Dynamic response of high and thin rock slope under blasting vibration: a case study

3.1. Blasting practice

The Chiwan parking lot, an important part of Shenzhen Metro Line 12, is intended for parking metro vehicles. The foundation of the parking lot is designed to be at an elevation of approximately 6.0 m above the global mean sea level, and the original ground surface of the parking lot has a topography that ranges from 17.0 m to 20.0 m in elevation, as shown in Fig. 3. Therefore, large scale excavation works for foundation construction were carried out in the parking lot areas by drilling and blasting.

Fig. 3Location of blasting area and arrangement of monitoring points

Double-hole blasts were widely carried out for the foundation excavation of the parking lot. Blasting parameters and charging structures commonly adopted in the blasting practices were broadly similar. The detailed drilling and blasting parameters are summarized in Table 2, and the typical charging structure is presented in Fig. 4. Non-electric millisecond detonators MS3 and MS9 were used to set up different initiation networks fired in a zigzag sequence. Detonators MS3 were used between the delays including two vertical blastholes. Detonators MS9 were used for downhole delays. The typical initiation network for the blasting area shown in Fig. 3 (BA) is plotted in Fig. 5.

3.2. Blasting vibration monitoring system

The vibration induced by the blasting practices for foundation construction of the parking lot might have adverse effects on the safety of the case slope, so the blasting vibration monitoring for the slope was conducted. The monitoring systems, composed of triaxial velocity transducers and intelligent monitors Blast-UM as shown in Fig. 6, were used in the blasting vibration monitoring programs. The monitoring range in vibration amplitude and frequency of the systems are 0.001-35.0 cm/s and 5-300 Hz respectively, which cover an overwhelming majority of blasting vibration recordings. The monitoring systems were arranged at the toe of the slope’s benches at different elevations, as shown in Fig. 2 and Fig. 3.

Fig. 4Typical charging structure

Fig. 5Typical initiation network for BA

Fig. 6Blasting vibration monitoring system. Photo was taken by the authors on the Chiwan Mountain slope (the case slope in the manuscript) when the blasting vibration monitoring was conducted

4.1. Peak particle velocity

The peak particle velocities (PPVs) of all monitoring points are shown in Fig. 8. The maximum PPVs in longitudinal, transverse, and vertical directions are 1.65 cm/s, 0.70 cm/s, and 0.87 cm/s, respectively. The minimum PPVs in longitudinal, transverse, and vertical directions are 0.23 cm/s, 0.16 cm/s, and 0.20 cm/s, respectively, and all of them occur at 3# monitoring point. Among PPVs in the three directions, the longitudinal PPVs are the largest, and the transverse and vertical PPVs are close. Meanwhile, PPVs in all directions first decrease and then increase with increasing elevation of monitoring points on the southern slope, indicating that the common attenuation law of blasting vibration works for the region within 1# – 3# monitoring point range, while the elevation amplification effect of blasting vibration occur from 3# monitoring point. The remarkable elevation amplification effect of the blasting vibration is observed from 4# monitoring point.

Fig. 7Typical blasting vibration waveforms of 1# monitoring point

Fig. 8PPVs of different monitoring points on the southern slope

4.2. Time-frequency characteristics

The wavelet transform was adopted to get insight into the time-frequency characteristics of the recorded blasting vibration. Based on the principle of wavelet transformation [31, 32], for a given function $\psi \left(t\right)\in L^2\left(R\right)$, where $L^2\left(R\right)$ is a signal space with limited energy, its Fourier transform can be written as $\widehat{\psi }\left(\omega \right)$. When $\widehat{\psi }\left(\omega \right)$ satisfies the following condition:

The continuous wavelet transform (CWT) for any function $f\left(t\right)\in L^2\left(R\right)$ is as follows:

The continuous wavelet inverse transform is $f\left(t\right)=\frac{1}{C_{\psi }}{\int }_{R^{+}}{\int }_R\frac{1}{a^2}{W}_f\left(a,b\right)\psi \left(\frac{t-b}{a}\right)dadb$, where a and b are the scaling factor and translation factor, respectively, and $\overline{\psi \left(\frac{t-b}{a}\right)}$ is the conjugate function of $\psi \left(\frac{t-b}{a}\right)$.

