**1. Introduction**

Concrete has been the most significant construction material throughout history. Ultra-high performance concrete (UHPC) was developed as a cementitious composite material with far higher strength, durability, and resistance to external environments than traditional construction materials [1]. With the addition of fibres, the mechanical properties of ultra-high performance fibre reinforced concrete (UHPFRC) can be further improved especially the performance under tensile loading [2]. Moreover, the resistance to impact, chemical degradation, abrasion, and fire can also be improved [3]. UHPFRC has been widely used in airports [4], bridges, roofs, and cladding [5].

Comparing to the ascending branch of the stress-strain curve, the fibre addition always has a more significant impact on the cracking stage of UHPFRC. The fibre acts as a 'bridge' to bond the separated cracked concrete matrix and then being pulled out gradually, which is also known as the 'bridging effect' [6]. It compensates for the disadvantage of UHPC of being too brittle. However, if the fibres are distributed poorly inside concrete, for example, perpendicular to the loading direction, the bridging effect will be adversely affected.

Many factors can influence the fibre distribution, for example, fibre volume fraction [7,8], external vibration [9], and mixing sequence [10]. Previous researchers have developed various testing

methods to test the fibre distribution inside the concrete matrix, for example, image analysis [10–12], AC-IS (AC-Impedance Spectroscopy) [13–15], X-ray scanning [16,17], and the magnetic core inductive method [18,19]. Compared to these testing methods, the C-shape ferromagnetic probe method is more economical and convenient, so this method is chosen to be the detecting method in this research.

The C-shape ferromagnetic probe test was proposed by Faifer et al. [20]. However, the main focus of their research was on the theoretical background modelling and derivation of the probe, for example, the simulation of flux density of the probe [21]. The application of this probe was also limited to the basic monitoring of fibre volume content and fibre orientation angle of steel fibre reinforced concrete [21]. In 2016, Nunes et al. made a simple and cheap C-shaped ferrite core with coils wound on it [22]. They further applied this testing method on thin UHPFRC plates. By placing the probe on 300 mm × 300 mm × 30 mm specimens in different directions and testing the inductance, the corresponding fibre distribution conditions can be identified. According to Nunes et al., a relative magnetic permeability μr,mean, which reflects fibre content, can be calculated as:

$$
\mu\_{\rm r,op} = \frac{\mathcal{L}\_{\rm op}}{\mathcal{L}\_{\rm air}} \tag{1}
$$

$$\text{tr}^{\text{r},(90^\circ - \Phi)} = \frac{\text{L}\_{\text{air}}}{\text{L}\_{\text{air}}} \tag{2}$$

$$
\mu\_{\text{r,mean}} = \frac{\mu\_{\text{r,op}} + \mu\_{\text{r,(\theta0^\circ - \phi)}}}{2} \tag{3}
$$

where Lϕ, L(90◦−<sup>ϕ</sup>), and Lair represent the magnetic inductance at ϕ, (90◦ − ϕ), and in the air.

The relationship between the orientation indicator ρ<sup>Δ</sup> and the relative magnetic permeability can be expressed as:

$$
\rho\_{\Delta} = \rho\_{(90^\circ - \text{q})} - \rho\_{\text{q}} = \frac{\mu\_{\text{r},(90^\circ - \text{q})} - \mu\_{\text{r},\text{q}}}{2\left(\mu\_{\text{r,mean}} - 1\right)}\tag{4}
$$

Nune et al.'s later research was mainly emphasizing on the detection of uniformly distributed UHPFRC by aligning the fibres with an intense magnetic field [23,24], but the intrinsic heterogeneity of fibre distribution had been proved to largely affect the tensile response of fibre reinforced concrete [25]. Moreover, recent studies were mainly focusing on the relationship between fibre content and flexural performance [26,27]. Therefore, both the local fibre spatial and orientation distribution should be taken into consideration, but this area still needs further investigation.

