Evaluation of polyethylene pipe–sandy soil interaction subjected to strike-slip faulting

C_c

coefficient of curvature of soil, used in the Unified Soil Classification System

C_u

coefficient of uniformity of soil, used in the Unified Soil Classification System

D

external pipe diameter (m)

D₁₀

size of soil particles for which 10% of the particles are finer (mm)

D₃₀

size of soil particles for which 30% of the particles are finer (mm)

D₆₀

size of soil particles for which 60% of the particles are finer (mm)

D_i

reference displacement of point i (m)

d_i

analysed displacement of point i for the iteration (m)

E

modulus of elasticity

E_i

iteration error of displacement i

Er_n

overall error

E_s

slope of the linear portion (N/m)

F₁

force of the spring in the non-linear phase (N)

F₂

force of the spring in the non-linear phase (N)

F_y

yield stress (MPa)

H

burial depth to the centre line of the pipe (m)

I

moment of inertia

K₀

stiffness of the pipe–soil equivalent spring per unit length of the pipe (kN/m/m)

M

bending moment

N_qh

horizontal bearing capacity factor

P_u

lateral pipe–soil interaction force per unit length of pipe (kN/m), according to American Society of Civil Engineers (ASCE) guidelines

P_y

yield force capacity of the equivalent pipe–soil spring per unit length of pipe (kN/m), determined by the pipe–soil interaction optimisation software

t

wall thickness of the pipe (mm)

V

shear force

w

load

y″

second derivative of the vertical deformation of the beam

y⁽³⁾

third derivative of the vertical deformation of the beam

y⁽⁴⁾

fourth derivative of the vertical deformation of the beam

y_u

ultimate relative displacement capacity (m), defined by ASCE guidelines

y_y

yield relative displacement capacity (m)

γ

soil unit weight (kN/m³)

γ_average

average density of the soil profile (kg/m³)

Φ

internal friction angle of sandy soil (°)

Polyethylene pipes are predominantly used to transfer different types of liquids and gas. These pipes can be a good alternative to other types of pipes, including steel pipes, due to their ease of transportation, the fact that they neither corrode nor rust, their thermal insulation, high flexibility and so on. Buried pipelines are constantly at a risk of being damaged by different geological hazards such as landslides, faulting movements and seismic events (Ariman and Muleski, 1981; Guo and Stolle, 2005; O’Rourke and Liu, 1999; Rahman and Taniyama, 2015). The seismic analysis of buried pipelines should be carried out by correctly estimating the interaction that takes place between the soil and the pipe. In recent earthquakes, buried pipelines are reported to have sustained significant damage (Ariman and Muleski, 1981; Bruneau et al., 2000; O’Rourke, 1992; O’Rourke et al., 2014; Petak and Elahi, 2001; Trautmann and O’Rourke, 1985). Many studies have been carried out on the interaction between buried pipelines and the soil surrounding them (Abdoun et al., 2009; Ariman, 1983; Ariman and Muleski, 1981; Guha et al., 2013; Guo and Stolle, 2005; Wagner et al., 1989; Xie et al., 2011; Yoshizaki et al., 2003; Zheng et al., 2012). However, the complexities associated with the modelling of soil–pipe interaction have made the introduction of a reliable analytical method difficult. Proposing simple methods with acceptable accuracy for engineering evaluations has been the aim of many investigations (Karamitros et al., 2007; Liu et al., 2017, 2018; O’Leary and Datta, 1985; Trautmann and O’Rourke, 1985; Villarraga et al., 2014; Xie, 2008; Zhou and Murray, 1996). The way that a buried pipeline is commonly modelled is shown in Figure 1. This model is a two-dimensional (2D)/three-dimensional representation of a pipe that is constructed using beam elements. The surrounding soil with which the beam interacts is modelled using a series of bilinear longitudinal, transverse horizontal or vertical springs (ASCE, 1984).