Since the recorded blasting vibration waveforms are discrete data composed of a series of time sampling points, the parameters a and b need to be discretized. All data and parameters need to be divided by a binary grid (for example, $a=2^j$, $b=2^{j}k$, $j,k\in Z$) to obtain the binary wavelet transform in the following formula:

The CWT time-frequency spectra of all monitoring points are shown in Fig. 9. The red dotted lines in Fig. 9 depict the trends in dominant frequencies of the blasting vibration. On the whole, the dominant frequencies tend to decay with the increase in elevation. From 1# to 3# monitoring point, the dominant frequencies are reduced from above 50 Hz to around 35 Hz. From 4# to 7# monitoring point, the dominant frequencies remain largely the same at about 30 Hz. In particular, part of the dominant frequencies of 7# monitoring point is within 20 Hz which occurs at the end of the waveform, and part of the dominant frequencies of 5# to 7# monitoring points in the vertical direction are beyond 50 Hz.

4.3. Energy-frequency characteristics

The wavelet packet transform (WPT) was adopted to get insight into the energy-frequency characteristics of the recorded blasting vibration. Based on the principle of WPT [33]. The WPT reconstruction algorithm can be indicated as the following equation:

where $d_{i,j,k}$ is the coefficient of WPT, ${\left\{h_k\right\}}_{k\in Z}$ and ${\left\{g_k\right\}}_{k\in Z}$ are the low-pass and high-pass filter coefficients, respectively. The filter coefficients ${\left\{h_k\right\}}_{k\in Z}$ and ${\left\{g_k\right\}}_{k\in Z}$ in orthogonal multi-resolution analysis shall satisfy the below equation:

Then, Eq. (4) can be reconstructed as ${\left|d_{i,j,k}\right|}^2={\left|\sum _{k\in Z}{h}_{k-2l}{d}_{i+1,j,k}+\sum _{k\in Z}{g}_{k-2l}{d}_{i+1,j+1,k}\right|}^2$. The wavelet packet coefficients can be calculated by definition as $d_{j,k}=⟨f_{i,j},{\phi }_{j,k}\left(t\right)⟩$, where ${\phi }_{j,k}\left(t\right)$ and ${\phi }_{j+1,k}\left(t\right)$ are standard orthogonal bases in the wavelet packet spaces $U_{i+1,j}$ and $U_{i+1,j+1}$, respectively. The wavelet packet spaces $U_{i+1,j}$ and $U_{i+1,j+1}$ are mutually orthogonal, that is, mutually orthogonal ${\phi }_{j,k}\left(t\right)$ and ${\phi }_{j+1,k}\left(t\right)$ exist. As a result, $\mathrm{c}\mathrm{o}\mathrm{s}⟨{\phi }_{j,k}\left(t\right),{\phi }_{j+1,k}\left(t\right)⟩=0$ and thus $\mathrm{c}\mathrm{o}\mathrm{s}⟨\sum _{k\in Z}{h}_{k-2l}{d}_{i+1,j,k},\sum _{k\in Z}{h}_{k-2l}{d}_{i+1,j,k}⟩=0$.

Then, the relationship between decomposition coefficients at level $i$ and level $i+1$ are shown as Eq. (6):

where ${\left\{h_k\right\}}_{k\in Z}$ and ${\left\{g_k\right\}}_{k\in Z}$ can be regarded as standard orthogonal bases of $d_{i+1,j,k}$, $d_{i+1,j+1,k}$ in the frequency domain, respectively. Therefore, the results of ${\left\{h_k\right\}}_{k\in Z}$ and ${\left\{g_k\right\}}_{k\in Z}$ transformation do not change the modular value, that is, one can write:

Fig. 9CWT time-frequency spectra in three directions for different monitoring points

The signal $S\left(t\right)$ in Hilbert space $L^2\left(R\right)$ can be indicated as:

where $f_{i,j}\left(t_j\right)=\sum _{k=1}^m{d}_{i,j}^{\left(k\right)}{\phi }_{j,k}\left(t\right)$ is the projection of the signal $S\left(t\right)$ in the wavelet packet space $U_{i,j}$, $i$ is the scale parameter (i.e., decomposition level of wavelet packet), $j$ is the $j$th wavelet subspace of decomposing wavelet packet to level $i$, and $k$ is displacement parameter, and $m$ is the signal discretization sampling point.

The proportion of the energy of the frequency band corresponding to $f_{i,j}\left(t_j\right)$ in the wavelet packet space $U_{i,j}$ accounting for that of the signal $S\left(t\right)$ is $P_{i,j}$:

where $E_{i,j}=\sum _{k=1}^m{\left|d_{i,j}^{\left(k\right)}\right|}^2$ is the frequency band energy corresponding to $f_{i,j}\left(t_j\right)$ in the wavelet packet space and $E=\sum _{j=0}^{2^i-1}{E}_{i,j}$ is the overall energy of the signal.

