An AI-based auto-design for optimizing RC frames using the ANN-based Hong– Lagrange algorithm

Artificial neural networks (ANNs)-based objective functions such as costs and weights of reinforced concrete (RC) frames with four-by-four bays and four floors are optimized simultaneously based on big datasets of 330,000 designs according to ACI 318-19, whereas corresponding design parameters, which minimize objective functions, are also obtained. The Pareto frontier verified by big datasets shows reductions up to 44.983% and 33.111% in costs and weights, respectively, compared with probable designs based on averages of 688 (0.1%) best designs among 688,000 samples. Optimized designs meeting requirements imposed by codes and architects are achieved using the ANN-based Hong–Lagrange algorithm in which complex analytical objective functions are replaced by ANN-based objective functions. 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 parameters. 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.


Literature review
Building optimization is always an ultimate goal of structural engineers.However, it is difficult to explicitly derive analytical objective functions to optimize complex reinforced concrete (RC) frames that meet all code requirements simultaneously.This is a complex task, especially, when multiple constraining conditions are to be imposed.Zou et al. (2007) formulated the life-cycle costs of a building as a multi-objective optimization (MOO) problem consisting of material costs and expected damages due to seismic actions.The MOO problem, then, was solved using a ε-constraint method, providing a Pareto set of optimal building designs.Paya-Zaforteza et al. (2009) presented a method based on a similated annealing algorithm, optimizing CO 2 emissions and costs of buildings designed based on the Spanish code.Results indicated a close relationship between two objective functions where environmental and economical efficiencies of a design minimizing CO 2 emissions were relatively similar to those obtained from a design with optimized costs.On the other hand, Yeo and Potra (2015) reported a difference of 5% to 10% between CO 2 emissions of a design minimizing CO 2 emissions and those of the cheapest design.Camp and Huq (2013) implemented the Big Bang-Big Crunch algorithm in reducing CO 2 emissions and costs of RC frames, resulting in improvements compared with genetic and annealing algorithms.The study by Sharafi, Hadi, and Teh (2012) applied a colony optimization algorithm in minimizing a cost of a 3D RC frame, resulting in a further cost reduction of 4.8% compared with a study by Sahab, Ashour, and Toropov (2005) which combined an exhaustive search algorithm, a genetic algorithm, and a Hook and Jeeves method.Esfandiari et al. (2018Esfandiari et al. ( -2017) ) introduced an algorithm that combined multi-criterion decisionmaking and particle swarm optimizations, accelerating convergences in finding optimal solutions for 3D RC frames subjected to lateral seismic forces.Bai, Jin, and Ou (2020) maximized seismic resistance of RC structures by an iterative analysis-and-redesign scheme, substantially reducing story drifts while slightly increasing material costs.Hysteresis behaviors of structures are predicted based on Bouc-Wen models in studies by Sirotti et al. (2021), Pelliciari et al. (2020Pelliciari et al. ( , 2018)).
The majority of previous studies evaded complexities of explicit objective functions in structural designs by using metaheuristic methods such as genetic algorithm, pattern search algorithm, and colony optimization algorithm.Examples of artificial neural networks (ANNs)based structural designs were found in studies by Srinivas and Ramanjaneyulu (2007), Shin et al. (2020), Asteris et al. (2016), andGarcía-Segura, Yepes, andFrangopol (2017).Behaviors of bridge decks were predicted by ANNs, and designs are optimized using genetic algorithms, in the study of Srinivas and Ramanjaneyulu (2007).Hazards of seismically deficient RC frames were assessed and mitigated using ANNs by Shin et al. (2020), aiding retrofit designs in buildings.Asteris et al. (2016) predicted fundamental periods of infilled RC structures using ANNs, showing good accuracies when verified by analytical investigations.García-Segura, Yepes, and Frangopol (2017) conducted an optimization of post-tensioned concrete road bridges using ANNs, minimizing total expected costs while achieving required levels of safety and durability.However, the implementation of ANNs in optimizations of RC frames were not common even though ANNs have been successfully used in in many areas such as medical, auto pilot, financial, etc.The present study provided a novel procedure to calculate design parameters while minimizing both single and multi-objective functions for structural frames using the ANN-based Hong-Lagrange algorithm.ANN-based Hong-Lagrange algorithm was successfully implemented in optimizing beam and column designs in the studies of Hong and Nguyen (2021) and Hong, Nguyen, and Nguyen (2021).It was anticipated that the optimizations for beams, columns, and frames can assist human engineers in enhancing design accuracies and reducing their labor while offering optimized designs based on ANNs-based Lagrange optimization that are not available currently.Accuracies were verified by conventional structural designs, resulting in the basis for data-centric engineering which is not based on structural mechanics.

