·Table of Contents
An Investigation of the Transfer Function of the Impact-Echo Response and its Application
C. C. Cheng
Associate Professor,chaoyang University of Technology,Taichung,Taiwan
C. P. Yu
Assistant Professor,chaoyang University of Technology,Taichung,Taiwan
The impact-echo method, developed in mid-80's , has been found to be a portable and effective way to detect voids and cracks inside the concrete structures. In the method, transient waves were generated by the impact of a spherical steel ball, 3 mm to 20 mm in diameter, on the surface of the test object. Vertical displacements are recorded by the point-receiver on the surface near the impact-position and are usually analyzed in the frequency domain as an amplitude spectrum. The benefit of testing concrete using the impact-echo method is the low frequency contents, usually below 80 kHz, generated by the impact of a steel ball. Only the defects or inclusions larger than the scale of several centimeters, which is major concerned for the structural safety, can be detected and the noises and scatters in the response due to small air bubbles or imperfections in concrete can be minimized.
As a crack creates an interface in a concrete structure below the testing position, the waves reflected repeatedly between the top surface and the interface produce a high amplitude peak at the frequency corresponding to the depth of the crack. This peak-amplitude is not identical in amplitude while multiple tests were carried out at the same position because the different impact duration and strength in each test inject various frequency contents into concrete. Thus, the peak in the amplitude spectra can only reveals the presence of the crack without any indication of the size and the bond condition of the interface.
This kind of problem can be potentially improved by looking at the corresponding transfer function instead of the frequency amplitude spectrum. The transfer function is the Fourier transfer of the response corresponding to an impulse where the force-time history is a delta function. Because the amplitude of the frequency contents for an impulse is always a constant, the peak related to the interface only corresponds to the geometrical condition of the interface. The effect of impact strength and contact-time of individual test can be eliminated.
To evaluate the transfer function of an impact-echo response, the time history of the impact force or the related frequency content will be required. However, the force-time history can not be recorded during the test using present instrumentation. One can only reasonably evaluate the duration and maximum amplitude of the impact from the R-wave records.
In this paper, the background section shows the theoretical surface response due to a step-force applied on a semi-infinite media as well as the theory of deriving the transfer function. Based on the former theory, the analytical solution of the impact response on a semi-infinite concrete-based media is obtained. The relation between the amplitude of the R-wave and the force-amplitude is established as functions. The theoretical relations are used to derive the transfer function of the real impact-echo responses which are performed on concrete slabs embedded with artificial crack. Finally, the effects of the duration and amplitude of R-wave to the derived transfer functions are studied followed by a discussion of the potential application of this analysis scheme to common deterioration of concrete.
Surface response due to a step-force applied on a semi-infinite media
For the impact response on a semi-infinite medium, the analytical solution can be obtained by deriving either the Green function (in time domain) or the transfer function (in frequency domain). The surface vertical response due to a step force was first published by Pekeris for the special case n = 0.25 (l= m). Following the methodology of Pekeris, the surface vertical response corresponding to an arbitrary value of n
is derived as shown in Eq.(1)
a, b, c are roots of the characteristic equation F(x) = 0
r is radial distance
m is shear modulus
Z is the amplitude of the unit force
t is time
r is mass density
In equation (1), the solution is expressed in terms of the dimensionless time variable t and is separated into four parts by t equals to b, 1 and g. These three values of t stand for the first arrivals of P-wave, S-wave, and R-wave, respectively. It is clear that the solution indicates non-dispersive forms for the three fundamental waves. Therefore, the travelling wave on semi-infinite media retains its shape as it propagates. It is also interesting to note that the solution exhibits a singular point at t = g which results in the R-wave dominated phenomenon. The R-wave portion of the surface response is of particular interest in the following study.
According to the elastic wave-theory, an impact-echo response (u(r, t)) is the result of the impulse response (G(r, t)) convoluted with the actual impact force-time function (F(t)) as shown in Eq. (2).
Follow the rule of convolution, the transfer function, which is the Fourier transfer of the impulse response, can be obtained by dividing the amplitude in the impact-echo spectrum over the one in the impact force-spectrum (Eq. (3)).
whereF() is the Fourier transfer
Using present instrumentation, the real transfer function can not be obtained because the force history is not recorded during the impact-echo test. Thus, the force-time function is substituted by the time history of R-wave.
The theoretical impact response on a concrete based semi-infinite medium
In this study, a semi-infinite media with common concrete properties such as shear modulus G = 1.38x1010 kg/m/s2, mass density r= 2300 kg/m 3, and Poisson's ratio n=0.18, was subjected to a point vertical load P(0,t) = p0f(t) at the origin r = 0. As a result, the shear wave velocity Cs is about 2450 m/s, the P-wave velocity is about 3873 m/s (1.5811Cs), and the Rayleigh wave velocity is near 2252 m/s (0.9194Cs).
To investigate the relation between the surface vertical responses and the impact duration and amplitude, the surface displacement was evaluated using the convolution integral between equation (1) with n=0.18 and specific impulse functions. The vertical responses corresponding to the near-source and far-away positions are computed, in which a unit load (p0=1) with a time variation f(t) given by a half cycle sinusoidal pulse with duration td are applied. The half cycle sinusoidal pulse is most likely occurred for steel ball impacting on a elastic surface . The time step used in the numerical integration is Dt = 0.1 ms.
