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    Towards reliability-based design for geotechnical engineering

    K. K. Phoon

    National University of Singapore, Singapore

    ABSTRACT: This paper presents an overview of the evolution in structural and geotechnical design

    practice over the past half a decade or so in relation to how uncertainties are dealt with. For the general

    reader who is encountering reliability-based design (RBD) for the first time, this would provide a

    valuable historical perspective of our present status and important outstanding issues that remain to beresolved. The key elements of RBD are briefly discussed and the availability of statistics to provide

    empirical support for the development of simplified RBD equations is highlighted. Several important

    implementation issues are presented with reference to an EPRI study for reliability-based design of

    transmission line structure foundations. Reliability-based design, simplified or otherwise, provides a

    more consistent means of managing uncertainties, but it is by no means a perfect solution. Engineering

    judgment still is indispensable in many aspects of geotechnical engineering reliability analysis merely

    removes the need for guesswork on how uncertainties affect performance and is comparable to the use of

    elasto-plastic theory to remove the guesswork on how loads induce stresses and deformations.

    INTRODUCTION

    The basis for making a geotechnical design decision is not as well studied nor subjected to the same

    degree of formal scrutiny as structural design. Goble (1999) noted in an NCHRP Synthesis Report on

    Geotechnical Related Development and Implementation of Load and Resistance Factor Design (LRFD)

    Methods that the "education of geotechnical engineers strongly emphasizes the evaluation of soil and rock

    properties" and "the design process does not receive the emphasis that it does in structural engineering

    education". In fact, the first time the topic "Codes and Standards" was selected for formal discussion in a

    major ISSMGE conference was in 1989 (Ovesen 1989). Examination of current practice shows that

    procedures for selecting nominal soil strengths are not well-defined or followed uniformly. Some

    engineers use the mean value, while others use the most conservative of the measured strengths (Whitman,

    1984). Different calculation methods are preferred in different localities or even by different engineers in

    the same locality (Goble 1999). The manner in which the factor of safety is incorporated in the design

    equation also is highly varied (Kulhawy 1984, 1996). Green & Becker (2001) made similar observationsin a National Report on Limit State Design in Geotechnical Engineering in Canada. Golder (1966) noted

    quite aptly in a discussion of the second Terzaghi Lecture by Arthur Casagrande that: "We do not know

    how we make a decision."

    Ironically, this view is rarely acknowledged publicly in the geotechnical engineering profession. The

    predominant view ranges from "If it is not broke, why fix it" (Green & Becker 2001) to a general feeling

    that conventional practice is perfectly adequate to do optimal design (Committee on Reliability Methods

    1995, Kulhawy 1996). Although existing practice has undoubtedly served the profession well for many

    years, this paper argues that genuine improvements are possible if our practice were to be complemented

    by reliability-based design (RBD) methodologies. No one is advocating total abandonment of existing

    practice for something entirely new. In fact, the reverse is probably closer to the truth - many aspects of

    current design practice would still appear in new RBD codes, albeit in a modified form (Kulhawy &

    Phoon 1996). It is also important to highlight that geotechnical design would be subjected to increasing

    codification as a result of code harmonization across material types and national boundaries. It is also

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    clear that regulatory pressure eventually would bring geotechnical design within an umbrella framework

    predominantly established by structural engineers. In the United States, this process is already well

    underway for highway bridge design. In reference to AASHTO LRFD Specifications (AASHTO 2002),

    Withiam (2003) noted that the leverage to drive implementation was the intention to sunset (i.e. no

    longer publish) the long-standing AASHTO Standard Specifications that have provided nationalrequirements for highway bridge superstructure and substructure design since the 1930s. The difficulty

    of maintaining status quo was highlighted about 10 years ago in the National Research Council Report on

    Reliability Methods for Risk Mitigation in Geotechnical Engineering (Committee on Reliability Methods

    1995).

    A fundamental change in mindset, similar to what has taken place in the structural community since

    the 1970s, is needed for the profession to take the next step. It is accurate to say that this change has not

    taken place in the geotechnical community in North America (Goble 1999, DiMaggio et al. 1999, Green

    & Becker 2001), although significant initiatives have been launched by major agencies in recent years

    such as OHBDC3 (Ministry of Transportation Ontario 1992), CAN/CSA-S472-92 (CSA 1992a), API RP

    2A-LRFD (API 1993), EPRI (Phoon et al. 1995), NBCC (National Research Council of Canada 1995),

    AASHTO LRFD Bridge Code (AASHTO 2002), and CHBDC (CSA 2000).Currently, the geotechnical community is mainly preoccupied with the transition from working or

    allowable stress design (WSD/ASD) to Load and Resistance Factor Design (LRFD). The term "LRFD" is

    used in a loose way to encompass methods that require all limit states to be checked using a specific

    multiple-factor format involving load and resistance factors. This term is used most widely in the United

    States and is equivalent to "Limit State Design (LSD)" in Canada. Both LRFD and LSD are

    philosophically akin to the partial factors approach commonly used in Europe, although a different

    multiple-factor format involving factored soil parameters is used. The emphasis in LRFD or its

    equivalent in Canada and Europe is primarily on the re-distribution of the original global factor safety in

    WSD into separate load and resistance factors (or soil parameter partial factors). The absence of strong

    analytical calibration and verification in Eurocode 7 (CEN/TC250 1994) and OHBDC3 (Ministry of

    Transportation Ontario 1992) is noted by DiMaggio et al. (1999) in an FHWA Report on "Geotechnical

    Engineering Practices in Canada and Europe". Paikowsky & Stenersen (2000) also noted a similar lackof data supporting current AASHTO LRFD specifications.

    There are strong practical reasons to consider geotechnical LRFD as a simplified reliability-based

    design procedure, rather than an exercise in rearranging the original global factor of safety. This calls for

    a willingness to accept the fundamental philosophy that: (a) absolute reliability is an unattainable goal in

    the presence of uncertainty and (b) probability theory can provide a formal framework for developing

    design criteria that would ensure that the probability of "failure" (used herein to refer to exceeding of any

    prescribed limit state) is acceptably small. In other words, geotechnical LRFD should be derived as the

    logical end-product of a philosophical shift in mindset to probabilistic design in the first instance and a

    simplification of rigorous reliability-based design into a familiar look and feel design format in the

    second. The need to draw a clear distinction between accepting reliability analysis as a necessary

    theoretical basis for geotechnical design and downstream calibration of simplified multiple-factor designformats, with emphasis on the former, was emphasized by Phoon et al. (2003a).

