# Random Dynamic Response Characteristics of Pipeline Subjected to Blasting Cylindrical SH Waves

Authors: Shiwei Lu; Heng Zhang; Ling Ji
Journal: Shock and Vibration
 (2021)

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## Abstract

It is essential to investigate the influence of blasting vibrations on pipelines, and the dynamic response is the crux in the safety issues. At present, the blasting seismic wave is usually regarded as a plane wave. However, there is little research about the dynamic response characteristics of underground structures subjected to nonplane waves. The analytical solution to dynamic stress concentration factor (DSCF) of pipelines subjected to cylindrical SH wave was derived. Besides, the randomness of the shear modulus of soil was considered, and the statistical analysis of DSCF was carried out by the Monte Carlo simulation method. Results show that the variability of the shear modulus of soil has a significant influence on the probability distribution of DSCF. The larger the variation coefficient of the shear modulus is, the more obvious the skewness of DSCF is. The influence of low-frequency wave on pipeline increases with the reducing normalized distancer∗, while the influence of high-frequency wave reduces and the variation amplitude of DSCF increases. Compared with the DSCF of pipe subjected to a plane wave, a lower dominant frequency or larger normalized distance for the cylindrical SH wave will generate a more similar statistical characteristic of DSCF.


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## Body

## 1. Introduction

With the rapid development of transportation construction, the drilling and blasting method has been widely used in the excavation of underground space due to its efficiency and economy. However, blasting seismic waves induced by the structures excavation will undermine the safety of nearby pipelines which are important channels for energy transmission and even cause serious economic losses and successive disasters.A large number of research studies have been implemented to study the dynamic response characteristics of underground structures subjected to blasting seismic waves. Pao and Mow [1] investigated the dynamic stress concentration factors (DSCFs) of a cavity in an unbounded elastic space under the incident plane P, SH, and SV waves. Lee and Trifunac [2–4] studied the dynamic response of tunnels and cavities induced by plane P, SH, and SV waves. Thambirajah et al. [5] considered the effect of DSCF and studied the scattering of SH waves by two caverns. Liu and Wang [6] revealed the dynamic response law of cavities subjected to plane waves. Wang et al. [7] used the wave function expansion method to give an analytical solution to the dynamic response of a deep-buried soft rock circular tunnel under the incident plane SH wave. Fu et al. [8] studied the interaction between the soil and the tunnel when the blasting plane SH wave was considered. Yi et al. [9] proposed an analytical solution to the dynamic response of a circular tunnel when the imperfect contact exists between surrounding rock and lining subjected to incident plane P waves. Liu et al. [10] used the complex functions and multilevel coordinate method to analyze the influence of P wave on pipelines in saturated soil. He et al. [11] studied the scattering of plane SH waves by underground caverns with arbitrary cross-sectional shapes. Zhang et al. [12] derived the analytical solution to the dynamic responses of deep-water foundation sites when both incident plane P and SV waves were considered. Liang et al. [13] studied the diffraction of plane SH waves by a semicircular cavity in half-space by using the wave function expansion method. Xu et al. [14] deduced a series solution of dynamic stress for a circular lining tunnel subjected to incident plane P waves in an elastic half-space. Xu [15] derived the blasting safety criterion of an unlined circular tunnel subjected to plane P waves. Qi et al. [16, 17] carried out the dynamic analysis for circular inclusions near interface impacted by SH waves using the complex method and Green’s function method. Fan et al. [18] predicted the dynamic response of a circular lined tunnel with an imperfect interface to plane SV waves based on the wave function expansion method and the linear spring model. Lu et al. [19] studied the dynamic stress concentration and vibration velocity scaling factors of an unlined circular tunnel subjected to a triangular P wave. Xia et al. [20] calculated the vibration response of buried flexible HDPE pipes under impact loads based on the Winkler model and the Timoshenko beam theory. Recently, Jiang et al. [21–23] studied the dynamic failure behavior of buried cast iron gas pipeline subjected to blasting vibration. In order to simplify the analysis, most of these studies approximately regard blasting seismic waves as plane waves. However, the assumption is not reliable when the explosion source is close to the underground structure because the curvature of incident blasting waves cannot be ignored. Although some research studies about cylindrical waves have been carried out, the random dynamic response of underground structures under cylindrical waves is still rare in literature [24–26].At present, He and Liang [27] used the Monte Carlo simulation method to study the influence of the shape variability of the outer wall on the peak value of DSCF around the inner wall. However, due to the effects of multiple factors, such as sedimentation, weathering, chemical action, transportation processes, and different loading history, the physical and mechanical properties of rock and soil vary spatially. Therefore, it is more suitable to study the DSCF of pipelines under cylindrical SH waves when the shear modulus of soil is considered as random parameter.


## 2. Random Dynamic Responses of Pipeline Subjected to Cylindrical SH Waves

### 2.1. Simplified Model

A linear wave source is assumed to locate atO1, and the axis of a circular pipe whose inner and outer radius are a and b, respectively, coincides with O2. The distance between O1 and O2 is r0, and the coordinate systems O1x1y1 and O2x2y2 are established on O1 and O2, respectively, as shown in Figure 1. P is an arbitrary point in the soil, and the distances away from O1 and O2 are r1 and r2, respectively. The displacement function of the cylindrical SH wave generated at O1 can be expressed in the following form [28, 29]:(1)Wi=H01β1r1exp−iωt,where H01 is the 0th-order Hankel function, i is the imaginary unit, β1 is the wave number of SH wave in the soil, and β1 = ω/cs1, in which ω is the frequency of the incident wave and cs1=G/ρ is the speed of SH wave in the soil. ρ is the density of the soil. The shear modulus G of the soil is considered as a random parameter, which can be described as(2)Gξ=G¯1+αξ,where G¯ is the mean value of the shear modulus of soil and ξ is a random variable with standard normal distribution N(0,1). As a result, the shear modulus of the soil G and the SH wave-number β1 will be a function of the random variable ξ. Then, the coefficient of variation (Cov) of the shear modulus δG of the soil can be calculated as(3)δG=DGξEGξ=α.Figure 1
Incident cylindrical SH wave.Generally, when a cylindrical SH wave reaches the pipe, it will generate a reflected SH wave (Wr) propagating outward in the soil and a refracted SH wave (Wf) propagating outward and inward in the pipe. Their displacement can be expressed as follows:(4)Wr=∑n=0∞Hn1β1ξr2Bncosnθ2+Ensinnθ2exp−iωt,Wf=∑n=0∞Hn1β2r2Cncosnθ2+Fnsinnθ2exp−iωt+∑n=0∞Hn2β2r2Dncosnθ2+Gnsinnθ2exp−iωt,where β2 = ω/cs2 is the wave number of SH waves in the pipe and cs2=G2/ρ2 is the speed of SH wave in the pipe, in which G2 is the shear modulus of the pipe and ρ2 is the density of the pipe.In order to obtain the displacement function of the incident wave, the reflected wave, and the refracted wave, the incident wave potential function in theO1x1y1 coordinate system must be converted to the expression form in the O2x2y2 coordinate system. The conversion formula is expressed as follows [30, 31]:(5)Hn1β1ξr1cosnθ1=∑m=−∞∞−1mHm1β1ξr0Jm+nβ1ξr2cosm+nθ2,where Hn1 is the nth-order Hankel function and Jn is the nth-order Bessel function.Substituting equation (5) into equation (1), we can express the displacement function of the cylindrical SH wave as follows:(6)Wi=∑n=0∞A0nJnβ1ξr2cosnθ2exp−iωt,where A0n=−1n∈nHn1β1ξr0; if n = 0, ∈0=1; when n > 0, ∈0=2.


### 2.2. Boundary Conditions

LetW1=Wi+Wr and W2=Wf; considering the interface between the pipe and the soil as an ideal contact interface, the boundary conditions can be expressed as follows:(7)τrz1=τrz2,W1=W2,ifr2=b,τrz2=0,ifr2=a.The relationship between the displacement and the stress of a cylindrical SH wave can be expressed as follows:(8)τrz=μ∂W∂r,τθz=μr⋅∂W∂θ.According to equations (5) and (8), the value of Bn, Cn, Dn, En, Fn, and Gn can be obtained.


### 2.3. Solution of Random DSCF Based on the Monte Carlo Simulation Method

In order to obtain general results, the dimensionless parameters of DSCF and normalized distancer∗ are defined as follows:(9)DSCF=maxτθz2τrz1i,(10)r∗=r0b,where τrz1i=−μ1β1ξH11β1ξr1.When the randomness of structural parameters is considered, the dynamic response of the structure will also be random. In order to obtain the random response of the structure, a series of various strategies such as the Monte Carlo simulation method [32], stochastic finite element method [33], and the probability density evolution method [34] are proposed. The Monte Carlo simulation method has a high calculation accuracy in solving complex functions. Besides, the calculation accuracy is not affected by the variability of random parameters. Therefore, it is widely used in theoretical research studies. Owing to the given explicit formula of DSCF, the Monte Carlo simulation method is adopted to obtain the statistical results of DSCF. The main steps are as follows:Step 1. Use equation (2) to describe the randomness of the shear modulus of the soil and generate a sufficient number of samples ξ which obeys the standard normal distributionStep 2. According to the derivation process of DSCF, substitute each sample into the expression of the shear modulus of soil and obtain the corresponding result of DSCFStep 3. Perform statistical analysis on the samples of DSCF obtained in Step 2


## 2.1. Simplified Model

A linear wave source is assumed to locate atO1, and the axis of a circular pipe whose inner and outer radius are a and b, respectively, coincides with O2. The distance between O1 and O2 is r0, and the coordinate systems O1x1y1 and O2x2y2 are established on O1 and O2, respectively, as shown in Figure 1. P is an arbitrary point in the soil, and the distances away from O1 and O2 are r1 and r2, respectively. The displacement function of the cylindrical SH wave generated at O1 can be expressed in the following form [28, 29]:(1)Wi=H01β1r1exp−iωt,where H01 is the 0th-order Hankel function, i is the imaginary unit, β1 is the wave number of SH wave in the soil, and β1 = ω/cs1, in which ω is the frequency of the incident wave and cs1=G/ρ is the speed of SH wave in the soil. ρ is the density of the soil. The shear modulus G of the soil is considered as a random parameter, which can be described as(2)Gξ=G¯1+αξ,where G¯ is the mean value of the shear modulus of soil and ξ is a random variable with standard normal distribution N(0,1). As a result, the shear modulus of the soil G and the SH wave-number β1 will be a function of the random variable ξ. Then, the coefficient of variation (Cov) of the shear modulus δG of the soil can be calculated as(3)δG=DGξEGξ=α.Figure 1
Incident cylindrical SH wave.Generally, when a cylindrical SH wave reaches the pipe, it will generate a reflected SH wave (Wr) propagating outward in the soil and a refracted SH wave (Wf) propagating outward and inward in the pipe. Their displacement can be expressed as follows:(4)Wr=∑n=0∞Hn1β1ξr2Bncosnθ2+Ensinnθ2exp−iωt,Wf=∑n=0∞Hn1β2r2Cncosnθ2+Fnsinnθ2exp−iωt+∑n=0∞Hn2β2r2Dncosnθ2+Gnsinnθ2exp−iωt,where β2 = ω/cs2 is the wave number of SH waves in the pipe and cs2=G2/ρ2 is the speed of SH wave in the pipe, in which G2 is the shear modulus of the pipe and ρ2 is the density of the pipe.In order to obtain the displacement function of the incident wave, the reflected wave, and the refracted wave, the incident wave potential function in theO1x1y1 coordinate system must be converted to the expression form in the O2x2y2 coordinate system. The conversion formula is expressed as follows [30, 31]:(5)Hn1β1ξr1cosnθ1=∑m=−∞∞−1mHm1β1ξr0Jm+nβ1ξr2cosm+nθ2,where Hn1 is the nth-order Hankel function and Jn is the nth-order Bessel function.Substituting equation (5) into equation (1), we can express the displacement function of the cylindrical SH wave as follows:(6)Wi=∑n=0∞A0nJnβ1ξr2cosnθ2exp−iωt,where A0n=−1n∈nHn1β1ξr0; if n = 0, ∈0=1; when n > 0, ∈0=2.


