Life-cycle reliability-based optimization of civil and aerospace structures

Review Article

Life-cycle reliability-based optimization of civiland aerospace structures

Dan M. Frangopol a,*, Kurt Maute b

a Department of Civil, Environmental and Architectural Engineering, University of Colorado, Campus Box 428,

Boulder, CO 80309-0428, USAb Department of Aerospace Engineering Sciences, Center for Aerospace Structures, University of Colorado,

Boulder, CO 80309-0429, USA

Received 5 November 2002; accepted 11 December 2002

Abstract

Today, it is widely recognized that optimization methodologies should account for the stochastic nature of engi-

neering systems and that concepts and methods of life-cycle engineering should be used to obtain a cost-effective design

during a specified time horizon. The recent developments in life-cycle engineering of civil and aerospace structures based

on system reliability, time-dependent reliability, life-cycle maintenance, life-cycle cost and optimization constitute an

important progress. The objective of this study is to present a brief review of the life-cycle reliability-based optimization

field with emphasis on civil and aerospace structures.

� 2003 Published by Elsevier Science Ltd.

Keywords: Life-cycle engineering; Optimization; Life-cycle cost; Structural systems; Simulation; Structures; Aerospace structures; Civil

structures; Maintenance; Reliability-based design; System reliability

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

2. Basic concepts in stochastic structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

3. Formulations of RBDO problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

3.1. Life-cycle costs and utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

4. State of knowledge in RBDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

5. Current applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

5.1. Civil applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

5.2. Aerospace applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

5.2.1. Composite structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

5.2.2. Multi-disciplinary problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

5.2.3. Reliability assessment and RBDO in aerospace practice . . . . . . . . . . . . . . . . . 403

*Corresponding author. Tel.: +1-303-492-7165; fax: +1-303-492-7317.

E-mail address: [email protected] (D.M. Frangopol).

0045-7949/03/$ - see front matter � 2003 Published by Elsevier Science Ltd.

doi:10.1016/S0045-7949(03)00020-8

Computers and Structures 81 (2003) 397–410

www.elsevier.com/locate/compstruc

mail to: [email protected]

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Appendix A. Introduction to references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

1. Introduction

The general goals of engineering design are maximizing the utility of a system or a device while simultaneously

minimizing its life-cycle costs, which include the costs for developing, manufacturing, and maintaining the system. This

task is further complicated by inherently non-deterministic nature of system itself and the conditions under which it

operates. Since the design goals in general conflict, a compromise needs to be found balancing the risk of utility and cost

over the life-time of the system.

Often a design, which meets the above goals, is labeled with the terms ‘‘robust’’ and ‘‘reliable’’. These terms have

different flavors in the context of design. On a conceptual level, the robustness or reliability of a system can be

improved through identifying, understanding, and, if possible, eliminating basic failure mechanisms. Focusing on

stochasticity, the reliability can be associated with the probability that a failure occurs, and optimizing for ro-

bustness can be understood as minimizing the probability of failure. While identifying and understanding failure

mechanisms belong to the classical problems of physics-oriented disciplines, such as structural mechanics, optimizing

for minimum failure probability of non-deterministic systems is an increasingly important field in design optimi-

zation.

In recent decades, numerous optimization methods have been developed to address the above design challenge

on the whole or special aspects of it. At the beginning of this development the non-deterministic, uncertain na-

ture of engineering design problems was often ignored and a perfect system under clearly predictable opera-

tion conditions assumed. Designs found by deterministic approaches are often prone to be highly sensitive to

variations of systems and operating parameters, and therefore of limited value for the solution of practical

problems.

Today, it is widely recognized that optimization methodologies should account for the stochastic nature of engi-

neering systems. These methodologies can be classified into two groups: robust design optimization (RDO) and reli-

ability-based design optimization (RBDO). Based on purely deterministic analysis RDO methods attempt to maximize

the deterministic performance and simultaneously to minimize the sensitivity of the performance with respect to ran-

dom parameters. This approach leads to a multi-objective optimization problem capturing the impact of uncertainties

only in a qualitative sense.

