Evaluation of a novel Asymmetric Genetic Algorithm to optimize the structural design of 3D regular and irregular steel frames

Abstract

We propose a new algorithm, named Asymmetric Genetic Algorithm (AGA), for solving optimization problems of steel frames. The AGA consists of a developed penalty function, which helps to find the best generation of the population. The objective function is to minimize the weight of the whole steel structure under the constraint of ultimate loads defined for structural steel buildings by the American Institute of Steel Construction (AISC). Design variables are the cross-sectional areas of elements (beams and columns) that are selected from the sets of side-flange shape steel sections provided by the AISC. The finite element method (FEM) is utilized for analyzing the behavior of steel frames. A 15-storey three-bay steel planar frame is optimized by AGA in this study, which was previously optimized by algorithms such as Particle Swarm Optimization (PSO), Particle Swarm Optimizer with Passive Congregation (PSOPC), Particle Swarm Ant Colony Optimization (HPSACO), Imperialist Competitive Algorithm (ICA), and Charged System Search (CSS). The results of AGA such as total weight of the structure and number of analyses are compared with the results of these algorithms. AGA performs better in comparison to these algorithms with respect to total weight and number of analyses. In addition, five numerical examples are optimized by AGA, Genetic Algorithm (GA), and optimization modules of SAP2000, and the results of them are compared. The results show that AGA can decrease the time of analyses, the number of analyses, and the total weight of the structure. AGA decreases the total weight of regular and irregular steel frame about 11.1% and 26.4% in comparing with the optimized results of SAP2000, respectively.

Keywords

optimization / steel frame / Asymmetric Genetic Algorithm / constraints of ultimate load / constraints of serviceability limits / penalty function

Introduction

Several new materials and structures have been developed in recent years, and they have been used in big cities such as Tehran. Today the steel frame structure plays an important role in structural and civil engineering because the steel frame structures have various advantages such as the high strength, easiness of shaping, and high ductility. They can help engineers to design tall buildings and cost-efficient structures [¹].

Designing low-cost structures is important that it has satisfied pre-defined objectives. Optimization methods have been improved to serve this aim by archiving an appropriate balance between the weight and safety of the structures [¹]. There are several optimization algorithms which can help decrease the structural weight and cost and significantly, although they ensure to satisfy the ultimate yield and serviceability limits. During past decades, the optimizing steel frame has been an important issue in structural engineering. Several related studies were carried out to optimize some structure frames; for example, in 2000, Pezeshk et al. [²] used Genetic Algorithm (GA) for designing the geometrical nonlinear steel frame. Camp et al. [³] used the Ant Colony Optimization (ACO) in partnership with penalty constraint handling technique for obtaining the optimum design of steel structure with using separate variables. Hasançebi et al. [⁴] employed the Bat Inspired method (BI) for solving the optimization problem. Harmony Search algorithm (HS) is another optimization method that was employed for solving the optimization problem in some previous studies [⁵–⁷]. Kaveh and Talatahari [⁸] employed an Improved Ant Colony (IAC) for the optimization of several steel frames. Aydoğdu and Saka [⁹] utilized IAC to optimize the weight of the irregular steel frames. Some researchers used Big Bang Big Crunch algorithm (BBBC) for optimization of the steel frames [¹⁰,¹¹]. In these methods, the constraints choose based on the practically accepted standard, for example, British Standard (BS5950-1990) or Specification for Structural Steel Buildings (ANSI/AISC-1994), (ANSI/AISC-2010) [¹²].

Le et al. [¹] employed SLDM and IDE for solving the optimization problem. According to the previous studies, in all of the optimization problems of steel frames, the weight and cost are the main objective functions, although a certain level of safety is another important parameter in the most optimization problems. In recent years, these methods and theories have improved for optimizing the structure [¹³]. Papadrakakis et al. [¹⁴] combined the Monte Carlo simulation method with the Latin hypercube sampling technique to handle probabilistic constraints of 2D steel frames under seismic loads. Ghasemi and Yousefi [¹⁵] employed a GA for solving the probabilistic design of steel frames. Shayanfar et al. [¹⁶] utilized a combination of GA and OpenSees to solve an optimization problem.

On the other hand, OpenSees was employed in association with GA to optimize the weight of a steel frame [¹⁶]. Li et al. [¹⁷] have presented the Single-Loop Deterministic Method (SLDM), although other researchers were used SLDM for solving optimization problem from 1998 to 2004. They show the computational cost is reduced by this method [¹⁸–²³]. Storn and Price [²⁴] presented the Differential Evolution (DE) for solving the optimization problems. They showed that this method is an efficient approach to reduce computational cost. This method was developed in different versions; therefore, it was used in the various optimization problem such as composite plates optimization problems, pattern recognition, damage detection and, truss structure optimization problems [²⁵–³⁵]. The review of the previous studies shows that the optimization steel frame is widespread.

There are several useful studies regarding the optimization composite sections, steel sections and steel frame [³⁶–⁴⁰]. In addition, many developed optimization methods have been utilized to optimize steel frame and steel section in recent years [⁴¹–⁴⁴].

