Reverse design-based optimizations for reinforced concrete columns encasing H-shaped steel section using ANNs

ABSTRACT This study aims to propose artificial neural networks (ANNs) to solve structural engineering problems and to provide optimal designs for reinforced concrete columns with H-shaped steel sections. A reverse scenario based on preassigned safety factor ( ) for a factored biaxial load, steel ratio ( ), and aspect ratio of columns ( ) is presented, meeting code requirements. A back-substitution (BS) method using a chained training scheme with a revised sequence (CRS) is implemented to optimize training. Effects of rebar and steel ratios on objective functions, cost index ( ), emission, and column weight ( ) are identified. Three-dimensional interaction diagrams are obtained based on optimized , emission, and based on preassigned equivalent to 1.0. Predictions from ANNs are ascertained using structural mechanics, demonstrating a significance of an accuracy of the optimized design obtained by the proposed ANNs. This study provides a useful and practical design for SRC columns by lessening engineer’s effort while increasing design accuracies. GRAPHICAL ABSTRACT


Introduction
Reinforced concrete columns encasing steel sections (SRC columns) have been popular in building infrastructures, bridge piers, and earthquake-resistant constructions. The most common type of steel section used in SRC columns is H-shaped steel sections. SRC columns are highly efficient and economical structures due to the rigidity and stiffness of concrete and the strength and ductility of steel sections being used together. Another advantage of SRC columns is the durability and cost-effectiveness of concrete and its high resistance to fire. Several studies have investigated the experimental behavior of SRC columns under an axial load (Chicoine et al. 2002;Wang 1999), under uniaxial bending and axial compressive load (Mirza, Hyttinen, and Hyttinen 1996;Ricles and Paboojian 1994;El-Tawil and Deierlein 1999;Li and Matsui 2000), and under biaxial bending and axial compressive load (Munoz and Hsu 1997a;Morino, Matsui, and Watanabe 1984). Some analytical studies have also been conducted by Chen and Lin (Chen and Lin 2006), Nguyen and Hong (Nguyen and Hong 2020), Virdi and Dowling (Virdi and Dowling 1973), Roik and Bergmann (Roik and Bergmann 1990), Dundar et al. (Dundar et al. 2008), and Munoz and Hsu (Munoz and Hsu 1997b) to investigate the performance of SRC columns. Foraboschi (Foraboschi 2019(Foraboschi , 2020(Foraboschi , 2016 investigated theoretical designs of reinforced concrete columns and beams, which can be used to generate large datasets for ANN trainings. Despite these recent analytical and experimental studies on SRC columns, complexities in their behavioral analysis have not attracted enough attention, unlike concrete and steel structures. Therefore, this study proposes artificial neural networks (ANNs) to predict the capacities of SRC columns including design axial strength (ϕP n ), design moment strengths (ϕM nX and ϕM nY ) about X and Y directions, respectively. The capacities of ANNs to learn a trend of large datasets have been considered as an alternative tool for predicting the capacities of complex structures. Some studies have applied ANNs successfully to predict capacities of complex structures such as the one-way slabs by Abambres and Lantsoght (Abambres and Lantsoght 2020), FRP confined rectangular concrete columns by Sharifi et al. (Sharifi, Lotfi, and Moghbeli 2019), and reinforced concrete beams by Asteris et al. , Armaghani et al. (Armaghani et al. 2019), and Hong et al. Hong, Pham, and Nguyen 2021).
An ANN for optimal designs of SRC columns with preassigned safety factor (SF) equivalent to 1.0 is presented in this study. This means that factored loads are equal to design strengths. Table 1 summarizes terminologies implemented in the study. Dimensional sections of columns and steel sections are obtained as outputs of ANN. A theory of strain compatibility proposed by Nguyen and Hong (Nguyen and Hong 2020) is used to generate large datasets of SRC columns under a biaxial load. A reverse design is introduced in this study to optimize SRC column designs corresponding to minimized cost (CI c ), CO 2 emission, and weight (W c ). Column and steel dimensions are determined accordingly. Constraints required by ANSI/AISC 360-16 (ANSI 2016) and ACI 318-19 (ACI Committee 2019) are satisfied when safety factor, rebar, and steel ratios as shown in Table 2(b) are preassigned on an input-side.

