# Multiscale Numerical Analysis of TRM-Reinforced Masonry under Diagonal Compression Tests

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Campaign

#### 2.1. Building and Strengthening of Masonry Panels

^{3}high, 1000 mm

^{3}wide and 120 mm

^{3}thick, characterised by 14 rows of clay bricks (CB) of average size 250 × 55 × 120 mm

^{3}, as illustrated in Figure 1.

#### 2.2. Mechanical Characterisation of Materials

^{2}for the compression modulus. For the purpose of our numerical analysis, the GM elastic modulus was estimated using the following relationship (cf. [43]), Equation (1):

_{c}to the compression strength f

_{c}. Equation (1) can also be empirically applied to mortars, as suggested in [44]. For the mortars used in our TRM application, the coefficient k

_{c}can be estimated as equal to 3146, by using E

_{c}and f

_{c}measured for LM (cf. Table 3). Substituting this value and the f

_{c}of GM into Equation (1), the compression elastic modulus E

_{c}of GM was estimated as equal to 2489 N/mm

^{2}. Poisson’s ratios for LM, GM and BM were assumed to be the same and all equal to 0.15. The assumed values of E and ν for LM, GM and BM are summarised in Table 5.

#### 2.3. Characterisation of Mortar–Brick Interfaces

_{v0}, was estimated by using triplet tests, according to the standard UNI EN 1052-3 (2007) [45]. The main results are summarised in Table 6 in terms of mean values and coefficients of variation (CV).

_{v0}can be assumed as the mortar–brick interface bond shear strength.

#### 2.4. In-Plane Diagonal Compression Tests

_{p}), vertical peak displacements (δ

_{p}), their relative average values and failure modes are summarised in Table 7.

_{p,av}increased only slightly (+18%).

_{k}and δ

_{k}(cf. Table 8).

_{k}and δ

_{k}have not been defined for BTRM specimens (cf. Table 8). As shown in Figure 6d, diagonal cracks were observed both in bricks and in the TRM layer matrix, demonstrating the reinforcement’s high efficiency.

_{p}and δ

_{p}(Table 7), while for GTRM it is estimated using P

_{k}and δ

_{k}. An important increase in average stiffness can be noticed for both strengthening systems, especially for BTRM (+128%).

## 3. Numerical Investigation of Interface Properties

_{b}), defined as the interface bond shear strength at zero normal stress, is an important requested parameter for the implementation of the numerical model. According to the standard UNI EN 1052-3 (2007) [45], the average shear stress measured in triplet tests to estimate the interface bond strength of a single specimen is calculated using Equation (2):

_{max}

_{,i}is the maximum shear load and A

_{i}the specimen’s transversal area parallel to the mortar joint. The average bond strengths for the three tested brick–mortar interfaces are listed in Table 6. It is believed that f

_{v0}is affected by non-zero normal stress distribution and thus cannot be considered as providing a realistic estimate of the interface bond cohesion. To provide a more realistic estimation of c

_{b}, we adopted the following original strategy. The triplet shear test was modelled in Abaqus for each of the three used mortars (LM, GM and BM), according to the standardised configuration, cf. UNI EN 1052-3 (2007) [45]. All materials are considered isotropic and infinitely elastic, and perfect contact was assumed between bricks and mortar. A thin layer of mortar elements, hereafter denoted as IL (interface layer), was used to represent the mortar–brick interface where debonding was experimentally observed, i.e., the fracture surface (Figure 3a,c,d). During the simulation, we monitored the numerical average shear stress in the IL. The simulation was stopped when the monitored shear stress equalled the measured bond strength f

_{v0}of the studied mortar (cf. Table 6). This allows an analysis of the numerical normal stress and shear stress distributions in the IL at the instant of debonding. Finally, a new c

_{b}estimate is obtained with an iterative analytical procedure based on the Mohr–Coulomb failure criterion [48,49], as a function of the numerical normal and shear stresses in the IL.

#### 3.1. Triplet Numerical Model and Shear Test Simulation

_{v0}(0.29 N/mm

^{2}for the interface CB-LM, cf. Table 6). It can be seen that the distribution of ${\mathsf{\tau}}_{\mathrm{yz}}$ (Figure 7c) is not uniform and there is also a non-zero normal stress contribution (${\mathsf{\sigma}}_{\mathrm{yy}}$, Figure 7b).

