The reliability analysis and reliability-based optimization of laminated circular cylinders under axial buckling instability are studied. Structural reliability is measured in terms of Hasofer-Lind reliability index. The response surface models are used in both the calculation of the reliability index and the reliability-based optimization. In reliability analysis, both deterministic and probabilistic sensitivity factors are investigated; the results show that the reliability index is most sensitive to the applied load and Young's modulus of the material. Two cases are considered in the optimization study. In the first case, the cylinder weight is minimized subject to a reliability constraint whereas in the second case, cylinder reliability is maximized subject to a weight constraint. In addition, two different optimization techniques are studied. In the first technique, a global response surface model of the buckling response based on 3000 Monte Carlo simulations is used for the design optimization whereas in the second technique, multiple local regression models, with each based on approximately 20 simulations, are used in sequential search of an optimum design. An optimum design is found. The results based on sequential application of multiple local regression models are close to those from global optimization while the former is much more efficient in terms of computational cost.

A two part study is performed to investigate the application of reliability-based design optimization (RBDO) approach to design elastic-plastic stiffener beams with Tsection. The objectives of this study are to evaluate the benefits of reliability-based optimization over deterministic optimization, and to illustrate through a practical design example some of the difficulties that a design engineer may encounter while performing reliability-based optimization. Other objectives are to search for a computationally economic RBDO method and to utilize that method to perform RBDO to design an elastic-plastic T-stiffener under combined loads and with flexural-torsional buckling and local buckling failure modes. First, a nonlinear elastic-plastic T-beam was modeled using a simple 6 degree-of-freedom non-linear beam element. To address the problems of RBDO, such as the high non-linearity and derivative discontinuity of the reliability function, and to illustrate a situation where RBDO fails to produce a significant improvement over the deterministic optimization, a graphical method was developed. The method started by obtaining a deterministic optimum design that has the lowest possible weight for a prescribed safety factor (SF), and based on that design, the method obtains an improved optimum design that has either a higher reliability or a lower weight or cost for the same level of reliability as the deterministic design. Three failure modes were considered for an elastic-plastic beam of T cross-section under combined axial and bending loads. The failure modes are based on the total plastic failure in a beam section, buckling, and maximum allowable deflection. The results of the first part show that it is possible to get improved optimum designs (more reliable or lighter weight) using reliability-based optimization as compared to the design given by deterministic optimization. Also, the results show that the reliability function can be highly non-linear with respect to the design variables and with discontinuous derivatives. Subsequently, a more elaborate 14-degrees-of-freedom beam element was developed and used to model the global failure modes, which include the flexural-torsional and the out-of-plane buckling modes, in addition to local buckling modes. For this subsequent study, four failure modes were specified for an elasticplastic beam of T-cross-section under combined axial, bending, torsional and shear loads. These failure modes were based on the maximum allowable in-plane, out-ofplane and axial rotational deflections, in addition, to the web-tripping local buckling. Finally, the beam was optimized using the sequential optimization with reliabilitybased factors of safety (SORFS) RBDO technique, which was computationally very economic as compared to the widely used nested optimization loop techniques. At the same time, the SOPSF was successful in obtaining superior designs than the deterministic optimum designs (either up to12% weight savings for the same level of safety, or up to six digits improvement in the reliability for the same weight for a design with Safety Factor 2.50). / Ph. D.

