# Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Limit Analysis of Buttressed Arches

- 1.
- Masonry does not have any tensile strength;
- 2.
- Masonry has infinite compressive strength;
- 3.
- Masonry has infinite shear strength, i.e., sliding between masonry parts cannot occur.

- (I)
- Local mechanism (L), with the formation of four hinges A, B, C and D within the thickness of the arch, whose locations are identified through the angles α
_{A}, α_{B}, α_{C}and α_{D}, measured with respect the horizontal line; - (II)
- Global mechanism (G), characterised by the presence of two hinges at the pier basis A (internal) and D (external) and two hinges B and C within the thickness of arch, whose positions are defined by the angles α
_{B}and α_{C}, respectively; - (III)
- Semi-global mechanism (S); characterised by the presence of one hinge D at the base of the pier (external) and three hinges A, B and C within the thickness of the arch, whose positions are identified by the angles α
_{A}, α_{B}and α_{C}.

_{i}, i = 1, 2, 3. While centres C

_{1}and C

_{3}are easily located as the points A and D, respectively, C

_{2}comes from the sufficient and necessary condition of collinearity between absolute and relative centres of rotation for all blocks.

- Elastic behaviour. In this case, the internal work reads ${W}_{int}^{k}={\int}_{\mathsf{\Omega}}^{}\mathit{\sigma}:\mathit{\epsilon}d\mathsf{\Omega}$ where $\Omega $ is the domain volume. Since at collapse plastic deformations are usually much larger than elastic, this contribution on the overall energy is usually disregarded and the blocks are assumed to be rigid.
- Perfectly plastic hinges. In this case a constant distribution of stresses is assumed along the crack, equal to tensile strength ${\sigma}_{t}$ (Figure 2a); the displacement profile is linear and equal to $u\left(x\right)=\frac{{u}_{0}}{t}x$, where ${u}_{0}$ is the maximum crack opening, ${u}_{0}=\gamma t$ with $\gamma $ angular opening of the hinge, t the member thickness and x the local axis parallel to the crack. The internal work of the i-th hinge becomes:$${W}_{int,i}^{k}=\frac{1}{2}{\sigma}_{t}t{u}_{0}\left(\vartheta \right)$$
- Not resisting hinge. This is the usual case in the analysis of masonry structures at collapse. The internal work of the i-th hinge is null, ${W}_{int,i}^{k}=0$.
- Perfectly plastic reinforcement at intrados or extrados (Figure 2b). In this case, it is possible to assume that upon crack opening the reinforcement is able to provide a force equal to ${F}_{i}={f}_{reinf}\cdot {t}_{reinf}$ with ${f}_{reinf}$ tensile strength of the reinforcement and ${t}_{reinf}$ its thickness. The virtual work is ${W}_{int,i}^{k}={F}_{i}\cdot {u}_{0}\left(\vartheta \right)$ if the hinge opens on the side of the reinforcement, zero otherwise, since the contribution of FRM reinforcement can usually be neglected in compression.

#### 2.2. An Automatic Tool for Limit Analysis of Buttressed Arches

- the geometrical and material features of the structure including the reinforcement, in the form of a perfectly plastic layer at intrados, extrados or both sides. It is possible to define the tensile strength, the thickness and the ultimate displacement of the reinforcement layer, whose contribution is evaluated by introducing the relevant internal work in Equation (3) as per Section 2.1.
- the type of hinge: Not resisting or Fully plastic. Following the discussion in Section 2.1, the hinge constitutive behaviour modifies the internal work contribution;
- the type of variable horizontal forces: Concentrated in a point or Proportional to masses;
- optional additional static vertical forces;
- the type of analysis: Evaluate, to evaluate a multiplier for a specific hinge position; Minimise, to evaluate the collapse multiplier solving the minimisation problem (4);
- the discretisation Δα for the hinges, used in the collapse multiplier exploration;
- the user-defined keystone displacement;

^{*}that activates the identified collapse mechanisms, related to the specific value of λ, can be therefore calculated through:

#### 2.3. Parametric Analysis Settings

- -
- span (S), or, alternatively, half-span (s);
- -
- arch thickness (t);
- -
- eccentricity (e);
- -
- arch rise (h);
- -
- pillars base (B);
- -
- pillar height (H).

- -
- e/s = 0 (Semicircular arch);
- -
- e/s = 0.5 (Drop pointed arch);
- -
- e/s = 1 (Equilateral pointed arch);
- -
- e/s = 1.5 (Lancet pointed arch).

- -
- t/s = 0.125;
- -
- t/s = 0.15;
- -
- t/s = 0.175;
- -
- t/s = 0.2.

