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KSCE Journal of Civil Engineering (2011) 15(7):1185-1196
DOI 10.1007/s12205-011-1254-1
1185
www.springer.com/12205
Geotechnical Engineering
Resistance Factor Calculations for LRFD of Axially Loaded Driven Piles in Sands
Dongwook Kim*, Moonkyung Chung**, and Kiseok Kwak***
Received June 23, 2010/Revised November 11, 2010/Accepted December 30, 2010
Abstract
This paper presents the development of Load and Resistance Factor Design (LRFD) of axially-loaded driven piles in sands. Theresistance factors of base and shaft resistances were calculated separately to account for their different uncertainty levels. The ratiosof dead-to-live load and ultimate base resistance to limit shaft resistance change the uncertainty levels of total load and total pilecapacity, respectively; thus, those ratios should be reflected in the calculation of base and shaft resistance factors. For thedevelopment of LRFD for axially-loaded driven piles in sands, the ultimate limit state for an axially-loaded driven pile wasestablished based on the Imperial College Pile (ICP) design method; the uncertainties of loads and resistance were accessed;reliability analyses were performed using the First-order Reliability Method (FORM); and finally, reasonable resistance factors ofbase and shaft resistances were calculated based on the results of reliability analyses for different target reliability index levels. Theload factors used for the calculation of resistance factors are the ones proposed by AASHTO and ASCE/SEI 7-05. From the results ofextensible reliability analyses using FORM, the resistance factors for base and shaft resistances were found to be highly dependent onthe ratios of the dead-to-live load and the ultimate base resistance to the limit shaft resistance. Resistance factors are proposed fordifferent combinations of these ratios within their possible ranges.
Keywords: load and resistance factor design, driven pile, imperial college pile design method, reliability analysis, resistance factor,
load factor, first-order reliability method
1. Introduction
Load and Resistance Factor Design (LRFD) is conceptually a
more advanced design method than the existing Allowable Stress
Design (ASD). Successful implementation of LRFD on geotech-nical structures contributes to an economical and safe design.
Recently, many countries, such as the United States, Canada,
China, Japan, and Korea, commence to replace or already have
replaced the ASD with the LRFD. For example, the Federal
Highway Administration (FHWA) in the United States mandated
the use of LRFD for new bridge designs after October 1, 2007.
LRFD in structural engineering was developed in the mid-1980s
and has been successfully implemented in design practice.
However, LRFD in geotechnical engineering has not been fully
developed for most geotechnical structures. There has been
research into LRFD for driven piles (Zhang et al., 2001; Paikowsky
et al., 2004; Allen, 2005), and these research results wereobtained based on reliability analyses and are reflected in the
AASHTO LRFD bridge design specifications (2007). For geo-
technical structures other than piles, such as mechanically stabilized
earth walls, slopes, and shallow foundations, the values of
Resistance Factors (RFs) proposed in the AASHTO LRFD bridge
design specifications (2007) are not based on reliability analyses,
but rather are back-calculated from the existing Factor of Safety
(FS) values. A rational framework for LRFD development shouldbe established for the replacement of these back-calculated RF
values with reasonable RF values calculated based on reliability
analyses.
A large number of studies were done for the calibration of
resistance factors of geotechnical structures. Phoon and Kulhawy
(2003) corroborated the importance of implementing the multiple
resistance factor design concept for foundations. Phoon and
Kulhawy (2002) performed FORM for the calculation of multiple
resistance factors of drilled shafts for a given target reliability
level. The multiple resistance factors for uplifts of drilled shafts
include the resistance factors for side resistance and resistance
from the self-weight of the drilled shaft. For shallow transmis-sion line structures, Phoon et al. (2003) also calibrated different
uplift resistance factors for uplift side resistance, uplift tip resist-
ance and dead weight of foundation against uplift force. Honjo et
*Member, Post-doctoral Researcher, Geotechnical Engineering and Tunnelling Research Division, Korea Institute of Construction Technology, Goyang
411-712, Korea (Corresponding Author, E-mail: [email protected])
**Member, Research Fellow, Geotechnical Engineering and Tunnelling Research Division, Korea Institute of Construction Technology, Goyang 411-712,
Korea (E-mail: [email protected])
***Member, Research Fellow, Geotechnical Engineering and Tunnelling Research Division, Korea Institute of Construction Technology, Goyang 411-712,
Korea (E-mail: [email protected])
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al. (2002) established a procedure for the calculation of partial
factors for dead load, seismic load, base resistance and shaft
resistance of axially-loaded case-in situ piles. Kim et al. (2005)
calibrated local resistance factors of driven piles based on 140
Pile Driving Analyzer (PDA) data and 35 static load test data
from North Carolina Department of Transportation in the United
States.
Most of the research performed to date regarding resistance
factor calculations of driven piles have focused on calibrating a
single RF value of the total pile capacity, which is the sum of the
ultimate base resistance Qb,ultand the limit shaft resistance QsL,
for a given target reliability index (McVay et al., 2000; Paikowsky
et al., 2004; Titi et al., 2004; Kwaket al., 2010; Allen, 2005).
The RF values proposed in the current AASHTO LRFD bridge
design specifications for driven piles were also developed for the
total pile capacity calculated based on reliability analysis for the
given target reliability index. In other words, the current AASHTO
LRFD bridge design specifications (2007) also use a single RF
value for both Qb,ultandQsL for driven piles. However, Foye (2005)
and Foye et al. (2009) proposed different RF values forQb,ultand
QsL for open-ended and closed-ended driven piles, but those RF
values were calculated for a conservative single dead-to-live load
ratio (DL/LL) and a single target reliability index (T) of 3.0.
The uncertainty levels (bias factors, coefficients of variation,
and distribution types) ofQb,ultand QsL are quite different. Ac-
cordingly, it is reasonable to assign different RF values for Qb,ultand QsL reflecting their own uncertainties. The uncertainty levels
ofQb,ultand QsL are highly dependent on the method of prediction
ofQb,ult and QsL (Jardine et al., 2005; Schneider, 2007). If the
physics underlying the Ultimate Limit State (ULS) of driven piles
are reasonably incorporated in the prediction methods of Qb,ultand QsL, the prediction methods will have less uncertainty. In this
paper, the uncertainties ofQb,ultand QsL are assessed based on the
qualitative pile load test database constructed for the develop-
ment of Imperial College Pile (ICP) design methods (Jardine et al,
2005). Using the Load Factors (LFs) given in the design
specifications (AASHTO, 2007; ASCE/SEI 7-05, 2005), the RF
values ofQb,ult and QsL are calculated from extensive reliability
analyses using the First-order Reliability Method (FORM) for
different combinations ofDL/LL and Qb,ult/QsL. The FORM, as
set forth by Hasofer and Lind (1974) and Low and Tang (1997),
is used to calculate reliability index in this paper.
