O R I G I N A L A R T I C L E
Optimum structural design of spatial steel frames
via biogeography-based optimization
Serdar C¸ arbas¸1
Received: 16 August 2015 / Accepted: 21 December 2015 / Published online: 8 January 2016 Ó The Natural Computing Applications Forum 2016
Abstract Metaheuristic algorithms have provided an efficient tool for designers by which discrete optimum design of real-size steel space frames under design code requirements can be obtained. In this study, the optimum sizing design of steel space frames is formulated according to provisions of Load and Resistance Factor Design— American Institute of Steel Construction. The weight of the steel frame is taken as objective function. The design algo-rithm selects the appropriate W sections for members of the steel frame such that the frame weight is the minimum and design code limitations are satisfied. The biogeography-based optimization algorithm is utilized to find out the optimum solution of the discrete programming problem. This algorithm is one of the recent additions to metaheuristic techniques which are based on theory of island biogeogra-phy where each habitat is assumed to be potential solution for the design problem. The performance of the biogeogra-phy-based optimization algorithm is compared with other recent metaheuristic algorithms such as adaptive firefly algorithm, teaching and learning-based optimization, artifi-cial bee colony optimization, dynamic harmony search algorithm, and ant colony algorithm. It is shown that bio-geography-based optimization algorithm outperforms other metaheuristic techniques in the design examples considered. Keywords Structural design optimization Spatial steel frames Metaheuristic techniques Biogeography-based optimization LRFD–AISC
1 Introduction
Design optimization of steel space frames provides an economical solution which not only comes up with the optimum W sections for the members of the frame, but also helps to reduce carbon dioxide emission due to the saving achieved in the required amount of steel for the construction of the frame. In practice, design of a steel space frame requires selection of W sections for its beams and columns such that Load and Resistance Factor Design—American Institute of Steel Construction (LRFD–AISC) [1] design code limitations are satisfied and the weight or cost of the material of the frame is the minimum. Naturally, formula-tion of such decision-making problem turns out to be dis-crete programming problem because W sections are available as discrete values in the steel sections table. Although in early attempts for finding the solution of design optimization problems of steel space frames, mathematical programming techniques are used, these techniques were only able to provide optimum designs for small-size steel space frames under stress and displacement constraints [2]. These techniques run into convergence difficulties when the size of steel frame become large and the constraints are implemented according to design code provisions. There-fore, researchers had to come up with different types of solution techniques to find the optimum solution of steel space frame design problems. Metaheuristic algorithms have provided a tool for designer to fulfill this objective. These techniques are based on natural phenomena such as swarm intelligence, music improvisation, survival of fittest, and others [2]. They are quite efficient particularly finding the solutions of discrete programming problems [3–8]. Notable progress has been achieved in broad array of nat-ure-inspired algorithms and their applications. The sphere of activity in metaheuristics-based structural optimization is
& Serdar C¸arbas¸ scarbas@kmu.edu.tr
1 Department of Civil Engineering, Karamanoglu Mehmetbey
University, Karaman, Turkey DOI 10.1007/s00521-015-2167-6
constantly developing with the improvement in novel techniques [9–18]. Metaheuristics and swarm intelligence are turning out to be more preferred for optimum design of steel frame structures [19–21].
In the formulation of the design optimization of steel space frames, the sequence numbers of W sections in the steel profiles table is taken as design variables. There are 272 W sections available in the list. Members of steel space frame can be grouped together for practical reasons. The total number of groups in the frame is the total number of design variables in the design problem. These variables can have any values from 1 to 272 during the design process. Once the sequence number of the W section is selected from the list for any design variable, the complete cross-sectional properties of the section become available which can be used in the analysis of the steel frame. With the treatment of sequence number of W sections as design variable, the mathematical modeling of the design optimization problem turns out to be discrete programming problem [22]. As mentioned in the preceding section, metaheuristic tech-niques are shown to be quite efficient in obtaining the solution of such programming problems. Several reviews are available in the literature which comprehensively summarizes the use of metaheuristic algorithms in the design optimization of steel skeleton structures [23–26]. Among these algorithms, biogeography-based optimization (BBO) [27] technique reveals itself due to its capacity of obtaining a near-global optimum especially in problems with large amount of design variables. Recently, this tech-nique has been featly implemented a broad array of engi-neering and mathematical optimization problems [28–31]. In this paper, the performance of BBO algorithm in finding the solution of design optimization problems of steel space frames is investigated. For this purpose, design optimization problem of two real-size steel space frames is formulated according to design code provisions and they are solved by BBO and the obtained results are compared to the previ-ously reported results attained by other metaheuristic techniques such as adaptive firefly algorithm, teaching and learning-based optimization, artificial bee colony opti-mization, dynamic harmony search algorithm and ant col-ony algorithm. The BBO algorithm has two control parameters called migration and mutation operators that are critical for the successful application of the algorithm. The initial values selected for these parameters affect the per-formance of the algorithm dearly.