This paper proposed a new algorithm to solve the optimization problems of steel frames with separate variables, which is named Asymmetric Genetic Algorithm (AGA). The novelty of this study related to a penalty function of AGA, which can reduce computational cost and weight of the steel frame against some published GAs. In addition, this method does not evaluate constraints for each element which this issue causes that time of analyses, and the number of analyses is decreased against some algorithms. On the other hand, this algorithm can simplify solving the optimization problems by using its developed penalty function; therefore, this algorithm is an improved version of GA. In the present study, results of AGA are compared with the results of some previous GAs such as Particle Swarm Optimization (PSO), Particle Swarm Optimizer with Passive Congregation (PSOPC), Particle Swarm Ant Colony Optimization (HPSACO), Imperialist Competitive Algorithm (ICA), and Charged System Search (CSS). According to this comparison, AGA can reduce the computational cost and weight of steel frame; therefore, it can solve the complex problems better than these algorithms in a lower time that this issue is the second novelty of this study. As a result, AGA obtains two improvements against these algorithms. In the present investigation, the variables are cross-sectional areas of beams and columns that are presumed as discrete variables. Many conditions such as constraints of ultimate load and serviceability limits are specified based on the characteristics for Structural Steel Building by the American Institute of Steel Construction (AISC).

Formulation of the optimization problem of steel frames

The original mathematical form of an optimization problem of steel structures is formulated as

\min \mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}W\left(\mathbf{X}\right)={\displaystyle \sum _{i=1}^{n_{\mathrm{e}}}{A}_i}·L_i·{\rho }_i,

\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{var}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}:DV=\left\{X_i\right\},

(1)\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{int}:\{\begin{array}{cc}{C}_k^{\mathrm{F}}\le 0,\hfill & k=1,\mathrm{...},n_{\mathrm{e}},\\ C_k^{\mathrm{S}}\le 0,\hfill & k=1,\mathrm{...},n_{\mathrm{e}},\\ C_k^{\mathrm{\Delta }}\le 0,\hfill & k=1,\mathrm{...},n_{\mathrm{e}},\\ X_i\mathrm{\in }\mathit{G},\hfill & k=1,\mathrm{...},n_{\mathrm{e}}\mathrm{.}\end{array}

The objective function W(A) in Eq. (1) is the weight of the whole frame; n_e is the number of structural elements in the frame; A_i, r_i, and L_i are the cross-sectional areas, weight density, and length of ith structural components, respectively, where X_i is related to the ith section that is selected from the population G. On the other hand, A_i is selected from discrete modules set G of thin wing steel (W), and Box shaped column and I-shaped beam provided by AISC; are constraints related to ultimate load following AISC-LRFD 2010 [³⁶]. Generally, C_k^{\mathrm{F}} is calculated as

(2)C_k^{\mathrm{F}}=R_{\mathrm{u}}-\varphi R_{\mathrm{n}},

where C_k^{\mathrm{F}} are constraints related to the strength design for element number k and R_{\mathrm{u}} is the required strength using LRFD load combination, R_{\mathrm{n}} denotes the normal strength, \varphi is the resistance factor and \varphi R_{\mathrm{n}} the design strength. After analyzing frame, axial forces and bending moments are calculated for each element. To check ultimate loading limits for each element, the constraints should be evaluated for each element under its axial forces and bending moments, as shown in the following equations.

(3)\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\frac{P_{\mathrm{r}}}{P_{\mathrm{c}}}\ge 0.2,\frac{P_{\mathrm{r}}}{P_{\mathrm{c}}}+\frac{8}{9}(\frac{M_{\mathrm{r}x}}{M_{\mathrm{c}x}}+\frac{M_{\mathrm{r}y}}{M_{\mathrm{c}y}})\le 1.0,

(4)\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\frac{P_{\mathrm{r}}}{P_{\mathrm{c}}}\ge 0.2,\frac{P_{\mathrm{r}}}{{2P}_{\mathrm{c}}}+(\frac{M_{\mathrm{r}x}}{M_{\mathrm{c}x}}+\frac{M_{\mathrm{r}y}}{M_{\mathrm{c}y}})\le 1.0,

P_c and P_r denoting the design and required axial strength, respectively. M_c and M_r are the design and required flexural strength, respectively. P_r and M_r are computed based on the AISC-LRFD 2010. P_c = Ø × P_n and M_c = Ø_b× M_n are calculated based on the property of each element; P_n and M_c are the nominal compression and flexural strength, respectively. P_c and M_c are computed for each element based on chapter E and F of the AISC, respectively. Ø = 0.9 and Ø_b = 0.9 are defined as the resistance factor for compression/tensile and bending, respectively. y and x are subscript symbols that are related to weak and strong axis bending, respectively.

C_k^{\mathrm{\Delta }} are constraints related to the ultimate load limits based on the AISC-LRFD 2010. In addition, C_k^{\mathrm{S}} are constraints related to design for stability based on AISC-LRFD 2010 [³⁶,⁴⁴]. On the other hand, the ultimate load limits of the overall structure, and the individual members and connections and checked based on this standard in this study [³⁶,⁴⁴].

