A practical engineering design problem is used to examine the over-conservativeness of designs obtained using deterministic optimization with worst-case parameter assumptions and a safety factor. Additionally, an attempted application of reliability-based design optimization (RBDO) demonstrates the limits of RBDO for practical problems. The design problem considered here is TESCO's Internal Casing Drive System (ICDS), which is used in feeding pipeline, or casing, into predrilled holes. After developing a finite element model of the ICDS, experimental data is used to successfully validate modeling methods and assumptions. The validated model is then subjected to multiple analyses to determine an appropriate design configuration to be used as the starting point for optimization. Worst-case, safety factor-based design optimization (SFBDO) is then applied considering two and three design variables, and is successful in increasing the critical load of the ICDS, Pcrit, by 35% and 45%, respectively. An efficient and recognized RBDO method, Sequential Optimization and Reliability Assessment, is selected for application to the design problem to determine an optimum design based on reliability. Due to the optimization formulation, however, SORA cannot be applied. The ICDS design problem represents a practical example that demonstrates the difficulties and limits in applying RBDO to practical engineering design problems. To evaluate the over-conservativeness of worst-case SFBDO, structural reliability analysis is performed on the deterministic optimum designs. It is found that the value of Pcrit for both the two and three variable optimum designs can be increased by 53% while maintaining acceptable probability of failure, demonstrating the over-conservativeness of the worst-case SFBDO. / Thesis (Master, Mechanical and Materials Engineering) -- Queen's University, 2008-09-05 10:51:26.273

[pt] Este trabalho compara o projeto ótimo determinístico (DDO) com o projeto ótimo baseado em confiabilidade (RBDO) de pórticos planos de concreto armado. A estrutura é modelada por uma malha de elementos finitos usando elementos de barras e considerando a não-linearidade geométrica e dos materiais. Na formulação do problema de otimização proposto as variáveis de projeto são definidas para cada elemento finito da malha. Elas são as armaduras superior e inferior das seções transversais de extremidade do elemento, a altura da seção do elemento, as áreas de armadura transversal e o parâmetro D usado para descrever os estados limites últimos de acordo com a norma brasileira NBR 6118 (ABNT, 2004). Os algoritmos de otimização utilizados são os de programação quadrática seqüencial (PQS), programação linear seqüencial (PLS) e o método das direções viáveis (MDV). As variáveis randômicas do problema de RBDO são a resistência à compressão do concreto, as resistências à tração e à compressão do aço, assim como as cargas aplicadas. As funções de comportamento são de dois tipos, a primeira é relativa à carga crítica da estrutura e a segunda ao controle de deslocamento para o estado limite de utilização. Para o cálculo da probabilidade de falha de uma função de comportamento, em cada iteração do problema de RBDO, o método FORM (PMA) utilizará o algoritmo HMV para obtenção do ponto de projeto. Análise de sensibilidade é feita pelo método analítico. / [en] This work compares the Deterministic Design Optimization (DDO) with the Reliability-Based Design Optimization (RBDO) of reinforced concrete plane frames. The structure is modeled by a finite element mesh using bar elements and considering both geometric and material nonlinearities. In the formulation of the proposed optimization problem the design variables are defined for each element of the finite element mesh. They are the areas of tensile and compressive reinforcement at the element ends, the depth of the element rectangular cross-section, the areas of shear reinforcement, and the parameter D used to describe the deformation limit sates for the element cross-sections defined according to the Brazilian code for the design of concrete structures NBR 6118 (ABNT, 2004). The optimization algorithms used are the Sequential Linear Programming (SLP), the Sequential Quadratic Programming (SQP) and the Method of Feasible Direction (MFD). The random variables of the RBDO problem are the concrete compressive strength, the steel compressive and tensile strength, as well as some applied loads. The performance functions are of two types, the first relates to the critical load of the structure and the second to the control of displacements in the serviceability state. For performing the calculation of the probability of failure for the associated performing function in each iteration of the RBDO problem, the method FORM (PMA) will be used in connection with the HMV algorithm for obtaining the project point. The sensitivity analyses are carried out by the analytical method.