The remaining of the paper is arranged as follows: Sect.2 formulates the optimum discrete design of spatial steel frames, while the BBO algorithm is described in Sect.3. In Sect.4, the numerical examples are presented to demonstrate the efficiency of the developed algorithm. The conclusions deducted from this study are briefly discussed in Sect.5.
2 Description of the design optimization problem
The design of spatial steel frames necessitates the selection of steel sections for its columns and beams from a standard steel section tables such that the frame satisfies the ser-viceability and strength requirements specified by the code of practice, while the economy is observed in the overall or material cost of the frame. When the design constraints are implemented from LRFD–AISC [1], the following non-linear discrete programming problem is obtained.
2.1 The objective function
The objective function is taken as the minimum weight of the frame which is expressed as in the following.
Minimize W¼X ng r¼1 mr Xtr s¼1 ls ð1Þ
where W defines the weight of the frame and mris the unit weight of the steel section selected from the standard steel sections table that is to be adopted for group r. tris the total number of members in group r, and ng is the total number of groups in the frame. ls is the length of members which belong to group r.
2.2 Strength constraints
For the case where the effect of warping is not included in the computation of the strength capacity of W sections that are selected for beam–column members of the frame, the following inequalities given in Chapter H of LRFD–AISC are required to be satisfied.
for Pu /Pn 0:2; gs;i¼ Pu /Pn þ8 9 Mu /bMnx þ Mu /bMny 1:0 ð2Þ for Pu uPn \0:2; gs;i¼ Pu 2/Pn þ Mu /bMnx þ Mu /bMny 1:0 ð3Þ where Mnx is the nominal flexural strength at strong axis
(x-axis), Mny is the nominal flexural strength at weak
axis (y-axis), Mux is the required flexural strength at
strong axis (x-axis), Muy is the required flexural strength
at weak axis (y-axis), Pn is the nominal axial strength
(tension or compression), and Pu is the required axial
strength (tension or compression) for member i. l repre-sents the loading case. The values of Mux and Muy are
obtained by carrying out P D analysis of the steel frame. This is an iterative process which is quite time-consuming. In Chapter C of LRFD–AISC, an alternative procedure is suggested for the computations of Mux and
Muy values. In this procedure, two first-order elastic
analyses are carried out. In the first, frame is analyzed under the gravity loads only where the sway of the frame is prevented to obtain Mnt values. In the second,
the frame is analyzed only under the lateral loads to find Mlt values. These moment values are then combined
using the following equation as given in the design code.
Mu¼ B1Mntþ B2Mlt ð4Þ
where B1is the moment magnifier coefficient and B2is the
sway moment magnifier coefficient. The details of how these coefficients are calculated are given in Chapter C of LRFD– AISC. Equations (2) and (3) represent strength constraints for doubly and singly symmetric steel members subjected to axial force and bending. If the axial force in member k is tensile force, the terms in these equations are given as: Pukis
the required axial tensile strength, Pnkis the nominal tensile
strength, / becomes /t in the case of tension and called strength reduction factor which is given as 0.90 for yielding in the gross section and 0.75 for fracture in the net section, /bis the strength reduction factor for flexure given as 0.90,
Muxk and Muykare the required flexural strength, and Mnxk
and Mnykare the nominal flexural strength about major and
minor axis of member k, respectively. It should be pointed out that required flexural bending moment should include second-order effects. LRFD suggests an approximate pro-cedure for computation of such effects which is explained in Chapter C1 of LRFD. In the case the axial force in member k is compressive force, the terms in Eqs. (2) and (3) are defined as: Pukis the required compressive strength, Pnk is
the nominal compressive strength, and / becomes /cwhich is the resistance factor for compression given as 0.85. The remaining notations in Eqs. (2) and (3) are the same as the definition given above.
The nominal tensile strength of member k for yielding in the gross section is computed as Pnk¼ FyAgkwhere Fyis the
specified yield stress and Agkis the gross area of member k.
The nominal compressive strength of member k is computed as Pnk¼ AgkFcr where Fcr¼ 0:658k 2 c Fy for kc 1:5 and Fcr¼ 0:877=k2c Fyfor kc[ 1:5 and kc¼Klrp ffiffiffiffiffiffiffiffiffi Fy E q : In these expressions, E is the modulus of elasticity, and K and l are the effective length factor and the laterally unbraced length of member k, respectively.
2.3 Displacement constraints
The lateral displacements and deflection of beams in steel frames are limited by the steel design codes due to ser-viceability requirements. According to the ASCE Ad Hoc Committee report [32], the accepted range of drift limits in
the first-order analysis is 1/750 to 1/250 times the building height H with a recommended value of H/400. The typical limits on the inter-story drift are 1/500 to 1/200 times the story height. Based on this report, the deflection limits recommended are proposed in [33–35] for general use which is repeated in Table 1.
