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

Pareto frontier for steel-reinforced concrete beam developed based on ANN-based Hong-Lagrange... JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING https://doi.org/10.1080/13467581.2023.2193621 BUILDING STRUCTURES AND MATERIALS Pareto frontier for steel-reinforced concrete beam developed based on ANN- based Hong-Lagrange algorithm Won-Kee Hong and Dinh Han Nguyen Department of Architectural Engineering, Kyung Hee University, Yongin, Republic of Korea ABSTRACT ARTICLE HISTORY Received 14 November 2022 Multi-objective optimization (MOO) is always a challenging issue for engineers in the field of Accepted 17 March 2023 structural engineering, where several objective functions must be satisfied under equality and inequality constraints to meet requirements imposed by engineers and decision-makers. This KEYWORDS study proposes a novel approach to solve MOO problems for steel-reinforced concrete (SRC) ANN-based Hong-Lagrange beams using an artificial neural network (ANN)-based Hong-Lagrange algorithm. Proposed algorithm; artificial neural method in this paper optimizes three specific objective functions, including cost (CI ), CO networks; multi-objective b 2 emissions, and beam weight (W), simultaneously. Neural networks are trained by 200,000 optimization; steel- reinforced concrete beams; samples, which are randomly generated by structural mechanics-based calculations, to derive unified objective function three specific objective functions. Unified objective function is, then, proposed based on weight fractions of each objective function. An ANN-based Hong-Lagrange technique identi- fies 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 algo- rithm 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. 1. Introduction and signification of current concrete (RC) beams is presented by Shariat et al. study (2018) using a computational Lagrangian Multiplier method. However, calculations of steel-reinforced con- 1.1. Introduction crete (SRC) members are much more complex than In the area of structural engineering, an optimization common RC structures due to the contribution of a design has always been a challenging issue over the H-shaped steel section embedded inside concrete last few decades. Four major categories of structural material, thus, being expensive for calculations of optimization can be classified, such as cost reduction, strengths, deflections, and flexibility. environmental impact reduction, structural perfor- SRC members have been used in various types of mance improvement, and multi-objective optimization infrastructure facilities, such as buildings, parking, and (MOO), according to a study by Mei and Wang (2021). A transportation. A sustainability of SRC structures study of an optimization of rectangular reinforced increases significantly due to the protection of CONTACT Won-Kee Hong hongwk@khu.ac.kr Department of Architectural Engineering, Kyung Hee University, Yongin 17104, Republic of Korea © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the Architectural Institute of Japan, Architectural Institute of Korea and Architectural Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 W.-K. HONG AND D. H. NGUYEN concrete outside, preventing high temperature and are optimized for cost and CO emissions (Kaveh, corrosive agents of the surrounding environment for Mottaghi, and Izadifard 2022), and optimal design of H-shaped steel sections. Steel, rebar, and concrete RC frames considering CO emissions is achieved via work simultaneously to prevent damage to the struc- metaheuristic algorithms by Kaveh, Izadifard, and tures when SRC members are subjected to imposed Mottaghi (2020). A Pareto frontier derived based on load. Many studies have been performed to investigate an ANN-based Hong-Lagrange algorithm to simulta- the behaviors of SRC members under applied loads (Li neously optimize designs for three objective functions and Matsui 2000; El-Tawil and Deierlein 1999; Bridge (CI , CO , W) is well compared with the lower boundary b 2 and Roderick 1978; Mirza, Hyttinen, and Hyttinen 1996; of large datasets of random designs generated by Ricles and Paboojian 1994; Furlong 1974; Mirza and structural mechanics-based calculations Skrabek 1992; Chen and Lin 2006; Dundar et al. 2008; (AutoSRCbeam). This research provides a powerful Munoz and Hsu 1997; Rong and Shi 2021; Virdi, tool for the next generation in optimizing multiple Dowling, and BS 449, & BS 153 1973, Roik and objective functions for SRC beams that meet any Bergmann 1990; Brettle 1973). SRC members’ capacity loads and design standards. Both equalities and can be calculated by conventional structural inequalities are taken into consideration based on mechanics-based calculations, whereas an artificial engineers’ requirements and regional design neural network (ANN)-based design method has been standards. developed to design structural members that does not Large datasets used to train neural networks are require complex calculations. Many researchers suc- obtained by structural mechanics-based calculations, cessfully applied ANNs in structural analysis, including called AutoSRCbeam, which was developed by authors notable studies of Abambres and Lantsoght 2020, in previous studies (Nguyen and Hong 2019; Hong Sharifi, Lotfi, and Moghbeli (2019), Asteris et al. 2019). The design accuracy of SRC beams using an (2019), and Armaghani et al. (2019). The potential of ANN-based Hong-Lagrange algorithm is verified both ANNs in civil engineering is also demonstrated by the through structural analysis and large structural data- studies of Kaveh et al. (Kaveh and Khalegi 1998; Kaveh sets as shown in the previous study (Hong and Nguyen and Servati 2001), which show that ANNs can predict 2022a). concrete strength and design double layer grids with This study requires a large amount of data, and results comparable to traditional methods. hence, neural networks require a large amount of The authors perform several studies for an optimiza- data to train effectively, which can be a challenge tion of either single objective functions or multi-objec- when using small computers. Computational com- tive functions for RC members in previous studies (Hong plexity is also challenging in that training neural and Nguyen 2022b; Hong, Nguyen, and Nguyen 2022; networks can be computationally intensive, requiring Hong, Nguyen, and Pham 2022; Hong and Le 2022; high-performance hardware. However, this study has Hong 2021; Nguyen and Hong 2019). The authors also developed methods to mitigate some of the limita- published a study “An AI-based auto-design for optimiz- tions listed above. The main limitation of this study ing RC frames using the ANN-based Hong–Lagrange is computational complexity as high-performance algorithm” in which ANNs-based objective functions computers are needed to generate large datasets such as costs and weights of RC frames with four-by- and train ANNs. four bays and four floors are optimized simultaneously based on big datasets of 330,000 designs in accordance 1.2. Research significances with ACI 318-19, whereas corresponding design para- meters which minimize objective functions are also A forward design includes 15 input parameters (L, d, b, f , obtained (Hong and Pham 2023). f’ , ρ , ρ , h , b , t , t , f , Y , M , M ) and 11 outputs (ϕM , c sc st s s f w yS s D L n Multi-objective functions for SRC beams including ε , ε , Δ , Δ , µ , CI , CO emission, W, X , SF), as rt st imme long ϕ b 2 s cost (CI ), CO emissions, and weight (W) are presented shown in Table 1. No studies were found to present b 2 in this study, presenting a key value of an ANN-based based on ANNs to optimize the three objective functions Hong-Lagrange algorithm for an application to a struc- including CI , CO emission, and W at the same time for b 2 tural design for engineers. MOO is always a challenging an SRC beam. In this study, the authors present an ANN- issue for engineers in the field of structural engineer- based Hong-Lagrange algorithm, which is used to opti- ing, where several objective functions must be satis- mize three objective functions (CI , CO emission, and W) b 2 fied under equality and inequality constraints to meet simultaneously for an SRC beam. requirements imposed by engineers and decision- This study presents a hybrid network using an ANN makers. Kaveh et al. proposed a genetic algorithm for and Hong-Lagrange algorithm to optimize an SRC optimal design of RC retaining walls (Kaveh, Kalateh- beam, capable of simultaneously optimizing three Ahani, and Fahimi-Farzam 2013), and a vibrating parti- objective functions (CI , CO emission, and W) with b 2 cles system (VPS)-based algorithm for MOO (Kaveh and significant accuracies. A unified objective function Ilchi Ghazaan 2020). RC bridge columns and bent caps (called UFO) is developed for an SRC beam in this JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 3 Table 1. Forward design scenario for SRC beam. Forward design scenario Input parameters Output parameters 1 L 9 b 1 ϕM 9 W s n 2 d 10 t 2 ε 10 X f rt s 3 b 11 t 3 ε 11 SF w st 4 f 12 f 4 Δ y yS imme. 5 f ’ 13 Y 5 Δ c s long 6 14 M 6 µ ρ D ϕ rt 7 ρ 15 M 7 CI rc L b 8 h 8 CO s 2 study (Hong et al. 2022). UFO is formulated based on shown in Table 1. Fifteen input parameters include three objective functions via weighted fractions, inte- length of beam (L), beam dimensions (b, d), material grating them into one objective function to simulta- strengths of concrete (f’ ), rebar (f ), steel (f ), com- c y yS neously optimize all objective functions (CI , CO pressive and tensile rebar ratio (ρ , ρ ), steel height (h ), b 2 c t s emission, and W). A Pareto frontier (also called Pareto steel flange (b ), steel web thickness (t ), steel flange s f front) for an SRC beam is constructed based on a thickness (t ), moment due to dead load (M ), and w D combination of MOO results. A contribution of each moment due to live load (M ). Eleven output para- objective function is represented by the trade-off of meters include design moment capacity (ϕM ) exclud- their weight fractions, selected by an interest of engi- ing beam weight, tensile strains of steel and rebar (ε , st neers and decision-makers. This study is a stepping- ε ), immediate and long-term deflections (Δ , Δ ), rt imme long stone for a design of next generation, not based on curvature ductility (µ ), materials and manufacture cost structural mechanics but based on ANNs. An ANN- (CI ) per 1 m length, CO emission per 1 m length, b 2 based Hong-Lagrange algorithm simultaneously opti- beam weight (W) per 1 m length, horizontal clearance mizes multi-objective functions for engineers and deci- (X ), and safety factor (SF). A cost (CI ) for materials and s b sion-makers. manufacture, CO emissions, and beam weights (W) are selected as multiple objective functions for an optimization of an SRC beam in the present study. 