2. Procedure of Resistance Factor Calculation
For the successful calculation of reasonable RF values ofQb,ultand QsL for axially-loaded driven piles, a rational framework
should be established. In this paper, the following steps are used
for the calculation of RF values forQb,ultand QsL.
1. Select a reasonable equation representing the ULS of driven
piles;
2. Select a LF for each load type (e.g. dead load or live load)
from the design specifications and investigate the uncertainty
of each load;
3. Assess the uncertainties ofQb,ult and QsL using a high-quality
database including the results of pile load tests and site investi-
gations;
4. Determine an appropriate target reliability index (or target
probability of failure) considering the importance of the struc-
ture;
5. Perform reliability analysis based on the uncertainties of the
loads and resistances;
6. Calculate the RF values of the base and shaft resistances.
3. LRFD of Axially-Loaded Driven Piles in Sands
3.1 Identification of the ULS Equation using Imperial Col-
lege Pile Design Method
In order to identify the ULS of an axially-loaded driven pile in
sands, the ICP design method is selected for the calculation of
Qb,ult and QsL. This design method has been proven to predict
Qb,ult and QsL accurately by many researchers (Jardine, 1985;
Lehane, 1992; Chow, 1997). The ICP design method predicts
Qb,ult and QsL using the tip resistance qc values measured from
cone penetration tests (CPTs). The ICP design method has been
successfully implemented worldwide for offshore, marine, and
onshore sites (Jardine et al, 2005). The structures designed using
ICP methods are offshore platforms, large bridges, and relatively
small piles supporting light industrial facilities (Jardine et al,
2005).
The ultimate pile resistance (or total pile capacity) Qultis repre-
sented by the sum of the ultimate base resistance Qb,ult and the
limit shaft resistance QsL:
(1)
The ultimate base resistance Qb,ult, which is the base capacity
mobilized at a pile head settlement of 10% of its pile diameter, is
the product of the unit ultimate base resistance qb,ult and the pile
base areaAb:
(2)
If the layer along the pile shaft is divided into n sublayers, the
limit shaft resistance QsL is the sum of the unit limit shaft resis-
tances qsL,imultiplied by the pile shaft areas As,iof n sublayers:
(3)
The proposed qb,ult for closed-ended piles in the literature
(Jardine et al., 2005) are as follows:
(4)
where Dpileis the outer diameter of the pile, DCPTis the diameter
of cone (typicalDCPT= 0.036 m), and qcb,avg is the average cone
resistance at the location of pile base. For open-ended piles, qb,ultdepends on the plugged mode inside the pile. For fully plugged
Qult Qb u lt , QsL+=
Qb u lt , qb u lt , Ab=
QsL qsL i,As i,i 1=
n
= i 1 n, ,=( )
qb u lt ,1 0.5ln Dpile DCPT( )[ ]qc b a vg , ifDpile 0.9m
max 1 0.5ln Dpile DCPT( ) 0.3,[ ]qc b a vg , ifDpile 0.9m>
=
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mode, qb,ultof open-ended piles is assumed as the half ofqb,ult of
closed-ended piles:
(5)
For unplugged open-ended piles, qb,ult is calculated by thefollowing equation:
(6)
The same equation is proposed for the unit limit shaft resis-
tances of closed-ended piles and open-ended piles:
(7)
where qcis the cone resistance, 'v is the vertical effective stress,
pA is the reference stress (100 kPa), h is the distance along thepile shaft from the pile base, G is the shear stiffness calculated by
qc[0.0203 + 0.00125qc(pAv)0.51.216 106qc
2(pAv)1]1, and r
is the radial displacement during pile loading (typically assumed
to be 0.02 mm), and cv is the critical-state interface friction
angle between pile and soils contacted to the pile shaft.
3.2 Load Factors and Uncertainties of Loads
In our analysis, two types of axial load are assumed: dead load
DL and live loadLL on a pile head. According to ASCE/SEI 7-
05 (2005), the load factors (LFDLandLFLL) ofDL andLL for
building structures are 1.2 and 1.6, respectively. The uncertainties
ofDL andLL for building structures are summarized in Table 1.The uncertainties of the loads include specifying bias factors (),
Coefficients of Variation (COV), and distribution types. In our
analysis, the distribution types and the values of bias factors and
COVs in Table 1 are used for the simulation of the uncertainties
ofDL andLL for building structures. The bias factorx of variable
x is defined as the ratio of its mean valuexto nominal valuexn.
(8)
The coefficient of variation COVxof variable x is the ratio of
its standard deviation to mean value.
(9)
The AASHTO LRFD bridge design specifications (2007) pro-
pose that the load factors forDL andLL for transportation facilities
are 1.25 and 1.75, respectively. The uncertainties ofDL andLL
reported in the AASHTO LRFD bridge design specifications
(2007) are summarized in Table 2. The assessment of load
uncertainties are carried out based on the work done by Nowak
(1999). The bias factor and the coefficient of variation (COV) of
DL varies with the material used in bridge construction, while
those ofLL change depending on a bridges span length and the
number of lanes. ForDL, the COV value depends on the material
property as shown in Table 2. Considering that the dead load
induced by the asphaltic wearing surfaces is generally a small
portion of the totalDL, in our analysis, we conservatively used a
COV of 0.10 forDL. The bias factor of 1.05 is also conservati-
vely selected and used in our analysis. Likewise, the LL bias
factor of 1.2 and the COV of 0.205 were conservatively chosen.
