Nonlinear analysis of monolithic beam-column connections for reinforced concrete frames

Alva, Gerson; Tsutake, Alexandre

doi:10.1590/S1983-41952020000500015

Gerson Alva

Universidade Federal de Uberlândia – UFU, Faculdade de Engenharia Civil, Uberlândia, MG, Brasil

https://orcid.org/0000-0002-2528-5757

Alexandre Tsutake

Universidade Federal de Uberlândia – UFU, Faculdade de Engenharia Civil, Uberlândia, MG, Brasil

https://orcid.org/0000-0001-6005-8640

Corresponding author: Gerson Alva. E-mail: alva_gerson@yahoo.com.br

Conflict of interest: Nothing to declare.

Author contributions: Gerson Alva (G.A.): conceptualization, data curation, formal analysis, investigation, methodology, validation, writing. Alexandre Tsutake (A.T.): investigation, methodology, validation, writing.

Editors: Osvaldo Luís Manzoli, José Luiz Antunes de Oliveira e Sousa, Guilherme Aris Parsekian.

Thumbnail

Figure 1
Deformation mechanisms: Alva and El Debs model [⁸8 G. Alva and A. El Debs, "Moment–rotation relationship of RC beam-column connections – experimental tests and analytical model," Eng. Struct., vol. 56, pp. 1427–1438, 2013, http://dx.doi.org/10.1016/j.engstruct.2013.07.016.
http://dx.doi.org/10.1016/j.engstruct.20... ]

Thumbnail

Figure 2
Bond-slip model proposed by Sezen and Moehle [³3 H. Sezen and J. Moehle, “Bond-slip behavior of reinforced concrete members,” in Proc. Fib Symp. (CEB-FIP) – Concr. Struct. Seismic Reg., 2003, pp. 1-10.]: (a) bond stress distribution; (b) axial stress distribution; (c) axial strain distribution.

Thumbnail

Figure 3
Mechanism B: slips caused by flexural cracks - Alva and El Debs model [⁸8 G. Alva and A. El Debs, "Moment–rotation relationship of RC beam-column connections – experimental tests and analytical model," Eng. Struct., vol. 56, pp. 1427–1438, 2013, http://dx.doi.org/10.1016/j.engstruct.2013.07.016.
http://dx.doi.org/10.1016/j.engstruct.20... ]

Thumbnail

Figure 4
Definition of effective area

A_{c, e f f}

- Eurocode 2 [⁹9 European Committee for Standardization, Design of Concrete Structures – Part 1: General Rules and Rules for Building, Eurocode 2, 2004.]

Thumbnail

Figure 2
Bond-slip model proposed by Sezen and Moehle [³3 H. Sezen and J. Moehle, “Bond-slip behavior of reinforced concrete members,” in Proc. Fib Symp. (CEB-FIP) – Concr. Struct. Seismic Reg., 2003, pp. 1-10.]: (a) bond stress distribution; (b) axial stress distribution; (c) axial strain distribution.

Thumbnail

Figure 3
Mechanism B: slips caused by flexural cracks - Alva and El Debs model [⁸8 G. Alva and A. El Debs, "Moment–rotation relationship of RC beam-column connections – experimental tests and analytical model," Eng. Struct., vol. 56, pp. 1427–1438, 2013, http://dx.doi.org/10.1016/j.engstruct.2013.07.016.
http://dx.doi.org/10.1016/j.engstruct.20... ]

Thumbnail

Figure 4
Definition of effective area

A_{c, e f f}

- Eurocode 2 [⁹9 European Committee for Standardization, Design of Concrete Structures – Part 1: General Rules and Rules for Building, Eurocode 2, 2004.]

Thumbnail

Figure 5
Generic rectangular section - strains, stresses in concrete and internal resultant forces

Thumbnail

Figure 6
Idealized stress-strain curves for materials – NBR 6118 [¹1 Associação Brasileira de Normas Técnicas, Projeto de Estruturas de Concreto – Procedimento, NBR 6118, 2014.]

