Although the strengthening systems considered are very different, all of them have shown to improve the masonry shear resistance. In the Reticulatus method, the additional shear resistance (*V**R*) is obtained by the reinforcement embedded in the mortar joints. It can be postulated that the contribution of the repointing mortar in the shear resistance is negligible. In the CP system, the matrix contributes to the gain in shear resistance (*V**M*) and no additional reinforcement (*V**R*) is incorporated. Finally, in WWM, the gain in shear resistance (*V**R* + *V**M*) was attributed by reinforcement and matrix applied on the surface of the walls. In order to determine the enhancement of shear resistance in different strengthening systems considered, the term *V**RM* defined in Eq. (1), can be generalised as given in Eq. (2), where, the shear strength contribution is either from the matrix (i.e. CP/*V**M*) or reinforcement (i.e. Reticulatus/ *V**R*), or from both components (i.e. in the case of WWM/*V**R* + *V**M*).

Subsequently, it can noted that the shear resistance of the strengthening system depends on the shear resistance of individual contributions from matrix (*V**M*) and reinforcement (*V**R*). In the following sections, the methods to evaluate the contribution of matrix and reinforcement in the shear resistance are verified.

## 3.2.1 Contribution of matrix strength

Cementitious matrix is commonly used in FRCM, WWM and CP strengthening systems. Specifically, the CP strengthening systems only use cementitious matrix to plaster on the masonry surface. Primarily, it is accepted that the matrix cracks and de-bonds, once its tensile resistance or interface shear bond resistance is attained under diagonal compression loading. Few different formulations have been used in the past to account the contribution of matrix in the diagonal shear resistance of masonry assemblages. Table 4 summaries these literature formulations to account the contribution of surface matrix/mortar plastering into the shear resistance of masonry. Especially, the study from Lin et al. (2014) used the formulation specified in JSCE (2008) to compute the shear resistance of ECC plastered masonry panels, where the resistance is determined from the minimum between the tensile strength of ECC matrix and bond strength between matrix and substrate. Almeida et al. (2015) adopted the same formulation given in Table 1 (i.e. *V**dt*) to determine the tensile resistance of URM and to compute the resistance of matrix (*f**t,max*) with relevant cross section area (*A**mx*). It can be noted that Ferretti and Mazzotti (2021) used the matrix flexural strength (*f**fl,mx*), where other studies used matrix tensile strength.

Determining the direct tensile strength of matrix is a quite difficult task, however, most of the experimental studies invariably reported the compressive strength of matrix along with the diagonal compression testing data. Thus the matrix tensile strength has to be derived from the compressive strength values. Consequently, the formulation given in EN 1992-1-1 (2004) to determine the direct tensile strength of concrete from the compressive strength is used in this study. The formulation is given in Eq. (3). This formulation is recommended for concrete grades less than 50 MPa, thus it could be also applicable to most of the cementitious matrixes used for masonry strengthening.

$${f}_{t,mx=}0.3{f}_{c,mx}^{0.67}$$

3

In order to assess the contribution of matrix to the strengthened masonry, the established database was used in this study. Especially, the CP database comprised only the assemblages strengthened with mortar matrixes, and it was used to verify the applicability of available design provisions in the literature. It has to be mentioned that, if the CP strengthening is applied only to one side of the masonry, the predicted resistance is reduced by 0.7 factor as recommended in CNR-DT-200 (2013) and CNR-DT-215 (2018)for FRP and FRCM strengthened assemblies. Consequently, the predictive formulations suggested by Almeida et al. (2015) and Donnini et al. (2021) were taken to predict the contribution of matrix strength to the strengthened systems. The other two formulations were not considered for comparison, because (1) the Lin et al. (2014) formulation involves bond characteristics between masonry and matrix (but not experimental data were found) and (2) the Ferretti and Mazzotti (2021) uses the matrix flexural strength (most of the literature studies do not provide the matric flexural strength, however only the corresponding compressive strength).

Figure 3 shows the experimental and predicted shear resistances of the strengthened masonry assemblages (i.e. CP). It has to be mentioned that the experimental shear resistances (on the horizontal axis) are the overall strengths of the strengthened assemblages. The predicted reistance is given by the contribution of URM (from previous section) and matrix incorporated. This exercise was implemented to verify the contribution from the matrix to the overall strength of strengthened masonry. In general, it can be noted that the formulations considered conservatively predicted the contribution of matrix to the shear resistance of strengthened masonry. The basic statistical parameters, derived in the analyses, are given in Table 5 for comparison purpose. However, the formulation trilled by Almeida et al. (2015) (i.e. also used to predict the tensile resistance of URM assemblages) conservatively predict the resistance than the formulation used by Donnini et al. (2021).

