# Prediction of Joint Shear Deformation Index of RC Beam–Column Joints

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Test Program

#### 2.1. Test Specimens

#### 2.2. Material Properties

#### 2.3. Test Setup and Loading History

#### 2.4. Instrumentation

## 3. Test Results

#### 3.1. Lateral Load–Drift Ratio Relationships and Failure Modes

#### 3.2. Strains of Beam Reinforcement

- At the peak strength of the specimens ($\delta =2.0\%$), the strains in flexural beam bars measured at the beam end (i.e., the joint face) were lower than the yield strains (${\epsilon}_{yt}$ = 2604 μ) or equal to the yield strains. In the subsequent loading, the strains at the beam end did not increase. This indicates that the specimens failed owing to joint shear.
- The amount of shear damage in the joint significantly decreased when the moderate amount of joint shear reinforcement was provided. Thus, the beam reinforcement strains at the beam end increased (compare the strains in Figure 10a–c). This indicates that the joint shear reinforcement successfully reduced the joint shear damage and increased the maximum lateral loads.
- The beam reinforcement at the column right face under positive loading was subjected to compressive stress at the beginning of the test. However, above $\delta =1.0\%$, the bond deterioration of the beam-reinforcing bar passing through the joint occurred. Consequently, the transition of the compressive stress in the beam bar on the compression side at the beam end section to the tensile stress occurred (Figure 10a–c). The concrete took this compressive stress in the beam bar on the compression side, and the height of the compressive zone for the concrete may have increased, resulting in a smaller moment of the lever arm. Therefore, the moment in the beam section at the column face may have decreased. This phenomenon may have caused lateral strength degradation and stiffness degradation in all the test specimens (Figure 8). Our experiments are consistent with previous findings in the literature (Hakuto et al. [5] and Shiohara [12]).

#### 3.3. Joint Shear Stress

#### 3.4. Decomposition of Lateral Drift

## 4. Analytical Studies of Interior Beam–Column Joints

#### 4.1. Finite-Element Modeling and Verification

#### 4.2. Parametric Studies

^{2}, 550 × 550 mm

^{2}, and 600 × 600 mm

^{2}, respectively. The following section presents the results of the parametric study. Additionally, the regression and correlation analysis were performed to investigate the relationship between the selected design parameters and joint shear behavior. Based on the analysis, three simple equations are proposed to predict the contribution of the joint deformation to the total deformation of beam–column joint connections.

#### 4.3. Shear Stress and Shear Deformation of Interior Joints

_{0}, and all the specimens exhibited B–Failure.

_{0}is shown in Figure 16d. According to the results, joint failure did not occur even when the ${v}_{ju}/\sqrt{{f}_{c}^{\prime}}$ was relatively high. This means that joint failure does not always result from a lack of shear strength of the joint but rather insufficient moment resisting capacity. In that case, the joint shear deformation increases, although the joint maintains its shear strength. Thus, the joint acts similar to a hinge. The increase in the proposed coefficient K

_{0}, together with the joint shear reinforcement shifts the failure plane to a beam flexural hinge yield mechanism. This is because the moment resisting capacity of the joints increases with respect to the increasing amount of joint hoop and column-to-beam flexural strength ratio. These results are inconsistent with the ASCE 41 provisions for the shear-strength evaluation of interior joints, i.e., ${v}_{ju}=0.83\sqrt{{f}_{c}^{\prime}}$ for $s>0.5{h}_{c}$ and ${v}_{ju}=1.245\sqrt{{f}_{c}^{\prime}}$ for $s\le 0.5{h}_{c}$. Therefore, the “joint shear” is a useful index for the induced force level but is not suitable for defining joint failure. In addition to the joint shear index, it is necessary to consider the “shear deformation index” (SDI), which can be used to define the failure of beam–column connections.

_{1}and K

_{2}= ρ

_{j}K

_{1}is shown in Figure 17a–c for two expected structural deformations (in ASCE 41-13, the lateral story drift ratio of moment frames should be ≥4.0% for collapse prevention and 2.0% for life safety). The joint shear reinforcement ratio, area ratio of adjoining members, and column-to-beam flexural strength ratio (${\rho}_{j}$, ${A}_{c}/{A}_{b}$, and $\sum {M}_{c}/\sum {M}_{b}$, respectively) were considered as the parameters. In this study, the structural deformations were assumed to be 3.5% for collapse prevention and 2.5% for life safety. Assuming that the joint deformation of the beam column joint contributes to the total lateral drift ratio, the SDI is defined as the ratio of the lateral drift ratio ${R}_{sps}$ (Equation (2)) due to the joint deformation to the total lateral drift ratio ${R}_{s}$.

_{1}and K

_{2}= ρ

_{j}K

_{1}were proposed.

- $\sum {M}_{c}/\sum {M}_{b}<1.5$
- $({A}_{s,top}{f}_{y})/({b}_{b}d{f}_{c}^{\prime})>0.25$

- $1.5\le \sum {M}_{c}/\sum {M}_{b}<2.0$
- $0.15<({A}_{s,top}{f}_{y})/({b}_{b}d{f}_{c}^{\prime})\le 0.25$
- $0.25\%\le {\rho}_{j}<0.5\%$.

- $2.0\le \sum {M}_{c}/\sum {M}_{b}<3.0$
- $0.05<({A}_{s,top}{f}_{y})/({b}_{b}d{f}_{c}^{\prime})\le 0.15$
- $0.5\%\le {\rho}_{j}<1.0\%$.

#### 4.4. Verification of Proposed Equations

## 5. Conclusions

- With regard to the strength and stiffness, the performance of the test specimens was satisfactory up to $\delta $ = 2.0% (lateral drift ratio), beyond which the strength and stiffness generally degraded. The maximum loads occurred at $\delta $ = 2.0–2.5%, after which concrete cracking and spalling became severe at $\delta $ = 3.5–4.0%. Among the eight specimens, S13-N exhibited the worst performance. This was mainly due to the absence of joint shear reinforcement and the smaller flexural strength ratio.
- Experimental and finite-element investigations indicated that throughout the lateral loading, the joint shear stress increased, while the width of the diagonal shear cracks on the joint core surfaces increased. The maximum joint shear stress values exceeded the limits of ${v}_{ju}=0.83\sqrt{{f}_{c}^{\prime}}$ for $s>0.5{h}_{c}$ and ${v}_{ju}=1.245\sqrt{{f}_{c}^{\prime}}$ for $s\le 0.5{h}_{c}$ suggested by ASCE 41. The joint shear deformation contributed approximately 40% of the total lateral drift at the maximum shear stress ${v}_{j}$, and the corresponding lateral drift ratio was approximately 3.0–3.5%. This indicates that the “joint shear” is a useful index for the beam–column joints with high shear stress levels of ${v}_{j}\ge 1.7\text{}\sqrt{{f}_{c}^{\prime}}$ but is unsuitable for defining the shear failure of beam–column joints with medium or low shear stress levels of ${v}_{j}\approx 1.25\u20131.5\sqrt{{f}_{c}^{\prime}}$ and ${v}_{j}\approx 1.0\sqrt{{f}_{c}^{\prime}}$.
- Using the results of parametric studies (39 specimens), the shear stress of the interior joints was investigated considering the design variables ${\rho}_{j}$, ${h}_{b}/{h}_{c}$, and $\sum {M}_{c}/\sum {M}_{b}$. The maximum shear stress ${v}_{ju}$ varied between 0.85$\sqrt{{f}_{c}^{\prime}}$ and 1.65$\sqrt{{f}_{c}^{\prime}}$. However, joint failure did not occur in some specimens, even if ${v}_{ju}/\sqrt{{f}_{c}^{\prime}}$ was relatively large. This is because the increase in the proposed coefficient K
_{0,}together with the joint shear reinforcement shifts the failure plane to a beam flexural hinge yield mechanism. - The shear deformation of interior joints was examined with consideration of the design variables ${\rho}_{j}$, ${A}_{c}/{A}_{b}$ and $\sum {M}_{c}/\sum {M}_{b}$. The contribution of the joint shear deformation to the total deformation ranged from 10.0% to 80.0% depending on the values of the proposed coefficients ${K}_{1}$ and ${K}_{2}={\rho}_{j}{K}_{1}$. The joints with smaller values of ${K}_{1}$ and ${K}_{2}$ performed poorly, exhibiting wide inclined cracks and deformations that accounted for up to 80% of the overall lateral deformation ($\delta $ = 2.5% and $\delta $ = 3.5%). Larger values of ${K}_{1}$ and ${K}_{2}$ yielded smaller deformation of the joint region.
- The design based on limiting the joint shear stress can be used for safety. However, it should be supplemented with consideration of the corresponding joint shear deformation to define the joint failure clearly. According to the results, three simple equations were proposed for predicting the joint deformation contribution to the total story drift of beam–column joints under critical structural deformations. The equations were able to predict SDI values with reasonable agreement with the experimental data. Compared with previously proposed models and theories, our method does not require complex nonlinear numerical analyses of the structure or sub-assemblage.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Current Design Methods

#### Appendix A.1. Flexural Yielding of Beam or Column

**Figure A1.**Internal forces and reaction forces acting on an interior beam–column connection under a lateral load.

#### Appendix A.2. Shear Strength of Beam–Column Joint

#### Appendix A.3. Bond Strength of Beam Reinforcement

#### Appendix A.4. Strength Predictions for Test Specimens

Specimen | Column Yielding | Beam Yielding | Joint Shear Failure | $\mathbf{Predicted}\text{}\mathbf{Strength},\text{}{\mathit{P}}_{\mathit{n}}\text{}\left(\mathbf{kN}\right)$ | Failure Mode | Anchorage Length | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{M}}_{\mathit{n}\mathit{c}},\text{}(\mathbf{kN}\xb7\mathbf{m})$ | ${\mathit{P}}_{\mathit{n}\mathit{c}},\text{}\left(\mathbf{kN}\right)$ | ${\mathit{M}}_{\mathit{n},\mathit{b}\mathit{l}},\text{}\left(\mathbf{kN}\text{\xb7}\mathbf{m}\right)$ | ${\mathit{M}}_{\mathit{n},\mathit{b}\mathit{r}},\text{}(\mathbf{kN}\xb7\mathbf{m})$ | ${\mathit{P}}_{\mathit{n}\mathit{b}},\text{}\left(\mathbf{kN}\right)$ | Conforming | Nonconforming | |||||||

${\mathit{V}}_{\mathit{j}\mathit{n}}^{\mathit{C}},\text{}\left(\mathbf{kN}\right)$ | ${\mathit{P}}_{\mathit{j}\mathit{n}}^{\mathit{C}},\text{}\left(\mathbf{kN}\right)$ | ${\mathit{V}}_{\mathit{j}\mathit{n}}^{\mathit{N}\mathit{C}},\text{}\left(\mathbf{kN}\right)$ | ${\mathit{P}}_{\mathit{j}\mathit{n}}^{\mathit{N}\mathit{C}},\text{}\left(\mathbf{kN}\right)$ | ${\mathit{l}}_{\mathit{a}\mathit{l}}\text{}\left(\mathbf{mm}\right)$ | $\frac{{\mathit{l}}_{\mathit{a}\mathit{l}}}{{\mathit{l}}_{\mathit{d}\mathit{h}}}$ | ||||||||

S16-N | 78.0 | 125 | 51.6 | 51.6 | 76.4 | - | - | 247.93 | 39.9 | 39.9 | J | 250 | 0.78 |

S16-32 | 77.7 | 124 | 51.5 | 51.5 | 76.3 | 367 | 59.1 | - | - | 59.1 | J | 250 | 0.78 |

S16-34 | 78.2 | 125 | 51.6 | 51.6 | 76.4 | 375 | 60.2 | - | - | 60.2 | J | 250 | 0.78 |

S13-N | 62.8 | 100 | 57.4 | 57.4 | 85.0 | - | - | 254 | 40.9 | 40.9 | J | 250 | 0.96 |

S13-32 | 63.0 | 101 | 57.8 | 57.8 | 85.6 | 389 | 62.7 | - | - | 62.7 | J | 250 | 0.96 |

S13-34 | 63.1 | 101 | 57.8 | 57.8 | 85.6 | 391 | 62.8 | - | - | 62.8 | J | 250 | 0.96 |

U13-N | 79.9 | 128 | 56.9 | 42.1 | 73.3 | - | - | 264 | 42.4 | 42.4 | J | 250 | 0.96 |

U13-34 | 79.9 | 128 | 56.9 | 42.1 | 73.3 | 396 | 63.6 | - | - | 63.6 | BJ | 250 | 0.96 |

## Appendix B. Constitutive Model for Nonlinear Fe Analysis of Test Specimens

#### Appendix B.1.Constitutive Law for Concrete

#### Appendix B.1.1. Tensile Behavior

**Figure A2.**Model for concrete: (

**a**) predefined tensile stress–strain curve; (

**b**) predefined compressive stress–strain curve.

#### Appendix B.1.2. Compressive Behavior

#### Appendix B.2. Constitutive Law for Reinforcement

**Figure A3.**Model for the reinforcement: (

**a**) stress–strain curve for the steel reinforcement; (

**b**) bond–slip law based on CEB-Figure 1990.

#### Bond–Slip Law

#### Appendix B.3. Geometry Modeling

## Appendix C. Analysis Variables of Parametric Study

Specimens | Geometric Properties | Top Rebar of Beam | Bottom Rebar of Beam | Joint Hoop Ratio | Joint Aspect Ratio | Area Ratio | Strength Ratio | Failure Mode | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(a) | (b) | (c) | (d) | |||||||||||||||

L | H | h_{c} | b_{c} | h_{b} | b_{b} | d_{b} | A_{s} | f_{y} | d_{b} | A_{s} | f_{y} | ρ_{j}, (%) | ${\mathit{h}}_{\mathit{b}}/{\mathit{h}}_{\mathit{c}}$ | ${\mathit{A}}_{\mathit{c}}/{\mathit{A}}_{\mathit{b}}\text{}$ | $\frac{\sum \mathit{M}\mathit{c}}{\sum {\mathit{M}}_{\mathit{b}}}$ | FE Prediction | ||

Group 1 | M1 | 6000 | 3000 | 600 | 600 | 500 | 400 | 22 | 1548 | 431 | 19 | 1146 | 431 | - | 0.83 | 1.8 | 2.5 | B |

M2 | 0.3 | 0.83 | 1.8 | 2.5 | B | |||||||||||||

M3 | 0.6 | 0.83 | 1.8 | 2.5 | B | |||||||||||||

M4 | 25 | 2026 | - | 0.83 | 1.8 | 2.2 | B | |||||||||||

M5 | 0.3 | 0.83 | 1.8 | 2.2 | B | |||||||||||||

M6 | 0.6 | 0.83 | 1.8 | 2.2 | B | |||||||||||||

M7 | 600 | 400 | 22 | 1548 | - | 1.0 | 1.5 | 2.0 | BJ | |||||||||

M8 | 0.24 | 1.0 | 1.5 | 2.0 | B | |||||||||||||

M9 | 0.48 | 1.0 | 1.5 | 2.0 | B | |||||||||||||

M10 | 25 | 2026 | - | 1.0 | 1.5 | 1.7 | BJ | |||||||||||

M11 | 0.24 | 1.0 | 1.5 | 1.7 | B | |||||||||||||

M12 | 0.48 | 1.0 | 1.5 | 1.7 | B | |||||||||||||

M13 | 700 | 400 | 22 | 1548 | - | 1.17 | 1.29 | 1.7 | BJ | |||||||||

M14 | 0.2 | 1.17 | 1.29 | 1.7 | B | |||||||||||||

M15 | 0.4 | 1.17 | 1.29 | 1.7 | B | |||||||||||||

M16 | 25 | 2026 | - | 1.17 | 1.29 | 1.5 | BJ | |||||||||||

M17 | 0.2 | 1.17 | 1.29 | 1.5 | B | |||||||||||||

M18 | 0.4 | 1.17 | 1.29 | 1.5 | B | |||||||||||||

Group 2 | M19 | 6000 | 3000 | 500 | 500 | 500 | 400 | 22 | 1548 | - | 1.0 | 1.25 | 1.7 | B | ||||

M20 | 0.36 | 1.0 | 1.25 | 1.7 | B | |||||||||||||

M21 | 0.72 | 1.0 | 1.25 | 1.7 | B | |||||||||||||

M22 | 25 | 2026 | - | 1.0 | 1.25 | 1.5 | BJ | |||||||||||

M23 | 0.36 | 1.0 | 1.25 | 1.5 | B | |||||||||||||

M24 | 0.72 | 1.0 | 1.25 | 1.5 | B | |||||||||||||

M25 | 600 | 400 | 22 | 1548 | - | 1.2 | 1.04 | 1.4 | BJ | |||||||||

M26 | 0.29 | 1.2 | 1.04 | 1.4 | BJ | |||||||||||||

M27 | 0.57 | 1.2 | 1.04 | 1.4 | B | |||||||||||||

M28 | 25 | 2026 | - | 1.2 | 1.04 | 1.2 | J | |||||||||||

M29 | 0.29 | 1.2 | 1.04 | 1.2 | BJ | |||||||||||||

M30 | 0.57 | 1.2 | 1.04 | 1.2 | B | |||||||||||||

M31 | 700 | 400 | 22 | 1548 | - | 1.4 | 0.89 | 1.2 | J | |||||||||

M32 | 0.24 | 1.4 | 0.89 | 1.2 | BJ | |||||||||||||

M33 | 0.48 | 1.4 | 0.89 | 1.2 | B | |||||||||||||

M34 | 25 | 2026 | - | 1.4 | 0.89 | 1.0 | J | |||||||||||

M35 | 0.24 | 1.4 | 0.89 | 1.0 | BJ | |||||||||||||

M36 | 0.48 | 1.4 | 0.89 | 1.0 | B | |||||||||||||

M37 | 550 | 550 | 600 | 400 | 25 | 2026 | - | 1.1 | 1.26 | 1.46 | BJ | |||||||

M38 | 0.26 | 1.1 | 1.26 | 1.46 | B | |||||||||||||

M39 | 0.52 | 1.1 | 1.26 | 1.46 | B |

**a**) L = beam length (mm); H = column height (mm); h

_{c}= column depth (mm); b

_{c}= column width (mm); h

_{b}= beam depth (mm); b

_{b}= beam width (mm); (

**b**) d

_{b}= rebar diameter (mm); A

_{s}= area of rebar (mm

^{2}); and f

_{y}= yield strength of rebar (MPa); (

**c**) yield strength of joint hoop is assumed to be 345 MPa; (

**d**) B = beam failure; BJ = beam joint failure (joint failure after beam yield); and J = joint failure (joint failure before beam yield.

## Appendix D. Summary of Beam–Column Connection Tests

Research Team | Specimens | Joint Hoop Ratio | Joint Aspect Ratio | Area Ratio | Strength Ratio | Mechanical Reinforcement Ratio | SDI at 2.5% | SDI at 3.5% | Failure Mode |
---|---|---|---|---|---|---|---|---|---|

ρ_{j}, (%) | $\frac{{\mathit{h}}_{\mathit{b}}}{{\mathit{h}}_{\mathit{c}}}$ | $\frac{{\mathit{A}}_{\mathit{c}}}{{\mathit{A}}_{\mathit{b}}}$ | $\frac{\sum {\mathit{M}}_{\mathit{c}}}{\sum {\mathit{M}}_{\mathit{b}}}$ | $\frac{{\mathit{A}}_{\mathit{s},\mathit{t}\mathit{o}\mathit{p}}{\mathit{f}}_{\mathit{y}}}{{\mathit{b}}_{\mathit{b}}\mathit{d}{\mathit{f}}_{\mathit{c}}^{\prime}}$ | $\frac{{\mathit{R}}_{\mathit{s}\mathit{p}\mathit{s}}}{{\mathit{R}}_{\mathit{s}}}$ | $\frac{{\mathit{R}}_{\mathit{s}\mathit{p}\mathit{s}}}{{\mathit{R}}_{\mathit{s}}}$ | Test Result | ||

Fuji and Morita (1991) [17] | A1 | 0.52 | 1.14 | 1.21 | 1.24 | 0.50 | 0.62 | 0.69 | J |

A2 | 0.52 | 1.14 | 1.21 | 2.02 | 0.19 | 0.41 | 0.62 | J | |

A3 | 0.52 | 1.14 | 1.21 | 1.24 | 0.50 | 0.65 | 0.72 | J | |

A4 | 0.69 | 1.14 | 1.21 | 1.24 | 0.50 | 0.69 | 0.72 | J | |

Joh et al. (1991 [16]) | HL | 1.27 | 1.17 | 1.29 | 2.41 | 0.09 | 0.03 | 0.04 | B |

MH | 0.55 | 1.17 | 1.29 | 2.41 | 0.09 | 0.03 | 0.05 | B | |

B9 | 1.1 | 1.17 | 1.29 | 2.41 | 0.10 | 0.05 | 0.07 | B | |

B10 | 1.1 | 1.17 | 1.29 | 2.41 | 0.10 | 0.06 | 0.09 | B | |

B11 | 1.1 | 1.17 | 1.29 | 2.41 | 0.10 | 0.07 | 0.10 | B | |

Noguchi and Kashiwazaki (1992) [38] | J1 | 0.66 | 1.00 | 1.50 | 1.53 | 0.24 | 0.40 | 0.48 | BJ |

J3 | 0.66 | 1.00 | 1.50 | 1.48 | 0.17 | 0.37 | 0.41 | J | |

J4 | 0.66 | 1.00 | 1.50 | 1.53 | 0.24 | 0.37 | 0.48 | BJ | |

J5 | 0.66 | 1.00 | 1.50 | 1.37 | 0.26 | 0.31 | 0.40 | J | |

J6 | 0.66 | 1.00 | 1.50 | 1.47 | 0.27 | 0.30 | 0.33 | J | |

Oka and Shiohara (1992) [39] | J1 | 0.46 | 1.00 | 1.25 | 1.72 | 0.15 | 0.40 | 0.60 | BJ |

J7 | 0.46 | 1.00 | 1.25 | 2.12 | 0.13 | 0.10 | 0.13 | B | |

J10 | 0.46 | 1.00 | 1.25 | 1.35 | 0.34 | 0.44 | 0.58 | J | |

Kimamura et al. (2000) [18] | No.1 | 0.15 | 1.00 | 1.39 | 1.7 | 0.24 | 0.47 | 0.53 | J |

No.2 | 0.31 | 1.00 | 1.39 | 1.7 | 0.24 | 0.51 | 0.44 | J | |

No.3 | 0.62 | 1.00 | 1.39 | 1.7 | 0.24 | 0.42 | 0.44 | J | |

No.4 | 0.31 | 1.00 | 1.39 | 2.56 | 0.16 | 0.11 | 0.11 | B | |

No.5 | 0.62 | 1.00 | 1.39 | 2.56 | 0.16 | 0.08 | 0.08 | B | |

Li and Leong (2014) [40] | NS1 | 0.71 | 1.11 | 1.08 | 2.11 | 0.07 | N/A | 0.09 | B |

AS1 | 0.71 | 1.11 | 1.08 | 3.59 | 0.07 | N/A | 0.03 | B | |

NS2 | 0.48 | 1.11 | 1.08 | 1.86 | 0.03 | N/A | 0.10 | B | |

AS2 | 0.48 | 1.11 | 1.08 | 3.9 | 0.03 | N/A | 0.03 | B | |

NS3 | 0.71 | 1.11 | 1.08 | 2.35 | 0.07 | N/A | 0.09 | B | |

AS3 | 0.71 | 1.11 | 1.08 | 3.83 | 0.07 | N/A | 0.03 | B | |

NS4 | 0.57 | 1.11 | 1.08 | 2.82 | 0.03 | N/A | 0.11 | B | |

AS4 | 0.57 | 1.11 | 1.08 | 5.15 | 0.03 | N/A | 0.03 | B | |

Hwang et al. (2014) [41] | C1-400 | 1.34 | 0.91 | 1.57 | 1.67 | 0.31 | 0.18 | 0.20 | BJ |

C2-600 | 1.34 | 0.91 | 1.57 | 1.68 | 0.27 | 0.14 | 0.18 | BJ | |

C3-600 | 1.34 | 0.91 | 1.41 | 1.22 | 0.27 | 0.15 | 0.19 | BJ | |

C4-600 | 1.34 | 0.91 | 1.57 | 1.88 | 0.27 | 0.16 | 0.19 | BJ | |

Melo et al. (2014) [42] | IPA-1 | 0 | 1.67 | 0.60 | 0.94 | 0.06 | 0.60 | 0.82 | J |

IPA-2 | 0 | 1.67 | 0.60 | 0.99 | 0.04 | 0.40 | 0.75 | J | |

IPB | 0 | 1.67 | 0.60 | 0.96 | 0.06 | 0.50 | 0.62 | J | |

IPE | 0 | 1.67 | 0.60 | 1.23 | 0.06 | 0.36 | 0.75 | J | |

ID | 0 | 1.67 | 0.60 | 0.88 | 0.07 | 0.52 | 0.77 | J | |

Alaee and Li (2017) [43] | IN80 | 0.71 | 1.11 | 1.08 | 1.71 | 0.05 | 0.14 | 0.11 | B |

IH80 | 0.57 | 1.11 | 1.08 | 2.18 | 0.06 | 0.26 | 0.20 | BJ | |

IH80A | 0.57 | 1.11 | 1.08 | 3.75 | 0.06 | 0.05 | 0.05 | B | |

IN100 | 0.71 | 1.11 | 1.08 | 1.72 | 0.04 | 0.26 | 0.20 | BJ | |

IH100 | 0.71 | 1.11 | 1.08 | 2.19 | 0.05 | 0.31 | 0.29 | BJ | |

IH60 | 0.57 | 1.11 | 1.08 | 2.29 | 0.06 | 0.31 | 0.25 | BJ | |

IH60A | 0.57 | 1.11 | 1.08 | 3.23 | 0.06 | 0.09 | 0.07 | B | |

Yang and Zhao (2018) [44] | CL1 | 1.17 | 1.25 | 0.91 | 1.23 | 0.13 | 0.27 | 0.61 | BJ |

CL2 | 1.54 | 1.25 | 1.07 | 1.24 | 0.19 | 0.37 | 0.73 | BJ | |

CL3 | 1.60 | 0.89 | 1.58 | 1.37 | 0.23 | 0.41 | 0.73 | BJ | |

CL4 | 1.54 | 1.25 | 1.07 | 1.14 | 0.16 | 0.34 | 0.61 | BJ |

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**Figure 1.**Dimensions and reinforcement details of the specimens. (

**a**) Group 1 and Group 3 specimens, (

**b**) Group 2 specimens.

**Figure 6.**(

**a**) Measurement system for the joint deformation and fixed end rotation; (

**b**) strain-gauge locations (unit in mm).

**Figure 7.**Measurement mechanism of the system: (

**a**) For chord rotations and fixed end rotations; (

**b**) For deformation of joint.

**Figure 9.**Failure modes at the end of the test: (

**a**) S16-N; (

**b**) S13-N; (

**c**) U13-N; (

**d**) S16-32; (

**e**) S13-32; (

**f**) S16-34; (

**g**) S13-34; (

**h**) U13-34.

**Figure 10.**Strain distributions of the beam flexural reinforcement: (

**a**) S13-N; (

**b**) S13-32; and (

**c**) S13-34.

**Figure 12.**Decomposition of the lateral drift of the interior beam–column joint: (

**a**) percentage of deformation for Groups 1, 2, and 3 at a lateral drift of 3.5%; (

**b**) deformation components.

**Figure 16.**Maximum shear stress at the interior joints for different combinations of design variables: (

**a**) no joint reinforcement ratio; (

**b**) low joint reinforcement ratio; (

**c**) moderate joint reinforcement ratio; (

**d**) total design variables.

**Figure 17.**Shear deformation index (SDI) of interior connections for different combinations of design variables: (

**a**) low ductility demand (<2); (

**b**) moderate ductility demand (2–4); and (

**c**) high ductility demand (>4).

**Figure 21.**Experimental shear deformation index (SDI) versus the SDI predicted by proposed equations: (

**a**) J-mode predicted by Equation (3); (

**b**) BJ-mode predicted by Equation (4); and (

**c**) B-mode predicted by Equation (5).

Specimens | S16-N | S16-32 | S16-34 | S13-N | S13-32 | S13-34 | U13-N | U13-34 | |
---|---|---|---|---|---|---|---|---|---|

Concrete Strength, ${f}_{c}^{\prime}$ (MPa) | 28.2 | 27.5 | 28.6 | 29.7 | 30.9 | 31.1 | 32.1 | 31.9 | |

Axial Load Ratio, (${N}_{u}/{f}_{c}^{\prime}{A}_{g}$) | 0.07 | 0.08 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | |

Beam | Width × Depth in mm | 200 × 250 | 200 × 250 | 200 × 250 | |||||

Top Rebar, % | 1.35 | 1.44 | 1.44 | ||||||

Bottom Rebar, % | 1.35 | 1.44 | 0.88 | ||||||

Column | Width × Depth in mm | 250 × 250 | 250 × 250 | 250 × 250 | |||||

Reinforcing Bar Ratio (%) | 2.43 | 1.62 | 2.43 | ||||||

Joint | Hoop Ratio ${\rho}_{j}$ (%) | - | 0.36 | 0.72 | - | 0.36 | 0.72 | - | 0.72 |

Flexural Strength Ratio | 1.5 | 1.5 | 1.5 | 1.1 | 1.1 | 1.1 | 1.6 | 1.6 | |

Joint Demand Ratio (${V}_{jn}/{V}_{j}$) | 0.82 | 0.76 | 0.91 | ||||||

Anchorage Length Ratio (${h}_{c}/{d}_{b})$ | 15.6 | 19.2 | 19.2 |

Diameter | Grade | $\mathbf{Yield}\text{}\mathbf{Strength},\text{}{\mathit{f}}_{\mathit{y}},\text{}\left(\mathbf{MPa}\right)$ | $\mathbf{Yield}\text{}\mathbf{Strain},\text{}{\mathit{\epsilon}}_{\mathit{y}}\text{}\left(\text{\xb5}\right)$ | $\mathbf{Ultimate}\text{}\mathbf{Strength},\text{}{\mathit{f}}_{\mathit{u}},\text{}\left(\mathbf{MPa}\right)$ | $\mathbf{Elastic}\text{}\mathbf{Modulus},\text{}{\mathit{E}}_{\mathit{s}}\text{}\left(\mathbf{GPa}\right)$ |
---|---|---|---|---|---|

D6 | SD345 | 363 | 1998 | 542 | 182 |

D13 | SD390 | 498 | 2602 | 669 | 192 |

D16 | 440 | 2449 | 618 | 180 |

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## Share and Cite

**MDPI and ACS Style**

Gombosuren, D.; Maki, T.
Prediction of Joint Shear Deformation Index of RC Beam–Column Joints. *Buildings* **2020**, *10*, 176.
https://doi.org/10.3390/buildings10100176

**AMA Style**

Gombosuren D, Maki T.
Prediction of Joint Shear Deformation Index of RC Beam–Column Joints. *Buildings*. 2020; 10(10):176.
https://doi.org/10.3390/buildings10100176

**Chicago/Turabian Style**

Gombosuren, Dagvabazar, and Takeshi Maki.
2020. "Prediction of Joint Shear Deformation Index of RC Beam–Column Joints" *Buildings* 10, no. 10: 176.
https://doi.org/10.3390/buildings10100176