Research significance based on novelty and innovation
The steel reinforced concrete columns have been wildly used in practical engineering; however, most of research focused on conventional theorybased design for the steel reinforced concrete columns. In addition, applications of SRC columns to practical designs have gained less attention than concrete and steel structures due to complexities in their behavior. Engineers also find it challenging to determine an accurate size of H-shaped steel sections for optimal designs. A trial-and-error technique that is timeconsuming and laborious is the only conventional design method for obtaining optimal designs. It is also challenging for engineers to determine the proper column sizes and steel sections in a preliminary design stage and to obtain optimized designs with minimum cost index (CI c ), CO 2 emission, and column weights (W c ). The proposed method optimizes designs of SRC columns using a reverse design with sufficient accuracies without repeating based on trial-and-error technique that the conventional design method must use.
In this study, ANNs are proposed as an alternative tool for designing SRC columns under a biaxial load when SF s, steel ratio (ρ s ), and   Figure 1(a,b), which illustrates the strain compatibility of SRC columns under a biaxial load. As shown in Table 2, ANNs are introduced to assist engineers in deciding sizes of columns and steel sections with preassigned SF s in an input layer of the ANN. A reverse design is proposed to optimize SRC columns using ANNs. Optimized designs obtained by ANNs based on reverse design are verified using structural mechanics, proving that accuracies of the proposed method are adequate for use in practical designs. Engineers can minimize the three objective parameters, which are cost index (CI c ), CO 2 emissions, and W c , yielding various rebar (ρ rX and ρ rY ) and steel ratios (ρ s ) accordingly. Column height (h) and width (b) of concrete column sections as well as height (h s ) and width (b s ) of H-shaped steel sections are also calculated on an output-side of the ANN when minimizing cost index (CI c ), CO 2 emissions, and W c . Engineers, thus, determine optimized column sizes and steel sections in a preliminary design stage while minimizing cost index (CI c ), CO 2 emission, and column weights (W c ). The study applied a back-substitution (BS) method with a chained training scheme and revised sequence (CRS) proposed by Hong et al.  2019) to train ANNs. This study contributes to novel and innovative AI-based designs of SRC columns under a biaxial load. An algorithm for optimizing designs of SRC columns using ANN-based reverse designs is illustrated in Figure 3.

Generation of large structural datasets
Strain compatibility proposed by Nguyen and Hong (Nguyen and Hong 2020) is used to develop a structural software to study an analytical behavior of SRC columns as shown in Figure 2. Mander curve (Mander, Priestley, and Park 1988) is used for modelling concrete model. An elastoplastic model is used for simulating rebar and steel materials. Figure 4 shows an algorithm of strain compatibility used to create large datasets. A large dataset of 100,000 is generated based on strain compatibility proposed by Nguyen and Hong (Nguyen and Hong 2020) to train networks.   SF; ε r ; ε s ; ρ s ; D r;max ; Y s ; X s ; b=h; CI c ; CO 2 ; W c À � . Table 2(b) demonstrates a reverse scenario, where SF, ρ s , and b=h are exchanged to an input layer. Table 3 summarizes        0.004 to prevent rebars from yielding under sustained service loads. The maximum rebar ratio is restricted to 0.08 according to Section 10.6.1, ACI 318-19 (ACI Committee 2019) to make sure that concrete is consolidated sufficiently around rebars. Distributions of rebar ratios from 100,000 datasets are exhibited in Figure 5. Skew distributions of rebar ratios along X and Y directions are found in Figure 5. This is because total rebar ratios are constrained to be greater than 0.004 and smaller than 0.08 according to ANSI/AISC 360-16 (Ansi 2016) and ACI 318-19 (ACI Committee 2019) even if each rebar ratio is generated uniformly in a range of 0.001 to 0.08. Reverse networks are established for the SRC columns as shown in Figure 6, where input layers consist of 13 parameters: three reverse inputs (SF, ρ s , and b=h) and ten parameters (t w ; t f ; f 0 c ; f yr ; f ys ; ρ rX ; ρ rY ; P u ; M uX ; M uY ).
Eight parameters ε r ; ε s ; D r;max ; À Y s ; X s ; CI c ; CO 2 ; W c Þ are considered in an output layer as shown in Step 2 of Figure 7. Reverse scenario is conducted to design SRC columns, where SF is optimized to equal 1.0, resulting in design strengths equivalent to a factored biaxial load.