Asymmetric Genetic Algorithm for solving optimization problems of steel frames

Genetic Algorithm for solving an optimization problem

GA is used to generate high quality solutions in search and optimization problems. GA was introduced by Holland [⁴⁵,⁴⁶], but a successful result of this method owes serious effort of Goldberg [⁴⁶,⁴⁷]. This method has been used efficiently science the early 1990s for solving the structural optimization problems [⁴⁷].

GA is related to Darwin’s theory of natural selection. Reducing the weight of steel frame and computational cost are usually the main aim of using GA in the structural optimization problem. When the algorithm can find the optimal solution, the process of solving the optimization problem is stopped [⁴⁸,⁴⁹]. When all elements of the structure should satisfy the constraints in the structural optimization problems, the problem is classified as a complex optimization problem. Generally, the basic GA often is not useful to solve this kind of complex optimization problem [⁵⁰].

Asymmetric Genetic Algorithm

AGA is an improved GA, which can solve the complex problem easily. Moreover, solving the optimization problem is simplified by this Algorithm. A good optimization algorithm solver can help save a huge time of computational cost in the complex optimization problems. The assessment of constraints is carried out by numerical simulation methods such as the finite element method (FEM). To evaluate ability of the developed algorithm, its results should be compared with results of the conventional GA and some improved GAs [⁴⁵], the AGA has two improvements, which are presented in the next two subsections.

Evaluating constraints just for the best solution from population

After selecting population, members of the population are sorted by weight function, and just the best members are evaluated by the constraints. If the lowest weight of the population satisfies the constraints, AGA does not evaluate other members based on the constraints. If else, the penalty function is applied to the best member, and then the population is sorted, and the new best solution is evaluated by the constraints again. This procedure continues to obtain the lowest weight member that satisfied by whole constraints. After achieving this purpose, AGA starts the next step.

As a result, assessment of constraints is not done for whole members of the population in this algorithm; therefore, the number of structural analyses will be less than weight calculation. Moreover, some members with lower weight are not evaluated by constraints; therefore, this issue causes that convergence speed of AGA is increased against GA, which is the basic form of GA.

Asymmetric Genetic Algorithm for handling probabilistic constraints

Although minimizing cost-effective is the first aim of GA, this aim in AGA is the second goal. The first goal in AGA is finding members with a lower weight, which can satisfy the constraints. The probabilistic constraints in Eq. (1) can be rewritten in the general form as:

(5)C\left({\mathit{X}}^k\right)=\{\begin{array}{ll}W\left({\mathit{X}}^k\right)+P\left({\mathit{X}}^k\right),\hfill & \mathrm{if}\hspace{1em}W\left({\mathit{X}}^k\right)=\min \left\{W\left({\mathit{X}}^i\right)|i\in \left\{\mathrm{1},\mathrm{2},...,n_{\mathrm{pop}}\right\}\right\}\hfill \\ W\left({\mathit{X}}^k\right),\hfill & \mathrm{if\; else,}\hfill \end{array}

(6)P\left({\mathit{X}}^k\right)=W\left({\mathit{X}}^k\right)\times V\left({\mathit{X}}^k\right),

where V\left({\mathit{X}}^k\right) is a function that is defined as the violation of constraints in this study. P\left({\mathit{X}}^k\right)is the penalty function.

(7)V\left({\mathit{X}}^k\right)=\{\begin{array}{ll}{\displaystyle \frac{w_{\max }}{w_{\min }}}+\mathrm{1},\hfill & \begin{array}{l}\mathrm{if\; the\; problem\; is\; not\; passed}\hfill \\ \mathrm{constraints\; of\; ultimates\; load}\hfill \\ \mathrm{and\; serviceability\; limits,}\hfill \end{array}\hfill \\ \\ \mathrm{0},\hfill & \mathrm{if\; else,}\hfill \end{array}

where w_max and w_min are a maximum and minimum weight of elements, respectively. According to above-mentioned equations, after sorting members based on their weights, the constraints of ultimate load and serviceability limits for each element is evaluated for the lowest weight elements, and then the penalty function is applied. If the constraints are satisfied, the value of the penalty function is zero and, if the constraints are not satisfied, a value of penalty function is calculated and applied. After applying the value of the penalty function, the solutions are sorted again. If the solution is changed, the previous process is repeated for finding the best solution. This process continues to find the optimal solution.

According to the AGA procedure, just the constraints for the solutions are evaluated that they have the lowest weight in population in one time of cycles. This algorithm is not evaluated constraints of ultimate load limits for all of the members.

The steps of AGA

To explain the difference of AGA with the previous GA, the steps of this method are classified and described in this subsection. This method includes the following steps.

Random selection of an initial population

Calculating weight function for all elements of the population and sorting all of them

Evaluation of the constraints of ultimate load limits for the best solution of an initial population

If the optimal solution of the population satisfies constraints, the algorithm continues and go to the next step. If else, AGA uses the penalty function to solve the optimization problem, which is explained in the previous subsection.