## 2.2. Boundary Conditions

LetW1=Wi+Wr and W2=Wf; considering the interface between the pipe and the soil as an ideal contact interface, the boundary conditions can be expressed as follows:(7)τrz1=τrz2,W1=W2,ifr2=b,τrz2=0,ifr2=a.The relationship between the displacement and the stress of a cylindrical SH wave can be expressed as follows:(8)τrz=μ∂W∂r,τθz=μr⋅∂W∂θ.According to equations (5) and (8), the value of Bn, Cn, Dn, En, Fn, and Gn can be obtained.


## 2.3. Solution of Random DSCF Based on the Monte Carlo Simulation Method

In order to obtain general results, the dimensionless parameters of DSCF and normalized distancer∗ are defined as follows:(9)DSCF=maxτθz2τrz1i,(10)r∗=r0b,where τrz1i=−μ1β1ξH11β1ξr1.When the randomness of structural parameters is considered, the dynamic response of the structure will also be random. In order to obtain the random response of the structure, a series of various strategies such as the Monte Carlo simulation method [32], stochastic finite element method [33], and the probability density evolution method [34] are proposed. The Monte Carlo simulation method has a high calculation accuracy in solving complex functions. Besides, the calculation accuracy is not affected by the variability of random parameters. Therefore, it is widely used in theoretical research studies. Owing to the given explicit formula of DSCF, the Monte Carlo simulation method is adopted to obtain the statistical results of DSCF. The main steps are as follows:Step 1. Use equation (2) to describe the randomness of the shear modulus of the soil and generate a sufficient number of samples ξ which obeys the standard normal distributionStep 2. According to the derivation process of DSCF, substitute each sample into the expression of the shear modulus of soil and obtain the corresponding result of DSCFStep 3. Perform statistical analysis on the samples of DSCF obtained in Step 2


## 3. Engineering Background

The underground passage of Baotong Temple passes through the concrete sewage pipe at a short distance and is excavated by drilling and blasting. The top of the passage is only 0.69 m away from the sewage pipe. The density and shear modulus of the pipeline are 2400 kg/m3 and 12.61 GPa, respectively. The inner radius a and the outer radius b are 400 mm and 465 mm, respectively. The density of the surrounding soil is 1980 kg/m3, and the average shear modulus of the soil G¯ is 150 MPa. According to the construction information, the dominant frequency of blasting seismic waves is below 200 Hz. Consequently, the dominant frequency f and r0 are considered as 10∼200 Hz and 2b∼10b, respectively. δG is selected as 0.1 and 0.2, respectively, in the following analysis.


## 4. Results and Discussion

### 4.1. Convergence Analysis

The Monte Carlo simulation method is employed to obtain the statistical results of the DSCF. First, a certain number of samples which obey the standard normal distribution are generated, then the random shear modulus is expressed using equation (2), and finally the sample values of the DSCF are calculated through equations (4)–(9).Before the implementation of the Monte Carlo simulation progress, it is necessary to study the influence of the number of the samples on the convergence of the solution. Figure2 exhibits the first four statistic moments (mean, standard deviation, skewness, and kurtosis) of DSCF generated by the Monte Carlo simulation method with the number of the samples increasing from 1 × 103 to 5 × 105 when the four dominant frequencies are 10 Hz, 50 Hz, 100 Hz, and 200 Hz, respectively. It is found that in the aspects of mean and standard deviation, only a small number of samples is needed to obtain stable results regardless of f. However, in the aspects of skewness and kurtosis, a large number of samples are required. It indicates that a sufficient large number of samples are necessary to obtain stable high-order statistical moments of DSCF to ensure the convergence. For example, the calculation results of the first four statistic moments tend to be stable when more than 100,000 samples are selected, and thus 100,000 samples should be used in the subsequent analysis.Figure 2
First four statistic moments of DSCF obtained with different numbers of samples: (a) mean; (b) standard deviation; (c) skewness; (d) kurtosis.
(a)(b)(c)(d)Figure3 shows the mean and standard deviation of the DSCF when different values of r∗ are considered. It is observed from Figure 3 that when δG is equal to 0.1 and 0.2, the mean of the DSCF decreases significantly when the dominant frequency increases. When the dominant frequency is equal to 50 Hz, 100 Hz, and 200 Hz, the standard deviation of DSCF decreases with the increasing r∗, especially when 2 ≤ r∗ ≤ 5. However, the standard deviation of DSCF is almost unchanged when f = 10 Hz.Figure 3
Mean and standard deviation of DSCF for different values ofr∗: (a) mean (δG = 0.1); (b) standard deviation (δG = 0.1); (c) mean (δG = 0.2); (d) standard deviation (δG = 0.2).
(a)(b)(c)(d)


### 4.2. Analysis of the Pipeline Dynamic Response Characteristics

Figures4 and 5 show the probability density function (PDF) curve and cumulative distribution function (CDF) curve of DSCF when δG = 0.1 and 0.2, respectively. DSCF of the pipeline under the plane wave is also provided for comparison.Figure 4
PDF curves (left) and CDF curves (right) of DSCF whenδG = 0.1: (a) PDF f = 10 Hz; (b) CDF f = 10 Hz; (c) PDF f = 50 Hz; (d) CDF f = 50 Hz; (e) PDF f = 100 Hz; (f) CDF f = 100 Hz; (g) PDF f = 200 Hz; (h) CDF f = 200 Hz.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 5
PDF curves (left) and CDF curves (right) of DSCF whenδG = 0.2: (a) PDF f = 10 Hz; (b) CDF f = 10 Hz; (c) PDF f = 50 Hz; (d) CDF f = 50 Hz; (e) PDF f = 100 Hz; (f) CDF f = 100 Hz; (g) PDF f = 200 Hz; (h) CDF f = 200 Hz.
(a)(b)(c)(d)(e)(f)(g)(h)It can be seen from Figure4 that when δG is small (i.e., δG = 0.1), the PDF of DSCF approximately obeys the Gaussian distribution when f = 10 Hz. With the increase in r∗, the PDF curve integrally shifts to the left, which suggests that the mean of the DSCF reduces. However, the mean of DSCF increases with the increase in r∗ when f is equal to 50 Hz, 100 Hz, and 200 Hz. The PDF curve when f = 50 Hz is obviously different from that of other cases. When the dominant frequency is relatively small (10 Hz or 50 Hz), the PDF and CDF of DSCF generated by the cylindrical SH wave are very similar to the results of the plane wave. As the dominant frequency increases, the distinctions between the cylindrical SH wave and plane wave will appear; however, the PDF and CDF of DSCF are getting close to the results of the plane wave as r∗ increases.It is found from Figure5 that the variation of the mean of DSCF with r∗ is similar when δG is equal to 0.1 and 0.2, respectively. However, all the PDF curves of DSCF become leftward, and they do not obey the normal distribution any more, especially in the case of f = 50 Hz. A comparison between Figures 4 and 5 shows that the larger δG is, the more obvious the skewness of PDF is, which indicates that the relationship of DSCF and shear modulus of soil is significantly nonlinear. Compared with the DSCF under the plane wave, it has a same trend as δG = 0.1; a lower dominant frequency or larger normalized distance under the cylindrical SH wave will obtain a closer result.


## 4.1. Convergence Analysis

The Monte Carlo simulation method is employed to obtain the statistical results of the DSCF. First, a certain number of samples which obey the standard normal distribution are generated, then the random shear modulus is expressed using equation (2), and finally the sample values of the DSCF are calculated through equations (4)–(9).Before the implementation of the Monte Carlo simulation progress, it is necessary to study the influence of the number of the samples on the convergence of the solution. Figure2 exhibits the first four statistic moments (mean, standard deviation, skewness, and kurtosis) of DSCF generated by the Monte Carlo simulation method with the number of the samples increasing from 1 × 103 to 5 × 105 when the four dominant frequencies are 10 Hz, 50 Hz, 100 Hz, and 200 Hz, respectively. It is found that in the aspects of mean and standard deviation, only a small number of samples is needed to obtain stable results regardless of f. However, in the aspects of skewness and kurtosis, a large number of samples are required. It indicates that a sufficient large number of samples are necessary to obtain stable high-order statistical moments of DSCF to ensure the convergence. For example, the calculation results of the first four statistic moments tend to be stable when more than 100,000 samples are selected, and thus 100,000 samples should be used in the subsequent analysis.Figure 2
First four statistic moments of DSCF obtained with different numbers of samples: (a) mean; (b) standard deviation; (c) skewness; (d) kurtosis.
(a)(b)(c)(d)Figure3 shows the mean and standard deviation of the DSCF when different values of r∗ are considered. It is observed from Figure 3 that when δG is equal to 0.1 and 0.2, the mean of the DSCF decreases significantly when the dominant frequency increases. When the dominant frequency is equal to 50 Hz, 100 Hz, and 200 Hz, the standard deviation of DSCF decreases with the increasing r∗, especially when 2 ≤ r∗ ≤ 5. However, the standard deviation of DSCF is almost unchanged when f = 10 Hz.Figure 3
Mean and standard deviation of DSCF for different values ofr∗: (a) mean (δG = 0.1); (b) standard deviation (δG = 0.1); (c) mean (δG = 0.2); (d) standard deviation (δG = 0.2).
(a)(b)(c)(d)


## 4.2. Analysis of the Pipeline Dynamic Response Characteristics

Figures4 and 5 show the probability density function (PDF) curve and cumulative distribution function (CDF) curve of DSCF when δG = 0.1 and 0.2, respectively. DSCF of the pipeline under the plane wave is also provided for comparison.Figure 4
PDF curves (left) and CDF curves (right) of DSCF whenδG = 0.1: (a) PDF f = 10 Hz; (b) CDF f = 10 Hz; (c) PDF f = 50 Hz; (d) CDF f = 50 Hz; (e) PDF f = 100 Hz; (f) CDF f = 100 Hz; (g) PDF f = 200 Hz; (h) CDF f = 200 Hz.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 5
PDF curves (left) and CDF curves (right) of DSCF whenδG = 0.2: (a) PDF f = 10 Hz; (b) CDF f = 10 Hz; (c) PDF f = 50 Hz; (d) CDF f = 50 Hz; (e) PDF f = 100 Hz; (f) CDF f = 100 Hz; (g) PDF f = 200 Hz; (h) CDF f = 200 Hz.
(a)(b)(c)(d)(e)(f)(g)(h)It can be seen from Figure4 that when δG is small (i.e., δG = 0.1), the PDF of DSCF approximately obeys the Gaussian distribution when f = 10 Hz. With the increase in r∗, the PDF curve integrally shifts to the left, which suggests that the mean of the DSCF reduces. However, the mean of DSCF increases with the increase in r∗ when f is equal to 50 Hz, 100 Hz, and 200 Hz. The PDF curve when f = 50 Hz is obviously different from that of other cases. When the dominant frequency is relatively small (10 Hz or 50 Hz), the PDF and CDF of DSCF generated by the cylindrical SH wave are very similar to the results of the plane wave. As the dominant frequency increases, the distinctions between the cylindrical SH wave and plane wave will appear; however, the PDF and CDF of DSCF are getting close to the results of the plane wave as r∗ increases.It is found from Figure5 that the variation of the mean of DSCF with r∗ is similar when δG is equal to 0.1 and 0.2, respectively. However, all the PDF curves of DSCF become leftward, and they do not obey the normal distribution any more, especially in the case of f = 50 Hz. A comparison between Figures 4 and 5 shows that the larger δG is, the more obvious the skewness of PDF is, which indicates that the relationship of DSCF and shear modulus of soil is significantly nonlinear. Compared with the DSCF under the plane wave, it has a same trend as δG = 0.1; a lower dominant frequency or larger normalized distance under the cylindrical SH wave will obtain a closer result.


## 5. Conclusions

(1)
The shear modulus of soil is defined as a random parameter, and the analytical expression of the maximum DSCF of pipe subjected to an incident cylindrical SH wave is established based on the Fourier–Bessel expansion and the Monte Carlo simulation method.(2)
When the dominant frequency is 10 Hz, the mean and median of DSCF decrease as the normalized distance increases. However, this is contrary to those when the frequency of incident wave is 50 Hz, 100 Hz, and 200 Hz, which indicates that when the normalized distance is small, the lower-frequency wave has a greater impact on the pipeline than the higher-frequency wave.(3)
With the increase in normalized distance, the PDF of DSCF gradually becomes narrow, indicating that the variability of DSCF decreases with the increase in normalized distance.(4)
Compared with the DSCF of pipe subjected to a plane wave, a lower dominant frequency or larger normalized distance for the cylindrical SH wave will generate much closer PDF and CDF of DSCF.



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*Source: 5592858-2021-07-19.xml*