The accuracy of the results of the analyses depends largely on how well the behaviours of the equivalent springs are defined. The American Society of Civil Engineers (ASCE) guidelines (ASCE, 1984) use elastic-perfectly plastic bilinear springs to obtain the soil–pipe interaction curve (Figure 2). The elasto-plastic model is fully characterised by two parameters: (a) the maximum resistance t_u, p_u or q_u in the axial, transverse horizontal and transverse vertical directions, respectively, and (b) the maximum elastic deformation x_u, y_u or z_u (O’Rourke and Liu, 1999). The ASCE guidelines (ASCE, 1984) suggest using as the effective stiffness (units of force per unit area) twice the ratio of the ultimate resistance to the maximum ‘elastic’ deformation (x_y, y_y or z_y) – for example, 2p_u/y_u for a transverse horizontal spring. Note that for elasto-plastic idealisation, this spring coefficient is effective only for relative displacements less than half the maximum values of x_u, y_u and z_u, beyond which the resistance is assumed constant.

The first experimental assessment of pipe–soil interaction was conducted by Trautman and O’Rourke (1985). The following relationships were proposed so that the force applied to the pipe unit length and also the yield and ultimate displacements of the pipe could be calculated:

The diameter of the pipe, the thickness of the pipe, the pipe burial depth the specific weight of the soil and the horizontal bearing capacity factor of the soil are represented by D, t, H, γ and N_qh, respectively. Even though the ease of use of the equations proposed by Trautman and O’Rourke (1985) has been the major factor behind their widespread use and despite the fact that these equations are the basis of many of the criteria recommended by ASCE, many studies have revealed multiple uncertainties. Abdoun et al. (2009) investigated different factors affecting the interaction that takes place between a buried pipe and the soil that surrounds it when the entire system is under the action of a strike-strip fault. The authors concluded that the transverse spring presented by the ASCE guidelines for the studied H/D ratios does not suitably conform to the results obtained from the centrifuge tests. Also, for H/D = 6, a 300% discrepancy was observed between the force recorded during the experiment and the value given by the ASCE guidelines. Abolmaali and Kararam (2013) studied the deformation of buried polyethylene pipes by individually investigating the different deformation modes of the pipes. Almahakeri et al. (2014) assessed the bending of buried steel pipes along their lengths in a full-scale setting. The results of the experiment indicated that the H/D ratio exerts a significant influence on the forces that are applied to the pipe. In a study conducted in 2015, an analytical stress analysis approach was introduced by Rahman and Taniyama (2015) for low-depth steel pipelines under the influence of fault displacements. Zeng et al. (2019) introduced a method for evaluating the deformation and the strain of buried pipelines under the influence of faults. The nature of equivalent springs makes them a suitable tool for modelling pipe–soil interaction. However, the proposed relationships might not possess suitable accuracy to evaluate this interaction, particularly for polyethylene pipes. The aim of the present work is to employ a novel approach to investigating buried polyethylene pipe–soil interaction by taking into account the effect of a strike-slip fault. To do this, using a full-scale experimental set-up, two 8 m polyethylene pipes with diameters of 120.5 and 214 mm were tested under the action of a strike-slip fault with a displacement of 600 mm. Throughout loading, the lateral displacement along the length of the pipe was recorded at any moment using linear potentiometer transducers (LPTs). Thus, the absolute deformation curve of the pipe at any moment was drawn with appropriate accuracy, which was then taken as the criterion upon which to determine the soil–pipe interaction. The only remaining ambiguity was the properties of the equivalent springs that needed to be calculated. Using a finite-element software program and an optimisation algorithm developed in it, the properties of the equivalent springs were computed so that the deformations of the pipes in the numerical models would have the highest level of congruence with the results of the experiments. In fact, the specifications of the equivalent springs were optimised based on the lateral deformations of the tested pipes. The results reported in this study differ from what is proposed by the guidelines of ASCE (1984) and American Lifelines Alliance (ALA, 2001) in terms of maximum pipe–soil interaction force and initial stiffness.

The experimental investigation consisted of two full-scale tests performed on polyethylene pipes under the influence of a strike-slip fault.

The first test was performed on a pipe with an outer diameter of 120.5 mm and a length of 8 m. The pipe used in the second test had an outer diameter of 214 mm and a length of 8 m. Additional information regarding the two is presented in Table 1.

The two tests were carried out using the same dense sandy soil (with an average relative density of 90%). The gradation curve of the soil was obtained based on the American Society for Testing and Materials (ASTM) Unified Soil Classification System (ASTM, 2011a) and is presented in Figure 3. In accordance with the ASTM standard, if more than 50% of the coarse fraction is passing a number 4 sieve, then it is a sandy soil. Otherwise, the soil is gravel. In accordance with the presented curve, 93% of the soil mass passed through a sieve with an opening diameter of 4.75 mm (sieve number 4). Therefore, the soil used in the experiments was a sandy soil.

