M-N interaction curves for rectangular concrete-filled steel tube columns subjected to uniaxial bending moments

Fernandes, Matheus; De Nardin, Silvana; Almeida Filho, Fernando Menezes de

doi:10.1590/S1983-41952022000100008

Matheus Fernandes

Universidade Federal de São Carlos – UFSCar, Programa de Pós-graduação em Engenharia Civil, São Carlos, SP, Brasil

https://orcid.org/0000-0003-1408-1027

Silvana De Nardin

Universidade Federal de São Carlos – UFSCar, Programa de Pós-graduação em Engenharia Civil, São Carlos, SP, Brasil

https://orcid.org/0000-0002-8736-4987

Fernando Menezes de Almeida Filho

Universidade Federal de São Carlos – UFSCar, Programa de Pós-graduação em Engenharia Civil, São Carlos, SP, Brasil

https://orcid.org/0000-0002-6927-6377

Corresponding author: Matheus Fernandes. E-mail: matheustkful@gmail.com

Conflict of interest: Nothing to declare.

Author contributions: MF: conceptualization, writing, formal analysis, methodology; SDN and FMAF: conceptualization, supervision, writing, formal analysis, methodology.

Editors: Bernardo Horowitz, Guilherme Aris Parsekian.

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Figure 1
Models for M-N interaction: ABNT NBR 8800 [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].] and ABNT NBR 16239 [²¹21 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings Using Hollow Sections, NBR 16239:2013, 2013. [in Portuguese].].

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Figure 2
Model II for M-N interaction. Adapted from EN 1994-1-1 [²²22 European Committee for Standardization, Eurocode 4: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1994-1-1, 2004.].

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Figure 3
Resulting forces on components of rectangular composite cross-section.

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Figure 4
Strain limits for strain regions of composite cross-section.

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Figure 5
Reduction of plastic resistance to axial force due to flexural buckling.

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Figure 6
Iterative process of computational code.

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Figure 7
Modified interaction curves for Model II [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].] and Eurocode [²²22 European Committee for Standardization, Eurocode 4: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1994-1-1, 2004.].

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Figure 8
Comparison of parabolic and simplified M-N interaction curves with results of Du et al. [¹⁸18 Y. Du, Z. Chen, Y.-B. Wang, and J. Y. Richard Liew, "Ultimate resistance behavior of rectangular concrete-filled tubular beam-columns made of high-strength steel," J. Construct. Steel Res., vol. 133, pp. 418–433, 2017.].

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Figure 9
Comparison of parabolic and simplified M-N interaction curves with results of Melo [¹⁷17 R. A. Melo, “Numerical analysis of the square concrete filled steel tube composite columns and under eccentric loads,” M.S. thesis, Civ. Eng. Dep., Fed. Univ. São Carlos, São Carlos, 2018. [in Portuguese].].

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Figure 10
Effects of concrete strength on M-N interaction curves.

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Figure 11
Effect of yielding strength of steel on M-N interaction curves.

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Figure 12
Effect of steel ratio on M-N interaction curves.

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Figure 13
Effect of variables on ultimate values of force and moment.

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Figure 14
Behavior of the M-N interaction curves for the highest values of concrete strength and yielding strength of steel.

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Table 1
M-N pair of cross-section strength.

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Table 2
Ultimate values-concrete compressive strains. Adapted from ABNT NBR 6118 [²⁹29 Associação Brasileira de Normas Técnicas, Design of Concrete Structures – Procedure, NBR 6118:2014, 2008. [in Portuguese].].

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Table 3
Strains in cross-section according to strain region.

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Table 4
M-N pair of strength including the reduction effects due to buckling mode.

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Table 5
Validation of computational code: specimens’ details of refs [¹⁷17 R. A. Melo, “Numerical analysis of the square concrete filled steel tube composite columns and under eccentric loads,” M.S. thesis, Civ. Eng. Dep., Fed. Univ. São Carlos, São Carlos, 2018. [in Portuguese].], [¹⁸18 Y. Du, Z. Chen, Y.-B. Wang, and J. Y. Richard Liew, "Ultimate resistance behavior of rectangular concrete-filled tubular beam-columns made of high-strength steel," J. Construct. Steel Res., vol. 133, pp. 418–433, 2017.].

