Optimal design of steel skeletal structures using the enhanced genetic algorithm methodology

Abstract

This study concerns with the design optimization of steel skeletal structures thereby utilizing both a real-life specification provisions and ready steel profiles named hot-rolled I sections. For this purpose, the enhanced genetic algorithm methodology named EGAwMP is utilized as an optimization tool. The evolutionary search mechanism of EGAwMP is constituted on the basis of generational genetic algorithm (GGA). The exploration capacity of EGAwMP is improved in a way of dividing an entire population into sub-populations and using of a radial basis neural network for dynamically adjustment of EGAwMP’s genetic operator parameters. In order to improve the exploitation capability of EGAwMP, the proposed neural network implementation is also utilized for prediction of more accurate design variables associating with a new design strategy, design codes of which are based on the provisions of LRFD_AISC V3 specification. EGAwMP is applied to determine the real-life ready steel profiles for the optimal design of skeletal structures with 105, 200, 444, and 942 members. EGAwMP accomplishes to increase the quality degrees of optimum designations Furthermore, the importance of using the real-life steel profiles and design codes is also demonstrated. Consequently, EGAwMP is suggested as a design optimization tool for the real-life steel skeletal structures.

Keywords

design optimization / genetic algorithm / multiple populations / neural network

Introduction

Humankind’s endeavors to protect themselves from the various natural and/or non-natural damages have led to the considerably important progressions in the evolution of structural systems. Therefore, the idealization of structural systems is sustainably improved or developed in a way of utilizing an optimization tool for finding out a structural system with both a lower cost and a higher load-carrying capacity against any structural failure. While the load-carrying capacity of structural systems is increased through the various arrangements in their vertical, horizontal and brace members, constructing these structural forms, so-called regular or irregular frames, grillage systems, domes, vaults etc., using the ready hot-rolled steel profiles leads to an increase in the economic profit. Therefore, these constructive structural systems named skeletal structures are widely utilized as benchmark examples in the area of difficult optimal design problems due to the large numbers of their design variable and constraints [¹]. In this regard, the design of steel skeletal structures has been optimized by use of either gradient-based classical optimization techniques or heuristic search methods based on imitation of biological or physical events [²–⁶]. In the last decade, the bio-inspired approaches (Genetic Algorithm, Swarm, Ant Colony, Grey Wolf Optimizer, Whale Optimization Algorithm (WOA), Enhanced Whale Optimization Algorithm (EWOA) etc.) and physic-inspired approaches (Simulating Annealing, Harmony Search, Bin-bang, Colliding Bodies Optimization (CBO), Vibrating Particles System etc.) have been preferably utilized for the design optimization and extended by improving or developing their optimization procedures [⁷–¹⁵]. Furthermore, it is noted that the alternative optimization strategies, such as level-sets are also successfully utilized in the different engineering-related design problems [¹⁶–¹⁸]. Particularly, generational genetic algorithm (GGA) based on Darwinian’s natural selection theorem, has attracted the attention of designers all over the world. GGA maintains a population of designations throughout a genetic search, which is managed by genetic operators like mutation, crossover, selection etc. GGA has a simple genetic search mechanism. But, the genetic search is easily stagnated after some generations. One of the main reason behind this obstacle is regarded with the assignment of an appropriate values to the genetic operator parameters. In order to deal with this obstacle mentioned, an entire population may be separated into the small parts in other words, sub-populations. In fact, the use of GGA with sub-populations was evaluated in Ref. [¹⁹] and demonstrated how that it suffered from lack of its exploiting and exploring capabilities for the optimal design of steel structures. It was also shown that the use of genetic operator parameters without dynamically adjusted values prevents accurately adoption of the genetic search to the current genetic environment and causes an increase in the computational cost of optimization procedures to explore the promising regions of solution space. The other important obstacle in evolutionary search of GGA is related to its exploration capacity due to a tendency to the stagnation situation in the evolutionary search. In order to overcome these barriers mentioned above, it is inspired from the usage of neural network algorithms widely utilized in the approximation of mathematical functions. In fact, the mathematical function approximation has been already solved thereby using the regression models. Particularly, the regression models such as moving least-squares (MLS), Kriging or penalized spline (nonparametric) regression model are successfully utilized to solve various engineering problems [²⁰–²⁵]. Thus, a new genetic algorithm methodology named EGAwMP was developed thereby dividing the entire populations into small ones and implementing a neural network for both the assignment of different genetic operator parameter values to sub-populations and the prediction of more accurate design variables along with using a new design strategy based on provisions of LRFD_AISC V3 specification (Load and Resistance Factor Design American Institute of Steel Construction, Version 3). The architecture of neural network is constructed utilizing a radial basis network with two layers which is utilized for approximation of any mathematical function. Thus, it is possible to predict both the design variables and the values of genetic operator parameter. Furthermore, the proposed design strategy has a big importance in decreasing problem-dependent complexity. The major factor behind the design complexity is the increased number of members which are utilized to construct the real-life steel structures. Particularly, the variety in the cross-sectional properties of these members and requirement of using the provisions of any well know specification for more accurate and reliable design causes correspondingly an increase in the computational cost of the proposed optimization procedure. Although it has been shown that one of the recent design approaches, called “grouping method”, obtains better optimum designations by decreasing the problem-dependent complexity, a designer faces difficulty about determination of which design variable can be used as a basis for grouping. Because, the member-related design constraints not only depend on their cross-sectional properties but also their stress, lengths, locations in steel construction etc. [²⁶]. The proposed design strategy is controlled by a feasible solution pool. Each member of feasible solution pool has a different un-penalization degree of designs which is defined using a ratio of available strength of related member to the allowable nominal strength called “unity”. Thus, it is possible to determine the maximum values of unity values for each members of skeletal structure.

