# Prediction of Rubble-Stone Masonry Walls Response under Axial Compression Using 2D Particle Modelling

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Particle Models

#### 2.1. General Formulation

#### 2.2. Contact Stiffness and Resistance Models

#### 2.3. Generation of the Numerical Models

#### 2.4. Model Parameters and Calibration Requirements

## 3. Experimental Work

#### 3.1. Rubble-Stone Masonry Specimens

#### 3.2. Materials and Specimens’ Properties

^{3}[21]. The average compressive strength, measured from 14 specimens with nearly 60 × 60 × 60 mm

^{3}, prepared and tested according to [22], is 47.8 MPa. Besides these results, six stone specimens with almost 70 × 70 × 70 mm

^{3}were tested within the scope of this work, aiming to obtain the modulus of elasticity in axial compression [22], of 5.89 GPa, required for the numerical analysis. The hydrated air-lime used in the mortar of the rubble-stone masonry specimens [1] is a binder resulting from the decomposition under the temperature effect, of limestones with a percentage of calcium carbonate or calcium and magnesium higher than 95% [23]; according to NP EN 459-1:2011, this hydrated air-lime belongs to class CL90 S. The dimensions of the sands used in the experimental specimens mortar varies between 0.149 mm (minimum dimension of both River sand and Yellow pit sand) and 2.38 mm (maximum dimension of River sand).

^{3}, the following are used in this work: compressive strength and tensile flexural strength [24] of 0.650 MPa and 0.300 MPa, respectively. Under the scope of this work, additional tests were performed on mortar prismatic samples of 160 × 40 × 40 mm

^{3}, and the following results were obtained: compressive strength of 0.633 MPa and modulus of elasticity in axial compression of 74.95 MPa [24].

#### 3.3. Experimental Analysis of the URM Specimens

^{2}/0.32 m

^{2}). According to Table 1, the following average values were obtained as mechanical parameters of the reference (URM) specimens M43, M21, and M32: ${\sigma}_{c,max}=0.43$ MPa; ${\epsilon}_{v,{F}_{max}}=4.9\u2030$ and $E=0.305$ GPa.

## 4. Numerical Modelling

#### 4.1. Model Generation

^{3}[1]. Figure 6 shows an image of the frontal view of specimen M43 that was used to define the stone contours required to build the 2D-PM model, see Section 2.3.

#### 4.2. Parameter Calibration–Uniaxial Testing

## 5. Particle Model Prediction

#### 5.1. Frontal Model

#### 5.2. Lateral Model

**Table 7.**Elastic properties and peak-strength—numerical and experimental values [1].

Model | ${\mathit{F}}_{\mathit{m}\mathit{a}\mathit{x}}$ [kN] | E [GPa] | ${\mathit{\sigma}}_{\mathit{c},\mathit{m}\mathit{a}\mathit{x}}$ [MPa] | ${\mathit{d}}_{\mathit{v},{\mathit{F}}_{\mathit{m}\mathit{a}\mathit{x}}}$ [mm] |
---|---|---|---|---|

m-refined | 270 | 0.254 | 0.56 | 3.38 |

m-refined-elastic | 325 | 0.254 | 0.68 | 4.50 |

[1] | 168 (238) | - | 0.49 (0.73) | 5.97 (6.39) |

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

URM | Unreinforced masonry |

FEM | Finite element method |

DEM | Discrete element method |

DDA | Discontinuous deformation analysis |

RVE | Representative volume element |

2D-PM | 2D Particle model |

PM | Particle model |

## References

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**Figure 6.**Numerical model generation: (

**a**) Frontal wall photography with stone contour and (

**b**) final stone units representative polygons discretized with inner particles.

**Figure 7.**2D-PM numerical models that are representative of the masonry walls facets with a zoom of the particle assembly including the inter-particle contacts.

**Figure 8.**Final crack patterns: compression tests (specimen size 0.04 × 0.04 m

^{2}) and bending tests (specimen size 0.14 × 0.04 m

^{2}).

**Figure 9.**(

**a**) Stress-displacement diagram for compression test and (

**b**) Force-displacement diagram for bending test.

**Figure 12.**Contact crack evolution for frontal numerical model with 25% mortar–Yield plateau of 0.80 MPa (m-refined) and elastic behaviour under compression (m-refined (elastic)).

**Figure 13.**Numerical stress-displacement diagrams for frontal numerical models with 25% mortar volume–Frontal model–Refined and coarse particle assemblies.

**Figure 14.**Frontal model (m-refined)-Evolution of the damage pattern. Instances (

**a**–

**d**) are identified on Figure 10 as “Representative points”.

**Figure 15.**Frontal model (m-refined (elastic))-Evolution of the damage pattern. Instances (

**a**–

**d**) are identified on Figure 10 as “Representative points”.

**Figure 16.**Comparison of the experimental [1] and numerical stress-displacement diagrams for frontal numerical models with 25% mortar volume–Lateral model–Refined particle assembly.

**Figure 18.**Lateral Model m-refined-Evolution of the damage pattern. Instances (

**a**–

**d**) are identified on Figure 16 as “Representative points”.

**Figure 19.**Numerical stress-displacement diagrams for lateral numerical models with 25% mortar volume–Lateral model–Refined and coarse particle assemblies.

