1 Introduction

Design of the structures with minimum weight or minimize an objective function value based on the minimal cost of the structures, regarding the design criteria, is the main purpose for structural optimization [1,2,3,4,5]. Mathematical programming methods have widely been applied in solid and structural optimization [6,7,8,9]. During the past few decades, several optimization algorithms have been improved for different structural systems like truss and steel frame (SF) structures [10,11,12,13,14,15,16,17,18]. As well as, the performance-based design of SFs utilizing meta-heuristic optimization algorithms has been developed by the researchers [19,20,21]. The meta-heuristic algorithms have remarkable characteristics that vary from the gradient-based methods. This class of optimization methods not only demands no gradient computations but also they are straightforward for computer programming. Employing meta-heuristic algorithms allows exploration of a further fraction of the design space in comparison with gradient-based optimization methods. The meta-heuristics illustrate the effectiveness of numerous optimized structural issues like SFs or truss [22,23,24,25,26]. Many researchers employed popular meta-heuristics such as genetic algorithms (GA), particle swarm optimization (PSO), ant colony optimization (ACO) and harmony search (HS) for the optimization of structural systems. In the present study, the firefly algorithm (FA) [27], biogeography-based optimization (BBO) [28] and their combination are focused.

FA is a recently developed nature-inspired meta-heuristic based on the flashing patterns and behavior of fireflies. Additionally, FA is more efficient in comparison with other design optimization strategies since requires a lower output of evaluations for function. Moreover, FA has certain defects, for instance trapping to the local optimization results in the case of a complex searching space as well as a disability to do an excellent local search near the local optimum. The common knowledge that the BBO may carry out superior to the other meta-heuristic optimization strategies through the initial iterations, but in the case of increasing the number of iterations, it does not available competitive. BBO was encouraged based on biogeography, which refers to geographical distribution. In the current work, an unused optimization algorithm is represented by a combination of FA and BBO for the optimization of SF structures. The optimization procedure is performed by BBO around the optimum alternative found by FA to finely investigate the design space. As long as the algorithm is a serial integration of FA and BBO it is denoted as FA–BBO meta-heuristic. In this paper, the design variables are areas for the cross section of the structural elements and the design constraints are imposed on the nodal displacements and element stresses. The numerical output demonstrates the effectiveness and productiveness of FA–BBO against the FA and BBO.

2 Optimization Problem Formulation

For optimal design of an SF including ne members accumulated in ng design sets, the design variables of any design set are usually chosen based on the standard profile list. In this case, the optimization problem can be formulated in the following way:

$$ {\text{Minimize: }}w (X )= \sum\limits_{i = 1}^{ng} {\rho_{i} A_{i} \sum\limits_{j = 1}^{nm} {L_{j} } } $$
(1)
$$ {\text{Subject}}\;{\text{to:}}\; \, g_{k} (X )\le 0,\quad k = 1 ,2 ,\ldots ,nc $$
(2)
$$ X = \{ \begin{array}{*{20}c} {x_{1} } & {x_{2} } & \ldots & {x_{i} } & \ldots & {x_{ng} } \\ \end{array} \}^{\text{T}} $$
(3)

in this equations; xi presents an integer number to express the sequence of steel cross sections for the ith group; w is the frame weight, the weight per unit volume and area of the cross sections are presented by ρi, and Ai, respectively, for the ith group section; the number of the elements is presented by nm that accumulated through the ith set; for the jth element in the ith set Lj presented as a length of the elements; the kth behavioral constraint represented by gk.

The lateral inter-story drift ratio can be considered as follows:

$$ g_{d ,l} = \frac{{\theta_{l} }}{{R_{I} }} - 1 \le 0,\quad l = 1, \ldots ,ns $$
(4)

where Ɵl is the drift ratio for the inter-story; the inter-story drift ratio index represented by Rl authorized based on the code of practice and ns is the overall number of stories.