2. ANN-based design scenarios for steel- reinforced concrete beams 3. Generation of large datasets Figure 1 demonstrates the geometry of an SRC beam, A structural mechanics-based calculation called including beam section (h, b) and steel H-shaped sec- AutoSRCbeam is used to generate 200,000 datasets. tion (h , b , t and t ), which is encased in concrete AutoSRCbeam is established based on an algorithm as s s f w material. A forward design scenario for an SRC beam indicated in Figure 2. This program was developed by is presented based on 15 inputs and 11 outputs, as the authors in the previous study (Nguyen and Hong Figure 1. Geometry of SRC beams (Hong, Nguyen, and Nguyen 2022). 4 W.-K. HONG AND D. H. NGUYEN Figure 2. Flowchart for generating big data of SRC beams used to train network (Nguyen and Hong 2019). 2019). Fifteen input parameters for AutoSRCbeam are According to ACI 318–19 (Standard 2019), deflec - randomly selected in designated ranges, randomly pro- tions are limited to L/360 for immediate deflection viding 11 output parameters. Ranges for dimensions of (Δ ) and L/240 for long-term deflection (Δ ). In imme. long SRC beams are designated from 6000 to 12,000 mm, 500 the preliminary design stage, beam sections are to 1500 mm, and 0.3d to 0.8d for beam length (L), beam unknown; thus, the design moment capacity of a height (h), and beam width (b), respectively, where d is beam (ϕM ) is formulated by excluding a self-weight effective beam depth in a range of 406.1–1444.5 mm, as of a beam when generating large datasets. Factored referred to in Table 2. The dimensions of H-shaped steel moment (M ) represents a magnitude of externally section are randomly selected in appropriate ranges of applied moments, calculated by load combination 0.4–0.6d for steel section height (h ), 0.3–0.6b for steel of M and M with load factors (M = 1.2M + s D L u D section width (b ), and 5–25 mm for both steel web and 1.6M ). A safety factor represents how safe beam is s L flange thickness (t and t ), as shown in Table 2. Material against applied loads, which is calculated as a ratio w f strengths of beam components are chosen in ranges of between a design moment strength and factored 30–50 MPa, 500–600 MPa, and 275–325 MPa for con- moment (SF =ϕM /M ), and the safety factor must n u crete (f’ ), rebar (f ), and steel (f ), respectively. not be smaller than 1.0. c y yS Compressive rebar ratio is randomly chosen in a range Original and normalized large datasets of 200,000 are of 1/400 ~ 1.5ρ , where ρ is tensile rebar ratio with a shown in Table 3(a) and (b), respectively, generated by rt rt pffiffiffiffi � minimum ρ ¼ max 0:25 f =f ; 1:4=f , following AutoSRCbeam. Inputs of structural mechanics-based cal- y y rt;min c ACI standard (Standard 2019). Notations and ranges of culation including 15 parameters (L, d, b, f , f’ , ρ , ρ , h , b , y c sc st s s fifteen input parameters defining an SRC beam are indi- t , t , f , Y , M , M ) are randomly selected, yielding eleven f w yS s D L cated in Table 2. ACI 319-18 code requests tensile rebar corresponding output parameters (ϕM , ε , ε , Δ , Δ , n rt st imme. long strain (ε ) greater than 0.003 + ε (ε is yield strain of μ , CI , CO , W, X , SF) for design SRC beams. Mean, maxima, rt ty ty ϕ b 2 s rebar) to ensure enough ductility of beam. and minima of overall 26 parameters based on 200,000 This study focuses on SRC beams with fixed-fixed datasets are indicated in Table 3. All input and output end conditions, as illustrated in Figure 3. SRC beams parameters are normalized in a range from −1 to 1 by are subjected to uniform loads, including dead and using MAPMINMAX function of MATLAB (MathWorks live loads, yielding M and M , respectively. 2022a), as shown in Table 3. D L JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 5 Table 2. Nomenclatures and ranges of parameters defining SRC beams (Hong and Nguyen 2022a). Notation Range Input L (mm) Beam length [6000 ~12000] parameters d (mm) Effective beam depth [406.1 ~1445.5] b (mm) Beam width [0.3 ~0.8]d h (mm) H-shaped steel height [0.4 ~0.6]d b (mm) H-shaped steel width [0.3 ~0.6]b t (mm) H-shaped steel flange thickness [5 ~25] t (mm) H-shaped steel web thickness [5 ~25] f’ (MPa) Concrete strength [30 ~50] f (MPa) Rebar strength [500 ~600] f (MPa) Steel strength [275 ~325] yS � � pffiffiffi ρ Tensile rebar ratios rt 0:25 f c 1:4 ρ ¼ max ; rt;min f f y y ρ Compressive rebar ratios [1/400 ~1.5] ρ rc rt Ys Vertical clearance h i M (kN·m) Moment due to dead load ~ 1 0:2 M (M = 1.2M + u D 1:2 1.6M ) M (kN·m) Moment due to service live load ðM 1:2M Þ L u D 1:6 (M = 1.2M + 1.6M ) u D L Output ϕM (kN·m) Design moment without considering effect of self-weight at ε = 0.003 n c parameters μ Curvature ductility, μ ¼ ϕ =ϕ ϕ ϕ u y Where, ϕ : Curvature at ε = 0.003 ϕ : Curvature at tensile rebar yield ε Tensile rebar strain at ε = 0.003 ε ≥0.003+ ε rt c rt ty ε Compressive rebar strain at ε = 0.003 rc c Δ Immediate deflection due to M service live load (Δ ≤ L/360, ACI 318– imme. L imme. 19) Δ Sum of long-term deflection due to sustained loads and immediate deflection due to (Δ ≤ L/240, ACI 318–19) long long additional live load CI (KRW/m) Cost index per 1 m length of beam CO (t-CO / CO emission per 1 m length of beam 2 2 2 m) W (kN/m) Beam weight per 1 m length of beam X Horizontal clearance SF Safety factor (ϕM /M ) n u Figure 3. Fixed-fixed steel reinforced concrete (SRC) beams optimized using ANN (Hong, Nguyen, and Nguyen 2022). networks are formulated independently when 15 4. Multi-objective optimization using ANN- input parameters (L, d, b, f , f’ , ρ , ρ , h , b , t , t , based Hong-Lagrange Algorithm for steel- y c sc st s s f w f , Y , M , M ) are mapped to each of 11 output reinforced concrete beams yS s D L parameters (ϕM , ε , ε , Δ , Δ , μ , CI , CO , W, n rt st imme. long ϕ b 2 4.1. Training artificial neural networks based on X , SF), as demonstrated in. Table 4 presents the parallel training method training results of eleven networks using PTM, based on 200,000 datasets that are divided into In this study, ANNs are trained using a parallel three distinct portions, covering 70% (140,000 data- training technique (PTM) (Hong 2021; Hong, Pham, sets) for training, 15% (30,000 datasets) for and Nguyen 2022). As shown in 11 training 6 W.-K. HONG AND D. H. NGUYEN Table 3. Large datasets of SRC beams generated by AutoSRCbeam. Parameter Data 1 Data 2 Data 3 Data 4 Data 200,000 Mean (μ) Maxima Minima (a) Non-normalized data 200,000 datasets generated by AutoSRCbeam 15 Input parameters for structural mechanics-based (AutoSRCbeam) 1 L (mm) 8400 11900 9900 11050 9850 9995.3 12000.0 8000.0 2 d (mm) 786.0 1097.8 972.1 680.5 910.5 1016.8 1445.5 394.8 3 b (mm) 470 850 430 290 765 620.3 1200.0 150.0 4 h (mm) 160 645 270 155 450 400.3 1180.0 5.0 5 b (mm) 170 395 160 130 230 272.1 720.0 45.0 6 t (mm) 6 7 28 9 11 16.1 30.0 5.0 7 t (mm) 15 27 16 9 5 16.5 30.0 5.0 8 f’ (MPa) 41 22 29 43 23 40.0 60.0 20.0 9 f (MPa) 513 548 423 412 518 500.1 600.0 400.0 10 f (MPa) 296 358 267 289 379 300.1 400.0 200.0 yS 11 ρ 0.01957 0.01855 0.04355 0.01324 0.00337 0.01976 0.0637 0.0024 rt 12 ρ 0.02295 0.02714 0.04649 0.01401 0.00097 0.01944 0.0894 8.16E–06 rc 13 Ys 110.8 137.4 242.4 267.9 231.6 236.7 944.2 60.0 14 M (kN·m) 290.9 4956.5 1809.1 426.3 630.3 2866.8 36670.7 11.3 15 M (kN·m) 691.0 3166.6 1469.9 272.3 133.3 1430.7 23431.2 1.1 11 Output parameters for structural mechanics-based (AutoSRCbeam) 16 ϕM (kN·m) 2865.8 10794.1 6286.4 795.7 1512.6 7284.7 46755.1 56.2 17 ε 0.0107 0.0074 0.0056 0.0114 0.0094 0.0082 0.0462 0.0050 rt 18 ε 0.0081 0.0052 0.002 0.006 0.0064 0.0049 0.0348 −0.0005 st 19 μ 3.45 2.34 2.39 4.76 3.32 2.96 18.62 1.75 20 Δ (mm) 2.15 4.57 1.94 7.00 0.87 2.35 26.15 0.10 imme. 21 Δ (mm) 3.73 18.63 5.85 28.90 6.35 10.81 88.29 0.83 long 22 CI (KRW/m) 226654.0 754831.5 515783.1 114789.7 199231.1 501586.6 2295688.1 26576.4 23 CO (t-CO /m) 0.45 1.40 1.06 0.21 0.34 0.936 4.782 0.044 2 2 24 W (kN/m) 11.23 28.97 15.50 5.81 18.57 20.60 60.78 1.99 25 Xs (mm) 150 227.5 135 80 267.5 174.1 415.0 30.0 26 SF 1.97 0.98 1.39 0.84 1.56 1.38 2.00 0.75 (b) Normalized data 200,000 datasets generated by AutoSRCbeam 15 Input parameters for structural mechanics-based (AutoSRCbeam) 1 L (mm) −0.800 0.950 −0.050 0.525 −0.075 −0.002 1.0 −1.0 2 d (mm) −0.255 0.338 0.099 −0.456 −0.018 0.184 1.0 −1.0 3 b (mm) −0.390 0.333 −0.467 −0.733 0.171 −0.104 1.0 −1.0 4 h (mm) −0.729 0.093 −0.542 −0.737 −0.237 −0.322 1.0 −1.0 5 b (mm) −0.630 0.037 −0.659 −0.748 −0.452 −0.327 1.0 −1.0 6 t (mm) −0.600 −0.533 0.867 −0.400 −0.267 0.072 1.0 −1.0 7 t (mm) −0.200 0.760 −0.120 −0.680 −1.000 −0.083 1.0 −1.0 8 f’ (MPa) 0.050 −0.900 −0.550 0.150 −0.850 −0.0003 1.0 −1.0 9 f (MPa) 0.130 0.480 −0.770 −0.880 0.180 0.001 1.0 −1.0 10 f (MPa) −0.040 0.580 −0.330 −0.110 0.790 0.001 1.0 −1.0 yS 11 ρ −0.440 −0.473 0.342 −0.647 −0.969 −0.434 1.0 −1.0 rt 12 ρ −0.487 −0.393 0.040 −0.687 −0.979 −0.565 1.0 −1.0 rc 13 Ys −0.885 −0.825 −0.587 −0.530 −0.612 −0.600 1.0 −1.0 14 M (kN·m) −0.985 −0.730 −0.902 −0.977 −0.966 −0.844 1.0 −1.0 15 M (kN·m) −0.941 −0.730 −0.875 −0.977 −0.989 −0.878 1.0 −1.0 (Continued) JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 7 Table 3. (Continued). 11 Output parameters for structural mechanics-based (AutoSRCbeam) 16 ϕM (kN·m) −0.880 −0.540 −0.733 −0.968 −0.938 −0.690 1.0 −1.0 17 ε −0.723 −0.883 −0.971 −0.689 −0.786 0.001 1.0 −1.0 rt 18 ε −0.513 −0.677 −0.858 −0.632 −0.609 0.092 1.0 −1.0 st 19 μ −0.799 −0.930 −0.924 −0.643 −0.814 −0.857 1.0 −1.0 20 Δ (mm) −0.835 −0.650 −0.852 −0.465 −0.933 −0.820 1.0 −1.0 imme. 21 Δ (mm) −0.934 −0.593 −0.885 −0.358 −0.874 −0.772 1.0 −1.0 long 22 CI (KRW/m) −0.824 −0.358 −0.569 −0.922 −0.848 −0.581 1.0 −1.0 23 CO (t-CO /m) −0.827 −0.428 −0.571 −0.929 −0.876 −0.623 1.0 −1.0 2 2 24 W (kN/m) −0.686 −0.082 −0.540 −0.870 −0.436 −0.367 1.0 −1.0 25 Xs (mm) −0.377 0.026 −0.455 −0.740 0.234 −0.251 1.0 −1.0 26 SF 0.952 −0.632 0.024 −0.856 0.296 0.001 1.0 −1.0 8 W.-K. HONG AND D. H. NGUYEN Figure 4. ANN-based Hong-Lagrange optimization algorithm of five steps based on unified functions of objectives (UFO) (Hong and Le 2022). Table 4. Training results based on PTM method (Hong and Nguyen 2022a). Training with an Validation Suggested Best Stopped R at best No. output Data Layers Neurons epoch epoch epoch epoch Test MSE epoch 1 ϕM 200,000 4 64 1000 50,000 48,865 49,865 8.25E–06 1.0000 2 ε 200,000 4 64 1000 50,000 34,638 35,638 5.30E–05 0.9987 rt 3 ε 200,000 4 64 1000 50,000 49,997 50,000 4.17E–05 0.9992 st 4 Δ 200,000 4 64 1000 50,000 49,997 50,000 9.96E–06 0.9997 imme. 5 Δ 200,000 4 64 1000 50,000 49,748 50,000 1.80E–05 0.9997 long 6 μ 200,000 4 64 1000 50,000 35,898 36,898 4.22E–05 0.9989 7 SF 200,000 4 64 1000 50,000 27,140 27,141 1.69E–07 1.0000 8 X 200,000 4 64 1000 50,000 15,079 16,079 2.08E–06 1.0000 9 CI 200,000 4 64 1000 50,000 32,795 33,795 1.47E–06 1.0000 10 CO 200,000 4 64 1000 50,000 21,661 21,661 2.15E–08 1.0000 11 W 200,000 4 64 1000 50,000 49,944 50,000 2.77E–04 0.9996 validation, and 15% (30,000 datasets) for testing strain of tensile rebar (ε ) and several parameters rt data. The lowest and highest test MSE are 2.77E– (ϕM , SF, X , CI , CO ), respectively, as indicated in n s b 2 04 and 1.69E–07 for training weight (W) and safety Table 4. Neural networks are trained based on factor (SF), respectively. The lowest and highest 50,000 and 1000 for suggested and validation regression (R) are 0.9987 and 1.0000 for the training epochs, respectively. Trainings are implemented by 0 0 11 0 0 1 1 B B CC B B B B C C CC D 4 4 3 3 1 1 N 1 3 4 B B B B C C CC CI ¼ g f W f W . . . f W g ðXÞþ b . . .