Table 4

Formulations given to predict the contribution of matrix strength

References | Formulation | Remarks |

Lin et al. (2014) | \(Min \left({t}_{m}0.72L{f}_{t,mx}, 0.18\sqrt{{f}_{c,mx}}L{t}_{m}\right)\) | Taken from JSCE [], for ECC strengthened elements, where the minimum contribution of ECC matrix and their bond between the substrate is accounted. *f**c,mx* is the compressive strength of matrix. |

Almeida et al. (2015) | \(\frac{\text{tan}\theta +\sqrt{21.2+{tan}^{2}\theta }}{10.6}{f}_{t,mx}{A}_{mx}\) | Similar to the tensile resistance of URM, however instead of tensile strength of URM, the tensile strength of matrix is used. |

Ferretti and Mazzotti (2021) | \(\frac{{f}_{fl,mx}{A}_{mx}}{0.5}\) | *f**fl,mx* and *A**mx* are the flexural strength the net cross section of matrix. |

Donnini et al. (2021) | \(\frac{L}{0.707}{f}_{t,mx}n{t}_{m}\) | *L* is the width of the panel, *f**t,mx* and *t**m* are the tensile strength and thickness of matrix. Then *n* is the number of layers of strengthening FRCM. |

Table 5

Basic statistical parameters of ME computed for matrix strengthened assemblages.

Strengthening system | Formulation considered | Mean of ME | Minimum ME | Maximum ME | COV (%) |

CP | Donnini et al (2021) | 1.97 | 0.68 | 8.33 | 85.1 |

Almeida et al (2015) | 3.84 | 0.75 | 14.66 | 81.9 |

## 3.2.2 Contribution of the reinforcement

The contribution of the reinforcement to the shear resistance of strengthened masonry (WWM and Reticulatus) involves complex mechanisms. However, different rational approaches have been developed to account the contribution of fabric/fibre in the FRCM and FRP strengthened masonry assemblages. A similar method can be used to establish the contribution of reinforcement in WWM and Reticulatus techniques. However, unlike FRCM and FRP, the WWM and Reticulatus methods involve the use of steel meshes and wires, which are anchored to the walls. For these strengthening methods, slippage mechanisms between the matrix and masonry substrate are relatively rare to occur compared to FRCM systems. Especially, due to the debonding mechanism in FRP, the term “effective/design tensile strain” and “effective/design tensile stress” limits are considered in FRP systems to account for the fibre contribution to the shear resistance (Vaculik et al. 2018; Porta et al. 2008). Due to the slippage and debonding mechanisms in FRCM systems, commonly referred as “telescopic effect”, similar analogy is considered to deduce the “conventional/design tensile strain” and “conventional/design tensile stress” limits (Thermou et al. 2021; Araya-Letelier et al. 2019). However, it can be hypothesized that such debonding and slippage would not be the major failure modes in WWM and Reticulatus methods as reinforcement bars/meshes are stiffer, and matrix cracking (due to failure of matrix/mortar under tensile stress) was more frequently recorded in the experimental studies. Further matrix impregnation in fabrics, and their debonding observed in FRCM, are not the mechanism observed in steel bars/wire in Reticulatus and WWM systems. Subsequently, the cracking of matrix depicts as debonding in WWM and Reticulatus reinforced, however it is not primarily due to the relative slippage between reinforcement and matrix.

Since the WWM and Reticulatus systems involve the use of steel meshes, bars and wires, the design approach of Reinforced Masonry (RM), which is widely used in North America and Australasia for concrete block walls will be considered (Araya-Letelier et al. 2019; Zahra et al 2021), where the reinforcing steel bars are inserted in the vertical cores of the hollow blocks and additional bars are embedded in the horizontal bed joints. In this RM, the reinforcement bars (horizontal and vertical) are placed mainly at the center of the walls (hollow cores), however in WWM and Reticulatus, the reinforcement is provided on the surface or near the surface (i.e. for Reticulatus). Figure 4 illustrates the reinforcement arrangement in different systems considered. Thus, similar approach can be drawn from the design concepts provided for Near Surface Mounted (NSM) FRP bars in masonry (ACI 440.2R-17). The available design formulations are given in Table 6 with their references.

It can be noted that the formulations have similar forms, where the contribution of reinforcement is accounted using the effective horizontal cross-sectional area of reinforcement and its yield strength. It is widely understood that the horizontal reinforcement provides resistance to the in-plane shear action, and the vertical reinforcement is primarily effective in resisting the in-plane flexural action. Therefore, the generalized formulation to predict the contribution of reinforcement in resisting the shear action can be written:

$${V}_{R}=C{A}_{r}{f}_{r}\frac{d}{s}$$

4

Where, *A**r* and *f**r* are the area and yield strength of reinforcement, respectively. *S* and *d* are the spacing of reinforcement and effective depth of shear resistance (it is equal to the width of the panels tested). *C* is a coefficient taken to account the contribution/efficiency of horizontal reinforcement in resisting shear. It has been well established that the horizontal reinforcement does not fully contribute to resisting the shear effects. It was highlighted, that the shear effect is initially carried by masonry, and the reinforcement is fairly unstressed. The horizontal reinforcement only starts contributing to resist shear once the cracks appeared and opned in masonry. Therefore, the contribution of shear reinforcement is reduced to conservatively predict the shear resistance (Voon et al 2007; Augenti et al. 2010) in RM assemblages. A similar analogy was used to verify the established methodology to predict the shear resistance of WWM and Reticulatus strengthened masonry.

