Calibration concepts for load and resistance factor design (LRFD) of

Calibration concepts for load and resistancefactor design (LRFD) of reinforced soil walls

Richard J. Bathurst, Tony M. Allen, and Andrzej S. Nowak

Abstract: Reliability-based design concepts and their application to load and resistance factor design (LRFD or limit statesdesign (LSD) in Canada) are well known, and their adoption in geotechnical engineering design is now recommended formany soil–structure interaction problems. Two important challenges for acceptance of LRFD for the design of reinforcedsoil walls are (i) a proper understanding of the calibration methods used to arrive at load and resistance factors, and(ii) the proper interpretation of the data required to carry out this process. This paper presents LRFD calibration principlesand traces the steps required to arrive at load and resistance factors using closed-form solutions for one typical limit state,namely pullout of steel reinforcement elements in the anchorage zone of a reinforced soil wall. A unique feature of thispaper is that measured load and resistance values from a database of case histories are used to develop the statistical pa-rameters in the examples. The paper also addresses issues related to the influence of outliers in the datasets and possibledependencies between variables that can have an important influence on the results of calibration.

Key words: load and resistance factor design (LRFD), limit states design (LSD), reliability-based design, closed-form solu-tions, reinforced soil walls, steel reinforcement, pullout.

Resume : On connaıt bien les concepts de conception bases sur la fiabilite et leur application au facteur de conception decharge et de resistance (LRFD ou la conception aux etats limites au Canada), et on recommande maintenant leur adoptionen conception d’ingenierie geotechnique pour plusieurs problemes d’interaction sol-structure. Deux importants defis pourl’acceptation du LRFD pour le calcul de murs en sol arme sont : (i) une bonne comprehension des methodes de calibrageutilisees pour arriver aux facteurs de charge et de resistance, et (ii) la bonne interpretation des donnees requises pour reali-ser ce processus. Cet article presente les principes de calibrage du LRFD et trace les etapes requises pour arriver aux fac-teurs de charge et de resistance en utilisant des solutions exactes pour un etat limite typique d’arrachement d’elementsd’armature en acier dans la zone d’ancrage d’un mur en sol arme. Une caracteristique unique de cet article est que des va-leurs de charges et de resistances mesurees provenant d’une base de donnees d’histoires de cas sont utilisees pour develop-per les parametres statistiques dans les exemples. Cet article traite aussi des problemes relies a l’influence des donneesetrangeres dans les ensembles de donnee et des dependances possibles rentre les variables qui peuvent avoir une influenceimportante sur les resultats du calibrage.

Mots-cles : facteurs de conception de charge et de resistance « LRFD » conception d’etat limite « LSD », conception ba-see sur la fiabiliite, solutions exactes, murs en sol arme, armature d’acier, arrachement.

[Traduit par la Redaction]

Introduction

Use of reliability theory to determine load and resistancefactors for limit states design of civil engineering structures(called load and resistance factor design, or LRFD in theUSA, and limit states design, or LSD in Canada) is nowwell developed, and its adoption in geotechnical engineeringdesign is now recommended for many soil–structure interac-tion problems. The fundamentals of reliability theory, as

they apply to structural design, can be found in the textbookby Nowak and Collins (2000). An excellent overview of thedevelopment of reliability-based design as it applies to geo-technical engineering has been provided by Kulhawy andPhoon (2002). Goble (1999) and Becker (1996a, 1996b)also have provided a useful background to the developmentof limit states design practice for foundation engineering inNorth America.

The American Association of State Highway and Trans-portation Officials (AASHTO) has committed to the LRFDapproach for all structures, including reinforced soil walls(AASHTO 2007). Currently, an allowable (working) stressdesign (ASD or WSD) method is used for reinforced soilwalls (AASHTO 2002). In Canada, the Canadian foundationengineering manual (Canadian Geotechnical Society 2006)recommends an LRFD approach using the North Americanfactored resistance approach. The current Canadian highwaybridge design code (CSA 2006) will also be updated to in-clude reinforced soil walls within an LRFD framework.

For geotechnical engineers, the implementation of LRFDis understood using prescribed limit state equations and load

Received 21 June 2007. Accepted 5 June 2008. Published on theNRC Research Press Web site at cgj.nrc.ca on 26 September2008.

R.J. Bathurst.1 GeoEngineering Centre at Queen’s-RMC,Department of Civil Engineering, Royal Military College ofCanada, Kingston, ON K7K 7B4, Canada.T.M. Allen. State Materials Laboratory, Washington StateDepartment of Transportation, Olympia, WA 98504-7365, USA.A.S. Nowak. Department of Civil Engineering, University ofNebraska, W181 Nebraska Hall, Lincoln, NE 68588-0531, USA.

1Corresponding author (e-mail: [email protected]).

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Can. Geotech. J. 45: 1377–1392 (2008) doi:10.1139/T08-063 # 2008 NRC Canada

and resistance factors taken from tables in design codes(NRCC 2005; CSA 2006; AASHTO 2007). However, the re-casting of retaining wall allowable stress design within anLRFD framework has been problematic due to a lack ofstatistical data suitable for probabilistic analysis of load andresistance parameters. Furthermore, both the load and resist-ance (or strength) components in limit state formulations in-clude soil unit weight and (or) strength as input parametersthat also have inherent variability. For reinforced soil walls,the soil reinforcement elements add an additional level ofcomplication. Calibration for reinforced soil retaining walldesign (called mechanically stabilized earth walls in USAterminology) has until now been restricted to comparisonwith ASD practice (often called calibration by fitting).

There remain two important challenges for acceptance ofLRFD for the design of reinforced soil walls by geotechnicalengineers: (i) an understanding of the calibration methodsused to arrive at load and resistance factors, and (ii) theproper interpretation of the data required for calibration.This paper addresses these two challenges, which have notreceived the level of attention in the literature that is re-quired to give geotechnical engineers (i) confidence in thegeneral LRFD approach, (ii) confidence that load and resist-ance factors found in design codes are reasonable, and(iii) the tools to carry out the calibration themselves usingthe current North American approach to geotechnical LRFD.

