2. Formulation of the optimization problem

An optimization process via genetic algorithms using MATLAB-SAP2000 Open Application Programming Interface (OAPI) is presented for optimum design of

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

space frames with semirigid connections in Artar and Daloğlu [4].

Truss and Frames - Recent Advances and New Perspectives

loadings and designed according to the LRFD-AISC design criteria.

optimization problems of steel trusses and frames.

this subject were conducted by Pavlovčič et al. [17].

et al. in [20].

46

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

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 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

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

The optimization problem deals with the weight minimization of 3D steel frames consisting of N members, under constraint of natural frequencies of vibration and allowable displacements due to design loads.

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 different search spaces are adopted for columns and beams, containing 29 and 56 available profiles, respectively, provided in Table 1.

$$\mathbf{x} = \{I\_1, I\_2, \dots, I\_i\} \tag{1}$$

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

$$\omega w(\mathbf{x}) = \sum\_{i=1}^{N} \rho\_i A\_i L\_i \tag{2}$$

The maximum horizontal displacement and the first natural frequency of vibration are the constraints written as


#### Table 1.

Comparison of results of the 10-bar truss.

Truss and Frames - Recent Advances and New Perspectives

$$\begin{aligned} \frac{\delta\_{\text{max}}(\mathbf{x})}{\overline{\delta}} - \mathbf{1} &\le \mathbf{0} \\ \mathbf{1} - \frac{f\_1(\mathbf{x})}{\overline{f}\_1} &\le \mathbf{0} \end{aligned} \tag{3}$$

The following flowchart represents the scheme of DE:

Select randomly rand ∈ [0, 1]

xj,i,<sup>G</sup> xj, <sup>r</sup>1,<sup>G</sup>

where NPOP represents the number of population.

The fitness function W(x) is defined by Eq. (5).

8 ><

>:

wð Þ¼ x

Wð Þ¼ x

tion of the APM can be found in [19].

4. Numerical examples

49

4.1 Preliminary experiment

xj,i,<sup>G</sup> xj, <sup>r</sup>1,<sup>G</sup> þ F xj, <sup>r</sup>2,<sup>G</sup> � xj, <sup>r</sup>3,<sup>G</sup>

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

To handle the constraints, the APM is adopted, proposed by Lemonge and

wð Þþ x ∑ nc jj¼1

�

current population. The penalty parameter kjj is defined in Eq.(7):

kjj ¼ j j h i wð Þ x

wð Þ x , if x is feasible

where wð Þ x is the objective function of the candidate solution without penaliza-

where h i wð Þ x is the average value of the objective function of the solutions of the

∑nc

where vjjð Þ <sup>x</sup> � � means the violation of the jj-th constraint averaged over the current population considering only infeasible individuals. The complete formula-

To validate the proposed search mechanism, a well-known benchmark 10-bar truss, shown in Figure 2, is considered. Two cases are analyzed: in the first one, the problem consists of weight minimization considering displacements and stresses as constraints and, in the second case, minimization of its weight considering natural frequencies as constraints; for both cases the design variables are the cross-sectional areas of the bars (totalizing 10 variables). The truss has Young's modulus equal to

wð Þ x , if wð Þ x >h i wð Þ x h i wð Þ x , if wð Þ x ≤ h i wð Þ x

vjjð Þ <sup>x</sup> � �

ll¼<sup>1</sup>½ � vllð Þ <sup>x</sup>

2

, (7)

kjjvjjð Þ x , otherwise

(5)

(6)

� �

Initialize pseudorandom population

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

Else

end if

For j = 1: NPOP

Select randomly J ∈ {1,…, NPOP}

If rand < Pcr or j = J

For i = 1: NPOP

end For

end For

Barbosa [19].

tion Eq.(6)

where δmax is the maximum displacement at the top of the structure, δ is the maximum allowable displacement, f <sup>1</sup> is the first natural frequency of vibration, and f <sup>1</sup> is the minimum allowed frequency by the standard codes.

### 3. Differential evolution algorithm and the adaptive penalty scheme

The algorithm used in this study is the traditional DE algorithm, which was introduced in 1995 by Storn and Price [18]. It is based on evolution of population of vectors in the search space. This algorithm has been showing robustness in solving structural mono- and multi-objective optimization problems.

The first step of the algorithm consists of generating a pseudorandom population in the search space. Then, the evolution of the vectors is governed by Eq. (4):

$$\mathbf{x}\_{\mathbf{j},i,G} \gets \mathbf{x}\_{\mathbf{j},r1,G} + F(\mathbf{x}\_{\mathbf{j},r2,G} - \mathbf{x}\_{\mathbf{j},r3,G}) \tag{4}$$

where xj,i,<sup>G</sup> represents the new individual of the new generation; xj, <sup>r</sup>1,G, xj, <sup>r</sup>2,G, and xj, <sup>r</sup>3,<sup>G</sup> represent, respectively, the base vector and two other vectors from the previous generation (both three vectors are randomly selected and different among them); and F represents the scale factor of the difference between vectors. This expression (Eq. 4) is illustrated in Figure 1.

