Optimal Design of Planar Steel Frames Using the Hybrid Teaching–Learning and Charged System Search Algorithm

Abstract

In this study, a novel effective algorithm called HTC which results from a hybridization process between the teaching–learning-based optimization (TLBO) and charged system search (CSS) algorithms is proposed and utilized to optimize planar steel frames. Among the features of the proposed algorithm, simplicity and low number of setting parameters can be mentioned. The main goals of this hybridization process are to establish a balance between exploration and exploitation, increase the rate of convergence, and avoid getting trapped in the local optimal point. Moreover, the purpose of optimizing steel frames is to calculate their minimum weight by choosing suitable sections as well as considering the stress and displacement constraints in accordance with the requirements of the AISC design regulations using the load and resistance factor design (LRFD) method. To evaluate the performance of the HTC algorithm, standard benchmark steel frames are examined, and the results of the proposed algorithm are compared with other metaheuristic optimization methods. According to the outcomes of the comparison, the proposed algorithm has better performance than the other methods.

Introduction

Optimization methods are divided into two categories. The first category is gradient-based methods that use results in accurate optimization, but in the cases of complex problems, they have low speed and cannot calculate the derivative of the required functions. Given the new needs of humans and the complexity of optimization problems, researchers need powerful methods to solve them. That is why many metaheuristic methods have been proposed in the last decade that are inspired by nature and the laws of physics and chemistry. These methods do not require basic calculations such as calculating the gradient of functions, so they are faster at solving optimization problems than conventional algorithms. In metaheuristic approaches, there are two essential components, which are exploration and exploitation. These components are directly related to the searchability of an algorithm. The exploration component leads to further exploration of the search space to find better solutions and improve their diversity. The exploitation component also improves the solutions’ quality by searching more in local areas around the solutions.

Different optimization problems are divided into two categories. The first category consists of unconstrained optimization problems. In these problems, the goal is to maximize or minimize the objective function without any constraint on the design variables. Various methodologies have recently been developed to solve this class of problems. For example, Goodarzimehr et al. (2022a, e, b, c, d) developed a novel metaheuristic method based on the theory of relativity physics called special relativity search (SRS); Talatahari et al. (2021a, b) introduced the crystal structure algorithm (CryStAl) for solving unconstrained problems; and Dastan et al. (2022) introduced a new hybrid method for solving unconstrained numerical problems. The second class is constrained optimization problems, which are more challenging for researchers. In most real-life problems, including the analysis of structures in civil engineering, optimization is done with some constraints. The constraints on the behavior and performance of a system are called behavioral constraints, and the constraints on the physics and geometry of the problem are called geometric or lateral constraints. Recently, Goodarzimehr et al. (2023) implemented SRS for solving some of the most challenging problems in structural and mechanical sciences. Topal et al. (2022) proposed a hybrid method for maximization of the fundamental frequency of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) quadrilateral plates. Goodarzimehr et al. (2022a, e, b, c, d) used a new strategy to develop chaos game optimization (CGO) for solving structural problems with dynamic constraint, which are highly nonlinear. Momeni et al. (2022) developed a novel soft computing-based method for predicting flexural strength of concrete beams made of recycled aggregates. Goodarzimehr et al. (2022a, e, b, c, d) optimized the geometrically nonlinear space structures using generalized displacement control analysis, and Goodarzimehr et al. (2022a, e, b, c, d) developed a hybrid method to optimize space structures.

In the last three decades, several methods have been proposed by researchers, including the work of Goldberg and Holland (1988), who proposed the genetic algorithm (GA). This algorithm uses Darwin's theory and generates a new generation using the mutation mechanism. Simon (2008) proposed the biogeography-based optimization (BBO) algorithm based on the distribution of vital species in different regions. Storn and Price (1997) generated a new population by using the mutation process and adding the weight difference between two population vectors to the third vector. Inspired by the corona virus, which is an unknown animal virus, Salehan and Deldari (2021) proposed a new algorithm called the corona virus optimization (CVO) algorithm, which was used to solve a number of discrete and continuous benchmark functions and problems. Moreover, Li and Tom (2020) introduced the virus-spreading optimization (VSO) algorithm, which is inspired by the spread of viruses between individuals. Inspired by the collective behavior of fish and birds, Kennedy and Eberhart (1995) proposed the particle swarm optimization (PSO) algorithm. Dorigo et al. (1999) proposed the ant colony optimization (ACO) algorithm by observation and inspiration from the collective behavior of ants to find the closest path to food resulting from a chemical substance called pheromone. Karaboga and Basturk (2007) presented the artificial bee colony (ABC) algorithm using the relationship between worker, guard, and queen bees to find food sources. Chu et al. (2006) proposed the cat swarm optimization (CSO) algorithm using the behavior of cats in searching, tracking, and finding baits. Moreover, Mirjalili et al. (2018) proposed a metaheuristic algorithm called the grasshopper optimization algorithm (GOA), which is inspired by the influence and social behavior of grasshoppers on their environment. Inspired by the unique mating behavior of certain species of spiders, Hayyolalam and Kazem (2020) proposed a new algorithm called the black widow optimization (BWO) algorithm. The behavior of these spiders results in the elimination of inappropriate solutions that causes the convergence of the new algorithm.