Research significance
Derivation-based approaches are unable to efficiently optimize large structure designs because optimizations and sensitivity analysis are complex due to various requirements imposed by codes and a large number of input and output variables.Metaheuristic methods such as genetic algorithms, similated annealing algorithms, and colony optimization algorithms are widely applied, as discussed in the literature review.However, there are debates about their accuracies and robustness, for example, results provided by genetic algorithms can be unstable and converge to the local minima because procedures initializations, crossover, and mutations heavily rely on randomness (Blum and Roli 2003).Achieving optimal solutions, hence, in day-to-day engineering practices is still challenging even if numerous research has been proposed in the field.The present study offers a novel algorithm that systematically and conveniently optimizes building frames, bridging state-of-the-art artificial intelligence (AI) technologies and practical engineering.
This study uses an ANN-based Hong-Lagrange algorithm with constraints imposed by codes and architects to holistically optimize RC frames, recognizing that it is difficult to explicitly derive analytical objective functions when optimizing complex RC frames that meet all code requirements simultaneously.The approach provided in this study optimizes RC frames based on ANN-based objective functions, solving non-linear optimization problems under strict constraints imposed by design codes and interests of engineers.The use of ANN-based objective functions eliminates complex derivations of explicit mathematical formulations which hinders applications of optimizations for practical designs.
This study considers RC frames where the ANNbased auto-designs for optimizing RC frames are developed based on the ANN-based Hong-Lagrange algorithm.Additional structural systems such as dual frames, steel RC frames, and prestressed frames are under development.AI-based auto-designs proposed in this study perform design optimization while optimizing design targets.Design parameters are produced automatically, whereas it is challenging to achieve auto-designs by conventional approaches.Villarrubia et al. (2018) proposed a method of approximating objective functions by ANNs and minimizing constrained ANN-based objective functions by Lagrange functions and Karush-Kuhn-Tucker (KKT) conditions.Theoretically, this method can provide accuracy and robustness equivalent to derivativebased methods while eliminating complex derivation processes when good training is achieved.The ANNbased Hong-Lagrange algorithm illustrated in Figure 1 is developed referring to the method proposed by Villarrubia et al. (2018) Optimization consists of four steps as follows as shown in Figure 2.

