# Experimental and Numerical Investigation on the Size Effect of Ultrahigh-Performance Fibre-Reinforced Concrete (UHFRC)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Investigation

#### 2.1. Material Preparation and Geometry of the Examined Specimens

^{3}) and 3% per volume (234 kg/m

^{3}) for mixtures 1 and 2, respectively. For the development of UHPFRC-1, a combination of shorter (i.e., 6 mm) and longer (i.e., 13 mm) brass coated steel fibres were added, whereas for the development of UHPFRC-2, only one length of fibres (13 mm) was incorporated in the mixture. All fibres had a diameter of 0.16 mm, a tensile strength of 3000 MPa and Modulus of Elasticity 200 GPa.

#### 2.2. Flexural Prism Tests

_{t}is calculated using Equation (1):

M | is the bending moment; |

I | is the moment of inertia; |

y | is the distance of the centroid from the extreme fibre. |

${\sigma}_{t-3p}and{\sigma}_{t-4p}$ | are the flexural strength values calculated from the three-point and four-point bending tests (MPa); |

$P$ | is the peak load (N); |

$L$ | is the effective span length (mm); |

$b$ | is the width of specimen (mm); |

$d$ | is the depth of the specimens (mm). |

## 3. Constitutive Modelling and Numerical Analysis

#### 3.1. Direct Tensile Tests

#### 3.2. UHPFRC Constitutive Modelling

_{t}) followed by a bilinear descending branch (Figure 9). This model will be used for the numerical modelling of both UHPFRC-1 and UHPFRC-2 to evaluate the reliability of the model for these two different types of UHPFRC.

_{ch}) equal to 2 mm and finite elements size (l

_{t}) equal to 65 mm [17]. This model has been found to be able to accurately predict the behaviour of UHPFRC-2, however the reliability of this model is highly dependent on the values of the characteristic size and the mesh size of the elements of the numerical models, which significantly affect the results in the post-crack region. Therefore, it is important to develop models which can accurately predict the behaviour of UHPFRC independently of the size of the Finite Element Models. This crucial aspect is addressed in this study with the development of a model which takes into consideration the size of the elements and can be used to accurately predict the behaviour of various geometries of UHPFRC specimens. The proposed tensile stress-strain characteristics are defined each time depending on the size of the finite elements (l

_{t}), following the model presented in Figure 10.

_{t new}) was 14.7 mm while the respective value for the specimen of Figure 12a was 15 mm. Following the procedure described in Figure 10, the UHPFRC constitutive model was altered by adjusting the characteristic size value (l

_{ch}= 2 mm) which was initially proposed for l

_{t}= 65 mm, multiplying it with the ratio $\frac{{l}_{tnew}}{{l}_{t}}$ (i.e., for l

_{t new}= 14.7 mm, $\frac{{l}_{tnew}}{{l}_{t}}$ = 0.2 and l

_{ch new}$=0.2\xb7l$

_{ch}$=0.4\mathrm{mm}$).

