Process Simulation of Pseudo-Static Seismic Loading Effects on Buried Pipelines: Finite Element Insights Using RS2 and RS3

Abstract

Buried pipelines represent critical lifeline infrastructure whose seismic performance is governed by complex soil–structure interaction mechanisms. In this study, a process-based numerical framework is developed to evaluate the pseudo-static seismic response of buried steel pipelines installed within a trench. A comprehensive parametric analysis is conducted using the finite-element software Rocscience RS2 (version 11.027) to examine the influence of burial depth, pipeline diameter, slope angle, groundwater level, soil type, and permanent ground deformation. The seismic loading was represented using a pseudo-static horizontal acceleration, which approximates permanent ground deformation rather than full dynamic wave propagation. Therefore, the results represent simplified lateral seismic demand and not the complete dynamic soil–structure interaction response. To verify the reliability of the 2D plane–strain formulation, a representative configuration is re-simulated using the fully three-dimensional platform Rocscience RS3. The comparison demonstrates excellent agreement in shear forces, horizontal displacements, and cross-sectional distortion patterns, confirming that RS2 accurately reproduces the dominant load-transfer and deformation mechanisms observed in three-dimensional (3D) models. Results show that deeper burial and stiffer soils increase shear demand, while higher groundwater levels and larger permanent ground deformation intensify lateral displacement and cross-sectional distortion. The combined 2D–3D evaluation establishes a validated computational process for predicting the behavior of buried pipelines under a pseudo-static lateral load and provides a robust basis for engineering design and hazard mitigation. The findings contribute to improving the seismic resilience of lifeline infrastructure and offer a validated framework for future numerical investigations of soil–pipeline interaction.

1. Introduction

According to ASME B31.4, a pipeline includes all physical components used to transport liquids or gases, such as pipes, valves, fittings, flanges (with bolting and gaskets), regulators, pressure vessels, pulsation dampeners, relief valves, pipe-mounted appurtenances, pump units, metering systems, pressure-regulating, pressure-limiting, and pressure-relief stations, as well as fabricated assemblies. The term “underground pipelines” (U/G) refers to pipelines that are buried or installed within trenches [1]. Due to their wide spatial extent and constant interaction with surrounding subsurface materials, these systems are inherently exposed to numerous geotechnical and seismic hazards that may compromise their structural integrity.

Pipelines transporting oil or gas from remote production areas to demand centers can span hundreds of miles, crossing varied terrain and frequently intersecting active fault zones [2]. Buried pipelines are particularly susceptible to seismic hazards because they experience forces and deformations transmitted through soil–pipe interaction; essentially, ground motion directly leads to pipeline deformation [3]. A rupture at even a single point can disable the entire system, underscoring the importance of designing pipelines that can withstand significant fault displacements [2]. Historical earthquake data further show that seismic events not only cause direct structural damage but can also trigger secondary hazards such as fires and explosions. Thus, the safety and operational continuity of lifeline infrastructure are strongly dependent on the seismic response of buried pipelines [4]. Seismic hazards associated with such failures can generally be classified based on the underlying ground movement mechanisms and the nature of the resulting damage.

Seismic hazards affecting buried pipelines can generally be classified into four categories based on the type of damage induced [5]: permanent ground deformation (PGD) related to soil failures, including longitudinal and transverse PGD as well as landslides; buoyancy due to liquefaction; permanent ground deformation related to faulting; and seismic wave propagation.

Understanding these hazard mechanisms is fundamental for establishing appropriate seismic design criteria and performance objectives, which are determined by regional regulations and the anticipated intensity of ground motion. For seismic design purposes, the design basis earthquake (DBE) is generally defined as an event with a 10% probability of exceedance in 50 years—equivalent to a return period of roughly 475 years—and is designated as DD-2 in the current Turkish Earthquake Code. In terms of functional importance, pipelines are typically divided into four criticality classes [5]. Class I comprises essential pipelines that must remain fully operational during and after a design earthquake, such as high-pressure oil and gas lines or emergency water pipelines needed for post-earthquake response. Class II includes critical pipelines serving large populations, where temporary service interruption is acceptable until minor repairs are completed. This category typically covers medium-pressure oil and gas systems, as well as major water supply lines. Class III consists of ordinary pipelines whose temporary failure does not pose a significant risk—examples include low-pressure oil and gas pipelines and general water distribution lines. Class IV encompasses low-importance pipelines whose failure has minimal societal impact and can be repaired after the seismic event, usually local or small-scale water networks [5]. Beyond functional classification, the mechanical design and construction of pipelines are also key factors in determining their seismic performance.

