Optimization of slender reinforced concrete columns subjected to biaxial bending using genetic algorithms

Susana de Lima Pires

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Departamento de Estruturas, Campinas, SP, Brasil

Maria Cecilia Amorim Teixeira da Silva

Universidade Estadual de Campinas – UNICAMP, Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Departamento de Estruturas, Campinas, SP, Brasil

Corresponding author: Maria Cecilia Amorim Teixeira da Silva. E-mail: mcatds07@gmail.com

Author contributions: SLP: conceptualization, methodology, formal analysis, development of the computational tools, writing; MCATS: conceptualization, methodology, data curation, writing, supervision.

Integer	Binary number				b	dicam13	ncam13	ncam24
					h	dicam24
					(cm)	(mm)
0	0	0	0	0	20	10	2	0
1	0	0	0	1	25	12.5	3	1
2	0	0	1	0	30	16	4	2
3	0	0	1	1	35	20	5	3
4	0	1	0	0	40	22	6	4
5	0	1	0	1	45	25	7	5
6	0	1	1	0	50	32	8	6
7	0	1	1	1	55	40	9	7
8	1	0	0	0	60
9	1	0	0	1	65
10	1	0	1	0	70
11	1	0	1	1	75
12	1	1	0	0	80
13	1	1	0	1	85
14	1	1	1	0	90
15	1	1	1	1	95

Col.	b	h	ρ	L	f_c	f_y	P_u,t	P_u,a	Δ
Col.	(cm)	(cm)	ρ	(cm)	(MPa)	(MPa)	(kN)	(kN)	Δ
10L4-1	8	8	3.95%	24	25.5	387	109.5	98	0.89
10L4-2	8	8	3.95%	24	25.5	387	109.3	98	0.90
60L2-1	8	8	1.98%	144	25.5	387	63.7	62	0.97
60L2-2	8	8	1.98%	144	25.5	387	65.7	62	0.94
100L2-1	8	8	1.98%	237	25.5	387	38.2	38	0.99
100L2-2	8	8	1.98%	237	25.5	387	35.0	38	1.09
100L4-1	8	8	3.95%	237	25.5	387	49.0	48	0.98
100L4-2	8	8	3.95%	237	25.5	387	47.0	48	1.02

TOTAL SEARCH Results - Processing time: 18 minutes
C_global		Width		Height		Diameter		Diameter	Number of bars	Number of bars
(R$)		(cm)		(cm)		[layers 1 and 3]		[layers 2 and 4]	[layers 1 and 3]	[layers 2 and 4]
						(mm)		(mm)	(un)	(un)
439,79		40		20		10		0	3	0
GENETIC ALGORITHM Results (using Penalty Method) - Processing time - 12 seconds
C_min	E_rel		Width		Height		Diameter	Diameter	Number of bars	Number of bars
(R$)			(cm)		(cm)		[layers 1 and 3]	[layers 2 and 4]	[layers 1 and 3]	[layers 2 and 4]
							(mm)	(mm)	(un)	(un)
439.79	0%		40		20		10	10	2	1
447.94	2%		35		25		10	16	3	0
439.79	0%		40		20		10	40	3	0
447.94	2%		35		25		10	10	2	1
453.87	3%		30		30		12.5	0	2	0
439.79	0%		40		20		10	16	3	0
439.79	0%		40		20		10	0	3	0
439.79	0%		40		20		10	20	3	0
439.79	0%		40		20		10	10	3	0
439.79	0%		40		20		10	0	3	0
GENETIC ALGORITHM Results (using Death Penalty Method) - Processing time: 10 seconds
C_min	E_rel		Width		Height		Diameter	Diameter	Number of bars	Number of bars
(R$)			(cm)		(cm)		[layers 1 and 3]	[layers 2 and 4]	[layers 1 and 3]	[layers 2 and 4]
							(mm)	(mm)	(un)	(un)
443.00	1%		40		20		12.5	0	2	0
439.79	0%		40		20		10	12.5	3	0
447.94	2%		35		25		10	0	3	3
439.79	0%		40		20		10	10	3	0
443.00	1%		40		20		12.5	0	2	0
439.79	0%		40		20		10	0	3	0
439.79	0%		40		20		10	0	3	0
439.79	0%		40		20		10	0	3	0
450.66	2%		30		30		10	12.5	3	0
439.79	0%		40		20		10	0	3	0