Unlike RDO approaches, RBDO methods allow the design for a specific risk and target reliability level accounting

for the various sources of uncertainty in a quantitative sense. In order to optimize for minimum life-cycle cost,

probabilistic deterioration models and reliability-based maintenance, inspection, and repair methods can be incorpo-

rated into the framework of RBDO. However, RBDO approaches are based on stochastic analysis methods and,

therefore, algorithmic-wise more challenging and computationally more expensive compared to deterministic ap-

proaches.

The potential of RBDO methods for life-cycle reliability optimization is promising but still in its infancy. Past and

current research efforts are focusing on the efficient formulation and solution techniques of RBDO problems for static

and time-varying systems with linear and non-linear deterministic behavior. The objective of this paper is to present a

short survey on the most important approaches and recent developments in RBDO methods focusing on the appli-

cation to civil and aerospace structures. The list of references contains the items that the authors consider particularly

relevant for this subject.

The remaining of this paper is organized as follows: In Section 2 the basic concepts and analysis methods to evaluate

the stochastic structural response are introduced. In Section 3 basic formulations of RBDO problems are presented.

Previous accomplishment in RBDO are summarized in Section 4 and the potential of RBDO for civil and aerospace

engineering design is illustrated in Section 5. Finally, after the conclusions in Section 6, an introduction into recom-

mended literature on RBDO methods is presented.

398 D.M. Frangopol, K. Maute / Computers and Structures 81 (2003) 397–410

2. Basic concepts in stochastic structural analysis

A key component of RBDO methods is the evaluation of the stochastic structural response. In this section, the basic

concepts and formulations in stochastic structural analysis are outlined. A detailed presentation of structural reliability

concepts can be found in several books, such as [1–4], review articles [5–7], and conference proceedings, such as the

International Conference on Structural Safety and Reliability [8–12], the International Conference on Applications of

Statistics and Probability [13–17], ASCE specific Conferences on Probability Mechanics and Structural Reliability [18–

22], and the Working Conference on Reliability and Optimization of Structural Systems [23–31].

The stochastic nature of the random structural resistance R and the random load effects S can be described by theirprobability density functions (PDFs) fRðrÞ and fSðsÞ, respectively, where r and s are random parameters describing thestructural resistance and load effects. The probability of failure Pf is defined as the probability of occurrence of the eventR6 S and can be evaluated by solving the following convolution integral:

Pf ¼Z 1

0

FRðsÞfSðsÞds ð1Þ

where FR is the cumulative distribution function of R. This convolution integral has a closed-form solution if both, Rand S, are normal (Gaussian) or log-normal. For example, for normal independent distributed variables the probabilityof failure is:

Pf ¼ 1� UðbÞ and b ¼ lR � lSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2R þ r2S

p ð2Þ

where U is the standard normal distribution function and b the reliability index; lð�Þ denotes the mean value and rð�Þ the

standard deviation.

In the general case, the probability of failure Pf is defined by the limit state gjðyÞ ¼ 0 with gj 6 0 defining the failurestate and y denoting the vector of ny random variables. The convolution integral (1) takes on the following form [1]:

Pfj ¼Zgj 6 0

fY ðyÞdy ð3Þ

where fY ðyÞ is the joint PDF. The reliability level of a system can also be characterized by the performance function Ppj :

PpjðpjÞ ¼Zgj 6 pj

fY ðyÞdy ð4Þ

where pj is the performance measure.In most cases Pfj cannot be evaluated by analytical means and numerical methods are employed, such as:

• Monte Carlo simulation methods [32–35] feature generality, simplicity, and effectiveness on problems that are highly

non-linear with respect to the uncertainty parameters. The most serious drawback is the computational costs, in par-

ticular when the reliability level is high, that is the failure probability low.

• Importance sampling (IS) methods [35,36] reduce the number of samples required for a simulation based probabi-

listic analysis. Variants of IS methods are Adaptive IS [37,38] and Radial IS [39].

• First- and second-order reliability methods (FORM/SORM) approximate the reliability index [2,35,40,41]. These

methods require a search for the most probable point (MPP) on the failure surface (gj ¼ 0) in the standard normalspace. FORM employs a linear approximation of the limit state function at the MPP and is considered accurate as

long as the curvature is not too large. SORM features an improved accuracy by using a quadratic approximation.