In previous research, the magnetic probe method was only applied on thin plates. If this method is going to be applied for the purpose of checking the structural safety, it is necessary to determine the effective depth of the probe. In order to fill the gap, the effective depth of the magnetic probe was firstly examined and an attenuation factor was obtained to further correct the fibre volume content data. Apart from this, a simplified analytical solution was derived to determine the relationship between the fibre orientation angle and orientation indicator. In addition, compressive tests and flexural tests had been conducted and the correlated equation between mechanical performance and the tested fibre distributions was derived. The corresponding theoretical explanation of strain-hardening behaviour was clarified.

#### **2. Material Preparation and Concrete Mixing**

#### *2.1. Materials and Specimen List*

Premix powder material provided from third party company (Beirong Circular Materials Co., Ltd., Yingtan, Jiangxi, China) were used to ensure the concrete grade in each batch. The premix powders contained the basic components needed for UHPFRC (cement, fine sand, silica fume, and quartz). Straight steel fibres (as shown in Figure 1) covered with a brass coating were added in the mix. The diameter and length of fibres were 0.25 mm and 12.5 mm respectively, given an aspect ratio of 50. The uniaxial tensile strength of the fibres can go up to 2850 MPa. Polycarboxylate superplasticizer was added to ensure the workability. Detailed mix proportions are presented in Table 1.

**Figure 1.** Straight steel fibres used in research.

**Table 1.** Mix proportion for pre-mix ultra-high performance fibre reinforced concrete (UHPFRC; provided by third party company).


Table 2 lists the dimensions and casting purpose of all specimens. Cubes were used to determine the compressive strength. Plates with different thicknesses were designed to detect the fibre orientation and spatial distribution, and the following flexural performances.


**Table 2.** Specimen list and casting purpose for different types of specimens.

#### *2.2. Concrete Mixing Process and Curing Condition*

Table 3 shows the suggested mixing procedure and mixing time provided from the supply company. Considering the differences of the mixing environment and fibre amount, the mixing time varied by ±30 s.

**Table 3.** Mixing procedure and time for premix UHPFRC.


All the cube specimens were cast in three layers with minor hand vibration to expel the air trapped in the fresh concrete. For plates, concrete was poured from the centre and let it flow freely towards the four edges (see Figure 2). To avoid the segregation of fibres, no vibration was applied.

**Figure 2.** Plate casting direction.

All the fresh concrete specimens were covered with a plastic film to prevent the early shrinkage due to water evaporation. After 24 h of hardening, all specimens were demolded and moved to a steam curing tank. The curing duration included 2 h for increasing the curing temperature to 90 ◦C and then another 48 h constant curing at 90 ◦C.

### **3. C-Shape Magnetic Probe Test**

### *3.1. Probe Specification*

A magnetic probe was manufactured based on Nunes' research [22]. The probe (Figure 3a) was made of a high frequency inductive Mn-Zn ferrite core wrapped by 350 turns of 0.9 mm diameter enameled copper wire, then tightened by black insulated rubber tape to protect the safety of users. The ferrite core used in this research was 76 mm tall, 93 mm long, and 30 mm wide. Detailed dimensions of the magnetic probe can be seen in Figure 3b.

**Figure 3.** (**a**) Appearance of the magnetic probe and (**b**) detailed dimensions of the ferrite core.

The inductive test was carried out after curing. The magnetic probe was placed on a smooth surface of the UHPFRC specimen and connected with a LCR meter with two clips. Tonghui TH2830 LCR meter was used to measure the magnetic inductance under 1 kHz with a test signal of 1 V. The variation of inductance of a single object was lower than ±0.01 mH under this testing condition.

### *3.2. E*ff*ective Depth Test Method*

Non-ferromagnetic object like wood, plastic, and glass did not show any impact on the inductance data. By placing 100 mm × 150 mm non-ferromagnetic acrylic plates (Figure 4a) with different thicknesses (0, 2, 4, 8, 12, 15, 17, 20, 24, 28, 32, and 36 mm) between the specimens and the magnetic probe (Figure 4b), the magnetic inductance data can be measured and the relative magnetic permeability (RMP) μ can be calculated as:

$$
\mu = \frac{\mathcal{L}\_0}{\mathcal{L}\_{\text{air}}} \tag{5}
$$

#### where

L0 magnetic inductance when placing the magnetic probe directly on the specimen;

Lair magnetic inductance when placing the magnetic probe in the air and far from any conductive object.

**Figure 4.** (**a**) Acrylic plates used in the experiment and (**b**) experiment set up of an effective depth test.

However, since the initial relative magnetic permeability for each group was different, it was difficult to reflect on how the relative magnetic permeability decayed with the increase of thickness. Thus, an attenuation factor (AF) was introduced to describe the residual proportion of magnetic permeability. It can be calculated as:

$$\text{AF}\_{\text{t}} = \frac{\mu\_{\text{t}} - 1}{\mu\_0 - 1} \times 100\% \tag{6}$$

where


#### *3.3. E*ff*ective Depth Test Results*

The effective depth testing was conducted on 2% and 2.5% vol. UHPFRC. In total, 4 points (2 points for each group) were tested. Based on Equation (6), AF data of each testing point were calculated and the results are shown in Table 4. The relative magnetic permeability decreased with the increase of plate thickness. For specimens with lower fibre content, the relative magnetic permeability tended to drop quicker. All groups of AF data dropped below 10% for depths greater than 24 mm. This proved that the fibres 24 mm away from the testing surface had little effect on the relative magnetic permeability.


**Table 4.** Average attenuation factors calculated for each group.

The AF data presented in Figure 5 shows very similar exponential trends for all groups of testing points. It proved that the fibres closer to the testing surface had a more significant effect on the relative magnetic permeability, especially within the top 6 mm of the testing surface. After the top 6 mm, the AF dropped below 50%.

**Figure 5.** Attenuation factors of 2% and 2.5% vol. specimens at different thicknesses.

To get a theoretical expression of this relationship, a curve fitting analysis was conducted using MATLAB. The relationship between the testing depth and attenuation factor can be expressed by Equation (7) with an R-squared value equal to 0.995.

$$\mathbf{A}\mathbf{F} = \mathbf{e}^{-0.115 \times \mathbf{t}} \tag{7}$$

This test revealed the effective depth when using this particular probe. If any thin specimens are going to be tested in the future, AF can be used for correcting and calculating the real fibre volume content value.

#### *3.4. Plate Test Method*

The magnetic probe test of all the plates were tested right after finishing curing. The bottom surfaces during casting were used as the testing surface as they were smooth and flat. Considering the length of the magnetic probe was 93 mm, the 500 mm × 500 mm plates were labelled on a 9 × 9 grid from A1 to I9 at equal distances of 50 mm (Figure 6a). A paper testing map (Figure 6b) with 81 points highlighted in red was made to accurately locate the testing points.

**Figure 6.** Testing area divisions of the UHPFRC plate. (**a**) Schematic graph (unit: mm) and (**b**) practical experimental design.

As shown in Figure 7, by placing the magnetic probe in two orthogonal directions (horizontally and vertically), the spatial distribution and orientation distribution at each red point were measured.

**Figure 7.** Testing directions. (**a**) Horizontal and (**b**) vertical.

The magnetic inductances of the red points in Figure 6 were recorded directly from the LCR meter. In total, 81 data points were collected for each plate. Air inductance was labelled as Lair. The magnetic inductance values measured in the horizontal and vertical directions were labelled as Lij,x and Lij,y. All the magnetic inductance values were divided by the air inductance to get the relative magnetic permeability μ.

The average of relative magnetic permeability measured in two orthogonal directions is the indication of fibre volume content. Based on Equation (3), the average relative magnetic permeability on each red point can be calculated as Equation (8). Symbols i and j are used to represent the point location, e.g., μ11,x represents the horizontally measured relative magnetic permeability at the point on the top left corner.

$$
\mu\_{\text{ij,ave}} = \frac{\mu\_{\text{ij,x}} + \mu\_{\text{ij,y}}}{2} \tag{8}
$$