3.3 Uncertainties of Ultimate Base Resistance Qb,ult and
Limit Shaft Resistance QsLThe measured values (Qb,ult, meas. and QsL,meas.) of the ultimate
base resistances and the limit shaft resistances are obtained from
the compression pile load test results, whereas their predicted
values (Qb,ult, pred. and QsL,pred.) are calculated using the ICP design
methods introduced in the earlier section. The ratios (Qb,ult, meas./
Qb,ult, pred. and QsL, meas./QsL, pred.) of the measured-to-predicted ultimate
base resistance Qb,ult and limit shaft resistance QsL are recon-
structed based on the data available (data are obtained by accur-
ately digitizing the figures provided in the book) from Jardine et
al. (2005). The mean and COV of the digitized data were the
same as the values proposed in the literature. Jardine et al. (2005)
stated that the database was constructed based on pile load test
results reported by Chow (1997), Willliams et al. (1997), Zuidberg
and Vergobii (1996), CUR (2001), Jardine and Standing (2000),
Jardine et al. (2001), and Jardine et al. (1998). The piles used in
the pile load tests were closed-ended or open-ended driven piles
made of steel or concrete.
qb u lt ,0.5 0.25ln Dpile DCPT( )[ ]qc b a vg , ifDpile 0.9m
max 0.5 0.25ln Dpile DCPT( ) 0.15,[ ]qc b a vg , ifDpile 0.9m>
=
qb u lt , qcb a vg ,=
qsL 0.029qc'vpA------
0.13
maxh
Ab --------------- 8,
0.38 2Gr
Ab ---------------
+
tancv=
xxxn-----=
COVx xx-----=
Table 1. Bias Factors, Coefficients of Variation, and Distribution
Types ofDL and LL for Building Structures (Ellingwood,
1999)
Load typeBias factor
()Coefficient of
variation (COV)Distribution
type
DL 1.05 0.10 Normal
LL 1.0 0.25Type I based onlargest extreme
Table 2. Bias Factors, Coefficients of Variation, and Distribution Types ofDL and LL (AASHTO, 2007 and Nowak, 1999)
Load type Bias factor COV Distribution type Note
DL 1.0-1.05 0.08-0.25 NormalFactory-made: =1.03, COV=0.08Cast-in-place: =1.05, COV=0.10Asphaltic wearing surface: =1.0, COV=0.25
LL 0.6-1.2 0.17-0.205 Lognormal
=0.6 for No. of lanes 4 in one direction.=1.2 for No. of lanes = 1 in one direction.COV=0.17 for span 9 m and No. of lanes 4 in one direction.COV=0.205 for span = 3 m and No. of lanes = 1 in one direction.
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The measured-to-predicted ratios (Qb,ult, meas./Qb,ult, pred. andQsL, meas./
QsL, pred.) ofQb,ult and QsL were calculated and were plotted as
distributions in Fig. 1. The forty eight Qb,ult, meas./Qb,ult, pred.values
and forty QsL, meas./QsL, pred. values in Fig. 1 were used for the as-
sessment of the uncertainties ofQb,ultand QsL predictions, respec-
tively. The driven piles used in the assessment of Qb,ult uncer-
tainty had various pile geometries (pile length = 1.1-47 m; pile
diameter = 0.07-2.0 m) and a wide range of relative density at the
pile bases (25-96%). Qb,ult, meas./Qb,ult, pred. values were not biased
with respect to the relative densities near pile bases and a pile
diameters. The QsL uncertainty was assessed based on pile tests
with diverse pile dimensions (pile length = 1.8-47 m; pile diameter
= 0.1-2.0m) and different average relative densities (31-100%)
along the pile shaft. QsL, meas./QsL, pred. values were also no biased
with respect to the relative densities along pile shafts and
slenderness ratio (ratio of pile length to pile diameter).
The distribution ofQb,ult, meas./Qb,ult, pred. is well-fitted to a lognor-
mal distribution with a mean (bias factor ofQb,ult, meas./Qb,ult, pred.) of
1.023 and a COV of 0.201, while that of QsL, meas./QsL, pred. nearly
follows a lognormal distribution with a mean of 1.088 and a
COV of 0.287. These uncertainties (bias factors, COVs, and dis-
tribution types) ofQb,ultand QsL are used in the reliability analyses.
3.4 Target Reliability Index
Conceptually, structures designed using LRFD methods guar-
antee a certain level of reliability, which is called the target
reliability index T. In other words, the reliability of a system
designed using LRFD is greater than or equal to the target
reliability index. A target reliability index of 3.0 is generally used
for foundation design practice (Paikowsky et al., 2004; Allen,
2005; Foye et al., 2006; and Foye et al., 2009). In general, the
AASHTO LRFD bridge design specifications (2007) propose to
use a Tof 3.5 (an approximate probability of failure of 0.0002)
in the designs of the main elements and components, the failure
of which may ultimately cause bridge failure. Lower levels ofTcan be used for less important elements and components of
bridges. For group piles, which allow redundancy, a lowerT(2.33) could be assumed (AASHTO, 2007 and Zhang et al.,
2001). In this paper, three levels (2.5, 3.0, and 3.5) of target
reliability index are considered and the corresponding resistance
factors are calculated.
3.5 Resistance Factor Calculation
For the reliability analysis using the FORM, the assessment of
uncertainties of total resistance and total load (sum of loads) are
required. The uncertainty of total resistance is a function of the
Qb,ult/QsL ratio and the uncertainties ofQb,ult, and QsL while that of
total load is a function of theDL/LL ratio and the uncertainties of
DL andLL. In addition to the uncertainties ofDL, LL,Qb,ult, and
QsL summarized in Table 3, the ratios ofDL/LL and Qb,ult/QsL are
required to account for the uncertainty of the total resistance and
the sum of the loads. Reliability analyses are performed reflect-
ing the uncertainties ofDL, LL, Qb,ult, and QsL with different
ratios ofDL/LL and Qb,ult/QsL. The results of a reliability analysis
include the calculations of the reliability index and the most
probable failure point (or the most probable ULS values ofDL,
LL, Qb,ult, and QsL).
To account for the different uncertainty levels of the ultimate
base resistance Qb,ult and the limit shaft resistance QsL, the resis-
tance factors forQb,ult and QsL are calibrated separately. Accord-
ingly, the LRFD criterion for axially-loaded driven piles can be
mathematically expressed as the following inequality:
(10)
where RFbaseand Qb,ult,n are the resistance factor for base resis-
tance and the nominal Qb,ult, RFshaft and QsL,n are the resistance
factor for shaft resistance and the nominal QsL, LFi is the load
factors from the design specifications, such as AASHTO (2007)
and AISC/SEI 7-05 (2005), and Qi,n is the nominal applied load
(or design load).
