Stochastic Finite Element and Satellite Structure Design - Finite Element Analysis for Satellite Structures - page 204

i ¼ X i l Xi
r Xi

X 0 ð 5 : 6 Þ

where X 0

i is a random variable with zero mean and unit standard deviation.
Equation (7) is used to transform the original limit state g(X) = 0 to the reduced
limit state g(X 0 ) = 0. The safety index b is defined as the minimum distance from
the origin of the axes in the reduced coordinate system to the limit state surface
(failure surface). It can be expressed as

q

x 0 ð t b ¼ x 0 ðÞ

ð 5 : 7 Þ

The minimum distance point on the limit state surface is called the design point.
It is defined by the vector x in the original coordinate system and by vector x 0 in
the reduced coordinate system. It is obvious that the nearer x is to the origin,
the larger is the failure probability. Thus, the minimum distance point on the limit
state surface is also the most probable failure point. This point represents the worst
combination of the stochastic variables and is named the design point or the
most probable point (MPP) of failure. For nonlinear limit states, the computation
of the MPP becomes as an optimization problem,

p
Subject to the constraint g ðÞ¼ gX 0 ¼ 0

D ¼ Minimize x 0

t

x 0

ð 5 : 8 Þ

where x 0 represents the coordinates of the checking point on the limit state
equation in the reduced coordinates to be estimated. Using the method of Lagrange
multipliers, we can obtain the minimum distance as,

P

n

x 0
i

og
oX 0
i

i ¼ 1

b ¼ s

P

ð 5 : 9 Þ

2

n

og
oX 0
i

i ¼ 1

og oX 0
i

is the i th

where partial derivative evaluated at the design point with

. The disadvantage in the Hasofer-Lind method that is
applicable only to normal variables. If not all the variables are normally distrib-
uted, it is necessary to transform the non-normal variables into equivalent normal
variables. Rackwitz and Fiessler [ 5 ] estimated the parameters of the equivalent
normal distribution, l Xi and r Xi , by imposing two conditions. The cumulative
distribution functions and the probability density functions of the actual variables
and

coordinates x 0 1 ; x 0 1 ; ... ; x 0

n

the equivalent normal variables should be equal at the design point

on the failure surface

x 0 1 ; x 0 1 ; ... ; x 0

n