Optimized Interaction P-M diagram for Rectangular Reinforced Concrete Column based on Artificial Neural Networks

ABSTRACT This study proposes an artificial neural network for a design of reinforced concrete (RC) columns for structural engineers who are interested in performing reverse designs, exploring influences of structural parameters (e.g., , and ) or code requirements on structural performances. The proposed networks enable both forward and reverse designs for an RC column, which is challenging to be achieved using conventional designs. An AI-based surrogate model of RC columns with sufficient training accuracy can comprehensively replace conventional design software, exhibiting excellent productivity for both forward and reverse designs. In addition, useful reverse design models based on neural networks can be established by relocating preferable control parameters, including safety factor (SF = ) and an aspect ratio of column sections, into the input region. All associated design parameters, including , , and , are computed on an output side. Design charts, such as the P–M diagram, are constructed, demonstrating design moment strength equal to the factored moment demand by specifying SF = 1. The design scenarios can be extended as further as possible to meet the requirements of engineers. Graphical Abstract


Introduction
So far, various artificial intelligence (AI) applications have been developed in structural engineering to perform specialized design tasks. Anderson et al. (1997) demonstrated the potential of artificial neural networks (ANNs) to predict the design response of a steel connection, indicating that the performance of a structural framework is not sensitive to the connection response. Sharifi, Lotfi, and Moghbeli (2019) developed a formula to predict the compressive capacity of a fiber reinforcement polymer structure based on ANNs. Abambres and Lantsoght (2020) applied ANNs to predict the shear capacity of one-way slabs under concentrated loads. Lu et al. (2020) implemented a tree predictive algorithm and a novel feature selection to predict the punching shear capacity of steel fiber-RC flat slabs. Another application of ANNs for predicting the shear strength of RC beams was developed by Asteris et al. (2019) and Armaghani et al. (2019). In addition, Dahou et al. (2009) proposed an ANN-based model of a bond between conventional ribbed steel bars and concrete. They investigated the influence of various input parameters on predicting the ultimate pull-out load between the rebar and concrete. Charalampakis and Papanikolaou (2021) explored the effect of training datasets on the accuracies for the training and design of RC columns and bridge piers by applying ANNs. Recently, Hong, Pham, and Nguyen (2021) and  designed an ANN-based doubly RC beam. They implemented ANNs to design ductile concrete beams in detail and establish design charts, which yielded an accurate and rapid prediction of multiple design parameters.
This paper presents an ANN that is implemented with design issues of RC columns. Conventionally, RC columns were designed based on a complex structural analysis procedure, including numerical simulation and analytical calculation, which can computationally be expensive, thus severely limiting the product design and efficiency. In this paper, an AI-based surrogate model is developed to comprehensively replace conventional software, providing excellent efficiency and design productivity for both forward and reverse designs. ANN networks trained on large data of RC columns demonstrate sufficient training accuracies to provide reverse designs. Structural datasets for training purposes are randomly generated using the structural software AutoCol, developed by Nguyen and Hong (2019a, 2019b. Seven ordinary inputs, including column configuration (b � h), rebar ratio (ρ s ), material properties (f 0 c ; f y ), and factored load pair (P u ; M u ), are required to evaluate the structural performances in terms of design axial force (ϕP n ) and moment (ϕM n ), safety factor (SF), aspect ratio (b/h), rebar strain (ε s ), design efficiency indexes of cost (CI c ), CO 2 emission, and structural weight (W c ). In addition, useful reverse design models are introduced, where preferable control parameters, such as SF and b/h, are relocated into the input region.
Performances of proposed method would be less dependent on types of structure and material such as column, beam, frame, bridge, etc., but depend on characteristics of big datasets of considered problem. This methodology has a potential to be extended to other design fields, such as prestressed concrete beam Pham and Hong 2021), sandwich panels (CoDyre, Mak, and Fam 2018;Sharaf and Fam 2011;Tomlinson, Teixeira, and Fam 2016), composite cladding wall panel (Shawkat, Honickman, and Fam 2008), or even effect of corrosion on RC structures under seismic (Bossio et al. 2019), etc., as long as large datasets can be collected from these studies.