This paper illustrates LRFD calibration using an examplefor the limit state associated with steel reinforcement ele-ments in the anchorage zone of a reinforced soil wall. Theexample is based on material presented in a larger LRFDcalibration guidance document by Allen et al. (2005). Forbrevity and to emphasize important issues, the example isrestricted to calibration using closed-form solutions. Aunique feature of this paper is that actual data for load andresistance terms are available (Allen et al. 2001, 2004).However, a number of important points regarding the selec-tion and statistical treatment of the data are presented in thecurrent paper to qualify the general approach proposed byAllen et al. (2005). The more adaptable Monte Carlo ap-proach to estimate resistance factors is not reviewed here,but points made regarding statistical treatment and interpre-tation of load and resistance data are equally applicable forthis approach.

BackgroundIt is necessary to provide a brief background on the con-

cepts behind LRFD in North American practice to under-stand the treatment of data presented later in the paper. Thebasic concept behind LRFD in North American practice isillustrated in Fig. 1a. Here, uncorrelated distributions of ran-dom load (Q) and resistance (R) values are shown as normal(Gaussian) frequency distributions. For a prescribed limitstate (g), expressed as

½1� g ¼ R� Q

the probability of failure is equal to the area of the fre-quency distribution in Fig. 1b for which g < 0, where g iscalculated using random values of R and Q. Clearly, as themean value (u) of the distribution for g moves to the rightin Fig. 1b, the probability of failure (pf) becomes less. For

a normal distribution of g values, the probability of failurecan be equated explicitly to the value of the reliability index� = u/�, where � is the standard deviation of g. The non-linear relationship between probability of failure and relia-bility index for a normal distribution of g is shown inFig. 2. For example, a probability of failure of 1 in 1000corresponds to a reliability index � = 3.09. The calculationof pf can be easily carried out using the standard normal cu-mulative function (NORMSDIST) in Microsoft Excel:

½2� pf ¼ 1� NORMSDIST ð�Þ

If load and resistance values are normally distributed andthe limit state function is linear (e.g., eq. [1]), then � is thereciprocal of the coefficient of variation (COV) of g = R –Q and can be calculated exactly as follows (e.g., Nowak andCollins 2000):

½3� � ¼ R � Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

R þ �2Q

qwhere R is the mean value of the resistance R, Q is themean value of the load Q, �R is the standard deviation forthe resistance R, and �Q is the standard deviation for theload Q. If distributions for R and Q deviate from normaldistributions, then eq. [3] is only an approximation.

If both the load and resistance distributions are log-normal and the limit state function is a product of randomvariables, then � can be calculated using a closed-form solu-tion reported by Withiam et al. (1998) and Nowak (1999).Specifically, expressing the limit state function as g = R/Q –1, then � can be determined as follows:

½4� � ¼ln½R=Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ COV2

QÞ=ð1þ COV2RÞ

q�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln½ð1þ COV2QÞð1þ COV2

RÞ�q

where COVR and COVQ are the coefficients of variation forthe resistance and load values, respectively. Expanded ver-sions of these expressions for LRFD calibration purposesare presented later in the paper.

Example limit state for reinforced soil wallsThe first step in LRFD design is to identify each possible

ultimate or serviceability limit state and to express it using adeterministic function as in conventional allowable stressdesign. For example, consider the ultimate limit state againstpullout of steel grid reinforcement (steel bar mat and weldedwire) as illustrated in Fig. 3. The resistance term R usingAASHTO (2002, 2007) is

½5� R ¼ Tpo ¼ 2Le�vF� ¼ 2Le�szF�

where Tpo is the pullout capacity (kN/m), Le is the ancho-rage length (m), �v = �s z is the vertical stress (kPa), �s isthe soil unit weight (kN/m3), and F* is the dimensionlesspullout resistance factor (in this case, F* is a function ofthe thickness and horizontal spacing of the reinforcementtransverse bars and depth z of the reinforcement). The an-chorage length is illustrated in Fig. 3a.

The load for the limit state calculation in this example isassumed to be due to the self-weight of the wall backfill

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(i.e., no live load or other types of loads). The load is calcu-lated using the AASHTO simplified method:

½6� Q ¼ Tmax ¼ Sv�vKr

where Tmax is the maximum tensile load in the reinforce-ment (kN/m), Sv is the tributary spacing of the reinforce-ment layer (m), and Kr is the dimensionless lateral earthpressure coefficient acting at the reinforcement layer depth.For steel grid reinforced soil walls, Kr varies from 2.5Ka to1.2Ka at the top of the wall to a depth of 6 m below the walltop, respectively, and remains at 1.2Ka below 6 m (Fig. 3b),where Ka = f(�) is the dimensionless coefficient of activelateral earth pressure, where f is the soil friction angle,which is constrained to � £ 408 according to AASHTO(2002, 2007). This restriction is applied in calculations forpredicted load values that are described later in the paper.

For the same reinforcement layer in a very large sampleof nominally identical walls, we expect that the same equa-tions would yield a range of computed values of R and Q

due to variability in the values of �s and � and accuracy ofthe measurements taken to back-calculate F* in laboratorypullout tests. Variability in the magnitude of soil-related in-put parameters is inherent to these materials in field con-struction and laboratory pullout tests. In the field, additionalvariability will result from quality of construction and arange of different equipment employed. All other input pa-rameters can be taken as deterministic (i.e., assigned con-stant values) for practical LRFD calibration and design.

Adopting the limit state format in eq. [1], the ultimatelimit state for reinforcement pullout is expressed as

½7� g ¼ Tpo � Tmax ¼ 2Le�vF� � Sv�vKr

The probability of Tmax exceeding Tpo must be kept belowan acceptable level. Hence, for design the values of Tpo andTmax must be decreased and increased, respectively, by mul-tiplying them by suitably selected resistance and load fac-tors to achieve an acceptable probability of failure, wherefailure is defined as g £ 0.

In North America, LRFD (or LSD in Canada) is based ona factored resistance approach. The general approach can beexpressed as

½8� ��iQni � ’Rn

where Qni is the nominal (specified) load, Rn is the nominal(characteristic) resistance, �i is the load factor, and 4 is theresistance factor. Here, to simplify the general approach andbe consistent in the calculation of nominal load and resis-tance values, we assume that mean values of soil unitweight and strength in eqs. [5] and [6] will be used.

In the current North American approach, uncertainty inthe calculation of the resistance side of the equation is cap-tured by a single resistance factor while load contributionsare assigned (typically) different load factors. Load factorterms have values �i ‡ 1, and the resistance term shouldhave a value 4 £ 1.

Using eqs. [1] and [8], the design limit state equation forreinforcement pullout with a single load term can be ex-pressed as

½9� �QTmax � ’Tpo � 0

where the load factor notation �Q is now adopted for the

Fig. 1. Probability of failure in reliability-based design. (a) Frequency distributions for random values of load and resistance terms.(b) Probability of failure.

