AI-based Lagrange optimization for designing reinforced concrete columns

Figures & data

Figure 1. Tansig activation function and its derivatives.

Figure 2. Size configuration of output vector, Jacobian, and Hessian of hidden layer l.

Figure 3. AI-based LaGrange optimization flowchart.

Table 1. Summary of RC column parameters.

RC COLUMN PARAMETERS
Seven inputs for engineering mechanic	$b$	- Column section width
	$h$	- Column section height
	${\rho }_s$	- Geometric rebar ratio
	$f_c^{\mathrm{\prime }}$	- Concrete strength
	$f_y$	- Rebar yield strength
	$P_u$	- Factored axial force
	$M_u$	- Factored bending moment
Nine corresponding outputs	$\varphi P_n$	- Design axial force
	$\varphi M_n$	- Design moment
	$SF$	- Safety factor ($\varphi M_n/M_u$)
	$b/h$	- Aspect ratio of column section
	${\epsilon }_s$	- Rebar strain
	$C{I}_c$	- Cost index of column per height
	$C{O}_2$	- CO2 emission per column height
	$W_c$	- Weight of column per column height
	${\alpha }_{e/h}$	- Angular eccentricity of applied load

Figure 4. RC column.

Table 2. Training accuracies based on PTM.

Training table: Forward – Lagrange 7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\mathrm{\prime }}$, $f_y$, $P_u$, $M_u$) – 9 Outputs ($\varphi P_n$, $\varphi M_n$, $SF$, $b/h$, ${\epsilon }_s$, $C{I}_c$, $C{O}_2$, $W_c$, ${\alpha }_{e/h}$)
No.	Data	Layers	Neu-rons	Best Epoch	MSE T. Perf	R at Best Epoch
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($\varphi P_n$) – PTM
1a	100,000	1	80	53,763	2.9.E-5	0.998
1b	100,000	2	80	74,692	2.3.E-6	1.000
1 c	100,000	5	80	57,688	7.7.E-7	1.000
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($\varphi M_n$) – PTM
2a	100,000	1	80	40,253	5.8.E-5	0.998
2b	100,000	2	80	38,900	4.6.E-6	1.000
2 c	100,000	5	80	34,897	2.2.E-6	1.000
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($SF$) – PTM
3	100,000	1	80	73,206	4.1.E-3	0.975
3a	100,000	2	80	86,964	4.5.E-4	0.999
3b	100,000	5	80	117,296	1.3.E-4	0.999
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($b/h$) – PTM
4a	100,000	1	80	100,000	1.8.E-7	1.000
4b	100,000	2	80	52,526	4.8.E-9	1.000
4 c	100,000	5	80	160,349	2.2.E-9	1.000
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output (${\epsilon }_s$) – PTM
5a	100,000	1	80	47,339	2.0.E-4	0.992
5b	100,000	2	80	150,505	3.7.E-5	0.999
5 c	100,000	5	80	45,505	2.4.E-5	0.999
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($C{I}_c$) – PTM
6a	100,000	1	80	100,000	1.3.E-7	1.000
6b	100,000	2	80	105,702	4.3.E-8	1.000
6 c	100,000	5	80	48,163	5.4.E-7	1.000
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($C{O}_2$) – PTM
7a	100,000	1	80	45,900	3.5.E-7	1.000
7b	100,000	2	80	19,721	5.4.E-7	1.000
7 c	100,000	5	80	26,948	1.3.E-6	1.000
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output ($W_c$) – PTM
8a	100,000	1	80	100,000	3.8.E-8	1.000
8b	100,000	2	80	48,972	6.6.E-8	1.000
8 c	100,000	5	80	21,920	6.7.E-7	1.000
7 Inputs ($b$, $h$, ${\rho }_s$, $f_c^{\prime }$, $f_y$, $P_u$, $M_u$) – 1 Output (${\alpha }_{e/h}$) – PTM
9a	100,000	1	80	53,951	6.3.E-4	0.996
9b	100,000	2	80	178,258	4.2.E-6	1.000
9 c	100,000	5	80	168,078	1.9.E-7	1.000

Table 3. Column design scenarios for optimization.