The objective of this study is to develop a new modified Bayesian Kriging (MBKG) surrogate modeling method that can be used to carry out confidence-based reliability-based design optimization (RBDO) for problems in which simulation analyses are inherently noisy and standard Kriging approaches fail. The formulation of the MBKG surrogate modeling method is presented, and the full conditional distributions of the unknown MBKG parameters are derived and coded into a Gibbs sampling algorithm. Using the coded Gibbs sampling algorithm, Markov chain Monte Carlo is used to fit the MBKG surrogate model. A sequential sampling method that uses the posterior credible sets for inserting new design of experiment (DoE) sample points is proposed. The sequential sampling method is developed in such a way that the new DoE sample points added will provide the maximum amount of information possible to the MBKG surrogate model, making it an efficient and effective way to reduce the number of DoE sample points needed. Therefore, it improves the posterior distribution of the probability of failure efficiently. Finally, a confidence-based RBDO method using the posterior distribution of the probability of failure is developed. The confidence-based RBDO method is developed so that the uncertainty of the MBKG surrogate model is included in the optimization process. A 2-D mathematical example was used to demonstrate fitting the MBKG surrogate model and the developed sequential sampling method that uses the posterior credible sets for inserting new DoE. A detailed study on how the posterior distribution of the probability of failure changes as new DoE are added using the developed sequential sampling method is presented. Confidence-based RBDO is carried out using the same 2-D mathematical example. Three different noise levels are used for the example to compare how the MBKG surrogate modeling method, the sequential sampling method, and the confidence-based RBDO method behave for different amounts of noise in the response. A comparison of the optimization results for the three different noise levels for the same 2-D mathematical example is presented. A 3-D multibody dynamics (MBD) engineering block-car example is presented. The example is used to demonstrate using the developed methods to carry out confidence-based RBDO for an engineering problem that contains noise in the response. The MBD simulations for this example were done using the commercially available MBD software package RecurDyn. Deterministic design optimization (DDO) was first done using the MBKG surrogate model to obtain the mean response values, which then were used with standard Kriging methods to obtain the sensitivity of the responses. Confidence-based RBDO was then carried out using the DDO solution as the initial design point.

The objectives of this study are (1) to develop an efficient variable screening method for reliability-based design optimization (RBDO) and (2) to develop a new RBDO method incorporated with the confidence level for limited input data problems. The current research effort involves: (1) development of a partial output variance concept for variable screening; (2) development of an effective variable screening sequence; (3) development of estimation method for a confidence level of a reliability output; and (4) development of a design sensitivity method for the confidence level. In the RBDO process, surrogate models are frequently used to reduce the number of simulations because analysis of a simulation model takes a great deal of computational time. On the other hand, to obtain accurate surrogate models, we have to limit the dimension of the RBDO problem and thus mitigate the curse of dimensionality. Therefore, it is desirable to develop an efficient and effective variable screening method for reduction of the dimension of the RBDO problem. In this study, it is found that output variance is critical for identifying important variables in the RBDO process. A partial output variance, which is an efficient approximation method based on the univariate dimension reduction method (DRM), is proposed to calculate output variance efficiently. For variable screening, the variables that has larger partial output variances are selected as important variables. To determine important variables, hypothesis testing is used so that possible errors are contained at a user-specified error level. Also, an appropriate number of samples is proposed for calculating the partial output variance. Moreover, a quadratic interpolation method is studied in detail to calculate output variance efficiently. Using numerical examples, performance of the proposed variable screening method is verified. It is shown that the proposed method finds important variables efficiently and effectively. The reliability analysis and the RBDO require an exact input probabilistic model to obtain accurate reliability output and RBDO optimum design. However, often only limited input data are available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the RBDO optimum to lose its confidence level. Therefore, it is necessary to consider the reliability output, which is defined as the probability of failure, to follow a probability distribution. The probability of the reliability output is obtained with consecutive conditional probabilities of input distribution type and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to practically calculate the probability of the reliability output. A confidence-based RBDO (C-RBDO) problem is formulated using the derived probability of the reliability output. In the C-RBDO formulation, the probabilistic constraint is modified to include both the target reliability output and the target confidence level. Finally, the design sensitivity of the confidence level, which is the new probabilistic constraint, is derived to support an efficient optimization process. Using numerical examples, the accuracy of the developed design sensitivity is verified and it is confirmed that C-RBDO optimum designs incorporate appropriate conservativeness according to the given input data.

Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Recent studies have shown that considering proof test data in a Reliability Based Design Optimization (RBDO) environment can result in design improvement. Proof testing involves the physical testing of each and every component before it enters into service. Considering the proof test data as part of the RBDO process allows for improvement of the original design, such as weight savings, while preserving high reliability levels. Composite Over-Wrapped Pressure Vessels (COPV) is used as an example application of achieving weight savings while maintaining high reliability levels. COPVs are light structures used to store pressurized fluids in space shuttles, the international space station and other applications where they are maintained at high pressure for extended periods of time. Given that each and every COPV used in spacecraft is proof tested before entering service and any weight savings on a spacecraft results in significant cost savings, this thesis put forward an application of RBDO that accounts for proof test data in the design of a COPV. The method developed in this thesis shows that, while maintaining high levels of reliability, significant weight savings can be achieved by including proof test data in the design process. Also, the method enables a designer to have control over the magnitude of the proof test, making it possible to also design the proof test itself depending on the desired level of reliability for passing the proof test. The implementation of the method is discussed in detail. The evaluation of the reliability was based on the First Order Reliability Method (FORM) supported by Monte Carlo Simulation. Also, the method is implemented in a versatile way that allows the use of analytical as well as numerical (in the form of finite element) models. Results show that additional weight savings can be achieved by the inclusion of proof test data in the design process. / AFRIKAANSE OPSOMMING: Onlangse studies het getoon dat die gebruik van ontwerp spesifieke proeftoets data in betroubaarheids gebaseerde optimering (BGO) kan lei tot 'n verbeterde ontwerp. BGO behels vele aspekte in die ontwerpsgebied. Die toevoeging van proeftoets data in ontwerpsoptimering bring te weë; die toetsing van 'n ontwerp en onderdele voor gebruik, die aangepaste en verbeterde ontwerp en gewig-besparing met handhawing van hoë betroubaarsheidsvlakke. 'n Praktiese toepassing van die BGO tegniek behels die ontwerp van drukvatte met saamgestelde materiaal bewapening. Die drukvatontwerp is 'n ligte struktuur wat gebruik word in die berging van hoë druk vloeistowwe in bv. in ruimtetuie, in die internasionale ruimtestasie en in ander toepassings waar hoë druk oor 'n tydperk verlang word. Elke drukvat met saamgestelde materiaal bewapening wat in ruimtevaartstelsels gebruik word, word geproeftoets voor gebruik. In ruimte stelselontwerp lei massa besparing tot 'n toename in loonvrag. Die tesis beskryf 'n optimeringsmetode soos ontwikkel en gebaseer op 'n BGO tegniek. Die metode word toegepas in die ontwerp van drukvatte met saamgestelde materiaal bewapening. Die resultate toon dat die gebruik van proeftoets data in massa besparing optimering onderhewig soos aan hoë betroubaarheidsvlakke moontlik is. Verdermeer, die metode laat ook ontwerpers toe om die proeftoetsvlak aan te pas om sodoende by ander betroubaarheidsvlakke te toets. In die tesis word die ontwikkeling en gebruik van die optimeringsmetode uiteengelê. Die evaluering van betroubaarheidsvlakke is gebaseer op 'n eerste orde betroubaarheids-tegniek wat geverifieer word met talle Monte Carlo simulasie resultate. Die metode is ook so geskep dat beide analitiese sowel as eindige element modelle gebruik kan word. Ten slotte, word 'n toepassing getoon waar resultate wys dat die gebruik van die optimeringsmetode met die insluiting van proeftoets data wel massa besparing kan oplewer.