The demand-capacity ratio (DCR) constraints of structural elements exposed to axial load as well as flexural stresses are computed in the following way [29]:

$${\rm DCR}= \left\{ {\begin{array}{*{20}c} {\left[ {\frac{{P_{u} }}{{2\phi_{\text{c}} P_{\text{n}} }} + \left( {\frac{{M_{ux} }}{{\phi_{\text{b}} M_{nx} }} + \frac{{M_{uy} }}{{\phi_{\text{b}} M_{ny} }}} \right)} \right]} & {\quad {\text{if }}\frac{{P_{u} }}{{\phi_{\text{c}} P_{\text{n}} }} < 0. 2} \\ {\left[ {\frac{{P_{u} }}{{\phi_{\text{c}} P_{\text{n}} }} + \frac{8}{9}\left( {\frac{{M_{ux} }}{{\phi_{\text{b}} M_{nx} }} + \frac{{M_{uy} }}{{\phi_{\text{b}} M_{ny} }}} \right)} \right]} & {\quad {\text{if }}\frac{{P_{u} }}{{\phi_{\text{c}} P_{\text{n}} }} \ge 0. 2} \\ \end{array} } \right. $$
(5)
$$ g_{{{\text{DCR}},m}} = {\text{DCR}}_{m} - 1 \le 0,\quad m = 1, \ldots ,ne $$
(6)

in these equations Pu presents the required strength; Pn presents the nominally axial compression or tension strength; Øc presents the resistance coefficient; Mux and Muy present the essential flexural strengths through the x and y directions; Mnx and Mny present the nominal flexural strengths through the x and y directions, and Øb = 0.9 is the reduction factor of flexural resistance.

In the present work, the exterior penalty function method (EPFM) [30] is used to handle the design constraints. Using EPFM converts the main constrained optimization problem into an unconstrained one. Therefore, the pseudo unconstrained objective function may be presented through the following equation:

$$ \varPsi (X,r) = w(X)\left( {1 + r\sum\limits_{l = 1}^{ns} {(\hbox{max} \{ 0,g_{{{\text{d}}, \, l}} \} )^{2} } + r\sum\limits_{m = 1}^{ne} {(\hbox{max} \{ 0,g_{{{\text{DCR}}, \, m}} \} )^{2} } } \right) $$
(7)

in this equation \( \varPsi \) present, the pseudo-objective function and r present a penalty parameter. For minimizing the above-mentioned pseudo-objective function, several popular meta-heuristics are employed in the present study. The theoretical background of the meta-heuristics is explained in the next section.

3 Meta-heuristic Algorithms

The main idea that supports designing the meta-heuristic algorithms is to tackle complex optimization issues when the other optimization methods have failed to be effective. Meta-heuristics are applied to a very wide range of problems and they mimic natural metaphors to solve complex optimization problems. In this study, FA and BBO are utilized for discovering the optimum design of SFs subject to static loading.

3.1 Firefly Algorithm

The FA is a relatively new meta-heuristic optimization algorithm that was proposed by Yang at Cambridge University in 2008 according to the inspired by the flashing behavior of fireflies [31]. Fireflies communicate, search for prey and find mates employing bioluminescence with varied flashing models [32]. By considering the previous investigations, it may be considered that in almost all areas of engineering FA can be employed as a powerful optimization tool [33,34,35,36,37,38,39]. To develop the FA, natural flashing specifications of fireflies have been employed based on the following rules [27]:

  1. a.

    The fireflies are unisex; consequently, one firefly will be attracted to other fireflies irrespective of the sex.

  2. b.

    The attractiveness of any firefly is proportional to its brightness; therefore, for every two flashing fireflies, the less brightness of the firefly will move toward the brighter one. The attractiveness is proportional to the brightness and by increasing the distance the brightness will decrease. It will move randomly when there is no brighter firefly than the particular one.

  3. c.

    The brightness for a firefly is obtained based on the nature of the objective function.

Firefly attractiveness is obtained based on its brightness or light intensity that is determined following the objective function of the optimization problem. Nevertheless, β (the attractiveness) that depends on the judgment of the beholder defers based on the distance between two fireflies. β can be presented in the following way [31]:

$$ \beta = \beta_{0} e^{{ - \gamma \cdot r^{2} }} $$
(8)

in this equation r is the distance between two fireflies, β0 is the attractiveness for r = 0, and γ is the light absorption factor.