When a driven pile is axially loaded to the ULS, many possible
combinations ofQb,ult, QsL and Qi(in this paper,DL andLL) exist
that satisfy the ULS. The surface consisting of these combina-
RFbase( )Qb ult n, , RFshaf t( )QsL n,+ LFi( )Qi n,
Fig. 1. Distributions of (a) Ultimate Base Resistance Qb,ult and (b)
Limit Shaft Resistance QsL
Table 3. Summary of the Bias Factors, COVs, and Distribution Types of the Resistances and Loads Used in the Reliability Analysis
Force Bias factor () COV Distribution type Note
ResistanceQb,ult 1.023 0.201 Lognormal (Jardine et al., 2005)
QsL 1.088 0.287 '' ''
Load
ASCE-7DL 1.05 0.1 Normal Building and other structures
(Ellingwood, 1999)LL 1.0 0.25 Type I
AASHTODL 1.05 0.1 Normal Bridge substructures
(AASHTO, 2007; Nowak, 1999)LL 1.2 0.205 Lognormal
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tions ofQb,ult, QsL and Qi values can be defined as the failure
surface. Out of these numerous combinations ofQb,ult, QsL and Qivalues, probabilistically a combination of the most probable ULS
(or the most probable failure) values of Qb,ult, QsL, and Qi is
unique and these values are denoted as Qbase,LS, Qshaft,LS, and Qi,LS,
respectively.
These ULS values (Qbase,LS, Qshaft,LS, and Qi,LS) are obtained from
the results of reliability analysis based on the well-defined ULS
equation, which in turn yield well-assessed uncertainties associ-
ated with the ULS equation, and theDL/LL and Qb,ult/QsL ratios.
The optimum base and shaft resistance factors (RF*base and
RF*shaft) and optimum load factors (LF*DL andLF
*LL) are defined as
the ratios of the most probable ULS to the nominal values as
follows:
and (11)
and (12)
Then,RFbase andRFshaftcan be obtained without violating the
LRFD criterion [Inequality (10)] using the following equations,
proposed by Foye (2005):
(13)
(14)
4. Parametric Studies
As we mentioned in an earlier section, theDL/LL and Qb,ult/QsLratios are important in assessing the uncertainties of total pile
capacity and total load in addition to the uncertainties ofDL,LL,
Qb,ult, and QsL. From a designers point of view, the nominal ratios
(QDL,n/QLL,n and Qb,ult,n/QsL,n) ofDL/LL and Qb,ult/QsL are more
meaningful than the mean values ofDL/LL and Qb,ult/QsL because
calculation of the nominal values (QDL,n, QLL,n, Qb,ult,n, and QsL,n)
ofDL,LL, Qb,ult, and QsL are necessary for pile designs, but the
calculation of their mean values are not required.
In this paper, by performing a series of reliability analyses
varying the ratio (QDL,n/QLL,n) of the nominal dead load to the
nominal live load and the ratio (Qb,ult,n/QsL,n) of the nominal ulti-
mate base resistance to the nominal limit shaft resistance, tenta-
tiveRFbase andRFshaftvalues were calibrated that are compatible
with the ASCE/SEI 7-05 (2005) load factors and the AASHTO
(2007) load factors following a flow chart shown in Fig. 2.
Before a direct calculation of RF values is conducted, it is
worthwhile to examine the changes of optimum factors with
changes in QDL,n/QLL,n and Qb,ult,n/QsL,n. Due to the different uncer-
tainty levels ofDL andLL, the change ofQDL,n/QLL,n results in a
change of uncertainty of total load. Similarly, Qb,ult,n/QsL,ndetermines
the uncertainty level of total pile capacity. An optimum factor
implies a relative distance between the most probable ULS value
and its nominal value [Eqs. (11) and (12)]. Optimum factors of
loads are generally higher than a unity while those of resistances
are less than a unity. To reach the ULS of a pile, it is likely that
the loads are maximized and the resistances are minimized;
therefore, the Qi,LS values are likely to be maximized (greater
than Qi,n) while Qbase,LS and Qshaft,LS tend to be minimized (less
than Qb,ult,n and QsL,n).
For building structures, the QDL,n/QLL,n ratio may vary with the
materials used for building construction, the building dimensions
(length, width, and height), the building use types (residential, com-
mercial, public, or industrial). The QDL,n/QLL,n ratio for bridge
structures changes with the span lengths for bridges (Hansell and
Viest, 1971; AASHTO, 2007; Withiam et al., 2001). The QDL,n/
QLL,nratio is calculated as 0.5 for a bridge with a span length of
10 m using the empirical equation proposed by Hansell and Viest
(1971), while the calculated QDL,n/QLL,n is nearly equal to 4.0 with
span length of 70 m. In our analysis, the range ofQDL,n/QLL,n is
assumed 0.5-4 for both building and bridge structures.
The ratio ofQb,ult,n/QsL,n is determined based on the Qb,ult and
QsL equations proposed in the design specifications considering
the pile characteristics (pile diameter, pile length, and roughness
along pile shaft) and the foundation conditions (soil profiles, and
strengths along pile shaft and at bearing layer). In our analysis, to
account for numerous possible design cases, we assumed a wide
range ofQb,ult,n/QsL,n, from 0 to 10. However, a Qb,ult,n/QsL,n ratio of 0
may not be possible to happen in practice because we do not expect
Qb,ult,n = 0 for sand layers near the pile base. However, the Qb,ult,n/
QsL,n ratio of 0 is included in our paper for illustration purpose.
Due to the different levels of load uncertainties between build-
ing structures (Ellingwood, 1999) and bridge structures (Nowak,
RFbase*
Qbase LS,Qb ult n, ,----------------= RFshaf t
*Qshaf t LS,
QsL n,-----------------=
LFDL*
QDL LS,QDL n,--------------= LFLL
*QLL LS,QLL n,--------------=
RFbase minLFDLLFDL
*------------
LFLLLFLL
*-----------,
RFbase*=
RFshaf t minLFDLLFDL
*------------
LFLLLFLL
*-----------,
RFshaf t*=
Fig. 2. Flow Chart of Resistance Factor Calculation using FORM
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Dongwook Kim, Moonkyung Chung, and Kiseok Kwak
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1999), the optimum factors ofDL, LL, Qb,ult, and QsL for both
cases are examined. The optimum factors ofDL,LL, Qb,ult, and
QsL are functions of the bias factors, COVs, and distribution
types ofDL,LL, Qb,ult, and QsL as well as QDL,n/QLL,n, and Qb,ult,n/
QsL,n. Three different target reliability index values are used;T=
2.5, 3.0, and 3.5. For each T, a series of reliability analyses were
performed, varying QDL,n/QLL,n, and Qb,ult,n/QsL,n, for which the cal-
culated reliability indices were equal to the T. Optimum factors
of driven piles used as building substructures are calculated and
summarized in Table 4, and the optimum LFs and RFs are
plotted in Figs. 3 and 4, respectively.
