Design Oriented Model for the Assessment of T-Shaped Beam-Column Joints in Reinforced Concrete Frames

Abstract

Beam-column joints represent very important elements of reinforced concrete (RC) structures. In fact, beams and columns, at the boundary, generate internal forces acting on concrete core and on reinforcement bars with a very high gradient. To fully understand the seismic performances and the failure modes of T-shaped beam-column joints (external corner-positioned) in RC structures, a simplified analytical model of joint behaviour is proposed and theoretical simulations have been performed. The model is based on the solution of a system of equilibrium equations of cracked joint portions designed to evaluate internal stresses at different values of column shear forces. The main aim of the proposed model is to identify the strength hierarchy. Limit values of different internal stresses allow us to detect the occurrence of different failure modes (namely the failure of the cracked joint, the bond failure of passing through bars, and the flexural/shear failures of columns or beams) associated with column shear forces; the smaller one represents the capacity of the joint. The present work, focusing on T-shaped joints, could represent a useful tool for designers to quantify the performance of new structures or of existing ones. In fact, such a tool allows us to push an initial undesired failure mode to a more appropriate one to be evaluated. Finally, some experimental results of tests available in literature are reported, analysed, and compared to the predictions of the proposed model (by means of a worked example) and of some international codes. The outcomes confirm that failure modes and corresponding joint capacities require an analytical model, like the proposed one, to be accurately predicted.

1. Introduction

Reinforced concrete has been used for more than a century. The early buildings were surely designed by simply considering vertical loads. During the following decades, different codes have followed, so nowadays it is very common to find buildings designed according to old seismic codes or without considering any seismic provision. These buildings usually present a lack of capacity in beam-column joints. Although joints have a restricted area, they are strongly involved in the transfer of forces, mostly under earthquakes. In particular, they often present low shear strength in the joints, leading to brittle shear failures [1,2]. An inadequate design can lead to a shear failure of beam-column joints and to a consequent collapse of the structure [3], particularly in the case of earthquakes [4,5,6,7], even if they are expected to be safe according to classical seismic evaluations that neglect the failure of the joints. For example, referring to L’Aquila earthquake (2009) [8,9], some observations of damaged structures showed that mid-90s Italian structures or older [10,11,12] could present structural lack in joints due to the design approach or insufficient reinforcement, especially in cases of lack of transverse reinforcements (or due to the reduction of the reinforcement cross section due to corrosion [13,14,15,16,17], which should be accounted for in the assessments). In addition, under a seismic action, the beam-column joint is subjected to bending moments (clockwise or counter clockwise) and bars are pushed or pulled at the sides of the joint panel. Under such loading, and if concrete strength and quality are low or the column is not wide enough, debonding of steel bars could occur [18,19] and beams could lose their load capacity.

In previous works available in scientific literature, a number of beam-column joint models have been proposed. For instance, rotational hinges have been calibrated [20,21,22], or macro-element models have been defined [23,24,25], to simulate failed joint behaviour; in a few cases, finite element (FE) models have been proposed [26,27]; however, FE models are usually complex and require significant resources; consequently, they are usually not preferable for practitioner-oriented analyses. Each model has limitations and advantages, but there is no wide consensus on a preferable modelling technique, as many of them are more oriented to a research approach, requiring a comparably detailed and refined modelling of the frames. To obtain a good balance between simplicity and accuracy, this paper proposes a simplified approach suitable for analyzing frames and detecting beam-column joint failure that does not require particular calibrations for each joint.

2. Research Significance

The aim of the present paper is to investigate the corner-positioned RC beam-column joints in order to study their seismic behaviour and to establish the strength hierarchy in the case of failure of beam-column joints, particularly focusing on corner-positioned joints. It aims to contribute to strength hierarchy principles and good practice in their assessment. Modern design codes are strictly linked to these design principles, because the beam-column joint panels’ behaviour influences the ductility of the global structure. Presented theoretical analytical outcomes have been discussed and compared to experimental results.

3. Basis of Theoretical Model

Many experimental data, related to external corner-positioned joints and present in the scientific literature, were gathered and analysed. These data are the result of experimental laboratory tests performed on joints with various geometries, subjected to cyclic load, for which the load levels leading to joint’s failure have been recorded. The evaluation model used to analyse the joints and failure modes is an improvement on the model proposed by Shiohara [28], partially modified in Bossio et al. [29] to account for internal joints. In the case of external joints, the simplifications due to symmetry are no longer valid; therefore, the verse of the shear (positive or negative) should be accounted for and equations modified accordingly. Several geometries of joints have been considered and the shear force has been applied to columns; the so called column shear, V_c, values are assumed as the main force parameter in frame analysis, and the related failure modes, involving the joint assemblage, were evaluated for each of them. Starting from the basic configuration (typical of experimental laboratory test), some geometrical and internal reinforcement modifications were considered and column shear, V_c, values were calculated, so it was possible to detect the relevant failure mode variations, too.

3.1. Mathematical Formulation

The scheme of T-shaped joint was identified by defining the geometric parameters and stresses, as shown in Figure 1 and Figure 2, in the case of negative or positive column shear, respectively. Because of the asymmetry of this kind of joint, two different verses of column shear, V_c, need to be considered (positive and negative). In fact, depending on the column shear verse, the forces acting on the joint panel change (Figure 1a and Figure 2a). The internal forces of the joint, acting on each of the four rigid portions cut by typical diagonal cracks in joints, need to be defined. In the case of positive column shear, the tensile forces of longitudinal bars that cross the joint panel are named F₁, F₂, F₇, and F₈, and the compressive forces (assumed as acting at the position of the bars) are named F₃, F₄, F₅, and F₆, while the F₉, F₁₀ are the resultant force carried by stirrups and/or eventual external reinforcement, and C is the concrete compressive resultants on half of the diagonal (i.e., $0.5{\mathrm{H}}_{\mathrm{b}}/\sin \mathsf{\vartheta }$) of the joint panel. Particularly, F₉ represents the resultant force exerted by stirrups or horizontally-placed externally bonded reinforcements (EBR), and F₁₀ represents the resultant force exerted by vertically-placed EBR (e.g., made of FRP with fibers aligned in the vertical direction). In order to simplify the calculation procedure, the resulting compressive forces of the concrete strut, C, are assumed to act normally on the cross section and are centred with respect to the diagonal of the joint. Conversely, in the case of negative column shear, the tensile strain forces of longitudinal bars that cross the joint panel are named F₃, F₄, F₅, and F₆, and the compressive forces are named F₁, F₂, F₇, and F₈, while the resultant force absorbed by stirrups and/or externally bonded reinforcements is named F₉, and F₁₀ and C are the same, as previously defined.

Referring to Figure 1, in case of positive column shear, the internal forces are obtained by solving Equation (1):

$${\mathrm{F}}_{\mathrm{i}}={\mathrm{C}}_{\mathrm{i}}+{\mathrm{S}}_{\mathrm{i}}\mathrm{for}\mathrm{i}=3,4,5,6$$

(1)

while referring to Figure 2, in case of negative column shear, internal forces are obtained by solving Equation (2):

$${\mathrm{F}}_{\mathrm{i}}={\mathrm{C}}_{\mathrm{i}}+{\mathrm{S}}_{\mathrm{i}}\mathrm{for}\mathrm{i}=1,2,7,8$$

(2)

It is possible to evaluate each internal force by solving a system of equations involving the three equilibrium equations (vertical translation, horizontal translation, and rotation) for each of the four rigid portions of the joint. Of the twelve equations, nine only are independent. If the values for F₉, F₁₀, N_c, and N_b are fixed, and considering Equation (3), it is possible to reduce the unknowns to nine.

$${\mathrm{V}}_{\mathrm{b}}=\frac{2·{\mathrm{L}}_{\mathrm{c}}}{{\mathrm{L}}_{\mathrm{b}}}·{\mathrm{V}}_{\mathrm{c}}=2·\mathrm{a}·{\mathrm{V}}_{\mathrm{c}}$$

(3)

and aspect ratio is introduced as a = L_c/L_b.

In the case of positive column shear, by solving the system of Equations (4)–(12), all internal forces are evaluated as a function of V_c.

$${\mathrm{F}}_1+{\mathrm{F}}_4-\mathrm{C}·\sin \mathsf{\vartheta }-{\mathrm{V}}_{\mathrm{c}}=0$$

(4)

$${\mathrm{F}}_1-{\mathrm{F}}_6+{\mathrm{F}}_9-\mathrm{C}·\sin \mathsf{\vartheta }+{\mathrm{N}}_{\mathrm{b}}=0$$

(5)

$${\mathrm{F}}_6+{\mathrm{F}}_7-\mathrm{C}·\sin \mathsf{\vartheta }-{\mathrm{V}}_{\mathrm{c}}=0$$

(6)

$${\mathrm{F}}_3-{\mathrm{F}}_2-{\mathrm{F}}_{10}+\mathrm{C}·\cos \mathsf{\vartheta }-{\mathrm{N}}_{\mathrm{c}}=0$$

(7)

$${\mathrm{F}}_2+{\mathrm{F}}_5-\mathrm{C}·\cos \mathsf{\vartheta }-2·\mathrm{a}·{\mathrm{V}}_{\mathrm{c}}=0$$

(8)

$${\mathrm{F}}_8-{\mathrm{F}}_5+{\mathrm{F}}_{10}-\mathrm{C}·\cos \mathsf{\vartheta }+{\mathrm{N}}_{\mathrm{c}}+2·\mathrm{a}·{\mathrm{V}}_{\mathrm{c}}=0$$

(9)

$${\mathrm{h}}_{\mathrm{b}}^{*}·\left({\mathrm{F}}_1+{\mathrm{F}}_4\right)+{\mathrm{h}}_{\mathrm{c}}^{*}·\left({\mathrm{F}}_2+{\mathrm{F}}_3\right)-\frac{{\mathrm{C}}^2}{\mathrm{B}·{\mathrm{f}}_{\mathrm{c}}}-{\mathrm{L}}_{\mathrm{c}}·{\mathrm{V}}_{\mathrm{c}}=0$$

(10)

$${\mathrm{h}}_{\mathrm{b}}^{*}·\left({\mathrm{F}}_1+{\mathrm{F}}_6\right)+{\mathrm{h}}_{\mathrm{c}}^{*}·\left({\mathrm{F}}_2+{\mathrm{F}}_5\right)-\frac{{\mathrm{C}}^2}{\mathrm{B}·{\mathrm{f}}_{\mathrm{c}}}-2·{\mathrm{L}}_{\mathrm{c}}·{\mathrm{V}}_{\mathrm{c}}=0$$

(11)

$${\mathrm{h}}_{\mathrm{b}}^{*}·\left({\mathrm{F}}_6+{\mathrm{F}}_7\right)+{\mathrm{h}}_{\mathrm{c}}^{*}·\left({\mathrm{F}}_5+{\mathrm{F}}_8\right)-\frac{{\mathrm{C}}^2}{\mathrm{B}·{\mathrm{f}}_{\mathrm{c}}}-{\mathrm{L}}_{\mathrm{c}}·{\mathrm{V}}_{\mathrm{c}}=0$$

(12)

while in the case of negative column shear, the system to solve is represented by Equations (10)–(18), because rotational equilibriums are the same, while translational equilibriums change.

$${\mathrm{F}}_1+{\mathrm{F}}_4-\mathrm{C}·\sin \mathsf{\vartheta }-{\mathrm{V}}_{\mathrm{c}}=0$$

(13)

$${\mathrm{F}}_1-{\mathrm{F}}_6-{\mathrm{F}}_9+\mathrm{C}·\sin \mathsf{\vartheta }-{\mathrm{N}}_{\mathrm{b}}=0$$

(14)

$${\mathrm{F}}_6+{\mathrm{F}}_7-\mathrm{C}·\sin \mathsf{\vartheta }-{\mathrm{V}}_{\mathrm{c}}=0$$

(15)

$${\mathrm{F}}_2-{\mathrm{F}}_3-{\mathrm{F}}_{10}+\mathrm{C}·\cos \mathsf{\vartheta }-{\mathrm{N}}_{\mathrm{c}}=0$$

(16)

$${\mathrm{F}}_2+{\mathrm{F}}_5-\mathrm{C}·\cos \mathsf{\vartheta }-2·\mathrm{a}·{\mathrm{V}}_{\mathrm{c}}=0$$

(17)

$${\mathrm{F}}_8-{\mathrm{F}}_5-{\mathrm{F}}_{10}+\mathrm{C}·\cos \mathsf{\vartheta }-{\mathrm{N}}_{\mathrm{c}}+2·\mathrm{a}·{\mathrm{V}}_{\mathrm{c}}=0$$

(18)

All the symbols introduced in previous equations are depicted in Figure 1 and Figure 2 and in the notation list; it is worth noting that in Equations (10)–(12), the width B of the joint can be equal either to width B_b of beam or width B_c of column, or it can even be different. By solving the system of Equations (4)–(12) or (10)–(18), it is possible to correlate each internal force to column shear values, e.g., as F₁(V_c). Then, the limit values of internal forces generating the internal failure mechanisms of the joint can be defined and, consequently, the corresponding column shear, V_c, values are strictly related to the activation of each failure mode, according to

Table 1. Ultimate column shear, V_c, for each considered failure mode.

Column Shear	Failure Mode
Vc1	Bending for beam
Vc2	Bending for column
Vc3	Shear for beam
Vc4	Shear for column
Vc5	Joint involving horizontal bars
Vc6	Joint involving vertical bars (upper column)
Vc7	Joint involving vertical bars (lower column)
Vc8	Joint due to maximum bond exceedance
Vc9	Joint due to medium bond exceedance
Vc10	Joint due to minimum bond exceedance
Vc11	Joint due to crushing of concrete strut

. The capacity of the joint is represented by the minimum of the V_c values.

3.2. Failure Modes

Focusing on joint panel, three failure modes were considered: (i) joint failure with tensile yielding or failure of longitudinal horizontal or vertical reinforcements (related to V_c5 to V_c7 respectively, according to Table 1); (ii) de-bonding of longitudinal reinforcements (related to V_c8 to V_c10, according to Table 1); and (iii) failure of concrete strut (related to V_c11, according to Table 1). The former failure mode is related to the conventional failures of reinforcements in tension (breakage, or yielding on safe side). Considering different layers of bars, i, having cross section, A_k, and characteristic yielding (or eventually ultimate) stress, f_y,k, each F_i, can be calculated according to Equation (19) and limited to the axial strength, F_max.

$${\mathrm{F}}_{\max ,\mathrm{i}}={\displaystyle \sum }{\mathrm{A}}_{\mathrm{k}}·{\mathrm{f}}_{\mathrm{y},\mathrm{kc}}$$

(19)

When F_max is reached in a horizontal or vertical reinforcement layer, the corresponding V_c is the ultimate column shear V_c5 or V_c6 and V_c7 (for upper or lower columns that have a different axial load), respectively.

Referring to a failure mode due to de-bonding of longitudinal reinforcements (related to V_c8, V_c9, or V_c10), three different de-bonding conditions were considered according to Model Code 2010 and given concrete compressive strength f_c [30]:

$${\mathsf{\tau }}_{\max }=2.5·\sqrt{{\mathrm{f}}_{\mathrm{c}}}$$

(20)

$${\mathsf{\tau }}_{\mathrm{med}}=1.25·\sqrt{{\mathrm{f}}_{\mathrm{c}}}$$

(21)

$${\mathsf{\tau }}_{\min }=0.3·\sqrt{{\mathrm{f}}_{\mathrm{c}}}(\mathrm{e}.\mathrm{g}.,\text{ smooth bars})$$

(22)

The maximum bond force is provided by the bond strength $\mathsf{\tau }\_$ multiplied by the lateral surface of the bars only for the length inside the rigid portions cut by diagonal cracks in joints. Three conditions of adhesion can be identified, and corresponding bond strength values are ${\mathsf{\tau }}_{\max }$, ${\mathsf{\tau }}_{\mathrm{med}}$, and ${\mathsf{\tau }}_{\min }$ (but it is worth noting that bond force could reach values up to six times lower in some cases [30]). In the case of positive shear, at the upper layer of bars, the value of bond force is F₁ + S₄, while, at the bottom side, the bond force is F₇ + S₆; in the case of negative shear, at the upper layer of bars, the value of bond force is F₄ + S₁, while, at the bottom side, the bond force is F₆ + S₇, where S_s are the steel contributions. The resultant compressive force needs to be split into steel contribution, S_, and concrete contribution, C_ (Figure 3a,b shows S₄ and S₆ in the case of positive column shear depicted in Figure 1b). In particular, as shown in Figure 1b (for S₄ and S₆) and Figure 2b (for S₁ and S₇), there are four steel contributions that can be calculated, for instance, according to Equations (23)–(26).

$${\mathrm{S}}_4{=\mathrm{F}}_7·\frac{{\mathrm{A}\prime }_{\mathrm{s}}·(2\mathrm{c}-{\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*})}{{\mathrm{A}}_{\mathrm{s}}·{(\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*}-2\mathrm{c})}$$

(23)

$${\mathrm{S}}_6{=\mathrm{F}}_1·\frac{{\mathrm{A}}_{\mathrm{s}}·(2\mathrm{c}-{\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*})}{{\mathrm{A}\prime }_{\mathrm{s}}·{(\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*}-2\mathrm{c})}$$

(24)

$${\mathrm{S}}_1={\mathrm{F}}_6·\frac{{\mathrm{A}\prime }_{\mathrm{s}}·\left(2·\mathrm{c}-{\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*}\right)}{{\mathrm{A}}_{\mathrm{s}}·\left({\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*}-2·\mathrm{c}\right)}$$

(25)

$${\mathrm{S}}_7={\mathrm{F}}_4·\frac{{\mathrm{A}}_{\mathrm{s}}·\left(2·\mathrm{c}-{\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*}\right)}{{\mathrm{A}\prime }_{\mathrm{s}}·\left({\mathrm{H}}_{\mathrm{b}}+{\mathrm{h}}_{\mathrm{b}}^{*}-2·\mathrm{c}\right)}$$

(26)

To calculate the neutral axis depth, c, recalling linearity of strain diagram of cross section (before attainment of debonding), it is possible to offset to zero the static moment, with respect to the neutral axis, considering the homogenized cross section. When maximum bond force is reached, the corresponding V_c is the ultimate column shear V_c8 to V_c10, depending on the considered bond condition.

The latter considered failure mode is the crushing of concrete strut (related to V_c11). It is possible to obtain the maximum value of concrete contact force, C, by limiting its value to the value of compression strength (of concrete f_c) in half strut by solving Equation (27):

$${\mathrm{C}}_{\max }=\mathrm{B}·f_c·\frac{{\mathrm{H}}_{\mathrm{b}}}{2·\sin \vartheta }$$

(27)

If shear friction is assumed along cracks, the concrete contact force, C, can be assumed inclined, θ, not as the diagonal crack (i.e., ϑ ≠ θ), thus requiring a shear friction check, in which deviation from diagonal (i.e., │ϑ − θ│) should be limited to a threshold value.

When maximum concrete contact force is reached, the corresponding V_c is the ultimate column shear V_c11.

Even if present paper focuses on T-shaped joint failure only, the evaluation of the beam and column failure modes is also recommended, according to classical structural analysis approaches (Cosenza et al., 2008) [31]. Considering the failure modes potentially occurring outside the joint that are related to bending moment of beam (ultimate bending moment of beam, M_b) or column (ultimate bending moment of column, M_c, and note that upper and lower columns have different axial loads, and hence strength), and to shear failure of beam (ultimate shear of beam, V_b), it is possible to express them in relation to the column shear, V_c, by solving Equations (28)–(30), respectively.

$${\mathrm{V}}_{\mathrm{c}1}=\frac{{\mathrm{M}}_{\mathrm{b}}·{\mathrm{L}}_{\mathrm{b}}}{\left[\left({\mathrm{L}}_{\mathrm{b}}{-\mathrm{H}}_{\mathrm{c}}\right)·{\mathrm{L}}_{\mathrm{c}}\right]}$$

(28)

$${\mathrm{V}}_{\mathrm{c}2}=\frac{2·{\mathrm{M}}_{\mathrm{c}}}{{\mathrm{L}}_{\mathrm{c}}-{\mathrm{H}}_{\mathrm{b}}}$$

(29)

$${\mathrm{V}}_{\mathrm{c}3}=\frac{{\mathrm{V}}_{\mathrm{b}}·{\mathrm{L}}_{\mathrm{b}}}{2·{\mathrm{L}}_{\mathrm{c}}}$$

(30)

Finally, the column ultimate shear value leading to column shear failure is, obviously, directly equal to the ultimate column shear, V_c4. The minimum value of calculated column shear values corresponds to the failure mode involving the joint panel.

Please note that even if ultimate bending moment of beam is related to reinforcement yielding, like as the condition provided by Equation (19) for the joint failure, the two failure modes are totally different, as flexural failure of beam is the preferred ductile failure of the system, while joint failure is in any case a brittle failure (associated with reinforcement failure, or simply yielding).

4. Behaviour of Proposed Model and Validation

Scientific literature presents numerous experimental test results; among them, tests by El-Amoury and Ghobarah, [32], Masi et al. [27], and Russo and Pauletta, [33], performed on different kinds of geometry of T-shaped joint, were selected to evaluate stress values leading to failure of the system. By using the proposed theoretical model for each geometry and each considered failure modes of the joints, the values of column shear, V_c, were calculated (and named V_c1 to V_c11 for each considered failure mode, according to Table 1).

Differently from internal joints (Bossio et al. [29]), T-shaped joints are lacking in symmetry simplifications, and positive sign of shear needs to be distinguished from negative one. In order to explain proposed model application, a worked example is provided in the Appendix A and some model predictions were compared to experimental tests. Actual failure mode is represented by the minimum value of column shear load and is limited by bond conditions. The main idea is to create an interesting tool to prevent undesired failure modes (brittle shear failure of the joint, beams, or columns) or to push the failure mode to a more desired ductile beam flexural failure (in case of retrofit or design review). In fact, according to “capacity design” approach, once the desired failure mode is decided, it is possible to use external strengthening (on existing structures), or the design can be calibrated to let ultimate column shear of undesired failure modes exceed the ultimate column shear of the desired failure mode to prevent those undesired failure modes.

Geometrical dimensions and other features of considered specimens are reported in

Table 2. Geometrical and mechanical properties of specimens of experimental programs.

Specimen ID Number	Concrete Charcteristics		Concrete Section		Concrete Cover		Beam						Column						Stirrups
							Upper Side			Lower Side			Upper Side			Lower Side			Beam		Column
	Nc	fc	Beam	Column	Beam	Column	Bar Diameter	A′s,b	fy	Bar Diameter	As,b	fy	Bar Diameter	A′s,c	fy	Bar Diameter	As,c	fy	Legs, Diameter	fy	Legs, Diameter	fy
	kN	MPa	cm2	cm2	mm	mm	mm	cm2	MPa	mm	cm2	MPa	mm	cm2	MPa	mm	cm2	MPa	N°, mm	MPa	N°, mm	MPa
El-Amoury and Ghobarah (2002) [32]
T0	600	30.6	25 × 40	25 × 40	40	40	20	12.56	425	20	12.56	425	20	9.42	425	20	9.42	425	2, 10	425	2, 15	425
Masi et al. (2009) [27]
T1	290	17.9	30 × 50	30 × 30	30	30	12	2.26	478	12	2.26	478	14	3.08	478	14	2.08	478	2, 8	478	2, 8	478
Russo and Pauletta (2012) [33]
12_6	90	22.16	30 × 24	30 × 30	2.7	2.7	12	2.26	315.4	12	2.26	315.4	12	2.26	315.4	12	2.26	315.4	2, 6	315.4	2, 6	315.4
12_8	90	22.16	30 × 24	30 × 30	2.9	2.9	12	2.26	315.4	12	2.26	315.4	12	2.26	315.4	12	2.26	315.4	2, 8	315.4	2, 8	315.4

. An overview of predictions of experiments provided by the proposed model is in

Table 3. Theoretical prediction of shear capacity of considered specimens related to failure modes.

Experimental Program
Column Shear at Failure	Positive Shear				Negative Shear
	El-Amoury and Ghobarah (2002) [32]	Masi et al. (2009) [27]	Russo and Pauletta (2012) [33]		El-Amoury and Ghobarah (2002) [32]	Masi et al. (2009) [27]	Russo and Pauletta (2012) [33]
	T0	T1	12_6	12_8	T0	T1	12_6	12_8
Vc1 [kN]	65.71	17.75	9.11	9.11	65.71	17.75	9.11	9.11
Vc2 [kN]	172.76	56.76	35.59	35.26	172.76	56.76	35.59	35.26
Vc3 [kN]	175.65	156.78	24.32	42.82	175.65	156.78	24.32	42.82
Vc4 [kN]	270.24	145.89	73.00	72.46	270.24	145.89	73.00	72.46
Vc5 [kN]	59.07	13.59	8.52	8.33	59.07	13.59	8.52	8.33
Vc6 [kN]	125.75	86.40	25.09	24.54	125.75	86.40	25.09	24.54
Vc7 [kN]	137.08	100.45	30.40	29.57	114.68	74.68	21.28	20.88
Vc8 [kN]	110.93	25.62	24.44	23.37	110.93	25.62	24.44	23.37
Vc9 [kN]	60.49	13.00	12.40	11.86	60.49	13.00	12.40	11.86
Vc10 [kN]	15.28	3.15	3.01	2.87	15.28	3.15	3.01	2.87
Vc11 [kN]	160.93	160.93	102.04	102.04	160.93	160.93	102.04	102.04

Bold numbers indicate failure considering maximum bond exceedance. Italic numbers indicates failure considering minimum bond exceedance.

(and a worked example for T1 test [27] prediction for positive shear is in the Appendix A). All considered specimens present an experimental failure mode due to joint failure, with yielding of longitudinal bars both considering positive or negative shear (Table 3).

5. Simulation of Research by El-Amoury and Ghobarah

The joint is predicted to suffer a joint failure with yielding of horizontal longitudinal bars with a value of V_c5 equal to 59.07 kN both in case of positive or negative shear, close to experimental test’s value [32], which was 58.5 kN (+1% with respect to experimental value). It is worth noting that the desired failure mode of flexural bending of beams is quite close in terms of ultimate column shear (i.e., V_c1). This means that a slight improvement of the joint capacity (e.g., by means of two additional stirrups V_c5 increases up to 66 kN) would let it move from undesired brittle joint failure to a desirable ductile flexural failure of beams. In fact, V_c1 would be still 65.7 kN, higher than new value of V_c5, and hence the lowest column shear controls the failure mode.

6. Simulation of Research by Masi et al.

In this case, the failure mode is represented again by the joint failure involving horizontal longitudinal bars. The value of V_c5 is 13.59 kN, considering that the yielding of bars (and 16.7 kN considering the tensile failure of the bars), both in case of positive or negative shear −24% (−7% considering the rupture of the bars), is lower than experimental value of 18.0 kN [27]. In this case, V_c1 is equal to 17.75 kN, both in positive or negative shear for all cases; hence, an improvement of the joint design would require that V_c1 is the lowest ultimate column shear. In

Table 4. Comparison of experimental results [27] and simplified model failure predictions to validate simplified proposed model.

Failure Mode	No Retrofit	No Retrofit	2 Stirrups	2 Stirrups	4 Stirrups	4 Stirrups
	Pos. Shear	Neg. Shear	Pos. Shear	Neg. Shear	Pos. Shear	Neg. Shear
Vc1 [kN]	17.75	17.75	17.75	17.75	17.75	17.75
Vc2 [kN]	56.76	56.76	56.76	56.76	56.76	56.76
Vc3 [kN]	156.78	156.78	156.78	156.78	156.78	156.78
Vc4 [kN]	145.89	145.89	145.89	145.89	145.89	145.89
Vc5 [kN]	13.59	13.59	19.51	19.51	25.35	25.35
Vc6 [kN]	86.40	86.40	86.40	86.40	86.40	86.40
Vc7 [kN]	100.45	74.68	100.45	74.68	100.45	74.68
Vc8 [kN]	25.62	25.62	31.77	31.77	37.82	37.82
Vc9 [kN]	13.00	13.00	19.34	19.34	25.59	25.59
Vc10 [kN]	3.15	3.15	9.64	9.64	16.03	16.03
Vc11 [kN]	160.93	160.93	160.93	160.93	160.93	160.93

, some predictions are provided if two or four stirrups are added in the joint panel.

In the case of minimum bond, the failure mode never moves from minimum bond (V_c10 is the smallest ultimate column shear) to other failure modes (apart from adding five stirrups of diameter D8 with two legs, but this is not shown in Table 4). In the case of medium bond, the force that F₉ has to provide is equal to 72 kN, reachable by applying two stirrups of diameter D8 with two legs. Figure 4 shows the results obtained for both cases.

7. Simulation of Research by Russo and Pauletta

Considering experimental test results in Russo and Pauletta [33], the failure modes of considered specimens (12_6 and 12_8) were represented by the joint failure with yielding of horizontal longitudinal bars with values of 6.8 kN and 8.18 kN, respectively. For the specimen 12_6, the theoretical value of V_c5 is 7.52 kN (+10% with respect to experimental value), while for the specimen 12_8, the value of V_c5 is 8.33 kN (+2% with respect to experimental value).

8. Parametric Analysis

The proposed model was used to understand the effect of geometric variations in comparison to reference condition in terms of column shear and failure modes. The failure mode and the related values of column shear (both in the case of positive or negative sign of shear) related to eleven considered cases were shown in

Table 5. Variability of considered parameters: failure modes and column shear capacity (both in case of positive or negative shear) of considered joint configurations.

ID Number	Nc	fc	Stirrups	FRP	Beam Dimension	Failure Mode (Pos. Column Shear)	Failure Mode (Neg. Column Shear)
--	kN	MPa	--	--	cm2	--	kN
1	0	20	--	--	(50 × 30)	Vc2	37.1
2	315	20	--	--	(50 × 30)	Vc5	57.5
3	630	20	--	--	(50 × 30)	Vc5	57.5
4	315	40	--	--	(50 × 30)	Vc5	59.5
5	315	60	--	--	(50 × 30)	Vc5	60.1
6	315	20	2D8	--	(50 × 30)	Vc2	62.2
7	315	20	4D8	--	(50 × 30)	Vc2	62.2
8	315	20	8D8	--	(50 × 30)	Vc2	62.2
9	315	20	--	--	(70 × 30)	Vc2	66.8
10 1	315	20	--	--	(50 × 30)	Vc2	62.2
11 2	315	20	--	--	(50 × 30)	Vc8	32.6

Failure due to: V_c2—Bending for column, V_c5—Joint with yielding of horizontal bars, V_c8—Joint due to maximum bond exceedance. ¹ ϑ = −20°, ² ϑ = +20°.

. Analyzed reference joint had a beam of dimensions 30 cm × 50 cm (increased in a case to 30 cm × 70 cm), while the column always had the dimensions 30 cm × 30 cm. The column height was 340 cm, reinforced by using four D16 longitudinal ribbed bars; considered beam had a length of 165 cm, reinforced asymmetrically, by using five D16 and tree D16 longitudinal ribbed bars, for upper and lower sides, respectively. Both column and beam were transversally reinforced by using D8 stirrups (two legs) at a distance of 20 cm each other. Concrete cover was 26 mm. Figure 5 shows the geometry of the considered beam-column joint. In order to evaluate the behavior of proposed simplified model, a parametric analysis was performed to highlight peculiarities of the joint. Main parameters were: axial load on column, N_c; concrete strength, f_c; number and spacing of stirrups in the joint; longitudinal reinforcement ratio in beams and columns; beam height, and hence joint panel dimensions. Table 5 shows main geometrical characteristics, material properties, and failure modes in the case of positive or negative shear, related to considered configurations 1 to 11.

Table 6. Column shear, Vc, for each considered failure mode, related to each configuration.

Column Shear at Failure		Shear Sign	Vc1 (kN)	Vc2 (kN)	Vc3 (kN)	Vc4 (kN)	Vc5 (kN)	Vc6 (kN)	Vc7 (kN)	Vc8 (kN)	Vc9 (kN)	Vc10 (kN)	Vc11 (kN)
Configuration ID Number	1	Pos.	75.98	37.14	135.21	161.06	57.48	58.48	77.28	75.34	40.00	9.99	145.68
		Neg.	46.09	37.14	135.21	161.06	35.46	58.48	46.27	48.62	25.08	6.15	145.68
	2	Pos.	75.98	62.18	135.21	161.06	57.48	93.70	114.64	75.34	40.00	9.99	145.68
		Neg.	46.09	62.18	135.21	161.06	35.46	93.70	76.97	48.62	25.08	6.15	145.68
	3	Pos.	75.98	79.46	135.21	161.06	57.48	120.52	136.73	75.34	40.00	9.99	145.68
		Neg.	46.09	79.46	135.21	161.06	35.46	120.52	103.34	48.62	25.08	6.15	145.68
	4	Pos.	76.99	65.21	135.21	161.06	59.48	103.95	139.14	110.50	57.46	14.17	291.36
		Neg.	46.72	65.21	135.21	161.06	36.13	103.95	81.79	70.05	35.77	8.71	291.36
	5	Pos.	77.70	66.88	135.21	161.06	60.10	107.21	147.97	137.37	70.85	17.38	437.04
		Neg.	47.29	66.88	135.21	161.06	36.35	107.21	83.27	86.47	43.97	10.68	437.04
	6	Pos.	75.98	62.18	135.21	161.06	62.76	93.70	114.64	81.11	46.69	17.28	145.68
		Neg.	46.09	62.18	135.21	161.06	41.10	93.70	76.97	54.68	31.59	12.97	145.68
	7	Pos.	75.98	62.18	135.21	161.06	67.93	93.70	114.64	86.70	53.24	24.45	145.68
		Neg.	46.09	62.18	135.21	161.06	46.65	93.70	76.97	60.62	37.99	19.68	145.68
	8	Pos.	75.98	62.18	135.21	161.06	77.98	93.70	114.64	97.29	65.84	38.35	145.68
		Neg.	46.09	62.18	135.21	161.06	57.49	93.70	76.97	72.09	50.42	32.78	145.68
	9	Pos.	109.79	66.79	192.27	161.06	80.32	164.51	242.60	109.83	57.07	14.06	145.68
		Neg.	66.38	66.79	192.27	161.06	49.06	164.51	116.51	69.82	35.64	8.68	145.68
	10	Pos.	75.98	62.18	135.21	161.06	63.64	64.11	74.91	125.00	85.96	23.38	145.68
		Neg.	46.09	62.18	135.21	161.06	39.58	64.11	55.48	100.54	56.26	14.35	145.68
	11	Pos.	75.98	62.18	135.21	161.06	51.64	131.88	131.88	32.55	11.77	2.86	145.68
		Neg.	46.09	62.18	135.21	161.06	31.91	131.88	117.37	82.06	44.42	11.20	145.68

Failure due to: V_c1—Bending for beam, V_c2—Bending for column, V_c3—Shear for beam, V_c4—Shear for column, V_c5—Joint with yielding of horizontal bars, V_c6—Joint with yielding of vertical upper bars, V_c7—Joint with yielding of vertical lower bars, V_c8—Joint due to maximum bond exceedance, V_c9—Joint due to medium bond exceedance, V_c10—Joint due to minimum bond exceedance, V_c11—Joint due to crushing of concrete strut. Bold numbers indicate failure considering maximum bond. Italic numbers indicate failure considering minimum bond.

provides an insight into the values of ultimate column shear, V_c, in the case of positive or negative shear, leading to each considered failure mode, related to considered configurations 1 to 11.

Table 5 and Table 6 show a significant effect of column axial load, mainly because the bending failure of the column is strictly related to it; once the column failure is prevented, failure of the joint involves horizontal longitudinal bars (and this failure depends on shear sign, e.g., IDs 2 to 5). Adding stirrups (i.e., IDs 6 to 8) to the joint panel improves its capacity, and impacts on joint capacity if joint fails; however, even if joint failure is prevented by enough reinforcement (i.e., IDs 7 and 8), the bending failure of the column or beam depends only on their longitudinal reinforcement and column shear sign. Obviously, bending moment failure of beam is the preferred mode. A higher beam (i.e., ID 9) increases the dimensions of the joint, and the beam capacity too, hence the global capacity increases, but it could push the failure in the column. Friction at joint cracks impacts on the capacity (i.e., IDs 10–11 vs. ID 2) and should be considered as a benefit for the joint behavior; hence, maximum capacity could be considered, but neglecting such friction provides safe side results.

9. Comparison between International Codes and Proposed Model

Joints have been analyzed experimentally and numerically in the past; however, there is still no general agreement in the research community about the assessment of joint behaviour and resistant mechanism, frequently resulting in different design details and amounts of shear reinforcement in the joints provided by international codes and guidelines to assess the column-beam joint behaviour. Particularly, in present paper, the American standard ACI 352, (2002) [34], the Japanese code AIJ, (1999) [35], the European Code, EC8, (2005) [36], and the Italian code (NTC2008 D.M. 14/01/2008—[37]) were considered and their evaluations were compared to the proposed simplified model. Geometrical parameters and material characteristics of each previously considered experimental test were used, and results corroborate the idea that such topic is still an open issue. The comparison should be in terms of columns shear, leading to failure of the beam-column joints. However, referring to Figure 6, the value of the so-called joint shear, V_ij, can be calculated according to Equation (31), as it is the simple parameter commonly adopted in guidelines to evaluate joint capacity.

$${\mathrm{V}}_{\mathrm{ij}}=\mathrm{T}-{\mathrm{V}}_{\mathrm{c}}$$

(31)

In the case of yielded steel bars, tension forces T are calculated by multiplying the bars’ cross section by the yield stress. Prediction models of joint shear capacity proposed by the ACI 352:2002 [34] and the AIJ:1999 [35] are based on the assumption that the failure of joint occurs in correspondence to the maximum value of joint shear. According to the American code [34], the maximum available joint shear value, V_j,av, is calculated according to the Equation (32):

$${\mathrm{V}}_{\mathrm{j},\mathrm{av}}=0.083·\mathsf{\gamma }·\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\prime }}·{\mathrm{A}}_{\mathrm{j}}$$

(32)

where the factor γ considers the confinement effect due to the elements converging into the joint, f′_c is concrete compressive strength, and A_j is the horizontal section of the joint. The evaluation of γ depends on two considerations: (i) joints can dissipate the energy by means of inelastic strains (Type 1) or by plastic strains (Type 2); and (ii) the code defines many joint categories, according to the external confinement. In this case, this kind of joint corresponds to Type 2 and category A.3 (not previously mentioned cases), so the value of γ is equal to 12. According to the Japanese code [35], the maximum available joint shear value, V_j,av, is calculated according to the Equation (33):

$${\mathrm{V}}_{\mathrm{j},\mathrm{av}}=\mathrm{k}·\mathsf{\varphi }·0.8·{\mathrm{f}\prime }_{\mathrm{c}}^{0.7}·{\mathrm{A}}_{\mathrm{j}}$$

(33)

where the factor k considers the number of resisting elements converging into the joint in the direction of the stress, the factor ϕ considers the confinement effect due to the beam, while other symbols have the same meaning as before. In this case, k = 0.7 and ϕ = 0.85 were assumed.

According to the European code [36] and Italian code [37], to check the joint design, the Equation (34) should be satisfied:

$${\mathrm{V}}_{\mathrm{j},\mathrm{bd}}\le \mathsf{\eta }·{\mathrm{f}}_{\mathrm{cd}}·{\mathrm{b}}_{\mathrm{j}}·{\mathrm{h}}_{\mathrm{jc}}·\sqrt{1-\frac{{\mathsf{\nu }}_{\mathrm{d}}}{\mathsf{\eta }}}$$

(34)

where η is equal to

$$\mathsf{\eta }={\mathsf{\alpha }}_{\mathrm{j}}·\left(1-\frac{{\mathrm{f}}_{\mathrm{ck}}}{250}\right)$$

(35)

where f_ck is the characteristic compressive strength of concrete in (MPa) and α_j = 0.48, f_cd is the design compressive strength of concrete, b_j is the width of the joint, equal to the MIN[MAX (B_c; B_b); MIN (B_c + H_c/2; B_b + H_c/2)], where Bc is the width of the column, Bb is the width of the beam, H_c is the height of the column, h_jc is the distance between the most external layers of reinforcements into the column, and $\mathsf{\nu }$_d is the axial load ratio in the column above the joint, accounting for the axial capacity of the concrete only. In addition, European Code [36] and Italian Code [37] prescribe a minimum value of transverse reinforcement according to Equation (36) in order to avoid diagonal cracking of the joint due to the attainment in diagonal of design tensile strength of concrete, f_ctd.

$$\frac{{\mathrm{A}}_{\mathrm{sh}}·{\mathrm{f}}_{\mathrm{ywd}}}{{\mathrm{b}}_{\mathrm{j}}·{\mathrm{h}}_{\mathrm{jw}}}\ge \frac{{\left(\frac{{\mathrm{V}}_{\mathrm{jbd}}}{{\mathrm{b}}_{\mathrm{j}}·{\mathrm{h}}_{\mathrm{jc}}}\right)}^2}{{\mathrm{f}}_{\mathrm{ctd}}+{\mathsf{\nu }}_{\mathrm{d}}·{\mathrm{f}}_{\mathrm{cd}}}-{\mathrm{f}}_{\mathrm{ctd}}$$

(36)

where A_sh is the whole section of the stirrups, f_ywd is the design yielding stress of stirrups, b_j is the width of the joint, and h_jw is the distance between upper and lower layers of reinforcement into the beam. Furthermore, Italian Code [37] provides two expressions to perform checks in compression and tension for existing nodes, which allows the calculation of the maximum values of diagonal compression and tensile stresses inside the concrete, which must be lower than percentages of the concrete compressive strength f_c. Equations (37) and (38) are used to calculate the values of maximum diagonal compressive stress σ_nc and the maximum diagonal tension stress σ_nt, respectively.

$${\mathsf{\sigma }}_{\mathrm{nc}}=\frac{{\mathrm{N}}_{\mathrm{c}}}{2·{\mathrm{A}}_{\mathrm{g}}}+\sqrt{{\left(\frac{{\mathrm{N}}_{\mathrm{c}}}{2·{\mathrm{A}}_{\mathrm{g}}}\right)}^2+{\left(\frac{{\mathrm{V}}_{\mathrm{n}}}{{\mathrm{A}}_{\mathrm{g}}}\right)}^2}\le 0.5·{\mathrm{f}}_{\mathrm{c}}$$

(37)

$${\mathsf{\sigma }}_{\mathrm{nt}}=\left|\frac{{\mathrm{N}}_{\mathrm{c}}}{2·{\mathrm{A}}_{\mathrm{g}}}+\sqrt{{\left(\frac{{\mathrm{N}}_{\mathrm{c}}}{2·{\mathrm{A}}_{\mathrm{g}}}\right)}^2+{\left(\frac{{\mathrm{V}}_{\mathrm{n}}}{{\mathrm{A}}_{\mathrm{g}}}\right)}^2}\right|\le 0.3·\sqrt{{\mathrm{f}}_{\mathrm{c}}}$$

(38)

where N_c is the axial load on the upper column, A_g is the horizontal section of the joint, V_n is the joint shear defined as V_ij in Equation (31), and f_c is the concrete compressive strength (in MPa in (38)).

Joint Shear Failure

As a first step, the value of column shear, V_c, has been expressed as joint shear, and it was compared to the predictions by codes and guidelines in

Table 7. Comparison between proposed model, codes, and experimental results.

	Experimental Joints
	Joint Shear (kN)	Limit Joint Shear (kN)	Joint Shear (kN)	Joint Shear (kN)	Limit Joint Shear (kN)
	Model	ACI	AIJ	EC8	Experimental
El-Amoury and Ghobarah (2002) [32] T0	474.73	550.96	521.93	793.31	475.30
Masi et al. (2009) [27] T1	91.36	378.72	322.10	462.26	90.07
Russo and Pauletta (2012) [33]	63.78	421.98	374.76	702.67	64.50

. As shown there, proposed model is comparable to experimental outcomes, but it would be interesting to understand the differences in terms of failure predictions with the codes.

Figure 7 shows the comparison between the joint shear for experimental tests, codes, and the proposed model. The European code [36] presents the higher values of joint shear limit for each considered case, while proposed model presents the lower ones.

All the considered codes overestimate the experimental outcomes in terms of joint shear; hence, they could seem not on safe side, but according to Equation (31) they provide, conversely, extremely reduced and over conservative ultimate column shear values related to joint failure. This apparent contradiction can be explained, since it has been shown in many research papers [4,5,11] that the so-called “joint shear” is an inappropriate index for the evaluation of vulnerability of beam-column joints due to joint shear failure, despite the fact it is usually assumed to be the main, simple parameter controlling this failure mode. In fact, the joint shear stress increases even after apparent beginning of joint shear failure, as well as after deterioration of column shear. The reason of the degradation in column shear is due also to the anchorage capacity of longitudinal reinforcement in the beams passing through the joint panel [4,5]. When the anchorage stress attains its capacity, the internal moment lever arms reduce, followed by a deterioration of flexural strength (and variation of forces T, too, hardly quantifiable by design equations in codes). All these phenomena acting together lead to a global reduction of the beam-column joint performances [11].

10. Conclusions

Present paper provides a theoretical yet simplified formulation accounting for failure mechanisms of beam-column joints. The possible different failure modes of beam-column joints are various, but some of them can be considered preferable in respect to others. As is known, a ductile-type failure is preferable to a brittle-type failure; therefore, a design can be performed considering the strength hierarchy. This principle allows for the identification of a better way to design a new joint or eventually retrofit an existing one (e.g., adopting traditional or innovative solutions like as FRP, FRCM, or even newer technologies [38,39,40,41,42,43,44]) to achieve the desired behaviour and failure mode.

The proposed formulation requires the solution of a system of nine equations: the column shear force V_c is the driving parameter, and the equations allow us to correlate each internal stress to the corresponding V_c, and each considered failure mode is associated with a threshold value of internal stresses yielding to the evaluation of V_c1 to V_c11. The minimum value of V_{c_} is the capacity of the beam-column joint and provides the associated failure mode, too.

In relation to the type and the amount of reinforcement used, for instance, it is possible to switch from a brittle failure to a ductile failure. According to the strength hierarchy, this can be obtained by letting the V_c associated with a ductile failure of the flexural beam failure (e.g., the flexural beam failure associated with V_c1) be the lowest V_c compared to other brittle failures (e.g., a joint failure associated with V_c5 to V_c11). However, it also appeared clearly that significant errors at the original design phase or very low bond in the smooth bars could jeopardize the success of retrofit interventions for existing joints, requiring excessive amounts of strengthening.

The proposed theoretical model was compared to experimental tests and to international code provisions. The proposed model showed its advantages, as it accurately indicated the failure mode and capacity, dissimilarly from the codes. The reason for this is believed to be that the codes consider, implicitly, only a few potential failure modes. Often this led to poor correlations, when considering the codes, which were over-conservative, thus enforcing the need to develop effective, yet simplified, models, considering not only joint shear but the whole joint assembly behaviour, as in the case of the proposed model.