An AI-based Lagrange optimization for a design for concrete columns encasing H-shaped steel sections under a biaxial bending

ABSTRACT shaped steel sections subjected to biaxial loads. The Lagrange multiplier method is used to optimize cost index (CIc), and CO2 emission of the columns. ANNs can be implemented to generalize functions for objective parameters CIc , and CO2 emission. Generalized functions can replace complex analytical functions that are difficult to derive when optimizing objective functions. In the AI-based Lagrange multiplier method, dimensions of columns and steel sections are calculated as output parameters corresponding to minimized CIc , and CO2 emission. Note that 3D interaction diagrams of SRC columns subjected to biaxial bending and concentric axial loads are also formulated based on optimal results. An accuracy of the ANN-based optimal designs is demonstrated using structural mechanics based on a strain compatibility. A hybrid network based on both ANNs and Lagrange multiplier method identified design parameters which reduced CIc , and CO2 emission of a column by 30.7% and 40.4% respectively, when compared with those of a conventionally designed column. Graphical abstract


Introduction and background
ANNs have been used successfully by Abambres and Lantsoght (2020), Sharifi, Lotfi, and Moghbeli (2019), Asteris et al. (2019), and Armaghani et al. (2019) for structural analysis. Hong, Pham, and Nguyen (2021),  have performed optimization studies of reinforced concrete beams based on ANN.  also successfully applied the Lagrange multiplier method to obtain optimal designs of reinforced rectangular concrete columns based on ANN.
Steel-encased reinforced concrete columns (SRC) are being used extensively to enhance a strength and ductility of steel sections with an increased rigidity and stiffness of concrete. Steel sections can be protected from fire and corrosion by encasing them in concrete columns. SRC members have been recently used in high-rise buildings and transportation facilities. Mirza, Hyttinen, and Hyttinen (1996), Ricles and Paboojian (1994), El-Tawil and Deierlein (1999), Li and Matsui (2000), Brettle (1973), Bridge and Roderick (1978), Furlong (1974), and Mirza and Skrabek (1992) have experimentally investigated a behavior of SRC columns when subjected to uniaxial bending and axial compressive loads whereas Munoz and Hsu (1997a) and Morino, Matsui, and Watanabe (1984) investigated a performance of SRC columns under biaxial bending and axial compressive loads. Chen and Lin (2006), Dundar et al. (2008), Munoz and Hsu (1997b), Furlong (1968), Kato (1996), Virdi and Dowling (1973), and Roik and Bergmann (1990) also investigated behaviors of SRC columns. Effects of confined concrete on performance of SRC columns have been investigated by Rong, Shi, and Wang (2021) and . In the conventional method, however, optimal designs of SRC columns can be obtained only by trial-anderror, which requires considerable effort on the part of engineers. In this study, a hybrid network using ANNs based on Lagrange multipliers was proposed with an acceptable accuracy to offer optimal designs of reinforced concrete columns encasing H-shaped steel sections subjected to biaxial bending and concentric axial loads. Nguyen and Hong (2020) and Hong (2019) investigated a numerical and experimental behavior of such SRC columns based on a strain compatibility to generate large datasets for use in a training of ANNs.

Research objectives and innovations
Optimizing designs of SRC columns based on conventional Lagrange multipliers are complex, requiring analytical objective functions. In this study, ANNs are used to generalize differentiable objective and constraint functions to implement the Lagrange optimization. Generalized functions replace analytical functions when formulating Lagrange functions. Solutions under KKT conditions based on inequality constraints defined by design requirements are, then, obtained based on Newton-Raphson iterations. This study minimizes cost index (CI c ), and CO 2 emission of SRC columns under a biaxial load. For this task, a hybrid network of ANNs is adopted to obtain generalized functions for objective and constraint functions because analytical objective functions and constraint constraints are difficult to derive for an application of a conventional Lagrange optimization. A large dataset of 100,000 is used to derive ANN-based objective functions for CI c , and CO 2 emission of SRC columns and their constraints as functions of input parameters. Design parameters including column dimensions and rebar ratios are obtained as outputs while minimizing CI c , and CO 2 emission. Cost index (CI c ), and CO 2 emission) are good objective targets for optimizing a design of SRC columns. Cost (CI c ) is an interest of structural engineers, whereas CO 2 emission is the interests of governments and contractors, respectively. Any design target can be adopted as an objective function. The accuracy of the minimized objective functions (CI c , and CO 2 emission) and corresponding design parameters is verified by using a structural mechanicsbased software (AutoSRCHCol) developed by Nguyen and Hong (Nguyen and Hong 2020). Errors are insignificant, indicating that the proposed network could be implemented in practical designs to help engineers optimize SRC column designs.