Also, examples of the methods that follow the laws of physics and chemistry are as follows: Kaveh and Talatahari (2010) presented the charged system search (CSS) algorithm based on the laws of Newtonian mechanics and Coulomb’s laws. Hatamlou (2013) introduced the black hole optimization algorithm which is inspired by the black hole phenomenon in physics. In this method, the best particle is selected as the black hole, and the stars that get too close to the black hole will be swallowed by it. Rashedi et al. (2009) proposed the gravitational search algorithm (GSA) based on Newton's laws of gravitation. Kirkpatrick et al. (1983) proposed the simulated annealing (SA) algorithm based on the laws of physics. Erol and Eksin (2006) presented the big-bang big crunch algorithm, which is inspired by the big bang and big crunch theories. The chemical reaction optimization (CRO) algorithm was introduced by Lam and Li (2010), which is inspired from a natural process of transforming unstable substances to stable ones.

Nowadays, researchers have developed new algorithms for solving problems using two or more algorithms in a hybrid process. The purpose of doing so is to identify the strengths and weakness points in the original algorithms and to balance the ability to explore and exploit. Some examples of these algorithms are given in the following: Maheri et al. (2016) developed the hybrid genetic and particle swarm optimization (GA-PSO) algorithm to optimize the size and geometry of structures. Shojaee et al. (2013), introduced the method of moving asymptotes/improved discrete particle swarm optimization (MMA-IDPSO) algorithm, which optimizes the topology and size of truss structures. Talatahari et al. (2020) proposed the hybrid teaching–learning-based optimization and harmonic search (TLBO-HS) algorithm for optimizing large-scale structures. Moreover, Talatahari et al. (2021a, b) used the hybrid symbiotic organisms search (SOS) and harmony search algorithm to optimize the size of structures. Omidinasab and Goodarzimehr (2020) proposed the hybrid particle swarm optimization and genetic algorithm (PSO-GA) for the optimal design of truss structures with discrete variables.

Optimization is used in most engineering fields such as mechanics, economics, electronics, civil engineering, geophysics, and molecular modeling. It has been widely used in civil engineering. Construction management and transportation engineering are among the main branches of civil engineering that rely heavily on optimization. Steel structures, frames, and trusses are among the most common and widely used structures in civil engineering and whose economic and safe design is of great importance. For the first time, Camp et al. (2005) used the ant colony optimization algorithm to optimize steel frames in a discrete manner. The objective function in this study was the total weight. Also, Kaveh and Shojaee (2007) utilized the ACO algorithm to optimize several applicable problems. In another study, Degertekin (2008) used the harmony search (HS) algorithm for the optimal design of steel frames. This method is based on the analogy between the process of achieving natural harmony and finding solutions for optimization problems. Ghatte (2021) proposed a hybrid firefly algorithm and biogeography-based optimization (FA-BBO) for the optimal design of steel frames based on flashing patterns and biogeography-based optimization. Babaei and Sanaei (2016) used hybrid genetic and ant colony optimization (HGAC) for optimal design of braced frames. Kaveh et al. (2020) utilized seven metaheuristic population-based algorithms for the optimal design of two-dimensional (planar) frame structures. Tan and Lahmer (2019) used robust design optimization (RDO) for shape optimization design of arch dams. Dehghani et al. (2023) proposed a population-based method called modified adolescent identity search algorithm (MAISA) to optimize the weight of skeletal structures. Goodarzimehr et al. (2022a, e, b, c, d) utilized an improved chaos game optimization algorithm for optimal frequency prediction of the variable stiffness a curvilinear composite plate.

In this study, by considering the strengths and weaknesses of the TLBO and CSS algorithms, we propose a new and unique algorithm that results from the hybridization of these algorithms. The balance between exploration and exploitation, the increase in the convergence rate, and not being trapped in the local optimal point are among the main aims of this combination process. The performance of the proposed algorithm is evaluated through three benchmark optimization steel frame problems with certain strength and displacement constraints. Then the results of the proposed algorithm’s performance are compared with some other methods presented previously by other researchers.

