# Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

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## Abstract

**:**

## 1. Introduction

## 2. The Modelling of Historical Masonry Structures

## 3. The Modeling of FRP Reinforcing

_{f}), characterized by a Young’s modulus (E

_{f}) and tensile strength (f

_{t}), incapable to resist to compression loads. A schematic layout of the modelling approach of a masonry element reinforced with a FRP strip is reported in Figure 2.

## 4. Retrofitting and Restoration of Curved Masonry Structures by FRP Materials

#### 4.1. Circular Arch

_{1}= 866 mm, which corresponds to a prototype tested in the laboratory, subjected to an unsymmetrical vertical static load [26]; then, two additional values of the radius (R

_{2}= 1500 mm and R

_{3}= 2500 mm) are considered in order to investigate the effect of the scale factor on the response of the unreinforced and reinforced systems.

_{0}), as represented in Figure 3, increased until the complete collapse of the structure. The results of the push-over analyses are presented both in terms of capacity curves, and collapse mechanisms. The capacity curves report the maximum lateral displacement of the arch vs. the base shear coefficient (base shear normalized by the own weight).

_{t}and σ

_{c}the tensile and compressive strengths, G

_{t}and G

_{c}the corresponding values of fracture energy, c the cohesion, μ the friction factor, and w the specific self-weight of masonry.

_{b}(Figure 5a) and in terms of base shear coefficient C

_{b}= V

_{b}/W (Figure 5b), being W the total weight of the arch. It can be observed that, as the radius of the arch increases, the global resistance of the arch increases as well (Figure 5a). On the contrary, in terms of the base shear coefficient, as the radius increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b).

_{y}= 200 Mpa. The diameters of the tie-rods are empirically chosen among commercial diameters, keeping constant the ratio between the radius of the arch and the diameter of the tie-rod. In the considered models the tie-rods’ heights h

_{r}with respect to the base of the arch is about R/4 (Figure 6). The yielding stress of the steel has been chosen among widely adopted steel typologies, and large enough to keep the tie-rods in the elastic field. The other two strategies consist of the introduction of FRP strips, at the intrados and at the extrados surfaces respectively (Figure 6). The reinforcement is constituted by strips arranged over the entire width and length of the arch made of glass fiber composite material (GFRP) and organic matrix. The adopted mechanical properties have been set according to [27], and reported in Table 2, in which E

_{f}and f

_{t}are the tensile module and the ultimate tensile strength of the reinforcement, and t

_{f}is the equivalent thickness. The bond-slip behaviour is described by the initial shear stiffness of the matrix k

_{s}, the ultimate debonding stress t

_{f}, the fracture energy G

_{s}, and the friction factor μ

_{s}.

_{1}in Figure 7a and point A

_{2}in Figure 7b) or at the intrados surface, closer the support of the arch, in the case of the extrados reinforcing (point B

_{1}in Figure 7a and point B

_{2}in Figure 7b). It is worth to note that, although the arches reported in Figure 7 are not scaled according to the relevant radius, they refer to different size of the arch, as better specified in the caption.

_{s}) and the global compression on the masonry (R

_{c}). The ultimate equilibrium of the section is imposed by considering the ultimate value of F

_{s}and evaluating the corresponding value of x under the hypothesis of linear elastic behaviour of the masonry (confirmed by the numerical simulations). Once the internal forces are computed, the ultimate moment (M

_{u}) and the ultimate eccentricity ${e}_{\mathrm{lim}}(N)={M}_{u}/N$ can be easily inferred.

_{f}/ε

_{fu}, being ε

_{f}and ε

_{f}

_{u}the current and the ultimate tensile strains of the textile, respectively. These rates are useful to identify the achievement of the tensile rupture of the reinforcement, which is here identified at the left end of the arch for the model reinforced at the extrados, and at the right end of the arch for the model reinforced at the intrados. These ruptures produce the sudden drops of the global resistance, as observed in the global capacity curves.

_{y}(N) at the same step (dashed lines), depending on the current compression force on the interface (N). The figures refer to the arches with R = 866 mm reinforced at the extrados (Figure 11a) and at the intrados (Figure 11b). In both cases, the tangential stress is lower than the corresponding yielding value confirming that the debonding mechanism does not occur. The latter results are apparently in contrast with other experimental and numerical results available in the literature, obtained considering similar FRP reinforced prototypes subjected to a vertical eccentric force [27]. The fact that no delamination phenomenon occurs for the treated cases might be due, in part, to the geometry and in part to the horizontal mass-proportional load distribution considered.

_{b},

_{max}) and increment of resistance (ΔV

_{b}), is reported in Table 3. The benefits in terms of strength resistance are higher in the models with the lowest radius (R = 866 mm) and the beneficial effects decrease as the radius increases. Furthermore, the comparison of the effects of the extrados and intrados arrangements of the FRP strips demonstrates that the scale effect observed in Figure 8 is confirmed for all of the cases investigated: in addition, for small radius models the application of FRP strips to the intrados and to the extrados provides similar effects (see the first column of Table 3), while in the case of large radius models the benefit associated to the extrados FRP reinforcement is significantly higher if compared to the intrados reinforcing.

_{f}= 0.298 mm) is investigated.

_{f}= 0.149 mm) are reported for comparison. An increment of strength and ductility is associated to the model with t

_{f}= 0.298 mm if compared to the standard thickness model. However in this case the ultimate lateral capacity is limited by the activation of the delamination as demonstrated by tangential stress distribution, which overlaps the yielding stress close to the right end of the arch (Figure 12b). At the peak load, the opening of the cylindrical hinges at the intrados is significantly delayed by the presence of FRP reinforcement (Figure 12c), causing a significant delamination in the post-peak branch (Figure 12d).

#### 4.2. Hemisperical Dome

_{0}) is applied until collapse in order to investigate a typical load scenario in seismic conditions.

_{b}= 0.6) and by a significant residual resistance as well. It is worth to note that the horizontal displacements of the monitored points decrease as the height of the control point increases.

_{1}, shows the effectiveness of the FRP retrofitting technique, which leads to a significant improvement in terms of resistance without implying any global stiffness alteration, thus guaranteeing that no significant change of the seismic demand for the structure occurs. On the other hand, the presence of FRP strips, not only increases the peak load of the arch, but significantly delays the loss of resistance in the post-peak branch (from around 2.5 mm for the unreinforced dome, to around 10 mm for the case of the strongly retrofitted dome), thus guaranteeing to the dome a larger ductility as well. In Table 5 the ultimate lateral resistance (C

_{b},

_{max}) and the percentages of strength increment (ΔC

_{b}) are reported, highlighting the enhancement associated to the FRP reinforcement application. The softly retrofitted model presents a residual lateral strength close to that relative to the unreinforced model, whereas the strongly retrofitted model presents a higher value of residual resistance due to a larger spreading of the damage at the ultimate condition, as shown in Figure 18.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Layout of the macro-element adopted for masonry at its three stages: (

**a**) plane element, (

**b**) spatial regular element and (

**c**) three-dimensional element for curved structures.

**Figure 2.**Schematic layout of the interaction between masonry elements and discrete fabric-reinforced polymeric (FRP) reinforcement elements.

**Figure 5.**Capacity curves of the unreinforced arches, expressed in terms of (

**a**) global base shear, and (

**b**) base shear coefficient.

**Figure 9.**Contribution of the FRP reinforcement at the peak load: normalized abscissa versus eccentricity of the acting force of the models with R = 866 reinforced at the (

**a**) extrados and (

**b**) intrados. Cross section internal equilibrium for extrados (

**c**) and intrados (

**d**) reinforcing.

**Figure 10.**Contribution of the FRP reinforcement at the peak load: normalized abscissa versus working rates of the reinforcement for the models reinforced at the (

**a**) extrados and (

**b**) intrados.

**Figure 11.**Tangential stress at the interface between masonry and FRP reinforcement in correspondence of the peak load for the models reinforced at the (

**a**) extrados; and (

**b**) intrados.

**Figure 12.**Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile (t

_{f}= 0.298 mm): (

**a**) capacity curve; (

**b**) tangential stress at the interface between masonry and FRP reinforcement in correspondence of the peak load, damage pattern at (

**c**) the peak load; and (

**d**) collapse.

**Figure 14.**Response of the unreinforced dome in terms of (

**a**) failure mechanism; and (

**b**) capacity curves.

**Figure 16.**Failure mechanisms of the reinforced models: dome with (

**a**) soft; and (

**b**) strong reinforcement.

E (Mpa) | G (Mpa) | σ_{t} (Mpa) | σ_{c} (Mpa) | G_{t} (N/mm) | G_{c} (N/mm) | c (Mpa) | μ (-) | W (kN/m^{3}) |
---|---|---|---|---|---|---|---|---|

2700 | 1080 | 0.30 | 8.53 | 0.01 | 0.30 | 0.26 | 0.6 | 18 |

Tensile | Bond-Slip | |||||
---|---|---|---|---|---|---|

E_{f} (GPa) | f_{t} (MPa)
| t_{f} (mm) | k_{s} (N/mm^{3}) | τ_{f} (MPa) | G_{s} (N/mm) | μ_{s} (-) |

450 | 1473 | 0.149 | 20 | 1.3 | 2.5 | 0.75 |

**Table 3.**Ultimate strength of the arches and increment of the ultimate load with respect to the unreinforced configuration.

Model | R = 866 mm | R = 1500 mm | R = 2500 mm | |||
---|---|---|---|---|---|---|

V_{b},_{max} (kN) | ΔV_{b} (%) | V_{b},_{max} (kN) | ΔV_{b} (%) | V_{b},_{max} (kN) | ΔV_{b} (%) | |

Unreinforced | 1.23 | - | 3.43 | - | 9.26 | - |

Tie rod | 1.23 | 0 | 3.43 | 0 | 9.26 | 0 |

Intrados FRP | 10.83 | 780 | 19.14 | 458 | 34.10 | 268 |

Extrados FRP | 10.72 | 772 | 21.93 | 539 | 43.00 | 364 |

E (Mpa) | G (Mpa) | σ_{t} (Mpa) | σ_{c} (Mpa) | G_{t} (N/mm) | G_{c} (N/mm) | c (Mpa) | μ (-) | w (kN/m^{3}) |
---|---|---|---|---|---|---|---|---|

1200 | 480 | 0.15 | 2.50 | 0.10 | 0.5 | 0.15 | 0.7 | 25 |

Model | C_{b},_{max} (-) | ΔC_{b} (%) |
---|---|---|

Unreinforced | 0.60 | - |

Softly retrofitted | 0.75 | 25 |

Strongly retrofitted | 1.00 | 67 |

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**MDPI and ACS Style**

Pantò, B.; Cannizzaro, F.; Caddemi, S.; Caliò, I.; Chácara, C.; Lourenço, P.B.
Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing. *Buildings* **2017**, *7*, 79.
https://doi.org/10.3390/buildings7030079

**AMA Style**

Pantò B, Cannizzaro F, Caddemi S, Caliò I, Chácara C, Lourenço PB.
Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing. *Buildings*. 2017; 7(3):79.
https://doi.org/10.3390/buildings7030079

**Chicago/Turabian Style**

Pantò, Bartolomeo, Francesco Cannizzaro, Salvatore Caddemi, Ivo Caliò, César Chácara, and Paulo B. Lourenço.
2017. "Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing" *Buildings* 7, no. 3: 79.
https://doi.org/10.3390/buildings7030079