Design of concrete columns
The conventional design programs require input structural properties, such as f y ,f c ', b, and h, to evaluate output parameters, such as design axial, moment strength, and rebar strains ϕP n ; ϕM n ; andε s ð Þ: Such a sequence for the design computation is considered a forward design. In contrast, in a reverse design, parameters calculated on the output side of the forward design, such as M n , SF, and b/h, can be reversely preassigned on the input side. The input parameters for AutoCol, including the seven inputs b; h; ρ s ; f 0 c ; f y ; P u ; M u À � , are shown with column configuration ( Figure 1). Table 1 presents the seven input and nine output parameters used to design the RC column shown in Figure 1. Axial loads (P u ) and moments (M u ) are applied to the column sections with eccentricity of e/h. The nine outputs ϕP n ; ϕM n ; SF; b=h; ε s ; CI c ; CO 2 ; W c ; α e=h À � are illustrated in Figure 1 and Table 1. The parameters CI c ,CO 2 emissions, and W c are calculated as part of large datasets by using AutoCol. In addition, the interaction diagram of axial loads (P) and moments (M) is evaluated based on a concrete strain of 0.003 for all regions -compressive, transition, and tensile -as shown in Figure 1. In a transition region, reduction factor, ϕ, varied from 0.65 to 0.9, presenting a significant change in strength capacity as shown in Figure 1.

Generation of big data
The algorithm for AutoCol used to generate large structural datasets of an RC column section is shown in Figure 2. The design results of AutoCol, which are similar to those obtained by MacGregor et al. (1997), are shown in Figure 3. A code is written to generate large datasets for an RC column. A total of 100,000 datasets for an RC column is randomly generated by AutoCol (Figure 2), some of which are listed in Table 2(a), where few non-normalized datasets with mean values of 16 parameters are listed with maximum and minimum ranges. It is noted that cost index of column per height (CI c ) is calculated based on local material price in Korea as summarized in Figure 4. As shown in Table 2(b), randomly generated data are normalized based on the min-max normalization technique, which is one of the most common methods to normalize datasets. The minimum and maximum values of the datasets are transformed into −1 and 1 for all features, respectively, whereas all other values are transformed into a decimal value between −1 and 1. Mean values of some parameters (e.g., ϕP n , ϕM n , ε s , etc.) are not zero (data skewness) because they are automatically calculated based on random input vectors. Data normalization is essential to obtain a well-defined surface for gradients of cost functions.

Column design by ANN based on TED, PTM, CRS: one forward design and two reverse designs
Two types of concrete column designs are developed: forward and reverse. In the reverse design, parameters obtained on the output side of the forward analysis are preassigned on the input side in a reverse manner. This is challenging for conventional design methods because a reverse analysis having higher order with multiple inputs and outputs is mathematically complicated. Seven inputs b; h; ρ s ; f 0 c ; f y ; P u ; M u À � and nine outputs ðϕP n ; ϕM n ; SF; b=h; ε s ; CI c ; CO 2 ; W c ; α e=h Þ for one forward and two reverse design scenarios are presented in Table 3. Three training techniques of TED, PTM, CRS mentioned by Hong, Pham, and Nguyen (2021) are conducted in this study to map input parameters to output parameters for both forward and reverse design scenarios.