En raison des propriétés physiques, électriques, mécaniques et chimiques exceptionnelles, les nanotubes de carbone(CNTs) sont considérés comme l'un des nanomatériaux les plus importants aujourd'hui. CNTs peuvent être classées comme des nanotubes de carbone à paroi simple (SWCNTs) ou des nanotubes de carbone à parois multiples (MWCNTs). La structure d'un SWCNT peut être vue comme une couche d’un atome de graphite de cylindres laminés. Les propriétés des SWCNTs sont fondamentales pour la recherche et pour de nombreuses applications en médecine, en électronique, en environnement, pollution... Le but de cette thèse est d'étudier les propriétés mécaniques et électromécaniques des matériaux SWCNTs avec différentes morphologies. Nous avons étudié les propriétés élastiques des SWCNTs individuels en utilisant la méthode des éléments finis (FE). Nous avons montré que les modules élastiques de SWCNTs dépendent du diamètre, chiralité et la longueur. Les modules élastiques augmentent significativement selon les plus petites valeurs du rayon. Lorsque le rayon devient plus grand, tous les modules élastiques convergent vers une valeur constante asymptotique. En outre, les modules de zigzag et SWCNT chirales sont plus sensibles à la variation du rayon par rapport à SWCNT fauteuil. Nous avons testé le module élastique de SWCNT film mince par nanoindentation. Nous avons trouvé que le module d'Young E et la dureté H de SWCNT film mince sont séparément E= 192.831± 13.922 GPa, H=12.57719 ± 0.759 GPa. Nous avons élaboré un modèle par éléments finis qui représente le comportement des SWCNT film mince dans le processus de nanoindentation. Dans cette étude nous avons élaboré un modèle élasto-plastique pour décrire le comportement mécanique du matériau au cours de l'indentation. Ensuite, nous avons évalué les propriétés élasto-plastiques de SWCNT film mince par l’interaction de la simulation par éléments finis avec les données statistiques du test expérimental. Le modèle bilinéaire proposé approxime bien la performance de SWCNT film mince dans la nanoindentation. Finalement, nous avons étudié les propriétés électrostrictives du composite (P (VDF-TrFE) / SWCNT) à base de SWCNTs par la méthode des éléments finis. Les résultats numériques trouvés montrent que l'électrostriction du composite SWCNT / P (VDP-TrFE) est considérablement dépendant de la fraction volumique de SWCNT et du rapport des constantes diélectriques (SWCNT et P (VDP-TrFE)). Dans ce travail, nous avons constaté que les propriétés des CNTs obtenues théoriquement ou expérimentalement impliquent des inévitables incertitudes. Donc la prise en compte des incertitudes des propriétés des CNTs devient nécessaire. Pour l’analyse de ces incertitudes, nous avons appliqué la méthode de l'optimisation basée sur la fiabilité (RBO). Cette méthode est un outil efficace pour assurer la fiabilité des résultats expérimentaux et numériques dans le processus d'estimation du comportement élasto-plastique pour SWCNT film mince. / Carbon nanotubes (CNTs) as one of the most important nanomaterials today have been demonstrated a combination of exceptional physical, electrical, mechanical, and chemical properties, which have resulted in their great potential of industrial application in many fields. CNTs can be categorized as single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The structure of a SWCNT can be viewed as one-atom-thick layer of graphite rolled cylinder. Well understand the property of SWCNTs is fundamental in the exploration of research and applications of CNTs based products.The present research is focus on evaluating the mechanical and electromechanical properties of SWCNTs materials with different material morphology. Under the hierarchical (or bottom-up) ideal, the elastic properties of the individual SWCNTs were studied by using Finite element (FE) method. Effects of the diameter, chirality and length on the elastic moduli of SWCNTs are discussed based on numerical calculations. Furthermore, the ultra-thin SWCNTs film (~200nm) is prepared by spin coating method. The elastic modulus of SWCNT thin film is estimated by nanoidentation test. Its elasto-plastic properties were then determined by FE simulation combined with the statistics constraints of the experimental results. The results showed that the mechanical performance of SWCNTs thin film during indentation can be approximately represented by a bilinear model. The mechanical parameters of SWCNTs thin film obtained by experiment and numerical calculation are : the Young’s modulus E=192.83± 13.922 Gpa, the tangent modulus Et ≈ 42GPa, and the yield stress Oy ≈ 8.4GPa, respectively. The electrostrictive properties of SWCl\lT- based composite (P(VDF-TrFE)/SWCNT) were also investigated by FE method. Numerical results show that the electrostriction of the SWCNT/P(VDP-TrFE) composite is greatly dependent on the volume fraction of SWCNT and the difference of dielectric constant between SWCNT and P (VDP-TrFE) copolymer. In this work, we found that the properties of CNTs obtained either by theory or by experiments involve inevitable uncertainty, and some are relatively large. Therefore, uncertainty analysis for the predicted properties of CNTs becomes necessary with the increasing product performance demands. The application of Reliability-Based Optimization (RBO) method in the process of elasto-plastic behavior estimation for SWCNT thin film indicates that RBO method should be an effective tool to ensure the reliability of experimental and numerical results.

This thesis discusses a methodology for probabilistic design optimization of aircraft structures subject to a multidisciplinary set of requirements originating from the desire to minimize structural weight while fulfilling the demands for quality, safety, producibility, and affordability. With this design methodology as the framework, a software is developed, which is capable of performing design optimization of metallic built-up beam structures where the material properties, external load, as well as the structural dimensions are treated as probabilistic random variables. The structural and failure analyses are based on analytical and semi-empirical methods whereas the component reliability analysis is based on advanced first-order second moment method. Metrics-based analytical models are used for the manufacturability analysis of individual parts with the total manufacturing cost estimated using models derived from the manufacturing cost / design guide developed by the Battelle¡¯s Columbus Laboratories. The resulting optimization problem is solved using the method of sequential quadratic programming. A wing spar design optimization problem is used as a demonstrative example including a comparison between non-buckling and buckling web design concepts. A sensitivity analysis is performed and the optimization results are used to highlight the tradeoffs among weight, reliability, and manufacturing cost.

