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(16.38)
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In this equation, the superscript or subscript star indicates that the term or derivative is evaluated at the nearest point on the failure criterion.
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16.3.2 The Taylor Series Approach
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We start from the starred point on the failure criterion even though we have not developed a method for calculating the location of this point. The Taylor series for nding the value of g at some other point is
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g(x1 , x2 , . . . , xn ) = g(x1 , x2 , . . . , xn ) +
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(16.39)
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This expression is in terms of the unprimed variables. After removing the higher order terms and recognizing that g = 0 at the failure criterion, we get g(x1 , x2 , . . . , xn ) = = (xi xi ) g xi
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The primed variables are, by de nition, zero when the unprimed variables have their mean values. Hence, g g xi (16.41) xi and
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2 x i
g xi
2 x i
g xi
1 x i g xi g xi
g xi
(16.42)
Therefore, g = = g xi
2
(16.43)
which is identical to the result found using the Lagrangian Multiplier.
16.3.3 Solving the Equations
Both the above approaches give the same nal equation for the reliability index, but they do not describe how to nd the starred point at which Equations (16.38) or (16.43) are to be evaluated. As in any other non-linear minimization problem, the choice of algorithm depends on the speci c function to be minimized and the constraints, and these will vary from problem to problem. There is a large literature on minimization problems; Press et al. (1992) give a good general introduction to the subject. Lin and Der Kiureghian (1991) examine a number of algorithms that have been proposed for the reliability problem. Rackwitz and his colleagues (Rackwitz 1976; Rackwitz and Fiessler 1978) proposed a technique that remains widely used. Ang and Tang (1990) included it in the second volume of their book, and it is described below. Equation (16.26) in Section 16.3.1 de nes G as the gradient of the failure function with respect to the primed variables. This can be normalized into a unit vector : g xi g xi
G (GT G)1/2
i =
(16.44)
We can use a superscript star to indicate that the unit vector is evaluated at the failure point. It then follows that the coordinates of the failure point must be xi = i (16.45)
The Rackwitz algorithm then proceeds in six iterative steps (Ang and Tang 1990): 1. Assume initial values of xi and compute the corresponding values of xi .
LINEAR OR NON-LINEAR FAILURE CRITERIA AND UNCORRELATED VARIABLES
2. 3. 4. 5. 6.
Compute G and at xi . Form the expressions for the new xi = xi i xi . Substitute these expressions into g(x1 , x2 , . . . , xn ) = 0, and solve for . With this value of , calculate new values of xi = i . Repeat steps 2 through 5 until the process converges.
The process is best understood from examples, two of which follow. The rst is the failure of the vertical cut with which the chapter started, and the second is the Culmann single plane slope failure analysis discussed in 15. Example 16.1 Reliability of Vertical Cut Using Factor of Safety The failure criterion based on the factor of safety is F 1 = 0, so g(c, ) = 4c 1=0 H 4 g = c c H 4c g = 2 H
g 4 = c H g 4c = 2 H First iteration:
Step 1 assume c* = c = 100 and = = 20, so c = = 0. g (4)(30) (4)(100)(2) g = 0.6 = 0.2 = = Step 2 c (20)(10) (20)2 (10) 0.948683 0.6 = 0.316228 0.2 Step 3 c = 100 (0.948683)(30) = 100 28.460490 G= = 20 + (0.316228)(2) = 20 + 0.632456 Step 4 [4(100 28.460490 )]/[10(20 + 0.632546 )] = 1 = 1.664357 Step 5 c = (1.664357)(0.948683) = 1.578947