The energy-frequency distributions in three directions for different monitoring points are presented in Fig. 10. The cyan boxes in Fig. 10 depict the trends in dominant frequency bands, in which the blasting vibration energy ratio is overwhelming.

On the whole, the dominant frequency bands tend to decay with the increase in elevation. From 1# to 3# monitoring point, the dominant frequency bands, in which the energy rations are more than 25%, are reduced from above 60-80 Hz to around 35 Hz. From 4# to 7# monitoring point, the dominant frequency bands, in which the energy rations are more than 30 %, remain largely the same within 20-40 Hz. In particular, part of the dominant frequency bands of 7# monitoring point is within 20 Hz, in which the energy rations are above 30 %. The above results are consistent with those presented in Fig. 9, but the difference is that the energy ratios within the dominant frequencies beyond 50 Hz of 5# to 7# monitoring points in the vertical direction are relatively lower, among which energy ratios in single frequency band are mostly less than 10 %.

Fig. 10Energy-frequency distribution in three directions for different monitoring points

5.1. Model and parameters

To further reveal the dynamic response characteristics of the thin and high rock slope under blasting vibration, a numerical study was implemented. As shown in Fig. 11, a numerical model including the whole slope was developed in the software LS-DYNA. The model is meshed with hexahedron-shaped brick elements, SOLID 164 elements, and each SOLID 164 element has 8 nodes. Kuhlemeyer and Lysmer [34] suggested the mesh size should be shorter than 1/8 – 1/10 of the wave length to properly reduce any wave distortion. According to this requirement, the element size adopted in the model varies from 0.1 m near the slope surface to 1.0 m inside the slope. There are a total of 15854 SOLID 164 elements and 32362 nodes contained in the model. Non-reflecting boundaries are enforced to the northern face, bottom face, and southern face below 1# monitoring point to eliminate the wave-reflecting effect. The southern face below 1# monitoring point is also the loading boundary that is enforced to the modified longitudinal and vertical velocities of 1# monitoring point (as shown in Fig. 7). Apart from seven monitoring points on the southern slope, the other six monitoring points on the northern slope are added to analyze the dynamic response characteristics of the northern slope.

Fig. 11Numerical model of case slope

The maximum PPV of the recorded blasting vibration is less than 1.7 cm/s, which is far less than the safety control standards for the rock slope [35], and the distance from the blasting source to the toe of the slope exceeds 100 m. Therefore, the blasting vibration of the slope can be approximately regarded as elastic vibration, and the linear elastic materials were adopted in the numerical model. The lithology for the case slope is mainly composed of slightly weathered granite, so the material parameters of rock masses were assigned the parameters of the slightly weathered granite listed in Table 1.

5.2. Model verification

The numerical results of blasting vibration were obtained when the loading boundary is enforced to the longitudinal and vertical velocities of 1# monitoring point. The numerical and monitoring results of blasting vibration for 1# to 7# monitoring points are listed in Table 3. There are small differences mostly within 10 % between the numerical and monitoring results. The deviations between monitoring and numerical results of vertical PPVs at 5# and 7# monitoring points are slightly above 10 %, but the absolute errors are 0.05 cm/s and 0.09 cm/s, which are quite small and can be ignored. The evolution laws of PPVs along with the elevation are kept the same for the numerical and monitoring results, both of which are that PPVs firstly decrease and then increase with increasing elevation. As a result, the numerical model can effectively simulate the blasting vibration response of the case slope to a certain extent.

6.1. Natural frequencies

Natural frequencies play an important role in the dynamic performance evaluation of structures [36-38]. Fig. 12 presents the first 10 natural frequencies and corresponding mode shapes of the whole slope, which were calculated through modal analysis in LS-DYNA. Higher modes of the modal parameters are not given here because they make nearly no contribution to the slope motion according to the theory of structural dynamics [39]. The fundamental frequency of the whole slope is 6.53 Hz corresponding to the first mode shape representing the overall horizontal motion. The second to sixth natural frequencies less than 20 Hz are all below the dominant frequencies in Fig. 9 or dominant frequency bands in Fig. 10. The second mode shape of the whole slope is characterized as overall vertical motion, while the third to sixth mode shapes are characterized as local motions of the whole slope. The seventh to tenth natural frequencies are within 20-30 Hz, and their corresponding mode shapes are also characterized as local motions of the whole slope.

