An optimization of round reinforced concrete columns subject to multiple loads using an artificial neural network (ANN)

Figures & data

Figure 1. Conventional design of RC columns.

Table 1. Design parameters’ nomenclature.

Notation	Nomenclature	Note
D	Column section diameter	Input parameters used in ANN to design RC columns
${\rho }_s$	Rebar ratio
$f_c^{\prime }$	Concrete strength
$f_y$	Rebar yield strength
$P_u$	Factored axial load
$M_u$	Factored bending moment
$\mathit{SF}$	Safety factor	Output parameters used in ANN to design RC columns
${\epsilon }_s$	Rebar strain
CIc	Cost index of column per height
CO2	CO2 emission per column height
Wc	Weight of column per column height
${\alpha }_{\mathit{e}/h}$	Angular eccentricity of applied load	Only applied to generate large dataset.

Figure 2. ANN-based design of RC columns.

Table 2. Ranges to generate large dataset using structural mechanics-based software AutoCC.

Parameters	Random range to generate large datasets
D	400 mm ~1200 mm
	0.01 ~ 0.08
	20 MPa ~ 70 MPa
	300 MPa ~ 550 MPa
SF	0.1 ~ 3.0
	0 ~Π/2

Figure 3. Topology of ANN to design RC columns.

Table 3. Training results and accuracies from using tansig activation function.

6 Inputs ($\mathit{D},{\rho }_s,f_c^{\prime },f_y,P_u,M_u$) − 5 Outputs ($\mathit{SF},{\epsilon }_s,\mathit{C}{I}_c,\mathit{C}{O}_2,W_c$)
PTM − 100,000 data − 100,000 suggested epochs – tansig activation function
Output	Layers	Neurons	Best Epoch	Stopped Epoch	Test MSE	R at Best Epoch
$\mathit{SF}$	10	20	98,390	99,390	4.0.E–04	0.9994
${\epsilon }_s$	10	80	66,717	66,817	2.0.E–05	0.9998
$\mathit{C}{I}_c$	10	20	13,224	14,224	7.6.E–07	1.0000
$\mathit{C}{O}_2$	5	20	43,211	44,211	6.8.E–08	1.0000
$W_c$	5	80	40,073	41,073	5.9.E–07	1.0000

Table 4. Training results and accuracies from using ReLU activation function.

6 Inputs ($\mathit{D},{\rho }_s,f_c^{\prime },f_y,P_u,M_u$) − 5 Outputs ($\mathit{SF},{\epsilon }_s,\mathit{C}{I}_c,\mathit{C}{O}_2,W_c$)
PTM − 100,000 data − 100,000 suggested epochs – ReLU activation function
Output	Layers	Neurons	Best Epoch	Stopped Epoch	Test MSE	R at Best Epoch
SF	5	20	76,790	77,790	0.0235	.9705
εs	10	20	33,183	34,183	0.0374	.5423
CIc	10	20	7,200	8,200	0.021	.9474
CO2	5	20	41,096	42,096	3.37E–05	.9999
Wc	5	20	10,684	11,684	1.62E–04	.9998

Figure 4. Two design cases.

Table 5. ANN-based design Case 1.

Table 6. ANN-based design Case 2.

Figure 5. ANN-based Lagrange optimization.

Table 7. Factored load and formulation of inequality and equality constraints.

Equality constraint $\mathit{c}\left(\mathit{x}\right)$	Inequality constraints $\mathit{v}\left(\mathit{x}\right)$
$c_1\left(\mathrm{x}\right)=f_c^{\prime }-40=0$	$v_1\left(\mathit{x}\right)={\rho }_s-0.01\ge 0$
$c_2\left(\mathrm{x}\right)=f_y^{-}500=0$	$v_2\left(\mathit{x}\right)=-{\rho }_s+0.08\ge 0$
$c_3\left(\mathrm{x}\right)=P_{u,1}^{-}25000=0$	$v_3\left(\mathit{x}\right)=\mathit{D}-400\ge 0$
$c_4\left(\mathrm{x}\right)=M_{u,1}^{-}4000=0$	$v_4\left(\mathit{x}\right)=-\mathit{D}+2000\ge 0$
$c_5\left(\mathrm{x}\right)=P_{u,2}^{-}5000=0$	$v_5\left(\mathit{x}\right)=\mathit{S}{F}_1^{-}1.0\ge 0$
$c_6\left(\mathrm{x}\right)=M_{u,2}^{-}4500=0$	$v_6\left(\mathit{x}\right)=\mathit{S}{F}_2^{-}1.0\ge 0$
$c_7\left(\mathrm{x}\right)=P_{u,3}^{-}10000=0$	$v_7\left(\mathit{x}\right)=\mathit{S}{F}_3^{-}1.0\ge 0$
$c_8\left(\mathrm{x}\right)=M_{u,3}^{-}3500=0$	$v_8\left(\mathit{x}\right)=\mathit{S}{F}_4^{-}1.0\ge 0$
$c_9\left(\mathrm{x}\right)=P_{u,4}^{-}22000=0$	$v_9\left(\mathit{x}\right)=\mathit{S}{F}_5^{-}1.0\ge 0$
$c_{10}\left(\mathrm{x}\right)=M_{u,4}^{-}5000=0$
$c_{11}\left(\mathrm{x}\right)=P_{u,5}^{-}27000=0$
$c_{12}\left(\mathrm{x}\right)=M_{u,5}^{-}2500=0$

Table 8. Design table for ANN-based Lagrange optimization.

Figure 6. Verification of Lagrange optimization by five million datasets for a P-M diagram capturing five load pairs shown in Figure 5b.

Table 9. Verification of AI-based optimization by large datasets.

Parameters	Model-LPs	Verified by large dataset	Error
CIc (KRW/m)	225,772	228,061	−1.01%
D (mm)	1272.8	1280.0	−0.57%
	0.0100	0.0100	0.00%

Table 10. Cost comparison between ANN-based hybrid Lagrange optimization and probable design.

Parameters	Model-LPs	Probable design	Saving
CIc (KRW/m)	225,772	470,508	51.53%