Mutation and crossover operators are considered in the differential evolution, and there is a predetermined probability of crossover (Pcr) as well as a probability of mutation between the new and the old individual.

Figure 1. Visual representation of DE scheme.

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

The following flowchart represents the scheme of DE:

Initialize pseudorandom population For i = 1: NPOP Select randomly J ∈ {1,…, NPOP} For j = 1: NPOP Select randomly rand ∈ [0, 1] If rand < Pcr or j = J xj,i,<sup>G</sup> xj, <sup>r</sup>1,<sup>G</sup> þ F xj, <sup>r</sup>2,<sup>G</sup> � xj, <sup>r</sup>3,<sup>G</sup> � � Else xj,i,<sup>G</sup> xj, <sup>r</sup>1,<sup>G</sup> end if end For

end For

δmaxð Þ x

<sup>1</sup> � <sup>f</sup> <sup>1</sup>ð Þ <sup>x</sup> f 1

f <sup>1</sup> is the minimum allowed frequency by the standard codes.

Truss and Frames - Recent Advances and New Perspectives

structural mono- and multi-objective optimization problems.

expression (Eq. 4) is illustrated in Figure 1.

Figure 1.

48

Visual representation of DE scheme.

of mutation between the new and the old individual.

where δmax is the maximum displacement at the top of the structure, δ is the maximum allowable displacement, f <sup>1</sup> is the first natural frequency of vibration, and

3. Differential evolution algorithm and the adaptive penalty scheme

in the search space. Then, the evolution of the vectors is governed by Eq. (4):

xj,i,<sup>G</sup> xj, <sup>r</sup>1,<sup>G</sup> þ F xj, <sup>r</sup>2,<sup>G</sup> � xj, <sup>r</sup>3,<sup>G</sup>

The algorithm used in this study is the traditional DE algorithm, which was introduced in 1995 by Storn and Price [18]. It is based on evolution of population of vectors in the search space. This algorithm has been showing robustness in solving

The first step of the algorithm consists of generating a pseudorandom population

where xj,i,<sup>G</sup> represents the new individual of the new generation; xj, <sup>r</sup>1,G, xj, <sup>r</sup>2,G, and xj, <sup>r</sup>3,<sup>G</sup> represent, respectively, the base vector and two other vectors from the previous generation (both three vectors are randomly selected and different among them); and F represents the scale factor of the difference between vectors. This

Mutation and crossover operators are considered in the differential evolution, and there is a predetermined probability of crossover (Pcr) as well as a probability

(4)

<sup>δ</sup> � <sup>1</sup>≤<sup>0</sup>

≤ 0

(3)

where NPOP represents the number of population.

To handle the constraints, the APM is adopted, proposed by Lemonge and Barbosa [19].

The fitness function W(x) is defined by Eq. (5).

$$\mathcal{W}(\mathbf{x}) = \begin{cases} \mathcal{w}(\mathbf{x}), & \text{if } \mathbf{x} \text{ is feasible} \\ \overline{\mathcal{w}}(\mathbf{x}) + \sum\_{\overline{j}=1}^{n\_c} k\_{\overline{j}l} v\_{\overline{j}}(\mathbf{x}), & \text{otherwise} \end{cases} \tag{5}$$

where wð Þ x is the objective function of the candidate solution without penalization Eq.(6)

$$w\overline{w}(\mathbf{x}) = \begin{cases} w(\mathbf{x}), & \text{if } w(\mathbf{x}) > \langle w(\mathbf{x}) \rangle \\ \langle w(\mathbf{x}) \rangle, & \text{if } w(\mathbf{x}) \le \langle w(\mathbf{x}) \rangle \end{cases} \tag{6}$$

where h i wð Þ x is the average value of the objective function of the solutions of the current population. The penalty parameter kjj is defined in Eq.(7):

$$k\_{\vec{\eta}} = |\langle w(\mathbf{x})\rangle| \frac{\langle v\_{\vec{\eta}}(\mathbf{x})\rangle}{\sum\_{l=1}^{n\_c} [v\_{l\vec{\eta}}(\mathbf{x})]^2} \tag{7}$$

where vjjð Þ <sup>x</sup> � � means the violation of the jj-th constraint averaged over the current population considering only infeasible individuals. The complete formulation of the APM can be found in [19].

### 4. Numerical examples

#### 4.1 Preliminary experiment

To validate the proposed search mechanism, a well-known benchmark 10-bar truss, shown in Figure 2, is considered. Two cases are analyzed: in the first one, the problem consists of weight minimization considering displacements and stresses as constraints and, in the second case, minimization of its weight considering natural frequencies as constraints; for both cases the design variables are the cross-sectional areas of the bars (totalizing 10 variables). The truss has Young's modulus equal to

Figure 2. The 10-bar truss.