EGAwMP is applied to the design optimization of the steel skeletal structures with 105, 200, 444, and 942 members utilized as the benchmark design problems in the literature. The spatial skeletal structure with 942-bar bears a big importance due to being both a large-scale and a real-life application problem with discrete type design variables. Particularly, the planar skeletal structures with 200-bar and 444-bar along with 105-bar obtained from the benchmark steel structures with continues and discrete type design variables are intentionally chosen and assigned as an example design optimization problem in order to i) evaluate the computing performance of EGAwMP thereby comparing it with its self-search implementation named “EGAwMP ignoring NN” and the other optimization algorithms ii) show the importance of using the real-life steel profiles (indicated with “W”) and design codes for the optimal design of skeletal structures.

This work begins with a brief introduction of the proposed optimal design procedure in Chapter 2. In Chapter 3, the some details about the working principle of EGAwMP are presented. The application of EGAwMP to the design examples and corresponding preliminary results are given in Chapter 4. The final remarks obtained from the optimal design of steel skeletal structure are summarized in Chapter 5.

Optimum design of steel skeleton structures

This study proposes to minimize the total weight of skeleton steel structure W for the design optimization of the skeleton steel structures considering the design constraints based on provisions of LRFD_AISC V3 specification (see Eqs. 1–4). The further details about the design constraints in Eq. 3 are given in Appendix. The weight minimization procedure in associated with the penalization procedure is formulated as

(1)min\mathrm{..}W={\displaystyle \sum _{k=\mathrm{1}}^m{(w*l)}_k+P}\mathrm{....}(k=\mathrm{1},\mathrm{...},m),

where,

(2)P={(r_0\cdot t)}^{\phi }\cdot ({\displaystyle \sum _{k=\mathrm{1}}^m{g}_{slend}^k+{\displaystyle \sum _{k=\mathrm{1}}^m{g}_{axial}^k}+{\displaystyle \sum _{k=\mathrm{1}}^m{g}_{mom}^k}+{\displaystyle \sum _{k=\mathrm{1}}^m{g}_{shear}^k}}+{\displaystyle \sum _{j=\mathrm{1}}^n{g}_{disp}^j})\cdot f,

(3)

and displacement constraint as,

(4)

The length of a member l and a unit weight w are utilized to compute the entire weight of structural system W. The joint displacements d are computed considering the degree of freedom i, the numbers of joint and member n and m. The slenderness of members is computed according to effective length factor k_effect, member length L and gyration radius r thereby limiting by an upper bound taken as 300.

