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Numerical simulation of attenuation and group velocity of guided ultrasonic wave in grouted rock bolts
Journal of Applied Geophysics 59 (2006) 337 – 344 www.elsevier.com/locate/jappgeo
Numerical simulation of attenuation and group velocity of guided ultrasonic wave in grouted rock bolts Y. Cui, D.H. Zou ⁎ Department of Civil and Resource Engineering, Dalhousie University, Sexton Campus, 1360 Barrington Street, P.O. Box 1000, Halifax, Nova Scotia, Canada, B3J 2X4 Received 8 July 2005; accepted 7 April 2006
Abstract In this paper, the guided ultrasonic wave propagating in grouted rock bolts was simulated with finite element method. An 800 mm partially grouted cylindrical rock bolt model was created. Dynamic input signals with frequency from 25 to 100 kHz were used to excite ultrasonic wave. The simulated waveform, group velocity and amplitude ratio matched well with the experimental results. This model made it possible to study the behaviour of the guided waves in the grouted bolt along its central axis. Analysis of the simulated results showed that the group velocity in grouted rock bolts is constant along the grouted length, and the boundary effect on the group velocity is negligible. This paper also presents methods to determine the attenuation coefficient from simulation and to determine the boundary effect on attenuation at the bolt ends. The analysis showed that the attenuation of the guided wave propagating inside the grouted bolts is similar to the theoretical solution in steel bar with infinite length. After correction for the boundary effects the grout length of a grouted rock bolt can be determined using the measured attenuation, with sufficient accuracy. © 2006 Elsevier B.V. All rights reserved. Keywords: Finite element; Simulation; Rock bolts; Guided wave; Attenuation
1. Introduction Rock bolts as ground reinforcement and stabilization system are widely used in underground and surface excavations in mining and civil engineering. Rock bolts, in many applications, are installed in the ground by grouting with cement or resin. To ensure the ground stability, it is often required to evaluate the grouting quality in the field. The traditional method such as pull-out test used for grout quality test is destructive, expensive and time consuming. ⁎ Corresponding author. Tel.: +1 902 494 3977; fax: +1 902 426 1037. E-mail address: [email protected] (D.H. Zou). 0926-9851/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jappgeo.2006.04.003
In recent years, there have been a number of research initiatives in non-destructive rock bolt tests (Thurner, 1988; Beard and Lowe, 2003; Madenga, 2004). The Geomechanics and Mining Innovations (GMI) research group at Dalhousie University has been very active for the past few years in research on guided ultrasonic waves to monitor rock bolts and has made significant contributions. We discovered that the behaviour of the guided waves in grouted rock bolts strongly depends on the wave frequency (Madenga et al., in press). A series of laboratory experiments were conducted using transmission-through setup, which is considered essential to understand the characteristics of the guided waves propagating along a bolt (Madenga, 2004). A clear relationship
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Fig. 1. Typical grouted rock bolt.
has been established (Madenga et al., in press) between the group velocity of the wave packet and the frequency at various curing times of the fresh grout. Further experiments led to the understanding of the effects of frequency and grout length on the wave attenuation (Zou et al., in press). These findings are significant break through in non-destructive testing of grouted rock bolts using guided ultrasonic waves. However, as in any experimental study, the number of experiments, which can be conducted, is limited by technical and economic factors. Thus, numerical simulation becomes an essential tool to supplement this research. Zhang et al. (2005) reported their initial work at GMI in numerical simulation of wave propagation in grouted rock bolts. They developed a finite element model using LSDYNA (Zhang et al., 2005) with proper mesh size and successfully reproduced the wave signals obtained from experiments. This paper presents the results of further research in this area by the GMI group. Before numerical simulation is discussed, it is important to understand some basic characteristics of guided wave in a rock bolt. One of the important characters of the guided wave is that its velocity is not only dependent on the material properties but also on the thickness of the material and frequency (Rose, 1999). The guided wave propagates as a packet, which is made up of a band of superimposed frequency components because of wave deformation, which is named dispersion. It is the group velocity that defines the speed at which the ‘envelope’ of the packet travels. It has been shown that in a rock bolt, the rate of energy transfer is identical to the group velocity (Achenbach, 1973). Although the analytical group velocity solution is valid for an infinite long steel bar, the boundary effects for a steel bar with finite length remain to be studied. Attenuation is another important character of the guided wave. In general it refers to the reduction in the signal strength after wave traveling some distance. The wave attenuation is defined by an attenuation coefficient. For example, the p-wave amplitude decay (one dimensional solution) can be expressed as a function of
the travel distance (Klimentos and McCann, 1990; Tavakoli and Evans, 1992). ln
Ab ¼ lnðRÞ ¼ −aL Aa
ð1Þ
Where Aa Ab α L R
is is is is is
the the the the the
amplitude at location a, amplitude at location b, attenuation coefficient, constant, distance from location a to b, amplitude ratio.
However, there has been little research on attenuation of guided wave in grouted rock bolts. For a finite length rock bolt, the attenuation inside the rock bolt is affected by the boundary condition and the validity of Eq. (1) for grouted rock bolts needs to be verified. The objective of this paper is to determine the effects of grouted length on the attenuation and group velocity along the grouted portion of rock bolts and to verify the above relationship using numerical modeling. Numerical simulation generally involves two major steps: a) to establish and verify a proper numerical model which can simulate experimental results, and b) to study the behaviour of the guided wave in various conditions, especially where it is difficult to conduct experiments or to find analytical solutions. In this paper, a grouted rock bolt model was simulated and the accuracy of simulated results was verified by comparing the simulated results with the experimental ones. The attenuation and group velocity along the simulated rock bolt model was studied and the effects of grouted length on the attenuation and group velocity were analyzed. 2. Simulation model and input parameters To simulate the experimental results of guided ultrasonic wave propagating in grouted rock bolts, a partially grouted rock bolt model was created as illustrated in Fig. 1.