The framework of this paper is as follows: In Sect. 2, the optimal design of frame structures based on the LRFD method is discussed. Section 3 describes the formulation of the basic algorithms (TLBO and CSS). Section 4 describes the formulation of the HTC algorithm and examines the basic principles and ideas of the proposed method. The efficiency of the proposed method is confirmed by the numerical examples in Sect. 5, and finally, the conclusions are summarized in Sect. 6.

Problem Formulation

Nowadays, one of the main goals in optimizing structures is weight optimization and minimization. One of the optimization methods is through the usage of metaheuristic algorithms, in which the initial population is first randomly generated, and then, in a repetitive process by establishing a balance between exploration and extraction, the population is promoted, and at the end of each iteration, the obtained solutions are compared with the limitations and constraints of the problem, which are mainly the displacements, stresses, and position of nodes in civil engineering problems. In Fig. 1a, the search space is displayed in two- and three-dimensional space. Figures 2, 3, 4, 5, 6, and 7 depict the search process of the particles in this space. In Fig. 1b, the purple and green dots represent the initial generated population, with the position of the best particle shown in green. With each iteration, the particles search and explore in this space, and in the end of each iteration, the particles move closer to the optimal solution, and finally, according to Fig. 1g, all particles converge at the optimal point.

Fig. 1.

a The search space in two- and three-dimensional space. b–g The search process of the particles

Fig. 2.

The flowchart of the teaching–learning-based optimization (TLBO) algorithm (Rao et al. 2011)

Fig. 3.

The flowchart of the charged search system (CSS) algorithm (Kaveh and Talatahari 2012)

Fig. 4.

The flowchart of the hybrid TLBO_CSS (HTC) algorithm

Fig. 5.

Geometry, grouping, and loading of the one-bay eight-story steel frame

Fig. 6.

Optimal weight of the one-bay eight-story frame in 30 independent designs

Fig. 7.

Convergence history diagram of the one-bay eight-story frame

The optimization problem can be set as follows:

$$\begin{array}{c}Minimizef(\text{X})=W({\mathit{A}}_i)=\sum _{i=1}^{\mathit{Ne}}{\rho }_i{A}_i{L}_i\\ Subjectto{g}_{Dj}=\frac{{\mathrm{\Delta }}_j(A_i)}{{\mathrm{\Delta }}_u}-1\le 0j=1,2,...,N_{\begin{array}{c}n\\ \\ \end{array}}\\ g_{Si}=\left\{\begin{array}{c}(\frac{P_u}{\varphi P_n})+\frac{8}{9}\left(\frac{M_{\mathit{ux}}}{{\varphi }_b{M}_{\mathit{nx}}},+,\frac{M_{\mathit{ux}}}{{\varphi }_b{M}_{\mathit{nx}}}\right)-1\text{for}\frac{P_u}{\varphi P_n}\ge 0.2\\ \left(\frac{P_u}{2\varphi P_n}\right)+\left(\frac{M_{\mathit{ux}}}{{\varphi }_b{M}_{\mathit{nx}}},+,\frac{M_{\mathit{ux}}}{{\varphi }_b{M}_{\mathit{nx}}}\right)-1\text{for}\frac{P_u}{\varphi P_n}<0.2\end{array}\right)i=1,2,...,N_e\\ {\mathit{A}}_i\in {\mathit{A}}_e=\{A_{e1},A_{e2},...,A_{ep}\}\\ \\ \end{array}$$ 1

in which W denotes the weight of the structures, and ${\rho }_i,A_i$, and $L_i$ represent the cross section, material density, and length of the ith member, respectively. $g_{Dj}$ and $g_{Si}$ are constraint conditions for steel frames. $g_{Dj}$ is the displacement violation, and $g_{Si}$ is stress violation based on LRFD requirements in the American Institute of Steel Construction (AISC) specification (Equation H1-1) (American Institute of Steel Construction 2010). ${\mathrm{\Delta }}_j$ and ${\mathrm{\Delta }}_u$ are the displacement of the jth floor and the allowable displacement (whose value is equal to 1/300 of the floor height), respectively. P_u is the required axial force, M_u represents the required bending moment, P_n denotes the nominal axial capacity, and M_n is the nominal bending capacity.${\varphi }_c$ and ${\varphi }_b$ are resistance factors (${\varphi }_c$ = 0.9 for tension and 0.85 for compression, and ${\varphi }_b$ = 0.9 for flexure). N_e denotes the number of members, and N_n indicates the number of stories in the structure. Moreover, A_e is the available profile list.

