Pareto frontier for steel-reinforced concrete beam developed based on ANN-based Hong-Lagrange algorithm

ABSTRACT Multi-objective optimization (MOO) is always a challenging issue for engineers in the field of structural engineering, where several objective functions must be satisfied under equality and inequality constraints to meet requirements imposed by engineers and decision-makers. This study proposes a novel approach to solve MOO problems for steel-reinforced concrete (SRC) beams using an artificial neural network (ANN)-based Hong-Lagrange algorithm. Proposed method in this paper optimizes three specific objective functions, including cost (CIb), CO2 emissions, and beam weight (W), simultaneously. Neural networks are trained by 200,000 samples, which are randomly generated by structural mechanics-based calculations, to derive three specific objective functions. Unified objective function is, then, proposed based on weight fractions of each objective function. An ANN-based Hong-Lagrange technique identifies optimal design parameters within the bounds constrained by 16 inequalities against external loads. The proposed method yields a set of optimal results, creating a Pareto frontier that optimizes multiple objectives. Pareto frontier using an ANN-based Hong-Lagrange algorithm is well compared with the lower boundary of large datasets of random designs which include 133,711 samples obtained by structural mechanics. A cost of an SRC beam is obtained as 219,279.1 KRW/m by an ANN-based Hong-Lagrange algorithm with an error of −0.14% verified by structural mechanics. GRAPHICAL ABSTRACT


Introduction
In the area of structural engineering, an optimization design has always been a challenging issue over the last few decades.Four major categories of structural optimization can be classified, such as cost reduction, environmental impact reduction, structural performance improvement, and multi-objective optimization (MOO), according to a study by Mei and Wang (2021).A study of an optimization of rectangular reinforced concrete (RC) beams is presented by Shariat et al. (2018) using a computational Lagrangian Multiplier method.However, calculations of steel-reinforced concrete (SRC) members are much more complex than common RC structures due to the contribution of a H-shaped steel section embedded inside concrete material, thus, being expensive for calculations of strengths, deflections, and flexibility.
SRC members have been used in various types of infrastructure facilities, such as buildings, parking, and transportation.A sustainability of SRC structures increases significantly due to the protection of concrete outside, preventing high temperature and corrosive agents of the surrounding environment for H-shaped steel sections.Steel, rebar, and concrete work simultaneously to prevent damage to the structures when SRC members are subjected to imposed load.Many studies have been performed to investigate the behaviors of SRC members under applied loads (Li and Matsui 2000;El-Tawil and Deierlein 1999;Bridge and Roderick 1978;Mirza, Hyttinen, and Hyttinen 1996;Ricles and Paboojian 1994;Furlong 1974; Mirza and Skrabek 1992;Chen and Lin 2006;Dundar et al. 2008;Munoz and Hsu 1997;Rong and Shi 2021;Virdi, Dowling, andBS 449, &BS 153 1973, Roik andBergmann 1990;Brettle 1973).SRC members' capacity can be calculated by conventional structural mechanics-based calculations, whereas an artificial neural network (ANN)-based design method has been developed to design structural members that does not require complex calculations.Many researchers successfully applied ANNs in structural analysis, including notable studies of Abambres andLantsoght 2020, Sharifi, Lotfi, andMoghbeli (2019), Asteris et al. (2019), and Armaghani et al. (2019).The potential of ANNs in civil engineering is also demonstrated by the studies of Kaveh et al. (Kaveh and Khalegi 1998;Kaveh and Servati 2001), which show that ANNs can predict concrete strength and design double layer grids with results comparable to traditional methods.