Fig. 2. Relationship between probability of failure and reliabilityindex for normal distributions.

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case of a single load term in the limit state function. Fromeq. [9], the minimum pullout capacity is

½10� Tpo ¼�Q

’Tmax

In other words, the nominal resistance value (Rn = Tpo)must always be greater than the nominal load value (Qn =Tmax) by a factor of �Q/4.

LRFD design versus LRFD calibration

The objective of LRFD design is to ensure that eq. [8] issatisfied (or in the example problem, eq. [9]). In design, thisis achieved by adjusting the values in the model eqs. [5] and[6] (for example) and (or) design geometry (e.g., wall heightand number of reinforcement layers) to satisfy eq. [8] usingprescribed values for the load and resistance factors.

The objective of LRFD calibration is to select values ofload and resistance factors to be used in design such thatthe computed probability of failure is below the target prob-ability of failure. This objective can be understood as the se-lection of factors to be applied to load and resistance termssuch that the idealized normal distributions illustrated inFig. 1a are far enough apart and the area of the distributionfunction for g < 0 in Fig. 1b does not exceed a prescribedvalue.

At the time of writing this paper, load and resistance fac-tors for internal stability design of reinforced soil walls us-ing a factored resistance approach are based on calibrationby fitting to ASD (D’Appolonia Engineering 1999;AASHTO 2007) (see the section titled Calibration using al-lowable stress design (ASD)). A description of a more rigor-ous approach for the calibration of the North AmericanLRFD method for internal stability of reinforced soil wallstructures is the main objective of this paper.

Selection of probability of failure for internalstability of reinforced soil walls

In general, resistance factors for the design of buildingand bridge structural components have been derived to pro-duce a probability of failure (pf) of about 1 in 5000 (thiscorresponds to � = 3.54). However, past geotechnical designpractice has resulted in an effective probability of failure forfoundations, in general, of approximately 1 in 1000(Withiam et al. 1998). For reinforced soil walls, the multiplelayers of horizontal reinforcement result in highly strength-redundant systems, and the failure or overstress of a singlereinforcement layer or strip will not result in failure of thewall. Hence, the internal stability of the reinforced wall as asystem has much greater reliability (i.e., system reliability)and much lower probability of failure than that of an indi-vidual reinforcement layer or element. Furthermore, the re-inforced soil is a deformable medium that helps redistributeloads through the composite structure (Allen et al. 2001). Auseful comparison is a piled foundation, where failure of asingle pile will result in load shedding to neighbouring piles(Zhang et al. 2001). For piled foundations, D’Appolonia En-gineering (1999) and Paikowsky (2004) have recommendedpf = 1 in 100 (� = 2.33). In the calculation examples to fol-low, a target probability of 1 in 100 is assumed to be rea-sonable for LRFD calibration of internal stability limitstates for reinforced soil walls for the reasons noted. Lastly,it should be noted that AASHTO (2007) recommends a75 year design life for all reinforced soil walls. Additionaldiscussion regarding the choice of � for reinforced soil walldesign is provided by Allen et al. (2005).

Analysis of load and resistance data

Model biasIdeally, the estimate of the probability of failure using

eqs. [3] and [4] for the example limit state should be basedon measurements of reinforcement load and pullout resist-

Fig. 3. AASHTO (2002, 2007) simplified method for steel bar mats and welded wire reinforcement pullout: (a) pullout model geometry andloads; (b) coefficient of lateral earth pressure. H, height of wall.


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ance for a large number of nominally similar structures.Statistical parameters needed to estimate the probability offailure can be taken from these measurements. To relate themeasured values of load and resistance from multiple loca-tions within a given structure or from multiple case histor-ies, a baseline of comparison is needed. The nominalprediction of load and resistance that corresponds to eachmeasured load or resistance value can be used as this base-line. It is reasonable to expect that there will be a differencebetween a predicted value and the corresponding measuredvalue due to model error, as well as variability in materialproperties. This difference can be expressed as a bias valuedefined as the ratio of the measured to predicted (nominal)value. Bias values are sometimes called ‘‘model factors’’ inthe literature (e.g., Phoon and Kulhawy 2003, 2005). Con-ceptually, each measured load and resistance value can bedivided by the appropriate predicted (nominal) load and re-sistance value to arrive at the distribution for nondimen-sional load and resistance terms in the calculation ofprobability of failure (Withiam et al. 1998; Allen et al.2005).

Continuing the example in the section titled Examplelimit state for reinforced soil walls, the resistance bias valueXR for a pullout resistance data point can be expressed as

½11� XR ¼Rmeasured

Rn predicted

¼ Tpo measured

Tpo predicted

and the load bias value XQ as

½12� XQ ¼Qmeasured

Qn predicted

¼ Tmax measured

Tmax predicted

A bias value of 1.0 means that the model predicts themeasured load or resistance exactly for the data point. Thiscan be expected to be an unusual occurrence in geotechnicalpractice. The requirement that there be no dependencies be-tween bias values and the corresponding load and resistance(predicted) values is discussed in the next section.

The mean and standard deviation of the measured load(Q, �Q) and resistance (R, �R) values in eqs. [3] and [4] canbe replaced by

½13a� R ¼ Rmeasured ¼ Rn predicted�R

½13b� Q ¼ Qmeasured ¼ Qn predicted�Q

½13c� �R ¼ COVRRmeasured

½13d� �Q ¼ COVQQmeasured

where Rmeasured and Qmeasured are the means of the measuredresistance and load values, respectively; �R and COVR arethe mean and coefficient of variation of the resistance biasvalues, respectively; and �Q and COVQ are the mean andcoefficient of variation of the load bias values, respectively.These equations enable the bias statistics to be used directlyin eqs. [3] and [4]. Specifically, the mean and standard de-viation of the bias values are scaled by the deterministicpredicted values to represent the statistics of the measuredload and resistance.

Rewriting eq. [10] as R ¼ �Q’Q and substituting eq. [13]

into eq. [3] leads to

½14� � ¼�Q

’�R � �Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

COVR�Q

’�R

� �2

þ ðCOVQ�QÞ2r

For the case of log-normal distributions, eq. [4] becomes

½15� � ¼ln

�Q

’�R

�Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ COV2

QÞ=ð1þ COV2RÞ

qh iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln½ð1þ COV2

QÞð1þ COV2RÞ�

qSee Allen at al. (2005) for additional details on the deri-

vation of eqs. [14] and [15] and the use of bias statistics toestimate the reliability index and the probability of failure.

