Summary for engineering paper

www.elsevier.com/locate/compstruc

Computers and Structures 85 (2007) 235–243

On the treatment of uncertainties in structural mechanics and analysis q

G.I. Schuëller *

Institute of Engineering Mechanics, Leopold-Franzens University Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria

Received 9 August 2006; accepted 31 October 2006 Available online 22 December 2006

Abstract

In this paper the need for a rational treatment of uncertainties in structural mechanics and analysis is reasoned. It is shown that the traditional deterministic conception can be easily extended by applying statistical and probabilistic concepts. The so-called Monte Carlo simulation procedure is the key for those developments, as it allows the straightforward use of the currently used deterministic analysis procedures.

A numerical example exemplifies the methodology. It is concluded that uncertainty analysis may ensure robust predictions of vari- ability, model verification, safety assessment, etc. � 2006 Elsevier Ltd. All rights reserved.

Keywords: Uncertainty; Monte Carlo simulaton; Finite elements; Response variability; Model verification; Robustness

1. Introduction

Structural mechanics analysis up to this date, generally is still based on a deterministic conception. Observed varia- tions in loading conditions, material properties, geometry, etc. are taken into account by either selecting extremely high, low or average values, respectively, for representing the parameters. Hence, this way, uncertainties inherent in almost every analysis process are considered just intuitively. Observations and measurements of physical processes, however, show not only variability, but also random char- acteristics. Statistical and probabilistic procedures provide a sound frame work for a rational treatment of analysis of these uncertainties. Moreover there are various types of uncertainties to be dealt with. While the uncertainties in mechanical modeling can be reduced as additional knowl- edge becomes available, the physical or intrinsic uncertain- ties, e.g. of environmental loading, can not. Furthermore,

0045-7949/$ - see front matter � 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.10.009

q Plenary Keynote Lecture presented at the 3rd MIT Conference on Computational Fluid and Solid Mechanics, Boston, MA, USA, June 14– 17, 2005.

* Tel.: +43 512 507 6841; fax: +43 512 507 2905. E-mail address: [email protected]

the entire spectrum of uncertainties is also not known. In reality, neither the true model nor the model parameters are deterministically known. Assuming that by finite ele- ment (FE) procedures structures and continua can be repre- sented reasonably well the question of the effect of the discretization still remains. It is generally expected, that an increase in the size of the structural models, in terms of degrees of freedom, will increase the level of realism of the model. Comparisons with measurements, however, clearly show that this expectation can not be confirmed. An ever refined FE model just decreases the discretization error, but all other aspects contributing to the discrepancy between prediction and measurement will not be improved. There are several reasons for this. Among them is the fact that the FE model, which is a mathematical idealization, represents the physical behaviour not exactly but with a certain accuracy only. Typical examples for this are strongly non-linear interactions in a linear model, ignoring flexi- bilities at joints, inaccurate modeling of the boundary conditions, ignoring the non-linear interaction, etc. Fur- thermore, even if it is assumed that the idealized mathemat- ical FE model represents the structural behavior, the model parameters do show uncertainties. As already stated above, these uncertainties refer to both loading – environmental

236 G.I. Schuëller / Computers and Structures 85 (2007) 235–243

loading such as water waves, wind, earthquakes, etc. are good examples for this – as well as to structural properties, such as imperfections of geometry, thickness, Young’s mod- ulus, material strength, fracture toughness, damping char- acteristics, etc. It is also well known, that the results of experimental measurements are subjected to uncontrollable random effects. This is the main reason, why they are so dif- ficult to reproduce. This fact leads directly to the claim as made before, i.e. that an increase in the number of degrees of freedom does not compensate for the insufficient model- ing of physical phenomena, such as not taking into account the uncertainties in the boundary conditions, etc. Needless to say that it is most important that the model reflects phys- ical phenomena. This of course includes the uncertainties in both the structural properties and loading conditions, respectively.

The structural analysis and design process as a whole is generally carried out in four steps, i.e. the load, material, structural and safety analysis, respectively. One has to admit that not in every step the same rationale with respect to the assessment of uncertainties can be observed. While for load – particularly for load combination – and safety analysis (i.e. for the determination of the safety factors) quite large uncer- tainties are generally accepted, the contrary is the case when it comes to structural, i.e. FE analysis. In this case, as already mentioned also above, even synthetic accuracy is sometimes generated by increased mesh refinement.