þ b þ b (1) CI lin CI t CI t CI CI CI CI CI bB B b @ b @ b b b A b A b CC |{z} |ffl{zffl} |ffl{zffl} |ffl{zffl} |fflfflffl{zfflfflffl} |{z} |{z} |{z} @ @ AA ½1� 1� ½1� 64� ½64� 64� ½64� 15� ½15� 1� ½64� 1� ½64� 1� ½1� 1� 0 0 0 0 1 1 11 B B B B C C CC D 4 4 3 3 1 1 N 1 3 4 B B B B C C CC CO ¼ g f W f W . . . f W g ðXÞþ b . . .þ b þ b (2) CO lin CO t CO t CO CO CO CO CO 2@ @ 2 @ 2 @ 2 2 2 A 2 A 2AA |{z} |ffl{zffl} |ffl{zffl} |ffl{zffl} |fflfflfflffl{zfflfflfflffl} |ffl{zffl} |ffl{zffl} |{z} ½1� 1� ½1� 64� ½64� 64� ½64� 15� ½15� 1� ½64� 1� ½64� 1� ½1� 1� JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 9 0 0 0 0 1 1 11 B B B B C C CC D L 4 3 3 1 1 N 1 3 4 W ¼ g @f @ W f @ W . . . f @ W g ðXÞþ b A . . .þ b Aþ b AA (3) W lin W t W t W W W W W |{z} |{z} |{z} |{z} |fflffl{zfflffl} |{z} |{z} |{z} ½1� 1� ½1� 64� ½64� 64� ½64� 15� ½15� 1� ½64� 1� ½64� 1� ½1� 1� i i i MATLAB Deep Learning Toolbox™ platform W ; W ; W ; weight matrices (i ¼ 1 4) CI CO W b 2 (MathWorks 2022b). obtained using training PTM. i i i b ; b ; b ; bias matrices (i ¼ 1 4) obtained CI CO W b 2 using training PTM. 4.2. Optimized objective functions using ANN- based Hong-Lagrange algorithm 4.2.2. Ten equality and 16 inequality constraints Table 5 describes 10 equality and 26 inequality 4.2.1. Derivation of objective functions based on constraints to optimize three objective functions forward neural networks (CI , CO , and W) simultaneously. Ten equality con- b 2 Derivation of objective functions including cost (CI ), CO b 2 straints include eight input parameters (L = emissions, and weight (W) based on forward ANNs is 10,0000 mm, d = 950 mm, f = 500 MPa, f’ = 30 y c described in this section, where the relationships among MPa, t = 12 mm, t = 8 mm, f = 500 MPa, Y = 70 f w yS s 15 input parameters (L, d, b, f , f’ , ρ , ρ , h , b , t , t , f , Y , y c sc st s s f w yS s mm) and two applied loads (M = 500 kN⋅m, M = D L M , M ) and 11 output parameters (ϕM , ε , ε , Δ , D L n rt st imme. 1500 kN⋅m). Besides 10 equalities selected, 16 Δ , μ , CI , CO , W, X , SF) are linked by weight and bias long ϕ b 2 s inequality conditions are also selected according matrices obtained by mapping entire input parameters to to ACI 318-19 standard (Standard 2019). A mini- each output parameter through an activation function mum requirement for rebar ratio is expressed by which yields nonlinear behaviors of the objective func- � � pffiffiffi tions as indicated in Equations (1)–(3)) (Hong and Nguyen 0 0:25 f c 1:4 equation ρ ¼ max ; ¼ 0:0028 rt;min f f 2022a; Krenker, Bešter, and Kos 2011; Villarrubia et al. y y 2018). ANNs with multilayer perceptron trained using according to ACI 318-19 code, and maximum PTM are based on 4 layers and 64 neurons, as shown in rebar ratio is selected at 0.05, which is indicated and IC in Table 5, respectively. A Table 4. An activation function tansig is used in this paper, 1 2 by inequality IC maximum rebar ratio of 0.05 is established arbitra- as expressed in Equation (4). rily for generating big data. ACI 318-19 also recom- mends that tensile rebar strain should not be less than 0.003 + f /200,000 = 0.0055 to ensure enough ductility of beams, which is indicated by inequality f ðxÞ ¼ tansigðxÞ ¼ 1 (4) t IC in Table 5. Immediate and long-term deflec - 2x 1þ e tions (Δ and Δ ) are limited by L/360 and L/ imme. long where 240 according to ACI 318-19, respectively, indi- X; input parameters, X = L; d; b; f ; f ; ρ ; ρ ; h ; b ; y sc st s s c cated by inequalities IC and IC , respectively. 13 14 t ; t ; f ; Y ; M ; M � Other equality constraints are also selected as f w yS s D L N D g ; g ; normalizing and de-normalizing functions. shown in Table 5. Ten equalities and 16 Table 5. Equality and inequality constraints imposed by ACI 318-19 for optimization designs (Hong and Nguyen 2022a). Equality conditions Inequality conditions � � pffiffiffi EC L = 10,000 mm IC ≤ ρ 1 1 0:25 f rt c 1:4 ρ ¼ max ; ¼ 0:0028 rt;min f f y y EC d = 950 mm IC ρ ≤ 0.05 2 2 rt EC f = 500 MPa IC ρ /400 ≤ ρ 3 y 3 rt rc EC f’ = 30 MPa IC ρ ≤ ρ 4 c 4 rc rt EC t = 12 mm IC 0.3b ≤ b 5 f 5 s EC t = 8 mm IC b ≤ 0.6b 6 w 6 s EC f = 325 MPa IC 0.4d ≤ h 7 yS 7 s EC Y = 70 mm IC h ≤ 0.6d 8 s 8 s EC M = 500 kN·m IC 0.3d ≤ b 9 D 9 EC M = 1500 kN·m IC b ≤ 0.8d 10 L 10 IC 0.003 + f /200,000 = 0.0055 ≤ ε 11 y rt IC 140 mm ≤ b 12 s IC Δ ≤ L/360 13 imme. IC Δ ≤ L/240 14 long IC 50 mm ≤ X 15 s IC 1.0 ≤ SF 16 10 W.-K. HONG AND D. H. NGUYEN inequalities are used to optimize multi-objective 4.3. Results of multi-objective optimization of functions simultaneously based on MATLAB steel-reinforced concrete beams Global Optimization Toolbox (MathWorks 2022, 4.3.1. Four specific cases for an optimization of 2022c, 2022d, 2022e). each objective function (CI , CO , and W, b 2 respectively) 4.2.3. Derivation of a unified function of objective A Pareto frontier is obtained based on a combination for a steel-reinforced concrete beam of multiple optimized designs for three objective func- A UFO for SRC beams is defined using algorithms tions (CI , CO , and W) with their weight fractions b 2 based on the weighted sum technique (Afshari, (w : w : w Þ for SRC beams in this study. These CI CO W b 2 Hare, and Tesfamariam 2019), which is created by weight fractions (w : w : w Þ represent trade-off CI CO W b 2 integrating three objective functions (CI , CO , and ratios contributed by each of three objective functions b 2 W) with their respective weight fractions to real-life optimizations for engineers and decision- w ; w ; w , as indicated in Equations (5) and makers. The 343 combinations of weight fractions are CI CO W b 2 (6). These fraction weights are in a range from 0 generated for constructing a Pareto frontier, including to 1, whose sum is 1 as shown in Equation (6). four specific cases, as indicated in Figure 5. It is noted Specific trade-offs among objectives are used to that the sum of three weight fractions of each combi- help engineers and decision-makers evaluate a nation is always equivalent to 1. Observation shows design project holistically. Single objective func- that four specific cases include an optimization of each tion for each of CI , CO , and W is a specific case objective function (CI , CO , and W, respectively), b 2 b 2 of UFO when w : w : w = 1:0:0; 0:1:0; 0:0:1, represented by Points 1, 2, and 3 on a Pareto frontier, CI CO W b 2 respectively. A Lagrange function of a UFO shown whereas Point 4 represents an evenly optimized design in Equations (7)–(9), which are based on three for CI , CO , and W, with their weight fraction b 2 objective functions with equality and inequality w : w : w = 1/3:1/3:1/3. Design parameters (b, CI CO W b 2 constraints shown in Table 5 being simultaneously h , b , ρ , ρ ) are obtained by solving MOO using an s s rc rt optimized. The Lagrange function is substituted ANN-based Hong-Lagrange algorithm with 10 equality into the built-in optimization toolbox of MATLAB and 16 inequality conditions, shown in Table 5. (MathWorks 2022) to obtain optimized design parameters. Figure 4 demonstrates a flowchart for 4.3.2. Pareto-efficient designs verified by structural five steps to solve MOO problems, providing datasets detailed descriptions of an algorithm that illus- A Pareto frontier includes 343 optimized designs for SRC trates the authors’ previous study for RC columns beams, indicated by red dots as shown in Figure 6(a–c). (Hong et al. 2022). Design parameters (b, h , b , ρ , ρ ) are calculated for four s s rc rt specific combinations while all 10 equalities (L = 10,000 mm, d = 950 mm, f = 500 MPa, f’ = 30 MPa, f = 325 ANN ANN ANN y c yS UFO ¼ w F ðxÞ ¼ w F ðxÞþ w F ðxÞ i 1 2 i 1 2 MPa, t = 12 mm, t = 8 mm, Y = 70 mm, M = 500 kN⋅m, f w s D ANN ANN ANN and M = 1500 kN⋅m) and 16 inequalities given in Table 5 þw F ðxÞ ¼ w CI ðxÞþ w CO ðxÞ 3 CI CO 3 b b 2 2 are satisfied. þw WANNðxÞ (5) where 4.3.2.1. A three-dimensional Pareto frontier for the three objective functions (CI CO , and W). Figure 6 b, 2 w þ w þ w ¼ 1 CI CO W (a) shows three-dimensional Pareto frontier for 0 � w ; w ; w � 1ðdimensionlessÞ (6) CI CO W three objective functions denoted by 343 red dots and 133,711 green design points generated by Lagrange function utilizing UFO function: structural mechanics-based calculations. Observation shows that a Pareto frontier using T T L ðx; λ ; λ Þ ¼ UFO λ ECðxÞ λ ICðxÞ (7) UFO c v ANN-based Hong-Lagrange algorithm is well located c v at the lower boundary of 133,711 random designs calculated by AutoSRCbeam. Four SRC beam designs ECðxÞ ¼ ½EC ðxÞ; EC ðxÞ; . . . ; EC ðxÞ� ; are shown in Figure 6(a) based on four specific 1 2 m m ¼ 10 : number of equality constraints (8) cases for optimizing three objective functions, where Design cases 1 and 2 only optimize SRC beams, yielding minimum cost (CI ) and CO emis- b 2 ICðxÞ ¼ ½IC ðxÞ; IC ðxÞ; . . . ; IC ðxÞ� ; sions, respectively. Design case 3 only optimizes 1 2 m m ¼ 16 : number of inequality constraints (9) weight (W), whereas Design case 4 evenly optimizes 2 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 11 Figure 5. Weight fractions generated for Pareto frontier. Figure 6. Pareto-efficient designs verified by structural datasets at EC9: MD = 500 kN⋅m; EC10: ML = 1500 kN⋅m. 12 W.-K. HONG AND D. H. NGUYEN Figure 6. (Continued). JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 13 Figure 6. (Continued). all three objective functions (CI , CO , and W). The 4.3.2.3. Design case 2. Design case 2 only opti- b 2 343 Pareto points are generated with different mizes CO emissions or single objective function trade-off ratios among three objective functions, CO , with corresponding weight factors also optimizing three objective functions simulta- (w : w : w ¼ 0 : 1 : 0) for three objective func- CI CO W b 2 neously, according to designated trade-off ratios. tions (CI , CO , and W), as illustrated in Figure 6(c,d). b 2 In Figure 6(b-d), Pareto frontiers are projected on Design case 2 considers a trade-off only contributed CI – W, CO – W, and CI – CO planes, respectively, by CO emissions, ignoring an interest in costs CI b 2 b 2 2 b where the trade-offs used for design points among and weights (W). Design parameters (b = 404.4 mm, three objective functions are shown. h = 380 mm, b = 150 mm, ρ = 0.0144, ρ = 0.0100) s s rt rc are obtained while all ten equalities and 16 inequal- ities given in Table 5 are satisfied. Optimized CO 4.3.2.2. Design case 1. Design case 1 only optimizes emission is obtained at 0.3920 t-CO /m, which is the costs (CI ) or single objective function CI , with corre- b b minimum CO emissions of all Pareto points, as sponding weight factors (w : w : w ¼ 1 : 0 : 0) CI CO W b 2 shown in Figure 6(c,d). Costs (CI ) and weights (W) assigned to three objective functions (CI , CO , and b 2 for Design case 2 correspond to 212,325 KRW/m W), as indicated in Figure 6(b,d). Design case 1 con- and 11.184 kN/m, respectively. The SRC beam siders a trade-off only contributed by costs (CI ), ignor- designed with Design cases 1 and 2 are largest in ing an interest in CO emissions and weights (W). volume to use more concrete, while reducing costs Design parameters (b = 383 mm, h = 380 mm, b = s s (CI ) and CO emissions compared with Design b 2 150 mm, ρ = 0.0152, ρ = 0.0114) are obtained while rt rc cases 3 and 4. all ten equalities and 16 inequalities given in Table 5 are satisfied. Optimized cost (CI ) is obtained at 211,987 KRW/m, which is the minimum cost of all 4.3.2.4. Design case 3. The minimum beam width (b = Pareto points, as clearly shown in Figure 6(b,d). CO 0.3d = 0.3 × 950 = 285 mm) is recommended for Design emissions and weights (W) for Design case 1 corre- case 3, which optimizes single objective function weights spond to 0.3927 t-CO /m and 10.698 kN/m, (W) based on an inequality constraint (IC ), shown in Table 2 9 respectively. 