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344
The rock bolt was partially embedded into a cylindrical concrete block to simulate an experimental sample with the same geometry as specified in Table 1. The input parameters of the model are listed in Table 2. The element size used in this model is the same as that suggested by Zhang et al. (2005), as listed in Table 3. Simulation was conducted using LS-DYNA, a commercial software package (Livermore Software Technology, 2001). To save computing time, axisymmetric condition of the sample was considered and only one quarter of the sample was created. The detail of the finite element model is shown in Fig. 2. For this research, a total of 3120 kinematic elasto-plastic rectangular elements were used to simulate the rock bolt, 64000 elastic rectangular elements to simulate the grouted concrete cylinder, and 75 239 nodes with 6 degrees to connect all the elements. It should be noted that at low load, the rock bolt is in elastic condition and elastic elements are used in the model. Plastic condition would apply only if the load exceeds the yielding strength of the bolt. Other detail of the finite element model can found from the LS-DYNA manual and Saeed (2003). The model was excited from the un-grouted end of the model by displacement signals with frequency in the range of 25 to 100 kHz. 3. Simulated results of ultrasonic guided wave Similar to the experiments, the signals were input from the un-grouted end of the sample and recorded from another end. The simulated axial displacement results at the receiving end of the model were compared with the experimental results to evaluate the accuracy of the simulation. Three typical simulated waveforms, at input frequencies of 25, 60 and 100 kHz respectively, are shown in Figs. 3a, 4a and 5a. The experimental results at the same input frequencies are illustrated in Figs. 3b, 4b and 5b. The results show that the simulated waveforms match the experimental ones very well at low frequency and have some scatters at higher frequency. Dispersion was observed from both the experimental and simulated results. It is also observed that the dispersion and the attenuation become more serious as the input frequency is increased. For simplicity, the first wave packet recorded at the receiving end is called the first arrival and the same wave packet recorded the 2nd time after being reflected back from the input end is called echo. The first arrival and the Table 1 The geometry of simulated grouted rock bolt Bolt length (mm)
Bolt diameter (mm)
Grouted length Diameter of concrete (mm) cylinder (mm)
800
19.5
750
160
339
Table 2 Material parameters used in simulation Density (g/cm3) Rock bolt 7.84 Concrete 2.405
Young's modulus (GPa)
Possion's ratio
210 38
0.3 0.25
echo are illustrated in Fig. 3. It can be observed in this figure that the ratio of the echo amplitude over the first arrival amplitude in the simulated waves is bigger than that in the experimental ones. The main reason of this phenomenon is that the fixed energy loss occurring at the end of the sample because of the equipment setup in experiments does not exist in the simulation. In the experiments, part of the wave energy is lost at the ends in contacting with the sensors, while in the simulation, there is no energy loss at the ends. This topic will be discussed further in the following section. 3.1. Comparison of group velocity It is observed that the group velocities determined directly from the raw waveforms were unreliable due to the dispersion effect (Madenga et al., in press). Only after filtering can reasonable results be obtained. This applies both to the experimental and simulated data. It indicates the difficulty of group velocity measurement of guided waves in grouted rock bolts because of the dispersion and other factors, such as, the noise in experimental background. The raw waveform data were filtered using a band filter of ±5 kHz centered at the input frequencies. The experimental and simulated results of group velocity are shown in Fig. 6. It can be seen that the results match each other very well with a correlation coefficient greater than 0.99. This can also be evaluated by the relative variation ratio: ev ¼
jvs −ve j ve
ð2Þ
where vs and ve are the simulated and experimental group velocities, respectively. The relative variation ratio is less than 7% in the results. 3.2. Comparison of attenuation Wave attenuation is usually measured by a coefficient, which as indicated in Eq. (1) has an inverse logarithm relation with the amplitude ratio. It is therefore more convenient to evaluate the amplitude ratio first. In this paper, a practical approach suggested by Zou et al. (in press) is used to calculate the ratio using the average amplitude: Rm ¼
A2 A1
ð3Þ
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Table 3 Element maximum size used in simulation (Zhang et al., 2005) 50 kHz
Steel bolt (Φ20 mm) Concrete
75 kHz
Radial (mm)
Axial (mm)
Radial (mm)
Axial (mm)
Radial (mm)
Axial (mm)
≤5 ≤3.5
≤5 ≤3.5
≤3 ≤2.1
≤3 ≤2.1
≤1.3 ≤1
≤2 ≤ 1.5
where, Rm A1 A2
100 kHz
is the measured amplitude ratio is the average amplitude of the first arrival is the average amplitude of the echo
A1 and A2 are determined within a time window with the same length centered around the maximum amplitudes of the first arrival and the echo, respectively. Zou et al. (in press) demonstrated that in laboratory experiments of grouted rock bolts, the main contributing factors to attenuation were the dispersion and spread (DISP), which vary with the frequency and grouted length. They further indicated that for a specific type of test equipment setup, there is a fixed amount of energy loss, called the setup energy loss, which accounts for all losses at the interface between the sample and the equipment. The amplitude drop caused by the setup energy loss is thus constant for a specific experimental setup. Therefore the measured amplitude ratio consists of two components: Rm ¼ R1 R2
corresponds to R1 in experiments. Fig. 7 shows the simulated and experimental results of attenuation. It is observed that the simulated results match the experimental results very well at low frequency range (below 60 kHz). At higher frequency there are some scatters in the experiment data. This phenomenon is mainly due to the fact that the echoes were weak at higher frequency range, inevitably aggravating the effects of the background noise. 3.3. Summary of finite element simulation results It is clear from the above results that the simulated waveforms matched the experimental data very well. The simulated group velocity in the frequency range from 25 to 100 kHz is very close to the experimental results, and the simulated attenuation is almost the same as that in
ð4Þ
where, R1 is the amplitude ratio after the DISP attenuation R2 is the amplitude ratio after the setup energy loss and it can be estimated by tests of free (un-grouted) rock bolts with the same experimental setup. In the finite element simulation, the setup energy loss does not exist. Therefore, the simulated amplitude ratio
Fig. 2. Finite element model of rock bolt (quarter size).