The authors perform several studies for an optimization of either single objective functions or multi-objective functions for RC members in previous studies (Hong and Nguyen 2022b;Hong, Nguyen, and Nguyen 2022;Hong, Nguyen, and Pham 2022;Hong and Le 2022;Hong 2021;Nguyen and Hong 2019).The authors also published a study "An AI-based auto-design for optimizing RC frames using the ANN-based Hong-Lagrange algorithm" in which ANNs-based objective functions such as costs and weights of RC frames with four-byfour bays and four floors are optimized simultaneously based on big datasets of 330,000 designs in accordance with ACI 318-19, whereas corresponding design parameters which minimize objective functions are also obtained (Hong and Pham 2023).
Multi-objective functions for SRC beams including cost (CI b ), CO 2 emissions, and weight (W) are presented in this study, presenting a key value of an ANN-based Hong-Lagrange algorithm for an application to a structural design for engineers.MOO is always a challenging issue for engineers in the field of structural engineering, where several objective functions must be satisfied under equality and inequality constraints to meet requirements imposed by engineers and decisionmakers.Kaveh et al. proposed a genetic algorithm for optimal design of RC retaining walls (Kaveh, Kalateh-Ahani, and Fahimi-Farzam 2013), and a vibrating particles system (VPS)-based algorithm for MOO (Kaveh and Ilchi Ghazaan 2020).RC bridge columns and bent caps are optimized for cost and CO 2 emissions (Kaveh, Mottaghi, and Izadifard 2022), and optimal design of RC frames considering CO 2 emissions is achieved via metaheuristic algorithms by Kaveh, Izadifard, and Mottaghi (2020).A Pareto frontier derived based on an ANN-based Hong-Lagrange algorithm to simultaneously optimize designs for three objective functions (CI b , CO 2 , W) is well compared with the lower boundary of large datasets of random designs generated by structural mechanics-based calculations (AutoSRCbeam).This research provides a powerful tool for the next generation in optimizing multiple objective functions for SRC beams that meet any loads and design standards.Both equalities and inequalities are taken into consideration based on engineers' requirements and regional design standards.
Large datasets used to train neural networks are obtained by structural mechanics-based calculations, called AutoSRCbeam, which was developed by authors in previous studies (Nguyen and Hong 2019;Hong 2019).The design accuracy of SRC beams using an ANN-based Hong-Lagrange algorithm is verified both through structural analysis and large structural datasets as shown in the previous study (Hong and Nguyen 2022a).
This study requires a large amount of data, and hence, neural networks require a large amount of data to train effectively, which can be a challenge when using small computers.Computational complexity is also challenging in that training neural networks can be computationally intensive, requiring high-performance hardware.However, this study has developed methods to mitigate some of the limitations listed above.The main limitation of this study is computational complexity as high-performance computers are needed to generate large datasets and train ANNs. 1.No studies were found to present based on ANNs to optimize the three objective functions including CI b , CO 2 emission, and W at the same time for an SRC beam.In this study, the authors present an ANNbased Hong-Lagrange algorithm, which is used to optimize three objective functions (CI b , CO 2 emission, and W) simultaneously for an SRC beam.