Statistical treatment of load and resistance bias valuesEquations [11]–[12] demonstrate that, for LRFD calibra-

tion purposes, statistical characterization of the distributionof actual load and resistance values is not used directly inthe approach adopted here. Rather, statistical parameters arecomputed using bias values (e.g., Withiam et al. 1998; Allenet al. 2005). The advantage of using bias values is that vari-ability in predicted load and resistance values resulting fromthe model selected for design is included explicitly in thesubsequent calculation of load and resistance factors. Fur-thermore, if the database of measured load and resistancevalues is taken from actual field measurements (for load)and laboratory measurements (for resistance), then inherentvariability in computed load and resistance terms will becaptured, provided that the data represent typical qualityand type of construction in the field, representative materialcomponents, consistent laboratory techniques, and the siteconditions for the structure being designed.

For brevity in this section, bias values are denoted as X inthe background equations. These values are plotted againstthe standard normal variable (z) for each data point. This isaccomplished in the following steps: (i) sort the bias valuesin the dataset from lowest to highest; (ii) calculate the prob-ability associated with each bias value in the cumulative dis-tribution function (CDF) as pf = i/(n +1), where i is the rankof each data point and n is the total number of points in thedataset; and (iii) calculate z = F–1(pf) using the inversestandard normal cumulative function. The corresponding Ex-cel function NORMSINV is

½16� z ¼ NORMSINV½i=ðnþ 1Þ�

To illustrate the procedure, the resistance data (X = XR)for steel grid reinforcement pullout tests in granular soils re-ported by Christopher et al. (1989) are plotted in Fig. 4. Thedata correspond to predicted and measured values from n =45 tests made up of five different reinforcement products incombination with 15 different granular soils. Clearly, thepredicted value for the resistance term Tpo predicted and thecorresponding resistance bias value will be influenced bythe choice of the characteristic value for the soil unit weightin eq. [5]. The calculations used here are based on valuesreported in the original data source. In general, the nominal(design) predictions used are determined from mean material


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Fig. 4. Cumulative distribution function (CDF) plots of resistance bias (XR) values for pullout capacity for steel grid reinforcement and fittedapproximations: (a) CDF plots with z and XR axes; (b) CDF plots with z and log XR axes.


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property values determined from project-specific measure-ments.

The plotted data are equivalent to a plot made on normalprobability paper (Fig. 4a). An important property of a CDFplot is that normally distributed data will appear as a straightline with a slope equal to 1/�, where � is the standard devi-ation, and the horizontal (bias) axis intercept as the mean(�) of the distribution, hence

½17� X ¼ �þ �z

The theoretical normal distribution based on normal dis-tribution statistics for the entire pullout resistance dataset(� = 1.48, and � = 0.817 or COV = 0.551) is plotted as astraight line in Fig. 4a (curve 1). Clearly, the entire datasetdoes not fit a normal distribution.

In general, the log-normal mean (�ln) and log-normalstandard deviation (�ln) can be calculated from the originalmean and standard deviation of normal statistics as follows(Benjamin and Cornell 1970):

½18� �ln ¼ ln�� 0:5�2ln

½19� �ln ¼ fln½ð�=�Þ2 þ 1�g0:5 ¼ ½ln ðCOV2 þ 1Þ�0:5

Note that ln is the natural logarithm (base e), and the co-efficient of variation COV = �/�. From these parameters,the log-normal distribution of the bias values as a functionof z can be calculated as follows:

½20� X ¼ exp ð�ln þ �ln zÞ

The predicted distribution using eq. [20] and the entiredataset are presented as curve 2 in Fig. 4a. Log-normallydistributed data will plot as a curve on normal probabilitypaper. The dataset in this case is reasonably well approxi-mated by a log-normal distribution. Note that a log-normallydistributed dataset can be made to plot as a straight line in az-bias plot (or normal probability paper) by plotting the nat-ural logarithm of each data point as illustrated in Fig. 4b.

Theoretically, eqs. [18] and [19] should yield the exactlog-normal mean and standard deviation for the dataset.However, these equations were derived for an idealized log-normal distribution, not a sample distribution from actualdata that will likely deviate from an idealized log-normaldistribution. Consequently, good agreement may not be ob-tained for the statistical parameters derived using the theo-retical equations versus determining the mean and standarddeviation directly from the natural logarithm of each datapoint in the distribution, especially if the COV of the datais greater than approximately 20%–30%. This difference isevident in Figs. 4a and 4b, where the log-normal distribu-tions are plotted using both approaches (curves 2 and 3).For the pullout normal statistics provided previously, �lnand �ln determined from eqs. [18] and [19] are 0.262 and0.515, respectively (curve 2). However, if these parametersare calculated directly by taking the mean and standard de-viation from the natural logarithm of all data points, �ln and�ln are equal to 0.273 and 0.480, respectively (curve 3).Nevertheless, normal statistics computed for curve 3 data us-ing log-normal statistics are, for practical purposes, the sameas the normal statistics for curve 2 data as shown in Table 1.

A necessary condition for load and resistance statistics isthat there is no statistical dependence between the bias val-ues and the corresponding nominal (predicted) values ofload or resistance. The bias statistics must be representativeof a random variable (i.e., no nonrandom influences affectthe bias values within the dataset). Phoon and Kulhawy(2003) examined a large database for drilled shafts andgave an example where undesirable dependencies existedbetween the bias and predicted values.

Possible dependency between the bias values and themagnitude of the nominal predictions can be quantified us-ing the Spearman rank correlation coefficient (�) (Walpoleand Myers 1978; Iman and Conover 1989; Phoon and Kul-hawy 2003). The meaning and computation of the Spearmanrank correlation coefficient are described in Appendix A.Tests for dependencies can also be carried out using Ken-dall’s � or Pearson’s correlation coefficient.

Figure 5 shows resistance (pullout) bias values plottedagainst predicted resistance values. A visual dependency be-tween the two parameters appears to be present for the en-tire dataset, with the bias values decreasing with an increasein the resistance value.

Using all n = 45 pullout resistance data points, the Spear-man rank correlation coefficient between XR and Rn (i.e.,Tpo predicted) is � = –0.374, which corresponds to a probabil-ity of p = 0.0065 that the two distributions are independent.Hence, the null hypothesis (the distributions are independ-ent) is rejected and the bias and predicted load values areconsidered correlated at a level of significance of 0.05 (Ap-pendix A).

