Research on the Vibration Response of High-Rise Buildings under Blasting Load

Abstract

The vibration caused by blasting load may result in damage to high-rise buildings. In view of this consideration, an investigation of the vibration law was conducted in the context of an actual engineering project. The objective of this study was to analyze the peak particle velocity (PPV), vibration frequency, and peak particle stress (PPS) of the buildings within a range of 50 m to 250 m from the epicenter, under the condition of a single-shot charge of 30 kg. To achieve this, a combination of theoretical analysis, field tests, and numerical experiments was employed. Sadovsky’s formula was used in combination with the least squares method to fit the propagation law of ground PPV. ANSYS 17.0/LS-DYNA and Origin 8.0 software were applied to study the amplification effect of building PPV and the relationship between PPV and PPS. Taking into account the difference between the height of the ground measuring point and the height of the explosive center, we investigated the PPV of high-rise buildings under three conditions of 36 m, 6 m, and −24 m drop from the explosive center, to strengthen the in-depth understanding of resonance effect. The following conclusions were reached: the ground PPV decreases with increasing horizontal distance from the explosive center, with the radial PPV being the largest. The vertical PPV of buildings exhibits a height amplification effect, with a magnification factor of 2.66. The radial and tangential PPVs of buildings demonstrate that the middle layer exhibits a relatively modest speed, whereas the low and high layers demonstrate considerably higher speeds. The greater the vertical distance from the explosion center is, the greater is the PPV. The vibration frequency is irregular, with an average of 10 Hz. The PPV of buildings is not proportional to the PPS, which is the highest at the bottom. It is recommended that the PPS of buildings be included in the criteria for safety allowances in blasting vibration.

1. Introduction

Blasting technology is a common practice in mining, foundation pit excavation, and tunnel excavation, playing a significant role in infrastructure construction. Nevertheless, the detrimental impacts of blasting technology, including the generation of blasting earthquakes, present a great challenge to the advancement of the blasting industry [1]. Blasting earthquakes propagate in the form of waves, causing ground vibration and building vibration.

Peak particle velocity (PPV) is one of the main factors in the vibration response of a blast. PPV of the ground is easy to obtain, and when only the elastic deformation is considered, PPV is directly proportional to the peak particle stress (PPS) [2,3]. Consequently, PPV is employed as an evaluation index in the initial phase. PPV gradually decays with the increase in the horizontal distance of the explosive center (HDEC). However, the decay law varies across different paths [4,5,6], with different speeds observed from the radial, vertical, and tangential direction [7]. Due to the existence of a slope elevation difference, the vertical distance of the explosive center (VDEC) also affects PPV, which is specifically shown to have an elevation amplification effect [8,9,10]. As a result of the unremitting efforts of experts and scholars, a number of new methods for vibration damping and noise reduction have emerged, including the parameter optimization method [11], the artificial intelligence method [12], the equivalent simulation method [13] and the acceleration equivalent transformation method [14].

The process of urbanization has resulted in a notable increase in the height of modern buildings. As concerns about seismic hazards caused by blasting operations gain increasing attention in society, a more detailed examination of the vibration response of high-rise buildings under blasting loads is warranted. The structural vibration response is influenced by both the height of a building and its structural form, as evidenced in references [15,16]. It has been observed that PPV of buildings does exhibit an amplification effect, which is not a universal phenomenon, as not all three-way velocities display such an effect [17,18]. To study the destruction, damage, and vibration response of buildings under blasting seismic waves, the duration of seismic wave action needs to be considered [19,20]. The frequency of blasting loads is greater than that of building’s self-oscillation frequency. As the HDEC increases, the loading frequency decreases. Additionally, PPV is observed to be high in the high-frequency band [21,22]. The idea of vibration damping from a frequency perspective is to prevent the generation of structural resonance [23,24]. The current literature on the vibration response of high-rise buildings to tunnel blasting is extensive [25,26], whereas that on the vibration response of high-rise buildings to mine blasting is comparatively limited. Tunnel-blasting single-explosive charges are much lower than mine-blasting charges. Furthermore, the propagation paths of seismic waves for the two are distinct. This paper employs mine blasting as a case study to investigate its vibration response to the surrounding high-rise buildings. The objective is to elucidate the vibration response law, gain a comprehensive understanding of the resonance effect, ascertain the structural damage mechanism, and provide a reference point for seismic isolation, seismic reduction, and seismic defense. The innovation of this work is to consider incorporating new factors into the safety standards for high-rise buildings under blasting vibration.

2. Theory of Blasting Vibration

2.1. Theory of Blasting Seismic Wave Propagation

Blasting earthquakes are a consequence of blasting operations and are characterized by a brief duration, a high vibration frequency, and a controlled release of energy. These operations are employed in engineering and construction for specific purposes. The occurrence of seismic waves is a consequence of blasting earthquakes. These waves do not directly cause damage to the rock itself; however, they induce vibration and fissure development in the rock, which can subsequently result in damage to the structure above the rock.

