As the second-order terms are small when the coeffi-

cients of variation are not exceedingly large (below

approximately 30 percent), the equation above is

sometimes approximated as:

The standard deviation of the log of the factor of

safety is obtained as:

Fln *FS *' F 2n *C *% F2n *D*

l

l

Fln *FS *'

2

2

ln 1 % *V*C % ln 1 % *V*D

Introducing an approximation,

2

2

ln 1 % *V*C ' . *V*C

the reliability index for lognormally distributed *C*, *D*,

Figure B-2. Alternative definitions of the reliability index

and *FS *and normally distributed ln *C*, ln *D*, and ln *FS*

can be expressed approximately as:

exceeds zero in terms of the standard deviation of

ln *FS*, it is:

ln

$'

$'

'

2

2

F ln *C *& ln *D*

Fln *C*/*D*

The exact expression is:

'

Fln *FS*

2

1 % *V*D

$ ' ln

distribution, the expected value of ln *C *is:

2

1 % *V*C

1 2

F

For many geotechnical problems and related deter-

2 ln *C*

ministic computer programs, the output is in the form

where:

of the factor of safety, and the capacity and demand

are not explicitly separated. The reliability index must

F2n *C *' ln 1 % *V*C

2

be calculated from values of *E[FS] *and FFS obtained

l

from multiple runs as described in the next section. In

this case, the reliability index is obtained using the

following steps:

FFS

safety is then:

Fln *FS *'

2

ln 1 % *V*FS

1

2

ln 1 % *V*C

2

1

2

ln 1 % *V*FS

1

2

ln 1 % *V*D

%

2

2

B-7

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