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4 of a structure needs to ensure that while being economically the AASHTO LRFD specifications applied to geotechnical viable it will suit the intended purpose during its working life. engineering utilized the work performed by Barker et al. (1991). Limit state (LS) is a condition beyond which the structure This code was mostly based on an adaptation of WSD to LRFD (i.e., a bridge in the relevant case), or a component, fails to and only marginally addressed the SLS. Continuous attempts fulfill in some way the intended purpose for which it was have been made since then to improve the scientific basis on designed. Limit state design (LSD) comes to meet the require- which the specifications were developed, including NCHRP ments for safety, serviceability, and economy. LSD most often Project 20-7 (Task 88), NCHRP Projects 12-35 and 12-55 for refers, therefore, to two types of limit states: the ULS, which earth pressures and retaining walls, NCHRP Project 12-24 for deals with the strength (maximum loading capacity) of the soil-nailing, and NCHRP Project 24-17 that calibrated for the structure, and the serviceability limit state (SLS), which deals first time the LRFD parameters for deep foundations based with the functionality and service requirements of a structure on extensive databases of deep foundation testing (Paikowsky to ensure adequate performance under expected conditions et al., 2004). NCHRP Project 12-66 addresses the needs of SLS (these can be, for example, under normal expected loads or in design of bridge foundations. The project's approach has extreme events like impact, an earthquake, etc.). required developing serviceability criteria for bridges based The ULS design of a structure and its components (e.g., a on foundation performance, defining methods for the eval- column or shallow foundation) depends upon the predicted uation of foundation displacements and establishing their loads and the capacity of the component to resist them (i.e., uncertainty, and calibrating the resistance factors assigned resistance). Both loads and resistance have various sources for the use of these methods based on the established SLS and levels of uncertainty. Engineering design has historically and target reliability. The backbone of the study has been compensated for these uncertainties by using experience and the development of databases to establish the uncertainty of subjective judgment. The new approach that has evolved aims the methods used to evaluate the horizontal and vertical dis- to quantify these uncertainties and achieve more rational en- placements of foundations. gineering designs with consistent levels of reliability. These Of the various AASHTO studies related to LRFD calibration uncertainties can be quantified using probability-based meth- and implementation, one important component remained ods resulting for example with the LRFD format, which allows deficient and that was the ULS of shallow foundations. The the separation of uncertainties in loading from uncertainties in topic is problematic because the ULS of coupled loading is not resistance, and the use of procedures from probability theory easily identified, and the current specifications (AASHTO, to assure a prescribed margin of safety. 2008), although providing the theoretical estimation of the The same principles used in LRFD for ULS can be applied to bearing resistance of soil (Section, contain specific the SLS, substituting the capacity resistance of the component language to exclude inclination factors (C10., noting with a serviceability limit, such as a quantified displacement, that the specified resistance factors are limited, varying for all crack, deflection or vibration. Since failure under the SLS conditions between = 0.45 to = 0.50. will not lead to collapse, the prescribed margin of safety can The combination of the foundation loads in the ULS frame- be smaller, i.e., the SLS can tolerate a higher probability of work is quite complex and needs to be addressed systemati- "failure" (i.e., exceedance of the criterion) compared with cally either via the existing nominal resistance calculation pro- that for the ULS. viding safety limits and appropriate resistance factors and/or a new methodology directly applicable to the evaluation of the 1.2.3 Geotechnical and AASHTO Perspective ULS under the desired load combinations. This issue is further explored in Section 1.6. The LSD and LRFD methods are becoming the standard methods for modern-day geotechnical design codes. In Europe (CEN, 2004; DIN EN 1997-1, 2008 including the National 1.3 Load and Resistance Annex, 1 draft 2009), Canada (Becker, 2003), China (Zhang, Factor Design 2003), Japan (Honjo et al., 2000; Okahara et al., 2003), the 1.3.1 Principles United States (Kulhawy and Phoon, 2002; Withiam, 2003; Paikowsky et al., 2004), and elsewhere, major geotechnical The intent of LRFD is to separate uncertainties in load- design codes are switching from ASD (or WSD) to LSD and ing from uncertainties in resistance and then to use proce- LRFD. dures from probability theory to ensure a prescribed margin A variation of LRFD was first adopted by AASHTO for the of safety. Sections 1.3 and 1.4 outline the principles of the design of certain types of bridge superstructures in 1977 methodology and present the common techniques used for under a design procedure known as Load Factor Design (LFD). its implementation. AASHTO LRFD Bridge Design Specifications was published Figure 1 shows probability density functions (PDFs) for in 1994 based on NCHRP Project 12-33. From 1994 to 2006, load effect (Q) and resistance (R). "Load effect" is the load cal-

