In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. When binomial variables are involved, traditional reliability methods, such as the first-order second moment (FOSM) method and the first-order reliability method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation (SPA) for bimodal variables and then employs SPA-based reliability methods with first-order approximation to predict the reliability. A limit-state function is at first approximated with the first-order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The SPA is then applied to estimate the reliability. Examples show that the SPA-based reliability methods are more accurate than FOSM and FORM.

References

References
1.
Choi
,
S.-K.
,
Grandhi
,
R.
, and
Canfield
,
R. A.
,
2006
,
Reliability-Based Structural Design
,
Springer Science & Business Media
, New York.
2.
Elishakoff
,
I.
,
Van Manen
,
S.
, and
Arbocz
,
J.
,
1987
, “
First-Order Second-Moment Analysis of the Buckling of Shells With Random Imperfections
,”
AIAA J.
,
25
(
8
), pp.
1113
1117
.
3.
Mahadevan
,
S.
, and
Haldar
,
A.
,
2000
,
Probability, Reliability and Statistical Method in Engineering Design
,
Wiley
,
New York
.
4.
Hasofer, A.
, and
Lind, N.
, 1973, “
An Exact and Invariant First-Order Reliability Format
,”
J. Eng. Mech.
,
107
(2), pp. 93–108.https://www.researchgate.net/publication/243758427_An_Exact_and_Invariant_First_Order_Reliability_Format
5.
Hohenbichler
,
M.
, and
Rackwitz
,
R.
,
1983
, “
First-Order Concepts in System Reliability
,”
Struct. Saf.
,
1
(
3
), pp.
177
188
.
6.
Breitung
,
K.
,
1984
, “
Asymptotic Approximations for Multinormal Integrals
,”
J. Eng. Mech.
,
110
(
3
), pp.
357
366
.
7.
Tvedt, L.
, 1983, “
Two Second-Order Approximations to the Failure Probability
,” Det Norske Veritas, Hovik, Norway, Report No. RDIV/20-004-83.
8.
Tvedt
,
L.
,
1990
, “
Distribution of Quadratic Forms in Normal Space-Application to Structural Reliability
,”
J. Eng. Mech.
,
116
(
6
), pp.
1183
1197
.
9.
Zhao
,
Y.-G.
, and
Ono
,
T.
,
1999
, “
New Approximations for SORM—Part 1
,”
J. Eng. Mech.
,
125
(
1
), pp.
79
85
.
10.
Madsen
,
H. O.
,
Krenk
,
S.
, and
Lind
,
N. C.
,
2006
,
Methods of Structural Safety
, Dover Publications, Mineola, NY.
11.
Ditlevsen
,
O.
, and
Madsen
,
H. O.
,
1996
,
Structural Reliability Methods
,
Wiley
,
New York
.
12.
Mansour
,
R.
, and
Olsson
,
M.
,
2014
, “
A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions
,”
ASME J. Mech. Des.
,
136
(
10
), p.
101402
.
13.
Laman
,
J. A.
, and
Nowak
,
A. S.
,
1996
, “
Fatigue-Load Models for Girder Bridges
,”
J. Struct. Eng.
,
122
(
7
), pp.
726
733
.
14.
Tallin
,
A.
, and
Petreshock
,
T.
,
1990
, “
Modeling Fatigue Loads for Steel Bridges
,”
Transp. Res. Rec.
,
1275
, pp.
23
26
.http://onlinepubs.trb.org/Onlinepubs/trr/1990/1275/1275-004.pdf
15.
Haider
,
S. W.
,
Harichandran
,
R. S.
, and
Dwaikat
,
M. B.
,
2008
, “
Estimating Bimodal Distribution Parameters and Traffic Levels From Axle Load Spectra
,”
Transportation Research Board 87th Annual Meeting
, Washington, DC, Jan. 13–17, Paper No.
08-0824
.https://trid.trb.org/view/847690
16.
Haider
,
S. W.
,
Harichandran
,
R. S.
, and
Dwaikat
,
M. B.
,
2009
, “
Closed-Form Solutions for Bimodal Axle Load Spectra and Relative Pavement Damage Estimation
,”
J. Transp. Eng.
,
135
(
12
), pp.