In a seminal contribution to the field, Soviet scientist Sadovsky proposed the following equation for peak velocity [27]:

$$V=K{({\displaystyle \frac{\sqrt[3]{Q}}{R}})}^{\alpha }$$

(1)

The Changjiang River Scientific Research Institute put forth an empirical formula for the variation in peak velocity with respect to HDEC and VDEC [28]:

$$V=K{({\displaystyle \frac{\sqrt[3]{Q}}{R}})}^{\alpha }{({\displaystyle \frac{\sqrt[3]{Q}}{H}})}^{\beta }$$

(2)

In Equations (1) and (2), V is the PPV of blasting vibration, cm/s; Q is the quantity of explosive charge for a single detonation, kg; R is the HDEC, m; H is the VDEC, m; K and α are parameters related to the terrain and geology and the blasting vibration attenuation index, respectively; and β is the elevation influence coefficient.

China’s Safety regulations for blasting (GB6722-2014) [29] recommends the use of the Sadowsky’s formula, Equation (1), to calculate the peak velocity. The peak velocity and frequency of the ground at the location of the protection object are used as joint criteria. It is also essential to consider the significance of a building, self-oscillation frequency, the distinction between new and old structures, the foundation, the relative position of the blasting point in relation to the building, and other pertinent factors. The permissible limits for blasting vibrations are delineated in

Table 1. Standard for safety allowance of blasting vibrations.

Serial No.	Category of Protection Object	Safe Permissible Particle Vibration Velocity (cm/s)
		f ≤ 10 Hz	10 Hz < f ≤ 50 Hz	f > 50 Hz
1	Earth kiln caves, adobe houses, and rubble houses	0.15–0.45	0.45–0.9	0.9–1.5
2	General civil buildings	1.5–2.0	2.0–2.5	2.5–3.0
3	Industrial and commercial buildings	2.5–3.5	3.5–4.5	4.2–5.0
…	…	…	…	…

Note: The table above is taken from Article 13.2.2 of the Safety regulations for blasting (GB6722-2014); f is frequency.

.

2.2. Resonance Theory

It is well established that when the loading frequency is in close proximity to the self-resonance frequency of a building structure, the likelihood of resonance within the structure is significantly increased [30,31]. In accordance with D’Alembert’s Principle, the differential equation of vibration for a single-degree-of-freedom structure is as follows:

$$m\ddot{y}+c\dot{y}+ky=P(t)$$

(3)

where $-m\ddot{y}$, $-c\dot{y}$, $-k\mathrm{y}$, and $P(t)$ are inertial, damping, elastic, and dynamic loads, respectively; m, c, and k are the mass, damping, and stiffness of the system, respectively; $\ddot{y}$, $\dot{y}$ and $y$ are acceleration, velocity, and displacement of the system, respectively.

Divide the left and right of Equation (3) simultaneously by the mass m and let $\omega =\sqrt{k/m}$, $\xi =c/2m\omega$. Then formula (3) can be written as:

$$\ddot{y}+2\xi \omega \dot{y}+{\omega }^{2}y=P(t)/m$$

(4)

where $\omega$ is the undamped self-oscillation frequency and $\xi$ is the damping ratio. Solving Equation (4), we obtain:

$$y=e^{-\xi \omega t}(y_0\cos w_{r}t+{\displaystyle \frac{v_0+\xi \omega y_0}{{\omega }_r}}\sin w_{r}t)+{\displaystyle \frac{1}{m{\omega }_r}}{\displaystyle {\int }_0^{t}P(\tau )e^{-\xi \omega (t-\tau )}\sin w_r(t-\tau )d\tau }$$

(5)

where $y_0$, $v_0$ are the initial displacement and initial velocity of the system, respectively. ${\omega }_r=\omega \sqrt{1-{\xi }^2}$ is the damped self-oscillating frequency. The parameter $\tau$ is the moment of dynamic loading. The t in the integral function is the moment of calculation of the displacement, and the upper limit of integration t is the time horizon of the action of the instantaneous impulse.

