Stochastic Finite Element and Satellite Structure Design - Finite Element Analysis for Satellite Structures - page 205

¼ F Xi x i

x i l Xi
r Xi

U

ð 5 : 10 Þ

F Xi x i

r Xi

l Xi ¼ x i U 1

in which U ðÞ is the CDF of the standard normal variable, l Xi and r Xi are the mean
and standard deviation of the equivalent normal variable at the design point.
Equating the PDFs of the original variable and the equivalent normal variable at
the design point results in

¼ f Xi x i

x i l Xi
r Xi

1
r Xi

/

F Xi x i

ð 5 : 11 Þ

r Xi ¼ / U 1

f Xi x ðÞ

in which / ðÞ and f Xi x i are the PDFs of the equivalent standard normal and
original non-normal random variable.
In cases of the performance function g(X) being a complicated, nonlinear or
implicit function, Rackwitz and Fiessler [ 6 ] have proposed using Newton-Raphson
type recursive algorithm. The algorithm linearizes the performance function at each
iteration point; however, instead of solving the limit state equation explicitly for b,it
uses the derivatives to find the next iteration point.
FORM Method can be described as follows:
• Define appropriate performance function g(X).
• Assume initial values of the design point x i ,i= 1, 2…, n, and compute the
corresponding value of the performance function g(). The initial point can be the
mean values of the random variables.
• Compute the mean and standard deviation at the design point of the equivalent
normal distribution for those variables that are nonnormal.

F Xi x i

r Xi

l Xi ¼ x i U 1

F Xi x i

r Xi ¼ / U 1

ð 5 : 12 Þ

f Xi x ðÞ

¼ x i l Xi
r Xi

y i

• Compute the partial derivative og = oX i evaluated at the design pointx i .
• Compute the partial derivatives og = oY i

in the equivalent standard normal space

by the chain rule of differentiation as