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uncorrelated variables. The remainder of the calculation proceeds as in the uncorrelated case, using the new variables. When the iterations have converged, the x variables are back-calculated.
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16.5.3 Low and Tang s Approach
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Low and Tang (1997a) proposed a very ef cient procedure that takes advantage of the optimization and solution techniques that are available in modern spreadsheets and mathematical software packages. They work directly with the correlated variables themselves without any rotations or transformations. If x is the vector of uncertain variables and C is their covariance matrix (not the correlation matrix) and is the vector of their means, they show that the Hasofer Lind reliability index is = min (x )T C 1 (x ) (16.56)
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subject to the constraint that x must satisfy the failure criterion that M = 0. The procedure is as follows: 1. Values of the means and covariance matrix are de ned and C 1 is calculated. 2. The functional form of the following equation is entered 2 = (x )T C 1 (x ) (16.57)
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3. The failure criterion (M = 0) is expressed as a constraint in terms of the variables in x. 4. The minimize or solve command (depending the spreadsheet or mathematical software used) is invoked to minimize 2 by changing the values of x subject to the constraint that the failure criterion is satis ed. 5. The result consists of values of the terms in x at the failure point and a corresponding value of 2 . Computing and the probability of failure follows easily. This technique is extremely easy to use. It can deal with correlated as well as uncorrelated variables. Low and Tang also show how distributions other than Normal can be employed.
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16.6 Non-normal Variables
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The Hasofer Lind approach is based on independent, Normal variables. We have shown how the procedure can be modi ed when the variables are correlated. When some of the variables are logNormally distributed, the problem can be reformulated in terms of the logarithms of those variables, which are Normally distributed. The mathematical manipulations will become more complicated, but the basic procedures will be those described in this chapter. For other distributions there is a procedure known as the Rosenblatt transformation (Rosenblatt 1952), which is also described by Ang and Tang (1990). This presentation follows Ang and Tang s. The transformation considers a set of n random variables
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THE HASOFER LIND APPROACH (FORM)
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(X1 , X2 , . . . , Xn ). If their joint CDF is F (x1 , x2 , . . . , xn ) and CDF, then we can develop a set of relations: (u1 ) = F1 (x1 ) (u2 ) = F2 (x2 |x1 ) . . . (un ) = Fn (xn |x1 , . . . , xn 1 )
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(u) is the standard normal
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(16.58)
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In words, this means that the u s are scaled so that their standard Normal CDFs correspond to the conditional CDFs of the x s. Equation (16.58) implies that u1 = u2 = . . . un =
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1 1 1
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[F1 (x1 )] [F2 (x2 |x1 )]
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[Fn (xn |x1 , . . . , xn )]
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(16.59)
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Figure 16.8 shows how this works in the case of one variable. On the right is the CDF of a function x that has an arbitrary distribution. For each value of x, there corresponds a value of the CDF. On the left is the CDF of the standard Normal variable, and for the particular value of the CDF there corresponds a value u. In most cases the failure functions and its derivatives are expressed in terms of the x s, but the analysis is driven by the u s. During the analysis one moves from a set of the u s to corresponding values of the x s by the following procedure: 1. 2. 3. 4. 5. For all the ui and nd (ui ). 1 Find x1 = F1 [ (u1 )]. 1 Find F2 [ (u2 )] and from the result and the value of x1 calculate x2 . 1 Find F3 [ (u3 )] and from the result and the values of x1 and x2 calculate x3 . Repeat for the rest of the variables.
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