2. Modeling of uncertainties

While in the deterministic conception a single value is considered to suffice for the representation of a particular variable, it is in fact a great number of values – each asso- ciated with a certain probability of occurrence of a partic- ular value – which is needed for a realistic description. As this overview is mainly addressed to deterministically ori- ented engineers, a few remarks concerning the mathemati- cal descriptions of uncertainties are in order. The variables in their basic form may be described as so called random variables X. Typical examples are the yield strength of materials, etc. The associated uncertainties are quantified by probability measures, e.g. described as probability den- sity functions. In other words, the probability that a parameter takes on values within an interval is,

Pða < X 6 bÞ¼ Z b

a fðxÞdx ð1Þ

This one-dimensional definition certainly can be expanded easily for multi-dimensional cases. The distribution of the occurrence of the various values, i.e. f(x), also denoted as the probability density function, is generally characterized by certain types of function such as the normal or Gaussian distribution, etc. The parameters, such as the central ten- dency or mean value as well as the variance, are estimated by statistical procedures.

For time variant processes, the probability density refers not only to one time instant, but to other time instances as

well, i.e. to a family of random variables X(t1), X(t2) more simply denoted by X(t). Again, if the distribution of X(t1), X(t2) are of Gaussian characteristics, such a process is denoted as Gaussian Process (see e.g. [1]). Typical examples are wind, water wave, earthquake records, etc. If the struc- tural randomness includes spatially correlated random fluc- tuations of systems or load parameters as well (see e.g. wind pressure fluctuations on area-like structures), the notion of a random field is used. A random field is gener- ally defined by its type of probability distribution and the associated distribution parameters, such as mean and var- iance, the autocorrelation function and other properties, such as homogeneity, etc. Homogeneity, for example means, that the statistical properties are independent from the specific location, which implies a constant mean value and a correlation function which depend only on the rela- tive distance in space or time. A schematic sketch of the above described quantification of randomness is given in Fig. 1.

When specifying uncertainties, it is advisable to consider all uncertain parameters instead of considering just some ‘‘most influential’’ uncertain parameters by a priori engi- neering judgement. This would prejudice the result and high sensitivities of parameters not labeled as ‘important’ would not be reflected by the response. It is crucial to note that the computational effort for a single finite element-run is basically independent of the number of random variables (uncertainties) introduced. Hence, considering a large number of uncertain parameters by using Monte Carlo simulation, cf. Section 3, entails no disadvantage, while considering all uncertainties ensures a robust prediction of the variability. As a side effect, the variability of the response might also serve as a tool to verify, e.g. the quality of the mathematical finite element (FE) model. A large scatter of the response reveals either an unfavorable high sensitivity on input parameters which are not sufficiently well known or indicates some modelling errors which require further improvements to arrive at robust pre- dictions.

A robust prediction means that the difference between prediction and actual response is within an acceptable range including all uncertainties. Robustness and accuracy (high fidelity) are antagonistic and hence a suitable com- promise has to be established. The acceptable deviation of the prediction and true system response must be large enough to account for all the variability included by the present uncertainties.

3. Methods and procedures

Finite element models generally contain quite a large number of parameters like elasticity constants, geometry specifications, loading parameters, boundary conditions, etc., (see e.g. [2]) of which most values are not perfectly known [3]. It was already stated above, that the so called ‘‘true’’ parameters can, if at all, be determined in excep- tional cases only, i.e. by experiments. Hence the values used

Fig. 1. Quantification of randomness.

G.I. Schuëller / Computers and Structures 85 (2007) 235–243 237

in deterministic FE analysis are so called nominal values which deviate to a certain extent from the unknown true value. The uncertainties within the input parameters natu- rally result in uncertainties of the output, i.e. the response. Since response predictions are the central goal of any FE analysis, and all predictions depend more or less on the uncertain input parameters, a rational approach has to include these unavoidable uncertainties. In other words, as shown in Fig. 2, the deterministic solution can be consid- ered a direct mapping of input and output. Uncertainty is defined as an input space, which has to be mapped into an output space.

Data are always scarce. Accessible statistical informa- tion, if any, might be restricted to the mean value, the stan- dard deviation, upper and lower fractile values or upper and lower bounds. However, with this concept, whatever information is available, it can be used and updated, as new information or data becomes available. Under these circumstances, it is reasonable to select the most convenient

Fig. 2. Deterministic conception versus concept including uncertainties.

distribution which reflects the known or assumed variabil- ity (uncertainty) and avoids realizations which are physi- cally not meaningful. For many reasons, the Gaussian normal distribution and log-normal distribution, respec- tively, are preferred. Since the uncertain input is specified mathematically by probability laws, the response follows also such laws, i.e. has a well defined unique distribution. Analytical methods to arrive at the distribution of the response, require very specialized knowledge, and, most important, are not generally applicable, i.e. limited in their application, and, in addition not straight forward. The alternatives to analytical approaches are those based on what is denoted as Monte Carlo simulation (see e.g. [4]). This approach is most generally applicable, and all deter- ministic analysis tools can be integrated to their full extent.