Dynamic loads are viewed as superimposed by a number of simple harmonic loads. Thus, it may be assumed that:

$$y_{\theta }(t)={\displaystyle \sum _{k=1}^n{A}_k\sin ({\theta }_{k}t)}$$

(6)

In this formula above, $y_{\theta }(t)$ represents the displacement of the system under a number of simple harmonic loads, $A_k$ denotes the static displacement under the amplitude of the kth simple harmonic load, and ${\theta }_k$ signifies the frequency of the kth simple harmonic load. Thus,

$$P(t)=m{\ddot{y}}_{\theta }(t)$$

(7)

By combining Equations (6) and (7) with Equation (5), we obtain:

$$y=e^{-\xi \omega t}(y_0\cos w_{r}t+{\displaystyle \frac{v_0+\xi \omega y_0}{{\omega }_r}}\sin w_{r}t)+{\displaystyle \frac{1}{{\omega }_r}}{\displaystyle \sum _{k=1}^n{\displaystyle {\int }_0^t{e}^{-\xi \omega (t-\tau )}{a}_k\sin ({\theta }_k\tau +{\psi }_k)\sin w_r(t-\tau )d\tau }}$$

(8)

where $a_k$ is the static acceleration under the kth simple harmonic load amplitude; ${\psi }_k$ is the phase difference between acceleration and displacement. The first term on the right-hand side of the above equation contains $e^{-\xi \omega t}$ and will gradually decay to zero. Only the second term remains in the final displacement, which is:

$$y={\displaystyle \frac{1}{{\omega }_r}}{\displaystyle \sum _{k=1}^n{\displaystyle {\int }_0^t{e}^{-\xi \omega (t-\tau )}{a}_k\sin ({\theta }_k\tau +{\psi }_k)\sin w_r(t-\tau )d\tau }}$$

(9)

The integration of the above equation results in:

$$\begin{array}{l}y(t)=e^{-\xi \omega t}{\displaystyle \sum _{k=1}^n\frac{{\omega }^2{A}_k\sin a_k}{\sqrt{{({\omega }^2-{{\theta }_{\mathrm{k}}}^2)}^2+4{\xi }^2{\omega }^2{{\theta }_{\mathrm{k}}}^2}}}\cos {\omega }_{r}t-e^{-\xi \omega t}{\displaystyle \sum _{k=1}^n\frac{{\omega }^2{A}_k(-\xi \frac{\omega }{{\omega }_r}\sin \alpha +\frac{{\theta }_{\mathrm{k}}}{{\omega }_r}\cos \alpha )}{\sqrt{{({\omega }^2-{{\theta }_{\mathrm{k}}}^2)}^2+4{\xi }^2{\omega }^2{{\theta }_{\mathrm{k}}}^2}}}\sin {\omega }_{r}t\\ +{\displaystyle \sum _{k=1}^n\frac{{\omega }^2{A}_k}{\sqrt{{({\omega }^2-{{\theta }_{\mathrm{k}}}^2)}^2+4{\xi }^2{\omega }^2{{\theta }_{\mathrm{k}}}^2}}}\sin ({\omega }_{k}t-\alpha )\end{array}$$

(10)

where $\alpha =\arctan \left({\displaystyle \raisebox{1ex}{$2\xi {\theta }_k/\omega $}\!\left/ \!\raisebox{-1ex}{$1-({\theta }_k/\omega {)}^2$}\right.}\right)$. Clearly, the first two terms of Equation (10) contain $e^{-\xi \omega t}$, which decays to zero. The final displacement is only the third item. If we consider only the effect of the kth simple harmonic load and deform the third term, we can obtain the displacement due to the kth simple harmonic load as:

$$y_k(t)=A_k\sin ({\omega }_{k}t-\alpha )/\sqrt{{[1-({\displaystyle \frac{{\theta }_{\mathrm{k}}}{\omega }}{)}^2]}^2+4{\xi }^2({\displaystyle \frac{{\theta }_{\mathrm{k}}}{\omega }}{)}^2}.$$

(11)

The maximum dynamic displacement produced by the kth simple harmonic load is:

$$Y_k(t)=A_k/\sqrt{{[1-({\displaystyle \frac{{\theta }_{\mathrm{k}}}{\omega }}{)}^2]}^2+4{\xi }^2({\displaystyle \frac{{\theta }_{\mathrm{k}}}{\omega }}{)}^2}.$$

(12)

Since $A_k$ is the static displacement under the kth simple harmonic load magnitude, it may be useful to suppose coefficient of dynamics:

$$\beta ={\displaystyle \frac{Y_k(t)}{A_k}}=1/\sqrt{{[1-({\displaystyle \frac{{\theta }_{\mathrm{k}}}{\omega }}{)}^2]}^2+4{\xi }^2({\displaystyle \frac{{\theta }_{\mathrm{k}}}{\omega }}{)}^2}.$$

(13)

And a frequency ratio coefficient: $\psi ={\theta }_k/\omega$. The derivation shows that as $\psi$ converges to 1, $\beta$ converges to $1/2\xi$. The damping ratio of the structure is less than 0.1, so the dynamic coefficient will increase to a value exceeding five times the initial value when resonance occurs.

