**4. Results**

### *4.1. Calibration of Ultrasound Pulse Velocity-Compression Strength Curves Based on the Strength Machine Tests*

After the measurements of ultrasonic pulse velocities of the tested samples, they were cut and tested in uniaxial loading on the strength machine. On that basis, hypothetical scaling curves were chosen.

Scaling Equation (3) was approximated for the measurements of the first concrete type shown in Section 4.2:

$$f\_c = 0.1983 \cdot \text{C}\_L^{4.3081} \text{ (MPa)} \tag{3}$$

The approximation of the scaling curve in Equation (3) was performed based on the results of destructive tests conducted on cylindrical samples with dimensions 10 cm × 10 cm. The results of destructive tests, measured pulse velocities, and strength values calculated using Equation (3) are presented in Table 1.

**Table 1.** Results of uniaxial destructive tests, measured pulse velocities, and strength values calculated with the use of the chosen hypothetical scaling curve.


Scaling curves for the remaining tested boreholes of the floors were approximated in the same way. Scaling curves established hypothetically were used to convert the rate of ultrasound wave in the given cross-section at the borehole height into concrete compression strength in this cross-section.

#### *4.2. Tested Types of Strength Distributions in Cross-Section of Concrete Borehole Materials*

The tested borehole materials provided an answer to the question of the distribution of concrete strength across the thickness of industrial floors in the tested facilities. Ignoring minor fluctuations of strength at different levels resulting from random positioning of aggregate or small defects in the concrete structure, or errors in measurement of the distance (the head ends go<sup>t</sup> inside surface pores), all results can be grouped according to the summary below.

4.2.1. Concrete "Homogeneous" across Its Thickness with Superficial Weakening

Passing times *t* (μs), wave velocities *CL* (km/s), and compressive strengths *fc* (MPa) in planes parallel to the surface of the boreholes are presented in Table 2. Calculation of compressive strength from ultrasound wave velocities was done with the use of Equation (3). The borehole materials were taken from an industrial floor 22 cm thick. Results are presented starting with the ordinal number 1 (5 mm from the top of the floor) in the direction of the bottom of the floor.


**Table 2.** Results of concrete ultrasound velocity test in borehole No. 1.

Distribution of compressive strength in the tested concrete at the sample height is presented in Figure 8.

Fluctuations across the average strength values (marked with the thin blue line in Figure 8) were relatively small. Its strong decrease begins at 19 cm from the bottom. Concrete is homogeneous across the floor thickness, only the top layer, approximately 10–20 mm, was weakened. The compressive strength of concrete in the top layer of the floor drops from the value of 25–30 MPa down to the value of 12 MPa at the measurement height of 10 mm.

4.2.2. "Homogeneous" Concrete with Weakening across the Top Layer 30–50 mm Thick

Calculation of compressive strength from ultrasound wave velocities was done with the use of Equation (3). The borehole materials were taken from an industrial floor 21 cm thick. Results are presented in Table 3 starting with the ordinal number 1 (5 mm from the top of the floor) in the direction of the bottom of the floor.

**Figure 8.** Concrete "homogeneous" across the floor thickness, only the thin top layer is considerably weakened.

**Table 3.** Results of concrete ultrasound velocity test in borehole No. 2.


The distribution of compressive strength for the tested concrete at the height of the sample is presented in Figure 9.

**Figure 9.** Concrete across the thickness of the floor relatively homogeneous, weakened thick top layer.

The concrete was relatively homogeneous across the floor thickness. The weakened top layer was 30–50 mm thick. Compressive strength in this layer of concrete drops from the value of 34–40 MPa down to the value of 24 MPa in the near-surface cross-section.

The bottom layer of the sample was significantly strengthened. Measurements made at a height of 2 mm from the bottom of the sample showed an increase in strength to 46 MPa from 35–40 MPa in close layers above.

4.2.3. Concrete Strength Is Changing across Entire Section with a Quick Weakening of the Top Layer

The borehole materials were taken from an industrial floor 15 cm thick. Calculation of compressive strength from ultrasound wave velocities was done with the use of Equation (4).

$$f\_c = 0.123 \cdot \text{C}\_L^{4.3081} \text{ (MPa)} \tag{4}$$

Results are presented in Table 4, starting with the ordinal number 1 (5 mm from the top of the floor) in the direction of the bottom of the floor.


**Table 4.** Results of concrete ultrasound velocity test in borehole No. 3.


**Table 4.** *Cont*.

The concrete across the floor thickness changed its strength along with the change in the measurement plane. The top layer with a high thickness (up to 50 mm) became weakened very quickly in the direction of the floor top. The distribution of compressive strength for the tested concrete at the height of the sample is presented in Figure 10.

**Figure 10.** The change of concrete strength across the floor thickness was continuous. The weakened top layer was very thick.

