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Topology optimization is a free-form design tool that allows identification of optimized material distribution within the design domain while engineering constraints are fulfilled. Topology optimization is used to arrive at high-performance designs in a variety of engineering disciplines (e.g., such as aerospace design, micro-structure design, photonic devices and structural engineering). Structural engineering has adopted topology optimization for recommending innovative structural layouts. However, most of the research in this area uses simplified mechanics, design objectives and assumes deterministic design conditions. The simplifications in mechanics is usually manifested with employing truss finite-element models that are only capable of transferring axial loads using members with simple geometry cross sectional shapes (e.g., squares), and simple objective functions include displacement-based metrics which are directly found using finite-element method. However real-world structures are capable of transferring loads via bending and shear in addition to axial forces, and are built from standard cross-sections such as wide flange I-beams. Many of these structures are mainly designed for stress constraints, and their applications are almost always accompanied with uncertainties. These uncertainties could be, for example, due to incomplete knowledge about external loads or a result of manufacturing errors, which is the focus of this thesis. However, due to numerical and theoretical challenges, incorporation of these necessary elements within structural topology optimization is not fully developed yet. This thesis addresses these challenges for taking the initial steps towards bringing topology optimization applications closer to realistic design conditions and constraints.This thesis begins with developing a methodology to efficiently incorporate members with standard cross sections such as I-beams in structural topology optimization. To this end, standard I-beams from the American Institute of Steel Construction Design manual are selected and a relationship between cross-sectional area and other section properties such as moment of inertia is developed through advanced regression analysis. This approach allows using member cross-sectional area as the independent design variable, and explicit derivation of gradients. These gradients are used in gradient-based optimizers for increasing computational efficiency. The next contribution of this thesis is implementation of an efficient methodology for controlling von Mises yield criterion within the structure. This criterion is suitable for stress-based design of steel frames. The proposed approach determines this criterion in a set of candidate points for each element, and then approximates their maximum with an analytical function. Therefore, stress-based performance of the structure was expressed analytically with a single scalar. Moreover, gradients were derived analytically, which resulted in computational efficiency. A comparison of stress-based designs and traditional compliance-based design demonstrated the significant changes in final topology and reduction of stress maximum. Moreover, designs for optimizing a combination of compliance and stress criteria were explored. The next contribution of this thesis the incorporation of global buckling constraints to stress-based design structural topology optimization to control instability and stress modes of failures simultaneously. Global buckling was controlled with determining the minimum value of buckling load factor that is given by solving the eigenvalue problem of linear buckling analysis. In the last part of this thesis, the focus is shifted to design under uncertainty. More specifically, geometric uncertainties were considered and were modeled with random variables that defined the geometry of the structure. These uncertainties were efficiently propagated to the structural response level, measured with both stress and displacement-based criteria, using stochastic perturbation method. The design objective was to arrive at topologies that are robust (insensitive) in the presence of these geometric uncertainties. Monte Carlo simulation was used to verify the predictions and designs from the proposed methodology, and an excellent agreement was observed. |