Column design scenario
In this study, an ANN-based forward network for a single biaxial load was investigated. Column dimensions including height (h) and width (b) are preassigned on an input-side when designing SRC columns. Steel height (h s ), flange width (b s ), flange (t f ), and web (t w ) thickness are defined as input parameters in forward design scenario. Material properties of concrete (f 0 c ), rebar (f yr ), and steel (f ys ) are also predetermined on an input-side. Factored biaxial loads including factored axial load (P u ), bending moments (M uX and M uY ) about X and Y axis shown in Figure 1(a) are chosen as inputs to design SRC columns. Factored biaxial loads are defined as P u , M uX , M uY in which P u is factored axial load, M uX and M uY are factored bending moments about X -and Ydirection, respectively. Figure 1(a) illustrates factored biaxial loads consisting of P u , M uX , M uY . Factored biaxial loads (P u , M uX , M uY ) are preassigned on an inputside in a forward design network, then engineers will design column section to obtain design strengths (ϕP n , ϕM nX , ϕM nY ) greater than preassigned factored biaxial loads (P u , M uX , M uY ).
Output parameters consist of safety factor (SF), rebar (ε r ) and steel (ε s ) strain, a maximum rebar diameter (D r;max ), vertical (Y s ) and horizontal (X s ) clearance, a column aspect ratio (b=h), and two objective functions: cost index (CI c ), and CO 2 emission. Table  1 presents nomenclatures for input and output parameters which are considered in this study. Forward design scenario is summarized in Table 2. Rebar ratios (ρ rX and ρ rY ) are preassigned on an input side in a forward design network, and hence, a maximum rebar diameter (D r;max ) is calculated accordingly on an output side based on column dimensions and rebar ratios. Steel sections are also designed for SRC columns, and hence, vertical (Y s ) and horizontal (X s ) shown in Figure 1(a) are determined on an output side based on a steel section and column dimensions.
Cost index (CI c ), and CO 2 emission are minimized on an output-side based on column dimensions, rebar ratios, and steel section. The corresponding design parameters are then calculated to minimize the three design targets. Both cost index (CI c ) and CO 2 are minimized as shown in Tables 7 and Table 9 Table 2. In the Lagrange multiplier method, cost index (CI c ), and CO 2 emission c of an SRC column are considered as objective functions whereas the material properties of concrete (f 0 c ), rebars (f yr ), and steel sections (f ys ) and the thickness of the flanges (t f ) and webs (t w ) of the steel sections are considered as equality constraints. The Lagrange multiplier method offers optimal designs of SRC columns with the lowest CI c , and CO 2 emission. Inequality constraints are implemented to control the optimization, which have to meet the code requirements of reinforced concrete columns encasing steel sections. Section 10.6.1 of ACI 318-19 (Standard 2019) requires the rebar ratios along X-and Y-axes ρ rX andρ rY ð Þ to be below 0.08. The minimum value allowed in Section I2.1a of ANSI/AISC 360-16 (American Institute of Steel Construction 2016) for the rebar ratio in SRC columns is 0.004. A minimum limit is required for the rebar ratio to reduce the effects of concrete creep and shrinkage, and to prevent the rebars from yielding when subjected to sustained service loads. A maximum limit of 0.08 is also suggested for the rebar ratios to ensure that concrete is consolidated sufficiently around the rebars. Bundled rebars also are considered when generating large datasets. The maximum diameter of a bundled rebars formed using four single rebars (32 mm in diameter) has to be less than 64 mm. According to Section I2.1a of ANSI/AISC 360-16 (American Institute of Steel Construction 2016), the steel ratios for reinforced concrete columns encasing steel sections have to be greater than or equal to 0.01. The design strength (ϕP n ; ϕM n ) of the columns has to be high enough to make them withstand the effects of factored biaxial loads P u ; M uX ; M uY ð Þ. Safety factor (SF) of a column is defined as a ratio of its design strength to factored loads, which has to be greater than or equal to 1.0.
Cost index (CI C ), and CO 2 emission of the columns are optimized individually by using the Lagrange multiplier method when designing SRC columns for biaxial loads. Based on Lagrange multiplier method, dimensions of columns and steel sections, and rebar ratios corresponding to optimized CI C , and CO 2 emission are obtained as outputs. Three-dimensional interaction diagrams of bending moment and axial load are presented based on the optimization.