Before making a testing box, the experimental model was validated by the study of Abdoun et al. (2009). Considering that the semi-infinite space of soil was replaced by the restricted space of the test box, the dimensions of the apparatus were chosen to have no influence on the deformation of the pipe. To determine the appropriate dimensions of the test box, the longitudinal absolute deformation of the pipe was considered the main parameter. For this purpose, changes were made in three dimensions of the box (length, width and height) such that the longitudinal absolute deformation of the pipe was not affected by the dimensions. In other words, as the size of the box increases, the results do not experience any change. By carrying out a large number of finite-element analyses, the length, width and height of the test box were selected as 8, 1.5 and 1.5 m, respectively. The device is made up of two individual 4 m parts, with one being stationary and the other capable of being displaced. To minimise the surface frictional forces, the movable part was equipped with a set of steel wheels rolling on a rail (see Figure 4(a)). Three steel stoppers were also used to fix the stationary part into place (see Figure 4(b)). An equivalent displacement of 600 mm was applied to the displaceable portion of the device (exactly at the level of the buried depth) through three hydraulic actuators with an overall capacity of 1050 kN. The movements of the actuators were monitored and controlled automatically by a 32-channel data logger. For better accessibility and to prevent the box from rotating, an LPT was placed on each actuator. These were controlled so that the difference in the relative displacements of the three actuators at any given moment would be less than 0.5 mm. Also, three compressive load cells were employed to record the forces applied by the three actuators (see Figure 4(c)). During the loading, the LPTs recorded the lateral displacement of the pipe at every moment. Prestressed cables were used to connect eight reference points (RP) on the pipe (0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5 and 7.5 m) to the LPTs located outside of the test device, making it possible to record the displacement of the pipe throughout the test with a maximum error of 0.2% (see Figure 4(d) and also S1(d) in the online supplementary material).

On the location of the fault and at two points flanking the fault at 1 m distances, two strain gauges were installed on both sides of the pipe (see S1(a) in the online supplementary material). To protect them and also to prevent them from separating from the pipe during the test, a plastic guard was placed on each of the strain gauges (see S1(b) in the online supplementary material). Finally, a rubber sheath was added to the entire ensemble (see S1(c) in the online supplementary material).

Figure 5(a) presents a schematic plan of the pipeline, the stationary and movable parts of the test apparatus, the displacements applied by the strike-slip fault, the placements of the actuators, the strain gauges and the LPTs. Using this set-up, the absolute deformation and the way in which the pipe has curved can be obtained at any given moment. The pipe was placed in the middle and 500 mm above the bottom of the box, meaning that the burial depth of the pipe was 1000 mm (see Figure 5(b)).

The different steps of preparing the test, including implementing the first and second layers of soil, creating the channel in which the pipe was laid, the laying of the pipe and creating the small grooves through which the LPT wires passed, are shown in Figure 6. To satisfy the boundary conditions, caps were installed on both ends of the pipe to prevent longitudinal, transverse and vertical displacements of the fixed end and also transverse and vertical displacements of the movable end (see Figure 6(b)). To prevent the soil from pouring outside of the apparatus, the small gaps between the modules were sealed using plastic tapes. Finally, after filling the box and levelling out the soil, a 200 × 200 mm mesh was drawn on the last layer of the soil using plaster (see S2 in the online supplementary material).

Before filling the box with sand, a modified Proctor test was performed based on the guidelines proposed by the ASTM standard to measure the density of the soil (ASTM, 2012). The results were then compared with those of an in situ test. For the soil, a maximum specific weight of 2220 kg/m³ was obtained for an optimum moisture of 10.7%. To reach a height of 1450 mm, six layers of soil were implemented inside the box. Each layer of the soil, with natural moisture and a relative thickness of 250 mm, was compacted using the 400 × 400 mm plate of an electric compactor. The in situ cone test was performed three times on each layer, and the average of the values obtained from the three tests was taken as the relative density of the layer (ASTM, 2015). The compaction percentage curves and the density of the soil against its depth in the two tests are illustrated in S3 and S4 in the online supplementary material, respectively. Also, in accordance with the procedure recommended by the ASTM standard guidelines, the relative density of each soil layer was obtained and recorded using the in situ sand cone test (ASTM, 2015). Each layer of the soil was compacted up to a compaction percentage of 85–95%. In each test, 40 t of soil was needed to fill the test box.

Based on the ASTM standard, a direct shear test was performed on samples with different percentages of relative density to obtain the internal friction angle of the soil (ASTM, 2011b). S5 in the online supplementary material shows the internal friction angle curves in terms of dry unit density and the percentage of compaction in the soil.

After the first test was finished, the soil was removed from the test device in its entirety and was replaced with new soil (taken from the same soil depot and with the same aggregate specifications) for the second test.

The cracking and disintegration of the surface of the soil shown in Figures 7 and 8 correspond to test 1 and test 2, respectively. The cracking initiated in tests 1 and 2 at displacements of 168 and 302 mm, respectively, accompanied by small cracks taking form on the edges of the box and at the location of the fault. As the displacement applied by the actuators increased, the cracks became more numerous and spread into both sides of the fault line in an almost symmetrical fashion. The soil surface failed predominantly in the fault location, with a maximum crack expansion of roughly 200 mm on both sides. Finally, by the end of the test, the highest soil disintegration was accompanied by the highest level of crack expansion.

By comparing Figures 7 and 8, it can be understood that the level of soil disintegration and crack expansion in test 2 is much more significant than in test 1, indicating a higher force concentration in the soil of the second test. This can be attributed to the deformation pattern along the length of the pipe. It can also be said that the pipe size influences the failure pattern of the soil in its surface and therefore the interaction between the soil and the pipe.

After the loading process finished, to evaluate the deformation of the pipe, the soil covering the pipe, which had a height of about 1 m, was carefully removed so as not to cause any inadvertent change to the deformed shape of the pipe. Figures 9(a) and 9(b) show the deformed shape of the pipe in tests 1 and 2, respectively. No part of the pipeline had sustained any damage, and no significant ovalisation could be seen in the cross-section of the pipe at any point along its length. In both tests, the pipe experienced deformation in its medial 1 m portion, which was split into two 0.5 m parts by the fault line. In a similar study on steel pipes, the authors reported that a larger segment of the pipe (the medial 2 m portion) had deformed (Ashrafy et al., 2020). Therefore, it can be concluded that polyethylene pipes exert a smaller force on the surrounding soil than steel pipes.

The curve of the absolute deformation of the pipe was drawn based on the displacements recorded by the LPTs connected to the pipe. The curve was then used for determining the soil–pipe interaction. Figures 10(a) and 10(b) present the absolute deformation of the pipe at fault displacements of 100, 200, …, 600 mm in tests 1 and 2, respectively.

The displacements obtained from the test confirmed that the deformation of the pipe possesses reverse symmetry. As can be seen, in pipes with larger diameters, a larger portion of the pipe deforms under the action of the fault, leading to a higher level of damage to the soil.

In total, six strain gauges were placed on three segments on the pipe (two strain gauges per segment). Each pair was installed at the same elevation, parallel to the longitudinal axis of the pipe (see Figure 5(a)). S6(a) and S6(b) in the online supplementary material present the axial strains of the polyethylene pipe in test 1 at 1 m distances from the fault in the stationary and mobile parts of the device, respectively. As can be seen, the magnitude of the bending moment in the polyethylene pipe in test 1 is insignificant, meaning that the force governing the behaviour of the pipe near the fault is the tensile force. Therefore, the largest deformations of the polyethylene pipe are concentrated in the fault location and its vicinities. This type of pipe deformation can be seen in Figures 9(a) and 10(a).

The tensile and compressive strains in the polyethylene pipe in test 2 at a 1 m distance from the fault in the stationary and mobile parts of the test apparatus are shown in S7(a) and S7(b) in the online supplementary material, respectively. As can be seen, the axial strains recorded by the two strain gauges installed on both sides of the fault have different signs, which is indicative of the presence of a considerable bending moment in the pipe, particularly on both sides of the fault. Figures 9(b) and 10(b) substantiate this assertion.

Over the course of the entire test, the force applied by the actuators to the box was recorded and was drawn based on the displacement of the box (see Figure 11). Given that all of the parameters (namely, the burial depth, the material of the pipe and the relative density of the soil) were the same in both tests, no noticeable change was seen in the applied total force by changing the diameter of the pipe from 120.5 to 214 mm. This means that the stiffness of the polyethylene pipe is not high enough to exert any kind of influence on the externally applied force. An important point to note here is that the total force given by the load cells cannot be taken as the soil–pipe interaction force. The reason for that is that the force has been applied by the actuators and is under the direct influence of parameters such as the shear force of the soil in the fault plane and the friction between the rails and the wheels attached to the box. Furthermore, the soil–pipe interaction force is related to the displacements of the soil and the pipe and their stiffnesses. Therefore, this force does not have a fixed value along the pipe length.

The next section will present a method for computing the pipe–soil interaction force and the behavioural parameters of the springs simulating the soil.

It is evident that the force exerted on the pipe obeys the pipe deformation and curvature – that is, if this curvature were to be drawn along the pipe length, the force acting on each point on the pipe could be obtained. Thus, the main aim, which is the evaluation of the force acting on the pipe, has to be achieved by calculating the stiffness of the springs that simulate the soil and obtain the deformation along the pipe length. For this, a pipe–soil interaction optimisation software program was developed. First, the results of the experiment, including the box displacement and the absolute deformation of the pipe at each loading stage, were categorised and fed into the optimisation software as input. The objective of the algorithm was then to obtain the stiffness of the springs in the numerical models as close to the experimental results as possible.

At small relative displacements, these parameters can be calculated without the need to take into account the effect of the longitudinal springs of the soil. Therefore, the longitudinal springs were not included in the modelling process. Also, the behavioural parameters of the equivalent springs – namely, the primary and secondary slopes in the force–displacement curves of the springs – and also the yield force of the horizontal springs were considered invariants, which means that they remain constant along the length of the pipe. Since the longitudinal relative displacement between the soil and the pipe changes, the force acting on the pipe is also variant.

The first step in using the developed optimisation algorithm was to construct a pipeline–spring model in the Abaqus software package (see Figure 12). The model was a 2D model based entirely on the conditions of the test. S8 in the online supplementary material presents a schematic view of the constructed pipe–spring model and the placements of the equivalent springs of the soil along the length of the pipe.

The relative displacement–force response of the equivalent springs is considered a bilinear curve by the optimisation algorithm (see Figure 13). In the pattern proposed for the equivalent springs, E_s is the slope of the linear portion, F₁ and F₂ represent the force of the spring in the non-linear phase and y_y and y_u are the yield and ultimate displacements, respectively. The algorithm would take an arbitrary value for each of the aforementioned parameters and proceed to obtain the optimum results.

In the next step, a displacement of 60 cm was applied to the base of the springs in the model constructed in the Abaqus software. The deformation values of the pipe at displacements of 5, 10, 15, …, 60 cm were then read by the software. Afterwards, the absolute deformations of the eight nodes on the pipe at displacements of 5, 10, 15, …, 60 cm (see Figure 14) were compared with the data recorded during the experiment. These nodes were the points to which the LPTs were attached during the tests.

Using the values recorded by the LPTs attached to points P1 to P8 on the pipe, eight separate errors for fault displacements of 5, 10, 15, …, 60 cm were obtained and then used to calculate the overall error. For example, at a displacement of 5 cm, an error (E_5cm) was calculated using Equation 4, in which D_i and d_i stand for the experimental and analytical displacements of point i, respectively. Similarly, the same procedure was carried out for displacements of 10, 15, …, 55, 60 cm to obtain E_10 cm, E_15 cm, …, E_55 cm, E_60 cm.

In the next step, the algorithm modifies the values of E_s, F₁ and F₂. Based on the defined optimisation algorithm, the code automatically determines whether to increase or decrease the values of these three parameters, leading to the error produced by Equation 5 becoming progressively smaller. This procedure continues until the error cannot be any smaller. In other words, there will come a point after which the error only increases, irrespective of the manner in which the values of E_s, F₁ and F₂ are modified. In this case, the algorithm has reached a local minimum. In the final step, the final values of E_s, F₁ and F₂ are displayed on the screen and the analysis is terminated. The resulting data are the force–displacement specification of the equivalent springs used to simulate the soil.

The algorithm performance was further validated using numerous examples. In all of the models, the optimised algorithm achieved an accuracy of more than 90% in estimating the relative force–displacement specifications of the springs simulating the soil.

This section lays out how the software calculated the specifications of the springs in test 1 and test 2. First, an 8 m polyethylene pipe was modelled in Abaqus using beam elements (similar to Figure 12). The stress–strain curves of both polyethylene pipes were obtained based on the instructions presented in the ISO 527-2:2012 standard (ISO, 2012) (see S9(a) and S9(c) in the online supplementary material). The displacement curvature of the tested sample was the same as the pipe from which it was extracted (see S9(b) in the online supplementary material).

The density and Poisson’s ratio of the polyethylene pipes used in the analyses are 3500 kg/m³ and 0.3, respectively. The geometrical properties of the two pipes are given in Table 1.

The absolute deformations of the pipe in both tests were given to the optimisation algorithm as input. By carrying out multiple analyses, the force–displacement information of the springs was derived. The results of tests 1 and 2 can be seen in Figures 15(a) and 15(b), respectively.

As is seen, in both tests, the yield displacement of the springs has moved too far to the right, rendering the behaviour of the spring practically linear. This means that under the fault displacement, the optimised springs do not transition from the linear phase to the non-linear phase. The optimisation algorithm, however, was designed to take into account the entire behavioural range of the equivalent springs. The force created between the soil and the pipe was not strong enough to cause the spring to enter the non-linear phase. It appears that since the pipe undergoes local deformations only, only a small segment of the pipe is affected by the applied forces. It is obvious that the pipe–soil interaction force is also dependent on the material of the pipe. In steel pipes, the high stiffness of the pipe is capable of inducing failure in the soil. Therefore, the soil easily transitions into the non-linear phase. However, as a consequence of their weaker material, polyethylene pipes cannot cause failure in the soil. Also, by increasing the pipe diameter from 120.5 to 214 mm (an increase of about 78%) the stiffness of the equivalent springs of the soil increases from 92.575 to 201.846 kN/m², indicating an increase of more than 100%. Given that in both tests the type of the soil (dense sandy soil) and the burial depth of the pipe were the same (H = 1 m), it can be inferred that the stiffnesses of the springs are entirely influenced by the pipe diameter. It can also be stated that due to the concentration of non-linear deformations of the pipe and the failure of the soil in the location of the fault, the non-linear behaviour of the springs has not exerted a significant influence on the pipe deformation, which is predominantly local. This can be clearly seen in S10 in the online supplementary material. In both tests, the maximum force created in the equivalent springs of the soil is concentrated in a limited section along the pipe length (the medial 2 m portion), at the location of the fault.

The absolute deformation diagrams of the pipe for tests 1 and 2, as well as the corresponding numerical models, are shown in Figures 16(a) and 16(b), respectively. Given the high level of agreement between the experimental and numerical results in both tests, the stiffnesses of the springs, which were computed by the optimisation algorithm, can be verified.

Here, the diagram of the equivalent springs obtained from the optimisation algorithm (see Figure 15) were compared to the relationships given by the ASCE (1984) and ALA (2001) guidelines. The results are shown in Figures 17(a) and 17(b) for tests 1 and 2, respectively.

As seen in both tests, the force capacity calculated for the springs is well below the values computed by way of the relationships given by ASCE and ALA. The stiffnesses computed for the springs in tests 1 and 2 deviate from the values recommended by the ASCE standard by factors of 16 and 9, respectively. The discrepancy is far more pronounced in test 1 than in test 2, which can be a corollary of the higher H/D ratio in test 1. This difference between the results was already reported by Abdoun et al. (2009). Also, according to Figures 17(a) and 17(b), the stiffness calculated for the equivalent springs diverges from the values proposed by Brinch-Hansen (1961) an even bigger margin than what is recommended by the ASCE guidelines. The lower limit of the values proposed by the ASCE guidelines is more congruent with the results calculated by the optimisation algorithm.

After verifying the absolute deformation diagrams obtained from the optimisation software for tests 1 (see Figure 16(a)) and 2 (see Figure 16(b)) and for a better assessment, these diagrams were compared with those obtained using stiffnesses calculated based on the ASCE and ALA guidelines (see Figures 17(a) and 17(b)). The results are compared in Figures 18 and 19.

Given the difference between the pipe deformation diagrams obtained from the optimisation algorithm and those obtained based on the ASCE and ALA guidelines for fault displacements of 30 and 60 cm, it appears that the optimisation algorithm did a better job at modifying the specifications of the equivalent springs.

In other words, the relationships recommended by ASCE and ALA guidelines, which are based on the studies carried out by O’Rourke (ASCE, 1984; O’Rourke and Liu, 1999) and Hansen (O’Rourke and Liu, 1999), calculate the specifications of the equivalent springs with a degree of error.

The important question is why is the relative displacement–force curve of the equivalent springs calculated by the optimisation software so drastically different from the curves obtained using the criteria proposed by the ASCE and ALA guidelines?

Assume that both the steel and the polyethylene pipe are subjected to a lateral displacement. If the pipe is considered a linear-elastic beam, the bending moment (M), shear force (V) and the load (w) applied to the pipe can be calculated using Equations 6–8, respectively.

In these relationships, E and I are the modulus of elasticity and the moment of inertia, respectively. The former depends on the material and the latter on the cross-section of the pipe. By assuming an elastic behaviour for the pipe, it can be stated that w is under the direct influence of the pipe deformation. Therefore, the main parameter in this relationship is EI – that is, the flexural rigidity. In addition to EI, the yield specification (F_y) of the pipe also influences its inelastic behaviour response. Therefore, the soil–pipe interaction force is affected not only by the soil but also by the geometrical properties and material of the pipe. If the modulus of elasticity and the yield stress of the polyethylene pipe are compared with those of a steel pipe, the large discrepancy among the forces acting on the pipe is reasonable. In the relationships provided by the ASCE and ALA guidelines, the effect of the pipe material is not directly taken into consideration. The material of the pipe is a very important and determining factor, since it is directly related to the pipe deformation. Another significant matter is that the interaction that takes place between the pipe and the soil depends on the type of excitation acting on the system. The interaction between the pipe and the soil for different excitations (wave propagation, faulting etc.) and pipe materials (steel, polyethylene etc.) is evaluated using the same set of equations. It is obvious that a different set of equations is needed for each case. In this study, a new method has been presented to evaluate soil–pipe interaction. Collecting comprehensive data requires the conducting of complementary empirical and numerical evaluations.

The main difficulty in analysing buried lifeline systems is the correct evaluation of the interaction taking place between the soil and the pipe using bilinear springs. Some investigations have cast doubt on the authenticity of the equations proposed by different guidelines such as those by ASCE and ALA to describe the behaviour of equivalent bilinear springs. The main objective of the current work has been to introduce a novel approach to studying the effect of strike-slip faulting on the interaction of polyethylene pipes with dense sandy soil. To do this, a full-scale laboratory set-up simulating the movement of a strike-slip fault was used to study two polyethylene pipes with diameters of 120.5 and 214 mm buried in dense sandy soil. In each stage of the experiments, the displacements of eight reference nodes on the pipe were recorded. A soil–buried pipe optimisation algorithm was programmed in Matlab and Abaqus software programs to carry out the calculations, making it possible to determine the non-linear response of the equivalent springs using the displacements of the mentioned reference points. The optimisation software progressively modified the stiffness of the springs until the highest agreement between the numerical and experimental results was achieved. Considering that the deformation of a beam element, the material and geometry of the pipe and the force applied thereto are interconnected, this type of optimisation can lead to an accurate estimation of the soil–pipe interaction force. The results presented herein show that despite what is stated in the ASCE guidelines, the force capacity of the equivalent springs simulating the soil around a polyethylene pipe (specifically those with high H/D ratios) is far smaller than the values computed using the ASCE-recommended relationships. The stiffnesses obtained for polyethylene pipes with diameters of 120.5 and 214 mm are respectively 16 and nine times smaller than the values put forth by ASCE. Furthermore, the yield force of the springs simulating the soil around polyethylene pipes had an even larger discrepancy with the criteria presented by the ALA guidelines. Also, the linear shape of the force–displacement curve of the equivalent springs indicated that the force generated by the pipe (and consequently by the springs) was not strong enough to force the spring to transition into the non-linear phase.