Fig. 3. Ultrasonic waveforms at 25 kHz in grouted rock bolt a) Simulated, b) Tested.
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344
Fig. 4. Ultrasonic waveforms at 60 kHz in grouted rock bolt bolt a) Simulated, b) Tested.
experiments at low frequency range. The difference in attenuation at higher frequencies is attributed to the experimental results which were affected by the inevitable errors in data analysis due to the weak signals and background noise. From these results, it can be concluded that the simulation is very successful and the finite element model can be used to study the behaviour of the guided waves propagating inside the rock bolt.
Fig. 5. Ultrasonic waveforms at 100 kHz in grouted rock bolts bolt a) Simulated, b) Tested.
decreases with the travel distance showing the effect of the DISP attenuation. 4.1. Group velocity inside the grouted rock bolt The processes of waveform filtering and group velocity calculation are the same as mentioned before. The group velocities along the length of the rock bolt
4. Study of guided wave inside the grouted rock bolt In laboratory tests or field measurements, a transducer can only be attached to the assessable parts of a bolt, which are often very limited. In numerical simulation, however, information at any location inside the simulated rock bolt model can be retrieved. In this research, the behaviour of the guided wave along the central axis is of interest and the displacements along the central axis of the bolt are recorded during simulation. The typical waveforms at 40 kHz frequency recorded at positions of 100 and 400 mm distance from the excitation end are shown in Fig. 8. It can be observed that the wave travel time increases with the travel distance, which coincides with the theoretical and experimental results. The amplitude
341
Fig. 6. Simulated and experimental results of group velocity.
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Fig. 7. Simulated and experimental results of amplitude ratio R1.
were calculated for various input frequencies from 25 to 100 kHz. Fig. 9 shows the results. Because the group velocity is calculated by dividing the travel distance by
Fig. 9. Simulated group velocity along the bolt.
the travel time, it is very difficult to determine the velocity at very short distance because of the difficulty in estimating the travel time. The distance where reasonable data can be obtained is 100 mm from the excitation end, which happens to be the starting point of the grouted length. It can be seen from Fig. 9 that the group velocity of the guided wave is frequency dependent. However for a specific frequency, the group velocity at any location along the bolt is the same as that at the end of the bolt. Thus the velocity change, if there is any, at the ends of the grouted rock bolts can be reasonably ignored in practice. 4.2. Attenuation inside the grouted rock bolt To evaluate the attenuation, the amplitudes along the central axis of the model in the grouted length were determined from the simulated waveforms. Analysis of
Fig. 8. Simulated waveforms inside the grouted rock bolt at 40 kHz input frequency a) at 100mm from the excitation end, b) at 400 mm from the excitation end.
Fig. 10. Relationship between wave travel length and ln(Ri) from simulation.
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344 Table 4 Attenuation coefficient and correction factor Frequency (kHz) 25 Attenuation coefficient (m− 1) Correction factor, K
30
35
40
45
50
0.3897 0.3969 0.4436 0.4614 0.4856 0.5162
1.4784 1.6265 1.6556 1.7749 1.8818 1.9958
the simulated amplitudes indicated that the amplitude change near the rock bolt ends is very complicated and the boundary effect at the bolt ends on the attenuation is significant. This phenomenon can be explained theoretically by the fact that the amplitude and attenuation are functions of material properties. At the bolt ends, the material properties change suddenly. This causes wave reflection. The combined effects make the condition near the bolt ends very complex. In the following we will first only consider the central portion of the bolt without considering the portion near the ends. 4.2.1. Attenuation in the central portion of the bolt The amplitude ratio at the central portion of the bolt is calculated at various locations using the following equation. Ri ¼
Ai A200
343
Zou et al., in press). The attenuation coefficient within the frequency range is calculated from the simulated data and the results are shown in Table 4. It can be seen that the attenuation coefficient increases with frequency but the change is small within the simulated frequency range. 4.2.2. Attenuation at the end As illustrated before, the boundary effect of attenuation at the bolt end is a very complex issue. It is necessary to make an assumption to simplify the problem: the whole grouted length has the same quality with the same attenuation coefficient. Thus, the boundary effect at the bolt ends can be accounted for by modifying amplitude ratio R1 by a correction factor, K: Rb ¼ R1 K
ð6Þ
where, Rb is the modified amplitude ratio for boundary effect. K will be independent of bolt length but vary with frequency. Hence, the modified amplitude ratio from the first arrival and echo at the bolt end has the following relation with the attenuation coefficient. aL ¼ −lnðR1 KÞ ¼ −lnðRb Þ
ð7Þ
or, ð5Þ
where,
aL ¼ −lnðR1 Þ−lnðKÞ
ð8Þ
where, R1 is defined in Eq. (4) for experiment data but can be calculated directly by Eq. (3) from the simulated data.