Reverse design-based optimization
Various reverse scenarios could be explored by exchanging the positions of the input and output parameters in ANN. ANN with its capacity to learn trends of datasets can provide solutions of reverse designs based on memorized trends from large datasets. In the reverse scenario as shown in Table 2(b), an input layer includes three reverse inputs, which are SF, steel ratio (ρ s ), and aspect ratio of columns (b=h) with 10 input parameters t w ; t f ; f 0 c ; f yr ; f ys ; ρ rX ; ρ rY ; P u ; M uX ; M uY À � . Output parameters in reverse scenario consist of 12 parameters including column dimensions (h and b), steel sections (h s and b s ), vertical (Y s ) and horizontal (X s ) clearances, rebar (ε r ) and steel (ε s ) strain, and three objective functions: cost CI c , CO 2 emission, and weight W c . Safety factor (SF) and an aspect ratio of columns (b=h) are preassigned on an input-side in reverse scenario to meet strength and architectural requirements, respectively. Steel (ρ s ) and rebar ratios (ρ rX and ρ rY ) vary in the ranges of 0.003-0.09 and 0.009-0.015, respectively, to minimize cost CI c , CO 2 emission, and weight W c . Column (h and b) and steel dimensions (h s and b s ) are, then, determined corresponding to minimized cost CI c , CO 2 emission, and weight W c . A trial-and-error technique based on conventional method may yield unacceptable designs with safety factor smaller than 1.0, whereas ANNs ensure to provide acceptable designs by preassigning safety factor equal to or greater than 1.0.

Formulation of back-substitution (BS) method
A BS method was proposed by Hong et al.  to solve a reverse design scenario of doubly reinforced concrete beams. A BS method consists of reverse CRS (BS-CRS) in the first step and forward structural mechanics in the second step. Figures 6 and 7 show the topology of the first step of a BS method. In reverse networks (Step 1), three reverse input parameters (SF, ρ s , and b=h) are preassigned at the input layer to predict four reverse outputs (h; b; h s ; and b s ). The training sequence of a BS-CRS is determined using three layers with 30 neurons, which results in a h ! b ! b s ! h s training sequence as shown in Figure 6. In Step 2 of BS, eight output parameters ε r ; ε s ; ð D r;max ; Y s ; X s ; CI c ; CO 2 ; W c Þ are calculated using structural mechanics proposed by Nguyen and Hong (Nguyen and Hong 2020)  software can be implemented in Step 2 of a BS to find forward outputs based on the reverse outputs obtained in Step 1 of the BS.

Network training
Arafa and Aleqdra (Arafa, Alqedra, and An-Najjar 2011) stated that there is no specific technique for finding optimal training parameters, which are a number of hidden layers and neurons. Figures 6 and 7   as presented in Table 4. The training is performed based on 100,000 datasets generated using structural algorithm suggested by Nguyen and Hong (Nguyen and Hong 2020 Þ, are preassigned. This reverse scenario aims to control safety factors equivalent to 1.0, which means constraining factored loads equal to design strengths. The minimum steel ratio (ρ s ) is limited to 0.01 according to Section I2.1a, ANSI/AISC 360-16 (Ansi 2016). The column aspect ratio (b=h) is also controlled by defining it in an input layer. According to Section I2.1a, ANSI/ AISC 360-16 (Ansi 2016), rebar ratios are constrained not to be smaller than 0.004. A maximum rebar ratio of 0.08 is suggested following Section 10.6.1, ACI 318-19 (ACI Committee 2019). Table 4 presents a training summary of the reverse scenario in Step 1 of the BS method. Figure 8(a) demonstrates the effect of rebar (ρ rX and ρ rY ) and steel ratio (ρ s ) on a maximum of rebar diameters. Two reverse inputs (SF and b=h) are each preassigned 1.0. The thicknesses of the flange and web of the H-shaped steel section are defined as 8 mm. Material properties, including compressive concrete strength (f 0 c ), rebar strength (f yr ), and steel strength (f ys ), are predetermined to be 30, 500, and 325 MPa, respectively. A factored biaxial load, including axial load (P u ), moment about the X-axis (M uX ), and moment about the Y-axis (M uY ), are defined as P u = 10,000 kN, M uX = 5000 kN·m, and M uY = 7000 kN·m. Rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) are varied in a range of 0.003-0.09 and 0.009-0.015, respectively. The valid range of rebar and steel ratios are 0.004-0.08 and greater than 0.01 as presented in Figure 8 (a). Bundled rebars are considered in 100,000 datasets. Section 25.6.1, ACI 318-19 (ACI Committee 2019) limits the use of any bundled rebar to a maximum of four single rebars. The maximum single rebars considered in this study is 32 mm. Therefore, a maximum rebar diameter of 64 mm is defined, equivalent to a bundled bar of four 32 mm rebars. The limits of a maximum rebar diameter, rebar ratios, and steel ratios establishing infeasible regions are shown in shadow blocks in Figure 8(a). The influence of rebar and steel ratios on column dimensions (h and b) are also exhibited in Figure 8(b).

