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

Transcription

1 1377 Calibration concepts for load and resistance factor 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 states design (LSD) in Canada) are well known, and their adoption in geotechnical engineering design is now recommended for many soil structure interaction problems. Two important challenges for acceptance of LRFD for the design of reinforced soil 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 principles and 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 this paper is that measured load and resistance values from a database of case histories are used to develop the statistical parameters in the examples. The paper also addresses issues related to the influence of outliers in the datasets and possible dependencies 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 solutions, reinforced soil walls, steel reinforcement, pullout. Résumé : On connaît bien les concepts de conception basés sur la fiabilité et leur application au facteur de conception de charge et de résistance (LRFD ou la conception aux états limites au Canada), et on recommande maintenant leur adoption en conception d ingéniérie géotechnique pour plusieurs problèmes d interaction sol-structure. Deux importants défis pour l acceptation du LRFD pour le calcul de murs en sol armé sont : (i) une bonne compréhension des méthodes de calibrage utilisées pour arriver aux facteurs de charge et de résistance, et (ii) la bonne interprétation des données requises pour réaliser ce processus. Cet article présente les principes de calibrage du LRFD et trace les étapes requises pour arriver aux facteurs de charge et de résistance en utilisant des solutions exactes pour un état limite typique d arrachement d éléments d armature en acier dans la zone d ancrage d un mur en sol armé. Une caractéristique unique de cet article est que des valeurs de charges et de résistances mesurées provenant d une base de données d histoires de cas sont utilisées pour développer les paramètres statistiques dans les exemples. Cet article traite aussi des problèmes reliés àl influence des données étrangères dans les ensembles de donnée et des dépendances possibles rentre les variables qui peuvent avoir une influence importante sur les résultats du calibrage. Mots-clés :facteurs de conception de charge et de résistance «LRFD» conception d état limite «LSD», conception basée sur la fiabiliité, solutions exactes, murs en sol armé, armature d acier, arrachement. [Traduit par la Rédaction] Introduction Use of reliability theory to determine load and resistance factors for limit states design of civil engineering structures (called load and resistance factor design, or LRFD in the USA, and limit states design, or LSD in Canada) is now well developed, and its adoption in geotechnical engineering design is now recommended for many soil structure interaction problems. The fundamentals of reliability theory, as Received 21 June Accepted 5 June Published on the NRC Research Press Web site at cgj.nrc.ca on 26 September R.J. Bathurst. 1 GeoEngineering Centre at Queen s-rmc, Department of Civil Engineering, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada. T.M. Allen. State Materials Laboratory, Washington State Department of Transportation, Olympia, WA , USA. A.S. Nowak. Department of Civil Engineering, University of Nebraska, W181 Nebraska Hall, Lincoln, NE , USA. 1 Corresponding author ( Bathurst-r@rmc.ca). they apply to structural design, can be found in the textbook by Nowak and Collins (2000). An excellent overview of the development of reliability-based design as it applies to geotechnical engineering has been provided by Kulhawy and Phoon (2002). Goble (1999) and Becker (1996a, 1996b) also have provided a useful background to the development of limit states design practice for foundation engineering in North America. The American Association of State Highway and Transportation Officials (AASHTO) has committed to the LRFD approach for all structures, including reinforced soil walls (AASHTO 2007). Currently, an allowable (working) stress design (ASD or WSD) method is used for reinforced soil walls (AASHTO 2002). In Canada, the Canadian foundation engineering manual (Canadian Geotechnical Society 2006) recommends an LRFD approach using the North American factored resistance approach. The current Canadian highway bridge design code (CSA 2006) will also be updated to include reinforced soil walls within an LRFD framework. For geotechnical engineers, the implementation of LRFD is understood using prescribed limit state equations and load Can. Geotech. J. 45: (2008) doi: /t08-063

