Abstract
JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING https://doi.org/10.1080/13467581.2022.2163174 BUILDING STRUCTURES AND MATERIALS An AI-based auto-design for optimizing RC frames using the ANN-based Hong– Lagrange algorithm Won-Kee Hong and Tien Dat Pham Department of Architectural Engineering, Kyung Hee University, Yongin, Republic of Korea ABSTRACT ARTICLE HISTORY Received 9 September 2022 Artificial neural networks (ANNs)-based objective functions such as costs and weights of Accepted 15 December 2022 reinforced concrete (RC) frames with four-by-four bays and four floors are optimized simulta- neously based on big datasets of 330,000 designs according to ACI 318-19, whereas corre- KEYWORDS sponding design parameters, which minimize objective functions, are also obtained. The ANN-based Hong–Lagrange Pareto frontier verified by big datasets shows reductions up to 44.983% and 33.111% in algorithm of RC frames; costs and weights, respectively, compared with probable designs based on averages of 688 Pareto frontier of RC frames; (0.1%) best designs among 688,000 samples. Optimized designs’ meeting requirements big datasets of RC frames; KKT solutions of RC frames; imposed by codes and architects are achieved using the ANN-based Hong–Lagrange algorithm weight fraction of MOO in which complex analytical objective functions are replaced by ANN-based objective func- problems tions. ANN is formulated to provide 32 forward outputs based on 18 forward inputs to minimize or maximize objective functions, such as costs and weights as a function of 18 input para- meters. When good training qualities are achieved, objective functions with equality and inequality constraints are implemented in the proposed method, which determines optimal design parameters for building with accuracies and robustness equivalent to derivation-based approaches, which are hard to obtain using metaheuristic methods. The proposed AI-based auto-designs perform optimization where design variables are produced automatically while optimizing design targets. 1. Introduction solved using a ε-constraint method, providing a Pareto set of optimal building designs. Paya-Zaforteza et al. 1.1. Literature review (2009) presented a method based on a similated anneal- Building optimization is always an ultimate goal of struc- ing algorithm, optimizing CO emissions and costs of tural engineers. However, it is difficult to explicitly derive buildings designed based on the Spanish code. Results analytical objective functions to optimize complex rein- indicated a close relationship between two objective forced concrete (RC) frames that meet all code require- functions where environmental and economical efficien - ments simultaneously. This is a complex task, especially, cies of a design minimizing CO emissions were relatively when multiple constraining conditions are to be imposed. similar to those obtained from a design with optimized Zou et al. (2007) formulated the life-cycle costs of costs. On the other hand, Yeo and Potra (2015) reported a building as a multi-objective optimization (MOO) pro- a difference of 5% to 10% between CO emissions of blem consisting of material costs and expected damages a design minimizing CO emissions and those of the due to seismic actions. The MOO problem, then, was cheapest design. Camp and Huq (2013) implemented CONTACT Won-Kee Hong hongwk@khu.ac.kr Room 104-1, Engineering building, Kyung Hee University Global Campus 1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 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. 2 W.-K. HONG AND T. D. PHAM the Big Bang-Big Crunch algorithm in were verified by conventional structural designs, result- reducing CO emissions and costs of RC frames, resulting ing in the basis for data-centric engineering which is not in improvements compared with genetic and annealing based on structural mechanics. algorithms. The study by Sharafi, Hadi, and Teh (2012) applied a colony optimization algorithm in minimizing 1.2. Research significance a cost of a 3D RC frame, resulting in a further cost reduc- tion of 4.8% compared with a study by Sahab, Ashour, Derivation-based approaches are unable to efficiently and Toropov (2005) which combined an exhaustive optimize large structure designs because optimizations search algorithm, a genetic algorithm, and a Hook and and sensitivity analysis are complex due to various Jeeves method.Esfandiari et al. (2018 – 2017) introduced requirements imposed by codes and a large number of an algorithm that combined multi-criterion decision- input and output variables. Metaheuristic methods such making and particle swarm optimizations, accelerating as genetic algorithms, similated annealing algorithms, convergences in finding optimal solutions for 3D RC and colony optimization algorithms are widely applied, frames subjected to lateral seismic forces. Bai, Jin, and as discussed in the literature review. However, there are Ou (2020) maximized seismic resistance of RC structures debates about their accuracies and robustness, for exam- by an iterative analysis-and-redesign scheme, substan- ple, results provided by genetic algorithms can be tially reducing story drifts while slightly increasing mate- unstable and converge to the local minima because rial costs. Hysteresis behaviors of structures are predicted procedures initializations, crossover, and mutations heav- based on Bouc–Wen models in studies by Sirotti et al. ily rely on randomness (Blum and Roli 2003). Achieving (2021), Pelliciari et al. (2020, 2018). optimal solutions, hence, in day-to-day engineering prac- The majority of previous studies evaded complexities tices is still challenging even if numerous research has of explicit objective functions in structural designs by been proposed in the field. The present study offers using metaheuristic methods such as genetic algorithm, a novel algorithm that systematically and conveniently pattern search algorithm, and colony optimization algo- optimizes building frames, bridging state-of-the-art artifi - rithm. Examples of artificial neural networks (ANNs)- cial intelligence (AI) technologies and practical based structural designs were found in studies by engineering. Srinivas and Ramanjaneyulu (2007), Shin et al. (2020), This