ANN-based Hong-Lagrange algorithm
Step 1: Any structural-based software including MIDAS or ETABS can be used to model frames.Graphical interfaces of MIDAS and ETABS are utilized so frames can be modeled accurately and conveniently even though there are geometrical irregularities.Information including frame geometry, frame supports, end-conditions of members, loading, story data, and structural groups are established using graphical tools  of MIDAS or ETAB as shown in descriptions from 1 to 6 in Step 1.1 of Figure 2. In Step 1.1, graphical interfaces of structural-based software (MIDAS or ETABS) are utilized to establish frame models.However, MIDAS or ETABS only serve as modelers, whereas models are not investigated by MIDAS or ETABS.The model is exported to external files such as MGT files by MIDAS.These MGT files are imported in the ABBA (AI-Based Build Analysis and design) frame generator in Step 1.2 to generate big data samples.The ABBA frame generator is a MATLABbased software.In this step, wind load parameters, seismic load parameters, load combinations, material properties, ranges of section sizes, and rebar ratios are assigned to the ABBA frame generator.In each data set, the generator randomly selects member sizes and rebars ratios based on ranges predefined by users to calculate output parameters including safety factors of members, story drifts, long-term deflections, etc. required for frame designs.
Step 2: ANNs are trained based on big datasets generated in Step 1 using MATLAB Deep Learning Toolbox (MathWorks 2022a).MATLAB Deep Learning Toolbox is a built-in application in the MATLAB platform where ANNs with different numbers of layers and neurons can be trained on big datasets.The toolbox produces regression models based on training, where weight and bias matrices are extracted to generalize functions including objective and constraining functions.The generalized functions are used to formulate Jacobi and Hessian matrices of Lagrange functions.Stationary points for Language functions are, then, identified using Newton-Raphson iteration (Upton and Cook 2014) in the SQP algorithm (MathWorks 2022a).
Step 3: ABBA-Frame optimization codes are developed to optimize frame designs based on single objective functions in Step 3.1.Design requirements are imposed by equalities and inequalities according to codes and architectural requirements.For example, all safety factors are greater than 1 and short-term deflections of beams should be less than 1/360 of a span length.An objective function for Lagrange optimization can be any design target, such as frame cost, CO 2 emissions, and weight of the frame.In Step 3.2, a Unified Function of Objectives (UFO) is established based on tradeoff ratios according to weighted sum methods (Zadeh 1963).
Step 4: The Lagrange optimization is implemented in UFO to determine optimized design parameters while optimizing multiple objective functions simultaneously.
The present study performs a data generating, training, and optimizing algorithms using MATLAB Deep Learning Toolbox (MathWorks 2022a), MATLAB Parallel Computing Toolbox (MathWorks 2022a), MATLAB Statistics and Machine Learning Toolbox (MathWorks 2022a), MATLAB Global Optimization Toolbox (MathWorks 2022a), MATLAB Optimization Toolbox (MathWorks 2022a), and MATLAB R2022a (MathWorks 2022a).The purpose of each toolbox is described in Table 1.Steps 3.1 and 4 are developed based on Villarrubia et al. (2018), and Step 3.2 is developed based on Zadeh (1963).
The present study includes five sections.Section 1 shows novelties and advantages of the proposed algorithm compared with available methods.Section 2 introduces configurations of example buildings, data generations, and design constraints.Section 3 explains formulations of objective functions based on weight and bias matrices of ANNs.Section 4 presents optimized designs obtained from the ANN-based Hong-Lagrange algorithm.Lastly, Section 5 summarizes the research with discussions and recommendations.

Building configurations
A four-by-four bay with four-story frame optimized in this study is shown in Figure 3, where the frame is divided into two groups for beam and column designs.The floor height and beam span are 4 and 8 m, respectively.An example of optimizing RC frames is carried out to demonstrate the effectiveness of the ANNbased Hong-Lagrange algorithm.However, the algorithm is not limited to any frame configurations, but it is possible for all frame layouts as long as these layouts can be modeled by either MIDAS or ETAB. Figure 3(a) shows a layout of beams and columns, while Figure 3 (b) illustrates slabs as dead loads.It is noted that weights and stiffnesses of slabs are considered in dynamic investigations even though slab weights are excluded in design tables.Weights of slabs are considered for dead loads to calculate structure masses, whereas frames are investigated with rigid diaphragms at each floor, taking into account stiffnesses of slabs.(C d ¼ 2:5), coefficients for calculating approximated fundamental periods C t ¼ 0:0466 ð Þ; spectral response acceleration parameter at a period of 1 s (S 1 ¼ 0:22 g), response modification coefficient (R ¼ 3), and gravity acceleration (9:81 =s).Material properties used in this study are also defined in Table 2, where material prices as Korean Won per cubic meters (KRW/m 3 ) are taken according to Korean markets in 2021.Units CO 2 emissions and energy consumptions are calculated according to studies of (Hong et al. (2010) and Kuk Kim et al.   (2013), respectively.It is noted that unit values of costs, weights, CO 2 emissions, and energy consumptions are not fixed, and they can be changed easily before generating big datasets.Optimized objective functions for cost index and a total weight of frames are calculated in Parameters 47 and 49 of Table 4, respectively.The objective functions of costs and weights in frames are calculated based on concrete volumes, rebar ratios, and stirrups areas in beams and columns.Costs and weights from slabs are excluded from objective functions; however, weights and stiffness of slabs are considered by the ABBA frame generator in dynamic investigations.Ranges of big datasets are shown in Table 4 where ranges of output parameters are obtained for preassigned ranges of input parameters.There are many more parameters calculated during bigdata generations, such as an upper limit fundamental period (C u T a ) for determining base shear, base shears, wind pressures, section strengths, etc.However, they are not printed on an output-side of the big datasets because they are not constrained by any code-based requirements.It is noted that F250,000 data samples are obtained by filtering from 330,000 big data samples.Some data such as deflections and stresses calculated when unfactored moments exceed nominal capacities of sections are removed.Other parameters, such as safety factors or story drifts, are generated correctly in all cases.