2.3.1 Deflection constraints
It is necessary to limit the mid-span deflections of beams in a spatial steel frame not to cause cracks in brittle finishes that they may support due to excessive displacements. Deflection constraints can be expressed as an inequality limitation as shown in the following.
gdj¼
djl
duj
1 0 j ¼ 1; . . .; nsm; l¼ 1; . . .; nlc ð5Þ
where djl is the maximum deflection of jth member under
the lth load case, duj is the upper bound on this deflection which is defined in the code as span/360 for beams carrying brittle finishers, nsm is the total number of members where
deflections limitations are to be imposed, and nlc is the
number of load cases. 2.3.2 Drift constraints
These constraints are of two types. One is the restriction applied to the top-story sway, and the other is the limitation applied on the inter-story drift.
2.3.2.1 Top-story drift constraint Top-story drift limita-tion can be expressed as an inequality constraint as shown in the following. gtdj ¼ Dtop jl H=Ratio 1 0 j¼ 1; . . .; njtop; l¼ 1; . . .; nlc ð6Þ where H is the height of the frame, njtopis the number of joints
on the top story, nlcis the number of load cases, and Dtop
jlis
the top-story drift of the jth joint under lth load case. 2.3.2.2 Inter-story drift In multi-story steel frames, the relative lateral displacements of each floor are required to
Table 1 Displacement limitations for steel frames
Item Deflection limit
1 Floor girder deflection for service live load L/360
2 Roof girder deflection L/240
3 Lateral drift for service wind load H/400 4 Inter-story drift for service wind load H/300
be limited. This limit is defined as the maximum inter-story drift which is specified as hsx/Ratio where hsx is the story
height and Ratio is a constant value given in ASCE Ad Hoc Committee report [32]. gidj ¼ Doh ð Þjl hsx=Ratio 1 0 j¼ 1; . . .; nst l¼ 1; . . .; nlc ð7Þ where nst is the number of story, nlcis the number of load
cases, and Dð ohÞjlis the story drift of the jth story under lth
load case.
2.4 Geometric constraints
In steel frames, it is desired that column section for upper floor should not have a larger section than the lower story column for practical reasons. Because having a larger section for upper floor requires a special joint arrange-ment which is neither preferred nor economical. The same applies to the beam-to-column connections. The W sec-tion selected for any beam should have a flange width smaller than or equal to the flange width of the W section selected for the column to which the beam is to be con-nected. These are shown in Fig.1 and named as geo-metric constraints. These limitations are included in the design optimization model to satisfy practical require-ments. Two types of geometric constraints are considered in the mathematical model. These are column-to-column geometric constraints and beam-to-column geometric limitations.
2.4.1 Column-to-column geometric constraints
The depth and the unit weight of W sections selected for the columns of two consecutive stores should be either equal to each other or the one in the upper story should be smaller than the one in the lower story. These limitations are included in the design problem as inequality constraints as shown in the following.
gcdi¼ Di Di1 1 0 i¼ 2; . . .; nj ð8Þ gcmi¼ mi mi1 1 0 i¼ 2; . . .; nj ð9Þ
where nj is the number of stories, mi is the unit weight of
W section selected for column story i, mi1 is the unit
weight of W section selected for of column story (i - 1), Diis the depth of W section selected for of column story i,
and Di1 is the depth of W section selected for of column
story (i - 1).
2.4.2 Beam-to-column geometric constraints
When a beam is connected to a flange of a column, the flange width of the beam should be less than or equal to the flange width of the column so that the connection can be made without difficulty. In order to achieve this, the flange width of the beam should be less than or equal to Dð 2tbÞ
of the column web dimensions in the connection where D and tbare the depth and the flange thickness of W section,
respectively, as shown in Fig.1. gbci¼ Bf bi Dci 2 tð Þbc i 1 0 i¼ 1; . . .; nj1 ð10Þ or gbbi¼ Bf bi Bf ci 1 0 i ¼ 1; . . .; nj2 ð11Þ
where nj1 is the total number of joints where beams are
connected to the web of a column, nj2is the total number of
joints where beams connected to the flange of a column, Dci is the depth of W section selected for the column at
joint i; tð Þb cjis the flange thickness of W section selected for
the column at joint i, Bf
ciis the flange width of W section
selected for the column at joint i, and Bf
bi is the flange
width of W section selected for the beam at joint i: The optimum design of spatial steel frames problem described in preceding sections where the objective func-tion is given in Eq. (1) and the constraints are depicted from Eqs. (2)–(11) is a nonlinear discrete optimization problem. It is apparent that in order to determine the optimum solution of this problem, steel designer has to find out the suitable combination of W sections that makes the
frame weight minimum in the same time the design code provisions are all satisfied. Here, the selection of a W section from an available steel profile list is carried out by choosing an integer number from a set which consist of integer numbers starting 1 to the total number of sections in the list. This integer number is the sequence number of that particular W section. Hence, the design solution is a set of integer numbers each of which represents the sequence number of W section in the design pool. This is a combi-natorial optimization problem.