#### 3.2. Analytical Procedure to Estimate c_{b}

_{b}as the bond cohesion estimated for these areas. The following procedure applies the Mohr–Coulomb failure criterion to locate the CEs in order to obtain an estimation for c

_{b}. Before illustrating the procedure, some assumptions must be made. According to the Mohr–Coulomb failure criterion [48,49], the bond strength ${\mathsf{\tau}}_{\mathrm{yz}}^{\mathrm{c}}$ (the superscript c indicates damage initiation) generally depends on normal stress ${\mathsf{\sigma}}_{\mathrm{yy}}^{}$ as follows, Equations (3) and (4):

_{t}(Table 3). The friction contribution was not investigated during the experimental campaign because the triplet bending tests were not conducted with increasing pressures. The procedure to estimate c

_{b}is as follows. A critical ratio (CR) is introduced to locate the CEs of the CB-LM IL. Equation (5) expresses the critical ratio ${\mathrm{CR}}_{\mathrm{i}}$ where the subscript i indicates the i

^{th}generic element of the CB-LM IL (Figure 7a), characterised by its ${\mathsf{\sigma}}_{\mathrm{yy},\mathrm{i}}$ and ${\mathsf{\tau}}_{\mathrm{yz},\mathrm{i}}$.

_{b}is still unknown. The higher the value of ${\mathrm{CR}}_{\mathrm{i}}$, the more critical the ith element is. Moreover:

- If ${\mathrm{CR}}_{\mathrm{i}}$< 1, the area corresponding to the ith element is not a crack initiation site.
- If ${\mathrm{CR}}_{\mathrm{i}}$= 1, the ith element is a CE, i.e., cracking initiates in the corresponding area.
- If ${\mathrm{CR}}_{\mathrm{i}}$> 1, the corresponding area is assumed to be already fractured.

_{b}when the maximum ${\mathrm{CR}}_{\mathrm{i}}$ is equal to 1. Figure 8 illustrates the iterative process to estimate c

_{b}through Equation (5).

_{v0}(Table 6). The increment $\Delta {\mathrm{c}}_{\mathrm{b}}^{\u2019}$ can be positive or negative, depending if the maximum ${\mathrm{CR}}_{\mathrm{i}}$ is higher or lower than 1, respectively. Figure 7b,c shows the location of the CEs for CB-LM IL with their ${\mathsf{\sigma}}_{\mathrm{yy}}$ and ${\mathsf{\tau}}_{\mathrm{yz}}$. The corresponding c

_{b}(0.53 N/mm

^{2}) is reported in Table 9.

_{b}estimation, can be repeated for CB-GM and CB-BM triplets, according to the following important points:

- For the CB-GM system, the IL was assumed to be located between the interior brick and the mortar joint because of the experimental evidence (cf. Figure 3c).

_{b}are reported in Table 9. It can be noticed that each one is greater than the corresponding measured f

_{v0}given in Table 6.

## 4. Numerical Model for the Diagonal Compression Test

#### 4.1. Reinforcement Layer Modelling

_{g}, provided by the constructors, and fibre mass density ${\mathsf{\rho}}_{\mathrm{f}}$ (Table 10):

#### 4.2. Constitutive Behaviour of Interface Joints

**t-δ**) law, with

**t**the traction vector and

**δ**the relative opening. This contact law exhibits an initial linear elastic response, followed by damage softening [48,49], as depicted in Figure 11.

**t**, stiffness

**K**and separation

**δ**:

_{nn}, K

_{ss}, K

_{tt}) depend on the elastic and shear moduli of the mortar (E

_{m}and G

_{m}, respectively). Many equations have been proposed in the literature for this purpose [48,49,53,54]. In this study, the stiffness components are obtained through Equations (10) and (11) proposed in [54]:

_{m}represents the effective thickness; for the contact surface between TRM and the wall, h

_{m}is the thickness of the cohesive element (1 mm, Figure 9), thus allowing an analysis of the two studied TRM techniques adopting the same mesh. Table 12 reports the elastic properties of the different interface joints.