5, resulting in lightest beam weights W = 8.628 kN/m, as 14 W.-K. HONG AND D. H. NGUYEN Table 6. Multiple-objective optimization using ANN-based Hong-Lagrange algorithm based on wðCI Þ : wðCO Þ : w ¼ 1=3 : 1=3 : 1=3. b 2 W shown in Figure 6(b,c). However, tensile and compressive Lagrange algorithm calculates b = 292.4 mm for Design rebar ratios (ρ , ρ ), and height of steel H-shaped (h ) are case 4 with three equal weight fractions rt rc s 0.0225, 0.0225, and 408.8 mm, respectively, larger than (w : w : w ¼ 1=3 : 1=3 : 1=3), which is the most h CI Weight those of the Design cases 1 and 2 in order to decrease favorable value of beam widths that a proposed method beam weights W. Beam capacity increases to meet the finds when three objective functions are optimized requirement of SF ≥ 1 when single objective function W is evenly. Beam width of b = 292.4 mm for Design case 4 is optimized as indicated in Figure 6(b,c). Costs and CO close to a minimum of b = 285 mm for Design case 3. emissions obtained in Design case 3 are 224,406 KRW/m and 0.4265 t-CO /m, respectively, larger than those 4.3.3. Verifications obtained in Design cases 1 and 2. Table 6 summarizes designs when an ANN-based Hong- Lagrange algorithm evenly minimizes three objective 4.3.2.5. Design case 4. Weight fractions w : w : functions CI , CO , and W based on w : w : w = 1/ h CI b b 2 CIb CO2 W w ¼ 1=3 : 1=3 : 1=3 are implemented evenly in 3:1/3:1/3. Green cells in Table 6 show 11 output para- Weight Design case 4, which optimizes all three objective func- meters obtained using structural mechanics-based calcu- tions at the same time with the equivalent contributions. lations (AutoSRCbeam) from 15 input parameters in red Observation shows that Design case 4 is located near cells that are calculated by an ANN-based Hong-Lagrange Design case 3 which only optimizes weight (W), indicating algorithm while all 10 equalities and 16 inequalities given median values of CI = 219,279 KRW/m and W = 8.757 kN/ in Table 5 are met. A cost of an SRC beam is obtained as m obtained between Design case 1 (or Design case 2) and 219,279.1 KRW/m by an ANN-based Hong-Lagrange algo- Design case 3, as shown in Figure 6. ANN-based Hong- rithm which is close to 219,583.6 KRW/m calculated by JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 15 structural mechanics-based calculation, indicating an This study provides a robust steppingstone for next insignificant error of −0.14%. Insignificant errors in CO generation in optimizing multiple objective functions emissions and weight are also caused at 2.38% and 0.23% with the trade-off ratios that are beneficial for engi- between an ANN-based Hong-Lagrange algorithm and neers and decision-makers. The current investigation structural mechanics-based calculations, respectively. An does not involve an experimentation but is based on insignificant error of 0.86% is observed when comparing ACI 318-19. It develops an AI-based program capable design moment (ϕM ) of 3031.9 kN⋅m calculated from of not only automating routine structural design tasks ANNs with 3005.8 kN⋅m obtained from structural but also improving design efficiency. To the best of the mechanics-based calculations, as presented in Table 6. authors’ knowledge, such a software is not yet avail- Tensile rebar strain (ε ) is calculated at 0.0055 and able for practical structural design applications; thus, 0.0054, obtained by ANNs and structural mechanics- the current design procedures are largely reliant on based calculations, respectively, leading to an insignifi - iterative methods. The following conclusions are cant error of 1.82%. ANNs and structural mechanics-based drawn from this study. calculation provide immediate deflections (Δ ) of 5.96 imme. mm and 5.67 mm, respectively, yielding a relative error of (1) This study aims to optimize multi-objective 4.87%, indicating only a difference of 0.29 mm. Long-term functions for SRC beams simultaneously, which deflection (Δ ) of 12.07 and 11.44 mm obtained by is interesting to engineers and decision-makers. long ANNs and structural mechanics-based calculation also The proposed method can be extended into indicate a negligible difference of 0.63 mm with an error various fields in real life. of 5.22%. It is noted that deflections limited in accordance (2) A unified function of objectives (UFO) is estab- with ACI 318-19 code are 27.8 mm (L/360) and 41.7 mm lished by integrating three objective functions (L/240) for immediate deflection (Δ ) and long-term consisting of costs (CI ), CO emissions, and imme. b 2 deflection (Δ ), respectively. Observation shows that weights (W) into one with their trade-offs of long immediate deflection (Δ ) and long-term deflection weight factions. MOO based on UFO is imple- imme. (Δ ) obtained by ANN-based Hong-Lagrange algorithm mented under 10 equality and 16 inequality long meet ACI 318–19 standard. Table 6 demonstrates a prac- constraints, which are used to impose design tical methodology for designing SRC beams using ANNs. requirements based on MATLAB Global It also presents an accuracy of the ANN-based Hong- Optimization Toolbox. Lagrange algorithm in calculating SRC beams with a max- (3) A Pareto frontier constructed based on 343 opti- imum error of −2.56% among all parameters except for mized designs for three objective functions (CI , errors of immediate (Δ ) and long-term deflection CO , W) is verified by large datasets, yielding a imme. 2 (Δ ) reaching 4.87% and 5.22%, respectively. good comparison with a lower boundary of long 133,711 designs randomly generated based on structural mechanics-based calculations (AutoSRCbeam) for preassigned external loads, 5. Conclusions M = 500 kN⋅m, M = 1500 kN⋅m. D L In this study, an ANN-based Hong-Lagrange algorithm (4) A robust and sustainable tool for engineers is is proposed for simultaneously optimizing multi-objec- proposed in this study to design SRC beams tive functions of an SRC beam that can be widely used using ANN-based Hong-Lagrange algorithm, by engineers. Engineers and decision-makers can use which optimizes multi-objective functions (CI , the method presented in this study as a guideline for CO emissions, and W) of an SRC beam simulta- optimizing multi-objective functions in the practical neously under external loads. Design case 4 of design of any structures. A UFO of SRC beams with a this study evenly minimizes three objective fixed-fixed condition is optimized, satisfying various functions CI , CO , and W (w : w : w = 1/ b 2 CIb CO2 W standard restrictions simultaneously, showing key 3:1/3:1/3) based on an ANN-based Hong- values of the proposed method. Equality and inequal- Lagrange algorithm. The ANN-based Hong- ity constraints can be established by an interest of Lagrange algorithm and structural mechanics- engineers to reflect their own design requirements. A based calculation yielded costs of 219,279.1 Pareto frontier consists of 343 optimized designs in KRW/m and 219,583.6 KRW/m, respectively, this study, minimizing three objective functions (CI , with an insignificant error of −0.14. Errors CO , and W) simultaneously with trade-off weight frac- between ANN-based Hong-Lagrange algorithm tions. A Pareto frontier is verified by the lower bound- and structural mechanics-based calculations ary of large datasets including 133,711 designs were also insignificant as 2.38% and 0.23% for randomly generated based on structural mechanics- CO emissions and weight, respectively. It is based calculations. The proposed ANN-based Hong- noted that the error for design moment (ϕM ) Lagrange algorithm can be extended to various fields was 0.86%, based on design moment (ϕM ) of in real-world designs. 3031.9 kN⋅m calculated from ANNs and 3005.8 16 W.-K. HONG AND D. H. NGUYEN kN⋅m calculated from structural mechanics- References based calculations, respectively. This study also Abambres, M., and E. O. Lantsoght. 2020. “Neural Network- demonstrated tensile rebar strains (ε ) which rt Based Formula for Shear Capacity Prediction of One-Way were calculated at 0.0055 and 0.0054, obtained Slabs Under Concentrated Loads.” Engineering Structures by ANNs and structural mechanics-based calcu- 211: 110501. doi:10.1016/j.engstruct.2020.110501. lations, respectively, leading to an insignificant Afshari, H., W. Hare, and S. Tesfamariam. 2019. “Constrained Multi-Objective Optimization Algorithms: Review and error of 1.82%. Comparison with Application in Reinforced Concrete (5) The main emphasis of this study is directed Structures.” Applied Soft Computing 83: 105631. doi:10. toward investigating the optimization of SRC 1016/j.asoc.2019.105631. beams. However, it is intended that future stu- Armaghani, D. J., G. D. Hatzigeorgiou, C. Karamani, A. dies will expand the optimization of all types of Skentou, I. Zoumpoulaki, and P. G. Asteris. 2019. “Soft structures including structural frames, thus pro- Computing-Based Techniques for Concrete Beams Shear viding a more comprehensive optimization Strength.” Procedia Structural Integrity 17: 924–933. doi:10. 1016/j.prostr.2019.08.123. designs for wider applications. Asteris, P. G., D. J. Armaghani, G. D. Hatzigeorgiou, C. G. (6) A limitation of this study is computational com- Karayannis, and K. 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Foundation of Korea (NRF) grant funded by the Korean gov- 1016/j.jcsr.2005.04.021. ernment [MSIT 2019R1A2C2004965]. Dundar, C., S. Tokgoz, A. K. Tanrikulu, and T. Baran. 2008. “Behaviour of Reinforced and Concrete-Encased Composite Columns Subjected to Biaxial Bending and Notes on contributors Axial Load.” Building and Environment 43 (6): 1109–1120. doi:10.1016/j.buildenv.2007.02.010. Dr. Won-Kee Hong is a Professor of Architectural Engineering El-Tawil, S., and G. G. Deierlein. 1999. “Strength and Ductility at Kyung Hee University. Dr. Hong received his Master’s and of Concrete Encased Composite Columns.” Journal of Ph.D. degrees from UCLA, and he worked for Englelkirk and Structural Engineering 125 (9): 1009–1019. doi:10.1061/ Hart, Inc. (USA), Nihhon Sekkei (Japan) and Samsung (ASCE)0733-9445(1999)125:9(1009). Engineering and Construction Company (Korea) before join- Furlong, R. W. 1974. “Concrete Encased Steel Columns— ing Kyung Hee University (Korea). He also has professional Design Tables.” Journal of the Structural Division 100 (9): engineering licenses from both Korea and the USA. Dr. Hong 1865–1882. doi:10.1061/JSDEAG.0003878. has more than 30 years of professional experience in struc- Hong, W. K. 2019. Hybrid Composite Precast Systems: tural engineering. His research interests include a new Numerical Investigation to Construction. United Kingdom: approach to construction technologies based on value engi- Woodhead Publishing, Elsevier. neering with hybrid composite structures. He has provided Hong, W. K. 2021. Artificial Intelligence-Based Design of many useful solutions to issues in current structural design Reinforced Concrete Structures. Daega. and construction technologies as a result of his research that combines structural engineering with construction technol- Hong, W. K., and T. A. Le. 2022. “ANN-Based Optimized ogies. 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Pareto frontier for steel-reinforced concrete beam developed based on ANN-based Hong-Lagrange algorithm

