Determination of damage properties of polyethylene pipes under high impact load

2.1. Theoretical solution of impact load and vibration velocity

Compared with the blasting vibration, the impact load generated by the blasting falling body has greater strength, longer duration and lower frequency, and it is more harmful to the protection target. Assuming that the soil is linear elastomer and the impact load of demolition blasting is considered as elastic collision, the Hertz collision theory can be used to obtain the maximum impact load $P_{max}$ under the action of the center of gravity. Combined with the theory of semi-infinite space elastomer, the Boussinesq equation is used to calculate the stress state formula of pipeline under impact load.

The impact load generated by blasting demolition body can be defined as the impact load generated by heavy weight falling from a height to touch the ground. The removed body and soil can be regarded as two elastic spheres $m_1$ and $m_2$ respectively. The impact load caused by falling weight m1 can be calculated by Hertz collision theory formula. The form is shown in Eq. (1) [14]:

where: $P_{max}$ – maximum surface impact load, MPa. $m_1$, $m_2$ – the mass of two bounces, kg; ${\upsilon }_0$ – the instantaneous velocity before the collision, m/s; $K$ – constant. It can be obtained from the following Eq. (2-4):

where: $r_1$, $r_2$ – the radius of the two bounces, m. $E_1$, $E_2$ – the modulus of elasticity of the two bounces, MPa; ${\mu }_1$, ${\mu }_2$ – the Poisson’s ratio of the two bounces.

Because the stiffness of the removed body is much greater than that of the soil, the soil is regarded as a semi-infinite elastic solid. Therefore, it can be regarded as $E_1$→ ∞, $m_1$→ ∞, $r_2$→ ∞ [15]. Therefore, Eqs. (2-4) can be expressed as:

2.2. Load analysis of pipeline

By using the semi-infinite space elastomer theory and the Boussinesq equations, the stress acting on the pipe in all directions can be calculated [16]. A schematic diagram of pipe pressure is shown in Fig. 1. The calculation formulas of tangential stress ${\sigma }_r$, axial stress ${\sigma }_{\theta }$ and $rz$ plane shear stress ${\tau }_{rz}$ under the impact load are respectively shown in Eq. (7-9):

where: $P_{max}$ – the maximum load on the soil surface, MPa; ${\sigma }_r$ – tangential stress of pipeline, MPa; ${\sigma }_{\theta }$ – axial stress of pipeline, MPa; ${\tau }_{rz}$ – shear stress of pipeline $rz$ plane, MPa; $z$ – distance from the top of the pipe to the surface, m; $r$ – horizontal distance from tube top to operation point, m. ${\mu }_2$ – Poisson’s ratio of bounce 2 (soil mass).

Fig. 1Schematic diagram of pipeline stress under impact load

Under the high-speed impact load of a falling object, the vertical force on the pipeline is not only the impact load, but also the dead weight stress map of the upper soil. By using the semi-infinite space elastomer theory and Boussinesq equation, the vertical (or annular) stress acting on the pipeline by impactor can be calculated, as shown in Fig. 1. The radial stress ${\sigma }_{z1}$ at the pipe vertex under impact load is shown in f Eq. (10):

where: ${\sigma }_{z1}$ represents radial stress of pipeline under impact load, MPa; $P_{max}$ represents the maximum load on the soil surface, MPa; $z$ represents distance from the top of the pipe to the surface, m; $r$ represents horizontal distance from tube top to operation point, m.

The position of the water line shall be taken into account in the calculation of the earth pressure acting on the pipeline. In this project, the water level line is under the pipe, and the earth pressure applied on the pipe is a constant load generated by the weight of the soil. The radial stress of the pipeline under the action of earth pressure can be calculated as follows:

where: ${\sigma }_{z2}$ represents radial stress of pipeline under the action of overburden pressure, MPa; $\gamma$ represents silty clay volumetric weight, MPa; $z$ represents distance from the top of the pipe to the surface, m.

Since the pipeline force is related to buried depth $z$, combined with Eq. (10-11), the radial force received by the pipeline is shown in Eq. (12) [17]:

where: $F^{\text{'}}$ represents perforation coefficient, which is a coefficient related to the buried depth of pipeline. It can be evaluated as shown in Table 1.