3. Field Test of Blasting Vibration

3.1. Overview of the Project

The study project in Guiyang City has a high degree of complexity, with numerous high-rise buildings, educational institutions, and major transportation routes in close proximity, which is illustrated in Figure 1. The rock fissures at the site was relatively short and dense, with poor connectivity and a fissure rate of between two and eight fissures per meter. The rock integrity was deficient, with a coefficient of solidity ranging from 9 to 11. A millisecond micro-differential time-delayed blasting method was adopted. Emulsion explosive of 70 mm diameter was selected. Maximum dosage for a single shot was 30 kg.

3.2. Monitoring Content and Methods

Blasting vibration monitoring was performed using the CBSD-VM-M01 Digital Network Vibrometer (Manufactured by Guangzhou Zhongbao Safety Net Technology Co., Ltd., Guangzhou, China). The instrument can simultaneously monitor radial, vertical, tangential peak velocity and vibration frequency.

From near to far, vibrometers 1* to 5* were installed on the ground near high-rise buildings No.1 to No.5 to monitor ground vibration. From the bottom up, vibrometers 1# to 5# were installed on the 1st, 5th, 9th, 13th, and 18th floors of each high-rise building to monitor the buildings’ vibration, as shown in Figure 2.

3.3. Monitoring Results

3.3.1. PPV and Ground Frequency

Blasting vibration monitoring of the ground where the high-rise buildings were located yielded 40 sets of valid data from eight blasts, as shown in

Table 2. PPV and frequency monitored on the ground. * Monitoring resulted in 40 data sets from eight blasts; only three blasts are shown here. Monitoring points are shown in Figure 2.

No. of Times	Monitoring Point	HDEC (m)	Radial PPV (cm/s)	Radial Frequency (Hz)	Vertical PPV (cm/s)	Vertical Frequency (Hz)	Tangential PPV (cm/s)	Tangential Frequency (Hz)
1	1*	47	0.278	13.5040	0.266	8.4686	0.212	15.3351
	2*	97	0.169	10.9863	0.176	8.6975	0.165	15.1062
	3*	147	0.113	10.5286	0.126	8.6975	0.113	8.6975
	4*	197	0.091	11.9019	0.082	7.3242	0.076	9.6130
	5*	247	0.047	11.9019	0.055	16.7084	0.041	13.2751
2	1*	43	0.298	8.6975	0.266	8.9264	0.218	17.3950
	2*	93	0.216	10.7574	0.186	8.9264	0.148	9.1553
	3*	143	0.109	9.3842	0.121	8.9264	0.113	9.8419
	4*	193	0.079	15.3351	0.079	20.8282	0.075	9.8419
	5*	243	0.048	13.5040	0.056	17.3950	0.048	6.6376
3	1*	71	0.253	10.0708	0.236	9.1553	0.206	16.4759
	2*	121	0.162	9.3842	0.163	9.3842	0.146	9.3842
	3*	171	0.116	9.3842	0.118	9.3842	0.101	9.3842
	4*	221	0.082	16.4795	0.076	7.3242	0.066	9.3842
	5*	271	0.046	9.3842	0.048	16.4795	0.046	7.0953
…	…	…	…	…	…	…	…	…

, of which, the 1st to 3rd data are shown.

Among the 40 sets of data, measuring point 1* of the second blast has the largest vibration velocity of 0.298 cm/s and a frequency of 8.6975 Hz. When the load frequency f ≤ 10 Hz, the permissible safe vibration velocity of general civil buildings is 1.5–2.0 cm/s in Table 1. It can be demonstrated that the safety of high-rise buildings is in accordance with the requisite standards.

3.3.2. PPV and Frequency of the Buildings

Blast vibration monitoring conducted on the 1st, 5th, 9th, 13th, and 18th floors of the high-rise buildings yielded a total of 15 sets of data from three blasts in building No.1, 20 sets of data from four blasts in building No.2, 15 sets of data from three blasts in building No.3, 20 sets of data from four blasts in building No.4, and 15 sets of data from three blasts in building No.5. Blast vibration data for building No.1 are shown in

Table 3. PPV and frequency monitored in high-rise building No.1 based on three blasts. # Monitoring points are shown in Figure 2.