    The former provides a consistent method for propagation of uncertainties and a unifying framework

    for risk assessment across disciplines (structural and geotechnical design) and national boundaries. Other

    competing frameworks have been suggested (e.g., -method by Simpson et al. 1981, worst attainablevalue method by Bolton 1989, Taylor series method by Duncan 2000) but none has the theoretical breadth

    and power to handle complex real-world problems that may require nonlinear 3-D finite element or other

    numerical approaches for solution. In the development of Eurocode 7, much attention has been focused

    on the geotechnical aspects of code harmonisation (Frank 2002, Ovesen 2002, Orr 2002). This clearly

    takes precedence over safety aspects, but the time is perhaps ripe to decide if a theoretical platform is

    necessary to rationalise risk assessment. If the platform is not reliability analysis, then what alternative is

    available?

    The need to derive simplified reliability-based design (RBD) equations perhaps is of practical

    importance to maintain continuity with past practice, but it is not necessary and increasingly fraught with

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    difficulties when sufficiently complex problems are posed. The limitations faced by simplified RBD have

    no bearing on the generality of reliability theory. This is analogous to arguing that limitations in closed-

    form elastic solutions are related to elasto-plastic theory. The use of finite element software on relatively

    inexpensive and powerful PCs (with gigahertz processors, gigabyte of memory, and hundreds of

    gigabytes - verging on terabyte - of disk) permit real world problems to be simulated on an unprecedentedrealistic setting almost routinely. Phoon et al. (2003a) presented some examples to clarify that limitations

    of the implementation (say LRFD) do not carry over to the underlying reliability framework. Attention

    should be focused on the more basic issue pertaining to the relevance of reliability theory in geotechnical

    design.

    This paper presents an overview of the evolution in structural and geotechnical design practice over

    the past half a decade or so in relation to how uncertainties are dealt with. For the general reader who is

    encountering reliability-based design for the first time, this would provide a valuable historical

    perspective of our present status and important outstanding issues that remain to be resolved. The key

    elements of RBD are briefly discussed and the availability of statistics to provide empirical support for

    the development of simplified RBD equations is highlighted. Several important implementation issues

    are presented with reference to an EPRI study for reliability-based design of transmission line structurefoundations (Phoon et al. 1995). Numerous examples of RBD calibration are given elsewhere (Phoon &

    Kulhawy 2002a, b; Phoon et al. 2003c; Phoon & Kulhawy 2004)

    HISTORICAL OVERVIEW

    Structural LRFD

    The classical structural reliability theory became widely known through a few influential publications

    such as Freudenthal (1947) and Pugsley (1955). The fundamental philosophy is that absolute reliability is

    an unattainable goal in the presence of uncertainty. Probability theory can provide a formal framework

    for developing design criteria that would ensure that the probability of "failure" (used herein to refer toexceeding of any prescribed limit state) is acceptably small. While the philosophy is elegant, the theory is

    mathematically intractable and numerically cumbersome. Cornell (1969) probably was the first to

    introduce the concept of a reliability index for simplified probabilistic design. Only second-moment

    information (mean and covariance) on uncertain parameters was needed and the computation was made

    simple by adopting the Gaussian model for random variables. However, the idea still was rather radical

    and could have been ignored if not for Lind (1971), who demonstrated that Cornell's reliability index

    could be used to derive load and resistance factors formally. The ability to repackage probabilistic design

    into a simplified multiple-factor design format with the same look and feel as existing design formats,

    while retaining theoretical rigor, is an important development from a practical point of view. To the

    authors' knowledge, LRFD was first implemented for steel building structures by Ravindra & Galambos

    (1978) using the theoretical basis established by Cornell (1969) and Lind (1971).In the meantime, serious theoretical difficulties were encountered with Cornell's index, with the most

    severe being the problem of invariance. Cornell's index was found to vary when certain simple limit

    states were reformulated in a mechanically equivalent way. Although second-moment reliability-based

    structural design was becoming widely accepted in the early seventies, the goal of developing simplified

    design criteria firmly founded on a rigorous reliability basis remained elusive. This unsatisfactory

    condition was resolved eventually by Hasofer & Lind (1974), when they proved mathematically that the

    nearest distance of the limit state function from the origin of a standard Gaussian space is an invariant

    measure of reliability. This major theoretical breakthrough enforces invariance while retaining the

    practical second-moment simplification of Cornell's index. The last piece of significant addition to the

    theoretical repertoire for solving time-invariant reliability problems was the algorithm of Rackwitz &

    Fiessler (1978), which provided a practical and computationally efficient recipe for computing this

    reliability index with no restriction on the number of random variables. The reliability method proposed

    by Hasofer & Lind (1974) and its subsequent generalizations to handle non-Gaussian and correlated

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    random variables commonly is called the First-Order Reliability Method (FORM). Ellingwood et al.

    (1980) were probably the first to apply FORM in a comprehensive way for simplified probabilistic design.

    Their study primarily presented load factors for buildings that were calibrated rationally using FORM and

    available statistical data.

    The above review may not do justice to the voluminous research conducted in structural reliabilityover the past forty or so years. However, it does provide an overview of the historical development of

    structural LRFD and the accompanying key theoretical advances supporting this development. In the

    aftermath of recent natural hazards (e.g., Northridge and Kobe earthquakes), the structural engineering

    profession currently is focusing on performance-based design aimed at meeting client-specific

    performance goals, in addition to complying with local building codes (Wen 2000, Buckle 2002). An

    example of a performance criteria matrix proposed by the California Department of Transportation

    (Caltrans) is shown in Table 1. Efficient techniques for solving time-dependent nonlinear system

    reliability problems are needed for such problems. Clearly, theoretical developments in structural

    reliability and applications to probabilistic design are still being pursued actively in the structural

    community.

    Geotechnical LRFD

    The development of geotechnical LRFD has taken a different track. One of the first efforts to rationalize

    foundation design can be attributed to Hansen (1965), who recommended separate checks for ultimate

    and serviceability limit states. In contrast, existing WSD often uses the global factor of safety for indirect

    control of serviceability. Hansen (1965) also recommended the use of partial factors for loads and soil

    parameters. These partial factors of safety were determined subjectively based on two guidelines: (a) a

    larger partial factor should be assigned to a more uncertain quantity, and (b) the partial coefficients should

    result in approximately the same design dimensions as that obtained from traditional practice. Ovesen

    (1989) highlighted the direct application of a partial factor to the source of uncertainty (soil parameter) as

    a notable improvement. The partial factors of safety suggested in 1965 were adopted in the Danish Code

    of Practice for Foundation Engineering (DGI 1978, 1985) with minor modifications. More recentimplementations include the Canadian Foundation Engineering Manual, CFEM, third edition (Canadian

    Geotechnical Society 1992), Geoguide 1, second edition (Geotechnical Engineering Office 1993), and

    Eurocode 7 (CEN/ TC250 1994).