**t**reaches its critical value, Equation (12):

_{t}(Table 3). The values of ${\mathrm{t}}_{\mathrm{n}}^{\mathrm{c}}$ are also reported in Table 12 for the different cohesive joints. According to the Mohr–Coulomb failure criterion [48,49], critical shear stresses in Equation (12) (${\mathrm{t}}_{\mathrm{s}}^{\mathrm{c}}$ and ${\mathrm{t}}_{\mathrm{t}}^{\mathrm{c}}$) will depend on the normal stress t

_{n}similarly to Equation (3), as expressed in Equation (13):

_{b}and f

_{t}. The values of c

_{b}and f

_{t}are reported in Table 9 and Table 3, respectively. A user subroutine USDFLD was developed to implement Equation (13) (cf. [52]). Damage propagation in the post-peak phase is taken into account as follows, Equation (14):

_{eff}should be introduced, defined as follows, Equation (15):

#### 4.3. Modelling of Anchors

^{2}, while numerical BA is characterised by an equivalent circular cross-section of area 14.5 mm

^{2}. Material properties, as provided by manufacturers, are listed in Table 4.

## 5. Results and Discussion

- symmetry plane YZ;
- FE analyses performed in a dynamic regime, in order to better deal with the nonlinearity of the constitutive cohesive law;
- imposed vertical displacements at the boundary (Figure 12a) in quasi-static conditions to simulate the low jack displacement rate of the experimental tests (0.08 mm/s).

_{k}and δ

_{k}) and also observed in [15]. The BTRM numerical curve (“num BTRM” in Figure 13) was obtained by assuming that no damage occurs in the bed and head interface joints (Figure 9a). Indeed, it was found that the numerical model omitted to reproduce the BTRM damage behaviour, showing an absence of bed joint sliding. With the assumption of neglecting bed and head joint sliding, the numerical BTRM linear elastic trend shown in Figure 13 predicts the experimental curve more accurately.

_{num}is calculated with P

_{p}and δ

_{p}for UM and BTRM, and with P

_{k}and δ

_{k}for GTRM. The numerical model slightly underestimates both P

_{k}(–18%) and δ

_{k}(–28%). GTRM stiffness is predicted with a good accuracy (error of +13%), although more important deviations are observed for UM and BTRM. However, the values of ΔK provided in Table 14 are compatible with the high dispersion of experimental results that typically characterises these materials (cf. [8]).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Observed triplet failure modes: (

**a**) CB-LM 7/8; (

**b**) CB-LM 1/8; (

**c**) CB-GM 3/3; (

**d**) CB-BM 3/3.

**Figure 4.**(

**a**) Test set-up schematisation; (

**b**) specimen GTRM-1 and (

**c**) specimen BTRM-1 in the testing device.

**Figure 7.**(

**a**) Triplet model set-up; (

**b**) normal stress and (

**c**) shear stress distributions in the CB-LM IL for$\overline{{\mathsf{\tau}}_{\mathrm{yz}}}={\mathrm{f}}_{\mathrm{v}0}$.

**Figure 12.**(

**a**) Numerical model layout; representation of the (

**b**) GTRM–wall and (

**c**) BTRM–wall interactions.

**Figure 13.**Comparison of experimental and numerical results: vertical load vs vertical displacement.

**Figure 14.**(

**a**) Numerical load vs displacement curves; (

**b**, part 1) damage distribution in masonry bed and head joints; (

**b**, part 2) damage distribution in wall–TRM interface joint.

Sample ID | Number of Samples | Matrix | Grid | Anchors |
---|---|---|---|---|

UM | 1 | --- | --- | --- |

GTRM | 2 | Hydraulic lime mortar (GM) sprayed in a single layer 30 mm thick on each side | Preformed GFRP square grid (GG), average thickness 3 mm, spacing 33 × 33 mm^{2} each side | Coupled L-shaped preformed GFRP anchors (GA) of 70 mm^{2} section, 100 mm long each |

BTRM | 2 | Mineral binder-added lime mortar (BM) sprayed in two layers 10 mm thick each side | Balanced basalt–stainless steel AISI 304 square grid (BG), average thickness 0.064 mm, spacing 8 × 8 mm^{2} each side | Helical stainless steel AISI 316 anchors (BA) of 14.5 mm^{2} section, 200 mm long each |

Test Type | Standard Reference | Specimen Description | Materials ID | Corresponding Symbols |
---|---|---|---|---|

Compression | UNI EN 772-1, 2015 [40] | Cubes 40 × 40 × 40 mm^{3} | CB, LM, GM, BM | f_{c} |