Research significances
This study presents a hybrid network using an ANN and Hong-Lagrange algorithm to optimize an SRC beam, capable of simultaneously optimizing three objective functions (CI b , CO 2 emission, and W) with significant accuracies.A unified objective function (called UFO) is developed for an SRC beam in this study (Hong et al. 2022).UFO is formulated based on three objective functions via weighted fractions, integrating them into one objective function to simultaneously optimize all objective functions (CI b , CO 2 emission, and W).A Pareto frontier (also called Pareto front) for an SRC beam is constructed based on a combination of MOO results.A contribution of each objective function is represented by the trade-off of their weight fractions, selected by an interest of engineers and decision-makers.This study is a steppingstone for a design of next generation, not based on structural mechanics but based on ANNs.An ANNbased Hong-Lagrange algorithm simultaneously optimizes multi-objective functions for engineers and decision-makers.

ANN-based design scenarios for steelreinforced concrete beams
Figure 1 demonstrates the geometry of an SRC beam, including beam section (h, b) and steel H-shaped section (h s , b s , t f and t w ), which is encased in concrete material.A forward design scenario for an SRC beam is presented based on 15 inputs and 11 outputs, as shown in Table 1.Fifteen input parameters include length of beam (L), beam dimensions (b, d), material strengths of concrete (f' c ), rebar (f y ), steel (f yS ), compressive and tensile rebar ratio (ρ c , ρ t ), steel height (h s ), steel flange (b s ), steel web thickness (t f ), steel flange thickness (t w ), moment due to dead load (M D ), and moment due to live load (M L ).Eleven output parameters include design moment capacity (ϕM n ) excluding beam weight, tensile strains of steel and rebar (ε st , ε rt ), immediate and long-term deflections (Δ imme , Δ long ), curvature ductility (µ ϕ ), materials and manufacture cost (CI b ) per 1 m length, CO 2 emission per 1 m length, beam weight (W) per 1 m length, horizontal clearance (X s ), and safety factor (SF).A cost (CI b ) for materials and manufacture, CO 2 emissions, and beam weights (W) are selected as multiple objective functions for an optimization of an SRC beam in the present study.2019).Fifteen input parameters for AutoSRCbeam are randomly selected in designated ranges, randomly providing 11 output parameters.Ranges for dimensions of SRC beams are designated from 6000 to 12,000 mm, 500 to 1500 mm, and 0.3d to 0.8d for beam length (L), beam height (h), and beam width (b), respectively, where d is effective beam depth in a range of 406.1-1444.5 mm, as referred to in Table 2.The dimensions of H-shaped steel section are randomly selected in appropriate ranges of 0.4-0.6dfor steel section height (h s ), 0.3-0.6bfor steel section width (b s ), and 5-25 mm for both steel web and flange thickness (t w and t f ), as shown in Table 2. Material strengths of beam components are chosen in ranges of 30-50 MPa, 500-600 MPa, and 275-325 MPa for concrete (f' c ), rebar (f y ), and steel (f yS ), respectively.Compressive rebar ratio is randomly chosen in a range of 1/400 ~ 1.5ρ rt , where ρ rt is tensile rebar ratio with a minimum ρ rt;min ¼ max 0:25 ffi ffi ffiffi

Generation of large datasets
, following ACI standard (Standard 2019).Notations and ranges of fifteen input parameters defining an SRC beam are indicated in Table 2. ACI 319-18 code requests tensile rebar strain (ε rt ) greater than 0.003 + ε ty (ε ty is yield strain of rebar) to ensure enough ductility of beam.
This study focuses on SRC beams with fixed-fixed end conditions, as illustrated in Figure 3. SRC beams are subjected to uniform loads, including dead and live loads, yielding M D and M L , respectively.
According to ACI 318-19 (Standard 2019), deflections are limited to L/360 for immediate deflection (Δ imme. ) and L/240 for long-term deflection (Δ long ).In the preliminary design stage, beam sections are unknown; thus, the design moment capacity of a beam (ϕM n ) is formulated by excluding a self-weight of a beam when generating large datasets.Factored moment (M u ) represents a magnitude of externally applied moments, calculated by load combination of M D and M L with load factors (M u = 1.2MD + 1.6M L ).A safety factor represents how safe beam is against applied loads, which is calculated as a ratio between a design moment strength and factored moment (SF =ϕM n /M u ), and the safety factor must not be smaller than 1.0.
Original and normalized large datasets of 200,000 are shown in Table 3 are randomly selected, yielding eleven corresponding output parameters (ϕM n , ε rt , ε st , Δ imme., Δ long , μ ϕ , CI b , CO 2 , W, X s , SF) for design SRC beams.Mean, maxima, and minima of overall 26 parameters based on 200,000 datasets are indicated in Table 3.All input and output parameters are normalized in a range from −1 to 1 by using MAPMINMAX function of MATLAB (MathWorks 2022a), as shown in Table 3.

Training artificial neural networks based on parallel training method
In this study, ANNs are trained using a parallel training technique (PTM) (Hong 2021;Hong, Pham, and Nguyen 2022).As shown in 11 training networks are formulated independently when 15 input parameters (L, d, b, f Moment due to service live load