3 Biogeography-based optimization (BBO)
Biogeography-based optimization algorithm is developed by Simon [27] which is based on the theory of island biogeography. Mathematical model of biogeography describes the migration and extinction of species between islands. An island is any area of suitable habitat which is isolated from the other habitats. Islands that are friendly to life are said to have high habitat suitability index (HSI). Features that correlate with HSI include such factors as
rainfall, diversity of vegetation, diversity of topographic features, land area, and temperature. The variables that characterize habitability are called suitability index vari-ables (SIVs). SIVs can be considered the independent variables of the habitat, and HSI can be considered the dependent variable. Naturally, habitats with a high HSI tend to have a large number of species, while those with a low HSI have a small number of species. Habitats with a high HSI have many species that emigrate to nearby habitats, simply by virtue of the large number of species that they host. Habitats with a high HSI have a low species immigration rate because they are already nearly saturated with species. Therefore, high HSI habitats are more static in their species distribution than low HSI habitats. This fact is used in biogeography-based optimization for carrying out migration. Relationship between species count, immi-gration rate, and emiimmi-gration rate is shown in Fig.2 [27], where I refers to the maximum immigration rate, E is the maximum emigration rate, S0is the equilibrium number of species, and Smax is the maximum species count.
The decision to modify each solution is taken based on the immigration rate of the solution. kkis the immigration
probability of independent variable xk. If an independent
variable is to be replaced, then the emigrating candidate solution is chosen with a probability that is proportional to the emigration probability lk which is usually performed using roulette wheel selection.
PðxjÞ ¼
lj
PN i¼1li
for j¼ 1; . . .; N ð12Þ
where N is the number of candidate solutions in the population.
Mutation is also another factor which is used to increase the species richness of islands. This increases the diversity among the population. Each candidate solution is associ-ated with a mutation probability defined by
Immigration Emigration I E Rate S0 Smax Number of species (λ) (μ)
Fig. 2 Species model of a single habitat where k is immigration rate and l is emigration rate
Biogeograpy Based Optimization Algorithm
For each solutionyk,k∈
{
1,....,N}
, define emigration probability μk fitness of yk,μk∈[ ]0,1For each solutionykdefine immigration probability λk=1−μk
y z
For each solution zk
For each solution feature s
Use λkto probabilistically decide whether to immigrate to zk
If immigrating then
Use { }μi to probabilistically select the emigrating solution yj ( )s y ( )s
zk j
end if next solution feature Probabilistically mutatezk next solution
z y Fig. 3 Pseudo-code for one
generation of biogeography-based optimization algorithm
3.5m 3.5m x 4 = 14 m 6m x 7 = 42m 6m x 10 = 60m 6m 6m (a) (b) (c) (d) Fig. 4 4-Story, 428-member
spatial steel frame, a 3-D view, bfront view, c plan view, dcolumn orientations
mðsÞ ¼ mmax
1 Ps
Pmax
ð13Þ mmaxis a user-defined parameter. Psis the species count of the habitat, and Pmax is the maximum species count. Mutation is carried out on the mutation probability of each habitat. The steps of the biogeography-based optimization algorithm can be listed as follows [36].
1. Set up initial population; define the migration and mutation probabilities.
2. Calculate the immigration and emigration rates for each candidate solution in the population.
3. Select the island to be modified based on the immi-gration rate.
4. Using roulette wheel selection on the emigration rate, select the island from which the SIV is to be immigrated.
5. Randomly select an SIV from the island to be emigrated.
6. Perform mutation based on the mutation probability of each island.
7. Calculate the fitness of each individual island. 8. If the fitness criterion is satisfied go to step 2.
The pseudo-code of biogeography-based optimization algorithm is given in Fig.3 [37].
