Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 137

are thought to be reasonable approximations of reality, and partly because they are conve-
nient for use in calculations of probability and reliability.

(a) Normal distributions for average s u = 14.36 kPa and COV = 5%, 15%, and 25%
(b) Lognormal distribution for average s u = 14.36 kPa and COV = 5%, 15%, and 25%

3.3.9 lognormal distribution
The lognormal distribution is an unsymmetrical bell-shaped curve, with the peak to the
left of the average value. The shape of the lognormal distribution is given by Equation 3.5 :

2

1
2

1
2

ln x − λ

Lognormal() px = exp − (3.5)

ς

ς x π

where
p ( x ) is the lognormal probability of a particular value of x
x is the variable
ζ (zeta) is the standard deviation of the natural logarithm of x

( )

2

(3.6)

ζ= ln 1 + COV

COV is the coefficient of variation, defined by Equation 3.3
λ (lambda) is the average value of the natural logarithm of x

1
2

= () −

2

(3.7)

λ ln x ζ

The lognormal distribution is based on the assumption that the logarithm of the variable
is normally distributed. The PDF does not extend to the left of zero (to negative values),
whereas the normal distribution extends infinitely far to the left and right of the average
value. Not extending to the left of zero seems more reasonable for quantities, such as shear
strength or factor of safety, which cannot be negative. Even though the normal distribution
extends to negative values, that part of the curve usually has negligible significance, and
the implication that negative values are possible is not a reason to reject use of the normal
distribution if it is otherwise reasonable and convenient.
The width of the bell-shaped lognormal curve is governed by the value of ζ, the standard
deviation of the logarithm of x. ζ is approximately equal to the COV. Figure 3.4b shows log-
normal distributions for average s u = 14.36 kPa, and values of ζ = 0.050, 0.149, and 0.246,
corresponding to COV = 0.050, 0.150, and 0.250, (values of standard deviation = 1, 2.14,
and 3.5 kPa).
As in the case of the normal distributions, the areas under all three of the lognormal
curves in Figure 3.4b are equal to unity, consistent with the fact that the total of all prob-
abilities is unity for any distribution.
A lognormal distribution of the measured variables in Table 3.1 (average s u = 14.36 kPa,
standard deviation = 2.14 kPa) is shown with the relative frequency diagram of measured
values in Figure 3.5. The use of the lognormal distribution to represent these data would
involve an assumption that the data follow a lognormal distribution. Judging solely by
the degree of conformance between the shape of the relative frequency diagram and the