Three-point bending | UNI EN 1015-11, 2007 [41] | Prisms 160 × 40 × 40 mm^{3} | LM, GM, BM | f_{t} |

Uniaxial compression | EN 1052-1, 2001 [42] | Single-leaf wall 645 × 630 × 120 mm^{3} (bricks: 250 × 55 × 120 mm^{3}) | CB + LM | E, ν |

Bending Tests | Compression Tests | |||||||
---|---|---|---|---|---|---|---|---|

Material ID | Number of Samples | f_{t} (N/mm^{2}) | CV (%) | Number of Samples | f_{c} (N/mm^{2}) | CV (%) | E (N/mm^{2}) | ν |

CB | --- | --- | --- | 20 | 19.37 | 12 | 20176 | 0.23 |

LM | 18 | 0.84 | 25 | 36 | 4.64 | 15 | 2499 | --- |

GM | 3 | 1.14 | 10 | 6 | 4.58 | 5 | --- | --- |

BM | 3 | 2.05 | 17 | 6 | 8.16 | 3 | --- | --- |

Material ID | Tensile Strength (N/mm^{2}) | Elastic Modulus (N/mm^{2}) | Ultimate Strain (%) |
---|---|---|---|

GG | ≥350 | ≥27 | ≥1.5 |

BG | ≥1700 | ≥70 | ≥1.9 |

GA | ≥440 | ≥26 | ≥1.7 |

BA | ≥700 (ε = 0.2%) | ≥150 | ≥3.0 |

Material ID | k_{c} | E (N/mm^{2}) | Ν |
---|---|---|---|

LM | --- | 2499 | 0.15 |

GM | 3146 | 2489 | 0.15 |

BM | --- | 9000 | 0.15 |

Interface ID | Number of Samples | f_{v0} (N/mm^{2}) | CV (%) |
---|---|---|---|

CB-LM | 8 | 0.29 | 23 |

CB-GM | 3 | 0.36 | 36 |

CB-BM | 3 | 0.37 | 68 |

Specimen ID | P_{p}(kN) | P_{p, av}(kN) | ΔP_{p, TRM/UM} (%) | δ_{p}(mm) | δ_{p, av}(mm) | Δδ_{p, TRM/UM}(%) | Failure Mode |
---|---|---|---|---|---|---|---|

UM | 71 | 71 | --- | 0.28 | 0.28 | --- | Bed joint sliding |

GTRM-1 | 146 | 154 | +117 | 0.65 | 0.72 | +157 | TRM debonding |

GTRM-2 | 162 | 0.78 | |||||

BTRM-1 | 198 | 183 | +158 | 0.35 | 0.33 | +18 | Diagonal traction |

BTRM-2 | 167 | 0.31 |

Sample ID | P_{k}(kN) | P_{k, av}(kN) | δ_{k}(mm) | δ_{k, av}(mm) | K (kN/mm) | K_{av}(kN/mm) | ΔK_{TRM/UM}(%) |
---|---|---|---|---|---|---|---|

UM | --- | --- | --- | 0.28 | 243 | 243 | --- |

GTRM-1 | 136 | 142 | 0.41 | 0.39 | 332 | 365 | +50 |

GTRM-2 | 147 | 0.37 | 397 | ||||

BTRM-1 | --- | --- | --- | --- | 566 | 553 | +128 |

Interface ID | c_{b} (N/mm^{2}) |
---|---|

CB-LM | 0.53 |

CB-GM | 0.64 |

CB-BM | 0.88 |

Material ID | ${\mathsf{\gamma}}_{\mathrm{g}}^{}\left(\mathrm{g}/{\mathrm{m}}^{2}\right)$ | ${\mathsf{\rho}}_{\mathrm{f}}\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\right)$ | ${\mathsf{\rho}}_{\mathrm{m}}\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\right)$ |
---|---|---|---|

GG | 1000 | 2600 | --- |

BG | 400 | 2750 | --- |

GM | --- | --- | 1400 |

BM | --- | --- | 1580 |

Ply | ${\mathrm{h}}_{\mathrm{f}}^{\mathrm{e}\mathrm{q}}\left(\mathrm{m}\mathrm{m}\right)$ | ${\mathrm{V}}_{\mathrm{f}}^{\mathrm{e}\mathrm{q}}$ | ${\mathsf{\rho}}_{\mathrm{g}\mathrm{m}}\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\right)$ | ${\mathrm{E}}_{\mathrm{r}\mathrm{p}}\left(\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}\right)$ | ${\mathsf{\nu}}_{\mathrm{r}\mathrm{p}}$ | ${\mathrm{G}}_{\mathrm{r}\mathrm{p}}\left(\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}\right)$ |
---|---|---|---|---|---|---|

