# RC Structures Strengthened by an Iron-Based Shape Memory Alloy Embedded in a Shotcrete Layer—Nonlinear Finite Element Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Program

## 3. Finite Element Simulation

#### 3.1. Material Model for Concrete in ABAQUS

#### 3.2. Modeling the Iron-Based Shape Memory Alloy in ABAQUS

#### 3.3. Finite Element Model Implementation

## 4. Results and Discussion

#### 4.1. Four-Point Bending Test on RC Beams Strengthened by Ribbed Fe-SMA Bars

#### 4.1.1. Concrete Damaged Plasticity (CDP) Parameters

_{b0}/f

_{c0}, which was set to 1.16 according to the ABAQUS manual [22] as there were no biaxial compression test results available for the concrete used in the beams. As observed in Figure 6 and Figure 7, variations in this parameter have a negligible effect on the results because the elements are not primarily under biaxial compression during loading.

_{c}and the plastic potential eccentricity e, were set as 2/3 and 0.1, respectively, as suggested by the ABAQUS manual [22]. The k

_{c}is determined from the results of the full triaxial test of concrete. However, these data were not available and the default values were assigned [22]. The results of the sensitivity analyses shown in Figure 8, Figure 9, Figure 10 and Figure 11 indicate no significant change in the load–displacement curves caused by variations in these two parameters; however, a small eccentricity value may lead to early convergence errors in the analysis.

#### 4.1.2. Load vs. Mid-Span Deflection Results

#### 4.2. Parametric Studies

#### 4.2.1. Effect of Fe-SMA Diameter

#### 4.2.2. Effect of Pre-Stressing Force

#### 4.2.3. Effect of Pre-Stressing Force Level

#### 4.3. Case Study, Numerical Modeling of Bridge Girder Pre-Stressed by Fe-SMA Bars in a Shotcrete Layer

^{2}cross-sectional area. A 1.25 m × 0.2 m upper concrete slab was also modeled. The cross-section schematic and longitudinal profile of the reference girder are illustrated in Figure 27 [33].