974
983
.
17.
Mones
,
E.
,
Araújo
,
N. A.
,
Vicsek
,
T.
, and
Herrmann
,
H. J.
,
2014
, “
Shock Waves on Complex Networks
,”
Sci. Reports
,
4
(
1
), p. 4949.
18.
Daniels
,
H. E.
,
1954
, “
Saddlepoint Approximations in Statistics
,”
Ann. Math. Stat.
,
25
(4), pp.
631
650
.https://projecteuclid.org/euclid.aoms/1177728652
19.
Hu, Z.
, and
Du, X.
, 2018, “
Saddlepoint Approximation Reliability Method for Quadratic Functions in Normal Variables
,”
Struct. Saf.
,
71
, pp. 24–32.
20.
Daniels
,
H. E.
,
1987
, “
Tail Probability Approximations
,”
Int. Stat. Rev.
,
1
, pp.
37
48
.
21.
Goutis
,
C.
, and
Casella
,
G.
,
1999
, “
Explaining the Saddlepoint Approximation
,”
Am. Stat.
,
53
(
3
), pp.
216
224
.
22.
Du
,
X.
, and
Sudjianto
,
A.
,
2004
, “
A Saddlepoint Approximation Method for Uncertainty Analysis
,”
ASME
Paper No. DETC2004-57269.
23.
Du
,
X.
,
2008
, “
Saddlepoint Approximation for Sequential Optimization and Reliability Analysis
,”
ASME J. Mech. Des.
,
130
(
1
), p.
011011
.
24.
Huang
,
B.
, and
Du
,
X.
,
2008
, “
Probabilistic Uncertainty Analysis by Mean-Value First-Order Saddlepoint Approximation
,”
Reliab. Eng. Syst. Saf.
,
93
(
2
), pp.
325
336
.
25.
Meng
,
D.
,
Huang
,
H.-Z.
,
Wang
,
Z.
,
Xiao
,
N.-C.
, and
Zhang
,
X.-L.
,
2014
, “
Mean-Value First-Order Saddlepoint Approximation Based Collaborative Optimization for Multidisciplinary Problems Under Aleatory Uncertainty
,”
J. Mech. Sci. Technol.
,
28
(
10
), pp.
3925
3935
.
26.
Du
,
X.
, and
Sudjianto
,
A.
,
2004
, “
First-Order Saddlepoint Approximation for Reliability Analysis
,”
AIAA J.
,
42
(
6
), pp.
1199
1207
.
27.
Dolinski
,
K.
,
1982
, “
First-Order Second-Moment Approximation in Reliability of Structural Systems: Critical Review and Alternative Approach
,”
Struct. Saf.
,
1
(
3
), pp.
211
231
.
28.
Lee
,
T. W.
, and
Kwak
,
B. M.
,
1987
, “
A Reliability-Based Optimal Design Using Advanced First Order Second Moment Method
,”
J. Struct. Mech.
,
15
(
4
), pp.
523
542
.
29.
Du
,
X.
, and
Chen
,
W.
,
2004
, “
Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design
,”
ASME J. Mech. Des.
,
126
(
2
), pp.
225
233
.
30.
Du
,
X.
,
Guo
,
J.
, and
Beeram
,
H.
,
2008
, “
Sequential Optimization and Reliability Assessment for Multidisciplinary Systems Design
,”
Struct. Multidiscip. Optim.
,
35
(
2
), pp.
117
130
.
31.
Lee
,
I.
,
Choi
,
K.
, and
Gorsich
,
D.
,
2010
, “
System Reliability-Based Design Optimization Using the MPP-Based Dimension Reduction Method
,”
Struct. Multidiscip. Optim.
,
41
(
6
), pp.
823
839
.
32.
McDonald
,
M.
, and
Mahadevan
,
S.
,
2008
, “
Design Optimization With System-Level Reliability Constraints
,”
ASME J. Mech. Des.
,
130
(
2
), p.
021403
.
33.
Rosenblatt
,
M.
