Because deterministic optimum designs obtained without taking uncertainty into account could lead to unreliable designs, a reliability-based approach to design optimization is preferable using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space $(X$-space) and a reliability analysis in an independent and standard normal random space $(U$-space). This process requires numerous nonlinear mappings between $X$- and $U$-spaces for various probability distributions. Therefore, the nonlinearity of the RBDO problem will depend on the type of distribution of random parameters, since a transformation between $X$- and $U$-spaces introduces additional nonlinearity into the reliability-based performance measures evaluated during the RBDO process. The evaluation of probabilistic constraints in RBDO can be carried out in two ways: using either the Reliability Index Approach (RIA), or the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity for RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of the highly nonlinear transformations that are involved. However, PMA is rather independent of probability distributions because it only has a small involvement with a nonlinear transformation.

1.
Youn
,
B. D.
,
Choi
,
K. K.
, and
Park
,
Y. H.
,
2003
, “
Hybrid Analysis Method for Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
125
(
2
), pp.
221
232
.
2.
Madsen, H. O., Krenk, S., and Lind, N. C., 1986, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ.
3.
Palle, T. C., and Michael, J. B., 1982, Structural Reliability Theory and Its Applications, Springer-Verlag, Berlin, Heidelberg.
4.
Rackwitz
,
R.
, and
Fiessler
,
B.
,
1978
, “
Structural Reliability Under Combined Random Load Sequences
,”
Comput. Struct.
,
9
, pp.
489
494
.
5.
Hohenbichler
,
M.
, and
Rackwitz
,
R.
,
1981
, “
Nonnormal Dependent Vectors in Structural Reliability
,”
J. Eng. Mech. Div.
,
107
(
6
), pp.
1227
1238
.
6.
Hasofer
,
A. M.
, and
Lind
,
N. C.
,
1974
, “
Exact and Invariant Second-Moment Code Format
,”
J. Eng. Mech. Div.
,
100
(
EMI
), pp.
111
121
.
7.
Liu
,
P. L.
, and
Kiureghian
,
A. D.
,
1991
, “
Optimization Algorithms for Structural Reliability
,”
Struct. Safety
,
9
, pp.
161
177
.
8.
Wu
,
Y. T.
,
Millwater
,
H. R.
, and
Cruse
,
T. A.
,
1990
, “
Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions
,”
AIAA J.
,
28
(
9
), pp.
1663
1669
.
9.
Wu
,
Y. T.
,
1994
, “
Computational Methods for Efficient Structural Reliability and Reliability Sensitivity Analysis
,”
AIAA J.
,
32
(
8
), pp.
1717
1723
.
10.
Enevoldsen
,
I.
, and
Sorensen
,
J. D.
,
1994
, “
Reliability-Based Optimization in Structural Engineering
,”
Struct. Safety
,
15
, pp.
169
196
.
11.
Wu, Y.-T., and Wang, W., 1996, “A New Method for Efficient Reliability-Based Design Optimization,” Probabilistic Mechanics & Structural Reliability: Proceedings of the 7th Special Conference, pp. 274–277.
12.
Yu
,
X.
,
Choi
,
K. K.
, and
Chang
,
K. H.
,
1997
, “
A Mixed Design Approach for Probabilistic Structural Durability
,”
Struct. Optim.
,
14
(
2-3
), pp.
81
90
.
13.
Grandhi
,
R. V.
, and
Wang
,
L. P.
,
1998
, “
Reliability-Based Structural Optimization Using Improved Two-Point Adaptive Nonlinear Approximations
,”
Finite Elem. Anal. Design
,
29
, pp.
35
48
.
14.
Tu
,
J.
, and
Choi
,
K. K.
,
1999
, “
A New Study on Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
121
(
4
), pp.
557
564
.
15.
Tu
,
J.
,
Choi
,
K. K.
, and
Park
,
Y. H.
,
2001
, “
Design Potential Method for Robust System Parameter Design
,”
AIAA J.
,
39
(
4
), pp.
667
677
.
16.
Arora, J. S., 1989, Introduction to Optimum Design, McGraw-Hill, New York, NY.