Crossover and mutation

In this step, the second generation of the population through a combination of genetic operators is created, which this step is named crossover and mutation. For each new breeding generation, a pair of a parent in the previous loop is chosen for breeding new generation; therefore, the new members of the population are created in each loop with mutation progress by this algorithm. The new members of the population typically share many characteristics of its parents. This random arrangement is carried out with a roulette wheel. In this research, the one point, two points, and several point methods are used for crossover progress. AGA is selected randomly one of these methods for the crossover progress.

Combination and creating a new population

This progress is carried out in three following steps.

1) Merge the main, new generation, the mutated population to create middle population (the next population).

2) Calculation weight function for all elements of the middle population and sorting them based on their weight.

3) Evaluating constraints of ultimate load limits for the best solution.

If the best solution of the middle population satisfies the constraints of ultimate load limits, the AGA run the next step. If else, the best solution of the middle population does not satisfy the constraints, the penalty function is applied to the best solution of the middle population then the solutions are sorted based on their weight, and the best solution is changed to the worst solution. Then the previous progress is carried out to find the new best solution. This progress continues to find the lowest weight members that can satisfy the constraints.

Producing a new population

In each loop, some members of the best members of the population are selected, and others are removed.

Termination

This generational process is repeated until a termination condition is reached. The flowchart of AGA is shown as Fig. 1.

Numerical examples for solving the optimization problems of steel frames

The goal of this section is to evaluate AGA in solving the optimization problems. In this study, six numerical examples are offered to demonstrate the applicability and robustness of the proposed AGA for solving the optimization problems. This section includes six subsections that a five-storey one-bay steel frame is optimized by AGA and GA in the first subsection and then, results of two algorithms are compared. Four regular and irregular steel frame in height and plan are proposed to optimize by AGA and optimization modules of SAP2000 in subsections 2.5, and the results of two methods are compared. Moreover, four steel frames include a regular nine-storey five-bay, a nine-storey six-bay (irregularities in the plan), a nine-storey five-bay steel frame (irregularities in height) and irregular nine-storey six-bay are optimized and analyzed in these subsections respectively. In section 6, a fifteen-storey three-bay planner frame is optimized by AGA which was optimized by the previous algorithm such as PSO, PSOPC, HPSACO, ICA, and CSS, results of these algorithms are compared in this subsection.

According to the previous section, AISC steel design standard (AISC-LRFD 2010) is utilized for designing elements of the steel frames. Load patterns are applied based on ASCE7 in this investigation. The property of soil is considered as type III. The indeterminate factor is applied one in these analyses. The deck is selected steel deck. Earthquake loading is applied by dynamic analysis method. The material properties are selected as Young’s modulus E = 2×10¹⁰ kgf/m²; effective yield strength f_y = 24×10⁶ kgf/m²; effective ultimate strength f_u = 37×10⁶ kgf/m²; weight density r = 7850 kg/m³. Table 1 shows the kind of loads, such as dead loads, live load, and lateral loads. All optimization problems are solved by Matlab coding and SAP2000 on a computer having configuration: CoreTM i5-7200U CPU@2.5 GHz 2.71, RAM 8.00 GB.

Five-storey one-bay steel frame

The first example is the regular five-storey one-bay steel frame. Figure 2 shows the geometry of this building. This steel frame has 40 elements (40 variables). In this subsection, this steel frame is optimized by AGA and GA, and the results of them are compared. The different kinds of load which are shown in Table 1, are applied to this steel frame. The relative parameters such as N_g (the number of generations), N_p (The population size), P_c (the crossover probability), and P_mut (the mutation probability), are considered same in two algorithms which amount of them are selected 1000, 50, 60, and 5, respectively. Design variables are cross sections for beams and columns. The cross sections of columns are selected from BOX200×10 to BOX600×45 within the standard sections presented by the AISC. The cross sections for beams are selected from IPE100 to IPE600 within the standard sections provided by the AISC.

After implementing the 20 independent runs, the optimal result of AGA (the best result among 20 results of AGA) is compared with the optimal result of GA (the best result among 20 results of GA), which are shown in Table 2. Although every run of GA and AGA has a different result in the optimization problem, the best result among 20 independent runs is reposted in this study. These results show that the reliability of AGA is more than GA, as shown in Fig. 3.

Figure 3 shows the number of generation of AGA and GA by a red and blue line, respectively. According to Fig. 3, the value of the objective function by AGA converges early at the 320th generation, although this value by the GA converges slower, until at the 582th generation. In addition, Fig. 4 shows the number of structural analyses in AGA and GA. Based on Fig. 4, the number of structural analyze in AGA is 80 times less than GA. On the other hand, the speed of convergence by AGA is more than the speed of convergence by GA; therefore, AGA can solve the optimization problem faster than GA.

According to the results of AGA and GA, the generation of both algorithms was 1000, although the number of structural analyses of AGA is lower than the number of structural analyses of GA. As a result, AGA finds optimal result earlier than GA in this problem.