study uses an ANN-based Hong–Lagrange algo- Asteris et al. (2016), and García-Segura, Yepes, and rithm with constraints imposed by codes and architects Frangopol (2017). Behaviors of bridge decks were pre- to holistically optimize RC frames, recognizing that it is dicted by ANNs, and designs are optimized using difficult to explicitly derive analytical objective functions genetic algorithms, in the study of Srinivas and when optimizing complex RC frames that meet all code Ramanjaneyulu (2007). Hazards of seismically deficient requirements simultaneously. The approach provided in RC frames were assessed and mitigated using ANNs by this study optimizes RC frames based on ANN-based Shin et al. (2020), aiding retrofit designs in buildings. objective functions, solving non-linear optimization pro- Asteris et al. (2016) predicted fundamental periods of blems under strict constraints imposed by design codes infilled RC structures using ANNs, showing good accura- and interests of engineers. The use of ANN-based objec- cies when verified by analytical investigations. tive functions eliminates complex derivations of explicit García-Segura, Yepes, and Frangopol (2017) conducted mathematical formulations which hinders applications of an optimization of post-tensioned concrete road optimizations for practical designs. bridges using ANNs, minimizing total expected costs This study considers RC frames where the ANN- while achieving required levels of safety and durability. based auto-designs for optimizing RC frames are However, the implementation of ANNs in optimizations developed based on the ANN-based Hong–Lagrange of RC frames were not common even though ANNs have algorithm. Additional structural systems such as dual been successfully used in in many areas such as medical, frames, steel RC frames, and prestressed frames are auto pilot, financial, etc. The present study provided under development. AI-based auto-designs proposed a novel procedure to calculate design parameters in this study perform design optimization while opti- while minimizing both single and multi-objective func- mizing design targets. Design parameters are pro- tions for structural frames using the ANN-based Hong– duced automatically, whereas it is challenging to Lagrange algorithm. ANN-based Hong–Lagrange algo- achieve auto-designs by conventional approaches. rithm was successfully implemented in optimizing beam and column designs in the studies of Hong and Nguyen 1.3. ANN-based Hong–Lagrange algorithm (2021) and Hong, Nguyen, and Nguyen (2021). It was anticipated that the optimizations for beams, columns, Villarrubia et al. (2018) proposed a method of approx- and frames can assist human engineers in enhancing imating objective functions by ANNs and minimizing design accuracies and reducing their labor while offer - constrained ANN-based objective functions by ing optimized designs based on ANNs-based Lagrange Lagrange functions and Karush–Kuhn–Tucker (KKT) optimization that are not available currently. Accuracies conditions. Theoretically, this method can provide JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 3 Figure 1. A graphical illustration of ANN-based Hong–Lagrange algorithm. Figure 2. Four steps of ANN-based Hong–Lagrange algorithm from generating big datasets to identifying Pareto frontiers. accuracy and robustness equivalent to derivative- functions are replaced by ANN-based objective func- based methods while eliminating complex derivation tions while multiple constraining conditions are processes when good training is achieved. The ANN- imposed by equality and inequality constraints in based Hong–Lagrange algorithm illustrated in Figure 1 Lagrange functions (Lagrange 1804). is developed referring to the method proposed by Optimization consists of four steps as follows as Villarrubia et al. (2018), where ANN-based Lagrange shown in Figure 2. optimizations are expanded to structural problems, Step 1: Any structural-based software including implanting constraints imposed by architecture and MIDAS or ETABS can be used to model frames. codes in Lagrange functions. This study performs Graphical interfaces of MIDAS and ETABS are utilized a holistic optimization of RC frames based on the ANN- so frames can be modeled accurately and conveniently based Hong–Lagrange algorithm, where ANN-based even though there are geometrical irregularities. objective functions such as cost (CI) and weight (W) Information including frame geometry, frame supports, of RC frames derived as a function of input parameters end-conditions of members, loading, story data, and which are, then, minimized based on 330,000 data structural groups are established using graphical tools samples, producing optimized design parameters as of MIDAS or ETAB as shown in descriptions from 1 to 6 in shown in Figure 1. Complex analytical objective Step 1.1 of Figure 2. In Step 1.1, graphical interfaces of 4 W.-K. HONG AND T. D. PHAM structural-based software (MIDAS or ETABS) are utilized The present study performs a data generating, train- to establish frame models. However, MIDAS or ETABS ing, and optimizing algorithms using MATLAB Deep only serve as modelers, whereas models are not inves- Learning Toolbox (MathWorks 2022a), MATLAB tigated by MIDAS or ETABS. The model is exported to Parallel Computing Toolbox (MathWorks 2022a), external files such as MGT files by MIDAS. These MGT MATLAB Statistics and Machine Learning Toolbox files are imported in the ABBA (AI-Based Build Analysis (MathWorks 2022a), MATLAB Global Optimization and design) frame generator in Step 1.2 to generate big Toolbox (MathWorks 2022a), MATLAB Optimization data samples. The ABBA frame generator is a MATLAB- Toolbox (MathWorks 2022a), and MATLAB R2022a based software. In this step, wind load parameters, seis- (MathWorks 2022a). The purpose of each toolbox is mic load parameters, load combinations, material prop- described in Table 1. Steps 3.1 and 4 are developed erties, ranges of section sizes, and rebar ratios are based on Villarrubia et al. (2018), and Step 3.2 is assigned to the ABBA frame generator. In each data developed based on Zadeh (1963). set, the generator randomly selects member sizes and The present study includes five sections. Section 1 rebars ratios based on ranges