Research On The Reliability And Sensitivity Analysis Of Structural Systems

Uncertainties are ubiquitous throughout the design and services of aeronautical structuralsystems. These uncertainties, on the one hand, lead to divergence of output performance ofstructural systems, on the other hand, result in occasionality of failures of structural systems.To study the propagation of these uncertainties in the structural systems so as to estimate thereliability and to explore the contributions of these uncertainties to the divergence of outputperformance as well as the failure probability can be very beneficial to the reliability androbust design of structural systems. Focusing on the reliability analysis, sensitivity analysisand uncertainty-based optimization of structural systems, this thesis presents the followingcontributions:1. For the reliability analysis of structural system with multiple failure modes, the additionlaws of failure probability are presented and proved. These addition laws indicate that thereliability problem of a structural system with m failure modes can be transformed into thereliability analysis of2m-1single failure modes, thus they make it possible to solve theproblem of reliability analysis of multiple failure modes by the methods developed for singlemode. Further, based on the addition laws of failure probability and linear programmingmodel, a bound method is proposed. With the constraints of several former order joint failureprobabilities, the proposed bound method can produce a narrow bound that contains the truevalue of the failure probability of structural system. At last, the addition laws and the boundmethod are applied to the reliability analysis of structural system with both random and fuzzyinput variables as well as multiple failure modes.2. As the classical variance-based sensitivity indices (also called Sobol’s indices) can notcorrectly reflect the relative importance of correlated inputs, a set of independent auxiliaryvariables is firstly introduced based on the Mahalanobis transformation, and the model outputvariance is then attributed to each correlated input through these auxiliary variables. With thisprinciple, the generalized variance-based sensitivity indices are proposed to correctly reflectthe relative importance of correlated inputs. Then, the variance-based sensitivity indices areapplied to reliability analysis, and the global reliability sensitivity analysis (GRSA) techniqueis developed. The GRSA technique can correctly detect the input variables that make majorcontributions to the failure probability and those inputs that make no contributions to thefailure probability. These sensitivity information are very valuable for reliability design ofstructural system and model simplification. For efficiently estimating the GRSA indices, threenumerical methods, i.e., single-loop Monte Carlo simulation, importance sampling andtruncated importance sampling, are developed. At last, the GRSA technique is applied to theflutter model of a two-dimensional airplane wing. 3. The moment ratio functions are developed and applied to sensitivity analysis. First, theclassical regional sensitivity analysis technique is improved. The regional moment ratiofunctions are introduced to measure the contribution of different distribution ranges of modelinputs to the moments of model output, and to quantify the changes of model output momentswhen the distribution ranges of model inputs are reduced. An efficient algorithm is alsodeveloped to estimate the regional moment ratio functions. This technique is ultimatelyapplied to a fault tree model that represents the unilateral asymmetric movement failure of anaircraft flap mechanism. Second, the parametric moment ratio functions are devised toquantify the sensitivity of model output moments to the distribution parameters of modelinputs, and the (progressively) unbiased Monte Carlo estimators are also derived. Third, theregional and parametric sensitivity analyses of the Sobol’s indices are introduced. Comparedwith the Sobol’s indices, the proposed techniques can provide much more plenty of sensitivityinformation without extra computational cost. Fourth, by reducing the distribution ranges ofmodel inputs, a new framework of variance-based sensitivity indices, called W-indices, aredefined, and three numerical algorithms are developed for computing the W-indices. Both theSobol’s indices and the W-indices are applied to a flap structure of an airplane. Results showthat, compared with the Sobol’s indices, the W-indices are more suitable for reducing themodel output uncertainty.4. The lack of efficient algorithm for computing the moment-independent sensitivityindices (also called delta indices) and inadequate knowledge of their physical interpretationshave prevented the delta indices from wide application. Thus, an efficient algorithm, calledsingle-loop Monte Carlo simulation, is proposed. This method estimates all the delta indiceswith only one set of samples, thus compared with classical algorithm, the efficiency issubstantially improved. Then, the Copula function is adopted to study the delta indices. It isfound that the delta indices can be interpreted as the dependence measures between modeloutput and inputs. Based on this observation, new moment-independent sensitivity indices(called extended delta indices) are introduced, and efficient numerical algorithms based onCopula function are also proposed for estimating the delta and extended delta indices. At last,the regional moment-independent sensitivity analysis as well as an efficient algorithm forimplementing it are proposed. The proposed method can inform the analysts with thecontributions of different distribution ranges of model inputs to the uncertainty of modeloutput without extra calling the model function, thus provides plenty of sensitivityinformation for enhancing the robustness of model prediction and output performance ofstructural systems.5. Inspired by the importance sampling, the extended Monte Carlo simulation (EMCS)method is developed to estimate probabilistic response functions (the functional relationshipsbetween the probabilistic responses and the distribution parameters of model inputs). With theEMCS method, only one set of samples is needed for estimating all the probabilistic responsefunctions, thus compared with the classical methods, the EMCS method is much more efficient. Then the EMCS method is applied to deal with the parametric global sensitivityanalysis (PGSA) of model inputs with both aleatory and epistemic uncertainties and theparametric optimization (PO) problem (such as robust design optimization). A sensitivityindex called R-index is also introduced to overcome the over-parameterized problemencountered in the optimization process. The results show that, by the EMCS methods, boththe PGSA and PO problems can be well solved with only one set of samples.