Investigating peak stresses in fitting and repair patches of buried polyethylene gas pipes

Abstract

Nowadays, polyethylene composes a large number of natural gas distribution pipelines installed under the ground. The focus of the present contribution is two fold. One of the objectives is to investigate the applicability of polyethylene fittings in joining polyethylene gas pipes which are electrofused onto the pipe ends and buried under the ground, by estimating stress distribution using finite element method. The second objective is to study the effectiveness of polyethylene repair patches which are used to mend the defected pipelines by performing a finite element analysis to calculate peak stress values. Buried polyethylene pipelines in the natural gas industry, can be imposed by sever loadings including the soil-structure interaction, traffic load, soil’s column weight, internal pressure, and thermal loads resulting from daily and/or seasonal temperature changes. Additionally, due to the application of pipe joints, and repair patches local stresses superimposed on the aforementioned loading effects. The pipe is assumed to be made of PE80 resin and its jointing socket, and the repair patch is PE100 material. The computational analysis of stresses and the computer simulations are performed using ANSYS commercial software. According to the results, the peak stress values take place in the middle of the fitting and at its internal surface. The maximum stress values in fitting and pipe are below the allowable stresses which shows the proper use of introduced fitting is applicable even in hot climate areas of Ahvaz, Iran. Although the buried pipe is imposed to the maximum values of stresses, the PE100 socket is more sensitive to a temperature drop. Furthermore, all four studied patch arrangements show significant reinforcing effects on the defected section of the buried PE gas pipe to transfer applied loads. Meanwhile, the defected buried medium density polyethylene gas pipe and its saddle fused patch can resist the imposed mechanical and thermal loads of 22°C temperature increase. Moreover, increasing the saddle fusion patch length to 12 inches reduces the maximum stress values in the pipe, significantly.

Keywords

Ansys software / polyethylene / buried pipelines / polyethylene joints / polyethylene patches / peak von Mises stress / soil-pipe interaction / temperature variation

Introduction

Worldwide developments in science and technology and the production of new materials have prompted the quality of lives. Since plastics are used to design many products, a complete understanding of these materials is of great importance. Specially, the increasing demands for polymeric materials in engineering applications require new methodologies to evaluate material capability to withstand applied loads. The most common plastics, polyethylene, is obtained from petroleum or natural gas mainly by polymerizing ethylene gas. High density polyethylene (HDPE) is sometimes referred to as linear polyethylene due to its very low level of branching. The principal advantages of polyethylene including its balance of physical properties in the solid state, its chemical inertness, its low cost and ready processability make it the material of choice for a wide range of applications. Polyethylene is a tough and flexible material with a high electrical resistance. The mechanical properties of polyethylene material may be categorized into two wide types including low strain properties such as yield stress and initial modulus, and high strain properties. Like other engineering materials, the modulus of elasticity of polyethylene sample, which is a measure of its rigidity, is normally estimated from the initial slope of the force versus elongation plot. Up to the yield point the deformation in polyethylene specimen is principally elastic. The tensile strength also known as ultimate tensile stress of this polymeric material, which is the force required to break it divided by its cross-sectional area, depends largely upon the draw ratio at break of a sample. For high density samples the true ultimate tensile stress is approximately inversely related to their molecular weight. The tensile properties of polyethylene specimens are greatly affected by temperature where the response of samples to temperature is affected strongly by their molecular characteristics [¹].

Although in recent two decades, worldwide research has been conducted by scientists into renewable energy resources, including solar, wind, and ocean energy to replace petroleum, natural gas and condensates, the demand for fossil fuels energy sources are still increasing [²]. The discovery of natural gas reservoirs, drilling into the projected depth in the reservoir, producing natural gas wells into pipelines and transferring it to production facilities and homes are troublesome and of great importance for gas companies [³–⁵]. Nowadays the use of polymers in load-bearing structural components is increasing. In particular, polyethylene composes a large number of natural gas distribution pipelines which are being installed under the ground. Polyethylene material has the advantage of being cost effective, light weight, easy to handle, install and repair. Additionally, the physical properties of polyethylene perfectly meet the requirement of natural gas distribution lines. These advantageous attributes consist of chemically inert, low flexural modulus of elasticity, which enables the pipe to be coiled and bent, high impact strength, abrasion resistant, heat fusible and ability to be squeezed off without damage [⁶]. Beside the mentioned advantages, there are some limitations in the application of polyethylene. These disadvantages include its high thermal expansion, poor weathering resistance, stress cracking effects, bond difficulties, poor temperature capability, and flammability. Polyethylene indicates a high tendency toward stress relaxation. Failures caused by stress relaxation consist of leakage from compression fitting and joints which are formed by forcing flexible polyethylene tubing over a rigid pipe. Therefore, polyethylene is rarely used in applications in which stress relaxation is likely to happen [⁷]. In addition to stress relaxation polyethylene can fail in a brittle manner when subjected to stress concentrators such as scratches, notches, joints formed by welding, incisions on the surface and inclusions or scratches introduced during installation, especially in gas and water distribution pipes [⁷]. Flame retardancy of polyethylene is becoming an important issue for its industrial application. Since polymer combustion includes a certain pattern of preheating, decomposing, ignation and combustion, flame retardants have been used to interrupt the cycle [⁸]. The design, analysis, and modeling of a structural component can be done successfully through an understanding of the mechanical properties of the material. Due to the vast usage of polyethylene lines in the natural gas industry, gas companies are repeatedly seeking ways to develop repair methods. Even with these properties polyethylene pipe lines are still expose to damage during their lifetime. Therefore, efficient repair methods have been in development since polyethylene gas pipelines were installed under the ground. Normally, butt fusion and electrofusion weld procedures are used to join polyethylene pipe sections together. The conventional repair methods for underground polyethylene pipelines which includes large excavating sites of the damaged pipeline, isolating and squeezing off a large section of the pipe, cutting out the defected piece and fusion welding in a replacement section of pipe, is very expensive and time consuming, and specially these repairs unpleasantly affect traffic during the repair procedures. Therefore, gas companies have been moving toward keyhole isolation and repair technologies.