The distance between fireflies i and fireflies j at corresponding Xi and Xj is obtained employing the following equation:

$$ r_{ij} = \left\| {X_{i} - X_{j} } \right\| = \sqrt {\sum\nolimits_{k = 1}^{d} { (x_{i ,k} { - }x_{j ,k} )^{2} } } $$
(9)

in this equation xi,k is the kth parameter of the spatial coordinate xi.

In the FA, the firefly i movement toward a brighter firefly j is obtained by the following equation:

$$ X_{i} = X_{i} + \beta_{0} e^{{ - \gamma .r_{ij}^{2} }} (X_{j} - X_{i} ) + \alpha ({\text{rand}} - 0.5) $$
(10)

in this equation, the second term belongs to the attraction. Additionally, the randomization parameter is represented by α where the rand is a randomly generated number that distributed in the uniform condition in [0, 1].

3.2 Biogeography-Based Optimization

Simon represented the BBO algorithm in 2008 as a certain feature algorithm in common with other biology-based algorithms [28]. In this algorithm, the main concept was inspired based on the biogeography, that relates to the biological organisms in terms of time and space. As a reliable approximation of many engineering issues, the BBO algorithm has been employed [40,41,42]. The different islands, centuries, lands, or even continents over decades, or millennia are the case studies of these investigations. The different ecosystems in terms of habitats or territories are evaluated to figure out the relationships between various species (habitants) in the case of immigration, emigration and mutation. The progress in ecosystems to achieve a stable situation based on the different types of predators and prey and the efficiency of migration and mutation was the major inspiration in BBO [28].

BBO employs several search agents called habitats. These habitats are analogous to fireflies in FA. The BBO algorithm assigns each habitat a vector of habitants suitability index (HSI) defines the overall fitness of a habitat. The higher the HSI, the more fit the habitat. The habitats evolve based on three main rules in the following ways [43]:

  1. 1.

    The high HSI habitats living in habitants that are more likely to emigrate to habitats by the low HSI.

  2. 2.

    The low HSI habitats more prone to attract new immigrant habitants from those with high HSI.

  3. 3.

    Regardless of their HSI values, the habitats might be facing the random changes in the habitants.

In nature, these concepts bring a balance between different ecosystems. In other words, nature tends to improve the overall stability of different geographical regions. The BBO algorithm utilizes these concepts to improve the HSI of all habitats, which results in changing the primary random solutions for particular problems. The employed BBO algorithm starts with a random set of habitats. Each habitat has ng different habitants that correspond to the design variables number of a particular problem. Besides, each habitat has its immigration, emigration and mutation rates. This mimics the characteristic of various geographically separated locations in nature.

The components of the biography model and their correspondence based on the BBO algorithm in terms of the optimization point of view represented in the following ways [44]:

  • Habitat (H): Habitat presents a solution within the search space of a d-dimensional numerical optimization problem.

  • Habitat Suitability Index (HSI): Throughout a biography pattern, the geographical areas, that are well investigated as residences for biological species are with a high value of HSI.

  • Suitability Index Variables (SIV): The computation of the HSI amount is affecting through the additional coefficients for instance rainfall, diversity of vegetation, land area as well as temperature, that they called the SIVs.

  • Immigration Rate: The criterion is devoted to controlling the immigration habitat. The maximum amount of the immigration amount in a habitat is achieved when the species are not inside of the habitat.

  • Emigration Rate: The µ criterion is controlling the habitat emigration. The emigration will be null when there are no species. By increasing the number of spaces, species can leave their habitat to explore other residences. The maximum emigration rate (E) is achieved through a habitat in the case of containing the maximum amount of species that can support it.

  • Migration model: Based on the various mathematical patterns for biogeography, different migration curves might be utilized. A linear model is represented in Fig. 1. In this figure, S0 is the equilibrium number of species, by consideration of an equal amount for the immigration and emigration rates. The maximum possible immigration rate represented by I, where the maximum possible emigration rate is represented by E, and the greatest possible amount of species that the habitat might support is represented by \( S_{ \hbox{max} } \).