COLUMN DESIGN SCENARIO
Column configuration	Squared column: $b=h$
Material properties	Concrete: $f_{c^{\prime }}$= 40 MPa Rebar: $f_y$ = 500 MPa
Factored load	Axial force: $P_u$ = 1000 kN Bending moment $M_u$ = 3000 kN⋅m
Safety factor	$SF$ = 1
Design target	Minimized $\text{b}C{\text{b}I}_{\text{b}c}$ (KRW/m)

Table 4. Summary of equality and inequality constraints.

Equality constraint$\mathbf{c}\left(\mathbf{x}\right)={\left[{\text{b}c}_1\left(\mathbf{x}\right),\dots ,{\text{b}c}_6\left(\mathbf{x}\right)\right]}^{\text{b}T}$	Inequality constraints$\mathbf{v}\left(\mathbf{x}\right)={\left[{\text{b}v}_1\left(\mathbf{x}\right),{\text{b}v}_2\left(\mathbf{x}\right)\right]}^{\text{b}T}$
${\text{b}c}_1\left(\mathbf{x}\right)=\text{b}b-\text{b}h=0$	$v_1\left(\mathbf{x}\right)={\rho }_s-0.01\ge 0$
${\text{b}c}_2\left(\mathbf{x}\right)={\text{b}f}_{\text{b}c}^{\text{b}\prime }-40=0$	$v_2\left(\mathbf{x}\right)=-{\rho }_s+0.08\ge 0$
${\text{b}c}_3\left(\mathbf{x}\right)={\text{b}f}_{\text{b}y}^{\text{b}\hspace{0.17em}}-500=0$
${\text{b}c}_4\left(\mathbf{x}\right)={\text{b}P}_{\text{b}u}^{\text{b}\hspace{0.17em}}-1000=0$
${\text{b}c}_5\left(\mathbf{x}\right)={\text{b}M}_{\text{b}u}^{\text{b}\hspace{0.17em}}-3000=0$
${\text{b}c}_6\left(\mathbf{x}\right)=\text{b}SF-1=0$

Table 5. Lowest cost index of columns (CI_c) for given constraints based on forward networks

(a) Based on forward training with 1 layer – 80 neurons.
AI-based Lagrange Optimization (1 layer – 80 neurons)
Parameters		Case 1	Case 2	Case 3
1	b (mm)	1022.1	629.1	980.1
2	h (mm)	1022.1	629.1	980.1
3	ρs	0.0100	0.0800	0.0118
4	fc’ (MPa)	40	40	40
5	fy (MPa)	500	500	500
6	Pu (kN)	1000	1000	1000
7	Mu (kN·m)	3000	3000	3000
Objective: $C{I}_c$ (KRW/m)		185,069.2	299,994.8	184,684.6
(b) Based on forward training with 2 layers – 80 neurons AI-based Lagrange Optimization (2 layers – 80 neurons)
Parameters		Case 1	Case 2	Case 3
1	$b$ (mm)	1068.5	615.5	924.8
2	$h$ (mm)	1068.5	615.5	924.8
3	${\rho }_s$	0.0100	0.0800	0.0162
4	fc’ (MPa)	40	40	40
5	$f_y$ (MPa)	500	500	500
6	$P_u$ (kN)	1000	1000	1000
7	$M_u$ kN·m)	3000	3000	3000
Objective: $C{I}_c$ (KRW/m)		201,942.3	286,645.1	195,507.9
(c) Based on forward training with 5 layers – 80 neurons AI-based Lagrange Optimization (5 layers – 80 neurons)
Parameters		Case 1	Case 2	Case 3
1	$b$ (mm)	1070.1	619.1	919.8
2	$h$ (mm)	1070.1	619.1	919.8
3	${\rho }_s$	0.0100	0.0800	0.0166
4	fc’ (MPa)	40	40	40
5	$f_y$ (MPa)	500	500	500
6	$P_u$ (kN)	1000	1000	1000
7	$M_u$ kN·m)	3000	3000	3000
Objective: $C{I}_c$(KRW /m)		202,275.4	289,955.2	195,671.1

Note:Case 1 – Inequality $v_1\left(\mathbf{x}\right)$ is active

Case 2 – Inequality $v_2\left(\mathbf{x}\right)$ is active

Case 3 – None of inequalities are active

Table 6. Accuracies of optimized CI_c based on forward Lagrange optimizations (a) Based on forward training with 1 layer – 80 neurons.