Column	l (cm)	h (cm)	b (cm)	$λ_{y}$	$λ_{z}$	Diameter Layers 1 and 3 (mm)	Number of bars 1+3 (un.)	Diameter Layers 2 and 4 (mm)	Number of bars 2+4 (un.)	Column Cost (R$)
2.1	300	25	40	23.09	41.57	10	10	0	0	546.40
2.2	400	25	45	30.75	55.43	16	16	0	0	767.34
2.3	500	25	50	34.74	69.28	10	10	10	4	972.20
2.4	600	25	55	37.79	83.14	10	10	10	4	1239.78
2.5	700	25	60	40.41	96.99	10	10	10	4	1531.73
2.6	800	25	60	46.19	110.85	10	10	12.5	6	1934.54
2.7	900	25	55	56.69	124.71	16	16	12.5	12	2667.97
2.8	1000	25	55	62.98	138.56	12.5	12.5	16	14	3524.93
2.9	1100	25	55	69.28	152.42	10	10	20	14	4720.39
2.1	1200	25	55	75.58	166.28	10	10	25	12	6125.08
2.11	1300	25	50	90.07	180.13	10	10	32	10	8001.14
2.12	1400	25	50	96.99	193.99	10	10	32	12	9843.42

Column	$λ_{Y}$	$λ_{Z}$	$e_{1 y}$	$e_{1 z}$	$α_{b}$	$λ_{1 Y}$	$λ_{1 Z}$	$\begin{matrix} A_{s t o t a l} \end{matrix}$	$A_{c}$	$ρ$	Group
Column	$λ_{Y}$	$λ_{Z}$	$e_{1 y}$	$e_{1 z}$	$α_{b}$	$λ_{1 Y}$	$λ_{1 Z}$	$(c m^{2})$	$(c m^{2})$	$ρ$	Group
2.1	23.09	41.57	0.57	1.05	1	35	35	4.71	1000	0.47%	2
2.2	30.75	55.43	0.57	1.05	1	35	35	8.04	1125	0.71%	2
2.3	34.74	69.28	0.57	1.05	1	35	35	6.28	1250	0.50%	2
2.4	37.79	83.14	0.57	1.05	1	35	35	6.28	1375	0.46%	2
2.5	40.41	96.99	0.57	1.05	1	35	35	6.28	1500	0.42%	3
2.6	46.19	110.85	0.57	1.05	1	35	35	10.50	1500	0.70%	3
2.7	56.69	124.71	0.57	1.05	1	35	35	22.77	1375	1.66%	3
2.8	62.98	138.56	0.57	1.05	1	35	35	33.06	1375	2.40%	3
2.9	69.28	152.42	0.57	1.05	1	35	35	47.12	1375	3.43%	4
2.1	75.58	166.28	0.57	1.05	1	35	35	62.05	1375	4.51%	4
2.11	90.07	180.13	0.57	1.05	1	35	35	83.57	1250	6.69%	4
2.12	96.99	193.99	0.57	1.05	1	35	35	99.65	1250	7.97%	4

Col.	b	h	L	dicam13	dicam24	ncam13	ncam24	c	f_c	f_y	P_u,t	P_u,a	Δ
Col.	(cm)	(cm)	(cm)	(mm)	(mm)	ncam13	ncam24	(cm)	(MPa)	(MPa)	(kN)	(kN)	Δ
23	12	12	240	12	0	2	0	1.5	43	684	320	305	0.95
24	12	12	240	12	0	2	0	1.5	43	636	288	305	1.06
27	20	20	300	16	0	2	0	1.5	33	636	990	1019	1.03
28	20	20	300	16	0	2	0	1.5	33	636	990	1019	1.03
32	20	20	400	16	0	2	0	1.5	37	636	920	981	1.07

Brasil