Visual inspection of the plot suggests that the four datapoints with the highest bias values are the source of depend-ency. Examination of the database of pullout tests revealsthat these data come from the same test series carried outunder low confining pressure (�v < 40 kPa) with dense com-pacted granular soils (compacted unit weight of 14.9 kN/m3)and a friction angle of 458 or more. Each of these test pa-rameters is at the limit of the range of values in the pullouttest database. Visual evidence that these tests can be consid-ered outliers is also apparent in Figs. 4a and 4b where thesame resistance bias values plot at the top right of the fig-ures. Removing these four points and recalculating theSpearman rank correlation coefficient with n = 41 gives� = –0.242, which corresponds to a probability of p = 0.063that the two populations are independent at a level of signif-icance of 0.05 (i.e., the null hypothesis that the two distribu-tions are independent cannot be rejected).

The approximation to the filtered data using n = 41 isshown as curve 5 in Fig. 4. Alternatively, the upper tail ofthe distribution can be ignored, and a visual fit to the lowertail was carried out as done by Allen et al. (2005) for thesame dataset (curve 4). The two curves are practically indis-tinguishable in Fig. 4, showing that both approaches give thesame result.

The removal of these points can also be justified when itis recognized that it is the distribution of resistance bias val-ues in the lower tail of the CDF plot that is important. It isthis region that corresponds to the overlap in the load andresistance distributions originally introduced in this paperand is related to the calculation of probability of failure.

An alternative strategy to avoid possible dependency of


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the bias and nominal prediction values is to divide the datainto intervals of the predicted value as illustrated in Fig. 5.This approach recognizes that the accuracy of the underlyingdeterministic model (i.e., eq. [5]) will vary depending on themagnitude of the input values. For example, it has beennoted that the four data points circled in Fig. 5 correspondto tests with a high friction angle soil conducted under lowconfining pressure. In a related study, the authors found thatloads generated in steel reinforcement systems are difficultto predict accurately for these conditions using current de-sign equations owing to complicated steel reinforcement –soil interaction effects (Bathurst et al. 2008a, 2008b).

In general and regardless of dependency concerns, it maybe appropriate to parse the bias data to reflect that differentprediction models apply to different test materials. In thiscase, separate bias statistics should be calculated for eachmaterial type. An example is given by Nowak and Szerszen

(2003), who computed bias statistics for concrete cylinderstrengths based on nominal (predicted) strength values fordifferent categories of concrete.

Reanalyses using the Spearman rank correlation methodshowed that the three data groups (without removal of anydata points) are independent at a level of significance ofabout 0.05. Cumulative distribution function plots are pre-sented in Fig. 6 for the resistance bias values in each datarange. Approximations to the data point sets are shown inthe same figure using log-normal predictions from normalstatistics. The datasets and the approximation curves inFig. 6 can be seen to move to the left and become steeperwith higher resistance values.

To illustrate this point quantitatively, the mean (�) andCOV of the resistance bias values are plotted against Rn inFig. 7. The plots (bold lines) show that both values decreasewith an increase in Rn value. This demonstrates that the

Table 1. Statistics for resistance bias data using n = 45 and 41 data points and best fit to tail.

Normal statistics Log-normal statistics

Curve numberin Fig. 4

No. of datapoints, n

Fittedrange

Statistics usedto generateapproximationsin Fig. 4 Mean, � COV, �/� Mean, �ln

Standarddeviation, �ln

1 45 All data Normal 1.48 0.5512 45 All data Normal 1.48 0.551 0.262a 0.515a

3 45 All data Log-normal 1.47b 0.509b 0.273 0.4804 45 Lower tail Normal 1.30 0.400 0.188a 0.385a

5 41 All data Normal 1.28 0.397 0.174a 0.383a

aComputed from normal statistics using eqs. [18] and [19].

bComputed from log-normal statistics using expressions � ¼ expð�ln þ 0:5�2lnÞ and COV ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpð�2

lnÞ � 1q

.

Fig. 5. Resistance bias factors (XR) versus predicted (nominal) resistance (Rn) values and grouped data ranges for Rn = 20–40, 40–60, and60–160 kN/m.


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underlying deterministic model expressed by eq. [5] is moreaccurate for test conditions that result in higher pullout loadvalues. The trend may not be unexpected when the historyof the calibration of the coefficient terms for eq. [5] is in-vestigated. The coefficient terms in eq. [5] were selected togive lower-bound estimates of pullout capacity for the entirerange of data available (Christopher et al. 1989). This is typ-ical practice in model calibration when the objective is todevelop conservatively safe allowable, working stress designequations. The result is that for conditions where the modelperforms badly (i.e., where there are fewer data and meas-ured pullout loads are high) the resulting bias factors arehigh. From a practical point of view, the back-fitted coeffi-cient F* used in eq. [5] could be adjusted to better capturethe influence of confining pressure and (or) friction angle onpredicted values. Alternatively, different mean and COVbias values can be assigned to different load ranges. A dis-advantage of this approach is that the number of data pointsis reduced, with the result that the spread (COV) in bias val-ues can become large.

Superimposed on Fig. 7 are the bias statistics using n = 45or 41 data points. Not unexpectedly, these data fall withinthe range of values using different subsets of the total data-set. Resistance bias statistics are summarized in Table 1 forthe entire dataset and in Table 2 for the parsed data.

Reinforcement load data can be analyzed using the sameapproach as that used for the resistance data. Figure 8 showsthe CDF data for load bias values (XQ). The data have beentaken from Allen et al. (2001, 2004), who reported a datasetof 20 well instrumented reinforced soil walls constructedwith bar mat and welded wire steel reinforcement (a total

of 34 data points from six different wall sections constructedwith compacted granular backfill). The measured loads werecalculated using readings from strain gages mounted directlyon reinforcement members. The friction angle used to calcu-late the lateral earth pressure value Kr in eq. [6] was takenas the reported value from triaxial or direct shear tests andcapped at 408 as required in AASHTO (2002, 2007) designcodes. No attempt was made to adjust the value upwards toaccount for plane-strain conditions that typically apply forthese walls (Allen et al. 2001, 2004).

Using the entire dataset, the distribution of data points isreasonably well approximated by log-normal distributionspresented as curves 9 and 10 in Fig. 8. However, it is thedata values in the upper tail of the distribution that are ofinterest, since it is this region that will contribute to the cal-culation of probability of failure for the same reason de-scribed for the distribution of resistance bias values. Anapproximation to the load distribution using a log-normal fitto the upper tail is plotted as curve 11 in Fig. 8 (Allen et al.2005).