10<sup>4</sup> ksi (68.95 GPa) and material density equal to 0.1 lb=in³ (2770 kg=m³). For case 1, the upper and lower bounds are equal to 0.1 and 33.50 in<sup>2</sup> , and two loads of 100 kips (444.82 kN) each are applied at nodes 2 and 4; for case 2, the bounds are equal to 0.1 and 7.75 in<sup>2</sup> , and an additional mass of 1000 lbs. (454.54 kg) is attached to free nodes (1–4). Constraints are set to 25 ksi (for both compression and tension) and 2 inches for case 1, and also f <sup>1</sup> ≥7 Hz, f <sup>2</sup> ≥ 15 Hz, and f <sup>3</sup> ≥20 Hz for case 2.

Vk ¼ V0S1S2S<sup>3</sup> (8)

<sup>k</sup> <sup>¼</sup> <sup>0</sup>:<sup>613</sup> � <sup>25</sup>:92 <sup>¼</sup> <sup>411</sup>:21 N=m² (10)

) are used; the steps are detailed in Eqs.(11)

9 m <sup>¼</sup> <sup>0</sup>:62 kN=<sup>m</sup> (11)

9 m <sup>¼</sup> <sup>1</sup>:23 kN=<sup>m</sup> (12)

, must

) and

, and the

) transfers

Vk ¼ 35 � 1:0 � 0:74 � 1:0 ¼ 25:9 m=s (9)

With the characteristic velocity, it is possible to determine the dynamic pressure

The dynamic pressure acting on the frame's larger facade, q = 0.411 kN/m<sup>2</sup>

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

and (12) and Figure 4, where Pc and PM are the uniform wind load on the corner

13:5 m²

27 m²

Two different types of gravity loads are considered: dead loads and live loads. The first one refers to self-weight of the structural elements, such as, the concrete slabs, which was adopted with a thickness equal to 10 cm, and the second one corresponds to the occupation. The dead load acting on the floor is 3 kN/m<sup>2</sup>

area) than the outer beams (OB), as can be observed in Figure 5. The blue area

) transfers its weight to the inner beams, and the red area (2.25 m<sup>2</sup>

. The inner beams (IB) would be more loaded (the largest influence

to columns, the influence area of corner columns (CC) (red area = 13.5 m<sup>2</sup>

kN m² �

> kN m² �

be transferred as a uniform load applied to the columns; to transfer the area loading

on the larger facade of the frame through Eq. (10).

The 39 members frame—joints and elements mapping.

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

<sup>q</sup> <sup>¼</sup> <sup>0</sup>:613V<sup>2</sup>

PC ¼ 0:411

PM ¼ 0:411

middle columns (MC) (blue area 27 m<sup>2</sup>

4.2.2 Gravity loads

live is 2 kN/m2

(4.5 m<sup>2</sup>

51

Figure 3.

columns and middle columns, respectively.

Table 2 shows the design variables (dv) (in<sup>2</sup> for case 1 and cm2 for case 2), the optimum weights (W) (lb for case 1 and kg for case 2) obtained for both cases, as well as a comparison with some results found in the literature where TS means the results obtained with this study. For both cases, 50 independent runs were performed; the population size is 50 which evolved in 200 generations leading to 10,000 function evaluations (nfe).

#### 4.2 Design loads

The first mutlistorey 3D steel frame is a simple three-storey steel frame, with 3 m of width, 6 m of length, and 9 m of height equally spaced. This 3D frame is subjected to lateral wind and gravity loads. The gravity and wind loads are defined based on the Brazilian technical codes NBR 6123 [2], detailed on the next subsections. The model in finite elements for the first 3D frame consists in 39 members and 24 joints depicted in Figure 3.

#### 4.2.1 Wind loads

To define the forces due to the wind on the columns, it demands to determine the dynamic pressure (q) acting on the area of the larger façade. For this purpose, two parameters, the wind basic velocity (V0) and the wind characteristic velocity (Vk), are necessary. The basic velocity V<sup>0</sup> is inherent of the region and assumed as the velocity of 3 s gust, exceeded in mean once in 50 years, 10 m above the ground on an open and plain field. For the city of Juiz de Fora, Minas Gerais State, Brazil, the basic velocity is equal to V<sup>0</sup> = 35 m/s. The characteristic velocity is defined by Eq. (8), where S<sup>1</sup> (topographic factor), S2(terrain roughness factor), and S3(statistic factor) are weighting coefficients resulting in 25.9 m/s as written in Eq. (9) ([2]).