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5 where the nominal (ultimate) resistance (Rn) multiplied by a mQ resistance factor () becomes the factored resistance (Rr), FS mR mQ which must be greater than or equal to the summation of Qn loads (Qi) multiplied by corresponding load factors (i) and a modifier (i). Load Effect (Q) i = D R l 0.95 (3) where i are factors to account for effects of ductility (D), fR(R), fQ(Q) Rn mR redundancy (R), and operational importance (I). Based on considerations ranging from case histories to exist- ing design practice, a prescribed value is chosen for probability of failure. Then, for a given component design (when applying Resistance (R) resistance and load factors), the actual probability for a fail- ure (the probability that the factored loads exceed the factored resistances) should be equal to or smaller than the prescribed value. In foundation practice, the factors applied to load effects are typically transferred from structural codes, and then resis- tance factors are specifically calculated to provide the pre- scribed probability of failure. R, Q The importance of uncertainty consideration regarding the Figure 1. An illustration of PDFs for load effect resistance and the design process is illustrated in Figure 1. In -- and resistance. this figure, the central factor of safety is FS = mR/mQ, whereas the nominal factor of safety is FSn = Rn/Qn. The mean factor of safety is the mean of the ratio R/Q and is not equal to the culated to act on a particular element (e.g., a specific shallow ratio of the means. Consider what happens if the uncertainty foundation), and the resistance is its bearing load capacity. in resistance is increased, and thus the PDF broadened, as In geotechnical engineering problems, loads are usually bet- suggested by the dashed curve. The mean resistance for this ter known than are resistances, so the Q typically has smaller curve (which may represent the result of another predictive variability than the R; that is, it has a smaller coefficient of method) remains unchanged, but the variation (i.e., un- variation (COV), hence a narrower PDF. certainty) is increased. Both distributions have the same In LRFD, partial safety factors are applied separately to the mean factor of safety one uses in WSD, but utilizing the dis- load effect and to the resistance. Load effects are increased by tribution with the higher variation will require the applica- multiplying characteristic (or nominal) values by load factors tion of a smaller resistance factor in order to achieve the same (); resistance (strength) is reduced by multiplying nominal prescribed probability of failure to both methods. values by resistance factors (). Using this approach, the fac- The limit state function g corresponds to the margin of safety, tored (i.e., reduced) resistance of a component must be larger i.e., the subtraction of the load from the resistance such that than a linear combination of the factored (i.e., increased) load (referring to Figure 2a): effects. The nominal values (e.g., the nominal resistance, Rn, and the nominal load, Qn) are those calculated by the specific g = R-Q (4) calibrated design method and the loading conditions, respec- tively, and are not necessarily the means (i.e., the mean loads, For areas in which g < 0, the designed element or structure mQ, or mean resistance, mR of Figure 1). For example, Rn is the is unsafe because the load exceeds the resistance. The proba- predicted value for a specific analyzed foundation, obtained bility of failure, therefore, is expressed as the probability (P) by using Vesic 's bearing capacity calculation, while mR is the for that condition: mean possible predictions for that foundation considering the various uncertainties associated with that calculation. p f = P ( g < 0) (5) This principle for the strength limit state is expressed in the AASHTO LRFD Bridge Design Specifications (AASHTO, 1994, In calculating the prescribed probability of failure (pf), 1997, 2001, 2006, 2007, 2008) in the following way: a derived probability density function is calculated for the margin of safety g(R,Q) (refer to Figure 2a), and reliability is Rr = Rn i i Qi (2) expressed using the "reliability index," . Referring to Figure 2b,