**Table 1.**Main experimental results obtained under compression loads of the rubble-stone’s masonry specimens.

Specimen | Age at Test | ${\mathit{F}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{\sigma}}_{\mathit{c},\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{d}}_{\mathit{v},{\mathit{F}}_{\mathit{m}\mathit{a}\mathit{x}}}$ | E |
---|---|---|---|---|---|

[days] | [kN] | [MPa] | [mm] | [GPa] | |

M47 ^{(1)} | 372 | 76.8 | 0.24 | 4.5 | 0.093 |

M43 ^{(2)} | 618 | 134.2 | 0.42 | 6.8 | 0.239 |

M21 ^{(3)} | 626 | 127.7 | 0.40 | 6.4 | 0.409 |

M32 ^{(4)} | 638 | 148.5 | 0.46 | 4.3 | 0.267 |

M36 ^{(5)} | 2866 | 238.3 | 0.74 | 6.4 | 0.212 |

M9 ^{(6)} | 3087 | 192.2 | 0.60 | 5.4 | 0.341 |

Model | Particles | Contacts | |||
---|---|---|---|---|---|

Stone (s) | Mortar (m) | m-m | m-s | s-s | |

Frontal-m-coarse | 14,990 | 88,738 | 40,337 | 247,943 | 21,053 |

Frontal-m-refined | 17,790 | 175,822 | 45,141 | 499,097 | 32,925 |

Lateral-m-coarse | 7501 | 45,381 | 20,034 | 127,176 | 10,154 |

Lateral-m-refined | 9363 | 90,012 | 23,327 | 255,619 | 16,275 |

(a) Experimental values | ||||

Material | E [GPa] | $\nu $ | ${\sigma}_{c}$ [MPa] | ${\sigma}_{t.fl}$ [MPa] |

Mortar | 0.075 | 0.16 | 0.65 | 0.3 |

Stone | 6.0 | 0.3 | 47.8 | - |

(b) Numerical predictions after calibration | ||||

Mortar (m-coarse) | 0.075 | 0.16 | 0.66 (0.69) | 0.15 (0.16) |

Mortar (m-refined) | 0.075 | 0.16 | 0.65 (0.67) | 0.25 (0.25) |

Stone | 6.0 | 0.3 | 47.8 | - |

Contacts | $\overline{\mathit{E}}$ | $\mathit{\alpha}$ | ${\mathit{\sigma}}_{\mathit{n}\mathit{t},\mathit{c}}$ | ${\mathit{\tau}}_{\mathit{c}}$ | ${\mathit{\mu}}_{\mathit{c}}$ | ${\mathit{G}}_{\mathit{f},\mathit{n}}$ | ${\mathit{G}}_{\mathit{f},\mathit{s}}$ |
---|---|---|---|---|---|---|---|

[GPa] | [–] | [MPa] | [MPa] | [–] | [N/m] | [N/m] | |

s-s | 8.60 | 0.11 | 8.90 | 35.7 | 1.0 | 0.3838 | 56.1403 |

m-m & m-s (m-coarse) | 0.09 | 0.43 | 0.16 | 0.16 | 1.0 | 0.0013 | 0.0030 |

m-m & m-s (m-refined) | 0.09 | 0.45 | 0.17 | 0.17 | 1.0 | 0.0020 | 0.0046 |

**Table 5.**Elastic properties and peak-strength—Frontal model-Numerical and experimental values [1].

Model | ${\mathit{F}}_{\mathit{m}\mathit{a}\mathit{x}}$ [kN] | E [GPa] | ${\mathit{\sigma}}_{\mathit{c},\mathit{m}\mathit{a}\mathit{x}}$ [MPa] | ${\mathit{d}}_{\mathit{v},{\mathit{F}}_{\mathit{m}\mathit{a}\mathit{x}}}$ [mm] |
---|---|---|---|---|

m-refined | 176 | 0.283 | 0.55 | 2.25 |

m-refined-elastic | 350 | 0.283 | 1.13 | 5.63 |

[1] | 168 (238) | – | 0.49 (0.73) | 5.97 (6.39) |

Loading Stage | ${\mathit{\sigma}}_{{\mathit{c}}_{\mathit{m}\mathit{a}\mathit{x}}}$ [MPa] | |
---|---|---|

m-m | m-s | |

3.00 mm/0.74 MPa | 5.94 | 5.21 |

4.50 mm/1.04 MPa | 10.36 | 10.26 |

5.63 mm/1.13 MPa | 14.27 | 15.13 |

4.50 mm/1.04 MPa | 11.79 | 11.38 |

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**MDPI and ACS Style**

Azevedo, N.M.; Pinho, F.F.S.; Cismaşiu, I.; Souza, M.
Prediction of Rubble-Stone Masonry Walls Response under Axial Compression Using 2D Particle Modelling. *Buildings* **2022**, *12*, 1283.
https://doi.org/10.3390/buildings12081283

**AMA Style**

Azevedo NM, Pinho FFS, Cismaşiu I, Souza M.
Prediction of Rubble-Stone Masonry Walls Response under Axial Compression Using 2D Particle Modelling. *Buildings*. 2022; 12(8):1283.
https://doi.org/10.3390/buildings12081283

**Chicago/Turabian Style**

Azevedo, Nuno Monteiro, Fernando F. S. Pinho, Ildi Cismaşiu, and Murilo Souza.
2022. "Prediction of Rubble-Stone Masonry Walls Response under Axial Compression Using 2D Particle Modelling" *Buildings* 12, no. 8: 1283.
https://doi.org/10.3390/buildings12081283