Effective Moduli Prediction For Infinite-level Self-similar Structures

This paper consists of two parts:the first part is focus on effective moduli prediction for infinite-level self-similar structure and the second part is mainly concentrate on application of NoLinear Semi-Definite Programming (NLSDP) in spatial truss structure optimization under global stability constraints.In the first part of this paper, a new infinite-level self-similar structure model is first proposed and in order to calculate its effective elastic moduli, a novel ultimate equivalency idea is applied to transform practical infinite-level structure into an equivalent model whose local structure moduli are unknown but equal to the moduli of global structure; and then the corresponding fixed point equations which can be solved by iteration are established via strain energy method. Considering boundary conditions effects, at last, a simple numerical example is presented to illustrate the performance of equivalent model for effective moduli prediction under several different uniform boundary conditions and the results are analyzed and compared with practical multi-level self-similar structure. The numerical results demonstrated the validity of the equivalent model for predicting effective moduli of the infinite-level self-similar structure.In the second part of this paper, the NLSDP formulation for truss structure optimization and design under global stability constraints is first established. Different from traditional formulation for structure optimization with global stability constraints, this formulation treats critical load factor as independent design variable and structure global stability constraints as semi-definite programming. So it not only can avoid difficulties both in solving and sensitivity analysis of critical load factor, but also can effectively tackle multiple eigenvalue problems. And then the mathematical equivalence between the bound formulation and NLSDP formulation is proved and the sensitivities for both objective function and constraint functions in NLSDP formulation are obtained; Final numerical examples and optimal result discussion demonstrated the superiority of the proposed approach.