15.1 Mathematical Background

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The basic mathematical problem is that of a random variable or variables X with probability distribution function (pdf) fX (x) and another variable Y , which is a deterministic

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Reliability and Statistics in Geotechnical Engineering Gregory B. Baecher and John T. Christian 2003 John Wiley & Sons, Ltd ISBN: 0-471-49833-5

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POINT ESTIMATE METHODS

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function of X, Y = g(X). X could be soil properties, geometrical parameters, seismic loadings, and so on; Y could be a factor of safety, settlement, quantity of ow, and so on. Assume that g(X) is well behaved and that the mth order moments of fX (x) exist. The question then is how to approximate the low-order moments of fY (y) using only the low-order moments of fX (x) and the function g(X). The approaches most commonly used in geotechnical reliability analysis (e.g. FOSM) start by approximating g(X) by a Taylor series, truncating it to low order terms, and computing the approximate moments from the result. This requires calculating or approximating the derivatives of g(X), which can involve signi cant algebraic or numerical effort, as well as iteration when g(X) is non-linear. Rosenblueth approached the problem by replacing the continuous random variable X with a discrete random variable whose probability mass function (pmf) pX (x) has the same moments of order m as does fX (x). He transformed pX (x) through Y = g(X) to obtain another discrete function with a pmf denoted pY (y). He then used this pmf to calculate the moments, which were assumed to approximate the moments of Y in the continuous case. The rst moment of fX (x) about the origin is the mean, X : X = x fX (x) dx (15.1)

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The higher-order central moments of fX (x) of order m are Xm = (x X )m fX (x) dx (15.2)

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We use Xm for the mth central moment to distinguish it from the mth power of X . The second central moment, X2 , is the variance, and its square root is the standard deviation, X . The discrete pmf has non-zero probabilities at a limited number of discrete points only. The corresponding moments of the discrete pmf pX (x) are Xm = (x X )m pX (x) (15.3)

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Then, equating the moments of fX (x) and pX (x) yields Xm = (x X )m fX (x) dx = (x Xm ) pX (x) (15.4)

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It is well known that any probability density function can be represented to any desired degree of accuracy by taking moments of high enough order. In principle, there are in nitely many pmf s of X satisfying the low-order moments of Equation (15.4). To limit these to a unique representation, Rosenblueth considered only pmf s with two, three, or some other small number of discrete masses, the number chosen depending on the order of the moments in Equation (15.4). This approach falls within a long-standing practice of approximating complicated functions by a series of simpler functions. As Equations (15.1) and (15.2) show, the moments of a pdf are integrals, and the large body of experience with numerical quadrature as an approximation to integration is relevant to the present problem. In particular, Gaussian quadrature procedures are concerned with choosing the optimal values of the coordinates at

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ROSENBLUETH S CASES AND NOTATION

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which to evaluate the integrand (called the Gauss points in the nite element literature) and the corresponding weights. Rosenblueth s method is in fact an application of the Gaussian quadrature procedures to the problem of nding the moments of a random function.

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15.2 Rosenblueth s Cases and Notation

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Rosenblueth (1975) deals with three cases: (1) when Y is a function of one variable X, whose mean, variance, and skewness are known; (2) when Y is a function of one variable X whose distribution is symmetrical and approximately Gaussian; and (3) when Y is a function of n variables X1 , X2 , . . . , Xn , whose distributions are symmetric and which may be correlated. In most cases the calculations are made at two points, and Rosenblueth uses the following notation: m m E[Y m ] P+ y+ + P y (15.5) In this equation, Y is a deterministic function of X, Y = g(X), E[Y m ] is the expected value of Y raised to the power m, y+ is the value of Y evaluated at a point x+ , which is greater than the mean, x , y is the value of Y evaluated at a point x , which is less than x , and P+ , P are weights. The problem is then to nd appropriate values of x+ , x , P+ , and P .

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