    Table 1. Caltrans performance criteria matrix (adapted from Buckle 2002)

    Bridge Type 1

    (Ordinary bridges)

    Bridge Type 2

    (Important bridges)

    Function evaluation earthquake

    (Frequent earthquake about 200-year

    event)

    PL1

    DL2

    PL1

    DL1

    Safety evaluation earthquake(Rare earthquake about 1000 to 2000-

    year event)

    PL2DL3

    PL1DL2

    where:

    PL1 - Performance Level 1, characterized by immediate and full access to normal traffic almost

    immediately after the earthquake

    PL2 - Performance Level 2, characterized by limited access possible in days, full access within months

    DL1 - Damage Level 1, characterized by minimal damage with essentially elastic performance

    DL2 - Damage Level 2, characterized by repairable damage which may executed with minimum loss of

    functionality

    DL3 - Damage Level 3, characterized by significant damage which may result in closure but not

    collapse

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    In North America, the factored resistance approach is the preferred design format. LRFD procedures

    for bridge superstructures and substructures were introduced in Canada in 1979 as part of the first edition

    of the Ontario Highway Bridge Design Code, OHBDC1 (Ministry of Transportation Ontario 1979, Green

    1991). Green (1991) further noted that these procedures were "basically a simple rearrangement of factor

    of safety design provisions". The design of deep foundations for power generating stations (OntarioHydro 1985) broadly followed the second edition, OHBDC2 (Ministry of Transportation Ontario 1983,

    Klym & Lee 1989). OHBDC is currently in its third edition (OHBDC3), but the foundation resistance

    factors are not based on reliability calibrations (Green & Becker 2001). In contrast, OHBDC for

    superstructures was calibrated using reliability theory in its second edition (Grouni & Nowak 1984). The

    target reliability indices selected were 3.5 for ultimate limit state and 1.0 for serviceability limit state

    (Nowak & Lind 1979).

    For fixed offshore platforms, the Canadian standard for foundations, CAN/CSA-S472-92 (CSA

    1992a), contains no specification of resistance factors for foundation design, although reliability-based

    LRFD is available for structural design (Been et al. 1993). Been et al. (1993) further noted that resistance

    factors were calibrated to the global factor of safety in an earlier 1989 draft commentary (CSA 1989) but

    were dropped in the 1992 version (CSA 1992b).The main geotechnical design manual in Canada is the Canadian Foundation Engineering Manual,

    CFEM (Canadian Geotechnical Society 1992). As noted previously, the 1992 version (third edition) is

    based on the partial factors of safety approach, although the fourth edition currently under preparation is

    expected to be revised to be consistent with the factored resistance format (Green & Becker 2001). The

    partial factors of safety in the third edition were calibrated so that they result, on average, in overall

    factors of safety that are in agreement with existing practice (Meyerhof 1984). To the authors

    knowledge, CFEM is the only design guide that indirectly recognizes the difficulty of using a single

    partial factor for each soil parameter to cover the wide range of design equations in which the same soil

    parameter can appear. Resistance modification factors and performance factors were recommended to

    ensure more reasonable agreement with existing practice. However, this procedure is not entirely

    successful, as noted by Baike (1985) and Valsangkar & Schriver (1991). The conflict between the need

    for simplicity or using small numbers of partial factors of safety, and the need to produce designscomparable with existing practice, does not appear to lend itself readily to simple solutions.

    The development of LRFD for foundations in the 1995 National Building Code of Canada (NBCC)

    followed a semi-analytical approach (Becker 1996b). Becker (1996b) opined that a full reliability-based

    LRFD is difficult to apply because of a lack of statistical data and it is time-consuming and expensive.

    Therefore, the resistance factors for foundation design were calibrated to fit WSD and to be consistent

    with a lumped parameter lognormal reliability formula. A target reliability index of 3.5 was used in

    NBCC for foundation design. As a reference, the NBCC for structural design was calibrated using a

    target of 3.5 for ductile behavior with normal consequence of failure and a target of 4.0 if either theconsequence of failure is severe or the failure mode is brittle (Becker 1996b).

    In the United States, the resistance factors for design of foundations in the AASHTO LRFD Bridge

    Code (AASHTO 2002) were derived from NCHRP Report 343 (Barker et al. 1991). The main rationaleis to remove the inconsistency between load factor design for superstructures and allowable stress design

    for foundations, which has resulted in duplication of design efforts because two sets of loads must be

    evaluated (Rojiani et al. 1991). The resistance factors appear to be determined using a mixture of

    judgment, calibration with WSD, and reliability analysis. Reliability analysis seems to be used in a

    limited way (Rojiani et al. 1991, Yoon & O'Neill 1997). The risk levels implied by an extensive range of

    existing calculation procedures (e.g., rational methods, semi-empirical methods, in-situ methods) formed

    the basis for the target reliability indices. Paikowsky & Stenersen (2000) further noted that the current

    AASHTO specifications were developed using insufficient data, hence they utilized mostly back-

    calculated factors. Most interestingly, the target reliability index for bridge superstructures is 3.5 (Grubb

    1997), which is much higher than the target reliability indices quoted in NCHRP Report 343 (2.0 to 2.5,

    2.5 to 3.5, and 3.5 for driven piles, drilled shafts and spread footings, respectively). Project NCHRP 24-

    17 was initiated to provide: (a) recommended revisions to the driven pile and drilled shaft portions of

    section 10 of AASHTO Specifications and (b) detailed procedure for calibrating deep foundation

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    resistance factors. The present recommendation is to design for a target reliability of 2.33 if there are five

    or less piles in a group and 3.00 otherwise (Paikowsky 2002).

    For API RP 2A-LRFD (API 1993), foundations are treated as one of the structural elements in the

    RBD calibration process (Moses & Larrabee 1988). A lumped resistance parameter with a bias of 1.0 and

    a coefficient of variation (COV) of 20% was assumed for pile capacity. The foundation resistance factorwas adjusted to achieve an average reliability index of 2.2 for pile axial capacity. A lumped resistance

    model also was assumed for transmission line structure foundations in the ASCE Manual & Report 74

    (Task Committee on Structural Loadings 1991) to preserve a common reliability calibration scheme for

    structural and foundation components. Resistance factors for a range of lumped resistance COVs (20 to

    50%) and target probabilities of failure (0.25% to 1%) were presented. The range of probability of failure

    corresponds to reliability indices between 2.3 and 2.8. Geotechnical considerations were marginalized

    because API RP 2A-LRFD and ASCE Manual & Report 74 were focused on structural design. The first

    attempt to develop simplified RBD specifically for transmission line structure foundations with primary

    emphasis on geotechnical considerations was described in EPRI Report TR-105000 (Phoon et al. 1995).

    A number of geotechnical aspects in this study are of general applicability and will be discussed in detail

    later.

    RELIABILITY CALIBRATION

    Basic theory

    The principal difference between RBD and the traditional or partial factors of safety design approaches

    lies in the application of reliability theory, which allows uncertainties to be quantified and manipulated

    consistently in a manner that is free from self-contradiction. A simple application of reliability theory is

    shown in Figure 1 to define some of the key terms used in RBD. Uncertain design quantities, such as the

    load (F) and the capacity (Q), are modeled as random variables, while design risk is quantified by the

    probability of failure (pf). The basic reliability problem is to evaluate pf from some pertinent statistics ofF and Q, which typically include the mean (mF or mQ) and the standard deviation (sF or sQ), and possibly

    the probability density function.