Ri is amplitude ratio at the ith location A200 is the average amplitude of the wave packet at location 200 mm. Ai is the average amplitude at the ith location (i = 200 to 700 mm measured from the excitation end). Fig. 10 shows the results of Ri at frequency 25, 50 and 100 kHz, which has a logarithmic relationship with the travel length. It should be pointed out that almost the same results are obtained when Ri is calculated from two consecutive locations. Fig. 10 indicates that the attenuation of guided wave propagating in grouted rock bolts follows the one dimensional solution as defined in Eq. (1). The relationship between Ri and the wave travel distance is linear. Therefore, the attenuation coefficient, which is traditionally defined as the measured attenuation can be easily determined from the slope of the line. The steeper the slope, the higher the attenuation. In practice, the suitable range of the input signal frequency during guided wave tests is 25 to 50 kHz to ensure the quality of the recorded signals (Madenga, 2004;
Fig. 11. The tested, simulated, and calculated amplitude ratio.
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The attenuation ratio R1 is shown in Fig. 7 for the 750 mm grouted rock bolt. The total wave travel distance is twice of the grouted length for a round trip. The correction factor can then be calculated and the results of K are also shown in Table 4. To verify the above assumption regarding the correction factor of the boundary effects, the amplitudes ratio R1 for two grouted rock bolts with 300 and 500 mm grouted length, respectively, are calculated using Eq. (8) and the known values of K and α in Table 4. The results are presented in Fig. 11 together with the experimental results (Zou et al., in press) of the two bolts. The results in Fig. 11 show that the calculated amplitude ratio R1 matches the experimental results quite well with correlation coefficient of 0.98 for all the three curves. For the 300 mm grouted bolt, there is a small scatter. The possible reason for the scattering of this bolt is the testing errors because of the short length. The accuracy of the simulated amplitude ratio can also be estimated by the relative variation ratio: jRs −Re j er ¼ Re
ð9Þ
where Rs and Re are the simulated and experimental amplitude ratios, respectively. The calculated relative variation ratio is less than 10% except the 300 mm grouted bolt. One can conclude that the above assumption is reasonable and the correction factor can adequately account for the boundary effects in experiments. The calculated results are accurate enough for use in grout quality test with guided ultrasonic waves. 5. Conclusion and discussion This paper presented a finite element model, which has adequate accuracy to simulate the behaviour of the guided wave propagating in grouted rock bolts. The simulated results, including the waveform, group velocity, and amplitude ratio, match the experimental results well. The simulated attenuations are almost the same as those of experimental results at the low frequency range (≤50 kHz). The small scatter at the high frequency range (50–100 kHz) is possibly due to the background noise and measurement errors in experiments. With this numerical model, it becomes possible to study the wave behaviour inside the bolt along its central axis, which is very difficult if not impossible in experiments. Results of the simulation model indicate that the group velocity in the grouted rock bolts is
constant along the grouted bolt length and the boundary effects at the bolt ends on the group velocity can be ignored. This model also allows for the determination of the attenuation coefficient for a specific type of bolt conditions. The analysis also shows that the attenuation of guided wave propagating in rock bolts is similar to that of infinite length one dimensional solution. This paper has proposed a method to determine the boundary effects on the attenuation at the bolt ends. Based on this method, the calculated amplitude ratio matches well with the experimental results. Acknowledgement This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. References Achenbach, J.D., 1973. Wave Propagation in Elastic Solids. Applied Mathematics and Mechanics. North-Holland Publishing Company. ISBN: 0 444 10465 8. Beard, M.D., Lowe, M.J.S., 2003. Non-destructive testing of rock bolts using guided ultrasonic waves. International Journal of Rock Mechanics and Mining Science 40, 527–536. Klimentos, T., McCann, C., 1990. Relationships between compressional wave attenuation, porosity, clay content and permeability of sandstones. Geophysics 55, 998–1014. Livermore Software Technology Corporation, 2001. LS-DYNA, Version: PC-DYNA _ 960. Madenga, V., 2004. Application of Guided Waves to Grout Quality Testing of Rock Bolts, M.A.Sc. Thesis, Dalhousie University. Madenga, V., Zou, D.H., Zhang, C., in press. Effects of Curing Time and Frequency on Ultrasonic Wave Velocity in Grouted Rock Bolts. Journal of Applied Geophysics (available online at http://dx. doi.org/10.1016/j.jappgeo.2005.08.001). Rose, J.L., 1999. Ultrasonic Waves in Solid Media. Cambridge University Press, Cambridge. Saeed, M., 2003. Finite Element Analysis: Theory and Application with ANSYS, 2nd ed. Pearson Education, Upper Saddle River, N.J. Tavakoli, M.B., Evans, J.A., 1992. The effect of bone structure on ultrasonic attenuation and velocity. Ultrasonics 30, 389–395. Thurner, H.F., 1988. Instrument for Non-destructive Testing of Grouted Rock Bolts. 2nd International Symposium on Filed Measurements in Geomachanics. Balkema, Rotterdam, pp. 135–143. Zhang, C.H., Zou, D.H., Madenga, V., 2005. Numerical simulation of wave propagation in grouted rock bolts and the effects of mesh density and wave frequency. International Journal of Rock Mechanics and Mining Sciences 43 (4), 634–639. Zou, D.H., Cui, Y., Madenga, V., Zhang, C., in press. Effects of Frequency and Grouted Length on the Behaviour of Guided Ultrasonic Wave Propagating in Rock Bolts. International Journal of Rock mechanics and Mining Science.