The fundamental reliability analysis problem introduced above involves a single failure mode of a single component.

The reliability of a structural system can only be correctly assessed by considering the full structural system as a single

entity and can be accurately computed only if all its failure models are taken into account. Depending on the system

topology and geometry, material behavior, statistical correlation, and variability in loads and strengths the reliability of

a system can be vastly different from the reliability of its components [42,43].

The simplest models for system reliability are series and parallel failure modes. Practical expressions for system

reliability include lower and upper bounds for both series and parallel systems [1]. Some of these bounds consider

correlation between pairs of potential failure modes. Also, more complex system models involving series systems of

parallel systems can be used [44,45].

D.M. Frangopol, K. Maute / Computers and Structures 81 (2003) 397–410 399

The reliability of such systems can be approximated by advanced reliability techniques incorporated in computer

programs such as CALREL [46], STRUREL [47], PROBAN [48] and RELSYS [49].

In most reliability-based structural engineering studies loadings and resistances have been idealized as time invariant.

The loadings, however, fluctuate in time. Also, under environmental stressors, resistances are time-variant. Therefore,

structural reliability problems are time-variant. The study of time-variant structural system reliability is still in its in-

fancy. In order to perform a realistic reliability analysis of a structural system, the best approach is to find the

probability of system failure over a period of time t, that is the time-to-failure probability Pf ðtÞ given as:

Pf ðtÞ ¼ PfRðtÞ < SðtÞ in ð0; tg ð5Þ

where both the load-effect S and the resistance R are functions of time t. This approach requires complex integration.It was used by several researchers for both highly idealized and realistic systems [50–55].

An alternative approach, which is computationally less difficult, is to find the probability of failure at specific points

in time, the so-called point-in-time failure probability [56,57].

3. Formulations of RBDO problems

A standard optimization problem is formulated in terms of a cost function c, constraints gj, j ¼ 1; . . . ; nG, andoptimization variables x 2 Rnx .

minx

cðxÞ

s:t: gjðxÞ6 0 j ¼ 1; . . . ; nG ð6Þ

In structural optimization, the objective and constraints are generally functions of the structural response u, which inturn is a function of the optimization variables x. Typical optimization criteria, that are the objective and constraints,include weight, stiffness, displacements, stresses, eigenfrequencies, and buckling loads.

While typically the optimization criteria and the optimization variables are deterministic, in RBDO the criteria and

the variables may be subject to stochastic variations of random structural parameters and loadings. In the case of a

random design parameter, the optimization variable represents its mean value.

Several formulations of RBDO problems have been proposed in the literature. The most basic one accounts for

uncertainties through probabilistic constraints. In the reliability index approach (RIA) the lower limits �bbj for the as-

sociated limit states are prescribed leading to the following formulation [58–60]:

minx

cðxÞ

s:t: gj ¼ bjðxÞ � �bbj P 0 j ¼ 1; . . . ; nb ð7Þ

where nb is the number of probability constraints. The constraints in (7) can be also formulated in terms of probabilistic

performance measure (4). This formulation is called the probabilistic performance measure approach (PMA) or target

performance approach [61,62].

Most RIA and PMA approaches are based on FORM, where either standard gradient-based optimization methods,

such as sequential quadratic programming and the method of feasible directions, or tailored MPP search algorithms,

such as the HL-RF algorithm, are employed to determine the reliability index or the performance measure, respectively

[6,63]. FORM is in particular attractive for RBDO, since it allows for efficient computation of the gradients dbj=dxiwith respect to the optimization variables [64,65]. However, the overall numerical effort for solving RBDO problems

with FORM is still considerable, since at each iteration of the design optimization loop the MPP search calls for the

solution of another, inner optimization problem.

Alternatively, the cost can be prescribed maximizing one or multiple reliability indices [66–68]:

maxx

fb1; . . . ; bnbg

s:t: g ¼ C � C6 0 ð8Þ

where C denotes the maximum tolerable cost. Formulations and algorithms in RBDO have been summarized in severalreview papers [42,43,66,69–71].