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

Figure 3. The 39 members frame—joints and elements mapping.

$$V\_k = V\_0 \mathbf{S}\_1 \mathbf{S}\_2 \mathbf{S}\_3 \tag{8}$$

$$V\_k = \mathbf{35} \times \mathbf{1.0} \times 0.74 \times \mathbf{1.0} = \mathbf{25.9 m/s} \tag{9}$$

With the characteristic velocity, it is possible to determine the dynamic pressure on the larger facade of the frame through Eq. (10).

$$q = 0.613V\_k^2 = 0.613 \times 25.9^2 = 411.21 \text{ N/m}^2 \tag{10}$$

The dynamic pressure acting on the frame's larger facade, q = 0.411 kN/m<sup>2</sup> , must be transferred as a uniform load applied to the columns; to transfer the area loading to columns, the influence area of corner columns (CC) (red area = 13.5 m<sup>2</sup> ) and middle columns (MC) (blue area 27 m<sup>2</sup> ) are used; the steps are detailed in Eqs.(11) and (12) and Figure 4, where Pc and PM are the uniform wind load on the corner columns and middle columns, respectively.

$$P\_C = 0.411 \frac{\text{kN}}{\text{m}^2} \times \frac{13.5 \text{ m}^2}{9 \text{ m}} = 0.62 \text{ kN/m} \tag{11}$$

$$P\_M = 0.411 \,\frac{\text{kN}}{\text{m}^2} \times \frac{27 \,\text{m}^2}{9 \,\text{m}} = 1.23 \,\text{kN/m} \tag{12}$$

#### 4.2.2 Gravity loads

Two different types of gravity loads are considered: dead loads and live loads. The first one refers to self-weight of the structural elements, such as, the concrete slabs, which was adopted with a thickness equal to 10 cm, and the second one corresponds to the occupation. The dead load acting on the floor is 3 kN/m<sup>2</sup> , and the live is 2 kN/m2 . The inner beams (IB) would be more loaded (the largest influence area) than the outer beams (OB), as can be observed in Figure 5. The blue area (4.5 m<sup>2</sup> ) transfers its weight to the inner beams, and the red area (2.25 m<sup>2</sup> ) transfers

10<sup>4</sup> ksi (68.95 GPa) and material density equal to 0.1 lb=in³ (2770 kg=m³). For case

100 kips (444.82 kN) each are applied at nodes 2 and 4; for case 2, the bounds are

Table 2 shows the design variables (dv) (in<sup>2</sup> for case 1 and cm2 for case 2), the optimum weights (W) (lb for case 1 and kg for case 2) obtained for both cases, as well as a comparison with some results found in the literature where TS means the

The first mutlistorey 3D steel frame is a simple three-storey steel frame, with 3 m of width, 6 m of length, and 9 m of height equally spaced. This 3D frame is subjected to lateral wind and gravity loads. The gravity and wind loads are defined based on the Brazilian technical codes NBR 6123 [2], detailed on the next subsections. The model in finite elements for the first 3D frame consists in 39 members

To define the forces due to the wind on the columns, it demands to determine the dynamic pressure (q) acting on the area of the larger façade. For this purpose, two parameters, the wind basic velocity (V0) and the wind characteristic velocity (Vk), are necessary. The basic velocity V<sup>0</sup> is inherent of the region and assumed as the velocity of 3 s gust, exceeded in mean once in 50 years, 10 m above the ground on an open and plain field. For the city of Juiz de Fora, Minas Gerais State, Brazil, the basic velocity is equal to V<sup>0</sup> = 35 m/s. The characteristic velocity is defined by Eq. (8), where S<sup>1</sup> (topographic factor), S2(terrain roughness factor), and S3(statistic factor) are weighting coefficients resulting in 25.9 m/s as written in Eq. (9) ([2]).

to free nodes (1–4). Constraints are set to 25 ksi (for both compression and tension) and 2 inches for case 1, and also f <sup>1</sup> ≥7 Hz, f <sup>2</sup> ≥ 15 Hz, and f <sup>3</sup> ≥20 Hz for

results obtained with this study. For both cases, 50 independent runs were performed; the population size is 50 which evolved in 200 generations leading to

, and an additional mass of 1000 lbs. (454.54 kg) is attached

, and two loads of

1, the upper and lower bounds are equal to 0.1 and 33.50 in<sup>2</sup>

Truss and Frames - Recent Advances and New Perspectives

equal to 0.1 and 7.75 in<sup>2</sup>

4.2 Design loads

4.2.1 Wind loads

50

10,000 function evaluations (nfe).

and 24 joints depicted in Figure 3.

case 2.

Figure 2. The 10-bar truss.

Figure 4. Wind loads on columns.

Po ¼ 1:4 � 3

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

Discrete search spaces for columns and beams.

observed in the scale of Figure 6.