The penalty method replaces a constrained optimization problem with a set of unconstrained problems. These problems are created by adding a condition to the objective function, which consists of a penalty parameter and a degree of violation of the constraints and limitations. As an example of the application of the penalty parameters, they were used in the Rayleigh–Ritz method in 1999 in modeling rigid boundaries. In this research, the penalized weight is calculated based on the penalty function as follows:

$$\gamma (x)=W(x){[1+\mu ]}^{\epsilon }$$ 2

where ε denotes the power of the penalty function, and μ is the function of violation of constraints, which is

$$\mu =\sum _j^{N_n}{g}_{Dj}+\sum _i^{N_e}{g}_{Si}$$ 3

in which $g_{Dj}$ and $g_{Si}$ are the violation of the displacement constraints and the story drifts and the violation of the stress constraints, respectively, which are in accordance with the requirements of the LRFD approach. The amount of the penalty of $g_{Si}$ is equal to

$$g_{Si}=\left\{\begin{array}{c}0if{g}_{Si}\le 0\\ g_{Si}if{g}_{Si}>0\end{array}\right)$$ 4

Equation (5) is also true for $g_{Dj}$.

Review of TLBO and CSS

The proposed algorithm in the present study is the result of a hybridization process between the TLBO and CSS algorithms, which are metaheuristic. Each of these algorithms has advantages and disadvantages that can be mitigated by combining them. In the following, a summary of each of the algorithms that forms the proposed HTC algorithm is provided.

Teaching–Learning-Based Optimization (TLBO) Algorithm

In 2011, Rao et al. proposed the teaching–learning-based optimization (TLBO) algorithm (Rao et al. 2011). This algorithm is inspired by the effect of the teacher on the performance of students in a classroom, and the output is considered in terms of their grades. The teacher is generally considered to be a highly educated person who shares his/her knowledge with the students. The quality of a teacher affects the outcomes of students (i.e., their grades). Obviously, a good teacher teaches the students in a way that they can get better results. The TLBO process is divided into two parts: the first part includes the "teacher stage" and the second part includes the "learner stage." The teacher stage means learning from the teacher, and the learner stage means learning through the interaction between other learners. In general, in this method, the teacher tries to spread the knowledge among the students, which in turn increases the level of knowledge of the whole class and helps the students to get good grades. Therefore, a teacher increases the average of the class by their teaching ability. In optimization algorithms, the population is design variables. Like other nature-inspired algorithms, TLBO is also a population-based approach that uses a population of solutions to obtain the global solution. According to TLBO, the population is considered as a group or level of students. Figure 2 shows the TLBO diagram.

Charged System Search (CSS) Algorithm

The CSS algorithm was introduced by Kaveh and Talatahari in (2010) and is one of the newest population-based metaheuristic methods used nowadays to solve various optimization problems, including structural problems. The CSS algorithm uses Coulomb’s laws of electrostatics and Newtonian laws of mechanics. In this algorithm, each factor is a charged particle with a predetermined radius. The amount of charge of the particles is considered based on their quality. The quantity of force produced is determined using the laws of electrostatics, and the quality of motion is determined using the laws of Newtonian mechanics. The charged particle system method is a multifactorial method, each of which is called a charged particle (CP), which has an electric charge equal to q_i. Therefore, these particles will create an electric field proportional to q_i. The value of q_i must be defined such that particles with good solutions have a larger electric field and, therefore, more force. The important point is that the initial position of the particle is chosen randomly, and the initial velocity of all charged particles is considered zero. The weaknesses of this algorithm include algorithm complexity, large number of regulatory parameters, low convergence rate, and poor performance in solving structural problems. Figure 3 shows the diagram of the CSS algorithm.

Hybrid TLBO_CSS (HTC) Algorithm

Nowadays, due to the progress and complexity of optimization problems, researchers are trying to use more powerful algorithms to solve them. Therefore, the utilization of hybrid algorithms has attracted a lot of attention. The purpose of hybridization of two or more algorithms is to improve the performance of each algorithm and reduce their disadvantages in a hybrid process. This research is intended to present a new and powerful algorithm for solving optimization problems by hybridization of two novel algorithms, the teaching–learning-based optimization (TLBO) and charged system search (CSS) algorithms, which have been proposed in recent years. Both of these algorithms are population-based. One of the disadvantages of the TLBO algorithm is that it gets stuck in the local optimal point due to the lack of a system for measuring the distance between the learner and the optimal point. Disadvantages of the CSS algorithm include high dependence on the number of iterations, large number of initial parameters, large number and complexity of functions, and physical and mathematical relationships. However, one of the most important features of the TLBO algorithm is that it does not depend on parameters, because this algorithm has the least possible number of parameters, and it can be considered as a special advantage for this algorithm. The purpose of hybridization these two algorithms is to propose a new algorithm that has a high convergence rate and has a good control over exploration and exploitation compared to the CSS and TLBO algorithms. The main body of the proposed algorithm (i.e., HTC algorithm) is made by the TLBO algorithm, and to eliminate the weaknesses of this algorithm, the Newton and Coulomb operators of the CSS algorithm are used. This strengthens the operating parameter and makes the algorithm search better among the local points. Figure 4 shows the diagram of the Hybrid TLBO_CSS (HTC) algorithm. The optimization steps of the HTC algorithm are as follows:

Step 1: Generation of the initial population randomly and similar to other population-based methods and selection of the best particle or solution.