Generation of large structural datasets
In the design of SRC columns, the 14 input and 11 output parameters listed in Table 2 are used for large dataset generation. The 14 input parameters include dimensions of columns (height h and width b) and steel sections (height h s , width b s , and flange and web thicknesses t f and t w , respectively); material properties of concrete (f 0 c ), rebars (f yr ), and steel sections (f ys ); rebar ratios (ρ rX and ρ rY ); and biaxial loads (P u ; M uX ; M uY ). Output parameters include SF; strains of rebar (ε r ) and steel (ε s ); aspect ratio of the columns (b=h); steel ratio (ε s ); maximum rebar diameter (D r;max ); vertical (Y s ) and horizontal (X s ) clearances; and the cost index (CI c ), CO 2 emission, and weight (W c ) of the columns. Figure 1 illustrates the cross-section and strain compatibility of SRC columns. As shown in Figure  2, structural analysis software AutoSRC HCol developed based on the strain compatibility algorithm proposed by Nguyen and Hong (2020) is used to create large datasets for the forward training of SRC column designs. A structural design software (AutoSRCHCol) was developed by Nguyen and Hong (2020) using strain compatibility to investigate performance of SRC columns. Nguyen and Hong (2020) uses a constitutive model for concrete proposed by Mander. An elasto-plastic model is used for simulating rebar and steel materials. Accuracies of an AutoSRCHCol software are ascertained using Abaqus simulation and experimental data introduced by Nguyen and Hong (2020). An AutoSRCHCol software is used to verify accuracies of AI-based Lagrange optimization for use in practical designs. Figure 3 illustrates the flow chart showing a large dataset generation. Column height (h) and width (b) are in the 500-2000 mm range.
Compressive concrete strength (f 0 c ) is in the 30-50 MPa range. Strengths of rebar (f yr ) and (f ys ) are created randomly to fall in the 400-600 MPa and 275-325 MPa ranges, respectively. The biaxial load pair (P u ; M uX ; M uY ) is first investigated. A large dataset consisting of 100,000 data points is thereafter generated for designing the SRC columns. The statistical parameters of the input parameters, including maxima, means, minima, standard deviations, and variances, are summarized in Table 3. In Figure 4(a) to (n), the histograms of randomized input parameters are presented describing their distribution.

Network training based on PTM
The parallel training method (PTM) introduced by Hong et al. (Hong, Pham, and Nguyen 2021) is used in the forward network to map the 14 input parameters Figure 5. The optimal training parameters including a number of hidden layers and neurons of each output parameter are found among the nine training networks (8, 9, and 10 hidden layers with 40, 50, and 60 neurons). Table 4 presents the optimal forward training results for each output parameter. Mean square error (MSE) is used to evaluate accuracy of testing datasets when training ANNs as shown in Table 4.

Formulation of the Lagrange function
In this study, generalized functions are derived for objective functions (CI c , and CO 2 emission), and equality and inequality constraints using ANNs as shown in Equation (1).
where x is an input vector; N is a number of layers, including hidden layers and output layer; W n is a weight matrix between layer n À 1 and layer n; b n is a bias matrix of layer n; and g N and g D are normalization and de-normalization functions, respectively. Activation functions f n t at layer n are implemented to formulate nonlinear relationships of the networks; a linear activation function f n lin is selected for an output layer because output values are unbounded. An artificial intelligence (AI)-based objective function as shown in Equation (1) is used for Lagrange optimization.  presented the Lagrange function L as a function of input variables x ¼ x 1 ; x 2 ; . . . ; x n ½ � T , equality ½c x ð Þ� and inequality ½v x ð Þ� constraints, and Lagrange multipliers of equality and inequality constraints λ c ¼ λ 1 ; λ 2 ; . . . ; λ m ½ � T and λ v ¼ λ 1 ; λ 2 ; . . . ; λ l ½ � T , respectively, as shown in Equation (2). Vectors of equality ½c x ð Þ� and inequality ½v x ð Þ� constraints are given in Equations (3) and (4), respectively. Hong and Nguyen (2021) implemented Newton-Raphson method to solve nonlinear equations and find saddle points of Lagrange functions based on the Jacobian and Hessian matrix as shown in Equations (5) to (7). The Newton-Raphson approximation was repeated until convergence was achieved.
vation of the Lagrange function based on Karush-Kuhn-Tucker conditions (Kuhn and Tucker 1951;Karush 1939). The derivations of AI-based Hessian matrix were developed by the MathWorks Technical Support Department (MathWorks 2021).
ANN-based functions for cost (CI c ), and CO 2 emission of the SRC columns and equality, inequality constraints meeting design requirements are used for the Lagrange optimization. These ANN-based functions replace analytical functions due to the difficulty of deriving and differentiating analytical functions. Cost index (CI c ), and CO 2 emission of SRC columns are minimized by minimizing the Lagrange functions. ANNs provide weight and bias matrices presented in Table  5 by which functions for cost (CI c ), and CO 2 emission of the SRC columns are generalized.