Load bias values are plotted against predicted load valuesin Fig. 9 to investigate possible dependency between loadbias and predicted load populations. The Spearman rank cor-relation coefficient with n = 34 gives � = 0.015, which cor-responds to p = 0.534. The two distributions are clearlyuncorrelated, and no further treatment of the load dataset isrequired.

The statistics for load bias data are summarized in Table 3.The values in the table are reasonably similar, which is con-sistent with visual observation of the corresponding approx-imations in Fig. 8.

Fig. 6. Cumulative distribution function (CDF) plots of resistance bias (XR) values for steel grid pullout capacity but with the datasetgrouped into ranges of nominal (predicted) values.


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Estimating the load factorIt is best to have an estimate of the load factor before be-

ginning the final calibration process. There are many combi-nations of load and resistance factors that will yield thedesired probability of failure (pf) or reliability index (�) fora given limit state and set of statistics for the random varia-bles involved. It is desirable to set the load factors so thatthey are greater than 1.0 and the resistance factors so thatthey are less than 1.0 for LRFD as noted earlier. This maynot be possible in some cases, however, depending on howconservative or nonconservative (i.e., how biased) the pre-diction method is for load or resistance.

The following equation can be used as a starting point toestimate the load factor, if load statistics are available:

½21� �Q ¼ �Qð1 þ n�COVQÞ

where �Q is the load factor; �Q is the bias factor (i.e., meanof the bias) for the reinforcement load due to dead load, de-fined as the mean of the ratio of the measured to predictedload; COVQ is the coefficient of variation of the ratio of themeasured load to predicted reinforcement load; and n� is aconstant. For a given value of n�, the probability of exceed-ing any factored load is about the same. The greater the va-

lue of n�, the lower the probability the measured load willexceed the predicted nominal load. A value of n� = 2 forthe strength limit state was used in the development of theCanadian highway bridge design code and AASHTO LRFDbridge design specifications (Nowak 1999; Nowak and Col-lins 2000). This value is used in the example computationsto follow.

Computed load factors using the data in Table 3 with n� =2 range from 1.73 to 1.87. A value of 1.75 is selected here.A visual check on the reasonableness of the selected loadfactor can be carried out by plotting predicted load valuesagainst ‘‘measured’’ values. Figure 10 shows unfactored andfactored (predicted) load values for steel grid reinforced soilwalls, using the AASHTO simplified method, plotted againstmeasured values. Figure 10 shows that many of the originaldata points are below the 1:1 correspondence line. Applyinga load factor �Q = 1.75 moves almost all of the data pointsabove the 1:1 line, which is a desirable end result.

Note that a single set of load factors is often used formultiple limit states. As resistance factors are developed foreach limit state, it may be necessary to make minor adjust-ments to the load factors to ensure that, for a single set ofload factor values, the same target � value is achieved forall limit states.

Fig. 7. Mean and COV of bias values for pullout resistance as a function of the magnitude of the predicted (nominal) resistance.

Table 2. Statistics for resistance bias data using selected measured pullout capacity ranges.

Curve numberin Fig. 6


Pullout range,Rn (kN/m)

Statistics used to generatelog-normal approximationsin Fig. 6 Mean, � COV, �/�

6 13 20–40 Normal 2.07 0.5897 13 40–60 Normal 1.48 0.3068 19 60–160 Normal 1.09 0.286


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Fig. 8. Cumulative distribution function (CDF) plots of load bias (XQ) values for reinforcement loads for steel grid reinforced soil walls andfitted approximations.

Fig. 9. Load bias values versus predicted load values.


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Estimating the resistance factorOnce the load factor is selected, the resistance factor can

be estimated through iteration to produce the desired magni-tude for �, using eqs. [14] and [15] (as applicable), a designpoint method based on the Rackwitz–Fiessler procedure(Rackwitz and Fiessler 1978), or the more adaptable and rig-orous Monte Carlo method (e.g., Allen et al. 2005). Here,the example described earlier in the paper is continued usingeq. [15] rewritten as follows:

½22� ’ ¼�Q

�R

�Q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ COV2

QÞ=ð1þ COV2RÞ

qexp f�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln½ ð1þ COV2

QÞð1þ COV2RÞ�

qg

Equation [22] together with data from Tables 1 and 3have been used to calculate 4 for a target reliability indexvalue of � = 2.33 (probability of failure pf � 1=100) andload factor value of �Q = 1.75. The results of these compu-tations are presented in Table 4. The calculations are pre-sented in two groups: calculations using all data points forthe resistance bias values and those using parsed datasetswith different ranges of predicted pullout capacity.

The computed resistance factors using the larger datasetsrange from 0.57 to 0.67. A value of 4 = 0.61 is calculatedusing statistics for best fit to tail of the resistance bias CDF(curve 4, Table 1) and the upper tail of the load bias values(curve 11, Table 3).

For the parsed datasets, the computed resistance factorsrange from 0.62 to 0.89. Using the best fit to tail for theload data and the middle range of parsed data (i.e., pulloutload predictions in the range of 40–60 kN/m), the computedresistance factor is 0.81.

Typically, it is the pullout capacity of the reinforcementlayers at the top of the wall that controls design in conven-tional practice. Hence, the resistance bias statistics for curve6 in Fig. 6 may be argued to be the best set for design. Ifthis argument is accepted, then the combination of best fitto tail of the load curve (curve 11 in Fig. 8) gives a resist-ance factor of 0.69. However, this resistance dataset is influ-enced by possible outliers owing to high shear strength soilsas discussed earlier. Clearly, judgement is required in the fi-nal selection of bias statistics to be used in calibration exer-cises of the type described in this paper.

Note that the outlier (filtered) data points previously iden-

Table 3. Statistics for load bias data.

Curvenumber inFig. 8


Fittedrange

Statistics used to generatelog-normal approximationsin Fig. 8 Mean, � COV, �/� Mean, �ln

Standarddeviation, �ln

9 34 All data Normal 0.954 0.406 –0.124a 0.391a

10 34 All data Log-normal 0.959b 0.434b –0.128 0.41511 34 Upper tail Normal 0.973 0.462 –0.124a 0.440a

aComputed from normal statistics using eqs. [18] and [19].

bComputed from log-normal statistics using expressions � ¼ expð�ln þ 0:5�2lnÞ and COV ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpð�2

lnÞ � 1q

.