Numerical simulation of attenuation and group velocity of guided ultrasonic wave in grouted rock bolts Y. Cui, D.H. Zou ⁎ Department of Civil and Resource Engineering, Dalhousie University, Sexton Campus, 1360 Barrington Street, P.O. Box 1000, Halifax, Nova Scotia, Canada, B3J 2X4 Received 8 July 2005; accepted 7 April 2006
Abstract In this paper, the guided ultrasonic wave propagating in grouted rock bolts was simulated with finite element method. An 800 mm partially grouted cylindrical rock bolt model was created. Dynamic input signals with frequency from 25 to 100 kHz were used to excite ultrasonic wave. The simulated waveform, group velocity and amplitude ratio matched well with the experimental results. This model made it possible to study the behaviour of the guided waves in the grouted bolt along its central axis. Analysis of the simulated results showed that the group velocity in grouted rock bolts is constant along the grouted length, and the boundary effect on the group velocity is negligible. This paper also presents methods to determine the attenuation coefficient from simulation and to determine the boundary effect on attenuation at the bolt ends. The analysis showed that the attenuation of the guided wave propagating inside the grouted bolts is similar to the theoretical solution in steel bar with infinite length. After correction for the boundary effects the grout length of a grouted rock bolt can be determined using the measured attenuation, with sufficient accuracy. © 2006 Elsevier B.V. All rights reserved. Keywords: Finite element; Simulation; Rock bolts; Guided wave; Attenuation
1. Introduction Rock bolts as ground reinforcement and stabilization system are widely used in underground and surface excavations in mining and civil engineering. Rock bolts, in many applications, are installed in the ground by grouting with cement or resin. To ensure the ground stability, it is often required to evaluate the grouting quality in the field. The traditional method such as pull-out test used for grout quality test is destructive, expensive and time consuming. ⁎ Corresponding author. Tel.: +1 902 494 3977; fax: +1 902 426 1037. E-mail address: [email protected] (D.H. Zou). 0926-9851/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jappgeo.2006.04.003
In recent years, there have been a number of research initiatives in non-destructive rock bolt tests (Thurner, 1988; Beard and Lowe, 2003; Madenga, 2004). The Geomechanics and Mining Innovations (GMI) research group at Dalhousie University has been very active for the past few years in research on guided ultrasonic waves to monitor rock bolts and has made significant contributions. We discovered that the behaviour of the guided waves in grouted rock bolts strongly depends on the wave frequency (Madenga et al., in press). A series of laboratory experiments were conducted using transmission-through setup, which is considered essential to understand the characteristics of the guided waves propagating along a bolt (Madenga, 2004). A clear relationship
338
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344
Fig. 1. Typical grouted rock bolt.
has been established (Madenga et al., in press) between the group velocity of the wave packet and the frequency at various curing times of the fresh grout. Further experiments led to the understanding of the effects of frequency and grout length on the wave attenuation (Zou et al., in press). These findings are significant break through in non-destructive testing of grouted rock bolts using guided ultrasonic waves. However, as in any experimental study, the number of experiments, which can be conducted, is limited by technical and economic factors. Thus, numerical simulation becomes an essential tool to supplement this research. Zhang et al. (2005) reported their initial work at GMI in numerical simulation of wave propagation in grouted rock bolts. They developed a finite element model using LSDYNA (Zhang et al., 2005) with proper mesh size and successfully reproduced the wave signals obtained from experiments. This paper presents the results of further research in this area by the GMI group. Before numerical simulation is discussed, it is important to understand some basic characteristics of guided wave in a rock bolt. One of the important characters of the guided wave is that its velocity is not only dependent on the material properties but also on the thickness of the material and frequency (Rose, 1999). The guided wave propagates as a packet, which is made up of a band of superimposed frequency components because of wave deformation, which is named dispersion. It is the group velocity that defines the speed at which the ‘envelope’ of the packet travels. It has been shown that in a rock bolt, the rate of energy transfer is identical to the group velocity (Achenbach, 1973). Although the analytical group velocity solution is valid for an infinite long steel bar, the boundary effects for a steel bar with finite length remain to be studied. Attenuation is another important character of the guided wave. In general it refers to the reduction in the signal strength after wave traveling some distance. The wave attenuation is defined by an attenuation coefficient. For example, the p-wave amplitude decay (one dimensional solution) can be expressed as a function of
the travel distance (Klimentos and McCann, 1990; Tavakoli and Evans, 1992). ln
Ab ¼ lnðRÞ ¼ −aL Aa
ð1Þ
Where Aa Ab α L R
is is is is is
the the the the the
amplitude at location a, amplitude at location b, attenuation coefficient, constant, distance from location a to b, amplitude ratio.
However, there has been little research on attenuation of guided wave in grouted rock bolts. For a finite length rock bolt, the attenuation inside the rock bolt is affected by the boundary condition and the validity of Eq. (1) for grouted rock bolts needs to be verified. The objective of this paper is to determine the effects of grouted length on the attenuation and group velocity along the grouted portion of rock bolts and to verify the above relationship using numerical modeling. Numerical simulation generally involves two major steps: a) to establish and verify a proper numerical model which can simulate experimental results, and b) to study the behaviour of the guided wave in various conditions, especially where it is difficult to conduct experiments or to find analytical solutions. In this paper, a grouted rock bolt model was simulated and the accuracy of simulated results was verified by comparing the simulated results with the experimental ones. The attenuation and group velocity along the simulated rock bolt model was studied and the effects of grouted length on the attenuation and group velocity were analyzed. 2. Simulation model and input parameters To simulate the experimental results of guided ultrasonic wave propagating in grouted rock bolts, a partially grouted rock bolt model was created as illustrated in Fig. 1.
