Numerical and experimental evaluation of sonic resonance against ultrasonic pulse velocity and compression tests on concrete core samples

1.1. Ultrasonic pulse velocity (UPV)

The UPV method is an important non-destructive technique that provides reliable results based on rapid measurements using cost effective equipment. The Dynamic Elastic Modulus is associated with Ultrasonic velocity by density and Poisson’s ratio of the materials as shown in Eq. (1) and therefore, it is possible to associate it with concrete’s strength. The dynamic Young’s modulus ($E_d$) can be determined as follows:

where $\rho$ is mass density of the material, $V_p$ is ultrasonic pulse velocity and $v$ is Poisson’s ratio [2]. Fig. 1 shows that the velocity $V_p$, relative to the compressive strength $f_c$, is highly dispersed, indicating a strong dependence of the results on Poisson’s ratio [3].

Fig. 1Correlation diagram of compressive strength fc, velocity Vp and Poisson’s ratio

Fig. 2Correlation diagram of compressive strength fc, velocity Vp and Poisson’s ratio as resulted from the present study

1.2. Sonic resonance (SR)

The SR method is a dynamic acoustic method associated with a frequency analysis of the short-time response of a media to an impact load [4]. The principle of the method is mainly based on the calculation of the propagation velocity in finite length circular cross section rods. The propagation velocity $V_c$ is calculated from the resonance frequency during free oscillation of the specimen following an excitation as in Eq. (2):

where $V_c$ is wave propagation velocity in a cylinder with ratio $L/D$ > 5, $f$ is fundamental frequency of the first harmonic, $L$ is specimen’s length, $D$ is specimen’s diameter. Young’s modulus, $E_d$, is directly related to the $V_c$ wave propagation velocity through the Eq. (3):

where $\rho$ is mass density of the material. Eq. (3) can be considered as the solution for a slender solid rod with a sufficient long length $L$. In this condition the longitudinal vibration becomes a one-dimensional wave equation irrelevant to Poisson’s ratio [5].

The main drawback of SR is that in order to calculate $V_c$ from Eq. (2), it is necessary to have specimens with ratio $L/D$> 5. In cylinders whose lengths are of the same order of magnitude as their diameters, a contraction or an extension causes non-negligible radial displacements, which depend on Poisson’s ratio [6].

When the $L/D$> 5 condition is not met, a correction factor must be considered. In similar studies the dynamic Young’s modulus is calculated through the fundamental transverse frequency of a rectangular bar using Eq. (4). A correction factor is taken into account depending on the radius of gyration to the length of the specimen ratio and on Poisson’s ratio, as stated in ASTM C215 [7]:

where $E_d$ is Young’s modulus (Pa), $m$ is mass of the bar (kg), $b$, $t$ are dimensions of cross section of prism (m), $L$ is length of the specimen (m), $T$ is correction factor that depends on the radius of gyration to the length of the specimen ratio $L$ and on Poisson’s ratio, $f$ is fundamental transverse frequency (Hz).

2.1. Numerical analysis

A finite element (FEM) numerical analysis has been carried out in order to overcome the shape influence of the cores on the results. Plaxis 2D software was used to create simulations of concrete cores using axisymmetric (Free-Free) models.

A total number of 72 simulations were created varying the following parameters: (a) Ratio $L/D$ = 0.7, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.5, 3.0, 4.0, 5.0, (b) Poisson’s Ratio $v$= 0.25, 0.30, 0.36, 0.40, 0.45.

The longitudinal resonance frequencies during the free oscillation were determined from the Fast Fourier Transformation (FFT) spectra. Fig. 3 illustrates an example of a free oscillation spectrum with the values of the first and second longitudinal resonance frequencies.

The influence of the $L/D$ ratio of the models on the measured rod velocity can be calculated by taking a correction factor equal to the ratio of the theoretical ($f_{1,th}$) to the calculated ($f_{1,calc}$.) 1st mode through FEM analysis. In Fig. 4 the variation of the above ratio as a function of the $L/D$ ratio for each value of the Poisson’s ratio is shown. A non-linear multivariable regression analysis was performed according to the Levenberg and Marquardt algorithm as can be seen in Fig. 4.

Fig. 3Fourier Transform of accelerations’ timeline with first and second longitudinal resonance frequencies marked

Fig. 4Theoretical to calculated 1st frequency ratio as function of L/D ratio and Poisson ratio diagram

The best-fit curve that represents the variation of correction factor vs $L/D$ ratio has the following equation Eq. (5):

where $y$ is shape’s correction factor, $x=L/D$ ratio, $y_o$, ${{\rm A}}_1$, $A_2$, $t_1$,$t_2$ coefficients that are depended on Poisson’s ratio, $v$, derived from the same analysis and can be seen in Eqs. (6-10):

The shape’s correction factor can be calculated using Eqs. (5-10) by using the $L/D$ ratio and the Poisson’s ratio values.

2.2. Experimental analysis

In the experimental analysis the cores were initially tested with the NDT methods. For the UPV tests, the Proceq -Tico apparatus with a pair of 54 kHz P-wave transducers was used. The travel time of the ultrasonic pulse through the cores of known length was measured and thus the ultrasonic pulse velocity $V_p$ was calculated. Then the constrained dynamic modulus, $M$ through UPV was calculated using Eq. (11):

where $\rho$ is mass density of concrete (kN/m³), $V_p$ is ultrasonic pulse velocity (km/sec).

For the SR tests 54 kHz Kistler’s Accelerometer was used with a measuring range of ±50 g, sensitivity of 100.3 mV/g and weight of 10.6 grams. The signal from the accelerometers was analyzed by a Digital Signal Analyzer (DSA), resulting in the frequencies’ spectrum. From the FFT spectrum the frequency of the 1st mode, $f_1$ was then determined. Based on the results of the numerical analysis and applying the empirical Eq. (5) for the shape correction, the value of the $f_1$ was corrected. The rod velocity is calculated from the following Eq. (12):

where $L$ is specimen’s length (m) and $f_{cor}$ is corrected frequency (Hz).

The Young’s modulus from Eq. (3) was then calculated using the $V_c$ value from Eq. (12). The Poisson’s ratio was indirectly determined from the calculated values of the two moduli (Constrained and Young’s) according to Eqs. (13-14) [8].

where $S=\pm \sqrt{E^2+9M^2-10EM}$, $E$ is Young’s dynamic modulus and $M$ is constrained dynamic modulus.

Finally, the uniaxial compression test was conducted and the compressive strength of concrete was calculated. In order to determine the final compressive strength, a correction was made due to the variation of the specimens’ size. The correction was based on Greek Bulletin E7 which is applicable when evaluating the strength class of existing concrete structures [9].