Design requirements as inequality constraints
Table 5 presents 70 inequalities imposed during an optimizations of RC frames.A number of inequalities is greater than that of forward input parameters because multiple inequalities are applied to one parameter.For example, Inequalities V 1 and V 13 shown in Table 5(a) are applied to HC 1 (size of columns in Column Group 1).In Tables 5(a, b), Inequalities from V 1 to V 43 are imposed on 18 forward input parameters when optimizing designs.Ranges for 14 forward input parameters out of 18 forward input parameters indicated in 26 inequalities from V 3 to V 6 , from V 9 to V 12 , V 17 , from V 22 to V 29 , V 31 , and from V 36 to V 43 are taken slightly narrower than ranges of big datasets to avoid sparse data at the edge range of the big datasets.It is noted that big datasets should be wide enough to cover expected magnitudes, for example, a safety factor of 1 should appear inside ranges of big datasets for training if a safety factor of 1 is desired in optimal designs.Constructability of a frame is considered in Inequalities from V 13 to V 15 where sizes and rebars of the upper column groups are constrained to be smaller than or equal to those of the lower column group.Inequalities from V 44 to V 47 in  5(g) constrains the maximum story drift ratios of all stories to be ≤0:015.The limitation of story drift ratios in four-story frames with a Risk Category II is 0.02 according to Table 12.12-1 in ASCE 7-16 (Loads and Structures 2017).However, the present example uses a conservative limit of 0.015 (a default limitation in MIDAS) for story drift ratios.Inequality V 69 in Table 5 (g) constrains the stability coefficient to be smaller than or equal to θ max ¼ 0:5 βC d ¼ 0:5 2:5 ¼ 0:2 according to Section 12.8.7 in ASCE 7-16 (Loads and Structures 2017).Inequality V 70 in Table 5(g) constrains the lateral deflections due to wind loads to be smaller than or equal to H500 ¼ 32 mm according to Section CC.2.2 in ASCE 7-16 (Loads and Structures 2017).
In summary, the optimization determines 18 design variables listed in Table 4(a) including sizes and rebar ratios of beams and columns to minimize costs and weights simultaneously.Designs are also constrained by 70 inequalities constraints presented in Table 5, ensuring the safety, stability and constructability of optimized frames.

