with the partial derivatives taken at the expansion point

2

ln *E FS*

1 % *V*FS

(in this case the mean or expected value). Using the

$'

'

factor of safety as an example performance function,

Fln *FS*

2

ln 1 % *V*FS

the variance is obtained by finding the partial deriva-

tive of the factor of safety with respect to each random

variable evaluated at the expected value of that varia-

ble, squaring it, multiplying it by the variance of that

random variable, and summing these terms over all of

the random variables:

Methods such as direct integration, Taylor's series

method, point estimate method, and Monte Carlo

2

M *FS*

simulation are available for calculating the mean and

M *X*i

standard deviation of the performance function. For

direct integration, the mean value of the function is

obtained by integrating over the probability density

The standard deviation of the factor of safety is then

function of the random variables. A brief description

simply the square root of the variance.

of the other methods follows. The references should

be consulted for additional information.

(a) Having the expected value and variance of the

factor of safety, the reliability index can be calculated

as described earlier in this appendix. Advantages of

method is one of several methods to estimate the

the Taylor's series method include the following:

moments of a performance function based on moments

of the input random variables. It is based on a Taylor's

The relative magnitudes of the terms in the

series expansion of the performance function about

above summation provide an explicit indica-

some point. For the Corps' navigation rehabilitation

tion of the relative contribution of uncertainty

studies, the expansion is performed about the expected

of each variable.

values of the random variables. The Taylor's series

method is termed a first-order, second-moment

The method is exact for linear performance

(FOSM) method as only first-order (linear) terms of

functions.

the series are retained and only the first two moments

(mean and the standard deviation) are considered. The

Disadvantages of the Taylor's series method include

method is summarized below.

the following:

(1) Independent random variables. Given a

It is necessary to determine the value of

function *Y = g(X*1, X2, ... Xn), where all *X*i values are

derivatives.

independent, the expected value of the function is

obtained by evaluating the function at the expected

The neglect of higher order terms introduces

values of the random variables:

errors for nonlinear functions.

(b) The required derivatives can be estimated

numerically by evaluating the performance function at

two points. The function is evaluated at one increment

For a function such as the factor of safety, this implies that

above and below the expected value of the random

the expected value of the factor of safety is calculated

variable *X*i and the difference of the results is divided

using the expected values of the random variables:

by the difference between the two values of *X*i.

Although the derivative at a point is most precisely

evaluated using a very small increment, evaluating the

derivative over a range of plus and minus one standard

The variance of the performance function is taken as:

deviation may better capture some of the nonlinear

behavior of the function over a range of likely values.

Thus, the derivative is evaluated using the following

2

M*Y*

approximation:

M *X*i

B-8

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