predefined by users to shows novelties and advantages of the proposed algo- calculate output parameters including safety factors of rithm compared with available methods. Section 2 members, story drifts, long-term deflections, etc. introduces configurations of example buildings, data required for frame designs. generations, and design constraints. Section 3 explains Step 2: ANNs are trained based on big datasets gen- formulations of objective functions based on weight erated in Step 1 using MATLAB Deep Learning Toolbox and bias matrices of ANNs. Section 4 presents opti- (MathWorks 2022a). MATLAB Deep Learning Toolbox is mized designs obtained from the ANN-based Hong– a built-in application in the MATLAB platform where Lagrange algorithm. Lastly, Section 5 summarizes the ANNs with different numbers of layers and neurons can research with discussions and recommendations. be trained on big datasets. The toolbox produces regres- sion models based on training, where weight and bias 2. Frame investigations matrices are extracted to generalize functions including objective and constraining functions. The generalized 2.1. Building configurations functions are used to formulate Jacobi and Hessian A four-by-four bay with four-story frame optimized in matrices of Lagrange functions. Stationary points for this study is shown in Figure 3, where the frame is Language functions are, then, identified using Newton– divided into two groups for beam and column designs. Raphson iteration (Upton and Cook 2014) in the SQP The floor height and beam span are 4 and 8 m, respec- algorithm (MathWorks 2022a). tively. An example of optimizing RC frames is carried Step 3: ABBA-Frame optimization codes are developed out to demonstrate the effectiveness of the ANN- to optimize frame designs based on single objective based Hong–Lagrange algorithm. However, the algo- functions in Step 3.1. Design requirements are imposed by equalities and inequalities according to codes and rithm is not limited to any frame configurations, but it architectural requirements. For example, all safety factors is possible for all frame layouts as long as these layouts are greater than 1 and short-term deflections of beams can be modeled by either MIDAS or ETAB. Figure 3(a) should be less than 1/360 of a span length. An objective shows a layout of beams and columns, while Figure 3 function for Lagrange optimization can be any design (b) illustrates slabs as dead loads. It is noted that target, such as frame cost, CO emissions, and weight of weights and stiffnesses of slabs are considered in the frame. In Step 3.2, a Unified Function of Objectives dynamic investigations even though slab weights are (UFO) is established based on tradeoff ratios according to excluded in design tables. Weights of slabs are consid- weighted sum methods (Zadeh 1963). ered for dead loads to calculate structure masses, Step 4: The Lagrange optimization is implemented whereas frames are investigated with rigid diaphragms in UFO to determine optimized design parameters at each floor, taking into account stiffnesses of slabs. while optimizing multiple objective functions Weights presented in design tables only represent the simultaneously. total weight of beams and columns which are Table 1. Purposes of toolboxes used in the ANN-based Hong–Lagrange algorithms. Toolbox Purpose MATLAB R2022a (MathWorks 2022a) A platform to develop codes MATLAB Parallel Computing Toolbox (MathWorks 2022a) A toolbox for performing calculations parallelly on all CPUs, enhancing speeds of data generations and design optimizations MATLAB Statistics and Machine Learning Toolbox (MathWorks 2022a) A toolbox for normalizing big datasets before training MATLAB Deep Learning Toolbox (MathWorks 2022a) A toolbox for training ANN models from big datasets MATLAB Global Optimization Toolbox (MathWorks 2022a) Toolboxes for determining solutions for constrained optimization problem MATLAB Optimization Toolbox (MathWorks 2022a) JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 5 Figure 3. Illustration of an RC frame under optimization: (a) Beam and column layouts of an RC frame under optimization, (b) RC frame under optimization with slabs considered as dead loads, (c) A frame model in MIDAS. optimized in the present study. Frame geometries, factor (K ), exposure class, and enclosing condition, zt loading, support conditions, end-conditions of mem- whereas seismic load parameters include spectral bers, and member grouping are modeled by MIDAS as response acceleration parameter at short periods shown in Figure 3(c). Parameters defining vertical and (S = 0.55 g), deflection amplification factor: horizontal loads are presented in Table 2. Wind load (C ¼ 2:5), coefficients for calculating approximated parameters include basic wind speed, topographic fundamental periods ðC ¼ 0:0466Þ; spectral response Figure 4. Five steps to optimize multiple objective functions simultaneously by the ANN-based Hong–Lagrange algorithm (MathWorks 2022a). 6 W.-K. HONG AND T. D. PHAM Figure 5. The Pareto frontier when minimizing CI and W, comparing optimized designs points with big datasets. Table 2. Loads and material properties of an RC frame under optimizations. Vertical loads Live load ¼ 6kN=m Self-weight of 200 mm-thickness slab ¼ 4:8kN=m Finishing layer ¼ 1:2kN=m Wind load parameters Basic wind speed Topographic factor (K ) Exposure class Enclosing condition zt 26 m/s 1 C Enclosed frame Seismic load parameters Spectral response Deflection amplification factor: Coefficients for calculating approximated fundamental acceleration parameter at short C ¼ 2:5 periods: C ¼ 0:0466 and x ¼ 0:9 d t periods: S = 0.55 g Spectral response Response modification coefficient: R ¼ 3 Gravity acceleration: 9:81m=s acceleration parameter at a period of 1 s: S ¼ 0:22 g Long-period transition period: T ¼ 8 s The importance factor: I ¼ 1 Site class: D L e Overstrength factor: Damping ratio: 5% Ω ¼ 3 Material properties Concrete Rebar Stirrup Strength (MPa) 40 600 500 Elastic modulus (GPa) 29.75 200 200 Thermal coefficient 9.9E-6 1E-5 1E-5 Poisson’s ratio 0.2 0.3 0.3 Unit weight (kN/m ) 25 78.5 78.5 Unit price (Korean Won – KRW/m ) 94E3 8.5E6 8.3E6 Unit CO emission (T-CO /m ) 0.168 19.73 19.73 2 2 Unit energy consumption (MJ/m ) 2.4E3 25E4 25E4 