    A simple closed-form solution for pf is available if both Q and F are normally distributed. For this

    condition, the safety margin (M = Q - F) also is normally distributed with the following mean (mM) and

    standard deviation (sM):

    FQM mmm = (1a)2F

    2Q

    2M sss += (1b)

    Once the probability distribution of M is known, the probability of failure (pf) can be evaluated as:

    pf = Prob(Q < F) = Prob(Q - F < 0) = Prob(M < 0) =

    M

    M

    s

    m(2)

    in which Prob() = probability of an event and () = standard normal cumulative function. Numericalvalues for () can be obtained easily using the function NORMSDIST(-) in MS Excel. The

    probability of failure is usually very small for civil infrastructures. A more convenient measure of design

    risk is the reliability index (), which is defined as:

    = --1(pf) =2F

    2Q

    FQ

    ss

    mm

    +

    (3)

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    Figure 1. Reliability assessment for capacity (Q) > load (F)

    in which -1() = inverse standard normal cumulative function. The function -1 () also can be obtainedeasily from MS Excel using NORMSINV(). The reliability indices for most geotechnical componentsand systems lie between 1 and 5, corresponding to probabilities of failure ranging from about 0.16 to 3 10

    -7, as shown in Figure 2. It is tempting to compare with the traditional factor of safety because both

    parameters lie in the same range. However, their relationship is actually non-unique, as shown below:

    ( ) 2F2QFSFS

    COVCOVm

    1m

    +

    = (4)

    in which mFS = mQ/mF = mean factor of safety and COVQ = sQ/mQ = coefficient of variation (COV) of

    capacity, COVF = sF/mF = COV of load. Different reliability indices can be obtained for the same mean

    factor of safety, depending on COVs of Q and F. In this sense, can be considered as an extension andmore complete version of FS that attempts to incorporate both deterministic and statistical information on

    Q and F.

    The problem of calculating pf for the general case in which Q is modeled as a nonlinear function of

    several non-normal random variables is more difficult than the simple case shown in Figure 1. A

    commonly used numerical technique that provides good approximate solutions for engineering

    applications is the First-Order Reliability Method (FORM). This technique can be easily implemented

    using MS Excel. Details are given by Low & Phoon (2002).

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    1E-07

    1E-06

    1E-05

    1E-04

    1E-03

    1E-02

    1E-01

    1E+00

    0 1 2 3 4 5

    Reliability index

    Probability

    offailur

    High

    Poor

    Hazardous

    Above average

    Good

    Below average

    Unsatisfactory

    310-7

    0.023

    0.16 0.07

    610-3

    310-5

    10-3

    Figure 2. Relationship between reliability index () and probability of failure (pf) (adapted from TableUS Army Corps of Engineers 1997, Table B-1)

    Load and resistance factors

    The basic objective of RBD is to ensure that the probability of failure of a component does not exceed an

    acceptable threshold level. Based on this objective, an economical design would be one in which the

    probability of failure does not depart significantly from the threshold. For the design problem shown inFigure 1, the RBD objective can be formally stated as follows:

    pf = Prob(Q < F) pT (5)

    in which pT = acceptable target probability of failure. Hence, the only basic difference between RBD and

    existing practice is that one controls the probability of failure, rather than the factor of safety. In non-

    mathematical terms, this is equivalent to controlling the fraction of unacceptable solutions in a weighted

    parametric study, where the weights refer to the likelihood of a set of input parameters being correct.

    Reliability-based design, as exemplified by Equation 5, allows the engineer to make a conscious choice

    on an acceptable level of design risk and then proceed to a set of design dimensions that are consistent

    with that choice. In contrast to the traditional or partial factors of safety approach, logical consistency

    between the computed design risk and the uncertainties inherent in the design process is assured by

    probabilistic analysis, such as FORM.

    A log-normal probability model is commonly used in place of the simple normal probability model

    because most physical quantities are non-negative (e.g., Paikowsky 2002). The analytical solution is

    available but the following well-known approximation is typically used for reliability calibration (e.g.,

    Ravindra & Galambos 1978, Becker 1996a):

    ( )2Q

    2F

    FSe

    COVCOV

    mlog

    += (6)

    To derive the LRFD format, the denominator in Equation 6 must be linearized as follows (Lind 1971):

    ( )QF2

    Q

    2

    F COVCOV75.0COVCOV ++ (7)

    Using Equations 6 and 7, the following simple LRFD format can be obtained:

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    Qn = Fn (8)

    in which Qn, Fn = nominal capacity and load, and , = resistance and load factors given by:

    ( )Qn

    QCOV75.0exp

    Q

    m = (9a)

    ( )Fn

    F COV75.0expF

    m+= (9b)

    It can be seen that reliability-calibrated resistance and load factors include the target risk level () and theunderlying parametric uncertainties (COVQ and COVF) rationally into the design process.

    Reservations have been expressed about the increase in the complexity of the design calculations

    resulting from the use of reliability theory. Complicated reliability calculations are undesirable because:

    (a) statistical information is not sufficiently well-defined to warrant sophisticated treatment, (b) there is a

    greater risk of making computational errors, (c) a study in soil behavior prediction is reduced to a mere

    mathematical exercise, and (d) attention will be diverted from the proper characterization of the ground

    mass and appreciation of the physical, chemical, and mechanical processes taking place in it (Beal, 1979;

    Semple, 1981; Simpson, et al., 1981; Boden, 1981). These reservations are not without merits. Excessive

    preoccupation with maintaining simplicity would, however, ultimately be a disservice to the geotechnical

    engineering profession. Historical hindsight has shown clearly that the judicious use of rational methods,

    as initiated by Terzaghi in 1943, was the primary cause of most of the significant advances in soil

    mechanics following World War II. The cost to pay for rationality is that design calculations could

    become more complicated. However, this cost is more than offset by the benefits associated with the use

    of rational methods. For example, the improvement in soil behavior prediction allows less conservatism

    to be applied in the design. The use of reliability methods is the next logical step toward greater

    rationality in design, and their potential benefits should not be discarded heedlessly because of the

    reluctance to advance beyond the current level of complexity in design.

    GEOTECHNICAL UNCERTAINTIES

    Parametric uncertainty

    The evaluation of soil and rock properties is one of the key design aspects that distinguishes geotechnical

    from structural engineering. None of the current geotechnical LRFD implementations consider this

    important issue explicitly. The purpose of this section is to highlight two important observations: (a)

    geotechnical variability is a complex attribute that needs careful evaluation, and (b) extensive statistical

    data are available for use as first-order estimates in RBD calibration and application. Extensivecalibration studies by Phoon et al. (1995) indicated that foundation resistance factors in the RBD

    equations can be calibrated for broad categories of data quality (e.g., COV of undrained shear strength =

    10-30%, 30-50%, 50-70%) without compromising on the uniformity of reliability achieved. Hence, it is

    not true that there are insufficient statistical data to warrant realistic reliability calculations.