    Fig. 1
    figure 1

    The linear model of migration [44]

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Emigration (\( \mu_{k} \)) and immigration (\( \lambda_{k} \)) are formulated as functions of the habitants number in the following way [43]:

$$ \mu_{k} = \frac{E \cdot n}{N} $$
(11)
$$ \lambda_{k} = \left( {\frac{1 - n}{N}} \right) \cdot I $$
(12)

in this equation, n is the current number of habitants, N is the allowed maximum number of habitants.

These latter two rates are depicted in Fig. 2. It can be inferred from this figure that a high number of habitants coincides with a high probability of emigration and a low probability of immigration [43].

Fig. 2
figure 2

General steps of the BBO algorithm

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The third component in BBO, mutation, improves the exploration of BBO as well as maintains habitats. The component is illustrated in the following ways [43]:

$$ m_{n} = M \cdot \left( {\frac{{1 - P_{n} }}{{P_{ \hbox{max} } }}} \right) $$
(13)

in this equation, the initial value for mutation is M that is defined by the user \( p_{n} \) is the mutation probability belongs to nth habitat, and \( p_{ \hbox{max} } \) = argmax (\( p_{n} \)), n=1,2, …, N.

The global levels of the BBO algorithm are represented within Fig. 3 as a flowchart. Figure 2 demonstrates that the BBO algorithm initiates by random groups of habitats. Subsequently of obtaining the HSI of any habitat, the emigration, immigration and mutation ratios are upgraded. Based on these ratios, the non-elite habitants are migrated and mutated. For use as the next elites generation, a predefined amount of the best habitats is rescued. Lastly, the algorithm of BBO is concluded based on the satisfaction of a termination criterion. The elitism prevents the best solutions out to become corrupted with immigration. Therefore, some of the best habitats at any iteration are retained. Consequently, if their HSI is ruined based on the mutation, then the best solutions might be covered [43].

Fig. 3
figure 3

Flowchart of the proposed algorithm

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3.3 FA–BBO Algorithm

One of the defects known in FA is that all fireflies in final iterations converge to the more attractive firefly and therefore the algorithm converges to local optima and this is a reason that FA incapacitates for doing a proper local search. Considering the local search ability of FA is relatively poor and the local search ability of BBO is appropriate in comparison with FA. Therefore, FA and BBO are serially integrated to overcome the aforementioned difficulty.

To resolve the problem for improving the performance of the optimization process for finding a near-global solution, FA and BBO are serially integrated and this results in an algorithm termed an FA–BBO algorithm. The proposed FA–BBO meta-heuristic is a two-stage algorithm. In the first stage, a preliminary optimization is performed by FA to explore the design space by performing a limited number of iterations (say n1). The optimum solution found by FA is termed XFA. In the second stage, a normal finer search is implemented about the XFA by employing BBO again by performing a limited number of iterations (say n2). The initial population of the FA is randomly selected from all over the design space while the optimal initial population of BBO is selected as follows: XFA is directly transformed to the optimal initial population and the rests, say Xrndi, be chosen using the following equation:

$$ X_{j} = {\mathbf{\rm N}} (X_{\text{FA}} ,\gamma X_{\text{FA}} ) ,\quad j = 1,2, \ldots ,(n - 1) $$
(14)

where the randomly normal distributed vector is represented by N with a mean of XFA with a standard deviation of γXFA.

The γ coefficient has a critical role in the case of the convergence of the algorithm. In the current study, different values are investigated for γ where the optimal performance is achieved for the value of γ = 0.1. The flowchart of the proposed algorithm is represented in Fig. 3.

4 Numerical Results

For evaluation of the presented algorithm, three benchmarks, SF optimization problems are represented and the outputs are controlled based on the existing reports in the literature. W-shaped (from the standard profile list) cross sections were employed for all of the members. For each design example, 25 independent optimization runs have been performed and the best design is reported.

4.1 Example 1: 10-Story Planar SF

Figure 4 represents a 10-story SF, where the module of elasticity is 200 GPa and the yield stress of the material is 248.2 MPa. Moreover, the effective length factors of the members are obtained as Kx = 0 for a sway-permitted frame and the out-of-plane effective length factor is specified as Ky = 1. Any of the columns is considered unbraced through the length and one-fifth of the length of any span has been considered as unbraced length. The lateral inter-story drift ratio is limited by h/300 (h is the story height).