In the BBO, infeasible designs that violate some of the problem constraints are penalized using an external penalty function approach [38], and their objective function values are computed according to Eq. (14).
fc¼ W 1 þ Xnc i¼1 Ci !e ð14Þ
where W is the design weight of a solution calculated as per Eq. (1), fcis the constrained objective function value of the solution, Ciis the value of total constraint violations which is calculated by summing the violation of each individual constraint, and nc is the total number of constraints in the design optimization. Constraint functions for the steel frame are given through Eqs. (2)–(11). In addition, e = 2.0
Table 2 Member grouping of 4-story, 428-member spatial steel frame
Story Side beam Inner beam Corner beam Side column Inner column
1 1 2 9 10 11
2 3 4 12 13 14
3 5 6 15 16 17
4 7 8 18 19 20
Table 3 Final best design of 428-member spatial steel frame obtained with BBO
Size variables BBO Size variables BBO
Ready section Area (mm2) Ready section Area (mm2)
1 W360X32.9 41.7 11 W200X46.1 58.6 2 W250X32.7 41.7 12 W410X100 127 3 W460X52 66.3 13 W250X80 102 4 W310X32.7 41.8 14 W360X134 171 5 W530X66 83.7 15 W460X113 144 6 W460X52 66.3 16 W310X97 123 7 W360X32.9 41.7 17 W360X147 188 8 W460X52 66.3 18 W920X201 256 9 W410X53 68.1 19 W840X193 247 10 W250X49.1 62.5 20 W530X150 192 1250 1750 2250 2750 3250 3750 4250 0 10000 20000 30000 40000 50000
Feasible Best Design (kN)
Number of Iteration
BBO
Fig. 5 Optimization history of 4-story, 428-member spatial steel frame using BBO
is the penalty coefficient used to tune the intensity of penalization as a whole.
4 Numerical examples
This section covers performance evaluation of the BBO in discrete sizing optimization of spatial steel frames. To this end, the optimization algorithm of BBO is coded in Intel Visual Fortran [39] for analysis and design of structural
systems sampled during the course of optimization. The investigated examples include minimum weight design of two spatial steel frame structures. The optimum designs to these frames with the BBO are sought by implementing the algorithm over a predefined number of iterations. In order to evaluate the accuracy of the final solutions obtained with the BBO, the optimum solutions are compared to those previously reported in the literature by some other robust metaheuristic algorithms, and the results are evaluated. Due to the stochastic nature of the BBO, each problem is
Table 4 Maximum constraint values and minimum frame weights for 4-story, 428-member spatial steel frame with different metaheuristic techniques
Algorithm BBO (present study) TLBO [44] ABC [44] DHS [44] ACO [44]
Minimum weight (kN) 1332.29 1503.91 1512.11 1526.01 1573.21
Maximum top-story drift (cm) 2.867 2.680 2.911 2.663 2.856
Maximum inter-story drift (cm) 0.875 0.869 0.875 0.874 0.873
Maximum strength constraint ratio 0.978 0.980 0.998 0.85 1.00
Maximum number of iterations 50,000 50,000 50,000 50,000 50,000
0 5 10 15 20 25 30 35 40 45 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
No. of top story joint
Top-story Drift (cm)
Existing value Max. top-story drift value
0 1 2 3 4 0.35 0.525 0.7 0.875 No. of story Inter-story Drift (cm)
Existing value Max. inter-story drift value
(a)
(b) Fig. 6 Drift outlines of the
4-story, 428-member spatial steel frame; a inter-story drift outline, b top-story drift outline
independently solved several times and the best results collected are used for comparisons. The population size is set to 50, and the number of elites that specify how many of the best solutions to keep from one generation to the next is set to 2.0 for both examples. The mutation probability per solution per independent variable is selected as 0.01 for first example and 0.005 for second example. These values are assigned to constant values that are arbitrarily chosen within their recommended ranges by Simon [40,41] based on the observed efficiency of the technique in different problem fields. It is obvious that best values of these parameters depend on the size of search space. Therefore, the effect of these parameters is investigated in all design examples for BBO algorithm. Each example is designed several times by considering different values of parameter sets. After conducting relatively enough sensitivity analysis on the preassigned parameter values, following conclusions can be drawn; a large population size has a better initial solution, but more generations between improvements, and smaller improvements. A small population size has a poorer initial solution, but fewer generations between improvements, and larger improvements [40]. Moreover, it is demonstrated in the literature that elitism parameter improves performance of the technique, but if too many elites are used, then performance worsens [41]. Addition-ally, it should be noted that considering constant value for mutation probability parameter has no major effect on the performance of the algorithm. Moreover, the large values for this parameter may lead to unnecessary computational effort and also bias the search process in the solution space [27,40,41].
The value of maximum number of analyses for each design example is considered as 50,000 and 75,000, respectively. The total number of structural analysis required to reach the optimum design is large similar to most of metaheuristic algorithms. This number can be
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 200 250 300 350 400 450
Strength contraints ratio
No. of member
Existing value Strength ratio limit
Fig. 7 Strength ratios of members in 4-story, 428-member spatial steel frame
(a)
(b)
(c)
Fig. 8 8-Story, 1024-member spatial steel frame, a 3-D view, b front view, c plan and column orientations view
reduced by carrying out some enhancement in the algo-rithm, such as adding upper bound strategy (UBS) [21,42,
43], as a future work. The idea behind UBS is to detect those candidate designs which have no chance to improve the search during the optimization process. After identi-fying the non-improving candidate designs, they are directly excluded from the structural analysis, thus reduc-ing the total computational effort [21].