All elements of the optimal solution which are optimized by AGA can satisfy all constraints of ultimate load limits, which are defined based on the specification for structural steel buildings by AISC. Figure 5 shows the conventional histories of \varphi R_{\mathrm{n}}/R_{\mathrm{u}} for each element in the best solution of the population which they are depended on C_k^{\mathrm{F}} in Eq. (1). According to section 2, C_k^{\mathrm{F}} should be less than zero, \varphi R_{\mathrm{n}}/R_{\mathrm{u}} should be less than one, which are shown obviously in Fig. 5.

According to the results, time of solving the optimization problem by GA is longer than AGA, because the number of elements which GA should be evaluated based on the constraints is more than AGA; therefore, the convergence time of GA in this optimization problem is more than those in AGA. The number of variables that should be evaluated based on the constraints by GA is more than those of the AGA. According to the results, GA is not proposed to solve the complex optimization problem. As a result, other examples of this study which include two nine-storey five-bay steel frame and two nine-storey six-bay steel frame, are optimized by AGA and optimization modulus of commercial software SAP2000, and then the results of them are compared. GA cannot optimize the complex problem such as these steel frame; therefore, the results of AGA are compared with the results of the optimization modulus of SAP2000 in these problems.

Regular nine-storey five-bay

In this example, a regular nine-storey five-bay which is shown in Fig. 6, is optimized by AGA and optimization modulus of SAP2000. There are 864 elements (beams and columns) in this structure which are selected independently by various cross-sections that are provided by the AISC from W4×13 to W44×335. Based on the details in Table 1, the dead, live, and lateral loads are applied in this problem.

The parameters relevant to AGA in this example are selected as given below: the number of generations N_g = 1000; the population size N_p = 200; the crossover probability P_c = 60; the mutation probability P_mut = 5.

In this example, the steel frame design is optimized by AGA and optimization modules of SAP2000, and then, the results of the two methods are compared, as shown in Table 3. According to the results of Table 3, the results of AGA show a better solution with respect to the weight per area and the total weight of the structure in this example. The total weight of the optimized structure by AGA is 11.6 percent lower than the total weight of the optimized structure by the optimization modules of SAP2000.

Figure 7 shows convergence histories of AGA for solving the deterministic optimization problem of the regular nine-storey five-bay steel frame. The number of weight calculations is 130200 in this example, although structural analyses are done just for 6117 of them. This issue causes reducing time analyses by AGA in solving the optimization problem. The best result of the AGA coverage much faster than those of the optimization modules of SAP2000.

All elements of the optimal solution in this example which are optimized by AGA satisfy all constraints of ultimate load limits, which are defined based on the specification for structural steel buildings by AISC. Figure 8 shows \varphi R_{\mathrm{n}}/R_{\mathrm{u}}for each element in the best solution in this example, which this parameter is related to C_k^{\mathrm{F}} in Eq. (1). Figure 8 shows that \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is lower than one for each element in solving the optimization problem by AGA and SAP2000. The mean of \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is calculated 0.84 and 0.72 for solving the optimization problem by AGA and SAP2000, respectively. Also, the standard deviation is calculated 0.07 and 0.15 in solving the optimization problem by AGA and SAP2000, respectively. According to the results, AGA is more strength than the optimization modules of SAP2000 in solving this optimization problem.

Nine-storey six-bay steel frame (irregularities in plan and regularities in height)

A nine-storey six-bay steel frame is considered as the optimization problem in this subsection that is irregularities in plan and regularities in height, as shown in Fig. 9. This frame is optimized by AGA and optimization modules of SAP2000, and the results of the two methods are compared. The frame consists of 873 members, which could be selected by various cross section and are defined as independent variables in this problem. The frame is subjected to lateral and vertical loads, as shown in Table 1. The cross sections of beams and columns are selected within the standard sections provided by the AISC from W4×13 to W44×335. The parameters for the AGA in this optimization problem are selected as given below: the number of generations N_g = 1000; the population size N_p = 250; the crossover probability P_c = 60; the mutation probability P_mut = 5.

This optimization problem is solved by AGA and optimization modules of SAP2000. Total weight of steel frame and the number of structural analyses which are obtained by AGA and optimization modules of SAP2000 are compared in this example. These results are shown in Table 4. The values of weight per area and the total weight in the optimal solution of AGA are lower than those of SAP2000. The total weight of the structure, which is obtained by AGA is 20.4 percent lower than the total weight of the structure that is obtained by the optimization modules of SAP2000. As a result, AGA can optimize this steel frame better than the optimization modules of SAP2000. Moreover, the optimal solution of AGA shows better results than the optimal solution of SAP2000 with respect to the values of weight per area and total weight of the structure.

Convergence histories of AGA for solving this optimization problem are shown in Fig. 10. The number of weight calculations is 162750 in this example, although structural analyses are done for 6918 of them; therefore, AGA uses about 4.2 percent of the whole elements of various generations for analyzing structure this example; thus, time analyses of AGA in the optimization problem reduces.

