Optimum design of prestressed steel beams via genetic algorithm

Netto, Protáze Mageveske; Calenzani, Adenílica Fernanda Grobério; Alves, Élcio Cassimiro

doi:10.1590/0370-44672022760010

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Civil Engineering • REM, Int. Eng. J. 76 (1) • Jan-Mar 2023 • https://doi.org/10.1590/0370-44672022760010 copy

Optimum design of prestressed steel beams via genetic algorithm

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

The objective of this article is to present an optimization problem formulation to reduce the total structural cost of prestressed doubly-symmetric and monosymmetric I-shaped steel simply supported beams with straight tendons. The optimization problem was implemented via MATLAB’s native Genetic Algorithm. The validation and evaluation processes adopted two examples from literature. The design method follows the Brazilian standard NBR 8800:2008 for the Ultimate and Serviceability Limit States. The best result was found for a monosymmetric case by up to 20.00% and 25.70%. Saving in material weight and installation of tendons, without exceeding the security limits, was also effective. Furthermore, the results presented an efficient alternative for structural engineering, providing a significant model for similar analyzes.

Keywords:
prestressing; steel beams; genetic algorithm; optimization

Diameter	9.5 mm	15.2 mm
Ct_s	R$ 12.88
Ct_t	R$ 12.69
Ct_ti	R$ 125.66	R$ 158.65
µ_t	0.004158 kN/m	0.012152 kN/m

Internal forces	Distributed load (q)	Concentrated load (P )	Pretension load (F )
Bending moment	$\frac{q L^{2}}{8}$	$\frac{P a}{2}, a < \frac{L}{2}$ $\frac{P (L - a)}{2}, a \leq \frac{L}{2}$	Fe
Shear force	$\frac{q L}{2}$	$\frac{P a}{L}$	-
Normal force	-	-	F

Variable	Lower limit	Upper limit
t_fs and t_fi [cm]	1.6	4.44
b_fs and b_fi [cm]	10	55
d [cm]	55	200
n_t [units]	0	20

Example	d (mm)	b_fi (mm)	b_fs (mm)	t_fi (mm)	t_fs (mm)	t_w (mm)	N_tendons	Ix(e+4mm⁴)	δ(mm)	σ_c (MPa)	σ_t (MPa)	Total Cost (R$)
Abbas et al. (2018)ABBAS, A. L.; MOHAMMED, A. H.; KHALAF, R. D.; ABDUL-RAZZAQ, K. S. Finite element analysis and optimization of steel girders with external prestressing. Civil Engineering Journal, v. 4, p. 1490-1500, 2018.	984.3	390.8	391.8	20.80	22.60	10.40	3	465231.00	56.83	-193.40	147.20	60240.10
MS0	1200	250	300	16.00	16.01	10.40	7	445518.29	38.68	-196.44	169.20	48291.32
MS5	1160	290	210	16.00	25.24	10.40	5	443415.54	46.24	-199.95	184.46	49198.32
MS10	1320	160	330	16.00	16.06	10.40	7	504286.07	32.51	-162.01	186.57	48970.94
MS15	1140	290	170	16.00	32.28	10.40	7	427262.86	44.74	-200.00	179.67	49475.77
MS20	1200	260	300	16.00	16.16	10.40	7	452928.76	42.21	-198.00	178.50	48745.42
DS0	1080	270	270	20.80	20.80	10.40	5	412110.28	56.86	-194.06	159.60	50226.59
DS5	1080	270	270	20.80	20.80	10.40	5	412110.28	57.54	-195.28	162.55	50226.59
DS10	1080	270	270	20.80	20.80	10.40	5	412110.28	58.22	-196.50	165.49	50226.59
DS15	1080	270	270	20.80	20.80	10.40	6	412110.28	56.60	-193.62	158.47	50470.70
DS20	1080	270	270	20.80	20.80	10.40	6	412110.28	57.42	-195.08	162.00	50470.70

Example	d (mm)	b_fi (mm)	b_fs (mm)	t_fi (mm)	t_fs (mm)	t_w (mm)	N_tendons	Ix(e+4mm⁴)	δ(mm)	σ_c (MPa)	σ_t (MPa)	Total Cost (R$)
Ferreira (2007)FERREIRA, A. C. Prestressed steel beams: static, modal and rupture analysis of the prestressing cable and computational application for projects. Tese (Doutorado em Engenharia Civil). Universidade de Brasília. Departamento de Engenharia Civil e Ambiental, 2007. ( em portugues).	1000	380	500	32.00	44.40	16.00	18	853611.00	58.11	-169.84	328.18	134317.52
MS01	1630	410	190	16.00	39.82	16.00	10	1425607.95	55.20	-344.96	271.90	104810.90
MS02	1620	410	180	16.00	44.40	16.00	9	1423223.71	59.51	-343.50	288.69	104755.85
MS03	1640	550	230	16.00	39.82	12.50	8	1581053.44	57.71	-331.97	260.59	99801.48
MS04	1680	530	200	16.00	44.40	12.50	8	1623991.92	55.56	-330.14	257.95	99407.54
MS05	1620	410	210	16.00	37.50	16.00	9	1426673.17	59.34	-342.80	287.76	104739.17
MS06	1640	540	240	16.00	37.50	12.50	9	1563784.41	54.45	-331.21	250.88	99613.63
MS07	1610	420	250	16.00	31.50	16.00	9	1425142.96	59.65	-342.41	286.06	104981.83
MS08	1660	530	280	16.00	31.50	12.50	9	1594697.76	52.95	-327.36	248.46	99575.71
MS09	1630	410	300	16.00	25.00	16.00	9	1444331.46	58.50	-342.90	283.31	104701.26
MS10	1660	530	350	16.00	25.00	12.50	9	1599878.18	52.74	-326.19	247.61	99604.15
MS11	1610	420	340	16.00	22.40	16.00	9	1423455.23	59.78	-344.34	285.02	104695.19
MS12	1670	540	390	16.00	22.40	12.50	8	1636304.76	55.25	-326.73	254.04	99827.13
DS01	1210	310	310	39.82	39.82	16.00	18	1038055.18	69.61		108.77	118056.96
DS02	1170	300	300	44.40	44.40	16.00	19	1012765.33	69.49	-248.03	98.74	121553.88
DS03	1260	320	320	39.82	39.82	12.50	16	1120211.86	69.95	-257.31	120.02	110545.52
DS04	1190	310	310	44.40	44.40	12.50	18	1042745.55	70.17	-251.09	100.59	114313.04
DS05	1240	310	310	37.50	37.50	16.00	17	1051585.88	71.34	-262.31	122.19	115270.03
DS06	1290	330	330	37.50	37.50	12.50	15	1157792.70	70.48	-260.81	131.13	109230.62
DS07	1320	330	330	31.50	31.50	16.00	15	1127893.83	71.20	-270.58	143.96	111668.59
DS08	1390	350	350	31.50	31.50	12.50	13	1260938.21	68.85	-267.51	151.34	104840.53
DS09	1400	360	360	25.00	25.00	16.00	14	1178925.00	68.99	-276.61	154.55	107825.48
DS10	1520	390	390	25.00	25.00	12.50	10	1420564.06	68.98	-278.78	187.63	101257.04
DS11	1470	370	370	22.40	22.40	16.00	12	1254444.70	70.95	-287.36	182.15	106163.28
DS12	1580	410	410	22.40	22.40	12.50	9	1491047.92	68.26	-282.45	199.72	99903.72

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