The member responses, P_u, M_u and V_u are computed using a ready software named ANSYS [²⁷]. Then, the strengths of members are checked according to the proposed constraints in Eq. 3. Also, the displacement of joints are checked against the serviceability of structural system (see Eq.4). In case of exceeding one of constraint-related upper limit values, this unsatisfactory result is penalized by a penalty value P. The constants used in the penalization formulation in Eq.2 are taken as r₀ = 0.50, j = 2, f = 10, and t = current generation number as given in Reference [²⁸]. Furthermore, the resistance factors in Eq. (3–4) are taken as {\phi }_t=0.90,{\phi }_c=0.85,{\phi }_b=0.90 and {\phi }_s=0.90.

Neural network integrated genetic algorithm (EGAwMP)

As a neural network, a radial basis network which is utilized for approximation of any mathematical function, is utilized to predict both the design variables and the values of genetic operator parameter (see a pseudo code in Fig. 1 and genetic operators along with their parameters in Table 1). The proposed neural network with two layers has no neurons at initial stage and adds neurons to its hidden layer until a specified mean squared error goal (default is 0.00) is met. In the first step, the radial basis network is designed for a new network thereby using a command newrb in MATLAB [³⁰]. In the design of this new radial basis, the feasible genetic operator parameter and design variable values, {Par}_{\mathrm{A}\mathrm{l}\mathrm{l}}^{\mathrm{F}\mathrm{e}\mathrm{a}\mathrm{s}}−P^{\mathrm{F}\mathrm{e}\mathrm{a}\mathrm{s}} and F^{\mathrm{F}\mathrm{e}\mathrm{a}\mathrm{s}} , which are obtained in the previous generation and collected in the feasible solution pool, are utilized as input and output. In the second step, the new genetic operator parameter and design variable values {Par}_{\mathrm{A}\mathrm{l}\mathrm{l}}−P are generated using a command sim in MATLAB [³⁰]. Then, the predicted design variables and genetic operator parameter values are used to generate sequentially new populations in associated with a new design strategy. The proposed design strategy is managed feasible candidate solutions, penalization degrees of which is computed according to the provisions of LRFD_AISC V3 specification. This design strategy provides a big support for an increase in the exploration capability of EGAwMP due to a stagnation situation in the evolutionary search. Following the implementation of the design strategy proposed here, the population P is re-created according to the feasible solution pool (see Fig. 1).

Design examples

A planar skeletal structure with 105, 200-bar and two spatial skeletal structures with 444 and 942-bar are optimized according to provisions of LRFD_AISC V3 specification. Although these design examples with continuous and discrete type design variables were optimized by use of various optimization approaches presented in literature, the number of their constraints is increased by including stability and tension-compression-flexural dependent strength constraints, based on the provisions of LRFD_AISC V3 specification. The hot rolled steel profiles with W cross-sectionals are chosen among 283 different cross-sections, properties of which are taken from the current database named “aisc-shapes-database-v15.0” and easily obtained using the official web-page of American Institute of Steel construction (AISC) (see Ref. [³¹]).

EGAwMP is executed 10 times for each design example and obtained a number of optimum designations with different convergence degree. The Number of Penalized Joints (NPJ) due to Displacement Constraints and Number of Penalized Members (NPM) due to Design Constraints are included into each of related Tables. An optimum designation stored in a whole evolutionary search, called best result, is determined among the designations coming from these executions. The feasible solutions with higher quality which is utilized depending on the activation of design strategy is also tabulated. The variation in both fitness values of feasible solutions and average of these fitness values including average of fitness values coming from 10 runs through outer generation number Par_OGN are displayed. The maximum of unity values corresponding to the related design constraints and joint displacement values are also depicted in order to show the activation of design strategy and determining the members with higher sensitivity. The values of genetic operator parameters adjusted through neural network implementation are only visualized for competition and migration operators due to the number of sub-populations. In order to demonstrate whether there exists any relationship among parameters of these genetic operators, figures are drawn by use of two axes, each of which represents a different genetic operator parameter. While the material belongs to the left axes is depicted with a continuous line, a dot line is utilized for the representation of material belongs to the right axes. It is also noted that a long format is preferred in the unit conversion. Thus, it is assumed that 1 mm= 0.039370079 inch, 1 mm² = 0.0015500037 inch², 1 mm³= 0.000061024 inch³ etc.