- -
- B/s = 0.4;
- -
- B/s = 0.5;
- -
- B/s = 0.6;
- -
- h/H = 0.5;
- -
- h/H = 0.75;
- -
- h/H = 1.

^{3}. The 144 configurations identified were analysed in unreinforced conditions (UN), with reinforcement at intrados (RI), at extrados (RE), or on both sides (RI-E). In the latter three cases, the reinforcement was characterised by a thickness t

_{reinf}= 30 mm, and a tensile strength f

_{reinf}= 0.70 MPa, which are typical though conservative design values for FRM mortar applications [17,32]. Given the difficulty of defining a typical ultimate displacement capacity of ductile mortar (δ

_{u,m}), the results are then provided as dimensionless ductility of the arch ${\delta}^{*}$, according to Equation (6).

## 3. Results

#### 3.1. Validation of the Automatic Tool

- specific weight w;
- elastic behaviour: Young’s modulus E, shear modulus G;
- shear behaviour of the block: elastic-plastic Turnsek–Cacovic criterion with shear strength τ
_{0}and ultimate shear strain γ_{u}(Figure 9b); - sliding behaviour: elastic-plastic Mohr–Coulomb criterion with cohesion c, friction coefficient μ and fracture energy G
_{f}, assumed very high to simulate the elastic-perfectly plastic behaviour as an approximation of the frictional behaviour of the joints (Figure 9c).

_{t}and tensile fracture energy G

_{t}. In particular, three levels of f

_{t}= 0.05, 0.10, 0.15 MPa with fixed G

_{t}= 0.01 N/mm, and three levels of G

_{t}= 0.01, 0.025, 0.05 N/mm with fixed f

_{t}= 0.10 MPa were considered for both weak and strong masonry models. Even though fracture energy is usually related to tensile strength (as in the simplified formulations reported in [34]), in this section tensile strength and fracture energy were investigated separately to isolate the individual contribution to the force-displacement plot. Variable distributed loads proportional to mass were applied to the structure, while the self-weight was maintained constant. The nonlinear static problem is solved in Histra by means of an implicit solver based on modified Newton–Raphson iterative procedure, enhanced by arc-length integration method.

_{LA}= 0.189 provided by limit analysis. This is reasonable, since once the residual strength of the crack approaches zero, masonry behaves like an assemblage of rigid blocks connected at the hinges, following thus Heyman’s hypotheses. The only exception is represented by the strong-masonry model with f

_{t}= 0.1 MPa, G

_{t}= 0.05 N/mm, in the following referred to as Model 1, which shows a larger multiplier even at u = 15 mm (λ = 0.215). It is clear that increasing the crack tensile fracture energy, failure occurs with a much more diffused damage within the arch, and the rigid-block approximation becomes less realistic.

_{t}and G

_{t}, the same deformed configuration displayed in Figure 11a was observed, with the exception of Model 1, which exhibited the deformed shape shown in Figure 11b. The comparison between Figure 11a and c shows that the position of hinges is globally captured by limit analysis, even though some differences are observed for the hinges on the left, which are respectively placed at α

_{A}= 15° and α

_{B}= 75°. This configuration provides a force multiplier λ

_{min}= 0.189. The position of cracks in the DMEM model would correspond to α

_{A}= 0° and α

_{B}= [90°]

^{−}, where the minus indicates a hinge position immediately on the left of α

_{B}= 90°. This configuration, evaluated by the automatic tool, provide λ = 0.193, which is very close to λ

_{min}. Given the small sensitivity of the load multiplier to variation in α

_{A}and α

_{B}, the position of the hinges given by limit analysis can be considered accurate. Incidentally, the configuration of Figure 11b corresponds to the hinge distribution provided by the automatic tool in case of fully resisting plastic hinge (Case 2 in Section 2.1) and any value of f

_{t}.

_{t}= 0.1 MPa and respectively G

_{t}= 0.01 N/mm and G

_{t}= 0.05 N/mm are displayed. While for low fracture energy values (Figure 11d) the plastic strain contour plot is mostly coincident with the final position of the cracks (Figure 11a), with two some other hinges yet to open, the model with higher G

_{t}shows a plastic strain concentration, i.e., the onset of cracking, at both pier bases (Figure 11e). This crack layout is not matched by the final position of the hinges (Figure 11a), and thus implies a redistribution of plastic strain during loading, which in turn is responsible of the overall higher strength shown by high-G

_{t}models. The crack at left base may eventually close with increasing keystone displacement, determining the final collapse configuration displayed in Figure 11a, or remain active as in the configuration of Figure 11b, which represents Model 1 collapse configuration.