    There are three primary sources of geotechnical uncertainties: (a) inherent variabilities, (b)

    measurement uncertainties, and (c) transformation uncertainties. The first results primarily from the

    natural geologic processes that produced and continually modify the soil mass in-situ. The second is

    caused by equipment, procedural and/or operator, and random testing effects. Equipment effects result

    from inaccuracies in the measuring devices and variations in equipment geometries and systems

    employed for routine testing. Procedural and/or operator effects originate from the limitations in existing

    test standards and how they are followed. In general, tests that are highly operator-dependent and have

    complicated test procedures will have greater variability than those with simple procedures and littleoperator dependency, as described in detail elsewhere (Kulhawy & Trautmann 1996). Random testing

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    error refers to the remaining scatter in the test results that is not assignable to specific testing parameters

    and is not caused by inherent soil variability. The third component of uncertainty is introduced when

    field or laboratory measurements are transformed into design soil properties using empirical or other

    correlation models (e.g., correlating the standard penetration test N value with the undrained shear

    strength). Obviously, the relative contribution of these components to the overall uncertainty in thedesign soil property depends on the site conditions, degree of equipment and procedural control, and

    quality of the correlation model. Therefore, soil property statistics that are determined from total

    variability analyses only can be applied to the specific set of circumstances (site conditions, measurement

    techniques, correlation models) for which the design soil properties were derived. Useful guidelines on

    typical coefficients of variation of many common soil properties are summarized in Tables 2 to 5. Others

    are reported by Jones et al. (2002).

    Model uncertainty

    A similar effort is underway to quantify uncertainties associated with geotechnical calculation models.

    Although many geotechnical calculation models are simple, reasonable predictions of fairly complexsoil-structure interaction behavior still can be achieved through empirical calibrations. Because of our

    geotechnical heritage that is steeped in such empiricisms, model uncertainties can be significant. Even a

    simple estimate of the average model bias is crucial for reliability-based design. If the model is

    conservative, it is obvious that the probabilities of failure calculated subsequently will be biased, because

    those design situations that belong to the safe domain could be assigned incorrectly to the failure domain,

    as a result of the built-in conservatism.

    Robust model statistics can only be evaluated using: (a) realistically large-scale prototype tests, (b) a

    sufficiently large and representative database, and (c) reasonably high quality testing where extraneous

    uncertainties are well-controlled. With the possible exception of foundations, insufficient test data are

    available to perform robust statistical assessment of the model error in many geotechnical calculation

    models. The development of a fully rigorous reliability-based design code that can handle the entire

    range of geotechnical design problems is currently impeded by the scarcity of these important statistics.Sidi (1986) was among the first to report model statistics that were established firmly using a large load

    test database assembled by Olson & Dennis (1982). The focus of the study was on friction piles in clay

    subjected to axial loading. Briaud & Tucker (1988) conducted a similar study using a 98-pile load test

    database obtained from the Mississippi State Highway Department. Recent literature includes estimation

    of model statistics for the calibration of deep foundation resistance factors for AASHTO [American

    Association of State Highway and Transportation Officials] (Paikowsky 2002). A substantial part of the

    study pertains to the evaluation of driven pile axial capacity using dynamic methods. None of these

    studies addresses the applicability of model statistics beyond the conditions implied in the database. This

    question mirrors the same concern expressed previously on the possible site-specific nature of soil

    variabilities.

    Phoon & Kulhawy (2003) presents a critical evaluation of model factors using an extensive databasecollected as part of an EPRI (Electric Power Research Institute) research program on transmission line

    structure foundations (Chen & Kulhawy 1994). This study considers only free-head rigid drilled shafts,

    using the databases summarized in Table 6. Three common models for lateral soil resistance, by Reese,

    Hansen, and Broms, are used to compute the theoretical lateral capacity under undrained and drained

    loading modes. Details are given elsewhere (Chen & Kulhawy 1994). Also, there are several methods to

    interpret lateral capacity from load tests: displacement limit, rotation limit, lateral or moment limit, and

    hyperbolic capacity (Hirany & Kulhawy 1988). All theoretical models are based on failure mechanisms

    requiring full mobilization of soil strength. Therefore, the hyperbolic capacity is probably closest to this

    ultimate state assumed in the analyses. A plausible and common method of correcting for model error is

    to assume the following multiplicative model (e.g., Ang & Tang 1984, Sidi 1986):

    Hh = M Hu (10)

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    Table 2. Approximate guidelines for inherent soil variability (Source: Phoon and Kulhawy 1999a).

    Test type Propertya

    Soil type Mean COV(%)

    Lab strength su(UC) Clay 10-400 kN/m2 20-55

    su(UU) Clay 10-350 kN/m2 10-30su(CIUC) Clay 150-700 kN/m

    2 20-40

    ' Clay & sand 20-40o 5-15

    CPT qT Clay 0.5-2.5 MN/m2 < 20

    qc Clay 0.5-2.0 MN/m2 20-40

    Sand 0.5-30.0 MN/m2 20-60

    VST su(VST) Clay 5-400 kN/m2 10-40

    SPT N Clay & sand 10-70 blows/ft 25-50

    DMT A reading Clay 100-450 kN/m2 10-35

    Sand 60-1300 kN/m2 20-50

    B reading Clay 500-880 kN/m2 10-35

    Sand 350-2400 kN/m2 20-50

    ID Sand 1-8 20-60

    KD Sand 2-30 20-60

    ED Sand 10-50 MN/m2 15-65

    PMT pL Clay 400-2800 kN/m2 10-35

    Sand 1600-3500 kN/m2 20-50

    EPMT Sand 5-15 MN/m2

    15-65

    Lab index wn Clay & silt 13-100 % 8-30

    wL Clay & silt 30-90 % 6-30

    wP Clay & silt 15-25 % 6-30

    PI Clay & silt 10-40 % b

    LI Clay & silt 10 % b

    , d Clay & silt 13-20 kN/m3 < 10

    Dr Sand 30-70 % 10-40c

    50-70d

    a - su = undrained shear strength; UC = unconfined compression test; UU = unconsolidated-undrained

    triaxial compression test; CIUC = consolidated isotropic undrained triaxial compression test; ' =effective stress friction angle; qT = corrected cone tip resistance; qc = cone tip resistance; VST = vaneshear test; N = standard penetration test blow count; A & B readings, ID, KD, & ED = dilatometer A & B

    readings, material index, horizontal stress index, & modulus; pL & EPMT = pressuremeter limit stress &

    modulus; wn = natural water content; wL = liquid limit; wP = plastic limit; PI = plasticity index; LI =

    liquidity index; & d = total & dry unit weights; Dr= relative densityb - COV = (3-12%) / mean

    c - total variability for direct method of determination

    d - total variability for indirect determination using SPT values

    in which Hh = measured hyperbolic capacity, Hu = computed lateral capacity, and M = model factor,

    typically assumed to be a log-normal random variable. The empirical distributions of M for model-scale

    laboratory tests and full-scale field tests are summarized in Figure 3.