A higher value for an optimum load factor (LF*DL orLF*LL)
represents that the most probable ULS value of the load (QDL,LSor QLL,LS) is determined as being relatively greater than its
nominal value (QDL,n orQLL,n). As shown in Fig. 3,LF*DL andLF
*LL
increase with an increasing target reliability index T.LF*DL and
LF*LL are determined not only from the uncertainties of the loads
(DL andLL) and the QDL,n/QLL,n ratio, but also from the uncer-
tainties of the pile resistances (Qb,ult and QsL) and the Qb,ult,n/QsL,nratio. Overall, LF*DL tends to increase with an increasing QDL,n/
QLL,nand with an increasing Qb,ult,n/QsL,n while LF*LL is likely to
increase with a decreasing QDL,n/QLL,n and with an increasing
Qb,ult,n/QsL,n. Fig. 3 shows that, for a given Qb,ult,n/QsL,n,, the increase
ofLF*DL and the decrease ofLF*LL coupled with an increasing
QDL,n/QLL,n mean that the unit contribution of the DL (DL at the
most probable ULS normalized by its nominal value) to pile
failures (attainment of ULS of piles) increases with an increasing
QDL,n/QLL,n while that (LL at the most probable ULS normalized
by its nominal value) ofLL to pile failures decreases with an
increasing QDL,n/QLL,n.
In Fig. 4, a higher RF value indicates that the resistance at the
ULS is determined closer to its nominal resistance. Similarly to
Table 4. Optimum Factors (LF*DL,LF*LL,RF
*base, and RF
*shaft) of Axially-loaded Single Driven Piles Used as Foundations of Building Structures
TQDL,n/QLL,n
LF*DL LF*LL RF
*base RF
*shaft
Qb,ult,n/QsL,n Qb,ult,n/QsL,n Qb,ult,n/QsL,n Qb,ult,n/QsL,n
0 0.5 1 5 10 0 0.5 1 5 10 0 0.5 1 5 10 0 0.5 1 5 10
2.5
0.5 1.07 1.08 1.08 1.08 1.08 1.34 1.48 1.52 1.52 1.50 N/A 0.86 0.81 0.73 0.71 0.62 0.70 0.76 0.95 1.02
1 1.09 1.10 1.10 1.10 1.10 1.26 1.39 1.43 1.42 1.41 N/A 0.85 0.79 0.70 0.69 0.59 0.67 0.73 0.93 1.00
2 1.10 1.12 1.13 1.13 1.13 1.15 1.25 1.28 1.27 1.26 N/A 0.83 0.77 0.67 0.66 0.58 0.64 0.70 0.91 0.99
4 1.12 1.14 1.15 1.15 1.15 1.07 1.12 1.14 1.13 1.13 N/A 0.82 0.76 0.66 0.65 0.57 0.63 0.69 0.90 0.99
3.0
0.5 1.08 1.08 1.08 1.08 1.08 1.44 1.63 1.67 1.67 1.65 N/A 0.84 0.78 0.69 0.67 0.55 0.65 0.71 0.92 1.00
1 1.09 1.10 1.11 1.11 1.10 1.33 1.51 1.56 1.56 1.54 N/A 0.81 0.75 0.66 0.64 0.52 0.61 0.68 0.89 0.97
2 1.11 1.14 1.14 1.14 1.14 1.20 1.33 1.37 1.36 1.34 N/A 0.79 0.73 0.63 0.61 0.51 0.58 0.64 0.87 0.96
4 1.13 1.16 1.17 1.17 1.17 1.09 1.16 1.18 1.18 1.17 N/A 0.78 0.71 0.61 0.59 0.50 0.57 0.63 0.85 0.95
3.5
0.5 1.08 1.08 1.08 1.08 1.08 1.55 1.79 1.85 1.84 1.82 N/A 0.81 0.75 0.65 0.63 0.49 0.60 0.67 0.89 0.97
1 1.10 1.11 1.11 1.11 1.11 1.42 1.65 1.71 1.71 1.67 N/A 0.78 0.72 0.62 0.60 0.46 0.56 0.63 0.86 0.95
2 1.12 1.15 1.15 1.15 1.15 1.25 1.42 1.46 1.46 1.44 N/A 0.76 0.69 0.58 0.56 0.44 0.53 0.59 0.83 0.93
4 1.14 1.18 1.19 1.19 1.18 1.12 1.20 1.22 1.22 1.21 N/A 0.74 0.67 0.56 0.55 0.44 0.51 0.58 0.81 0.91
Fig. 3. Optimum LFs for (a) DL and (b) LL for Driven Piles Used as
Foundations of Building Structures, Resulting from Reliabil-
ity Analyses using the Load Uncertainties Proposed by
Ellingwood (1999) (the optimum LFs in the figures are
given as values in Table 4)
Fig. 4. Optimum RFs for (a) Qb,ultand (b) QsL for Driven Piles Used
as Foundations of Building Structures, Resulting from Reli-
ability Analyses Corresponding to the Load Uncertainties
Proposed by Ellingwood (1999) (the optimum RFs in the
figures are given as values in Table 4)
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the load cases, for a given QDL,n/QLL,n in Fig. 4, the decrease of
RF*base and the increase ofRF*shaftwith an increasing Qb,ult,n/QsL,n
imply that the unit contribution of Qb,ult (Qb,ult at the most pro-
bable ULS normalized by its nominal value) to Qultdecreases with
an increasing Qb,ult,n/QsL,n while that (QsL at the most probable
ULS normalized by its nominal value) of QsL to Qult increases
with an increasing Qb,ult,n/QsL,n.
Since the uncertainty levels of loads for bridge substructures
are different from those for building substructures, in addition to
the reliability analyses performed for driven pile cases of building
substructures, the reliability analyses are carried out for calculation
of the optimum load and resistance factors (LF*DL, LF*LL, RF
*base,
andRF*shaft) for the driven piles used as the bridge substructures.
The optimum factors are summarized in Table 5, and the opti-
mum LFs and RFs are plotted in Figs. 5 and 6, respectively.
The optimum factors (Table 5) for the driven piles used as
substructures of bridges are slightly different from those calcu-
lated for the piles used as building substructures (Table 4) because
the uncertainty levels ofDL, Qb,ult, and QsL are the same for both
the building and bridge cases, but only the uncertainty levels of
LL for both cases are slightly different (Table 3). Even though the
COV value (0.25) ofLL for building substructures is greater than
that (0.205) for bridge substructures, the effect of the COV
difference on the RF values is reduced because of the greater bias
factor (1.2) ofLL for bridge substructures compared to that (1.0)
for building structures. These optimum factors are used for cal-
culation of the resistance factors of the base and shaft resistance
of axially-loaded driven piles using Eqs. (13) and (14).