## 5. Conclusions

_{c}and the ratio of the biaxial compressive strength to the uniaxial compressive strength did not significantly affect the load–displacement response of the models; different concrete dilation angles yielded different results in the post-cracking stage. Adopting a proper viscosity parameter had a positive effect on the convergence and accuracy of the analysis results.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Janke, L. Applications of shape memory alloys in civil engineering structures—Overview, limits and new ideas. Mater. Struct.
**2005**, 38, 578–592. [Google Scholar] [CrossRef] - Cladera, A.; Weber, B.; Leinenbach, C.; Czaderski, C.; Shahverdi, M.; Motavalli, M. Iron-based shape memory alloys for civil engineering structures: An overview. Constr. Build. Mater.
**2014**, 63, 281–293. [Google Scholar] [CrossRef] - 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] - Schranz, B.; Czaderski, C.; Vogel, T.; Shahverdi, M. Bond investigations of prestressed, near-surface-mounted, ribbed memory-steel bars with full bond length. Mater. Des.
**2020**, 196, 109145. [Google Scholar] [CrossRef] - 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] - 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] - Michels, J.; Shahverdi, M.; Czaderski, C.; El-Hacha, R. Mechanical performance of iron-based shape-memory alloy ribbed bars for concrete prestressing. ACI Mater. J.
**2018**, 115, 877–886. [Google Scholar] - 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] - 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] - Ghafoori, E.; Neuenschwander, M.; Shahverdi, M.; Czaderski, C.; Fontana, M. Elevated temperature behavior of an iron-based shape memory alloy used for prestressed strengthening of civil structures. Constr. Build. Mater.
**2019**, 211, 437–452. [Google Scholar] [CrossRef] - Yamauchi, K.; Ohkata, I.; Tsuchiya, K.; Miyazaki, S. Shape Memory and Superelastic Alloys: Technologies and Applications; Woodhead Publishing: Sawston, UK, 2011; pp. 1–208. [Google Scholar]
- Youssef, M.A.; Alam, M.S.; Nehdi, M. Experimental Investigation on the Seismic Behavior of Beam-Column Joints Reinforced with Superelastic Shape Memory Alloys. J. Earthq. Eng.
**2008**, 12, 1205–1222. [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] - 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] - Ghassemieh, M.; Mostafazadeh, M.; Sadeh, M.S. Seismic control of concrete shear wall using shape memory alloys. J. Intell. Mater. Syst. Struct.
**2012**, 23, 535–543. [Google Scholar] [CrossRef] - Ruiz-Pinilla, J.G.; Montoya-Coronado, L.A.; Ribas, C.; Cladera, A. Finite element modeling of RC beams externally strengthened with iron-based shape memory alloy (Fe-SMA) strips, including analytical stress-strain curves for Fe-SMA. Eng. Struct.
**2020**, 223, 111152. [Google Scholar] [CrossRef] - Alam, M.; Youssef, M.; Nehdi, M. Analytical prediction of the seismic behaviour of superelastic shape memory alloy reinforced concrete elements. Eng. Struct.
**2008**, 30, 3399–3411. [Google Scholar] [CrossRef] - Auricchio, F.; Taylor, R.L.; Lubliner, J. Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behavior. Comput. Methods Appl. Mech. Eng.
**1997**, 146, 281–312. [Google Scholar] [CrossRef] - Abdulridha, A.; Palermo, D.; Foo, S.; Vecchio, F.J. Behavior and modeling of superelastic shape memory alloy reinforced concrete beams. Eng. Struct.
**2013**, 49, 893–904. [Google Scholar] [CrossRef] - Malagisi, S.; Marfia, S.; Sacco, E.; Toti, J. Modeling of smart concrete beams with shape memory alloy actuators. Eng. Struct.
**2014**, 75, 63–72. [Google Scholar] [CrossRef] [Green Version] - Khalil, W.; Mikolajczak, A.; Bouby, C.; Ben Zineb, T. A constitutive model for Fe-based shape memory alloy considering martensitic transformation and plastic sliding coupling: Application to a finite element structural analysis. J. Intell. Mater. Syst. Struct.
**2012**, 23, 1143–1160. [Google Scholar] [CrossRef] - Hibbitt, K.; Sorensen, I. ABAQUS/Standard User’s Manual; Dassault Systèmes Simulia Corp.: Providence, RI, USA, 2011. [Google Scholar]
- Rezazadeh, M.; Costa, I.; Barros, J.A.O. Influence of prestress level on NSM CFRP laminates for the flexural strengthening of RC beams. Compos. Struct.
**2014**, 116, 489–500. [Google Scholar] [CrossRef] [Green Version] - Domingo, J.C.; Kuang-Han, C. Stress-Strain Relationship for Plain Concrete in Compression. ACI J. Proc.
**1985**, 82. [Google Scholar] [CrossRef] - ACI. Building Code Requirements for Structural Concrete; ACI 318-08; ACI: Farmington Hills, MI, USA, 2008. [Google Scholar]
- Comité Euro-International Du Beton. CEB-FIP, Model Code 1990; Comité Euro-International Du Beton: Paris, France, 1991; pp. 87–109. [Google Scholar]
- Omran, H.Y.; El-Hacha, R. Nonlinear 3D finite element modeling of RC beams strengthened with prestressed NSM-CFRP strips. Constr. Build. Mater.
**2012**, 31, 74–85. [Google Scholar] [CrossRef] - Hillerborg, A.; Modéer, M.; Petersson, P.-E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res.
**1976**, 6, 773–781. [Google Scholar] [CrossRef] - Gopalaratnam, V.S.; Shah, S.P. Softening Response of Plain Concrete in Direct Tension. ACI J. Proc.
**1985**, 82, 310–323. [Google Scholar] [CrossRef] - Mercan, B.; Schultz, A.E.; Stolarski, H.K. Finite element modeling of prestressed concrete spandrel beams. Eng. Struct.
**2010**, 32, 2804–2813. [Google Scholar] [CrossRef] - Szczecina, M.; Winnicki, A. Calibration of the CDP Model Parameters in Abaqus. In Proceedings of the 2015 World Congress on Advances in Structural Engineering and Mechanics (ASEM15), Incheon, Korea, 25–29 August 2015. [Google Scholar]
- Yapar, O.; Basu, P.; Nordendale, N. Accurate finite element modeling of pretensioned prestressed concrete beams. Eng. Struct.
**2015**, 101, 163–178. [Google Scholar] [CrossRef] - Michels, J.; Staskiewicz, M.; Czaderski, C.; Kotynia, R.; Harmanci, Y.E.; Motavalli, M. Prestressed CFRP Strips for Concrete Bridge Girder Retrofitting: Application and Static Loading Test. J. Bridg. Eng.
**2016**, 21, 04016003. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Cross-sections of Beams 9, 10 and 11 (in mm) [14].