Step 2: Calculation of the average of each design variable.

Teacher phase:

Step 3: Selection of the best solution as the teacher and upgrading the particles in the teacher phase according to Eqs. (5) and (6).

$$X_{\text{new}}^k=X_{\text{old}}^k+r(X^{\text{teacher}}-T_F\ast M(j))$$ 5

$$M(j)=\frac{{\sum }_{K=1}^N\frac{X^K(j)}{F^k}}{{\sum }_{K=1}^N\frac{1}{F^k}}$$ 6

where X^k (j) represents the jth design variable, T_F is used as the teaching factor, r is a random number in the range [0,1], and M (j) indicates the mean of the class. Moreover, F^k denotes the penalty fitness function.

Step 4: Controlling the boundary conditions according to the upper and lower limits specified in the initial parameters of the algorithm.

Step 5: Evaluation of the solution and comparison between the new solution and the old ones for selecting the best solution.

Learner phase:

Step 6: Choosing two solutions (students) randomly and generation of new solutions according to Eqs. (7) and (8).

In this step, the people of the population (who are classmates) increase their knowledge by working together. This is similar to what really happens among friends and classmates. This step is formulated as follows:

Students p and q are randomly chosen from the class such that they are not equal, so we have

$$If\underset{}{}{X}_p<X_q$$

$$X_{\text{new}}^p=X_{\text{old}}^p+r(X_{\text{old}}^p(j)-X^q(j))$$ 7

otherwise,

$$X_{\text{new}}^p=X_{\text{old}}^p+r(X^q(j)-X_{\text{old}}^p(j))$$ 8

in which r is a random number in the range [0,1], and X^p (j) represents the jth design for the pth design vector.

Step 7: Quality assessment of the solution and comparison between the new solution and the old ones for selecting the best solution.

CSS operator:

Step 8: Calculation of f, q, and r using Coulomb's law and according to Eqs. (9–11).

$$q_i=\frac{fit(i)-fitworst}{fitbest-fitworst},{}_{}\underset{}{}i=1,2,...,N$$ 9

where “fitbest” and “fitworst” denote the best and worst fitness between all of the particles, respectively. fit(i) represents the value of the objective function, and N is the total number of CPs. The separation distance, r_ij, between two charged particles is also defined as

$$r_{\mathit{ij}}=\frac{∥{\mathit{X}}_i-{\mathit{X}}_j∥}{∥\left({\mathit{X}}_i-{\mathit{X}}_j\right)/2-{\mathit{X}}_{\mathit{best}}∥+\epsilon }$$ 10

in which X_i and X_j are the positions of CPs, X_best represents the position of the best recent CP, and ε denotes a positive number.

The value of the resultant electric force on each CP is calculated using Eq. (11). Also, since each particle is considered as a sphere, the parameter a is the radius of this sphere, which is calculated using Eq. (12).

$$F_j=q_j\sum _{i,i\ne j}\left(\frac{q_i}{a^3}{r}_{\mathit{ij}}·i_1+\frac{q_i}{r_{\mathit{ij}}^2}·i_2\right)p_{\mathit{ij}}(X_i-X_j),{}_{}{}_{}\left\{\begin{array}{c}j=1,2,...,N,\\ i_1=1,{}_{}{i}_2=0\iff r_{\mathit{ij}}<a,\\ i_1=0,{}_{}{i}_2=1\iff r_{\mathit{ij}}\ge a,\end{array}\right)$$ 11

$$a=0.1\times max\left(\left\{x_i,max-x_i,min\left|\right)i=1,2,...,n\left(\right\}\right)\right)$$ 12

Step 9: Calculation of the new particle $(X_{j,new})$ and the new velocity $(V_{j,new})$ based on Newton's law.

The new position and velocity of each CP are calculated using Eqs. (13) and (14).

$$V_{j,\text{new}}=\frac{X_{j,\text{new}}-X_{j,\text{old}}}{\mathrm{\Delta }t}$$ 13

$$X_{j,\text{new}}={\text{rand}}_{j1}·k_a·\frac{F_j}{m_j}·\mathrm{\Delta }{t}^2+{\text{rand}}_{j2}·k_v·V_{j,\text{old}}·\mathrm{\Delta }t+X_{j,\text{old}}$$ 14

where k_a is the acceleration coefficient, and k_v denotes the velocity coefficient for controlling the previous velocity, which are calculated using Eqs. (15) and (16). rand_j1 and rand_j2 are two random numbers that are evenly distributed in the range (0,1). m_j represents the mass of the jth CP, which is equal to q_j. Also, $\mathrm{\Delta }t$ indicates the time step.

$$k_v=0.5\left(1-\text{iter}/{\text{iter}}_{max}\right)$$ 15

$$k_a=0.5\left(1+\text{iter}/{\text{iter}}_{max}\right)$$ 16

in which iter represents the actual number of iterations, and iter_max is the maximum number of iterations.