Derivation of AI-based objective functions and Lagrange functions to optimized CI c , and CO 2 emission of the columns
Lagrange function shown in Equation (2) is formulated in terms of ANN-based objective functions, cost index (CI c ), and CO 2 emission) of SRC columns. Objective functions are generalized by ANNs as shown in Table 6 where fourteen input parameters h; b; h s ; b s ; t w ; t f ; f 0 c ; f yr ; f ys ; ρ rX ; ρ rY ; P u ; M uX ; M uY À � are used to define objective functions of CI c , and CO 2 emission of SRC columns when being subjected to biaxial loads (P u ,M uX ,M uY ). Table 4 shows best training accuracies of 11 output predictions including objective functions, cost index (CI c ), and CO 2 emission of   SRC columns based on PTM using 8, 9, and 10 hidden layers with 40, 50, and 60 neurons. Equality and inequality constraints are preassigned to govern the Lagrange optimization. As shown in Table 6, the material properties of concrete, rebars, and steel sections are set as equality constraints (f 0 c = 30 MPa, f yr = 500 MPa, and f ys = 325 MPa, respectively), whereas the flange (t f ) and web (t w ) thicknesses of the steel sections are taken as 8 mm as shown in Table 6. SF denoted as v 1 in Table 6 is considered as an inequality constraint with a value greater than 1.0. Steel ratio ρ s ð Þ presented as v 2 in Table 6 is set at 0.01 or above in accordance with Section I2.1a of ANSI/AISC 360-16 (American Institute of Steel Construction 2016). Rebar ratios are limited to the range 0.004-0.08 in accordance with ANSI/AISC 360-16 (American Institute of Steel Construction 2016) and ACI 318-19 (Standard 2019), respectively, and are shown in Table 6 as v 3 and v 4 . The maximum, minimum allowable strains in rebar (ε r ) and steel (ε s ) are not specified in ACI 318-19 (Standard 2019) and ANSI/AISC 360-16 (American Institute of Steel Construction 2016), and thus could be of any value. Dimensions of SRC columns are constrained by inequality conditions as shown in v 7 , v 8 and v 17 .

Formulation of active and inactive conditions
Equality constraints include nine equality functions as shown in Table 6. These equality constraints are used in the Lagrange optimization to reduce a complexity of a network. In Table 6, 17 inequality constraints (v i ) are established based on requirements by design codes and architects when optimizing SRC columns. A matrix for inequality constraints (S) shown in Equation (8) is proposed to designate activating inequality  Table 5. Weight and bias matrices of a forward network for objective functions (CI c ; and CO 2 ) considering one biaxial load.     Table 6. Inequality v 2 is active when its corresponding inequality factor (s 2 ) is equal to 1.0, indicating that rebar ratio is equivalent to its allowable minimum value (0.004), and leading to Equation (9) in which inequality constraint v 2 is treated as equality constraint during optimization.  (2) and (9).

EQUALITY CONSTRAINTS INEQUALITY CONSTRAINTS
. . . where f x ð Þ are the AI-based generalized functions of CI c , and CO 2 emission.

Optimized cost (CI c ) verified by structural design software (AutoSRCHCol)
As shown in Table 7, cost (CI c ) of an SRC column is minimized based on a biaxial bending (M uX ¼ 6500 kN·m, and M uY ¼ 5000 kN·m) with an axial load (P u ¼ 12000 kN). Fourteen input parameters minimizing cost (CI c ) of an SRC column are used in the structural design software (AutoSRCHCol) developed by Nguyen and Hong (Nguyen and Hong 2020) to calculate 11 output parameters, verifying the design accuracy of AI-based Lagrange forward designs. In Table 7, the biggest error of −3.41% is presented in the rebar strain (ε r ) when optimized cost based on AI-based Lagrange forward design is compared with those obtained using structural software. The errors in SF (−0.43%) and optimized CI c (1.16%) are insignificant, and hence, the accuracy of the hybrid network which minimizes Lagrange function is acceptable, indicating that the network can be used in practical designs.

