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A pushover analysis method based on semirigid connection concept is developed and the colliding bodies optimization algorithm is employed to find optimum seismic design of frame structures. Two numerical examples from the literature are studied. The results of the new algorithm are compared to the conventional design methods to show the power or weakness of the algorithm.

The traditional seismic design practice entails specifying the desired performance objective, and subsequently the structure is designed to meet specific performance levels. Performance-based design is a more general approach which tries to design buildings with predictable loading-induced performance, rather than being based on prescriptive mostly empirical code specifications. The earthquakes and strong winds are the two major loading conditions imposed on buildings and the performance-based seismic design is becoming well accepted in professional practice for the design of buildings under seismic loading [

Performance-based design methodology allows a significantly different approach for formulating optimization problems, leading to the field of performance-based design optimization (PBDO) [

This paper presents a new developed method, the so-called colliding bodies optimization algorithm (CBO) [

In structural design, it is desirable to reach a proposed service-ability level with the least usage of the material [

Roof drift of 0.4%, 0.7%, 2.5%, and 5% of the height of structure is taken as allowable roof drift for OP, IO, LS, and CP performance levels in design optimization process, respectively [

The structural optimization problems can be expressed as minimizing the weight of structures as

To predict the seismic demands on building frameworks, a developed computer-based pushover analysis procedure is utilized. The analysis process is inspired of second-order inelastic analysis of semirigid framed structures where rigidity factor is replaced with plasticity factor in stiffness matrix. The detailed explanations are presented in [

In physics, collisions between bodies are governed by laws of momentum and energy. When a collision occurs in an isolated system, the total momentum of the system of objects is conserved.

Provided that there are no net external forces acting upon the objects, the momentum of all objects before the collision equals the momentum of all objects after the collision.

The conservation of the total momentum demands that the total momentum before the collision is the same as the total momentum after the collision and is expressed by the following equation:

The formulas for the velocities after a one-dimensional collision are

The colliding bodies optimization algorithm is one of the metaheuristic search methods recently developed [

The initial positions of CBs are determined with random initialization of a population of individuals in the search space:

The magnitude of the body mass for each CB is defined as

The arrangement of the CBs objective function values is performed in ascending order. The sorted CBs are equally divided into two groups.

The lower half of CBs (stationary CBs): these CBs are good agents which are stationary and the velocity of these bodies before collision is zero. Thus,

The upper half of CBs (moving CBs): these CBs move toward the lower half. Then, the better and worse CBs, that is, agents with upper fitness value of each group, will collide together. The change of the body position represents the velocity of these bodies before collision as

After the collision, the velocity of bodies in each group is evaluated using (

New positions of CBs are obtained using the generated velocities after the collision in position of stationary CBs. The new positions of each moving CB are

Also, the new position of each stationary CB is obtained as

The optimization is repeated until a termination criterion, specified as the maximum number of iteration, is satisfied. It should be noted that a body’s status (stationary or moving body) and its numbering are changed in two subsequent iterations.

The initial positions of CBs are determined randomly in the search space.

The magnitude of the body mass for each CB is defined.

The arrangement of the CBs objective function values is performed in ascending order. The sorted CBs are equally divided into two groups:stationary and moving CBs.

After the collision, the velocity of bodies in each group is evaluated and the new positions of CBs are evaluated using the generated velocities after the collision in position of stationary moving CBs.

Steps

Since the algorithm is a continuous algorithm, for solving the discrete problem like the performance-based design of frames it is necessary to make some modifications. Here, we use a rounding function which changes the continuous value of a result to the nearest discrete value. Although this change is simple and efficient, it may reduce the exploration of the algorithm, [

Two building frameworks are selected for seismic optimum design using the metaheuristic algorithm [

The expected yield strength of steel material used for column members is

The configuration, grouping of the member,s and applied loads of the four-bay three-story framed structure are shown in Figure

Three-story steel moment frame.

The optimum results for the CBO, a hybrid CSS [

The statistical information of performance-based optimum designs for the 4-bay 3-story frame.

Algorithm | This work | Hybrid CSS [ |
PSACO [ |
PSO [ |
ACO [ |
GA [ |
A conventional design [ |
---|---|---|---|---|---|---|---|

Best weight (kN) | 280.32 | 273.7 | 279.2 | 286.3 | 283.4 | 303.9 | 412.9 kN |

Average weight (kN) | 292.36 | 286.7 | 290.4 | 302.4 | 294.3 | 321.5 | — |

Worst weight (kN) | 308.63 | 297.8 | 298.5 | 310.7 | 303.2 | 339.7 | — |

Std. Dev. (kN) | 5.786 | 5.651 | 6.453 | 10.453 | 7.566 | 14.332 | |

Average number of analyses | 4,500 | 4,500 | 4,500 | 8,500 | 3,900 | 6,800 | — |

Convergence history of the CBO algorithm for the 4-bay 3-story frame.

A five-bay nine-story steel frame is considered as shown in Figure

Nine-story steel moment frame.

The statistical results obtained by the metaheuristic algorithms are presented in Table

The statistical information of performance-based optimum designs for the 4-bay 9-story frame.

Algorithm | This work | Hybrid CSS [ |
PSACO [ |
PSO [ |
ACO [ |
GA [ |
---|---|---|---|---|---|---|

Best weight (kN) | 1600.25 | 1568.66 | 1601.32 | 1682.63 | 1631.83 | 1723.1 |

Average weight (kN) | 1660.36 | 1626.32 | 1650.55 | 1725.36 | 1696.2 | 1791.4 |

Worst weight (kN) | 1780.62 | 1725.36 | 1759.65 | 1813.25 | 1786.94 | 1943.2 |

Std. Dev. (kN) | 31.02 | 30.35 | 38.52 | 66.35 | 49.33 | 78.33 |

Average number of analyses | 5,500 | 5,000 | 6,000 | 12,500 | 5,600 | 9,700 |

Convergence history of the CBO algorithm for the 4-bay 9-story frame.

Performance-based design is a general approach which tries to design buildings with predictable loading-induced performance. In performance-based seismic design, the criteria are expressed in terms of achieving a set of performance objectives while the structure is under levels of seismic hazard. In this paper, the performance-based design of frame structures is formulated to be optimized by the new algorithm, the colliding bodies' optimization. In order to control the lateral drift of building frameworks under seismic loading, a nonlinear analysis is utilized. The analysis method is based on a second order analysis of members including geometrical nonlinearly (using semirigid steel framework concept). The best, average, worst, and standard deviations of minimum weights are obtained by some metaheuristic algorithms as well as the CBO. Although the CBO cannot find the best results, the differences between the results of the CBO algorithm and the best one were small, and the new algorithm did not improve the results significantly.

The author declares that there is no conflict of interests regarding the publication of this paper.