Fig. 10. Predicted loads versus measured loads for steel grid reinforced soil walls using the AASHTO simplified method.


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tified could be removed first. If correlations between thebias and nominal resistance values still exist, the datasetcould still be broken up into ranges of predicted nominal re-sistance to eliminate any unwanted dependencies. However,removing the outliers first could eliminate much of the datafor the lower range subset of nominal resistance values.Hence, this approach was not used in the example here.

It is of practical interest to compare the results of the cali-bration exercise described here with current recommendedload and resistance factors for a steel grid reinforced soilwall. For example, a value of 4 = 0.9 is currently recom-mended by AASHTO (2007). This value is close to the valueof 0.85 computed using the middle range of predicted resist-ance values with �Q = 1.75. However, the current load factorrecommended by AASHTO (2007) is �Q = 1.35. Resistancefactors computed using this value are summarized in Table 5.The largest resistance factor in this table is 4 = 0.69, which isless than 0.9. In fact, the best values can be argued to be in therange of (say) 4 = 0.44–0.63. Hence, the combination of 4 =0.9 and �Q = 1.35 is not consistent with the available steel-gridreinforced soil wall case study data if a target pf = 1/100 isspecified. Using the curve 11 load statistics and resistance sta-tistics for curves 4 and 7 together with 4 = 0.9 and �Q = 1.35gives a pf in the range of about 1/10 to 1/20 (eqs. [2] and [15]).These probability values are unreasonably large, particularlyas pullout failure of these systems seldom if ever occurs inthe field. The reasons for this contradiction and possible ex-planations are discussed in the following section.

Calibration using allowable stress design(ASD)

Conventional practice in North America for the design of

reinforced soil walls is based on allowable stress design(ASD), which assumes a single global factor of safety (FS)expressed as

½23� FS ¼ Rn

�Qni

Combining eqs. [8] and [23] gives the following expres-sion for resistance factor:

½24� ’ ¼ ��iQni

FS �Qni

However, eq. [24] cannot accurately account for variabil-ity in load and resistance terms, nor is there a quantitativeestimate of the probability of failure associated with thevalue of FS assumed. Nevertheless, as a check on the esti-mate of resistance factor value, the equation can be arguedto provide a link to past experience with successful ASDpractice, thus providing a benchmark that reflects manyyears of safe design.

For the pullout limit state for welded wire and bar mat re-inforced walls analyzed previously, the safety factor used inASD practice is FS = 1.5 (AASHTO 2002). If the currentlyprescribed load factor value of �Q = 1.35 recommended byAASHTO for internal wall stability for earth pressure dueto soil self-weight is used, then 4 = 1.35/1.5 = 0.9 (the cur-rent recommended value in AASHTO (2007)). This value islarger than 4 = 0.57–0.67 using statistics for the data plottedin Figs. 4 and 8.

An initial conclusion is that LRFD calibration using thestatistical data presented in this paper results in a more con-servative design with respect to past practice, or the factorof safety used in current ASD for reinforcement pullout

Table 4. Computed resistance factor 4 for � = 2.33 and �Q = 1.75.

Resistance bias (normal statistics)

Load bias (normal statistics)

CDF curve 2 inFig. 4 (m = 1.47,COV = 0.509)

CDF curve 4a inFig. 4 (m = 1.30,COV = 0.400)



CDF curve 8 inFig. 6 (m = 1.09,COV =0.286)

CDF curve 9 in Fig. 8(m = 0.954, COV = 0.406)

0.615 0.666 0.739 0.890 0.677


0.595 0.641 0.718 0.854 0.648

CDF curve 11a in Fig. 8(m = 0.973, COV = 0.462)

0.571 0.612 0.690 0.812 0.616

aBest fit to tail.

Table 5. Computed resistance factor 4 for � = 2.33 and �Q = 1.35.

Resistance bias (normal statistics)

Load bias (normal statistics)


CDF curve 4a inFig. 4 (m = 1.30,COV = 0.400)





0.440 0.474 0.570 0.687 0.522


0.427 0.459 0.554 0.659 0.500

CDF curve 11a in Fig. 8(m = 0.973, COV = 0.462)

0.410 0.440 0.532 0.626 0.475

aBest fit to tail.


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should be increased. However, it is important to note thatthe good performance observed in the past regarding thislimit state could be the result of other conservative practicesthat have contributed to the safety of these walls. For exam-ple, a minimum reinforcement length of 70% of the wallheight or 2.4 m, whichever is greater, is currently specified(AASHTO 2007). These empirical criteria likely result in anexcessive reinforcement length to contain and anchor the ac-tive zone for the internal stability failure model assumed(see Fig. 3a). Conservative selection of soil shear strengthfor design could also contribute to an additional safety mar-gin, as could load sharing between the upper layers of soilreinforcement. In fact, rewriting eq. [24] to isolate FS andusing (say) 4 = 0.60 and �Q = 1.75 gives FS = 1.75/0.60 =2.9. Hence, based on the limited available dataset for rein-forcement pullout for steel-grid reinforced soil walls, itcould be argued that the true FS against pullout in currentASD practice is closer to 3 rather than to 1.5. Clearly, thechoice of which resistance factor to use in LRFD practice isa judgment call that must be made by those who approveLRFD-based design codes and specifications.

Additional considerations for LRFDcalibration

The original bias data plotted in Figs. 4 and 8 have beenshown to deviate from idealized log-normal distributions fit-ted to the entire datasets. It is tempting to remove outliersbased on visual inspection or possibly rigorous statisticaltests. However, this should be done with caution, since it isoften the case that the outliers appear at the tails of the dis-tributions. Removing apparent outliers in the tails of theCDF plots often removes data that should be used to definethe probability of failure for the limit state under examina-tion, i.e., the typically limited number of extreme wall caseswhere loads have been excessively high and (or) resistancehas been excessively low.

It is important that the statistical data used to characterizea given random variable truly represent random processes. Ifnot, the statistics will be erroneous. This is especially impor-tant when attempting to group data together from multiplesources to generate the dataset used to characterize the ran-dom variable in question.

For the resistance data provided in this paper, a possiblereason for the treatment of some data points as outliers isthat these points are from tests carried out at low confiningpressures on soils with a high friction angle. Hence, thesedata points are at or beyond the applicable limits of the pull-out design model. Data that are beyond the limits of the de-sign model used in LRFD calibration will likely introducenonrandom effects.

