# Finite Element Modeling of GFRP-Reinforced Concrete Interior Slab-Column Connections Subjected to Moment Transfer

^{*}

## Abstract

**:**

## 1. Introduction

_{c}is the factored shear stress resistance of concrete, λ is the concrete density factor, ϕ

_{c}is the concrete resistance factor, β

_{c}is the ratio of longer to shorter sides of the column, E

_{f}is the elastic modulus for the FRP flexural reinforcement, ρ

_{f}is the FRP flexural reinforcement ratio, ${f}_{c}^{\prime}$ is the concrete compressive strength (up to 60 MPa), α

_{s}= 4.0 for interior connections, d is the slab average depth, and b

_{0.5d}is the critical section perimeter.

_{c}is the nominal shear strength provided by concrete, ${f}_{c}^{\prime}$ is the concrete compressive strength, c is the depth of the neutral axis of the cracked transformed section, k is ratio of depth of neutral axis to reinforcement depth, ρ

_{f}is the FRP flexural reinforcement ratio, and n

_{f}is the modular ratio.

## 2. Summary of Experimental Program

#### 2.1. Test Specimens

Specimens | Flexural Reinforcement | Type of Reinforcing Bars | Concrete Compressive Strength, ${\mathit{f}}_{\mathit{c}}^{\prime}$ (MPa) | Failure Loads, ${\mathit{V}}_{{exp}.}$ (kN) | Flexural Capacities (Yield Line) | ||
---|---|---|---|---|---|---|---|

Layout | Ratio, ρ (%) | ${\mathit{V}}_{{flex}.}$ (kN) | ${\mathit{V}}_{{exp}.}/{\mathit{V}}_{{flex}}$ | ||||

SN-0.65 | No. 15M @192 mm | 0.65 | Steel | 42 ± 0.5 | 486 | 575 | 0.84 |

GN-0.65 | No. 16 @192 mm | GFRP | 42 ± 0.9 | 363 | 536 | 0.67 | |

GN-0.98 | No. 16 @128 mm | 0.98 | 38 ± 0.7 | 378 | 591 | 0.64 | |

GN-1.30 | No. 16 @96 mm | 1.30 | 39 ± 0.6 | 425 | 663 | 0.64 | |

GH-0.65 | No. 16 @192 mm | 0.65 | 70 ± 1.1 | 380 | 675 | 0.56 |

#### 2.2. Test Setup and Instrumentations

#### 2.3. Material Properties

Bar Material | Diameter (mm) | Area (mm^{2}) | Tensile Modulus^{§} (GPa) | Ultimate Strength^{§} (MPa) | Ultimate Strain^{§} (%) |
---|---|---|---|---|---|

GFRP—No. 16 | 15.9 | 198^{§} | 68.0 ± 0.3^{§} | 1398 ± 34^{§} | 2.05 ± 0.04^{§} |

Steel—No. 15M | 15.9 | 200 | 200 | f_{y} = 480** | ɛ_{y} = 0.24** |

^{§}Values are based on the nominal cross-sectional area of GFRP bars; ** Steel yield stress and strain.

#### 2.4. Main Findings

## 3. Finite Element Modeling

#### 3.1. Reinforcing Bars

#### 3.2. Concrete Material

- (1)
- Non-linearity in compression including softening and hardening;
- (2)
- Cracking of concrete in tension based on the non-linearity fracture mechanics;
- (3)
- Biaxial strength failure criterion;
- (4)
- Reducing the compressive strength after cracking;
- (5)
- Tension stiffening effect;
- (6)
- Reduction of the stiffness in shear after cracking.

**Figure 3.**Uniaxial stress-strain law for concrete, reproduced with permission [11].

#### 3.3. Bearing Plates

#### 3.4. Reinforcement-Concrete Interface

#### 3.5. Model Geometry and Boundary Conditions

**Figure 5.**ATENA-3D model. (

**a**) Model geometry; (

**b**) Reinforcement configuration; (

**c**) Loading and supporting plates (meshed); (

**d**) Slab and column (meshed).

#### 3.6. Model Verification

## 4. Parametric Study

_{b}and 4.0 ρ

_{b}, with 0.5 ρ

_{b}increments, where ρ

_{b}is the balanced reinforcement ratio, defined as the ratio at which concrete crushes and GFRP bars rupture simultaneously); (2) shear perimeter-to-depth ratio (11.5, 15.25, and 19, corresponding to the square column cross-section with side dimensions of 300, 450, and 600 mm and effective slab depth of 160 mm); (3) column aspect ratios (between 1.0 and 5.0). For each parameter, a comparison has been performed in terms of load-deflection curve, load-strain curve in the GFRP bars, and failure capacity.

#### 4.1. Flexure Reinforcement Ratio

_{f}E

_{f}, of the models. Generally, increasing the axial stiffness from half the balanced reinforcement ratio to four times the balanced reinforcement ratio increased the post-cracking stiffness of the model, which in turn decreased the deflection at the same load level.

