3.7. THE RACKWITZ AND FIESSLER (R-F) METHOD 143
From Equations (3.62), (a), and (c), we can have the equivalent standard deviation
X
eq
:
X
eq
D
1
f
X
.
x
/
z
X
D
NORM:S:DIST
.
0:277430
/
7:68993 10
4
D
0:383881
7:68993 10
4
D 499:2: (d)
From Equations (3.63), (c), and (d), we can have the equivalent mean
X
eq
:
X
eq
D X
z
X
X
eq
D 2600 0:277430 499:20 D 2461:5: (e)
(b) Use the MATLAB program to determine
X
eq
and
X
eq
at the design point X
D 2600.
From Equations (3.57), (3.60), and (3.61), we can compile a short program as listed here.
% Calculate an equivalent mean and standard deviation
% Input the lognormal distribution parameter
mln=7.81; % Log mean
sln=0.192; % Log standard deviation
xs=2600; % Design point x*
fx=lognpdf (xs, mln, sln); % PDF of log-normal distribution
Fx=logncdf (xs, mln, sln); % CDF of log-normal distribution
zs=norminv (Fx); % The inverse value of the normal
% distribution per Equation (3.61)
seq=normpdf (zs)/fx; % Equivalent standard deviation per
% Equation (3.64)
meq=xs-seq*zs; % Equivalent mean, Equation (3.65)
display ('Equivalent mean')
meq
display ('Equivalent standard deviation')
seq
MATLAB gives the following results:
Equivalent mean D 2461:5.
Equivalent standard deviation D 499:2.
e R-F method is an iterative process. e procedure is very similar to the H-L method.
However, at the beginning of each iterative process, any non-normally distributed random vari-
able will be converted into an equivalent normal distribution with the equivalent mean and the
equivalent standard deviation. Following is the general procedure for the R-F method.
Step 1: Calculate the mean for non-normal distributed random variables.
For a clear description of the R-F method, we can rearrange the limit state function, as
shown in Equation (3.64). In Equation (3.64), the first r random variables are non-normally