Formulating ANN-based objective functions
Table 6 presents training accuracies of the ANNs in which a combination of the two types of hidden layers including 5 and 10 layers and a combination of the two types of neurons including 80 and 128 neurons are implemented.A number of layers and neurons are shown in Table 6 with epochs that provide the best training accuracies.Validation indicates both designated number of epochs and terminating epochs to prevent over-fitting.For example, training proceeded up to 50,000 epochs for Story 1 of column safety factor as designated, however, training terminated at Epoch 45,506 to prevent over-fitting for story 3 of column safety factor.
The best training accuracy for rebar strain corresponding to concrete strain of 0.003 at beam end of Group 1 is obtained with 5 layers and 128 neurons, yielding Test mean square errors (MSE) = 1.2727E-04 and Regression = 0.9991.Training accuracies in terms of Test MSE and Regression are presented to judge training results.Detailed descriptions of ANN training on structural data can be found in the books by Hong (2019Hong ( , 2021)).

ANN-based Hong-Lagrange algorithm
The ANN-based optimization of multi-objective functions for RC frames shown in Figure 3 is performed in 5 steps as presented in   Design parameters identified for Design P1, Design P5, and Design P9 based on nine fractions are presented in Table 7. Design P1 (w CI : w W ¼ 1 : 0), Design P5 (w CI : w W ¼ 0.5 : 0 .5),and Design P9 (w CI : w W ¼ 0:1) are indicated in Table 7. Design parameters of Design P1 based on a fraction of w CI : w W ¼ 1 : 0 in which CI is only minimized are obtained.Design P9 identifies an optimized design parameters where weight of the RC frame, W, is only minimized based on (w CI : w W ¼ 0:1) in which a Pareto frontier similar to 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 CI : w W ¼ 0.5 : 0 .5).Tables 7(a It is noted that, in Table 7(a), the upper tensile rebar ratio ρ ts B 1 = 0.0065 at beam end is obtained as shown in Design P1 when a beam cost CI is minimized at 71,174,081 KRW, whereas the upper tensile rebar ratio ρ ts B 1 = 0.0090 is obtained as shown in Design P9 when a beam weight is minimized to 7,307 kN from 8,585 kN of Design P1, resulting increased rebar ratio (ρ st B 1 Þ by 0:0090À 0:0065 0:0065 � 0:39% ¼ 39% to reduce the beam weight, but sacrificing cost which increased from Probable designs are determined based on averages of the top 688 (0.1%) designs among 688,000 designs randomly generated based on the ABBA generator shown in Figure 5, resulting in probable design values of 129,367,608 KRW and 10,924 kN for CI and W, respectively.Optimized designs, and hence, produces a cost savings up to 44.983% and a weight reduction up to 33.111% over probable designs.

Conclusions
A resilient design capable of optimizing RC frames has been performed beyond human efficiency.It is difficult for engineers to pre-assign constraining conditions on an input-side for a conventional design.The present study replaced complex analytical objective functions by ANNbased objective functions.Any type of objective function and design target of interest can be implanted as artificial neural genes to govern an optimization process minimizing design targets for RC frames.Design codes and constraining requirements include economies, environments, and serviceability.ANN is formulated to provide 32 forward output parameters based on 18 forward input parameters to simultaneously optimize objective functions of a cost CI and a weight W of RC frames.In this study, pure RC frames are optimized where the AIbased auto-designs are developed using the ANN-based Hong-Lagrange algorithm.Additional structural systems such as dual frames, SRC frames, and PT frames are under developments.AI-based auto-designs proposed in this study replace conventional design optimization processes by automatically producing design parameters while optimizing design targets.It is challenging to achieve auto-designs by conventional approaches.
In this study, ANN-based Hong-Lagrange algorithm is introduced for a holistic optimization of RC frames.Objective functions with equality and inequality constraints are implanted in ANN-based Lagrange functions to control optimizations of the designs.Design parameters are obtained based on 330,000 data samples to minimize ANN-based objective functions for costs CI and weights W c of RC frames.Structural optimality is verified by 668,000 data samples, resulting in cost savings up to 44.983% and weight reductions up to 33.111% compared with probable designs based on the average of 688 (0.1%) best designs among 688,000 big data samples.Design accuracies are enhanced and verified by conventional structural designs while reducing labors of human engineers in optimizing designs for beams, columns, and frames.ANN-based optimized designs were not available before the present research.Now it is possible to autonomously optimize design targets to calculate design parameters which is challenging to obtain using conventional design methods.This study is a steppingstone for the next step in structural analysis and design research with the advent of AIbased Data-centric Engineering which is not based on structural mechanics (Hong W. K. 2023).

