Design-driven harmony search (DDHS) in steel frame optimization

Engineering Structures 59 (2014) 798–808

Contents lists available at ScienceDirect

Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Design-driven harmony search (DDHS) in steel frame optimization Patrick Murren a, Kapil Khandelwal b,⇑ a b

University of Notre Dame, Notre Dame, IN 46556, United States Dept. of Civil & Env. Engg. & Earth Sci., University of Notre Dame, United States

a r t i c l e

i n f o

Article history: Received 4 December 2012 Revised 18 September 2013 Accepted 3 December 2013 Available online 31 December 2013 Keywords: Steel moment frames Optimization Harmony search Stochastic search Genetic algorithm Ant-colony optimization

a b s t r a c t Recent research efforts in the field of steel frame optimization have demonstrated the successful implementation of stochastic methods such as genetic algorithms, simulated annealing, ant colony optimization, particle swarm, and harmony search in obtaining least-weight designs. The stochastic nature of these methods is effective in scouring the large design space of discrete variables intrinsic to steel frame optimization problems. However as the randomness of these methods can limit the optimality of the frame designs obtained, the frequency with which these designs are obtained, and the computational effort required to obtain them, no one algorithm has distinguished itself as most effective. This paper presents a design-driven harmony search (DDHS) algorithm for optimization of steel moment frames. Based on the harmony search method, DDHS incorporates intelligence in the stochastic search DDHS by using constraint satisfaction or violation data from previous trial solutions to steer the optimization in the direction of larger or smaller sections as sorted by an appropriate section parameter. Thus, DDHS generates random trial solutions within intelligently specified search neighborhoods, resulting in improved performance in steel frame optimization as measured by the optimality of the designs obtained, the consistency with which successive runs obtain these designs, and the number of structural analyses required to obtain them. This paper demonstrates the efficient performance of DDHS as applied to three benchmark problems – a 15-member, a 105-member, and a 168-member planar frame optimization. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Optimization of steel frames is an important open problem that has been addressed by various researchers during the last few decades. The overall optimization objective is usually to minimize frame weight, subject to code-specified strength and drift constraints. The design variables are the member shapes obtained from steel tables provided in the steel construction manual. Thus, the design space is discrete, and the optimization can be executed by relaxing the design variables or working directly in the discrete variable spaces [4]. However, algorithms that operate directly on discrete design spaces are more useful in practical steel frame optimizations. Various metaheuristic algorithms for optimization of steel frames based on stochastic search methods that directly operate on discrete spaces have been proposed. Representative examples include: genetic algorithms (GA) [3,16,17,22–24]; ant colony optimization (ACO) [5,20]; tabu search [18]; simulated annealing [2,14]; harmony search [6,7,8,13]; teaching–learning based optimization [25] and hybrid algorithms [10,19], among others. These various algorithms generally function as a form of computational

⇑ Corresponding author. Tel.: +1 5746312655. E-mail address: [email protected] (K. Khandelwal). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.12.003

Darwinism – survival of the fittest solutions, the retention of desirable and shedding of undesirable solution components, and the breeding of optimal solutions from a bank of the fittest population. Two important components of metaheuristic search algorithms are diversification and intensification. The performance of the algorithm depends on the appropriate incorporation of these two counterbalancing components. Diversification ensures that the entire design variable domain is adequately searched. Weak diversification can result in poor exploration of the search space, leading to sub-optimal results. Strong diversification, on the other hand, can slow down the convergence. Conversely, intensification attempts to exploit the history and experience of the search process to rapidly converge to the best design [26]. Strong intensification results in quick convergence to local optima, while weak intensification may prevent an algorithm from properly converging. Algorithmic components such as parent crossovers and elitism (genetic algorithms), cooling schedules and the Boltzmann constant (simulated annealing), and pheromone matrix parameters (ant colony optimization) are examples of means by which wellestablished methods have implemented diversification and intensification. Harmony search uses the memory consideration rate and pitch adjustment rate to control both diversification and intensification, by generating and mutating solutions randomly or by generating and mutating solutions based on prior intelligence. All

799

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

the above algorithms are capable of obtaining optimal designs in steel frame optimization problems formulated on large, discrete variable spaces. However, no one algorithm stands out in the literature as clearly ascendant in terms of three key metrics – (1) optimality, as measured by the cost of the best design obtained; (2) robustness, as measured by the consistency with which optimal or near-optimal designs are obtained; and (3) efficiency, as measured by the amount of computational cost required to obtain an optimal design. This paper presents a new stochastic search algorithm for optimization of steel frames termed design-driven harmony search (DDHS). DDHS is derived from harmony search, using the convenient representation of best-existing solutions in the form of the harmony matrix as a basis for improving algorithmic performance by incorporating more intelligence into the mutation step of this stochastic search method. This is primarily achieved through: (a) integrating member design information available from previous structural analyses and design steps into the stochastic search, and (b) incorporating parameter-specific search neighborhoods into the stochastic search. These two modifications result in a more intelligent direction of the stochastic search, a conversion of the mutation step from a random diversification process on a local variable space to a more directed intensification process. Quantitatively, this results in improved optimality of designs obtained, algorithmic robustness, and computational efficiency when compared to other stochastic algorithms implemented in the same benchmark optimizations. The effectiveness of the DDHS algorithm is verified through its application to benchmark 15-member, 168member and 105-member planar steel frames previously studied using harmony search, GA, ACO, and hybrid optimization methods. The paper is organized as follows: Section 2 describes the discrete design optimization problem for steel frame optimization. In Section 3, the harmony search optimization algorithm is presented. Section 5 describes the proposed design-driven harmony search algorithm, and its application to the benchmark problems is presented in Section 6. Finally, the important conclusions of this study are summarized in Section 7.

violation, and pd = penalty coefficient for drift violation. In the unconstrained formulation Eq. (2), the penalty coefficients pS > 0 and pd > 0 can be independently specified to penalize designs that violate strength and drift requirements, respectively. The strength constraint violation function, Cs(x), is defined using AISC-LRFD [1] design specifications as follows:

8   > < /PPu n þ 89 /MMuxnx þ /MMuyny c b b   DCRi ðxÞ ¼ > : 2/PuP þ /MMux þ /MMuy nx ny c n b

g i ðxÞ ¼ DCRi ðxÞ  1 ai ðxÞ ¼

C s ðxÞ ¼

mg nk X X min WðxÞ ¼ q Ak ðxÞ Li x

k¼1

ð1Þ

i¼1

subjected to design constraints of AISC-LRFD [1] and drift constraints. In Eq. (1), x is the design vector of W-shapes selected from steel tables (consisting of integer shape IDs, each of which corresponds to a wide-flange shape), mg is the total number of member groups in the frame design, nk is the number of members in group k with area Ak, Li is the member length, and q is the weight density of steel (equal to 76.9 kN/m3). Note that each member in a member group has the same assigned W-shape. The design examples presented in this study are taken from Degertekin [7] and Kaveh and Talatahari [19]. For the sake of comparison, the same design constraints as presented in these papers are used in this study. The design is subjected to two constraints: (1) AISC-LRFD [1] beam/ column design strength constraints and (2) maximum allowable inter-story drift constraints. To incorporate these constraints, the original constrained optimization formulation Eq. (1) is penalized and an unconstrained formulation is used as follows:

min FðxÞ ¼ WðxÞð1 þ ps C s ðxÞ þ pd C d ðxÞÞ x

ð2Þ

where Cs(x) = strength constraint violation function, Cd(x) = drift constraint violation function, ps = penalty coefficient for strength

0

< 0:2

ð3Þ

g i ðxÞ 6 0

g i ðxÞ g i ðxÞ > 0

nele X

ai ðxÞ

ð4Þ

In the above definition of the constrained violation function Cs(x) (Eq. (4)), DCRi(x) is the demand-to-capacity ratio of the ith member that results from the frame design specified in x. Similarly, all force values refer to the demands on the ith member that result from the frame design specified in x: Pu = required axial strength (compression or tension); Pn = nominal axial strength (compression or tension); Mux = required flexural strength about the major axis; Muy = required flexural strength about the minor axis; Mnx = nominal flexural strength about the major axis; Mny = nominal flexural strength about the minor axis; /c = axial resistance factor (compression 0.85, tension 0.90); /b = flexural resistance factor (equal to 0.9); and nele = number of beam and column members in frame. The out-of-plane effective length factors, Ky, for the columns are set equal to 1.0. The in-plane effective length factors, Kx, are obtained using the approximate method proposed by Dumonteil [9]. The beam lateral–torsional buckling modification factor, Cb, is conservatively assumed to be 1.0. The drift constraint violation function, Cd(x), is defined as follows:

2. Optimization problem

!



P 0:2

Pu /c P n

i¼1

hi ðxÞ ¼

The discrete design optimization problem considered in this study is as follows:

b

Pu /c P n

C d ðxÞ ¼

di ðxÞ  1 bi ðxÞ ¼ da ns X



0

hi ðxÞ  0

hi ðxÞ hi ðxÞ > 0

bi ðxÞ

ð5Þ

i¼1

where di(x) = story drift at ith floor, da = allowable story drift, and ns = number of stories. The steel frame design optimization problem presented above is a discrete optimization problem, as the design variable x is chosen from a list W-shapes available in steel tables. Therefore, both the objective and the constraint functions are non-differentiable and non-convex, and stochastic methods are typically used for solving such problems [21]. Moreover, the objective and the constraint functions can be evaluated either using linear structural analysis or via nonlinear structural analyses. In this study, only linear static analysis is used to evaluate the objective and constraint functions; however, the presented harmony search method can also be used together with nonlinear structural analyses as shown in Murren [21]. 3. Harmony search (HS) method Harmony search (HS) – the name derived from the practice of disparate musicians contributing various notes to produce an optimal harmony – is a stochastic algorithm that uses a bank of bestexisting solutions to generate new trial solutions in an effort to heuristically search large design variable spaces. In this section the important steps in the harmony search method as applied to

800

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

optimization of steel frames are summarized. Further details of HS implementation can be found in Degertekin [7] and Geem [11].

as a new design vector and the harmony matrix is again re-sorted from most to least optimal. No repetition of design vectors is allowed in the HM matrix.

3.1. Step 1: Initialize the harmony matrix (HM) 3.4. Step 4: Convergence Harmony search begins with the generation of random design vectors (harmonies) from the discrete variable space and storing them in the harmony matrix (HM). The number of designs stored is the harmony memory size (HMS). HMS is an algorithmic parameter that depends on the size of the optimization problem under consideration, generally increasing with problem complexity. The goal of the harmony matrix is to provide a bank of the most optimal designs known, to be used as a basis in generating new trial designs. The matrix in Eq. (6) below depicts how the harmony design vectors are arranged.

2

x1;1

6 6 6 x2;1 HM ¼ 6 6 . 6 .. 4 xhms;1

x1;2 x2;2 .. . xhms;2

... .. . .. .

x1;ng x2;ng .. .

. . . xhms;ng

3 7 7 7 7 7 7 5

!

Fðx1 Þ

! Fðx2 Þ .. .. . . ! Fðxhms Þ

ð6Þ

Fðx1 Þ 6 Fðx2 Þ 6    6 Fðxhms Þ In the harmony matrix, a typical entry xi,j represents the selected section for members in the member group j in the ith design, with ng being the total number of member groups in the frame, and F(xi) is the fitness of design vector xi evaluated using Eq. (2). In this study, the HM is initialized by randomly generating 2  HMS initial design vectors and then retaining the top HMS designs in the harmony matrix. This is done to increase the likelihood of beginning the optimization with feasible designs, with the aim of improving the rate of convergence. 3.2. Step 2: Improvise a new harmony In this step a new trial design vector (harmony) is generated (improvised) using the available HM. There are two possibilities for generating a new design: (a) A new trial design, xnew, is randomly generated from the available discrete variable design space with probability 1  MCR, where MCR is the memory consideration rate. (b) With probability equal to the memory consideration rate (MCR), a new trial design xnew, is generated using randomly selected components from the HM. After the trial solution is generated, its components are mutated with probability equal to the pitch adjustment rate (PAR). In a harmony search algorithm if a solution component is selected for mutation, the designated discrete variable value is altered in a random search direction by step size of 1. However, for steel frame optimization Degertekin [7] suggested a step size of 3 for better performance. MCR values typically fall within the range of 0.7–0.95 [12], indicating that 70–95% of the time the trial solution will be generated from the harmony matrix. PAR values are typically in the range 0.3–0.7 [12], indicating that 30–70% of a trial solution’s components will be mutated on average. 3.3. Step 3: Evaluate the new harmony Once generated and mutated, the trial design vector is checked for fitness through evaluation of the objective function. If the trial design vector is found to be more optimal than the least optimal design in the harmony matrix, then the trial design takes its place

