Numerical analysis on displacements of steel-fiber-reinforced concrete beams

Mateus Fereira Pontes

Universidade Federal de Catalão – UFCAT, Departamento de Engenharia Civil, Catalão, GO, Brasil

Wanderlei Malaquias Pereira Junior

Universidade Federal de Catalão – UFCAT, Departamento de Engenharia Civil, Catalão, GO, Brasil

José Júlio de Cerqueira Pituba

Universidade Federal de Catalão – UFCAT, Departamento de Engenharia Civil, Catalão, GO, Brasil

Corresponding author: Wanderlei Malaquias Pereira Junior. E-mail: wanderlei_junior@ufg.br

Author contributions: J. J. C. Pituba: conceptualization, funding acquisition, supervision, writing, methodology, formal analysis; M. F. Pontes: writing, data curation, formal analysis; W. M. Pereira Junior: conceptualization, writing, methodology, data curation, formal analysis.

E (MPa)	A_t	A_c	B_t (MPa^-1)	B_c (MPa^-1)	Y_0t (MPa)	Y_0c (MPa)	β_t	β_c
31900	15	0.70	1290	2.50	0.000086	0.00455	0.00000045	0.000030

Fiber volume percentage V_f (%)	Peak Stress (MPa)	Yield stress (MPa)	Ultimate strain (m/m)
2.00	525.00	420.00	0.025
2.50	550.00	440.00	0.025

Test	h (cm)	b (cm)	L (cm)	As (cm²)	h/b
h0	25.00	12.50	230	4.022	2.00
h1	26.25	12.50	230	4.022	2.10
h2	28.13	12.50	230	4.022	2.25
h3	31.25	12.50	230	4.022	2.50
h4	34.38	12.50	230	4.022	2.75
h5	37.50	12.50	230	4.022	3.00

Test	h (cm)	bw (cm)	L (cm)	As (cm²)	Reinforcement area reduction (%)	1st load position (cm)	Distance between loads (cm)
L0	25.00	12.50	230.00	200.00	4.022	80.00	70.00
L1	26.25	12.50	280.00	250.00	4.022	97.50	85.00
L2	25.13	12.50	330.00	300.00	4.022	115.00	100.00
L3	31.25	12.50	380.00	350.00	4.022	130.00	120.00
L4	34.38	12.50	430.00	400.00	4.022	147.50	135.00
L5	37.50	12.50	480.00	450.00	4.022	165.00	150.00

Test	h (cm)	b (cm)	L (cm)	As (cm²)	Reinforced area reduction (%)
as0	25.00	12.50	230.00	4.02	-
as1	25.00	12.50	230.00	3.62	10%
as2	25.00	12.50	230.00	3.22	20%
as3	25.00	12.50	230.00	2.82	30%
as4	25.00	12.50	230.00	2.41	40%
as5	25.00	12.50	230.00	2.01	50%

Test	Maximum load (kN)	SLS Load (kN)	Increasing of SLS load (kN)	Increasing of SLS load (%)
h0-2.0%	186.17	74.46	----	----
h1-2.0%	200.42	80.16	14.25	7.66%
h2-2.0%	223.02	89.20	36.85	19.79%
h3-2.0%	262.35	104.94	76.18	40.92%
h4-2.0%	294.47	117.79	108.30	58.18%
h5-2.0%	338.00	135.20	151.83	81.56%
h0-2.5%	195.38	78.15	----	----
h1-2.5%	211.38	84.55	15.99	8.19%
h2-2.5%	236.59	94.63	41.21	21.10%
h3-2.5%	280.67	112.26	85.29	43.65%
h4-2.5%	327.76	131.10	132.38	67.76%
h5-2.5%	377.69	151.07	182.31	93.31%

Test	Load at 4 mm (kN)	Load at 8 mm (kN)	Load at 12 mm (kN)
h0-2.0%	99.50	178.42	185.24
h1-2.0%	112.76	193.42	199.72
h2-2.0%	134.86	217.36	221.62
h3-2.0%	177.09	258.02	258.18
h4-2.0%	214.35	290.77	285.80
h5-2.0%	269.41	334.93	324.97
h0-2.5%	97.68	177.79	193.73
h1-2.5%	111.30	199.52	211.06
h2-2.5%	133.98	226.77	235.82
h3-2.5%	177.59	274.22	279.33
h4-2.5%	228.45	324.22	321.53
h5-2.5%	288.17	374.58	363.91

Test	Maximum load (kN)	SLS load (kN)	Decreased load on SLS (kN)	Decreased load on SLS (%)
as0-2.0%	186.17	74.46	----	----
as1-2.0%	176.34	70.53	3.93	5.28%
as2-2.0%	165.71	66.28	8.18	10.99%
as3-2.0%	155.75	62.30	12.16	16.34%
as4-2.0%	144.96	57.98	16.48	22.14%
as5-2.0%	134.21	53.68	20.78	27.91%
as0-2.5%	195.38	78.15	----	----
as1-2.5%	185.89	74.35	3.79	4.86%
as2-2.5%	176.47	70.58	7.56	9.68%
as3-2.5%	167.13	66.85	11.30	14.46%
as4-2.5%	156.62	62.64	15.50	19.84%
as5-2.5%	147.19	58.87	19.27	24.66%

Test	Load at 4 mm (kN)	Load at 8 mm (kN)	Load at 12 mm (kN)
as0-2.0%	99.50	178.43	185.24
as1-2.0%	95.60	168.76	175.06
as2-2.0%	91.65	158.40	165.16
as3-2.0%	87.61	148.68	155.35
as4-2.0%	83.31	139.58	144.67
as5-2.0%	78.91	129.15	132.97
as0-2.5%	97.68	177.79	193.73
as1-2.5%	94.28	170.35	184.85
as2-2.5%	90.80	163.88	175.36
as3-2.5%	87.28	156.24	166.13
as4-2.5%	83.44	146.81	155.95
as5-2.5%	79.60	137.37	145.77

