# Fatigue Analysis of FRP Strengthened RC Slabs Reinforced with Plain Bars Under Moving Load

## Abstract

In Japan, old RC bridge-deck slabs were economically designed by small thickness without considering the fatigue resistance. These slabs are subjected to a huge repetition of moving loads. Therefore, they are suffering from fatigue damage. This damage is more significantly observed than that of slabs reinforced with deformed bars. To extend their fatigue life, a suitable strengthening technique is required such as externally bonded FRP sheets. Using numerical method for predicting the improvement in their fatigue life is strongly beneficial to take full advantage of this strengthening technique. This study presents a proposed numerical method based on bridging stress degradation concept to analyze two full scale RC slabs reinforced with plain bars under moving load. One of them is strengthened with externally bonded FRP sheets in longitudinal and transverse directions on the slab bottom surface. The interfacial bond behavior between FRP sheet and concrete surface with its degradation due to fatigue loading is implemented. This study provides the propagation of cracked elements, center displacement evolution, cracking pattern and FRP strain. For the strengthened RC slab, the major crack opening is restricted by the contribution of FRP sheets in longitudinal and transverse directions. Therefore, the strengthened RC slab shows longer fatigue life and smaller deformation. By comparing these numerical results with the experimental results, the current numerical method provides a good agreement.

## 1 Introduction

Reinforced concrete (RC) bridge is one of the most common structural members, which is subjected to a large number of load repetitions due to heavy traffic resulting a fatigue damage. This damage is significantly observed in the RC slab than other member. Therefore, it is important to consider the fatigue resistance in the design. According to previous studies of Matsui (1987); and Perdikaris and Beim (Perdikaris and Beim 1988), the most common mode of fatigue failure was by punching shear mode, which is significant in RC slab with small thickness.

Old slabs in Japan are suffering from two main problems as the follows. First problem is that these slabs were economically designed with small thickness. Second problem is that the plain bars were used as reinforcement for some of these slabs. Therefore, a significant fatigue damage was observed in theses slabs than those reinforced with deformed bars (Shakushiro et al. 2011). Therefore, a suitable strengthening technique is required to extend their fatigue life such as externally bonded fiber reinforced polymer (FRP), which have been extensively used for external strengthening of RC structure members. Mitamura et al. (2011) provided an experimental study to evaluate the effect of FRP strengthening on the fatigue durability improvement of RC slabs reinforced with plain bars under moving load fatigue. But, non-perfect-strengthening design causes a loss in the functions of the structures over time. Therefore, the main target of this study is developing numerical method to evaluate this strengthening and predict the extended fatigue life for RC slabs reinforced with plain bars.

For RC slabs reinforced with plain bars, the proposed numerical method by Drar et al. (2015); and Drar and Matsumoto (2016) was employed to predict their fatigue life and behaviors. Therefore, this study can be extended to examine the effect of FRP strengthening on these slabs. The experimental study by Mitamura et al. (2011) can be used to verify the numerical results.

## 2 Method

In this section, the developed numerical method will be described. This method was developed based on the finite element method (FEM) considering the different materials behaviors under the fatigue loading. Therefore, the following explanation shows the numerical model for all components of the FRP composite structure.

### 2.1 Concrete

Constitutive laws for concrete (Maekawa et al. 2003)

Compression | Tension | ||
---|---|---|---|

\( \varepsilon_{m} \le \varepsilon \le 0 \) | \( \sigma = f_{c} \frac{\varepsilon }{{\varepsilon_{m} }}\left( {2 - \frac{\varepsilon }{{\varepsilon_{m} }}} \right) \) | \( 0 \le \varepsilon \le \varepsilon_{t} \) | \( \sigma = E_{c} \varepsilon \) |

\( \varepsilon_{u} \le \varepsilon < \varepsilon_{m} \) | \( \sigma = f_{c} \frac{{\varepsilon_{u} - \varepsilon }}{{\varepsilon_{u} - \varepsilon_{m} }} \) | \( \varepsilon_{t} < \varepsilon \) | \( \sigma = f_{t} \left( {\frac{{\varepsilon_{t} }}{\varepsilon }} \right)^{0.4} \) |

*l*is cracked element size, \( \varepsilon_{t\hbox{max} } \) maximum tensile strain,

*N*is number of cycles, \( \sigma_{N} \) and \( \sigma_{1} \) are bridging stress at the

*N*th and the first cycle, respectively.

### 2.2 Reinforcing Bar

## 3 FRP

A linear elastic stress-strain relation is employed to model FRP elements. This assumption is good enough to simulate FRP behavior because reaching to its maximum tensile strength is assumed to be very difficult. The reason is that the maximum tensile strength of the FRP is much larger than other materials. Moreover, the authors assumed that there is no strength degradation due to fatigue loading (Loo et al. 2012).

### 3.1 Interfacial Bond Element

Adhesive material is used to bond FRP sheets with the facing concrete surface, which is considered as the weakest part of FRP composite. Therefore, the interfacial bond behavior between FRP sheet and concrete surface and its degradation due to fatigue loading are carefully integrated to obtain accurate numerical results. Their numerical model can be explained as follows.

*τ*, and the relative slip displacement,

*S*. The interfacial fracture energy, \( G_{f} \), can be represented by the area under the

*τ*−

*S*curve, which corresponds to the energy per unit bond area required for complete debonding. A bilinear bond-slip relation (Lu et al. 2005, 2004) is used in the present study as shown in Fig. 3. This relationship can be divided to three parts; (1) before debonding, (2) debonding initiation, and (3) complete debonding. The maximum bond strength, \( \tau_{max} \), the corresponding slip, \( S_{0} \), and the total fracture energy, \( G_{f} \), are governed by the tensile strength of the concrete, \( f_{t} \), and a width ratio parameter,

*B*, as follows;

*τ*, versus slip displacement,

*S*, relationship by Dai et al. (2005) shows a decrease of interfacial bond stiffness and an increase of the slip displacement. In this study, the model by Loo et al. (2010) is employed by reducing the interfacial bond stiffness according to the following equation.

*N*, \( E_{b0} \) is the interfacial bond stiffness at the first cycle, Δ

*τ*is the bond stress range, \( \tau_{f} \) is the bond stress at failure, and the constants

*α*,

*β*and

*γ*are parameters to fit the experimental data of Dai et al. and Yun et al. (Dai et al. 2005; Yun et al. 2008).

*α*,

*β*and

*γ*equal to −190.3, 0.990 and 8,797, respectively.

*τ*. This simplification can be considered as the worst case to simulate the fatigue life. Figure 4 shows the degradation of interfacial bond stiffness due to fatigue loading.

## 4 Fatigue Analysis

Details of analyzed RC slabs

Slab id | Slab dimensions (mm) | FRP sheets | |||
---|---|---|---|---|---|

Width (mm) | Thickness (mm) | Longitudinal spacing (mm) | Transverse spacing (mm) | ||

S0 | 3300 × 2650 × 160 | Without strengthening | |||

S330 | 100 | 1.2 | 330 | 330 |

Moving load levels for analyzed RC slabs

Number of cycles (×10 | >1 | 1–2 | 2–3 | 3–4 | 4–7.8 | >7.8 |
---|---|---|---|---|---|---|

Moving load (kN) | 120 | 130 | 150 | 170 | 200 | 230 |

Materials properties (Mitamura et al. 2011)

Material | Young’s modulus (GPa) | Poisson’s ratio | Strength (MPa) | |
---|---|---|---|---|

Tension | Compression | |||

Concrete | 26.133 | 0.2 | 3.3 | 45.26 |

Reinforcing bar | 187 | 0.3 | 235 (Yield) | |

FRP | 171 | 0.3 | 3320 | – |

Adhesive | 2.66 | 0.3 | 35 | 56 |

### 4.1 Propagation of Cracked Elements

^{3}to the fatigue life by cycles. The average degradation ratios for all analyzed slabs are listed in Table 5. Strengthened RC slab, S330, shows a slower degradation ratio than the non-strengthened RC slab, S0. The reason is that the FRP strengthening plays an important role to restrict the major crack opening. This leads to decrease the maximum tensile strain of concrete than the non-strengthened RC slab resulting in slower bridging stress degradation according to Eq. (1).

The percentages of cracked elements volumes

Slab id | Cracked elements volume % | Average degradation ratio (mm | ||
---|---|---|---|---|

| | | ||

S0 | 22% | 49.1% | 9479.6 | |

S330 | 10.7% | 33.6% | 50.5% | 618.8 |

### 4.2 Center Displacement Evolutions

Strengthened RC slab, S330, shows a smaller center displacement than the non-strengthened RC slab, S0, at the same number of cycles. The reason is that FRP strengthening works as an additional reinforcement for strengthened RC slab. This leads to a decrease of concrete strain. Moreover, center displacement evolution of non-strengthened RC slab, S0, shows a higher slope than that in strengthened RC slab, S330. Therefore, the reduction ratio of slab stiffness for non-strengthened RC slab, S0, is larger. The possible reason can be explained as that the FRP strengthening leads to a significant slower degradation ratio than the non-strengthened RC slab as shown in Table 5. According to this explanation, strengthened RC slab, S330, shows a longer fatigue life than the non-strengthened RC slab.

The comparison between numerical and experimental center displacement evolution provides similar values for fatigue life and center displacement, indicating an acceptable agreement between them.

### 4.3 Transverse FRP Sheet Strain

The maximum numerical and experimental strain values located at the loading position. Moreover, increasing moving load level leads to increase numerical and experimental FRP sheet strain. This increasing is significant at the loading position than other parts. The reason is that the bridging stress degradation of concrete is influenced by the maximum tensile strain, which locates in this zone. This leads to crack opening localization at slab center resulting in a higher strain of surrounding FRP sheets. Therefore, the improvement of FRP strengthening is significant at slab center, which contains a wide crack opening for the major cracks.

## 5 Conclusions

A proposed numerical method based on bridging stress degradation concept was presented in this paper to study the effect of FRP strengthening on the fatigue behaviors of RC slabs reinforced with plain bars. The interfacial bond behavior between FRP sheet and facing concrete surface with its degradation due to fatigue loading was considered in this method to obtain accurate numerical results. The comparison between numerical and experimental results provided an acceptable agreement between them.

Strengthened RC slab provided a longer fatigue life and smaller deformations than the non-strengthened RC slab. To investigate this improvement, the numerical analysis was conducted in this study. FRP strengthening leads to restrict the major crack opening, which is distributed due to moving load fatigue. This results in a slower degradation ratio than the non-strengthened RC slab.

FRP sheet strain showed a localized higher strain at slab center, which contain a wide crack opening due to significant bridging stress degradation in this zone. Therefore, an extensive FRP strengthening at slab center is recommended.

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