2

4.4 Experiment 2

the matrix [(f

53

4.3 Experiment 1

Table 2.

kN

<sup>δ</sup> <sup>¼</sup> <sup>H</sup>

<sup>400</sup> <sup>¼</sup> <sup>9000</sup>

Five independent runs and 100 generations with a population of 50 individuals are the parameters set for DE. The best solution found is detailed in Figure 5. It is possible to note that the algorithm reached both the lightest profile for beams and columns, which are W 150 � 13 (red) and W 150 � 22.5 (blue), respectively, leading to a final weight of 2050 kg. For the best solution found, rigorously feasible in the evolutionary process, the maximum displacement is 19.6 mm as can be

The second experiment has the same characteristics as the first one. However, a constraint concerning the first natural frequency of vibration is added (Eq. (16)) that must be at least 4 Hz, according to the dynamic comfort values prescribed by NBR 8800 [3]. The natural frequencies are obtained calculating the eigenvalues of

nf x M + K)] [23], where M and K are the mass and stiffness matrices,

m² <sup>þ</sup> <sup>1</sup>:<sup>5</sup> � <sup>2</sup>

Profiles for columns Profiles for beams

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

W 150 � 22.5 W 250 � 89 W 150 � 13 W 310 � 21 W 410 � 38.8 W 530 � 66 W 150 � 29.8 W 250 � 101 W 150 � 18 W 310 � 23.8 W 410 � 46.1 W 530 � 72 W 150 � 37.1 W 250 � 115 W 150 � 24 W 310 � 28.3 W 410 � 53 W 530 � 74 W 200 � 35.9 W 310 � 79 W 200 � 15 W 310 � 32.7 W 410 � 60 W 530 � 82 W 200 � 41.7 W 310 � 93 W 200 � 19.3 W 310 � 38.7 W 410 � 67 W 530 � 85 W 200 � 46.1 W 310 � 97 W 200 � 22.5 W 310 � 44.5 W 410 � 75 W 530 � 92 W 200 � 52 W 310 � 107 W 200 � 26.6 W 310 � 52 W 410 � 85 W 530 � 101 W 200 � 53 W 310 � 110 W 200 � 31.3 W 360 � 32.9 W 460 � 52 W 530 � 109 W 200 � 59 W 310 � 117 W 250 � 17.9 W 360 � 39 W 460 � 60 W 610 � 101 W 200 � 71 W 310 � 125 W 250 � 22.3 W 360 � 44 W 460 � 68 W 610 � 113 W 200 � 86 W 360 � 91 W 250 � 25.3 W 360 � 51 W 460 � 74 W 610 � 125 W 250 � 62 W 360 � 101 W 250 � 28.4 W 360 � 57.8 W 460 � 82 — W 250 � 73 W 360 � 110 W 250 � 32.7 W 360 � 64 W 460 � 89 — W 250 � 80 W 360 � 122 W 250 � 38.5 W 360 � 72 W 460 � 97 — W 250 � 85 — W 250 � 44.8 W 360 � 79 W 460 � 106 —

m² �

kN

The first experiment consists of the three-storey steel frame to minimize the weight as depicted in Figure 3, considering the loads discussed in the previous subsection subjected to the maximum horizontal displacements as the constraint that occur on the top of the frame. The frame has 9 m of height leading to an allowable displacement at the top equal to 22.5 mm as Eq. (15) shows:

2:25 m²

3 m <sup>¼</sup> <sup>5</sup>:4 kN=m (14)

<sup>400</sup> <sup>¼</sup> <sup>22</sup>:5 mm (15)

Figure 5. Influence area for beams—design loads.

its weight to the outer beams. The design factor used for dead loads was 1.4 and for the live loads 1.5, according to the Brazilian Technical Standard code [2]. Eqs. (13) and (14) summarize the calculus of the design loads adopted in the experiments, according to the Brazilian Technical Standard code, where Pi and Po are the uniform loading on inner beams and outer beams, respectively [3].

$$P\_i = \left(1.4 \times 3 \frac{\text{kN}}{\text{m}^2} + 1.5 \times 2 \frac{\text{kN}}{\text{m}^2} \right) \times \frac{4.5 \text{ m}^2}{3 \text{ m}} = 10.8 \text{ kN/m} \tag{13}$$

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


#### Table 2.

Discrete search spaces for columns and beams.

$$P\_o = \left(1.4 \times 3 \frac{\text{kN}}{\text{m}^2} + 1.5 \times 2 \frac{\text{kN}}{\text{m}^2} \right) \times \frac{2.25 \text{ m}^2}{3 \text{ m}} = 5.4 \text{ kN/m} \tag{14}$$

#### 4.3 Experiment 1

The first experiment consists of the three-storey steel frame to minimize the weight as depicted in Figure 3, considering the loads discussed in the previous subsection subjected to the maximum horizontal displacements as the constraint that occur on the top of the frame. The frame has 9 m of height leading to an allowable displacement at the top equal to 22.5 mm as Eq. (15) shows:

$$\overline{\delta} = \frac{H}{400} = \frac{9000}{400} = 22.5 \text{ mm} \tag{15}$$

Five independent runs and 100 generations with a population of 50 individuals are the parameters set for DE. The best solution found is detailed in Figure 5. It is possible to note that the algorithm reached both the lightest profile for beams and columns, which are W 150 � 13 (red) and W 150 � 22.5 (blue), respectively, leading to a final weight of 2050 kg. For the best solution found, rigorously feasible in the evolutionary process, the maximum displacement is 19.6 mm as can be observed in the scale of Figure 6.