acceleration parameter at a period of 1 s (S ¼ 0:22 g), according to Korean markets in 2021. Units CO emis- 1 2 response modification coefficient (R ¼ 3), and gravity sions and energy consumptions are calculated accord- acceleration (9:81 =s). Material properties used in this ing to studies of (Hong et al. (2010) and Kuk Kim et al. study are also defined in Table 2, where material prices (2013), respectively. It is noted that unit values of costs, as Korean Won per cubic meters (KRW/m ) are taken weights, CO emissions, and energy consumptions are Table 3. Load combinations considered in the present study. Dead load Live load Wind load Seismic load Service combinations 1 1 0 0 1 0 0.6 0 1 0.75 0 0.525 Strength combinations 1.2 1.6 0.8 0 0.9 0 1.6 0 1.275 1 0 1 A combination for calculating masses in dynamic analyses 1 0.25 0 0 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 7 not fixed, and they can be changed easily before gen- Optimized objective functions for cost index and a total weight of frames are calculated in erating big datasets. Parameters 47 and 49 of Table 4, respectively. The objective functions of costs and weights in frames 2.2. Data generations are calculated based on concrete volumes, rebar ratios, and stirrups areas in beams and columns. A frame generator is developed, referring to open-source Costs and weights from slabs are excluded from software LESM (Fernando Martha et al. 2022). The ABBA objective functions; however, weights and stiffness generator produces 330,000 data samples of an RC frame of slabs are considered by the ABBA frame genera- (Figure 3) according to ACI 318-19 (Building Code tor in dynamic investigations. Ranges of big data- Requirements for Structural Concrete (ACI 318-19) sets are shown in Table 4 where ranges of output Commentary on, & Building Code Requirements for parameters are obtained for preassigned ranges of Structural Concrete (ACI 318R-19) 2019) and ASCE 7-16 input parameters. There are many more parameters (Loads and Structures 2017) as presented in Table 4. Each calculated during bigdata generations, such as an data sample includes 18 forward input parameters and 32 upper limit fundamental period (C T ) for determin- forward output parameters. The present example uses u a ing base shear, base shears, wind pressures, section three service load combinations and three strength load strengths, etc. However, they are not printed on an combinations shown in Table 3 which influences the output-side of the big datasets because they are design significantly. Table 4. Ranges of big data, (a) Forward design input parameters in 330,000 samples (18 varied inputs), (b) Forward design output parameters for big data (32 output parameters). No. Parameters Range Column Group 1 1 HC Size (mm) (assuming square columns) 400–1000 2 ρ C Rebar ratio in X-direction 0.01–0.04 X 1 3 ρ C Rebar ratio in Y-direction 0.01–0.04 Y 1 Column Group 2 4 HC Size (mm) (assuming square columns) 400–1000 5 ρ C Rebar ratio in X-direction 0.01–0.04 X 2 6 ρ C Rebar ratio in Y-direction 0.01–0.04 Y 2 Beam Group 1 7 BB Section width (mm) 200–1000 8 HB Section height (mm) 400–1000 9 ρ B Top rebar ratio at beam end 0.001–0.025 ts 1 10 ρ B Top rebar ratio at mid-span 0.001–0.025 tm 1 11 ρ B Bottom rebar ratio at beam end 0.001–0.025 bs 1 12 ρ B Bottom rebar ratio at mid-span 0.001–0.025 bm 1 Beam Group 2 13 BB Section width (mm) 200–1000 14 HB Section height (mm) 400–1000 15 ρ B Top rebar ratio at beam end 0.001–0.025 ts 1 16 ρ B Top rebar ratio at mid-span 0.001–0.025 tm 1 17 ρ B Bottom rebar ratio at beam end 0.001–0.025 bs 1 18 ρ B Bottom rebar ratio at mid-span 0.001–0.025 bm 1 Column safety factors 19 SFC S Column Group 1 on Story 1 0.097–4.924 330,000 1 1 20 SFC S Column Group 1 on Story 2 0.134–6.815 330,000 1 2 21 SFC S Column Group 2 on Story 3 0.121–8.238 330,000 2 3 22 SFC S Column Group 2 on Story 4 0.149–10.400 330,000 2 4 Beam Group 1 on Story 1 23 SFB S Safety factor 0.049–3.799 330,000 1 1 24 Δ B S Short-term deflection (mm) 0.282–20 F250,000 s 1 1 25 Δ B S Long-term deflection (mm) 0.489–34.999 F250,000 L 1 1 26 σ B S Rebar stress under service loads 6.773–599.9 F250,000 s 1 1 Beam Group 1 on Story 2 27 SFB S Safety factor 0.05–3.596 330,000 1 2 28 Δ B S Short-term deflection (mm) 0.282–20 F250,000 s 1 2 29 Δ B S Long-term deflection (mm) 0.522–34.999 F250,000 L 1 2 30 σ B S Rebar stress under service loads 6.956–599.9 F250,000 s 1 2 Beam Group 2 on Story 3 31 SFB S Safety factor 0.046–3.751 330,000 2 3 32 Δ B S Short-term deflection (mm) 0.276–20 F250,000 s 2 3 33 Δ B S Long-term deflection (mm) 0.505–35 F250,000 L 2 3 34 σ B S Rebar stress under service loads 7.668–599.899 F250,000 s 2 3 Beam Group 2 on Story 4 35 SFB S Safety factor 0.045–3.581 330,000 2 4 36 Δ B S Short-term deflection (mm) 0.345–20 F250,000 s 2 4 37 Δ B S Long-term deflection (mm) 0.623– 35 F250,000 L 2 4 38 σ B S Rebar stress under service loads 7.016–599.899 F250,000 s 2 4 Rebar strain when concrete strain reaches 0.003 39 ε B Rebar strain of Beam Group 1 at supports 0.002–0.079 330,000 sp 1 40 ε B Rebar strain of Beam Group 1 at mid-span 0.002–0.077 330,000 sm 1 41 ε B Rebar strain of Beam Group 2 at supports 0.002–0.081 330,000 sp 2 42 ε B Rebar strain of Beam Group 2 at mid-span 0.002–0.083 330,000 sm 2 Dynamic analyses 43 T Fundamental period (s) 0.195–2.361 330,000 44 D Maximum wind deformations (mm) 0.275–11.961 330,000 max 45 Drift Maximum story drift ratio of all stories 0.002–0.06 330,000 max 46 θ Stability coefficient 0.005–0.222 330,000 Effective indexes 47 CI Cost indexðKoreanWon KRWÞ 39,962,969–565,522,773 330,000 48 CO CO emission ðT CO Þ 82–1251 330,000 2 2 2 49 W Total weight of beams and columns ðkNÞ 4296–37,570 330,000 50 Energy consumption 1,103,363–16,222,172 330,000 8 W.