GG + GM | 0.384 | 0.384 | 1862 | 8464 | 0.12 | 2072 |

BG + BM | 0.145 | 0.145 | 1750 | 14507 | 0.13 | 4863 |

Interface Joint | Mortar | ${\mathrm{h}}_{\mathrm{m}}\left(\mathrm{m}\mathrm{m}\right)$ | ${\mathrm{K}}_{\mathrm{n}\mathrm{n}}\left(\mathrm{N}/\mathrm{m}{\mathrm{m}}^{3}\right)$ | ${\mathrm{K}}_{\mathrm{s}\mathrm{s}}\left({\mathrm{K}}_{\mathrm{t}\mathrm{t}}\right)\left(\mathrm{N}/\mathrm{m}{\mathrm{m}}^{3}\right)$ | ${\mathrm{t}}_{\mathrm{n}}^{\mathrm{c}}\left(\mathrm{N}/\mathrm{m}{\mathrm{m}}^{2}\right)$ |
---|---|---|---|---|---|

Bed joint in UM | LM | 17 | 149 | 61 | 0.84 |

Head joint in UM | LM | 10 | 264 | 108 | 0.84 |

TRM–wall | GM | 1 | 2640 | 1087 | 1.14 |

TRM–wall | BM | 1 | 9503 | 3913 | 2.05 |

**Table 13.**Numerical results (peak load, peak displacement, failure mode) and comparison with the experimental values.

Specimen ID | P_{p, num}(kN) | P_{p, exp,av}(kN) | ΔP_{p, num/exp}(%) | δ_{p, num}(mm) | δ_{p, exp,av}(mm) | Δδ_{p, num/exp}(%) | Numerical Failure Mode |
---|---|---|---|---|---|---|---|

UM | 78 | 71 | +9 | 0.24 | 0.28 | −14 | Bed joint sliding |

GTRM | 161 | 154 | +5 | 0.65 | 0.72 | −10 | TRM debonding |

BTRM | 206 | 183 | +13 | 0.49 | 0.33 | +48 | TRM debonding |

**Table 14.**Numerical results (knee load, knee displacement, stiffness) and comparison with the experimental values.

Specimen ID | P_{k, num}(kN) | P_{k,exp, av} (kN) | ΔP_{k, num/exp}(%) | δ_{k, num}(mm) | δ_{k, exp,av}(mm) | Δδ_{k, num/exp}(%) | K_{num}(kN/mm) | K_{exp, av}(kN/mm) | ΔK _{num/exp}(%) |
---|---|---|---|---|---|---|---|---|---|

UM | --- | --- | --- | --- | --- | --- | 325 | 243 | +34 |

GTRM | 116 | 142 | −18 | 0.28 | 0.39 | −28 | 414 | 365 | +13 |

BTRM | --- | --- | --- | --- | --- | --- | 420 | 553 | −24 |

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## Share and Cite

**MDPI and ACS Style**

Gulinelli, P.; Aprile, A.; Rizzoni, R.; Grunevald, Y.-H.; Lebon, F.
Multiscale Numerical Analysis of TRM-Reinforced Masonry under Diagonal Compression Tests. *Buildings* **2020**, *10*, 196.
https://doi.org/10.3390/buildings10110196

**AMA Style**

Gulinelli P, Aprile A, Rizzoni R, Grunevald Y-H, Lebon F.
Multiscale Numerical Analysis of TRM-Reinforced Masonry under Diagonal Compression Tests. *Buildings*. 2020; 10(11):196.
https://doi.org/10.3390/buildings10110196

**Chicago/Turabian Style**

Gulinelli, Pietro, Alessandra Aprile, Raffaella Rizzoni, Yves-Henri Grunevald, and Frédéric Lebon.
2020. "Multiscale Numerical Analysis of TRM-Reinforced Masonry under Diagonal Compression Tests" *Buildings* 10, no. 11: 196.
https://doi.org/10.3390/buildings10110196