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Table 6
Estimated values of M-N pairs of strength vs. results of Du et al. [¹⁸18 Y. Du, Z. Chen, Y.-B. Wang, and J. Y. Richard Liew, "Ultimate resistance behavior of rectangular concrete-filled tubular beam-columns made of high-strength steel," J. Construct. Steel Res., vol. 133, pp. 418–433, 2017.].

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Table 7
Estimated values of M-N pairs of strength vs. results of Melo [¹⁷17 R. A. Melo, “Numerical analysis of the square concrete filled steel tube composite columns and under eccentric loads,” M.S. thesis, Civ. Eng. Dep., Fed. Univ. São Carlos, São Carlos, 2018. [in Portuguese].].

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Table 8
Influence of concrete strength on M-N pair of strength.

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Table 9
Influence of yielding strength of steel on M-N pair of strength.

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Table 10
Influence of steel ratio on M-N pair of strength.

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Table 11
M-N pairs of strength for the highest values of concrete strength and yielding strength of steel.

(1)

(2)

(5)

(6)

(7)

(8)

(9)

Plastic resistance to compression, N_pl,Rd
$N_{p l, R d} = λ (x - t) (b_{2} - 2 t) (- α_{c} f_{c d}) - σ_{a_{1}} A_{a_{1}} + σ_{a_{2}} A_{a_{2}} - \sum_{i = 1}^{n} 2 σ_{a_{i c}} A_{a_{i c}} + \sum_{i = 1}^{n} 2 σ_{a_{i t}} A_{a_{i t}}$	(3)
Resistance moment, M_Rd
$M_{R d} = λ (x - t) (b_{2} - 2 t) (\frac{b_{1}}{2} - \frac{λ (x - t)}{2} - t) (- α_{c} f_{c d}) - σ_{a_{1}} A_{a_{1}} y_{a}_{1} + σ_{a_{2}} A_{a_{2}} y_{a_{2}} - \sum_{i = 1}^{n} 2 σ_{a_{i c}} A_{a_{i c}} y_{a}_{i c} + \sum_{i = 1}^{n} 2 σ_{a_{i t}} A_{a_{i t}} y_{a}_{i t}$	(4)

Concrete strength classes	Concrete strain
C20-C50	$ε_{c u} = - 0,35 %$ ; $ε_{c 2} = - 0,2 %$
C55-C90	$ε_{c u} = - (0,26 % + 3,5 % {[\frac{90 - f_{c}}{100}]}^{4})$
C55-C90	$ε_{c 2} = - (0,2 % + 0,0085 % {(f_{c} - 50)}^{0,53})$

Cross-section		Strain in the layer i
Region I		$ε_{i} = \frac{1 % (d_{i} - x)}{d_{s} - x}$
Region II		$ε_{i} = \frac{- ε_{c u} (d_{i} - x)}{x - t}$
Region III		$ε_{i} = \frac{- ε_{c 2} (x - d_{i})}{x - t - \frac{ε_{c u} - ε_{c 2}}{ε_{c u}} (h - t)}$

Resistance to axial force, N_Rd
$N_{R d} = [λ (x - t) (b_{2} - 2 t) (- α_{c} f_{c d}) - σ_{a_{1}} A_{a_{1}} + σ_{a_{2}} A_{a_{2}} - \sum_{i = 1}^{n} 2 σ_{a_{i c}} A_{a_{i c}} + \sum_{i = 1}^{n} 2 σ_{a_{i t}} A_{a_{i t}}] χ$	(10)
Resistance moment, M_Rd
$M_{R d} = [λ (x - t) (b_{2} - 2 t) (\frac{b_{1}}{2} - \frac{λ (x - t)}{2} - t) (- α_{c} f_{c d}) - σ_{a_{1}} A_{a_{1}} y_{a}_{1} + σ_{a_{2}} A_{a_{2}} y_{a_{2}} - \sum_{i = 1}^{n} 2 σ_{a_{i c}} A_{a_{i c}} y_{a}_{i c} + \sum_{i = 1}^{n} 2 σ_{a_{i t}} A_{a_{i t}} y_{a}_{i t}] μ$	(11)