**Figure 2.**Test set-up of the beams (in mm) [14].

**Figure 27.**Longitudinal profile and cross-section of the girders (in cm) [33].

**Figure 29.**Effect of non-activated Fe-SMA bars on load–deflection behavior of girders compared to the reference.

**Figure 30.**Effect of activated Fe-SMA bars on load–deflection behavior of girders compared to the reference.

Fe-SMA Bar Diameter | Displacement Ductility Index |
---|---|

6 mm | 6.6 |

8 mm | 4.9 |

10 mm | 4.4 |

12 mm | 3.6 |

Parameter | Non-Activated Fe-SMA Bars | Activated Fe-SMA Bars |
---|---|---|

Cracking load (kN) | 5.6 | 9.2 |

Load at 4 mm mid-span deflection (kN) | 7.5 | 11.3 |

Bottom steel rebar yielding load (kN) | 10.7 | 16.3 |

Ultimate load (kN) | 20.7 | 21.8 |

Displacement ductility Index | 7.9 | 5 |

**Table 3.**Mechanical properties of concrete, longitudinal steel rebar and tendons [33].

Parameter | Value |
---|---|

Cubic compressive strength of concrete—girder (MPa) | 64.6 |

Cubic compressive strength of concrete—slab (MPa) | 50.0 |

Elastic modulus of concrete—girder (GPa) | 34.7 |

Elastic modulus of concrete—slab (GPa) | 32.1 |

Elastic modulus of steel rebar (GPa) | 200 |

Elastic modulus of steel tendon (GPa) | 210.3 |

Yield stress of steel rebar (MPa) | 462 |

Yield stress of steel tendon (MPa) | 1660 |

Tensile strength of steel rebar (MPa) | 545 |

Tensile strength of steel tendon (MPa) | 1810 |

Strain at failure of steel rebar (%) | 10.6 |

Strain at failure of steel tendon (%) | 3.76 |

Girder Name | Total Cross-Sectional Area of Fe-SMA Bars (mm^{2}) |
---|---|

Reference | none |

G1 | 235.6 |

G2 | 339.3 |

G3 | 461.8 |

G4 | 615.8 |

G5 | 804.2 |

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

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/ma13235504

**AMA Style**

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(23):5504.
https://doi.org/10.3390/ma13235504

**Chicago/Turabian Style**

Dolatabadi, Neda, Moslem Shahverdi, Mehdi Ghassemieh, and Masoud Motavalli.
2020. "RC Structures Strengthened by an Iron-Based Shape Memory Alloy Embedded in a Shotcrete Layer—Nonlinear Finite Element Modeling" *Materials* 13, no. 23: 5504.
https://doi.org/10.3390/ma13235504