This paper proposes an effective couple method for solving reliability-based multi-objective optimization problems of truss structures with static and dynamic constraints. The proposed coupling method integrates a single-loop deterministic method (SLDM) into the nondominated sorting genetic algorithm II (NSGA-II) algorithm to give the so-called SLDM-NSGA-II. Thanks to the advantage of SLDM, the probabilistic constraints are treated as approximating deterministic constraints. And therefore the reliability-based multi-objective optimization problems can be transformed into the deterministic multi-objective optimization problems of which the computational cost is reduced significantly. In these reliability-based multi-objective optimization problems, the conflicting objective functions are to minimize the weight and the displacements of the truss. The design variables are cross-section areas of the bars and contraints include static and dynamic constraints. For reliability analysis, the effect of uncertainty of parameters such as force, added mass in the nodes, material properties and cross-section areas of the bars are taken into account. The effectiveness and reliability of the proposed method are demonstrated through three benchmark-type truss structures including a 10-bar planar truss, a 72-bar spatial truss and a 200-bar planar truss. Moreover, the influence of parameters on the reliability-based Pareto optimal fronts is also carried out.

The paper deals with the multi-objective optimization problems of laminated composite beam structures. The objective function is to minimize the weight of the whole laminated composite beam and maximize the natural frequency. In particular, the simultaneous use of all the design variables such as fiber volume fractions, thickness and fiber orientation angles of layers is conducted, in which the fiber volume fractions are taken as continuous design variables with the constraint on manufacturing process while the thickness and fiber orientation angles are considered as discrete variables. The beam structure is subjected to the constraint in the natural frequency which must be greater than or equal to a predetermined frequency. For free vibration analysis of the structure, the finite element method is used with the two-node Bernoulli-Euler beam element. For solving the multi-objective optimization problem, the nondominated sorting genetic algorithm II (NSGA-II) is employed. The reliability and effectiveness of the proposed approach are demonstrated through three numerical examples by comparing the current results with those of previous studies in the literature.