Author	Specimen	H × B × t (mm)	f_y (MPa)	f_c (MPa)	L_e (mm)	λ_o	e (mm)	N_exp (kN)	M_exp (kNcm)
Du et al. [¹⁸18 Y. Du, Z. Chen, Y.-B. Wang, and J. Y. Richard Liew, "Ultimate resistance behavior of rectangular concrete-filled tubular beam-columns made of high-strength steel," J. Construct. Steel Res., vol. 133, pp. 418–433, 2017.]	D1	150×150×8,28	488,4	55,3	1180	0,33	0	2947	0
	D2						45	1672,5	8770
	D3						150	807,1	12970
	D4						1200	108,5	13220
Melo [¹⁷17 R. A. Melo, “Numerical analysis of the square concrete filled steel tube composite columns and under eccentric loads,” M.S. thesis, Civ. Eng. Dep., Fed. Univ. São Carlos, São Carlos, 2018. [in Portuguese].]	M1	150×150×3	250	40	1240	0,29	20	966	2445,9
	M2						30	858,3	3116,8
	M3						40	765,8	3716,0
Le: effective length

Model	D1		D2		D3		D4
Model	N (kN)	M (kNm)	N (kN)	M (kNm)	N (kN)	M (kNm)	N (kN)	M (kNm)
Du et al. [¹⁸18 Y. Du, Z. Chen, Y.-B. Wang, and J. Y. Richard Liew, "Ultimate resistance behavior of rectangular concrete-filled tubular beam-columns made of high-strength steel," J. Construct. Steel Res., vol. 133, pp. 418–433, 2017.]	2947	0	1672,5	87,7	807,1	129,7	108,5	132,2
Proposed model	2855,41 (- 3,1%)	0	1451,03 (- 13,2%)	76,05 (- 13,3%)	671,17 (- 16,8%)	107,61 (- 17%)	93,15 (- 14,1%)	115,48 (- 12,6%)
Model II [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].]	3107,24 (+5,4%)	0	1628,8 (- 2,6%)	85,1 (-3%)	819,18 (+1,5%)	131,71 (+1,5%)	107,67 (- 0,8%)	132,82 (+0,5%)
Eurocode [²²22 European Committee for Standardization, Eurocode 4: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1994-1-1, 2004.]	3251,03 (+10,3%)	0	1711,07 (+2,3%)	89,39 (+1,9%)	834,62 (+3,4%)	134,2 (+3,5%)	108,75 (+0,2%)	134,19 (+1,5%)
Model I [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].]	2969,18 (+0,8%)	0	1365,34 (- 18,4%)	71,98 (- 17,9%)	645,05 (- 20,1%)	104,55 (- 19,4%)	91,9 (- 15,3%)	116,93 (- 11,6%)
Modified Model II [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].]	2969,18 (+0,8%)	0	1505,33 (- 10%)	79,35 (- 9,5%)	733,11 (- 9,2%)	118,85 (- 8,4%)	93,36 (- 14%)	118,85 (- 10,1%)
Modified EN-1994-1-1 [²²22 European Committee for Standardization, Eurocode 4: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1994-1-1, 2004.]	3136 (+6,4%)	0	1588,31 (- 5%)	83,73 (- 4,5%)	739,68 (- 8,4%)	119,93 (- 7,5%)	94,02 (- 13,3%)	119,93 (- 9,3%)
ABNT NBR 16239 [²¹21 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings Using Hollow Sections, NBR 16239:2013, 2013. [in Portuguese].]	3098,30 (+5,1%)	0	1539,70 (- 7,9%)	80,70 (- 7,9%)	733,11 (- 9,2%)	118,85 (- 8,4%)	93,36 (- 14%)	118,85 (- 10,1%)