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6 4 f(g) mQ g Load effect (Q) mg(=mR-mQ) 3 Probability density function g < 0 (failure) Failure region area = pf Qn mg Performance (g) (=mR-mQ) mR 2 Rn Resistance (R) 1 Performance (g) 0 0 1 2 3 -0.5 0 0.5 1 1.5 R, Q g(R,Q) = R- Q (a) (b) Figure 2. An illustration of probability density function for (a) load, resistance, and performance function and (b) the performance function (g(R,Q)) demonstrating the margin of safety (pf ) and its relation to the reliability index, (g = standard deviation of g). the reliability index is the number of standard deviations of the g = ln ( R ) - ln (Q ) = ln ( R Q ) (8) derived PDF of g, separating the mean safety margin from the nominal failure value of g being zero: If R and Q follow lognormal distributions, log R and log Q follow normal distributions, thus the safety margin, g, follows = mg g = (mR - mQ ) Q 2 + R 2 (6) a normal distribution. As such, the relationship obtained in Equation 7 is still valid to calculate the failure probability. where mg and g are the mean and standard deviation of the Figure 2b illustrates the limit state function, g, for normal dis- safety margin defined in the limit state function Equation 4, tributed resistance and load, the defined reliability index, respectively. (also termed target reliability, T), and the probability of fail- The relationship between the reliability index () and the probability of failure (pf) for the case in which both R and Q follow normal distributions can be obtained based on Equa- Table 1. Relationship between tion 6 as the following: reliability index and probability of failure. p f = ( - ) (7) Reliability index Probability of failure pf 1 where is the error function defined as ( z ) = - z 1.0 0.159 e 1.2 0.115 u 2 2 exp - du. The relationship between and pf is provided 1.4 0.0808 2 1.6 0.0548 1.8 0.0359 in Table 1. The relationships in Table 1 remain valid as long 2.0 0.0228 as the assumption is that the reliability index () follows a 2.2 0.0139 normal distribution. 2.4 0.00820 2.6 0.00466 As the performance of the physical behavior of engineer- 2.8 0.00256 ing systems usually cannot obtain negative values (load and 3.0 0.00135 resistance), it is better described by a lognormal distribution. 3.2 6.87 E-4 3.4 3.37 E-4 The margin of safety is taken as log R - log Q, when the resis- 3.6 1.59 E-4 tances and load effects follow lognormal distributions. Thus, 3.8 7.23 E-5 the limit state function becomes the following: 4.0 3.16 E-5

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7 ure, pf. For lognormal distributions, these relations will relate 4 mQ to the function g = ln(R /Q) as explained above. The values provided in Table 1 are based on series expan- Qn sion and can be obtained by a spreadsheet (e.g., NORMSDIST Rn Probability density function in Excel) or standard mathematical tables related to the stan- 3 dard normal probability distribution function. It should be Load noted, however, that previous AASHTO LRFD calibrations effect (Q) Qn Rn and publications for geotechnical engineering, notably Barker 2 mR et al. (1991) and Withiam et al. (1998), have used an approx- imation relationship proposed by Rosenblueth and Esteva (1972), which greatly errs for < 2.5, the typical zone of inter- est in ULS design calibration ( = 2 to 3) and errs even more 1 Resistance (R) in the zone of interest for SLS calibrations ( < 2.0). For lognormal distributions of load and resistance one can show (e.g., Phoon et al., 1995) that Equation 6 becomes the 0 following: 1 2 3 R, Q mRN - mQN = Figure 3. An illustration of the LRFD factors QN 2 + 2 RN determination and application (typically > 1, (mR mQ ) (1 + COVQ2 ) (1 + COVR2 ) ln < 1) relevant to the zone in which load is = (9) greater than resistance (Q > R). ln [(1 + COVR2 )(1 + COVQ 2 )] where 1.3.3 First Order Second Moment mQN, mRN = the mean of the natural logarithm of the load and the resistance, The First Order Second Moment (FOSM) method of cali- QN, RN = the standard deviations of the natural log- bration was proposed originally by Cornell (1969) and is based arithm of the load and the resistance, and on the following. For a limit state function g(m): mQ, mR, = the simple means and the coefficients of COVQ, COVR variation for the load and the resistance of mean mg g (m1 , m2 , m3 , . . . , mn ) (12) the normal distributions. These values can be transformed from the lognormal distri- n 2 g g 2 i i 2 bution using the following expressions for variance (13) i =1 i the load and similar ones for the resistance: 2 n + g - g i- 2 = ln (1 + COV 2 ) or i i i2 i QN Q (10) i =1 and mQN = ln (mQ ) - 0.5 QN 2 (11) where m1 and i = the means and standard deviations of the basic variables (design parameters); 1.3.2 The Calibration Process i, i = 1,2, . . . , n; The problem facing the LRFD analysis in the calibration g+ - i = mi + mi, and g i = mi - mi for small increments process is to determine the load factor () and the resistance mi; and factor () such that the distributions of R and Q will answer xi is a small change in the basic variable to the requirements of a specified . In other words, the and value, xi. described in Figure 3 need to answer to the prescribed tar- get reliability (i.e., a predetermined probability of failure) Practically, the FOSM method was used by Barker et al. described in Equation 9. Several solutions are available and (1991) to develop closed-form solutions for the calibration of are described below, including the recommended procedure the geotechnical resistance factors () that appeared in the for the research reported herein (see Section 1.3.5). previous AASHTO LRFD specifications.