Fig. 12First 10 modal parameters of the whole slope

Considering the elevation amplification effect of blasting vibration exits from 4# monitoring point and dominant frequencies for 4# to 7# monitoring points remain largely the same, the first 10 natural frequencies and mode shapes of the slope above 4# monitoring point (the upper slope) were extracted, as presented in Fig. 13. The mode shapes of the upper slope are nearly the same as those of the whole slope, while their natural frequencies are rather different. The fundamental frequency of the upper slope is 13.51 Hz, which is within the dominant frequency band below 20 Hz of 7# monitoring point. The second to sixth natural frequencies are within 20-40 Hz, which fall into the unchanged dominant frequencies of 4# to 7# monitoring points. The seventh to tenth natural frequencies are within 40-60 Hz, which lie in the part of dominant frequencies beyond 50 Hz of 5# to 7# monitoring points. The close correspondence between the natural frequencies and dominant frequency characteristics of the upper slope indicates that the natural frequencies play a key role in the dynamic response of the slope.

Fig. 13First 10 modal parameters of the slope above monitoring point 4#

Fig. 14PPV distribution under vibration with different frequency characteristics

a) 0-20 Hz

b) 20-40 Hz

c) 40-60 Hz

d) 60-100 Hz

e) 100-150 Hz

6.2. Effects of vibration frequency on dynamic response of slope

To investigate the relationship between blasting vibration frequency characteristics and natural frequencies of the slope, five numerical cases corresponding to five loading conditions were carried out. Each loading condition corresponds to a set of modified blasting vibration waveforms with a specific frequency band, and the five different frequency bands are 0-20 Hz, 20-40 Hz, 40-60 Hz, 60-100 Hz, and 100-150 Hz, respectively, which are selected based on the frequency distribution characteristics and cover the dominating frequency components of the blasting vibration on the case slope. The modified blasting vibration waveform can be obtained as:

where $v\left(t\right)$ is the original blasting vibration waveform of the 1# monitoring point, and $v_{bp}\left(t\right)=Bandpass\left(v\left(t\right),f_{low},f_{high}\right)$ can be got through bandpass filtering ranging from low frequency $f_{low}$ to high frequency $f_{high}$ of $v\left(t\right)$.

PPV distributions of the slope under five different vibration loading conditions are presented in Fig. 14.

As presented in Fig. 14(a), PPVs decline in a fluctuation way with the increase of horizontal distance from the blasting source when the frequency range of the loading vibration is within 0-20 Hz, and PPVs of monitoring points are less than 0.4 cm/s except those of 1# monitoring point.

When the frequency ranges of the loading vibration are within 20-40 Hz and 40-60 Hz, PPVs decline firstly and then increase on the southern slope with the increase of horizontal distance from the blasting source, while PPVs on the northern slope fluctuate and are less than 0.25 cm/s, as presented in Fig. 14(b) and Fig. 14(c). The elevation amplification effect exits from 4# monitoring point for both the two conditions shown in Fig. 14(b) and Fig. 14(c), which matches the monitoring results, and the elevation amplification effect for the case of 20-40 Hz is more apparent.

For the case of 60-100 Hz presented in Fig. 14(d), PPVs also decline in a fluctuation way as in the case in Fig. 14(a), while the PPVs for the 4# and 7# monitoring points are beyond 0.4 cm/s, which are larger than the PPVs of their adjacent monitoring points.

For the last case of 100-150 Hz presented in Fig. 14(e), PPVs decline in a fluctuation way on the southern slope while increase on the northern slope with the increase of horizontal distance from the blasting source, and the maximum PPV on the northern slope exceeds 0.8 cm/s.

6.3. Effects of vibration amplitude on dynamic response of slope

The effects of vibration amplitude on the dynamics response of the slope were also investigated, and three numerical cases corresponding to each specific loading condition within a certain frequency range were conducted. The three cases are $V_m=\mathrm{m}\mathrm{a}\mathrm{x}\left(V\left(t\right)\right)$= 5.0 cm/s, 10.0 cm/s, and 15.0 cm/s. For blasting vibration with the frequency range within 0-20 Hz, Fig. 15 gives the typical PPV distributions of the slope under vibration with different amplitudes.

Fig. 15PPV distribution under vibration with different amplitudes (0-20 Hz)

a) Horizontal

b) Vertical

The applied vibration amplitudes have significant importance on the values of PPVs but no effects on the distribution of PPVs, and the values of PPVs are approximately proportional to the applied vibration amplitudes. For example, the horizontal PPVs at 2# monitoring point under $V_m=$ 5.0 cm/s, 10.0 cm/s, and 15.0 cm/s are about 1.5 cm/s, 3.0 cm/s, and 4.5 cm/s, indicating that the PPV is approximately proportional to the applied vibration amplitude. As for blasting vibration with other frequency ranges, the effects of vibration amplitude are kept the same.