In the approaches for structural design, it is essential to keep structures working safe during their lifetime that needs obtaining a fundamental knowledge of the material failure [⁹–¹⁶]. For polyethylene materials, the rupture properties involves the generation of fresh surfaces under the effect of suddenly imposed tensile or flexural stresses. The main type of polyethylene rupture in thick specimens is crack propagation through material, such as the wall of a pipe. Inelastic deformation and the formation of new surface areas are the two processes that absorb energy in the rupture of polyethylene. The items manufactured from polyethylene are not stable under the application of the prolonged stresses. Morphological instability can display itself as creep, stress relaxation, brittle failure, and environmental stress cracking. Furthermore, well organized methods to computationally model failure and fracture include extended finite element method, phantom node methods, meshfree methods [¹⁷], the extended meshfree methods, phase-field modeling method [¹⁸–²⁰], efficient remeshing techniques and cracking particles methods, to name a few [²¹–³⁰]. Also, the thermal characteristics of polyethylene, particularly its somewhat low melting and softening temperatures, are among the primary factors that specify its realm of application [¹].

The simulation and analysis of the functional failure in simple composite pipes under the effect of internal pressure were investigated by Rafiee and Reshadi [³¹]. Generally, polyethylene pipelines are regarded as flexible. Underground polyethylene pipes deflect under loads, including over-burden and traffic loads. Gas company officials and engineers must provide the soil to establish an envelope of supporting material around the pipe and its related socket or patch, for the deflection to be preserved at an allowable tolerance. The pipe deflection range is determined by the aforementioned enveloping support, the burial depth, traffic loading from the surface, the pipe internal pressure, the pipe standard dimension ratio (SDR), and any local discontinuities resulting from different joints, patches or other accessible causes [³²]. Mainly, the supporting envelop which is called embedment is constructed by encompassing the pipe with dense and secure materials. The portion of the reinforcement by the embedment is straightly relative to its stiffness. Consequently, the embedment material is usually compacted [³³]. Significant load decrease may happen because of arching, which is, load decrease in the soil at the top of the pipe and around it which lead to load carrying away from the pipe. Therefore, while installing the polyethylene gas pipe under the ground, to obtain the needed standards, suitable materials must be employed in each layer around the pipe [³⁴]. Developing better understanding of the performance of structures, like buried polyethylene pipes will result in its safe operation under static, dynamic and environmental loads. Therefore, several researchers have attempted to perform experimental and analytical study on soil-pipe interaction to estimate loads, deflections and stresses [³⁵–³⁷]. Afterward, flexible pipe materials such as polyethylene were employed in underground applications. However, because of the complicated pipe-soil interaction, where the large amount of the imposed load is carried by the soil around the pipe, it is awkward to estimate the deflections and stresses in the soil and pipe materials. Therefore, numerical approaches were employed by researchers to investigate buried polyethylene pipes responses to the imposed loading conditions. Specially, Ansys, the finite element modeling software, has been used by authors for the simulation of underground gas pipe systems [³⁸–⁴²]. In the present contribution, the methodology described in Refs. [³⁸–⁴²] on finite element modeling of buried polyethylene gas pipes is applied to further estimate the maximum stress values in underground polyethylene gas pipe problems which are imposed to the local geometry changes in the form of socket joints and repair patches. The concentration of the present contribution is mainly divided to two fold. One of the objectives is to investigate the case of joining buried polyethylene pipes by electrofusion socket joints made of HDPE material and to determine the effectiveness range of this material for underground applications. The second objective of the research is to study the effectiveness of HDPE repair patches including four different patch arrangements which are suggested to repair damaged underground polyethylene gas pipes, and to obtain maximum stress values in the pipe and patch materials. In all computer simulations, the polyethylene gas pipe, socket joint, repair patches, the soil grades surrounding the pipe, socket, patch, and the underground installation methods and burial depth are all chosen according to the standards commonly used in gas distribution companies. For all parts of the research, the simultaneous effects of mechanical loads including pipe internal pressure, vehicle traffic load, soil column load, and more critical thermal loads of temperature variations in polyethylene pipe and its socket/patch on the induced stresses in PE80 pipe and PE100 socket/patch materials are investigated using Ansys software. The results show that under the application of the aforementioned loading conditions, stress concentrations will appear in the pipe and its socket/patch due to local changes in the pipe geometry compared to the case of simple buried pipe.