No. of Times	Motoring Point	HDEC (m)	Radial PPV (cm/s)	Radial Frequency (Hz)	Vertical PPV (cm/s)	Vertical Frequency (Hz)	Tangential PPV (cm/s)	Tangential Frequency (Hz)
1	5#	50	0.228	9.6130	0.602	9.1553	0.185	20.5994
	4#	50	0.215	9.6130	0.482	9.1553	0.18	11.4441
	3#	50	0.186	9.3842	0.464	9.1553	0.169	12.3596
	2#	50	0.183	9.6130	0.356	9.1553	0.164	11.4441
	1#	50	0.243	9.3842	0.253	9.1553	0.195	10.9863
2	5#	50	0.235	11.2152	0.613	8.6975	0.196	11.4441
	4#	50	0.202	11.2152	0.512	8.6975	0.196	11.4441
	3#	50	0.182	14.6484	0.451	8.6975	0.189	12.3596
	2#	50	0.176	11.2152	0.382	8.6975	0.156	11.4441
	1#	50	0.232	11.4441	0.235	8.6975	0.185	11.2152
3	5#	50	0.201	9.6130	0.612	9.3842	0.183	10.9863
	4#	50	0.199	9.6130	0.501	9.3842	0.182	11.2152
	3#	50	0.148	9.3842	0.446	9.3842	0.186	12.3596
	2#	50	0.16	9.6130	0.354	9.3842	0.175	11.2152
	1#	50	0.223	9.6130	0.198	9.3842	0.182	11.2152
Mean value	5#	50	0.221	10.147	0.609	9.079	0.188	14.343
	4#	50	0.205	10.147	0.498	9.079	0.186	11.368
	3#	50	0.172	11.139	0.454	9.079	0.181	12.360
	2#	50	0.173	10.147	0.364	9.079	0.165	11.368
	1#	50	0.233	10.147	0.229	9.079	0.187	11.139

.

3.4. Analysis of Monitoring Results

3.4.1. PPV

In accordance with the test data presented in Table 2, scatter plots of the radial, vertical, and tangential PPV of the ground with respect to HDEC are provided in Figure 3a–c. As illustrated in the figures, the ground radial, vertical, and tangential PPV demonstrate a decline with an increase in the burst center distance. The results of the calculation demonstrate that, in general, radial PPV > vertical PPV > tangential PPV.

The propagation law of the ground radial PPV as a function of HDEC is explored using Sadowsky’s formula as follows [32,33].

Taking logarithms for the left and right sides of Equation (1):

$$\lg V=\lg K+\alpha \lg (\sqrt[3]{Q}/R)$$

(14)

Suppose $y=\lg V$, $a=\lg K$, $b=\alpha$, $x=\lg (\sqrt[3]{Q}/R)$

Then Equation (14) becomes:

$$\mathrm{y}=a+bx$$

(15)

Bringing HDEC R, PPV V, and single-shot charge Q (30 kg) from Table 2 into the expression set for $x,y$, the data set ($x_i,y_i$) then emerges, where $i=1,2,3,\dots ,40$. A linear fitting of the least squares method to the 40 data sets yielded values of $a,b$. The topographic geological coefficient and attenuation coefficient are calculated using $\alpha =b$ and $K={10}^a$. Eventually, according to the form of Equation (1), the propagation law of radial PPV with HDEC is:

$$V_r=6.87{(\sqrt[3]{30}/R)}^{1.07}$$

(16)

Similarly, the propagation laws of the vertical and tangential PPV of the ground with HDEC are:

$$V_v=4.93{(\sqrt[3]{30}/R)}^{0.99}$$

(17)

$$V_t=3.33{(\sqrt[3]{30}/R)}^{0.92}$$

(18)

Based on the test data in Section 3.3.2, the variation in radial, vertical, and tangential PPV with floor is plotted in Figure 3d–f. From the figures, the radial and tangential PPVs show a trend of decreasing and then increasing with the increase in floor height. The PPV is observed to be high in the lower and upper floors of the buildings, and low in the middle-low floors. The vertical PPV shows an increasing trend with elevation amplification effect. PPV of the top floor is 2.66 times that of the bottom floor in building No.1, which is located at a horizontal distance of 50 m from the center of the explosion. In general, the vertical PPV is greater than the radial and tangential PPV.

3.4.2. Frequency

In accordance with the test data in Table 2, a scatter plot of ground vibration frequencies with HDEC is plotted in Figure 4a. Based on the experimental data in Section 3.3.2, line plots of radial, vertical, and tangential vibration frequencies with respect to the floor are plotted in Figure 4b–d. It can be seen that the radial, vertical, and tangential vibration frequencies of the ground have no obvious pattern in the range of 50 m to 250 m from the explosive center. The frequencies exhibit a range between 8 Hz and 20 Hz, with a mean value of 10 Hz [34,35]. In the high-rise buildings, radial, vertical, and tangential frequencies also show no obvious pattern, with an average value of 10 Hz.

4. Numerical Test of Blasting Vibration

4.1. Model and Parameters

4.1.1. Model

The “Rock–High-rise Buildings” model was created using the ANSYS 17.0/ LS-DYNA software. The model was divided into an underground part and an above-ground part. The underground part included rock and explosive. The above-ground part, or the high-rise buildings, included columns, beams, and floor slabs. The HDEC of high-rise buildings No.1 to No.5 were 50 m, 100 m, 150 m, 200 m, and 250 m, respectively. The height of all buildings was 54 m, as shown in Figure 5.