2 1378 Can. Geotech. J. Vol. 45, 2008 and resistance factors taken from tables in design codes (NRCC 2005; CSA 2006; AASHTO 2007). However, the recasting of retaining wall allowable stress design within an LRFD framework has been problematic due to a lack of statistical data suitable for probabilistic analysis of load and resistance parameters. Furthermore, both the load and resistance (or strength) components in limit state formulations include soil unit weight and (or) strength as input parameters that also have inherent variability. For reinforced soil walls, the soil reinforcement elements add an additional level of complication. Calibration for reinforced soil retaining wall design (called mechanically stabilized earth walls in USA terminology) has until now been restricted to comparison with ASD practice (often called calibration by fitting). There remain two important challenges for acceptance of LRFD for the design of reinforced soil walls by geotechnical engineers: (i) an understanding of the calibration methods used to arrive at load and resistance factors, and (ii) the proper interpretation of the data required for calibration. This paper addresses these two challenges, which have not received the level of attention in the literature that is required to give geotechnical engineers (i) confidence in the general LRFD approach, (ii) confidence that load and resistance factors found in design codes are reasonable, and (iii) the tools to carry out the calibration themselves using the current North American approach to geotechnical LRFD. This paper illustrates LRFD calibration using an example for the limit state associated with steel reinforcement elements in the anchorage zone of a reinforced soil wall. The example is based on material presented in a larger LRFD calibration guidance document by Allen et al. (2005). For brevity and to emphasize important issues, the example is restricted to calibration using closed-form solutions. A unique feature of this paper is that actual data for load and resistance terms are available (Allen et al. 2001, 2004). However, a number of important points regarding the selection and statistical treatment of the data are presented in the current paper to qualify the general approach proposed by Allen et al. (2005). The more adaptable Monte Carlo approach to estimate resistance factors is not reviewed here, but points made regarding statistical treatment and interpretation of load and resistance data are equally applicable for this approach. Background It is necessary to provide a brief background on the concepts behind LRFD in North American practice to understand the treatment of data presented later in the paper. The basic concept behind LRFD in North American practice is illustrated in Fig. 1a. Here, uncorrelated distributions of random load (Q) and resistance (R) values are shown as normal (Gaussian) frequency distributions. For a prescribed limit state (g), expressed as ½1Š g ¼ R Q the probability of failure is equal to the area of the frequency distribution in Fig. 1b for which g < 0, where g is calculated using random values of R and Q. Clearly, as the mean value (u) of the distribution for g moves to the right in Fig. 1b, the probability of failure (p f ) becomes less. For a normal distribution of g values, the probability of failure can be equated explicitly to the value of the reliability index = u/, where is the standard deviation of g. The nonlinear relationship between probability of failure and reliability index for a normal distribution of g is shown in Fig. 2. For example, a probability of failure of 1 in 1000 corresponds to a reliability index = The calculation of p f can be easily carried out using the standard normal cumulative function (NORMSDIST) in Microsoft Excel: ½2Š p f ¼ 1 NORMSDIST ðþ If load and resistance values are normally distributed and the limit state function is linear (e.g., eq. [1]), then is the reciprocal of the coefficient of variation (COV) of g = R Q and can be calculated exactly as follows (e.g., Nowak and Collins 2000): ½3Š ¼ q R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 2 R þ 2 Q where R is the mean value of the resistance R, Q is the mean value of the load Q, R is the standard deviation for the resistance R, and Q is the standard deviation for the load Q. If distributions for R and Q deviate from normal distributions, then eq. [3] is only an approximation. If both the load and resistance distributions are lognormal and the limit state function is a product of random variables, then can be calculated using a closed-form solution 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: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½4Š ¼ ln½r=q q ð1 þ COV 2 Q Þ=ð1 þ COV2 R Þ Š ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln½ð1 þ COV 2 Q Þð1 þ COV2 R ÞŠ where COV R and COV Q are the coefficients of variation for the resistance and load values, respectively. Expanded versions of these expressions for LRFD calibration purposes are presented later in the paper. Example limit state for reinforced soil walls The first step in LRFD design is to identify each possible ultimate or serviceability limit state and to express it using a deterministic function as in conventional allowable stress design. For example, consider the ultimate limit state against pullout of steel grid reinforcement (steel bar mat and welded wire) as illustrated in Fig. 3. The resistance term R using AASHTO (2002, 2007) is ½5Š R ¼ T po ¼ 2L e v F ¼ 2L e s zf where T po is the pullout capacity (kn/m), L e is the anchorage length (m), v = s z is the vertical stress (kpa), s is the soil unit weight (kn/m 3 ), and F* is the dimensionless pullout resistance factor (in this case, F* is a function of the thickness and horizontal spacing of the reinforcement transverse bars and depth z of the reinforcement). The anchorage length is illustrated in Fig. 3a. The load for the limit state calculation in this example is assumed to be due to the self-weight of the wall backfill

3 Bathurst et al 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 reliability index for normal distributions. (i.e., no live load or other types of loads). The load is calculated using the AASHTO simplified method: ½6Š Q ¼ T max ¼ S v v K r where T max is the maximum tensile load in the reinforcement (kn/m), S v is the tributary spacing of the reinforcement layer (m), and K r is the dimensionless lateral earth pressure coefficient acting at the reinforcement layer depth. For steel grid reinforced soil walls, K r varies from 2.5K a to 1.2K a at the top of the wall to a depth of 6 m below the wall top, respectively, and remains at 1.2K a below 6 m (Fig. 3b), where K a = f() is the dimensionless coefficient of active lateral 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 for predicted load values that are described later in the paper. For the same reinforcement layer in a very large sample of nominally identical walls, we expect that the same equations would yield a range of computed values of R and Q due to variability in the values of s and and accuracy of the measurements taken to back-calculate F* in laboratory pullout tests. Variability in the magnitude of soil-related input parameters is inherent to these materials in field construction and laboratory pullout tests. In the field, additional variability will result from quality of construction and a range of different equipment employed. All other input parameters can be taken as deterministic (i.e., assigned constant values) for practical LRFD calibration and design. Adopting the limit state format in eq. [1], the ultimate limit state for reinforcement pullout is expressed as ½7Š g ¼ T po T max ¼ 2L e v F S v v K r The probability of T max exceeding T po must be kept below an acceptable level. Hence, for design the values of T po and T max must be decreased and increased, respectively, by multiplying them by suitably selected resistance and load factors to achieve an acceptable probability of failure, where failure is defined as g 0. In North America, LRFD (or LSD in Canada) is based on a factored resistance approach. The general approach can be expressed as ½8Š i Q ni R n where Q ni is the nominal (specified) load, R n is the nominal (characteristic) resistance, i is the load factor, and 4 is the resistance factor. Here, to simplify the general approach and be consistent in the calculation of nominal load and resistance values, we assume that mean values of soil unit weight and strength in eqs. [5] and [6] will be used. In the current North American approach, uncertainty in the calculation of the resistance side of the equation is captured by a single resistance factor while load contributions are assigned (typically) different load factors. Load factor terms have values i 1, and the resistance term should have a value 4 1. Using eqs. [1] and [8], the design limit state equation for reinforcement pullout with a single load term can be expressed as ½9Š Q T max T po 0 where the load factor notation Q is now adopted for the

