·Table of Contents ·Civil Engineering | An Investigation of the Transfer Function of the Impact-Echo Response and its ApplicationC. C. ChengAssociate Professor,chaoyang University of Technology,Taichung,Taiwan C. P. Yu Assistant Professor,chaoyang University of Technology,Taichung,Taiwan Contact |
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[2] 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.
Transfer Function
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).
Eq.(2) |
Eq.(3) |
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 = 1.38x10^{10} kg/m/s^{2}, mass density r= 2300 kg/m^{ 3}, and Poisson's ratio n=0.18, was subjected to a point vertical load P(0,t) = p_{0}f(t) at the origin r = 0. As a result, the shear wave velocity C_{s} is about 2450 m/s, the P-wave velocity is about 3873 m/s (1.5811C_{s}), and the Rayleigh wave velocity is near 2252 m/s (0.9194C_{s}).
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 (p_{0}=1) with a time variation f(t) given by a half cycle sinusoidal pulse with duration t_{d} are applied. The half cycle sinusoidal pulse is most likely occurred for steel ball impacting on a elastic surface [1]. 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) |
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 (A_{0}) can be calculated from the maximum R-wave amplitude (A_{peak}) 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 A_{0} 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 (t_{d})_{,} impact-receiver distance (r), and shear wave speed (C_{s}) 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 (A_{0})_{n} can be calculated from the maximum R-wave amplitude (A_{p}) corresponding to the present case dividing by a correlation factor F_{n} as shown in Eq.(5).
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 |
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 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
t_{d} | 34 | 25.5 | 27 | 22.5 | 12 | 12 | 19.5 | 28 | 38 | 40 | 17 | 19.5 | 16 |
A_{p} | 1.164 | 1.021 | 0.906 | 0.842 | 1.325 | 1.237 | 1.002 | 1.502 | 1.338 | 1.368 | 1.531 | 0.283 | 1.724 |
F_{n} | 0.942 | 0.97 | 0.967 | 0.977 | 1.044 | 1.044 | 1.003 | 0.975 | 0.942 | 0.942 | 1.019 | 1.003 | 1.027 |
Amp_{am} | 21300 | 23400 | 17300 | 14300 | 15300 | 14200 | 15200 | 27300 | 20200 | 18400 | 31600 | 4580 | 31800 |
Amp_{tr} | 1.09 | 1.11 | 0.93 | 0.85 | 0.87 | 0.88 | 0.82 | 0.94 | 1.10 | 1.13 | 1.22 | 0.88 | 1.14 |
Table 1: test results for 10´10 cm void at depth 7 cm |
Fig 5: (a) the distribution of the void amplitude, and (b) the void transferred amplitude according to the test numbers | |
Fig 6: The relations of the transferred amplitude (a) to impact duration and (b) the maximum R-wave amplitude |
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