Lors du fonctionnement des packages électroniques, ceux ci sont exposés à diverses sollicitations d'ordres thermiques et mécaniques. De même, la combinaison de ces sources de contraintes constitue l'origine de la quasi majorité des défaillances des packages électroniques. Pour s'assurer de la bonne résistance des packages électroniques, les fabricants pratiquent des tests de fiabilité et des analyses de défaillance avant toute commercialisation. Toutefois, les essais expérimentaux, lors de la phase de conception et de l'élaboration des prototypes, s'avèrent contraignants en termes de temps et de ressources matérielles. En revanche, la simulation numérique à l'aide de la méthode des éléments finis constitue une option alternative en termes de temps et de ressources. Les objectifs dévolus aux travaux de recherche visent à élaborer quatre modèles éléments finis en 3D, validés/calibrés par des essais expérimentaux, intégrant les recommandations JEDEC (1) en vue de : - Procéder à la caractérisation thermique et thermomécanique des packages électroniques ; - Et prédire la durée de vie en fatigue thermique des joints de brasures et ce, en lieu et place de la caractérisation expérimentale normalisée. Or, la mise en œuvre des modèles éléments finis présente certains inconvénients liés aux incertitudes au niveau de la géométrie, des propriétés matériaux, les conditions aux limites ou les charges. Ceux ci ont une influence sur le comportement thermique et thermomécanique des systèmes électroniques. D'où la nécessité de formuler le problème en termes probabilistes et ce, dans le but de mener une étude de fiabilité et d’optimisation des packages électroniques. Pour remédier au temps de calcul énorme généré par les méthodes d’analyse de fiabilité classiques, nous avons développé des méthodologies spécifiques à cette problématique, via des méthodes d’approximation basées sur le krigeage avancé,qui nous ont permis de bâtir un modèle de substitution, qui rallie efficacité et précision. Par conséquent, une analyse de fiabilité a été menée avec exactitude et dans un temps extrêmement court, via les méthodes de simulation Monte Carlo et FORM/SORM, couplées avec le modèle de krigeage avancé. Ensuite, l’analyse de fiabilité a été associée dans le processus d’optimisation, en vue d’améliorer la performance et la fiabilité de la conception structurelle des packages électroniques. A la fin, nous avons procédé à l’applicabilité des dites méthodologies d’analyse de fiabilité aux quatre modèles éléments finis ainsi développés. Il résulte que les analyses de fiabilité menées se sont avérées très utiles pour prédire les effets des incertitudes liées aux propriétés matériaux. De même, l’analyse d’optimisation de fiabilité ainsi réalisée nous a permis d’améliorer la performance et la fiabilité de la conception structurelle des packages électroniques. (1) JEDEC (Joint Electron Device Engineering Council) est un organisme de normalisation des semi-conducteurs. / During operation, electronic packages are exposed to various thermal and mechanical solicitations. These solicitations combined are the source for most of electronic package failures. To ensure electronic packages robustness, manufacturers perform reliability testing and failure analysis prior to any commercialization. However, experimental tests, during design phase and prototypes development, are known to be constraining in terms of time and material resources. This research aims to develop four finite element models in 3D, validated/calibrated by experimental tests, integrating JEDEC recommendations to : - Perform electronic packages thermal and thermomechanical characterization ; - Predict the thermal fatigue life of solder joints in place of the standardized experimental characterization.However, implementation of the finite element model has some disadvantages related to uncertainties at the geometry, material properties, boundary conditions or loads. These uncertainties influence thermal and electronic systems thermomechanical behavior. Hence the need to formulate the problem in probabilistic terms, in order to conduct a reliability study and a electronic packages reliability based design optimization.To remedy the enormous computation time generated by classical reliability analysis methods, we developed methodologies specific to this problem, using approximation methods based on advanced kriging, which allowed us to build a substitution model, combining efficiency and precision. Therefore reliability analysis can be performed accurately and in a very short time with Monte Carlo simulation (MCS) and FORM / SORM methods coupled with the advanced model of kriging. Reliability analysis was associated in the optimization process, to improve the performance and electronic packages structural design reliability. In the end, we applied the reliability analysis methodologies to the four finite element models developed. As a result, reliability analysis proved to be very useful in predicting uncertainties effects related to material properties. Similarly, reliability optimization analysis performed out has enabled us to improve the electronic packages structural design performance and reliability. In the end, we applied the reliability analysis methodologies to the four finite element models developed. As a result, reliability analysis proved to be very useful in predicting uncertainties effects related to material properties. Similarly, reliability optimization analysis performed out has enabled us to improve the electronic packages structural design performance and reliability.