-K. HONG AND T. D. PHAM Table 5. Equality and inequality constraints. (a) Inequalities imposed on forward input parameters related to column configurations V : HC ≤ 1000 mm V : ρ C � 0:039 V : ρ C � 0:01 1 1 1 6 1 11 2 Y Y V : HC � 400 mm V : HC � 1000 mm V : ρ C � 0:039 2 1 7 2 12 Y 2 V : ρ C � 0:01 1 V : HC � 400 mm V : HC � HC 3 1 8 2 13 2 1 V : ρ C � 0:039 V : ρ C � 0:01 1 V : ρ C � ρ C 4 1 9 2 14 2 1 X X X X V : ρ C � 0:01 1 V : ρ C � 0:0 39 V : ρ C � ρ C 5 1 10 2 15 2 1 Y X Y Y (b) Inequalities imposed on forward input parameters related to beam configurations V : BB � 1000 mm V : ρ B � 0:015 V : ρ B � 0:0 15 16 1 26 1 36 2 tm ts V : BB � 250 mm V : ρ B � 0:0025 V : ρ B � 0:0025 17 1 27 1 37 2 tm ts V : HB � 1000 mm V : ρ B � 0:015 V : ρ B � 0:015 18 1 28 bm 1 38 bs 2 V : HB � 400 mm V : ρ B � 0:0025 V : ρ B � 0:0025 19 1 29 bm 1 39 bs 2 V : BB � HB V : BB � 1000 mm V : ρ B � 0:015 20 1 1 30 2 40 2 tm V : BB � 0:3HB V : BB � 250 mm V : ρ B � 0:0025 21 1 1 31 2 41 2 tm V : ρ B � 0:0 15 V : HB � 1000 mm V : ρ B � 0:015 22 ts 1 32 2 42 bm 2 V : ρ B � 0:0025 V : HB � 400 mm V : ρ B � 0:0025 23 ts 1 33 2 43 bm 2 V : ρ B � 0:015 V : BB � HB 24 1 34 2 2 bs V : ρ B � 0:0025 V : BB � 0:3HB 25 1 35 2 2 bs (c) Inequalities imposed on forward output parameters related to safety factors V : SFC S � 1 V : SFC S � 1 V : SFB S � 1 : SFB S � 1 44 1 1 46 2 3 48 1 1 50 2 3 V : SFC S � 1 V : SFC S � 1 V : SFB S � 1 V : SFB S � 1 45 1 2 47 2 4 49 1 2 51 2 4 (d) Inequalities imposed on forward output parameters related to rebar strains V : ε B � 0:006 V : ε B � 0:006 V : ε B � 0:006 V : ε B � 0:006 52 sp 1 53 sm 1 54 sp 2 55 sm 2 (e) Inequalities imposed on forward output parameters related to deflections Short-term deflection Long-term deflection V : Δ B S � L360 ¼ 22:22 mm V : Δ B S � L240 ¼ 33:33 mm 56 s 1 1 60 L 1 1 V : Δ B S � L360 ¼ 22:22 mm V : Δ B S � L240 ¼ 33:33 mm 57 s 1 2 61 L 1 2 V : Δ B S � L360 ¼ 22:22 mm V : Δ B S � L240 ¼ 33:33 mm 58 s 2 3 62 L 2 3 V : Δ B S � L360 ¼ 22:22 mm V : Δ B S � L240 ¼ 33:33 mm 59 s 2 4 63 L 2 4 (f) Inequalities imposed on forward output parameters related to rebar stresses under service loads V : σ B S � 23f ¼ 400 MPa V : σ B S � 23f ¼ 400 MPa 64 s 1 1 y 66 s 2 3 y V : σ B S � 23f ¼ 400 MPa V : σ B S � 23f ¼ 400 MPa 65 s 1 2 y 67 s 2 4 y (g) Inequalities imposed on forward output parameters related to lateral deformations V : Drift � 0:015 V : θ � 0:2 V : D � H500 ¼ 32 mm 68 max 69 70 max not constrained by any code-based requirements. It training if a safety factor of 1 is desired in optimal is noted that F250,000 data samples are obtained designs. Constructability of a frame is considered in by filtering from 330,000 big data samples. Some Inequalities from V to V where sizes and rebars of 13 15 data such as deflections and stresses calculated the upper column groups are constrained to be smaller when unfactored moments exceed nominal capaci- than or equal to those of the lower column group. ties of sections are removed. Other parameters, Inequalities from V to V in Table 5(c) constrain 44 47 such as safety factors or story drifts, are generated column and beam safety factors in all four stories correctly in all cases. (from Stories 1 to 4) to be ≥1, ensuring safeties of optimized designs. Inequalities V to V in Table 5 52 55 (d) constrain beam rebar strains to be ≥0.006 when 2.3. Design requirements as inequality concrete strains reach 0.003, avoiding brittle failure in constraints beams according to Section 9.3.3.1 of ACI 318=19 (Building Code Requirements for Structural Concrete Table 5 presents 70 inequalities imposed during an (ACI 318-19) Commentary on, & Building Code optimizations of RC frames. A number of inequalities Requirements for Structural Concrete (ACI 318R-19) is greater than that of forward input parameters 2019). Inequalities from V to V in Table 5(e) con- 56 63 because multiple inequalities are applied to one para- strain beam immediate deflections to be smaller than meter. For example, Inequalities V and V shown in 1 13 or equal to and beam long-term deflections to be Table 5(a) are applied to HC (size of columns in smaller than or equal to according to Table 24.2.2 Column Group 1). In Tables 5(a, b), Inequalities from 240 in ACI 318-19 (Building Code Requirements for V to V are imposed on 18 forward input parameters 1 43 Structural Concrete (ACI 318-19) Commentary on, & when optimizing designs. Ranges for 14 forward input Building Code Requirements for Structural Concrete parameters out of 18 forward input parameters indi- (ACI 318R-19) 2019), guaranteeing the serviceability cated in 26 inequalities from V to V , from V to V , 3 6 9 12 of optimized frames. Inequalities from V to V in 64 67 V , from V to V , V , and from V to V are taken 17 22 29 31 36 43 Table 5(f) constrain rebar tensile stresses under service slightly narrower than ranges of big datasets to avoid loads to be smaller than or equal to according to sparse data at the edge range of the big datasets. It is Table 24.3.2.1 in ACI 318-19 (Building Code noted that big datasets should be wide enough to Requirements for Structural Concrete (ACI 318-19) cover expected magnitudes, for example, a safety fac- Commentary on, & Building Code Requirements for tor of 1 should appear inside ranges of big datasets for JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 9 Structural Concrete (ACI 318R-19) 2019) for controlling including 5 and 10 layers and a combination of the two crack widths. Inequality V in Table 5(g) constrains the types of neurons including 80 and 128 neurons are maximum story drift ratios of all stories to be ≤0:015. implemented. A number of layers and neurons are The limitation of story drift ratios in four-story frames shown in Table 6 with epochs that provide the best with a Risk Category II is 0.02 according to training accuracies. Validation indicates both desig- Table 12.12-1 in ASCE 7-16 (Loads and Structures nated number of epochs and terminating epochs to 2017). However, the present example uses prevent over-fitting. For example, training proceeded a conservative limit of 0.015 (a default limitation in up to 50,000 epochs for Story 1 of column safety factor MIDAS) for story drift ratios. Inequality V in Table 5 as designated, however, training terminated at Epoch (g) constrains the stability coefficient to be smaller 45,506 to prevent over-fitting for story 3 of column 0:5 0:5 than or equal to θ ¼ ¼ ¼ 0:2 according to safety factor. max βC 2:5 The best training accuracy for rebar strain corre- Section 12.8.7 in ASCE 7-16 (Loads and Structures sponding to concrete strain of 0.003 at beam end of 2017). Inequality V in Table 5(g) constrains the lateral Group 1 is obtained with 5 layers and 128 neurons, deflections due to wind loads to be smaller than