Disclosure statement
No potential conflict of interest was reported by the authors.
, where ANN-based Lagrange optimizations are expanded to structural problems, implanting constraints imposed by architecture and codes in Lagrange functions.This study performs a holistic optimization of RC frames based on the ANNbased Hong-Lagrange algorithm, where ANN-based objective functions such as cost (CI) and weight (W) of RC frames derived as a function of input parameters which are, then, minimized based on 330,000 data samples, producing optimized design parameters as shown in Figure 1.Complex analytical objective functions are replaced by ANN-based objective functions while multiple constraining conditions are imposed by equality and inequality constraints in Lagrange functions (Lagrange 1804).

Figure 1 .
Figure 1.A graphical illustration of ANN-based Hong-Lagrange algorithm.

Figure 2 .
Figure 2. Four steps of ANN-based Hong-Lagrange algorithm from generating big datasets to identifying Pareto frontiers.
Weights presented in design tables only represent the total weight of beams and columns which are optimized in the present study.Frame geometries, loading, support conditions, end-conditions of members, and member grouping are modeled by MIDAS as shown in Figure3(c).Parameters defining vertical and horizontal loads are presented in Table2.Wind load parameters include basic wind speed, topographic factor (K zt ), exposure class, and enclosing condition, whereas seismic load parameters include spectral response acceleration parameter at short periods (S s = 0.55 g), deflection amplification factor:

Figure 4 .
Figure 4. Five steps to optimize multiple objective functions simultaneously by the ANN-based Hong-Lagrange algorithm (MathWorks 2022a).

Figure 3 .
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.

Figure 5 .
Figure 5.The Pareto frontier when minimizing CI and W, comparing optimized designs points with big datasets.

Figure 4 .
Steps 1 to 3 calculate ANN-based Lagrange functions for single objective function by which a UFO is derived in Step 4 to minimize it in Step 5, leading to an identification of a Pareto frontier.A set of MOO results is a Pareto frontier or a Pareto set which is obtained by solving KKT ), 5(b), and 5(c) present design parameters optimized by ANN-based Hong-Lagrange algorithm of UFO while insignificant errors are verified by structural mechanics-based ABBA frame generator.Design accuracies of ANNs for Design P1, Design P5, and Design P9 are also shown in Tables 7(a), 5(b), and 5(c) where differences between design parameters and the ABBA frame generator are negligible.

Table 1 .
Purposes of toolboxes used in the ANN-based Hong-Lagrange algorithms.

Table 3 .
Load combinations considered in the present study.

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).

Table 5
Inequalities from V 64 to V 67 in Table5(f) constrain rebar tensile stresses under service loads to be smaller than or equal to according to

Table 6 .
Training accuracy, showing test MSE and regression.
In this study, design parameters are optimized simultaneously based on two objective functions (cost index and weight of the RC frame).Designs P1 to P9 corresponding to nine fractions (w CI : w W ¼ 1 : 0, w CI : w W ¼ 0.875 : 0 .125,wCI: w W ¼ 0.75 : 0 .25,wCI: w W ¼ 0.625 : 0 .375,wCI: w W ¼ 0.5 : 0 .5,wCI: w W ¼ 0.375 : 0 .625,wCI: w W ¼ 0.25 : 0 .75,wCI: w W ¼ 0.125 : 0 .875,wCI: w W ¼ 0:1) on Pareto frontier are indicated in Figure5which also demonstrates that big datasets hardly show minimized data in the lowest data range, however, it does not mean that big datasets do not cover the data in the lowest data zone.The gaps between Pareto frontier and big datasets can be filled if more big datasets are generated.It is the proposed method that, more efficiently, optimizes designs and predicts the lowest bound of the big datasets with one run.