Fig. 3 depicts the basic principles of Monte Carlo sam- pling where the laws of statistics are exploited to derive information on the variability of the response. By using a suitable number generator (see e.g. [5]) statistically inde- pendent samples of the input are generated by a type of game of chance, and which follow the prescribed probabil- ity distributions of the uncertain parameters.

Suppose that all introduced random variables are repre- sented by a component Xj of the vector X ¼fX jg

n j¼1.

Hence, the input distribution f(x1, x2, . . . , xn) is represented according to statistical laws by a finite number N of inde- pendent samples fxðkÞgNk¼1. Each vector x

(k) specifies for each uncertain parameter a deterministic discrete value and consequently defines deterministically the response which might be represented by the vector r(k) = r(x(k)). Hence, traditional deterministic FE analysis procedures can be used to provide the mapping r(k) = r(x(k)) between input and response.

In the simplest case, it might be justified to assume that all uncertainties are independent. Such an assumption is reasonable as long as this assumption does not contradict experience and physical properties. When the components are considered as independent, each component can be generated by available random number generators where distribution and parameters must be supplied. One can approximate the expectation or the mean of each response quantity ri by,

Fig. 3. Stochastic analysis based on Monte Carlo sampling.

238 G.I. Schuëller / Computers and Structures 85 (2007) 235–243

li ¼ Efrig� 1

N

XN k¼1

riðxðkÞÞ ð2Þ

and the variance by,

Varfrig¼ r2i ¼ Efðri � liÞ 2g

� 1

N � 1 XN k¼1 ðriðxðkÞÞ� liÞ

2 ; ð3Þ

where E{Æ} denotes the expectation operator. In addition, the linear correlation between different response quantities can be computed:

qil ¼ Efðri � liÞðrl � llÞg=ðrirlÞ

� PN

k¼1ðriðx ðkÞÞ� liÞðrlðxðkÞÞ� llÞPN

k¼1ðriðxðkÞÞ� liÞ 2 � PN

k¼1ðrlðxðkÞÞ� llÞ 2

h i1=2 : ð4Þ

The above estimates (right hand side expression) are ran- dom variables themselves, which fluctuate randomly. The fluctuation diminishes with an increasing sample size N, where the mean li and standard deviation ri ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varfrig

p have a coefficient of variation of �

ffiffiffiffiffiffiffiffiffi 1=N

p . For assessing

the variability of the response, a sample size N in the order of 30–100 suffices for the first step to obtain estimates on the mean and the variance of the response. Hence the esti- mates will vary approximately 20% and 10% when using a sample size N of 25 and 100, respectively. This accuracy generally suffices to get a good picture on the variability of the response.

The above simple and straight forward procedure becomes more involved for cases where correlations between random variables need to be considered. Uncer- tain structural properties like Young’s modulus in a contin- uous beam or plate, the thickness of a plate, geometric imperfections of shells, fluctuation of pressure due to wind loading and earthquake ground motion are continuous with respect to space or time. Hence, uncertain properties in the close neighborhood are generally strongly correlated, while the mutual dependency diminishes with the distance in space and/or time. Such continuous uncertain properties are described mathematically by random fields where the

correlation coefficient tends to one as the distance tends to zero. When the uncertain properties are modeled by ran- dom variables, these variables Xj, are now a function of time t (e.g. earthquake acceleration at a specific site), of the spatial position p (e.g. geometric imperfection of shells) or of both (p, t) as for wind pressure which varies continu- ously with respect to time and space. The minimal informa- tion needed to describe a random field is the specification of the mean,

ljðpÞ¼ EfX jðpÞg ð5Þ

and the covariance,

Cjðp1; p2Þ¼ EfðX jðp1Þ� ljðp1ÞÞðX jðp2Þ� ljðp2ÞÞg: ð6Þ

For the case where the f(xj(p)) is normally distributed, the random field can be approximated very conveniently by the so called Karhunen–Loève expansion [6,7],