4.1.2. Material Parameters

The rock and explosive parameters are shown in

Table 4. Parameters of rock.

Density (kg/m3)	Elastic Modulus (GPa)	Poisson’s Ratio	Yield Strength (MPa)	Tangent Modulus (MPa)
2090	23	0.22	20	250

and

Table 5. Parameters of explosive.

Density (kg/m3)	Detonation Velocity (m/s)	Detonation Pressure (GPa)	A (GPa)	B (GPa)	R1	R2	$\mathit{\omega }$	Internal Energy per Unit Volume (GPa)	Relative Volume
1200	5500	10	214.4	0.182	4.2	0.9	0.15	4.19	1

.

A, B, $R_1$, $R_2$, and $\omega$ in the table are material constants, which obey the following JWL equation of state for burst pressure P, internal energy per unit volume E, and relative volume V:

$$P=A\left(1-{\displaystyle \frac{\omega }{R_{1}V}}\right)e^{-R_{1}V}+B\left(1-{\displaystyle \frac{\omega }{R_{2}V}}\right)e^{-R_{2}V}+{\displaystyle \frac{\omega E}{V}}$$

(19)

4.2. Numerical Test Results

After the process of generating the K-file, modifying the K-file, and solving the K-file, the LS-PrePost 4.0 software was used to view the velocity response of the nodes, which were located on the ground floor and floors of the building. The vertical PPV is shown in

Table 6. Vertical PPV of high-rise buildings (cm/s).

Floor	No.1	No.2	No.3	No.4	No.5	Floor	No.1	No.2	No.3	No.4	No.5
18	0.562	0.326	0.223	0.112	0.076	8	0.358	0.223	0.162	0.085	0.061
17	0.553	0.302	0.221	0.106	0.072	7	0.359	0.202	0.157	0.086	0.063
16	0.520	0.268	0.213	0.100	0.074	6	0.356	0.186	0.156	0.085	0.063
15	0.501	0.268	0.201	0.105	0.072	5	0.332	0.185	0.152	0.082	0.062
14	0.485	0.263	0.192	0.103	0.068	4	0.312	0.176	0.130	0.079	0.052
13	0.462	0.260	0.190	0.101	0.071	3	0.302	0.172	0.136	0.078	0.056
12	0.452	0.265	0.182	0.096	0.068	2	0.286	0.165	0.113	0.072	0.053
11	0.416	0.240	0.167	0.094	0.065	1	0.265	0.168	0.112	0.075	0.052
10	0.412	0.230	0.169	0.093	0.065	ground	0.256	0.169	0.116	0.076	0.051
9	0.396	0.225	0.168	0.092	0.063

.

4.3. Analysis of Numerical Test Results

In accordance with the test data presented in Section 4.2, we constructed a line graph of PPV versus HDEC, as illustrated in Figure 6a. As HDEC increases, the ground PPV tends to decrease. The radial PPV is greater than the vertical PPV, which is greater than the tangential PPV. Using the same calculation method as in Section 3.4.1, we fit the radial, vertical, and tangential PPV propagation laws, respectively, as follows.

$$V_r=5.66{(\sqrt[3]{30}/R)}^{1.02}$$

(20)

$$V_v=4.36{(\sqrt[3]{30}/R)}^{0.97}$$

(21)

$$V_t=3.29{(\sqrt[3]{30}/R)}^{0.93}$$

(22)

In accordance with the test data presented in Section 4.2, a line graph was constructed to illustrate the variation in PPV with respect to the floor number, from the ground to the 18th floor (Figure 6b–d). As the floor height increases, the radial and tangential PPVs show a decreasing and then increasing trend. The vertical PPV tends to increase, with a clear elevation amplification effect. The amplification effect is more pronounced for high-rise buildings with smaller HDEC. The PPV of the top floor is 2.2 times that of the bottom floor in the No.1 building, with a burst center distance of 50 m. Overall, the vertical PPV is greater than the radial and tangential PPVs.

4.4. Comparative Analysis of Numerical and Field Tests

4.4.1. Comparison of Ground PPV

The data obtained from the ground PPV in the field test (Section 3.3.1) and the ground PPV in the numerical test (Section 4.2) were subjected to a comparative analysis, the results of which are presented in

Table 7. Comparison of ground PPV.

HDEC (m)	Field Test				Numerical Experiment				Relative Error
	V2 (cm/s)				V1 (cm/s)				$\left|{\mathit{V}}_2/{\mathit{V}}_1-1\right|\times 100\%$
	Radial	Vertical	Tangential	Combined	Radial	Vertical	Tangential	Combined	Radial	Vertical	Tangential	Combined
50	0.274	0.247	0.203	0.421	0.292	0.256	0.218	0.445	6.2	3.5	6.9	5.5
100	0.185	0.165	0.148	0.289	0.189	0.169	0.156	0.298	2.1	2.4	5.1	3.0
150	0.121	0.116	0.104	0.197	0.123	0.116	0.102	0.197	1.6	0.0	2.0	0.1
200	0.082	0.074	0.071	0.131	0.085	0.076	0.069	0.133	3.5	2.6	2.9	1.5
250	0.048	0.050	0.045	0.083	0.053	0.051	0.048	0.088	9.4	2.0	6.3	5.9

Note: The experimental data for the field test above were averaged from the data in Table 2.

. As shown in the table, the relative error of PPV is within 10%.

4.4.2. Comparison of Building PPV

A comparison of the data obtained from the building PPV in the field test (Section 3.3.2) and the building PPV in the numerical test (Section 4.2) is presented in

Table 8. Comparison of PPV of high-rise buildings.

Building	Floor	Field Test				Numerical Experiment				Relative Error
		V2 (cm/s)				V1 (cm/s)				$\left|{\mathit{V}}_2/{\mathit{V}}_1-1\right|\times 100\%$
		Radial	Vertical	Tangential	Combined	Radial	Vertical	Tangential	Combined	Radial	Vertical	Tangential	Combined
NO.1	18	0.221	0.609	0.188	0.675	0.248	0.562	0.196	0.645	10.8	8.4	4.1	4.6
	13	0.205	0.498	0.186	0.570	0.235	0.462	0.182	0.549	12.6	7.9	2.2	3.8
	9	0.172	0.454	0.181	0.518	0.206	0.396	0.168	0.477	16.5	14.6	7.9	8.6
	5	0.173	0.364	0.165	0.435	0.203	0.332	0.175	0.427	14.8	9.6	5.7	2.1
	1	0.233	0.229	0.187	0.376	0.263	0.265	0.201	0.424	11.5	13.7	6.8	11.3
NO.2	18	0.209	0.344	0.166	0.435	0.186	0.326	0.163	0.409	12.5	5.4	1.8	6.4
	13	0.190	0.265	0.157	0.362	0.185	0.260	0.152	0.353	2.7	1.9	3.3	2.4
	9	0.177	0.230	0.147	0.325	0.172	0.225	0.142	0.317	2.9	2.2	3.5	2.7
	5	0.157	0.190	0.133	0.280	0.152	0.185	0.128	0.272	3.3	2.7	3.9	3.2
	1	0.173	0.183	0.131	0.283	0.168	0.168	0.148	0.280	3.0	8.6	11.7	1.3
NO.3	18	0.149	0.242	0.123	0.310	0.136	0.223	0.118	0.287	9.8	8.7	4.5	8.2
	13	0.137	0.195	0.114	0.264	0.132	0.190	0.109	0.256	3.5	2.5	4.9	3.2
	9	0.129	0.173	0.111	0.242	0.124	0.168	0.106	0.234	3.8	2.8	5.0	3.5
	5	0.121	0.157	0.085	0.215	0.116	0.152	0.085	0.209	4.0	3.1	0.4	2.9
	1	0.126	0.109	0.086	0.187	0.119	0.112	0.102	0.193	5.6	2.7	15.4	2.7
…	…	…	…	…	…	…	…	…	…	…	…	…	…

Note: The experimental data for the field test above were averaged from the data in Table 2.

. As can be seen from the table, the relative error is, in general, within 15%.

In summary, the numerical simulation test results are in good agreement with the field test results.

4.5. Blasting Vibration Tests Considering Different Slope Height Differences

4.5.1. Numerical Model

The aforementioned tests were implemented with VDEC = 6 m, and VDEC was defined as the height of the ground level of the building minus the height of the center of the explosive. Considering the effect of slope height difference, in order to get the vibration velocity response of blasting under different VDECs, three finite element models of “Rock–High-rise Building” were established. The VDECs were −24 m, 6 m, and 36 m, respectively, while the HDEC was 50 m. The remaining parameters were consistent with those of Section 4.1, as illustrated in Figure 7a–c. Figure 7d illustrates the tangential vibration velocity cloud at a moment for a Rock–High-rise Building with a 36 m burst center pendant distance.

4.5.2. Numerical Test Results and Analysis

According to the test results, in the VDEC = 36 m model, the vertical vibration velocity waveform of the 18th floor of the high-rise building is shown in Figure 8a. The figure uses the cm–g–us unit system. Draw line graphs Figure 8b–d showing the variation of peak vibration velocity with floor level based on numerical experiment results.