3.1. Life-cycle costs and utility

One of the most general objectives is the expected life-cycle cost of a structural system. According to [72] the life-cycle

costs of systems under hazard risk, such as bridges, can be divided into the following four major categories:

cLC ¼ cð1ÞLC þ cð2ÞLC þ cð3ÞLC þ cð4ÞLC ð9Þ

where cLC is the total life-cycle costs, cð1ÞLC the planned and owner costs, c

ð2ÞLC the user costs associated with cð1ÞLC, c

ð3ÞLC the

unplanned and owner costs, and cð4ÞLC the user costs associated with cð3ÞLC. An alternative formulation of the expected life-

cycle costs has been introduced [73]:

cLC ¼ cðIÞLC þ cðPMÞLC þ cðINSÞLC þ cðREPÞLC þ cðFÞLC ð10Þ

where cðIÞLC is the initial costs, cðPMÞLC the expected costs of preventive maintenance, cðINSÞLC the expected costs of inspection,

cðREPÞLC the expected costs of repair, and cðFÞLC the expected costs of failure. All cost component are total costs estimatedduring the life-span of the structure.

The total expected utility U has been expressed in monetary form [74] as follows:

U ¼ B� cðIÞLC � L ð11Þ

where B denotes the benefit derived from the existence of the system and L the expected loss due to failure. All quantitiesin (9)–(11) are expected present values. The concept of optimization as the basis of code making and reliability veri-

fication and the life quality index were recently treated in [75,76].

4. State of knowledge in RBDO

First steps in formulating practical design as an optimization problem were undertaken by Forssell [77] in the

early 1920s. Along with the introduction of formal design optimization procedures for deterministic problems by

Schmit [78] and the establishment of the reliability bases of structural safety and design by Freudenthal et al.

[79], Cornell [80], Ang and Cornell [81], and Ellingwood and Ang [82], Moses and Kinser [83] applied the concept of

RBDO to the weight minimization of structures subject to reliability and cost constraints. Related approaches and

applications have been reported for example in [83,84]. In the mid 1970s RBDO was extended onto systems level design

[42,74].

During the past three decades developments have continued in both structural reliability analysis and struc-

tural optimization. Some of the most significant results in the context of RBDO methods can be summarized as

follows:

1. Creation of robust, efficient and general optimization algorithms and their integration into design software for the

application to complex large-scale structural and multi-disciplinary problems [85–90].

2. Development of efficient analytical and semi-analytical parameter sensitivity analysis for non-linear mechanical

problems including coupled multi-field problems [91–99].

3. Identification of failure modes of structural systems and their incorporation into the evaluation of the overall sys-

tem reliability [3,44].

4. Introduction of stochastic analysis methods for engineering problems including probabilistic finite element methods

and random process and field models [7,100–106].

5. Development of first- and second-order reliability methods improved by optimization algorithms which are tailored

to the MPP search [2,37,40,63,107–110].

6. Development of time-dependent reliability models for analysis, design and optimization of structural systems [4,50–

55,111].

7. Integration of reliability analysis methods into design optimization frameworks including RIA and PMA [58–

61,110,112].

8. Extension of reliability-based optimization capabilities upward into the design variable hierarchy, such as shape,

material, and topological optimization: sizing [43,113], shape [112,114,115], topology [116–119].

9. Use of expected life-cycle cost models for optimal design of new structural systems and the optimal operation of

existing ones [73,120–137].

10. Dissemination of reliability and optimization methods and applications for practicing engineers [60,138].


The progress in stochastic analysis and reliability assessment and optimization of structural systems is also published

in conference proceedings [8–31]. In addition, several ASCE publications [123–125] and special issues of international

journals [126,139–141] are dedicated to this subject.