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    Table 3. Summary of total measurement error of some laboratory tests (Phoon & Kulhawy 1999a)

    No. No. Tests / Property Value Property

    data Group (unitsb) COV (%)

    Propertya

    Soil type groups Range Mean Range Mean Range Mean

    su(TC) Clay, silt 11 - 13 7-407 125 8-38 19

    su(DS) Clay, silt 2 13-17 15 108-130 119 19-20 20

    su(LV) Clay 15 - - 4-123 29 5-37 13

    (TC) Clay, silt 4 9-13 10 2-27o

    19.1o 7-56 24

    (DS)Clay, silt 5 9-13 11 24-40

    o33.3

    o 3-29 13

    Sand 2 26 26 30-35o

    32.7o 13-14 14

    tan (TC) Sand, silt 6 - - - - 2-22 8

    tan (DS) Clay 2 - - - - 6-22 14

    wn Fine-grained 3 82-88 85 16-21 18 6-12 8

    wL Fine-grained 26 41-89 64 17-113 36 3-11 7

    wP Fine-grained 26 41-89 62 12-35 21 7-18 10

    PI Fine-grained 10 41-89 61 4-44 23 5-51 24

    Fine-grained 3 82-88 85 16-17 17.0 1-2 1

    a - su = undrained shear strength; = effective stress friction angle; TC = triaxial compression test;UC = unconfined compression test; DS = direct shear test; LV = laboratory vane shear test; wn = natural water

    content; wL = liquid limit; wP = plastic limit; PI = plasticity index; = total unit weightb - units of su = kN/m

    2; units of wn, wL, wP, and PI = %; units of = kN/m

    3

    The model-scale load tests were conducted in uniform kaolinite clay and filter sand deposits prepared

    under controlled laboratory conditions. Hence, uncertainties arising from evaluation of soil parameters

    are minimal. In addition, construction variabilities and measurement errors associated with load tests also

    are minimal. Therefore, model uncertainties computed from laboratory tests should be an accurate

    indicator of errors arising from the use of simplified calculation models. The main concern is whether the

    model factors are applicable beyond the uniform profile and specific soil type used in the laboratory.

    Model factors from field tests are expected to be more general because they are computed from load tests

    conducted in more diverse site environments. However, it is reasonable to query if the statistics of such

    model factors are lumped statistics, in the sense that extraneous sources of uncertainties (e.g.,

    construction variabilities, measurement errors incurred during load test, uncertainties in soil parameter

    evaluation) are inextricably included in the computation.

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    Table 4. Summary of measurement error of common in-situ tests (Kulhawy & Trautmann 1996)

    Coefficient of variation, COV (%)

    Test Equipment Procedure Random Totala

    Rangeb

    Standard penetration test (SPT) 5c - 75d 5c - 75d 12 - 15 14c - 100d 15 - 45

    Mechanical cone penetration test

    (MCPT)

    5 10e -15f 10e -15f 15e -22f 15 - 25

    Electric cone penetration test

    (ECPT)

    3 5 5e

    -10f

    7e

    - 12f

    5 - 15

    Vane shear test (VST) 5 8 10 14 10 - 20

    Dilatometer test (DMT) 5 5 8 11 5 - 15

    Pressuremeter test, pre-bored (PMT) 5 12 10 16 10 - 20g

    Self-boring pressuremeter test

    (SBPMT)

    8 15 8 19 15 - 25g

    a - COV(Total) = [COV(Equipment)2

    + COV(Procedure)2

    + COV(Random)2]

    0.5

    b - Because of limited data and judgment involved in estimating COVs, ranges represent probable magnitudes of

    field test measurement error

    c, d - Best to worst case scenarios, respectively, for SPT

    e, f - Tip and side resistances, respectively, for CPT

    g - It is likely that results may differ for po, pf, and pL, but the data are insufficient to clarify this issue

    A comparison between laboratory and field data such as that shown in Figure 3 is illuminating.

    Laboratory and field results are plotted as white and grey histograms in the first row of each cell,

    respectively. Visual inspection and simple statistics [mean, standard deviation (S.D.), coefficient of

    variation (COV)] show that the histograms are similar. The p-values from the Mann-Whitney test

    formally show that the null hypothesis of equal medians cannot be rejected at 5% significance level, with

    the exception of the drained factor for the Hansen model. Therefore, it is reasonable to argue that the

    results presented in Figure 3 have wider applicability beyond the conditions implied by the underlying

    databases, and the model uncertainties are mainly caused by errors intrinsic to the respective simplified

    calculation models. A more robust estimate of the empirical distribution is obtained by combining the

    laboratory and field data, as shown in the second row of each cell in Figure 3. For rigid drilled shafts

    subjected to lateral-moment loading, the COV of the model factor appears to fall within a narrow range of

    25 to 40%. Another detailed example on cantilever walls in sand is discussed by Phoon et al. (2003d).

    PRACTICAL IMPLEMENTATION ISSUES

    Conceptual basis for EPRI study

    Although existing geotechnical LRFD codes look the same as their structural counterparts, they are

    incompatible with structural RBD on closer inspection because one or more of the following key elements

    are missing (Kulhawy & Phoon 2002):

    a. The primary objective in structural RBD is to achieve a minimum target reliability index across aspecified domain of interest (e.g., foundation geometries and types, loading modes, soil conditions,

    etc.). Structural RBD requires deliberate and explicit choices to be made on the target reliability

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    Table 5. Approximate guidelines for design soil property variability (Phoon & Kulhawy 1999b)

    Design Point Spatial avg. Correlation

    propertya

    Testb Soil type COV (%) COV

    c(%) equation

    su(UC) Direct (lab) Clay 20-55 10-40 -su(UU) Direct (lab) Clay 10-35 7-25 -

    su(CIUC) Direct (lab) Clay 20-45 10-30 -

    su(field) VST Clay 15-50 15-50 14

    su(UU) qT Clay 30-40e

    30-35e 18

    su(CIUC) qT Clay 35-50e

    35-40e 18

    su(UU) N Clay 40-60 40-55 23

    sud KD Clay 30-55 30-55 29

    su(field) PI Clay 30-55e - 32

    Direct (lab) Clay, sand 7-20 6-20 -)TC( qT Sand 10-15

    e10

    e 38

    cv PI Clay 15-20e

    15-20e 43

    Ko Direct (SBPMT) Clay 20-45 15-45 -

    Ko Direct (SBPMT) Sand 25-55 20-55 -

    Ko KD Clay 35-50e

    35-50e 49

    Ko N Clay 40-75e - 54

    EPMT Direct (PMT) Sand 20-70 15-70 -

    ED

    Direct (DMT) Sand 15-70 10-70 -

    EPMT N Clay 85-95 85-95 61

    ED N Silt 40-60 35-55 64

    a - su = undrained shear strength; UU = unconsolidated-undrained triaxial compression test;