As the floor elevation is increased, the radial and tangential PPVs demonstrate a pattern of initial decline, followed by an upward trajectory. The PPV is observed to be relatively high in the lower and upper floors of the building, in comparison to the middle floors, where it is found to be relatively low. The vertical PPV exerts an amplification effect in elevation, with the greatest amplification observed in high-rise building of VDEC = 36 m. The vertical PPV exceeds the radial and tangential PPV. For radial and tangential velocity, the PPV for VDEC = 36 m is approximately equal to the PPV for VDEC = 6 m, and greater than the PPV for VDEC = −24 m. In regard to vertical velocity, PPV for VDEC of 36 m > PPV for VDEC of 6 m > PPV for VDEC of −24 m.

4.6. Relationship between PPV and PPS

As stated in the introduction, when only elastic deformation is considered, PPV is proportional to PPS. Whether PPV and PPS of high-rise building still satisfy this condition under blasting loads is discussed as below. PPS of the 1st to 18th floors of high-rise building No.1 in Section 4.1 are shown in

Table 9. PPS of building No.1.

Value of PPS (MPa)
Floor	1	2	3	4	5	6	7	8	9
Value	21.0	17.8	11.7	10.2	14.0	12.3	15.8	20.3	13.7
Floor	10	11	12	13	14	15	16	17	18
Value	15.3	15.6	14.0	12.5	16.4	9.7	10.6	12.0	7.2

.

The stress waveform of the 2nd floor is shown in Figure 9a, where the cm–g–us unit system was used [36]. A line graph plotting the variation in PPS along the floor height is shown in Figure 9b. The combined PPVs of field measurements and numerical tests for the No.1 high-rise building, as detailed in Table 8 of Section 4.4, are plotted as line graphs along the floor height in Figure 9c.

PPS exhibits a fluctuating pattern along the height of the floor, displaying an overall decreasing trend. The first floor has the highest PPS of 21 MPa, which exceeds the standard value of 20.1 MPa for axial compressive strength of C30 concrete. The combined PPV tends to increase along the layer height. PPV and PPS do not have a linearly proportional relationship.

5. Discussion

In the context of blasting vibration tests for high-rise buildings, it is clear that the radial and tangential PPVs at the low and high levels are relatively high, while those at the middle level are comparatively low. This may stem from the high load excitation on the lower floors of the building and the low constraints on the upper floors. It can be observed that there is no elevation amplification effect for radial and tangential PPVs. However, an elevation amplification effect is present for vertical PPV. This may be explained by the resonance theory of structure.

The 18-story high-rise building was modeled and a modal analysis was performed as shown in Figure 10. As illustrated in the figure, the 1–10 orders of self-oscillation frequencies for the high-rise building span a range of 0.78 to 6.59 Hz [37]. The first step mode is mainly generated by radial motion with a frequency of 0.78 Hz. The second step mode is mainly generated by tangential motion with a frequency of 0.84 Hz. In the fourth to tenth step mode, the weight of the modes generated by the vertical motion increases gradually, with a frequency range of 2.47–6.59 Hz. The radial and tangential natural frequencies of the building are low, far from the blasting load frequency of 10 Hz. This results in the absence of resonance, as well as the lack of an elevation amplification effect. The vertical self-oscillation frequency of the structure is high, and there is an elevation amplification effect when the frequency of the blasting load is in proximity to that of the structure.

6. Conclusions

In this paper, a mine blasting project is used as an example, and a variety of analytical methods are employed, including field tests, numerical tests, and other analytical techniques. These are used to study the vibration response of high-rise buildings within a range of 50–250 m horizontal distance from the center of the blasting loads. The analysis is conducted using software such as ANSYS and Origin. The results of this analysis lead to the following conclusions.

(1)

In the context of blasting operations, the ground PPV is observed to decline as HDEC increases. This phenomenon is evidenced by the observation that the tangential PPV is less than that of the vertical PPV, which in turn is less than that of the radial PPV. The vertical PPV in high-rise buildings is observed to increase with floor elevation, exhibiting an elevation amplification effect. The magnification factor for an 18-story building with HDEC = 50 m can reach a value of 2.66. The tangential and radial PPVs of the building demonstrate a proclivity for decline and subsequent augmentation with the ascent of the floor height. The vertical vibration velocity of the building is greater than the radial and tangential vibration velocities.

(2)

VDEC exerts an influence on PPV. The vertical PPV of the building increases with increasing VDEC. Both the radial and tangential PPVs of the building increase with increasing VDEC, if VDEC has a negative value.

(3)

There is no significant pattern in the ground frequency and building frequency, which span a range of 8 Hz to 20 Hz, with a mean value of 10 Hz.

(4)

PPS is not proportional to PPV. As the height of a structure increases, PPS tends to decrease and fluctuate, while PPV tends to increase. The highest PPS is observed at the first floor level. It is therefore recommended that buildings be designed with an increased ground floor column section. Meanwhile, it is recommended that PPS of buildings should be included in the blasting vibration safety allowance.