Column interaction diagram corresponding to an optimized CI c
A three-dimensional P-M interaction diagram is plotted based on the minimized cost (CI c ) obtained based on AI-based Lagrange forward design as shown in Figure 6. The cost CI c of the SRC column is optimized when biaxial loads are set equivalent to a design strength of (1) Calculated on an output-side when optimizing CI c .
(2) Equality constraints. the SRC columns (SF ¼ 1) as illustrated in Figure 6. In Table 7, the dimensions of columns and steel sections, and the rebar ratios are obtained from the AI-based Lagrange forward design corresponding to a minimized CI c . Figure 7 illustrates the column dimensions obtained when Lagrange function is minimized.

Verification of optimized CI c based on ANN by large datasets
A large dataset containing 5,750,000 data points is generated with preassigned equality constraints as shown in Table 6 to verify the minimized cost index (CI c ) obtained by ANN-based Lagrange multipliers. A set of 96,902 observations is obtained when filtering a 5,750,000 dataset through SF set at a value greater than 1.0 to ensure that the columns are strong enough to resist factored loads. The maximum rebar diameter (D r;max ) is set at 64 mm, which is equivalent to a bundle of four 32 mm single rebars. SF and D r;max are the output parameters; thus, they are not controlled initially when generating large datasets. The relationship between the axial load (P u ) and cost index of a column (CI c ) is illustrated in Figure 8(d) based on 96,902 observations. Moment of two different axes (M uX and M uY ) are set at 6500 kN·m and 5000 kN·m as shown in equality constraints c 6 and c 7 of Table 6, respectively, and axial load (P u ) is set to 12,000 kN as shown in equality constraint c 5 of Table 6. As shown in Table 7 and Figure 8(d), the minimized CI c obtained based on ANN is 569,126 KRW/ m at a preassigned factored moments (M uX and M uY ) of 6500 kN·m, 5000 kN·m and axial load (P u ) of 12,000 kN, which shows only 1.16% error when being compared with structural calculations. The minimum CI c obtained from 96,902 observations which are extracted from a 5,750,000 dataset is 572,670 KRW/m, whereas the proposed method based on ANN offers a column design with a minimized CI c of 569,126 KRW/m, which is less than 572,670 KRW/m obtained from big datasets by 0.6% as presented in Figure 8(d) and Table      (1) Calculated on an output-side when optimizing CO 2 .
(2) Equality constraints. 8. It is also noted that the minimized cost of the column based on ANN is less than that of a probable column design of 821,730 KRW/m by 30.7% determined based on trendline functions ("polyfit" and "polyval" commands) provided in MATLAB (MathWorks 2021) as shown in Figure 8(d). Table 8 compares the minimized CI c obtained from the proposed method with those based on four different large datasets, 650,000 datasets shown in Figure 8(a), 1,750,000 datasets shown in Figure 8(b), 3,750,000 datasets shown in Figure 8(c), and 5,750,000 datasets shown in Figure 8(d). The minimum cost CI c of 621,669 KRW/m found from 650,000 and 1,750,000 datasets are identical, demonstrating a difference of −9.2% compared with the minimized cost of 569,126 KRW/m based on the proposed method. The difference between the minimum CI c obtained using large datasets and the proposed method is reduced by −0.6% when a number of the datasets increases beyond 3,750,000 as shown in Table 8 and Figure 8(c).

Optimized CO 2 emission verified by structural design software (AutoSRCHCol)
As shown in Table 9, a biaxial bending (M uX ¼ 6500 kN·m, M uY ¼ 5000 kN·m) with an axial load (P u ¼ 12000 kN) are considered in the design of an SRC column, minimizing CO 2 emission. Fourteen input parameters minimizing CO 2 emission of an SRC column are used in the structural design software (AutoSRCHCol) proposed by Nguyen and Hong (Nguyen and Hong 2020) to ascertain the design accuracy of AI-based Lagrange forward designs. The trend of the CO 2 emission similar to the optimized cost index (CI c ) is found. Concrete volume increases to compensate rebar ratio reduction when a reduction of rebar ratios of the SRC columns causes their cost indices and CO 2 emissions to decrease simultaneously. The rebar strain (ε r ) demonstrates an error of −3.41%. However, the difference in the rebar strain (ε r ) obtained is only 0.0001, being negligible in practical designs. Errors in SF of −0.43% and minimized CO 2 emission of 0.61% are also insignificant proving that the accuracy of a hybrid network minimizing Lagrange functions is adequate for use in practical designs.