## 4. Results and Discussion

## 5. Conclusions

- The flexural strength of the examined prisms is reduced as the depth of the specimens is increased, which confirms the so called “size effect”.
- This reduction is attributed to the uneven distribution of fibres in thicker elements (e.g., 100 mm), as opposed to specimens with smaller thickness where there is a more even distribution of the fibres and therefore increased flexural strength is achieved.
- The reduction rate of the flexural strength values is more pronounced in case of UHPFRC-1, which is linked to its higher percentage of steel fibres (UHPFFRC-1 has 6% steel fibres while UHPFRC-2 has 3%).
- The proposed numerical modelling approach can accurately predict all the examined types (with different fibre volume fractions and different geometries), confirming the applicability of the proposed method for the simulation of UHPFRC specimens with different dimensions, eliminating the impact of the size effect. The proposed methodology can be used to accurately simulate the response of relatively thin UHPFRC layers and with thicknesses no more than 100 mm. Further research is required for the simulation of UHPFRC specimens with thickness higher than 100 mm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Nicolaides, D. Fracture and Fatigue of CARDIFRC. Ph.D. Thesis, Cardiff University, Wales, UK, 2004. [Google Scholar]
- Nicolaides, D.; Kanellopoulos, A.; Karihaloo, B.L. Investigation of the effect of fibre distribution on the fatigue performance and the autogenous shrinkage of CAR-DIFRC
^{®}. In Measuring, Monitoring and Modelling Concrete Properties; Konsta-Gdoutos, M.S., Ed.; Springer: Dordrecht, The Netherlands, 2006; pp. 3–16. [Google Scholar] - Farhat, F.A.; Nicolaides, D.; Kanellopoulos, A.; Karihaloo, B.L. CARDIFRC
^{®}—Performance and application to retrofitting. J. Eng. Fract. Mech.**2007**, 74, 151–167. [Google Scholar] [CrossRef] - Nicolaides, D.; Kanellopoulos, A.; Petrou, M.; Savva, P.; Mina, A. Development of a new Ultra High Performance Fibre Reinforced Cementitious Composite (UHPFRCC) for impact and blast protection of structures. Constr. Build. Mater.
**2015**, 95, 667–674. [Google Scholar] [CrossRef] - Paschalis, S.; Lampropoulos, A. Fiber content and curing time effect on the ten-sile characteristics of ultra high performance fiber reinforced concrete. Struct. Concr.
**2017**, 18, 577–588. [Google Scholar] [CrossRef] - Hannawi, K.; Bian, H.; Prince-Agbodjan, W.; Raghavan, B. Effect of different types of fibers on the microstructure and the mechanical behavior of ultra-high performance fiber-reinforced concretes. Compos. Part B Eng.
**2016**, 86, 214–220. [Google Scholar] [CrossRef][Green Version] - Abbas, S.; Soliman, A.M.; Nehdi, M.L. Exploring mechanical and durability properties of ultra-high performance concrete incorporating various steel fiber lengths and dosages. Constr. Build. Mater.
**2015**, 75, 429–441. [Google Scholar] [CrossRef] - Gesoglu, M.; Güneyisi, E.; Muhyaddin, G.F.; Asaad, D.S. Strain hardening ultrahigh performance fiber reinforced cementitious composites: Effect of fiber type and concentration. Compos. B Eng.
**2016**, 103, 74–83. [Google Scholar] [CrossRef] - Kazemi, S.; Lubell, A.S. Influence of specimen size and fiber content on mechanical properties of ultra-high-performance fiber-reinforced concrete. ACI Mater. J.
**2012**, 109, 675–684. [Google Scholar] - Wu, Z.; Shi, C.; He, W.; Wu, L. Effects of steel fiber content and shape on mechanical properties of ultra high performance concrete. Constr. Build. Mater.
**2016**, 103, 8–14. [Google Scholar] [CrossRef] - Wu, Z.; Khayat, K.H.; Shi, C. How do fiber shape and matrix composition affect fiber pullout behavior and flexural properties of UHPC? Cem. Concr. Compos.
**2018**, 90, 193–201. [Google Scholar] [CrossRef][Green Version] - Yoo, D.Y.; Kang, S.T.; Yoon, Y.S. Enhancing the flexural performance of ultrahigh-performance concrete using long steel fibers. Compos. Struct.
**2016**, 147, 220–230. [Google Scholar] [CrossRef] - Mahmud, G.; Yang, Z.; Hassan, A. Experimental and numerical studies of size effects of ultra high performance steel fibre reinforced concrete (UHPFRC) beams. Constr. Build. Mater.
**2013**, 48, 1027–1034. [Google Scholar] [CrossRef] - An, M.; Zhang, L.; Yi, Q. Size effect on compressive strength of reactive powder con-crete. J. China Univ. Min. Technol.
**2008**, 18, 279–282. [Google Scholar] [CrossRef] - Awinda, K.; Chen, J.; Barnett, S. Investigating geometrical size effect on the flexural strength of the ultra high performance fibre reinforced concrete using the cohesive crack model. Constr. Build. Mater.
**2015**, 105, 123–131. [Google Scholar] [CrossRef][Green Version] - Hassan, A.M.T.; Jones, S.W.; Mahmud, G.H. Experimental test methods to determine the uniaxial tensile and compressive behaviour of ultra high performance fibre reinforced concrete (UHPFRC). Constr. Build. Mater.
**2012**, 37, 874–882. [Google Scholar] [CrossRef] - Lampropoulos, A.; Paschalis, S.; Tsioulou, O.; Dritsos, S. Strengthening of rein-forced concrete beams using ultra high performance fibre reinforced concrete (UHPFRC). Eng. Struct.
**2016**, 106, 370–384. [Google Scholar] [CrossRef][Green Version] - Bastien-Masse, M.; Brühwiler, E. Contribution of R-UHPFRC Strengthening layers to the shear resistance of RC elements. Struct. Eng. Int.
**2016**, 4, 365–374. [Google Scholar] [CrossRef] - Paschalis, S.; Lampropoulos, A.; Tsioulou, O. Experimental and numerical study of the performance of ultra high performance fiber reinforced concrete for the flexural strengthening of full scale reinforced concrete members. Constr. Build. Mater.
**2018**, 186, 351–366. [Google Scholar] [CrossRef] - Paschalis, S.; Lampropoulos, A. Developments in the use of Ultra High Perfor-mance Fiber Reinforced Concrete as strengthening material. Constr. Build. Mater.
**2021**, 233, 111914. [Google Scholar] - Bastien-Masse, M.; Brühwiler, E. Experimental investigation on punching re-sistance of R-UHPFRC–RC composite slabs. Mat. Struct.
**2016**, 49, 1573–1590. [Google Scholar] [CrossRef] - Bastien-Masse, M.; Brühwiler, E. Composite model for predicting the punching re-sistance of R-UHPFRC–RC composite slabs. Eng. Struct.
**2016**, 117, 603–616. [Google Scholar] [CrossRef] - Benson, S.D.P.; Karihaloo, B.L. CARDIFRC
^{®}—Development and mechanical properties. Part III: Uniaxial tensile response and other mechanical properties. Mag. Conc. Res.**2005**, 57, 433–443. [Google Scholar] [CrossRef][Green Version] - JSCE-SF4 III P. Method of Tests for Steel Fiber Reinforced Concrete; Concrete library of JSCE, The Japan Society of Civil Engineering: Tokyo, Japan, 1984. [Google Scholar]

**Figure 2.**Schematic presentation of (

**a**) the three-point bending test (UHPFRC-1) and (

**b**) the four-point bending test (UHPFRC-2) (dimensions in mm).

**Figure 4.**Characteristic failures of selected typical prisms for (

**a**) 25 mm, (

**b**) 50 mm, (

**c**) 75 mm and (

**d**) 100 mm depths.

**Figure 12.**Numerical models for UHPFRC-1 prisms (

**a**) 25 × 100 × 500 mm, (

**b**) 50 × 100 × 500 mm, (

**c**) 75 × 100 × 500 mm, and (

**d**) 100 × 100 × 500 mm.

**Figure 13.**Load deflection results: Numerical vs. Experimental for UHPFRC-1 prisms with (

**a**) 25 and (

**b**) 100 mm depth values.

**Figure 14.**Load deflection results: Numerical vs. Experimental for UHPFRC-2 prisms with (

**a**) 25 mm, (

**b**) 50 mm, (

**c**) 75 mm and (

**d**) 100 mm depth values.

Material | Mix Proportions (kg/m^{3}) | |
---|---|---|

UHPFRC-1 | UHPFRC-2 | |

Cement (52.5N) | 855 | 657 |

GGBS | 418 | |

Silica Fume | 214 | 119 |

Silica Sand | 940 | 1051 |

Superplasticizers | 28 | 59 |

Water | 188 | 185 |

Steel Fibres | 468 | 234 |

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**

Lampropoulos, A.; Nicolaides, D.; Paschalis, S.; Tsioulou, O. Experimental and Numerical Investigation on the Size Effect of Ultrahigh-Performance Fibre-Reinforced Concrete (UHFRC). *Materials* **2021**, *14*, 5714.
https://doi.org/10.3390/ma14195714

**AMA Style**

Lampropoulos A, Nicolaides D, Paschalis S, Tsioulou O. Experimental and Numerical Investigation on the Size Effect of Ultrahigh-Performance Fibre-Reinforced Concrete (UHFRC). *Materials*. 2021; 14(19):5714.
https://doi.org/10.3390/ma14195714

**Chicago/Turabian Style**

Lampropoulos, Andreas, Demetris Nicolaides, Spyridon Paschalis, and Ourania Tsioulou. 2021. "Experimental and Numerical Investigation on the Size Effect of Ultrahigh-Performance Fibre-Reinforced Concrete (UHFRC)" *Materials* 14, no. 19: 5714.
https://doi.org/10.3390/ma14195714