5. Results
Based on the results of the reliability analyses for building sub-
structures, theRFbase andRFshaftvalues compatible with ASCE/
SEI 7-05 LFs are calculated for different QDL,n/QLL,n and Qb,ult,n/
QsL,n ratios and different target reliability indices using Eqs. and .
The RFbase and RFshaft values are summarized in Table 6 and
plotted in Fig. 7. For driven piles used as foundations of building
structures, the calculatedRFbase andRFshaftvalues varied within
ranges of 0.68-0.93 and 0.62-1.03 forT= 2.5, 0.61-0.86 and
0.55-0.96 forT= 3.0, and 0.55-0.79 and 0.49-0.91 forT= 3.5,
Table 5. Optimum Factors (LF*DL,LF*LL,RF
*base, and RF
*shaft) of Axially-loaded Single Driven Piles Used as Foundations of Bridge Structures
TQDL,n/QLL,n
LF*DL LF*LL RF
*base RF
*shaft
Qb,ult,n/QsL,n Qb,ult,n/QsL,n Qb,ult,n/QsL,n Qb,ult,n/QsL,n
1.07 1.08 1.08 1.08 1.08 1.49 1.62 1.65 1.65 1.63 N/A 0.85 0.80 0.71 0.69 0.56 0.64 0.70 0.88 0.95
2.5
0.5 1.09 1.10 1.10 1.10 1.10 1.42 1.54 1.57 1.57 1.55 N/A 0.84 0.78 0.69 0.67 0.55 0.62 0.68 0.87 0.93
1 1.10 1.12 1.13 1.13 1.12 1.34 1.42 1.45 1.45 1.43 N/A 0.83 0.76 0.67 0.65 0.54 0.60 0.66 0.85 0.92
2 1.12 1.14 1.15 1.15 1.14 1.27 1.32 1.33 1.33 1.33 N/A 0.82 0.76 0.66 0.65 0.53 0.59 0.64 0.85 0.92
4 1.07 1.08 1.08 1.08 1.08 1.57 1.73 1.77 1.77 1.75 N/A 0.82 0.76 0.67 0.65 0.50 0.59 0.65 0.85 0.92
3.0
0.5 1.09 1.10 1.11 1.11 1.10 1.48 1.64 1.67 1.67 1.65 N/A 0.80 0.74 0.64 0.62 0.48 0.56 0.63 0.83 0.91
1 1.11 1.13 1.14 1.14 1.14 1.38 1.49 1.51 1.51 1.50 N/A 0.79 0.72 0.62 0.60 0.47 0.54 0.60 0.81 0.90
2 1.13 1.16 1.17 1.17 1.16 1.29 1.35 1.36 1.36 1.35 N/A 0.78 0.71 0.61 0.59 0.47 0.53 0.59 0.80 0.89
4 1.08 1.08 1.08 1.08 1.08 1.65 1.86 1.91 1.91 1.88 N/A 0.79 0.72 0.62 0.60 0.44 0.54 0.61 0.82 0.91
3.5
0.5 1.10 1.11 1.11 1.11 1.11 1.55 1.74 1.79 1.79 1.76 N/A 0.77 0.70 0.60 0.58 0.43 0.51 0.58 0.80 0.88
1 1.12 1.15 1.15 1.15 1.15 1.42 1.56 1.59 1.59 1.57 N/A 0.75 0.68 0.57 0.55 0.41 0.49 0.55 0.77 0.86
2 1.14 1.18 1.18 1.18 1.18 1.31 1.38 1.40 1.40 1.39 N/A 0.74 0.66 0.56 0.54 0.41 0.48 0.54 0.76 0.86
4 1.07 1.08 1.08 1.08 1.08 1.49 1.62 1.65 1.65 1.63 N/A 0.85 0.80 0.71 0.69 0.56 0.64 0.70 0.88 0.95
Fig. 5. Optimum LFs for (a) DL and (b) LL for Driven Piles Used as
Bridge Substructures, Resulting from Reliability Analyses
Using the Load Uncertainties Proposed by Nowak (1999)
(the optimum LFs in the figures are given as values in Table5)
Fig. 6. Optimum RFs for (a) Qb,ultand (b) QsL for Driven Piles Used
as Bridge Substructures, Resulting from Reliability Analy-
ses Corresponding to the Load Uncertainties Proposed by
Nowak (1999) (the optimum RFs in the figures are given as
values in Table 5)
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respectively, depending on the QDL,n/QLL,n and Qb,ult,n/QsL,n ratios
(QDL,n/QLL,n and Qb,ult,n/QsL,n ranges of 0.5-4 and 0.5-10, respecti-
vely). TheRFshaftcorresponding to Qb,ult,n/QsL,n = 0 are not highli-
ghted in Table 6 and are excluded in the tentative RFshaft recom-
mendations because it is unlikely to find Qb,ult,n/QsL,n= 0 cases in
practice (Qb,ult,n = 0 may not be possible for a pile base located at
a sand layer).
TheRFbase andRFshaftcorresponding to specific QDL,n/QLL,n and
Qb,ult,n/QsL,n ratios for the piles used as bridge substructures are
summarized in Table 7 and are represented in Fig. 8. The
calculatedRFbase andRFshaftvalues for driven piles used as bridge
substructures are not much different from those values for
building cases. For driven piles used as bridge substructures, the
specific values ofRFbase andRFshaft are within ranges of 0.71-
0.96 and 0.65-1.06 forT= 2.5, 0.64-0.87 and 0.57-0.99 forT=
3.0, 0.56-0.82 and 0.51-0.94 forT= 3.5, respectively, depending
on the QDL,n/QLL,n and Qb,ult,n/QsL,n ratios (RFshaftvalues forQb,ult,n/
QsL,n= 0 are excluded).