Network formulation
A wide variety of structural software, such as AutoCol developed by Nguyen and Hong (2019a, 2019b, has been developed to help designers design an RC column considering the design requirements. A traditional design process requires huge computational cost as shown in Figure 2, even RC flexural design is unambiguous. In contrast, AI-based model formulates multilayer perceptron functions as shown in Equations (1), (2), and (3), which can calculate all output parameters rapidly as shown in Table 4. The forward design presents nine output parameters (ϕP n ; ϕM n; SF, b/h, ε s , Cost, CO 2 , Weight, α e=h ) for the given seven input parameters b; h; ρ s ; f 0 c ; f y ; P u ; M u À � . An AI-based surrogate model trained on large datasets generated by AutoCol develops structural designs to improve an analysis efficiency and design productivity ( Figure 5). The AI-based model performs accurately and rapid for both forward and reverse designs. For AI-based forward designs, the relations between the seven input variables b; h; ρ s ; f 0 c ; f y ; P u ; M u À � and nine output variables ϕP n ; ϕM n ; SF; b=h; ε s ; CI c ; CO 2 ; W c ; α e=h À � are approximated based on an ANN by mapping the input parameters to the output parameters by using three types of training networks, which are (TED, PTM, and CRS). The training summaries of the forward design based on TED, PTM, and CRS are presented in Table A1 according to the combinations of two and five hidden layers with three types of neurons (20, 50, and 80), which are trained on 100,000 datasets. Basic concepts such as hidden layers, neurons, epochs, etc. can be found in Hong, ch10 (2019) and Goodfellow, Bengio, and Courville (2016). In TED (  (b)) maps the input vector on each output parameter, such as SF, through individual neural networks (Equation (2)).
Multilayer perceptron function using TED: where y i denotes one parameter among the output variables, such as SF; g N and g D are the normalization functions of the input vector (based on min-max normalization) and de-normalization function of the considered output vector, y i: ; L is the number of layers, including hidden layers and an output layer; W l is the weight matrix between layers l À 1 and l; and b l is the bias matrix of layer l. The activation functions (tansig and tanh), f l t at layer l, are implemented to formulate nonlinear relations of the networks, whereas a linear activation function, f L lin , is selected for the output layer because the output values are unbounded.
Multilayer perceptron functions using PTM: Another training method, chained revised sequence (CRS), is used for network training. CRS allows the predecessor output parameters to be simultaneously used as the input feature indexes of the successor output variables. Better training accuracies can be obtained with a well-determined training sequence of (b=h ! W c ! CO 2 ! CI c ! ϕP n ! ε s ! ϕM n ! α e=h ! SF). Output parameters are relocated on an input side for training next outputs once they are used to train network. A sequence of input feature indexes is arranged such that the smallest MSE for each prediction of the remaining output parameters is obtained. The weights and biases of multilayer perceptron are adjusted in Equation (3) based on CRS for forward scenarios during the training. Multilayer perceptron functions using CRS:  is the output vector arranged in a well-determined training sequence.

Training accuracies based on the numbers of layers and neurons
The training accuracies of the networks obtained by the abovementioned three methods are evaluated based on the following mean square errors ( (20,50,80)] to identify the best training and design accuracies. As shown in Table A1, CRS demonstrates training accuracies similar to those of PTM, whereas the training accuracies of TED are weaker than those of both PTM and CRS (Table A1(a)). However, Table  A1 shows that the training accuracies of only SF, CI c , and α e=h are enhanced with CRS, whereas the test MSEs for the remaining output parameters b=h; W c ; CO 2 ; ϕP n ; ε s ; ϕM n ð Þ similar to or slightly less accurate than those yielded by PTM (Table  A1(b)) are obtained by CRS (Table A1(c)). In some cases, the training accuracies of PTM are better than those of CRS, which indicates that the best training methods are discretely selected to achieve the best design accuracies. A comparison between CRS and PTM indicates that considering the training sequence does not help increase the training accuracies of the forward design scenario for an RC column, indicating that the output parameters of an RC column exhibit low correlation among Table 3. Design scenarios for one forward and four reverse designs.
themselves. However, CRS outperforms the training networks on complex datasets for reverse designs, as shown for the training results in Section 4.3.2.

