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Abstract

Polyethylene (PE) gas pipelines are often buried underground, and natural settlement of the earth’s ground can cause hard objects in the soil to press against the surface of the pipeline, forming point load. This results in stress concentration and indentation of the pipeline, thereby increasing the risk of failure. Given the rate-dependent nature of PE materials, the Suleiman model is utilized to fit uniaxial tensile test data, thereby obtaining the material’s elastic and plastic parameters. To investigate the denting behavior of buried gas PE pipes under point load, a finite element model of the pipe-soil system under the influence of hard objects is established, and its rationality is verified through point load simulation experiments. This study analyzes the effects of hard object size, internal pipeline pressure, and pipeline diameter on pipeline denting behavior. The deformation rate $δ$ of the pipe is used as a quantitative metric to evaluate the denting deformation of PE pipes. The results indicate that significant stress concentration and pipeline denting occur at the location of the hard object. Apart from the location of the hard object, the stress and strain variations along the pipeline’s axial direction remain relatively small. Moreover, when the soil settlement remains constant, the deformation rate and its range of variation in pipeline denting remain largely unchanged. The influence of hard object size and internal pressure on pipeline denting deformation is relatively minor, while pipeline diameter has a more significant effect. The results may be referable for safety assessment of PE pipes due to point load.

Keywords

Polyethylene (PE) gas pipeline rate correlation point load buried pipe indentation behavior

Introduction

PE pipes are crucial carriers for oil and gas transportation, accounting for over 90% of the plastic pipe network usage in the United States. With a large urban population, any leaks or explosions in PE pipes can lead to severe consequences. According to the 2021 statistics from the Plastic Pipe Data Collection (PPDC) of the United States, point load are the third leading cause of PE pipe failures, after installation errors and material defects. Point load are caused by the mutual compression between hard objects in the soil and the pipeline, resulting in indentations and stress concentrations in the pipe, as shown in Figure 1 Therefore, it is necessary to conduct research on PE pipes under point load.

Figure 1.

Schematic diagram of PE pipe denting caused by point load.

With regard to the buried PE pipes, scholars have conducted numerous research studies. For instance, Tafreshi et al.^1–3 carried out laboratory tests of small-diameter High Density Polyethylene material (HDPE) pipes buried in reinforced sand under repeated-load and obtained the values of the vertical diametral strain (VDS) and the settlement of the soil surface (SSS) under different parameters. The effects of various vital parameters such as, relative density of sand, number of reinforced layers and embedded depth of pipe on VDS and SSS were studied by ANNs, it was found that the predicted was in line with the measured values. Wang⁴ studied the mechanical properties of buried PE pipeline with different defect parameters on the condition of surface subsidence and surface load through ABAQUS. Luo et al.^5,6 investigated the deformation behavior of PE pipe subjected to seismic landslide and foundation settlement through ABAQUS and proposed a related failure criterion due to yielding. Zhang et al.⁷ studied the mechanical behaviors and failure mechanisms of buried PE pipes under fault movement, and discussed the effects of gas pressure, fault dislocation, soil and pipe size on the mechanical behavior of PE pipes. The study indicates that gas pressure has a less effect on the mechanical behavior of PE pipe. Chen et al.⁸ studied the deformation process of PE pipe and the position transfer law of failure location under the combined loading of non-uniform settlement and landslide. The results showed the damage caused by the landslide to the pipe is more serious than that by the settlement. Zhou et al.⁹ conducted a series of tests to investigate the structural responses of HDPE double-wall corrugated pipes subjected to the localized ground subsidence and proposed empirical equations to correlate the volume of pipe displacement profile with the volume of settlement trough at the backfill surface. Wu et al.^10,11 conducted numerical simulation analysis of mechanical behavior of buried PE pipe under different parameters when the land subsidence occurred and carried out nonlinear regression analysis to fit the relationship between the stress and parameters. Through the coupling of test and finite element, Shi et al.¹² established a steel wire fracture failure model of buried PSP pipe under foundation settlement, and revealed the coupling law of internal pressure and settlement on failure location and load threshold. Lee et al.¹³ compared the failure probability of three design criteria for corrugated polyethylene pipes based on the first-order reliability method (FORM), and pointed out that the deflection criterion had the highest failure probability, which increased with the increase of the coefficient of variation of parameters such as buried depth and pipe diameter.