Model	M1		M2		M3
Model	N (kN)	M (kNm)	N (kN)	M (kNm)	N (kN)	M (kNm)
Melo [¹⁷17 R. A. Melo, “Numerical analysis of the square concrete filled steel tube composite columns and under eccentric loads,” M.S. thesis, Civ. Eng. Dep., Fed. Univ. São Carlos, São Carlos, 2018. [in Portuguese].]	966	24,46	858,3	31,17	765,8	37,16
Proposed model	763,56 (-21%)	19,45 (-20,5%)	663,48 (-22,7%)	24,05 (-22,8%)	575,56 (-24,8%)	27,91 (-24,9%)
Model II [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].]	824,52 (-14,6%)	20,98 (-14,2%)	737,48 (-14,1%)	26,74 (-14,2%)	632,35 (-17,4%)	30,71 (-17,4%)
Eurocode [²²22 European Committee for Standardization, Eurocode 4: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1994-1-1, 2004.]	917,43 (-5%)	23,29 (-4,8%)	815,33 (-5%)	29,63 (-4,9%)	677,68 (-11,5%)	32,91 (-11,4%)
Model I [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].]	564,95 (-41,5%)	14,32 (-41,5%)	466,03 (-45,7%)	16,97 (-45,6%)	389,93 (-49,1%)	18,98 (-48,9%)
Modified Model II [²⁰20 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings, NBR 8800:2008, 2008. [in Portuguese].]	780,82 (-19,2%)	19,83 (-18,9%)	692,34 (-19,3%)	25,20 (-19,2%)	560,31 (-26,8%)	27,34 (-26,4%)
Modified EN-1994- 1- 1 [²²22 European Committee for Standardization, Eurocode 4: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1994-1-1, 2004.]	868,98 (-10%)	22,11 (-9,6%)	745,36 (-13,2%)	27,17 (-12,8%)	594,73 (-22,3%)	29,02 (-21,9%)
ABNT NBR 16239 [²¹21 Associação Brasileira de Normas Técnicas, Design of Steel and Composite Structures for Buildings Using Hollow Sections, NBR 16239:2013, 2013. [in Portuguese].]	801,57 (-17,0%)	20,46 (-16,3%)	708,64 (-17,4%)	25,97 (-16,7%)	539,73 (-29,5%)	26,37 (-29,0%)

Concrete (MPa)	Point A	Point B	Point D
Concrete (MPa)	N (kN)	M (kNm)	N (kN)	M (kNm)
C20	606,47	22,25	105,67	23,62
C30 (+50%)	716,69 (+18,2%)	22,91 (+2,9%)	163,7 (+54,9%)	25,62 (+8,4%)
C40 (+100%)	825,34 (+36,1%)	23,38 (+5,1%)	217,39 (+105,5%)	27,62 (+16,9%)
C50 (+150%)	932,64 (+53,8%)	23,72 (+6,6%)	272,19 (+157,6%)	29,63 (+25,4%)
C60 (+200%)	1006,68 (+66%)	23,93 (+7,5%)	304,6 (+188,2%)	30,85 (+30,6%)
C70 (+250%)	1070,19 (+76,4%)	24,07 (+8,2%)	330,86 (+213,1%)	31,89 (+35%)
C80 (+300%)	1123,42 (+85,2%)	24,18 (+8,6%)	356,98 (+237,8%)	32,75 (+38,6%)
C90 (+350%)	1165,05 (+92,1%)	24,23 (+8,9%)	383,56 (+262,9%)	33,42 (+41,5%)

Yielding strength of steel (MPa)	Point A	Point B	Point D
Yielding strength of steel (MPa)	N (kN)	M (kNm)	N (kN)	M (kNm)
250	606,46	22,24	106,82	23,63
300 (+20%)	675,71 (+11,4%)	26,31 (+18,3%)	102,08 (-4,4%)	27,44 (+16,1%)
350 (+40%)	743,83 (22,6%)	30,29 (+36,2%)	96,11 (-10%)	31,21 (+32,1%)
400 (+60%)	810,83 (+33,7%)	34,20 (53,8%)	85,26 (-20,2%)	34,91 (+47,7%)
450 (+80%)	876,73 (+44,6%)	38,03 (+71%)	78,32 (-26,7%)	38,55 (+63,1%)

Steel ratio	Point A	Point B	Point D
Steel ratio	N (kN)	M (kNm)	N (kN)	M (kNm)
0,12	722,46	28,62	101,65	29,67
0,15 (+25%)	836,63 (+15,8%)	34,74 (+21,4%)	99,40 (-2,2%)	35,55 (+19,8%)
0,18 (+50%)	948,92 (+31,3%)	40,63 (+41,9%)	93,40 (-8,1%)	41,27 (+39,1%)

Point A	Point B	Point D
N (kN)	M (kNm)	N (kN)	M (kNm)
1417,4	41,5	300,8	47,07

[1] Corresponding author: Matheus Fernandes. E-mail: matheustkful@gmail.com

Brasil

Brasil

M-N interaction curves for rectangular concrete-filled steel tube columns subjected to uniaxial bending moments

Curvas de interação M-N para pilares mistos preenchidos de seção retangular sujeitos a flexão composta reta

abstract: