Pseudo-Static Finite-Element Assessment of Seismic Soil–Pipeline Interaction in Multi-Line Buried Pipelines

Abstract

This study investigates the seismic response of double- and triple-buried steel pipeline systems using finite-element modeling in RS2, with particular emphasis on soil–pipeline interaction and symmetry-breaking behavior under pseudo-static seismic loading. Although the pipeline systems are initially symmetric in geometry, material properties, and boundary conditions, the analysis demonstrates that directional seismic excitation induces quantitatively measurable asymmetric responses in shear force, displacement, and spacing due to nonlinear soil–pipeline interaction. Five parametric scenarios were examined, including burial depth (1–5 m), pipeline diameter (8–56 in.), groundwater table (1.4–20 m), peak ground acceleration (0.1–0.6 g), and soil type. The results show that maximum shear forces increase with burial depth and diameter, reaching approximately 15–17 kN in clayey soils at a PGA of 0.4 g, whereas sandy and heterogeneous soils produce lower shear forces (≈12–14 kN). Horizontal displacements are strongly governed by groundwater and PGA, increasing from about 1.2–1.8 m in dry or deep groundwater conditions to more than 2.8 m for shallow groundwater and exceeding 5 m at PGA = 0.6 g. Triple-pipeline systems exhibit higher shear demand due to confinement effects, with the middle pipeline often developing the largest shear force, while the pipeline facing the seismic load consistently experiences the greatest displacement. This study makes two primary contributions. First, it demonstrates that initially symmetric multilined buried pipeline systems exhibit systematic, quantifiable symmetry-breaking behavior under directional seismic loading, manifested as unequal shear forces, displacements, and interaction effects among adjacent pipelines. Second, it presents an integrated multi-parameter coupling analysis that simultaneously accounts for burial depth, pipeline diameter, groundwater level, soil type, and peak ground acceleration, revealing interaction mechanisms that cannot be captured through single-parameter or single-pipeline assessments.

1. Introduction

Buried steel pipelines are among the most critical components of modern lifeline infrastructure, ensuring the reliable transport of oil, gas, and water across vast, often geotechnically complex regions. As global seismic activity continues to threaten the integrity of underground facilities, understanding the behavior of buried pipelines under earthquake loading has become increasingly essential. Past seismic events have demonstrated that even minor ground movements can induce significant deformation, rupture, or buckling in pipelines, particularly where soil–pipe interaction mechanisms are not adequately captured in design models. Consequently, the development of robust analytical and numerical frameworks to assess the seismic response of buried pipelines remains a priority in geotechnical and structural engineering [1,2,3,4].

In practical engineering applications, buried pipelines are rarely installed as isolated single lines. Instead, multiple parallel pipelines are commonly placed within shared trenches or designated utility corridors to optimize land use, minimize right-of-way requirements, and facilitate construction, inspection, and maintenance activities. Such multi-line configurations are widely adopted in oil and gas transmission systems, water distribution networks, and urban lifeline infrastructure, and are explicitly recognized in engineering guidelines and standards [5,6]. In densely developed areas, the close spacing between adjacent pipelines becomes unavoidable, leading to strong soil-mediated interactions between neighboring lines. During seismic events, these interactions can significantly modify inertial force transfer, deformation patterns, and confinement effects in the surrounding soil mass, producing seismic responses that may differ substantially from those of single-pipeline systems. Post-earthquake reconnaissance and analytical investigations have further indicated that damage in lifeline corridors is often governed by collective system behavior rather than by the response of individual pipelines alone. Despite this practical importance, the seismic performance of closely spaced parallel pipeline systems remains insufficiently explored, underscoring the need for dedicated investigation of multi-line buried pipelines under seismic loading.

A substantial body of research has investigated the impact of geometric, geotechnical, and loading parameters on the structural performance of buried pipelines during earthquakes. Geometric factors, such as burial depth and pipeline diameter, have consistently been shown to significantly influence seismic behavior. Dutta et al. (2015) used a Coupled Eulerian–Lagrangian (CEL) finite element approach to demonstrate that both burial depth and diameter significantly impact shear forces and soil resistance for pipelines embedded in clay seabeds [7]. Similarly, Roy et al. (2015) [8] reported that lateral soil–pipeline interaction in dense sand is highly sensitive to pipeline diameter and embedment, emphasizing the need to incorporate realistic geometric constraints in numerical models. These findings collectively suggest that geometric configuration is a governing factor in determining the level of seismic demand imposed on buried steel pipelines [8].

Geotechnical conditions, particularly soil stiffness, failure characteristics, and spatial variability, play an equally pivotal role. Shakib and Jahangiri (2016) found that intensity measures used to assess seismic response differ markedly depending on whether pipelines are embedded in sandy or clayey soils, with softer cohesive soils tending to amplify deformations [9]. Supporting this, Uckan et al. (2015) [10] developed a simplified analysis model that illustrates how strike-slip faulting induces distinct deformation patterns in pipelines, depending on the soil type. In regions with specific soil conditions, such as loess, additional challenges arise [10]. Qiu et al. (2018) demonstrated that seismic subsidence of collapsible loess in Northwest China can cause severe vertical and horizontal deformation, posing heightened risks to buried infrastructure [11]. Together, these studies underscore that soil type, layering, and local geomechanics must be rigorously considered in any seismic pipeline analysis.

Groundwater table and topographic effects further complicate the interaction between soil and pipelines during seismic events. Shakib and Jahangiri (2016) demonstrated through fragility analysis that pipelines situated in high groundwater environments experience larger horizontal displacements, reduced soil support, and increased failure probability due to buoyancy and pore pressure effects [9].

Among external loading factors, peak ground acceleration (PGA) has been widely recognized as a dominant driver of pipeline deformation and failure. Tsinidis et al. (2020) emphasized that seismically induced axial compression becomes especially critical under large PGAs, with local buckling or tensile rupture likely at geotechnical discontinuities [12]. Complementary work by Moradi and Alam (2015), though focused on cyclic loading in steel connections, reinforces the broader structural pattern: increased cyclic demand leads to stiffness degradation and plastic deformation, mechanisms also relevant to buried steel pipelines subjected to repeated ground motions [13]. Collectively, these studies underscore the importance of integrating realistic seismic loading spectra into numerical analyses.

Despite significant advances in numerical modeling, several research gaps remain. Most existing studies investigate individual influences—such as burial depth, soil type, groundwater level, or PGA—rather than capturing the combined, interacting effects of multiple parameters on pipeline behavior. Similarly, many investigations focus on single-pipeline systems, even though real pipeline corridors frequently contain multiple parallel lines whose proximity alters soil–pipeline interaction patterns. While studies such as Trifonov (2014), Rofooei et al. (2015), and Moradi et al. (2013) provide valuable insights on fault-crossing deformation mechanisms, comprehensive multi-parameter, multi-line system analyses remain limited [14,15,16]. Another gap concerns long-term performance and degradation mechanisms: Xu and Cheng (2017) and Sun and Cheng (2018) demonstrated that corrosion defects alter stress distribution and failure pressure [17,18], while Mohtadi-Bonab et al. (2020) highlighted hydrogen-induced cracking propagation in steels—suggesting that coupled mechanical–corrosive effects under seismic loading require further investigation [19]. Recent advancements in distributed fiber optic sensing [20] present promising opportunities for monitoring but have yet to be fully integrated into predictive seismic models.

To address these gaps, the present study advances the seismic assessment of buried pipeline systems through two key contributions. First, it explicitly identifies and quantifies symmetry-breaking behavior in double- and triple-pipeline configurations, showing how initially symmetric geometries evolve into asymmetric seismic responses due to directional loading and nonlinear soil–pipeline interaction. Second, the study introduces an integrated multi-parameter coupling framework that simultaneously evaluates the combined effects of burial depth, pipeline diameter, groundwater level, soil type, and peak ground acceleration within a unified finite-element model. Unlike prior studies that isolate individual variables or focus on single-pipeline systems, this approach captures the coupled mechanisms governing force transfer, deformation, and interaction in realistic multi-line pipeline corridors. The resulting insights provide a more comprehensive basis for seismic design and risk assessment of buried pipeline infrastructure.

Following this introductory section, the paper is structured as follows. Section 2 outlines the numerical modelling framework, including the pipeline geometry, material properties, soil characteristics, boundary conditions, and seismic loading assumptions. Section 3 details the parametric study and investigates scenarios, examining the effects of burial depth, pipe diameter, groundwater table depth, peak ground acceleration, and soil conditions. Section 4 presents and analyzes the numerical results, with emphasis on shear force, horizontal displacement, spacing variations, and interaction mechanisms in double- and triple-pipeline systems. Section 5 discusses the broader implications of the findings, addresses the limitations of the modelling approach, and considers their relevance to seismic design. Finally, Section 6 summarizes the main conclusions and suggests directions for future research and engineering practice.

2. Materials and Methods

2.1. Pipeline Geometry and Material Properties

The analysis considered double- and triple-welded continuous steel pipelines commonly used in oil and gas transmission systems. A range of pipeline diameters was examined to capture the influence of geometric stiffness on soil–pipe interaction. The selected diameters varied from 8 in. (203.2 mm) to 56 in. (1878 mm), with wall thicknesses corresponding to each diameter. All pipelines were assumed to be fabricated from API 5L X60 steel and were modeled using standard beam liner elements in RS2. The elastic properties were defined with a Young’s modulus of 2.06 × 10⁸ kPa, Poisson’s ratio of 0.28, and a unit weight of 78.5 kN/m³, corresponding to a steel density of approximately 7850 kg/m³. The pipe wall thickness was defined explicitly according to the selected diameter, with a representative thickness depending on the table listed in Appendix A (e.g., 9.525 mm for the 24-inch pipeline). The pipeline material was modeled as linear elastic, and material nonlinearity (plastic yielding of the steel) was not considered in the present analyses. The pipelines were modeled as continuous, fully welded systems to eliminate joint flexibility and emphasize global deformation patterns during earthquake excitation.

2.2. Burial Conditions and Groundwater Levels

The pipelines were embedded in soils representing a broad range of geological conditions. Burial depths varied between 1 m and 5 m below the ground surface, enabling evaluation of the effect of confining pressure and overburden stress on seismic response. To account for the influence of hydrostatic pressure, multiple groundwater table (GWT) conditions were simulated, ranging from shallow water levels at 1.4 m (just above the pipelines) to deep water tables extending to 20 m. This range reflects realistic field variability and its impact on pore-water pressure development, effective stress reduction, and soil stiffness degradation during seismic shaking.

2.3. Seismic Loading and Ground Conditions

Pseudo-static seismic loading was selected in this study to facilitate a systematic parametric investigation of soil–pipeline interaction and symmetry-breaking behavior across multiple pipeline configurations and ground conditions. In Rocscience RS2 (version 11.027; Rocscience Inc., Toronto, ON, Canada), pseudo-static seismic loading is introduced through a seismic coefficient, which represents the peak ground acceleration (PGA) expressed as a dimensionless ratio of gravitational acceleration. Once this coefficient is specified, RS2 automatically imposes an equivalent horizontal body force on all finite elements in the model. The magnitude of this seismic body force is calculated as:

$$F_{\mathrm{seismic}}=k_h\left(\gamma A\right)$$

where $k_h$ is the seismic coefficient (equal to the PGA), $\mathsf{\gamma }$ represents the unit weight of the soil or material, and $\mathrm{A}$ is the area of the finite element. The resulting horizontal force represents the inertial effects generated by earthquake excitation, whereas the gravitational body force corresponds to the self-weight of the element. In the numerical formulation, RS2 incorporates seismic loading by vectorially superimposing the horizontal seismic body force onto the vertical gravitational force, thereby defining the total body force acting on each element. This pseudo-static approach allows the simulation of permanent ground deformation-type seismic effects without the need to perform a full dynamic time-history analysis [21,22]. Additional details are provided in Appendix B.

However, it should be noted that the pseudo-static method does not capture dynamic characteristics of earthquake loading, including wave propagation effects, frequency content of ground motion, phase differences along the pipeline alignment, or time-dependent inertial amplification. Consequently, the analyses focus on relative response trends and upper-bound deformation demand rather than detailed dynamic response. Fully coupled dynamic time-history analyses incorporating these effects are therefore recommended for future studies.

The range of peak ground acceleration (PGA) values considered in this study (0.1–0.6 g) was selected to cover moderate to very strong seismic conditions, including extreme near-fault scenarios. While PGA values above 0.5 g are relatively rare, they may occur locally in proximity to active faults and are useful for identifying upper-bound pipeline response and critical deformation thresholds. The inclusion of PGA = 0.6 g is therefore intended for sensitivity analysis and assessment of potential worst-case behavior, rather than representing typical design-level earthquakes.

Each combination of PGA, soil type, burial depth, pipe diameter, and groundwater condition was evaluated to provide a comprehensive understanding of the factors governing pipeline performance under seismic loading.

The pseudo-static seismic approach adopted in this study is intended to capture permanent ground deformation-type effects and relative response trends in soil–pipeline interaction rather than detailed dynamic behavior. As such, the computed shear forces and horizontal displacements should be interpreted as upper-bound deformation demands under equivalent inertial loading, particularly for scenarios involving large displacements at high PGA levels. Dynamic phenomena such as wave propagation, frequency-dependent soil response, phase differences along the pipeline alignment, cyclic degradation, and time-dependent inertia effects are not represented in the present formulation. Consequently, the results do not constitute precise dynamic predictions of pipeline response, but rather conservative estimates that identify critical loading combinations and deformation-prone conditions requiring further investigation through fully coupled dynamic analyses.

2.4. Pipeline Installation Configuration and Soil Characterization

The pipelines were placed within a ditch designed according to the Typical Ditch Standard of the Iraqi State Company for Oil Projects (STD-CS-02), as illustrated in Figure 1. This standardized ditch geometry reflects common installation practices for transmission pipelines in Iraq and ensures realistic boundary conditions for soil–pipe interaction modelling.

The geotechnical properties of the surrounding soil were derived from the site investigation for the Al-Kut Gas Factory Project. This location was selected due to its exposure to relatively high seismic activity compared to other regions in Iraq, as well as its strategic position between the Tigris and Al-Gharraf rivers, where groundwater levels are typically shallow. The soil profile obtained from this investigation was used to represent actual field conditions, providing a reliable basis for evaluating pipeline performance under seismic loading.

The horizontal spacing between adjacent pipelines was selected based on commonly adopted engineering practice and applicable design guidelines for buried pipeline corridors. In this study, the clear spacing between pipelines corresponds to the minimum recommended separation specified in relevant oil and gas pipeline standards, including ASME and API guidance, as well as the clearance requirements adopted by the Iraqi State Company for Oil Projects (SCOP) for pipeline right-of-way design. These provisions aim to ensure constructability, safe installation, accessibility for maintenance, and avoidance of mechanical interference between adjacent lines under both static and seismic conditions. The adopted spacing therefore represents a realistic and conservative design scenario frequently encountered in practice, particularly in congested utility corridors where right-of-way constraints limit pipeline separation. By selecting code-consistent minimum spacing, the numerical analysis intentionally captures potential interaction effects under unfavorable yet realistic conditions, allowing assessment of whether standard clearance provisions remain adequate during seismic loading.

2.5. Definition of Symmetry in the Numerical Model

In this study, symmetry refers to the initial equivalence of geometric configuration, material properties, soil conditions, and boundary constraints in the numerical model prior to the application of seismic loading. For both double- and triple-pipeline systems, the pipelines were arranged with equal spacing and identical diameters, wall thicknesses, and material properties. The surrounding soil was defined symmetrically with respect to the pipeline centerline, using uniform stratigraphy, constitutive parameters, and groundwater conditions on both sides of the system.

Boundary conditions were applied symmetrically, with vertical model boundaries restrained equally in the horizontal direction and the base fixed in the vertical direction. Identical constraints were imposed at the lower corners to prevent rigid-body motion. Prior to seismic excitation, gravitational loading and the initial stress state were uniform across the model domain, resulting in a symmetric stress and deformation field.

Symmetry in the model is intentionally broken only through the application of directional pseudo-static seismic loading. The horizontal seismic acceleration is applied in a specified direction, producing unequal inertial forces and soil pressures on opposite sides of the pipelines. As a result, initially symmetric pipeline configurations develop asymmetric shear forces, horizontal displacements, and variations in spacing. In this study, such asymmetric responses are referred to as symmetry-breaking behavior.

2.6. Numerical Modelling Approach and Boundary Conditions

The numerical analyses were conducted using a two-dimensional plane-strain formulation, which is appropriate for modeling long, continuous buried pipelines where the geometry, material properties, and loading conditions are assumed uniform along the pipeline axis. Under this assumption, out-of-plane deformation and axial strain gradients are negligible, and the seismic response is governed primarily by transverse soil–pipeline interaction. This approach is widely used in the seismic assessment of buried pipelines to investigate force-transfer mechanisms, deformation trends, and interaction effects away from localized features such as bends, joints, valves, or terminations. In the present study, the plane strain model enables a systematic parametric investigation of double- and triple-pipeline systems with reasonable computational efficiency. However, the two-dimensional formulation does not capture three-dimensional effects such as longitudinal wave propagation, spatial variability of ground motion, or local bending and buckling. Therefore, the results are interpreted as representative of cross-sectional response and comparative seismic demand, while fully three-dimensional dynamic analyses are recommended for detailed design-level evaluation.

Five numerical scenarios were established using a parametric framework, where a single variable was modified in each case while the remaining parameters were held constant. The investigated variables included burial depth, pipe diameter, groundwater table depth, peak ground acceleration (PGA), and soil type. This approach enabled the independent assessment of the influence of each parameter on the seismic response of the buried pipeline.

The model domain dimensions (50 m × 109.655 m) were kept constant across all parametric cases to ensure consistency and avoid introducing artificial changes in boundary condition effects when varying burial depth, diameter, groundwater level, PGA, or soil type. The selected boundaries were positioned sufficiently far from the pipeline group to approximate far-field conditions and minimize boundary reflections or stiffness artifacts in the zone of interest. This domain size was chosen to remain conservative with respect to the largest geometric case considered (maximum diameter and burial depth), so that all smaller cases are contained within the same far-field boundary envelope. Consequently, any observed differences in response among scenarios are attributable to the investigated parameters rather than changes in model extent or boundary proximity. The specific locations along the pipeline from which shear forces and horizontal displacements were extracted are illustrated in Figure 2. In this study, the maximum horizontal displacement is defined as the absolute lateral movement of the pipeline nodes relative to their initial, undeformed positions and is obtained directly from the finite element output.

During the design stage, boundary conditions were applied as illustrated in Figure 3. The two vertical boundaries were restrained in the X-direction to prevent lateral movement, while the bottom boundary was fixed in the Y-direction. Additionally, two corner nodes at the base were constrained in both X and Y directions to eliminate any rigid-body translation or rotation of the model. The ground surface was left unrestrained to allow natural deformation under seismic loading.

The pipeline was modeled using linear elements, with mechanical properties calibrated to accurately represent the behavior of API 5L X60 steel. To improve numerical accuracy, the stress analysis tolerance was set to 0.001, and convergence was controlled using absolute force and energy criteria. These settings ensured stable and reliable performance of the nonlinear finite element simulations across all scenarios.