4 1380 Can. Geotech. J. Vol. 45, 2008 Fig. 3. AASHTO (2002, 2007) simplified method for steel bar mats and welded wire reinforcement pullout: (a) pullout model geometry and loads; (b) coefficient of lateral earth pressure. H, height of wall. case of a single load term in the limit state function. From eq. [9], the minimum pullout capacity is ½10Š T po ¼ Q T max In other words, the nominal resistance value (R n = T po ) must always be greater than the nominal load value (Q n = T max ) by a factor of Q /4. LRFD design versus LRFD calibration The objective of LRFD design is to ensure that eq. [8] is satisfied (or in the example problem, eq. [9]). In design, this is achieved by adjusting the values in the model eqs. [5] and [6] (for example) and (or) design geometry (e.g., wall height and number of reinforcement layers) to satisfy eq. [8] using prescribed values for the load and resistance factors. The objective of LRFD calibration is to select values of load and resistance factors to be used in design such that the computed probability of failure is below the target probability of failure. This objective can be understood as the selection of factors to be applied to load and resistance terms such that the idealized normal distributions illustrated in Fig. 1a are far enough apart and the area of the distribution function for g < 0 in Fig. 1b does not exceed a prescribed value. At the time of writing this paper, load and resistance factors for internal stability design of reinforced soil walls using a factored resistance approach are based on calibration by fitting to ASD (D Appolonia Engineering 1999; AASHTO 2007) (see the section titled Calibration using allowable stress design (ASD)). A description of a more rigorous approach for the calibration of the North American LRFD method for internal stability of reinforced soil wall structures is the main objective of this paper. Selection of probability of failure for internal stability of reinforced soil walls In general, resistance factors for the design of building and bridge structural components have been derived to produce a probability of failure (p f ) of about 1 in 5000 (this corresponds to = 3.54). However, past geotechnical design practice has resulted in an effective probability of failure for foundations, in general, of approximately 1 in 1000 (Withiam et al. 1998). For reinforced soil walls, the multiple layers of horizontal reinforcement result in highly strengthredundant systems, and the failure or overstress of a single reinforcement layer or strip will not result in failure of the wall. Hence, the internal stability of the reinforced wall as a system has much greater reliability (i.e., system reliability) and much lower probability of failure than that of an individual reinforcement layer or element. Furthermore, the reinforced soil is a deformable medium that helps redistribute loads through the composite structure (Allen et al. 2001). A useful comparison is a piled foundation, where failure of a single pile will result in load shedding to neighbouring piles (Zhang et al. 2001). For piled foundations, D Appolonia Engineering (1999) and Paikowsky (2004) have recommended p f = 1 in 100 ( = 2.33). In the calculation examples to follow, a target probability of 1 in 100 is assumed to be reasonable for LRFD calibration of internal stability limit states for reinforced soil walls for the reasons noted. Lastly, it should be noted that AASHTO (2007) recommends a 75 year design life for all reinforced soil walls. Additional discussion regarding the choice of for reinforced soil wall design is provided by Allen et al. (2005). Analysis of load and resistance data Model bias Ideally, the estimate of the probability of failure using eqs. [3] and [4] for the example limit state should be based on measurements of reinforcement load and pullout resist-

