In this paper, a safety envelope concept for load tolerance is introduced. This shows the capacity of the current design as a future reference for design upgrade, maintenance, and control. The safety envelope is applied to estimate the load tolerance of a structural part with respect to the fatigue reliability. First, the dynamic load history is decomposed into the average value and amplitude, which are modeled as random variables. Second, through fatigue analysis and uncertainty propagation, the reliability is calculated. Last, based on the implicit function evaluation for the reliability, the boundary of the safety envelope is calculated numerically. The effect of different distribution types of random variables is then investigated to identify the conservative envelope. In order to improve the efficiency of searching the boundary, probabilistic sensitivity information is utilized. When the relationship between the safety of the system and the load tolerance is linear or mildly nonlinear, the linear estimation of the safety envelope turns out to be accurate and efficient. During the application of the algorithm, a stochastic response surface of logarithmic fatigue life with respect to the load capacity coefficient is constructed, and the Monte Carlo simulation is utilized to calculate the reliability and its sensitivities.

1.
Kwak
,
B. M.
, and
Kim
,
J. H.
, 2002, “
Concept of Allowable Load Set and Its Application for Evaluation of Structural Integrity
,”
Mech. Struct. Mach.
0890-5452,
30
(
2
), pp.
213
247
.
2.
Isukapalli
,
S. S.
,
Roy
,
A.
, and
Georgopoulos
,
P. G.
, 2000, “
Efficient Sensitivity/Uncertainty Analysis Using the Combined Stochastic Response Surface Method and Automated Differentiation: Application to Environmental and Biological Systems
,”
Risk Anal.
0272-4332,
20
(
5
), pp.
591
602
.
3.
Isukapalli
,
S. S.
,
Roy
,
A.
, and
Georgopoulos
,
P. G.
, 1998, “
Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems
,”
Risk Anal.
0272-4332,
18
(
3
), pp.
351
363
.
4.
Wu
,
Y.-T.
, 1994, “
Computational Methods for Efficient Structural Reliability and Reliability Sensitivity Analysis
,”
AIAA J.
0001-1452,
32
(
8
), pp.
1717
1723
.
5.
Liu
,
H.
,
Chen
,
W.
, and
Sudjianto
,
A.
, 2004, “
Probability Sensitivity Analysis Methods for Design under Uncertainty
,”
Tenth AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
,
Albany, NY.
, August 30-September 1.
6.
Safe Technology, 2004, FE-SAFE, Software Package, Ver. 5.1, Sheffield, England.
7.
Yu
,
X. M.
,
Chang
,
K. H.
, and
Choi
,
K. K.
, 1998, “
Probabilistic Structural Durability Prediction
,”
AIAA J.
0001-1452,
36
(
4
), pp.
628
637
.
8.
Matsuishi
,
M.
, and
Endo
,
T.
, 1968, “
Fatigue of Metals Subjected to Varying Stress-Fatigue Lives Under Random Loading
,”
Proceedings of the Kyushu District Meeting
,
JSEM
,
Fukuoka, Japan
, pp.
37
40
.
9.
Miner
,
M. A.
, 1945, “
Cumulative Damage in Fatigue
,”
J. Appl. Mech.
0021-8936,
12
, pp.
A159
A164
.
10.
Ghanem
,
R. G.
, and
Spanos
,
P. D.
, 1991,
Stochastic Finite Elements: A Spectral Approach
,
Springer-Verlag
,
New York
.
11.
Xiu
,
D.
,
Lucor
,
D.
,
Su.
,
C.-H.
, and
Karniadakis
,
G. E.
, 2002, “
Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos
,”
J. Fluids Eng.
0098-2202,
124
(
1
), pp.
51
59
.
12.
Choi
,
S. K.
,
Grandhi
,
R. V.
,
Canfield
,
R. A.
, and
Pettit
,
C. L.
, 2003, “
Polynomial Chaos Expansion With Latin Hypercube Sampling for Predicting Response Variability
,”
44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference
,
Norfolk, VA
, April 7-10.
