5. Analysis of results

Two numerical experiments discussed concerning a three-storey 3D frame, in which it was possible to observe the importance of the natural frequency of vibration considered as a constraint. In general, it is neglected in the great majority of the structural optimization problems. The best solution found for Experiment 2 was heavier than the best solution found in Experiment 1. It can be justified since the first natural frequency of vibration was included in the problem formulation resulting in a heavier optimized structure.

A set of three experiments concerning a six-storey 3D frame were conducted with the members linked in three different groups. The constraints for these experiments are the maximum displacement at the top of the frame. The members were linked in two, four, and eight groups, and, as expected, the weights decrease as the number of linked bars increases. It is important to note from the results of the case where the members were linked in eight groups (Experiment 3.3) that the algorithm found only five distinct profiles.

Table 13 summarizes the results of Experiment 3, and the graphic in Figure 19 is a curve of the tradeoff presenting a comparison of each one of the best solutions and their corresponding number of distinct profiles used.

Another important point was the fact that no feasible solutions were found for the six-storey frame with no bracings, considering the constraint concerning the first natural frequency of vibration. This fact indicated the conception of a new model increasing the stiffness of the structure to make possible a feasible optimized solution. Thus, bracings were considered in the new model increasing the total number of members. The result of Experiment 4 was very interesting leading to a lighter structure than the three other experiments (3.1, 3.2, and 3.3), even if presenting a more complex geometry with more members after the inclusion of the bracings in the model. The importance of the bracings in 3D steel frames was shown not only concerning their stability and stiffness but also improving its dynamic behavior.

#### Truss and Frames - Recent Advances and New Perspectives


Acknowledgements

DOI: http://dx.doi.org/10.5772/intechopen.87022

Conflict of interest

Author details

and Patricia H. Hallak

Juiz de Fora, Brazil

65

FAPEMIG.

The authors wish to thank the reviewers that helped the quality of the chapter, Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq (grant 306186/2017-9), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais,

Design Optimization of 3D Steel Frameworks Under Constraints of Natural Frequencies…

The authors declare that they have no conflicts of interest.

Cláudio H.B. Resende, José P.G. Carvalho, Afonso C.C. Lemonge\*

\*Address all correspondence to: afonso.lemonge@ufjf.edu.br

provided the original work is properly cited.

Graduate Program in Civil Engineering, Federal University of Juiz de Fora,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

#### Table 13.

Analysis of results of Experiment 3.

Figure 19. Tradeoff curve of Experiment 3.

### 6. Conclusions and extensions

The study conducted in this chapter focused on the minimization of the weight of 3D steel frames, subjected to constraints concerning horizontal displacements and natural frequencies of vibration. It is interesting to note the importance of a structural optimization study before the design is conceived, which leads to more competitive and sometimes counterintuitive.

In the experiments addressed in this chapter, it is easy to conclude that the natural frequency of vibration is an essential characteristic to be considered in the formulation of the structural optimization problems.

As future works the approaches will extend to multi-objective optimization problems with more constraints, such as, stress, stability, geometry, and interstorey drifts, introducing more real aspects to the optimization problems in engineering. Strategies should be considered for automatic grouping of members without the need for preliminary analysis by the designer. For this, special encodings will be used via cardinality constraints as can be seen in the structural optimization problems discussed in the references [20, 24, 25].

### 7. Remark

The codes used to solve the optimization problems presented in this chapter are written in Matlab® language, and the final results, as well as the figures, are checked by the SAP 2000®.

Design Optimization of 3D Steel Frameworks Under Constraints of Natural Frequencies… DOI: http://dx.doi.org/10.5772/intechopen.87022