Derivation of a unified function of objective for a steel-reinforced concrete beam
A UFO for SRC beams is defined using algorithms based on the weighted sum technique (Afshari, Hare, and Tesfamariam 2019), which is created by integrating three objective functions (CI b , CO 2 , and W) with their respective weight fractions w CI b ; w CO 2 ; w W , as indicated in Equations ( 5) and ( 6).These fraction weights are in a range from 0 to 1, whose sum is 1 as shown in Equation ( 6).Specific trade-offs among objectives are used to help engineers and decision-makers evaluate a design project holistically.Single objective function for each of CI b , CO 2 , and W is a specific case of UFO when w CI b : w CO 2 : w W = 1:0:0; 0:1:0; 0:0:1, respectively.A Lagrange function of a UFO shown in Equations ( 7)-( 9), which are based on three objective functions with equality and inequality constraints shown in Table 5 being simultaneously optimized.The Lagrange function is substituted into the built-in optimization toolbox of MATLAB (MathWorks 2022) to obtain optimized design parameters.Figure 4 demonstrates a flowchart for five steps to solve MOO problems, providing detailed descriptions of an algorithm that illustrates the authors' previous study for RC columns (Hong et al. 2022).
where Lagrange function utilizing UFO function:

Four specific cases for an optimization of each objective function (CI b , CO 2 , and W, respectively)
A Pareto frontier is obtained based on a combination of multiple optimized designs for three objective functions (CI b , CO 2 , and W) with their weight fractions (w CI b : w CO 2 : w W Þ for SRC beams in this study.These weight fractions (w CI b : w CO 2 : w W Þ represent trade-off ratios contributed by each of three objective functions to real-life optimizations for engineers and decisionmakers.The 343 combinations of weight fractions are generated for constructing a Pareto frontier, including four specific cases, as indicated in Figure 5.It is noted that the sum of three weight fractions of each combination is always equivalent to 1. Observation shows that four specific cases include an optimization of each objective function (CI b , CO 2 , and W, respectively), represented by Points 1, 2, and 3 on a Pareto frontier, whereas Point 4 represents an evenly optimized design for CI b , CO 2 , and W, with their weight fraction w CI b : w CO 2 : w W = 1/3:1/3:1/3.Design parameters (b, h s , b s , ρ rc , ρ rt ) are obtained by solving MOO using an ANN-based Hong-Lagrange algorithm with 10 equality and 16 inequality conditions, shown in Table 5.

Pareto-efficient designs verified by structural datasets
A Pareto frontier includes 343 optimized designs for SRC beams, indicated by red dots as shown in Figure 6(a-c).Design parameters (b, h s , b s , ρ rc , ρ rt ) are calculated for four specific combinations while all 10 equalities (L = 10,000 mm, d = 950 mm, f y = 500 MPa, f' c = 30 MPa, f yS = 325 MPa, t f = 12 mm, t w = 8 mm, Y s = 70 mm, M D = 500 kN⋅m, and M L = 1500 kN⋅m) and 16 inequalities given in Table 5 are satisfied.shown in Figure 6(b,c).However, tensile and compressive rebar ratios (ρ rt , ρ rc ), and height of steel H-shaped (h s ) are 0.0225, 0.0225, and 408.8 mm, respectively, larger than those of the Design cases 1 and 2 in order to decrease beam weights W. Beam capacity increases to meet the requirement of SF ≥ 1 when single objective function W is optimized as indicated in Figure 6(b,c).Costs and CO 2 emissions obtained in Design case 3 are 224,406 KRW/m and 0.4265 t-CO 2 /m, respectively, larger than those obtained in Design cases 1 and 2.

Verifications
Table 6 summarizes designs when an ANN-based Hong-Lagrange algorithm evenly minimizes three objective functions CI b , CO 2 , and W based on w CIb : w CO2 : w W = 1/ 3:1/3:1/3.Green cells in Table 6 show 11 output parameters obtained using structural mechanics-based calculations (AutoSRCbeam) from 15 input parameters in red cells that are calculated by an ANN-based Hong-Lagrange algorithm while all 10 equalities and 16 inequalities given in Table 5 are   kN⋅m calculated from structural mechanicsbased calculations, respectively.This study also demonstrated tensile rebar strains (ε rt ) which were calculated at 0.0055 and 0.0054, obtained by ANNs and structural mechanics-based calculations, respectively, leading to an insignificant error of 1.82%.(5) The main emphasis of this study is directed toward investigating the optimization of SRC beams.However, it is intended that future studies will expand the optimization of all types of structures including structural frames, thus providing a more comprehensive optimization designs for wider applications.(6) A limitation of this study is computational complexity, as high-performance computers are needed to generate large datasets and train ANNs.
(a) and (b), respectively, generated by AutoSRCbeam.Inputs of structural mechanics-based calculation including 15 parameters (L, d, b, f y

Figure 2 .
Figure 2. Flowchart for generating big data of SRC beams used to train network (Nguyen and Hong 2019).