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8 1 + COVQ 2 etc.). The calibration process is presented in Figure 4 and R ( i Qi ) detailed by Paikowsky et al. (2004). 1 + COVR2 = (14) { } Each limit state (ultimate or serviceability) can be repre- mQ exp T ln [(1 + COVR2 )(1 + COVQ 2 )] sented by a performance function of the form: where g ( X ) = g ( X1 , X 2 , . . . , X n ) (16) R = resistance bias factor, mean ratio of measured in which X = (X1, X2, . . . , Xn) is a vector of basic random resistance over predicted resistance; variables of strengths and loads. The performance function COVQ = coefficient of variation of the load; g(X), often called the limit state function, relates random vari- COVR = coefficient of variation of the resistance; and ables to either the strength or serviceability limit state. The T = target reliability index. limit is defined as g(X) = 0, implying failure when g(X) 0 (but strictly g(X) < 0) (see Figures 2 and 4). Referring to Figure 4, When just dead and live loads are considered, Equation 14 the reliability index, , is the distance from the origin (in stan- can be rewritten as dard normal space transformed from the space of the basic random variables) to the failure surface at the most probable Q 1 + COVQD 2 + COVQL2 point on that surface, that is, at the point on g(X) = 0 at which R D D + L QL 1 + COVR 2 the joint probability density function of X is greatest. This is = (15) QD ln [(1 + COVR2 ) sometimes called the design point, and is found by an itera- + QL exp T tive solution procedure (Thoft-Christensen and Baker, 1982). QD QL (1 + COVQD2 + COVQD 2 )] This relationship can also be used to back calculate represen- tative values of the reliability index, , from current design where practice. The computational steps for determining using D, L = dead and live load factors, FORM are provided by Paikowsky et al. (2004). QD/QL = dead to live load ratio, In developing code provisions, it is necessary to follow QD, QL = dead and live load bias factors, current design practice to ensure consistent levels of reliabil- COVQD = coefficient of variation for dead load, and ity over different evaluation methods (e.g., pile resistance or COVQL = coefficient of variation for live load. displacement). Calibrations of existing design codes are needed to make the new design formats as simple as possible and to The probabilistic characteristics of the foundation loads put them in a form that is familiar to designers. For a given are assumed to be those used by AASHTO for the superstruc- reliability index, , and probability distributions for resis- ture (Nowak, 1999); thus D, L, QD and QL are fixed, and a tance and load effects, the partial safety factors determined by resistance factor can be calculated for a resistance distribution the FORM approach may differ with failure mode. For this rea- (R, COVR) for a range of dead load to live load ratios. son, calibration of the calculated partial safety factors (PSFs) is important in order to maintain the same values for all loads at different failure modes. In the case of geotechnical codes, 1.3.4 First Order Reliability Method the calibration of resistance factors is performed for a set of LRFD for structural design has evolved beyond FOSM load factors already specific in the structural code. Thus, the to the more invariant First Order Reliability Method (FORM) load factors are fixed. A simplified algorithm was used in approach (e.g., Ellingwood et al., 1980; Galambos and NCHRP Project 24-17 to determine resistance factors: Ravindra, 1978), while geotechnical applications have lagged behind (Meyerhof, 1994). In order to be consistent with the 1. For a given value of the reliability index, , probability previous structural code calibration and the load factors to distributions and moments of the load variables, and the which it leads, the calibration of resistance factors for deep coefficient of variation for the resistance, compute mean foundations in NCHRP Project 24-17 used the same method- resistance, mR, using FORM. ology (Paikowsky et al., 2004). The LRFD partial safety fac- 2. With the mean value for R computed in Step 1, the PSF, , tors were calibrated using FORM as developed by Hasofer and is revised as Lind (1974). FORM can be used to assess the reliability of a n component with respect to specified limit states and provides imLi i =1 a means for calculating partial safety factors and i for resis- = (17) mR tance and loads, respectively, against a target reliability level, . FORM requires only first and second moment information on where mLi and mR are the mean values of the load and resistances and loads (i.e., means and variances) and an as- strength variables, respectively, and i, i = 1, 2, . . . , n, are sumption of distribution shape (e.g., normal, lognormal, the given set of load factors.