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Hoje em dia, é amplamente reconhecido que o projeto de estruturas otimizadas deve ser robusto em relação a incertezas nas forças, geometria e propriedades do material. Entretanto, existem diversas alternativas para considerar tais incertezas em problemas de otimização estrutural. Esta tese apresenta quatro formulações para lidar com incertezas no problema de otimização topológica com restrição de tensão. As três primeiras são desenvolvidas para lidar com incertezas na intensidade e direção das forças aplicadas: 1) formulação robusta probabilística, onde substituem-se as restrições de tensão originais por uma soma ponderada entre os seus valores esperados e desvios padrão, obtidos por meio do método de perturbação de primeira ordem; 2) formulação baseada em confiabilidade, onde consideram-se restrições de tensão probabilísticas; o problema é formulado por meio de uma abordagem acoplada de primeira ordem; 3) formulação robusta não probabilística, onde considera-se o pior cenário possível para as restrições de tensão; o problema é formulado com uma abordagem acoplada de otimização com anti-otimização. A quarta formulação não segue o padrão das três primeiras; diferente das demais, esta é desenvolvida para lidar com incerteza uniforme de manufatura: 4) formulação robusta de três campos, onde três topologias são consideradas de forma simultânea durante o processo de otimização, de forma a simular possíveis imperfeições que possam ocorrer devido a erros de manufatura. As quatro abordagens são bastante diferentes na forma de lidar com as incertezas; no entanto, o procedimento de solução é o mesmo: a abordagem baseada em densidade é empregada na parametrização material, enquanto que o método do Lagrangiano aumentado é empregado para solucionar o problema resultante, de forma a lidar com o elevado número de restrições de tensão. Diversos exemplos são solucionados para mostrar a aplicabilidade das formulações propostas. Os exemplos são posteriormente verificados através da Simulação de Monte Carlo e comparados com os resultados determinísticos. Os resultados mostram que as estruturas obtidas com a abordagem tradicional determinística são extremamente sensíveis a incertezas. As formulações desenvolvidas nesta tese, por outro lado, mostraram-se alternativas válidas a formulação determinística, fornecendo resultados robustos e confiáveis na presença de incertezas. / It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads, geometry and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This thesis addresses four formulations to handle uncertainties in topology optimization with stress constraint. The first three are developed to handle uncertainties in magnitude and direction of applied loads: 1) probabilistic robust formulation, where the original stress constraints are replaced by a weighted sum between their expectations and standard deviations; these are obtained by first-order perturbation approach; 2) reliability-based formulation, where probabilistic stress constraints are considered; the problem is formulated by a coupled first order approach; 3) non-probabilistic robust formulation, where the worstcase scenario for the stress constraints is considered; the problem is formulated by a coupled approach called optimization with anti-optimization. The fourth formulation is quite different from the first three; it is developed to handle uniform boundary variation: 4) three-field robust approach, where three topologies are simultaneously considered during the optimization process, in order to simulate imperfections which may occur due to manufacturing errors. These four formulations are quite different in handling with uncertainties; however, the solution rocedure is the same: the density approach is employed to material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Several examples are solved to demonstrate applicability of proposed formulations. Numerical examples are further verified via Monte Carlo Simulation and compared to deterministic results. The results show that the structures obtained with raditional deterministic formulation are extremely sensitive to uncertainties. On the other hand, the formulations developed in this thesis are shown to be valid alternatives to the deterministic formulation, providing robust and reliable results in the presence of uncertainties.