5 Bathurst et al ance for a large number of nominally similar structures. Statistical parameters needed to estimate the probability of failure can be taken from these measurements. To relate the measured values of load and resistance from multiple locations within a given structure or from multiple case histories, a baseline of comparison is needed. The nominal prediction of load and resistance that corresponds to each measured load or resistance value can be used as this baseline. It is reasonable to expect that there will be a difference between a predicted value and the corresponding measured value due to model error, as well as variability in material properties. This difference can be expressed as a bias value defined as the ratio of the measured to predicted (nominal) value. Bias values are sometimes called model factors in the literature (e.g., Phoon and Kulhawy 2003, 2005). Conceptually, each measured load and resistance value can be divided by the appropriate predicted (nominal) load and resistance value to arrive at the distribution for nondimensional load and resistance terms in the calculation of probability of failure (Withiam et al. 1998; Allen et al. 2005). Continuing the example in the section titled Example limit state for reinforced soil walls, the resistance bias value X R for a pullout resistance data point can be expressed as ½11Š X R ¼ R measured R n predicted ¼ T po measured T po predicted and the load bias value X Q as ½12Š X Q ¼ Q measured Q n predicted ¼ T max measured T max predicted A bias value of 1.0 means that the model predicts the measured load or resistance exactly for the data point. This can be expected to be an unusual occurrence in geotechnical practice. The requirement that there be no dependencies between 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] can be replaced by ½13aŠ ½13bŠ ½13cŠ ½13dŠ R ¼ R measured ¼ R n predicted R Q ¼ Q measured ¼ Q n predicted Q R ¼ COV R R measured Q ¼ COV Q Q measured where R measured and Q measured are the means of the measured resistance and load values, respectively; R and COV R are the mean and coefficient of variation of the resistance bias values, respectively; and Q and COV Q are the mean and coefficient of variation of the load bias values, respectively. These equations enable the bias statistics to be used directly in eqs. [3] and [4]. Specifically, the mean and standard deviation of the bias values are scaled by the deterministic predicted values to represent the statistics of the measured load and resistance. Rewriting eq. [10] as R ¼ Q Q and substituting eq. [13] into eq. [3] leads to ½14Š Q ¼ R Q rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 COV Q R R þðcovq Q Þ 2 For the case of log-normal distributions, eq. [4] becomes h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii ln Q R ð1 þ COV 2 Q Q Þ=ð1 þ COV2 R Þ ½15Š ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln½ð1 þ COV 2 Q Þð1 þ COV2 R ÞŠ See Allen at al. (2005) for additional details on the derivation of eqs. [14] and [15] and the use of bias statistics to estimate the reliability index and the probability of failure. Statistical treatment of load and resistance bias values Equations [11] [12] demonstrate that, for LRFD calibration purposes, statistical characterization of the distribution of actual load and resistance values is not used directly in the approach adopted here. Rather, statistical parameters are computed using bias values (e.g., Withiam et al. 1998; Allen et al. 2005). The advantage of using bias values is that variability in predicted load and resistance values resulting from the model selected for design is included explicitly in the subsequent calculation of load and resistance factors. Furthermore, if the database of measured load and resistance values is taken from actual field measurements (for load) and laboratory measurements (for resistance), then inherent variability in computed load and resistance terms will be captured, provided that the data represent typical quality and type of construction in the field, representative material components, consistent laboratory techniques, and the site conditions for the structure being designed. For brevity in this section, bias values are denoted as X in the background equations. These values are plotted against the standard normal variable (z) for each data point. This is accomplished in the following steps: (i) sort the bias values in the dataset from lowest to highest; (ii) calculate the probability associated with each bias value in the cumulative distribution function (CDF) as p f = i/(n +1), where i is the rank of each data point and n is the total number of points in the dataset; and (iii) calculate z = F 1 (p f ) using the inverse standard normal cumulative function. The corresponding Excel function NORMSINV is ½16Š z ¼ NORMSINV½i=ðn þ 1ÞŠ To illustrate the procedure, the resistance data (X = X R ) for steel grid reinforcement pullout tests in granular soils reported by Christopher et al. (1989) are plotted in Fig. 4. The data correspond to predicted and measured values from n = 45 tests made up of five different reinforcement products in combination with 15 different granular soils. Clearly, the predicted value for the resistance term T po predicted and the corresponding resistance bias value will be influenced by the choice of the characteristic value for the soil unit weight in eq. [5]. The calculations used here are based on values reported in the original data source. In general, the nominal (design) predictions used are determined from mean material

6 1382 Can. Geotech. J. Vol. 45, 2008 Fig. 4. Cumulative distribution function (CDF) plots of resistance bias (X R ) values for pullout capacity for steel grid reinforcement and fitted approximations: (a) CDF plots with z and X R axes; (b) CDF plots with z and log X R axes.