Design results
The design accuracy of the TED model is weaker than that of the PTM model, as shown in Table A2, because TED uses a single multilayer perceptron function to map the input vector to multiple outputs, whereas PTM implements an independent network to map each output parameter. Table A2 presents AI-based design results based on combinations of two and five layers for an RC column dimensions (b � h) of 500 × 700 and a rebar ratio (ρ s ) of 0.02 against axial load (P u ) and moment (M u ).
Smaller errors of −0.80% for ϕP n and −0.31% for ϕM n based on TED, −0.05% for ϕP n and 0.70% for ϕM n based on PTM, and 0.20% for ϕP n and −1.66% for ϕM n based on CRS are observed for the design in Table A2(a) when the factored axial load (P u ) and moment (M u ) are set to 5000 kN and 1000 kN m, respectively. CRS yields acceptable accuracies with a maximum error of −1.66% for the design moment (ϕM n ). The largest error of 2.01% in rebar strain, ε s , is obtained using TED, which is reduced to −0.95% when PTM is implemented against an axial load (P u ) of 5000 kN and moment (M u ) of 1000 kN m, as shown in Table A2(a). A similar design is presented in Table A2(b), where only the preassigned value of axial force (P u ) is changed from 5000 to 500 kN. This  design also indicates that the design accuracy of the model trained by PTM is better than that of the model trained by TED. The overall design accuracies yielded by the PTM model and CRS model are greater than those yielded by the TED model as shown in Table A2(b).
The training accuracies of AI-based models using TED, PTM, and CRS neural networks increase with the enhancement of the training quality, such as adjusting the training parameters (e.g., numbers of hidden layers, neurons, and required epochs). Another suggestion to improve the training quality is to enrich the training datasets for a sparse region where the current prediction is considered uncertain.

Reverse design
In the reverse scenarios shown in Table 3, the design parameters calculated in the output of forward designs are preassigned in the input to determine the parameters for design backward. A forward AI-based model with higher analysis efficiency has been proposed to replace the structural software AutoCol, introduced in Section 4.2.1. The principle of a forward model is to map the input parameters b; h; ρ s ; f 0 c ; f y ; P u ; M u À � and output parameters ðϕP n ; ϕM n ; SF; b=h; ε s ; CI c ; CO 2 ; W c ; α e=h Þ over the entire space. However, designers are often interested in the socalled reverse scenario, in which structural performances (e.g., ϕP n ; ϕM n , and ε s ) or code verification (SF) are often controlled as inputs, whereas the associated design parameters, such as b, h, and ρ s , are computed on the output side.
In other words, the reverse surrogate model aims to explore the input space by mapping ordinary output parameters (reversely placed on the input side for reverse design) to ordinary input parameters (including ordinary inputs that are reversely placed on the input side for reverse design). A complete reverse scenario includes exchanging all ordinary input parameters b; h; ρ s ; f 0 c ; f y ; P u ; M u À � and ordinary output parameters ϕP n ; ϕM n ; SF; b=h; ε s ; CI c ; CO 2 ; W c ; α e=h À � ; in contrast, in a partial reverse design, part of the output vector is exchanged with that of the input vector. However, many reverse models typically have inappropriate cross-relations among ordinary and reverse input parameters, whereby no such output vector satisfies the given input conditions. For this reason, the number of reverse inputs should be appropriately selected to avoid the input conflicts, such as Reverses 1, 2, 3 and 4, as shown in Table 3.