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JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING https://doi.org/10.1080/13467581.2023.2193621 BUILDING STRUCTURES AND MATERIALS Pareto frontier for steel-reinforced concrete beam developed based on ANN- based Hong-Lagrange algorithm Won-Kee Hong and Dinh Han Nguyen Department of Architectural Engineering, Kyung Hee University, Yongin, Republic of Korea ABSTRACT ARTICLE HISTORY Received 14 November 2022 Multi-objective optimization (MOO) is always a challenging issue for engineers in the field of Accepted 17 March 2023 structural engineering, where several objective functions must be satisfied under equality and inequality constraints to meet requirements imposed by engineers and decision-makers. This KEYWORDS study proposes a novel approach to solve MOO problems for steel-reinforced concrete (SRC) ANN-based Hong-Lagrange beams using an artificial neural network (ANN)-based Hong-Lagrange algorithm. Proposed algorithm; artificial neural method in this paper optimizes three specific objective functions, including cost (CI ), CO networks; multi-objective b 2 emissions, and beam weight (W), simultaneously. Neural networks are trained by 200,000 optimization; steel- reinforced concrete beams; samples, which are randomly generated by structural mechanics-based calculations, to derive unified objective function three specific objective functions. Unified objective function is, then, proposed based on weight fractions of each objective function. An ANN-based Hong-Lagrange technique identi- fies 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 algo- rithm 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. 1. Introduction and signification of current concrete (RC) beams is presented by Shariat et al. study (2018) using a computational Lagrangian Multiplier method. However, calculations of steel-reinforced con- 1.1. Introduction crete (SRC) members are much more complex than In the area of structural engineering, an optimization common RC structures due to the contribution of a design has always been a challenging issue over the H-shaped steel section embedded inside concrete last few decades. Four major categories of structural material, thus, being expensive for calculations of optimization can be classified, such as cost reduction, strengths, deflections, and flexibility. environmental impact reduction, structural perfor- SRC members have been used in various types of mance improvement, and multi-objective optimization infrastructure facilities, such as buildings, parking, and (MOO), according to a study by Mei and Wang (2021). A transportation. A sustainability of SRC structures study of an optimization of rectangular reinforced increases significantly due to the protection of CONTACT Won-Kee Hong hongwk@khu.ac.kr Department of Architectural Engineering, Kyung Hee University, Yongin 17104, Republic of Korea © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the Architectural Institute of Japan, Architectural Institute of Korea and Architectural Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 W.-K. HONG AND D. H. NGUYEN concrete outside, preventing high temperature and are optimized for cost and CO emissions (Kaveh, corrosive agents of the surrounding environment for Mottaghi, and Izadifard 2022), and optimal design of H-shaped steel sections. Steel, rebar, and concrete RC frames considering CO emissions is achieved via work simultaneously to prevent damage to the struc- metaheuristic algorithms by Kaveh, Izadifard, and tures when SRC members are subjected to imposed Mottaghi (2020). A Pareto frontier derived based on load. Many studies have been performed to investigate an ANN-based Hong-Lagrange algorithm to simulta- the behaviors of SRC members under applied loads (Li neously optimize designs for three objective functions and Matsui 2000; El-Tawil and Deierlein 1999; Bridge (CI , CO , W) is well compared with the lower boundary b 2 and Roderick 1978; Mirza, Hyttinen, and Hyttinen 1996; of large datasets of random designs generated by Ricles and Paboojian 1994; Furlong 1974; Mirza and structural mechanics-based calculations Skrabek 1992; Chen and Lin 2006; Dundar et al. 2008; (AutoSRCbeam). This research provides a powerful Munoz and Hsu 1997; Rong and Shi 2021; Virdi, tool for the next generation in optimizing multiple Dowling, and BS 449, & BS 153 1973, Roik and objective functions for SRC beams that meet any Bergmann 1990; Brettle 1973). SRC members’ capacity loads and design standards. Both equalities and can be calculated by conventional structural inequalities are taken into consideration based on mechanics-based calculations, whereas an artificial engineers’ requirements and regional design neural network (ANN)-based design method has been standards. developed to design structural members that does not Large datasets used to train neural networks are require complex calculations. Many researchers suc- obtained by structural mechanics-based calculations, cessfully applied ANNs in structural analysis, including called AutoSRCbeam, which was developed by authors notable studies of Abambres and Lantsoght 2020, in previous studies (Nguyen and Hong 2019; Hong Sharifi, Lotfi, and Moghbeli (2019), Asteris et al. 2019). The design accuracy of SRC beams using an (2019), and Armaghani et al. (2019). The potential of ANN-based Hong-Lagrange algorithm is verified both ANNs in civil engineering is also demonstrated by the through structural analysis and large structural data- studies of Kaveh et al. (Kaveh and Khalegi 1998; Kaveh sets as shown in the previous study (Hong and Nguyen and Servati 2001), which show that ANNs can predict 2022a). concrete strength and design double layer grids with This study requires a large amount of data, and results comparable to traditional methods. hence, neural networks require a large amount of The authors perform several studies for an optimiza- data to train effectively, which can be a challenge tion of either single objective functions or multi-objec- when using small computers. Computational com- tive functions for RC members in previous studies (Hong plexity is also challenging in that training neural and Nguyen 2022b; Hong, Nguyen, and Nguyen 2022; networks can be computationally intensive, requiring Hong, Nguyen, and Pham 2022; Hong and Le 2022; high-performance hardware. However, this study has Hong 2021; Nguyen and Hong 2019). The authors also developed methods to mitigate some of the limita- published a study “An AI-based auto-design for optimiz- tions listed above. The main limitation of this study ing RC frames using the ANN-based Hong–Lagrange is computational complexity as high-performance algorithm” in which ANNs-based objective functions computers are needed to generate large datasets such as costs and weights of RC frames with four-by- and train ANNs. four bays and four floors are optimized simultaneously based on big datasets of 330,000 designs in accordance 1.2. Research significances with ACI 318-19, whereas corresponding design para- meters which minimize objective functions are also A forward design includes 15 input parameters (L, d, b, f , obtained (Hong and Pham 2023). f’ , ρ , ρ , h , b , t , t , f , Y , M , M ) and 11 outputs (ϕM , c sc st s s f w yS s D L n Multi-objective functions for SRC beams including ε , ε , Δ , Δ , µ , CI , CO emission, W, X , SF), as rt st imme long ϕ b 2 s cost (CI ), CO emissions, and weight (W) are presented shown in Table 1. No studies were found to present b 2 in this study, presenting a key value of an ANN-based based on ANNs to optimize the three objective functions Hong-Lagrange algorithm for an application to a struc- including CI , CO emission, and W at the same time for b 2 tural design for engineers. MOO is always a challenging an SRC beam. In this study, the authors present an ANN- issue for engineers in the field of structural engineer- based Hong-Lagrange algorithm, which is used to opti- ing, where several objective functions must be satis- mize three objective functions (CI , CO emission, and W) b 2 fied under equality and inequality constraints to meet simultaneously for an SRC beam. requirements imposed by engineers and decision- This study presents a hybrid network using an ANN makers. Kaveh et al. proposed a genetic algorithm for and Hong-Lagrange algorithm to optimize an SRC optimal design of RC retaining walls (Kaveh, Kalateh- beam, capable of simultaneously optimizing three Ahani, and Fahimi-Farzam 2013), and a vibrating parti- objective functions (CI , CO emission, and W) with b 2 cles system (VPS)-based algorithm for MOO (Kaveh and significant accuracies. A unified objective function Ilchi Ghazaan 2020). RC bridge columns and bent caps (called UFO) is developed for an SRC beam in this JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 3 Table 1. Forward design scenario for SRC beam. Forward design scenario Input parameters Output parameters 1 L 9 b 1 ϕM 9 W s n 2 d 10 t 2 ε 10 X f rt s 3 b 11 t 3 ε 11 SF w st 4 f 12 f 4 Δ y yS imme. 5 f ’ 13 Y 5 Δ c s long 6 14 M 6 µ ρ D ϕ rt 7 ρ 15 M 7 CI rc L b 8 h 8 CO s 2 study (Hong et al. 2022). UFO is formulated based on shown in Table 1. Fifteen input parameters include three objective functions via weighted fractions, inte- length of beam (L), beam dimensions (b, d), material grating them into one objective function to simulta- strengths of concrete (f’ ), rebar (f ), steel (f ), com- c y yS neously optimize all objective functions (CI , CO pressive and tensile rebar ratio (ρ , ρ ), steel height (h ), b 2 c t s emission, and W). A Pareto frontier (also called Pareto steel flange (b ), steel web thickness (t ), steel flange s f front) for an SRC beam is constructed based on a thickness (t ), moment due to dead load (M ), and w D combination of MOO results. A contribution of each moment due to live load (M ). Eleven output para- objective function is represented by the trade-off of meters include design moment capacity (ϕM ) exclud- their weight fractions, selected by an interest of engi- ing beam weight, tensile strains of steel and rebar (ε , st neers and decision-makers. This study is a stepping- ε ), immediate and long-term deflections (Δ , Δ ), rt imme long stone for a design of next generation, not based on curvature ductility (µ ), materials and manufacture cost structural mechanics but based on ANNs. An ANN- (CI ) per 1 m length, CO emission per 1 m length, b 2 based Hong-Lagrange algorithm simultaneously opti- beam weight (W) per 1 m length, horizontal clearance mizes multi-objective functions for engineers and deci- (X ), and safety factor (SF). A cost (CI ) for materials and s b sion-makers. manufacture, CO emissions, and beam weights (W) are selected as multiple objective functions for an optimization of an SRC beam in the present study. 