X jðpÞ¼ x ½0� j ðpÞþ

Xm k¼1

nk � x ½k� j ðpÞ ð7Þ

where fx½k�j g m k¼0 are deterministic functions with respect to

time and space. This representation introduces just m inde- pendent standard normal random variables nk with the property of zero mean and unit standard deviation:

Efnkg¼ 0; Efnknlg¼ dkl ð8Þ with dk=l = 1 and dk5l = 0. The Karhunen–Loève expan- sion is related to the required specification of the mean and covariance by the following relations:

ljðpÞ¼ x ½0� j ðpÞ ð9Þ

Cjðp1; p2Þ¼ Xm k¼1

x½k�j ðp1Þx ½k� j ðp2Þ ð10Þ

A schematic sketch of a random field and its autocorrela- tion function is shown in Fig. 4. Frequently an exponential autocorrelation function is used as a model. The distance between two points of the field, e.g. xi and xj is generally defined by a correlation length, lf. For the case where no continuous random field is involved, but just discrete ran- dom variables which are correlated, a quite similar repre- sentation exists as shown above for random fields, where

–1 –0.5

0 0.5

1

–1

0

1 0

0.5

1

xi xj

C ff

( x i

,y j)

a b

Fig. 4. (a) Characterization of random fluctuations by (b) its exponential autocorrelation function.

G.I. Schuëller / Computers and Structures 85 (2007) 235–243 239

the free continuous vector p will be replaced by discrete values.

It is generally recognized that direct Monte Carlo simu- lation (MCS) is the most general and versatile method to process uncertainties. Moreover, as mentioned before, it allows the direct use of the deterministic FE structural models and hence the respective general purpose software (see e.g. [8]). However, a serious drawback is the enormous computational efforts, which are already required even for medium size problems. Hence, various techniques have already been developed to reduce these computational efforts, i.e. by trying to guide the samples into the domain of interest, which may be – e.g. for reliability problems – the region of structural failure, i.e. g(x) 6 0, where g(x) = 0 represents the limit state function of a structure. If the domain of interest in the input space is either known or can be computed by gradient based optimization proce- dures (for linear uncertain structures and responses) direct approaches such as importance sampling (see e.g. [9]) and line sampling (see e.g. [10]) may be applied. For cases where the domain of interest is too complex to be deter- mined efficiently (e.g. for strongly non linear responses) evolutionary strategies, such as double and clump [11], Russian roulette and splitting (see e.g. [12]) and subset sim- ulation [13] are applicable. A recent benchmark study [14] has shown that for highly non linear problems, subset sim- ulation appears to be the most versatile and efficient proce- dure. Subset simulation overcomes the computational inefficiency of direct Monte Carlo simulation for estimating small probabilities, by expressing such an event as a prod- uct of larger, conditional probabilities. This is achieved by defining a decreasing sequence of events denoted as subsets.

4. Practical applications

In this section it is shown that the procedures as described in the above sections allow rational uncertainty and reliability analysis of large scale engineering structures.

The necessary computational efforts remain within accept- able limits.

4.1. Coupled load analysis of launcher-mounted satellite

In this example a large-scale FE model of an aerospace application is analyzed, specifically the dynamic response of a satellite structure mounted on a launcher rocket. This is accomplished by reducing the problem with the sub- structuring technique known as ‘‘Craig–Bampton’’ method [15]. Clearly, in a FE model with this complexity and size, the uncertainty about its parameters is significant and unavoidable. In the present example, this random scatter is modeled in a probabilistic framework, by treating the parameters as random variables and estimating the result- ing uncertainty in the response via advanced Monte Carlo simulation.

The currently available computational procedures for the uncertainty and reliability analysis of dynamical sys- tems are assessed and compared, e.g. in [16]. It is concluded that the Karhunen–Loève expansion is advantageous when uncertainty estimation is required, while (advanced) Monte Carlo simulation procedures are recommended when reli- ability estimates are sought.

Fig. 5 shows the FE model of the launcher-satellite assemblage. It is divided into sub-structures, specifically the two lateral solid propellant boosters, the main stage and the upper composite at the top. The latter contains the payload, in the present case the satellite. The displayed sub-division reflects the decomposition of the numerical model applying the Craig–Bampton method. The entire model contains approximately 170,000 degrees of freedom, where the satellite contains roughly 120,000 DOFs, and the solid propellant boosters, the main stage and the upper composite approximately 12,000, 8800 and 16,000 DOFs, respectively. The FE model of the satellite is significantly more refined than that of the launcher components, since the response quantities of interest are exclusively located in the satellite. After the Craig–Bampton reduction the

Fig. 5. (a) Ariane rocket (Foto: courtesy of ESA); (b) FE model coupled launcher-satellite structure (courtesy of ESA).