Reverse scenario 1
In Reverse scenario 1, nine design parameters (b, h, ϕP n , ϕM n ,ε s , CI c , CO 2 , W c , α e=h in yellow output cell of Table 3) are calculated in the output when preassigning seven design parameters (ρ s ; f 0 c ; f y ; P u ; M u ; SF; b=h in pink input cell of Table 3), including reverse input parameters (SF and b=h). Reverse 1 aims to control the SF and b/h under the given load pair of factored loads (P u ; M u ) by placing these variables on the input side. Table A3 presents the training summary of Reverse design 1 based on CRS by using a combination of hidden layers (two and five layers) and three types of neurons (20, 50, and 80) trained on 100,000 datasets. The best training accuracies are demonstrated by CRS with awell-determined training sequence of (ϕP In this section, the reverse model provided by the CRS method is examined to solve two practical design scenarios of an RC column, as summarized in Table 5. Figures 6 and 7 show that many parameters are available for defining the section configurations b � h, rebar ratio ρ s , and concrete strength f 0 c satisfying the design criteria given in Design Scenarios 1 and 2 for Reverse 1 scenario. An optimal design of each scenario can be obtained by varying the preassigned values of ρ s and f 0 c , as shown in Figures 6 and 7. Five out of seven outputs (f y ; P u ; M u ; SF; b=h) are known according to design requirements while preassigned values of ρ s and f 0 c ranging from 0.01 to 0.08 and from 30 to 50 MPa, respectively, are implemented in both Design scenarios 1 and 2 to evaluate effects of rebar ratio and concrete strength on design efficiency and CI c under different loading conditions. Note that safety factor (SF) was defined as ϕM n =M u in Table 1, implying that designs of columns with preassigned safety factor (SF) of 1.0 are to have design moment strength (ϕM n Þ which is the same with factored moment (M u ).
For Design 1 with load demands of P u = 6000 kN and M u = of 1500 kN m, five design results with arbitrarily preassigned ρ s and f 0 c . One optimal design  Figure 6 demonstrates effects of ρ s and f 0 c on a section configuration b � h and a cost efficiency CI c , indicating that a lower rebar ratio leads to a greater section and lower cost index. In addition, axial force P u = 6000 kN is relatively big compared to bending moment M u = of 1500 kN m, causing rebar strains (from 0.00123 to 0.00195) smaller than yield strain of rebars (0.0025) as shown in Figure 8 and Table A4. Considered load point are located in a compression-controlled region as a result of small rebar strains. Therefore, Figure 7. Effect of preassigned rebar ratio and concrete strength on design efficiency (CI c ) in Design 2 ðf y ¼ 500MPa; P u ¼ 500kN; M u ¼ 2500kN; SF ¼ 1; b=h ¼ 1Þ.
a greater concrete strength is preferable for compression region, helps reducing not only b � h but also CI c as indicated in Figure 6. As a result, an optimal solution for Design 1 is obtained at a minimum rebar ratio of ρ s ¼ 0:01 and maximum concrete strength of f 0 c ¼ 50 MPa, providing a minimum CI c of 87124:4 KRW/m and section dimension of b ¼ h ¼ 683:3. Using the same ρ s of 0.01 but smaller f 0 c of 30 MPa, Design case 5 requires not only greater section dimension (b ¼ h ¼ 772:6) but also 15% more expensive compared to an optimal solution as shown in Table A4 and Figure 6.  Table A4. It is noted that comparisons are made based on local material price in Korea as summarized in Figure 4.
In Design scenario 2, P u is reduced to 500 kN (from 6000 kN in Design scenario 1), whereas M u is increased to 2500 kN⋅m (from 1500 kN⋅m in Design scenario 1). A purpose of changing in Design scenario 2 is to investigate an effect of rebar ratio, ρ s , and concrete strength, f 0 c , on design efficiency, CI c , when load demands are located in a tension-controlled region as illustrated in Figure 9. In Figure 7, the AI-based reverse model, indicated with the pink contour, agrees with the structural calculation, where a smaller rebar ratio is required for a greater section to resist a given load pair of (P u ,M u ) = (500 kN, 2500 kN⋅m). However, unlike Design scenario 1 (Figure 6), Figure 7 shows that an optimal design of Design scenario 2 does not result from a minimum rebar ratio (ρ s;min ¼ 0:01) and maximum concrete strength (f 0 c ¼ 50 MPa). Design cases 1 and 2 use the same rebar ratio of ρ s ¼ 0:02 but different in concrete strength (f  Table A5. However, Figure 9 witnesses a big difference in strength capacity at compression zone between Cases 1 and 2, in which pure compression capacity of Case 1 (f 0 c ¼ 40 MPa) reaches 15,800 kN, whereas one of Case 2 (f 0 c ¼ 30 MPa) is smaller than 13,000 kN. A similar result comparison is obtained from Cases 3 and 4, indicating that an effect of concrete strength on strength capacity at a tensioncontrolled region is minor. An optimal design of Design scenario 2 is obtained at a rebar ratio of ρ s ¼ 0:0152 and minimum concrete strength of f 0 c ¼ 30 MPa, providing a minimum CI c of 174673:5 KRW/m and section dimension of b ¼ h ¼ 910 (mm). In Figure 9, all P-M diagrams pass through a red dot (P u ,M u ; 500 kN, 2500 kN m), indicating suitable design accuracies of the proposed AI-based reverse model in Design scenario 2. The training sequence of the output parameters based on CRS is represented under all design tables (8→9→13→1→2→15→14→12→16 for  Table A5).
AI-based results of Design scenarios 1 and 2 indicate that influences of design parameters (ρ s , f 0 c , etc.) on design efficiency (CI c ) are different depending on characteristics of load demands and material prices (Figure 4). Concrete strength significantly impacts on column capacity when loading eccentricity is small (compression controlled). On the other hand, strength capacity in a tension-controlled region is almost unchanged when concrete strength is increased. Rebar ratio is also a critical parameter contributing to strength capacity and design efficiency of an RC section as presented in Design scenarios 1 ( Figure 6) and 2 (Figure 7). By relocating a safety factor (SF) into an input region, the proposed method helps designers effectively select material properties, section configuration per desired load demands, optimize design efficiency, and save computational effort.