Table 6. RFbase and RFshaft Values of Axially-loaded Single Driven Piles Used as Foundations of Building Structures; Compatible with
ASCE/SEI 7-05 LFs (ASCE, 2005)
T QDL,n/QLL,n
RFbase RFshaft
Qb,ult,n/QsL,n Qb,ult,n/QsL,n
0 0.5 1 5 10 0 0.5 1 5 10
2.5
0.5 N/A 0.93 0.86 0.77 0.76 0.65 0.71 0.76 0.94 1.02
1 N/A 0.93 0.86 0.77 0.75 0.61 0.69 0.75 0.96 1.03
2 N/A 0.89 0.82 0.72 0.70 0.59 0.65 0.70 0.91 0.99
4 N/A 0.86 0.79 0.69 0.68 0.57 0.62 0.67 0.88 0.96
3.0
0.5 N/A 0.82 0.75 0.66 0.65 0.57 0.60 0.65 0.83 0.90
1 N/A 0.86 0.77 0.68 0.67 0.54 0.61 0.66 0.87 0.96
2 N/A 0.84 0.76 0.66 0.64 0.51 0.58 0.64 0.86 0.95
4 N/A 0.81 0.73 0.63 0.61 0.50 0.55 0.61 0.82 0.92
3.5
0.5 N/A 0.72 0.65 0.56 0.56 0.47 0.51 0.55 0.73 0.80
1 N/A 0.76 0.67 0.58 0.57 0.48 0.52 0.56 0.76 0.86
2 N/A 0.79 0.71 0.61 0.59 0.45 0.52 0.58 0.81 0.91
4 N/A 0.76 0.67 0.57 0.55 0.43 0.49 0.55 0.78 0.87
Fig. 7. RFs for (a) Qb,ultand (b) QsL for Building Structures Compat-
ible with ASCE-7 LFs
Table 7. RFbase and RFshaft Values of Axially-loaded Single Driven Piles Used as Foundations of Bridge Structures; Compatible with
AASHTO LFS (AASHTO, 2007)
T QDL,n/QLL,n
RFbase RFshaft
Qb,ult,n/QsL,n Qb,ult,n/QsL,n
0 0.5 1 5 10 0 0.5 1 5 10
2.5
0.5 N/A 0.92 0.85 0.75 0.74 0.66 0.69 0.74 0.94 1.01
1 N/A 0.96 0.87 0.77 0.76 0.63 0.70 0.75 0.97 1.06
2 N/A 0.92 0.85 0.74 0.73 0.61 0.67 0.73 0.95 1.03
4 N/A 0.90 0.82 0.72 0.71 0.60 0.65 0.70 0.92 1.01
3.0
0.5 N/A 0.83 0.75 0.66 0.65 0.56 0.59 0.64 0.84 0.92
1 N/A 0.86 0.77 0.67 0.66 0.55 0.60 0.65 0.87 0.96
2 N/A 0.87 0.79 0.68 0.66 0.53 0.60 0.66 0.89 0.99
4 N/A 0.84 0.76 0.65 0.64 0.52 0.57 0.63 0.86 0.96
3.5
0.5 N/A 0.74 0.66 0.57 0.56 0.47 0.51 0.56 0.75 0.84
1 N/A 0.77 0.69 0.59 0.58 0.48 0.52 0.57 0.78 0.88
2 N/A 0.82 0.73 0.62 0.60 0.46 0.53 0.60 0.84 0.94
4 N/A 0.79 0.70 0.59 0.57 0.45 0.51 0.57 0.81 0.91
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The equivalentRF(RFeq) values of total capacities were cal-
culated fromRFbase andRFshaftusing the following equation:
(15)
For three different levels ofT(2.5, 3.0, and 3.5),RFeq values
were calculated for different ratios of QDL,n/QLL,nand Qb,ult,n/QsL,n,
and they are summarized in Table 8. As mentioned earlier, a
lower value of Qb,ult,n/QsL,n (0-0.5) may not be encountered in
practice; therefore,RFeq values corresponding to Qb,ult,n/QsL,n = 0
are excluded in theRFeq comparison for the givenT. It is found
that, for a given target reliability index, there is a substantial
change ofRFeq with respect to QDL,n/QLL,n and QDL,n/QLL,n.
The current LRFD design specifications and other literature
propose a single RF value for total capacity for a given target
reliability index. If we were to propose a single value of RF for
the total pile capacity without differentiating QDL,n/QLL,n and
Qb,ult,n/QsL,nfor a given target reliability index, we may suggest
the lowestRFeq value among theRFeq values (Table 8) calculated
for the target reliability index. Designs of driven piles in sands
using these lowestRFeq values result in uneconomical designs by
imposing an extra margin of safety in pile designs, which can be
represented by the difference between the RFeq values for eachRFeq RFbaseQb ult n, , RFsha ftQsL n,+
Qb ult n, , QsL n,+--------------------------------------------------------------=
Fig. 8. RFs for (a) Qb,ult and (b) QsL for Bridge Structures Compati-
ble with AASHTO LFs
Table 8. Equivalent Resistance Factor (RFeq) of Total Pile Capacity Qult of Axially-loaded Single Driven Piles Used as Foundations of
Building and Bridge Structures
T QDL,n/QLL,n
Driven piles used for building substructures Driven piles used for bridge substructures
Qb,ult,n/QsL,n Qb,ult,n/QsL,n
0 0.5 1 5 10 0 0.5 1 5 10
2.5
0.5 0.65 0.78 0.81 0.80 0.78 0.66 0.77 0.79 0.78 0.77
1 0.61 0.77 0.81 0.80 0.78 0.63 0.79 0.81 0.80 0.79
2 0.59 0.73 0.76 0.75 0.73 0.61 0.75 0.79 0.78 0.76
4 0.57 0.70 0.73 0.72 0.71 0.60 0.73 0.76 0.75 0.73
3.0
0.5 0.57 0.67 0.70 0.69 0.67 0.56 0.67 0.70 0.69 0.67
1 0.54 0.69 0.72 0.71 0.70 0.55 0.69 0.71 0.70 0.69
2 0.51 0.67 0.70 0.69 0.67 0.53 0.69 0.72 0.71 0.69
4 0.50 0.64 0.67 0.66 0.64 0.52 0.66 0.69 0.69 0.67
3.5
0.5 0.47 0.58 0.60 0.59 0.58 0.47 0.58 0.61 0.60 0.59
1 0.48 0.60 0.62 0.61 0.60 0.48 0.60 0.63 0.62 0.60
2 0.45 0.61 0.65 0.64 0.62 0.46 0.63 0.67 0.66 0.63
4 0.43 0.58 0.61 0.61 0.58 0.45 0.60 0.63 0.63 0.60
Table 9. Equivalent Factor of Safety (FSeq) of Axially-loaded Single Driven Piles Used as Foundations of Building and Bridge Structures
T QDL,n/QLL,n
Driven piles used for building substructures Driven piles used for bridge substructures
Qb,ult,n/QsL,n Qb,ult,n/QsL,n
0 0.5 1 5 10 0 0.5 1 5 10
2.5
0.5 2.26 1.87 1.81 1.84 1.87 2.41 2.06 2.00 2.02 2.06
1 2.30 1.82 1.74 1.75 1.81 2.38 1.91 1.85 1.87 1.912 2.26 1.83 1.75 1.77 1.84 2.32 1.89 1.80 1.83 1.88
4 2.25 1.83 1.75 1.77 1.81 2.26 1.85 1.77 1.80 1.84
3.0
0.5 2.57 2.18 2.10 2.13 2.18 2.85 2.36 2.28 2.31 2.36
1 2.59 2.02 1.96 1.97 2.01 2.72 2.18 2.10 2.13 2.18
2 2.61 2.00 1.90 1.92 2.00 2.68 2.07 1.96 1.99 2.05
4 2.56 2.01 1.91 1.93 2.01 2.61 2.04 1.94 1.97 2.03
3.5
0.5 3.12 2.53 2.44 2.49 2.52 3.38 2.71 2.60 2.63 2.70
1 2.92 2.33 2.28 2.30 2.35 3.12 2.49 2.39 2.42 2.48
2 2.96 2.19 2.07 2.07 2.15 3.08 2.26 2.13 2.15 2.23
4 2.98 2.21 2.10 2.12 2.21 3.02 2.25 2.13 2.16 2.23
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QDL,n/QLL,n and Qb,ult,n/QsL,n and the lowest value ofRFeq within
each target reliability index.