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344
The rock bolt was partially embedded into a cylindrical concrete block to simulate an experimental sample with the same geometry as specified in Table 1. The input parameters of the model are listed in Table 2. The element size used in this model is the same as that suggested by Zhang et al. (2005), as listed in Table 3. Simulation was conducted using LS-DYNA, a commercial software package (Livermore Software Technology, 2001). To save computing time, axisymmetric condition of the sample was considered and only one quarter of the sample was created. The detail of the finite element model is shown in Fig. 2. For this research, a total of 3120 kinematic elasto-plastic rectangular elements were used to simulate the rock bolt, 64000 elastic rectangular elements to simulate the grouted concrete cylinder, and 75 239 nodes with 6 degrees to connect all the elements. It should be noted that at low load, the rock bolt is in elastic condition and elastic elements are used in the model. Plastic condition would apply only if the load exceeds the yielding strength of the bolt. Other detail of the finite element model can found from the LS-DYNA manual and Saeed (2003). The model was excited from the un-grouted end of the model by displacement signals with frequency in the range of 25 to 100 kHz. 3. Simulated results of ultrasonic guided wave Similar to the experiments, the signals were input from the un-grouted end of the sample and recorded from another end. The simulated axial displacement results at the receiving end of the model were compared with the experimental results to evaluate the accuracy of the simulation. Three typical simulated waveforms, at input frequencies of 25, 60 and 100 kHz respectively, are shown in Figs. 3a, 4a and 5a. The experimental results at the same input frequencies are illustrated in Figs. 3b, 4b and 5b. The results show that the simulated waveforms match the experimental ones very well at low frequency and have some scatters at higher frequency. Dispersion was observed from both the experimental and simulated results. It is also observed that the dispersion and the attenuation become more serious as the input frequency is increased. For simplicity, the first wave packet recorded at the receiving end is called the first arrival and the same wave packet recorded the 2nd time after being reflected back from the input end is called echo. The first arrival and the Table 1 The geometry of simulated grouted rock bolt Bolt length (mm)
Bolt diameter (mm)
Grouted length Diameter of concrete (mm) cylinder (mm)
800
19.5
750
160
339
Table 2 Material parameters used in simulation Density (g/cm3) Rock bolt 7.84 Concrete 2.405
Young's modulus (GPa)
Possion's ratio
210 38
0.3 0.25
echo are illustrated in Fig. 3. It can be observed in this figure that the ratio of the echo amplitude over the first arrival amplitude in the simulated waves is bigger than that in the experimental ones. The main reason of this phenomenon is that the fixed energy loss occurring at the end of the sample because of the equipment setup in experiments does not exist in the simulation. In the experiments, part of the wave energy is lost at the ends in contacting with the sensors, while in the simulation, there is no energy loss at the ends. This topic will be discussed further in the following section. 3.1. Comparison of group velocity It is observed that the group velocities determined directly from the raw waveforms were unreliable due to the dispersion effect (Madenga et al., in press). Only after filtering can reasonable results be obtained. This applies both to the experimental and simulated data. It indicates the difficulty of group velocity measurement of guided waves in grouted rock bolts because of the dispersion and other factors, such as, the noise in experimental background. The raw waveform data were filtered using a band filter of ±5 kHz centered at the input frequencies. The experimental and simulated results of group velocity are shown in Fig. 6. It can be seen that the results match each other very well with a correlation coefficient greater than 0.99. This can also be evaluated by the relative variation ratio: ev ¼
jvs −ve j ve
ð2Þ
where vs and ve are the simulated and experimental group velocities, respectively. The relative variation ratio is less than 7% in the results. 3.2. Comparison of attenuation Wave attenuation is usually measured by a coefficient, which as indicated in Eq. (1) has an inverse logarithm relation with the amplitude ratio. It is therefore more convenient to evaluate the amplitude ratio first. In this paper, a practical approach suggested by Zou et al. (in press) is used to calculate the ratio using the average amplitude: Rm ¼
A2 A1
ð3Þ
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Table 3 Element maximum size used in simulation (Zhang et al., 2005) 50 kHz
Steel bolt (Φ20 mm) Concrete
75 kHz
Radial (mm)
Axial (mm)
Radial (mm)
Axial (mm)
Radial (mm)
Axial (mm)
≤5 ≤3.5
≤5 ≤3.5
≤3 ≤2.1
≤3 ≤2.1
≤1.3 ≤1
≤2 ≤ 1.5
where, Rm A1 A2
100 kHz
is the measured amplitude ratio is the average amplitude of the first arrival is the average amplitude of the echo
A1 and A2 are determined within a time window with the same length centered around the maximum amplitudes of the first arrival and the echo, respectively. Zou et al. (in press) demonstrated that in laboratory experiments of grouted rock bolts, the main contributing factors to attenuation were the dispersion and spread (DISP), which vary with the frequency and grouted length. They further indicated that for a specific type of test equipment setup, there is a fixed amount of energy loss, called the setup energy loss, which accounts for all losses at the interface between the sample and the equipment. The amplitude drop caused by the setup energy loss is thus constant for a specific experimental setup. Therefore the measured amplitude ratio consists of two components: Rm ¼ R1 R2
corresponds to R1 in experiments. Fig. 7 shows the simulated and experimental results of attenuation. It is observed that the simulated results match the experimental results very well at low frequency range (below 60 kHz). At higher frequency there are some scatters in the experiment data. This phenomenon is mainly due to the fact that the echoes were weak at higher frequency range, inevitably aggravating the effects of the background noise. 3.3. Summary of finite element simulation results It is clear from the above results that the simulated waveforms matched the experimental data very well. The simulated group velocity in the frequency range from 25 to 100 kHz is very close to the experimental results, and the simulated attenuation is almost the same as that in
ð4Þ
where, R1 is the amplitude ratio after the DISP attenuation R2 is the amplitude ratio after the setup energy loss and it can be estimated by tests of free (un-grouted) rock bolts with the same experimental setup. In the finite element simulation, the setup energy loss does not exist. Therefore, the simulated amplitude ratio
Fig. 2. Finite element model of rock bolt (quarter size).
