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Otimização de Estruturas Reticuladas com Constrangimentos de Tensão, Encurvadura e Compliance
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A presente dissertação tem como objetivo apresentar uma metodologia para resolver
problemas de minimização de massa de estruturas reticuladas (treliças), que incluem constran-
gimentos de tensão, encurvadura,
compliance e densidades intermédias. Neste estudo, é apli-
cada uma otimização baseada no gradiente, utilizando o algoritmo de otimização MMA e fun-
ções contínuas e diferenciáveis. A metodologia utiliza dois tipos de variáveis: uma topológica,
que considera a densidade artificial das barras, e uma dimensional, que consiste nas áreas das
secções transversais das mesmas.
A presente dissertação visa solucionar problemas relacionados com o aparecimento de
mecanismos em estruturas reticuladas, pois a sua deteção e resolução são essenciais para ga-
rantir a estabilidade e integridade estrutural.
A abordagem proposta inclui o uso de dois métodos de otimização topológica, um cal-
cula o campo de deslocamentos para as cargas aplicadas e o outro calcula diretamente as
forças instaladas nas barras através da formulação plástica. Com base nestes dois métodos
averigua-se estabilidade cinemática da estrutura. Também foram investigadas soluções para
garantir a conectividade adequada entre os elementos da estrutura, surgindo a função de pe-
nalização da densidade proposta no final desta dissertação.
Adicionalmente, foi realizada uma verificação da formulação de Euler para análise de
encurvadura em elementos de barra à compressão. Concluiu-se que o programa prevê de ma-
neira aceitável a carga e a tensão crítica, fortalecendo a confiança na precisão das análises
estruturais realizadas pelo programa para os exemplos apresentados nesta dissertação.
This dissertation presents a methodology for solving mass minimization problems in truss structures, which include constraints such as stress, buckling, compliance, and intermedi- ate densities. The study employs a gradient-based optimization approach using the MMA op- timization algorithm and continuous and differentiable functions. The methodology utilizes two types of variables: topological, considering the artificial density of bars, and dimensional, consisting of the cross-sectional areas of the bars. The main aim of this dissertation is to address issues related to the occurrence of mech- anisms in structures, as their precise detection and effective resolution are crucial for ensuring structural stability and integrity. The proposed approach involves the use of two topological optimization methods, one of which calculates the displacement field for applied loads, and the other directly computes the forces exerted on the bars through plastic formulation. Based on these two methods, the kinematic stability of the structure is assessed. Additionally, solutions were investigated to en- sure adequate connectivity among the elements of the structure, leading to the proposed den- sity penalization function outlined at the end of this dissertation. Furthermore, a verification of the Euler formulation for buckling analysis in compressive bar elements was conducted. It was concluded that the program adequately predicts the load and critical stress, thereby strengthening confidence in the accuracy of the structural analyses performed by the program for the examples presented in this dissertation
This dissertation presents a methodology for solving mass minimization problems in truss structures, which include constraints such as stress, buckling, compliance, and intermedi- ate densities. The study employs a gradient-based optimization approach using the MMA op- timization algorithm and continuous and differentiable functions. The methodology utilizes two types of variables: topological, considering the artificial density of bars, and dimensional, consisting of the cross-sectional areas of the bars. The main aim of this dissertation is to address issues related to the occurrence of mech- anisms in structures, as their precise detection and effective resolution are crucial for ensuring structural stability and integrity. The proposed approach involves the use of two topological optimization methods, one of which calculates the displacement field for applied loads, and the other directly computes the forces exerted on the bars through plastic formulation. Based on these two methods, the kinematic stability of the structure is assessed. Additionally, solutions were investigated to en- sure adequate connectivity among the elements of the structure, leading to the proposed den- sity penalization function outlined at the end of this dissertation. Furthermore, a verification of the Euler formulation for buckling analysis in compressive bar elements was conducted. It was concluded that the program adequately predicts the load and critical stress, thereby strengthening confidence in the accuracy of the structural analyses performed by the program for the examples presented in this dissertation
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Palavras-chave
Otimização Topológica Compliance Treliças Mecanismos Cinemática MMA