The developments in optimal design and reliability-based design optimization are also documented in numerous

proceedings of conferences, such as ISSMO�s World Congress of Structural and Multidisciplinary Optimization [142–145] and AIAA�s Symposium on Multidisciplinary Analysis and Optimization [146–149].The above advances have found their way into several academic and commercial computers codes. An overview of

optimization tools algorithms for general and design applications is given in [150]. Commercial optimization software

packages tailored to engineering design optimization are, for example, VisualDoc [151], ISight [152] and BOSS quatro

[153]. In addition optimization modules are included into finite element analysis packages, for example, in OPTI-

STRUCT [154], ANSYS [155], and GENESIS [156]. Reliability analysis and RBDO methods are implemented, for

example, in PROBAN [48], CALREL [46], RELTRAN [157], STRUREL [47], RELSYS [49], and RELTSYS [111].

5. Current applications

Reliability-based optimization methods have been applied to a broad range of structural design and maintenance

problems in civil, mechanical and aerospace engineering. In the following sections the most significant developments in

the areas of civil and aerospace engineering are summarized.

5.1. Civil applications

The applications of reliability-based optimization methods in civil engineering as a tool for day to day design is still

not advanced as it would be desirable [158]. However, these methods have been successfully disseminated into several

areas of civil applications, such as buildings, bridges, nuclear and off-shore structures. These methods have been also

successfully used in reliability-based code calibration procedures for design and assessment of buildings and bridges

[159,160] and also in life-cycle maintenance procedures for existing structures. Since the literature on the above ap-

plications is too abundant to be summarized in this study the reader is referred to [1,3,42,50–54,57,68,69,73,113–

115,122–135,139,161,162], among others.

5.2. Aerospace applications

Aerospace systems inherently operate in extreme and variable environments, causing great uncertainty in their op-

erating conditions. Reliability requirements for aerospace components and systems are severe due to the often costly

and fatal consequences of failure. Simultaneously, aerospace systems are also subject to tough requirements on min-

imum weight and maximum performance. RBDO methods are an appealing approach for solving the resulting,

complex design problems in aerospace engineering.

The analysis of aerospace structures is in general complicated by complex loading and operating conditions, such as

aerodynamic pressure and thermal loads. Often the interaction of one or multiple fields needs to be taken into account

for predicting the load acting on the structure. Furthermore, aerospace systems are often actively controlled leading to

another level of non-linearity in the structural response. This complexity is amplified in RBDO requiring a stochastic

analysis of multi-field and multi-disciplinary systems.

5.2.1. Composite structures

Composite materials allow for minimum weight design by tailoring the material composition to the loading condi-

tions and are, therefore, frequently used in aerospace structures. However, it is crucial to account for uncertainties in

material and geometric parameters and loading conditions since composite structures typically fail abruptly.

Therefore, numerous approaches have been presented applying RBDO methods to the design of composite thin-

walled structures. Probabilistic load conditions and material properties including manufacturing uncertainties have

been considered for example in [161,163–167]. Composites structures with degradation models have been considered in

[168] and buckling instabilities in [169]. The difference between deterministic optimization and RBDO results shows the

importance of accounting for uncertainties in particular in the design of composite structures [170].


5.2.2. Multi-disciplinary problems

The performance of aircraft is dominated by coupled multi-physics phenomena, such as aeroelasticity or thermo-

aeroelasticity. While aeroelastic instabilities, such as divergence and flutter, lead to failure, fluid–structure interaction in

general needs to be accounted for to achieve optimum performance.

Traditionally, only the deterministic case has been considered first using linear flow theories and simplified structural

models [171–174], and more recently employing non-linear Euler and Navier–Stokes flows and detailed structural finite

element models [99,175–177]. Only few approaches have been presented that account for uncertainties.

In addition to structural parameters, an important source of uncertainties are the operating conditions. Stochastic

variations in gust loads have been considered for RBDO of wings in [178–180] using elementary structural and gust

models. In [181,182] a method for preliminary weight optimization of a wing structure for mean-gust response and

aileron effectiveness reliability has been presented. Failure due to flutter has been treated in [183,184] considering

uncertainties in material parameters and the fluid loads predicted by a Doublet–Lattice method.