Step 10: Quality evaluation of the solution and comparison between the new solution and the old ones for selecting the best solution.

Step 11: Checking to see whether the termination rule can be applied.

Step 12: Reporting the best solution.

Numerical Problems

The problems addressed in this study include the optimal design of skeletal structures, such as frames. The purpose of using and solving these problems is to examine the performance and show the capabilities of the HTC algorithm in solving practical problems. Therefore, the solution of different steel frame cases, including one-bay and eight-story, three-bay and 15-story, and three-bay and 24-story frames, is considered, which are fully reviewed, and their analysis results are presented. Also, to evaluate the stability of the algorithm, each of these structures are designed 30 times independently, and the standard deviation (Std) results provided in the tables show the degree of stability.

One-Bay Eight-Story Frame

The one-bay eight-story steel frame considered in the first example is subjected to the lateral load shown in Fig. 5. The modulus of elasticity of steel is 200 Gpa, and its specific weight is 76.8 $\text{kN}/{\text{m}}^3$. The members of this two-dimensional frame are divided into eight groups, which are shown in Fig. 5. The sections of the members of this structure are also chosen from 267 W-shaped rolled sections existing in AISC. The only constraint is the movement of the roof floor, which is limited to 5.08 cm.

Comparison of the performance results of the HTC hybrid algorithm with some metaheuristic ones such as GA, ACO, EBA, and PSOPC listed in Table 1. These results show that the proposed algorithm has some extent performance compared to the other algorithms, and it has reduced the frame weight to 31.098 kip. It reduced the weight by 1.872% compared to the ACO algorithm, which is in the second place, and by 10% compared to the PSOPC algorithm, which has the weakest performance among these algorithms. This frame has been designed 30 times independently, and the number of iterations and particles of this example are 300 and 100, respectively. Optimal weights are shown in Fig. 6. The value of the standard deviation (Std) obtained from 30 independent designs is 0.2589, which indicates the stability of the proposed algorithm. The convergence history diagram shown in Fig. 7 indicates the high convergence rate of the algorithm, especially in the initial 50 iterations.

Table 1.

Performance comparison for the one-bay eight-story frame

Element group	GA, Camp (1998)	ACO, Kaveh and Shojaee S (2007)	PSOPC, Kaveh A, Talatahari S (2008)	EBA, Kaveh A, Zakian P (2014)	HTC
1	W18 × 46	W21 × 50	W18 × 35	W18 × 35	W18 × 40
2	W16 × 31	W16 × 26	W14 × 26	W16 × 31	W18 × 35
3	W16 × 26	W16 × 26	W16 × 26	W16 × 31	W14 × 22
4	W12 × 16	W12 × 14	W14 × 26	W12 × 14	W12 × 14
5	W18 × 35	W16 × 26	W24 × 62	W18 × 35	W18 × 35
6	W18 × 35	W18 × 40	W18 × 35	W18 × 35	W18 × 35
7	W18 × 35	W18 × 35	W16 × 31	W18 × 35	W18 × 35
8	W14 × 26	W14 × 22	W12 × 30	W16 × 31	W14 × 22
Best weight (kip)	32.83	31.68	34.21	31.86	31.098
Average weight (kip)	N/A	N/A	N/A	N/A	31.4352
Std	N/A	N/A	N/A	N/A	0.2589

Three-Bay 15-Story Frame

This steel frame has three bays and 15 stories and consists of 64 joints and 105 members. The columns are grouped into ten distinct groups, while all of the beams form only one group. The column groups are formed in a way that the outer columns are grouped into separate column groups each three consecutive stories (starting from the foundation), and the inner columns form another separate column group. Therefore, the problem has 11 discrete design variables. The member groups are shown in Fig. 8. All design variables (member groups) are selected from 267 W sections. The modulus of elasticity of steel is E = 200 Gpa, and the yield stress of steel is F_y = 248.2 MPa. The unbraced length of each beam is considered to be one fifth the length of the span of the frame. It is assumed that the columns are not braced along their length. In this example, the resistance and displacement constraints are considered based on the requirements of AISC-LRFD, and the maximum displacement of the last floor is limited to 23.5 cm.

Fig. 8.