The objective of this study is to propose an efficient sampling-based RBDO using a new classification method to reduce the computational cost. In addition, accuracy improvement strategies for the Kriging method are proposed to reduce the number of expensive computer experiments. Current research effort involves: (1) developing a new classification method that is more efficient than conventional surrogate modeling methods while maintaining required accuracy level; (2) developing a sequential adaptive sampling method that inserts samples near the limit state function; (3) improving the efficiency of the RBDO process by using a fixed hyper-spherical local window with an efficient uniform sampling method and identification of active/violated constraints; and (4) improving the accuracy of the Kriging method by introducing several strategies. In the sampling-based RBDO, only accurate classification information is needed instead of accurate response surface. On the other hand, in general, surrogates are constructed using all available DoE samples instead of focusing on the limit state function. Therefore, the computational cost of surrogates can be relatively expensive; and the accuracy of the limit state (or decision) function can be sacrificed in return for reducing the error on unnecessary regions away from the limit state function. On the contrary, the support vector machine (SVM), which is a classification method, only uses support vectors, which are located near the limit state function, to focus on the decision function. Therefore, the SVM is very efficient and ideally applicable to sampling-based RBDO, if the accuracy of SVM is improved by inserting virtual samples near the limit state function. The proposed sequential sampling method inserts new samples near the limit state function so that the number of DoE samples is minimized. In many engineering problems, expensive computer simulations are used and thus the total computational cost needs to be reduced by using less number of DoE samples. Several efficiency strategies such as: (1) launching RBDO at a deterministic optimum design, (2) hyper-spherical local windows with an efficient uniform sampling method, (3) filtering of constraints, (4) sample reuse, (5) improved virtual sample generation, are used for the proposed sampling-based RBDO using virtual SVM. The number of computer experiments is also reduced by implementing accuracy improvement strategies for the Kriging method. Since the Kriging method is used for generating virtual samples and generating response surface of the cost function, the number of computer experiments can be reduced by introducing: (1) accurate correlation parameter estimation, (2) penalized maximum likelihood estimation (PMLE) for small sample size, (3) correlation model selection by MLE, and (4) mean structure selection by cross-validation (CV) error.

Internal combustion (IC) engines that consume hydrocarbon fuels have dominated the propulsion systems of air-vehicles for the first century of aviation. In recent years, however, growing concern over rapid climate changes and national energy security has galvanized the aerospace community into delving into new alternatives that could challenge the dominance of the IC engine. Nevertheless, traditional aircraft sizing methods have significant shortcomings for the design of such unconventionally powered aircraft. First, the methods are specialized for aircraft powered by IC engines, and thus are not flexible enough to assess revolutionary propulsion concepts that produce propulsive thrust through a completely different energy conversion process. Another deficiency associated with the traditional methods is that a user of these methods must rely heavily on experts experience and advice for determining appropriate design margins. However, the introduction of revolutionary propulsion systems and energy sources is very likely to entail an unconventional aircraft configuration, which inexorably disqualifies the conjecture of such connoisseurs as a means of risk management. Motivated by such deficiencies, this dissertation aims at advancing two aspects of aircraft sizing: 1) to develop a generalized aircraft sizing formulation applicable to a wide range of unconventionally powered aircraft concepts and 2) to formulate a probabilistic optimization technique that is able to quantify appropriate design margins that are tailored towards the level of risk deemed acceptable to a decision maker. A more generalized aircraft sizing formulation, named the Architecture Independent Aircraft Sizing Method (AIASM), was developed for sizing revolutionary aircraft powered by alternative energy sources by modifying several assumptions of the traditional aircraft sizing method. Along with advances in deterministic aircraft sizing, a non-deterministic sizing technique, named the Probabilistic Aircraft Sizing Method (PASM), was developed. The method allows one to quantify adequate design margins to account for the various sources of uncertainty via the application of the chance-constrained programming (CCP) strategy to AIASM. In this way, PASM can also provide insights into a good compromise between cost and safety.

The objective of this study is to develop an accurate surrogate modeling method for construction of the surrogate model to represent the performance measures of the compute-intensive simulation model in reliability-based design optimization (RBDO). In addition, an assessment method for the confidence level of the surrogate model and a conservative surrogate model to account the uncertainty of the prediction on the untested design domain when the number of samples are limited, are developed and integrated into the RBDO process to ensure the confidence of satisfying the probabilistic constraints at the optimal design. The effort involves: (1) developing a new surrogate modeling method that can outperform the existing surrogate modeling methods in terms of accuracy for reliability analysis in RBDO; (2) developing a sampling method that efficiently and effectively inserts samples into the design domain for accurate surrogate modeling; (3) generating a surrogate model to approximate the probabilistic constraint and the sensitivity of the probabilistic constraint with respect to the design variables in most-probable-point-based RBDO; (4) using the sampling method with the surrogate model to approximate the performance function in sampling-based RBDO; (5) generating a conservative surrogate model to conservatively approximate the performance function in sampling-based RBDO and assure the obtained optimum satisfy the probabilistic constraints. In applying RBDO to a large-scale complex engineering application, the surrogate model is commonly used to represent the compute-intensive simulation model of the performance function. However, the accuracy of the surrogate model is still challenging for highly nonlinear and large dimension applications. In this work, a new method, the Dynamic Kriging (DKG) method is proposed to construct the surrogate model accurately. In this DKG method, a generalized pattern search algorithm is used to find the accurate optimum for the correlation parameter, and the optimal mean structure is set using the basis functions that are selected by a genetic algorithm from the candidate basis functions based on a new accuracy criterion. Plus, a sequential sampling strategy based on the confidence interval of the surrogate model from the DKG method, is proposed. By combining the sampling method with the DKG method, the efficiency and accuracy can be rapidly achieved. Using the accurate surrogate model, the most-probable-point (MPP)-based RBDO and the sampling-based RBDO can be carried out. In applying the surrogate models to MPP-based RBDO and sampling-based RBDO, several efficiency strategies, which include: (1) using local window for surrogate modeling; (2) adaptive window size for different design candidates; (3) reusing samples in the local window; (4) using violated constraints for surrogate model accuracy check; (3) adaptive initial point for correlation parameter estimation, are proposed. To assure the accuracy of the surrogate model when the number of samples is limited, and to assure the obtained optimum design can satisfy the probabilistic constraints, a conservative surrogate model, using the weighted Kriging variance, is developed, and implemented for sampling-based RBDO.