A planar frame structure with 105-bar

This steel planar frame with 105-bar is one of the popular skeletal design benchmarks utilized in the area of the optimal structural design (see Refs. [⁶,³²,³³]). The proposed benchmark design problem has a three bay, each of which has 15-story (see the member and node numbers of the steel planar frame in Fig. 2(a)). The elasticity modules and yield stress of steel material is taken as 29 ksi (200000N/mm²) and 36 ksi (248.2 N/mm²). While the design constraints are based on the provisions of LRFD_AISC V3 specification, the lateral displacements of nodes constrained to an upper value, which is computed as \frac{\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{.}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{m}\mathrm{ent}}{\mathrm{H}}\le \mathrm{R}=\frac{1}{300}. Thus, the upper value of nodal lateral displacements for a better serviceability purpose is limited to \frac{53065\mathrm{m}\mathrm{m}(174.1\mathrm{f}\mathrm{t})}{300}=176\mathrm{m}\mathrm{m}(0.580\mathrm{f}\mathrm{t}). It is noted that the complexity degree of current design constraints is higher compared to the ones utilized for the optimization of this planar frame (see Reference [⁶]). Member of the planar frame with 105-bar are categorized into 11 groups (Par_ND=11):

A1_{(1,4,5,11,13,18)}, A2_{(2,3,7,10,14,17)}, A3_{(20,25,26,32,34,39)}, A4_{(21,23,28,31,36,38)}, A5_{(41,46,48,53,55,60)}, A6_{(42,45,50,51,57,58)}, A7_{(62,67,68,74,76,81)}, A8_{(64,65,70,73,77,79)}, A9_{(83,88,89,95,96,102)}, A10_{(84,87,92,94,98,101)}, A11_{(6,8,9,,12,15,16,19,22,24,27,29,30,33,35,37,40,43,44,47,49,52,54,56,59,61,63,66,69,71,72,75,78).}

Due to the dynamically adjustment of genetic operators of EGAwMP, the main parameter set of EGAwMP are initialized as ({Par}_{\mathrm{O}\mathrm{G}\mathrm{N}}=10,{Par}_{\mathrm{I}\mathrm{G}\mathrm{N}}=50,{Par}_{\mathrm{S}\mathrm{P}\mathrm{N}}=10,{Par}_{\mathrm{S}\mathrm{P}\mathrm{S}}=1\le \le 10).

The variation in the parameter values genetic operator utilized in the migration ({Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{R}}and{Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{T}}) and competition ({Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{C}\mathrm{R}}−{Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{N}\mathrm{S}\mathrm{C}}) procedures of EGAwMP is depicted in Figs. 3(a) and 3(b). It is observed that there exists no relationship between the parameters of migration and competition-related operators. However, the evolutionary search mechanic of EGAwMP achieves to obtain an optimal design with the lowest weight, 87624.3105 kg (39745.7186 lb) (see Table 2). Due to the existence of evolutionary search without a stagnation problem, the design strategy is not activated (see Fig. 3(c)). The variation in the average of best result and 10 runs is presented in Fig. 4. The view of optimal design for the planar frame is schematized in Fig. 1(b). It is mentioned that the unity values are used to indicate stress ratios according to the provisions of LRFD_AISC V3 specification. Thus, the maximum unity values corresponding to the proposed design constraints and the maximum joint displacements are depicted in Figs. 5(a) and 5(b). Considering the maximum unity values in Fig. 5(b), the members with no of 65 and 68 are the most sensible ones regarding to the unity values. Nevertheless, EGAwMP success to obtain an optimal design with the lightest weight and relatively lower unity values compared to the other optimization algorithms.