Scholars have conducted some research on PE pipes under point loads. Hessel J.¹⁴ predicted the lifespan of unburied PE pipes based on point load test results. Fenk et al.¹⁵ performed finite element simulations on unburied PE pipes, identifying stress concentrations on the pipe wall caused by point load and establishing a method for calculating long-term static hydraulic pressure strength. Majer Z et al.^16–18 analyzed the influence of parameters such as circumferential stress, hard object size, elastic modulus, and Poisson’s ratio on fracture parameters like crack stress intensity factor and crack size parameters of unburied PE pipes through numerical simulations, and calculated the lifespan of the PE pipes. Wee J-W et al.¹⁹ utilized a crack layer model to simulate the discontinuous crack propagation in buried PE pipes under point load conditions. Studies have shown that the presence of point load affects the crack propagation behavior of the pipes, accelerating their failure and significantly reducing their service life.

Although scholars have conducted extensive research on PE pipe, these studies have primarily focused on the stress response, lifespan calculations, methods for calculating crack stress intensity factors and other geological disasters such as landslides and earthquake. There is limited attention paid to the indentation behavior of buried PE pipes under point load, and existing research rarely considers the interaction between the pipe and the soil, which is also a major factor affecting the deformation and stress of the pipes.

Therefore, this paper collected data through uniaxial tensile tests and utilized the Suleiman model to fit and obtain the constitutive parameters of PE materials. Considering the interaction between the pipe and soil, a finite element method was employed to simulate and study the denting behavior of gas polyethylene (PE) pipelines under point load. The study analyzed the influence of factors such as the size of the hard object, internal pressure within the pipeline, and the diameter of the pipeline, and conducted corresponding safety evaluations.

Experiment

PE uniaxial tensile test

PE is a rate-dependent material, and this study focuses on buried PE pipes where the rate of ground settlement can affect the material properties of the PE pipes. The Suleiman model is related to the strain rate,²⁰ as shown in equation (1). Therefore, considering the rate-dependency of PE materials, the Suleiman model was chosen to obtain the relationship between the PE material parameters and the rate. Based on the rate of ground settlement, the stress-strain relationship of the material can be determined. In urban areas, the rate of ground settlement is relatively low, equivalent to quasi-static loading, with rates ranging from 1 × 10⁻⁶ s⁻¹∼1 × 10⁻⁵ s⁻¹.²¹ In this study, a ground settlement rate of 1 × 10⁻⁶ s⁻¹ was selected to calculate the material parameters of PE.

σ = \frac{ε}{a + b \cdot ε}

(1)

Where σ is the true stress, ε is the true strain, and m and n are fitting constants.

Selecting the PE 100 gas pipe under the SDR 11 series as the research object, SDR is the ratio of external diameter to wall thickness of the pipe, the external diameter is 110 mm and the wall thickness is 10 mm. With reference to ISO 6259,²² a dumbbell type specimen is taken from the PE pipe. The specimen is made mechanically and the thickness of the specimen is the wall thickness of the pipe. The information about the specimen is shown in Figure 2.

Figure 2.

PE material tensile test.

To obtain the PE material parameters using the Suleiman model, the specimen needs to be subjected to tensile tests at different rates, with tensile rates set at 0.5, 1, 5, 10, 15, 50 mm/min.

The experimental data and processing of this paper are quoted from another paper of Hu et al.²³ The load-displacement data obtained from the tests at different tensile rates are converted into the true stress-true strain curves of the material. As shown in Figure 3, PE material has very obvious nonlinear behavior, and the elastic modulus and yield strength tend to increase with the increase of tensile rate. Considering that the Suleiman model is related to the strain rate and can well describe the nonlinear behavior of polyethylene materials, the model is used for parameter fitting.

Figure 3.

True stress-strain curves of PE materials at different tensile rates.

Based on the true stress-strain data, the values of m and n at different rates were fitted to obtain the data shown in Table 1. Then the relationship between m and n and the tensile rate was obtained by logarithmic fitting as Equation (2) and Equation (3).

m = 0.000479788 - 0.47916 \cdot \ln v

(2)

n = 0.03229 - 0.00139 \cdot \ln v

(3)

Table 1.

The fitted m and n values at different stretching speeds.

Tensile rate (mm/min)	Strain rate (S⁻¹)	m (×10⁻⁴)	n	Correlation coefficient R²
0.5	1.5 × 10⁻⁴	5.13752	0.03443	0.978
1.0	3.0 × 10⁻⁴	5.7026	0.03047	0.998
5.0	1.5 × 10⁻³	2.75815	0.03148	0.997
10.0	3.0 × 10⁻³	3.60066	0.02779	0.995
15.0	4.5 × 10⁻³	3.96927	0.02893	0.994
50.0	1.5 × 10⁻²	3.28691	0.02693	0.995

Based on Equation (2) and Equation (3) the Suleiman model expression for PE at different tensile rates can be obtained:

σ = \frac{ε}{0.000479788 - 0.47916 \cdot \ln v + (0.03229 - 0.00139 \cdot \ln v) \cdot ε}

(4)

Furthermore, the Suleiman model expression for the PE material at a strain rate of 1 × 10-6 s⁻¹ can be derived from equation (4) as:

σ = \frac{ε}{0.000278398 + 0.04239 \cdot ε}

(5)

Finally, the relationship between true stress-strain and tensile rate of PE material can be determined by equation (5). Based on the ground settlement rate of 1 × 10-6 s⁻¹, the parameters of elastic modulus and yield strength of PE material can be obtained, as shown in and Table 2 and Table 3.

