Page 513 - Handbook of Properties of Textile and Technical Fibres

486 Handbook of Properties of Textile and Technical Fibres

For C > 3:6, the Weibull distribution is nearly symmetrical and has a form very
similar to a normal distribution. Very interesting is the expression for the coefficient
of variation, which is in case of B ¼ 0 the function of parameter C only.

2
9 1 =

8
2

> 2 1 >
>
p ffiffiffiffiffiffiffiffiffiffiffiffi >G 1 þ G 1 þ >
>
< =
Dðs f Þ C C
CV ¼ ¼ (13.70)
Eðs f Þ > 2 1 >
G
> >
> 1 þ >
C
: ;
For small C it is valid CV ¼ C 0:92 for 0:05 C 0:5. These approximations
are interesting for an interpretation of Weibull parameters as well.
An estimation of the Weibull parameters is based on the experimental strength
; i ¼ 1.N. There exist a lot of methods for the estimation of parameters
values s f i
A, B, and C (Militký, 1996). Standard maximum likelihood method leads to the solution
of three nonlinear equations (Meloun et al., 1992). If there is a possibility to estimate
B; i ¼ 1.N
independently the parameter B, the reduced strength data s rf i ¼ s f i b
can be simply created. In this case, the moment method based on the comparison of the
first two moments leads to the estimation of the parameter C as the solution of the
nonlinear equation (Militký, 1996):
1

Gð1 þ 2=CÞ 2
s f 2 1 s f ¼ 0 (13.71)
G ð1 þ 1=CÞ
where s f is the standard deviation of strength data. The parameter A is then estimated
1
from Eqn. Eðs f Þ¼ s f þ B, i.e., A ¼ s f B G 1 þ where s f is the mean
b
C
b
b
strength value of the original data.
For a quick estimation of parameters A, C (in case of known B) it is attractive to use
b
the so-called Q-Q graph to check the Weibull distribution acceptability for experi-
mental data (Meloun et al., 1992). This graph is simply derived from the distribution
Þ by order statistics P i z i=ðN þ 1Þ for
function defined by Eq. (13.69) replacing Fðs f i
by ordered values (empirical quantiles) s s .
rfðiÞ rfðiþ1Þ
i ¼ 1.N and replacing s rf i
After substitution and rearrangements the final linear form

lnð lnð1 P i ÞÞ ¼ lnðAÞþ C ln s (13.72)
fðiÞ

is obtained. The Q-Q graph then depends on lnð lnð1 P i ÞÞ on ln s . In the case
fðiÞ
of validity of the Weibull distribution with a lower limit B, this dependence should be a
b
straight line with slope C and intercept ln(A)(Meloun et al., 1994). By using standard
linear regression, the parameters A and C can be roughly estimated (Meloun et al.,
1992). Due to the empirical quantiles s roughness and their nonconstant variances
rfðiÞ
the special treatment for refining of P i can be used (Tiryakioglu and Hudak, 2008).