Furthermore, for all the examples, the wide-flange (W) profile list of ready sections is used to size the structural members. The material properties of steel are taken as fol-lows: modulus of elasticity (E) = 208 GPa (30,167.84 ksi) and yield stress (Fy) = 250 MPa (36.26 ksi), and unit
weight of the steel (q) = 7.85 ton/m3.
4.1 Example 1: 4-story, 428-member spatial steel frame
The first design example of this study is a 4-story, 428-member spatial steel frame. This design instance is formerly tackled by Aydogdu and Akin [44] using different metaheuristic techniques. Here, it is intended to evaluate the efficiency of the BBO technique compared to the contemporary sizing optimization algorithms. Three-di-mensional, front, and plan views of this building as well as column orientations are illustrated in Fig.4. The frame has 172 joints and 428 members which are collected in 20 independent member groups. The member grouping of the frame is illustrated in Table2. For design purpose, the frame is subjected to gravity loads as well as lateral loads
that are computed according to ASCE 7-05 [45]. The design dead and live loads are taken as 2.88 and 2.39 kN/ m2, respectively. Basic wind speed is considered as 85 mph (38 m/s) for the wind load. The following load combina-tions are considered in the design of the frame according to the code specification LRFD and ASCE 7-05: 1.2D ? 1.6L ? 0.5S, 1.2D ? 0.5L ? 1.6S, 1.2D ? 1.6WX ? L ? 0.5S and 1.2D ? 1.6WZ ? L ? 0.5S where D is the dead load, L represents the live load, S is the snow load, and WX and WZ are the wind loads in the global X- and Z-axes, respectively. The drift ratio limits for this example are taken as 0.875 cm for inter-story drift and 3.5 cm for top-story drift. Maximum deflection of beam members is restricted as 2.0 cm.
The optimization history showing the variation of the best feasible weight during the cycles of the BBO algo-rithm is depicted in Fig.5. The section designations attained for each member group by BBO algorithm are tabulated in Table3. In Table4, minimum frame weight located by the BBO algorithm is compared with the available results reported in the literature based on a teaching and learning-based optimization (TLBO), an artificial bee colony (ABC) algorithm, a dynamic harmony search (DHS) algorithm, and an ant colony optimization (ACO) [44]. Also, maximum constraint values for each algorithm are illustrated in this table. According to these results, the BBO algorithm locates an optimum design weight of 1332.29 kN, which is lighter than the design weights obtained by the other techniques. The optimum design produced by BBO is 11.41, 11.89, 12.69, and
Table 5 Member grouping of 8-story 1024-member spatial steel frame
Story Side beam Inner beam Corner column Side column Inner column
1 1 2 17 18 19 2 3 4 20 21 22 3 5 6 23 24 25 4 7 8 26 27 28 5 9 10 29 30 31 6 11 12 32 33 34 7 13 14 35 36 37 8 15 16 38 39 40
Table 6 Minimum design weights and the maximum constraint values for 8-story, 1024-member spatial steel frame with different metaheuristic techniques
Algorithm BBO (present study) AFFA [46] ABC [47] DHS [46,47] ACO [46,47]
Minimum weight (kN) 6462.79 6748.99 6761.47 7210.12 7689.51
Maximum top-story drift (cm) 6.508 6.675 6.858 6.577 6.239
Maximum inter-story drift (cm) 0.875 0.875 0.875 0.874 0.871
Maximum strength constraint ratio 1.0 0.993 0.995 0.986 0.989
Table 7 Final best designs of 8-story, 1024-member spatial steel frame obtained with different metaheuristic algorithms # Group type BBO AFFA [ 46 ] ABC [ 47 ] DHS [ 46 , 47 ] ACO [ 46 , 47 ] # Group type BBO AFFA [ 46 ] ABC [ 47 ] DHS [ 46 , 47 ] ACO [ 46 , 47 ] 1 Beam W310X21 W310X38.7 W360X32.9 W310X158 W200X35.9 21 Column W360X196 W460X128 W610X195 W360X216 W690X265 2 Beam W310X21 W200X46.1 W250X32.7 W610X195 W200X46.1 22 Column W250X58 W530X165 W250X58 W360X162 W760X147 3 Beam W410X53 W310X79 W530X92 W760X134 W410X75 23 Column W360X162 W690X240 W1000X321 W360X216 W1100X390 4 Beam W250X49.1 W310X117 W250X58 W760X134 W360X32.9 24 Column W360X237 W1000X258 W760X284 W690X217 W1100X343 5 Beam W610X101 W460X82 W360X101 W460X82 W360X110 25 Column W250X67 W530X165 W360X162 W840X299 W1000X321 6 Beam W250X58 W460X82 W200X71 W460X106 