Table 2.

Elastic parameters of PE material.

Elastic modulus (MPa)	Poisson’s ratio	Yield stress (MPa)	Density (kg/m³)
1216.13	0.45	14.18	950

Table 3.

Plastic parameters of PE material.

Plastic stress (MPa)	Plastic strain	Plastic stress (MPa)	Plastic strain
14.18	0	21.56	0.069
19.01	0.026	22.32	0.114
20.78	0.047	23.09	0.299

Point load experiment and material parameter validation

To investigate whether PE pipes can function normally under point load conditions, it is necessary to conduct point load experiments on PE pipes. This paper presents a simulation experiment on PE pipes under point load, where a spherical indenter is used to mimic the effect of a hard object on the pipe, as shown in Figure 4. The behavior of PE pipes under point load is analyzed, and the experimental steps are as follows:

(a) Firstly, a PE pipe with a width consistent with the wooden baseplate is cut, and the pipe is securely fixed to the wooden baseplate using a specially designed clamp.

(b) The spherical indenter is positioned directly above the axis of the pipe, and then metal blocks are placed around the pipe to further secure it and prevent rolling during the loading process.

(c) Once the pipe is securely fixed, a constant loading rate of 1 mm/min is applied, with a maximum displacement of 50 mm.

(d) The load force-displacement data is collected for subsequent processing and analysis.

Figure 4.

Point load experiment test setup and instrumentation.

The finite element model of the point load experiment was established in Finite Element Analysis (FEA) software, as shown in Figure 5. The maximum loading displacement of the spherical pressure head was 50 mm. The load-displacement curve of the PE pipe was extracted and compared with the actual experimental data. The simulation and experimental results are shown in Figure 6. The pipeline has obvious concave deformation, and the load force on the PE pipe increases with the increase of loading displacement. The simulation result curve is roughly consistent with the experimental result curve, and the maximum error is 6.21%, which is within the acceptable range. The material parameters used in modeling, such as elastic and plastic parameters, are the experimental data. Therefore, the reliability of the measured material parameters can be considered.

Figure 5.

Finite element model for point load experiment.

Figure 6.

Point load experiment and simulation load force-displacement curve.

Finite element model

Pipe-soil modeling under point load

In this study, a buried PE pipe model under point load was established using FEA software. A displacement load was applied to the upper surface of the soil to simulate the natural settlement process of the ground, and under this displacement load, the hard object and the pipe were compressed to form a point load. The following assumptions were considered during model establishment: (1) The welding of the pipe was not considered; (2) Geological settlement was considered as uniform; (3) The hard object causing the point load was simplified as a regular spherical stone. PE pipe, soil and hard objects are all face-to-face contact, which is full-circle support.

To facilitate analysis and calculation, a 1/2 three-dimensional point load-pipe-soil model was established. A settlement displacement was set on the upper surface of the model, while symmetric constraints were applied to the half section of the model, and fixed constraints were set on the lower surface, left and right sides. The boundary conditions and mesh division of the model are shown in Figure 7.

Figure 7.

Schematic diagram of boundary condition setting for pipe-soil model.

The soil model selected for this study is the Drucker-Prager (D-P) model, which is capable of accurately representing the soil state and providing stable calculations. The soil parameters are presented in Table 4, while the rock parameters are shown in Table 5.

Table 4.

Material parameters of soil.²⁰

Elastic modulus (MPa)	Poisson’s ratio	Density (kg/m³)	Friction angle (°)	Dilation angle (°)	Flow stress ratio
32.5	0.4	1867	36.5	0	1

Table 5.

Material parameters of rock.¹⁷

Elastic modulus (GPa)	Poisson’s ratio	Density (kg/m³)
50.7	0.28	2670

Grid independence verification

To ensure the accuracy and reliability of the calculation results, a grid independence verification is required for the model. Therefore, the grid sizes are set to 1.0, 1.5, 2.0, 4.0, and 6.0 mm, and different internal pressures are applied to analyze the Mises stress conditions of the PE pipe. Figure 8 shows the maximum Mises stress of PE pipe with different grid sizes under different internal pressures. It can be observed that the maximum Mises stress of the PE pipe differs significantly when the grid size ranges from 2.0 to 6.0 mm, while stress values are relatively stable when the mesh size is set to 1.0 to 2.0 mm. Therefore, to ensure both high calculation accuracy and relatively fast computation time, the mesh size was set to 2.0 mm. This sizing balance was chosen to capture the necessary details of the model while maintaining an efficient simulation process.