240 G.I. Schuëller / Computers and Structures 85 (2007) 235–243

number of DOFs of the coupled system amounts to approximately 800.

For the Monte Carlo simulation all material and geome- try parameters specified in the model input files have been treated as random variables. This results in a total of around 1300 random variables both for the satellite and the launcher. In order to separately assess the effects of the uncertainties in the launcher on one hand and in the satellite on the other hand, two sets of analyses have been per- formed. In the first set, the launcher properties have been varied randomly and the satellite set deterministically, and

Fig. 6. (a) Load case 1: lift-off – time domain; (b) load case 2: end-of- booster pressure – frequency domain.

vice-versa for the second set. The magnitude of the scatter in terms of the coefficient of variation is based on experimen- tal data and on experience with similar structures and varies in the range of 4–6%. The distribution type is mostly Gauss- ian, with the exception of the viscous damping ratios, which are log-normal; for these the C.o.V. has been set signifi- cantly higher, namely 40%, since considerable uncertainty is usually associated with damping. For a cross-section of the uncertainties of the types of material and geometrical properties it is referred to Table A.1. The load cases consid- ered in this example are the so-called Lift-off (LO, Figs. 6a and 7) and end-of-booster-pressure oscillation (EBP, Figs. 6b and 8), which takes place about 2 min after the ignition of the boosters at lift off. In essence, this load case simulates the pressure oscillation inside the combustion chamber of the solid propellant booster, cf. Fig. 6b. The linear fre- quency response analysis covers the frequency range from 30 to 54 Hz, with intervals of 0.25 Hz and has been per- formed with the commercial FE code MSC.Nastran.

The purpose of the analysis is to estimate the effect of the uncertainties in the launcher and the satellite on the structural response. The considered critical response quan- tity is the von Mises stress in the beam connecting the solar panel to the satellite structure. The location of this beam is depicted in Fig. 9.

The results of the probabilistic analysis for the von Mises stress in the solar panel connector are synthesized in Fig. 10. In this case the results of the advanced simula- tion procedures, such as line sampling and subset simula- tion are compared to direct Monte Carlo simulation (for pF P 10

�2). The result of the various methods are in very good agreement. Fig. 10 also shows the very low number of samples required by the advanced simulation methods in order to match the computationally demanding direct

40000

45000

50000

55000

60000

65000

70000

75000

80000

85000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

N

time in seconds

99.8% 99.0% 90.0% 80.0% 50.0% 20.0% 10.0% 1.0%

nominal

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 4

4.5

5

5.5

6

6.5

7

7.5

8 x 104

Time t

In te

rf a ce

f o rc

e [

N ]

Mean μ μ–σ μ+σ

Fig. 7. Monte Carlo simulation, lift-off load case – interface force.

0

100

200

300

400

500

600

700

30 35 40 45 50 55

N

Frequency in [Hz]

99.8% 99.0% 90.0% 80.0% 50.0% 20.0% 10.0%

1.0% nominal

30 35 40 45 50 55 –50

0

50

100

150

200

250

300

350

Frequency ω [rad/s]

In te

rf a ce

f o rc

e [ N

]

Mean μ μ–σ μ+σ

Fig. 8. Monte Carlo simulation, EPB load case – interface force.

Fig. 9. Location of beam connecting the solar panel.

G.I. Schuëller / Computers and Structures 85 (2007) 235–243 241

Monte Carlo simulation. This shows that, e.g. line sampling and subset simulation enable estimation of small failure probabilities (pF � 10�6) with sustainable compu- tational efforts, i.e. by requiring about 400 simulations only.

5. Conclusions and outlook

It was in the pre computer times and its early stages of development when the computational efforts to process uncertainties were prohibitive. In those times the use of

Table A.1 Assumed coefficients of variation of uncertain finite element model parameters

Element/material type Property C.o.V. (r/l)

Isotropic material Young’s modulus 8% Poisson’s ratio 3% Shear modulus 12% Mass density 4%

Orthotropic shell element material

Young’s modulus 8% Shear modulus 12% Mass density 4%

Solid element an-isotropic material

Mat. property matrix 12% Mass density 4%

Simple beam Section dimension 5% Non-structural mass 8%

Layered composite Non-structural mass material

8%

Thickness of plies 12% Orientation angle r = 1.5�

Shell element Membrane thickness 4% Non-structural mass 8%

Spring element Stiffness 10% Concentrated mass Mass 3% Damping Modal damping 20%

Fig. 10. Reliability analysis, static load case – shear force solar panel.