Fig. 4
figure 4

10-story steel frame

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Fabrication conditions are required for the element constructions of the frame. The same cross section for beams was employed for three beginning stories from the foundation. Furthermore, the same cross section for columns is employed for any two consecutive stories. Four cross sections for beams and five cross sections for columns resulting in the element grouping in nine design variables. The cross section for beam element sets is selected from the 267 W-shaped cross sections. The column sections are restricted to W14 and W12 (from the standard profile list). For all of the applied meta-heuristics, the population size is 40 and the upper limitation for iterations was 100. In the framework of FA–BBO, the number of iterations for preliminary optimization by FA and BBO is equal to 50 (i.e., n1 = n2 = 50). The outputs for the optimization are presented in Table 1.

Table 1 Optimal results of 10-story steel frame
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The results show that FA–BBO outperforms the algorithms applied in [46, 47]; moreover, the optimal solution found by FA–BBO is 2.06% and 1.17% lighter than those of FA and BBO. Figure 5 shows the convergence histories of FA, BBO and FA–BBO indicating that the FA–BBO possesses a better convergence rate compared with FA and BBO.

Fig. 5
figure 5

Convergence histories of FA, BBO and FA–BBO for 10-story steel frame optimization

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For the optimal solution found by FA–BBO, the drift ratio for inter-story and element stress ratios are shown in Figs. 6 and 7, respectively. The outputs present a feasible solution.

Fig. 6
figure 6

Inter-story drift profile for optimal 10-story steel frame based on FA–BBO

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Fig. 7
figure 7

Element group stress ratios for optimal 10-story steel frame based on FA–BBO

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4.2 Example 2: 24-Story Planar SF

A 24-story frame with a grouping of the details of the elements is considered and represented in Fig. 8. In this example, E = 205 GPa and Fy = 230.3 MPa. The factors for the effective length of the members are obtained as Kx = 0 for a sway-permitted frame and the out-of-plane effective length factor is considered as Ky = 1. All of the members in terms of beams and columns are specified as unbraced elements along the lengths.

Fig. 8
figure 8

24-story steel frame

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A H/300 limitation was considered for the maximum lateral displacement and h/300 was considered as a limitation of the maximum inter-story drift ratio. In this formulations, the height of the building represented by H and the height of one story represents by h. Fabrication conditions are imposing the construction of the 168-element frame demands the same cross section of the beam utilized through the first as well as the third bay on any floors (exclude the roofs), resulting in the scope of four groups of beams. The exterior columns teaming up into one group at the beginning of the foundation, where the interior columns teaming up in another set over three consecutive stories. 16 cross sections for columns and 4 cross sections for beams (20 design variables overall) were the groupings for the results. W-shaped cross sections for the beam element groups are chosen, where the cross section of columns groups is limited to W14 (from the standard profile list). The population size and the upper limitation for the iterations are considered to be 40 and 150, respectively. In the framework of FA–BBO, the number of iterations for preliminary and final optimization processes is equal to 75 (i.e., n1 = n2 = 75). Table 2 presents the results of the optimization.

Table 2 Optimal results of 24-story steel frame
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The results indicate that the optimal solution found based on the FA–BBO is better than those found in [27,28,29]. The weight of the optimal structure found by FA–BBO is 4.31% and 3.34% lighter than those of FA and BBO, respectively.

For the optimal solution found by FA–BBO, the drift ratio for inter-story and element stress ratios are shown in Figs. 9 and 10, respectively. Similarly, in this example, the outputs present a feasible solution.

Fig. 9
figure 9

Inter-story drift profile for optimal 24-story steel frame based on FA–BBO

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Fig. 10
figure 10

Element group stress ratios for optimal 24-story steel frame based on FA–BBO

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4.3 Example 3: 20-Story 3D Steel Braced Frame

A 20-story 3D braced SF (see Fig. 11) with 1040 elements are selected based on the study of Hasancebi et al. [48]. In this study, the columns are classified into three-member sets in a story; a corner, interior columns and exterior columns. Additionally, the beams are classified into two sets; interior beams and exterior beams.