The designed elements in optimal solution satisfy all of the constraints of ultimate load limits, which are defined based on the specifications of structural steel buildings provided by the AISC. Figure 11 shows \varphi R_{\mathrm{n}}/R_{\mathrm{u}}for each element in the best design. It is specified that this parameter is related to C_k^{\mathrm{F}}, which is presented in Eq. (1). Figure 11 shows that \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is less than one for each element in solving the optimization problem by AGA and SAP2000. The mean of \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is 0.74 and 0.63 for solving this optimization problem by AGA and SAP2000, respectively. In addition, the standard deviation is 0.12 and 0.17 in solving this optimization problem by AGA and SAP2000, respectively. According to the results, the results of AGA are better than the results of SAP2000 with respect to the values of weight per area and the total weight in the optimal solution. On the other hand, the AGA can optimize this structure with the lower weight elements against optimization modules of SAP2000 and also, this algorithm can reduce the time of analyses based the results of Table 4; therefore, the efficiency of AGA is better than those of optimization modules of SAP2000.

Nine-storey five-bay steel frame (regularities in plan and irregularities in height)

The fourth example, which is considered in this paper is a nine-storey five-bay frame, which is regularities in plan and irregularities in height, as shown in Fig. 12. The frame includes 864 members, which could be selected by various cross sections. The frame is subjected to lateral and vertical loads, as shown in Table 1. The cross sections of beams and columns are chosen within the standard sections provided by the AISC from W4×13 to W44×335. The parameters for AGA in this optimization problem are selected as given below: the number of generations N_g = 1000; the population size N_p = 200; the crossover probability P_c = 60; the mutation probability P_mut = 5.

First, this optimization problem is solved by AGA and optimization modules of SAP2000, and then, the results of them are compared. Table 5 shows the results of solving this optimization problem such as total weight, weight per area, and the number of analyses, which are obtained by AGA and optimization modules of SAP2000. AGA shows better results with respect to the weight per area and the total weight of the structure in comparing with results of SAP2000. According to the results of Table 5, the total weight of the structure in AGA is 23.5 percent lower than the total weight of the structure in the optimization modules of SAP2000. In addition, AGA uses 17 cross sections for solving this optimization problem, although optimization modules of SAP2000 use 32 cross sections for solving this problem; therefore, time of analyses in AGA is less than the time of analyze in optimization modules of SAP2000.

Figure 13 shows convergence histories of AGA and SAP2000 for solving this structural optimization problem. The results show that the number of weight calculation is 130200 in this example, although structural analyses are done just for 8873 of them; therefore, AGA uses about 6.8 percent of the elements of all generations in the structural analyses process. According to this issue, AGA can reduce the time of analyses in the optimization problem against optimization modules of SAP2000.

For evaluating the constraints of ultimate load limits for each element, \varphi R_{\mathrm{n}}/R_{\mathrm{u}} should be checked in the optimization problem. All elements should satisfy the constraints of ultimate load limits, which are defined based on the specifications of structural steel buildings by the AISC. In this problem, \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is calculated for each element of the optimal solution, and this result is shown in Fig. 14. According to the results of Fig. 14, \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is less than one for each element in solving this optimization problem by AGA and SAP2000. Also, the mean of \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is reached to 0.76 and 0.65 for solving this optimization problem by AGA and SAP2000, respectively. In addition, the standard deviation is 0.15 and 0.12 in solving this problem by AGA and SAP2000, respectively. Based on the results in this subsection, AGA is more strength than optimization modules of SAP2000 in solving this optimization problem.

Irregular nine-storey six-bay steel frame

An irregular nine-storey six-bay steel frame is considered as the fifth example in this study, as shown in Fig. 15. The frame is subjected to lateral and vertical loads, as shown in Table 1. The vertical load includes live load and dead load. The frame consists of 873 members. The cross sections of beams and columns are chosen within the standard sections provided by the AISC from W4×13 to W44×335. The parameters relevant to AGA in this optimization problem are selected as given below: the number of generations N_g = 1000; the population size N_p = 250; the crossover probability P_c = 60; the mutation probability P_mut = 5.

In this subsection, this steel frame is optimized by AGA and optimization modules of SAP2000 and, the results of them are compared. Table 6 shows the results of solving this optimization problem such as total weight, weight per area, and the number of selective cross section, which are obtained by AGA and optimization modules of SAP2000. Based on the results of Table 6, the efficiency of AGA is better than the efficiency of optimization modules of SAP2000 with respect to the weight per area and the total weight of the structure. On the other hand, the total weight of the optimized steel frame by AGA is 26.38 percent lower than the total weight of the optimized steel frame by the optimization modules of SAP2000.

Convergence histories of the irregular nine-storey six-bay steel frame are calculated for solving the optimization problem by AGA and SAP2000, as shown in Fig. 16. The results show that the weight of 162750 elements is calculated in this example, although structural analyses are done just for 6215 of them. AGA utilizes about 3.8 percent elements of all generations in this optimization problem; therefore, this issue causes reducing the time of analyses in the optimization problem.