13.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, 1972, “
Orthogonal Polynomials
,”
Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables
,
9th ed.
,
Dover
,
New York
, pp.
771
802
.
14.
Myers
,
R. H.
, and
Montgomery
,
D. C.
, 1995,
Response Surface Methodology: Process and Product Optimization Using Designed Experiments
,
Wiley
,
New York
.
15.
Khuri
,
A. I.
, and
Cornell
,
J. A.
, 1996,
Response Surface: Design and Analysis
,
Dekker
,
New York
.
16.
Wang
,
G.
, 2003, “
Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points
,”
ASME J. Mech. Des.
1050-0472,
125
(
2
), pp.
210
220
.
17.
Villadsen
,
J.
, and
Michelsen
,
M. L.
, 1978,
Solution of Differential Equation Models by Polynomial Approximation
,
Prentice–Hall
,
Englewood Cliffs, NJ
.
18.
Gautschi
,
W.
, 1996,
Orthogonal Polynomials: Applications and Computation, Acta Numerica
,
Cambridge University Press
, Cambridge, pp.
45
119
.
19.
Cameron
,
R. H.
, and
Martin
,
W. T.
, 1947, “
The Orthogonal Development of Nonlinear Functionals in Series of Fourier-Hermite Functionals
,”
Ann. Math.
0003-486X,
48
, pp.
385
392
.
20.
Kim
,
N. H.
,
Wang
,
H.
, and
Queipo
,
N. V.
, 2006, “
Efficient Shape Optimization Under Uncertainty Using Polynomial Chaos Expansions and Local Sensitivities
,”
AIAA J.
0001-1452,
44
(
5
), pp.
1112
1116
.
21.
Webster
,
M. D.
,
Tatang
,
M. A.
, and
McRae
,
G. J.
, 1996, “
Application of the Probabilistic Collocation Method for an Uncertainty Analysis of a Simple Ocean Model
,” MIT Joint Program on the Science and Policy of Global Change.
22.
Kim
,
N. H.
,
Wang
,
H.
, and
Queipo
,
N. V.
, 2005, “
Adaptive Reduction of Design Variables Using Global Sensitivity in Reliability-Based Optimization
,” Int. J. Reliab. Safety (in press).
23.
Metropolis
,
N.
, and
Ulam
,
S.
, 1949, “
The Monte Carlo Method
.”
J. Am. Stat. Assoc.
0003-1291,
44
(
247
), pp.
335
341
.
24.
Rubinstein
,
R. Y.
, 1981,
Simulation and the Monte Carlo Method
,
Wiley
,
New York
.
25.
Hasofer
,
A. M.
, and
Lind
,
N. C.
, 1974, “
Exact and Invariant Second-Moment Code Format
.”
J. Eng. Mech.
0733-9399,
100
(
1
), pp.
111
121
.
26.
Rackwitz
,
R.
, 2000, “
Reliability Analysis-Past, Present and Future
.”
Eighth ASCE Specialty Conference on Probabilistic Mechanics and Structure Reliability
,
Notre Dame, IN
, July 24-26.
27.
Chiralaksanakul
,
A.
, and
Mahadevan
,
S.
, 2005. “
First-Order Approximation Methods in Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
1050-0472,
127
(
5
), pp.
851
857
.
28.
Karamchandani
,
A.
, and
Cornell
,
C. A.
, 1992, “
Sensitivity Estimation Within First and Second Order Reliability Methods
,”
Struct. Safety
0167-4730,
11
, pp.
95
107
.
29.
Devroye
,
L.
, 1986,
Nonuniform Random Variate Generation
,
Springer-Verlag
,
New York
.
30.
Allgower
,
E. L.
, and
Georg
,
K.
, 1990,
Numerical Continuation Methods: An Introduction
,
Springer-Verlag
,
Berlin
.
You do not currently have access to this content.