    UC = unconfined compression test; CIUC = consolidated isotropic undrained triaxial compression

    test; su(field) = corrected su from vane shear test; = effective stress friction angle; TC = triaxialcompression; cv = constant volume ; Ko = in-situ horizontal stress coefficient; EPMT =pressuremeter modulus; ED = dilatometer modulus

    b - VST = vane shear test; qT = corrected cone tip resistance; N = standard penetration test blow count;

    KD = dilatometer horizontal stress index; PI = plasticity index

    c - averaging over 5 meters

    d - mixture of su from UU, UC, and VSTe - COV is a function of the mean; refer to COV equations in Phoon & Kulhawy (1999b) for details

    Table 6. Databases on laterally-loaded rigid drilled shafts (Phoon & Kulhawy 2003)

    Description No. tests D/B e/D

    Undrained loading:

    Model-scale lab tests 47 3.0 8.0 0.03 4.0

    Full-scale field tests 27 2.3 10.5 0.03 6.8

    Drained loading:

    Model-scale lab tests 55 2.6 9.0 0.06 5.0

    Full-scale field tests 22 2.5 7.0 0 5.4

    Note: D = depth; B = diameter; e = load eccentricity

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    Model Undrained Drained

    Undrained:

    Reese (1958)

    Drained:Reese et al. (1974)

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Reese)

    0

    5

    10

    15

    20

    Freq

    uency

    Mean =S.D. =COV =

    n =

    p =

    1.430.370.2647

    0.315

    Lab

    1.400.470.3327

    Field

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Reese)

    0

    5

    10

    15

    20

    Freq

    uency

    Mean =S.D. =COV =

    n =

    p =

    1.190.570.4855

    0.433

    Lab

    1.190.360.3022

    Field

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Reese)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =

    1.420.410.2974

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Reese)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =

    1.190.510.4377

    Hansen (1961)

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Hansen)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =p =

    1.950.550.28470.296

    Lab

    1.850.570.3127

    Field

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Hansen)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =p =

    1.050.330.31550.002

    Lab

    0.830.250.3022

    Field

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Hansen)

    0

    5

    10

    15

    20

    Fre

    quency

    Mean =S.D. =COV =

    n =

    1.920.560.2974

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Hansen)

    0

    5

    10

    15

    20

    Frequ

    ency

    Mean =S.D. =COV =

    n =

    0.980.320.33

    77

    Undrained:

    Broms (1964a)

    Drained:

    Broms (1964b)

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Broms)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =p =

    2.280.800.35470.875

    Lab

    2.290.950.4127

    Field

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(simplified Broms)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =p =

    1.310.530.40550.844

    Lab

    1.270.410.3222

    Field

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(Broms)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =

    2.280.850.3774

    0.4 1.2 2.0 2.8 3.6

    Hh/Hu(simplified Broms)

    0

    5

    10

    15

    20

    Frequency

    Mean =S.D. =COV =

    n =

    1.300.500.3877

    Figure 3. Model factors for rigid drilled shafts under undrained and drained lateral-moment loading

    modes (Phoon & Kulhawy 2003)

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    index, scope of calibration domains, and representative designs populating each domain. This is

    philosophically different from the objective of achieving designs comparable to working stress

    design.

    b. The secondary objective in structural RBD is to increase uniformity of reliability across the domain

    of interest, which is rarely emphasized and verified in geotechnical LRFD. In fact, the typical use ofa single resistance factor for each loading mode is not adequate for this task, as elaborated in item 6

    below.

    c. Soil variability is the most significant source of uncertainty, but it is not quantified in a robust way (if

    at all) and incorporated explicitly in the code calibration process.

    d. Probabilistic load models compatible with the relevant structural codes are not spelled out clearly. It

    is unclear if the original structural load models have been used for code calibration. Load

    combinations are not amenable to simplified lognormal reliability analysis (in contrast to the more

    general first-order reliability method or FORM highlighted in item 5) unless they are approximated as

    some lumped load parameters.

    e. Rigorous reliability analysis using FORM is not used as the main tool to integrate loads, basic soil

    parameters, and calculation models in a realistic and self-consistent way, both physically andprobabilistically. The commonly adopted approach of simplifying geotechnical capacity as a single

    lognormal random variable has limitations (Phoon et al. 2003b).

    f. No guidelines on selection of nominal or characteristic soil parameters are usually given. It is also

    unclear how resistance factors will be affected by the site conditions, measurement techniques, and

    correlation models used to derive the relevant design parameters.

    There are no technical and/or practical difficulties in addressing these incompatibilities directly. The

    basic solution is to follow a more general calibration procedure as outlined below.

    Achieving uniform reliability

    In EPRI Report TR-105000, uniform reliability is realized by partitioning the design domains and using a

    Multiple Resistance Factor Design (MRFD) format. The design of drilled shafts (bored piles) for upliftunder undrained loading will be used as an example. Two simple design formats were selected for

    reliability calibration:

    LRFD: F50uQun (11)MRFD: F50suQsun + tuQtun + wW (12)

    in which F50 = 50-year return period load, Qun = nominal uplift capacity, Qsun = nominal side resistance,

    Qtun = nominal tip resistance, W = weight of foundation, and u, su, tu and w = resistance factors.Figure 4a show the reliability levels implicit in existing ULS designs. Note that the existing WSD format

    is essentially the same as the LRFD format (Equation 11), because the reciprocal of the traditional factor

    of safety (FS) is equal to the resistance factor (u). Therefore, the variation in the reliability index () or

    probability of failure (pf) at a fixed FS is indicative that the LRFD format will produce fairly pooruniformity in reliability when it is applied over the entire design domain. In EPRI Report TR-105000, the

    uniformity in reliability was improved by using the following general calibration procedure:

    a. Perform a parametric study on the variation of the reliability level with respect to each deterministic

    and statistical parameter in the design problem. Examples of deterministic parameters that control the

    design of drilled shafts include the diameter (B) and depth to diameter (D/B) ratio. Examples of

    statistical parameters include the mean and coefficient of variation (COV) of the undrained shear

    strength (su).

    b. Partition the parameter space into several smaller domains. An example of a simple parameter space

    is shown in Figure 5. The reason for partitioning is to achieve greater uniformity in reliability over

    the full range of deterministic and statistical parameters. For those parameters identified in Step (a)

    as having a significant influence on the reliability level, the size of the partition clearly should be

    smaller. In addition, partitioning ideally should conform to existing geotechnical conventions.