Fig. 11
figure 11

20-story 3D steel frame

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The corner, interior and exterior columns as well as interior and exterior beams, and bracings are categorized with the same cross section through the two adjacent stories. Therefore, 60 distinct member sets are in the example. ASD-AISC [49] specification is used to control the stress constraints. Additionally, the displacements in x and y directions at joints are limited to 18.29 cm where the maximum amount of inter-story drifts ratio is considered as 0.91 cm. The elastic modulus was considered as 2.039 × 1010 kg/m2 and the yield stress was considered as 2.531 × 107 kg/m2 for the used steel. The applied dead and live loads considered from the first to 19th floors are considered 2.88 kN/m2 and 2.39 kN/m2, respectively. In the case of the roof, a dead load of 2.88 kN/m2 and a snow load of 1.20 kN/m2 were considered. The gravitational loads are utilized as distributed uniform loads through the beams based on the distribution formulations improved for slabs [48]. The wind loads for design are calculated based on ASCE 7-05 [50] as follows [48]:

$$ p_{\text{w }} = \left( {0.613 K_{z} K_{zt} K_{\text{d}} V^{2} I} \right)\left( {G C_{\text{p}} } \right) $$
(15)

in this equation the design wind pressure is represented by pw in terms of kN/m2, the velocity exposure coefficient is represented by Kz, the topographic factor represented by Kzt, the direction factor of wind reprinted by Kd, the basic speed for wind represented by V, the gust factor represented by G and the coefficient for external pressure represented by Cp. Based on the parameters that have been used by Hasancebi et al. [48], the buildings are considered in a flat terrain location with V = 46.94 m/s and exposure category B and the following values: \( K_{zt} = 1.0 \), Kd = 0.85, I = 1.0, G = 0.85 and Cp = 0.8 for windward facing and 0.5 for leeward facing. After the calculation of the wind loads, the load is acting in the uniform condition at each level of the floors. A combination of the gravity and wind loads is considered into two types of loading conditions based on the wind loads acting along the x-axis or the y-axis [48].

To implement the optimization process, the population size is chosen to be 50 and the maximum number of iterations is limited to 400. For FA–BBO meta-heuristic, the number of iterations for each of FA and BBO meta-heuristics is limited to 200 (i.e., n1 = n2 = 200). Table 3 presents the optimization results.

Table 3 Optimal results of 20-story 3D steel frame
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The optimum solution for the weight based on FA–BBO is 406.13 ton performing 20000 analyses, which is lighter than 412.91 ton reported in [48] conducting 60,000 analyses. Moreover, the results show that FA–BBO finds a solution which is 6.17% and 5.1% lighter than the solutions found by FA and BBO, respectively. Figures 12 and 13 depict the inter-story drift ratio profile as well as the element stress ratios for the optimal solution of FA–BBO, respectively. These results demonstrate the feasibility of the optimal solution.

Fig. 12
figure 12

Inter-story drift profile for optimal 20-story 3D steel frame based on FA–BBO

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Fig. 13
figure 13

Element group stress ratios for optimal 20-story 3D steel frame based on FA–BBO

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5 Conclusions

In the present study, a combined meta-heuristic algorithm is represented for the design optimization of SF structures. In this case, the main characteristics of FA and BBO optimization algorithms are combined and the resulted algorithm is termed as FA–BBO meta-heuristic. In this algorithm, FA acts as an explorer while the exploitation task is achieved by BBO. In other words, BBO performs a finer search around the solution found by FA. Three SF optimization examples are represented to demonstrate the computational superiority of the presented meta-heuristic algorithm. In each design examples, the results of FA–BBO are compared with other algorithms proposed in other studies. In all examples, the weight of the optimal structure found by FA–BBO is lighter compared with those of the other algorithms. Furthermore, the rate for the convergence of the FA–BBO is better than those of FA and BBO. Finally, it can be carried out that FA–BBO is an efficient meta-heuristic algorithm for the optimization of SF structures.