Fig. 3. Ultrasonic waveforms at 25 kHz in grouted rock bolt a) Simulated, b) Tested.
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344
Fig. 4. Ultrasonic waveforms at 60 kHz in grouted rock bolt bolt a) Simulated, b) Tested.
experiments at low frequency range. The difference in attenuation at higher frequencies is attributed to the experimental results which were affected by the inevitable errors in data analysis due to the weak signals and background noise. From these results, it can be concluded that the simulation is very successful and the finite element model can be used to study the behaviour of the guided waves propagating inside the rock bolt.
Fig. 5. Ultrasonic waveforms at 100 kHz in grouted rock bolts bolt a) Simulated, b) Tested.
decreases with the travel distance showing the effect of the DISP attenuation. 4.1. Group velocity inside the grouted rock bolt The processes of waveform filtering and group velocity calculation are the same as mentioned before. The group velocities along the length of the rock bolt
4. Study of guided wave inside the grouted rock bolt In laboratory tests or field measurements, a transducer can only be attached to the assessable parts of a bolt, which are often very limited. In numerical simulation, however, information at any location inside the simulated rock bolt model can be retrieved. In this research, the behaviour of the guided wave along the central axis is of interest and the displacements along the central axis of the bolt are recorded during simulation. The typical waveforms at 40 kHz frequency recorded at positions of 100 and 400 mm distance from the excitation end are shown in Fig. 8. It can be observed that the wave travel time increases with the travel distance, which coincides with the theoretical and experimental results. The amplitude
341
Fig. 6. Simulated and experimental results of group velocity.
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Fig. 7. Simulated and experimental results of amplitude ratio R1.
were calculated for various input frequencies from 25 to 100 kHz. Fig. 9 shows the results. Because the group velocity is calculated by dividing the travel distance by
Fig. 9. Simulated group velocity along the bolt.
the travel time, it is very difficult to determine the velocity at very short distance because of the difficulty in estimating the travel time. The distance where reasonable data can be obtained is 100 mm from the excitation end, which happens to be the starting point of the grouted length. It can be seen from Fig. 9 that the group velocity of the guided wave is frequency dependent. However for a specific frequency, the group velocity at any location along the bolt is the same as that at the end of the bolt. Thus the velocity change, if there is any, at the ends of the grouted rock bolts can be reasonably ignored in practice. 4.2. Attenuation inside the grouted rock bolt To evaluate the attenuation, the amplitudes along the central axis of the model in the grouted length were determined from the simulated waveforms. Analysis of
Fig. 8. Simulated waveforms inside the grouted rock bolt at 40 kHz input frequency a) at 100mm from the excitation end, b) at 400 mm from the excitation end.
Fig. 10. Relationship between wave travel length and ln(Ri) from simulation.
Y. Cui, D.H. Zou / Journal of Applied Geophysics 59 (2006) 337–344 Table 4 Attenuation coefficient and correction factor Frequency (kHz) 25 Attenuation coefficient (m− 1) Correction factor, K
30
35
40
45
50
0.3897 0.3969 0.4436 0.4614 0.4856 0.5162
1.4784 1.6265 1.6556 1.7749 1.8818 1.9958
the simulated amplitudes indicated that the amplitude change near the rock bolt ends is very complicated and the boundary effect at the bolt ends on the attenuation is significant. This phenomenon can be explained theoretically by the fact that the amplitude and attenuation are functions of material properties. At the bolt ends, the material properties change suddenly. This causes wave reflection. The combined effects make the condition near the bolt ends very complex. In the following we will first only consider the central portion of the bolt without considering the portion near the ends. 4.2.1. Attenuation in the central portion of the bolt The amplitude ratio at the central portion of the bolt is calculated at various locations using the following equation. Ri ¼
Ai A200
343
Zou et al., in press). The attenuation coefficient within the frequency range is calculated from the simulated data and the results are shown in Table 4. It can be seen that the attenuation coefficient increases with frequency but the change is small within the simulated frequency range. 4.2.2. Attenuation at the end As illustrated before, the boundary effect of attenuation at the bolt end is a very complex issue. It is necessary to make an assumption to simplify the problem: the whole grouted length has the same quality with the same attenuation coefficient. Thus, the boundary effect at the bolt ends can be accounted for by modifying amplitude ratio R1 by a correction factor, K: Rb ¼ R1 K
ð6Þ
where, Rb is the modified amplitude ratio for boundary effect. K will be independent of bolt length but vary with frequency. Hence, the modified amplitude ratio from the first arrival and echo at the bolt end has the following relation with the attenuation coefficient. aL ¼ −lnðR1 KÞ ¼ −lnðRb Þ
ð7Þ
or, ð5Þ
where,
aL ¼ −lnðR1 Þ−lnðKÞ
ð8Þ
where, R1 is defined in Eq. (4) for experiment data but can be calculated directly by Eq. (3) from the simulated data.
