18.2.1 Calibration to allowable stress design

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Determinating load and resistance factors for a particular case is done either by calibrating Equation (18.6) to past practice, that is, to typical or codi ed factors of safety used in working stress design, or by optimizing Equation (18.6) for a chosen reliability index

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LOAD AND RESISTANCE FACTOR DESIGN

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using the methods of reliability theory of the previous chapters. Typically, this optimization has used either rst-order second-moment reliability (FOSM) or Hasofer Lind reliability (FORM). Calibration through tting to the ASD method is used when insuf cient statistical data exist to perform a calibration by optimization (Withiam et al. 1997). The method of calibrating by tting to ASD is a simple process in which the LRFD Equation (18.6) is divided by the fundamental ASD Equation (18.1) to determine an equation for the resistance factor ( ). If an average value of 1.0 is used for the load modi er , then i Qi FS Qi (18.8)

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If only the dead loads and live loads are considered, this becomes = D QD + L QL FS (QD + QL ) (18.9)

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Dividing both the numerator and denominator by QL (O Neill 1995) yields D QD + L QL = QD FS +1 QL

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(18.10)

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Equation (18.10) can be used to determine the resistance factors that need to be used in the LRFD equations to obtain a factor of safety equal to that of the ASD method. While the load and resistance factor design method represented by Equation (18.6) as calibrated using Equation (18.10) provides a clearer separation of load and resistance uncertainties than does a single factor of safety, the use of the load and resistance factors based on earlier, non-reliability-based procedures does little to improved design. Table 18.1 shows values of the resistance factors using Equation (18.10) for factors of safety varying between 1.5 to 4 and average dead and live load factors of 1.25 and 1.75, respectively. The table gives resistance factors that would be used to obtain factors of safety of 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0 for dead to live load ratios of 1, 2, 3 and 4. Clearly,

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Table 18.1 Values of resistance factors from Equation (18.10) corresponding to different values of safety factor and dead to live load ratios for D = 1.25 and L = 1.75 (after Withiam et al. 1998) Resistance factor, Safety factor 1.5 2.0 2.5 3.0 3.5 4.0 QD /QL = 1 1.00 0.75 0.60 0.50 0.53 0.38 QD /QL = 2 0.94 0.71 0.57 0.47 0.40 0.35 QL /QD = 3 0.92 0.69 0.55 0.46 0.39 0.34 QL /QD = 4 0.90 0.68 0.54 0.45 0.39 0.34

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calibrating the LRFD method to existing practice leads to a design much the same as that obtained using ASD.

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18.2.2 Calibration using FOSM reliability

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The resistance factor chosen for a particular limit state must take into account (i) the variability of the soil and rock properties, (ii) the reliability of the equations used for predicting resistance, (iii) the quality of the construction workmanship, (iv) the extent of soil exploration (little versus extensive), and (v) the consequences of failure. Replacing the ultimate or nominal resistance Rn in Equation (18.6) by E[R]/ R (the product of the expected value of resistance E[R] and the statistical bias in the resistance R ) and ignoring the correction terms leads to a resistance factor that satis es the inequality i Qi (18.11) R ( R /E[R]) For the case in which R and Q are each logNormally distributed and probabilistically independent (an assumption often made in LRFD codes for geotechnical design), setting Equation (18.11) to an equality produces a simple relationship between resistance factor and the uncertainties in resistance and loads as expressed in coef cients of variation. For logNormal variables, it is convenient to de ne the limiting condition on the safety margin involving the logarithms, lnR lnQ, such that g(R, Q) = ln R ln Q = 0 (18.12)

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Then the reliability index is the number of standard deviations of the derived random variable (lnR lnQ) separating its mean value from the limiting state or failure condition of 0 E[ln R] E[ln Q] (18.13) = Var[ln R ln Q] That standard deviation is related to the marginal standard deviations by the sum of variances Var[ln R ln Q] = Var[ln R] + Var[ln Q] (18.14) Recall from Appendix A that the rst two moments of a logNormal variable can be simply related to the rst two moments of the logarithms of the variable by E[R] E[ln R] = ln 1+ 2 R Var[ln R] = ln(1 +

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2 R)

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(18.15) (18.16)

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in which is the coef cient of variation; similarly for Q. Substituting into Equation (18.13) yields the expression (Barker et al. 1991) R R = E(Q) exp{ T i Qi 1 + COV (Q)2 1 + COV (R)2

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ln[(1 + COV (R)2 )(1 + COV (Q)2 )]}

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(18.17)

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