2.3. Surface vibration velocity

There is no unified formula to calculate the collapse vibration velocity caused by the collapse of a structure. At present, a researcher at the institute of mechanics, Chinese academy of sciences, is generally used for calculation [18]. According to the regression analysis of the measured vibration monitoring data of similar projects in Wuhan by Wuhan blasting company, $K_t=$ 3.37 and $\beta =$ 1.66 were calculated:

where: $V_t$ – collapse vibration velocity, cm/s; $M$ – mass of falling member, kg; $G$ – gravitational acceleration, m/s²; $H$ – height of gravity center of collapse, m; ${\sigma }_t$ – failure strength of surface medium, MPa; $R$ – the distance from the measuring point to the center of the impact, m; $K_t$, $\beta$ –attenuation parameter.

3.1. Test parameters

The blasting chimney is a brick structure. The total height of the chimney is 50 m. The outer radius of the bottom is 2.55 m, the inner radius is 1.5 m, and the wall thickness is 1.05 m. Chimney density ${\rho }_c=$ 1600 kg·m^-3, collapse mass $m_1=$7×105 kg, collapse height $H=$20 m. Next to the chimney is an HDPE sewage pipe buried in the ground. The soil layer embedded in the pipe is a typical silty clay layer of Wuhan stratum, with a depth of 2 m. The vertical distance between the chimney and the pipeline axis is 9 m. The direction of the chimney collapse is parallel to the direction of the pipeline axis. The outside diameter $D$ is 90 cm and the inside diameter $d$ is 78 cm. The total length of the pipeline is 18 m, and the length of each of the three sections is 6 m. The pipe interface is socket type, and the thickness of rubber ring at the pipe interface is 0.5 cm. The spatial position relationship between pipe and chimney is shown in Fig. 4. The dimensions of pipes and chimneys are shown in Fig. 5.

Fig. 3Demolish collapse flow

a) Before initiating

b) 0.3 sec

c) 2.1 sec

d) 4.4 sec

e) 5.2 sec

f) 5.9 sec

g) 6.6 sec

g) 8.5 sec

k) Falling objects

Fig. 4Schematic diagram of the relationship between pipe and chimney position

The temperature in the pipe is 20 ℃. The environment temperature in the pipe is 20 ℃. Pipeline physical parameters: Poisson’s ratio $\mu =$0.46; density $\rho =$936 kg·m^-3, Young’s modulus $E=$834.9 MPa, ring stiffness $SN=$8kN·m^-2, strength limit ${\sigma }_u=$31.6 MPa, elongation 116 %.

Because the chimney adopts the directional collapse scheme and the collapse distance is allowed, the blasting cut is arranged at the bottom of the chimney. The incision shape was positive trapezoid, the central angle of the incision was 216°, and the incision height was 1.5 m. The 2# rock emulsion explosive is adopted, the hole diameter is 40 mm, the depth is 70 cm, the spacing is 50 cm, the row spacing is 50 cm. The single hole charge is 300 g.

Fig. 5Chimney and pipe size and space position, a) planform, b) sectional view

a)

b)

3.2. Monitoring scheme and arrangement of measuring points

The purpose of this experiment is to obtain the change of earth pressure, the change of velocity of the pipe and the soil above the pipe, and the change law of dynamic strain of the pipe under the impact load caused by demolition blasting and collapse of the chimney. For the purposes mentioned above, the main test items include: pipeline dynamic strain ($\epsilon$), pipeline particle vibration velocity ($V_P$) and earth pressure ($P$), and particle velocity above the pipeline ($V_G$). The point space diagram of each monitoring item is shown in Fig. 6.

Fig. 6Schematic diagram of monitoring points

3.2.1. Soil pressure

In order to study the change characteristics of earth pressure caused by the collapse of chimney demolition to the ground, three earth pressure boxes were embedded under the center of gravity of the collapsed object. Three earth pressure boxes were buried at equal distances, with a spacing of 2.5 m. The middle earthen box is at the center of the chimney collapse. The line of the earth pressure cells is perpendicular to the established collapse axis, and the line is 20 m away from the chimney. The buried earth pressure box is shown in Fig. 7. The whole bridge of the earth pressure box is connected, and DH5956 dynamic signal acquisition instrument is used for data acquisition.

Fig. 7Soil pressure test point

3.2.2. Dynamic strain

In order to study the stress and strain state of pipeline under blasting vibration, a monitoring section is set in the numerical model. The monitoring section is located at the shortest distance from the chimney, 4.7 m from the pipe ends. Two measuring points are arranged along the annular direction of the pipeline, and the annular strain gauge and axial strain gauge were arranged at each measuring point. There were 4 strain gauges in total. As shown in Fig. 8. The strain gauge with length of 80 mm and resistance 120 Ω is instrumented on the pipeline and connected with a quarter of the bridge to DH5956 with a measuring accuracy of 10 microstrain and sampling rate 20000 Hz/channel.

Fig. 8Dynamic strain test point

3.2.3. Vibration velocity

In order to study the vibration velocity change of the particle, including particles on the surface and in the pipeline. The tc-4850 blasting vibration meter was selected as the vibration velocity measurement system. Due to the influence of soft soil properties on vibration propagation and monitoring, a special support device is used to fix the sensor o TC-4850, as is shown in Fig. 7. The device can connect vibration sensors to soil particles, and the sensor will not be loose and unable to collect data during the process of vibration wave propagation.

Three vibration velocity measuring points are arranged in the pipeline, which are located at the pipe end (measuring point 5) inside the pipeline, the closest position to the center of the chimney axis (measuring point 4) and the pipe joint interface (measuring point 3). Two measuring points are placed on the surface above the pipeline, namely, ground measurement point 1 is directly above measurement point 3, and ground measurement point 2 is directly above measurement point 4. All points are shown in Fig. 9. Because there is a sewage well at the position of the pipe opening, the vibration velocity measuring point is not arranged on the ground directly above the pipe opening.

Fig. 9Vibration velocity test point

4.1. Soil pressure

Combined with the principle of soil pressure test, the soil pressure measured in the test is the product of the strain of the soil pressure box and the constant coefficient $K$, the unit is MPa. Therefore, the earth pressure curve obtained by data processing is shown in Fig. 10.

Fig. 10Soil pressure test results

It can be known that the impact load at the center of gravity of chimney collapse is the largest, with a value of 590.3 MPa. There is little difference in the peak value of impact load on both sides, and its value is smaller than the center of the pipeline. All monitoring points are subjected to compressive stress. According to curve C, the acting time of impact load is about 2 seconds and reaches its peak value when it is close to 2.5 seconds. However, curve AB shows that at the chimney edge, the impact load is a three-stage change law, which first reaches the peak value and then gradually decreases and finally tends to equilibrium.

4.2. Dynamic strain

Due to the low frequency and long duration of impact load, the collected data need to be de-noised. The dynamic strain of each measurement point of the pipeline after denoising is shown in Fig. 11. According to the results shown in the Fig. 11, the point dynamic strain at A above the pipeline is smaller than that at B on the explosion-facing side of the pipeline, and the annular strain at both points is larger than the axial strain. Annular is compressive strain, axial is tensile strain. The maximum microstrain of the circumferential strain at point A is 1403, and the maximum microstrain of the circumferential strain at point B is 1065. Due to the propagation characteristics of caving impact load, its propagation law under the surface is similar to blasting seismic wave, both of which are dominated by P wave incidence. Combined with the practical engineering situation, the vibration shock wave first propagates to the explosion-facing side of the pipeline, and the effect on the top of the pipeline is relatively weak.

Fig. 11Dynamic strain test results, a) measuring point A, b) measuring point B

4.3. Vibration velocity

As a reflection of the energy when the blasting seismic wave passes through the medium, the vibration velocity is the most intuitive measurement data in vibration monitoring. According to the experimental protocol mentioned above, the monitoring data of collapse vibration velocity of demolition blasting is obtained. The vibration velocity and dominate frequency of each direction of the 5 measuring points are listed as shown in Table 2. According to the statistical results in the table, it can be seen that: 1) The internal vibration velocity of the pipeline is greater than the surface vibration velocity; 2) Main frequencies in all directions are concentrated between 5 Hz-20 Hz; 3) The vibration velocity and dominate frequency of the particle in the $Z$ direction in the pipeline are greater than $X$ and $Y$ direction, while the vibration velocity of the particle in the $Z$ direction at the surface is smaller than $X$ and $Y$ and the frequency is greater than $X$ and $Y$. This is because the impact vibration wave operating point is located at the surface, so the surface is easier to be affected by the surface wave, so the horizontal direction of the vibration speed is greater than the vertical direction. However, the shock vibration wave propagates the body wave in the soil, so the particle vibration velocity has more influence in the vertical direction.

The vibration velocity waveforms of surface measurement point 1 and pipeline measurement point 4 are shown in Fig. 12(a) and 12(b). The vibration shock wave generally consists of three stages in the propagation process, namely the initial phase, the main phase and the aftershock phase. Combined with Fig. 12(a), it can be seen that the collected data of vibration shock wave conforms to the theory.

Fig. 12Vibration velocity results, a) measuring point 1, b) measuring point 4

4.4. Validation for theoretical analysis by example and numerical simulation

The data of the project case is put into the theoretical calculation formula mentioned above. This is equivalent to a sphere with a radius ($R_2$) of 4.71m, depending on the mass and density of the chimney. The initial velocity of the ball ($v_0$) at the moment of free fall is 98 m/s. The elastic modulus ($E_2$) of silty clay is 5 MPa.

According to Eq. (6), the maximum impact load generated by this project is 684.3 MPa. Combined with the pipeline stress state Eq. (7-12), the different stresses on the pipeline can be obtained, as shown in Table 3. According to the surface vibration velocity prediction formula, the combined vibration velocity at the actual surface monitoring point 1 and 2 can be calculated as 0.451 cm/s and 0.178 cm/s, respectively.

The process of numerical simulation is as follows. Firstly, the numerical model of pipeline collapse is adopted to obtain the pressure of pipeline collapse to the ground. Secondly, the surface pressure is obtained according to the location of pipeline collapse, and the pressure is applied to the pipeline model for calculation. The specific numerical model parameters are shown as follows.

The pipe is high-density polyethylene, and it is a viscous-elastic material. This material can be modeled by *MAT_ PLASTICITY_POLYMER. This is an elastic-plastic material model that can define an arbitrary stress-strain curve and arbitrary strain rate dependency. Furthermore, the model can simulate the exact brittle response of the polymer at high strain. The stress-strain curve used in this study is obtained by [19]. The equation can be expressed as:

The physical and mechanical parameters are determined by experiments, other parameters in the equation are determined by Reference and shown in Table 4.

The calculation parameters used in the numerical simulation are based on laboratory mechanical tests. All materials in the model are simplified to be isotropic and homogeneous without considering the material fracture, rock mass joints, and discontinuities. *MAT_DRUCKER_PRAGER is used to model the silty clay [20]. The frictional angle and cohesion determine the yield surface. The modified Drucker-Prager yield surface is used in this material model enabling the shape of the surface to be distorted into a more realistic definition for soils. The yield surface is expressed as:

where $T$ is the shear strength, ${\sigma }_m$ is the mean stress, $\phi$ is the internal friction angle, $c$ is the cohesion.

*MAT_PLASTIC_KINEMATIC material model is used for strong weathered sandstone and stemming. To manage the material failure, the PK material model is used in conjunction with the secondary created material failure criterion model *MAT_ADD_EROSION of LS-DYNA. The constitutive formula is as follows:

where ${\sigma }_y$ is the yield stress, ${\sigma }_0$ is the initial yield stress, $C$ and $P$ are the strain rate parameters, $\epsilon$ is the strain rate, ${\epsilon }_p^{eff}$ is the effective plastic strain, $E_p$ is the plastic hardening modulus, $\beta$ is the hardening coefficient:

where $E_{tan}$ is the tangent modulus and $E$ is Young’s modulus. The physical and mechanical parameters of soil, rock, and stemming are shown in Table 5. The numerical simulation process of chimney collapse is shown in Fig. 13.

Fig. 13Numerical simulation process of chimney collapse

a) 0 sec

b) 4.4 sec

c) 5.2 sec

d) 6 sec

e) 6.6 sec

f) Falling objects

The maximum impact load and surface combined vibration velocity obtained by theoretical calculation are compared with the actual engineering test results and numerical simulation results, as shown in Table 6. It can be found that the value obtained by theoretical calculation is slightly larger than the actual monitoring data. The reason is that the discontinuity of soil during the propagation of shock wave is not considered in the theoretical calculation, which results in energy loss. So the vibration amplitude is reduced. And in the actual blasting process, the chimney collapse form is not the whole collapse. Therefore, theoretical calculation is larger than actual monitoring data. However, the error between them is less than 20 %, so the theoretical results are reliable.

It needs to be explained that: a. Combined with the actual situation, the amplitude of impact load decreases rapidly with the increase of horizontal distance from the pipe, and the frequency also decreases with the increase of distance from the action center of impact load. Therefore, the probability of failure due to resonance in the far zone is small. b. In the near region, the shock load frequency is similar to the natural vibration frequency of the pipeline, so when the amplitude of energy reaches the strength limit of the pipeline, the pipeline is easy to be destroyed. Therefore, the impact load calculated by Eq. (6-12) is reasonable. 3. It is assumed that the ground plane of the project chimney collapse is a horizontal semi-infinite space elastomer.