or yielding Test mean square errors (MSE) = 1.2727E-04 equal to H500 ¼ 32 mm according to Section CC.2.2 and Regression = 0.9991. Training accuracies in terms in ASCE 7–16 (Loads and Structures 2017). of Test MSE and Regression are presented to judge In summary, the optimization determines 18 design training results. Detailed descriptions of ANN training variables listed in Table 4(a) including sizes and rebar on structural data can be found in the books by Hong ratios of beams and columns to minimize costs and (2019, 2021). weights simultaneously. Designs are also constrained by 70 inequalities constraints presented in Table 5, ensuring the safety, stability and constructability of 3.2. ANN-based Hong–Lagrange algorithm optimized frames. The ANN-based optimization of multi-objective func- 3. Formulating ANN-based objective tions for RC frames shown in Figure 3 is performed in 5 functions and ANN-based Hong–Lagrange steps as presented in Figure 4. Steps 1 to 3 calculate algorithm ANN-based Lagrange functions for single objective function by which a UFO is derived in Step 4 to mini- 3.1. Formulating ANN-based objective functions mize it in Step 5, leading to an identification of a Pareto Table 6 presents training accuracies of the ANNs in frontier. A set of MOO results is a Pareto frontier or which a combination of the two types of hidden layers a Pareto set which is obtained by solving KKT Table 6. Training accuracy, showing test MSE and regression. Number of Best Test Parameter samples Layer Neuron epoch Validation MSE Regression Column safety factor SFC S 330,000 10 80 48,817 50,000 3.4E-05 0.9999 1 1 SFC S 330,000 5 128 48,621 50,000 3.6E-05 0.9999 1 2 SFC S 330,000 5 128 40,506 45,506 7.0E-05 0.9994 2 3 SFC S 330,000 5 128 43,156 48,156 7.8E-05 0.9995 2 4 Beam at Story 1 SFB S 330,000 5 128 49,999 50,000 3.1E-05 1.0000 1 1 Δ B S F250,000 10 80 11,298 16,298 2.9E-03 0.9978 s 1 1 Δ B S F250,000 10 80 11,794 16,794 3.7E-03 0.9970 L 1 1 σ B S F250,000 10 80 10,347 15,347 6.0E-03 0.9966 s 1 1 Beam at Story 2 SFB S 330,000 5 128 49,524 50,000 3.3E-05 1.0000 1 2 Δ B S F250,000 5 128 13,961 18,961 5.8E-03 0.9962 s 1 2 Δ B S F250,000 10 80 21,970 26,970 3.8E-03 0.9976 L 1 2 σ B S F250,000 10 80 9806 14,806 7.7E-03 0.9957 s 1 2 Beam at Story 3 SFB S 330,000 5 128 49,987 50,000 3.9E-05 1.0000 2 3 Δ B S F250,000 5 128 11,813 16,813 3.7E-03 0.9972 s 2 3 Δ B S F250,000 5 128 5491 10,491 7.1E-03 0.9899 L 2 3 σ B S F250,000 10 80 7932 12,932 7.5E-03 0.9956 s 2 3 Beam at Story 4 SFB S 330,000 10 80 49,986 50,000 4.2E-05 1.0000 2 4 Δ B S F250,000 10 80 7698 12,698 5.4E-03 0.9960 s 2 4 Δ B S F250,000 10 80 11,754 16,754 3.9E-03 0.9972 L 2 4 σ B S F250,000 10 80 20,762 25,762 3.1E-03 0.9983 s 2 4 Rebar strain when concrete strain reaches ε B 330,000 5 128 36,056 41,056 1.2E-04 0.9991 sp 1 0.003 ε B 330,000 5 128 35,736 40,736 1.4E-04 0.9991 sm 1 ε B 330,000 5 128 30,896 35,896 1.2E-04 0.9991 sp 2 ε B 330,000 5 128 47,476 50,000 1.1E-04 0.9991 sm 2 T ðsÞ 330,000 5 128 48,670 50,000 2.6E-05 1.0000 Lateral deflection D 330,000 5 128 48,756 50,000 5.7E-07 1.0000 max Drift 330,000 5 128 47,621 50,000 1.3E-06 1.0000 max θ 330,000 10 80 49,784 50,000 7.4E-07 1.0000 Objective function CI 330,000 5 128 49,992 50,000 1.2E-06 1.0000 CO2 330,000 5 128 49,987 50,000 2.1E-06 1.0000 W 330,000 5 128 49,993 50,000 6.4E-06 1.0000 Energy consumption 330,000 5 128 49,875 50,000 5.7E-07 1.0000 10 W.-K. HONG AND T. D. PHAM conditions (Kuhn and Tucker 1951) with a Newton– Design parameters identified for Design P1, Design Raphson method in the SQP algorithm (MathWorks P5, and Design P9 based on nine fractions are pre- 2022a). A Pareto frontier investigating particular trade- sented in Table 7. Design P1 (w : w ¼ 1 : 0), CI W off ratios estimates how much sacrifice is made by Design P5 (w : w ¼ 0.5 : 0 .5), and Design P9 CI W each objective function to UFO based on the proposed (w : w ¼ 0:1) are indicated in Table 7. Design para- CI W ANN-based Hong–Lagrange algorithm. A design exam- meters of Design P1 based on a fraction of w : w ¼ CI W ple of RC frames is performed in this study in which 1 : 0 in which CI is only minimized are obtained. Design two objective functions, cost, and weight, are simulta- P9 identifies an optimized design parameters where neously minimized using ANNs, leading to optimizing weight of the RC frame, W , is only minimized based on UFO. Design parameters minimizing UFO of RC frames (w : w ¼ 0:1) in which a Pareto frontier similar to CI W are also obtained. that of Design P1 is obtained. Design P5 identifies an optimized design parameters with an equal tradeoff between the two objective functions based on (w : w ¼ 0.5 : 0 .5). Tables 7(a), 5(b), and 5(c) pre- CI W 4. Optimal designs sent design parameters optimized by ANN-based In this study, design parameters are optimized simul- Hong–Lagrange algorithm of UFO while insignificant taneously based on two objective functions (cost index errors are verified by structural mechanics-based ABBA and weight of the RC frame). Designs P1 to P9 corre- frame generator. Design accuracies of ANNs for Design sponding to nine fractions (w : w ¼ 1 : 0, w : CI W CI P1, Design P5, and Design P9 are also shown in w ¼ 0.875 : 0 .125, w : w ¼ 0.75 : 0 .25, w : W CI W CI Tables 7(a), 5(b), and 5(c) where differences between w ¼ 0.625 : 0 .375, w : w ¼ 0.5 : 0 .5, w : w ¼ W CI W CI W design parameters and the ABBA frame generator are 0.375 : 0 .625, w : w ¼ 0.25 : 0 .75, w : w ¼ CI W CI W negligible. 