The design guides or specifications (AASHTO, 2002; USACE,
1993) regarding driven piles propose to use a FS range of 2.5-3.0
for pile designs using the ASD. The equivalent FS (FSeq) values
are calculated using the following equation:
(16)
For each T (2.5, 3.0, and 3.5), FSeq values were calculated
using Eq. (16) for different ratios of QDL,n/QLL,n and Qb,ult,n/QsL,n(Table 9). As shown in Table 9, the FSeq values for driven piles
used for building substructures were less than those values for
bridge substructures. The FSeq values were within ranges of 1.74-
2.06 forT= 2.5, 1.90-2.36 forT= 3.0, and 2.07-2.71 forT=
3.5. From the comparison between theFSeq values in Table 9 and
the FS values (2.5-3.0) proposed in ASD design guides and
specifications, it is inferred that driven piles designed using the
RF values (LRFD) proposed in this paper could lead to more
economical designs of axially-loaded driven piles in sands than
those designed using FS (ASD).
6. Conclusions
In this study, we separated the RF for total pile capacity into
two resistance factors, RFbase forQb,ult and RFshaft forQsL. The
equations used for Qb,ult and QsLpredictions of axially-loaded
driven piles in sands are the ones proposed in the ICP design
methods. The uncertainties of Qb,ult and QsL are assessed by
identifying their bias factors, coefficients of variation, and
distribution types obtained from the existing database used for
ICP design method development. For three different Tvalues
(2.5, 3.0, and 3.5), RFbase and RFshaft are calculated based on
reliability analyses for different combinations ofQDL,n/QLL,n, and
Qb,ult,n/QsL,n, within their possible ranges (QDL,n/QLL,n = 0.5-4 and
Qb,ult,n/QsL,n = 0.5-10). As a result, for a givenT,RFbase andRFshaftwere highly dependent on QDL,n/QLL,n, and Qb,ult,n/QsL,n.
The benefit of selecting RF values (RFbase andRFshaft) with dif-
ferentiating QDL,n/QLL,n, and Qb,ult,n/QsL,n was evaluated by com-
paring the corresponding RFeq values for different QDL,n/QLL,n,
and Qb,ult,n/QsL,n ratios. For a givenT, among the calculatedRFeq
values for different QDL,n/QLL,n, and Qb,ult,n/QsL,n ratios, a difference
between the minimum to the maximumRFeq was more than 10%
of the maximumRFeq.
In this paper, tentativeRFbase andRFshaftvalues are calculated
and suggested in Table 6 (resistance factors for driven piles used
as building substructures) and Table 7 (resistance factors for
driven piles used as bridge substructures) for LRFD of axially-
loaded driven piles in sands. From the comparison between the
FS values proposed for driven pile ASDs and the FSeq values
(Table 9) calculated from RFeq (Table 8), it is inferred that the
designs of driven piles using the tentative RF values proposed in
this paper could contribute economical designs. For an economi-
cal driven pile design with a given target reliability index, rather
than using a single conservative resistance factor of total pile
capacity, designers may use the RF values that reflect different
QDL,n/QLL,n and Qb,ult,n/QsL,n ratios.
Notations
Ab: Pile base area
As,i: Pile shaft area of ith sublayer
ASD: Allowable stress design
COV: Coefficient of variation
DCPT: Diameter of cone
Dpile: Outer diameter of pile
DL: Dead load
FORM: First-order reliability method
: Factor of safety
FSeq: Equivalent factor of safety
G: Shear stiffness
h: Distance along pile shaft from pile base
ICP: Imperial College Pile
LF: Load factor
LF*DL: Optimum dead load factor
LF*LL: Optimum live load factor
LL: Live load
LRFD: Load and resistance factor design
RF: Resistance factor
RFbase: Resistance factor for base resistance
RFshaft: Resistance factor for shaft resistance
RF*base: Optimum resistance factor for base resistance
RF*shaft: Optimum resistance factor for shaft resistance
RFeq: Equivalent resistance factor
pA: Reference stress (100kPa)
QDL,n: Nominal dead load
QLL,n: Nominal live load
Qi,n: Nominal load
Qi,LS: Load corresponding to the most probable ultimate limit
state
Qb,ult: Ultimate base resistance
Qb,ult,n: Nominal ultimate base resistance
Qbase,LS: Base resistance corresponding to the most probable
ultimate limit state
Qshaft,LS: Shaft resistance corresponding to the most probable
ultimate limit state
QsL: Limit shaft resistance
QsL,n: Nominal limit shaft resistance
qb,ult: Unit ultimate base resistance
qc: Cone resistance
qcb,avg: Average cone resistance at the location of pile base
qsL: Unit limit shaft resistance
ULS: Ultimate limit state
: Reliability index
T: Target reliability index
r: Radial displacement during pile loading
cv: Critical-state interface friction angle between pile shaft
and surrounding soils
FSeq Qb ult n, , QsL n,+QDL n, QLL n,+------------------------------- LFDLQDL n, LFLLQLL n,+
RFeq
QDL n, QLL n,+( )-----------------------------------------------------= =
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Vol. 15, No. 7 / September 2011 1195
: Bias factor
: Mean
eq: Equivalent mean
: Standard deviation
eq: Equivalent standard deviation
'v: Vertical effective stress
Acknowledgements
The research presented in this paper was performed as parts of
the Super long span bridge research project funded by the
Ministry of land, transport and maritime affairs of South Korea
and the Development of hybrid large-scale foundation with high
efficiency project funded by Korea Institute of Construction
Technology. The authors acknowledge the support from these
two organizations.
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