Point Estimate Method — Several Random Variables

16.6 Point Estimate Method — Several Random Variables

Rosenblueth [1975] generalized the methodology for any number of correlated variables. For example, for a function of three random variables — say, y = y[x (1), x (2), x (3)] — where r(i, j) is the correlation coefficient between variables x(i) and x( j),

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r= − 1.0 x

r = 0.80 x

r= 0 x

Ey N [] = p ( +++ )y ( +++ )p++– + ( )y ( ++– ) + L + p ( ––– )y ( ––– ) (16.31a) where

y ( ± ± ± ) = yx [ () 1 ± s x1 [ ], x 2 () ± s x2 [ ], x 3 () ± s x3 [] ] (16.31b) p ( +++ ) = p ( ––– ) = ---- 1 3 [ 1 + r12 ( , ) + r23 ( , ) + r31 ( , ) ]

p ( ++– ) = p ( ––+ ) = ---- 1 3 [ 1 + r12 ( , ) – r23 ( , ) – r31 ( , ) ]

p ( +–+ ) = p ( –+– ) = ---- 1 3 [ 1 – r12 ( , ) – r23 ( , ) + r31 ( , ) ]

p ( +–– ) = p ( –++ ) = ---- 1 3 [ 1 – r12 ( , ) + r23 ( , ) – r31 ( , ) ] (16.31c)

where s[xi] is the standard deviation of x (i). The sign of r(i, j) is determined by the multiplication rule of i and j; that is, if the sign of i = (–), and of j = (+), then (i)( j) = (–)(+) = (–). Equation (16.31a) has 2 3 = 8 terms, all permutations of the three + s and – s. In general for M variables there are 2 M terms and M(M – 1)/2 correlation coefficients, the number of combinations of M objects taken two at a time. The coefficient on the right-hand side of Eqs. (16.31c), in general, is (1/2) M .

Example 16.11

The recommendation of the American Concrete Institute [Galambos et al., 1982] for the design of reinforced concrete structures is (in simplified form)

R ≥ 1.6D + 1.9L

where R is the strength of the element, D is the dead load, and L is the lifetime live load. (a) If —

D = 10, L = 8, V(D) = 10%, V(L) = 25%, and r(D, L) = 0.75, find the expected value and standard deviation of R for the case R = 1.6D + 1.9L. (b) If the results in part (a) generate a normal variate and the maximum strength of the element R is estimated to be 40, estimate the implied probability of failure.

Solution. The solution is developed in Fig. 16.9 . Generalizations of the PEM to more than three random variables are given by Harr [1987]. The PEM procedure yields the first two moments of the dependent random functions under consideration. Func- tional distributions must then be obtained and statements must be made concerning the probabilities of events. Inherent in the assumption of the form of a particular distribution is the imposition of the limits or range of its applicability. For example, for the normal it is required that the variable range from – • to + •; the range of the lognormal and the exponential is 0 to + •. Such assignments may not be critical if knowledge of distributions is desired in the vicinity of their expected values and their coefficients

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Variable, x

x x(+) σ [x] x(–) D 10 1 11 9

8 2 10 6 r(D,L)=+0.75

R(ij) R(ij) 2 R(+ +): 36.6 1340 R(+ –): 29.0 841 R(– +): 33.4 1116 R(– –): 25.8 666

p(+ +)= 1 _ (1+r)=0.44

E[R]=R – = Σ R(ij)p(ij)

4 =0.44(36.6+25.8)+0.06(29.0+33.4) p(+ –)= _ (1–r)=0.06 1 =31.20

4 1 E[R p(– +)= 2 ]= Σ R(ij) 2 _ (1–r)=0.06 p(ij) 4 =0.44(1340+666)+0.06(841+1116) =1000.06

p(– –)= 1 _ (1+r)=0.44 4

u[R]=E[R 2 ] – (E[R]) 2 =1000.06 – (31.20) 2 =26.62

s [R]=5.16;

V(R)=16.5%

From Eq. (15.29a), the exact solution for E[R]=1.6 D – +1.9L – =31.20

Eq. (15.29d), the exact solution for n [R]=(1.6) 2 n [D]+(1.9) 2 n [L] +2(1.6)(1.9)(1)(2)(0.75)=26.12

Of course, for this example the exact solution is easier to obtain. This is not generally the case.

b) P f =P[R ≥ 40]= 1 _ − y= 40–31.20 _ 1 − 2 y [1.71]=0.044 (Table 3) 5.16 2 The exact solution is 0.043

FIGURE 16.9 Solution to Example 16.11. of variation are not excessive (say, less than 25%). On the other hand, estimates of reliability (and of the

probability of failure) are vested in the tails of distributions. It is in such characterizations that the beta distribution is of great value. If the limits are known, zero is often an option, and probabilistic statements can readily be obtained. In the event that limits are not defined, the specification of a range of the mean plus or minus three standard deviations would generally place the generated beta distribution well within the accuracy required by most geotechnical engineering applications (see Table 16.2 ).