Geometry, grouping, and loading of the three-bay 15-story steel frame

The HTC algorithm performs better than the other methods listed in Table 2. This algorithm has reduced the weight of the structure about 0.91% compared to the PF-SSOA method, which is in the second place in terms of the best answer. It also obtained about 0.51% lower average value than the PF-SSOA algorithm in 30 independent designs. The obtained standard deviation indicates that the answers obtained from this algorithm are more stable. The proposed algorithm, with the standard deviation (Std) of 4.6739, is in the first place in this regard, and the PF-SSOA, TLBO, and ES-DE algorithms with the Std values of 4.81, 11.346, and 14.65 are in the next positions, respectively. The value of this parameter has been reduced by 2.91%, 58.8%, and 68.12%, respectively, compared to the best and worst algorithms in the table, which is given in the form of bar graphs according to Fig. 9 for better demonstration. the number of iterations and particles of this example are 250 and 100, respectively.

Table 2.

Performance comparison of the three-bay 15-story frame

Element group	ES-DE, Talatahari et al. (2015)	EWOA, Kaveh (2016)	PF-SSOA, Kaveh et al. (2021)	TLBO, Kaveh et al. (2020)	HTC
1	W18 × 106	W14 × 99	W14 × 99	W12 × 96	W24 × 104
2	W36 × 150	W27 × 161	W27 × 161	W27 × 161	W36 × 150
3	W12 × 79	W27 × 84	W27 × 84	W27 × 84	W16 × 77
4	W27 × 114	W24 × 104	W24 × 104	W24 × 104	W24 × 104
5	W30 × 90	W21 × 68	W16 × 67	W10 × 68	W14 × 74
6	W10 × 88	W18 × 86	W18 × 86	W30 × 90	W27 × 94
7	W18 × 71	W21 × 48	W8 × 48	W8 × 48	W18 × 60
8	W18 × 65	W14 × 68	W12 × 65	W24 × 68	W18 × 60
9	W8 × 28	W8 × 31	W8 × 31	W8 × 28	W8 × 24
10	W12 × 40	W10 × 45	W10 × 39	W10 × 39	W16 × 36
11	W21 × 48	W21 × 44	W21 × 44	W21 × 50	W21 × 44
Best weight (KN)	415.06	392	389.24	390.42	385.709
Average weight (KN)	438.26	403.99	395.3	423.67	393.2624
Std	14.65	N/A	4.81	11.346	4.6739

Fig. 9.

Bar chart standard deviation (std) diagram of the algorithms

Figure 10 shows the optimal solution in each of the designs. Drift is defined as the ratio of the displacement difference between the above and bottom story to the height of the story. According to codes and regulations, the drift value is limited because excessive displacement of the structure damages the nonstructural components and disturbs its residents. The obtained stress ratios of the members in the case of most optimal design are shown in Fig. 11, According to this figure, the highest and lowest value of the stress ratios occurred in members 25 and 101 with the values of 0.994 and 0.246, respectively. Figure 12 shows the optimal design drift values obtained for each story, most of which is related to the fourth floor. The convergence history diagram shown in Fig. 13 indicates the high convergence rate of the algorithm, especially in 200 iterations.

Fig. 10.

The optimal weight of the three-bay 15-story frame per independent design

Fig. 11.

The stress ratio of each member in the optimal response for the three-bay 15-story frame

Fig. 12.

Story drift diagram resulting from the performance of the HTC algorithm in solving the three-bay 15-story frame

Fig. 13.

The convergence history diagram of the three-bay 15-story frame

Three-Bay 24-Story Frame

In the third example, the three-bay 24-story steel frame shown in Fig. 14 is examined. This frame consists of 168 members, including 96 columns and 72 beams. This structure is one of the important structures considered in the field of structural optimization with discrete variables. All members in this structure are divided into 20 groups, four of which are related to beams, whose sections are selected from 267 W-shaped rolled sections, and 16 groups consist of columns that are limited to W-14 sections (37 W-shaped rolled sections). The modulus of elasticity of the materials (E) is 29,732 ksi, and the yield stress (F_y) is 33.4 ksi. In this frame, the effective length coefficients of the members for in-plane lateral buckling are considered to be k_x ≥ 0, and for out-of-plane buckling, it is considered to be one (k_y = 1). All beams and columns are not braced along their length. In this example, the considered strength and displacement constraints are based on the AISC-LRFD requirements (Fig. 15).

Fig. 14.

Geometry, grouping, and loading of the three-bay 24-story steel frame

Fig. 15.

