Keywords

1 Introduction

Municipal pipe network is an important part of municipal infrastructure. However, due to lack of testing and maintenance, the pipes are commonly suffering from corrosion, leakage and other diseases. Erosion void (as Fig. 1) is caused by the pipe's own leakage, rainwater scouring, external disturbance and other factors, it makes the pipe structural stress change, which has a greater impact on the safety and stability of pipe operation, and the continued development of void may lead to secondary problems such as ground subsidence or landslides, posing a serious safety risk. For soil erosion void, there have been related repair techniques, such as grouting, polymer [1,2,3], but the force characteristics of erosion void are not known, resulting in the repair design is not based on evidence, and the operation is not prudent and may lead to the secondary damage of the pipe. Therefore, it is of great engineering significance to study the force distribution characteristics and mechanical response of eroded pipes to provide theoretical and data support for the design of pipe rehabilitation.

Fig. 1.
figure 1

Sketch of soil erosion void

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Scholars have carried out a lot of research on soil erosion void. Jiang L derived the erosion direction and erosion shape of the soil around the pipe through the phenomenon of solid-liquid coupling around the pipe after seepage from a pressure pipe in a model test [4]. Li H J designed a model test to investigate the mechanism of pipe-soil interaction induced by the ground subsidence, and proposed a limit equilibrium solution for the soil pressure at the top of the pipe with ground settlement [5]. Tan Z found that erosion void had a large effect on the bending moment of the pipe and the degree of the effect would increase greatly when it exceeded 45° [6]. Meguid M A investigated the soil pressure redistribution around the pipe of the eroded pipes [7]. Peter J M found that the void produces a large change in the mechanical state of the buried pipe, but the grouting can effectively alleviate this phenomenon [8, 9]. At present, a large number of studies have been carried out on the quantitative analysis of the causes of soil erosion void and the load distribution around the pipe with soil erosion void, but most of the studies are focused on the axial force of the pipe, and are mainly aimed at the rigid pipe, there is a lack of circumferential force analysis of the flexible pipeline. The force and deformation characteristics of flexible pipes and rigid pipes are different, which leads to more uncertainty in parameter selection and difficulty in precise control of construction parameters when trenchless repairing and designing for flexible pipes with soil erosion void.

This paper takes PE pipe as the research object, adopts ABAQUS finite element software to establish the numerical model of buried PE pipe. Based on the test results, we analyze the influence the width and location of the erosion void on the annular force of the pipe under loading, provide theoretical and data support for rehabilitation design of eroded pipes.

2 Analysis of Force Characteristics of Eroded Pipes

2.1 Impact of Erosion Void on Buried Pipes

When void occurs, the pipe body above the void loses the support force of the soil, which destroys the original force equilibrium state of the pipe-soil, the soil pressure around the pipe is redistributed, and the soil pressure on the pipe body around the void is increased and the load concentration phenomenon occurs, as Fig. 2.

Fig. 2.
figure 2

Sketch of soil pressure around the pipe

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According to the equivalent load, the soil pressure in the void is transferred to void surroundings, and comparing with the force of the pipe when it is not eroded, the problem of redistributing the soil pressure around the pipe with soil erosion void can be simplified to the problem of additional loading, and the formula is obtained:

$${p}_{\text{t}}={p}_{\text{d}}+{p}_{\text{f}}$$
(1)

Where, \({p}_{\text{t}}\) is the soil pressure on the bottom of the pipe with soil erosion void, kPa; \({p}_{\text{d}}\) is the soil pressure on the bottom of the pipe without soil erosion void, kPa; \({p}_{\text{f}}\) is the additional load on the bottom of the pipe with soil erosion void, kPa.

2.2 Calculation Method of Additional Load on Buried PE Pipe with Soil Erosion Void

This paper analyzes the soil pressure of buried PE pipe with soil erosion void on the basis of Spangler soil pressure model [10]. When the soil erosion void occurs, the soil pressure around the void will show obvious redistribution and load concentration phenomenon. Referring to the modified tunnel lining load-structure model under void conditions proposed by Ying [11], the Spangler soil pressure model is modified, and in order to simplify the model, the soil pressure model is obtained without considering that the soil erosion void of smaller width makes the pipe produce larger lateral deformation, which leads to larger changes of lateral soil pressure of the pipe, as Fig. 3.

Assuming that a soil erosion void with width l at the bottom of the pipe; x is the ratio of the influence range of the void on the soil pressure to the width of the void, and the length of the influence range of the void is xl, m; y is the ratio of the gradually increasing section of the additional load to the influence range of void; \({p}_{\text{f},\text{max}}\) is the maximum additional load, kN/m2; m is the length of the additional stress direction downward, m; pm is the soil pressure at the top of the pipe, kN/m2; p0 is the horizontal soil pressure on the side of the pipe, kN/m2; and pd is the soil pressure at the bottom of the pipe at the initial stage, kN/m2.

Fig. 3.
figure 3

Soil pressure model of buried PE pipe with soil erosion void

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When the width of the soil erosion void is small, the soil pressure at the void is transferred to the surrounding of the void, from which it can be seen that the two shaded areas in the figure are equal; at the same time, according to the principle of similarity of triangles, the following formula can be introduced:

$${S}_{1}={S}_{2}$$
(2)
$${S}_{1}={p}_{\text{f},\text{max}}\left(xl/2-m\right)/2$$
(3)
$${S}_{2}={p}_{\text{d}}\left(l+m\right)/2$$
(4)
$$\frac{{p}_{\text{f},\text{max}}}{{p}_{\text{d}}}=\frac{xyl/2-m}{m}$$
(5)

The association leads to the equation:

$${p}_{\text{f},\text{max}}={p}_{\text{d}}\left(\frac{2}{x}+y\right)$$
(6)

As a result, the formula for calculating the maximum additional load \({p}_{\text{f},\text{max}}\) for buried PE pipe with soil erosion void is derived. However, the above formula only applies to the small width of erosion void, its influence range does not exceed or just reach the edge of the pipe base.

When the width of void is large, and the influence range of evacuation theory exceeds the edge of pipe base, i.e. l + xl > Dsinα, the above model is no longer applicable. According to previous research, with the gradual increase of the width of the void, the influence range of the void also increases, the load concentration is gradually shifted to both sides, and the trend of the load concentration gradually disappears, and the additional stress tends to be evenly distributed.

Therefore, when Dsinα/(1 + x) < l < Dsinα, the buried PE pipe soil pressure model shown in Fig. 4.

Fig. 4.
figure 4

Soil pressure model of pipe when the soil void grows

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Based on the equivalent loads the equation can be obtained:

$${p}_{\text{f}}=\frac{{p}_{\text{d}}l}{\left(D\text{sin}\alpha -l\right)}$$
(7)

3 Finite Element Analysis

3.1 Finite Element Model

In order to further verify the analysis results, this paper uses ABAQUS finite element software to establish a numerical model of pipe-soil interaction of PE pipe with soil erosion void.

Since this study mainly analyzes the circumferential force of the pipe, it can be simplified to a plane strain problem and establish a two-dimensional model. The pipe material is 400 mm PE100 grade polyethylene pipe, which is commonly used in municipal projects. The width of the fill on the side of pipe is 3 times the radius of the pipe, and the depth of the pipe is set at 4 times radius. The size of the soil is 1.6 m × 1.6 m, and the outer diameter of the PE pipe is 0.4 m with a wall thickness of 19.1 mm. As shown in Fig. 5.

Fig. 5.
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2D model

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The density of PE pipe is 0.965 g · cm−3, elastic principal model. The density of soil is 0.965 g · cm−3, Moore Coulomb model, angle of internal friction is 37.8° and cohesion is taken as 0. The basic properties of the modeled materials are shown in Table 1.

Table 1. Material properties of the model
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The surface mutual contact is normal hard contact, tangential penalized contact, and the friction coefficient between PE pipe and sandy soil can be taken as 0.4. The boundary condition at the bottom of the model is set to be completely fixed, while the left and right sides are fixed to have the horizontal displacement as 0.

In this simulation, the analysis process includes four analysis steps, which are ground stress equilibrium, pipe activation, setting the soil erosion void around the pipe and loading. The model change function is used to make the soil at the void around the pipe in the unactivated state to set the soil erosion void around the pipe.

3.2 Simulation Program

Taking the erosion void width of 10 cm at the bottom of the pipe as the control group, and considering width and position of void, the simulation results are compared with the control group to study the influence law of different parameter characteristics on the force characteristics of buried PE pipes. The specific program is shown in Table 2.

Table 2. Numerical simulation program
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4 Analysis of Simulation Results

4.1 Width of the Soil Erosion Void

  1. (1)

    Pipe deformation

Table 3 shows the pipe deformation and vertical displacement of pipe top and bottom after 50 kN load is added to the model.

Table 3. Deformation of pipes
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When the width of the void is 5 cm, the deformation of the pipe is the same as the deformation of the pipe without evacuation, which indicates that the degree of influence of the void on the pipe at this time is small, and it is not enough to change the deformation of the pipe. When the width of the void increases to 10 cm, the deformation of the pipe changes, the vertical deformation of the pipe decreases by 14.1%, and the lateral deformation decreases by 6.3%. When the width of the void increases to 20 cm, the deformation of the pipe changes more obviously, the vertical deformation of the pipe decreases by 51.2%, the lateral deformation decreases by 29.2%, and the lateral deformation is larger than the vertical deformation of the pipe.

Fig. 6.
figure 6

Sketch of pipe shape (deformation factor is 10)

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When the soil erosion void at the bottom of pipe, the vertical displacements of the top and bottom of the pipe are increased. The vertical displacement of the bottom of the pipe increases by 0.01 mm, 0.69 mm and 2.52 mm with the increase of width of void. The vertical displacement of the top of the pipe is less affected by void, with the increment of 0.06 mm, 0.15 mm and 0.56 mm. Combined with the shape of the deformed pipe (Fig. 6), it can be found that the bottom of the pipe loses the support of the soil and produces a bulge to the bottom. Soil support part will produce a bulge to the bottom, and the void mainly affects the vertical displacement of the pipe bottom by affecting the vertical displacement of the pipe bottom, making the distance between the top of the pipe and the bottom of the pipe increase, which in turn leads to a reduction in the vertical deformation of the pipe.

  1. (2)

    Pipe stress

Extract the Mises stress of the outer and inner walls of the buried pipe after the addition of 50 kN load, and draw a comparative diagram of the pipe stress (Fig. 7).

Fig. 7.
figure 7

Stress curve of pipe

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The impact of the void on the pipe stress is mainly located in the bottom and waist of the pipe. The influence of void on the stress distribution of pipe is as follows: the stress on the outer wall of pipe bottom decreases, and the stress on the inner wall increases; the stress on the outer wall of pipe waist increases, and the stress on the inner wall slightly decreases. The maximum stress is 1671.51 kPa, located at the top outer wall of the pipe, when the void does not occur. When the width of 5cm of the void appeared, only the bottom and waist of the tube stress produces a small change, the other positions of the stress is not affected, and the location and value of the maximum stress remain unchanged. When the width of the void increases to 10 cm, the stress at the outer wall of pipe bottom and the inner wall of pipe waist decreases, and the stress at the inner wall of the pipe bottom and the outer wall of the pipe waist increases, but the maximum stress remains unchanged. When the width of the void is increased to 20 cm, the location of the maximum stress is shifted from the outer wall of the top of the pipe to the outer wall of the pipe waist, which is 1660.10 kPa.

  1. (3)

    Vertical soil pressure at the bottom of the pipe

The initial vertical soil pressure at the bottom of the pipe in the model is plotted in Fig. 8. The vertical soil pressure around the void will produce the load concentration phenomenon, with the increase of the void, the load concentration is shifted to both sides, and the concentration trend is gradually reduced.

Fig. 8.
figure 8

Vertical soil pressure at the bottom of the pipe

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Under the action of soil gravity load, the average value of vertical soil pressure at the bottom of the pipe is 15.62 kPa when the void does not appear, and the maximum additional load is 12.64 kPa and 16.53 kPa when the width of the void is 5 cm and 10 cm respectively, and the maximum additional load calculated according to Eq. 6 is 12.51 kPa, with an error of 1.04% and 18.14 kPa, with an error of 8.87%. When the width of void is 20 cm, the average additional load is 22.28 kPa, and the additional load calculated by Eq. (7) is 24.41 kPa, with an error of 9.56%.

There is a certain error between the simulation results and theoretical calculations, and the calculation error increases with the increase of the width of the void, which may not take into account the effect of the void on the width of the base of the pipe bottom. However, the above additional load calculation model for the bottom of eroded pipes can be used for reference.

4.2 Location of the Soil Erosion Void

  1. (1)

    Pipe deformation

The deformation of the pipe after loading is shown in Table 4. When the void appears at the bottom or top of the pipe, the pipe deformation is reduced, but when the void appears at the top of the pipe, the pipe deformation is smaller. The pipe deformation increases when the void appears on the side of the pipe. Combined with the displacement of the pipe and the shape diagram of the pipe after deformation (Fig. 9), the void affects the pipe deformation by changing the displacement of pipe at the place where void is located, and the influence of void at different locations on the deformation of pipe is as follows: the pipe produces an obvious bulge towards the place where the void is located, while no obvious deformation occurs in other parts. In the case of the same width of the voids, the degree of influence of the location of the voids on the pipe deformation for the top of the pipe > side > bottom.

Table 4. Deformation of pipes
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Fig. 9.
figure 9

Sketch of pipe shape (deformation factor is 10)

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  1. (2)

    Pipe stress

Figure 10 shows the comparison of pipe stresses after loading. Under external loading, the maximum stress of the pipe is located in the outer wall of the top of the pipe when there is no void, and the value is 1671.51 kPa. When the void is located in the bottom of the pipe, the maximum stress of the pipe basically remains unchanged. When the void is located in the left side of the pipe, the stress in the left inner wall of the pipe increases to 2409.87 kPa, an increase of 44.17%, which becomes the maximum stress of the pipe. When the void is located at the top of the pipe, the stress at the bottom and waist of the pipe remains unchanged, the stress at the outer wall of the pipe shoulder increases, the stress at the inner wall of the pipe shoulder and the top of the pipe decreases, and the maximum stress of the pipe decreases. It is hypothesized that the reason for this is that when there is a void at the top of the pipe, the top stress decreases due to the loss of part of the soil load at the top.

Fig. 10.
figure 10

Stress curve of pipe

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When the void at side, the scope and degree of influence on the pipe waist is greater than that on the pipe shoulder. In the case of the same width of voids, the degree of influence of the location of the voids on the stress of the pipe is side ＞ bottom ＞ top, but voids at top can cause ground collapse, its harm can’t be ignored.

5 Conclusions

This paper takes buried PE pipes in municipal engineering as the research object, establishes the calculation method of additional load at the bottom of the pipe with soil erosion void. Using ABAQUS to establish a 2D numerical model, the influence of the width and location of the void on the annular force characteristics of the buried PE pipe with soil erosion void was investigated by the control variable method.

  1. (1)

    As a kind of pipeline disease, soil erosion void has a greater impact on the safety and stability of pipeline operation. The pipe at the void loses the restraining effect of soil, and load concentration effect around void results in changes in pipe structural stress, which has a greater impact on the safety and stability of pipeline operation.

  2. (2)

    The scope and degree of influence of the void on the pipes are positively correlated with the width of the void, and when the void is small (less than 5 cm), the deformation of the pipe and the structural stress do not change significantly. With the increase of the width of the void, the bottom of the pipe produces a bulge downward, the location of the maximum strain and maximum stress of the pipe changes, and the value exceeds the initial maximum value, and the pipe bearing capacity is reduced.

  3. (3)

    The influence of the location of the void on the deformation of the pipes is that the pipe bulges significantly to the place where the void is located, while no significant deformation occurs in other parts of the pipe. The degree of influence of the location of the void on the pipe stress is side ＞ bottom ＞ top.