0.125 : 0 .875, w : w ¼ 0:1) on Pareto frontier are CI W It is noted that, in Table 7(a), the upper tensile indicated in Figure 5 which also demonstrates that big rebar ratio ρ B = 0.0065 at beam end is obtained ts datasets hardly show minimized data in the lowest as shown in Design P1 when a beam cost CI is data range, however, it does not mean that big data- minimized at 71,174,081 KRW, whereas the upper sets do not cover the data in the lowest data zone. The tensile rebar ratio ρ B = 0.0090 is obtained as ts gaps between Pareto frontier and big datasets can be shown in Design P9 when a beam weight is mini- filled if more big datasets are generated. It is the mized to 7,307 kN from 8,585 kN of Design P1, proposed method that, more efficiently, optimizes resulting increased rebar ratio (ρ B Þ by st 0:0090 0:0065 designs and predicts the lowest bound of the big � 0:39% ¼ 39% to reduce the beam 0:0065 datasets with one run. weight, but sacrificing cost which increased from Table 7. Optimal designs based on fraction combinations P1, P5, and P9 in Figure 5. (a) Design P1; minimizing CI (W = 1 & W = 0) (refer to Fig. 5) CI W (Continued) JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 11 Table 7. (Continued). (b) Design P5; minimizing CI and W (W = 0.5 & W = 0.5) (refer to Fig. 5) CI W (c) Design P9; minimizing CI and W (W = 0 & W = 1) (refer to Fig. 5) CI W 71,174,081 to 104,003,171 KRW with 46% increase. (w : w ¼ 1 : 0), Design P5 (w : w ¼ 0.5 : 0 CI W CI W It is noted, in Table 7(b), that the upper tensile .5), and Design P9 (w : w ¼ 0 : 1) can also be CI W rebar ratio ρ B = 0.0092 at beam end is obtained obtained from Pareto curve. Input parameters ts as shown in Design P5 when weight of the RC including preassigned input parameters and 18 frame is minimized based on an equal tradeoff varied input parameters shown in Table 4 are between the two objective functions at Design P5 used for an ANN and a structural mechanics- (w : w ¼ 0.5 : 0.5). A cost CI and a beam weight based ABBA generator to obtain 32 corresponding CI W obtained with Design P5 are 74,914,364 KRW and outputs shown in Table 4. Design accuracies 7,845 kN which are mid-range of those obtained obtained using equation Error ¼ between Design P1 and Design P9. Other design ge ne rator AN N Output Output � 100% based on ANN and AN N Output parameters than those shown with Design P1 12 W.-K. HONG AND T. D. PHAM structural mechanics-based software ABBA genera- parameters which is challenging to obtain using tor seen in Tables 7(a), (b), and (c) are as large conventional design methods. This study is as −7.5%. a steppingstone for the next step in structural ana- Probable designs are determined based on averages lysis and design research with the advent of AI- of the top 688 (0.1%) designs among 688,000 designs based Data-centric Engineering which is not based randomly generated based on the ABBA generator on structural mechanics (Hong W. K. 2023). shown in Figure 5, resulting in probable design values of 129,367,608 KRW and 10,924 kN for CI and W, respec- Disclosure statement tively. Optimized designs, and hence, produces a cost savings up to 44.983% and a weight reduction up to No potential conflict of interest was reported by the authors. 33.111% over probable designs. Funding 5. Conclusions This work was supported by the National Research A resilient design capable of optimizing RC frames has Foundation of Korea (NRF) grant funded by the Korean gov- been performed beyond human efficiency. It is difficult ernment [MSIT 2019R1A2C2004965]. for engineers to pre-assign constraining conditions on an input-side for a conventional design. The present study References replaced complex analytical objective functions by ANN- based objective functions. Any type of objective function Artificial Neural Network-based Optimized Design of Reinforced and design target of interest can be implanted as artificial Concrete Structures (under preparation). Taylor and Francis. Asteris, P. G., A. K. Tsaris, L. Cavaleri, C. C. Repapis, A. Papalou, neural genes to govern an optimization process minimiz- F. Di Trapani, and D. F. Karypidis. 2016. “Prediction of the ing design targets for RC frames. Design codes and con- Fundamental Period of Infilled Rc Frame Structures Using straining requirements include economies, Artificial Neural Networks.” Computational Intelligence and environments, and serviceability. ANN is formulated to Neuroscience 2016: 1–12. doi:10.1155/2016/5104907. provide 32 forward output parameters based on 18 for- Bai, J., S. Jin, and J. Ou. 2020. “An Efficient Method for ward input parameters to simultaneously optimize objec- Optimizing the Seismic Resistance of Reinforced Concrete Frame Structures.” Advances in Structural Engineering 23 (4): tive functions of a cost CI and a weight W of RC frames. In 670–686. doi:10.1177/1369433219878856. this study, pure RC frames are optimized where the AI- Blum, C., and A. Roli. 2003. “Metaheuristics in Combinatorial based auto-designs are developed using the ANN-based Optimization: Overview and Conceptual Comparison.” Hong–Lagrange algorithm. Additional structural systems ACM Computing Surveys 35 (3): 268–308. doi:10.1145/ such as dual frames, SRC frames, and PT frames are under 937503.937505. developments. AI-based auto-designs proposed in this Building Code Requirements for Structural Concrete (ACI 318-19) Commentary on, & Building Code Requirements study replace conventional design optimization pro- for Structural Concrete (ACI 318R-19). 2019. “An ACI cesses by automatically producing design parameters Standard.” ACI Committee 318. while optimizing design targets. It is challenging to Camp, C. V., and F. Huq. 2013. “CO2 and Cost Optimization of achieve auto-designs by conventional approaches. Reinforced Concrete Frames Using a Big bang-big Crunch In this study, ANN-based Hong–Lagrange algo- Algorithm.” Engineering Structures 48: 363–372. doi:10. rithm is introduced for a holistic optimization of 1016/j.engstruct.2012.09.004. Esfandiari, M. J., G. S. Urgessa, S. Sheikholarefin, and RC frames. Objective functions with equality and S. H. D. Manshadi. 2018. “Optimum Design of 3D inequality constraints are implanted in ANN-based Reinforced Concrete Frames Using DMPSO Algorithm.” Lagrange functions to control optimizations of the Advances in Engineering Software 115 (July 2017): designs. Design parameters are obtained based on 149–160. doi:10.1016/j.advengsoft.2017.09.007. 330,000 data samples to minimize ANN-based Fernando Martha, L., R. Lopez Rangel, P. Cortez Lopes, and objective functions for costs CI and weights W of C. Horta. 2022. “LESM.” PONTIFICAL CATHOLIC UNIVERSITY RC frames. Structural optimality is verified by OF RIO DE JANEIRO. García-Segura, T., V. Yepes, and D. M. Frangopol. 