_{b}), 0.75 (2.5 ρ

_{b}), 0.9 (3.0 ρ

_{b}), 1.05 (3.5 ρ

_{b}), and 1.2% (4.0 ρ

_{b}) were 9980, 6620, 5300, 4110, and 3510 με, respectively, which represent 48%, 32%, 25%, 20%, and 17% of the ultimate tensile capacity of the GFRP bar. In addition, the models with 0.15% (0.5 ρ

_{b}), 0.3% (1.0 ρ

_{b}), and 0.45% (1.5 ρ

_{b}) reinforcement ratios failed before reaching the calculated service load of 295 kN.

**Figure 11.**Relationship between the normalized punching shear stress at failure and the axial stiffness of the slabs.

#### 4.2. Shear Perimeter-to-Depth Ratio (Σo/d)

**Figure 14.**Relationship between the punching shear stress at failure and the perimeter-to-depth ratio.

#### 4.3. Column Aspect Ratio

## 5. Summary and Conclusions

- (1)
- The constructed model was able to predict the behavior of the slab-column connections in terms of ultimate capacity, load-deflection curve, and load-strain curve with a reasonable accuracy. The average experimental-to-FEM shear strength ratio was approximately 1.03. In addition, Equation (6) resulting from the parametric study is similar to that adapted by the Canadian standards [6] (Equation (3)), which emphasize the accuracy of the finite element model.
- (2)
- Increasing the reinforcement ratio from half to four times the balanced reinforcement ratio reduced the strain and deflection. Furthermore, the shear strength of the slabs increased by 93% due to the increase in the reinforcement ratio by sevenfold from 0.15 (0.5 ρ
_{b}) to 1.2% (4.0 ρ_{b}). - (3)
- Deflection and strains decreased due to the increase in the side length of the square column. In addition, the punching shear stresses at failure decreased by 26% and 34% due to the increase in the perimeter-to-depth ratio by 33% and 65%, respectively. Moreover, a modification factor for the current governing equation (Equation (3)) in the Canadian standard [6] was introduced (Equation (6)) to consider this effect; however, further investigation is still needed to address this issue.
- (4)
- Considerable enhancement in the post-cracking stiffness can be noticed due to the increase in the column aspect ratio. In addition, increasing the column aspect ratio from one to five increased the punching strength by approximately 95%.
- (5)
- The effect of column rectangularity vanished after a value greater than three, which agrees with the results reported in the literature for the steel-RC slabs.
- (6)
- At the service load stage, increasing the reinforcement ratio and column dimensions reduced the strain and deflection. In addition, the models with 1.05% and 1.2% reinforcement ratios and the ones with column aspect ratios of 4.0 and 5.0 satisfied the Canadian standard [6] and the American guideline [7] serviceability limits.

## Acknowledgments

## Author Contribution

## Conflicts of Interest

## Notations

${b}_{0.5d}$ | = the critical shear perimeter (at d/2 from the column face, mm) |

c | = cracked transformed section neutral axis depth (mm) |

${c}_{s}$ | = the side length of the square column (mm) |

d | = the effective slab depth (mm) |

ɛ | = strain at a given point |

${\epsilon}_{c}$ | = strain at the peak stress ${f}_{c}^{\prime ef}$ |

${E}_{O}$ | = initial elastic modulus |

${E}_{C}$ | = secant elastic modulus at the peak stress, ${E}_{C}=\frac{{f}_{c}^{\prime ef}}{{\epsilon}_{c}}$ |

E_{f} | = modulus of elasticity of longitudinal FRP reinforcement (MPa) |

${f}_{C}^{\prime}$ | = compressive strength of concrete (MPa) |

${f}_{c}^{\prime ef}$ | = maximum compressive stresses |

${f}_{t}^{\prime ef}$ | = maximum tensile stresses |

k | = ratio of depth of neutral axis to reinforcement depth |

L_{n} | = clear span (distance between the inner faces of the supports) |

${n}_{f}$ | = ratio of modulus of elasticity of FRP bars to modulus of elasticity of concrete |

${s}_{1}$ | = bond slippage |

t | = bond stresses |

t_{max} | = the maximum bond strength of the steel bar embedded in concrete |

ν_{c} | = the nominal shear strength provided by concrete |

α_{s} | = 4.0 for interior column, 3.0 for edge column and 2.0 for corner column |

β_{c} | = the ratio of long side to short side of the column |

λ | = factor to account for low-density concrete (1.0 for normal weight concrete) |

ϕ_{c} | = resistance factor for concrete (0.65) |

ρ_{f} | = longitudinal reinforcement ratio for FRP |

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**MDPI and ACS Style**

Gouda, A.; El-Salakawy, E.
Finite Element Modeling of GFRP-Reinforced Concrete Interior Slab-Column Connections Subjected to Moment Transfer. *Fibers* **2015**, *3*, 411-431.
https://doi.org/10.3390/fib3040411

**AMA Style**

Gouda A, El-Salakawy E.
Finite Element Modeling of GFRP-Reinforced Concrete Interior Slab-Column Connections Subjected to Moment Transfer. *Fibers*. 2015; 3(4):411-431.
https://doi.org/10.3390/fib3040411

**Chicago/Turabian Style**

Gouda, Ahmed, and Ehab El-Salakawy.
2015. "Finite Element Modeling of GFRP-Reinforced Concrete Interior Slab-Column Connections Subjected to Moment Transfer" *Fibers* 3, no. 4: 411-431.
https://doi.org/10.3390/fib3040411