Basic design theory

The design and analysis of an underground pipe line is completely unlike the design and analysis of plant piping. Particular difficulties are included as the result of the special characteristics of a pipe line, codes needed and the expertises required in design. The study of mechanical response of a buried pipe line begins with the estimation of imposed loads. For a pipe line which is buried underground, any pipe motion has to overcome soil force, which can be categorized into two parts: friction force generated by sliding and pressure force developing from pushing. Therefore the main problem in pipe line design and analysis is to study soil-pipe interaction, which means the pipe and its surrounding soil act with each other to rule the pipe performance. The external loads appied on an underground polyethylene pipe are divided into two parts. The permanent load from the soil weight is called the dead load. The loads imposed at the surface such as vehicle live loads, which may not be permanent are called the surcharge load. In calculations for the earth load on flexible polyethylene pipes an easy way is to assume that the overburden load imposed to the pipe crown is equal to the soil column weight projecting above the pipe [⁴³]. The true load applied on a buried polyethylene pipe might in fact be much lower that the aforementioned applied load, because shear resistance shifts a portion of the earth load to trench sidewalls and embedment. To consider this consequence, the imposed pressure at the crown of the flexible pipe is estimated via Eq. (1) which is known as Marston equation:

(1)P_{\mathrm{M}}=C_{\mathrm{D}}W{B}_{\mathrm{D}},

where P_M corresponds to vertical soil pressure, B_D is the trench width at pipe crown, C_D is load coefficient, and W corresponds to the unit weight of the soil. In Eq. (1), which calculates the pressure at the pipe crown, it is assumed that the obtained stress is the same all around the pipe. Clearly, because of the pipe flexibility and pipe-soil interaction, this approximation is not true and therefore, local changes in stress values should be included in the calculations. Additionally, a significant load in buried flexible pipe design is surcharge load, mostly in the form of truck wheel load. Based on the American association of state highway and transportation officials (AASHTO), this loading condition which is superimposed to the soil pressure on the pipe can be calculated by using H20 truck highway loading [⁴³]. Also, soil classification systems for underground flexible pipe installation depend on the mechanical and physical properties of the soil and are based on AASHTO and American society for testing and material (ASTM) standards. The calculation of the applied loads on underground gas pipe lines which is based on the method of pipe installation in the trench exists in the ASTM D2321 standard [⁴⁴–⁴⁷]. In the present research, the procedure discussed in Refs. [³⁶–³⁹], is employed in the problems involved buried pipe stress estimation and the loading condition, problem dimension and also, mechanical properties of the soil grades used for different soil layers including bedding, haunching, initial and final back fill are selected according to the aforementioned references.

With reference to the mechanical behavior of polyethylene, a reduced strength can be used in stress calculations and then consider the material as linear elastic, even though, polyethylene pipes are actually viscoelastic in nature, where the creep response is of great importance. This method is performed either by defining an effective modulus or by employing the minimum required strength (MRS) as below equation:

(2){\sigma }_{\mathrm{s}}={\displaystyle \frac{MRS}{\mathrm{C}}},

where s_s corresponds to the design stress, MRS is the minimum required strength, and C is a constant, called design coefficient [⁴³,⁴⁶,⁴⁸].

Modeling of underground polyethylene gas pipe

In this research, PE80 and PE100 materials are selected for buried gas pipe and its jointing socket respectively. According to study the peak stress values in an underground polyethylene gas pipe, subjected to different loading conditions, the value of 11 is chosen for both pipe and its socket SDR (Standard dimension ratio). Mechanical and physical properties of PE80 pipe and PE100 socket materials along with their dimensions are reported in Table 1.

Figure 1 illustrates the electro fused fitting dimensions used to connected polyethylene pipes. In this figure, D, L and H are Socket inside diameter, Socket length, and Socket height, respectively [³⁴]. Both PE80 and PE100 resins are identified as suggesting exceptional long-term performance as pressure pipes. Particularly, PE100 pipe material proposes several advantages for joining PE80 gas distribution pipes. PE100 material is the proper option to join PE80 pipes. PE100 material can bear the minimum hoop stress of 10 MPa for 50 years in 20°C. In contrast, the minimum hoop stress for PE80 material for a working life of 50 years in 20°C is as low as 8 MPa. The greater long-term creep resistance of PE100 materials results in the higher working pressure at identical wall thickness, or possibly, thinner wall pipe or fitting at identical working pressure, providing higher hydraulic capacity than PE80 materials. Additionally, PE100 materials shows better notch resistance compared with PE100 materials. This indicates they are surprisingly less influenced by surface defect while laying, which has apparent superiority for ploughing in and particularly trenchless methods. PE100 pipe materials have advanced resistance to fast crack growth. Consequently, if a rapid crack starts to grow in a pipe made from PE100, it will be stopped following just a short distance of propagation. This issue is of significant importance in the fuel gas distribution pipe lines because a rapid crack failure could results in catastrophic effects.

With respect to study the minimum stress values in the electrofusion fitting and buried pipe, the combination effects of soil column load, surcharged wheel load, polyethylene socket joint and pipe temperature variation and imposed internal pressure are investigated and the structural problem is accurately modeled in the commercial package of ANSYS V12 software. Additionally, as the result of similar influence of applying concentrated and distributed traffic wheel load in the assumed installation depth, the distributed traffic load is considered and studied for all created underground models. Furthermore, the soil-pipe interaction is also implemented in the models by applying face-to-face contact elements of conta 172 and target 170, existing in ANSYS modeling library.

Results and discussions

Figure 2 depicts a typical dimensional view of the pipe surrounding layers in the trench. For this research, the dimension of x = 1.5 m and z = 1.5 m is selected for the model upper surface. As clarified in the previous section, this contribution focus includes two separate folds of electrofused fittings and repair patches, where the conventional loading conditions including traffic wheel load, soil column weight, temperature daily and seasonal variations, internal pipe pressure, and the presence of the local changes in the pipe geometry are subjected to evaluate the stress distribution in the buried PE80 pipe and its related fitting and patch made of PE100 resins. With respect to the working temperature of 35°C for buried pipe lines installed at a depth of 125 cm, and based on minimum strength value for polyethylene PE100 resin, the maximum allowable stress for this contribution is 5.4 MPa. Furthermore, the design stress value is estimated to be 4.3 MPa for the PE80 pipe material, considering that the value of 1.5 is selected for the safety factor.

