# Natural Fibers for Out-of-Plane Strengthening Interventions of Unreinforced Masonry Buildings in Aggregate Configuration

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Out-of-Plane Behavior of Masonry Walls in URM Aggregate Buildings: The Effect of Strengthening

#### 2.1. Out-of-Plane Behavior of Masonry Walls in URM Aggregate Buildings

_{e}= L

_{i}

_{e}, is given by the contribution of vertical loads and horizontal forces as follows:

_{e}= 𝛼(θ)·𝑊·δ

_{x}

^{𝑤}′ − 𝑊·δ

_{y}

^{𝑤}′

_{x}

^{𝑤}′ and δ

_{y}

^{𝑤}′ are, respectively, the horizontal and vertical virtual displacements of the point of application of the weight (see Figure 1b).

_{i}, is zero and, consequently, the horizontal load multiplier as a function of the angle of rotation is provided by the following equation:

_{W}3sin θ + x

_{W}3cos θ)]/[𝑊3(x

_{W}3sin θ + y

_{W}3cos θ)].

_{uw1}+ W

_{uw2}+ W

_{s1}+ W

_{s2}+ 2·∑f

_{i})

- μ is the friction coefficient [26],
- W
_{uw}_{1}and W_{uw}_{2}are the weights of the transversal walls placed above the interconnection semi-blocks, respectively, on the right and on the left side of the façade (Figure 3a), - W
_{s}_{1}and W_{s}_{2}are the loads transmitted to the interconnection semi-blocks by the slabs, in the case of slabs parallel to the façade (Figure 3b).

_{i}= γ

_{m}·t·h

_{b}·l·i·(i + 1)/2,

_{i}, is no longer zero but equal to:

_{i}= F·δ

_{x}

^{𝐹}’

- F is the resultant of the friction forces evaluated through Equation (4);
- δ
_{x}^{𝐹}’ is the horizontal virtual displacement of the point of application of force F (Figure 4c).

_{e}, given by Equation (2), and the internal work L

_{i}, given by Equation (6), the horizontal load multiplier as a function of the angle of rotation in the case of considering the interaction, in terms of friction forces, becomes:

_{W}sin θ + x

_{W}cos θ) + F·(x

_{F}sin θ + y

_{F}cos θ)]/[𝑊·(x

_{W}sin θ + y

_{W}cos θ)]

#### 2.2. Strengthening Interventions toward Out-of-Plane Overturning Mechanisms

_{i}, becomes:

_{i}= F·δ

_{x}

^{𝐹}

^{’}+ 2·S·δ

_{x}

^{S’}

_{e}, given by Equation (2), and the internal work L

_{i}, given by Equation (8), Equation (1) becomes:

_{W}sin θ + x

_{W}cos θ)+ F·(x

_{F}sin θ + y

_{F}cos θ)+ 2·S·(x

_{S}sin θ + y

_{S}cos θ)]/[𝑊·(x

_{W}sin θ + y

_{W}cos θ)]

## 3. Case Study

#### Strengthening Solutions: Natural Fibers vs. Synthetic Fibers

_{f}

_{f}is the cross-section of the reinforcement accounted for in the case study. This datum has been obtained by means of a geometric proportion in terms of the width between the strip used for the accounted shear lap tests in [10,38] and the ones applied to the façade.

## 4. Discussion and Final Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Façade wall subjected to a global simple overturning; (

**b**) overturning façade in the deformed configuration with acting forces and virtual displacement without considering any interaction among the units (no aggregate effect).

**Figure 2.**(

**a**) Plan view of the structural unit where the interconnection semi-blocks between the overturning façade and the transverse walls shared with the adjacent units are evidenced; (

**b**) profile of the interconnection semi-blocks.

**Figure 3.**(

**a**) Weights of the transversal walls placed above the interconnection semi-blocks; (

**b**) loads transmitted to the interconnection semi-blocks by the slabs parallel to the façade.

**Figure 4.**(

**a**) FRCM strips arranged along the wall prone to the overturning mechanism and fixed to sidewalls; (

**b**) reinforcement contribution depending on the local adhesion behavior; (

**c**) overturning façade in the deformed configuration with acting forces and virtual displacement considering the mutual interaction among the units (aggregate effect) and the contribution provided by the FRCM reinforcement.

**Figure 5.**(

**a**) Borgo San Rocco’s plan (the analyzed façade is reported in red); (

**b**) prospectus of the analyzed S.U. (the portion of the overturning façade is reported in red).

**Figure 6.**Forces acting on the portion of overturning façade wall of the S.U. due to the mutual interaction between the units (aggregate effect).

**Figure 8.**Forces acting on the portion of façade wall prone to the overturning when a strengthening intervention with NFRMC strips is applied.

**Figure 10.**Horizontal loads multiplier α vs. displacement d of the overturning wall strengthened by sisal-FRCM.

**Figure 11.**Horizontal loads multiplier α vs. displacement d of the overturning wall strengthened by PBO-FRCM.

**Figure 12.**Horizontal loads multiplier α vs. displacement d of the overturning wall strengthened by FRCM with natural and synthetic fibers.

Dimension of the Overturning Wall | Dimension of the Interconnection Blocks | |||
---|---|---|---|---|

h (m) | L (m) | t (m) | h_{b} (m) | l (m) |

3.9 | 8 | 0.70 | 0.15 | 0.20 |

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**MDPI and ACS Style**

Cima, V.; Bartolomeo, C.; Grande, E.; Imbimbo, M.
Natural Fibers for Out-of-Plane Strengthening Interventions of Unreinforced Masonry Buildings in Aggregate Configuration. *Sustainability* **2022**, *14*, 9967.
https://doi.org/10.3390/su14169967

**AMA Style**

Cima V, Bartolomeo C, Grande E, Imbimbo M.
Natural Fibers for Out-of-Plane Strengthening Interventions of Unreinforced Masonry Buildings in Aggregate Configuration. *Sustainability*. 2022; 14(16):9967.
https://doi.org/10.3390/su14169967

**Chicago/Turabian Style**

Cima, Valentina, Chiara Bartolomeo, Ernesto Grande, and Maura Imbimbo.
2022. "Natural Fibers for Out-of-Plane Strengthening Interventions of Unreinforced Masonry Buildings in Aggregate Configuration" *Sustainability* 14, no. 16: 9967.
https://doi.org/10.3390/su14169967