The other approach demonstrated in this paper to dealwith nonrandom influences in the dataset for calibration pur-poses is to parse the data into ranges of predicted nominalvalues. Nevertheless, judgement must be exercised. Thequantity of data can have a strong effect on the estimationof the statistical parameters (mean value and coefficient ofvariation), depending on the required confidence level. Thehigher the confidence level desired, the larger the numberof data points required. For a given confidence level, the re-quired number of data points can be determined using the

formulae and tables provided in textbooks on statistics (e.g.,Lloyd and Lipow 1982). Additional discussion on the pointsraised here, including the treatment of outliers, can be foundin the guidance document by Allen et al. (2005).

SummaryThis paper has focused on calibration issues for load and

resistance factor design of geotechnical structures with sim-ple soil–structure interaction models. The general approachto determine the resistance factor using closed-form solu-tions has been demonstrated for the ultimate limit state de-scribing pullout of bar mat and welded wire reinforcementin reinforced soil walls. The following are the main pointsof this paper:

(1) Statistical treatment of load and resistance values shouldbe based on bias values of measured to predicted loadand resistance terms to capture the influence of theadopted design models on the values of estimated loadand resistance factors.

(2) Special attention to the distribution of the bias values inthe tails of CDF plots for input data is required.

(3) Removal of outliers in the lower tail and upper tail ofresistance and load bias data, respectively, in CDF plotsshould be undertaken with caution, since it is the data inthese region that controls the estimate of probability offailure.

(4) The best fit to tail technique described by Allen et al.(2005), which is essentially an extension of theRackwitz–Fiessler method, may give the same result asthe removal of outliers. For the example in this paper,both approaches resulted in a good fit to the lower tailof the resistance bias value distribution. However, thismay not be the case for other datasets because of the lo-cation of the outliers in the distribution. This highlightsthe importance of removing any outliers that can causenonrandom influences on statistical characterization ofthe random variables of interest.

(5) Hidden statistical dependencies between bias values andpredicted (nominal) values of load and resistance can ex-ist in calibration datasets. These dependencies can oftenbe detected visually by plotting the bias-predicted valuedata. However, a quantitative check can be carried outby applying the Spearman rank correlation test to thedata. Statistical dependencies may have physical signifi-cance related to the accuracy of the measurements at lowresistance values (or high load values). The underlyingempirical-based design models available today for soil–structure interaction problems have been influenced bythe tendency of their developers to fit a lower (or upper)bound to the data when scatter is large. A possible strat-egy to reduce or eliminate these statistical dependenciesis to parse the dataset into ranges of predicted nominalresistance. Alternatively, specific data points identifiedas outliers can be removed subject to the cautions raisedpreviously.

(6) Load factors that are required to estimate the resistancefactor should be based on statistics for load data applic-able to the type of structure and the design model to beused. For example, load factors that are currently recom-mended in codes for conventional retaining walls using


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conventional earth pressure theory may not be applicableto reinforced soil walls.

(7) Calibration of LRFD against current ASD practice is atbest an interim solution until more data are available forcalibration using simple reliability-based design meth-ods. However, calibration by fitting to ASD can still beuseful as a benchmark to check the results of LRFD ca-librations against successful past practice. Good judg-ment will always be required to decide how much toadjust load and resistance factors based on the results ofrigorous statistical calibration and values of load and re-sistance factors consistent with past successful designpractice. The quality of the statistical data for calibrationpurposes is a key factor in that decision.

Concluding remarksThis paper has demonstrated that load and resistance fac-

tor design (LRFD) calibration as an impediment to accept-ance of LRFD in geotechnical engineering practice shouldnot be an issue once the basic approach is presented in aform that is understandable to the vast majority of geotech-nical engineers who have no formal training in this area.Furthermore, an important outcome of the example calibra-tion exercise used in this paper is that simple models, evenif mechanistically crude, may be adequate for LRFD pro-vided they are calibrated using bias values computed usingthe same models. The result of more accurate deterministicmodels in LRFD computations will be load and resistancefactors that are closer to 1 and smaller coefficient of varia-tion (COV) values. Lastly, future emphasis to promote geo-technical design practice based on LRFD should be on thecollection of data suitable for statistical treatment within areliability-based design framework rather than the develop-ment of more eloquent deterministic models implementedwithin an allowable stress design (ASD) format.

AcknowledgementsThe writers are grateful to Mr. Bing Huang for reviewing

this manuscript and offering a number of useful suggestions.

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Appendix AThe Spearman rank correlation coefficient is a quantita-

tive measure of the strength of a monotonic relationship be-tween two datasets regardless whether or not the relationshipis linear. Consider the distribution of X and corresponding Yvalues (equivalent to data points for XR and Rn predicted in theexample). The Spearman procedure involves ranking in as-cending magnitude the values of X and Y in sequence from1 to n, where n is the number of data points. Hence, thesmallest values of X and Y are assigned the value of 1, andthe largest values in each set a value of n (i.e., the two setsof values are ranked independently). If there are no dupli-cate data points, the ranking of each pair of data points(RXi ; RYi) is then used to calculate the rank correlation coef-ficient � according to

½A1� � ¼ 1�6Xn

1

ðRXi� RYi

Þ2

nðn2 � 1ÞThe following test statistic (an approximation to the Stu-

dent’s t distribution for n £ 18) is used to determine if thenull hypothesis that X and Y are independent is rejected at alevel of significance of a and n–2 degrees of freedom:

½A2� T� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffin� 2

1� �2

s

The following three cases lead to a rejection of the nullhypothesis that the two distributions are independent:

(1) if both distributions increase or decrease together andT� > –t1–a, n–2;

(2) if X increases and Y decreases, or vice versa, and T� <t1–a, n–2;

(3) if there is a monotonic relationship between the two dis-tributions and T� > t1–a/2, n–2 or T� < –t1–a/2, n–2.

As n ? ?, the distribution of � values approaches a nor-mal distribution with a mean of zero and a standard devia-tion of 1=

ffiffiffiffiffiffiffiffiffiffiffin� 1

p. Hence,

½A3� z ¼ �� 0

1=ffiffiffiffiffiffiffiffiffiffiffin� 1

pThe probability associated with z can be computed as

½A4� p ¼ �ðzÞ ¼ NORMSDISTðzÞ

If p < a, then the null hypothesis is rejected at a level ofsignificance of a in a one-tailed test.


# 2008 NRC Canada

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Calibration concepts for load and resistance factor design (LRFD) of