[Report] Resistance Factor Calculations for LRFD of Axially Loaded Driven Piles in Sands

7/31/2019 [Report] Resistance Factor Calculations for LRFD of Axially Loaded Driven Piles in Sands

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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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Dongwook Kim, Moonkyung Chung, and Kiseok Kwak

1186 KSCE Journal of Civil Engineering

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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Vol. 15, No. 7 / September 2011 1187

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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Vol. 15, No. 7 / September 2011 1189

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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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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Vol. 15, No. 7 / September 2011 1191

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)


8/12



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


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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Vol. 15, No. 7 / September 2011 1193

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


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


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


10/12



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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[Report] Resistance Factor Calculations for LRFD of Axially Loaded Driven Piles in Sands