Figure 4 .
Figure 4. ANN-based Hong-Lagrange optimization algorithm of five steps based on unified functions of objectives (UFO) (Hong and Le 2022).
Pareto frontier for the three objective functions (CI b, CO 2 , and W).

Figure 6 (
a) shows three-dimensional Pareto frontier for three objective functions denoted by 343 red dots and 133,711 green design points generated by structural mechanics-based calculations.Observation shows that a Pareto frontier using ANN-based Hong-Lagrange algorithm is well located at the lower boundary of 133,711 random designs calculated by AutoSRCbeam.Four SRC beam designs are shown in Figure 6(a) based on four specific cases for optimizing three objective functions, where Design cases 1 and 2 only optimize SRC beams, yielding minimum cost (CI b ) and CO 2 emissions, respectively.Design case 3 only optimizes weight (W), whereas Design case 4 evenly optimizes
all three objective functions (CI b , CO 2 , and W).The 343 Pareto points are generated with different trade-off ratios among three objective functions, also optimizing three objective functions simultaneously, according to designated trade-off ratios.In Figure6(b-d), Pareto frontiers are projected on CI b -W, CO 2 -W, and CI b -CO 2 planes, respectively, where the trade-offs used for design points among three objective functions are shown.4.3.2.2.Design case 1. Design case 1 only optimizes costs (CI b ) or single objective function CI b , with corresponding weight factors (w CI b : w CO 2 : w W ¼ 1 : 0 : 0) assigned to three objective functions (CI b , CO 2 , and W), as indicated in Figure 6(b,d).Design case 1 considers a trade-off only contributed by costs (CI b ), ignoring an interest in CO 2 emissions and weights (W).Design parameters (b = 383 mm, h s = 380 mm, b s = 150 mm, ρ rt = 0.0152, ρ rc = 0.0114) are obtained while all ten equalities and 16 inequalities given in Table 5 are satisfied.Optimized cost (CI b ) is obtained at 211,987 KRW/m, which is the minimum cost of all Pareto points, as clearly shown in Figure 6(b,d).CO 2 emissions and weights (W) for Design case 1 correspond to 0.3927 t-CO 2 /m and 10.698 kN/m, respectively.4.3.2.3.Design case 2. Design case 2 only optimizes CO 2 emissions or single objective function CO 2 , with corresponding weight factors (w CI b : w CO 2 : w W ¼ 0 : 1 : 0) for three objective functions (CI b , CO 2 , and W), as illustrated in Figure 6(c,d).Design case 2 considers a trade-off only contributed by CO 2 emissions, ignoring an interest in costs CI b and weights (W).Design parameters (b = 404.4mm, h s = 380 mm, b s = 150 mm, ρ rt = 0.0144, ρ rc = 0.0100) are obtained while all ten equalities and 16 inequalities given in Table 5 are satisfied.Optimized CO 2 emission is obtained at 0.3920 t-CO 2 /m, which is the minimum CO 2 emissions of all Pareto points, as shown in Figure 6(c,d).Costs (CI b ) and weights (W) for Design case 2 correspond to 212,325 KRW/m and 11.184 kN/m, respectively.The SRC beam designed with Design cases 1 and 2 are largest in volume to use more concrete, while reducing costs (CI b ) and CO 2 emissions compared with Design cases 3 and 4. 4.3.2.4.Design case 3. The minimum beam width (b = 0.3d = 0.3 × 950 = 285 mm) is recommended for Design case 3, which optimizes single objective function weights (W) based on an inequality constraint (IC 9 ), shown in Table 5, resulting in lightest beam weights W = 8.628 kN/m, as
w h : w CI b : w Weight ¼ 1=3 : 1=3 : 1=3 are implemented evenly in Design case 4, which optimizes all three objective functions at the same time with the equivalent contributions.Observation shows that Design case 4 is located near Design case 3 which only optimizes weight (W), indicating median values of CI b = 219,279 KRW/m and W = 8.757 kN/ m obtained between Design case 1 (or Design case 2) and Design case 3, as shown in Figure 6.ANN-based Hong-Lagrange algorithm calculates b = 292.4mm for Design case 4 with three equal weight fractions (w h : w CI b : w Weight ¼ 1=3 : 1=3 : 1=3), which is the most favorable value of beam widths that a proposed method finds when three objective functions are optimized evenly.Beam width of b = 292.4mm for Design case 4 is close to a minimum of b = 285 mm for Design case 3.
met.A cost of an SRC beam is obtained as 219,279.1 KRW/m by an ANN-based Hong-Lagrange algorithm which is close to 219,583.6 KRW/m calculated by

