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FIGURE 12.1 Reliability-based design concept. (Adopted from Probability, Reliability and Statistical Methods in Engineering Design, by Haldar and Mahadevan, 2000, with permission from John Wiley & Sons, Inc.)

The concept is shown in Figure 12.1. In Figure 12.1, the nominal values of resistance and load effect, denoted as Rn and Sn, respectively, and the corresponding PDFs of R and S are shown. The overlapped (dashed area) between the two PDFs provides a qualitative measure of the probability of failure. Controlling the size of the overlapped area is essentially the idea behind reliability-based design. Haldar and Mahadevan [14] pointed out that the area could be controlled by changing the relative locations of the two PDFs by separating the mean values of R and S (mR and mS), the uncertainty expressed in terms of their standard deviations (aR and sS), and the shape of the PDFs [fR(r) and fs(s)].

Although conceptually simple, the evaluation of Equation 12.2 may not be easy except for some special cases. Consider S = S1 + S2 H-----H Sn, representing n statistically independent load effects (dead, live, wind loads, etc.), and R and S are independent normal random variables. Under these assumptions, the resistance and the ith load factors in Equation 12.1 can be shown to be [14]

where b is the reliability index, a measure of probability of failure, dR and dS are the coefficients of variation (COV) of R and S (a measure of uncertainty), kR is the number of standard deviations below the mean resistance in selecting the nominal value of R (a measure of underestimation of the resistance), kS is the number of standard deviations above the mean load in selecting the nominal value of the ith load (a measure of overestimation in the load), and

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