7 Bathurst et al property values determined from project-specific measurements. The plotted data are equivalent to a plot made on normal probability paper (Fig. 4a). An important property of a CDF plot is that normally distributed data will appear as a straight line with a slope equal to 1/, where is the standard deviation, and the horizontal (bias) axis intercept as the mean () of the distribution, hence ½17Š X ¼ þ z The theoretical normal distribution based on normal distribution statistics for the entire pullout resistance dataset ( = 1.48, and = or COV = 0.551) is plotted as a straight line in Fig. 4a (curve 1). Clearly, the entire dataset does not fit a normal distribution. In general, the log-normal mean ( ln ) and log-normal standard deviation ( ln ) can be calculated from the original mean and standard deviation of normal statistics as follows (Benjamin and Cornell 1970): ½18Š ln ¼ ln 0:5 2 ln ½19Š ln ¼fln½ð=Þ 2 þ 1Šg 0:5 ¼½ln ðcov 2 þ 1ÞŠ 0:5 Note that ln is the natural logarithm (base e), and the coefficient of variation COV = /. From these parameters, the log-normal distribution of the bias values as a function of z can be calculated as follows: ½20Š X ¼ exp ð ln þ ln zþ The predicted distribution using eq. [20] and the entire dataset are presented as curve 2 in Fig. 4a. Log-normally distributed data will plot as a curve on normal probability paper. The dataset in this case is reasonably well approximated by a log-normal distribution. Note that a log-normally distributed dataset can be made to plot as a straight line in a z-bias plot (or normal probability paper) by plotting the natural logarithm of each data point as illustrated in Fig. 4b. Theoretically, eqs. [18] and [19] should yield the exact log-normal mean and standard deviation for the dataset. However, these equations were derived for an idealized lognormal distribution, not a sample distribution from actual data that will likely deviate from an idealized log-normal distribution. Consequently, good agreement may not be obtained for the statistical parameters derived using the theoretical equations versus determining the mean and standard deviation directly from the natural logarithm of each data point in the distribution, especially if the COV of the data is greater than approximately 20% 30%. This difference is evident in Figs. 4a and 4b, where the log-normal distributions are plotted using both approaches (curves 2 and 3). For the pullout normal statistics provided previously, ln and ln determined from eqs. [18] and [19] are and 0.515, respectively (curve 2). However, if these parameters are calculated directly by taking the mean and standard deviation from the natural logarithm of all data points, ln and ln are equal to and 0.480, respectively (curve 3). Nevertheless, normal statistics computed for curve 3 data using log-normal statistics are, for practical purposes, the same as the normal statistics for curve 2 data as shown in Table 1. A necessary condition for load and resistance statistics is that there is no statistical dependence between the bias values and the corresponding nominal (predicted) values of load or resistance. The bias statistics must be representative of a random variable (i.e., no nonrandom influences affect the bias values within the dataset). Phoon and Kulhawy (2003) examined a large database for drilled shafts and gave an example where undesirable dependencies existed between the bias and predicted values. Possible dependency between the bias values and the magnitude of the nominal predictions can be quantified using the Spearman rank correlation coefficient () (Walpole and Myers 1978; Iman and Conover 1989; Phoon and Kulhawy 2003). The meaning and computation of the Spearman rank correlation coefficient are described in Appendix A. Tests for dependencies can also be carried out using Kendall s or Pearson s correlation coefficient. Figure 5 shows resistance (pullout) bias values plotted against predicted resistance values. A visual dependency between the two parameters appears to be present for the entire dataset, with the bias values decreasing with an increase in the resistance value. Using all n = 45 pullout resistance data points, the Spearman rank correlation coefficient between X R and R n (i.e., T po predicted )is = 0.374, which corresponds to a probability of p = that the two distributions are independent. Hence, the null hypothesis (the distributions are independent) is rejected and the bias and predicted load values are considered correlated at a level of significance of 0.05 (Appendix A). Visual inspection of the plot suggests that the four data points with the highest bias values are the source of dependency. Examination of the database of pullout tests reveals that these data come from the same test series carried out under low confining pressure ( v < 40 kpa) with dense compacted granular soils (compacted unit weight of 14.9 kn/m 3 ) and a friction angle of 458 or more. Each of these test parameters is at the limit of the range of values in the pullout test database. Visual evidence that these tests can be considered outliers is also apparent in Figs. 4a and 4b where the same resistance bias values plot at the top right of the figures. Removing these four points and recalculating the Spearman rank correlation coefficient with n = 41 gives = 0.242, which corresponds to a probability of p = that the two populations are independent at a level of significance of 0.05 (i.e., the null hypothesis that the two distributions are independent cannot be rejected). The approximation to the filtered data using n = 41 is shown as curve 5 in Fig. 4. Alternatively, the upper tail of the distribution can be ignored, and a visual fit to the lower tail was carried out as done by Allen et al. (2005) for the same dataset (curve 4). The two curves are practically indistinguishable in Fig. 4, showing that both approaches give the same result. The removal of these points can also be justified when it is recognized that it is the distribution of resistance bias values in the lower tail of the CDF plot that is important. It is this region that corresponds to the overlap in the load and resistance distributions originally introduced in this paper and is related to the calculation of probability of failure. An alternative strategy to avoid possible dependency of

8 1384 Can. Geotech. J. Vol. 45, 2008 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 number in Fig. 4 No. of data points, n Fitted range Statistics used to generate approximations in Fig. 4 Mean, COV, / Mean, ln Standard deviation, ln 1 45 All data Normal All data Normal a a 3 45 All data Log-normal 1.47 b b Lower tail Normal a a 5 41 All data Normal a a a Computed from normal statistics using eqs. [18] and [19]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b Computed from log-normal statistics using expressions ¼ expð ln þ 0:5 2 lnþ and COV ¼ expð 2 ln Þ 1. Fig. 5. Resistance bias factors (X R ) versus predicted (nominal) resistance (R n ) values and grouped data ranges for R n = 20 40, 40 60, and kn/m. the bias and nominal prediction values is to divide the data into intervals of the predicted value as illustrated in Fig. 5. This approach recognizes that the accuracy of the underlying deterministic model (i.e., eq. [5]) will vary depending on the magnitude of the input values. For example, it has been noted that the four data points circled in Fig. 5 correspond to tests with a high friction angle soil conducted under low confining pressure. In a related study, the authors found that loads generated in steel reinforcement systems are difficult to predict accurately for these conditions using current design equations owing to complicated steel reinforcement soil interaction effects (Bathurst et al. 2008a, 2008b). In general and regardless of dependency concerns, it may be appropriate to parse the bias data to reflect that different prediction models apply to different test materials. In this case, separate bias statistics should be calculated for each material type. An example is given by Nowak and Szerszen (2003), who computed bias statistics for concrete cylinder strengths based on nominal (predicted) strength values for different categories of concrete. Reanalyses using the Spearman rank correlation method showed that the three data groups (without removal of any data points) are independent at a level of significance of about Cumulative distribution function plots are presented in Fig. 6 for the resistance bias values in each data range. Approximations to the data point sets are shown in the same figure using log-normal predictions from normal statistics. The datasets and the approximation curves in Fig. 6 can be seen to move to the left and become steeper with higher resistance values. To illustrate this point quantitatively, the mean () and COV of the resistance bias values are plotted against R n in Fig. 7. The plots (bold lines) show that both values decrease with an increase in R n value. This demonstrates that the