Steps 2 and 3 are repeated, and the search is terminated when either: (a) the optimality of the best existing solution is not improved after a specified number of iterations; or (b) a finite number of overall iterations is achieved. 4. Performance of harmony search method The size of the harmony matrix (HMS), the memory consideration rate (MCR), and the pitch adjustment rate (PAR) play an important role in determining the effectiveness of a harmony search optimization. MCR and PAR control the diversification and intensification in the HS method. For obtaining the best performance, these parameters should be carefully specified. For better performance, an adaptive strategy wherein MCR is slowly increased as the algorithm proceeds can be adopted [12]. An adaptive method for dynamically varying MCR and PAR was also proposed by Hasanebi et al. [15]. As mentioned earlier, HMS is a subjective parameter that should be specified with consideration of the optimization problem size. For the examples presented, sensitivity studies and engineering judgment were used to specify reasonable values for HMS, MCR, and PAR. As there are no objective guidelines for which harmony search parameters consistently produce the best results, this study focuses primarily on the algorithm and not on deriving optimal values of HMS, MCR, and PAR. 5. Design-driven harmony search (DDHS) One of the important diversification/intensification steps in harmony search is the harmony mutation. Clearly, three important decisions in the mutation step are the search direction (up or down the sorted W-shapes table), the step size, and the neighborhood in which the mutation is conducted (i.e. the parameter by which discrete sections are sorted). In classic harmony search, trial solution component mutation is a purely stochastic process – mutation occurs at a rate equal to the PAR by a specified number of discrete steps in a randomized direction. Though harmony search is an effective algorithm, the efficiency and accuracy of the method can be improved substantially by mutating trial solutions intelligently based on the structural response data of the previously-analyzed solutions stored in the harmony matrix. More specifically, this data can be used to determine: (1) the logical search direction (toward larger or smaller steel sections); (2) the logical search neighborhood; and (3) the appropriate magnitude of the mutation for the trial solution components. These, along with a handful of other modifications, are incorporated into a design-driven harmony search (DDHS) algorithm, presented in this paper to produce lighter steel frame designs (high accuracy), produce those designs frequently (high robustness), and decrease computational time by requiring fewer structural analyses (high efficiency). The specifics of the DDHS are described in Sections 5.1–5.5. 5.1. Use of design data to intelligently search variable space Two important decisions in the mutation step are the search direction and the step size. DDHS employs an improved method to choose the search direction and step size, based on the information available from design feasibility results. To this end, existing constraint evaluation data, specifically member demand-to-capacity (DCR) values, are retained after each iteration. Then, when

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

harmony matrix components are chosen for trial design vectors, the DCR data is used to determine whether a particular member is overdesigned, under-designed, or close to optimal. Of course, this is not an exact representation of a variable’s fitness, as a member DCR cannot be isolated from a global trial design vector; however, the DCR provides a satisfactorily accurate gauge of a member’s fitness for use. As an anecdotal example, one can readily deduce that a W14  22 section would not be appropriate when specified as a column at the ground level of a high-rise steel frame structure. A purely stochastic algorithm, especially one that operates on one design vector (ex., simulated annealing) or a bank of closely related design vectors (ex., genetic algorithms) may take a significant amount of computation before improving upon or surpassing and discarding such a design. Storing the DCR data of each solution after the structural analysis allows DDHS to more easily obtain more optimal designs not only for vectors with such badly specified shapes in the anecdotal example but also for those with more appropriate yet sub-optimal shapes. Mutation in DDHS for a member group is based on the maximum demand-to-capacity ratio (DCRmax,k) in that group, where DCRmax,k is the maximum value of demand-to-capacity ratio in member group k. Then, if DCRmax,k P (1 + dcrtol), with dcrtol specified as 0.1, the section is deemed under-designed. Such sections are automatically mutated in the direction of larger, stronger sections. Based on the extent of the DCR violation, the under-designed sections are mutated probabilistically within a specified step range given by Eqs. (6) and (7); where Sr is the mutation range, DS is a random integer in the interval [1, Sr], and dcrinc = 0.05. Also, the function ceil(x) rounds the x to the nearest integer greater than or equal to x, and the function randi(N) generates a random integer between 1 and N. Hence by using Eqs. (7) and (8), the weakest sections are mutated by a larger magnitude; slightly infeasible sections are mutated by a smaller magnitude. An efficient section selection scheme is also incorporated for this case: if a section lighter than the one to which the algorithm initially mutated, yet further down the list of W-shapes as sorted by the appropriate parameter, is identified, this lighter section is instead chosen. Thus, the probabilistic mutation component of algorithm retains the benefits of stochastic solution diversification, while the intelligent specification of the step direction and range helps the algorithm to efficiently and appropriately intensify.

801

search, followed by a gradual refinement toward lighter, more optimal solutions. Therefore, the search direction is deterministically obtained, while the step size is probabilistically determined when DCRmax,k P (1 + dcrtol), or DCRmax,k 6 (1  dcrtol). Provisions, described in section 5.3, are made for those member-groups whose (1  dcrtol) 6 DCRmax,k 6 (1 + dcrtol). 5.2. Defining neighborhoods of discrete variables

