# Numerical Modeling of Unreinforced Masonry Walls Strengthened with Fe-Based Shape Memory Alloy Strips

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fe-Based Shape Memory Alloy

## 3. Masonry Wall: A Case Study

## 4. Numerical Modeling

#### 4.1. Material Model

_{m}. After that, the slope of the linear area depends on the type of mortar. In the macro-model of the masonry wall, a three-line simplified behavior proposed by Agnihotri et al. was used to define the tensile behavior of the homogenized material [42]. In this model, ${f}_{t}$ is the tensile strength of the mortar and the axial tensile response is shown in Figure 7b. A stress of 0.02 MPa was used in large deformation for convergence in finite element analysis [44].

#### 4.2. Damage Parameter

_{t}) and compression damage (d

_{c}). Figure 8 shows the axial stress-strain behavior diagram in tension and pressure for concrete materials. In this figure, the model stiffness in loading is reduced by a factor of (1 − d

_{c}) or (1 − d

_{t}) [47,48]. In this form, ε

_{t}

^{in}and ε

_{c}

^{in}are the residual tensile and compressive strains in the case of non-damage, respectively, and ε

_{t}

^{pl}and ε

_{c}

^{pl}are the residual tensile and compressive strains in the case of concrete damage, respectively. Equations (4) and (5) show the relationships between these strains.