#### 3.2. Unreinforced Buttressed Arches

#### 3.3. Retrofitted Buttressed Arches

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- De Matteis, G.; Zizi, M. Seismic Damage Prediction of Masonry Churches by a PGA-based Approach. Int. J. Archit. Herit.
**2019**, 13, 1165–1179. [Google Scholar] [CrossRef] - Brandonisio, G.; Mele, E.; De Luca, A. Limit analysis of masonry circular buttressed arches under horizontal loads. Meccanica
**2017**, 52, 2547–2565. [Google Scholar] [CrossRef] - Brandonisio, G.; Angelillo, M.; De Luca, A. Seismic capacity of buttressed masonry arches. Eng. Struct.
**2020**, 215, 110661. [Google Scholar] [CrossRef] - Dimitri, R.; De Lorenzis, L.; Zavarise, G. Numerical study on the dynamic behavior of masonry columns and arches on buttresses with the discrete element method. Eng. Struct.
**2011**, 33, 3172–3188. [Google Scholar] [CrossRef] - Dimitri, R.; Tornabene, F. A parametric investigation of the seismic capacity for masonry arches and portals of different shapes. Eng. Fail. Anal.
**2015**, 52, 1–34. [Google Scholar] [CrossRef] - Pulatsu, B.; Erdogmus, E.; Bretas, E.M.; Lourenco, P. In-Plane Static Response of Dry-Joint Masonry Arch-Pier Structures. In Proceedings of the AEI 2019, Tysons, WV, USA, 3–6 April 2019. [Google Scholar]
- Borri, A.; Castori, G.; Corradi, M. Intrados strengthening of brick masonry arches with composite materials. Compos. Part B Eng.
**2011**, 42, 1164–1172. [Google Scholar] [CrossRef] - Cancelliere, I.; Imbimbo, M.; Sacco, E. Experimental tests and numerical modeling of reinforced masonry arches. Eng. Struct.
**2010**, 32, 776–792. [Google Scholar] [CrossRef] - Foraboschi, P. Strengthening of Masonry Arches with Fiber-Reinforced Polymer Strips. J. Compos. Constr.
**2004**, 8, 191–202. [Google Scholar] [CrossRef] - Gattesco, N.; Boem, I.; Andretta, V. Experimental behaviour of non-structural masonry vaults reinforced through fibre-reinforced mortar coating and subjected to cyclic horizontal loads. Eng. Struct.
**2018**, 172, 419–431. [Google Scholar] [CrossRef] - Angiolilli, M.; Gregori, A.; Cattari, S. Performance of Fiber Reinforced Mortar coating for irregular stone masonry: Experimental and analytical investigations. Constr. Build. Mater.
**2021**, 294, 123508. [Google Scholar] [CrossRef] - Ferrara, G.; Caggegi, C.; Martinelli, E.; Gabor, A. Shear capacity of masonry walls externally strengthened using Flax-TRM composite systems: Experimental tests and comparative assessment. Constr. Build. Mater.
**2020**, 261, 120490. [Google Scholar] [CrossRef] - Alecci, V.; De Stefano, M.; Focacci, F.; Luciano, R.; Rovero, L.; Stipo, G. Strengthening Masonry Arches with Lime-Based Mortar Composite. Buildings
**2017**, 7, 49. [Google Scholar] [CrossRef] [Green Version] - Del Zoppo, M.; Di Ludovico, M.; Prota, A. Analysis of FRCM and CRM parameters for the in-plane shear strengthening of different URM types. Compos. Part B Eng.
**2019**, 171, 20–33. [Google Scholar] [CrossRef] - ICC Evaluation Service; AC434. Acceptance Criteria for Masonry and Concrete Strengthening Using Fiber-Reinforced Cementitious Matrix (FRCM) Composite Systems; ICC Evaluation Service: Brea, CA, USA, 2011. [Google Scholar]
- ACI 549.4R-13. Guide to Design and Construction of Externally Bonded Fabric-Reinforced Cementitious Matrix (FRCM) Systems for Repair and Strengthening Concrete and Masonry Structures; American Concrete Institute: Farmington Hills, MI, USA, 2013. [Google Scholar]
- Angiolilli, M.; Gregori, A.; Vailati, M. Lime-Based Mortar Reinforced by Randomly Oriented Short Fibers for the Retrofitting of the Historical Masonry Structure. Materials