2. ANN-based design scenarios for steel- reinforced concrete beams 3. Generation of large datasets Figure 1 demonstrates the geometry of an SRC beam, A structural mechanics-based calculation called including beam section (h, b) and steel H-shaped sec- AutoSRCbeam is used to generate 200,000 datasets. tion (h , b , t and t ), which is encased in concrete AutoSRCbeam is established based on an algorithm as s s f w material. A forward design scenario for an SRC beam indicated in Figure 2. This program was developed by is presented based on 15 inputs and 11 outputs, as the authors in the previous study (Nguyen and Hong Figure 1. Geometry of SRC beams (Hong, Nguyen, and Nguyen 2022). 4 W.-K. HONG AND D. H. NGUYEN Figure 2. Flowchart for generating big data of SRC beams used to train network (Nguyen and Hong 2019). 2019). Fifteen input parameters for AutoSRCbeam are According to ACI 318–19 (Standard 2019), deflec - randomly selected in designated ranges, randomly pro- tions are limited to L/360 for immediate deflection viding 11 output parameters. Ranges for dimensions of (Δ ) and L/240 for long-term deflection (Δ ). In imme. long SRC beams are designated from 6000 to 12,000 mm, 500 the preliminary design stage, beam sections are to 1500 mm, and 0.3d to 0.8d for beam length (L), beam unknown; thus, the design moment capacity of a height (h), and beam width (b), respectively, where d is beam (ϕM ) is formulated by excluding a self-weight effective beam depth in a range of 406.1–1444.5 mm, as of a beam when generating large datasets. Factored referred to in Table 2. The dimensions of H-shaped steel moment (M ) represents a magnitude of externally section are randomly selected in appropriate ranges of applied moments, calculated by load combination 0.4–0.6d for steel section height (h ), 0.3–0.6b for steel of M and M with load factors (M = 1.2M + s D L u D section width (b ), and 5–25 mm for both steel web and 1.6M ). A safety factor represents how safe beam is s L flange thickness (t and t ), as shown in Table 2. Material against applied loads, which is calculated as a ratio w f strengths of beam components are chosen in ranges of between a design moment strength and factored 30–50 MPa, 500–600 MPa, and 275–325 MPa for con- moment (SF =ϕM /M ), and the safety factor must n u crete (f’ ), rebar (f ), and steel (f ), respectively. not be smaller than 1.0. c y yS Compressive rebar ratio is randomly chosen in a range Original and normalized large datasets of 200,000 are of 1/400 ~ 1.5ρ , where ρ is tensile rebar ratio with a shown in Table 3(a) and (b), respectively, generated by rt rt pffiffiffiffi � minimum ρ ¼ max 0:25 f =f ; 1:4=f , following AutoSRCbeam. Inputs of structural mechanics-based cal- y y rt;min c ACI standard (Standard 2019). Notations and ranges of culation including 15 parameters (L, d, b, f , f’ , ρ , ρ , h , b , y c sc st s s fifteen input parameters defining an SRC beam are indi- t , t , f , Y , M , M ) are randomly selected, yielding eleven f w yS s D L cated in Table 2. ACI 319-18 code requests tensile rebar corresponding output parameters (ϕM , ε , ε , Δ , Δ , n rt st imme. long strain (ε ) greater than 0.003 + ε (ε is yield strain of μ , CI , CO , W, X , SF) for design SRC beams. Mean, maxima, rt ty ty ϕ b 2 s rebar) to ensure enough ductility of beam. and minima of overall 26 parameters based on 200,000 This study focuses on SRC beams with fixed-fixed datasets are indicated in Table 3. All input and output end conditions, as illustrated in Figure 3. SRC beams parameters are normalized in a range from −1 to 1 by are subjected to uniform loads, including dead and using MAPMINMAX function of MATLAB (MathWorks live loads, yielding M and M , respectively. 2022a), as shown in Table 3. D L JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 5 Table 2. Nomenclatures and ranges of parameters defining SRC beams (Hong and Nguyen 2022a). Notation Range Input L (mm) Beam length [6000 ~12000] parameters d (mm) Effective beam depth [406.1 ~1445.5] b (mm) Beam width [0.3 ~0.8]d h (mm) H-shaped steel height [0.4 ~0.6]d b (mm) H-shaped steel width [0.3 ~0.6]b t (mm) H-shaped steel flange thickness [5 ~25] t (mm) H-shaped steel web thickness [5 ~25] f’ (MPa) Concrete strength [30 ~50] f (MPa) Rebar strength [500 ~600] f (MPa) Steel strength [275 ~325] yS � � pffiffiffi ρ Tensile rebar ratios rt 0:25 f c 1:4 ρ ¼ max ; rt;min f f y y ρ Compressive rebar ratios [1/400 ~1.5] ρ rc rt Ys Vertical clearance h i M (kN·m) Moment due to dead load ~ 1 0:2 M (M = 1.2M + u D 1:2 1.6M ) M (kN·m) Moment due to service live load ðM 1:2M Þ L u D 1:6 (M = 1.2M + 1.6M ) u D L Output ϕM (kN·m) Design moment without considering effect of self-weight at ε = 0.003 n c parameters μ Curvature ductility, μ ¼ ϕ =ϕ ϕ ϕ u y Where, ϕ : Curvature at ε = 0.003 ϕ : Curvature at tensile rebar yield ε Tensile rebar strain at ε = 0.003 ε ≥0.003+ ε rt c rt ty ε Compressive rebar strain at ε = 0.003 rc c Δ Immediate deflection due to M service live load (Δ ≤ L/360, ACI 318– imme. L imme. 19) Δ Sum of long-term deflection due to sustained loads and immediate deflection due to (Δ ≤ L/240, ACI 318–19) long long additional live load CI (KRW/m) Cost index per 1 m length of beam CO (t-CO / CO emission per 1 m length of beam 2 2 2 m) W (kN/m) Beam weight per 1 m length of beam X Horizontal clearance SF Safety factor (ϕM /M ) n u Figure 3. Fixed-fixed steel reinforced concrete (SRC) beams optimized using ANN (Hong, Nguyen, and Nguyen 2022). networks are formulated independently when 15 4. Multi-objective optimization using ANN- input parameters (L, d, b, f , f’ , ρ , ρ , h , b , t , t , based Hong-Lagrange Algorithm for steel- y c sc st s s f w f , Y , M , M ) are mapped to each of 11 output reinforced concrete beams yS s D L parameters (ϕM , ε , ε , Δ , Δ , μ , CI , CO , W, n rt st imme. long ϕ b 2 4.1. Training artificial neural networks based on X , SF), as demonstrated in. Table 4 presents the parallel training method training results of eleven networks using PTM, based on 200,000 datasets that are divided into In this study, ANNs are trained using a parallel three distinct portions, covering 70% (140,000 data- training technique (PTM) (Hong 2021; Hong, Pham, sets) for training, 15% (30,000 datasets) for and Nguyen 2022). As shown in 11 training 6 W.-K. HONG AND D. H. NGUYEN Table 3. Large datasets of SRC beams generated by AutoSRCbeam. Parameter Data 1 Data 2 Data 3 Data 4 Data 200,000 Mean (μ) Maxima Minima (a) Non-normalized data 200,000 datasets generated by AutoSRCbeam 15 Input parameters for structural mechanics-based (AutoSRCbeam) 1 L (mm) 8400 11900 9900 11050 9850 9995.3 12000.0 8000.0 2 d (mm) 786.0 1097.8 972.1 680.5 910.5 1016.8 1445.5 394.8 3 b (mm) 470 850 430 290 765 620.3 1200.0 150.0 4 h (mm) 160 645 270 155 450 400.3 1180.0 5.0 5 b (mm) 170 395 160 130 230 272.1 720.0 45.0 6 t (mm) 6 7 28 9 11 16.1 30.0 5.0 7 t (mm) 15 27 16 9 5 16.5 30.0 5.0 8 f’ (MPa) 41 22 29 43 23 40.0 60.0 20.0 9 f (MPa) 513 548 423 412 518 500.1 600.0 400.0 10 f (MPa) 296 358 267 289 379 300.1 400.0 200.0 yS 11 ρ 0.01957 0.01855 0.04355 0.01324 0.00337 0.01976 0.0637 0.0024 rt 12 ρ 0.02295 0.02714 0.04649 0.01401 0.00097 0.01944 0.0894 8.16E–06 rc 13 Ys 110.8 137.4 242.4 267.9 231.6 236.7 944.2 60.0 14 M (kN·m) 290.9 4956.5 1809.1 426.3 630.3 2866.8 36670.7 11.3 15 M (kN·m) 691.0 3166.6 1469.9 272.3 133.3 1430.7 23431.2 1.1 11 Output parameters for structural mechanics-based (AutoSRCbeam) 16 ϕM (kN·m) 2865.8 10794.1 6286.4 795.7 1512.6 7284.7 46755.1 56.2 17 ε 0.0107 0.0074 0.0056 0.0114 0.0094 0.0082 0.0462 0.0050 rt 18 ε 0.0081 0.0052 0.002 0.006 0.0064 0.0049 0.0348 −0.0005 st 19 μ 3.45 2.34 2.39 4.76 3.32 2.96 18.62 1.75 20 Δ (mm) 2.15 4.57 1.94 7.00 0.87 2.35 26.15 0.10 imme. 21 Δ (mm) 3.73 18.63 5.85 28.90 6.35 10.81 88.29 0.83 long 22 CI (KRW/m) 226654.0 754831.5 515783.1 114789.7 199231.1 501586.6 2295688.1 26576.4 23 CO (t-CO /m) 0.45 1.40 1.06 0.21 0.34 0.936 4.782 0.044 2 2 24 W (kN/m) 11.23 28.97 15.50 5.81 18.57 20.60 60.78 1.99 25 Xs (mm) 150 227.5 135 80 267.5 174.1 415.0 30.0 26 SF 1.97 0.98 1.39 0.84 1.56 1.38 2.00 0.75 (b) Normalized data 200,000 datasets generated by AutoSRCbeam 15 Input parameters for structural mechanics-based (AutoSRCbeam) 1 L (mm) −0.800 0.950 −0.050 0.525 −0.075 −0.002 1.0 −1.0 2 d (mm) −0.255 0.338 0.099 −0.456 −0.018 0.184 1.0 −1.0 3 b (mm) −0.390 0.333 −0.467 −0.733 0.171 −0.104 1.0 −1.0 4 h (mm) −0.729 0.093 −0.542 −0.737 −0.237 −0.322 1.0 −1.0 5 b (mm) −0.630 0.037 −0.659 −0.748 −0.452 −0.327 1.0 −1.0 6 t (mm) −0.600 −0.533 0.867 −0.400 −0.267 0.072 1.0 −1.0 7 t (mm) −0.200 0.760 −0.120 −0.680 −1.000 −0.083 1.0 −1.0 8 f’ (MPa) 0.050 −0.900 −0.550 0.150 −0.850 −0.0003 1.0 −1.0 9 f (MPa) 0.130 0.480 −0.770 −0.880 0.180 0.001 1.0 −1.0 10 f (MPa) −0.040 0.580 −0.330 −0.110 0.790 0.001 1.0 −1.0 yS 11 ρ −0.440 −0.473 0.342 −0.647 −0.969 −0.434 1.0 −1.0 rt 12 ρ −0.487 −0.393 0.040 −0.687 −0.979 −0.565 1.0 −1.0 rc 13 Ys −0.885 −0.825 −0.587 −0.530 −0.612 −0.600 1.0 −1.0 14 M (kN·m) −0.985 −0.730 −0.902 −0.977 −0.966 −0.844 1.0 −1.0 15 M (kN·m) −0.941 −0.730 −0.875 −0.977 −0.989 −0.878 1.0 −1.0 (Continued) JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 7 Table 3. (Continued). 11 Output parameters for structural mechanics-based (AutoSRCbeam) 16 ϕM (kN·m) −0.880 −0.540 −0.733 −0.968 −0.938 −0.690 1.0 −1.0 17 ε −0.723 −0.883 −0.971 −0.689 −0.786 0.001 1.0 −1.0 rt 18 ε −0.513 −0.677 −0.858 −0.632 −0.609 0.092 1.0 −1.0 st 19 μ −0.799 −0.930 −0.924 −0.643 −0.814 −0.857 1.0 −1.0 20 Δ (mm) −0.835 −0.650 −0.852 −0.465 −0.933 −0.820 1.0 −1.0 imme. 21 Δ (mm) −0.934 −0.593 −0.885 −0.358 −0.874 −0.772 1.0 −1.0 long 22 CI (KRW/m) −0.824 −0.358 −0.569 −0.922 −0.848 −0.581 1.0 −1.0 23 CO (t-CO /m) −0.827 −0.428 −0.571 −0.929 −0.876 −0.623 1.0 −1.0 2 2 24 W (kN/m) −0.686 −0.082 −0.540 −0.870 −0.436 −0.367 1.0 −1.0 25 Xs (mm) −0.377 0.026 −0.455 −0.740 0.234 −0.251 1.0 −1.0 26 SF 0.952 −0.632 0.024 −0.856 0.296 0.001 1.0 −1.0 8 W.