Thumbnail

Figure 7
General scheme of strains and stresses in concrete along the section height

Thumbnail

Figure 8
Possible cases - RC sections subjected to axial compressive load and bending moment

Thumbnail

Figure 9
Equivalent stress-strain curve for concrete in tension (tension stiffening) – Torres et al. [¹¹11 L. I. Torres, F. López-Almansa, and L. Bozzo, "Tension-stiffening model for cracked flexural concrete members," J. Struct. Eng., vol. 130, no. 8, pp. 1242–1251, 2004, http://dx.doi.org/10.1061/(ASCE)0733-9445(2004)130:8(1242).
http://dx.doi.org/10.1061/(ASCE)0733-944... ]

Thumbnail

Figure 9
Equivalent stress-strain curve for concrete in tension (tension stiffening) – Torres et al. [¹¹11 L. I. Torres, F. López-Almansa, and L. Bozzo, "Tension-stiffening model for cracked flexural concrete members," J. Struct. Eng., vol. 130, no. 8, pp. 1242–1251, 2004, http://dx.doi.org/10.1061/(ASCE)0733-9445(2004)130:8(1242).
http://dx.doi.org/10.1061/(ASCE)0733-944... ]

Thumbnail

Figure 10
Moment-curvature curves generated using computational procedure (item 4.1) and using Branson´s expression (cracked stage) for LVP1: Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 11
Detail of the beam-column connections of the numerical simulations

Thumbnail

Figure 12
Loading history at the last stage of loading - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 13
Displacement transducers used to evaluate relative rotations - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 14
Experimental and theoretical moment-rotation curves for LVP1 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 17
Experimental and theoretical moment-rotation curves for LVP4 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 15
Experimental and theoretical moment-rotation curves for LVP2 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 16
Experimental and theoretical moment-rotation curves for LVP3 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 18
Structural model for obtaining the theoretical curves - Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]

Thumbnail

Figure 19
Experimental and theoretical force-displacement curves - Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]

Thumbnail

Figure 20
Experimental and theoretical moment-rotation curves - Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]

Thumbnail

Figure 12
Loading history at the last stage of loading - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 13
Displacement transducers used to evaluate relative rotations - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 14
Experimental and theoretical moment-rotation curves for LVP1 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 17
Experimental and theoretical moment-rotation curves for LVP4 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 15
Experimental and theoretical moment-rotation curves for LVP2 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 16
Experimental and theoretical moment-rotation curves for LVP3 - Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br
http://www.teses.usp.br... ]

Thumbnail

Figure 18
Structural model for obtaining the theoretical curves - Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]

Thumbnail

Figure 19
Experimental and theoretical force-displacement curves - Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]

Thumbnail

Figure 20
Experimental and theoretical moment-rotation curves - Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]

Thumbnail

Table 1
Mechanical properties of concrete and longitudinal steel reinforcement of the beam-column connections

Thumbnail

Table 2
Dimensions and area of the longitudinal reinforcement of the beam-column connections

Thumbnail

Table 3
Constants of Alva and El Debs [⁸8 G. Alva and A. El Debs, "Moment–rotation relationship of RC beam-column connections – experimental tests and analytical model," Eng. Struct., vol. 56, pp. 1427–1438, 2013, http://dx.doi.org/10.1016/j.engstruct.2013.07.016.
http://dx.doi.org/10.1016/j.engstruct.20... ] analytical model

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

Connection		f_c (MPa)	f_ct (MPa)	E_c (GPa)	f_y (MPa)	E_s (GPa)
Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br http://www.teses.usp.br... ]	LVP1	44,18	3,30	33,60	630	182
	LVP2	23,89	1,95	28,32	594	213
	LVP3	24,62	2,08	28,47	594	213
	LVP4	25,91	2,20	28,74	594	213
Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]	Specimen 2	28,94	2,89	25,91	335	200
	Specimen 5	24,80	2,48	23,70	351	200

Connections	L	H	h_b	b_b	h_c	b_c	A_s1	A_s2	A_s
Connections	(cm)	(cm)	(cm)	(cm)	(cm)	(cm)	(cm²)	(cm²)	(cm²)
Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br http://www.teses.usp.br... ] LVP1-LVP4	155,0	250,0	40,0	20,0	30,0	20,0	8,04	8,04	20,10
Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.] Specimen 2	118,1	152,4	25,4	20,3	27,9	20,3	5,70	4,00	11,35
Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.] Specimen 5	104,8	152,4	25,4	20,3	27,9	20,3	5,70	4,00	11,35