**2020**, 13, 3462. [Google Scholar] [CrossRef] [PubMed] - Bustos-García, A.; Moreno-Fernández, E.; Zavalis, R.; Valivonis, J. Diagonal compression tests on masonry wallets coated with mortars reinforced with glass fibers. Mater. Struct.
**2019**, 52, 60. [Google Scholar] [CrossRef] - Simoncello, N.; Zampieri, P.; Gonzalez-Libreros, J.; Pellegrino, C. Experimental behaviour of damaged masonry arches strengthened with steel fiber reinforced mortar (SFRM). Compos. Part B Eng.
**2019**, 177, 107386. [Google Scholar] [CrossRef] - Zhang, Y.; Tubaldi, E.; Macorini, L.; Izzuddin, B.A. Mesoscale partitioned modelling of masonry bridges allowing for arch-backfill interaction. Constr. Build. Mater.
**2018**, 173, 820–842. [Google Scholar] [CrossRef] [Green Version] - Cannizzaro, F.; Pantò, B.; Caddemi, S.; Caliò, I. A Discrete Macro-Element Method (DMEM) for the nonlinear structural assessment of masonry arches. Eng. Struct.
**2018**, 168, 243–256. [Google Scholar] [CrossRef] [Green Version] - Pantò, B.; Chisari, C.; Macorini, L.; Izzuddin, B. A macroscale modelling approach for nonlinear analysis of masonry arch bridges. In Proceedings of the SAHC 2021, Barcelona, Spain, 29 September–1 October 2021. [Google Scholar]
- Zizi, M.; Cacace, D.; Rouhi, J.; Lourenço, P.B.; De Matteis, G. Automatic Procedures for the Safety Assessment of Stand-alone Masonry Arches. Int. J. Archit. Herit.
**2021**, in press. [Google Scholar] [CrossRef] - Heyman, J. The Masonry Arch; Ellis Horwood Ltd.: Chichester, UK, 1982. [Google Scholar]
- De Luca, A.; Giordano, A.; Mele, E. A simplified procedure for assessing the seismic capacity of masonry arches. Eng. Struct.
**2004**, 26, 1915–1929. [Google Scholar] [CrossRef] - Gilbert, M.; Casapulla, C.; Ahmed, H. Limit analysis of masonry block structures with non-associative frictional joints using linear programming. Comput. Struct.
**2006**, 84, 873–887. [Google Scholar] [CrossRef] - Funari, M.F.; Spadea, S.; Lonetti, P.; Fabbrocino, F.; Luciano, R. Visual programming for structural assessment of out-of-plane mechanisms in historic masonry structures. J. Build. Eng.
**2020**, 31, 101425. [Google Scholar] [CrossRef] - Como, M. Statics of Historic Masonry Constructions; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Geuzaine, C.; Remacle, J.-F. Gmsh: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Ministero delle Infrastrutture e dei Trasporti. Istruzioni per l’applicazione dell’Aggiornamento Delle “Norme Tecniche per le Costruzioni” di cui al decreto ministeriale 17 gennaio 2018; Ministero delle Infrastrutture e dei Trasporti: Rome, Italy, 2019. [Google Scholar]
- Rondelet, J.B. Trattato Teorico e Pratico Dell’arte di Edificare. Prima Traduzione Italiana Sulla Sesta Edizione Originale Con Note e Giunte Importantissime; Caranenti: Mantova, Italy, 1832. (In Italian) [Google Scholar]
- Iucolano, F.; Liguori, B.; Colella, C. Fibre-reinforced lime-based mortars: A possible resource for ancient masonry restoration. Constr. Build. Mater.
**2013**, 38, 785–789. [Google Scholar] [CrossRef] - Alexakis, H.; Makris, N. Hinging Mechanisms of Masonry Single-Nave Barrel Vaults Subjected to Lateral and Gravity Loads. J. Struct. Eng.
**2017**, 143, 04017026. [Google Scholar] [CrossRef] - Lourenço, P.B. Recent advances in masonry modelling: Micromodelling and homogenisation. In Multiscale Modeling in Solid Mechanics; Galvanetto, U., Aliabadi, M.H.F., Eds.; Imperial College Press: London, UK, 2009; pp. 251–294. [Google Scholar]
- Aiello, M.A.; Cascardi, A.; Ombres, L.; Verre, S. Confinement of Masonry Columns with the FRCM-System: Theoretical and Experimental Investigation. Infrastructures
**2020**, 5, 101. [Google Scholar] [CrossRef]

