Abstract

Steel multistorey 3D frames are commonly used in business and residential buildings, industrial sheds, warehouses, etc. The design optimization of tall steel buildings is usually governed by horizontal loadings, such as, wind load, as well as its dynamic behavior, for which the structure must have the stiffness and stability in accordance with the safety criteria established by codes. This chapter deals with sizing structural optimization problems, concerning weight minimization of 3D steel frames, considering natural frequencies of vibration as well as allowable displacements as the constraints of the optimization problem. The discrete design variables are to be chosen from commercial profiles tables. A differential evolution (DE) is the search algorithm adopted coupled to an adaptive penalty method (APM) to handle the constraints. Three different 3D frames are optimized, presenting very interesting results.

Keywords: steel frame optimization, differential evolution, natural frequencies of vibration, wind load, adaptive penalty method

### 1. Introduction

Steel multistorey 3D frames are commonly used in business and residential buildings, industrial sheds, warehouses, etc. The design optimization of tall steel buildings is usually governed by horizontal loadings, such as, wind load, as well as its dynamic behavior, for which the structure must have the stiffness and stability following the safety criteria established by codes. The task of finding the most economical structures, that is, with the minimum weight and satisfying the constraints imposed by the codes, such as, ASD-AISC [1], NBR 6123 [2], and NBR 8800 [3], is not trivial. This may require an interactive process (trial and error) that may require very expensive or even impossible computational time. The behavior constraints include, for instance, combined bending and axial stress, shear stress, compression buckling, tension slenderness, drift ratio, multiple natural frequencies of vibration, elastic critical loads, etc.

This chapter is not an attempt to provide a survey of publications on structural optimization problems of multistorey 3D frames concerning many types of constraints. However, one can cite some papers where readers can find reviews regarding this issue.

An optimization process via genetic algorithms using MATLAB-SAP2000 Open Application Programming Interface (OAPI) is presented for optimum design of space frames with semirigid connections in Artar and Daloğlu [4].

and analysis of results are described in Sections 4 and 5, respectively. Finally, the

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

The optimization problem deals with the weight minimization of 3D steel frames consisting of N members, under constraint of natural frequencies of vibration and

The objective function w(x) (Eq. (2)) is the weight of the structure, in which L<sup>i</sup>

N i¼1

The maximum horizontal displacement and the first natural frequency of vibra-

dv [19] [21] TS [22] [20] TS A<sup>1</sup> 29.2257 30.520 30.268 5.5713 5.4870 5.6593 A<sup>2</sup> 0.1000 0.100 0.1018 2.4072 2.2475 2.2830 A<sup>3</sup> 24.1821 23.200 23.1493 5.4692 5.5000 5.3987 A<sup>4</sup> 14.9471 15.220 15.2456 2.3847 2.2320 2.3229 A<sup>5</sup> 0.1000 0.100 0.1001 0.1004 0.1000 0.1000 A<sup>6</sup> 0.3946 0.551 0.5546 0.7104 0.7285 0.7159 A<sup>7</sup> 7.4958 7.457 7.4902 3.6596 3.7976 3.6969 A<sup>8</sup> 21.9249 21.040 21.3433 3.6579 3.7820 3.7667 A<sup>9</sup> 21.2909 21.530 21.3958 2.0703 1.9840 1.9386 A<sup>10</sup> 0.1000 0.100 0.1001 1.9153 1.9065 1.9351 W 5069.09 5060.80 5061.45 532.390 532.124 532.03 nfe 28,0000 — 10,000 21,000 21,000 10,000

Case 1 Case 2

is the length, Ai is the cross-sectional area, and ρ<sup>i</sup> is the specific mass of the ith member. 7850 kg/m<sup>3</sup> is the specific mass of the steel used in the numerical experi-

wð Þ¼ x ∑

x ¼ f g I1;I2;…;Ii (1)

ρiAiLi (2)

The objective is to find an integer index vector x (Eq. (1)) which points to commercial steel profile where each index i points to a cross-sectional area (Ai), the inertias about the main axes (Ixi, Iyi) and the torsional constant (Iti). These properties are used to define a candidate solution in the evolutionary process. Two differ-

ent search spaces are adopted for columns and beams, containing 29 and 56

conclusions and extensions of this chapter are described in Section 6.

2. Formulation of the optimization problem

available profiles, respectively, provided in Table 1.

allowable displacements due to design loads.

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

ments presented in this chapter.

tion are the constraints written as

Table 1.

47

Comparison of results of the 10-bar truss.

An enhanced imperialist competitive algorithm for optimum design of skeletal structures is proposed by Maheri and Talezadeh [5]. In Aydoğdu et al. [6], an enhanced artificial bee colony algorithm is adopted to find the optimum design problem of steel space frames formulated according to the provisions of LRFD-AISC. Talatahari et al. [7] proposed the combination of an eagle strategy algorithm with the differential evolution (DE) which is implemented by interfacing SAP2000 structural analysis code and MATLAB mathematical software to find the optimum design of framed structures. Maheri et al. [8] proposed an enhanced honey bee mating optimization algorithm for the design of side sway steel frames. The robustness of the algorithm in terms of both solution quality and computational cost is proven by solving four design optimization problems of side sway steel frames. Optimal seismic design of three-dimensional steel frames is carried out in Kaveh and BolandGerami [9] with the structures subjected to gravity and earthquake loadings and designed according to the LRFD-AISC design criteria.

The harmony search metaheuristic is used as the search engine. Kaveh and BolandGerami [9] used a cascade-enhanced colliding body optimization to find the optimum design of large-scale space steel frames according to ASD-AISC. Jalili et al. [10] presented a modified biogeography-based optimization (MBBO) algorithm for the optimum design of skeletal structures with discrete variables. Gholizadeh and Poorhoseini [11] proposed a modified dolphin echolocation (MDE) algorithm proposed for the optimization of steel frame structures. Gholizadeh and Milany [12] used an improved fireworks algorithm (IFWA) to deal with the discrete structural optimization problems of steel trusses and frames.

Since the tall buildings present the need for in-depth analyses regarding their lateral stability, several studies are found in the literature on this subject. Cost efficiencies of various steel frameworks are investigated for the economical design of multistorey buildings by Hasançebi [13]. Braced and unbraced steel frames subjected to gravity and lateral seismic loads were studied by Memari and Madhkan [14]. Kameshki and Saka [15] compared pin-jointed frames considering several types of bracings with rigidly connected frames without bracings. Liang et al. [16] applied a performance-based design optimization method to discover optimum topologies of bracings for steel frames. In his paper Hasançebi [13] highlights important aspects regarding restrictions on the fabrication of structural elements. In this sense, it is essential. In this sense, it is imperative that construction costs of the resulting structures, rather than design weights only, must be evaluated. Studies on this subject were conducted by Pavlovčič et al. [17].

This chapter deals with sizing structural optimization problems, concerning weight minimization of 3D steel frames, considering natural frequencies of vibration as well as allowable displacements as the constraints of the optimization problem. The discrete design variables are to be chosen from commercial profile tables. A DE [18] is the search algorithm adopted coupled with an adaptive penalty method (APM) to handle the constraints [19]. An essential aspect of this chapter is the importance that must be given concerning the constraints regarding the first natural frequency of vibration of the frames. Often, they are neglected in the formulations of these structural optimization problems. A brief review of optimization problems considering natural frequencies of vibration as constraints is provided by Carvalho et al. in [20].

This chapter is as organized as follows: Section 2 presents the formulation of the optimization problem. Section 3 presents the basics concepts on differential evolution algorithm and the strategy to handle the constraints. Numerical experiments

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

and analysis of results are described in Sections 4 and 5, respectively. Finally, the conclusions and extensions of this chapter are described in Section 6.