Le présent travail traite l’optimisation des paramètres des amortisseurs à masses accordées (AMA) accrochés sur des structures, linéaires. Les AMAs sont des dispositifs de contrôle passif utilisés pour atténuer les vibrations induites par des chargements dynamiques (en particulier stochastiques) appliqués sur des structures. L’efficacité de tels dispositifs est étroitement liée aux caractéristiques dynamiques qu’on doit imposer à ces systèmes. Dans ce cadre, plusieurs stratégies d’optimisation peuvent être utilisées dans des contextes déterministes et non déterministes, où les paramètres de la structure à contrôler sont incertains. Parmi les différentes approches qu’on peut trouver dans la littérature, l’optimisation structurale stochastique (OSS) et l’optimisation basée sur la fiabilité (OBF) étaient particulièrement traitées dans le présent travail.Dans la première partie de ce travail, en plus de la nature stochastique des chargements extérieurs appliqués à la structure linéaire à contrôler, la présence de paramètres structuraux de type incertains mais bornés (IMB) est prise en considération et les bornes optimales des paramètres AMA ont été calculées. Le calcul de ces bornes a été fait en utilisant une technique basée sur un développement de Taylor suivi d’une extension aux intervalles. La technique, permettant l’obtention d’une approximation des bornes optimales, a été appliquée dans les cas d’un système à un degré de liberté (1DDL) et un autre à plusieurs degrés de libertés (nDDL). Les résultats obtenus ont montrés que la technique utilisée était bien adaptée pour la stratégie OSS et elle l’est moins pour l’approche OBF.Comme suite logique aux résultats de la première partie, la seconde partie de la présente dissertation est consacrée à la présentation de deux méthodes permettant l’obtention des bornes exactes et des bornes approximées des paramètres optimaux de l’AMA et ce, en présence de paramètres structuraux de type IMB. La première méthode est celle de la boucle d’optimisation continue imbriquée, la seconde est celle des extensions aux intervalles basées sur la monotonie. Les méthodes présentées, qui ont été appliquées avec l’approche OBF, sont valables pour n’importe quel problème d’optimisation faisant intervenir des paramètres de type IMB. Mis à part le calcul de bornes optimisées du dispositif AMA, la question de la robustesse, vis-à-vis des incertitudes structurales, a été également traitée et il a été prouvé que la solution optimale correspondante au contexte déterministe était la plus robuste.L’introduction d’une nouvelle stratégie OBF des paramètres AMA a fait l’objet de la troisième partie de cette dissertation. En effet, un problème OBF est toujours relié à un mode de défaillance caractérisé par le franchissement d’une certaine réponse, de la structure à contrôler, d’un certain seuil limite pendant une certaine durée de temps. Le nouveau mode de défaillance, correspondant à la nouvelle stratégie OBF, consiste à considérer qu’une défaillance ait lieu lorsque la puissance dissipée au niveau de la structure à contrôler, pendant une période de temps, excède une certaine valeur. Faisant intervenir l’approche par franchissement ainsi que la formule de Rice, la nouvelle stratégie a été appliquée dans le cas d’un système 1DDL et l’expression exacte de la probabilité de défaillance est calculée. En se basant sur une approximation mettant en œuvre la technique du minimum d’entropie croisé, la nouvelle stratégie a été, également, appliquée dans le cas d’un système à nDDL et les résultats obtenus ont montrés la supériorité de cette stratégie par rapports à deux autres tirées de la bibliographie. / The present work deals with the parameters optimization of tuned mass dampers (TMD) used in the control of vibrating linear structures under stochastic loadings. The performance of the TMD device is deeply affected by its parameters that should be carefully chosen. In this context, several optimization strategies can be found in the literature and among them the stochastic structural optimization (SSO) and the reliability based optimization (RBO) are particularly addressed in this dissertation.The first part of this work in dedicated to the calculation of the optimal bounds solutions of the TMD parameters in presence of uncertain but bounded (UBB) structural parameters. The bounds of the optimal TMD parameters are obtained using an approximation technique based on Taylor expansion followed by interval extension. The numerical investigations applied with one degree of freedom (1DOF) and with multi-degree of freedom (multi-DOF) systems showed that the studied technique is suitable for the SSO strategy and that it’s less appropriate for the RBO strategy.As immediate consequence of the obtained results in the first part of this work, in the second part a method, called the continuous-optimization nested loop method (CONLM), providing the exact range of the optimal TMD parameters is presented and validated. The numerical studies demonstrated that the CONLM is time consuming and to overcome this disadvantage, a second method is also presented. The second method is called the monotonicity based extension method (MBEM) with box splitting. Both methods have been applied in the context of the RBO strategy with 1DOF and multi-DOF systems. The issue of effectiveness and robustness of the presented optimum bounds of the TMD parameters is also addressed and it has been demonstrated that the optimum solution corresponding to the deterministic context (deterministic structural parameters) provide good effectiveness and robustness.Another aspect of RBO approach is dealt in the third part of the present work. Indeed, a new RBO strategy of TMD parameters based on energetic criterion is presented and validated. The new RBO approach is linked to a new failure mode characterized by the exceedance of the power dissipated into the controlled structure over a certain threshold during some interval time. Based on the outcrossing approach and the Rice’s formula, the new strategy is firstly applied to 1DOF system and exact expression of the failure probability is calculated. After that, a multi-DOF system is considered and the minimum cross entropy method has been used providing an approximation to the failure probability and then the optimization is carried out. The numerical investigations showed the superiority of the presented strategy when compared with other from the literature.