Optimized designs of the CI c and CO 2 emission
Cost index (CI c ) and CO 2 emission are considered as objective functions. Effects of rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) on the CI c and CO 2 emission are shown in Figures 9(a,b). Both SF and column aspect ratio (b=h) are assigned 1.0. Rebar ratios (ρ rX and ρ rY ) vary in a range of 0.003-0.09, and steel ratios are chosen in a range of 0.009-0.015. Factored loads are predetermined as P u = 10,000 kN, M uX = 5000 kN·m, and M uY = 7000 kN·m. Material properties are preassigned as f 0 c = 30 MPa, f yr = 500 MPa, and f ys = 325 MPa. Figure 9(a) shows the trends of cost index (CI c ) as a function of rebar (ρ rX and ρ rY ) and steel ratios (ρ s ). A minimized CI c is obtained when the rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) are minimized, which are 0.004 and 0.01, respectively. From Figure 9(a), the optimal CI c is obtained as 600,466 KRW/m when the rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) attain minima. Figure 9(b) shows similar effects of rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) on CO 2 emission. The minimum CO 2 emission is 1.00 t-CO 2 /m, with rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) minima of 0.004 and 0.01, respectively. Figure 8(b) shows the effects of rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) on column width (b) and height (h) with a 1.0-preassigned column aspect ratio (b=h). As shown in Figures 9(a) and (b), optimal designs of CI c and CO 2 emission are achieved when column width (b) and column height (h) attain their maxima, whereas rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) are minimized. It is observed that the trends to obtain optimal designs of CI c and CO 2 emission are similar because rebar and steel ratios are decreased to lower price and CO 2 emission, increasing the concrete volume to get a balanced section. The h, b, h s , and b s are defined as outputs in Step 1 of the BS method when rebar ratios (ρ rX and ρ rY ) and steel ratios (ρ s ) of 0.004 and 0.01, respectively, are defined as input parameters as shown in Table 5. A −0.73% SF error is observed in Table 5 when the SF predicted from the BS method is compared with the one calculated from AutoSRCHCol, indicating that optimal design accuracies of CI c and CO 2 emission are sufficient for use in practical designs. An interaction diagram in a three-dimensional view is established based on column and steel dimensions corresponding to the optimal design of CI c and CO 2 as shown in Figure 10.

