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Accelerated particle swarm optimization (APSO) is developed for finding optimum design of frame structures. APSO shows some extra advantages in convergence for global search. The modifications on standard PSO effectively accelerate the convergence rate of the algorithm and improve the performance of the algorithm in finding better optimum solutions. The performance of the APSO algorithm is also validated by solving two frame structure problems.

Optimum design of frame structures are inclined to determine suitable sections for elements that fulfill all design requirements while having the lowest possible cost. In this issue, optimization provides engineers with a variety of techniques to deal with these problems [

Particle swarm optimization (PSO), one of meta-heuristic algorithms, is based on the simulation of the social behavior of bird flocking and fish schooling. PSO is the most successful swarm intelligence inspired optimization algorithms. However, the local search capability of PSO is poor [

A number of studies have applied the PSO and improved it to be used in the field of structural engineering [

Optimum design of structures includes finding optimum sections for members that minimizes the structural weight

The maximum lateral displacement:

The interstory displacements:

LRFD interaction formula constraints (AISC 2001 [

For the proposed method, it is essential to transform the constrained optimization problem to an unconstraint one. A detailed review of some constraint-handling approaches is presented in [

The objective function that determines the fitness of each particle is defined as

The PSO algorithm, inspired by social behavior simulation [_{1} and rand_{2} represent random numbers between 0 and 1;

The standard PSO uses both the current global best

A simplified version which could accelerate the convergence of the algorithm is to use the global best only. Thus, in the APSO [

This simpler version will give the same order of convergence [

This section presents the numerical examples to evaluate the capability of the new algorithm in finding the optimal design of the steel structures. The final results are compared to the solutions of other methods to show the efficiency of the present approach. The proposed algorithm is coded in Matlab, and structures are analyzed using the direct stiffness method. The steel members used for the design consist of 267 W-shaped sections from the AISC database.

Figure

Topology of the 1-bay 8-story frame.

The APSO algorithm found the optimal weight of the one-bay eight-story frame to be 30.91 kN which is the best one compared to the other method. Table

Optimal design comparison for the 1-bay 8-story frame.

Element group | Optimal W-shaped sections | This study | ||
---|---|---|---|---|

GA [ |
ACO [ |
IACO [ | ||

1 | W18 × 35 | W16 × 26 | W21 × 44 | W21 × 44 |

2 | W18 × 35 | W18 × 40 | W18 × 35 | W16 × 26 |

3 | W18 × 35 | W18 × 35 | W18 × 35 | W14 × 22 |

4 | W18 × 26 | W14 × 22 | W12 × 22 | W12 × 16 |

5 | W18 × 46 | W21 × 50 | W18 × 40 | W18 × 35 |

6 | W16 × 31 | W16 × 26 | W16 × 26 | W18 × 35 |

7 | W16 × 26 | W16 × 26 | W16 × 26 | W18 × 35 |

8 | W12 × 16 | W12 × 14 | W12 × 14 | W16 × 26 |

Weight (kN) | 32.83 | 31.68 | 31.05 | 30.91 |

The configuration and applied loads of a 3-bay 15-story frame structure is shown in Figure

Topology of the 3-bay 15-story frame.

The effective length factors of the members are calculated as

The optimum design of the frame obtained by using APSO has the minimum weight of 411.50 kN. The optimum designs for PSO [

Optimal design comparison for the 3-bay 15-story frame.

Element group | Optimal W-shaped sections | This study | ||
---|---|---|---|---|

PSO [ |
HBB-BC [ |
ICA [ | ||

1 | W33 × 118 | W24 × 117 | W24 × 117 | W27 × 129 |

2 | W33 × 263 | W21 × 132 | W21 × 147 | W21 × 147 |

3 | W24 × 76 | W12 × 95 | W27 × 84 | W16 × 77 |

4 | W36 × 256 | W18 × 119 | W27 × 114 | W27 × 114 |

5 | W21 × 73 | W21 × 93 | W14 × 74 | W14 × 74 |

6 | W18 × 86 | W18 × 97 | W18 × 86 | W30 × 99 |

7 | W18 × 65 | W18 × 76 | W12 × 96 | W12 × 72 |

8 | W21 × 68 | W18 × 65 | W24 × 68 | W12 × 79 |

9 | W18 × 60 | W18 × 60 | W10 × 39 | W8 × 24 |

10 | W18 × 65 | W10 × 39 | W12 × 40 | W14 × 43 |

11 | W21 × 44 | W21 × 48 | W21 × 44 | W21 × 44 |

Weight (kN) | 496.68 | 434.54 | 417.46 | 411.50 |

The convergence history for the 3-bay 15-story frame.

The APSO algorithm, as an improved meta-heuristic algorithm, is developed to solve frame structural optimization problems. Optimization software based on the APSO algorithm was coded in the Matlab using object-oriented technology. A methodology to handle the constraints is also developed in a way that we first determine the weight of the structures generated by the particles, and if they become smaller than the so far best solution, then the structural analyses are performed. Two test problems were studied using the optimization program to show the efficiency of the algorithm. The comparison of the results of the new algorithm with those of other algorithms shows that the APSO algorithm provides results as good as or better than other algorithms and can be used effectively for solving engineering problems.