Este trabalho apresenta desenvolvimentos na modelagem mecânica de estruturas de barras em concreto armado, bem como no acoplamento entre modelos de confiabilidade e otimização do tipo RBDO para obtenção de dimensões ótimas, respeitando os requisitos de segurança especificados em projeto. Quanto à modelagem mecânica via Método dos Elementos Finitos (MEF), além do comportamento não-linear geométrico e dos materiais, foi considerada a contribuição dos mecanismos complementares de resistência ao cisalhamento, dados pelo engrenamento de agregados e efeito de pino das armaduras longitudinais. Além disso, um modelo simplificado que avalia a contribuição da armadura transversal também foi proposto. Foi desenvolvida uma formulação de otimização que deixa a posição da linha neutra livre, ao contrário de formulações existentes. Esta formulação resultou em projetos mais economicos dos que aqueles encontrados na literatura. Na questão do acoplamento de confiabilidade e otimização, foram exploradas melhorias no Método de Superfície de Resposta e no acoplamento direto via Método de Confiabilidade de Primeira Ordem e Técnica dos Gradientes Numéricos. Estas resultaram em maior precisão dos resultados e aumento na velocidade de convergência. Os modelos mecânicos, incluindo análise não-linear e mecanismos complementares, a formulação de otimização e as técnicas de confiabilidade foram implementados em um programa computacional para dimensionamento ótimo de elementos em concreto armado. O programa foi utilizado na resolução de vários problemas-exemplo. Verificou-se que a consideração dos mecanismos complementares de resistência ao cisalhamento produziram acréscimo na carga última, quando comparadas com as respostas sem tais efeitos. Verificou-se também que os mesmos mecanismos produziram um aumento, até mais significativo, nos índices de confiabilidade obtidos. As dimensões ótimas de elementos estruturais também foram comparadas, considerando-se modelos lineares e não-lineares dos materiais. O estudo mostrou que os custos da estrutura otimizada são menores, quando se considera os efeitos de comportamento não-linear dos materiais. / This work presents some developments in the mechanical modeling of reinforced concrete bar structures, as well in the coupling of reliability and RBDO optimization models, with the purpose of obtaining optimal dimensions considering the safety requirements specified in design. As for the mechanical modeling via Finite Element Method (FEM), in addition to geometrical and material nonlinear behaviors, the contribution of shear resistance complementary mechanisms (aggregate interlock and dowel action of longitudinal reinforcement) was taken into account. Moreover, a simplified model that evaluates the contribution of shear reinforcement was also proposed. In an improvement of existing formulations, an optimization scheme was developed which leaves the position of the neutral axis free. This improvement resulted in more economical cross-sections, than those found in the literature. With respect to the coupling of reliability and optimization methods, improvements were sought in the Response Surface Method and in the direct coupling via First Order Reliability and Numerical Gradients methods. These improvements resulted in greater precision and in increased convergence speed. The mechanical models, including non linear effects and complementary mechanisms , the optimization and reliability formulations were implemented in a computational code for the optimum design of reinforced concrete structures. The program was used to solve a number of example problems. It was found that the complementary mechanisms resulted in an increase of ultimate loads, when compared to the response obtained without these effects. These mechanisms also resulted in an even greater increase of the elements reliability. Optimal dimensions of the structural elements were also compared, considering linear and non-linear material models. The cost of the optimum structure was found to be smaller when non linear effects are taken into account.