LOAD AND RESISTANCE FACTOR DESIGN

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in which R is the overall bias for resistance, COV (Q) is coef cient of variation of the load, COV (R) is coef cient of variation of the resistance, T is target reliability index, and E(Q) is the expected value of total load. When dead and live loads are considered separately, Equation (18.17) becomes (Yoon and O Neill, 1997) R R = D E(QD ) + L E(QL ) (1 + COV (QD )2 + COV (QL )2 ) (1 + COV (R)2 ) QD E(QD ) + QL E(QL ) exp{ T ln[(1 + COV (R)2 )(1 + COV (QD )2 + COV (QL )2 )]} (18.18)

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in which R , QD and QL are bias terms for resistance, dead load (QD ) and live load (QL ), and D and L are dead load and live load factors, respectively. Because the levels of uncertainty in dead loads and live loads differ from one another, so, too, do their corresponding load factors. Table 18.2 shows typical bias terms and coef cients of variation for dead load components and for live loads on highway bridges. The AASHTO (1994) load factors combine dead loads into a single factor. A typical example of calibrated resistance factors is shown in Table 18.3, taken from Barker et al. (1991) for axially loaded piles, calibrated both against ASD and by using FOSM analysis with the speci ed target reliability indices. For use in codes, recommended values of load and resistance factors are typically rounded to the nearest increment of 0.05, and judgment on the historical development of earlier code factors of safety is taken into account in balancing differences between the two calibrations. Also, for most common situations, the dependence of reliability index, , on the ratio of dead load to live load, QD /QL , is relatively small. Thus, a representative value of QD /QL is typically used, and a single resistance factor chosen, irrespective of load ratio.

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18.2.3 Calibration using FORM reliability

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The currently preferred way to perform the calibration to obtain load and resistance factors is through rst-order reliability (FORM), that is, using the Hasofer Lind procedure discussed in 16. This procedure is based on choosing a checking point, called,

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Table 18.2 Statistics for structural load components for highway bridges. (Nowak, A. S., 1995, Calibration of LRFD Bridge Code, Journal of Structural Engineering, ASCE Vol. 121, No. 8, pp. 1245 1251, reproduced by permission of the American Society of Civil Engineers) AASHTO Load factor, 1.25 1.03 1.05 1.00 1.10 1.20 0.08 0.10 0.25 0.18

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Load component Dead load Factory-made Cast-in-place Asphaltic wearing surface Live load

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Bias,

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COV,

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LOAD AND RESISTANCE FACTOR DESIGN

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Table 18.3 Resistance factors for axial loaded driven piles in sand calibrated to ASD and FOSM (D/L = 3.7, D = 1.3 and = 2.17) (after Barker et al. 1991) Resistance factor , Soil test SPT SPT CPT CPT Pile length (m) 10 30 10 30 Factor of safety, FS 4.0 4.0 2.5 2.5 Target reliability, T 2.0 2.0 2.0 2.0 Reliability analysis, 0.48 0.51 0.59 0.62 Fitting with ASD 0.33 0.33 0.53 0.53 Recommended 0.45 0.45 0.55 0.55

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Notes: Dead load to live load ratio assumed to be, D/L = 3.7; load factors taken as D = 1.3 and = 2.17. Were these calculations performed with the current AASHTO (1997) load factors of D = 1.25 and = 1.75, the corresponding resistance factors would be 5 20% lower (Withiam et al. 1998).

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the design point, at a particular point on the limiting state surface, and calculating the reliability index, , separating that point from the joint mean of the uncertain load and resistance variables. The design point is taken as that point on the limiting state surface at which the probability density of the joint random variables is greatest. To simplify the procedure, the calculation of is made in a normalized space of derived random variables, in which each derived variable has unit variance and is independent of all other derived variables. In this derived space, the shortest distance between the joint mean of the derived variables and the state limit function identi es both the design point and the reliability index. The method yields partial safety factors for loads and resistances at a speci ed target reliability index, T . As discussed in 16, this procedure has the advantage over FOSM of greater invariance with respect to the mathematical de nition of limiting state. De ne the limit state function as, g(X) = g(X1 , X2 , . . . , Xn ) = 0 (18.19)

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in which X is a vector of random variables (X1 , X2 , . . . Xn ) of loads and resistances, for which the limit state is g(X) = 0. Failure occurs for g(X) < 0, non-failure otherwise (Figure 18.1). The n-dimensional space of the variables X is rst transformed to X by the normalization xi mxi xi = (18.20) x i to create zero-mean, unit-variance variables. The space X is then rotated to remove correlations among the derived variables, yielding zero-mean, unit-variance, independent variables. Finally, the space is transformed for non-Normal distributions (e.g. for logNormal variables the logarithm of each is taken), yielding a normalized space in which the shortest distance between the origin (i.e. the joint means of the derived random variables xi ) and the design point the most probably failure point on the transformed limit state gives the reliability index for the design (Figure 18.2). For a given target reliability index T , the mean values of basic variables can be used to compute partial safety factors required to provide the target reliability i = xi E[xi ] (18.21)

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