#### 4.3. Meshing and Boundary Conditions

#### 4.4. FE Models Verification and Mesh Sensitivity Analysis

## 5. Fe-Based Strips

## 6. Creating Post-Tension in Numerical Models by Fe-SMA Strips

## 7. Hysteresis Results

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sharda, A.; Manalo, A.; Ferdous, W.; Bai, Y.; Nicol, L.; Mohammed, A.; Benmokrane, B. Axial compression behaviour of all-composite modular wall system. Compos. Struct.
**2021**, 268, 113986. [Google Scholar] [CrossRef] - Ferdous, W.; Bai, Y.; Almutairi, A.D.; Satasivam, S.; Jeske, J. Modular assembly of water-retaining walls using GFRP hollow profiles: Components and connection performance. Compos. Struct.
**2018**, 194, 1–11. [Google Scholar] [CrossRef] - Da Porto, F.; Mosele, F.; Modena, C. In-plane cyclic behaviour of a new reinforced masonry system: Experimental results. Eng. Struct.
**2011**, 33, 2584–2596. [Google Scholar] [CrossRef] - Gouveia, J.P.; Lourenço, P.B. Masonry Shear Walls Subjected to Cyclic Loading: Influence of Confinement and Horizontal Reinforcement. In Proceedings of the North America Masonry Conference, St. Louis, MO, USA, 3–5 June 2007. [Google Scholar]
- Nateghi, F.; Alemi, F. Experimental Study of Seismic Behaviour of Typical Iranian URM Brick Walls. In Proceedings of the 14th World Conference on Earthquake Engineering, Beijing, China, 12–17 October 2008. [Google Scholar]
- Pujol, S.; Benavent-Climent, A.; Rodriguez, M.; Smith-Pardo, J. Masonry Infill Walls: An Effective Alternative for Seismic Strengthening of Low-Rise Reinforced Concrete Building Structures. In Proceedings of the 14th World Conference on Earthquake Engineering, Beijing, China, 12–17 October 2008. [Google Scholar]
- Maheri, M.; Najafgholipour, M.; Rajabi, A. The Influence of Mortar Head Joints on the In-Plane and Out-Of-Plane Seismic Strength of Brick Masonry Walls. In Proceedings of the 14th World Conference on Earthquake Engineering, Beijing, China, 12–17 October 2008. [Google Scholar]
- Petry, S.; Beyer, K. Limit states of modern unreinforced clay brick masonry walls subjected to in-plane loading. Bull. Earthq. Eng.
**2015**, 13, 1073–1095. [Google Scholar] [CrossRef] [Green Version] - Petry, S.; Beyer, K. Force-Displacement response of in-plane-loaded URM walls with a dominating flexural mode. Earthq. Eng. Struct. Dyn.
**2015**, 44, 2551–2573. [Google Scholar] [CrossRef] [Green Version] - Petry, S.; Beyer, K. Cyclic test data of six unreinforced masonry walls with different boundary conditions. Earthq. Spectra
**2015**, 31, 2459–2484. [Google Scholar] [CrossRef] [Green Version] - Beyer, K.; Tondelli, M.; Petry, S.; Peloso, S. Dynamic testing of a four-storey building with reinforced concrete and unreinforced masonry walls: Prediction, test results and data set. Bull. Earthq. Eng.
**2015**, 13, 3015–3064. [Google Scholar] [CrossRef] [Green Version] - Sadeghi Marzaleh, A. Seismic In-Plane Behavior of Post-Tensioned Existing Clay Brick Masonry Walls; ETH Zurich: Zurich, Switzerland, 2015. [Google Scholar] [CrossRef]
- Schultz, A.E.; Scolforo, M.J. Overview of prestressed masonry. Mason. Soc. J.
**1991**, 10, 6–21. [Google Scholar] - Kohail, M.; Elshafie, H.; Rashad, A.; Okail, H. Behavior of post-tensioned dry-stack interlocking masonry shear walls under cyclic in-plane loading. Constr. Build. Mater.
**2019**, 196, 539–554. [Google Scholar] [CrossRef] - Soltanzadeh, G.; Bin Osman, H.; Vafaei, M.; Vahed, Y.K. Seismic retrofit of masonry wall infilled RC frames through external post-tensioning. Bull. Earthq. Eng.
**2017**, 16, 1487–1510. [Google Scholar] [CrossRef] - Laursen, P.T.; Ingham, J.M. Structural testing of single-storey post-tensioned concrete masonry walls. Mason. Soc. J.
**2001**, 19, 69–82. [Google Scholar] - Farshchi, D.M.; Motavalli, M.; Schumacher, A.; Marefat, M.S. Nonlinear FE Modeling of In-Plane Behavior of Plain Masonry Walls and Investigation Effects of Post-Tensioning as a Parametric Study. In Proceedings of the 5th International Conference on Seismology and Earthquake Engineering, Tehran, Iran, 13–16 May 2007. [Google Scholar]
- Farshchi, D.M.; Motavalli, M.; Schumacher, A.; Marefat, M.S. Numerical modelling of in-plane behaviour of URM walls and an investigation into the aspect ratio, vertical and horizontal post-tensioning and head joint as a parametric study. Arch. Civ. Mech. Eng.
**2009**, 9, 5–27. [Google Scholar] [CrossRef] [Green Version] - Farshchi, D.M.; Marefat, M.; Motavalli, M.; Schumacher, A. A Numerical Investigation into the Brick size and Type of Mortar Effects on In-Plane Behavior of URM Walls and a Case Study. In Proceedings of the 8th International Masonry Conference, Dresden, Germany, 4–7 July 2010. [Google Scholar]
- Shahverdi, M.; Michels, J.; Czaderski, C.; Motavalli, M. Iron-based shape memory alloy strips for strengthening RC members: Material behavior and characterization. Constr. Build. Mater.
**2018**, 173, 586–599. [Google Scholar] [CrossRef] - Czaderski, C.; Shahverdi, M.; Brönnimann, R.; Leinenbach, C.; Motavalli, M. Feasibility of iron-based shape memory alloy strips for prestressed strengthening of concrete structures. Constr. Build. Mater.
**2014**, 56, 94–105. [Google Scholar] [CrossRef] - Shahverdi, M.; Czaderski, C.; Motavalli, M. Iron-based shape memory alloys for prestressed near-surface mounted strengthening of reinforced concrete beams. Constr. Build. Mater.
**2016**, 112, 28–38. [Google Scholar] [CrossRef] - Michels, J.; Shahverdi, M.; Czaderski, C. Flexural strengthening of structural concrete with iron-based shape memory alloy strips. Struct. Concr.
**2018**, 19, 876–891. [Google Scholar] [CrossRef] - Michels, J.; Shahverdi, M.; Czaderski, C. Mechanical performance of iron-based shape-memory alloy ribbed bars for concrete prestressing. ACI Mater. J.