A planar skeletal structure with 200-bar

The design of this planar skeletal structure, which of material elasticity module was taken as 206842.72 N/mm² (30000 ksi) was first optimized by use of continuous design variables using stress and displacement constraints limited their maximum values to 68.95 N/mm² (10 ksi) and ∓ 12.7 mm (0.5 in) (see Refs. [¹²,¹³,³⁴] along with Fig. 6(a)). While the main parameter values of EGAwMP are initialized as (Pa{r}_{\mathrm{O}\mathrm{G}\mathrm{N}}=10,Pa{r}_{\mathrm{I}\mathrm{G}\mathrm{N}}=100,Pa{r}_{\mathrm{S}\mathrm{P}\mathrm{N}}=10,Pa{r}_{\mathrm{S}\mathrm{P}\mathrm{S}}=1\le \le 10), the members of this planar skeletal structure are sorted into 29 groups (Par_ND=29): A1_(1,2,3,4), A2_{(5,8,11,14,17)}, A3_{(19,20,21,22,23,24)}, A4_{(18,25,56,63,94,101,132,139,170,177)}, A5_{(26,29,32,35,38)}, A6_{(6,7,9,10,12,13,15,16,27,28,30,31,33,34,36,37)}, A7_{(39,40,41,42)}, A8_{(43,46,49,52,55)}, A9_{(57,58,59,60,61,62)}, A10_{(64,67,70,73,76)}, A11_{(44,45,47,48,50,51,53,54,65,66,68,69,71,72,74,75)}, A12_{(77,78,79,80)}, A13_{(81,84,8790,93)}, A14_{(95,96,97,98,99,100)}, A15_{(102,105,108,111,114)}, A16_{(82,83,85,86,88,89,91,92,103,104,106,107,109,110,112,113)}, A17_{(115,116,117,118)}, A18_{(119,122,125,128,131)}, A19_{(133,134,135,136,137,138)}, A20_{(140,143,146,149,152)}, A21_{(120,121,123,124,126,127,129,130,141,142,144,145,147,148,150,151)}, A22_{(153,154,155,156)}, A23_{(157,160,163,166,169)}, A24_{(171,172,173,174,175,176)}, A25_{(178,181,184,187,190)}, A26_{(158,159,161,162,164,165,167,168,179,180,182,183,185,186,188,189)}, A27_{(191,192,193,194)}, A28_{(195,197,198,200)}, A29_(196,199).

In order to measure the quality of optimum designations under severe loading conditions, this planar truss was considered for two distinct loadings: firstly, three different loadings and, secondly, five different loadings. These two design cases are tackled to evaluate the quality of optimum designations obtained by EGAwMP in the following two sections.

Design case I

Three loading conditions are imposed to planar skeletal structure with 200-bar:

a) 4449.741 N (1000 lbf) acting in the positive x-direction at node points 1, 6, 15, 20, 29, 34, 43, 48, 57, 62 and 71,

b) 44497.412 N (10000 lbf) acting in the negative y-direction at node points 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 70, 71, 72, 73, 74, 75,