W310X107 26 Column W360X162 W690X240 W1000X321 W1100X390 W1100X390 7 Beam W410X100 W360X110 W530X92 W610X101 W460X106 27 Column W360X347 W1000X321 W840X299 W840X299 W1100X390 8 Beam W250X80 W360X110 W530X66 W760X134 W460X68 28 Column W310X97 W690X170 W360X262 W840X299 W1000X321 9 Beam W610X155 W360X110 W690X140 W360X162 W610X113 29 Column W610X174 W760X284 W1000X443 W1100X390 W1100X390 10 Beam W530X165 W460X52 W530X66 W760X220 W360X39 30 Column W1100X390 W1000X477 W1000X321 W920X365 W1100X499 11 Beam W690X125 W610X82 W760X134 W530X182 W690X125 31 Column W530X150 W760X173 W360X287 W840X299 W1100X390 12 Beam W310X107 W460X68 W310X28.3 W920X223 W460X52 32 Column W760X314 W1000X412 W1000X477 W1100X390 W1100X390 13 Beam W610X125 W840X176 W840X176 W310X86 W690X170 33 Column W1100X499 W1100X499 W1100X433 W1000X412 W1100X499 14 Beam W460X52 W360X32.9 W410X60 W690X125 W310X32.7 34 Column W530X150 W760X173 W1000X321 W1100X343 W1100X433 15 Beam W840X176 W760X147 W760X173 W310X74 W840X176 35 Column W920X446 W1100X433 W1000X477 W1100X499 W1100X499 16 Beam W310X28.3 W310X32.7 W410X38.8 W460X82 W310X28.3 36 Column W1100X499 W1100X499 W1100X499 W1100X499 W1100X499 17 Column W310X107 W200X35.9 W360X44 W410X149 W410X100 37 Column W920X201 W1000X296 W1000X321 W1100X343 W1100X433 18 Column W360X196 W360X101 W410X100 W760X161 W250X149 38 Column W1000X443 W1100X433 W1000X477 W1100X499 W1100X499 19 Column W200X41.7 W360X64 W200X52 W610X155 W250X89 39 Column W1100X499 W1100X499 W1000X477 W1100X499 W1100X499 20 Column W310X117 W530X165 W460X128 W610X155 W1100X390 40 Column W1000X258 W1100X343 W1000X321 W1100X343 W1100X499
15.31 % lighter than those attained by TLBO, ABC, DHS, and ACO, respectively.
Besides, in order to verify the satisfaction of the con-straints at the end of the optimization process, the inter-story drifts of the frame floors and the top-inter-story drifts which occur at the joints on the top story of the frame are compared to the allowable values as shown in Fig.6. These results demonstrate that the optimum solution obtained by BBO algorithm is feasible. From this figure, it is clearly seen that especially the inter-story drift lim-itation reached its upper bound of 0.875. This means that while inter-story drift constraint was active in the opti-mum design, the top-story sway limitation was relatively not active.
Additionally, the strength ratios of the members in the optimum design yielded by BBO are shown in Fig.7 to check whether the strength constraints are satisfied or not. It is noticed from this figure that the strength limitations are dominant in the design problem. In the optimum frame, the strength ratios of some members were very close to its upper bound of 1.0. Certainly in the optimum results, it is the strength constraints that govern the design.
4.2 Example 2: 8-story, 1024-member spatial steel frame
The second test problem is a three-dimensional, 8-story, 1024-member spatial steel frame (Fig.8). This problem was first studied by Saka et al. using an adaptive firefly algorithm (AFFA) [46], an artificial bee colony (ABC) algorithm [47], a dynamic harmony search (DHS) algo-rithm [46, 47], and an ant colony optimization [ACO] algorithm [46, 47] based metaheuristic techniques. The frame has 384 joints and 1024 members which are col-lected in 40 independent design variables. The member grouping of the frame is illustrated in Table5. The frame is subjected to gravity loads as well as lateral loads that are computed according to ASCE 7-05 [45]. Similar to the first example, the design dead and live loads are taken as 2.88 and 2.39 kN/m2, respectively. Basic wind speed is con-sidered as 85 mph (38 m/s). The following load combina-tions are considered in the design of the frame according to the code specification [1]: 1.2D ? 1.6L ? 0.5S, 1.2D ? 0.5L ? 1.6S, 1.2D ? 1.6WX ? L ? 0.5S and 1.2D ? 1.6WZ ? L ? 0.5S where D is the dead load, L represents the live load, S is the snow load, and WX and WZ are the wind loads in the global X- and Z-axes, respectively. Drift ratio limits for this example are taken as 0.875 cm for inter-story drift and 7 cm for top-story drift. Maximum deflection of beam members is restricted as 2.0 cm as in the first design example.