Standard deviation (Std) bar diagram of the algorithms

The HTC algorithm performed better than the other methods listed in Table 3. This algorithm has reduced the weight of the structure about 0.71% compared to the FA-BBO method, which is in the second place in terms of the best answer. The obtained standard deviation indicates that the answers obtained from this algorithm are more stable. The HTC algorithm with a standard deviation of 4.3254 is in the first place in this regard, and the ACO algorithms with a value of 4.56 is in the next positions. Figure 16 shows the optimal solution of each design. The obtained members’ stress ratios in case of the most optimal design are shown in Fig. 17. The highest and lowest stress ratios are 0.927 and 0.0996, which are related to members 130 and 162, respectively. Figure 18 shows the optimal design drift values obtained by considering the allowable drift for each story, and its maximum value is related to the fourth floor. The number of iterations and particles of this example are 250 and 100, respectively. The convergence history diagram shown in Fig. 19 indicates the high convergence rate of the algorithm, especially in 110 iterations.

Table 3.

Performance comparison of the three-bay 24-story frame

Element group	ACO, Camp et al. (2005)	HS, Degertekin (2008)	ES-DE, Talatahari et al. (2015)	FA-BBO, Ghatte (2021)	HTC
1	W30 × 90	W14 × 176	W30 × 90	W30 × 90	W30 × 90
2	W8 × 18	W14 × 145	W21 × 55	W12 × 14	W4 × 13
3	W24 × 55	W14 × 176	W24 × 48	W21 × 48	W21 × 48
4	W8 × 21	W14 × 132	W10 × 45	W6 × 9	W4 × 13
5	W14 × 145	W14 × 132	W14 × 145	W14 × 145	W14 × 159
6	W14 × 132	W14 × 109	W14 × 109	W14 × 120	W14 × 120
7	W14 × 132	W14 × 109	W14 × 99	W14 × 120	W14 × 132
8	W14 × 132	W14 × 82	W14 × 145	W14 × 74	W14 × 82
9	W14 × 68	W14 × 82	W14 × 109	W14 × 68	W14 × 74
10	W14 × 53	W14 × 61	W14 × 48	W14 × 53	W14 × 48
11	W14 × 43	W14 × 74	W14 × 38	W14 × 38	W14 × 30
12	W14 × 43	W14 × 48	W14 × 30	W14 × 22	W14 × 22
13	W14 × 145	W14 × 34	W14 × 99	W14 × 109	W14 × 99
14	W14 × 145	W14 × 30	W14 × 132	W14 × 109	W14 × 109
15	W14 × 120	W14 × 22	W14 × 109	W14 × 109	W14 × 90
16	W14 × 90	W14 × 22	W14 × 68	W14 × 90	W14 × 90
17	W14 × 90	W30 × 90	W14 × 68	W14 × 74	W14 × 68
18	W14 × 61	W10 × 22	W14 × 68	W14 × 68	W14 × 53
19	W14 × 30	W18 × 40	W14 × 61	W14 × 30	W14 × 34
20	W14 × 26	W12 × 16	W14 × 22	W14 × 22	W14 × 22
Best weight (kip)	220.47	214.86	212.39	202.90	201.544
Average weight (kip)	229.56	222.62	N/A	N/A	204.9118
Std	4.56	N/A	N/A	N/A	4.3254

Fig. 16.

Optimal weight of the three-bay 24-story frame per independent run

Fig. 17.

Stress ratio of each member in the optimal solution for the three-bay 24-story frame

Fig. 18.

Story drift diagram resulting from the performance of the HTC algorithm in solving the three-bay 24-story frame

Fig. 19.

Convergence history diagram of the three-bay 24-story frame

Conclusions

In this research, the combination of the TLBO and CSS metaheuristic algorithms was utilized, developing a new algorithm called HTC. The main purpose of this hybrid process was to reduce the weaknesses of the two primary algorithms, increase the convergence rate, and increase the balance between exploration and exploitation. In the proposed algorithm, the TLBO algorithm was used as the basic algorithm, and the Coulomb and Newton functions of the CSS algorithm were used to improve its performance. To evaluate the effectiveness and performance of the HTC algorithm in solving structural problems, three steel frame problems were solved and investigated, including one-bay eight-story, three-bay 15-story, and three-bay 24-story frames. The optimal design results obtained from this algorithm showed that this method can be used as a powerful algorithm in the optimal design of steel frames. According to the drift and allowable stress diagrams of the members, the ability of the proposed algorithm in optimization can be realized, because the story drift values are ​​close to the maximum allowable value, and the stress ratio of most members has values ​​close to one. By comparing the results of the proposed algorithm with some of the previous methods presented by other researchers, the results showed that the HTC algorithm has a high capability to optimize the weight of steel frames. The standard deviation (Std) obtained from 10 independent design times showed that the proposed algorithm has a high stability with a lower Std value compared to other compared algorithms. This new metaheuristic algorithm can be easily used in other complex mathematical and optimization problems.