Ri is amplitude ratio at the ith location A200 is the average amplitude of the wave packet at location 200 mm. Ai is the average amplitude at the ith location (i = 200 to 700 mm measured from the excitation end). Fig. 10 shows the results of Ri at frequency 25, 50 and 100 kHz, which has a logarithmic relationship with the travel length. It should be pointed out that almost the same results are obtained when Ri is calculated from two consecutive locations. Fig. 10 indicates that the attenuation of guided wave propagating in grouted rock bolts follows the one dimensional solution as defined in Eq. (1). The relationship between Ri and the wave travel distance is linear. Therefore, the attenuation coefficient, which is traditionally defined as the measured attenuation can be easily determined from the slope of the line. The steeper the slope, the higher the attenuation. In practice, the suitable range of the input signal frequency during guided wave tests is 25 to 50 kHz to ensure the quality of the recorded signals (Madenga, 2004;
Fig. 11. The tested, simulated, and calculated amplitude ratio.
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The attenuation ratio R1 is shown in Fig. 7 for the 750 mm grouted rock bolt. The total wave travel distance is twice of the grouted length for a round trip. The correction factor can then be calculated and the results of K are also shown in Table 4. To verify the above assumption regarding the correction factor of the boundary effects, the amplitudes ratio R1 for two grouted rock bolts with 300 and 500 mm grouted length, respectively, are calculated using Eq. (8) and the known values of K and α in Table 4. The results are presented in Fig. 11 together with the experimental results (Zou et al., in press) of the two bolts. The results in Fig. 11 show that the calculated amplitude ratio R1 matches the experimental results quite well with correlation coefficient of 0.98 for all the three curves. For the 300 mm grouted bolt, there is a small scatter. The possible reason for the scattering of this bolt is the testing errors because of the short length. The accuracy of the simulated amplitude ratio can also be estimated by the relative variation ratio: jRs −Re j er ¼ Re
ð9Þ
where Rs and Re are the simulated and experimental amplitude ratios, respectively. The calculated relative variation ratio is less than 10% except the 300 mm grouted bolt. One can conclude that the above assumption is reasonable and the correction factor can adequately account for the boundary effects in experiments. The calculated results are accurate enough for use in grout quality test with guided ultrasonic waves. 5. Conclusion and discussion This paper presented a finite element model, which has adequate accuracy to simulate the behaviour of the guided wave propagating in grouted rock bolts. The simulated results, including the waveform, group velocity, and amplitude ratio, match the experimental results well. The simulated attenuations are almost the same as those of experimental results at the low frequency range (≤50 kHz). The small scatter at the high frequency range (50–100 kHz) is possibly due to the background noise and measurement errors in experiments. With this numerical model, it becomes possible to study the wave behaviour inside the bolt along its central axis, which is very difficult if not impossible in experiments. Results of the simulation model indicate that the group velocity in the grouted rock bolts is
constant along the grouted bolt length and the boundary effects at the bolt ends on the group velocity can be ignored. This model also allows for the determination of the attenuation coefficient for a specific type of bolt conditions. The analysis also shows that the attenuation of guided wave propagating in rock bolts is similar to that of infinite length one dimensional solution. This paper has proposed a method to determine the boundary effects on the attenuation at the bolt ends. Based on this method, the calculated amplitude ratio matches well with the experimental results. Acknowledgement This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. References Achenbach, J.D., 1973. Wave Propagation in Elastic Solids. Applied Mathematics and Mechanics. North-Holland Publishing Company. ISBN: 0 444 10465 8. Beard, M.D., Lowe, M.J.S., 2003. Non-destructive testing of rock bolts using guided ultrasonic waves. International Journal of Rock Mechanics and Mining Science 40, 527–536. Klimentos, T., McCann, C., 1990. Relationships between compressional wave attenuation, porosity, clay content and permeability of sandstones. Geophysics 55, 998–1014. Livermore Software Technology Corporation, 2001. LS-DYNA, Version: PC-DYNA _ 960. Madenga, V., 2004. Application of Guided Waves to Grout Quality Testing of Rock Bolts, M.A.Sc. Thesis, Dalhousie University. Madenga, V., Zou, D.H., Zhang, C., in press. Effects of Curing Time and Frequency on Ultrasonic Wave Velocity in Grouted Rock Bolts. Journal of Applied Geophysics (available online at http://dx. doi.org/10.1016/j.jappgeo.2005.08.001). Rose, J.L., 1999. Ultrasonic Waves in Solid Media. Cambridge University Press, Cambridge. Saeed, M., 2003. Finite Element Analysis: Theory and Application with ANSYS, 2nd ed. Pearson Education, Upper Saddle River, N.J. Tavakoli, M.B., Evans, J.A., 1992. The effect of bone structure on ultrasonic attenuation and velocity. Ultrasonics 30, 389–395. Thurner, H.F., 1988. Instrument for Non-destructive Testing of Grouted Rock Bolts. 2nd International Symposium on Filed Measurements in Geomachanics. Balkema, Rotterdam, pp. 135–143. Zhang, C.H., Zou, D.H., Madenga, V., 2005. Numerical simulation of wave propagation in grouted rock bolts and the effects of mesh density and wave frequency. International Journal of Rock Mechanics and Mining Sciences 43 (4), 634–639. Zou, D.H., Cui, Y., Madenga, V., Zhang, C., in press. Effects of Frequency and Grouted Length on the Behaviour of Guided Ultrasonic Wave Propagating in Rock Bolts. International Journal of Rock mechanics and Mining Science.