Optimized designs of W c
Figure 9(c) shows variations of column weights (W c ) when the rebar and steel ratios varied from 0.003-0.09 and 0.009-0.015, respectively. Factored loads are predetermined as P u = 10,000 kN, M uX = 5000 kN·m, and M uY = 7000 kN·m. Material properties are preassigned as f 0 c = 30 MPa, f yr = 500 MPa, and f ys = 325 MPa. The SF and column aspect ratio (b=h) are assigned 1.0. Therefore, optimal designs with a minimum W c of 38.45 kN/m are achieved when the rebar (ρ rX and ρ rY ) and steel (ρ s ) ratios attain their maxima as shown in Figure 9(c). The maximum rebar (ρ rX and ρ rY ) is restricted to 0.044 because of a maximum rebar diameter of 64 mm as shown in Figure 8(a). Steel (ρ s ) ratios also yield their maximum of 0.015 to obtain an  optimal design with a minimized W c . Reverse outputs (h, b, h s , and b s ) are predicted as outputs from Step 1 of the BS method based on rebar (0.004) and steel (0.015) ratios obtained from Table 6. Table 6 presents an optimal design of W c with SF error of −1.58%, comparing results from the BS method and AutoSRCHCol software. Although errors of safety factor (SF) are greater than 1%, an absolute difference of SF between the BS method (1.00) and AutoSRCHCol (1.02) is only 0.02. Figure 11 shows an interaction diagram corresponding to optimal designs of minimum W c with 1.0 preassigned SF 1.0 when P u = 10,000 kN, M uX = 5000 kN·m, and M uY = 7000 kN·m are considered.
It is worth noting that designs with optimal W c are achieved when column dimensions decrease and rebar and steel ratios increase. Although it is unlikely to obtain optimal W c designs, optimal designs of CI c and CO 2 emission are achieved with increased column dimensions and decreased rebar and steel ratios.

Conclusion
This study develops an alternative approach to offer optimal designs of reinforced concrete with SRC columns under a biaxial load based on ANNs. Analytical behaviors of SRC columns are investigated by implementing ANNs with acceptable accuracy (errors smaller than 2%, compared with ANN predictions and structural mechanics). It is challenging for engineers to determine the proper column size and steel sections in a preliminary design stage to obtain optimized designs with minimum cost index (CI c ), CO 2 emission, and column weights (W c ). ANNs are proposed as an alternative tool for designing SRC columns under a biaxial load when SF and steel ratio (ρ s ) are prescribed as reverse inputs in a reverse scenario, which meets code requirements from ANSI/AISC 360-16 (Ansi 2016). A reverse design scenario is proposed to obtain designs of SRC columns, satisfying the requirements in ANSI/AISC 360-16 (Ansi 2016) and ACI 318-19 (ACI Committee 2019) with preassigned SF, steel ratio (ρ s ), and column aspect ratio (b=h) in an input layer.
The following conclusions can be further drawn for readers: (1) BS method with CRS training technique is applied to optimize training accuracies on four reverse outputs, which are column height (h), width (b), and dimensions of steel section (h s and b s ). These reverse outputs (h, b, h s , and b s ) are, then, back-substituted to calculate the other eight output parameters ε r ; ε s ; D r;max ; Y s ; X s ; CI c ; CO 2 ; W c À � .
(2) The effects of rebar and steel ratios on objective parameters, such as CI c , CO 2 emission, and W c , are illustrated in Figure 9. Rebar and steel ratios decrease to obtain minimized CI c and CO 2 emission, whereas rebar and steel ratios increase to yield minimized column weights (W c ). (3) It is worth noting that CI c and CO 2 emission are minimized similarly when rebar and steel ratios are reduced to their minima (0.004 and 0.01 for rebar and steel ratios, respectively, specified by ANSI/AISC 360-16 (Ansi 2016)). Designs with minimized column weight (W c ) are achieved in an opposite way when rebar and steel ratios increase. (4) Optimal design results of SRC columns based on the proposed ANNs are consistent with those calculated from structural mechanics. Observed errors are insignificantly smaller than 2% for SF when using the BS (reverse CRS -Forward AutoSRC HCol) method. (5) Three-dimensional interaction diagrams corresponding to optimized design with CI c , CO 2 emission, and W c are illustrated when safety factors (SF ¼ 1:0) for a factored biaxial load is preassigned.
(6) Simple, expeditious, and rapid but sufficiently accurate design tools of SRC columns are available to engineers to investigate a behavior of SRC columns in practical designs using the suggested ANNs. (7) Further studies to obtain optimal designs of SRC columns under multiple loads will be carried out by implementing derivative optimization methods, such as a Lagrange multiplier method, or non-derivative optimization, such as genetic algorithms. Multiobjective functions, luding CI c , CO 2 emission, and W c will be also investigated in future studies.