9 Bathurst et al Fig. 6. Cumulative distribution function (CDF) plots of resistance bias (X R ) values for steel grid pullout capacity but with the dataset grouped into ranges of nominal (predicted) values. underlying deterministic model expressed by eq. [5] is more accurate for test conditions that result in higher pullout load values. The trend may not be unexpected when the history of the calibration of the coefficient terms for eq. [5] is investigated. The coefficient terms in eq. [5] were selected to give lower-bound estimates of pullout capacity for the entire range of data available (Christopher et al. 1989). This is typical practice in model calibration when the objective is to develop conservatively safe allowable, working stress design equations. The result is that for conditions where the model performs badly (i.e., where there are fewer data and measured pullout loads are high) the resulting bias factors are high. From a practical point of view, the back-fitted coefficient F* used in eq. [5] could be adjusted to better capture the influence of confining pressure and (or) friction angle on predicted values. Alternatively, different mean and COV bias values can be assigned to different load ranges. A disadvantage of this approach is that the number of data points is reduced, with the result that the spread (COV) in bias values can become large. Superimposed on Fig. 7 are the bias statistics using n =45 or 41 data points. Not unexpectedly, these data fall within the range of values using different subsets of the total dataset. Resistance bias statistics are summarized in Table 1 for the entire dataset and in Table 2 for the parsed data. Reinforcement load data can be analyzed using the same approach as that used for the resistance data. Figure 8 shows the CDF data for load bias values (X Q ). The data have been taken from Allen et al. (2001, 2004), who reported a dataset of 20 well instrumented reinforced soil walls constructed with bar mat and welded wire steel reinforcement (a total of 34 data points from six different wall sections constructed with compacted granular backfill). The measured loads were calculated using readings from strain gages mounted directly on reinforcement members. The friction angle used to calculate the lateral earth pressure value K r in eq. [6] was taken as the reported value from triaxial or direct shear tests and capped at 408 as required in AASHTO (2002, 2007) design codes. No attempt was made to adjust the value upwards to account for plane-strain conditions that typically apply for these walls (Allen et al. 2001, 2004). Using the entire dataset, the distribution of data points is reasonably well approximated by log-normal distributions presented as curves 9 and 10 in Fig. 8. However, it is the data values in the upper tail of the distribution that are of interest, since it is this region that will contribute to the calculation of probability of failure for the same reason described for the distribution of resistance bias values. An approximation to the load distribution using a log-normal fit to the upper tail is plotted as curve 11 in Fig. 8 (Allen et al. 2005). Load bias values are plotted against predicted load values in Fig. 9 to investigate possible dependency between load bias and predicted load populations. The Spearman rank correlation coefficient with n = 34 gives = 0.015, which corresponds to p = The two distributions are clearly uncorrelated, and no further treatment of the load dataset is required. The statistics for load bias data are summarized in Table 3. The values in the table are reasonably similar, which is consistent with visual observation of the corresponding approximations in Fig. 8.

10 1386 Can. Geotech. J. Vol. 45, 2008 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 number in Fig. 6 No. of data points, n Pullout range, R n (kn/m) Statistics used to generate log-normal approximations in Fig. 6 Mean, COV, / Normal Normal Normal Estimating the load factor It is best to have an estimate of the load factor before beginning the final calibration process. There are many combinations of load and resistance factors that will yield the desired probability of failure (p f ) or reliability index () for a given limit state and set of statistics for the random variables involved. It is desirable to set the load factors so that they are greater than 1.0 and the resistance factors so that they are less than 1.0 for LRFD as noted earlier. This may not be possible in some cases, however, depending on how conservative or nonconservative (i.e., how biased) the prediction method is for load or resistance. The following equation can be used as a starting point to estimate the load factor, if load statistics are available: ½21Š Q ¼ Q ð1 þ n COV Q Þ where Q is the load factor; Q is the bias factor (i.e., mean of the bias) for the reinforcement load due to dead load, defined as the mean of the ratio of the measured to predicted load; COV Q is the coefficient of variation of the ratio of the measured load to predicted reinforcement load; and n is a constant. For a given value of n, the probability of exceeding any factored load is about the same. The greater the value of n, the lower the probability the measured load will exceed the predicted nominal load. A value of n = 2 for the strength limit state was used in the development of the Canadian highway bridge design code and AASHTO LRFD bridge design specifications (Nowak 1999; Nowak and Collins 2000). This value is used in the example computations to follow. Computed load factors using the data in Table 3 with n = 2 range from 1.73 to A value of 1.75 is selected here. A visual check on the reasonableness of the selected load factor can be carried out by plotting predicted load values against measured values. Figure 10 shows unfactored and factored (predicted) load values for steel grid reinforced soil walls, using the AASHTO simplified method, plotted against measured values. Figure 10 shows that many of the original data points are below the 1:1 correspondence line. Applying a load factor Q = 1.75 moves almost all of the data points above the 1:1 line, which is a desirable end result. Note that a single set of load factors is often used for multiple limit states. As resistance factors are developed for each limit state, it may be necessary to make minor adjustments to the load factors to ensure that, for a single set of load factor values, the same target value is achieved for all limit states.

11 Bathurst et al Fig. 8. Cumulative distribution function (CDF) plots of load bias (X Q ) values for reinforcement loads for steel grid reinforced soil walls and fitted approximations. Fig. 9. Load bias values versus predicted load values.

