Civil Engineering Reference
In-Depth Information
3.4.1 Weibull Statistical Analysis
The Weibull probability analysis involves fitting a data set with the cumulative
probability function:
0
x
< 0
F
(
x
) =
(3.29)
β
x
α
1−
e
−
x
≥ 0
where,
ʱ
> 0 is a scaled parameter of a curve measuring the spread of the data, and
β
is the shape parameter of the curve, indicating whether the failure rate is increas-
ing, remaining constant, or decreasing (Vardeman & Jobe 2001). Derivation of the
Weibull parameters (
) to fit the data (|
% error
|) to the aforementioned cu-
mulative distribution, using Weibull probability plots, was performed using the
procedure provided by Dorner (1999) on excel. The Weibull procedure can be
summarized as follows: First, the |% error| is arranged in ascending order. Second,
each |%error| is assigned a corresponding rank, which ranges from 1 to
n
, where
n
is the total number of data points. The median rank,
MR
, is found for each data
point by dividing
i
by
n
. A Weibull probability plot is a graph of ln(|% error|)
vs.
α
and
β
1
.
ln
ln
1
−
MR
A regression analysis is run to find an approximate linear equation to fit the
Weibull plot. Figure 3.9 shows the Weibull plots and corresponding regression
linear equation for each derived deflection equation.
The Weibull cumulative distribution function can be transformed to represent a
linear equation as shown below:
β
x
α
−
e
−
F
(
x
)
=
1
(3.30)
β
x
α
(
)
=−
ln 1−
F
(
x
)
(3.31)
β
1
1−
F
(
x
)
x
α
ln
=
(3.32)
1
1−
F
(
x
)
x
α
ln ln
= β ln
(3.33)