c) Loading conditions (a) and (b) acting together

It is clear that the parameters of genetic operators governed the evolutionary-related processes named migration ({Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{g}}and{Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{T}}) and competition ({Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{C}\mathrm{R}}−{Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{N}\mathrm{S}\mathrm{C}}) is not interacted with each other (see Figs. 7(a) and 7(b)). The design strategy is activated when the genetic operator parameter, Par_OGN equals to 7 (see Fig. 7(c)). The change in the average of best result and 10 runs is depicted in Fig. 8. When the evolutionary search is stagnated, the members of no 181, 187 and 196 are immediately removed from the related groups and taken a new group number (see the unity and displacement values corresponds to the related member and joints in Figs. 9(a)–9(d)). The activation of design strategy causes an increase in the group number (from 29 to 30), but leads to an increase in the number of feasible solution. In the end of evolutionary search performed by EGAwMP, the quality of optimum designations obtained by EGAwMP is higher than one obtained by EGAwMP ignoring NN implementation, but poorer compared with the ones obtained by the other algorithms presented in References [¹²,¹³,³⁴] (see Table 3). Firstly, the weight value, 11625.134 kg (25629.033 lb) used to represent the worst unfeasible design solution with discrete-type design variable is considered and compared it with the weight values 11542.610 kg (25447.10 lb), 11546.905 kg (25456.57 lb), 11542.456 kg (25446.76 lb) obtained by use of various optimization algorithms with the continuous-type design variable in related references. Then, it is clearly seen that any further improvement in the convergence degree of the optimal designation obtained using the continuous-type design variable is impossible for this benchmark steel structure problem (see Table 3). The steel profiles with w-shaped cross-sections corresponding to optimum designation are schematized in Fig. 6(b).

Design case II

The same planar skeletal structure tackled in design case I is optimized for minimum weight under five different loading conditions:

a) 4449.741 N (1000 lbf) acting in the positive x-direction at node points 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, and 71;

b) 44497.412 N (10,000 lbf) acting in the negative y-direction at node points 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 70, 71, 72, 73, 74, 75;

c) Loading conditions (a) and (b) acting together

d) 4449.741 N (i.e., 1000 lbf) (453.592 kgf) acting in the negative x-direction at node points 5, 14, 19, 28, 33, 42, 47, 56, 61, 70 and 75;

e) Loading conditions (b) and (d) acting together.

The variation in the parameter values of genetic operator used by migration ({Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{g}}and{Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{T}}) and competition ({Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{C}\mathrm{R}}−{Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{N}\mathrm{S}\mathrm{C}}) is presented in Figs. 10(a) and 10(b). Considering genetic operator parameter values, it is seen that there is no relationship between the parameters of both migration and competition operators. The neural network implementation achieves to adopt the evolutionary search into a varying genetic environment without activating the design strategy (see Fig. 10(c)). Although severe loading conditions cause a decrease in optimality quality compared to the design case I, optimum designation is more convergent than one obtained by GAwMp ignoring NN implementation but relatively poorer compared with one obtained using the optimization algorithm presented in Ref. [¹²] (see Table 4). The reason behind this result is same as the previous application problem’s one. When considering the weight value belong to the worst unfeasible design solution with discrete-type design variable and comparing the weight value belong to the related reference with continuous-type design variable in Table 4, it is clearly observed that any further improvement in the convergence degree of the optimal designation obtained by use of the continuous-type design variable is impossible for this benchmark steel structure problem. The change in average of both best result and 10 runs is presented in Fig. 11. The unity values of members and maximum displacement values of joints are depicted for each loading conditions in Figs. 12(a) and 12(b). A schematic view of steel profiles with W-shaped cross-sections corresponding to an optimum designation is presented in Fig. 6(c).

A spatial skeletal structure with 444-bar

Lamberti and Pappalettere [³⁵,³⁶] optimized this spatial skeletal structure, which has a material elasticity module 199947.961 N/mm² (29000ksi) and yielding point 68.95 N/mm² (10 ksi), considering a displacement constraint which of maximum value is limited to 6.35 mm (0.25 in.) (see Figs. 13(a) and 13(b)). Two loading conditions, 444822.1615 N (100000 lbf) at node 121 and 44482.2161 N (10000 lbf) at each other free node, are imposed to this spatial skeletal structure. While the main parameter values of EGAwMP are initialized as (Pa{r}_{\mathrm{O}\mathrm{G}\mathrm{N}}=10,Pa{r}_{\mathrm{I}\mathrm{G}\mathrm{N}}=100,Pa{r}_{\mathrm{S}\mathrm{P}\mathrm{N}}=10,Pa{r}_{\mathrm{S}\mathrm{P}\mathrm{S}}=1\le \le 10), the members of the spatial skeletal structure are sorted into 28 groups (Par_ND=28):