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    (a) (b)

    Figure 4. Drilled shafts in undrained uplift: (a) reliability levels implicit in existing ultimate limit state

    design and (b) performance of ultimate limit state RBD formats

    c. Select a set of representative points from each domain. Note that each point in the parameter space

    denotes a specific set of parameter values (Figure 5). Ideally, the set of representative points should

    capture the full range of variation in the reliability level over the whole domain.

    d. Determine an acceptable foundation design for each point and evaluate the reliability levels in the

    designs. Foundation design is performed using the set of parameter values associated with each

    point, along with a simplified RBD format and a set of trial resistance factors. The reliability of the

    resulting foundation design then is evaluated using the FORM algorithm.

    e. Quantify the deviations of the reliability levels from a pre-selected target reliability index, T. The

    following simple objective function can be used:

    n

    1=i

    2Tiwtusu )-(=),,(H (13)

    in which H() = objective function to be minimized, n = number of points in the calibration domain,and i = reliability index for the ith point in the domain.

    f. Adjust the resistance factors and repeat Steps (d) and (e) until the objective function is minimized.

    The set of resistance factors that minimizes the objective function (H) is the most desirable because

    the degree of uniformity in the reliability levels of all the designs in the domain is maximized. The

    following measure can be used to quantify the degree of uniformity that has been achieved:

    n

    H= (11)

    g. Repeat Steps (c) to (f) for the other domains.

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    Figure 5. Partitioning of parameter space for calibration of resistance factors.

    Table 7. Undrained uplift resistance/deformation factors for drilled shafts (Source: Phoon et al. 1995, pp.

    6-7 & 17).

    LRFD

    (ULS)

    MRFD

    (ULS)

    LRFD

    (SLS)Mean su

    (kN/m2)

    COV su(%)

    u su tu w u25 - 50 10 30 0.44 0.44 0.28 0.50 0.65

    Medium clay 30 50 0.43 0.41 0.31 0.52 0.63

    50 70 0.42 0.38 0.33 0.53 0.62

    50 - 100 10 30 0.43 0.40 0.35 0.56 0.64

    Stiff clay 30 50 0.41 0.36 0.37 0.59 0.61

    50 70 0.39 0.32 0.40 0.62 0.58

    100 - 200 10 30 0.40 0.35 0.42 0.66 0.61

    Very stiff clay 30 50 0.37 0.31 0.48 0.68 0.57

    50 70 0.34 0.26 0.51 0.72 0.52

    Note: Target = 3.2 for ultimate limit state (ULS) and 2.6 for serviceability limit state (SLS)

    The results of the RBD calibration exercise for drilled shafts in undrained uplift loading are shown in

    Table 7. Exact comparison with other LRFD resistance factors is difficult, but AASHTO recommends u= 0.55 for uplift capacity of drilled shaft (-method) in clay (NCHRP Report 343, Table 4.10.6-3) whileOHBDC/CHBDC and NBCC recommends u = 0.3 for tension capacity of deep foundations evaluatedusing static analysis (Green & Becker 2001). Note that the target reliability indices are 2.5 to 3.5 for

    AASHTO (probably closer to 2.5 as reported by Rojiani et al. 1991) and 3.5 for OHBDC/CHBDC/NBCC.

    The EPRI resistance factors vary from 0.34 to 0.44 depending on the quality of data, as shown in Table 7.

    They lie between AASHTO and OHBDC/CHBDC/NBCC factors, but are closer to the latter than the

    former, partially because the target reliability index is 3.2. More importantly, a single resistance factor

    cannot be expected to maintain uniform reliability over a wide and diverse range of design scenarios asshown in Figure 4a. The EPRI study shows that the use of a simple 3x3 partitioning on the mean and

    COV of undrained shear strength is sufficient to produce designs with distinctively more uniform

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    reliability (compare Figures 4a and 4b). The partitioning on the mean undrained shear strength also was

    selected to conform to existing geotechnical conventions, as noted previously.

    The EPRI study further recommended use of the Multiple Resistance Factor Design (MRFD) format

    for achieving a more consistent target reliability (Equation 12). The MRFD format is a natural

    generalization of the LRFD format that involves the application of one resistance factor to eachcomponent of the capacity rather than the overall capacity. MRFD is more physically meaningful for

    foundation design because the variability of each component can be significantly different. In addition, it

    achieves greater uniformity in reliability as shown in Figure 4b.

    CONCLUSIONS

    New reliability-based design (RBD) methodologies that are already adopted widely by the structural

    community are not accepted readily in the geotechnical community, partially because of the questionable

    robustness of the statistics used for code calibration and unfamiliarity with probabilistic concepts.

    However, maintaining status quo is increasingly untenable because of gathering momentum in code

    harmonization and broadening divergence between geotechnical and structural design. Becausegeotechnical design is only one component of harmonised codes, it is anticipated that structural reliability

    methods will eventually prevail in geotechnical design.

    There is a need to draw a clear distinction between accepting reliability analysis as a necessary

    theoretical basis for geotechnical design and downstream calibration of simplified multiple-factor design

    formats, with emphasis on the former. Simplified reliability-based design (RBD) equations in the form of

    LRFD/MRFD are probably required for routine design at present, but their limitations have no bearing on

    the generality of reliability theory. This paper argues that there is sufficient statistical support for the

    development of these simplified RBD formats. Reliability-based design, simplified or otherwise,

    provides a more consistent means of managing uncertainties, but it is by no means a perfect solution.

    Engineering judgment still is indispensable in many aspects of geotechnical engineering reliability

    analysis merely removes the need for guesswork on how uncertainties affect performance and is

    comparable to the use of elasto-plastic theory to remove the guesswork on how loads induce stresses anddeformations. Lacasse et al. (2004) observed that:

    Engineering depends on judgment, the exercise of which depends on knowledge derived from theoretical

    concepts, experiment, measurements, observations, and past experience. These building blocks have to

    be recognized, assembled, and evaluated collectively before judgment can be rendered.

    Historical hindsight has shown clearly that the judicious use of rational methods, as initiated by

    Terzaghi in 1943, was the primary cause of most of the significant advances in soil mechanics following

    World War II. The cost to pay for rationality is that design calculations could become more complicated.

    However, this cost is more than offset by benefits such as reduction in conservatism because of improved

    understanding and hence increased confidence in the design. The use of reliability methods is the nextlogical step toward greater rationality in design, and their potential benefits should not be discarded

    heedlessly because of the reluctance to advance beyond the current level of complexity in design.

    ACKNOWLEDGMENTS

    Financial support from the Site Investigation Committee, Korean Geotechnical Society; Hyundai

    Engineering & Construction Co., Ltd.; and Daelim Industry Co., Ltd are gratefully acknowledged. The

    author is also grateful to Dr Gil Lim Yoon for his kind invitation to deliver this special lecture and

    making all the arrangements for this visit.

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