Figure 17 shows \varphi R_{\mathrm{n}}/R_{\mathrm{u}} in solving the irregular nine-storey six-bay steel frame by AGA and SAP2000. The steel frame satisfies the constraints of ultimate load limits, which are defined based on the specification for structural steel building by AISC. For conducting this assessment, \varphi R_{\mathrm{n}}/R_{\mathrm{u}} should be checked in the optimization problem. \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is less than one for each element in solving this problem by AGA and SAP2000. The mean of \varphi R_{\mathrm{n}}/R_{\mathrm{u}} is reached to 0.76 and 0.43 for solving this optimization problem by AGA and SAP2000, respectively. In addition, the standard deviation is 0.13 and 0.30 in solving this problem by AGA and SAP2000, respectively. The results show that the AGA is more strength for solving this optimization problem.

Fifteen-storey three-bay planar steel frame

To compare results of AGA with results of some optimization algorithms, a 2-D steel frame is selected to optimize by AGA, which was optimized by other algorithms in the previous investigations. The constraints of this problem are divided into ultimate load limits, the maximum lateral displacement, and inter-storey displacement limits. On the other hand, other constraints of this optimization problem are defined as the maximum lateral displacement and the inter-storey displacements, which the formulation of them are shown in Eqs. (8) and (9), respectively.

(8){\vartheta }^{\Delta }={\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{T}}}{H}}-R\ge \mathrm{0},

(9){\vartheta }_j^{\mathrm{\Delta }}={\displaystyle \frac{d_j}{h_j}}-R_j\ge \mathrm{0},j=1,2,\mathrm{...},n_{\mathrm{s}},

where R_i, n_s, h_i, d_j, R, H, and {\mathrm{\Delta }}_{\mathrm{T}}are defined as the allowable inter-storey drift index, the total number of stories, the storey height of the jth floor, the inter-storey drift of the jth floor, the maximum drift index and the structure height respectively. {\mathrm{\Delta }}_{\mathrm{T}} is defined as the maximum lateral displacement of the roof, and d_j is utilized to give the comparative displacement of each floor in comparing with the below floor.

The allowable inter-storey drift index is considered as 1/300 based on AISC 2001 for controlling the optimum design of frame structure [¹⁰]. In addition, in this optimization problem, the constraints of ultimate load limits should be calculated and evaluated by the following equations:

(10){\vartheta }_j^I=\frac{P_{\mathrm{u}}}{2{\varphi }_{\mathrm{c}}{P}_{\mathrm{n}}}+(\frac{M_{\mathrm{u}\mathit{x}}}{{\varphi }_{\mathrm{b}}{M}_{\mathrm{n}\mathit{x}}}+\frac{M_{uy}}{{\varphi }_{\mathrm{b}}{M}_{\mathrm{n}y}})\le 1.0,

\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}{\displaystyle \frac{P_{\mathrm{r}}}{P_{\mathrm{c}}}}\ge \mathrm{0.2},

(11){\vartheta }_j^I=\frac{P_{\mathrm{u}}}{{\varphi }_{\mathrm{c}}{P}_{\mathrm{n}}}+\frac{8}{9}(\frac{M_{\mathrm{u}\mathit{x}}}{{\varphi }_{\mathrm{b}}{M}_{\mathrm{n}\mathit{x}}}+\frac{M_{\mathrm{u}y}}{{\varphi }_{\mathrm{b}}{M}_{\mathrm{n}y}})\le \mathrm{1.0.}

In these equations, P_u is defined as the required tensile or compressive strength; P_n is the nominal axial tensile or compressive strength; {\varphi }_{\mathrm{c}}is defined as the resistance factor which its values is selected 0.9 and 0.85 for tension and compression stress respectively; M_ux, M_uy and M_nx, M_ny are defined as the required and nominal flexural strengths in the x and y directions, respectively (for two-dimensional structures, M_ny = 0); and {\varphi }_{\mathrm{b}} is defined as the flexural resistance factor, and the value of it is 0.90.

Figure 18 demonstrates the geometry of 15-storey three-bay and also the loads which are applied to this frame structure. This frame includes 105 members and 64 joints. In this optimization problem, the constraints include the displacement and ultimate load limits that are provided by AISC. The optimal solution, which is solved by AGA, should satisfy the constraints. According to the constraints, the maximum displacement of each storey in each sway should be limited to 23.5 cm. The properties of the material such as modulus of elasticity and yield stress are applied 200 GPa and 248.2 MPa, respectively. Based on AISC, the effective factor (K_x) for each kind of component is calculated separately.

The out-of-plane effective length factor (K_y) for column member is computed one (K_y = 1). In addition, the out-of-plane effective length factor (K_y) for beam member is considered as 0.167 (K_y = 0.167). The unbraced length for all beam members are considered as one-fifth of the span length, and also, all columns are selected and specified as non-braced along its length.