Test	Maximum load (kN)	SLS load (kN)	Decreased load on SLS (kN)	Decreased load on SLS (%)
L0-2.0%	186.17	74.47	----	----
L1-2.0%	146.66	58.66	15.81	21.23%
L2-2.0%	121.03	48.41	26.06	34.99%
L3-2.0%	105.20	42.08	32.39	43.49%
L4-2.0%	88.84	35.54	38.93	52.28%
L5-2.0%	69.01	27.61	46.86	62.93%
L0-2.5%	195.38	78.15	----	----
L1-2.5%	160.51	64.20	13.95	17.85%
L2-2.5%	136.17	54.47	23.68	30.30%
L3-2.5%	118.27	47.31	30.85	39.47%
L4-2.5%	102.66	41.06	37.09	47.46%
L5-2.5%	85.71	34.28	43.87	56.13%

Test	Load at 4 mm (kN)	Load at 8 mm (kN)	Load at 12 mm (kN)
L0-2.0%	95.50	178.43	185.24
L1-2.0%	56.06	97.02	136.98
L2-2.0%	37.27	59.32	83.2
L3-2.0%	27.52	40.8	55.36
L4-2.0%	20.57	30.03	39.18
L5-2.0%	15.35	23.32	29.5
L0-2.5%	97.68	177.79	193.73
L1-2.5%	59.35	103.65	147.09
L2-2.5%	40.74	66.47	94.02
L3-2.5%	30.18	46.29	63.98
L4-2.5%	23.28	34.50	46.16
L5-2.5%	17.95	27.09	34.99

Test	Load at 4 mm (kN)	Load reduction (kN)	Load at 8 mm (kN)	Load reduction (kN)	Load at 12 mm (kN)	Load reduction (kN)
L0-2.0%	95.50	---	178.43	---	185.24	---
L1-2.0%	56.06	39.44	97.02	81.41	136.98	48.26
L2-2.0%	37.27	58.23	59.32	119.10	83.20	102.00
L3-2.0%	27.52	67.98	40.80	137.63	55.36	129.88
L4-2.0%	20.57	74.93	30.03	148.40	39.18	146.06
L5-2.0%	15.35	80.15	23.32	155.11	29.50	155.74
L0-2.5%	97.68	--	177.70	---	193.70	---
L1-2.5%	59.35	38.33	103.65	74.14	147.09	46.64
L2-2.5%	40.74	56.94	66.47	111.30	94.02	99.71
L3-2.5%	30.18	67.50	46.29	131.50	63.98	129.70
L4-2.5%	23.28	74.40	34.50	143.29	46.16	147.57
L5-2.5%	17.95	79.73	27.09	150.70	34.99	158.74

E_{T} =

λ_{11} [I \otimes I] + 2 μ_{1} [I \bar{\underline{\otimes}} I]

- λ_{22}^{+} (D_{1}, D_{4}, D_{5})

[A \otimes A]

- λ_{12}^{+} (D_{1}) [A \otimes I + I \otimes A]

- μ_{2} (D_{4}, D_{5})

[A \bar{\underline{\otimes}} I + I \bar{\underline{\otimes}} A]

E_{C} =

λ_{11} [I \otimes I] + 2 μ_{1} [I \bar{\underline{\otimes}} I]

- λ_{22}^{-} (D_{2}, D_{3}, D_{4}, D_{5})

- λ_{12}^{-} (D_{2}, D_{3})

[A \otimes I + I \otimes A] - λ_{11}^{-} (D_{3}) [I \otimes I] - \frac{(1 - 2 ν_{0})}{ν_{0}} λ_{11}^{-} (D_{3}) [I \otimes I] - μ_{2} (D_{4}, D_{5}) [A \bar{\underline{\otimes}} I + I \bar{\underline{\otimes}} A]

W_{e}^{*}_{+} = \frac{σ_{11}^{2}}{2 E_{0} {(1 - D_{1})}^{2}} + \frac{(σ_{22}^{2} + σ_{33}^{2})}{2 E_{0}} - \frac{ν_{0} (σ_{11} σ_{22} + σ_{11} σ_{33})}{E_{0} (1 - D_{1})} - \frac{ν_{0} σ_{22} σ_{33}}{E_{0}}

+ \frac{(1 + ν_{0})}{E_{0} {(1 - D_{4})}^{2} {(1 - D_{5})}^{2}} (σ_{12}^{2} + σ_{13}^{2}) + \frac{(1 + ν_{0})}{E_{0}} σ_{23}^{2}

W_{e}^{*}_{-} = \frac{σ_{11}^{2}}{2 E_{0} {(1 - D_{2})}^{2}} + \frac{(σ_{22}^{2} + σ_{33}^{2})}{2 E_{0} {(1 - D_{3})}^{2}} - \frac{ν_{0} (σ_{11} σ_{22} + σ_{11} σ_{33})}{E_{0} (1 - D_{2}) (1 - D_{3})} - \frac{ν_{0} σ_{22} σ_{33}}{E_{0} {(1 - D_{3})}^{2}}

+ \frac{(1 + ν_{0})}{E_{0} {(1 - D_{4})}^{2} {(1 - D_{5})}^{2}} (σ_{12}^{2} + σ_{13}^{2}) + \frac{(1 + ν_{0})}{E_{0}} σ_{23}^{2}

Brasil