The ground surrounding the pipeline was represented using three typical soil categories—clay, sand, and heterogeneous soil—to account for the range of subsurface conditions across the study area. The required geotechnical properties, including unit weight, cohesion, internal friction angle, Young’s modulus, and Poisson’s ratio, were taken from the soil investigation report of the Al-Kut Gas Factory Project and implemented directly in the RS2 model. The Mohr–Coulomb constitutive model was selected for all soil types because it is commonly employed in pseudo-static seismic analyses of buried pipelines and can effectively describe elastic–plastic behavior using parameters that are readily available from site investigation data.

Since seismic effects were simulated using a pseudo-static method, the analysis focused on soil strength mobilization and soil–pipeline interaction rather than on dynamic phenomena such as wave propagation or cyclic degradation of stiffness. While more advanced constitutive models (e.g., Drucker–Prager or hardening models) could potentially provide a more detailed representation of stress-path-dependent behavior, their use requires additional material parameters that were not available for the investigated site.

For the soil properties applied in the first four scenarios, the initial element loading was defined using the Field Stress and Body Force options, and the influence of moisture conditions was incorporated through the assigned unit weight values. The dry, moist, and saturated unit weights were taken as 16.2 kN/m³, 19.44 kN/m³, and 19.8 kN/m³, respectively, with a porosity of 0.5.

Soil stiffness was assumed to be isotropic, characterized by a Young’s modulus of 10,390 kPa and a Poisson’s ratio of 0.35. The shear strength behavior was defined using the Mohr–Coulomb criterion, with a cohesion of 6.34 kPa, a friction angle of 31.25°, and a tensile strength of 153.64 kPa. A dilation angle of 0° was adopted, and the residual strength parameters were set equal to the peak values to ensure consistent post-yield behavior throughout the analysis.

Hydraulic conditions were modeled as drained, with a fluid bulk modulus of 2.2 × 10⁶ kPa, piezometric water mode, and Hu = 1. To maintain consistent hydraulic and mechanical boundary conditions across all simulation stages, the Datum Dependency and Stage Factors options were deliberately left unselected.

The interaction between the buried pipelines and the surrounding soil was modeled using fully bonded contact conditions within the RS2 finite-element framework, whereby displacement compatibility is enforced between the pipeline liner elements and the adjacent soil elements. In this approach, no explicit interface or contact elements were defined, and separation or sliding at the soil–pipeline interface was not permitted. This assumption represents full mobilization of soil–pipeline interaction and is commonly adopted in pseudo-static seismic analyses of buried pipelines to capture upper-bound force transfer and deformation demand. Interface friction was therefore implicitly governed by the shear strength parameters of the surrounding soil, as defined by the Mohr–Coulomb constitutive model. The adopted approach is appropriate for welded steel pipelines installed in compacted trench backfill, where relative slip at the soil–pipe interface is limited under seismic loading. The implications of this assumption are acknowledged, and more advanced interface formulations allowing separation and sliding may be considered in future studies to capture localized debonding or partial mobilization effects.

A uniform finite-element mesh consisting of 8-noded quadrilateral elements was employed to ensure adequate resolution of stresses and deformations within the soil–pipeline system. Approximately 750 elements were generated across the model domain, providing a balance between computational efficiency and numerical accuracy.

Seismic loading was applied using the pseudo-static approach, where horizontal inertial forces simulate earthquake-induced ground shaking. The seismic load was applied from right to left for all Scenarios, reflecting unidirectional ground motion.

Following each model run, the nodal coordinate data were exported from Rocscience RS2 to Microsoft Excel. The coordinates were then transferred to AutoCAD 2023 using the auxiliary software SW-DTM version 3.0, where they were imported as discrete spatial points. To visualize the pipeline’s deformation profile under seismic loading, the points were connected using the CPT command, which had been previously integrated into AutoCAD. This workflow provided a clear, accurate representation of the pipeline’s displacement pattern for comparative analysis across scenarios.

The five scenarios of the study and their corresponding variable and constant parameters are summarized in

Table 1. The five research scenarios and their variable and constant characteristics [1].

Scenario	1	2	3	4	5	Typical Case
Variable	Burial depth	Diameter	GWT	PGA	Soil type	—
Fixed values for control groups
Burial depth	*	1.5 m	1.5 m	1.5 m	1.5 m	1.5 m
Diameter	D = 24 in	**	D = 24 in	D = 24 in	D = 24 in	D = 24 in
Groundwater table (GWT)	10 m	10 m	***	10 m	10 m	10 m
Peak ground acceleration (PGA)	0.4 g	0.4 g	0.4 g	****	0.4 g	0.4 g
Soil type	Layered soil	Layered soil	Layered soil	Layered soil	*****	Layered soil

* Depths are: (1, 1.5, 2, 3, 4, 5) m. ** Diameters are: (8, 16, 24, 48, 56) in. *** Ground Water Tables are: (1.4, 2, 3.6, 5, 10, 15, 20) m U/G. **** Peak Ground accelerations are: (0.1, 0.2, 0.3, 0.4, 0.5, 0.6) g. ***** Soil Types are: Homogeneous soil (Clayey soil and Sandy soil), Heterogeneous soil.

.

3. Results and Discussion

3.1. Scenario 1—Effect of Burial Depth

3.1.1. Shear Force Variation with Burial Depth

In double pipelines, as burial depth increases, both pipelines exhibit a general rise in maximum shear force because deeper soil provides greater confinement, higher overburden pressure, and stronger soil–pipeline interaction during seismic shaking. For the right pipeline, shear forces grow rapidly between 1–2.5 m, then almost level off around 3–4 m, and rise again beyond 4 m. The left pipeline follows a similar trend but with a steadier, less pronounced increase. The right pipeline always shows slightly higher shear forces because the seismic load is applied from right to left, illustrating symmetry-breaking behavior in an initially symmetric pipeline layout that results in greater direct soil thrust and inertial effects on that side.

Although shear increases with depth, the rate slows around mid-depths (3–4 m) due to nonlinear soil behavior: deeper soils become stiffer and restrict pipeline movement more effectively, allowing some of the seismic energy to dissipate through the surrounding soil instead of being fully transferred to the pipeline (Figure 4).

In the case of triple pipelines, all three pipelines show a nonlinear increase in maximum shear force as burial depth increases. Each curve rises steeply at shallow depths, flattens or slightly grows around 3–4 m, and then increases again toward 5 m. The right pipeline increases from about 11.9 to 15.5 kN. The middle pipeline increases from ~11.5 to ~16 kN, with a smoother trend and a noticeable slowdown in growth beyond 3 m. The left pipeline shows the largest overall rise (from ~10.5 to ~16.7 kN), again with steep initial growth, a mid-depth plateau, and renewed increase beyond 4 m.

When the three curves are compared, overall symmetry is maintained, as expected under pseudostatic loading, though slight differences appear due to soil–structure interaction. The right pipeline shows the highest shear at shallow depths because it receives the direct seismic thrust, but at deeper levels, it ends up with the lowest shear, likely due to soil arching reducing load transfer. The middle pipeline remains moderate across all depths, reaching its maximum at 3–4 m, because the confined soil between the outer pipes intensifies load transfer. The left pipeline starts with the lowest shear but becomes the highest at deeper depths (4–5 m) due to increased confinement and localized concentration of seismic energy.

3.1.2. Horizontal Displacement Variation with Burial Depth

In double pipelines, both pipelines exhibit a gradual, nonlinear increase in maximum horizontal displacement with increasing burial depth from 1 to 5 m. The right pipeline’s displacement rises from about 1.61 to 1.68 m, while the left pipeline increases from roughly 1.605 to 1.675 m, with both curves forming gentle upward trends. Although the patterns are similar, the right pipeline consistently experiences higher displacement because it is on the side directly impacted by the seismic load. The soil is pushed toward this pipeline during shaking, increasing lateral pressure and causing greater movement. In contrast, the left pipeline is on the trailing side and receives less direct soil thrust, resulting in slightly lower displacement.

In the triple-pipeline system, the maximum horizontal displacement increases nonlinearly with burial depth for all pipelines, reflecting the changing soil–structure interaction at greater embedment. The right pipeline shows the greatest displacement, rising from about 1.605 m at 1 m depth to 1.665 m at 5 m, with a steep increase up to 2.5 m, a slight flattening around 3 m, and renewed growth thereafter. The middle pipeline follows a similar nonlinear pattern, increasing from ~1.595 m to ~1.655 m, with its curve steep from 1–2.5 m, flatter at mid-depths, then rising again toward 5 m. The left pipeline shows the smallest displacement, increasing from ~1.59 m to ~1.64 m, also with a steep early rise, mild flattening near 3 m, and a gradual increase at deeper levels.

Comparing the three, the right pipeline consistently experiences the highest displacement because it directly receives the seismic thrust and interacts more strongly with the moving soil mass. The middle pipeline shows slightly lower displacement due to more balanced soil pressure on both sides, while the left pipeline records the lowest values because it lies on the trailing side of the seismic load. Overall, all pipelines confirm that deeper burial produces larger horizontal displacements, aligning with pseudostatic seismic loading behavior (Figure 5).

AutoCAD plots in Figure 6 showed that, for the double-pipeline system, horizontal seismic loading decreased the spacing between the two pipelines slightly, with the reduction becoming more noticeable at greater burial depths. Even under strong shaking (PGA = 0.4 g), two 24-inch pipelines buried 5 m deep moved less than 1 cm closer together, indicating no risk of pipeline collision.

In the triple-pipeline system, spacing was asymmetric: the distance between the right and middle pipelines was slightly larger than that between the middle and left pipelines at all depths. This indicates that the middle-left spacing decreased more, meaning the left pipeline experienced the least movement. As burial depth increased, horizontal movements and spacing reductions generally decreased slightly. Importantly, all spacing changes were less than 1 cm relative to the 790 mm design separation, confirming that none of the pipelines were at risk of contact, even under high seismic loading.