-K. HONG AND D. H. NGUYEN Figure 4. ANN-based Hong-Lagrange optimization algorithm of five steps based on unified functions of objectives (UFO) (Hong and Le 2022). Table 4. Training results based on PTM method (Hong and Nguyen 2022a). Training with an Validation Suggested Best Stopped R at best No. output Data Layers Neurons epoch epoch epoch epoch Test MSE epoch 1 ϕM 200,000 4 64 1000 50,000 48,865 49,865 8.25E–06 1.0000 2 ε 200,000 4 64 1000 50,000 34,638 35,638 5.30E–05 0.9987 rt 3 ε 200,000 4 64 1000 50,000 49,997 50,000 4.17E–05 0.9992 st 4 Δ 200,000 4 64 1000 50,000 49,997 50,000 9.96E–06 0.9997 imme. 5 Δ 200,000 4 64 1000 50,000 49,748 50,000 1.80E–05 0.9997 long 6 μ 200,000 4 64 1000 50,000 35,898 36,898 4.22E–05 0.9989 7 SF 200,000 4 64 1000 50,000 27,140 27,141 1.69E–07 1.0000 8 X 200,000 4 64 1000 50,000 15,079 16,079 2.08E–06 1.0000 9 CI 200,000 4 64 1000 50,000 32,795 33,795 1.47E–06 1.0000 10 CO 200,000 4 64 1000 50,000 21,661 21,661 2.15E–08 1.0000 11 W 200,000 4 64 1000 50,000 49,944 50,000 2.77E–04 0.9996 validation, and 15% (30,000 datasets) for testing strain of tensile rebar (ε ) and several parameters rt data. The lowest and highest test MSE are 2.77E– (ϕM , SF, X , CI , CO ), respectively, as indicated in n s b 2 04 and 1.69E–07 for training weight (W) and safety Table 4. Neural networks are trained based on factor (SF), respectively. The lowest and highest 50,000 and 1000 for suggested and validation regression (R) are 0.9987 and 1.0000 for the training epochs, respectively. Trainings are implemented by 0 0 11 0 0 1 1 B B CC B B B B C C CC D 4 4 3 3 1 1 N 1 3 4 B B B B C C CC CI ¼ g f W f W . . . f W g ðXÞþ b . . .þ b þ b (1) CI lin CI t CI t CI CI CI CI CI bB B b @ b @ b b b A b A b CC |{z} |ffl{zffl} |ffl{zffl} |ffl{zffl} |fflfflffl{zfflfflffl} |{z} |{z} |{z} @ @ AA ½1� 1� ½1� 64� ½64� 64� ½64� 15� ½15� 1� ½64� 1� ½64� 1� ½1� 1� 0 0 0 0 1 1 11 B B B B C C CC D 4 4 3 3 1 1 N 1 3 4 B B B B C C CC CO ¼ g f W f W . . . f W g ðXÞþ b . . .þ b þ b (2) CO lin CO t CO t CO CO CO CO CO 2@ @ 2 @ 2 @ 2 2 2 A 2 A 2AA |{z} |ffl{zffl} |ffl{zffl} |ffl{zffl} |fflfflfflffl{zfflfflfflffl} |ffl{zffl} |ffl{zffl} |{z} ½1� 1� ½1� 64� ½64� 64� ½64� 15� ½15� 1� ½64� 1� ½64� 1� ½1� 1� JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 9 0 0 0 0 1 1 11 B B B B C C CC D L 4 3 3 1 1 N 1 3 4 W ¼ g @f @ W f @ W . . . f @ W g ðXÞþ b A . . .þ b Aþ b AA (3) W lin W t W t W W W W W |{z} |{z} |{z} |{z} |fflffl{zfflffl} |{z} |{z} |{z} ½1� 1� ½1� 64� ½64� 64� ½64� 15� ½15� 1� ½64� 1� ½64� 1� ½1� 1� i i i MATLAB Deep Learning Toolbox™ platform W ; W ; W ; weight matrices (i ¼ 1 4) CI CO W b 2 (MathWorks 2022b). obtained using training PTM. i i i b ; b ; b ; bias matrices (i ¼ 1 4) obtained CI CO W b 2 using training PTM. 4.2. Optimized objective functions using ANN- based Hong-Lagrange algorithm 4.2.2. Ten equality and 16 inequality constraints Table 5 describes 10 equality and 26 inequality 4.2.1. Derivation of objective functions based on constraints to optimize three objective functions forward neural networks (CI , CO , and W) simultaneously. Ten equality con- b 2 Derivation of objective functions including cost (CI ), CO b 2 straints include eight input parameters (L = emissions, and weight (W) based on forward ANNs is 10,0000 mm, d = 950 mm, f = 500 MPa, f’ = 30 y c described in this section, where the relationships among MPa, t = 12 mm, t = 8 mm, f = 500 MPa, Y = 70 f w yS s 15 input parameters (L, d, b, f , f’ , ρ , ρ , h , b , t , t , f , Y , y c sc st s s f w yS s mm) and two applied loads (M = 500 kN⋅m, M = D L M , M ) and 11 output parameters (ϕM , ε , ε , Δ , D L n rt st imme. 1500 kN⋅m). Besides 10 equalities selected, 16 Δ , μ , CI , CO , W, X , SF) are linked by weight and bias long ϕ b 2 s inequality conditions are also selected according matrices obtained by mapping entire input parameters to to ACI 318-19 standard (Standard 2019). A mini- each output parameter through an activation function mum requirement for rebar ratio is expressed by which yields nonlinear behaviors of the objective func- � � pffiffiffi tions as indicated in Equations (1)–(3)) (Hong and Nguyen 0 0:25 f c 1:4 equation ρ ¼ max ; ¼ 0:0028 rt;min f f 2022a; Krenker, Bešter, and Kos 2011; Villarrubia et al. y y 2018). ANNs with multilayer perceptron trained using according to ACI 318-19 code, and maximum PTM are based on 4 layers and 64 neurons, as shown in rebar ratio is selected at 0.05, which is indicated and IC in Table 5, respectively. A Table 4. An activation function tansig is used in this paper, 1 2 by inequality IC maximum rebar ratio of 0.05 is established arbitra- as expressed in Equation (4). rily for generating big data. ACI 318-19 also recom- mends that tensile rebar strain should not be less than 0.003 + f /200,000 = 0.0055 to ensure enough ductility of beams, which is indicated by inequality f ðxÞ ¼ tansigðxÞ ¼ 1 (4) t IC in Table 5. Immediate and long-term deflec - 2x 1þ e tions (Δ and Δ ) are limited by L/360 and L/ imme. long where 240 according to ACI 318-19, respectively, indi- X; input parameters, X = L; d; b; f ; f ; ρ ; ρ ; h ; b ; y sc st s s c cated by inequalities IC and IC , respectively. 13 14 t ; t ; f ; Y ; M ; M � Other equality constraints are also selected as f w yS s D L N D g ; g ; normalizing and de-normalizing functions. shown in Table 5. Ten equalities and 16 Table 5. Equality and inequality constraints imposed by ACI 318-19 for optimization designs (Hong and Nguyen 2022a). Equality conditions Inequality conditions � � pffiffiffi EC L = 10,000 mm IC ≤ ρ 1 1 0:25 f rt c 1:4 ρ ¼ max ; ¼ 0:0028 rt;min f f y y EC d = 950 mm IC ρ ≤ 0.05 2 2 rt EC f = 500 MPa IC ρ /400 ≤ ρ 3 y 3 rt rc EC f’ = 30 MPa IC ρ ≤ ρ 4 c 4 rc rt EC t = 12 mm IC 0.3b ≤ b 5 f 5 s EC t = 8 mm IC b ≤ 0.6b 6 w 6 s EC f = 325 MPa IC 0.4d ≤ h 7 yS 7 s EC Y = 70 mm IC h ≤ 0.6d 8 s 8 s EC M = 500 kN·m IC 0.3d ≤ b 9 D 9 EC M = 1500 kN·m IC b ≤ 0.8d 10 L 10 IC 0.003 + f /200,000 = 0.0055 ≤ ε 11 y rt IC 140 mm ≤ b 12 s IC Δ ≤ L/360 13 imme. IC Δ ≤ L/240 14 long IC 50 mm ≤ X 15 s IC 1.0 ≤ SF 16 10 W.-K. HONG AND D. H. NGUYEN inequalities are used to optimize multi-objective 4.3. Results of multi-objective optimization of functions simultaneously based on MATLAB steel-reinforced concrete beams Global Optimization Toolbox (MathWorks 2022, 4.3.1. Four specific cases for an optimization of 2022c, 2022d, 2022e). each objective function (CI , CO , and W, b 2 respectively) 4.2.3. Derivation of a unified function of objective A Pareto frontier is obtained based on a combination for a steel-reinforced concrete beam of multiple optimized designs for three objective func- A UFO for SRC beams is defined using algorithms tions (CI , CO , and W) with their weight fractions b 2 based on the weighted sum technique (Afshari, (w : w : w Þ for SRC beams in this study. These CI CO W b 2 Hare, and Tesfamariam 2019), which is created by weight fractions (w : w : w Þ represent trade-off CI CO W b 2 integrating three objective functions (CI , CO , and ratios contributed by each of three objective functions b 2 W) with their respective weight fractions to real-life optimizations for engineers and decision- w ; w ; w , as indicated in Equations (5) and makers. The 343 combinations of weight fractions are CI CO W b 2 (6). These fraction weights are in a range from 0 generated for constructing a Pareto frontier, including to 1, whose sum is 1 as shown in Equation (6). four specific cases, as indicated in Figure 5. It is noted Specific trade-offs among objectives are used to that the sum of three weight fractions of each combi- help engineers and decision-makers evaluate a nation is always equivalent to 1. Observation shows design project holistically. Single objective func- that four specific cases include an optimization of each tion for each of CI , CO , and W is a specific case objective function (CI , CO , and W, respectively), b 2 b 2 of UFO when w : w : w = 1:0:0; 0:1:0; 0:0:1, represented by Points 1, 2, and 3 on a Pareto frontier, CI CO W b 2 respectively. A Lagrange function of a UFO shown whereas Point 4 represents an evenly optimized design in Equations (7)–(9), which are based on three for CI , CO , and W, with their weight fraction b 2 objective functions with equality and inequality w : w : w = 1/3:1/3:1/3. Design parameters (b, CI CO W b 2 constraints shown in Table 5 being simultaneously h , b , ρ , ρ ) are obtained by solving MOO using an s s rc rt optimized. The Lagrange function is substituted ANN-based Hong-Lagrange algorithm with 10 equality into the built-in optimization toolbox of MATLAB and 16 inequality conditions, shown in Table 5. (MathWorks 2022) to obtain optimized design parameters. Figure 4 demonstrates a flowchart for 4.3.2. Pareto-efficient designs verified by structural five steps to solve MOO problems, providing datasets detailed descriptions of an algorithm that illus- A Pareto frontier includes 343 optimized designs for SRC trates the authors’ previous study for RC columns beams, indicated by red dots as shown in Figure 6(a–c). (Hong et al. 2022). Design parameters (b, h , b , ρ , ρ ) are calculated for four s s rc rt specific combinations while all 10 equalities (L = 10,000 mm, d = 950 mm, f = 500 MPa, f’ = 30 MPa, f = 325 ANN ANN ANN y c yS UFO ¼ w F ðxÞ ¼ w F ðxÞþ w F ðxÞ i 1 2 i 1 2 MPa, t = 12 mm, t = 8 mm, Y = 70 mm, M = 500 kN⋅m, f w s D ANN ANN ANN and M = 1500 kN⋅m) and 16 inequalities given in Table 5 þw F ðxÞ ¼ w CI ðxÞþ w CO ðxÞ 3 CI CO 3 b b 2 2 are satisfied. þw WANNðxÞ (5) where 4.3.2.1. A three-dimensional Pareto frontier for the three objective functions (CI CO , and W). Figure 6 b, 2 w þ w þ w ¼ 1 CI CO W (a) shows three-dimensional Pareto frontier for 0 � w ; w ; w � 1ðdimensionlessÞ (6) CI CO W three objective functions denoted by 343 red dots and 133,711 green design points generated by Lagrange function utilizing UFO function: structural mechanics-based calculations. Observation shows that a Pareto frontier using T T L ðx; λ ; λ Þ ¼ UFO λ ECðxÞ λ ICðxÞ (7) UFO c v ANN-based Hong-Lagrange algorithm is well located c v at the lower boundary of 133,711 random designs calculated by AutoSRCbeam. Four SRC beam designs ECðxÞ ¼ ½EC ðxÞ; EC ðxÞ; . . . ; EC ðxÞ� ; are shown in Figure 6(a) based on four specific 1 2 m m ¼ 10 : number of equality constraints (8) cases for optimizing three objective functions, where Design cases 1 and 2 only optimize SRC beams, yielding minimum cost (CI ) and CO emis- b 2 ICðxÞ ¼ ½IC ðxÞ; IC ðxÞ; . . . ; IC ðxÞ� ; sions, respectively. Design case 3 only optimizes 1 2 m m ¼ 16 : number of inequality constraints (9) weight (W), whereas Design case 4 evenly optimizes 2 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 11 Figure 5. Weight fractions generated for Pareto frontier. Figure 6. Pareto-efficient designs verified by structural datasets at EC9: MD = 500 kN⋅m; EC10: ML = 1500 kN⋅m. 12 W.-K. HONG AND D. H. NGUYEN Figure 6. (Continued). JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 13 Figure 6. (Continued). all three objective functions (CI , CO , and W). The 4.3.2.3. Design case 2. Design case 2 only opti- b 2 343 Pareto points are generated with different mizes CO emissions or single objective function trade-off ratios among three objective functions, CO , with corresponding weight factors also optimizing three objective functions simulta- (w : w : w ¼ 0 : 1 : 0) for three objective func- CI CO W b 2 neously, according to designated trade-off ratios. tions (CI , CO , and W), as illustrated in Figure 6(c,d). b 2 In Figure 6(b-d), Pareto frontiers are projected on Design case 2 considers a trade-off only contributed CI – W, CO – W, and CI – CO planes, respectively, by CO emissions, ignoring an interest in costs CI b 2 b 2 2 b where the trade-offs used for design points among and weights (W). Design parameters (b = 404.4 mm, three objective functions are shown. h = 380 mm, b = 150 mm, ρ = 0.0144, ρ = 0.0100) s s rt rc are obtained while all ten equalities and 16 inequal- ities given in Table 5 are satisfied. Optimized CO 4.3.2.2. Design case 1. Design case 1 only optimizes emission is obtained at 0.3920 t-CO /m, which is the costs (CI ) or single objective function CI , with corre- b b minimum CO emissions of all Pareto points, as sponding weight factors (w : w : w ¼ 1 : 0 : 0) CI CO W b 2 shown in Figure 6(c,d). Costs (CI ) and weights (W) assigned to three objective functions (CI , CO , and b 2 for Design case 2 correspond to 212,325 KRW/m W), as indicated in Figure 6(b,d). Design case 1 con- and 11.184 kN/m, respectively. The SRC beam siders a trade-off only contributed by costs (CI ), ignor- designed with Design cases 1 and 2 are largest in ing an interest in CO emissions and weights (W). volume to use more concrete, while reducing costs Design parameters (b = 383 mm, h = 380 mm, b = s s (CI ) and CO emissions compared with Design b 2 150 mm, ρ = 0.0152, ρ = 0.0114) are obtained while rt rc cases 3 and 4. all ten equalities and 16 inequalities given in Table 5 are satisfied. Optimized cost (CI ) is obtained at 211,987 KRW/m, which is the minimum cost of all 4.3.2.4. Design case 3. The minimum beam width (b = Pareto points, as clearly shown in Figure 6(b,d). CO 0.3d = 0.3 × 950 = 285 mm) is recommended for Design emissions and weights (W) for Design case 1 corre- case 3, which optimizes single objective function weights spond to 0.3927 t-CO /m and 10.698 kN/m, (W) based on an inequality constraint (IC ), shown in Table 2 9 respectively. 5, resulting in lightest beam weights W = 8.628 kN/m, as 14 W.-K. HONG AND D. H. NGUYEN Table 6. Multiple-objective optimization using ANN-based Hong-Lagrange algorithm based on wðCI Þ : wðCO Þ : w ¼ 1=3 : 1=3 : 1=3. b 2 W shown in Figure 6(b,c). However, tensile and compressive Lagrange algorithm calculates b = 292.4 mm for Design rebar ratios (ρ , ρ ), and height of steel H-shaped (h ) are case 4 with three equal weight fractions rt rc s 0.0225, 0.0225, and 408.8 mm, respectively, larger than (w : w : w ¼ 1=3 : 1=3 : 1=3), which is the most h CI Weight those of the Design cases 1 and 2 in order to decrease favorable value of beam widths that a proposed method beam weights W. Beam capacity increases to meet the finds when three objective functions are optimized requirement of SF ≥ 1 when single objective function W is evenly. Beam width of b = 292.4 mm for Design case 4 is optimized as indicated in Figure 6(b,c). Costs and CO close to a minimum of b = 285 mm for Design case 3. emissions obtained in Design case 3 are 224,406 KRW/m and 0.4265 t-CO /m, respectively, larger than those 4.3.3. Verifications obtained in Design cases 1 and 2. Table 6 summarizes designs when an ANN-based Hong- Lagrange algorithm evenly minimizes three objective 4.3.2.5. Design case 4. Weight fractions w : w : functions CI , CO , and W based on w : w : w = 1/ h CI b b 2 CIb CO2 W w ¼ 1=3 : 1=3 : 1=3 are implemented evenly in 3:1/3:1/3. Green cells in Table 6 show 11 output para- Weight Design case 4, which optimizes all three objective func- meters obtained using structural mechanics-based calcu- tions at the same time with the equivalent contributions. lations (AutoSRCbeam) from 15 input parameters in red Observation shows that Design case 4 is located near cells that are calculated by an ANN-based Hong-Lagrange Design case 3 which only optimizes weight (W), indicating algorithm while all 10 equalities and 16 inequalities given median values of CI = 219,279 KRW/m and W = 8.757 kN/ in Table 5 are met. A cost of an SRC beam is obtained as m obtained between Design case 1 (or Design case 2) and 219,279.1 KRW/m by an ANN-based Hong-Lagrange algo- Design case 3, as shown in Figure 6. ANN-based Hong- rithm which is close to 219,583.6 KRW/m calculated by JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 15 structural mechanics-based calculation, indicating an This study provides a robust steppingstone for next insignificant error of −0.14%. Insignificant errors in CO generation in optimizing multiple objective functions emissions and weight are also caused at 2.38% and 0.23% with the trade-off ratios that are beneficial for engi- between an ANN-based Hong-Lagrange algorithm and neers and decision-makers. The current investigation structural mechanics-based calculations, respectively. An does not involve an experimentation but is based on insignificant error of 0.86% is observed when comparing ACI 318-19. It develops an AI-based program capable design moment (ϕM ) of 3031.9 kN⋅m calculated from of not only automating routine structural design tasks ANNs with 3005.8 kN⋅m obtained from structural but also improving design efficiency. To the best of the mechanics-based calculations, as presented in Table 6. authors’ knowledge, such a software is not yet avail- Tensile rebar strain (ε ) is calculated at 0.0055 and able for practical structural design applications; thus, 0.0054, obtained by ANNs and structural mechanics- the current design procedures are largely reliant on based calculations, respectively, leading to an insignifi - iterative methods. The following conclusions are cant error of 1.82%. ANNs and structural mechanics-based drawn from this study. calculation provide immediate deflections (Δ ) of 5.96 imme. mm and 5.67 mm, respectively, yielding a relative error of (1) This study aims to optimize multi-objective 4.87%, indicating only a difference of 0.29 mm. Long-term functions for SRC beams simultaneously, which deflection (Δ ) of 12.07 and 11.44 mm obtained by is interesting to engineers and decision-makers. long ANNs and structural mechanics-based calculation also The proposed method can be extended into indicate a negligible difference of 0.63 mm with an error various fields in real life. of 5.22%. It is noted that deflections limited in accordance (2) A unified function of objectives (UFO) is estab- with ACI 318-19 code are 27.8 mm (L/360) and 41.7 mm lished by integrating three objective functions (L/240) for immediate deflection (Δ ) and long-term consisting of costs (CI ), CO emissions, and imme. b 2 deflection (Δ ), respectively. Observation shows that weights (W) into one with their trade-offs of long immediate deflection (Δ ) and long-term deflection weight factions. MOO based on UFO is imple- imme. (Δ ) obtained by ANN-based Hong-Lagrange algorithm mented under 10 equality and 16 inequality long meet ACI 318–19 standard. Table 6 demonstrates a prac- constraints, which are used to impose design tical methodology for designing SRC beams using ANNs. requirements based on MATLAB Global It also presents an accuracy of the ANN-based Hong- Optimization Toolbox. Lagrange algorithm in calculating SRC beams with a max- (3) A Pareto frontier constructed based on 343 opti- imum error of −2.56% among all parameters except for mized designs for three objective functions (CI , errors of immediate (Δ ) and long-term deflection CO , W) is verified by large datasets, yielding a imme. 2 (Δ ) reaching 4.87% and 5.22%, respectively. good comparison with a lower boundary of long 133,711 designs randomly generated based on structural mechanics-based calculations (AutoSRCbeam) for preassigned external loads, 5. Conclusions M = 500 kN⋅m, M = 1500 kN⋅m. D L In this study, an ANN-based Hong-Lagrange algorithm (4) A robust and sustainable tool for engineers is is proposed for simultaneously optimizing multi-objec- proposed in this study to design SRC beams tive functions of an SRC beam that can be widely used using ANN-based Hong-Lagrange algorithm, by engineers. Engineers and decision-makers can use which optimizes multi-objective functions (CI , the method presented in this study as a guideline for CO emissions, and W) of an SRC beam simulta- optimizing multi-objective functions in the practical neously under external loads. Design case 4 of design of any structures. A UFO of SRC beams with a this study evenly minimizes three objective fixed-fixed condition is optimized, satisfying various functions CI , CO , and W (w : w : w = 1/ b 2 CIb CO2 W standard restrictions simultaneously, showing key 3:1/3:1/3) based on an ANN-based Hong- values of the proposed method. Equality and inequal- Lagrange algorithm. The ANN-based Hong- ity constraints can be established by an interest of Lagrange algorithm and structural mechanics- engineers to reflect their own design requirements. A based calculation yielded costs of 219,279.1 Pareto frontier consists of 343 optimized designs in KRW/m and 219,583.6 KRW/m, respectively, this study, minimizing three objective functions (CI , with an insignificant error of −0.14. Errors CO , and W) simultaneously with trade-off weight frac- between ANN-based Hong-Lagrange algorithm tions. A Pareto frontier is verified by the lower bound- and structural mechanics-based calculations ary of large datasets including 133,711 designs were also insignificant as 2.38% and 0.23% for randomly generated based on structural mechanics- CO emissions and weight, respectively. It is based calculations. The proposed ANN-based Hong- noted that the error for design moment (ϕM ) Lagrange algorithm can be extended to various fields was 0.86%, based on design moment (ϕM ) of in real-world designs. 3031.9 kN⋅m calculated from ANNs and 3005.8 16 W.-K. HONG AND D. H. NGUYEN kN⋅m calculated from structural mechanics- References based calculations, respectively. This study also Abambres, M., and E. O. Lantsoght. 2020. “Neural Network- demonstrated tensile rebar strains (ε ) which rt Based Formula for Shear Capacity Prediction of One-Way were calculated at 0.0055 and 0.0054, obtained Slabs Under Concentrated Loads.” Engineering Structures by ANNs and structural mechanics-based calcu- 211: 110501. doi:10.1016/j.engstruct.2020.110501. lations, respectively, leading to an insignificant Afshari, H., W. Hare, and S. Tesfamariam. 2019. “Constrained Multi-Objective Optimization Algorithms: Review and error of 1.82%. 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Journal

Journal of Asian Architecture and Building Engineering – Taylor & Francis

Published: Nov 2, 2023

Keywords: ANN-based Hong-Lagrange algorithm; artificial neural networks; multi-objective optimization; steel-reinforced concrete beams; unified objective function

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