Employing high-fidelity simulation tools an intermediate complexity wing is optimized under uncertainties in [185]

neglecting fluid–structure interaction. In order to account for uncertainties in predicting maneuver loads and the effect

of control surfaces, an optimization method for minimizing the weight and maximizing the invariance to random load

effects is presented in [186,187]. The uncertainties of the aerodynamic loads are quantified by comparing the results of

linear and non-linear aerodynamics. Considering uncertainties in structural parameters and flow conditions a RIA

approach has been recently presented, which is based on high-fidelity aeroelastic simulation and applied to the shape

and material optimization of 3-D flexible wing structures [112].

Today, only few optimization methodologies have been presented for the design of aerospace systems under un-

certainties. A general framework for the stochastic multi-disciplinary aircraft design has been presented in [188–190]

accounting for various source of uncertainties, such modeling and economic variability, and aiming for system affor-

dability. Uncertainty has been introduced primarily on the conceptual design level, where reliability analysis methods

are combined with system level deterministic analyses.

5.2.3. Reliability assessment and RBDO in aerospace practice

The need for the incorporation of reliability and uncertainty into the design process of aerospace systems has been

recognized by the commercial and industry sectors. This has been demonstrated through the development of com-

mercial codes to incorporate reliability analysis in the design process. Southwest Research Institute has developed a

modular computer software system, called NESSUS [191], to perform probabilistic analyses of structural or mechanical

components and systems. The program was initially developed for NASA to perform probabilistic analyses of space

shuttle main engine components and has been extended to general aerospace design problems, such as the design of

aircraft gas turbine rotors and disks [192]. Southwest Research Institute has also developed the computer software

DARWIN for prediction of the probability of failure of aircraft turbine rotor disks [193,194]. Applied Research As-

sociates uses techniques in stochastic mechanics such as FORM, SORM, Monte-Carlo simulation, adaptive IS, and

experimental design [195].

6. Conclusions

The state-of-the-art in life-cycle reliability-based optimization of civil and aerospace structures, which has been

briefly overviewed in this paper, leads to the following broad conclusions.

1. Uncertainties and their propagation over time need to be accounted for in the design of new structures and in the

planning of maintenance interventions on existing structures in order to obtain robust, reliable, and cost-efficient

solutions.

2. System reliability is required to be accounted for in life-cycle optimization as the overall performance indicator for

new and existing structures. This indicator has to include the interaction among potential failure modes. The effects

of structural deterioration due to mechanical loadings and environmental stressors on system reliability have also to

be accounted for. In addition, both ultimate and serviceability limit states and their time-dependent interaction

have to be considered in the life-cycle optimization formulation.

3. The minimum expected total cost has to be used as the life-cycle optimization criterion in the design of new struc-

tures. Alternatively, for planning maintenance interventions on existing structures over a prescribed time horizon,

the expected total intervention (i.e., inspection, maintenance, repair, replacement) cost has to be used as the opti-

mization criterion.


4. With the availability of ever faster computing platforms and the advent of parallel processing, high-fidelity, multi-

disciplinary simulation methods are becoming applicable for reliability analysis and reliability-based optimization

methodologies. Currently, these approaches are still in their infancy and the computational efficiency needs to be

further improved.

Acknowledgements

The first author gratefully acknowledges the financial support of the US National Science Foundation through grants

CMS-9506435, CMS-9522166, CMS-9912525, and CMS-0217290 and of the UK Highways Agency. Both authors

acknowledge the support of the Air Force Office of Scientific Research under grant no. F49620-01-1-052. The opinions

and conclusions presented in this paper are those of the writers and do not necessarily reflect the views of the sponsoring

organizations.

Appendix A. Introduction to references

The following references document the development and state-of-the-art in RBDO. Refs. [1–4,35,36,40,79–82] are

recommend to the reader as introduction into the concept of structural reliability analysis. An overview of generic

engineering optimization algorithms and methods is given in [85–90]. Refs. [42–45,55,58,60,63,66,67,69,70,74–

76,83,84,121] describe the concept of RBDO and numerical techniques used in RBDO. An overview of RBDO methods

applied to generic structural problems and to civil engineering structures is given in [1,3,42,50–54,57,68,69,73,113–

115,122–135,139,161,162]. Applications to aerospace structures including composite materials and multi-disciplinary

problems are described in [112,161,163–167,178–190,192–194].

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Documents

Life-cycle reliability-based optimization of civil and aerospace structures