Table 1 .
Forward design scenario for SRC beam.

Table 2 .
(Hong and Nguyen 2022aes of parameters defining SRC beams(Hong and Nguyen 2022a).

Optimized objective functions using ANN- based Hong-Lagrange algorithm 4.2.1. Derivation of objective functions based on forward neural networks
Derivation of objective functions including cost (CI b ), CO 2 emissions, and weight (W) based on forward ANNs is described in this section, where the relationships among 15 input parameters(L, d, b, fy , f' c , ρ sc , ρ st , h s , b s , t f , t w , f yS , Y s , M D , M L ) and 11 output parameters (ϕM n , ε rt , ε st , Δ imme., Δ long , μ ϕ , CI b , CO 2 , W, X (Hong and Nguyen 2022a;Krenker, Bešter, and Kos 2011;Villarrubia et al. 2018)ut parameters to each output parameter through an activation function which yields nonlinear behaviors of the objective functions as indicated in Equations (1)-(3))(Hong and Nguyen 2022a;Krenker, Bešter, and Kos 2011;Villarrubia et al. 2018).ANNs with multilayer perceptron trained using PTM are based on 4 layers and 64 neurons, as shown in Table 4.An activation function tansig is used in this paper, as expressed in Equation (4).where X; input parameters, X = L; d; b; f y ; f 0 c ; ρ sc ; ρ st ; h s ; b s ; � t f ; t w ; f yS ; Y s ; M D ; M L � T g N ; g D ; normalizing and de-normalizing functions.

Table 6 .
Multiple-objective optimization using ANN-based Hong-Lagrange algorithm based on w CI b ð Þ : w CO 2 ð Þ : w W ¼ 1=3 : 1=3 : 1=3.structural mechanics-based calculation, indicating an insignificant error of −0.14%.Insignificant errors in CO 2 emissions and weight are also caused at 2.38% and 0.23% between an ANN-based Hong-Lagrange algorithm and structural mechanics-based calculations, respectively.An insignificant error of 0.86% is observed when comparing design moment (ϕM n ) of 3031.9 kN⋅m calculated from ANNs with 3005.8 kN⋅m obtained from structural mechanics-based calculations, as presented in Table 6.Tensile rebar strain (ε n ) is calculated at 0.0055 and 0.0054, obtained by ANNs and structural mechanicsbased calculations, respectively, leading to an insignificant error of 1.82%.ANNs and structural mechanics-based calculation provide immediate deflections (Δ imme. ) of 5.96 mm and 5.67 mm, respectively, yielding a relative error of 4.87%, indicating only a difference of 0.29 mm.Long-term deflection (Δ long ) of 12.07 and 11.44 mm obtained by ANNs and structural mechanics-based calculation also indicate a negligible difference of 0.63 mm with an error of 5.22%.It is noted that deflections limited in accordance with ACI 318-19 code are 27.8 mm (L/360) and 41.7 mm (L/240) for immediate deflection (Δ imme. ) and long-term deflection (Δ long ), respectively.Observation shows that immediate deflection (Δ imme. ) and long-term deflection (Δ long ) obtained by ANN-based Hong-Lagrange algorithm meet ACI 318-19 standard.Table 6 demonstrates a practical methodology for designing SRC beams using ANNs.It also presents an accuracy of the ANN-based Hong-Lagrange algorithm in calculating SRC beams with a maximum error of −2.56% among all parameters except for errors of immediate (Δ imme. ) and long-term deflection (Δ long ) reaching 4.87% and 5.22%, respectively.