242 G.I. Schuëller / Computers and Structures 85 (2007) 235–243

the deterministic conception, which in fact is a simplifica- tion of the realistic situation, was absolutely necessary. But the breathtaking developments in the computer tech- nology (hardware) lay the ground for the reconsideration of the traditional structural analysis procedures. The possi- bility of parallel and distributed computing, the further development of Monte Carlo simulation procedures, etc., on the software side accelerate this process even more.

So far the following conclusions can be drawn:

• uncertainties in structural analysis and design can be treated quantitatively – even for larger types of engineer- ing structures;

• further efforts to increase computational efficiency are needed;

• quantitative uncertainty analysis should become an inherent part of engineering structural analysis;

• such uncertainty analysis ensures a robust prediction of variability, such as model verification, safety and reli- ability assessment, etc.;

• uncertainty and reliability analysis forms the basis for a rational reliability based optimization of structures.

It is envisaged, that not before long uncertainty analysis will be an inherent part of engineering structural analysis. Such uncertainty analysis ensures the robust prediction of variability, among many other possibilities, such as model verification, safety assessment, etc.

Acknowledgement

This research was partially supported by the Austrian Research Council (FWF) under contract number P16769- N12, which is gratefully acknowledged by the author.

The supply of the FE-model and the support by ESA/ES- TEC, Noordwijk is also deeply appreciated.

Appendix A

See Table A.1.

G.I. Schuëller / Computers and Structures 85 (2007) 235–243 243

References

[1] Lin YK. Probabilistic theory of structural dynamics. New York: Krieger; 1976.

[2] Bathe K-J. Finite element procedures. Englewood Cliffs (NJ): Pren- tice Hall; 1996.

[3] Oberkampf WL, Trucano TG, Hirsch C, Verification, validation and predictive capability in computational engineering and physics. Technical Report No. SAND2003-3769, Sandia National Laborato- ries, Albuquerque, NM, USA; 2003.

[4] Fishman GS, Monte Carlo: concepts, algorithms, and applications, Springer Verlag, Springer Series in Operations Research, New York; 1996.

[5] Brown BW, Lovato J, Russel K, Venier J, RANDLIB Library of Routines for Random Number Generation (C and Fortran).

[6] Loève M. Probability theory. 3rd ed. Princeton (NJ): D. Van Nostrand Company, Inc.; 1963.

[7] Ghanem RG, Spanos PD. Stochastic finite elements: a spectral approach. New York: Springer Verlag; 1991.

[8] Schuëller GI, editor. Structural reliability software. Struct Saf 2006;28(1–2):1–216 [special issue].

[9] Schuëller GI, Stix R. A critical appraisal of methods to determine failure probabilities. Struct Saf 1987;4:293–309.

[10] Koutsourelakis PS, Pradlwarter HJ, Schuëller GI. Reliability of structures in high dimensions, part I: algorithms and applications. Probabilistic Eng Mech 2004;19(4):409–17.

[11] Pradlwarter HJ, Kliemann W. First exit times in non-linear dynam- ical systems by advanced Monte Carlo simulation. In: Lin YK, Su TC, editors. Proceedings of the 11th ASCE engineering mechanics conference, vol. 1. New York (NY)/Fort Lauderdale: ASCE; 1996. p. 523–6.

[12] Pradlwarter HJ, Schuëller GI. Assessment of low probability events of dynamical systems by controlled Monte Carlo simulation. Prob- abilistic Eng Mech 1999;14:213–27.

[13] Au SK, Beck J. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Eng Mech 2001;16: 263–77.

[14] Schuëller GI, Pradlwarter HJ, Beck JL, Au SK, Katafygiotis LS, Ghanem R. Benchmark study on reliability estimation in higher dimensions of structural systems – an overview. In: Soize C, Schuëller GI, editors. Structural dynamics EURODYN 2005 – Proceedings of the 6th international conference on structural dynamics. Rotterdam (Paris, France): Millpress; 2005. p. 717–22.

[15] Craig RR, Bampton MCC. Coupling of substructures for dynamic analysis. AIAA J 1968;6(7):1313–9.

[16] Schuëller GI. Uncertainty and reliability analysis of structural dynamical systems. In: Soares CM, Martins J, Rodrigues H, Ambrósio J, editors. Computational mechanics – solids, structures and coupled problems. The Netherlands (EU): Springer; 2006. p. 541–54.