Electrofused fittings stress evaluation

For the first fold of the research, after simulating and finite element modeling of the socket joint buried polyethylene pipe problem, the aforementioned mechanical and thermal loading conditions which are imposed to the fitting and pipe sections during the lifetime are computationally applied to the finite element model as software inputs and finally stress distributions in fitting, pipe and surrounding materials which are the software outputs, are determined as well. The physical properties and dimensions used to prepare the model elements, including polyethylene pipe, fitting and surrounding soils are based on references [38–40]. As the result of geometry and loading conditions symmetry, the simulation contains the modeling of half of the physical problem which halves the solution time and computational costs. The curves on Figs. 3 and 4 indicate the maximum axial stress and the maximum von Mises stress values in the wall of buried polyethylene fitting used for gas distribution pipe joining, under the application of simultaneous mechanical loads including soil column load, 4 bar internal pressure, traffic load and thermal loads in the form of maximum temperature decrease of 15°C and rise of 5°C. For both figures, the value zero and ten on horizontal axis corresponds to the fitting internal and external surfaces positions, respectively, and stresses values are plotted from the internal to the external surface of the fitting in y-direction.

Furthermore, the curves on Figs. 5 and 6 indicate the concurrent results of the said mechanical and thermal loads on the maximum axial and von Mises stresses along the length of the fitting used to join two buried polyethylene gas pipes. For both figures, the horizontal axis is in x-direction and its origin is supposed to be on the fitting head. The upmost dashed lines are for the value of allowable stress of PE100 resin.

Similarly, the curves on Figs. 7 and 8 indicate the peak values of axial and von Mises stresses along one of the polyethylene gas pipes connected to the other pipe by polyethylene fitting, under the simultaneous effect of soil column weight, internal pressure, traffic load and temperature change, respectively. Because of the symmetry, the results are the same for the other pipe section, too.

The results of Figs. 3−8 show that+ 5°C temperature increase will cause around 31% and 63% decrease in the peak axial stress in the fitting and pipe, respectively, and the same temperature rise will result in about 19% and 3.8% decrease in the peak von Mises stress in the fitting and pipe, respectively. Also, the effect of a −15°C temperature decrease will be the increase of the peak axial stresses in the fitting and pipe to 64% and 235%, respectively, while the same temperature drop will result in about 73% and 12% increase in the peak von Mises stress in the fitting and pipe, respectively. The stress plot results of ANSYS show that the peak stress values take place in the middle of fitting and at its internal surface, while the peak stress value for the pipe is at the connection end. It can be implied from the results that under the assumed loading conditions the peak stress values are well below the allowable stresses and the application of this kind of fitting is appropriate for even hot calamite areas such as south-west of Iran.

Repair patch stress evaluation

In this section which is the second fold of the research, we use finite element method and continue the study of damaged buried polyethylene gas pipelines, which was described in Ref. [39], where the pipe crown was subjected to notch shape defects with specified sizes. Furthermore, we extend the research performed in Ref. [⁴⁰], where several patch arrangements of PE100 material were introduced to repair defected buried polyethylene gas pipes. The efficiency of partial circular, partial square, semi-cylindrical and saddle fusion patches can be well investigate through the stress analysis which can be obtained effectively by computer simulation tools such as ANSYS software, to find the peak stress values under sever imposed mechanical and thermal loads in real working conditions. Pipe, repair patches and surrounding soils were modeled and discretized using three dimensional solid95 elements from the element library of the commercial ANSYS v12 package. One of the main advantages of solid element is that, each node has three degrees of freedom (translations in the x, y, and z coordinate directions) that cause it to be a very appropriate choice for dealing with models containing curved boundary. Moreover, in order to obtain an optimum values of finite elements and consequently to minimize the computational expenses and required solution time, the regular pattern structured finite element meshes were generated in a careful manner. To compare the patch efficiencies, it is assumed that the wall thickness, inside diameter and material are identical for all introduced patches. The linear elastic behavior and isotropic physical properties of polyethylene material is introduced to the software.

Figure 9 depicts four patch arrangements including partial square, partial circular, semi-cylindrical and electrofusion patches and related dimensions. The pipe cross section is illustrated in the middle of the figure which is the same for all models. The patch thickness is selected to be identical and equal to 4.763 mm. Moreover, the material properties of pipe and patch resins are same as that presented in Table 1. In the following sections we extend the study of the explained simulations and investigate more aspects of the effects of thermal and mechanical loading conditions including, temperature changes, surcharge loads, soil weight load, soil-pipe interaction, and pipe internal pressure imposed to the repaired defected polyethylene pipe buried under the ground and its patch. For this part of the research the value of 1.25 is selected for the factor of safety regarding to 50 years life expectancy and 35°C design temperature. Therefore, the values of allowable stresses will be calculated as 5.2 and 6.5 MPa for PE80 pipe and PE100 patch materials, respectively. Prior to beginning to evaluate the complex problem of buried repaired polyethylene gas pipe and to investigate the effect of pipe-soil interaction on underground patch repaired polyethylene pipe lines, we execute the study on the validation of the results by finite element simulation of unburied patch repaired polyethylene pipe problem under the sole effect of the internal pressure and the results are presented in the next section.

Finite element modeling of unburied patch repaired pipe for stress evaluation

In this section, we determine the peak stress values in an unburied medium density polyethylene (MDPE) gas pipe under internal pressure subjected to notch shape damages and then repaired by attachment of a 3 inches saddle fusion patch. The obtained results can be helpful in the finite element model validity of the computational calculations and finally to find a computer model to design a proper patch. Consequently, the pipe and patch arrangements shown in Fig. 9 that is loaded to an internal pressure of 4 bar is investigated in ANSYS software. The created defects at the pipe crown are circular holes of 5 to 20 mm diameter. To obtain accurate results from computational modeling the pipe and patch finite elements are merged together to model a firm seamless elecrofusion connection. As the saddle fusion patch is concentric with the pipe itself, so the model geometry is created in ANSYS so that the origin of coordinate system to be aligned with the pipe and patch, while the z-axis to be aligned with the axis of the pipe and its related patch. Additionally, the proper boundary conditions can be selected for the finite element modeling in the present research is as Eqs. (3) and (4):

(3)z=\pm {\displaystyle \frac{l}{\mathrm{2}}},u_z=\mathrm{0},

(4)x=\mathrm{0},u_x=\mathrm{0.}

In the z-direction, plane strain boundary condition can be applied to the node located at both ends of the pipe model, as the pipe is assumed to be long pipe and is formulated as Eq. (3). Also, as can be seen in Fig. 9, we have geometry and loading symmetry in y-z plane, which is formulated by Eq. (4).

Peak stresses in unburied patch repaired pipes

In Table 2 the results of finite element method for maximum values of hoop and von Mises stresses in pipe and its related patch are presented for several hole sizes and pipe lengths.

Also, in order to obtain accurate results, different mesh conditions and number of elements are examined to evaluate the good performance of the present method. Considering the obtained solutions for the hoop and von Mises stresses from the finite element solution it can be concluded that:

1) For pipe models with identical hole sizes, increasing the pipe length above 0.5m will not affect the maximum hoop and von Mises stress values in the pipe and also in the patch, significantly.

2) The peak values of hoop and von Mises stresses in the patch will increase by increasing the hole size. Otherwise, increasing the hole size will not increase the maximum value of the aforementioned stresses in the pipe significantly because of the patch effect.

3) Comparing the results of similar problems with different number of elements indicates that if the distribution of the used finite elements is done properly, increasing the number of elements will not affect the results in pipe and patch significantly.

4) As can be seen in Fig. 10, the maximum values of hoop and von Mises stresses in the defected pipe occur on inside the pipe wall and around the hole at the sides of the defect location.

It can be implied that mesh type, finite elements, loading conditions, and applied boundary conditions in this section are selected appropriately and can be used in the next steps of the study and can be used to perform finite element modeling of the other three types of patch arrangements. Therefore, the value of 1.5m is selected for the pipe length in the next modeling stages.

Investigating different patch arrangements for unburied pipes

As we mentioned in the introduction, the problem of underground patch repaired pipes is very complicated because of the effect of soil-pipe interaction. Therefore, at first we study the simple case of patch repaired defected pipes which are not buried under the ground, that will help us in better understanding of the underground problem. Four different patch arrangements in the form of semi-cylindrical, circular-partial, square-partial, and saddle fusion patches are modeled and discussed. The same procedure described in previous section is followed and used to calculate the maximum stresses in the patch and pipe loaded to an internal gas pressure of 4 bar. The number of finite elements used in this study to model unburied pipe and its patch for various patch arrangements are presented in Table 3. The plotted curves in Fig. 11 show the variation of maximum von Mises stress in the defected patch repaired polyethylene pipe for each kind of patch arrangement and also in unrepaired defected pipe, versus different hole sizes in the pipe wall. Additionally, the decrease in peak von Mises stress values after applying a particular patch shape are depicted and compared in Table 4.

The data presented in Table 4 and the curves of Fig. 11 show that:

1)Square partial, circular partial, semi-cylindrical, and saddle fusion patches can reduce the peak von Mises stresses in the defected pipe, significantly.

2)The results of Table 4 show that the percentage of the maximum von Mises stress reduction in the pipe increases for larger hole diameters.

3)Saddle fusion patch is the most effective one among others and decreases the peak von Mises stresses about 48%. The minimum stress reduction is related to square-partial patch, which reduces the maximum von Mises stresses by about 8%.

4)Semi-cylindrical and saddle fusion patch arrangements have similar responses. Also, circular-partial patch and square-partial patches show similar responses.

5)The load-bearing effect of semi-cylindrical and saddle fusion patches are significantly more than the other two patch arrangements.

Investigating underground patch repaired pipe models

In this section we continue to investigate the studies in Refs. [³⁸,³⁹] with more details and under more critical loading conditions by superimposing various thermal loads in the form of temperature variations and the mechanical loads including surcharge loads, soil column weight, soil-pipe interaction and inside pressure of 4 bar applied to the repaired defected polyethylene pipe and its patch. The trench dimensions and pipe surroundings are selected based on Fig. 12. Also, soil grades which are selected based on ASTM standards along with model dimensions are depicted on the computer simulation model in Fig. 12. The results of previous section show that the maximum von Mises stress values are well below the allowable stresses (based on the design factor of 1.5) for working life of 50 years at 35°C. Additionally lower values of safety factor (design factor) are applicable to design underground gas pipes. By selecting the value of 1.25 for safety factor, the values of allowable stresses for working life of 50 years at 35°C will be 5.2 and 6.5 MPa for PE80 pipe and PE100 patch materials, respectively. We prefer to use these values for the following section of the research. To perform a proper finite element study on the underground structure, it is important to find an appropriate model dimension which the obtained results not to be dependent on the model size. Therefore, the finite element simulation of various model sizes are calculated and compared in Table 5. The results of maximum von Mises and hoop stresses in the pipe and its patch show that the upper surface dimension of z× x = 1.5 m × 1.5 m is appropriate to perform finite element modeling of this research, since larger models will not affect the stress values significantly. Also, the boundary conditions are selected as:

(5)z=\pm {\displaystyle \frac{l}{\mathrm{2}}},u_z=\mathrm{0},

(6)x=\mathrm{0},u_x=\mathrm{0},

(7)y=-\mathrm{207.15}\mathrm{mm}\mathrm{;}{{\displaystyle u}}_x={{\displaystyle u}}_y={{\displaystyle u}}_z=\mathrm{0.}

The simultaneous effects of thermo-mechanical loads on semi-cylindrical patch arrangement

Figure 13 shows the variations in the maximum values of von Mises stresses in the buried PE80 pipe that is repaired by a 76.2 mm long semi-cylindrical patch arrangement, versus defect sizes in the form of circular holes under simultaneous effects of mechanical loads in the form of soil load, 4 bar internal pressure, vehicle wheel load and various thermal loads in the form of daily and seasonal temperature variations. The upmost curve shows the values of maximum von Mises stress for the defected pipe before the application of patch repair and indicates that increasing hole diameter increases the maximum von Mises stress significantly. The comparison of this curve with allowable stress value for a working life of 50 years (dashed line) shows that the imposed stresses in defected pipe are significantly higher than allowable stress values. For the other four curves which show the maximum von Mises stress values in the defected polyethylene pipe repaired by semi-cylindrical patch, the stress values are well below the results of unrepaired defected pipe and also well below the allowable stress value. This means the aforementioned patch repair can strengthen the defected part of the pipe as well to transfer the gas fuel. Additionally, for the investigated defects at a constant temperature variation, the maximum von Mises stresses remain approximately constant even by increasing the hole size. The curves for −22°C, −15°C, 0°C (no temperature changes), 5°C, and 22°C temperature changes shows approximately similar trend with hole diameter increase where for these cases the minimum von Mises stresses are not increasing significantly for larger hole sizes. Generally stating, the maximum von Mises stresses in the pipe increases for higher temperature increases. That means the patch is more effective in reinforcing defected pipe for lower temperature changes. For example, for a fixed hole diameter of 20 mm, the stress reduction percentage in the pipe wall are 53%, 53%, 46%, 44%, and 35% for the temperature changes of −22°C, −15°C, 0°C, 5°C, and 22°C, respectively. Also, the patch is more effective in reinforcing defected pipe for larger hole diameters. The maximum stress values are related to the seasonal temperature increase of 22°C.

Additionally, Fig. 14 presents maximum values of von Mises stresses in the 3″ long semi-cylindrical patch arrangement versus pipe circular hole diameters under simultaneous effects of mechanical loads and various thermal loads in the form of daily and seasonal temperature variations. Based on the obtained results, the temperature variations have a significant effect on the maximum von Mises stresses in the semi-cylindrical patch itself. The lowest curve on Fig. 14 which shows the lowest values of maximum von Mises stresses belongs to the situation where no temperature change is imposed to the pipe and patch at the burial depth under the ground that means it is the case with the minimum valves of the induced maximum von Mises stresses compared to the other temperature changes. Additionally, the uppermost curve fits the data obtained for the maximum temperature change (22°C temperature increase, based on seasonal variations), which shows the case with highest values of induced maximum von Mises stresses among others. For the semi-cylindrical PE100 patch material it can be implied that higher temperature changes (both temperature increase and temperature decrease) impose higher maximum von Mises stresses. Also, for the cases of low temperature changes including 0°C, 5°C, the maximum von Mises stresses in the patch show an increase trend by increasing the pipe hole diameter, while for the cases of higher values of temperature changes including −15°C, −22°C, 22°C the maximum von Mises stresses will remain approximately constant even for larger hole diameters. Based on the calculated results, the semi-cylindrical patch can reinforce the proposed circular hole modeled defects efficiently. The only problem is that for higher temperature changes, the maximum von Mises stress values in the patch itself can be critical which requires more research and investigation on the other patch configurations.

The simultaneous effects of thermo-mechanical loads on circular partial patch arrangement

In this section in order to more investigate to find a proper patch, a circular partial patch arrangement is designed and its reinforcing effect on the damaged underground polyethylene gas pipe are being studied. The results of the FE simulation in the form of variations in the maximum values of von Mises stresses in the buried PE80 pipe that is repaired by the circular partial patch arrangement, versus defect sizes in the form of circular hole under simultaneous effects of the previously mentioned mechanical loads and various thermal loads are depicted in Fig. 15.

Similar to the previous case, as expected the upmost curve is related to the values of maximum von Mises stresses for the defected pipe prior to circular-partial patch application and shows that for larger hole diameters, the maximum von Mises stress increases significantly. Based on the results of the other five presented curves which shows the maximum von Mises stresses in the defected polyethylene pipe repaired by circular-partial patch and comparing these curves with dashed line that indicates allowable stress values, the imposed stress values are well below the results of unrepaired defected pipe but slightly above the allowable stress values. Considering the case of 20 mm hole diameter, the stress reduction percentage in the pipe wall are 31%, 29%, 26%, 24%, and 22% for the temperature changes of −22°C, −15°C, 0°C, 5°C, and 22°C, respectively. The results imply that for the application of the aforementioned patch repair to reinforce the damaged part of the pipe some stress relief mechanisms must be employed too. Additionally, for the investigated defects at a constant temperature variation, the maximum Mises stresses remain approximately constant even by increasing the hole size. The curves for −22°C, −15°C, 0°C, 5°C, and 22°C temperature changes shows approximately similar trend with hole diameter increase. The maximum stress values are related to the seasonal temperature increase of 22°C.

Additionally, the maximum values of von Mises stresses in the circular partial patch arrangement versus pipe circular hole diameters under simultaneous effects of aforementioned mechanical loads and various thermal loads in the form of daily and seasonal temperature variations are presented in Fig. 16. According to the obtained results, the temperature variations have significant effect on the maximum von Mises stress variations in the circular partial patch. The lowest curve on Fig. 16 belongs to the situation where no temperature change is imposed to the pipe and patch at the burial depth under the ground that means it is the case with the minimum valves of the induced maximum von Mises stresses. Additionally, the uppermost curve fits the data obtained for the maximum temperature change (22°C temperature decrease, based on seasonal variations), which shows the case with highest values of induced maximum von Mises stresses among others. For the circular partial PE100 patch material it can be implied that higher temperature changes (both temperature increase and temperature decrease) impose higher maximum von Mises stresses.

For higher temperature changes, including a 22°C temperature increase and a −22°C temperature decrease, the maximum von Mises stresses are above the allowable stress which means applying circular partial patches in these areas cannot be suggested. For lower temperature changes including −15°C, 0°C, and 5°C the maximum von Mises stresses are well below the allowable stress limit, which means the circular partial patch is applicable in the areas with the temperature changes up to the mentioned values.

The simultaneous effects of thermo-mechanical loads on square partial patch arrangement

The same procedure which was discussed for semi-cylindrical and circular partial patches in the two previous sections is used to investigate square partial patch arrangement by finite element method. The results of Ansys simulation for the variations in the maximum values of von Mises stresses in the buried PE80 pipe that is repaired by a square partial patch arrangement, versus defect sizes in the form of circular hole under simultaneous effects of mechanical loads in the form of soil load, 4 bar internal pressure, vehicle wheel load, and various thermal loads in the form of daily and seasonal temperature variations are depicted in Fig. 17. Comparing the curves showing the results of maximum von Mises stress values for various temperature changes and the upmost curve which is related to the defected unrepaired pipe, shows that square-partial patch arrangement plays an important role in decreasing maximum stress values and strengthening the defected part of the pipe. For more understanding, considering the case of the 20 mm hole diameter, the results show the stress reduction percentage in the pipe wall are 29.5%, 29.2%, 25.6%, 24.3%, and 21.7% for the temperature changes of −22°C, −15°C, 0°C, 5°C, and 22°C, respectively, which shows that patch has more reinforcing effects for lower temperature changes. Even though, the square-partial patch has significant effect on reinforcing the defected pipe and decreasing the maximum von Mises stress values, but comparing the results with the dashed line of allowable stress shows that the induced maximum von Mises stresses are slightly higher than allowable stress values. Therefore, if we decide to use this kind of patch, more researches should be conducted to obtain some stress relief mechanisms. Additionally, the curves show similar trends for different temperature changes. Furthermore, comparing these curves with the results obtained for circular-partial patch shows approximately similar trends between these two cases.

Additionally, the maximum values of von Mises stresses in the square partial patch arrangement versus pipe circular hole diameters under simultaneous effects of aforementioned mechanical loads and various thermal loads in the form of daily and seasonal temperature variations are presented in Fig. 18. As can be implied, the temperature variations have a significant effect on the variation of the maximum von Mises stresses in the square partial patch. The lowest curve on Fig. 16 belongs to the situation where no temperature change is imposed to the pipe and patch at the burial depth under the ground that means it is the case with the minimum valves of the induced maximum von Mises stresses. Additionally, uppermost curve fits the data obtained for the maximum temperature change (22°C temperature increase, based on seasonal variations), which shows the case with highest values of induced maximum von Mises stresses among others. It is clear that for low temperature changes, including 0°C and 5°C, the maximum von Mises stresses in the patch increases for larger hole diameters. For higher temperature changes including −15°C, −22°C, and 22°C the results will remain approximately constant even by increasing hole diameter. For the square partial PE100 patch material it can be implied that higher temperature changes (both temperature increase and temperature decrease) impose higher maximum von Mises stresses. The curves showing the data of patch stress results are below the allowable stress of the patch material except for the case of 22°C temperature increase.

The simultaneous effects of thermo-mechanical loads on saddle fusion patch arrangement

In the previous parts, three patch repair configurations were introduced and discussed in details. Finite element solutions for investigating the stress distribution in the aforementioned patch repairs to find the effectiveness of the proposed patch arrangements gave us the knowledge that semi-cylindrical patch configuration can effectively reinforce the defected part of the pipe. For the sake of finding a more reliable patch configuration we decide to investigate a full-cylindrical (called saddle fusion) patch repair. To verify the finite element model dimension, previously, the variation of maximum von Mises stresses in the repaired defected buried polyethylene gas pipe and its 3 inches long, saddle fusion patch under mechanical loads is well discussed. Figure 19 shows the variations in the maximum values of von Mises stresses in the buried PE80 pipe that is repaired by a saddle fusion patch arrangement, versus defect sizes in the form of circular hole under simultaneous effects of mechanical loads in the form of soil load, 4 bar internal pressure, vehicle wheel load, and various thermal loads in the form of daily and seasonal temperature variations. The comparison of the curves resulted from saddle fusion patch repair for various temperature changes of −22°C, −15°C, 0°C, 5°C, and 22°C with the upmost curve which shows the maximum von Mises stress values for defected pipe before repair shows that the saddle fusion patch repair effectively reinforces the damaged part of the pipe to reliably transfer the gas. Additionally, comparing the mentioned five curves with the dashed line that indicates allowable stress values for PE80 pipe material shows that the maximum von Mises stress values are well below the allowable stress. For more clarification considering the case of 20 mm hole diameter, the results show the stress reduction percentage in the pipe wall are 54%, 56%, 45%, 43%, and 36% for the temperature changes of −22°C, −15°C, 0°C, 5°C, and 22°C, respectively. All the five curves show similar trends.

Figure 20 presents maximum values of von Mises stresses in the 3 inch long, saddle fusion patch arrangement versus pipe circular hole diameters under simultaneous effects of previously mentioned mechanical and various discussed thermal loads. The obtained results show that, the temperature variations have significant effect on the maximum von Mises stresses in the saddle fusion patch. similar to the 3 previously discussed patch arrangements the lowest curve on Fig. 20 belongs to the situation where no temperature change is imposed to the pipe and patch at the burial depth under the ground that means it is the case with the minimum valves of the induced maximum von Mises stresses. Additionally, the uppermost curve fits the data obtained for the maximum temperature change (22°C temperature increase, based on seasonal variations), which shows the case with highest values of induced maximum von Mises stresses among others. For the saddle fusion PE100 patch material it can be implied that higher temperature changes (both temperature increase and temperature decrease) impose higher maximum von Mises stresses. For all curves, the maximum von Mises stress values in the patch will increase for larger hole diameters. For all cases, the maximum von Mises stresses are well below the allowable stress limit for PE100 material which means the saddle fusion patch is well applicable to repair the proposed defects in even hot areas. Therefore, the results show the application of 3 inches saddle fusion patch is advisable for the proposed loading condition.

Size effect investigation of saddle fusion patch

The results of Ref. [⁴⁰] and the data obtained in the previous sections show that the application of 3 inches saddle fusion patch is advisable for the proposed loading condition. To further investigate the size effect of saddle fusion patch and to design an optimum patch arrangement, a longer 12 inches saddle fusion patch is modeled and solved for stress determination. In Fig. 21 the maximum values of von Mises stresses are presented and compared for the short (3 inches) and long (12 inches), saddle fusion patch arrangements versus pipe circular hole diameters under simultaneous effects of mechanical loads in the form of soil column load, 4 bar internal pressure, vehicle wheel load and thermal loads in the form of 5°C seasonal temperature increase. Based on the results, the longer patch shows significant effect on the reduction of imposed maximum von Mises stresses in the defected pipe.

Conclusions

In this research an uncomplicated and practical computational simulation is proposed to solve real industrial fully three-dimensional complex problems of large buried gas pipelines. The finite element method is employed to estimate maximum stress values in buried gas pipeline imposed to thermal and mechanical loads and stress concentrations due to variations in the geometry. This research was performed in two fold. In the first section, stress values were estimated for the fitting joint of the buried polyethylene pipe so that by applying the proper pipe joints, the stress values to be reduced to levels below the allowable values. For the second section, the effects of the aforementioned thermal and mechanical loads on maximum stress values in patch repaired underground pipes are well studied. For this purpose, in this research, 3D finite element modeling of buried gas pipe and its socket/patch is performed using ANSYS software. Stress variations in the buried MDPE gas pipe and HDPE fitting and repair patch were fully studied in a hot climate region to estimate the critical stress values caused by stress concentrations at the pipe and fitting/patch, enabling us to find the applicable method for joining or repairing MDPE gas transportation piping in such areas. The optimum burial depth was found to be 1.25 m, while the maximum and minimum ground surface temperatures at this depth were calculated to be 35°C and 13°C, respectively. Furthermore, the soil column weight above the pipe, the surcharge loads in terms of traffic load, the gas pressure of 4 bar inside the pipe and the stress concentrations due to a local change in geometry (in the form of damage) were imposed on the pipe and its socket/patch resulting in the following conclusions: by considering the obtained plots for stress values in buried pipe and its socket and comparing it with allowable stress values for the pipe, the correct joining method can be investigated. Based on the results, maximum von Mises stresses occurs at the middle of the socket internal surface while the maximum values of the aforementioned stresses in the socket occur where the internal surface of the socket joins the pipe outer surface. In both pipe and socket the maximum values of the aforementioned stresses are well below the allowable stresses and therefore the introduced socket joint can be used under the described working condition. Additionally, for the problem of the patch repaired buried pipes, the results show that all four patch configurations have significant reinforcing effect on the defected section of the buried pipe under the aforementioned thermo-mechanical loads. Meanwhile, the maximum von Mises stresses in both pipe and saddle fusion patch are well below the allowable stress limit for polyethylene material which means the saddle fusion patch is well applicable to repair the proposed defects in even hot areas. Therefore, the results show the application of introduced socket and also saddle fusion patch is advisable for the proposed loading condition.

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