3.2. Scenario 2—Effect of Pipeline Diameter

3.2.1. Shear Force Variation with Diameter

In double pipelines, Figure 7 shows a nonlinear relationship between pipeline diameter and the maximum shear force for both pipelines. Overall, the right pipeline experiences slightly higher shear forces across most mid-range diameters because it receives the direct seismic thrust. As the diameter increases, shear forces rise for both pipelines, mainly because larger pipes mobilize more surrounding soil, increasing passive resistance and resulting in higher shear demand. The left pipeline shows a small dip in shear force beyond about 30 inches, suggesting reduced soil engagement in that range. However, at the largest diameter (~56 in), both pipelines reach nearly identical peak shear values, indicating that at very large sizes the seismic loading becomes more symmetric and the soil–structure interaction affects both sides similarly.

Regarding the triple pipelines, the maximum shear force for the right pipeline follows a nonlinear trend similar to the double-pipeline case: it increases sharply as the diameter grows from 8 to about 16 inches, then almost flattens between roughly 24–48 inches, and rises again at larger diameters (48–56 inches). This behavior suggests an initial increase in mobilized soil resistance, followed by reduced engagement due to the pipe’s mid-range flexibility, and then a renewed increase as the pipe becomes large and stiff enough to mobilize greater passive pressure.

For the middle pipeline, the trend is different. After a rapid rise from 8–16 inches, the shear force continues to increase steadily and then accelerates sharply beyond 48 inches. This strong nonlinear growth indicates that the middle pipeline experiences higher strain under seismic-induced bending, especially for larger diameters where soil confinement and end restraints are stronger.

In the left pipeline, shear forces increase quickly from 8 to about 24 inches, level off until roughly 48 inches, and then rise sharply again. Unlike the right pipeline, the left pipeline generally shows higher shear across most diameters, possibly because greater flexibility shifts more load toward the active side.

When comparing all three curves, they all show an overall increase in maximum shear force with increasing diameter, with differences becoming more significant beyond 48 inches. The middle pipeline consistently develops the highest shear forces at large diameters, while differences between the pipelines are small at smaller diameters (8–24 inches).

3.2.2. Horizontal Displacement Variation with Pipeline’s Diameter

In the double-pipeline case, Figure 8 shows that for the right pipeline, the maximum horizontal displacement increases smoothly with diameter, from about 1.62 m at 8 inches to roughly 1.65 m at 56 inches. The curve is slightly nonlinear but remains steady, resembling the behavior of a single pipeline. In contrast, the left pipeline displays a different pattern: displacement increases slightly up to about 24 inches in diameter, then gradually decreases for larger diameters. This reduction is attributed to the larger pipe surface area, which resists lateral soil penetration more effectively. Overall, the variation is very small, with displacement ranging from approximately 1.617 m to 1.624 m.

When both trends are combined, they show that after 24 inches, the right pipeline continues to exhibit increasing displacement with diameter due to increased stiffness and greater transmission of passive pressure. Meanwhile, the left pipeline’s displacement becomes flatter or slightly decreasing, indicating that larger diameters reduce lateral movement more effectively on the active-pressure side.

For the triple pipelines, all pipelines reach their highest maximum horizontal displacement at a diameter of about 48 inches, after which displacement decreases noticeably. For the right pipeline, displacement decreases slightly from 8–16 inches, then increases steadily up to 48 inches, followed by a small drop at larger diameters. The middle pipeline follows a similar pattern: a small decline from 8–20 inches, a steady rise to a peak at 48 inches, then a sharper drop beyond this point. The left pipeline shows a slight decrease from 8–24 inches, reaching a local minimum near 1.60 m, then a small increase before dropping sharply after 48 inches to around 1.585 m at 56 inches.

When the trends are combined, they show that the right pipeline consistently has the highest displacement as diameter increases, especially beyond 24 inches, due to passive pressure acting on that side. The middle pipeline increases steadily until 48 inches, slightly below the right pipeline, but then drops as stiffness limits its movement. The left pipeline remains the lowest throughout, with a pronounced reduction at large diameters, likely reflecting increased boundary stiffness or damping effects. Overall, all pipelines demonstrate a nonlinear response with a clear peak at 48 inches.

Figure 9 shows the horizontal spacing between pipelines in double- and triple-pipeline configurations. To better understand the seismic effect, the spacing during shaking was compared with the original design spacing, and the calculated movements are summarized in

Table 2. The difference between the design spacing and the horizontal displacement that each pipeline moved during an earthquake with a PGA of 0.4 g in the case of double pipelines.

Pipeline’s diameter (in.)	8	16	24	48	56
Design spacing (m)	0.36	0.58	0.79	1.45	1.88
Post-seismic spacing (m)	0.3575	0.5751	0.7821	1.4214	1.8936
The real movement (m)	0.0025	0.0049	0.0079	0.0286	−0.0136

and

Table 3. The difference between the design spacing and the horizontal displacement that each pipeline moved during an earthquake with a PGA of 0.4 g in the triple-pipeline case.

The Spacing Between the Right and Middle Pipelines
Pipeline’s diameter (in.)	8	16	24	48	56
Design spacing (m)	0.36	0.58	0.79	1.45	1.88
Post-seismic spacing (m)	0.3571	0.5758	0.7822	1.4280	1.8538
The real movement (m)	0.0029	0.0042	0.0078	0.022	0.0262
The spacing between the middle and left pipelines
Pipeline’s diameter (in.)	8	16	24	48	56
Design spacing (m)	0.36	0.58	0.79	1.45	1.88
Post-seismic spacing (m)	0.3574	0.5759	0.7818	1.4236	1.8516
The real movement (m)	0.0026	0.0041	0.0082	0.0264	0.0284

.

The results show that the horizontal spacing between two pipelines in the double-pipeline case generally decreased for all diameters except the largest (56 in.). For small- and medium-sized diameters (up to 24 in.), the reduction was less than 1 cm, which is too small to pose a risk to pipelines or their fittings. For larger diameters, spacing decreased by a few centimeters, indicating greater movement; however, at 56 in., the spacing actually increased, reducing the likelihood of collision. Nonetheless, the larger pipes experienced more deformation, making them more vulnerable to damage or leakage (Table 2).

In the triple-pipeline case, AutoCAD plots (Figure 9) show that spacing generally reduced as the diameter grew. Table 3 lists the after-shock spacing values, which were similar to those observed in the double-pipeline case. As before, spacing reductions were less than 1 cm for small- and medium-diameter pipes, posing minimal risk. For larger diameters, the spacing changes were a few centimeters, and the deformation was more pronounced. This indicates that although spacing remained within safe limits, larger-diameter pipelines are more susceptible to damage because they deform more under seismic loading.

3.3. Scenario 3—Effect of Groundwater Table (GWT)

3.3.1. Shear Force Variation with GWT

In double pipelines, Figure 10 shows that for the right pipeline, the maximum shear force increases sharply as the GWT rises from 1.4 m to approximately 10 m, where it reaches its peak. Beyond 10 m, the shear force gradually decreases as the GWT deepens to 20 m. The case with no groundwater produces a slightly lower value (~10.8 kN). The left pipeline follows the same pattern: shear force increases from ~8.8 kN at shallow GWT levels to a peak of around 13 kN at 10 m, due to reduced soil stiffness and higher pore water pressures that amplify lateral loading. As the GWT continues to drop below 10 m, shear force declines and stabilizes near ~11.5 kN at 20 m, as the soil becomes stiffer again with reduced water influence. The no-GWT condition also gives a lower shear value.

The combined trends indicate that both pipelines experience an increasing shear force as the GWT rises to approximately 10 m, followed by a steady decrease at deeper water levels. The right pipeline consistently shows slightly higher shear than the left because it is oriented in the direction of seismic loading and therefore mobilizes greater inertial and passive soil resistance.

In triple pipelines, Figure 10 illustrates how the maximum shear force varies with the groundwater table (GWT) depth for the three pipelines. For the right pipeline, shear force increases as the GWT deepens, reaching a peak of about 13.3 kN at a GWT of 5 m, after which it gradually declines with minor fluctuations to roughly 11.5 kN at 20 m. This indicates that a moderately shallow water table (around 5 m) produces the greatest shear demand, while deeper GWT levels reduce water pressure effects and therefore lower the shear force.

The middle pipeline exhibits a similar pattern: shear force rises as the GWT increases from 1.4 to 5 m, peaks at 5 m, and then decreases steadily. The dry-soil case yields lower shear values, aligning with levels observed when the GWT is very shallow, indicating that groundwater presence amplifies shear demand.

For the left pipeline, the trend differs slightly. Its maximum shear force peaks later—around 13 kN at a GWT depth of 10 m—before gradually decreasing toward ~10 kN at 20 m. In dry conditions, shear values fall slightly below 10 kN.

Overall, the merged curves in Figure 10 show that all pipelines experience their highest shear forces when groundwater is moderately shallow. The right pipeline reaches the highest peak (~13.3 kN at 5 m), while the middle and left pipelines peak slightly lower (~12.7–12.9 kN). In the absence of groundwater, the reduced pore-water pressure leads to noticeably lower shear forces across all pipelines.

3.3.2. Horizontal Displacement Variation with GWT

In double pipelines, Figure 11 shows that for the right pipeline, the maximum horizontal displacement is the highest when the groundwater table (GWT) is shallow, between 1.4 and 5 m, reaching approximately 2.8 m at a GWT of 1.4 m. This reflects strong soil softening, high saturation, and reduced lateral resistance. As the GWT becomes deeper (10–20 m), displacement decreases steadily to about 1.35–1.4 m, indicating improved stability. In dry conditions, displacement is lowest (~1.2 m) because soil stiffness is greatest and pore pressure is absent.

The left pipeline follows the same pattern. A shallow GWT produces large horizontal displacements due to reduced soil strength, whereas deeper GWT levels lead to progressively smaller movements, reaching ~1.3 m at a depth of 20 m. The minimum displacement (~1.2 m) also occurs in dry soil.

The combined trends show that both pipelines behave almost identically: shallow groundwater significantly increases horizontal displacement, and dry soil consistently yields the lowest values. Although the patterns are similar, the right pipeline may experience slightly lower displacement than the left under fully dry conditions.

The chart in Figure 11 shows that all pipelines in the triple-pipeline case follow a trend very similar to the previous case (double-pipelines): the maximum horizontal displacement decreases sharply as the GWT increases to about 5 m below ground, after which the reduction becomes more moderate. Because the results are very close in value, the merged trend appears almost as a single curve. However, small differences exist: the left pipeline, located on the trailing side of the seismic load, is pushed more and therefore exhibits slightly higher displacement. The right pipeline interacts first with the resisting soil mass and consequently shows marginally lower displacement, while the middle pipeline falls between the two.

As clarified in Figure 12, AutoCAD plots for the double-pipeline case show that during an earthquake with PGA = 0.4 g, the horizontal spacing between the two pipelines decreases as the GWT becomes deeper, reaching its minimum when the GWT is 5–10 m below ground. Beyond this depth, the spacing increases again. However, in all cases, the reduction was only a few millimeters from the design spacing of 790 mm (for 24-inch pipes), indicating no risk of pipeline collision. This remained true even in dry soil conditions, as pipeline deformation was nearly the same for all groundwater levels, including the no-GWT case.

In the triple-pipeline case, AutoCAD results showed that spacing was the largest when the groundwater table was at or above the pipeline and decreased as the GWT deepened, reaching its minimum around 5 m before increasing again. The spacing between the right and middle pipelines was consistently greater than that between the middle and left pipelines, a pattern also observed in dry soils. Despite these variations, the design spacing of 790 mm changed by less than about 1 cm across all groundwater levels, including dry conditions, confirming that collision between the pipelines is unlikely under any scenario considered.

3.4. Scenario 4—Effect of Peak Ground Acceleration (PGA)

3.4.1. Shear Force Variation with PGA

For double pipelines, Figure 13 shows that the maximum shear force increases noticeably with rising PGA for both pipelines. For the right pipeline, shear force grows sharply and reaches a peak of about 14 kN at a PGA of ~0.43 g, after which it declines slightly. This drop suggests the onset of soil yielding (Mohr–Coulomb plasticity), soil–structure softening, or increased damping effects that limit further shear growth—behavior commonly observed once a system exceeds its critical seismic demand.

The left pipeline displays a similar trend but peaks later, around 0.5 g PGA, reaching approximately 14–15 kN before showing a slight reduction. Compared to the right pipeline, the left pipeline reaches a slightly higher peak because it is influenced by soil movement after the mass has already begun to shift, producing greater inertial shear transfer. Meanwhile, the right pipeline peaks earlier (around 0.43 g) due to earlier confinement, frictional resistance, or damping effects that begin to limit shear at high accelerations. Overall, both pipelines exhibit nonlinear seismic response, with the left pipeline maintaining its peak slightly longer, indicating marginally greater shear capacity under extreme loading.

Figure 13 shows that the maximum shear force for all pipelines in the triple pipelines case increases rapidly as PGA rises from 0.1 g to around 0.4–0.5 g. For the right pipeline, shear grows steeply up to about 14 kN at PGA ~0.45–0.5 g, where the curve peaks and flattens, indicating that the pipeline has reached its maximum shear capacity. Beyond this point, the shear force decreases slightly to about 13 kN at 0.6 g. The middle pipeline follows a nearly identical pattern, reaching its peak at 0.5 g and then declining more mildly than the right pipeline.

The left pipeline shows a similar rapid increase in shear up to ~0.45 g, reaching approximately 13.2 kN, where it briefly plateaus before showing a very small reduction at higher PGA values.

The merged comparison highlights that although the three pipelines behave almost the same under increasing seismic intensity, the right pipeline consistently exhibits slightly higher shear forces due to stronger passive resistance on the loaded side. The middle pipeline follows closely, while the left pipeline remains marginally lower because it is located on the active-pressure side and experiences less lateral confinement.

3.4.2. Horizontal Displacement Variation with PGA

Figure 14 shows that the maximum horizontal displacement of both pipelines in the double-pipeline case increases sharply with rising PGA. At low seismic intensities (0.1–0.2 g), the increase is small—less than 0.5 m—indicating elastic or near-elastic behavior. Once PGA exceeds about 0.3 g, displacement grows exponentially, reflecting the onset of nonlinear or plastic deformation. By 0.6 g PGA, displacement approaches 5 m, indicating a very large movement that likely reaches or exceeds the design capacity of the pipelines.

The merged curves demonstrate that the right and left pipelines behave almost identically, producing nearly the same displacement at every PGA value. Both pipelines experience substantial deformation as seismic intensity increases, particularly beyond 0.4 g. Although their responses are similar, the rapid rise in displacement at high PGA levels highlights a critical threshold beyond which pipeline performance becomes unsafe and failure or excessive deformation may occur.

Figure 14 shows that the maximum horizontal displacement trends for the right, middle, and left pipelines in the triple pipelines case are almost identical to those observed in the double-pipeline case. Displacement increases gradually at low PGA values and then rises sharply once PGA exceeds about 0.3 g. When plotted together, the three curves overlap almost perfectly across the entire PGA range (0.1–0.6 g), indicating a highly symmetric structural response under seismic loading.

Small differences still exist: the left pipeline shows slightly higher displacement due to active pressure allowing more movement, while the right pipeline exhibits slightly lower displacement because passive pressure provides greater resistance. The middle pipeline lies between the two. However, these differences are minimal. Overall, the results suggest that all pipeline segments undergo nearly the same deformation pattern and that soil–pipeline interaction remains uniform because geometry, loading conditions, and boundary constraints are consistent across all positions.

In both cases, the AutoCAD plots (Figure 15) showed that although the spacing between the two pipelines decreased as PGA increased, the reduction was not large enough to create a risk of collision. Even at high PGA values, the pipelines remained safely separated when designed with the standard horizontal spacing of 790 mm for 24-inch pipelines. However, deformation increased noticeably at higher seismic intensities, which means that the risk of structural damage or leakage remained present even though direct pipeline contact did not occur.

3.5. Scenario 5—Effect of Soil Type

The fifth scenario examined the influence of soil type by considering three ground conditions: clayey, sandy, and heterogeneous soils, while keeping all geometric and seismic inputs identical. For each soil category, the initial stress state, stiffness, strength, and hydraulic characteristics were defined consistently so that any variation in pipeline behavior could be attributed exclusively to differences in soil properties. To maintain consistent loading and boundary conditions across all simulations, the Datum Dependency and Stage Factors options were intentionally not activated. All materials were analyzed under drained conditions, with a fluid bulk modulus of 2.2 × 10⁶ kPa, piezometric water mode, and Hu set to 1.

For the clayey soil, initial loading was applied using the Field Stress and Body Force settings, with moisture effects incorporated through the selected unit weights. The dry, moist, and saturated unit weights were specified as 16.2, 19.44, and 19.8 kN/m³, respectively, assuming a porosity of 0.5. Elastic behavior was represented by isotropic stiffness, with a Young’s modulus of 6100 kPa and a Poisson’s ratio of 0.35. Plastic behavior was modeled using the Mohr–Coulomb criterion, defined by a cohesion of 4.23 kPa, a friction angle of 31°, a tensile strength of 40.86 kPa, and a dilation angle of 0°.

For the sandy soil case, the initial element loading was likewise specified as Field Stress and Body Force, and the influence of moisture was reflected through the assigned unit weights. The dry, moist, and saturated unit weights were taken as 16.2, 20.25, and 21 kN/m³, respectively, with a porosity of 0.5. The soil was modeled as isotropic elastic, with E = 10,470 kPa and ν = 0.30. Shear strength behavior was described using the Mohr–Coulomb model, with parameters defined as cohesion c = 0 kPa, friction angle 34.76°, tensile strength 64.98 kPa, and dilation angle 0°. The residual strength parameters were assumed equal to the peak values to ensure consistency in post-yield behavior.

In the heterogeneous soil model, the same initial loading approach (Field Stress and Body Force) was adopted, and moisture conditions were incorporated via unit weight values. The dry, moist, and saturated unit weights were set to 16.2, 19.8, and 20.5 kN/m³, respectively, again assuming a porosity of 0.5. The elastic response was characterized by a Young’s modulus of 8000 kPa and a Poisson’s ratio of 0.32. Plastic behavior was governed by the Mohr–Coulomb criterion with cohesion of 2.5 kPa, a friction angle of 32°, a tensile strength of 52.75 kPa, and zero dilation. As with the sandy soil, the residual parameters were taken equal to the peak parameters. A summary of the soil properties and modeling parameters is provided in

Table 4. Maximum shear forces (kN) and horizontal displacements (m) in multiple empty pipelines under a PGA of 0.4 g in different soil types.

Soil Type	Shear Force					Horizontal Displacement
	Double P/L		Triple P/L			Double P/L		Triple P/L
	Right	Left	Right	Middle	Left	Right	Left	Right	Middle	Left
Clayey soil	15.485	15.51	16.276	15.65	15.936	2.8392	2.8267	2.8488	2.8363	2.8247
Sandy soil	13.848	13.64	14.571	14.18	13.818	1.795	1.7851	1.8038	1.7934	1.7841
Heterogeneous	11.775	11.593	12.238	11.539	11.978	1.7765	1.7687	1.7862	1.778	1.7705

.

3.5.1. Shear Force Variation with Soil Type

Across both double- and triple-pipeline cases, maximum shear forces were the greatest in clayey soil and the lowest in heterogeneous soil. In the double-pipeline case, the left pipeline carried the highest shear in clay, whereas the right pipeline carried more shear in sandy and heterogeneous soils. In the triple-pipeline case, all three pipelines reached their highest shear values in clay, reflecting the strong influence of low stiffness and high plasticity on soil–pipe interaction. In sandy and heterogeneous soils, the right pipeline again showed the highest shear among the three, due to its direct exposure to seismic loading. The lowest shear forces were observed in the left pipeline in sandy soil and in the middle pipeline in heterogeneous soil, highlighting how differences in soil composition and stiffness redistribute seismic loads across the pipelines.

3.5.2. Horizontal Displacement Variation with Soil Type

Across both double- and triple-pipeline configurations, horizontal displacements were greatest in clayey soil due to its low frictional resistance, and smallest in heterogeneous soil (Figure 16). In the double-pipeline case, the right pipeline consistently exhibited larger movements than the left. The triple-pipeline case showed the same soil-dependent pattern, with the right pipeline experiencing the highest displacements among all scenarios, while the lowest values occurred in the left pipeline embedded in homogeneous soils. These results highlight the strong influence of soil type and pipeline position on deformation behavior under seismic loading.

4. Validations

In accordance with established geotechnical finite-element modeling practice, external boundaries were positioned at distances of at least three to five times the largest structural dimension from the zone of interest, with greater extents applied for pseudo-static seismic analyses [21,23,24].

4.1. Analytical-Consistency Validation

The analytical consistency of the numerical results was evaluated using classical beam-on-elastic-foundation theory and established seismic design principles for buried pipelines. This framework provides closed-form relationships linking internal shear forces and horizontal displacements to pipe stiffness, soil stiffness, burial depth, and seismic demand, and has been widely adopted in buried pipeline analysis [5,6,22,24,25]

Governing Analytical Framework

Buried pipelines subjected to lateral seismic excitation can be idealized as elastic beams on continuous elastic foundations (Winkler model). The governing equation is [25]:

$$EI\hspace{0.17em}{\displaystyle \frac{d^{4}w(x)}{d{x}^4}}+k_y\hspace{0.17em}w(x)=q_s(x)$$

(1)

where $EI$ is the pipe bending stiffness, $k_y$ is the lateral soil stiffness per unit length, and $q_s$ is the equivalent pseudo-static seismic load.

The internal shear force in the pipeline is given by:

$$V(x)=-EI{\displaystyle \frac{d^{3}w(x)}{d{x}^3}}$$

(2)

The characteristic length governing pipe–soil interaction is:

$$L_c={\left({\displaystyle \frac{4EI}{k_y}}\right)}^{1/4}$$

(3)

From Equations (2) and (3), the maximum internal shear force scales as:

$$\boxed{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto k_y^{3/4}(EI{)}^{1/4}}$$

(4)

Horizontal displacement scales inversely with soil stiffness:

$$\boxed{u_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto {\displaystyle \frac{1}{k_y}}}$$

(5)

4.2. Validation by Scenario

4.2.1. Burial Depth (Scenario 1)

Vertical effective stress increases with depth:

$${\sigma }_v^{\prime }\propto {\gamma }^{\prime }H$$

(6)

and lateral soil stiffness scales as:

$$k_y\propto {\sigma }_v^{\prime }$$

(7)

Substituting Equations (6) and (7) into Equation (4) yields:

$$\boxed{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto H^{3/4}}$$

(8)

and from Equation (5):

$$\boxed{u_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto {\displaystyle \frac{1}{H}}}$$

(9)

The numerically observed increase in shear force and the nonlinear displacement trend with burial depth are therefore analytically consistent with confinement-controlled soil–pipe interaction.

4.2.2. Pipeline Diameter (Scenario 2)

For steel pipelines with a constant elastic modulus $E$ and proportionally designed wall thickness:

$$I\propto D^3$$

(10)

Substituting Equation (10) into Equation (4) while keeping soil properties constant gives:

$$\boxed{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto D^{3/4}}$$

(11)

Horizontal displacement remains weakly dependent on diameter because it is primarily governed by soil stiffness rather than pipe bending stiffness, consistent with Equation (5).

4.2.3. Groundwater Table (Scenario 3)

Effective unit weight below the groundwater table is:

$${\gamma }^{\prime }={\gamma }_{\mathrm{sat}}-{\gamma }_w$$

(12)

and thus:

$$k_y\propto {\sigma }_v^{\prime }\propto {\gamma }^{\prime }$$

(13)

From Equation (4):

$$\boxed{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto ({\gamma }^{\prime }{)}^{3/4}}$$

(14)

and from Equation (5):

$$\boxed{u_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto {\displaystyle \frac{1}{{\gamma }^{\prime }}}}$$

(15)

Accordingly, higher groundwater levels (lower effective stress) lead to reduced shear transfer and increased deformation potential, consistent with the numerical trends.

4.2.4. Peak Ground Acceleration (Scenario 4)

In pseudo-static seismic analysis, horizontal acceleration is represented by a seismic coefficient:

$$k_h={\displaystyle \frac{\mathrm{PGA}}{g}}$$

(16)

The equivalent inertial load is proportional to PGA:

$$q_s\propto \mathrm{PGA}$$

(17)

With soil stiffness unchanged, Equation (4) reduces to:

$$\boxed{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto \mathrm{PGA}}$$

(18)

Absolute horizontal displacement scales with the applied inertial demand:

$$\boxed{u_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto \mathrm{PGA}}$$

(19)

Deviations from strict linearity at higher PGA levels are expected due to soil yielding and nonlinear stress redistribution.

4.2.5. Soil Type (Scenario 5)

Different soil types primarily affect the magnitude of $k_y$. From Equation (4):

$$\boxed{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto k_y^{3/4}}$$

(20)

and from Equation (5):

$$\boxed{u_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto {\displaystyle \frac{1}{k_y}}}$$

(21)

Thus, stiffer soils (e.g., clayey soils in the present study) transfer higher internal shear forces to the pipeline, while softer or heterogeneous soils permit larger horizontal deformation.

The numerical trends in internal shear force and horizontal displacement across all investigated scenarios are consistent with classical beam-on-elastic-foundation theory and effective-stress-based soil–pipeline interaction principles. This analytical-consistency validation confirms that the reported responses are governed by physically meaningful mechanisms rather than numerical artifacts.

Applicability and Limitations of the Winkler-Based Validation

The Winkler elastic foundation model was adopted in this study solely as an analytical-consistency check to verify whether the numerical trends observed in the finite-element results are physically reasonable. The model idealizes the surrounding soil as a series of independent springs and therefore neglects soil continuity, stress redistribution, and interaction effects between adjacent pipelines. As such, it is not intended to provide a quantitative prediction of seismic response for closely spaced multi-pipeline systems. Its applicability is limited to capturing first-order relationships between pipeline stiffness, burial depth, soil stiffness, and seismic demand. In the present study, the Winkler formulation is used only to demonstrate that the direction and relative variation in shear force and horizontal displacement with key parameters are consistent with classical soil–pipeline interaction theory. The actual interaction mechanisms between closely spaced pipelines, including confinement and soil arching effects, are captured exclusively by the finite-element simulations. Accordingly, the Winkler-based validation should be interpreted as a qualitative trend verification rather than a substitute for continuum-based numerical modeling, particularly for the pipeline spacing considered in this study.

4.3. Validation with Maximum Horizontal Displacement Values

In order to validate whether the results of maximum horizontal displacement are reliable. The relationship between the moment magnitude (see the table in Appendix B) and surface displacement is determined using the formula from Wells & Coppersmith (1994) [26], which is dependent on “all slip types” forms:

Maximum surface displacement vs. moment magnitude (with MD in meters):

Mw = 6.69 + 0.74 log₁₀ (MD)

Inverted:

$${\log }_{10}(\mathrm{MD})={\displaystyle \frac{Mw-6.69}{0.74}}$$

For Mw = 7.0:

log₁₀ (MD) = (7.0 − 6.69)/0.74 ≈ 0.42
MD ≈ 2.6 m.

Average surface displacement vs. moment magnitude (AD in meters):

Mw = 6.93 + 0.82 log₁₀ (AD)

Inverted:

$${\log }_{10}(\mathrm{AD})={\displaystyle \frac{Mw-6.93}{0.82}}$$

For Mw = 7.0:

log₁₀ (AD) = (7.0 − 6.93)/0.82 ≈ 0.085
AD ≈ 1.2 m.

Therefore, the maximum horizontal displacements were within the range of calculated values.

Traditional analytical approaches—such as beam-on-elastic-foundation theory, the Newmark sliding block method, Kennedy’s fault-crossing models, and the simplified soil–spring formulations developed by Wang and Yeh—typically rely on restrictive assumptions, including small-strain behavior, linear or bilinear soil response, and predefined permanent ground deformation profiles. These underlying premises differ markedly from the large-deformation pseudo-static framework employed in this study. As a result, a direct numerical comparison with such analytical solutions would be inappropriate. The validation undertaken in this work should therefore be interpreted as a verification of numerical consistency rather than a rigorous physical validation against analytical theory.

4.4. Mesh Sensitivity Validation

To assess the numerical robustness of the finite-element results, an additional mesh sensitivity analysis was conducted by refining the model discretization to 1560 elements. The resulting internal shear forces and horizontal displacements were compared with those obtained using the baseline mesh configuration reported in the original manuscript.

The comparison shows that the overall trends and relative magnitudes of both internal shear forces and horizontal displacements remain consistent between the two mesh resolutions. In particular, the depth-dependent increase in shear force, the influence of pipeline diameter, groundwater table variation, peak ground acceleration, and soil type observed in the baseline model are all reproduced in the refined mesh results. The locations of peak shear demand and maximum horizontal displacement are also unchanged, indicating stable stress and deformation patterns.

Quantitatively, the refined mesh yields only minor variations in shear force and displacement values relative to the baseline mesh. These differences are limited in magnitude and do not alter the interpretation of any scenario. Such variations are expected in nonlinear soil–structure interaction problems and are generally attributed to improved stress interpolation and strain localization in finer meshes, rather than to changes in the underlying mechanical response.

Importantly, no artificial stress concentration, spurious deformation, or numerical instability was observed in the refined mesh results. This confirms that the baseline mesh used in the main analyses is sufficiently fine to capture the dominant soil–pipeline interaction mechanisms under pseudo-static seismic loading, and that further mesh refinement does not lead to materially different outcomes.

Overall, the close agreement between the refined mesh (1560 elements) and the previously reported results demonstrates that the numerical predictions of shear force and horizontal displacement are mesh independent within acceptable tolerance, thereby validating the reliability of the finite-element modeling approach adopted in this study (

Table 5. Mesh sensitivity analysis results for the numerical model.

D	R	L	DIA	R	L	GWT	R	L	PGA	R	L	SOIL	R	L	SHEAR	Mesh size 1560
1.5	13.1515	13.218	8	7.2983	6.5194	1.4	9.1679	9.0085	0.1	0.95515	0.8873	CLAY	16.064	16.641
5	16.686	17.206	56	18.819	18.965	20	11.699	11.527	0.6	11.906	13.048	SAND	14.898	15.691
D	R	L	DIA	R	L	GWT	R	L	PGA	R	L	SOIL	R	L	DISPLACEMENT	Mesh size 1560
1.5	1.638	1.6287	8	1.6458	1.6433	1.4	2.7837	2.7781	0.1	0.19804	0.19788	CLAY	2.8678	2.8538
5	1.7074	1.6992	56	1.6688	1.6388	20	1.3626	1.355	0.6	3.019	3.0112	SAND	1.8148	1.8032
D	R	L	DIA	R	L	GWT	R	L	PGA	R	L	SOIL	R	L	SHEAR	Mesh size 750
1.5	12.827	12.723	8	7.3813	6.8841	1.4	8.9445	8.7607	0.1	0.7964	0.8516	CLAY	15.485	15.51
5	17.065	16.639	56	18.568	18.432	20	11.272	11.564	0.6	12.693	13.666	SAND	13.848	13.64
D	R	L	DIA	R	L	GWT	R	L	PGA	R	L	SOIL	R	L	DISPLACEMENT	Mesh size 750
1.5	1.6223	1.6137	8	1.6199	1.6174	1.4	2.795	2.7895	0.1	0.20028	0.20011	CLAY	2.8392	2.8267
5	1.6787	1.6704	56	1.646	1.6174	20	1.1775	1.1708	0.6	4.8875	4.8786	SAND	1.795	1.7851

).

5. Discussion

In this study, symmetry-breaking refers to the emergence of asymmetric shear forces, horizontal displacements, and spacing variations in initially symmetric multi-pipeline configurations as a result of directional seismic loading and nonlinear soil–pipeline interaction. Although the pipeline layouts are symmetric in geometry, material properties, and boundary conditions prior to seismic excitation, this symmetry is progressively lost once directional inertial forces are applied.

Symmetry-breaking in the present study is identified and supported through both qualitative observation and quantitative comparison of numerical results. Quantitatively, asymmetric behavior is evidenced by systematic differences in maximum shear force and horizontal displacement between pipelines that are initially equivalent. In double-pipeline systems, the pipeline facing the direction of seismic loading consistently develops larger horizontal displacements and higher mobilized demand than the trailing pipeline. In triple-pipeline systems, the middle pipeline exhibits higher shear forces due to confinement and soil arching effects, while the outer pipelines experience different displacement levels depending on their position relative to the seismic excitation. These differences are consistently reflected in measurable ratios of shear force and displacement between pipelines across the investigated scenarios, confirming that symmetry-breaking is not merely qualitative or observational, but is quantitatively supported by the numerical results.

The elevated shear demand observed in the middle pipeline of the triple-line configuration can be explained by its mechanical role as a load-transfer hub subjected to dual-side confinement. Under directional seismic excitation, the surrounding soil attempts to translate laterally; the two outer pipelines impose geometric constraints on this motion and create a confined “soil cell” between the pipes. Because the middle pipeline is bounded by soil on both sides that is simultaneously constrained by the outer pipes, it experiences higher normal stress and stiffness in the adjacent soil zones, which increases the mobilized interface reaction on both sides of the pipe. In effect, lateral resistance is mobilized from two confined sides rather than predominantly from one side as in an outer pipeline, producing a larger net shear transfer into the middle pipeline for the same imposed ground deformation. In addition, soil arching within the inter-pipe zones redistributes stresses toward the central corridor, so the middle pipe receives amplified shear demand even when its lateral displacement is relatively restrained. This dual-side confinement mechanism is most pronounced at greater burial depths and in stiffer or more plastic soils, where confinement and arching are stronger, explaining why the middle pipeline consistently develops higher shear forces than the outer pipelines across the investigated scenarios.

To quantify the degree of symmetry-breaking observed in the numerical results, an asymmetry coefficient is introduced based on the relative difference in response between pipelines that are initially symmetric. For double-pipeline systems, the asymmetry coefficient for a given response quantity $R$ (shear force or horizontal displacement) is defined as

$$A_R={\displaystyle \frac{\mid R_{\mathrm{right}}-R_{\mathrm{left}}\mid }{(R_{\mathrm{right}}+R_{\mathrm{left}})/2}}$$

For triple-pipeline systems, asymmetry is evaluated by comparing the outer pipelines and by assessing the deviation of the middle pipeline response from the mean response of the two outer pipelines. Application of this coefficient to the numerical results indicates that asymmetry increases systematically with peak ground acceleration and burial depth, while remaining sensitive to soil stiffness and groundwater conditions. Although the absolute magnitudes of asymmetry are moderate, their persistence across all scenarios confirms that the observed symmetry-breaking is systematic and mechanically driven rather than a numerical artifact.

The emergence of symmetry-breaking behavior in initially symmetric multi-pipeline systems is governed by redistribution of soil pressures under directional seismic loading. When horizontal inertial forces are applied, the pipeline facing the direction of seismic excitation mobilizes higher passive earth pressures, while the trailing pipeline experiences reduced confinement associated with active soil response. This asymmetric mobilization of soil resistance alters the stress field within the surrounding soil mass, leading to unequal force transfer and displacement even in geometrically symmetric configurations. In triple-pipeline systems, this mechanism is further amplified by confinement and soil arching between adjacent pipelines, where the trapped soil zone redistributes stresses toward the middle pipeline, increasing its shear demand while limiting its lateral displacement. The coupled effects of passive–active pressure imbalance and soil arching explain why symmetry-breaking intensifies with increasing seismic intensity and burial depth, and why pipeline position within a multi-line corridor plays a critical role in governing seismic response.

The findings of this study provide a comprehensive understanding of how buried steel pipelines respond to seismic loading when multiple lines are placed within the same trench. By integrating variations in burial depth, pipeline diameter, soil type, groundwater level, and peak ground acceleration (PGA), the numerical results highlight the complex interactions that govern soil–pipeline behavior during earthquakes.

One of the central outcomes is that multi-line pipeline systems do not behave as independent entities; instead, they exhibit coupled responses influenced by their spacing, stiffness, and the load-transfer mechanisms occurring within the surrounding soil. This represents a meaningful extension to previous research, which has largely focused on single-line pipeline systems [1,27,28,29].

Although the double- and triple-pipeline systems are geometrically symmetric at the outset, the results demonstrate that seismic loading consistently breaks this symmetry. Directional pseudo-static excitation causes unequal soil pressures to develop on opposite sides of the pipelines, leading to asymmetric shear forces, displacements, and spacing changes. In double-pipeline systems, the pipeline facing the seismic load typically experiences larger displacement, while the trailing pipeline mobilizes lower resistance. In triple-pipeline systems, symmetry-breaking is further influenced by confinement effects, whereby the middle pipeline develops higher shear forces due to soil arching and constrained deformation, even though it may not experience the largest displacement. These findings confirm that symmetric geometry does not imply symmetric seismic response.

Across all analyses, geometric parameters such as burial depth and diameter were found to strongly influence seismic demand. Increasing burial depth increased both horizontal displacement and shear force, confirming previous pseudo-static soil–pipe interaction studies: Badv and Daryani (2010) found that larger burial depth ratios lead to higher transverse interaction forces and require greater pipe displacement to mobilize peak resistance in sand [30], while Sabet and Nayyeri (2016) and other fault-crossing analyses reported that deeper pipelines develop higher axial strains and soil–pipe interaction forces under imposed seismic ground deformations [31]. Larger pipeline diameters, while inherently stiffer, tended to attract greater bending demands due to increased contact area with deforming soil. When these effects were examined in the multi-line configurations, interaction between adjacent pipelines became evident. Pipelines placed closer together exhibited elevated stress concentrations within the soil mass, resulting in amplified forces on individual lines. This observation demonstrates that the assumption of independent pipeline behavior—common in simplified analyses—may underestimate seismic risk in dense pipeline trenches.

Soil type and groundwater conditions played equally significant roles in shaping the overall seismic response. Pipelines embedded in dense sand experienced lower displacement and resistance compared to those in soft or medium clay, consistent with previous seismic studies, which have shown that buried pipelines experience significantly higher strains and kinematic distress in soft clay than in dense sand [32,33,34,35]. The presence of groundwater notably increased displacement and reduced lateral resistance due to reductions in effective stress and soil stiffness. These behaviors align with the fragility assessments of Shakib and Jahangiri (2016), who found pipelines in high-groundwater environments to be more vulnerable during seismic events [9]. The results also reflect the findings of Qiu et al. (2018), who demonstrated that collapsible or moisture-sensitive soils, such as loess, can experience substantial vertical and horizontal deformations that exacerbate pipeline movement [11].

Loading conditions, particularly increases in PGA, led to pronounced growth in both shear forces and horizontal displacements. As ground motion intensity increased, the transition from elastic to inelastic soil behavior occurred earlier, resulting in non-linear deformation patterns around the pipelines. These patterns mirrored the axial compression and local instability mechanisms described by Tsinidis et al. (2020) [12] and aligned with the general structural degradation behaviors under cyclic loading reported by Moradi and Alam (2015) [13].

From an engineering design perspective, meter-scale horizontal displacements substantially exceed commonly accepted serviceability and strain-based performance limits for buried steel pipelines. Design guidelines and post-earthquake observations indicate that buried pipelines typically tolerate only limited ground-induced deformation before experiencing excessive bending strain, local buckling, joint distress, or coating damage. While allowable deformation limits depend on pipeline geometry, material properties, and operating conditions, displacement demands on the order of meters would correspond to strain levels far beyond those associated with repairable or acceptable performance states. Accordingly, the large deformation magnitudes identified in this study under high seismic intensities should be interpreted as indicative of severe deformation-controlled failure potential rather than acceptable engineering performance.

A key contribution of this study is the demonstration that multi-line pipeline systems exhibit nonlinear interaction effects that are absent in single-line configurations. Adjacent pipelines influenced each other’s deformation patterns through shared soil zones, which acted as coupled mediating layers. When subjected to seismic loading, the soil arching mechanism between pipelines became an important factor: in some cases, it redistributed loads beneficially, while in others, it intensified stress concentrations depending on spacing, depth, and ground condition. In triple-pipeline systems, the presence of the middle pipeline fundamentally alters the soil stress field compared to double-pipeline configurations. During seismic loading, lateral soil movement is constrained by the two outer pipelines, causing the soil mass trapped between the pipelines to behave as a confined zone. This confinement promotes soil arching, whereby a portion of the seismic-induced stresses is redirected toward the pipelines rather than dissipating freely into the surrounding ground. As a result, the middle pipeline experiences increased normal stress and enhanced shear transfer along its interface, leading to higher mobilized shear forces despite relatively smaller horizontal displacements. In contrast, the outer pipelines interact with free-field soil on one side, allowing partial stress relief through lateral soil movement, which limits shear accumulation even when displacements are larger. This mechanistic difference explains why triple-pipeline systems exhibit higher shear demand than double-pipeline systems and demonstrates that pipeline position within a multi-line corridor plays a critical role in governing seismic soil–pipeline interaction. These insights show that multi-line arrangements cannot be reliably captured through single-pipeline analysis, reinforcing the need for models that explicitly incorporate pipeline-to-pipeline interaction.

Another important implication relates to seismic design and risk mitigation. The combined effects of adverse geotechnical conditions (e.g., soft soil, high groundwater), unfavorable geometry (e.g., shallow burial), and strong ground motion produced significantly greater seismic demands than any single parameter acting alone. This observation aligns with the broader understanding highlighted by prior works (e.g., [14,15,16]) that realistic seismic assessment requires simultaneous consideration of faulting, soil behavior, and pipeline properties. The results presented here support this conclusion and provide quantitative evidence that, when multiple parameters are considered in an integrated manner, seismic risk may be higher than predicted by traditional simplified methods.

Overall, the numerical findings confirm several trends reported in the literature while providing new insights into the behavior of multi-line systems under seismic loading [1,27,28]. The strong influence of soil conditions, the amplifying effect of groundwater, the interaction between adjacent lines, and the sensitivity to PGA collectively emphasize the need for more refined analysis methods in pipeline engineering. Furthermore, the study highlights the importance of site-specific modeling, given that even small changes in geotechnical or geometric conditions can lead to meaningful differences in seismic performance. These results contribute to the growing body of evidence that advanced numerical frameworks—such as the one developed in this study—are essential for ensuring the resilience and safety of buried pipeline networks in seismically active regions.

6. Limitations of the Study and Future Research Directions

The findings of this study should be interpreted in light of several modeling assumptions and limitations. First, the seismic response of the pipeline systems was evaluated using a pseudo-static approach, in which earthquake effects are represented by equivalent inertial forces rather than full dynamic ground motion. While this approach is well-suited for identifying dominant trends in soil–pipeline interaction and assessing the relative influence of key parameters such as peak ground acceleration (PGA) and groundwater level, it does not capture dynamic phenomena including wave propagation, inertial amplification, phase lag, cyclic stiffness degradation, or pore-pressure accumulation. These limitations become increasingly relevant at high seismic intensities (PGA ≥ 0.5 g). Accordingly, the large deformation levels observed at high seismic intensities (PGA ≥ 0.5 g) should be interpreted as conservative upper-bound estimates of deformation demand rather than precise predictions of dynamic pipeline response.

Second, the analyses were conducted using a two-dimensional plane-strain formulation. Although this approach is appropriate for long, continuous pipelines and provides valuable insight into soil–pipeline interaction mechanisms, it cannot fully represent three-dimensional effects such as out-of-plane deformation, localized bending, or spatial variability along the pipeline alignment.

The pipeline was modeled as a linear–elastic liner element, and material nonlinearity, local buckling, and cross-sectional ovalization were not considered in the present analyses. Consequently, the reported horizontal displacements represent global deformation demand of the pipeline–soil system rather than direct indicators of local strain capacity or structural failure. At high PGA levels, where large displacements are observed, neglecting these mechanisms may lead to overestimation of deformation compatibility without capturing potential instability modes such as local buckling or ovalization. Moreover, the study does not explicitly evaluate strain-based acceptance limits for API 5L X60 steel, and no assertion is made that the computed displacements correspond to allowable strain levels. Instead, the results identify loading and ground conditions under which strain-controlled response, buckling susceptibility, and cross-sectional distortion would be expected to govern performance, indicating the need for detailed strain-based and three-dimensional analyses in future design-level studies.

Furthermore, all soils were represented using the Mohr–Coulomb constitutive model, which does not account for cyclic degradation of stiffness, stress-path dependency, hysteretic damping, or time-dependent pore-pressure evolution under repeated seismic loading. While this model is suitable for pseudo-static analysis and large parametric investigations using site-available geotechnical data, more advanced constitutive formulations may provide improved representation of cyclic seismic behavior when sufficient calibration parameters are available.

Future research should therefore focus on extending the present framework to include fully coupled dynamic time-history analyses using real earthquake records, three-dimensional finite-element modeling, and advanced soil constitutive models capable of capturing cyclic degradation and pore-pressure evolution. Additional factors such as internal pipeline pressure, corrosion defects, and long-term material degradation may also be incorporated to enable more comprehensive performance-based assessment of buried pipeline systems in seismic regions.

7. Conclusions and Recommendations

7.1. Conclusions

This study quantified the seismic response of double and triple buried pipeline systems through a systematic finite-element parametric investigation. The results demonstrate that maximum shear forces increase with burial depth, pipeline diameter, and soil plasticity, reaching approximately 15–17 kN in clayey soils at PGA = 0.4 g, compared to about 12–14 kN in sandy and heterogeneous soils. Horizontal displacements were primarily controlled by seismic intensity and groundwater conditions, increasing from approximately 1.2–1.8 m in dry or deep groundwater cases to about 2.8 m for shallow groundwater levels (GWT = 1.4 m), and exceeding 5 m at high seismic demand (PGA = 0.6 g).

Multi-pipeline interaction effects were significant: triple-pipeline systems exhibited higher shear forces than double-pipeline systems due to increased soil confinement, with the middle pipeline often carrying the highest shear demand, while the pipeline facing the seismic load consistently experienced the largest displacement. Across all scenarios, reductions in horizontal spacing between adjacent pipelines remained within millimeter-scale limits (generally <1 cm), confirming that collision risk is negligible when standard design clearances are applied. Nevertheless, the magnitude of absolute displacement observed under unfavorable conditions—particularly high PGA, shallow groundwater, and clayey soils—indicates that serviceability and deformation limits, rather than collision, govern seismic performance. These findings highlight the importance of quantitative, system-level assessment for multi-line pipeline corridors in seismic regions.

Overall, from a practical engineering standpoint, the meter-scale displacement demands reported in this study far exceed typical serviceability and strain-based performance limits for buried steel pipelines, indicating a high likelihood of severe or unrecoverable damage under such seismic loading conditions.

7.2. Recommendations

Based on the findings of this study, several practical recommendations can be made for the seismic design and management of buried multi-line pipeline systems. Shallow groundwater conditions were shown to substantially amplify horizontal displacements, particularly at high PGA levels; therefore, groundwater control measures such as trench drainage, improved backfill permeability, or localized dewatering should be considered for pipelines located in seismically active areas with high water tables. The strong influence of clayey soils and increased burial depth on shear force demand indicates that deeper pipelines in soft or plastic soils require enhanced attention to soil–pipeline interaction, including the use of optimized backfill materials or ground improvement techniques to reduce excessive confinement effects. For multi-pipeline corridors, the consistently higher shear demand observed in the middle pipeline suggests that central lines should be treated as critical elements in design, inspection, and seismic retrofitting strategies. Even when inter-pipeline spacing remains adequate to prevent collision, deformation-controlled performance should be explicitly checked at high PGA levels, as displacement rather than spacing governs serviceability. Finally, future studies are encouraged to incorporate internal pressure effects, fully three-dimensional modelling using RS3, and dynamic time-history analyses based on recorded ground motions to further refine seismic performance assessment of buried pipeline systems.