In Table6, the minimum design weights and the max-imum constraint values of the 8-story, 1024-member spa-tial steel frame obtained by the BBO algorithm is compared to the results previously reported by Saka et al. [46,47] with different metaheuristic techniques. The BBO algo-rithm produces a design weight of 6462.79 kN for the spatial steel frame. Relatively high design weights have been attained for the structure with other metaheuristic algorithms: 6748.99 kN by AFFA, 6761.47 kN by ABC, 7210.12 kN by DHS, and 7689.51 kN by ACO. From these results, it can be concluded that the BBO algorithm shows a very favorable performance and produces a final design weight which is 4.24, 4.42, 10.37, and 15.95 % lighter than those produced by AFFA, ABC, DHS, and ACO, respec-tively. Such a significant difference between the results clearly indicates the effectiveness and robustness of the proposed BBO algorithm on finding optimum solutions of the spatial steel frames. Also, the optimum steel section designations for each member group of the frame obtained by BBO algorithm and those obtained by different meta-heuristic techniques previously reported by Saka et al. [46,
47] are given in Table7.
The design history of optimum design obtained from BBO algorithm is shown in Fig.9. The 8-story, 1024-member spatial steel frame is the most challenging design example of this study owing to the large number of design variables considered. It is noted that not only the strength ratio constraints but also the drift constraints are active for this example. The strength ratios of members in the optimum design are shown in Fig.10. Inter-story drifts along the global X-axis are given in Fig.11a. The lateral displacement values of the joints at top story of the spatial steel frame considered for this design example along the
6250 7500 8750 10000 11250 12500 0 15000 30000 45000 60000 75000
Feasible Best Design (kN)
Number of Analyses
BBO
Fig. 9 Optimization history of 8-story, 1024-member spatial steel frame using BBO
global X-axis are illustrated in Fig.11b. The inter-story drifts and the sway of the top story along the global Z-axis are not given because they are much smaller than the ones shown in Fig.11. It is apparent from these figures that the
both strength and serviceability constraints were dominant in the optimum design. The strength ratios of some mem-bers were at its upper bound of 1.0, and some others were close to its upper bound of 1.0 as well as the inter-story
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 128 256 384 512 640 768 896 1024 Strength rontraints ratio No. of member
Existing value Strength ratio limit Fig. 10 Strength ratios of
members in 8-story, 1024-member spatial steel frame 0 1 2 3 4 5 6 7 8 0.650 0.725 0.800 0.875 No. of story Inter-story Drift (cm)
Existing value Max. inter-story drift value
0 4 8 12 16 20 24 28 32 36 40 44 48 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
No. of top story joint
Top-story Drift (cm)
Existing value Max. top-story drift value
(a)
(b)
Fig. 11 Drift outlines of the 8-story, 1024-member spatial steel frame; a inter-story drift outline, b top-story drift outline
drift which has upper bound of 0.875. The top-story drifts located at the joints on the top story of the frame were having values close their upper bounds of 7.0.
5 Conclusions
It is shown in this study that biogeography-based opti-mization algorithm is an efficient tool to develop structural design optimization techniques. It can effectively select optimum steel profiles (in this case W sections) for the members of a steel space frame under multiple load cases when steel design code provisions are required to be sat-isfied in the design. Two real-sized spatial steel frame structures which are a 4-story, 428-member steel frame and an 8-story, 1024-member steel frame subjected to strength and serviceability limitations according to LRFD– AISC specifications are designed by the proposed algo-rithm. These examples, which have been already reported in the literature as to be designed by other metaheuristic algorithms such as an adaptive firefly algorithm (AFFA), a teaching and learning-based optimization (TLBO), an artificial bee colony (ABC) algorithm, a dynamic harmony search (DHS) algorithm, and an ant colony optimization (ACO), are used for comparison. It is interesting to notice that the BBO algorithm locates an optimum design which is lighter than the design weights obtained by the other techniques. The optimum design produced by BBO is 11.41, 11.89, 12.69, and 15.31 % lighter than those attained by TLBO, ABC, DHS, and ACO, respectively, in the first design example. In the second, the BBO algorithm also shows a very favorable performance and produces a final design weight which is 4.24, 4.42, 10.37, and 15.95 % lighter than those produced by AFFA, ABC, DHS, and ACO, respectively. The superiority of the BBO algorithm basically derives from its more effective operators, the so-called migration and mutation operators. The migration operator shows the movement of species among different habitats to presume on the search space, while the mutation operator leads habitats to avoid trapping in a local opti-mum. It is apparent from the numerical results that the BBO algorithm is a reliable and robust technique which can be freely used by designers as a choice to develop a design optimization algorithm for steel structures.
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