Column interaction diagram corresponding to an optimized CO 2 emission
A three-dimensional P-M interaction diagram is plotted based on the minimized CO 2 emission obtained based on AI-based Lagrange forward design as shown in Figure 9.
The CO 2 emission of the SRC column is minimized when biaxial loads are made equivalent to a design strength of the SRC columns (SF ¼ 1) as illustrated in Figure 9. In Table 9, the dimensions of columns and steel sections, and the rebar ratios are obtained from the AI-based Lagrange forward design corresponding to a minimized CO 2 emission. Figure 10 illustrates the column dimensions obtained when Lagrange function is minimized.

Verification of an optimized CO 2 emission based on ANN by large datasets
A large dataset containing 5,750,000 data points is generated with preassigned equality constraints as shown in Table 6 to verify the minimized CO 2 emission obtained by ANN-based Lagrange multipliers. A set of 96,902 observations is obtained when filtering a 5,750,000 dataset through SF set at a value greater than 1.0 to ensure that the columns are strong enough to resist factored loads. The relationship between the axial load (P u ) and minimized CO 2 emission is illustrated in Figure 11(d) based on 96,902 observations. Moment of two different axes (M uX and M uY ) are set at 6500 kN·m and 5000 kN·m as shown in Figure 10. Section of an SRC column corresponding to minimized CO 2 emissions.   equality constraints c 6 and c 7 of Table 6, respectively, and axial load (P u ) is also set to 12,000 kN as shown in equality constraint c 5 of Table 6. As shown in Table 9 and Figure 11 (d), the minimized CO 2 emission obtained based on ANN is 0.946 t-CO 2 /m at a preassigned factored moments (M uX and M uY ) of 6500 kN·m, 5000 kN·m and axial load (P u ) of 12,000 kN, which shows only 0.61% error when being compared structural calculations. The minimum CO 2 emission obtained from 96,902 observations which are extracted from a 5,750,000 dataset is 0.955 t-CO 2 /m, whereas the proposed method based on ANN offers a column design with a minimum of CO 2 emission of 0.946 t-CO 2 /m, which is less than 0.955 KRW/m obtained from big datasets by −1.0% as presented in Figure 11(d) and Table 10. It is also noted that the minimized CO 2 emission of the column based on ANN is less than that of a probable column design of 1.587 t-CO 2 /m by 40.4% determined based on trendline functions ("polyfit" and "polyval" commands) provided in MATLAB (MathWorks 2021) as shown in Figure 11(d). Table 10 compares the minimized CO 2 emission obtained from the proposed method with those based on four different large datasets, 650,000 datasets shown in Figure 11(a), 1,750,000 datasets shown in Figure 11(b), 3,750,000 datasets shown in Figure 11(c), and 5,750,000 datasets shown in Figure 11(d). The minimum CO 2 emission of 1.122 t-CO 2 /m found from 650,000 and 1,750,000 datasets are identical, demonstrating a difference of −18.6% compared with the minimized cost of 0.946 t-CO 2 /m based on the proposed method. The difference between the minimum CO 2 emission obtained using large datasets and the proposed method is reduced by −1.0% when a number of the datasets increases beyond 3,750,000 as shown in Table 10 and Figure 11(c).

Conclusion
A hybrid network of ANNs minimizing Lagrange functions is proposed to optimize designs of SRC columns under a biaxial load. Cost index (CI c ), and CO 2 emission of SRC columns are objective functions. Some of the findings based on this study are highlighted as follows.
(1) Cost index (CI c ), and CO 2 emission of SRC columns with steel height (h s ), flange width (b s ), flange (t f ), and web (t w ) thickness under factored biaxial loads P u , M uX , M uY are minimized based on column dimensions, rebar ratios, and steel section. It was found that the corresponding design parameters are then calculated to minimize the three design targets. Both cost index (CI c ) and CO 2 are minimized when rebar ratios and a steel section are minimized while dimensions of the concrete section are maximized. On the other hand, column dimensions decrease to lower column weight, resulting in increased rebar and steel quantity to compensate concrete volume reduction.
(2) Material properties of concrete (f (6) The authors plan to perform ANN based-Lagrange optimization of SRC columns based on the two objective targets (CI c , and CO 2 emission) in a simultaneous way. An optimization curve will provide tradeoffs among the two objective targets, uncovering the inter-relationships of optimization between the two for decision makers.
(7) The proposed AI-based method heavily depends on fast computing facilities for generating large datasets, Lagrange-based training, and Newton-