12 1388 Can. Geotech. J. Vol. 45, 2008 Table 3. Statistics for load bias data. Curve number in Fig. 8 Statistics used to generate log-normal approximations Standard in Fig. 8 Mean, COV, / Mean, ln deviation, ln No. of data points, n Fitted range 9 34 All data Normal a a All data Log-normal b b Upper tail Normal a a a Computed from normal statistics using eqs. [18] and [19]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b Computed from log-normal statistics using expressions ¼ expð ln þ 0:5 2 lnþ and COV ¼ expð 2 ln Þ 1. Fig. 10. Predicted loads versus measured loads for steel grid reinforced soil walls using the AASHTO simplified method. Estimating the resistance factor Once the load factor is selected, the resistance factor can be estimated through iteration to produce the desired magnitude for, using eqs. [14] and [15] (as applicable), a design point method based on the Rackwitz Fiessler procedure (Rackwitz and Fiessler 1978), or the more adaptable and rigorous Monte Carlo method (e.g., Allen et al. 2005). Here, the example described earlier in the paper is continued using eq. [15] rewritten as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R Q ð1 þ COV 2 ½22Š ¼ Q Q Þ=ð1 þ COV2 R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ exp f ln½ ð1 þ COV 2 Q Þð1 þ COV2 R ÞŠ g Equation [22] together with data from Tables 1 and 3 have been used to calculate 4 for a target reliability index value of = 2.33 (probability of failure p f 1=100) and load factor value of Q = The results of these computations are presented in Table 4. The calculations are presented in two groups: calculations using all data points for the resistance bias values and those using parsed datasets with different ranges of predicted pullout capacity. The computed resistance factors using the larger datasets range from 0.57 to A value of 4 = 0.61 is calculated using 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 factors range from 0.62 to Using the best fit to tail for the load data and the middle range of parsed data (i.e., pullout load predictions in the range of kn/m), the computed resistance factor is Typically, it is the pullout capacity of the reinforcement layers at the top of the wall that controls design in conventional practice. Hence, the resistance bias statistics for curve 6 in Fig. 6 may be argued to be the best set for design. If this argument is accepted, then the combination of best fit to tail of the load curve (curve 11 in Fig. 8) gives a resistance factor of However, this resistance dataset is influenced by possible outliers owing to high shear strength soils as discussed earlier. Clearly, judgement is required in the final selection of bias statistics to be used in calibration exercises of the type described in this paper. Note that the outlier (filtered) data points previously iden-

13 Bathurst et al Table 4. Computed resistance factor 4 for = 2.33 and Q = Load bias (normal statistics) CDF curve 9 in Fig. 8 (m = 0.954, COV = 0.406) CDF curve 10 in Fig. 8 (m = 0.959, COV = 0.434) CDF curve 11 a in Fig. 8 (m = 0.973, COV = 0.462) Resistance bias (normal statistics) CDF curve 2 in Fig. 4 (m = 1.47, COV = 0.509) CDF curve 4 a in Fig. 4 (m = 1.30, COV = 0.400) CDF curve 6 in Fig. 6 (m = 2.07, COV = 0.589) CDF curve 7 in Fig. 6 (m = 1.48, COV = 0.306) CDF curve 8 in Fig. 6 (m = 1.09, COV =0.286) a Best fit to tail. Table 5. Computed resistance factor 4 for = 2.33 and Q = Load bias (normal statistics) CDF curve 9 in Fig. 8 (m = 0.954, COV = 0.406) CDF curve 10 in Fig. 8 (m = 0.959, COV = 0.434) CDF curve 11 a in Fig. 8 (m = 0.973, COV = 0.462) Resistance bias (normal statistics) CDF curve 2 in Fig. 4 (m = 1.47, COV = 0.509) CDF curve 4 a in Fig. 4 (m = 1.30, COV = 0.400) CDF curve 6 in Fig. 6 (m = 2.07, COV = 0.589) CDF curve 7 in Fig. 6 (m = 1.48, COV = 0.306) CDF curve 8 in Fig. 6 (m = 1.09, COV = 0.286) a Best fit to tail. tified could be removed first. If correlations between the bias and nominal resistance values still exist, the dataset could still be broken up into ranges of predicted nominal resistance to eliminate any unwanted dependencies. However, removing the outliers first could eliminate much of the data for 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 calibration exercise described here with current recommended load and resistance factors for a steel grid reinforced soil wall. For example, a value of 4 = 0.9 is currently recommended by AASHTO (2007). This value is close to the value of 0.85 computed using the middle range of predicted resistance values with Q = However, the current load factor recommended by AASHTO (2007) is Q = Resistance factors computed using this value are summarized in Table 5. The largest resistance factor in this table is 4 = 0.69, which is less than 0.9. In fact, the best values can be argued to be in the range of (say) 4 = Hence, the combination of 4 = 0.9 and Q = 1.35 is not consistent with the available steel-grid reinforced soil wall case study data if a target p f = 1/100 is specified. Using the curve 11 load statistics and resistance statistics for curves 4 and 7 together with 4 = 0.9 and Q = 1.35 gives a p f in the range of about 1/10 to 1/20 (eqs. [2] and [15]). These probability values are unreasonably large, particularly as pullout failure of these systems seldom if ever occurs in the field. The reasons for this contradiction and possible explanations 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 ½23Š FS ¼ R n Q ni Combining eqs. [8] and [23] gives the following expression for resistance factor: ½24Š ¼ iq ni FS Q ni However, eq. [24] cannot accurately account for variability in load and resistance terms, nor is there a quantitative estimate of the probability of failure associated with the value of FS assumed. Nevertheless, as a check on the estimate of resistance factor value, the equation can be argued to provide a link to past experience with successful ASD practice, thus providing a benchmark that reflects many years of safe design. For the pullout limit state for welded wire and bar mat reinforced walls analyzed previously, the safety factor used in ASD practice is FS = 1.5 (AASHTO 2002). If the currently prescribed load factor value of Q = 1.35 recommended by AASHTO for internal wall stability for earth pressure due to soil self-weight is used, then 4 = 1.35/1.5 = 0.9 (the current recommended value in AASHTO (2007)). This value is larger than 4 = using statistics for the data plotted in Figs. 4 and 8. An initial conclusion is that LRFD calibration using the statistical data presented in this paper results in a more conservative design with respect to past practice, or the factor of safety used in current ASD for reinforcement pullout