#### 4.2.4. Concrete Reinforced Superficially to the Expected Value Using the Mineral Powder

To improve the hardness of the top layer of the floor, powders made of hardening materials based on a cement binding agen<sup>t</sup> and strong filler were used. If the strength of the hardened concrete had characteristics in the vertical cross-section similar to the one described in Section 4.2.1 (only a thin 10–15 mm top layer of the floor was weakened), a correctly made hardening layer may balance the shortage of strength. The floor will achieve the designed quality on its top layer. In the presented example top layer of concrete floor has been reinforced using mineral powder to the average level in the vertical cross-section, in an expected way. After tests of the borehole samples of the examined floor on the strength testing machine, the scaling curve described using Equation (3) was adjusted.

The result of measurements regarding ultrasonic wave velocity and compressive strength in the sample's cross-section is presented in Table 5.


**Table 5.** Results of concrete ultrasound velocity test in borehole No. 4.

Distribution of strength in cross-section of the sample taken from the floor with the hardening layer made as expected (Table 5) is presented in Figure 11.

**Figure 11.** The strength of the floor reinforced superficially to the expected value using the mineral powder.

The superficially weakened floor, with the compressive strength of 25–30 MPa, strengthened in the near-surface zone up to the designed value of 30 MPa. Repair efficiency depends on the weakest cross-section under the strengthened layer and its ability to transfer shear stress arising because of the shrinkage of the strengthened layer. In the lower zone of the sample (2 mm from the bottom of the floor) the concrete was strengthened to a value of over 40 MPa. The concrete strengthening zone was about 20 mm thick.

4.2.5. Concrete Floors under Emergency Conditions with Top Layer Hardened using Mineral Powder

As far as concrete floors strengthened (and hardened) using the mineral powder under emergency conditions are concerned, three types of strength distribution were analyzed:


Table 6 presents the results of measurements concerning ultrasonic wave velocity in cross-section of the floor strengthened with too small a quantity of mineral powder. The borehole materials were taken from a floor 16.5 cm thick. Calculation of compressive strength from ultrasound wave velocities was done with the use of Equation (5):

$$f\_{\mathbb{C}} = 6.147 \cdot \mathbb{C}\_{L} \,^2 - 18.172 \cdot \mathbb{C}\_{L} + 10.786 \text{ (MPa)} \tag{5}$$


**Table 6.** Results of concrete ultrasound velocity test in borehole No. 5.

Distribution of strength in cross-section of a sample taken from the floor in which too small a quantity of hardening powder was used is presented in Figure 12.

**Figure 12.** Too weak ineffective strengthening of the industrial floor.

In this case, the compressive strength drops from a value close to 30 MPa at a depth of 5 cm from the top of the floor to 16 MPa in the zone close to the surface of destruction. The destruction occurred in the weakest layer of concrete, where strength reaches a value of not more than 16.3 MPa. In the reinforced surface plane, the strength increases to 19 MPa at a depth of 3 mm from the top of the floor. The mineral strengthening of the floor, in this case, should reach a depth of 25–30 mm, at which the concrete strength was at a minimum level of 20–25 <sup>N</sup>/mm2.

The strengthening will be ineffective also in the case in which concrete below the zone affected by the mineral powder is too weak and the entire strengthened layer is loosened. Measurements of ultrasonic wave velocity in cross-section of a sample taken from such a floor are presented in Table 7. The dependency between wave velocity *CL* and strength *fc* described using Equation (6) was used for calculations of the distribution of strength in the sample's cross-section.

$$f\_c = 112.880 \cdot \text{C}\_L \,^2 - 379.850 \cdot \text{C}\_L + 324.7 \, (\text{MPa}) \tag{6}$$


**Table 7.** Results of concrete ultrasound velocity test in borehole No. 6.


The distribution of strength in cross-section of the tested sample is presented in Figure 13.

**Figure 13.** Strengthening that was ine ffective because the concrete under the hardening layer had a very low strength of approximately 5 MPa.

In this situation, the strengthening was ine ffective because directly under the mineralized layer, with a strength of 20 MPa, the concrete was very weak, its strength decreases down to the value of approximately 5 MPa near the surface of destruction, and then increases up to the designed value of 30 MPa.

The floor reaches the highest strength value at the last measuring point, 1 cm deep from the bottom edge. The reinforcement was even at a thickness of 4 cm from the bottom, where the measured strength was 28 MPa and reaches 40 MPa in the plane at the bottom of the sample. The e ffect of concrete strengthening in the lower planes of industrial floors (Figures 9, 11 and 13) usually results from aggregate segregation, which occurs in the process of vibrating the concrete mix in gravitational field forces.

A similar e ffect of damage to the hardened floor occurs if excessive powder is used. Measurements regarding ultrasonic wave velocity for a borehole material from such a floor are presented in Table 8. The calibrated dependency of the approximation described using Equation (7) was used for calculation of compressive strength in the cross-section of the sample.

$$f\_c = 24.780 \cdot C\_L - 33.600 \text{ (MPa)} \tag{7}$$


**Table 8.** Results of concrete ultrasound velocity test in borehole No. 7.

The distribution of strength in the cross-section of the tested sample is presented in Figure 14.

**Figure 14.** Homogeneous concrete across the thickness of the floor was weakened in the top zone and was strengthened excessively (strength growth from 13 to 40 MPa).

In this case, the strength of the top layer exceeded the designed value. The measured strength of the borehole material in the middle of the sample's height has a value of approximately 30 MPa. Considering the possible weakening in the upper zone, the concrete was excessively strengthened with mineral powder. After strengthening, in the near-surface zone, the strength value increased up to 40 MPa, and below the strengthened zone it drops down to the value of 13.5 MPa. In the destruction zone, in which it was impossible to make a measurement, the value of strength was lower than 13.5 MPa—a value measured from its nearest surface (2 cm from the top of the floor). Considerable shrinkage in the place of contact between the layer with increased strength and the weakened layer lead to loosening (Figures 14 and 15).

**Figure 15.** Loosening of the floor from the weak concrete; (**a**) view of the entire sample; (**b**) close-up to the loosened zone.

The various situations involving floor concrete weakening in the top layer shown above, as well as successful and unsuccessful strengthening using hardening powder, explain the causes of the defects and the effects of the 'repairs'. The examination of concrete strength distribution across the thickness of the floor was possible thanks to the applied ultrasound method with spot heads.

The interpretation of the results obtained from tests of concrete strength distributions across the thickness of the floor presented above is satisfactory, however, no reference documents or admissible values regarding strength variability in a cross-section of concrete slab elements are available concerning this. The subsequent section presents a proposal of additional criteria for assessing concrete in floors and an attempt to indicate mathematical criteria ensuring the basis for evaluation of strength changes across the thickness of the tested concrete layer based on the strength values and the rate of change of strength values obtained from measurements of ultrasonic wave velocities.

#### *4.3. Proposal for Evaluating Concrete Quality in the Floor Based on Its Strength Gradient and Anti-Gradient*

The gradient of the scalar strength field indicates the direction of the quickest growth of strengths in individual points. The modulus (that is 'length') of each vector in such a field is equal to the rate of strength field growth in the given direction. The vector opposite to the gradient is sometimes called an anti-gradient. The strength gradient is a vector value and its unit in the SI system is pascal per meter (Pa/m). Due to the range of the measured strength gradient values for industrial floors, the unit MPa/cm is used for the description of them.

In the considered case of the strength field test, you can refer to a vertical strength gradient that corresponds to a strength change in line with the distance from the bottom of the slab (with the height of the measurement plane). The strength gradient between two measurement planes 1 and 2 can be expressed using Equation (8):

$$
\nabla f\_c = (f\_{c,2} - f\_{c,1}) / (Z\_2 - Z\_1) \text{ (MPa/cm)}\tag{8}
$$

where *fc,2* is concrete strength in plane 2 (closer to the top surface of the slab); *fc,1* is concrete strength in plane 1 (closer to the bottom surface of the slab); *Z*2—distance of the second measurement plane from the bottom of the slab; *Z*1—distance of the first measurement plane from the bottom of the slab.

Figure 16 shows the calculated gradients between various measurement planes for the borehole material for which measurement results are presented in Table 4.

**Figure 16.** Concrete strength gradients at various floor depths.

At a depth of 0 to 10 cm counting from the bottom of the slab, ∇*fc* is 0.7 MPa/cm apart from local fluctuations, and in layers located closer to the top surface it changes quickly (−3.0; −4.5; −8.0 MPa/cm). In the zone of powder-based hardening, the strength begins to increase (∇*fc* = +0.5 MPa/cm). Changes in the strength of concrete in the areas of fluctuation around the average value (marked with a blue line in Figure 16) are related to the arrangemen<sup>t</sup> of grains on the path of ultrasound waves in concrete and are not the subject of consideration. A zone in which there is a rapid decrease or increase in strength is analyzed, e.g., after strengthening and applied to a thin top layer, 20 to 50 mm, rarely thicker. When the anti-gradient is large, the decrease in concrete strength in the surface layer is very large and the floor will need repair. It is proposed to test the gradient in the following variants:


A summary showing described strength gradient parameters together with measured extreme and mean values of compressive strength of tested samples is presented in Table 9.



*Materials* **2020**, *13*, 118

Apart from the minimum, mean, and maximum values of compressive strength in cross-section of industrial floors, the strength variability dynamics can be, in this case, described using the value of strength gradient and anti-gradient, which is especially important for the distribution of strength for near-surface concrete layers. The floor should be made using concrete of a specified class and with strength variability gradient at as low a level as possible, guaranteeing that its performance features are maintained during use. The quality of concrete in an industrial floor can be in this case specified using not one but two parameters. The presented examples of strength distribution in the floors indicate that crumpling and cracking in industrial floors used in a standard way usually occur in the following cases:


Presented measurements of concrete floors require further research, also with the use of different measurement tools. A discussion of a wide pool of strength distribution results is needed, based on which it may be possible to determine the limit values of the measured concrete strength gradients that can be used to standardize the quality assessment of industrial concrete floors. In the future, it is also planned to carry out measurements of ultrasonic wave velocity and strength distribution in the different concrete constructions formed horizontally to compare them with the results presented in this article. The obtained values of reduced compressive strength in the upper zone of concrete elements are also important in the case of bent reinforced concrete slabs, in which it is advisable that the top zone is not weakened.