  jDCRmax;k  1:0j Sr ¼ ceil dcrinc

ð7Þ

DS ¼ randiðSr Þ

ð8Þ

In addition to using the DCR data of harmony vectors to direct trial solution mutation, DDHS also incorporates intelligence into the manner in which the variable space is arranged in the mutation step, creating logical ‘‘search neighborhoods’’ for solution components. In steel frame optimization it is not always clear what type of mutation neighborhoods should be employed. For instance, W-shapes can be pre-sorted in ascending (or descending) order by any one of the following fields: weight per unit length, strong axis plastic modulus (Zx), strong axis moment of inertia (Ix), or weak axis radius of gyration (ry). The search neighborhood will then depend on the initial sorting of the W-shapes. For the purpose of following discussion, the search neighborhoods are classified as NZ, NI and Nr, based on the initial sorting of the W-shapes in ascending order by Zx, Ix, or ry, respectively. Fig. 1 indicates the importance of identifying a proper neighborhood, as the discrete shapes surrounding a particular section can vary greatly depending on which parameter the sorting is based. The ability of the DDHS algorithm to assign the logical neighborhood of a discrete trial design vector component and mutate within that neighborhood is displayed in Fig. 2. If for a membergroup DCRmax,k > 1.1, its mutation is classified as strength-based, and the mutation for this member-group is carried out in NZ or Nr neighborhoods. Given the different nature of their critical design demands, beams and columns are mutated in different manners. The primary difference between the two is the capability of columns to be mutated with regard to their weak-axis radius of gyration, a buckling consideration. Beams requiring a strength-based mutation are automatically deemed as flexural-critical and mutated in NZ neighborhoods. For columns requiring a strength-based mutation, the axial and flexural components of the DCR data are analyzed to see if the member is dominated by its axial or bending capacity. If axial-dominated, the column is mutated in Nr neighborhood. If bending-dominated, it is mutated in NZ neighborhood. If for a member-group DCRmax,k 6 0.9, its mutation is classified as strength-based or drift-based. Since these members are adequately designed, there is no convenient way to determine whether drift or strength is the governing constraint. To

If DCRmax,k 6 (1.0  dcrtol), where dcrtol = 0.1, the section is deemed overdesigned and marked for mutation in the direction of smaller, lighter sections. The mutation of shapes toward lighter, more efficient sections is a solution refinement. In order to avoid mutating too coarsely, the magnitude of the mutation of overdesigned sections is considerably smaller than that for under-designed sections. Mutation step size in this case is given by Eq. (8), with Sr = 4 for columns and Sr = 6 for beams. Furthermore, this mutation is done by creating a member list with lower weight in the neighborhood of the original W-shape to be mutated, and sorting this list by weight in descending order. In DDHS implementation, this list is generated from 20 members above and 10 members below the current Wshape to be mutated. A member is then selected from this list based on the integer index given by Eq. (7). It should be noted that DDHS parameters such as dcrtol, dcrinc, and Sr, though subjectively specified, were determined based on the best observed performance of trying different values. The result of this mutation scheme is a general trend of quickly developing heavy, yet feasible solutions in the initial part of the

Fig. 1. Variation in neighborhoods for a W18  60 section, when sorted by Zx, ry, and Ix.

802

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

5.5. Significance of mutation scheme Mutation neighborhoods

Beam

Column

Strength based

Drift based

Strength based

Drift based

Mutate in Zx - neighborhood

Mutate in Ix - neighborhood

Mutate in Z x or ry neighborhood

Mutate in Ix - neighborhood

Fig. 2. Neighborhood definition for mutations in DDHS.

accommodate this mutation in DDHS, another algorithmic parameter termed drift mutation rate (DMR) is introduced. Then if DCRmax,k 6 0.9, the mutation is classified as drift-based with probability equal to the drift mutation rate (DMR), otherwise the mutation is strength-based. Strength-based mutations are performed in NZ or Nr neighborhoods, as discussed earlier. For both beams and columns, drift-based mutations are performed in NI neighborhoods. For drift-based mutations, the search direction is randomly chosen and the mutation step size in this case is given by Eq. (8), with Sr = 2 for both beam and column members. 5.3. Island hopping Once a member-group’s critical DCR approaches the optimum value of 1.0 the goal is to refine the design vectors by searching the immediately surrounding design space. An ‘‘island hopping’’ refinement is implemented for near optimum member-groups, designated as those member-groups with (1  dcrtol) 6 DCRmax,k 6 (1 + dcrtol), where dcrtol = 0.1. Member-groups whose critical DCRs fall between 0.9 and 1.1 are mutated in a random search direction with a small mutation step size, given by Eq. (8) with Sr = 2 for both beams and columns. This is important because the DCRs used to direct the mutation are not the exact measure of member fitness, as member DCRs cannot be isolated from the global trial vector. The random search direction of the island hopping scheme allows the mutation to occur in either direction, to correct any inaccuracies that may result from using prior axial–flexural interaction values to approximate the expected DCRs. This refinement also ensures that near-optimum solution components are capable of identifying globally optimum solutions, even if that requires mutating toward heavier sections.

The mutation schemes described in Sections 5.1–5.4 constitute the innovations of DDHS. By using existing structural response data to more intelligently define the appropriate neighborhood, direction, and magnitude of mutations, the mutation step converts from a stochastic, local diversification step to a guided, intensification step. To illustrate this point, consider the sample mutations presented in Figs. 3–5. The first of these figures presents the mutation of an over-designed section, where DCRmax,k 6 (1.0  dcrtol). These figures present the prospective DCRs for a particular section’s neighboring shapes, given the specified axial and flexural demands. The highlighted sections indicate the possible shapes to which the current frame member can be mutated. Using DDHS, if the mutation is designated as strength-based, one of four feasible sections, all of which are lighter than the existing W24  146 shape, will be selected. This intelligent mutation is significant, as DDHS in this instance will easily identify a more optimal section in the mutation step. If the DMR specifies a drift-based mutation using DDHS, a broader range of sections is available, allowing the algorithm to thoroughly explore the variable space. The standard harmony search mutation scheme, which mutates trial solution components randomly with step size ±1, would be able to identify feasible shapes but only have a 50% chance of mutating toward a more efficient section. In fact, standard HS would also have a 50% chance of mutating toward a significantly heavier section, thus decreasing optimality and/or slowing algorithmic convergence. The mutation of an under-designed section, where DCRmax,k P (1.0  dcrtol), is presented in Fig. 4. Here, standard harmony search again would be able to mutate to a broad range of sections, some feasible by strength and some not. Conversely, DDHS would direct the mutation toward one of four larger, yet feasible, sections based on the mutation scheme described in Section 5.1. Standard harmony search would only have a 50% chance of mutating to a feasible solution, thus delaying the identification of satisfactory designs and slowing convergence. Note that a drift-based mutation for DDHS is not permitted for under-designed sections. A sample island hopping mutation, for a section in which (1  dcrtol) 6 DCRmax,k 6 (1 + dcrtol), is presented in Fig. 5. In this case, DDHS and standard harmony search are practically identical, as both schemes mutate in a random direction. As described in

5.4. Duplicate trial solutions and adaptive MCR modifications In a structural optimization, the bulk of the computing time is expended in the evaluation of the objective function, in this case the structural analysis. The efficiency of the algorithm can be increased simply by storing all trial solutions that have been previously evaluated and preventing the repeat evaluation of a vector whose fitness is already known. This modification is implemented in DDHS. Also, as a harmony search optimization converges toward an optimum, it becomes less likely that a randomly generated solution will be fit enough to be stored in the harmony matrix. In this work, the random generation of trial solutions is closed off by setting the MCR equal to unity once the fitness of the lowest vector in the harmony matrix is within 8% of the fitness of the best harmony vector.

Fig. 3. Comparison of mutation possibilities for standard HS and DDHS for an overdesigned W24  146 column section.

803

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

Fig. 4. Comparison of mutation possibilities for standard HS and DDHS for an under-designed W30  90 section.

optimized using genetic algorithms (GA) [22], ant colony optimization (ACO) [5], an improved ant colony method (IACO) [20], harmony search method (HS) [7], and a hybrid algorithm of harmony search, particle swarm, and ant colony optimization (HPSACO) [19]. DDHS parameters, frame properties, loading, constraints, and design space are described in Table 1. For each of these optimizations using DDHS, the MCR was specified as 0.8 and the PAR was specified as 0.4. Two termination criteria were used: (1) the optimization was terminated if a maximum number of iterations (Nmax) is reached; or (2) the optimization was terminated if no new trial design was produced in a specified number (Nnew) of generations. Nnew is specified as 20% of Nmax. To evaluate the accuracy, robustness, and efficiency of the DDHS algorithm, optimization for each of the frames was conducted 100 times. Because of the stochastic nature of the algorithm, the global optimality of any obtained solution cannot be verified without an exhaustive search of the design space. Unless explicitly stated, the global optimality of the designs presented cannot be proven. The term ‘‘optimal design’’ refers to the least expensive design obtained for a particular example problem. Finite element analysis and design of frames is carried out in MatlabÒ computing environment on a personal computer with a Pentium 3.0 GHz microprocessor. The feasibility of optimum designs obtained using DDHS in this study were re-checked with SAP2000 v14.2.4 structural analysis program to confirm that all design strength requirements are satisfied. The MatlabÒ and SAP2000 analyses were consistent with the design code cited for each particular frame example. In the comparative studies presented in this paper, due diligence is taken to ensure that all constraints and problem data are equivalent to those stated in each particular benchmark study. All other design assumptions are explicitly stated. However, in investigating the optima reported in prior benchmark studies, some of these designs did not satisfy the strength and/or drift constraints. It is possible that other assumptions were used in benchmark studies, though not explicitly stated, that allowed prior designs – found to be infeasible here – to satisfy all design constraints.

6.1. Example 1: Two-bay three-story (2  3) frame The first example frame is the two-bay by three-story steel structure shown in Fig. 6. Frame members were split into two groups (ng = 2), beams and columns, and therefore, there were two variables in the problem. Optimization results are presented in Table 2. Note that for simplicity, all weights are presented as the sum of the products of member-group length multiplied by nominal member lineal weight. Average computing time for each optimization run was approximately 2 s. From Table 2 it can be

Fig. 5. Comparison of mutation possibilities for standard HS and DDHS in an island hopping scheme with a W12  179 section.

Section 5.4, this stochastic, less ‘‘intelligent’’ mutation mechanism in DDHS can account for slight discrepancies between stored DCRs and actual DCRs of the trial design vector. The island hopping scheme presented in Fig. 5 demonstrates that even though DDHS can intelligently prescribe effective mutations, a degree of randomness is still necessary for this heuristic algorithm to perform optimally. 6. Design examples The application of the DDHS algorithm to three steel frame optimization problems is presented here. These frames were earlier

Table 1 Design summary and optimization parameters. Design space

DDHS parameters

Loads/constraints

Example 1 Beam bracing: L/6 Beam: all WF-shapes Columns: W10

HMS = 25 ps = 10 Nmax = 2000

Fig. 6

HMS = 50 [ps, pd] = [4, 2] Nmax = 20,000

da = H/300 Fig. 11

HMS = 40 [ps, pd] = [8, 4] Nmax = 10,000

da = H/300 Fig. 15

Example 2 Beam: all WF-shapes Columns: W14 Example 3 Beam bracing: L/5 Beam and columns: all WF-shapes

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

seen that in each of the 100 simulations, the DDHS obtained the globally optimum frame with W24  62 beams, W10  60 columns, and a total weight of 83.6 kN (18,792 lb). For the best case this result was obtained in 8 iterations and an average of about 200 iterations are required to reach the global optimum. The global optimality of this solution was demonstrated by Pezeshk et al. [22], who performed an exhaustive search of all possible design combinations for this frame and concluded that the W24  62, W10  60 combination is the global optimum. It is important to note that the solution obtained by Degertekin [7] using HS is slightly infeasible within the specified constraints. The optimized design obtained from the DDHS algorithm is compared against those designs obtained previously using genetic algorithms (GA), ant colony optimization (ACO), and harmony search (HS) in Table 3. The GA obtained this optimum 5 times out of 30 simulations. The ACO obtained this design in 84% of its 100+ simulations. The HS reported the best of 30 simulations without indicating how many times this result was repeated. GA obtained the optimum after around 780 iterations, with 2200 iterations for ACO and 853 iterations for HS. Therefore, for this small frame example, DDHS demonstrates significantly improved robustness and efficiency over the methods in the benchmark studies.

Table 2 Summary of 100 optimizations runs for the 2  3 frame. Opt. wt. Avg. opt. wt. St. dev.

83.6 kN 83.6 kN 0 kN

No. of times opt. obtained Min. no. of iterations to opt. wt.

100/100 8

Table 3 Results comparison for the 2  3 frame.

a

2  3 Frame

Degertekin [6] – HS

DDHS/ACO/GA

Beam Column

W21  62 W10  54

W24  62 W10  60

Wt. (kN) Wt. (lb)

81.4a 18,292a

83.6 18,792

Reported design is infeasible.

w1

w1

2

4

w2

w3

1

3

P = 25.6 kN @ each floor

6.2. Example 2a: Three-bay twenty four-story (3  24) frame The second example frame is a three-bay by twenty-four-story steel structure shown in Fig. 7. Frame members were divided into twenty groups (ng = 20), and therefore there were twenty variables in the problem. Optimization results and the convergence history for the best and average designs for 100 optimization runs are shown in Table 4 and Fig. 8. The average computing time for each iteration was 20.5 min. In this example, DDHS obtained a best frame weight of 913.6 kN (205,386 lb) and average optimum weight of 921.4 kN (207,140 lb). The minimum optimum design was repeated only in 1 out of 100 iterations. The standard deviation of the optimum weights was 3.75 kN (843 lb). For the best case this result was obtained in 17,395 iterations. Figs. 9 and 10 show that both drift and DCR constraints are satisfied by the best optimum design for this frame. Drift values close to 1.0 indicate that the frame design is drift-dominated. The optimized designs obtained from DDHS, ACO, IACO, and HS are compared in Table 5. DDHS was able to obtain a lighter frame than ACO, IACO, and HS, by 6.8%, 5.7%, and 4.4% respectively. ACO obtained the optimum at 12,500 iterations. HS reached the optimum after 9924 iterations. Though DDHS required over 17,000 iterations to obtain its optimal design, Table 6 and Fig. 8 indicate how the algorithm was able to match the best results of the other algorithms very rapidly (within 3500 iterations). For the best optimum solution by DDHS, the large

w = 40.9 kN/m

w1 20 20

1

3 @ 3.05 m

22.2 kN 22.2 kN

2 @ 10.98 m Fig. 6. First example – 2  3 frame.

1

12 12

20

12

19

11

19

11

19

11

18

10

18

10

18

10

17

9

17

9

17

9

16

8

16

8

16

8

15

7

15

7

15

7

14

6

14

6

w3 = 6.92 kN/m

14

6

w4 = 5.95 kN/m

13

5

w3 3

13 13

6.10 m 3.66 m

11.12 kN

2

w4

Fy = 230 Mpa E = 205 GPa

w2

Fy = 248 MPa E = 200 GPa

18,792 lb 18,792 lb 0 lb

24 @ 3.66 m

804

w4 1

w1 = 4.38 kN/m w2 = 6.36 kN/m

5 5

8.53 m

Fig. 7. Second example – 3  24 frame.

majority of iterations in this large frame optimization occurred as the solution slowly refined itself. In this frame, DDHS demonstrated improved accuracy and efficiency over previous efforts. Though DDHS could only repeat its most optimal design only once out of 100 efforts, the standard deviation of the best designs in each simulation was less than 0.5% of the average optimum weight obtained.

805

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808 Table 4 Summary of 100 optimizations runs for the 3  24 frame. Opt. wt. Avg. opt. wt. St. dev.

913.6 kN 921.4 kN 3.75 kN

No. of times opt. obtained Min. no. of iterations to opt. wt.

1/100 17,395

Table 5 Results comparison for the 3  24 frame. 205,386 lb 207,140 lb 843 lb

Frame Weight (kN)

2900

2400

Average Optimum Weight 1900

Best Run Optimum Weight 1400

900

0

1000

2000

3000

4000

5000

Iteration Fig. 8. Convergence history of best and average solutions for the 3  24 frame.

Group No.

Camp et al. [5] – ACO

Degertekin [6] – HS

Kaveh and Talatahari [20] – IACO

DDHS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

W30  90 W8  18 W24  55 W8  21 W14  145 W14  132 W14  132 W14  132 W14  68 W14  53 W14  43 W14  43 W14  145 W14  145 W14  120 W14  90 W14  90 W14  61 W14  30 W14  26

W30  90 W10  22 W18  40 W12  16 W14  176 W14  176 W14  132 W14  109 W14  82 W14  74 W14  34 W14  22 W14  145 W14  132 W14  109 W14  82 W14  61 W14  48 W14  30 W14  22

W30  99 W16  26 W18  35 W14  22 W14  145 W14  132 W14  120 W14  109 W14  48 W14  48 W14  34 W14  30 W14  159 W14  120 W14  109 W14  99 W14  82 W14  53 W14  38 W14  26

W30  90 W12  26 W24  55 W6  8.5 W14  159 W14  109 W14  109 W14  74 W14  68 W14  48 W14  43 W14  26 W14  99 W14  109 W14  109 W14  99 W14  74 W14  61 W14  34 W14  22

Wt. (kN) Av. Wt. (kN) Wt. (lb) Av. wt. (lb)

980.7 1021.1 220,465 229,555

955.7 990.2 214,860 222,620

968.9 Not given 217,812 Not given

913.6 921.4 205,386 207,140

Interstory Drift Ratio

0.55

H / 300 Limits 0.5

Table 6 Average convergence history for the 3  24 frame.

0.45 Iteration number

0.4

DDHS-3 0.35 0.3 0

4

8

12

16

20

24

Story

1000 2000 3500 5000 10,000 15,000 20,000

DDHS average weight kN

lb

1067.1 981.2 950.3 940.6 928.5 924.1 921.4

239,890 220,590 213,620 211,450 208,730 207,760 207,140

Fig. 9. Inter-story drifts for the optimum 3  24 frame design obtained using DDHS.

Member DCRs

1

Table 7 Summary of 100 optimizations for the 3  24 frame with expanded column space.

0.8 0.6

Opt. wt. Avg. opt. wt. St. dev.

815.9 kN 851.6 kN 17.7 kN

0.4

No. of times opt. obtained Min. no. of iterations to opt. wt.

1/100 18,760

183,408 lb 191,440 lb 3,970 lb

0.2 0

24

48

72

96

120

144

168

Members Fig. 10. DCRs for the optimum 3  24 frame design obtained using DDHS.

6.3. Example 2b: Three-bay twenty four-story (3  24) with expanded column space In order to evaluate DDHS in larger optimization problems, the second example frame optimization is repeated here with a column variable space expanded to include not just the W14 sections, but all 267 shapes listed in the AISC steel construction manual [1]. The results of 100 optimization runs are presented in Table 7. The

best design obtained using DDHS weighed 815.9 kN (183,408 lb) with an average best weight of 851.6 kN (191,440 lb) and standard deviation of 17.7 kN. DDHS was only able to repeat this optimum once out of 100 runs. The convergence history, inter-story drifts, and DCRs of the best frame obtained using DDHS appear in Figs. 11–13, respectively. A comparison of the best design obtained using DDHS and the best design obtained using IACO is presented in Table 8. DDHS improved on the best design obtained by IACO by 7.9%. IACO required 3500 iterations to converge to an optimum. The optimality of the average DDHS weight surpassed that of the best design obtained with IACO in 12,000 iterations (Table 9). In IACO the search space is continuously reduced with iterations, and this lead to better intensification but poor diversification. As

806

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808 Table 8 Results comparison for the 3  24 frame with expanded column space.

Frame Weight (kN)

2800 Average Optimum Weight

2300 1800

Best Run Optimum Weight 1300 800

0

5000

10000

15000

20000

Iteration Fig. 11. Convergence history of best and average solutions for the 3  24 frame with expanded column space.

Interstory Drift Ratio

0.65

H / 300 Limits 0.55 0.45 0.35

Kaveh and Talatahari [20] – IACO

DDHS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

W30  99 W10  33 W18  35 W16  31 W36  170 W30  116 W30  116 W24  62 W24  62 W18  60 W16  36 W10  33 W24  76 W14  74 W24  62 W24  62 W18  46 W18  46 W18  35 W16  31

W27  84 W21  50 W21  44 W21  44 W30  116 W33  118 W30  90 W24  68 W24  76 W21  48 W16  40 W16  26 W30  99 W24  76 W24  76 W24  68 W21  48 W21  44 W16  31 W14  26

Wt. (kN) Av. wt. (kN) Wt. (lb) Av. wt. (lb)

886.0 916.9 199,176 206,119

815.9 851.6 183,408 191,440

DDHS 0.25 0.15 1

3

5

7

9

11 13 15 17 19 21 23

Story Fig. 12. Inter-story drifts for the optimum 3  24 frame (expanded column space) obtained using DDHS.

1 0.9

Optimum DCRs

Group No.

0.8 0.7 0.6 0.5 0.4 0

24

48

72

96

120

144

imposed. Optimization results and convergence history for the best optimum and average of optimum designs for 100 optimization runs is presented in Table 10 and Fig. 15. Average computing time for each optimization run was 7.2 min. For this frame the DDHS produced best frame with weight of 401.9 kN (90,340 lb) and average optimum weight of 411.2 kN (92,439 lb). The best optimum design was repeated only in 1 out of 100 iterations. The standard deviation of the optimum weights is 5.35kN (1204 lb). For the best case this result was obtained in 7518 iterations. Fig. 16 and 17 show that both drift and DCR constraints are satisfied by the best optimum design for this frame. DCR values close to 1.0 indicate that the frame design is primarily strength-dominated. A comparison of the optimized design obtained from DDHS and the HPSACO is presented in Table 11. The weight of the best design obtained by HPSACO was 426.4 kN (95,850 lb), produced in 6800 iterations. Therefore, there was about 6.1% improvement in the frame material weight using DDHS method. From Table 12, it can be seen that the average DDHS run improved upon the solution obtained by HPSACO within 5000 iterations. Thus, for this frame the DDHS algorithm produced better solutions than HSSACO with improved computational cost.

168

Member Fig. 13. DCRs for the optimum 3  24 frame design (expanded column space) obtained using DDHS.

a result quick convergence is achieved in IACO, however, with suboptimal performance. 6.4. Example 3: Three-bay fifteen-story (3  15) frame The third example frame is a three-bay by fifteen-story steel frame shown in Fig. 14. This frame was previously optimized by Kaveh and Talatahari [19] using hybrid harmony-search/particleswarm/ant-colony (HPSACO) optimization. Frame members are assigned to eleven groups ng = 11 as shown in Fig. 14. An additional constraint that limits the top story drift to 23.5 cm (9.25 in.) is also

7. Conclusions 1. In four examples investigated in this paper, DDHS performed effectively. For the 2  3 frame, the global optimum was consistently obtained with reduction in computational cost compared with benchmark studies. For the two larger 3  24 and 3  15 frames, DDHS produced better optimal designs than those obtained earlier. Though the global optimality of the DDHSobtained designs presented in this study is impossible to prove without an exhaustive search of all possible design combinations, these designs are more optimal than those feasible designs presented in prior benchmark studies. 2. The results demonstrate the broad applicability of the DDHS optimization method. DDHS produced results with comparably good accuracy, high robustness, and low computational cost for large and small problems, for drift and strength-dominated designs. It is important to note that no attempts were made to obtain best possible combinations of various algorithmic

807

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

600

Iteration number

DDHS average weight

1000 2000 5000 10,000 12,000 15,000 20,000

kN

lb

1248 1108 972 903 886 873 854

280,480 249,030 218,390 203,000 199,120 196,420 191,440

Frame Weight (kN)

Table 9 Average convergence history for the 3  24 frame with expanded column space.

Average Optimum Weight 550 500

Best Run Optimum Weight

450 400

0

2000

4000

6000

8000

10000

Iteration Fig. 15. Convergence history of best and average solutions for the 3  15 frame.

w = 50 kN/m

0.6

8

Interstory Drift Ratio

P = 30 kN at each floor

10

H / 300 Limits

9

7

0.5 0.4 0.3

Design Story Drifts 0.2 0.1 0

5

6

9

12

15

Story

14 @ 3.5 m

6

3

Fig. 16. Inter-story drift ratios for the 3  15 frame design obtained using DDHS.

Fy = 248 Mpa 1

E = 200 GPa

2

3

1

Member DCRs

4

4m

0.8 0.6 0.4 0.2 0

21

42

63

84

105

Members

3@5m

Fig. 17. DCRs for the 3  15 frame design obtained using DDHS.

Fig. 14. Third example – 3  15 frame.

Table 10 Summary of 100 optimizations runs for the 3  15 frame. Opt. wt. Avg. opt. wt. St. dev.

401.9 kN 411.2 kN 5.35 kN

No. of times opt. obtained Min. no. of iterations to opt. wt.

1/100 7518

90,340 lb 92,439 lb 1204 lb

parameters (other than harmony matrix size and penalty coefficients). This also indicates the robustness of the DDHS method, as the algorithm produced good results for the same set of parameters.

3. The fundamental reason for the effectiveness of the DDHS scheme is the incorporation of known design information into the mutation step. This leads to better search direction, improved step size determination in the discrete variable spaces, and the identification of appropriate search neighborhoods. These changes improve both the diversification and intensification properties of the search algorithm, helping DDHS to better search both the overall discrete variable space and local subspaces. It may be possible to improve the performance of other stochastic algorithms by incorporating such information in the search process. 4. The key to the performance of DDHS is the mutation scheme, the neighborhoods and manner in which intelligent mutations are specified. Only a single mutation scheme is investigated in

808

P. Murren, K. Khandelwal / Engineering Structures 59 (2014) 798–808

pressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

Table 11 Results comparison for the 3  15 frame. Group No.

DDHS

Kaveh and Talatahari [19]

1 2 3 4 5 6 7 8 9 10 11

W14  90 W40  183 W27  84 W30  116 W24  68 W14  90 W10  49 W12  65 W8  31 W21  50 W21  44

W21  111 W18  158 W10  88 W30  116 W21  83 W24  103 W21  55 W26  114 W10  33 W18  46 W21  44

Wt. (kN) Av. wt. (kN) Wt. (lb) Av. wt. (lb)

401.9 411.2 90,340 92,439

426.4 Not given 95,850 Not given

References

Table 12 Average convergence history for the 3  15 frame. Iteration number

1000 2000 3000 5000 7000 10,000

DDHS average weight kN

lb

552.6 469.8 444.4 425.3 417.2 411.2

124,230 105,610 99,905 95,611 93,793 92,428

this work. The scheme used in this study, as illustrated in Fig. 2, is an example of how constraint data can be used to improve the search. A more thorough examination of the efficacy of different mutation schemes may yield a more effective DDHS method. 5. The DDHS algorithm presented in this work has also been used together with three dimensional moment frames, braced frames, and also in conjunction with nonlinear static analysis [21]. However, further modifications may be required to the presented DDHS framework if a nonlinear dynamic analysis is to be considered for evaluation of a structural system performance. Moreover, the presented DDHS algorithm is only applicable for steel frame optimization; however, it can be modified and used for similar applications where design information can be used to steer the search intelligently in a large discrete space of design variables. The details of the implementation will indeed depend on the application itself though the general idea can be readily used.

Acknowledgments The presented work is supported in part by the US National Science Foundation through grants CMS-0928547 and CMS-1055314. Any opinions, findings, conclusions, and recommendations ex-

[1] AISC-LRFD. Manual of steel construction – load and resistant factor design. American Institute of Steel Construction, Chicago, IL; 1999. [2] Balling RJ. Optimal steel frame design by simulated annealing. J Struct Eng – ASCE 1991;117(6):1780–95. [3] Bel Hadj Ali N, Sellami M, Cutting-Decelle A-F, Mangin J-C. Multi-stage production cost optimization of semi-rigid steel frames using genetic algorithms. Eng Struct 2009;31(11):2766–78. [4] Burns SA. Recent advances in optimal structural design. Reston, VA: American Society of Civil Engineers; 2002. [5] Camp CV, Bichon BJ, Stovall SP. Design of steel frames using ant colony optimization. J Struct Eng 2005;131(3):369–79. [6] Degertekin SO. Harmony search algorithm for optimum design of steel frame structures: a comparative study with other optimization methods. Struct Eng Mech 2008;29(4):391–410. [7] Degertekin SO. Optimum design of steel frames using harmony search algorithm. Struct Multidiscip Optim 2008;36(4):393–401. [8] Degertekin SO, Hayalioglu MS. Harmony search algorithm for minimum cost design of steel frames with semi-rigid connections and column bases. Struct Multidiscip Optim 2010;42(5):755–68. [9] Dumonteil P. Simple equations for effective length factors. AISC Eng J 1992:111–5. [10] Fesanghary M, Mahdavi M, Minary-Jolandan M, Alizadeh Y. Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems. Comput Meth Appl Mech Eng 2008;197(33–40):3080–91. [11] Geem ZW. Harmony search algorithms for structural design optimization. Berlin, Heidelberg: Springer-Verlag; 2009. [12] Geem ZW. State-of-the-art in the structure of harmony search algorithm. In: Geem ZW, editor. Recent advances in harmony search algorithm. Berlin, Heidelberg: Springer-Verlag; 2010. [13] Hasancebi O, Erdal F, Saka MP. Adaptive harmony search method for structural optimization. J Struct Eng 2010;136(4):419–31. [14] Hasancebi O, Serdar C, Saka MP. Improving the performance of simulated annealing in structural optimization. Struct Multidiscip Optim 2010;41(2):189–203. [15] Hasanebi O, Erdal F, Saka MP. Adaptive harmony search method for structural optimization. J Struct Eng 2010;136(4):419–31. [16] Kameshki ES, Saka MP. Genetic algorithm based optimum bracing design of non-swaying tall plane frames. J Construct Steel Res 2001;57(10):1081–97. [17] Kameshki ES, Saka MP. Optimum design of nonlinear steel frames with semirigid connections using a genetic algorithm. Comput Struct 2001;79(17):1593–604. [18] Kargahi M, Anderson JC, Dessouky MM. Structural weight optimization of frames using tabu search. I: optimization procedure. J Struct Eng 2006;132(12):1858–68. [19] Kaveh A, Talatahari S. Hybrid algorithm of harmony search, particle swarm and ant colony for structural design optimization. In: Geem ZW, editor. Harmony search algorithms for structural design optimization. Berlin Heidelberg: Springer-Verlag; 2009. p. 59–198. [20] Kaveh A, Talatahari S. An improved ant colony optimization for the design of planar steel frames. Eng Struct 2010;32(3):864–73. [21] Murren P. Development and implementation of a design-driven harmony search algorithm in steel frame optimization. Notre Dame: University of Notre Dame; 2011. [22] Pezeshk S, Camp CV, Chen D. Design of nonlinear framed structures using genetic optimization. J Struct Eng, New York, NY 2000;126(3):387–8. [23] Ramires FB, Andrade SALd, Vellasco PCGdS, Lima LROd. Genetic algorithm optimization of composite and steel endplate semi-rigid joints. Eng Struct 2012;45(0):177–91. [24] Safari D, Maheri MR, Maheri A. Optimum design of steel frames using a multiple-deme GA with improved reproduction operators. J Construct Steel Res 2011;67(8):1232–43. [25] Tog˘an V. Design of planar steel frames using teaching–learning based optimization. Eng Struct 2012;34:225–32. [26] Yang XS. Harmony search as a metaheuristic algorithm. In: Geem ZW, editor. Music-inspired harmony search algorithm. Berlin, Heidelberg: SpringerVerlag; 2009.