*2.2. Ultrasonic Pulse Velocity (UPV)*

The ultrasonic pulse velocity was used as the on-site NDT testing method. In accordance with the theory of sound propagation in solids, the velocity of the ultrasonic signal depends on the density and elastic modulus of the material subjected to testing [48].

A calibration between compressive strength and ultrasonic pulse velocity for each concrete sample assures enough dependability for the two indicators [49]. Naik et al. [50] presented a full review of the method. ultrasonic pulse velocity can be determined with Equation (2) presented by Romanian Norm NP 137 [12].

$$\mathbf{V\_L = L/T} \tag{2}$$

where: VL—ultrasonic pulse velocity (km/s); L—path length in concrete (mm); and T transit time (μs).

#### *2.3. Theoretical Considerations*

Modulus of elasticity of concrete (E) is a property of concrete that estimates the potential deformation of a structural element under service conditions [51]. The factors influencing this property are the dosage of cement, concrete age and class, the binder characteristics, and proportions.

The static modulus of elasticity (Es) is a fundamental parameter that is defined by the stress–strain diagram under static loads [51] and it is generally estimated based on design code, not on direct measurements.

The dynamic modulus of elasticity (Ed), in comparison to Es, is defined by the ratio of stress–strain under vibratory conditions [52]. The most common techniques for determining Ed are resonance frequency or UPV [53], but a study conducted by Luo and Bungey [54] presented a new approach by using surface waves in order to determine Ed. For this study, Ed was determined accordingly to Romanian Guide GE 039 [55] via UPV using Equation (3).

$$\mathbf{E\_{d}} = \frac{(1+\Theta\_{\mathbf{d}}) \cdot (1-2\cdot\Theta\_{\mathbf{d}})}{1-\Theta\_{\mathbf{d}}} \cdot \frac{\mathbf{\mathcal{y}}}{\mathbf{g}} \cdot \mathbf{V\_{L}^{2}} \tag{3}$$

where: Ed—dynamic modulus of elasticity (MPa); Θd—dynamic Poisson's ratio; γ—air dry density (kg/m3); g—gravitational acceleration (m/s2); and VL—ultrasonic pulse velocity (km/s).

Romanian Guide GE 039 [55] presents a mathematical expression, Equation (4), for the determination of the dynamic Poisson's ratio, but for this study, the dynamic modulus of elasticity was assumed the value presented by the technical literature [55], namely Θ<sup>d</sup> = 0.25 (for concrete preserved in the air).

$$
\Theta\_{\rm d} = \frac{(2 \cdot \text{n} \cdot \text{l})^2}{\text{V}\_{\rm L}^2} \tag{4}
$$

where: n—fundamental resonant frequency (cycles/sec); l—length of specimen (m); and VL—ultrasonic pulse velocity (km/s).

Thereby, when considering the Θ<sup>d</sup> = 0.25 the values of the function depending on the dynamic Poisson's ratio becomes:

$$\mathbf{f}(\Theta\_{\mathbf{d}}) = \frac{(1 + \Theta\_{\mathbf{d}}) \cdot (1 - 2 \cdot \Theta\_{\mathbf{d}})}{1 - \Theta\_{\mathbf{d}}} = 0.83\tag{5}$$

Inserting Equation (5) in Equation (3) results the dynamic modulus of elasticity has the following expression:

$$\mathbf{E\_d} = 0.83 \cdot \frac{\mathcal{Y}}{\mathcal{g}} \cdot \mathbf{V\_L^2} \tag{6}$$

Regarding the air-dry density Salman [56] and Panzera et al. [57] conducted studies to find a linear correlation between air-dry density (γ) and UPV. In this study, Equation (7), presented by Salman [56] was used to determine the air-dry density of concrete.

$$\gamma = 114.8 \cdot \text{V}\_{\text{L}} + 1813 \tag{7}$$

where: γ—air-dry density (kg/m3) and VL—ultrasonic pulse velocity (km/s).

In order to establish the accuracy of the proposed equation, the air-dry density of concrete was experimentally determined. The samples were weighed and measured with the purpose of determining the apparent volume. Comparing the mean values of air-dry density obtained experimentally (γe) with the mean values of the predicted ones using Equation (7) (γt), it was shown it reached a precision rate of 98%.

Furthermore, the theoretical air-dry density was used in this study as it was proven to be efficient, thus the method remained completely non-destructive and depended only on UPV.

Romanian Guide GE 039 [55] stipulates that the ratio between Es and Ed ranges, in general, between 0.85–0.95. For this study, the correlation between the two moduli of elasticity was determined by experimentally. Therefore, each modulus of elasticity (Es and Ed) was calculated individually and then a direct link between them was established. Ed was determined via UPV (Equation (6)) and Es was determined via DT (Equation (8)).

For determining the static modulus of elasticity, with the air-dry density determined with Equation (7) and compressive strength obtained destructively (fis) determined with Equation (1), using the mathematical relationship presented by Noguchi et al. [51] (Equation (8)), a static modulus of elasticity could be determined.

$$\mathbf{E\_s} = 2.1 \cdot 10^5 \cdot \left(\frac{\gamma}{2.3}\right)^{1.5} \cdot \left(\mathbf{f\_c} / 200\right)^{1/2} \tag{8}$$

where: Es—static modulus of elasticity (MPa); fc = fis—concrete strength (MPa); and γ = γt—concrete air-dry density determined via UPV (kg/m3).

The dynamic modulus of elasticity was mathematically calculated with Equation (6), using the ultrasonic pulse velocity.

Comparing the values of the two moduli of elasticity, determined for each specimen separately, it was established a direct and linear link between them described in Equation (9).

$$\mathbf{E\_s} = 0.75 \cdot \mathbf{E\_d} \tag{9}$$

Using Equation (9), the static modulus of elasticity can now be determined only from the ultrasonic pulse velocity measurements and using Equation (7) the air-dry density can be obtained through the same measurements. Therefore, in Equation (8) the only unknown parameter remains concrete compressive strength (fc). Extracting that parameter and rewriting Equation (8) results in a relationship (Equation (10)) where the compressive strength value depends only on parameters that can be determined via UPV.

$$\mathbf{f\_c} = (\mathbf{E\_s^2} \cdot \mathbf{200}) \cdot \left[ 2.1 \cdot 10^5 \cdot (\gamma/2.3)^{1.5} \right]^2 \tag{10}$$