Connections		M_y (kN.cm)	ϕ (mm)	L_p (cm)	s_R (cm)	C₁ (kN.cm)^-2	C₂ (cm)
Alva [¹³13 G. Alva, “Estudo teórico-experimental do comportamento de nós de pórtico de concreto armado submetidos a ações cíclicas,” Ph.D. dissertation, Esc. Eng. São Carlos, Univ. São Paulo, São Carlos, 2004. [Online]. Available: http://www.teses.usp.br http://www.teses.usp.br... ]	LVP1	16577	16	30	12,9	9,526E-12	21,45
	LVP2	15645	16	30	12,9	7,861E-12	21,45
	LVP3	15369	16	30	12,9	1,132E-11	21,45
	LVP4	15380	16	30	12,9	1,114E-11	21,45
Lee et al. [¹⁴14 L. Lee, J. Wight, and R. Hanson, "RC beam-column joints under large load reversals," J. Struct. Div., vol. 103, no. 12, pp. 2337–2350, 1977.]	Specimen 2	3298	19	20,3	17,2	1,609E-10	18,76
	Specimen 5	3458	19	20,3	17,2	1,774E-10	18,73

y_{C G} = \frac{\int_{0}^{y_{c 2}} σ_{c} \cdot b \cdot y \cdot d y + \int_{y_{c 2}}^{x} 0,85 \cdot f_{c d} \cdot b \cdot y \cdot d y}{\int_{0}^{y_{c 2}} σ_{c} \cdot b \cdot d y + \int_{y_{c 2}}^{x} σ_{c} \cdot b \cdot d y} = \frac{b \cdot \int_{0}^{y_{c 2}} σ_{c} \cdot y \cdot d y + 0,85 \cdot f_{c d} \cdot b \cdot \int_{y_{c 2}}^{x} y \cdot d y}{R_{c c}}

y_{C G} = \frac{0,85 \cdot f_{c d} \cdot b}{R_{c c}} (\frac{k \cdot y_{c 2}^{3}}{3} - \frac{k^{2} \cdot y_{c 2}^{4}}{16} + \frac{k \cdot x^{2}}{2} - \frac{k \cdot y_{c 2}^{2}}{2})

y_{C G} = \frac{0,85 \cdot f_{c d} \cdot b \cdot k \cdot [\frac{x^{3}}{3} - \frac{{(x - h)}^{3}}{3} - \frac{k \cdot x^{4}}{16} + \frac{k \cdot {(x - h)}^{4}}{16}]}{R_{c c}}

R_{c c} = \int_{x - h}^{y_{c 2}} σ_{c} \cdot b \cdot d y + \int_{y_{c 2}}^{x} b \cdot σ_{c} \cdot d y = b \cdot (\int_{x - h}^{y_{c 2}} σ_{c} \cdot d y + 0,85 \cdot f_{c d} \cdot \int_{y_{c 2}}^{x} σ_{c} \cdot d y)

y_{C G} = \frac{\int_{x - h}^{y_{c 2}} σ_{c} \cdot b \cdot y \cdot d y + \int_{y_{c 2}}^{x} b \cdot σ_{c} \cdot y \cdot d y}{\int_{x - h}^{y_{c 2}} σ_{c} \cdot b \cdot d y + \int_{y_{c 2}}^{x} b \cdot σ_{c} \cdot d y} = \frac{b \cdot (\int_{x - h}^{y_{c 2}} σ_{c} \cdot y \cdot d y + \int_{y_{c 2}}^{x} σ_{c} \cdot y \cdot d y)}{R_{c c}}

R_{c c} = 0,85 \cdot f_{c d} \cdot b \cdot [\frac{k \cdot y_{c 2}^{2}}{2} - \frac{k \cdot {(x - h)}^{2}}{2} - \frac{k^{2} \cdot y_{c 2}^{3}}{12} + \frac{k^{2} \cdot {(x - h)}^{3}}{12} + x - y_{c 2}]

y_{C G} = \frac{0,85 \cdot f_{c d} \cdot b}{R_{c c}} [\frac{k \cdot y_{c 2}^{3}}{3} - \frac{k \cdot {(x - h)}^{3}}{3} - \frac{k^{2} \cdot y_{c 2}^{4}}{16} + \frac{k^{2} \cdot {(x - h)}^{4}}{16} + \frac{x^{2}}{2} - \frac{y_{c 2}^{2}}{2}]

[1] Corresponding author: Gerson Alva. E-mail: alva_gerson@yahoo.com.br

Brasil

Brasil

Nonlinear analysis of monolithic beam-column connections for reinforced concrete frames

Análise não linear de ligações viga-pilar monolíticas em pórticos de concreto armado

abstract: