# Masonry Walls Retrofitted with Vertical FRP Rebars

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Equation of Motion of the Retrofitted Masonry Wall

_{est}is the work of non-conservative forces and the Lagrangian parameter q

_{k}is taken to be equal to the rotation θ of the block.

_{est}have to be defined. Being not influenced by the presence of the vertical rebars, they can be expressed as:

_{o}the moment of inertia about the two centres of rotation O and O’, $\dot{\theta}$ is the angular velocity of the centre of mass, m is the mass of the element, ü(t) the time-dependent ground motion acceleration and θ is the rotation of the block, which is characterized by a slenderness α.

_{m}referring to the masonry block is:

_{r}regarding the vertical rebar is:

_{r}of the restraint from the centre of rotation O is equal to:

^{2}=mgR/I

_{o}is the frequency parameter of the block and r

_{p}is a parameter equal to 1 or 4 in case of centred or external rebars, respectively.

_{max}defined as:

_{max}represents the collapse strain of the composite material of which the rebars are made. According to [53], a reduction of the strength of the vertical restraints due to cyclic loads can be properly accounted for through a strain multiplier equal to 0.5 valid for all chosen fibres. Vice versa, in case of external retrofitting system, the collapse state depends on the analysed mechanism. If the wall undergoes to a bilateral motion, only one of the two systems of lateral rebars works for a clockwise or anticlockwise rotation of the block. Specifically, the rebars opposite to the centre of rotation elongate, contributing to the potential energy U

_{r}and influencing the dynamic motion of the rocking block. On the contrary, the reinforcement located near the centre of rotation exerts only a retaining action. Consequently, if one of the two systems of rebars breaks, the other one can continue to restrain the element. At the instant in which the reinforcement breaks, an instantaneous variation of the block velocity occurs. According to the Principle of Conservation of Energy, if in a system only conservative forces act, the sum of kinetic energy, gravitational potential energy and elastic potential energy remains constant, in the two time instants immediately before (-) and after (+) the failure of the rebar:

_{r}in Equation (10) becomes equal to zero and the motion of the block is governed by the classical equation of the rocking element.

## 3. Parametric Analyses

^{3}.

_{B}and T

_{C}. According to the Code provisions [54], seven artificial accelerograms are generated for each response spectrum, in order to match the elastic response spectra for 5% viscous damping, with a minimum duration of the stationary part of the accelerograms equal to 10 s. In the maximum range of periods between 0.15 s – 2.0 s and 0.15 s – 2T, where T is the fundamental elastic period of the structure in the direction along which the accelerogram is applied, no value of the mean 5% damping elastic spectrum of all the time histories, is less than 90% of the corresponding value of the 5% damping elastic response spectrum.

_{dyn}causing the collapse of the masonry element and the acceleration of activation of the rocking motion a

_{RM}:

_{dyn}, the equation of motion Equation (10) is numerically solved using a Newmark trapezoidal rule [55]. The seismic input ü(t), represented by a ground motion with a defined peak ground acceleration PGA, is increased by means of a multiplier C, up to the attainment of the overturning condition. The effectiveness of the adopted method and of the integration procedure for modelling the rocking behaviour of the masonry element has been demonstrated in [28], through a comparison with the outcomes of an experimental campaign performed by [24] on a masonry wall made of tuff units. The adopted procedure allows evaluating the force reduction factor as:

#### 3.1. Unrestrained Rocking Block

_{RM}is constant varying the height H of the element, while the dynamic collapse acceleration a

_{dyn}increases with increasing H. With reference to masonry elements characterized by the same geometry, the values of the q factor are higher in case of one-sided mechanism than in case of bilateral one.

#### 3.2. Force Reduction Factor of the Restrained Rocking Block

#### 3.3. Level of Seismic Improvement of the Restrained Rocking Block

_{R}and a

_{UR}, respectively). As it can be noted, for all considered values of the cross section of the GFRP retrofitting system, the level of seismic improvement decreases with increasing values of the height H of the masonry block. In fact, from the analysis of Equation (12), having fixed the angle of slenderness and the geometrical and mechanical properties of the vertical restraints (i.e., elastic modulus and area of the cross-section), the term related to the presence of the restraints decreases with increasing height H, i.e., with the element mass m.

## 4. Conclusions

_{R}) and unreinforced elements (a

_{UR}).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schemes of the investigated retrofitting techniques: (

**a**) external, (

**b**) centred vertical rebars.

**Figure 2.**Detail of the anchorages for the external retrofitting intervention (from [43]): (

**a**) real image of the external connection on the Chemistry building in Christchurch; (

**b**) schematic representation of the detail in (

**a**); (

**c**) retrofit cross-section in the Great Hall of the Canterbury College with a detail of the bottom anchorage.

**Figure 4.**Collapse mechanisms occurring in the out-of-plane behaviour: (

**a**) two-side rocking motion (images from https://www.dw.com/de/ein-beben-ersch%C3%BCttert-italiens-kultur/a-19515621-0); (

**b**) one-side rocking motion (images from http://www.lct-architettura.it/Restauri/&id=91).

H (m) | 3-4-5-6-7-8 |
---|---|

α | 5-7.5-10 |

ρ | 1‰-2‰-3‰ |

Material | GFRP-BFRP-CFRP |

Spectrum | A1–A2–A3 |

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**MDPI and ACS Style**

Coccia, S.; Di Carlo, F.; Imperatore, S.
Masonry Walls Retrofitted with Vertical FRP Rebars. *Buildings* **2020**, *10*, 72.
https://doi.org/10.3390/buildings10040072

**AMA Style**

Coccia S, Di Carlo F, Imperatore S.
Masonry Walls Retrofitted with Vertical FRP Rebars. *Buildings*. 2020; 10(4):72.
https://doi.org/10.3390/buildings10040072

**Chicago/Turabian Style**

Coccia, Simona, Fabio Di Carlo, and Stefania Imperatore.
2020. "Masonry Walls Retrofitted with Vertical FRP Rebars" *Buildings* 10, no. 4: 72.
https://doi.org/10.3390/buildings10040072