Fig 1: Typical vertical displacement response due to an impact on the surface of a semi-infinite media
Fig 2: Relation between amplitude ratio (Apeak/A0) and dimensionless impact duration (td) |
Figure 1 illustrates a typical surface response due to a point impact, in which r equals to 1 m and P(0,t) is a unit half cycle sine function. It is shown in figure 1 that the insignificant P-wave arrives at t = 0. 2547 ms, the more apparent S-wave arrives at t = 0.4083 ms, and finally the dominant R-wave takes over at t = 0.449 ms. The response amplitude is expressed in relation to the nominal displacement amplitude A0. In addition, symbol Apeak stands for the maximum R-wave (positive) amplitude and symbol E denotes the end of the non-dispersive R-wave action. It can be observed from the figure that the duration associated with the R-wave action (from R to E) is identical to the impact duration and the response curve associated with the R-wave action is somehow similar to the impact curve in shape.
Figure 2 summarizes the logarithmic relation between the peak displacement amplitude and the dimensionless impact duration td where td is Cs td /r (in terms of (Y = ln(Apeak/A0)) and (X = ln(td /g)). For the purpose of further application the relation shown in Figure 2 is approximated by three continuous functions as shown in Eq. (4).
Procedures to Derive the Transfer Function from the Impact-echo Response
In the impact-echo test, one can measure the waveform of vertical displacement at a distance r away from the impact position. As the R-wave traveling in concrete is likely to be non-dispersive at a close impact-receiver distance, the impact duration can be reasonably considered as the R-wave action. On the other hand, the maximum impact-force amplitude (A0) can be calculated from the maximum R-wave amplitude (Apeak) shown in the impact-echo response using Eq.(4).
However the prime objective of this study is to derive a normalized response using the idea of transfer function. The actual value of A0 is not necessary to be calculated. The relative impact-force amplitude is chosen to be the value of the maximum R-wave amplitude at the situation where the impact-duration (td), impact-receiver distance (r), and shear wave speed (Cs) equal to 20 ms, 0.03 m, and 2440 m/s, respectively. Thus, for the cases other than the default situation the normalized impact-force amplitude (A0)n can be calculated from the maximum R-wave amplitude (Ap) corresponding to the present case dividing by a correlation factor Fn as shown in Eq.(5).
The procedure to calculate the transfer function from an impact-echo response is demonstrated as the flow-chart shown in Figure 3. In the following sections, the peak amplitude in the transfer function is called the transferred amplitude
Fig 3: The procedure to calculate the transfer function from an impact-echo response is demonstrated as the flow-chart |
To confirm the theory of the transfer function, a 20 cm-thick concrete slab containing a 10 ´ 10 cm artificial void at the depth of 7 cm was cased. After 28 days of curing, the specimen is subjected to a series of impact-echo tests on the surface directly above the void. The testing parameters including impact duration generated by various sizes of the impactor and the impact strength, which can be demonstrated by the maximum R-wave amplitude in the displacement waveform. The impactor-receiver distance is 3 cm and the P-wave speed is about 4000 m/s. A representative testing result including the waveform, amplitude spectrum and transfer function is shown in Figure 4(a), (b), and (c), respectively. The impact duration shown as the R-wave action indicated from R to E in Figure 4(a) is about 22.5 ms. The maximum R-wave amplitude is 0.845 Volt. The amplitude of the peak at 27.8 kHz which corresponds to the depth of the void is 14300 in the amplitude spectrum (Figure 4(b)) and 0.952 in the transfer function (Figure 4(a)). The calculated correlation factor Fn is 0.952. Thirteen valid tests were performed on the specimen and the corresponding impact-duration (td), maximum R-wave amplitude (Ap), correlation factor (Fn), and the void-amplitude (Ampam) as well as the transferred amplitude (Amptr) were shown in Table 1.
|Fig 4: (a) The displacement waveform (b) the amplitude spectrum, and (c) the transferred spectrum for plate containing 10´10 void at depth of 10 cm|
Conclusion and Future Work
As the transferred amplitude has slight positive correlation with the maximum R-wave amplitude, the variation of the impact-force can still affect the transferred amplitude. More experimental studies are required to create a general relation for the maximum R-wave amplitude and the transferred void-amplitude to compensate this effect. This relation also needs to be examined for concrete specimens containing voids with different sizes and depths.
The method of transfer function has the potential to quantitatively measuring the degree of deterioration for some common problems in concrete structures, such as the interface between concrete and steel bar generated by corrosion of the steel bar, and the bond problem between old and new concrete. For the structures such as concrete tunnel, or mineshaft, the difference in acoustic impedance for concrete and the material underneath may also be characterized by this technique. Some of the experiments and numerical works related to these problems are studied underway.
- M.J. Sansalone and W.B. Street, Nondestructive Evaluation of Concrete and Masonry, BULLBRIER Press, Ithaca, pp. 99-103(1997)
- Pekeris, C. L. (1955), "The Seismic Surface Pulse", Proceedings of National Academy of Science, U.S.A., 41, pp 629-638.