Figure 8.

Maximum Mises stress of PE pipe with different grid sizes under different internal pressures.

Results and discussion

Evaluation criteria for pipeline cross-sectional deformation

Under the effect of ground settlement, the pipeline and hard objects squeeze each other to produce point load, which leads to the flattening or ovalization of the pipeline cross-section and local indentation at the hard objects, as shown in Figure 9(a) and (b). To ensure the normal operation of PE pipelines under point load, it is necessary to avoid significant deformation of the pipeline cross-section, which primarily involves a reasonable evaluation of cross-sectional deformation. A simple yet effective measurement method for cross-sectional deformation is ovalization, which assesses the deformation by comparing the diameters of the pipeline before and after deformation. Now it is introduced to similarly evaluate and quantify the dent deformation of PE pipes. The expression of the dent deformation rate $δ$ is:

δ = \frac{Δ D}{D_{0}}

(6)

Where, D₀ represents the outer diameter of the pipeline before deformation, ΔD = D₀ - D₁, indicates the change in diameter of the pipeline at the indentation, D₁ represents the outer diameter of the pipeline after deformation. In order to test the contribution of hard objects to the ovalization deformation and local depression of the pipeline, a comparative experimental group was set up, that is, only under the natural settlement of the ground (without the action of hard objects), the pipeline was squeezed by the soil layer to produce deformation. As shown in Figure 9(c), where D₂ represents the upper and lower diameters after deformation under the action of land subsidence only.

Figure 9.

Point load cause PE pipe deformation.

For pipeline systems made of most materials, a deformation of 5% can be considered as the maximum allowable design value. However, for PE pipes with strong toughness and resistance to deformation, it is generally believed that they will not be damaged even when the deformation reaches 20%. In²⁴ short-term deformation tests conducted in a laboratory, the deformation limit of PE pipes can reach up to 30%. However,²⁵ under the action of point load, PE pipes experience indentation deformation in addition to flattening. Therefore, to ensure safety, following the recommendation in literature,²⁶ when the safety factor of PE pipes is two and the deformation rate $δ$ reaches a value of 15%, the pipe is considered to have reached its cross-sectional deformation limit, a value also adopted by the Dutch specification NEN 3650.²⁷

The influence of hard object size on PE pipe

The influence of hard object size on stress and strain

The size of hard objects causing point load in soil is uncertain, thus it is necessary to investigate the indentation behavior of PE pipes under varying sizes of hard objects. In order to unify, the PE pipes used in the simulation below are SDR 11 series. The diameter/thickness of the pipe is equal to 11, that is, the thickness of the pipe changes with the diameter, and the wall thickness is known when the diameter is known. In this study, the radius R of the hard object is set to 0, 10, 20, 30, 40, 50, and 60 mm to explore the changes in pipeline performance with varying sizes of hard object. Figure 10 presents the local stress cloud contour of PE pipe with the different sizes of hard object, when the internal pressure is 0.4 MPa and the pipe diameter is 110 mm.

Figure 10.

Local stress cloud map of PE pipe when the size of hard objects changes.

As shown in Figure 10, stress concentration occurs in the PE pipe at the location of the hard object, accompanied by indentation of the pipe caused by the object. This is because the natural settlement movement of the soil alters the soil structure surrounding the pipe at the location of the hard object, leading to mutual compression between the pipe and the hard object, which causes indentation of the pipe. Additionally, this compression results in uneven stress distribution at the pipe location, causing stress concentration. In addition, the stress concentration area in the pipeline tends to increase with the increase of the size of the hard object. Under the action of stratum settlement, the contact area between the larger hard objects and the buried pe pipeline will be larger, so the stress concentration area of the pipeline will be larger.

As the size of the hard object changes, the stress variation curves along the axial direction of PE pipes with diameters D = 75, 90, 110, and 160 mm are depicted in Figure 11. The PE pipe exhibit significant stress concentration at the location of the hard object, accompanied by a stress mutation. The pipe stress at the location of the hard object fluctuates in a trifurcate shape. Furthermore, the maximum Mises stress in PE pipes of various diameters tends to gradually decrease with increasing size of the hard object.

Figure 11.

Stress variation of PE pipe along the axis direction when the size of hard object changes.

This is because the interaction between the pipe and the soil is a crucial factor affecting the stress and deformation of the pipe. When the size of the hard object is small, the Mises stress in the pipe is higher. On the contrary, as the size of the hard object increases, its contact area with the soil expands, which partially offsets the interaction force between the hard object and the pipeline. For example, in Figure 11, in the experimental group with a PE pipe diameter of 90 mm, when the radius of the hard object is 10 mm, the maximum Mises stress in the stress concentration area is close to 28 MPa. When the radius of the hard object gradually increases to 90 mm, the maximum values of different curves are also gradually decreasing, indicating that the Mises stress is gradually decreasing, and the experimental groups of other pipe diameters are similar.

The trifurcate shape of the stress fluctuation at the location of the hard object in the pipe is related to the deformation of the pipe at that point. The hard object causes indentation deformation in the pipe, with the indentation center located directly above the object, where the deformation is most severe and stress concentration is evident. Along the path from the indentation center to the edge of the indentation, there are also regions with significant pipe deformation, resulting in higher stress levels than those on either side of the indentation. Therefore, the pipe stress at the location of the hard object exhibits a trifurcate fluctuation pattern.

As the size of the hard object changes, the strain variation curves along the axial direction of PE pipes with diameters D = 75, 90, 110, and 160 mm are depicted in Figure 12. At the location of the hard object, the PE pipes exhibit significant strain concentration accompanied by a strain mutation. As the size of the hard object increases, the strain in PE pipes of various diameters initially decreases, then increases, and finally decreases again. When the object size is small, the strain at the location of the object fluctuates in a tooth-shaped pattern. Compared to pipes under point load, pipes without point load operate under normal conditions, where the stress distribution is uniform, resulting in minimal strain without fluctuations.

Figure 12.

Strain variation of PE pipe along the axis direction when the size of hard objects changes.

It can be seen from Figure 10 that when the size of the hard object or the size of the pipeline is small, its impact on the soil around the pipeline is limited. Since the hard sphere we set here does not consider its deformation, with the gradual increase of the contact area between the hard object and the PE pipe, it is easy to produce a small M-shaped deformation on the axial surface of the contact position, so the equivalent strain in the diagram shows obvious zigzag fluctuation. With the increase of the size of the hard object, the contact area with the soil increases, and the interaction force decreases, thus reducing the stress concentration and strain. However, when the size of the object continues to increase, the distribution of the interaction force becomes uneven at first, and then recovers to be uniform, resulting in the strain of the pipeline increasing first and then decreasing.

The influence of hard object size on indentation deformation

Figure 13 illustrates the positional changes along the radial direction of various points in the circumferential cross-section of PE pipes with different diameters at the indentation location as the size of the hard object varies. Figure 14(b) is the maximum equivalent strain under different hard size and different hard size/pipe diameter. From Figures 13 and 14, it can be observed that the positional change curves along the radial direction for the same pipeline under different hard object sizes almost coincide, with a pipeline indentation occurring at the location of the hard object (270°). Apart from the indentation location, the positions of other points in the circumferential cross-section along the radial direction remain largely unchanged. As the pipe diameter increases, the range of the pipeline indentation area gradually decreases. When not subjected to point load, the deformation rate $δ$ of PE pipes with different diameters do not differ significantly, fluctuating around 3.7%. However, when the size of the point load is fixed, the value of $δ$ decreases as the pipe diameter increases. The size of the point load has a more significant impact on smaller-diameter PE pipes, exacerbating their indentation deformation. Additionally, when the size of the hard object varies, the $δ$ values at the indentation locations of PE pipes with different diameters are less than 15%, indicating a safe state.

Figure 13.

The radial position variation of the circumferential section node at the indentation of the PE pipe when the size of hard objects changes.

Figure 14.

Effect of hard objects on indentation depth.

Soil movement is a crucial factor influencing pipeline indentation deformation, yet the soil settlement considered in this study is constant. Therefore, under a specific settlement value, as the size of the hard object changes, the pipeline indentation deformation and indentation range for the same pipe diameter remain relatively similar. As the pipe diameter increases, the contact area between the pipeline and the soil enlarges, allowing the interaction forces between them to distribute more evenly over a larger area, thereby reducing stress concentration on the pipe surface. This, in turn, minimizes the deformation on the pipe surface and reduces the range of the pipeline indentation area.

It can be seen that the local plastic deformation (indentation) of the pipeline and the overall elliptical deformation of the pipeline are caused by different factors. For example, when the diameter of the pipeline is 75 mm and the size of the hard object is 0, 10, 20, 30, 40, 50 and 60 mm respectively, pay attention to the deformation trend of the pipeline curve. When there is no hard object (the size of the hard object R is 0) and there are hard objects, the curve lines of the non-hard object action area basically coincide, and the cross-section shape also basically coincides with different hard object sizes. Other sizes of the pipeline have the same deformation law. This shows that the main contribution of hard objects is local plastic deformation (indentation), and the ovalization is caused by the subsidence of the stratum.

The influence of internal pressure on PE pipe

The influence of internal pressure on stress and strain

The internal pressure of fluid mediums within the pipeline is also a significant factor affecting pipeline deformation, and the maximum allowable working pressure for PE 100 gas pipelines is 1 MPa. This study explores the performance changes of pe pipe under different hard object sizes when the internal pressure P is set at 0, 0.2, 0.4, 0.6, 0.8, and 1 MPa. Figure 15 depicts the local stress contour plot of a PE pipe with a hard object radius R = 40 mm and a pipe diameter D = 110 mm under varying internal pressures.

Figure 15.

Local stress cloud map of PE pipe when internal pressure changes.

As seen in Figure 15, the PE pipe exhibits significant stress concentration and indentation at the location of the hard object. Additionally, the stress concentration area in the pipeline tends to decrease as the internal pressure increases. When the pipeline is subjected to internal pressure, the pressure distributes uniformly along the circumference of the pipe, providing a degree of resistance to external loads. This improves the stress distribution and deformation of the pipeline. As the internal pressure increases, it counteracts more external loads, thereby reducing the stress concentration in the pipeline. Therefore, the stress concentration area in the pipeline tends to decrease as the internal pressure increases.

Figure 16 shows the Mises stress variation curves along the axial direction of PE pipes under different hard object sizes as the internal pressure changes. It can be observed that the overall Mises stress of the PE pipe decreases as the internal pressure increases. At the location of the hard object, stress concentration occurs and exhibits a trifurcate shaped fluctuation pattern. Furthermore, the maximum Mises stress in the pipeline increases as the internal pressure rises.

Figure 16.

Stress variation of PE pipe along the axis direction when internal pressure changes.

The greater the internal pressure the pipeline is subjected to, the more external loads it counteracts, leading to a downward trend in overall stress of the pipeline. At the same time, the stress concentration area at the location of hard objects becomes more concentrated, resulting in a gradual increase in the maximum Mises stress of the pipeline with the increase of internal pressure. The stress at the location of the hard object also exhibits a trifurcate shaped fluctuation pattern, which is related to the larger deformation areas at the center and along the indentation path of the pipeline.

Figure 17 depicts the strain variation curves along the axial direction of PE pipe under different hard object sizes as the internal pressure changes. It is evident that as the internal pressure increases, the strain in the PE pipe also exhibits a downward trend. However, there is a significant strain mutation at the location of the hard object, where the maximum strain increases with increasing internal pressure, except when R = 30 mm, where the maximum strain varies inversely.

Figure 17.

Strain variation of PE pipe along the axis direction when internal pressure changes.

As the internal pressure in the pipeline increases, it becomes more resistant to external loads, resulting in an overall decrease in the pipeline’s strain. However, the more concentrated the stress concentration area is at the location of the hard object, the greater the maximum strain in the pipeline tends to increase with increasing internal pressure. When the size of the hard object is small, the effect of the internal pressure on the pipeline is greater than that of the point load, causing the maximum strain to increase with increasing internal pressure. Therefore, the internal pressure can serve as a protective factor to some extent, reducing both the deformation and stress on the pipeline.

The influence of internal pressure on indentation deformation

Figure 18 illustrates the positional changes along the radial direction of various points on the hoop section of the PE pipe at the indentation site under varying internal pressures, with different sizes of hard objects. The position change curves of the pipe along the radial direction under various sizes of hard objects and varying internal pressures almost coincide. The value of $δ$ slightly decreases as the internal pressure increases, with a reduction of approximately 0.55% in the pipe under 1 MPa internal pressure compared to that without internal pressure. Meanwhile, the hoop section without point load experiences minimal positional changes along the radial direction. The pipe exhibits indentations at the hard object locations (270°), and the extent of the indentation areas remains nearly consistent. Furthermore, under the same internal pressure, when the sizes of the hard objects are R = 40 mm, 50 mm, and 60 mm, the larger the point load size, the greater the value of $δ$ and the more significant the deformation of the pipe cross-section. However, when R = 30 mm, the relatively low value of $δ$ indicates that internal pressure can, to a certain extent, provide protection to PE pipes subjected to small-sized point load. When the internal pressure of the pipe varies and the size of the hard object changes, the value of $δ$ at the indentation of the PE pipe remains below 15%, ensuring the normal operation of the pipe Figure 19.

Figure 18.

The radial position variation of the circumferential section node at the indentation of the PE pipe when internal pressure changes.

Figure 19.

The influence of different pipe internal pressure on $δ$

The influence of diameter on PE pipe

The influence of diameter on stress and strain

This article also explores the impact of changes in PE pipe diameter on pipeline performance. Based on the SDR 11 series diameters of PE 100 pipes, the pipe diameters were set to D = 75, 90, 110, 160, and 200 mm. When the radius of the hard object was set to R = 40 mm, the local cloud contour of PE pipes under different internal pressures is shown in Figure 20. Stress concentration and pipe indentation have occurred at the hard object position of the PE pipe. When the internal and external load conditions of the pipe are the same, the smaller the diameter of the pipe, the more obvious the stress concentration phenomenon is. The smaller the diameter of the pipe, the smaller its cross-sectional area. Under the same load conditions, the stress of the pipe will be greater, so the stress concentration phenomenon of the smaller diameter pipe is more obvious.

Figure 20.

Local stress cloud map of PE pipe when the pipeline diameter changes.

Figure 21 shows the Mises stress variation curve along the axial direction of PE pipes under different internal pressures when the pipe diameter changes. As the pipe diameter increases, the Mises stress generally shows a trend of first increasing and then decreasing; while at the position of the hard object, the maximum Mises stress shows a trend of first decreasing and then increasing with the increase in pipe diameter. And the pipe stress at the position of the hard object also shows a trifurcate shaped fluctuation trend. The increase in pipe diameter increases the volume of the pipe and improves the overall strength and stiffness of the pipe. Within a certain range, increasing the pipe diameter will cause the Mises stress of the PE pipe to generally increase. At the position of the hard object, when the pipe diameter increases, the cross-sectional area of the pipe increases, reducing the degree of stress concentration at the position of the hard object, resulting in a decreasing trend of the maximum Mises stress at the position of the hard object. However, when the pipe diameter continues to increase beyond a certain range, the overall stiffness of the pipe increases, intensifying the stress concentration at the position of the hard object, leading to an increasing trend of the maximum Mises stress at the position of the hard object.

Figure 21.

Stress variation of PE pipe along the axis direction when the pipeline diameter changes.

Figure 22 depicts the strain variation curve along the axial direction of PE pipes under different internal pressures when the pipe diameter changes. It can be observed that there is significant strain concentration at the position of the hard object in the PE pipe, and the maximum strain shows a trend of first decreasing and then increasing as the pipe diameter increases; except for the area near the point load position, the strain curves of the PE pipe along the axial direction almost overlap. As the pipe diameter increases, its cross-sectional area also increases, reducing the stress concentration phenomenon at the point load position of the pipe. At the same time, the overall stiffness of the pipe increases, intensifying the strain concentration at the position of the hard object, leading to an increasing trend of the strain at the position of the hard object. Near the hard object, due to the stress concentration caused by the point load, the stress distribution in the pipe is uneven compared to other positions, resulting in a significant strain concentration phenomenon.

Figure 22.

Strain variation of PE pipe along the axis direction when the pipeline diameter changes.

The influence of diameter on indentation deformation

Figure 23 illustrates the radial displacement variations at various points along the circumferential cross-section of PE pipes at the indentation location under different internal pressures when the pipe diameter is altered. When the diameter changes, indentations occur at the hard object location (270°). As the pipe diameter increases under the same internal pressure, the deformation of the pipe’s cross-section gradually decreases, and the indentation deformation rate $δ$ significantly drops. For example, the deformation rate decreases by approximately 6.05% for a pipe with a diameter of 160 mm compared to one with a diameter of 75 mm. The shape of the contour curve representing the radial displacement variation remains largely unchanged under different internal pressure conditions. When the internal pressure increases while the pipe diameter remains constant, the indentation deformation rate $δ$ slightly decreases. For a pipe with a diameter of 160 mm, the value of $δ$ at 1 MPa internal pressure is 0.39% lower than that without any internal pressure. Additionally, regardless of the pipe diameter, the value o $δ$ $δ$ at the indentation location under different internal pressures is always less than the critical limit of 15%. As the pipe diameter increases, so does its cross-sectional area, which in turn reduces the deformation rate at the point load location.

Figure 23.

The radial position variation of the circumferential section node at the indentation of the PE pipe when the pipeline diameter changes.

Given that the pipes are made of the same material and subjected to the same external load, their deformation behavior under identical loads is similar. Therefore, the contour curves of radial displacement variations for pipes of different diameters exhibit similar shapes. Furthermore, under the same load conditions, the internal pressure has a minor impact on the deformation and indentation of the pipe, not significantly altering its deformation behavior. Consequently, the contour curves of radial displacement variations remain relatively unchanged under varying internal pressure conditions Figure 24.

Figure 24.

The influence of different pipe diameters on $δ$

Brief summary

When assessing the deformation and safety of PE pipe under point load, pipe diameter, hard object size, and internal pressure are all critical factors, albeit with varying degrees of influence. The influence of pipe diameter is particularly significant because larger diameters result in a larger cross-sectional area of the pipeline, which can more effectively distribute stress and reduce stress concentration. Additionally, pipelines with larger diameters exhibit greater stiffness, enabling them to more effectively resist external loads and decrease deformation rates. For instance, under the same internal pressure, the deformation rate δ of a 160 mm diameter pipeline is approximately 6.05% lower than that of a 75 mm diameter pipe. The size of hard objects also directly impacts pipeline deformation; smaller hard objects, due to their smaller contact area with the soil, lead to greater interaction forces with the pipeline, causing more pronounced stress concentration and deformation. However, the influence of hard object size is more significant on smaller diameter pipelines, exacerbating their dent deformation. Internal pressure positively affects pipeline deformation and stress; as internal pressure increases, the pipeline becomes better able to resist external loads, reducing overall strain and stress concentration. The presence of internal pressure also lowers the deformation rate of pipelines; for example, at an internal pressure of 1 MPa, the deformation rate δ of a 160 mm diameter pipeline is approximately 0.39% lower than when there is no internal pressure. Taken together, pipe diameter is the most important factor influencing pipeline deformation and safety, followed by hard object size and internal pressure.

Conclusions

This paper considers the rate-dependency of PE materials and conducts tensile tests on PE materials at different rates. A finite element model of the pipe-soil system under point load is established to investigate the effects of hard object size, internal pipeline pressure, and pipeline diameter on the stress and deformation of PE pipes. The following conclusions are drawn:

(1) Stress concentration occurs in the PE pipe at the location of the hard object, which simultaneously causes a dent in the pipeline. When the soil settlement remains unchanged, the amount and range of pipeline dent deformation under various influencing factors basically remain the same, and the position change curves of the pipeline along the radial direction almost coincide. Except for the dent location, the positions of other points along the circumferential cross-section show almost no change in the radial direction.

(2) The maximum Mises stress in the PE pipe gradually decreases as the size of the hard object increases, while the strain first decreases, then increases, and finally decreases again with the increase in the size of the hard object. The size of the hard object causing point load has no significant effect on the pipeline dent deformation. When not subjected to point load, the dent deformation ratio $δ$ values of PE pipes with different diameters do not vary significantly, fluctuating around 3.7%.

(3) The area of stress concentration in the pipeline decreases as the internal pressure in the pipeline increases, while the maximum Mises stress increases with the increase in internal pressure. When the size of the hard object is small, the influence of internal pressure on the pipeline is greater than that of point load, resulting in greater maximum strain in the pipeline as the internal pressure increases. The value of $δ$ slightly decreases with the increase in internal pressure, with a decrease of approximately 0.55% in $δ$ for the pipe with an internal pressure of 1 MPa compared to one without internal pressure.

(4) At the location of the hard object, the maximum Mises stress first decreases and then increases with the increase in pipeline diameter. Similarly, the maximum strain also first decreases and then increases with the increase in pipe diameter. When the pipe diameter increases with the same internal pressure, the deformation of the pipeline cross-section gradually decreases, and the value of $δ$ significantly drops; for example, the deformation ratio decreases by approximately 6.05% for a 160 mm diameter pipeline compared to a 75 mm diameter pipeline.

Footnotes

Author contributions

Gang Hu: Conceptualization, Methodology. Zhen Ma: Writing - original draft, Software, Revision. Peng Liu: Supervision. Guorong Wang: Supervision. Qiang Pu: Methodology, Data curation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by Sichuan Science and Technology Program (No. 2024NSFSC2009, No. 2023NSFSC1980).

ORCID iD

Gang Hu

Data Availability Statement

The authors do not have permission to share data*.

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Research on the indentation behavior and safety evaluation of PE gas pipeline under point load

Abstract

Keywords

Introduction

Experiment

PE uniaxial tensile test

Point load experiment and material parameter validation

Finite element model

Pipe-soil modeling under point load

Grid independence verification

Results and discussion

Evaluation criteria for pipeline cross-sectional deformation

The influence of hard object size on PE pipe

The influence of hard object size on stress and strain

The influence of hard object size on indentation deformation

The influence of internal pressure on PE pipe

The influence of internal pressure on stress and strain

The influence of internal pressure on indentation deformation

The influence of diameter on PE pipe

The influence of diameter on stress and strain

The influence of diameter on indentation deformation

Brief summary

Conclusions

Footnotes

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

Data Availability Statement

References