Table 7 shows the results of optimal solutions of this steel frame in the present study and the previous studies. In the previous studies, this steel planner frame was optimized by some optimization algorithms such as PSO, PSOPC, HPSACO, ICA, and CSS. In this study, this steel frame is optimized by AGA, and the results of AGA such as the number of analyses and the total weight of optimum designs are compared with the results of these algorithms. According to the previous study, the total weight of the optimum designs of this steel frame which were obtained by HPSACO, PSOPC, and PSO, were computed 426.36, 452.34, and 496.68 kN, respectively, and also number of analyses which were done by HPSACO, PSOPC, and PSO, were 6800, 5000, and 5000, respectively [⁵¹]. The total weight of the optimum designs of this steel planner frame was obtained by ICA and CSS, were computed 417.466 and 412.62 kN, respectively, and also the number of analyses of these algorithms are obtained 6000 and 5000, respectively [⁵²,⁵³]. After implementing the 20 independent runs, the optimal result of AGA (the best result among the 20 results of AGA) is reported in Table 7. Based on the results of Table 7, the total weight of the optimized steel frame that is achieved by AGA is computed 408.31 kN. This result is achieved after 500 generations and 4287 analyses. According to the comparison of the results of Table 7, the AGA can solve this optimization problem faster than other algorithms which were published in the previous studies. In addition, the optimal weight of this steel planner frame which is obtained by AGA is less than obtained optimal weight by other algorithms which were published in the previous studies and are available in Table 7. Convergence histories of AGA for solving this optimization problem are shown in Fig. 19. The constraints of ultimate limits and displacement limits of this problem are satisfied. The maximum stress ratio of elements is less than the maximum stress allowable, and also, the global sway at the top storey is less than the maximum allowable sway. In addition, maximum lateral displacement of this steel frame, which is obtained by AGA is less than allowable displacement, and the inter-storey drift of each storey satisfies allowable inter-storey drift index.

Conclusions

This study introduces a new GA, which is named AGA for solving the optimization problem of steel frames. This algorithm is constructed by defining the new objective function, which can help to create the best generation of the population with lower structural analyses. There are six optimization problems in this paper which are optimized by AGA and other optimization methods. On the other hand, the first problem is optimized by AGA and GA, and then, the result of them are compared. The problems 2-5 are referred to the regular/irregular nine-storey five/six-bay steel frames. These steel frames are optimized by AGA and optimization modules of SAP2000, and then, the result of them are compared. In addition, the constraints of ultimate load limits based on AISC are considered for solving these optimization problems (example 1-5). To verify and compare the performance of AGA, the results of AGA are compared with the results of PSO, PSOPC, HPSACO, ICA, and CSS. The important findings of this method are presented in the following.

1) The proposed method can solve the optimization problem with lower structural analyses in comparing to some optimization algorithms; therefore, the cost of computation is decreased significantly.

2) According to the number of variables and the constraints in problems 2-5, AGA can solve the complicated optimization problem.

3) AGA performs better than GA and optimization modulus of SAP2000 with respect to the lower obtained total weight and analysis time.

4) The total weight of the regular nine-storey five-bay steel frame is obtained from AGA about 11.1% less than the obtained total weight from optimization modulus of SAP2000 and also, the total weight of the irregular nine-storey five-bay steel frame is obtained from AGA 26.4% less than the obtained total weight from optimization modulus of SAP2000 approximately.

5) The total weight of the irregular nine-storey five-bay steel frame in the plan is obtained from AGA about 20.4.1% less than the obtained total weight from optimization modulus of SAP2000. In addition, the total weight of the irregular nine-storey five-bay steel frame in height is obtained from AGA 23.5% less the obtained total weight from optimization modulus of SAP2000 approximately.

6) According to the results, the ability and efficiency of AGA in the optimization of the irregular steel frame is better than those of AGA in the optimization of the regular steel frame.

7) In addition, the ability and performance of AGA in solving these optimization problems are better than the performance and ability of optimization modules of SAP2000 and GA.

8) The AGA can solve the optimization problem with a large number of variables and the constraints; therefore, the AGA could be developed to optimize other complicated problems and multidisciplinary optimization problems.

9) In future work, the novel AGA can be used to investigate fracture uncertainty of steel structures with a phase field model [⁵⁴–⁵⁹].

10) According to the results of problem 6, the performance and ability of AGA are better than those of several optimization algorithms such as PSO, PSOPC, HPSACO, ICA, and CSS with respect to the total weight of steel frame and number of analyses.

Replication of results

AGA algorithm is defined clearly in section 3. This algorithm consists of a developed penalty function, which helps to find the best population of elements. This penalty function is defined and described clearly in subsections 3.2 and 3.3. Figure 1 shows a flowchart which describes the AGA approach. Other important details of this approach, such as crossover and mutation, creating a new population, termination, and all steps of AGA are explained in section 3. The details of optimizations problems in this study, such as applied loads, details of the steel frame, and probabilistic constraints are explained clearly in this study. In addition, all optimization problems are solved by Matlab coding and SAP2000 on a computer having configuration: CoreTM i5-7200U CPU@2.5 GHz 2.71, RAM 8.00 GB. If the readers and reviewers request to get the code of AGA, we can send it to them.

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