Reverse scenario 2
In this section, Reverse Scenario 2 presents a scenario similar to Reverse Scenario 1 except for calculating b, h, and ρ s in the output. Reverse Scenario 2 is designed to determine 10 design parameters (b, h, ρ s , ϕP n , ϕM n , ε s , CI c , CO 2 , W c , α e=h in output yellow cell of Table 3) when six design parameters (f 0 c ; f y ; P u ; M u ; SF; b=h in input pink cell of Table 3) are preassigned in the input. Table  A6 presents the training summary of Reverse Scenario 2 based on TED, PTM, and CRS. The design results of Reverse model 2 implementing six input parameters of f 0 c ; f y ; P u ; M u ; SF; b=h � � ¼ 50; 500; 6000; 1500; 1; 1 ½ � are compared based on CRS when a b=h of 1 (square columns) and an SF of 1 are implemented to support a P u of 6000 kN and M u of 1500 kN m.
The training summary of Reverse Design 2 is compared based on TED, PTM, and CRS in Table A6. One million datasets are used to train ANNs with combination of two and five layers based on three types of neurons (20, 50, and 80) for TED (Table A6(a)), and the numbers of layers and neurons are shown in Table A6 (b) and (c) for PTM and CRS, respectively.
The output vectors (b, h, ρ s , ϕP n , ϕM n , ε s , CI c , CO 2 , W c , α e=h ) are mapped by the given input vectors (f 0 c ; f y ; P u ; M u ; SF; b=h), as shown in Table A6.   (b) shows that the trainings are difficult, demonstrating weak training accuracies of TED and PTM, because there can be multiple mapping output vectors for a given input vector for reverse designs, preventing neural networks from uniquely mapping the input and output parameters. Such problems do not exist for forward designs, where unique one-to-one mapping always exists.
The best training accuracies are demonstrated by CRS because the input parameters selected from the output parameters after each training of the networks are added as feature indexes to predict an output parameter, which helps address the nonunique mapping problem in training.
is added sequentially after each training of the networks to address the nonunique mapping uncertainty problem.
Reverse Scenario 2 is proposed to design columns including rebar ratios, column cost, CO 2 emissions, and α e=h without constraining the column width (b) in the input when load demands are located in transition (P u of 6000 kN and M u of 1500 kN m; Table A7) and tension-controlled (P u of 500 kN and M u of 2500 kN m; Table A8) regions; b=h of 1 (square columns) and SF of 1 are also implemented.
Tables A7 and A8 present design results of Design Scenarios 1 and 2 for Reverse 2 model based on TED, PTM, and CRS. The design accuracies yielded by the CRS model are much better than those yielded by the TED and PTM models. The largest errors of the CRS design table are À 3:95% and 4:60% when predicting ε s in Design scenarios 1 (Table A7) and 2 (Table A8) The CRS model for Reverse scenario 2 can provide an acceptable solution with reasonable accuracies, obtaining unique designs, such as those observed in Design Scenarios 1 and 2. However, the solution is not optimal, as indicated above. The optimization solutions for Designs 1 and 2 can be achieved through trial-and-error, as presented in Figures 6 and 7, respectively. Optimization charts identical to those obtained in Figures 6 and 7 and corresponding axial load-moment interaction diagrams can be obtained using ANNs.