,
1952
, “
Remarks on a Multivariate Transformation
,”
Ann. Math. Stat.
,
23
(
3
), pp.
470
472
.
34.
Hohenbichler
,
M.
, and
Rackwitz
,
R.
,
1981
, “
Non-Normal Dependent Vectors in Structural Safety
,”
J. Eng. Mech. Div.
,
107
(
6
), pp.
1227
1238
.http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0010649
35.
Stockholm
,
F. E.
,
1932
, “
On the Probability Function in the Collective Theory of Risk
,”
Skand. Aktuarietidskrift
,
15
(
3
), pp.
175
195
.
36.
Lugannani
,
R.
, and
Rice
,
S.
,
1980
, “
Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables
,”
Adv. Appl. Probab.
,
12
(
2
), pp.
475
490
.
37.
Zhang
,
J.
, and
Du
,
X.
,
2010
, “
A Second-Order Reliability Method With First-Order Efficiency
,”
ASME J. Mech. Des.
,
132
(
10
), p.
101006
.
38.
Alibrandi
,
U.
,
Alani
,
A.
, and
Koh
,
C.
,
2015
, “
Implications of High-Dimensional Geometry for Structural Reliability Analysis and a Novel Linear Response Surface Method Based on SVM
,”
Int. J. Comput. Methods
,
12
(
4
), p.
1540016
.
39.
Hurtado
,
J. E.
,
2012
, “
Dimensionality Reduction and Visualization of Structural Reliability Problems Using Polar Features
,”
Probab. Eng. Mech.
,
29
, pp.
16
31
.
40.
Katafygiotis
,
L. S.
, and
Zuev
,
K. M.
,
2008
, “
Geometric Insight Into the Challenges of Solving High-Dimensional Reliability Problems
,”
Probab. Eng. Mech.
,
23
(
2–3
), pp.
208
218
.
41.
Valdebenito
,
M.
,
Pradlwarter
,
H.
, and
Schuëller
,
G.
,
2010
, “
The Role of the Design Point for Calculating Failure Probabilities in View of Dimensionality and Structural Nonlinearities
,”
Struct. Saf.
,
32
(
2
), pp.
101
111
.
42.
Alibrandi
,
U.
,
Alani
,
A. M.
, and
Ricciardi
,
G.
,
2015
, “
A New Sampling Strategy for SVM-Based Response Surface for Structural Reliability Analysis
,”
Probab. Eng. Mech.
,
41
, pp.
1
12
.
43.
Hurtado
,
J. E.
, and
Alvarez
,
D. A.
,
2010
, “
An Optimization Method for Learning Statistical Classifiers in Structural Reliability
,”
Probab. Eng. Mech.
,
25
(
1
), pp.
26
34
.
44.
Gander
,
W.
, and
Gautschi
,
W.
,
2000
, “
Adaptive Quadrature—Revisited
,”
BIT Numer. Math.
,
40
(
1
), pp.
84
101
.
45.
Malcolm
,
M. A.
, and
Simpson
,
R. B.
,
1975
, “
Local Versus Global Strategies for Adaptive Quadrature
,”
ACM Trans. Math. Software (TOMS)
,
1
(
2
), pp.
129
146
.
46.
Song
,
S.
,
Lu
,
Z.
, and
Qiao
,
H.
,
2009
, “
Subset Simulation for Structural Reliability Sensitivity Analysis
,”
Reliab. Eng. Syst. Saf.
,
94
(
2
), pp.
658
665
.
47.
Zhao
,
H.
,
Yue
,
Z.
,
Liu
,
Y.
,
Gao
,
Z.
, and
Zhang
,
Y.
,
2015
, “
An Efficient Reliability Method Combining Adaptive Importance Sampling and Kriging Metamodel
,”
Appl. Math. Modell.
,
39
(
7
), pp.
1853
1866
.
48.
Hu
,
Z.
, and
Mahadevan
,
S.
,
2016
, “
Global Sensitivity Analysis-Enhanced Surrogate (GSAS) Modeling for Reliability Analysis
,”
Struct. Multidiscip. Optim.
,
53
(
3
), pp.
501
521
.
You do not currently have access to this content.