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9 A comparison between resistance factors obtained using FORM and resistance factors using FOSM for 160 calibra- Definition of Define Limit States Level: Failure Ultimate & Serviceability tions of axial pile capacity prediction methods is presented in Figure 5. The data in Figure 5 suggest that FORM results in resistance factors that are consistently higher than those Define Statistical Characteristics of Basic Random Variables obtained by FOSM. As a rule of thumb, FORM provided resis- Resistance Load tance factors for deep foundations approximately 10% higher than those obtained by FOSM. The practical conclusions that Determine Model Determine Load can be obtained from the observed data are that first evalua- Uncertainty for Uncertainties from tion of data can be done by the simplified closed-form FOSM Strength (from Superstructure to database) approach and the obtained resistance factors are on the low Foundation (from ST code) side (safe) for the resistance distributions obtained in the NCHRP 24-17 project (Paikowsky et al., 2004). MCS or Probability Calculation to Get Statistical Properties of Scalar R 1.3.5 Monte Carlo Simulation Reliability Assessment Monte Carlo Simulation (MCS) has become AASHTO's pre- Back-calculated Beta vs Load Ratio ferred calibration tool and is recommended for all AASHTO- Curves in Practice Assign Target Betas related calibrations. MCS is a powerful tool for determining the failure probability numerically, without the use of closed- Review Target Betas in the Literature and Practice form solutions such as those given by Equations 14 and 15. The objective of MCS is the numerical integration of the Calculate Load and Resistance Factors expression for failure probability, as given by the following equation: Select Load and Resistance Factors 1 N p f = P ( g 0) = I [ g i 0] N i =1 (18) Adjust for Mean/Nominal Parameters where I is an indicator function which is equal to 1 for gi 0, Case Study Designs for Comparison i.e., when the resulting limit state is in the failure region, and equal to 0 for gi > 0, when the resulting limit state is in the safe region. N is the number of simulations carried out. As Q g(x)=0 gL(x)=0 N, the mean of the estimated failure probability using Equation 18 can be shown to be equal to the actual failure Contours of Failure fRS=fx(x) probability (Rubinstein, 1981). Region Code calibration in its ideal format is accomplished in an Safe iterative process by assuming agreeable load () and resistance Region () factors and determining the resultant reliability index, . S When the desired target reliability index, T, is achieved, an acceptable set of load and resistance factors has been deter- mined. One unique set of load and resistance factors does not R exist; different sets of factors can achieve the same target reli- R ability index (Kulicki et al., 2007). Notes: ST = Structural The MCS process is simple and can be carried out as follows: MCS = Monte Carlo Simulation = mean Identify basic design variables and their distributions. Load g(x) = performance function of the limit state = limit state function is assumed to be normally distributed. g(x) = 0 = limit defining failure for g(x) < 0 Generate N number of random samples for each design vari- gL(x) = linearized performance function able based on its distributions, i.e., using the reported statis- Figure 4. Resistance factor analysis flow tics of load and resistance and computer-generated random chart using FORM (Ayyub and Assakkaf, numbers. 1999; Ayyub et al., 2000; Hasofer and Evaluate the limit state function N times by taking a set of Lind, 1974). the design variables generated above and count the number for which the indicator function is equal to 1.