Diversity of reverse scenarios
The excellence of reverse scenarios is to regulate structural performances (e.g., ϕP n ; ϕM n , and ε s ) or safety factor (SF) as inputs, which cannot be controlled by traditional method without expensive trial-and-error iterations. Diverse reverse scenarios can be extended as further as possible based on designing interest. For example, Reverse scenario 3 shown in Table 3 is an interesting reverse scenario, where rebar strain can be controlled to improve ductility and post-peak behavior of RC columns. Design scenario 2 shown in Table 5 is redesigned using Reverse scenario 3 to regulate rebar strains by preassigning it as an input as shown in Table A10. Two square column configurations of b � h ¼ 765 � 765 and 949 � 949 with rebar ratios of 0:0279 and 0:0131 are produced to resist load demands with high rebar strains of five and ten times of yield rebar strain, respectively.
In Reverse scenario 1, 2, and 3, column configuration (b � h) is calculated as output, causing decimal values, which is inconvenient for construction or might exceed architectural limitation. Therefore, a further design step, in which b and h are preassigned as inputs, is needed to round off decimal values of column configuration or satisfy architectural requirement as presented in Reverse scenario 4. For example, design results obtained from Table A10 are redesigned using Reverse scenario 4 with round off values of b and h as shown in Table A12."

Conclusions
(1) An ANN-based network was proposed for designing an RC column to meet the requirements of structural engineers who are interested in performing reverse designs, exploring influences of structural parameters (e.g., ϕP n ; ϕM n , and ε s ) or code requirements on structural performances. The proposed networks enable both forward and reverse designs for an RC column, which is challenging to be achieved using conventional designs. An AIbased surrogate model of an RC column with sufficient training accuracies can comprehensively replace conventional design software, providing design productivity with better efficiency for both forward and reverse designs.
(2) Useful reverse design models based on neural networks can be established by relocating the preferable control parameters, including SF and the aspect ratio of column sections, into the input region. All associated design parameters, including b, h, and ρ s , were computed on the output side. However, both the quantity and property of the reverse inputs should be carefully selected to avoid input conflicts. (3) Nonunique design problems can be addressed using an AI-based reverse model when CRSbased networks extend the input feature indexes as training continues. However, an optimal solution is not obtained using reverse networks without analysis in an iterative manner. In the future, optimal solutions will be explored.
(4) The input parameters need to be adjusted under the occurrence of an input conflict. The proposed algorithm adjusts the input parameters automatically to establish appropriate cross-relations among the input parameters. The input incompatibility can be fixed by adjusting the input parameters within the large datasets; however, the adjustment might not improve the design accuracies in the absence of considerable structural data sufficient to cover the adjusting range for the input cross-relations. The data ranges for training should be extended to saturate the sparse zone for all training networks.
(5) Design charts, such as P-M diagram, were constructed, which demonstrated design moment strength equal to the factored moment demand by specifying SF = 1. An AIbased surrogate model of an RC column demonstrated training accuracies sufficient enough to replace conventional design software, offering design productivity for both forward and reverse designs. The design scenarios can be extended as further as possible to satisfy the requirements of engineers.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT 2019R1A2C2004965).         Table A4. Influence of preassigned ρ s and f 0 c on five design cases in Design 1 (f y ¼ 500 MPa; P u ¼ 6000 kN; M u ¼ 1500 kN; SF ¼ 1; b=h ¼ 1).