Subsequently, using the formulations established to calculate the URM (section 3.1), matrix (section 3.2.1) and reinforcement contributions (section 3.2.2), the shear resistance of strengthened masonry (WWM and Reticulatus) was predicted. The generalised formula developed to evaluate the shear resistance of strengthened masonry is given in Eq. (5). Since the contribution of the URM and matrix have been already discussed in the previous sections, the accuracy of predicting the contribution of reinforcement in the shear resistance was verified in this section.

Table 6

Formulations to account the contribution of reinforcement.

Reference | Formulation | Remarks |

ACI 440.7R-10 (2017) | \({\rho }_{f}\frac{d}{s}\) | *s is the center-to-center spacing* *between the bars, and d is the effective masonry depth (i.e. minimum of length and width of wall/panel)* |

CSA S304.1-04 (2013) | \(0.6{A}_{r}{f}_{r}\frac{d}{s}\) | *A**r* *is the effective area of reinforcement, and f**r* *is the yield strength of reinforcement*, |

AS 3700 (2018) | \(0.8{A}_{sr}{f}_{r}\) | *A**sr* *is equal to* \({A}_{r}\frac{H}{L}\), *if* \(\frac{H}{L}\), *is less than 1, otherwise, A**sr* *should be considered minimum of horizontal and vertical reinforcement area.* |

MSJC (2013) | \(0.5{A}_{r}{f}_{r}\frac{d}{s}\) | *Similar terms are defined as per CSA S304.1-04* [70]. |

$${V}_{s}={V}_{URM}+\frac{\text{tan}\theta +\sqrt{21.2+{tan}^{2}\theta }}{10.6}{f}_{t,mx}{A}_{mx}+C{A}_{r}{f}_{r}\frac{d}{s}$$

5

Where, *V**URM* has to be determined from the set of formulations outlined in section 3.1. *θ*, *f**t,mx* and *A**mx* are the inclination angle between horizontal and diagonal of the masonry element, tensile strength of matrix and cross sectional area of matrix, respectively. Moreover, *A**r* and *f**r* are the area and yield strength of reinforcement, respectively. *S* and *d* are the spacing of reinforcement and effective depth of shear resistance. As mentioned, *C* is a constant incorporated to reduce the contribution of horizontal reinforcement to the shear resistance.

Consequently, the experimental database established for WWM and Reticulatus systems was utilised to verify the accuracy of the design approach. If the CP strengthening is applied only to one side of the masonry (WWM or Reticulatus), the predicted resistance is reduced by 0.7 factor as recommended, similar to other strengthened assemblies. Figure 5 shows the predicted and experimental shear capacity of WWM and Reticulatus strengthened masonry lateral capacities considered in the database. In order to conservatively predict the shear capacity of strengthened masonry, and to conservatively account for the contribution of reinforcement, *C* in the Eq. (5) was calibrated against the experimental database established. The value of *C* constant corresponding to the 95th percentile of *ME* was computed to achieve relatively conservative prediction. Thus, *C* was initially assumed as unity, then calibrated to achieve the 95th percentile of the data. Successively, the calibrated *C* values for WWM and Reticulatus systems were 0.51 and 0.37, respectively. It can be concluded that the proposed unified approach can be used to conservatively predict the shear resistance of the strengthened masonry, where the minimum *ME* is closed to 1 and the 95th percentile ME is fairly higher (> 4) for the cases considered in the analyses. Thus, it can be stated that the established formulation to predict the shear resistance of WWM, Reticulatus and CP strengthened masonry can conservatively be used.

Table 6

Statistical parameters obtained for the WWM and Reticulatus shear resistances.

Strengthening system | Mean of ME | Minimum ME | Maximum ME | COV (%) | 95th Percentile ME | Calibrated *C* value |

WWM | 2.75 | 1.04 | 5.35 | 38.0 | 4.48 | 0.51 |

Reticulatus | 2.47 | 0.94 | 6.01 | 51.7 | 4.85 | 0.37 |

Although the unified formulation developed has shown to predict the shear resistance of WWM, Reticulatus and CP strengthened masonry, the formulation could be further calibrated with more data from future research studies. Because masonry typologies varies between regions and counties, the applicability of the strengthening methods examined and their contribution to the shear resistance need further systematic experimental verification incorporating the variability in the constitutive properties, masonry geometry and strengthening types.