#### 4.4 Experiment 2

The second experiment has the same characteristics as the first one. However, a constraint concerning the first natural frequency of vibration is added (Eq. (16)) that must be at least 4 Hz, according to the dynamic comfort values prescribed by NBR 8800 [3]. The natural frequencies are obtained calculating the eigenvalues of the matrix [(f 2 nf x M + K)] [23], where M and K are the mass and stiffness matrices,

its weight to the outer beams. The design factor used for dead loads was 1.4 and for the live loads 1.5, according to the Brazilian Technical Standard code [2]. Eqs. (13) and (14) summarize the calculus of the design loads adopted in the experiments, according to the Brazilian Technical Standard code, where Pi and Po are the uniform

> kN m²

�

4:5 m²

3 m <sup>¼</sup> <sup>10</sup>:8 kN=m (13)

loading on inner beams and outer beams, respectively [3].

kN

m² <sup>þ</sup> <sup>1</sup>:<sup>5</sup> � <sup>2</sup>

Pi ¼ 1:4 � 3

Influence area for beams—design loads.

Figure 4.

Figure 5.

52

Wind loads on columns.

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Figure 6. The best solution and the displacement field for the Experiment 1.

respectively, and fnf is the equivalent eigenvectors concerning the nf natural frequencies of vibration of the structure. Also a member grouping is adopted, considering the symmetry of structure. In optimized structures, it can be attractive to use a reduced number of distinct cross-sectional areas to minimize the costs of fabrication, transportation, storing, checking, and welding, thereby providing labor savings.

Thus, the member grouping is conducted such as corner columns, middle columns, outer beams, and inner beams (IB) form four different groups which each group will have the same profile as defined in Figure 7 and Table 3.

$$\begin{aligned} \frac{\delta\_{\text{max}}(\mathbf{x})}{22.5} - \mathbf{1} &\le \mathbf{0} \\ \mathbf{1} - \frac{f\_1(\mathbf{x})}{4} &\le \mathbf{0} \end{aligned} \tag{16}$$

The DE parameters are the same in Experiment 1. The best solution found presented a final weight of 2587 kg, the maximum displacement of 12.9 mm, and the first natural frequency of vibration equal to 4.14 Hz corresponding, as expected, to a feasible solution. It is interesting to note that the algorithm distributed masses along the structure in a better way in order to satisfy the frequency constraint, in this case, decreasing the maximum displacement. Figure 8 and Table 4 show the

Group Characteristics Color Corner columns Red Middle columns Blue Inner beams Green Outer beams Black

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

This numerical experiment consists in to minimize the weight of a 3D steel frame with six-storeys and 78 members, as illustrated in Figure 9, and it is subjected to wind load and a constraint concerning the maximum horizontal displacement at the top of the frame. In this experiment, distinct member groupings are adopted to show how the final weights decrease as more different profiles are used. In this sense, the members are grouped in two, four, and eight groups independently. The wind and gravity loads are defined in the same way as the previous exper-

<sup>400</sup> <sup>¼</sup> <sup>18000</sup>

<sup>400</sup> <sup>¼</sup> 45 mm (17)

iment, and the displacement constraint is written as (Eq. (17)):

<sup>δ</sup> <sup>¼</sup> <sup>H</sup>

The best solution and the displacement field for the Experiment 2.

detailed results.

Member grouping for the Experiment 2.

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

Table 3.

4.5 Experiment 3

Figure 8.

55

Figure 7. Member grouping for the Experiment 2.

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


Table 3.

Member grouping for the Experiment 2.

The DE parameters are the same in Experiment 1. The best solution found presented a final weight of 2587 kg, the maximum displacement of 12.9 mm, and the first natural frequency of vibration equal to 4.14 Hz corresponding, as expected, to a feasible solution. It is interesting to note that the algorithm distributed masses along the structure in a better way in order to satisfy the frequency constraint, in this case, decreasing the maximum displacement. Figure 8 and Table 4 show the detailed results.

#### 4.5 Experiment 3

respectively, and fnf is the equivalent eigenvectors concerning the nf natural frequencies of vibration of the structure. Also a member grouping is adopted, considering the symmetry of structure. In optimized structures, it can be attractive to use a reduced number of distinct cross-sectional areas to minimize the costs of fabrication, transportation, storing, checking, and welding, thereby providing labor savings. Thus, the member grouping is conducted such as corner columns, middle columns, outer beams, and inner beams (IB) form four different groups which each

group will have the same profile as defined in Figure 7 and Table 3.

The best solution and the displacement field for the Experiment 1.

Truss and Frames - Recent Advances and New Perspectives

Figure 6.

Figure 7.

54

Member grouping for the Experiment 2.