14 1390 Can. Geotech. J. Vol. 45, 2008 should be increased. However, it is important to note that the good performance observed in the past regarding this limit state could be the result of other conservative practices that have contributed to the safety of these walls. For example, a minimum reinforcement length of 70% of the wall height or 2.4 m, whichever is greater, is currently specified (AASHTO 2007). These empirical criteria likely result in an excessive reinforcement length to contain and anchor the active zone for the internal stability failure model assumed (see Fig. 3a). Conservative selection of soil shear strength for design could also contribute to an additional safety margin, as could load sharing between the upper layers of soil reinforcement. In fact, rewriting eq. [24] to isolate FS and using (say) 4 = 0.60 and Q = 1.75 gives FS = 1.75/0.60 = 2.9. Hence, based on the limited available dataset for reinforcement pullout for steel-grid reinforced soil walls, it could be argued that the true FS against pullout in current ASD practice is closer to 3 rather than to 1.5. Clearly, the choice of which resistance factor to use in LRFD practice is a judgment call that must be made by those who approve LRFD-based design codes and specifications. Additional considerations for LRFD calibration The original bias data plotted in Figs. 4 and 8 have been shown to deviate from idealized log-normal distributions fitted to the entire datasets. It is tempting to remove outliers based on visual inspection or possibly rigorous statistical tests. However, this should be done with caution, since it is often the case that the outliers appear at the tails of the distributions. Removing apparent outliers in the tails of the CDF plots often removes data that should be used to define the probability of failure for the limit state under examination, i.e., the typically limited number of extreme wall cases where loads have been excessively high and (or) resistance has been excessively low. It is important that the statistical data used to characterize a given random variable truly represent random processes. If not, the statistics will be erroneous. This is especially important when attempting to group data together from multiple sources to generate the dataset used to characterize the random variable in question. For the resistance data provided in this paper, a possible reason for the treatment of some data points as outliers is that these points are from tests carried out at low confining pressures on soils with a high friction angle. Hence, these data points are at or beyond the applicable limits of the pullout design model. Data that are beyond the limits of the design model used in LRFD calibration will likely introduce nonrandom effects. The other approach demonstrated in this paper to deal with nonrandom influences in the dataset for calibration purposes is to parse the data into ranges of predicted nominal values. Nevertheless, judgement must be exercised. The quantity of data can have a strong effect on the estimation of the statistical parameters (mean value and coefficient of variation), depending on the required confidence level. The higher the confidence level desired, the larger the number of data points required. For a given confidence level, the required 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 points raised here, including the treatment of outliers, can be found in the guidance document by Allen et al. (2005). Summary This paper has focused on calibration issues for load and resistance factor design of geotechnical structures with simple soil structure interaction models. The general approach to determine the resistance factor using closed-form solutions has been demonstrated for the ultimate limit state describing pullout of bar mat and welded wire reinforcement in reinforced soil walls. The following are the main points of this paper: (1) Statistical treatment of load and resistance values should be based on bias values of measured to predicted load and resistance terms to capture the influence of the adopted design models on the values of estimated load and resistance factors. (2) Special attention to the distribution of the bias values in the tails of CDF plots for input data is required. (3) Removal of outliers in the lower tail and upper tail of resistance and load bias data, respectively, in CDF plots should be undertaken with caution, since it is the data in these region that controls the estimate of probability of failure. (4) The best fit to tail technique described by Allen et al. (2005), which is essentially an extension of the Rackwitz Fiessler method, may give the same result as the removal of outliers. For the example in this paper, both approaches resulted in a good fit to the lower tail of the resistance bias value distribution. However, this may not be the case for other datasets because of the location of the outliers in the distribution. This highlights the importance of removing any outliers that can cause nonrandom influences on statistical characterization of the random variables of interest. (5) Hidden statistical dependencies between bias values and predicted (nominal) values of load and resistance can exist in calibration datasets. These dependencies can often be detected visually by plotting the bias-predicted value data. However, a quantitative check can be carried out by applying the Spearman rank correlation test to the data. Statistical dependencies may have physical significance related to the accuracy of the measurements at low resistance values (or high load values). The underlying empirical-based design models available today for soil structure interaction problems have been influenced by the tendency of their developers to fit a lower (or upper) bound to the data when scatter is large. A possible strategy to reduce or eliminate these statistical dependencies is to parse the dataset into ranges of predicted nominal resistance. Alternatively, specific data points identified as outliers can be removed subject to the cautions raised previously. (6) Load factors that are required to estimate the resistance factor should be based on statistics for load data applicable to the type of structure and the design model to be used. For example, load factors that are currently recommended in codes for conventional retaining walls using