A1_(1-24), A2_(25-36), A3_(37-48), A4_(49-72), A5_(73-84), A6_(85-96), A7_(97-120), A8_(121-132), A9_(133-144), A10_(145-168), A11_(169-180), A12_(181-192), A13_(193-216), A14_(217-228), A15_(229-240), A16_(241-264), A17_(265-276), A18_(277-288), A19_(289-312), A20_(313-324), A21_(325-336), A22_(337-360), A23_(361-372), A24_(373-384), A25_(385-408), A26_(409-420), A27_(421-432), A28_(433-444)

The design of spatial skeletal structure with 444-bar is optimized by EGAwMP using discrete design variables. It is observed that there does not exist a relationship between genetic operator parameters of migration ({Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{g}}and{Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{T}}) and competition ({Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{C}\mathrm{R}}−{Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{N}\mathrm{S}\mathrm{C}}) (see Figs. 14(a) and 14(b)). The adaptation of evolutionary search using the neural network implementation continues until genetic operator parameter Par_OGN reaches to No 5. The design strategy is activated due to the stagnation problem occurred in evolutionary search (see Fig. 14(c)). The change in average of both best result and 10 runs depicted in Fig. 15. Following the activation strategy, the number of grouped member increases to 29 from 28 by discarding the members of No. 73, 78, and 83 from corresponding groups (see Figs. 16(a)–16(d)). Then, current population is re-created according to the design variables obtained by neural network implementation. The optimum designation obtained in the end of evolutionary search is more convergent than GAwMp ignoring NN implementation but relatively poorer compared with one obtained using the other algorithm presented in Reference Lamberti and Pappalettere [³⁵,³⁶]. (see Table 5). As a similar result obtained in the previous design example with 200-bars, it is clearly seen that any further improvement in the convergence degree of the optimal designation obtained by use of the optimization algorithm with the continuous-type design variable is impossible for this benchmark steel structure problem. The construction of the spatial skeletal structure according to optimum designation is schematized in Fig. 13(c).

A spatial skeletal structure with 942-bar

The design example with 942-bar and 224 nodes is tackled to evaluate the performance of EGAwMP for design optimization of large-scaled skeletal structures (see Figs. 17(a) and 17(b)). The material used to build this spatial skeletal structure has an elasticity module of 199947.961 N/mm² (29000ksi) and a yielding point of 248.211 N/mm² (36 ksi). Maximum value of joint displacement is limited to 381 mm (15.0 in). While the main parameter values of EGAwMP are initialized as (Pa{r}_{\mathrm{O}\mathrm{G}\mathrm{N}}=10,Pa{r}_{\mathrm{I}\mathrm{G}\mathrm{N}}=100,Pa{r}_{\mathrm{S}\mathrm{P}\mathrm{N}}=10,Pa{r}_{\mathrm{S}\mathrm{P}\mathrm{S}}=1\le \le 10), the members are sorted into 59 groups (Par_ND=59):

A1_(1-2), A2_(3-10), A3_(11-18), A4_(19-34), A5_(35-46), A6_(47-58), A7_(59-82), A8_(83-86), A9_(87-90), A10_(91-98), A11_(99-106), A12_(107–122), A13_(123-130), A14_(131-162), A15_(163-170), A16_(171-186), A17_(187-194), A18_(195-226), A19_(227-234), A20_(235-258), A21_(259-270), A22_(271-318), A23_(319-330), A24_(331-338), A25_(339-342), A26_(343-350), A27_(351-358), A28_(359-366), A29_(367-382), A30_(383-390), A31_(391-398), A32_(399-430), A33_(431-446), A34_(447-462), A35_(463-486), A36_(487-498), A37_(499-510), A38_(511-558), A39_(559-582), A40_(583-606), A41_(607-630), A42_(631-642), A43_(643-654), A44_(655-702), A45_(703-726), A46_(727-750), A47_(751-774), A48_(775-786), A49_(787-798), A50_(799-846), A51_(847-870), A52_(871-894), A53_(895-902), A54_(903-906), A55_(907-910), A56_(911-918), A57_(919-926), A58_(927-934), A59_(935-942).