OPTIMIMAL CIc DESIGN (based on 1 layer with 80 neurons)
Parameter		AI results	Check (AutoCol)	Error
1	$b$ (mm)	980.09	980.09	0.00%
2	$h$ (mm)	980.09	980.09	0.00%
3	${\rho }_s$	0.0118	0.0118	0.00%
4	$f_{c^{\prime }}$ (MPa)	40	40	0.00%
5	$f_y$ (MPa)	500	500	0.00%
6	$P_u$ (kN)	1000	1000	0.00%
7	$M_u$ (kN·m)	3000	3000	0.00%
8	$\varphi P_n$ (kN)	940.0	882.1	6.2%
9	$\varphi M_n$ (kN·m)	2922.1	2646.3	9.44%
10	$SF$	1.00	0.88	11.79%
11	$b/h$	1.00	1.00	0.00%
12	${\epsilon }_s$	0.0268	0.0270	−0.53%
13	$C{I}_c$ (KRW/m)	184,684.6	184,268.8	0.23%
14	$C{O}_2$ (t-CO2/m)	0.384	0.385	−0.21%
15	$W_c$ (kN/m)	2.263	2.263	−0.01%
16	${\alpha }_{e/h}$	0.344	0.316	8.09%
(b) Based on forward training with 2 layers – 80 neurons
OPTIMIMAL CIc DESIGN (based on 2 layers with 80 neurons)
Parameter		AI results	Check (AutoCol)	Error
1	$b$ (mm)	924.81	924.81	0.00%
2	$h$ (mm)	924.81	924.81	0.00%
3	${\rho }_s$	0.0162	0.0162	0.00%
4	$f_{c^{\prime }}$ (MPa)	40	40	0.00%
5	$f_y$ (MPa)	500	500	0.00%
6	$P_u$ (kN)	1000	1000	0.00%
7	$M_u$ (kN·m)	3000	3000	0.00%
8	$\varphi P_n$ (kN)	982.9	994.4	−1.2%
9	$\varphi M_n$ (kN·m)	3030.2	2983.1	1.55%
10	$SF$	1.00	0.99	0.56%
11	$b/h$	1.00	1.00	0.00%
12	${\epsilon }_s$	0.0220	0.0216	1.45%
13	$C{I}_c$ (KRW/m)	195,507.9	195,451.0	0.03%
14	$C{O}_2$ (t-CO2/m)	0.416	0.417	−0.35%
15	$W_c$ (kN/m)	2.015	2.015	−0.01%
16	${\alpha }_{e/h}$	0.299	0.299	−0.09%
(c) Based on forward training with 5 layers – 80 neurons
OPTIMIMAL CIc DESIGN (based on 2 layers with 80 neurons)
Parameter		AI results	Check (AutoCol)	Error
1	$b$ (mm)	919.75	919.75	0.00%
2	$h$ (mm)	919.75	919.75	0.00%
3	${\rho }_s$	0.0166	0.0166	0.00%
4	$f_{c^{\prime }}$ (MPa)	40	40	0.00%
5	$f_y$ (MPa)	500	500	0.00%
6	$P_u$ (kN)	1000	1000	0.00%
7	$M_u$ (kN·m)	3000	3000	0.00%
8	$\varphi P_n$ (kN)	1025.4	997.3	2.7%
9	$\varphi M_n$ (kN·m)	2998.7	2992.0	0.22%
10	$SF$	1.00	1.00	0.27%
11	$b/h$	1.00	1.00	0.00%
12	${\epsilon }_s$	0.0217	0.0213	1.93%
13	$C{I}_c$ (KRW/m)	195,671.1	195,792.1	−0.06%
14	$C{O}_2$ (t-CO2/m)	0.418	0.419	−0.15%
15	$W_c$ (kN/m)	19.924	19.930	−0.03%
16	${\alpha }_{e/h}$	0.297	0.297	−0.01%

Note: Seven inputs for AI design

Seven inputs for software check (AutoCol)

Figure 5. Verification of CI_c based on large datasets.

Figure 6. Axial force–moment (P–M) interaction diagrams for optimal CI_c.