**Figure 1.**Collapse mechanisms of a buttressed arch: (

**a**) Local mechanism; (

**b**) Global mechanism; (

**c**) Semi-global mechanism.

**Figure 2.**Stress distribution in an opening hinge: (

**a**) perfectly plastic, and (

**b**) hinge with perfectly plastic reinforcement.

**Figure 7.**Validation of the automatic tool against the analytical results in [33].

**Figure 9.**Typical cyclic constitutive laws used for nonlinear springs in DMEM: (

**a**) tension/compression (rocking), (

**b**) shear in the block and (

**c**) sliding.

**Figure 10.**λ-u plots for the analysed models: (

**a**,

**b**) weak masonry, (

**c**,

**d**) strong masonry. (

**a**,

**c**) plots at varying f

_{t}, (

**b**,

**d**) at varying G

_{t}.

**Figure 11.**Comparison between DMEM and limit analysis: (

**a**) DMEM deformed shape at failure for all models except Model 1, (

**b**) Model 1 DMEM deformed shape at failure, (

**c**) limit analysis hinge position, (

**d**) peak load plastic strain for weak masonry with f

_{t}= 0.1 MPa, G

_{t}= 0.01 N/mm, and (

**e**) peak load plastic strain for weak masonry with f

_{t}= 0.1 MPa, G

_{t}= 0.05 N/mm.

**Figure 12.**Values of a* for each considered eccentricity, each h/H configuration and B/s = 0.4, expressed with respect to the t/s ratio: (

**a**) h/H = 0.5, (

**b**) h/H = 0.75 and (

**c**) h/H = 1.0. Local mechanism (L) or semi-global mechanism (S) are highlighted.

**Figure 13.**Values of a* for each considered eccentricity, each h/H configuration and B/s = 0.5, expressed with respect to the t/s ratio: (

**a**) h/H = 0.5, (

**b**) h/H = 0.75 and (

**c**) h/H = 1.0. Local mechanism (L) or semi-global mechanism (S) are highlighted.

**Figure 14.**Values of a* for each considered eccentricity, each h/H configuration and B/s = 0.6, expressed with respect to the t/s ratio: (

**a**) h/H = 0.5, (

**b**) h/H = 0.75 and (

**c**) h/H = 1.0. Local mechanism (L) or semi-global mechanism (S) are highlighted.

**Figure 15.**Values of a* for each considered eccentricity, t/s ratio, h/H configuration and B/s value. Local (L) or semi-global mechanism (S) are highlighted.

**Figure 16.**Percentage of activation of Local or Semi-global mechanisms in (UN), (RI), (RE) and (RI-E) buttressed arches.

**Figure 17.**Values of a*[g] of (RI) buttressed arches, for each considered eccentricity, t/s ratio, h/H configuration and B/s value. Local mechanism (L) or Semi-global mechanism (S) are highlighted.

**Figure 18.**Values of a*[g] of (RE) buttressed arches, for each considered eccentricity, t/s ratio, h/H configuration and B/s value. Local mechanism (L) or Semi-global mechanism (S) are highlighted.

**Figure 19.**Values of a*[g] of (RI-E) buttressed arches, for each considered eccentricity, t/s ratio, h/H configuration and B/s value. Local mechanism (L) or Semi-global mechanism (S) are highlighted.

**Figure 20.**Percentage Δa* increment ± standard deviation of (RI), (RE) and (RI-E) buttressed arches, classified according to the variation of the activated mechanisms with respect to the (UN) buttressed arches.

Masonry Type | w (kN/m^{3}) | E (Mpa) | G (Mpa) | f_{c} (Mpa) | G_{c} (N/mm) | τ_{0} (Mpa) | γ_{u}(-) | c (Mpa) | μ (-) | G_{f} (N/mm) |
---|---|---|---|---|---|---|---|---|---|---|

Weak | 19 | 690 | 230 | 1.0 | 3.0 | 0.018 | 0.005 | 0.018 | 0.6 | 2.0 |

Strong | 19 | 1800 | 600 | 4.6 | 3.0 | 0.13 | 0.005 | 0.13 | 0.6 | 2.0 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chisari, C.; Cacace, D.; De Matteis, G.
Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches. *Buildings* **2021**, *11*, 406.
https://doi.org/10.3390/buildings11090406

**AMA Style**

Chisari C, Cacace D, De Matteis G.
Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches. *Buildings*. 2021; 11(9):406.
https://doi.org/10.3390/buildings11090406

**Chicago/Turabian Style**

Chisari, Corrado, Daniela Cacace, and Gianfranco De Matteis.
2021. "Parametric Investigation on the Effectiveness of FRM-Retrofitting in Masonry Buttressed Arches" *Buildings* 11, no. 9: 406.
https://doi.org/10.3390/buildings11090406