The values of joint loads at nodes (1,2,..,232) are 6672.232, 4448.221, 13344.665, 26689.329, 40033.995 N. These joint loads are distributed to three sections of spatial truss structure. Each section has different levels: 5 levels at section 1, 7 levels at section 2 and 11 levels at section 3. The distribution of these joint loads in x,y and z directions is tabulated according to node numbers in Table 6. In Table 6, node numbers are presented for each level of corresponding section. For example, 6672.232 N in x-direction is loaded in the node numbers (1 and 4) at the first level of section 1, (5 and 8) at the second level of section 1 and (21 and 24) at fifth level of section 1 sequentially .

The design optimization of this spatial skeletal structure was performed by Erbatur et al. [³⁷], Hasancebi and Erbatur [³⁸], Hasancebi [³⁹]. An optimum designation is obtained by use of discrete design variables. In this study, the same spatial skeletal structure is optimized to minimize its weight. The neural network implementation adopts the parameters of genetic operators (Par_All) to exploitation of current valuable genetic material for next generations. According genetic operator parameter values of migration ({Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{g}}and{Par}_{\mathrm{M}\mathrm{i}\mathrm{g}}^{\mathrm{M}\mathrm{T}}) and competition ({Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{C}\mathrm{R}}−{Par}_{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}}^{\mathrm{N}\mathrm{S}\mathrm{C}}), it is seen that there is no relationship between the parameters of these genetic operators (see Figs. 18(a) and 18(b)). Considering the variation in the number of feasible solutions through Par_OGN, it is observed that the success of neural network implementation is accelerated by the activation of design strategy at the point indicated by Par_OGN=7 (see Fig. 18(c)). Because a new feasible solution with higher quality is not obtained in the generation corresponding to Par_OGN=7, the design strategy is activated. Thus, the number of feasible solution is increased. The convergence history for average of both best result and 10 runs is presented in Fig. 19. Also, the unity values of members and displacement values of joints are depicted in Figs. 20(a)–20(d). In the end of evolutionary search, the optimum designation is more convergent than those obtained by both algorithms proposed in Literature [³⁷–³⁹] and EGAwMP ignoring NN implementation (see Table 7). It is noted that optimum designations obtained in References [³⁷–³⁹] are penalized due to the additional constraints proposed in this study (see Table 6). Steel profiles with w-shaped cross-section used to represent the optimum design are schematized for spatial structure in Fig. 17(c).

Final remarks

In this study, GGA is proposed for design optimization of steel skeletal structures. Its exploitation and exploration capabilities are improved by dividing an entire population into multiple populations and utilizing a neural network implementation for prediction of genetic operator parameter values and design variables. Thus, more-accurately-predicted design variables are utilized to re-crate the current population. Its exploration capability is also enhanced by use of a new design strategy for design variables. The proposed genetic algorithm named EGAwMP is applied for weight minimization of three steel skeletal structures with 105, 200, 444 and 942 bar. According to optimum designations obtained by EGAwMP, it is demonstrated that there does not exit a relationship among genetic operator parameters which of values are adjusted by neural network implementation. However, neural network implementation achieves to adopt genetic search into a varying genetic environment in a way of predicting both the values of genetic operator parameters and design variables. The activation of proposed design strategy causes to increase the capability of the neural network implementation. Using the real-life ready steel profiles along with the provisions of LRFD_AISC V3 specification for the optimal design of skeletal structures causes a divergence in the optimal designs, but an increase in the reliability degree for the application of resulted skeletal structures into the real-world. Nevertheless, the computing efficiency of EGAwMP is demonstrated with an increase in the quality degree of optimum designations obtained by use of both its self-search implementation named “EGAwMP ignoring NN” and available optimization algorithms.

As a next study, the evolutionary search capacity of EGAwMP will be evaluated thereby both including the other skeletal structures such as regular frames, the other large scaled skeletal structures etc. and utilizing the different regression models with an implementation into GGA.

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