δmaxð Þ x

<sup>1</sup> � <sup>f</sup> <sup>1</sup>ð Þ <sup>x</sup>

<sup>22</sup>:<sup>5</sup> � <sup>1</sup>≤<sup>0</sup>

<sup>4</sup> <sup>≤</sup> <sup>0</sup>

(16)

This numerical experiment consists in to minimize the weight of a 3D steel frame with six-storeys and 78 members, as illustrated in Figure 9, and it is subjected to wind load and a constraint concerning the maximum horizontal displacement at the top of the frame. In this experiment, distinct member groupings are adopted to show how the final weights decrease as more different profiles are used. In this sense, the members are grouped in two, four, and eight groups independently.

The wind and gravity loads are defined in the same way as the previous experiment, and the displacement constraint is written as (Eq. (17)):

$$\overline{\delta} = \frac{H}{400} = \frac{18000}{400} = 45 \text{ mm} \tag{17}$$

Figure 8. The best solution and the displacement field for the Experiment 2.


Table 4.

Optimum solution found for the Experiment 2.

Figure 9. The 78 members frame joints and elements mapping.

#### 4.5.1 Members linked in two groups

Firstly, the problem considers the possibility of using only two distinct profiles, one for the columns and other for the beams. The member grouping adopted is described in the Table 5 and in Figure 10.

4.5.2 Members linked in four groups

The best solution of Experiment 3.1.

Experiment 3.1—members linked in two groups according to Table 5.

4.5.3 Members linked in eight groups

Figure 13 and Table 8.

Figure 10.

Table 5.

Experiment 3.1—members linked in two groups.

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

Table 6.

57

found for Experiment 3.3.

In the second analysis of Experiment 3, the members are linked in four distinct groups as described in Table 7 and Figure 12, and the best result found is detailed in

Group Characteristics Color Cross section 1 Columns Red W 150 29.8 2 Beams Blue W 360 44 Maximum displacement 22.6 mm Total weight 8971 kg

Group Characteristics Color 1 Columns Red 2 Beams Blue

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

Finally, the same problem is optimized with the members linked in eight groups as shown in Table 9 and Figure 14. Table 10 and Figure 15 show the best solution

The DE parameters are the same as Experiment 2, and best solution found is detailed in Table 6 and in Figure 11.

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


Table 5.

Experiment 3.1—members linked in two groups.

Figure 10. Experiment 3.1—members linked in two groups according to Table 5.


#### Table 6.

The best solution of Experiment 3.1.

#### 4.5.2 Members linked in four groups

In the second analysis of Experiment 3, the members are linked in four distinct groups as described in Table 7 and Figure 12, and the best result found is detailed in Figure 13 and Table 8.

#### 4.5.3 Members linked in eight groups

Finally, the same problem is optimized with the members linked in eight groups as shown in Table 9 and Figure 14. Table 10 and Figure 15 show the best solution found for Experiment 3.3.

4.5.1 Members linked in two groups

The 78 members frame joints and elements mapping.

Figure 9.

56

Table 4.

Optimum solution found for the Experiment 2.

Truss and Frames - Recent Advances and New Perspectives

described in the Table 5 and in Figure 10.

detailed in Table 6 and in Figure 11.

Firstly, the problem considers the possibility of using only two distinct profiles, one for the columns and other for the beams. The member grouping adopted is

Group Characteristics Color Cross section 1 Corner columns Red W 150 22.5 2 Middle columns Blue W 200 46.1 3 Inner beams Green W 150 13 4 Outer beams Black W 200 15 Maximum displacement — 12.9 mm First natural frequency — 4.14 Hz Total weight — 2587 kg

The DE parameters are the same as Experiment 2, and best solution found is

#### Figure 11.

The best solution and displacement field of Experiment 3.1.


Figure 13.

Table 8.

Table 9.

59

The best solution of Experiment 3.2.

Experiment 3.3—members linked in eight groups.

The best solution and displacement field of Experiment 3.2.

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

Group Characteristics Color Cross section 1 Corner columns Blue W 200 35.9 2 Middle columns Red W 150 22.5 3 Inner beams Black W 150 13 4 Outer beams Green W 310 38.7 Maximum displacement 23 mm Total weight 7851 kg

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

Group Characteristics Color Corner columns—floors 1, 2, and 3 Blue Middle columns—floors 1, 2, and 3 Red Inner beams—floors 1, 2, and 3 Green Outer beams—floors 1, 2, and 3 Black Corner columns—floors 4, 5, and 6 Cyan Middle columns—floors 4, 5, and 6 Magenta Inner beams—floors 4, 5, and 6 Yellow Outer beams—floors 4, 5, and 6 Gray

#### Table 7.

Experiment 3.2—members linked in four groups.

Figure 12. Experiment 3.2–members linked in four groups according to Table 7.

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



#### Table 8.

Figure 11.

Table 7.

Figure 12.

58

The best solution and displacement field of Experiment 3.1.

