# Innovative Seismic Strengthening Techniques to Be Used in RC Beams’ Critical Zones

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Significant Research Studies

## 3. Experimental Program

#### 3.1. Specimens’ Description

#### 3.2. Strengthened Specimens’ Description

^{2}. The prestressing steel mechanical characteristics are a modulus of elasticity (E

_{p}) of 195 GPa, ultimate strength (f

_{u}) of 1860 MPa, and proof strength of 0.2% (f

_{pk,0.2}) of 1728 MPa.

#### 3.3. Test Setup

#### 3.4. Displacement History

_{0}, ±2xd

_{0}, ±3xd

_{0}, ±4xd

_{0}, ±5xd

_{0}, ±6xd

_{0}, and ±7xd

_{0}(where the base displacement, d

_{0}= 6 mm), with 3 cycles for each amplitude, starting from the displacement where the gravity load is attained. Failure was established when the beam does not resist the imposed gravity load (for more details see Gião et al., 2014 [11]). Figure 5a) illustrates a typical displacement cycle in the test procedure, starting from the force-controlled step until the pre-established value of the idealised gravity load was attained (stage 0). This was followed by the imposition of a required displacement +Δ (Stage I), force-controlled unloading until the gravity load value is restored (Stage II), imposition of a displacement −Δ (Stage III), and, finally, force-controlled loading until the gravity load value is re-established (Stage IV).

## 4. Test Results

#### 4.1. Failure Mechanisms

_{g}, induce to a deformation increase in the gravity direction, while the longitudinal bottom bars maintained an elastic behaviour in tension. This phenomenon is associated with the pre-established gravity negative moment. Thus, the bottom bars do not yield during the reverse loading cycle. The top cracks remained open and no significant “pinching” effects were observed.

#### 4.2. Experimental Results

#### 4.3. Performance Evaluation

_{r}), and the absorbed energy index (η).

_{u}) and yielding displacement (d

_{y}). The energy dissipation (W) corresponds to the area under the load-displacement diagram.

_{max}/F

_{S1}relation corresponds to the resistance capacity increase in comparison with the Specimen S1. The Specimen S2 (strengthened with external PT) (S2) presented a 22% strength increase. The strengthening solution with the additional UFRG jacketing (S3) reached 1.4 times the resistance capacity of the reference specimen.

_{S1}ratio represents the dissipated energy gain in the strengthening solutions. The strengthening solutions attained a considerable dissipated energy increase. The beam strengthened only with external PT (S2) had a 53% increase in the energy dissipation and, for the one with the additional UFRG jacketing (S3), a 100% energy dissipation increment was attained.

_{r}/d

_{rS1}ratio shows that a significant reduction in the residual deformation was attained with both strengthening solutions. The beam with external post-tensioning (S2) presented a decrease in deformation when compared with the reference specimen’s of, approximately, 40%. The beam with external PT and UFRG jacketing (S3) had a residual deformation decrease of, approximately, 50%. These observations indicate a more recentred behaviour.

_{PA}—Damage index;

_{u,mon}—ultimate displacement deformation under monotonic loading;

_{u,acum}—ultimate accumulated deformation;

_{y}—yield strength;

_{h}—dissipated hysteretic energy;

_{s}, and the hysteretic dissipated energy, E

_{d}(corresponding to the area contained by the inelastic force-displacement response curve), as shown in Figure 13 and given by Expression (2).

## 5. Theoretical Prediction

_{y}and M

_{max}, are determined from the imposition of the section force equilibrium and strain compatibility conditions. In Section 5.3, the determination of the deformation capacity is performed, assuming that the beam element exhibits approximately a rigid body plastic rotation around the plastic hinge centre. The idealised yielding displacement, d

_{y}*, is obtained based on the bilinear behaviour. Admitting to a bilinear moment-curvature relation and constant plastic curvature along the plastic hinge length, the ultimate displacement, d

_{u}, is estimated through Park and Paulay’s expressions [40].

#### 5.1. Determination of the Yield Bending Capacity

_{s}= ε

_{y}), the position of the neutral axis was determined through the section force equilibrium and strain’s compatibility conditions, as illustrated in Figure 16 and expressed in Equations (4) and (5).

_{u}—prestressing force;

_{s}—tensile force in the top reinforcement;

_{s}

_{1}—tensile force in the flange bottom reinforcement;

_{s}’—compression force in the compressed bottom reinforcement;

_{c}—compression force in the compressed concrete;

_{cf}—compression force in the CRFU jacketing.

_{c}), the yield bending capacity (M

_{y}), and the respective experimental values (M

_{y,exp}) for Specimens S1, S2, and S3.

#### 5.2. Determination of the Peak Bending Moment Capacity

_{max}, an equivalent rectangular stress block for compressed concrete was used—this is shown in Figure 17.

_{max,exp}) and the respective estimated value (M

_{max}). The position of the neutral axis (x) and the tensile strain in the reinforcement (ε

_{s}) are also shown.

#### 5.3. Determination of the Deformation Capacity

_{u}, was obtained by noting that the beam element exhibits approximately a rigid body plastic rotation around the plastic hinge centre (Figure 18) starting from the idealised yielding displacement, d

_{y}*, which was estimated by assuming elastoplastic behaviour.

_{u}—ultimate displacement;

_{p}—plastic hinge length, considered equal to 0.5 h;

_{u}—ultimate curvature;

_{y}—yield curvature.

_{su}) is 6% (Eurocode 8—part 3 [2]). The ultimate curvature can be obtained using Expression (12), and the position of the neutral axis (x) is shown in Table 5:

_{su}= 6%), eventually, due to the damage accumulation phenomenon associated with the repetition of successive cycles. Thus, the ultimate deformation obtained from the proposed analytical model corresponds to a value consistent with the imposed criterion, recommended in Eurocode 8—part 3 [2], which corresponds to limiting the level of strain in the longitudinal reinforcement to 6%.

#### 5.4. Proposed Multilinear Model

_{y}* (see Section 5.3); this is followed by stiffness degradation mostly due to cracking—branch II—from the first crack point (A) to the yield point (B); hardening stiffness due to steel hardening—branch III—until the peak point (C), when the maximum force is attained; finally, a decreasing slope or one near zero due to the steel softening—branch IV—until the failure point (D) corresponds to a strength loss until the conventional failure criterion of 85% of the strength capacity is reached.

_{I}—elastic stiffness (${K}_{I}={K}_{el}=\frac{{F}_{y}}{{d}_{y}^{*}}$, obtained considering a bilinear relation in Park and Paulay [41]—see Section 5.1 and Section 5.3);

_{cr}—cracking force;

_{cr}—displacement corresponding to cracking force (d

_{cr}= F

_{cr}/K

_{I});

_{II}—cracked stiffness (K

_{II}= C

_{II}.K

_{I});

_{y}—yielding force (taken as ${F}_{y}=\frac{{M}_{y}}{L}$, where L is the beam span and M

_{y}is determined from Equation (6));

_{y}—displacement corresponding to yielding force (obtained from $\left.{d}_{y}={d}_{cr}+\frac{{F}_{y}-{F}_{cr}}{{K}_{II}}\right)$;

_{III}—hardening stiffness (K

_{III}= C

_{III}.K

_{I});

_{max}—maximum force (taken as ${F}_{max}=\frac{{M}_{max}}{L}$, where L is the beam span and M

_{max}is determined obtained from Equation (7));

_{Fmax}—displacement corresponding to maximum force (obtained from $\left.{d}_{Fmax}={d}_{y}+\frac{{F}_{max}-{F}_{y}}{{K}_{III}}\right)$;

_{u}—failure force (85% Fmax);

_{u}—ultimate displacement (obtained from Equation (8));

_{IV}—softening stiffness (can be obtained from $\left.{K}_{IV}=\frac{{F}_{max}-{F}_{u}}{{d}_{Fmax}-{d}_{u}}\right)$.

_{II}and C

_{III}can be taken as 0.35 and 0.10, respectively, according to the experimental tests carried out.

## 6. Conclusions

_{su}, to 6% (the value proposed by Eurocode 8—part 3 [2]), is shown to be adequate.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Execution of the UFRG jacketing. (

**a**) Casting of the reinforced concrete beam; (

**b**) Concrete surface prepared with water and sandblasting, and casting holes; (

**c**) Application of the unidirectional and non-woven steel fibre mat; (

**d**) Formwork setting for gravity casting with external vibration; (

**e**) Specimen final aspect.

**Figure 5.**Experimental test setup and post-tensioning system with the typical imposed displacement cycle in the test procedure.

**Figure 11.**Diagram of post-tensioning force evolution versus imposed displacement: (

**a**) Specimen S2 and (

**b**) Specimen S3.

Specimen | Description | Initial Post-Tensioning Force (kN) | UFRG Jacketing |
---|---|---|---|

S1 | Reference | - | - |

S2 | Strengthened with external PT | 300 | - |

S3 | Strengthened with UFRG jacketing + external PT | 300 | 20 mm thickness |

Description | F_{max} (kN) | F_{max}/F _{S1} | Displacement Ductility | Residual Deformation | Energy Dissipation | ||||
---|---|---|---|---|---|---|---|---|---|

µ_{−} (*) | µ_{+} (*) | dr (mm) | dr/dr_{S1} | W (kNm) | W/W_{S1} | ||||

S1 | Reference | 212.50 | - | 10.40 | - | 126.20 | - | 28.60 | - |

S2 | PT | 260.10 | 1.22 | 7.70 | 2.40 | 71.20 | 0.56 | 43.80 | 1.53 |

S3 | PT + UFRG | 293.00 | 1.38 | 6.40 | 6.10 | 58.70 | 0.47 | 57.20 | 2.00 |

x (m) | ε_{c} (‰) | M_{y} (kNm) | M_{y, exp} (kNm) | D (%) | |
---|---|---|---|---|---|

S1 | 0.15 | 1.10 | 277.80 | 303.30 | 8 |

S2 | 0.18 | 1.50 | 353.70 | 375.30 | 6 |

S3 | 0.17 | 1.10 | 384.60 | 413.60 | 7 |

x (m) | ε_{S} (%) | M_{max} (kNm) | M_{max,exp} (kNm) | D (%) | |
---|---|---|---|---|---|

S1 | 0.07 | 1.90 | 311.70 | 318.80 | 2 |

S2 | 0.10 | 1.20 | 392.30 | 390.20 | −1 |

S3 | 0.08 | 2.40 | 425.00 | 439.50 | 3 |

1/r_{y} (m^{−1}) | 1/r_{u} (m^{−1}) | d_{y}* (m) | d_{plast} (m) | d_{u} (m) | θ = d/L (%) | θ_{exp} (%) | |
---|---|---|---|---|---|---|---|

S1 | 0.0075 | 0.155 | 0.005 | 0.060 | 0.065 | 4.4 | 8.0 (*) |

S2 | 0.0084 | 0.171 | 0.006 | 0.066 | 0.072 | 4.8 | 5.3 |

S3 | 0.0076 | 0.148 | 0.006 | 0.057 | 0.063 | 4.2 | 5.0 |

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## Share and Cite

**MDPI and ACS Style**

Gião, R.; Lúcio, V.; Chastre, C.
Innovative Seismic Strengthening Techniques to Be Used in RC Beams’ Critical Zones. *Buildings* **2023**, *13*, 95.
https://doi.org/10.3390/buildings13010095

**AMA Style**

Gião R, Lúcio V, Chastre C.
Innovative Seismic Strengthening Techniques to Be Used in RC Beams’ Critical Zones. *Buildings*. 2023; 13(1):95.
https://doi.org/10.3390/buildings13010095

**Chicago/Turabian Style**

Gião, Rita, Válter Lúcio, and Carlos Chastre.
2023. "Innovative Seismic Strengthening Techniques to Be Used in RC Beams’ Critical Zones" *Buildings* 13, no. 1: 95.
https://doi.org/10.3390/buildings13010095