2017. “Multi- 668,000 data samples, resulting in cost savings up objective Design of post-tensioned Concrete Road Bridges to 44.983% and weight reductions up to 33.111% Using Artificial Neural Networks.” Structural and compared with probable designs based on the aver- Multidisciplinary Optimization 56 (1): 139–150. doi:10. age of 688 (0.1%) best designs among 688,000 big 1007/s00158-017-1653-0. data samples. Design accuracies are enhanced and Hong, W. K. 2023. Artificial Neural Network-based Optimized Design of Reinforced Concrete Structures. United States: verified by conventional structural designs while CRC Press. reducing labors of human engineers in optimizing HONG, W.-K. 2019. Hybrid Composite Precast Systems: designs for beams, columns, and frames. ANN-based Numerical Investigation to Construction. United Kingdom: optimized designs were not available before the Woodhead Publishing, Elsevier. present research. Now it is possible to autono- Hong, W. K. 2021. Artificial intelligence-based Design of mously optimize design targets to calculate design Reinforced Concrete Structures. 978-89-6285-291-2. Korea. JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 13 Hong, W.-K., J.-M. Kim, S.-C. Park, S.-G. Lee, S.-I. Kim, K.-J. Yoon, Reinforced Concrete Structural Elements.” Structure and H.-C. Kim, and J. Tai Kim. 2010. “A New Apartment Infrastructure Engineering 16 (7): 917–930. doi:10.1080/ Construction Technology with Effective CO2 Emission 15732479.2019.1674893. Reduction Capabilities.” Energy 35(6. doi:10.1016/j. Pelliciari, M., G. C. Marano, T. Cuoghi, B. Briseghella, energy.2009.05.036. D. Lavorato, and A. M. Tarantino. 2018. “Parameter Hong, W. K., and M. C. Nguyen. 2021. “AI-based Lagrange Identification of Degrading and Pinched Hysteretic Optimization for Designing Reinforced Concrete Systems Using a Modified Bouc–Wen Model.” Structure Columns.” Journal of Asian Architecture and Building and Infrastructure Engineering 14 (12): 1573–1585. doi:10. Engineering 1–15. doi:10.1080/13467581.2021.1971998. 1080/15732479.2018.1469652. Hong, W. K., V. T. Nguyen, and M. C. Nguyen. 2021. Sahab, M. G., A. F. Ashour, and V. V. Toropov. 2005. “Cost “Optimizing Reinforced Concrete Beams Cost Based on Optimisation of Reinforced Concrete Flat Slab Buildings.” AI-based Lagrange Functions.” Journal of Asian Engineering Structures 27 (3): 313–322. doi:10.1016/j. Architecture and Building Engineering 1–18. doi:10.1080/ engstruct.2004.10.002. 13467581.2021.2007105. Sharafi, P., M. N. S. Hadi, and L. H. Teh. 2012. “Heuristic Kuhn, H. W., and A. W. Tucker. 1951. “Nonlinear Approach for Optimum Cost and Layout Design of 3D Programming.” Berkeley Symposium on Mathematical Reinforced Concrete Frames.” Journal of Structural Statistics and Probability Engineering 138 (7): 853–863. doi:10.1061/(asce)st.1943- Kuk Kim, S., W.-K. Hong, K. Hyo-Jin, and J. Tai Kim. 2013. “The 541x.0000508. Energy Efficient Expansion Remodeling Construction Shin, J., D. W. Scott, L. K. Stewart, and J. S. Jeon. 2020. Method of Bearing Wall Apartment Buildings with “Multi-hazard Assessment and Mitigation for pre-cast Composite Structural Systems.” Energy and seismically-deficient RC Building Frames Using Buildings 66: 714–723. doi:10.1016/j.enbuild.2013.07.080. Artificial Neural Network Models.” Engineering Lagrange, J. L. 1804. Leçons sur le calcul des fonctions. Paris, Structures 207 (February 2019): 110204. doi:10.1016/j. France: Imperiale. engstruct.2020.110204. Loads, M. D., and O. Structures. 2017. “Minimum Design Sirotti, S., M. Pelliciari, F. Di Trapani, B. Briseghella, G. Carlo Loads and Associated Criteria for Buildings and Other Marano, C. Nuti, and A. M. Tarantino. 2021. “Development Structures.” Minimum Design Loads and Associated and Validation of New Bouc–Wen Data-Driven Hysteresis Criteria for Buildings and Other Structures. doi:10.1061/ Model for Masonry Infilled RC Frames.” Journal of 9780784414248. Engineering Mechanics 147 (11): 1–15. doi:10.1061/(asce) MathWorks. 2022a. “Deep Learning Toolbox: User’s Guide em.1943-7889.0002001. (R2022a).” Accessed 26 July 2012. https://www.math Srinivas, V., and K. Ramanjaneyulu. 2007. “An Integrated works.com/help/pdf_doc/deeplearning/nnet_ug.pdf Approach for Optimum Design of Bridge Decks Using MathWorks. 2022a. “Global Optimization: User’s Guide Genetic Algorithms and Artificial Neural Networks.” (R2022a).” Accessed 26 July 2012. https://www.math Advances in Engineering Software 38 (7): 475–487. doi:10. works.com/help/pdf_doc/gads/gads.pdf 1016/j.advengsoft.2006.09.016. MathWorks. 2022a. “MATLAB (R2022a).” Upton, G., and I. Cook. 2014. A Dictionary of Statistics 3e. UK: MathWorks. 2022a. “Optimization Toolbox: Documentation Oxford university press. (R2022a).” Accessed July 26, 2022. https://uk.mathworks. Villarrubia, G., J. F. De Paz, P. Chamoso, and F. De la Prieta. com/help/optim/ 2018. “Artificial Neural Networks Used in Optimization MathWorks. 2022a. “Parallel Computing Toolbox: Problems.” Neurocomputing 272: 10–16. doi:10.1016/j.neu Documentation (R2022a).” Accessed 26 July 2022. com.2017.04.075. https://uk.mathworks.com/help/parallel-computing/ Yeo, D., and F. A. Potra. 2015. “Sustainable Design of MathWorks. 2022a. “Statistics and Machine Learning Reinforced Concrete Structures through CO2 Emission Toolbox: Documentation (R2022a).” Accessed 26 July Optimization.” Journal of Structural Engineering 141 (3). 2022. https://uk.mathworks.com/help/stats/ doi:10.1061/(asce)st.1943-541x.0000888. Paya-Zaforteza, I., V. Yepes, A. Hospitaler, and Zadeh, L. 1963. “Optimality and non-scalar-valued F. González-Vidosa. 2009. “CO2-optimization of Performance Criteria.” IEEE Transactions on Automatic Reinforced Concrete Frames by Simulated Control 8 (1): 59–60. doi:10.1109/TAC.1963.1105511. Annealing.” Engineering Structures 31 (7): 1501–1508. Zou, X. K., C. M. Chan, G. Li, and Q. Wang. 2007. doi:10.1016/j.engstruct.2009.02.034. “Multiobjective Optimization for Performance-Based Pelliciari, M., B. Briseghella, F. Tondolo, L. Veneziano, C. Nuti, Design of Reinforced Concrete Frames.” Journal of R. Greco, D. Lavorato, and A. M. Tarantino. 2020. Structural Engineering 133 (10): 1462–1474. doi:10.1061/ “A Degrading Bouc–Wen Model for the Hysteresis of (asce)0733-9445(2007)133:10(1462).
Journal
Journal of Asian Architecture and Building Engineering
– Taylor & Francis
Published: Sep 3, 2023
Keywords: ANN-based Hong–Lagrange algorithm of RC frames; Pareto frontier of RC frames; big datasets of RC frames; KKT solutions of RC frames; weight fraction of MOO problems