Truss and Frames - Recent Advances and New Perspectives

Experiment 3.2—members linked in four groups.

Experiment 3.2–members linked in four groups according to Table 7.

Group Characteristics Color Corner columns Red Middle columns Black Inner beams Green Outer beams Blue

The best solution of Experiment 3.2.


#### Table 9. Experiment 3.3—members linked in eight groups.

Figure 14. Experiment 3.3—members linked in eight groups according to Table 9.


#### Table 10.

The best solution of Experiment 3.3.

#### 4.6 Experiment 4

This experiment considers the first natural frequency of vibration as an additional constraint (Eq. (18)) to the six-storey 3D frame. However, the DE did not find any feasible solution. Thus, a new frame presenting 114 members (Figure 16) with bracings is proposed to stiffen the candidate solutions, rising the chances of reaching feasible solutions.

δmaxð Þ x

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

<sup>1</sup> � <sup>f</sup> <sup>1</sup>ð Þ <sup>x</sup>

described in Table 11 and in Figure 17.

One hundred fourteen members 3D frame—long and short façade.

Figure 15.

Figure 16.

61

The best solution and displacement field of Experiment 3.3.

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

<sup>45</sup> � <sup>1</sup>≤<sup>0</sup>

<sup>4</sup> <sup>≤</sup> <sup>0</sup>

For this improved structure, the members are linked in five different groups

(18)

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

Figure 15. The best solution and displacement field of Experiment 3.3.

Figure 16. One hundred fourteen members 3D frame—long and short façade.

$$\begin{aligned} \frac{\delta\_{\text{max}}(\mathbf{x})}{45} - \mathbf{1} &\le \mathbf{0} \\ \mathbf{1} - \frac{f\_1(\mathbf{x})}{4} &\le \mathbf{0} \end{aligned} \tag{18}$$

For this improved structure, the members are linked in five different groups described in Table 11 and in Figure 17.

4.6 Experiment 4

Table 10.

60

Figure 14.

Experiment 3.3—members linked in eight groups according to Table 9.

Truss and Frames - Recent Advances and New Perspectives

reaching feasible solutions.

The best solution of Experiment 3.3.

This experiment considers the first natural frequency of vibration as an additional constraint (Eq. (18)) to the six-storey 3D frame. However, the DE did not find any feasible solution. Thus, a new frame presenting 114 members (Figure 16) with bracings is proposed to stiffen the candidate solutions, rising the chances of

Group Characteristics Color Cross section Corner columns—floors 1, 2, and 3 Blue W 200 46.1 Middle columns—floors 1, 2, and 3 Cyan W 150 22.5 Inner beams—floors 1, 2, and 3 Red W 150 13 Outer beams—floors 1, 2, and 3 Black W 310 38.7 Corner columns—floors 4, 5, and 6 Cyan W 150 22.5 Middle columns—floors 4, 5, and 6 Cyan W 150 22.5 Inner beams—floors 4, 5, and 6 Red W 150 13 Outer beams—floors 4, 5, and 6 Gray W 250 32.7 Maximum displacement 26 mm Total weight 7421 kg

## Truss and Frames - Recent Advances and New Perspectives


Table 11.

Experiment 4—members linked in five groups.

The best result found presented the lightest structure of this set of experiments, even though presenting a greater number of members than the previous structural configuration. The maximum displacement at the top of the frame is 19 mm, and the first natural frequency of vibration is 4.38 Hz, leading to a feasible solution.

Group Characteristics Color Cross section 1 Corner columns Blue W 150 22.5 2 Middle columns Blue W 150 22.5 3 Inner beams Cyan W 150 13 4 Outer beams Cyan W 150 13 5 Bracers Cyan W 150 13 Maximum displacement — 19 mm First natural frequency — 4.38 Hz Total weight — 6091 kg

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

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

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 algo-

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

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

Figure 18 and Table 12 detail the best solution for this experiment.

5. Analysis of results

The best solution of Experiment 4.

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

Table 12.

resulting in a heavier optimized structure.

rithm found only five distinct profiles.

behavior.

63

their corresponding number of distinct profiles used.

Figure 17. Experiment 4—members linked in five groups according to Table 11.

Figure 18. The best solution and displacement field of Experiment 4.

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


Table 12.

Figure 17.

Table 11.

Experiment 4—members linked in five groups.

Truss and Frames - Recent Advances and New Perspectives

Figure 18.

62

Experiment 4—members linked in five groups according to Table 11.

Group Characteristics Color Corner columns Red Middle columns Gray Inner beams Black Outer beams Blue Bracers Cyan

The best solution and displacement field of Experiment 4.

The best solution of Experiment 4.

The best result found presented the lightest structure of this set of experiments, even though presenting a greater number of members than the previous structural configuration. The maximum displacement at the top of the frame is 19 mm, and the first natural frequency of vibration is 4.38 Hz, leading to a feasible solution. Figure 18 and Table 12 detail the best solution for this experiment.