**2018**, 115, 877–886. [Google Scholar] - Izadi, M.; Motavalli, M.; Ghafoori, E. Iron-based shape memory alloy (Fe-SMA) for fatigue strengthening of cracked steel bridge connections. Construct. Build. Mater.
**2019**, 227, 116800. [Google Scholar] [CrossRef] - Izadi, M.; Ghafoori, E.; Shahverdi, M.; Motavalli, M.; Maalek, S. Development of an iron-based shape memory alloy (Fe-SMA) strengthening system for steel plates. Eng. Struct.
**2018**, 174, 433–446. [Google Scholar] [CrossRef] - Izadi, M.; Ghafoori, E.; Motavalli, M.; Maalek, S. Iron-based shape memory alloy for the fatigue strengthening of cracked steel plates: Effects of re-activations and loading frequencies. Eng. Struct.
**2018**, 176, 953–967. [Google Scholar] [CrossRef] - Izadi, M.; Ghafoori, E.; Hosseini, A.; Motavalli, M.; Maalek, S.; Czaderski, C.; Shahverdi, M. Feasibility of Iron-Based Shape Memory Alloy Strips for Prestressed Strengthening of Steel Plates. In Proceedings of the Fourth International Conference on Smart Monitoring, Assessment and Rehabilitation of Civil Structures (SMAR 2017), Zurich, Switzerland, 13–15 September 2017. [Google Scholar]
- Shahverdi, M.; Czaderski, C.; Annen, P.; Motavalli, M. Strengthening of RC beams by iron-based shape memory alloy bars embedded in a shotcrete layer. Eng. Struct.
**2016**, 117, 263–273. [Google Scholar] [CrossRef] - Schranz, B.; Czaderski, C.; Vogel, T.; Shahverdi, M. Ribbed Iron-Based Shape Memory Alloy Bars for Pre-Stressed Strengthening Applications. In Proceedings of the IABSE 2019, Guimarães, Portugal, 27–29 March 2019. [Google Scholar]
- Schranz, B.; Czaderski, C.; Vogel, T.; Shahverdi, M. Bond behaviour of ribbed near-surface-mounted iron-based shape memory alloy bars with short bond lengths. Mater. Des.
**2020**, 191, 108647. [Google Scholar] [CrossRef] - Billah, A.M.; Alam, M.S. Seismic performance of concrete columns reinforced with hybrid shape memory alloy (SMA) and fiber reinforced polymer (FRP) bars. Constr. Build. Mater.
**2012**, 28, 730–742. [Google Scholar] [CrossRef] - Lagoudas, D.C. Shape Memory Alloys: Modeling and Engineering Applications; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Ghassemieh, M.; Rezapour, M.; Sadeghi, V. Effectiveness of the shape memory alloy reinforcement in concrete coupled shear walls. J. Intell. Mater. Syst. Struct.
**2017**, 28, 640–652. [Google Scholar] [CrossRef] - Karimi, A.H.; Karimi, M.S.; Kheyroddin, A.; Shahkarami, A.A. Experimental and Numerical Study on Seismic Behavior of An Infilled Masonry Wall Compared to An Arched Masonry Wall. Structures
**2016**, 8, 144–153. [Google Scholar] [CrossRef] - Ghassemieh, M.; Rezapour, M.; Taghinia, A. Predicting Low Cycle Fatigue Life through Simulation of Crack in Cover Plate Welded Beam to Column Connections. J. Comput. Appl. Mech.
**2017**, 48, 39–52. [Google Scholar] - Ajaei, B.B.; Ghassemieh, M. Reinforcing fillet welds preventing cracks in partial joint penetration welds. Int. J. Steel Struct.
**2015**, 15, 487–497. [Google Scholar] [CrossRef] - Abdulla, K.F.; Cunningham, L.S.; Gillie, M. Simulating masonry wall behaviour using a simplified micro-model approach. Eng. Struct.
**2017**, 151, 349–365. [Google Scholar] [CrossRef] - Rezapour, M.; Ghassemieh, M. Macroscopic modelling of coupled concrete shear wall. Eng. Struct.
**2018**, 169, 37–54. [Google Scholar] [CrossRef] - Lee, J.; Fenves, G.L. Plastic-damage model for cyclic loading of concrete structures. J. Eng. Mech.
**1998**, 124, 892–900. [Google Scholar] [CrossRef] - Lubliner, J.; Oliver, J.; Oller, S.; Onate, E. A plastic-damage model for concrete. Int. J. Solids Struct.
**1989**, 25, 299–326. [Google Scholar] [CrossRef] - Agnihotri, P.; Singhal, V.; Rai, D.C. Effect of in-plane damage on out-of-plane strength of unreinforced masonry walls. Eng. Struct.
**2013**, 57, 1–11. [Google Scholar] [CrossRef] - Kaushik, H.B.; Rai, D.C.; Jain, S.K. Stress-strain characteristics of clay brick masonry under uniaxial compression. J. Mater. Civ. Eng.
**2007**, 19, 728–739. [Google Scholar] [CrossRef] - Andreaus, U.; Ippoliti, L. Masonry panels under in-plane loading: A comparison between experimental and numerical results. WIT Trans. Model. Simul.
**1970**, 12. [Google Scholar] [CrossRef] - Sadeghi, V.; Hesami, S. Investigation of load transfer efficiency in jointed plain concrete pavements (JPCP) using FEM. Int. J. Pavement Res. Technol.
**2018**, 11, 245–252. [Google Scholar] [CrossRef] - Jankowiak, T.; Lodygowski, T. Identification of Parameters of Concrete Damage Plasticity Constitutive Model. Found. Civil Environ. Eng.
**2005**, 6, 53–69. [Google Scholar] - Bencardino, F.; Nisticò, M.; Verre, S. Experimental Investigation and Numerical Analysis of Bond Behavior in SRG-Strengthened Masonry Prisms Using UHTSS and Stainless-Steel Fibers. Fibers
**2020**, 8, 8. [Google Scholar] [CrossRef] [Green Version] - 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] - Najafi, F. Strengthening of an RC Beam Using Externally Un-Bonded Iron-Based Shape Memory Alloy Strips. Master’s Thesis, University of Tehran, Tehran, Iran, 2018. [Google Scholar]
- Abouali, S.; Shahverdi, M.; Ghassemieh, M.; Motavalli, M. Nonlinear simulation of reinforced concrete beams retrofitted by near-surface mounted iron-based shape memory alloys. Eng. Struct.
**2019**, 187, 133–148. [Google Scholar] [CrossRef] - Dolatabadi, N.; Shahverdi, M.; Ghassemieh, M.; Motavalli, M. RC Structures Strengthened by an Iron-Based Shape Memory Alloy Embedded in a Shotcrete Layer—Nonlinear Finite Element Modeling. Materials
**2020**, 13, 5504. [Google Scholar] [CrossRef] - Abaqus Documentation. 2017. Available online: abaqus-docs.mit.edu/2017/English/SIMACAEEXCRefMap/simaexc-c-docproc.htm (accessed on 3 June 2019).

**Figure 5.**The protocol of the applied lateral displacement, adapted from [35].

**Figure 6.**Finite element modeling approaches: (

**a**) detailed micro-model, (

**b**) simplified micro-model, (

**c**) macro-model.

**Figure 11.**Comparison of different models to validate sensibility (

**a**) different sizes (

**b**) different types.

**Figure 12.**Numerically studied walls, (

**a**) UMW2, (

**b**) UMW3, (

**c**) UMW4, (

**d**) UMW5, (

**e**) UMW6, and (

**f**) UMW7.

**Figure 13.**Stress distribution in numerically studied models, (

**a**) UMW2, (

**b**) UMW3, (

**c**) UMW4, (

**d**) UMW5, (

**e**) UMW6, and (

**f**) UMW7.

**Figure 15.**Hysteresis of Fe-SMA reinforced walls, (

**a**) UMW2, (

**b**) UMW3, (

**c**) UMW4, (

**d**) UMW5, (

**e**) UMW6, and (

**f**) UMW7.

**Figure 19.**Maximum strength in each cycle of the loading in the positive direction (

**a**) vertical-strip walls and (

**b**) cross-strip walls.

**Figure 20.**Permanent deformation during the cyclic loading for models, (

**a**) UMW2, (

**b**) UMW3, (

**c**) UMW4, (

**d**) UMW5, (

**e**) UMW6, and (

**f**) UMW7.

Dilation angle | 30° |

Flow potential eccentricity | 0.1 |

The ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress | 1.16 |

The ratio of second stress invariant | 0.67 |

Viscosity parameter | 0.001 |

**Table 2.**Mechanical properties of iron-based memory alloy [49].

Ultimate tensile strength (MPa) | 1000 |

Post-tension stress by heating the element to 160° C (Mpa) | 300–400 |

Pre-strain (%) | 4 |

Elasticity module after activation (Mpa) | 70,000 |

Models | Length (cm) | Height (cm) | Thickness (cm) | SMA Strip Thickness (mm) | Strip Width (mm) | Center to Center Distance of the Two Parallel Strips (cm) |
---|---|---|---|---|---|---|

UMW1 | 172 | 150 | 19.5 | - | - | … |

UMW2 | 172 | 150 | 19.5 | 1.5 | 120 | 86 |

UMW3 | 172 | 150 | 19.5 | 1.5 | 60 | 43 |

UMW4 | 172 | 150 | 19.5 | 1.5 | 30 | 34.5 |

UMW5 | 172 | 150 | 19.5 | 1.5 | 120 | … |

UMW6 | 172 | 150 | 19.5 | 1.5 | 60 | 57 |

UMW7 | 172 | 150 | 19.5 | 1.5 | 30 | 28.5 |

Cycle NO | UMW1 | UMW5 | UMW6 | UMW7 | ||||
---|---|---|---|---|---|---|---|---|

Permanent Deformation (mm) | Permanent Deformation (mm) | Percentage Reduction | Permanent Deformation (mm) | Percentage Reduction | Permanent Deformation (mm) | Percentage Reduction | ||

Positive deformations | 7 | 0.54 | 1.18 | −119.28 | 1.15 | −113.47 | 1.10 | −105.52 |

8 | 0.63 | 1.14 | −81.00 | 1.14 | −80.65 | 1.14 | −80.17 | |

9 | 0.67 | 1.16 | −73.18 | 1.15 | −72.33 | 1.14 | −71.05 | |

10 | 4.46 | 5.13 | −15.14 | 5.12 | −14.83 | 5.10 | −14.49 | |

11 | 6.17 | 5.14 | 16.65 | 5.03 | 18.44 | 4.40 | 28.71 | |

12 | 6.14 | 6.06 | 1.23 | 5.24 | 14.63 | 4.39 | 28.54 | |

13 | 10.65 | 9.56 | 10.26 | 9.35 | 12.16 | 9.12 | 14.33 | |

14 | 10.86 | 9.54 | 12.13 | 9.24 | 14.88 | 8.21 | 24.37 | |

Negative deformations | 7 | −0.65 | −1.22 | −86.32 | −1.14 | −74.67 | −1.03 | −57.97 |

8 | −0.70 | −1.53 | −118.65 | −1.20 | −72.00 | −1.10 | −57.82 | |

9 | −0.66 | −1.69 | −156.03 | −1.55 | −135.70 | −1.14 | −73.74 | |

10 | −5.36 | −5.02 | 6.36 | −4.47 | 16.54 | −4.42 | 17.51 | |

11 | −5.46 | −5.12 | 6.25 | −5.03 | 7.88 | −4.42 | 19.06 | |

12 | −5.45 | −5.37 | 1.55 | −5.14 | 5.60 | −4.40 | 19.20 | |

13 | −10.96 | −9.21 | 15.98 | −9.18 | 16.23 | −9.16 | 16.48 | |

14 | −10.69 | −9.48 | 11.35 | −9.16 | 14.37 | −8.10 | 24.26 |

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**

Rezapour, M.; Ghassemieh, M.; Motavalli, M.; Shahverdi, M.
Numerical Modeling of Unreinforced Masonry Walls Strengthened with Fe-Based Shape Memory Alloy Strips. *Materials* **2021**, *14*, 2961.
https://doi.org/10.3390/ma14112961

**AMA Style**

Rezapour M, Ghassemieh M, Motavalli M, Shahverdi M.
Numerical Modeling of Unreinforced Masonry Walls Strengthened with Fe-Based Shape Memory Alloy Strips. *Materials*. 2021; 14(11):2961.
https://doi.org/10.3390/ma14112961

**Chicago/Turabian Style**

Rezapour, Moein, Mehdi Ghassemieh, Masoud Motavalli, and Moslem Shahverdi.
2021. "Numerical Modeling of Unreinforced Masonry Walls Strengthened with Fe-Based Shape Memory Alloy Strips" *Materials* 14, no. 11: 2961.
https://doi.org/10.3390/ma14112961