Generally, oil and gas pipelines are designed and constructed as continuous systems, with joints that possess higher strength and stiffness than the pipe barrel itself [5]. The diameter of a pipeline is determined by several interrelated factors, including the operational requirements of the facility, the capital cost of both the pipe and auxiliary components such as pumping or compression stations, and the nature and volume of the transported fluid. Additional considerations include pressure ratings, which define the wall thickness and, consequently, the overall diameter, as well as the material and grade of the pipe, which influence its strength, toughness, and flexibility. Design codes and safety factors also play a significant role in ensuring compliance with relevant standards. Additionally, environmental constraints, terrain conditions, and fluid properties, such as temperature, velocity, and chemical composition, further guide material selection and sizing decisions.

Steel pipelines are manufactured in a broad range of commercial sizes, from as small as 1/8 inch (3.175 mm) to over 60 inches (1524 mm) in diameter. According to ASME B31.4 [1], the required wall thickness is calculated based on the fluid’s properties and must incorporate allowances for internal pressure, threading, grooving, corrosion, and erosion. For most pressure piping systems, steel is the preferred material, produced either as seamless or welded pipes. Seamless (SMLS) pipes, manufactured by piercing a billet followed by rolling or drawing, contain no welded seams and are typically supplied in standard lengths of 6, 9, or 12 m [6], making them ideal for diameters below 24 inches (609.6 mm). In contrast, welded pipes are fabricated by bending and joining steel plates or sheets into cylindrical forms and are categorized into two main types: longitudinal welded and spiral welded pipes. Larger diameters—generally exceeding 24 inches—are mostly produced using welded manufacturing methods.

Despite such design advancements, historical earthquake events have demonstrated that even well-engineered buried pipelines remain vulnerable to seismic damage, underscoring the need for continued assessment and modeling.

Breakage of continuous pipelines due to fault movement is the primary form of seismic damage. For instance, during the 1971 San Fernando earthquake, although only about 0.5% of the area was directly affected by fault movement, more than 1400 breaks occurred in water, natural gas, and sewer pipelines [7]. This highlights the vulnerability of buried lifeline systems to fault-induced ground deformation.

To better understand and mitigate pipeline failures during earthquakes, a wide range of analytical and numerical modeling approaches have been developed to simulate soil–pipeline interaction under seismic loading. The finite element (FE) method has become one of the most effective tools for this purpose, as it solves equilibrium equations with specified boundary conditions without relying on simplified deformation assumptions. Various FE programs have been used in past studies. For instance, Hart et al. [8] employed PIPLIN (1991) to design fault-crossing resistance for the All-American oil transmission pipeline crossing the San Andreas Fault, while Lee et al. analyzed similar problems using ANSYS (1994) [2]. Earlier research utilized one-dimensional (1D), three-dimensional (3D), and hybrid FE models [9,10,11,12,13,14] to validate analytical procedures, conduct parametric studies, and develop strain-based design criteria addressing tensile failure, beam buckling, local buckling, and ovalization. More recently, Demirci et al. [15] utilized ABAQUS (version 6.14)/EXPLICIT to simulate lateral pipe movement in clay, comparing the resulting soil resistance–lateral displacement behavior with the ASCE Guidelines (1984) for validation purposes. The latest work by Yousife et al. [2] utilized PLAXIS-2D V8.5, combined with shaking table experiments, to investigate the parameters influencing the seismic performance of buried unplasticized polyvinyl chloride (UPVC) pipeline flexible pipes subjected to seismic loading.

Although Rocscience RS2/RS3 are more commonly associated with slope and tunnel engineering, several applications in the literature have demonstrated their suitability for soil–structure interaction involving buried linear infrastructure. Díaz-Díaz et al. (2018) used RS2 to analyse ground response and surface settlements during pipe-jacking of a 1.5-m diameter underground collector [16], while the RS2 Thermal Verification Manual includes a benchmark freezing analysis of a buried pipeline [17]. More recently, Tyrer et al. (2023) and Indrawan et al. (2022) combined RS2 and RS3 to simulate tunnel- and deep-excavation-induced ground movements in 2D and 3D, validating RS2/RS3 predictions against PLAXIS and field data [18,19]. These examples indicate that the use of RS2 in the present study to model soil–pipe interaction and associated deformations is consistent with current practice.

Building upon these earlier works, the present study aims to provide a comprehensive numerical evaluation of the seismic behavior of buried steel pipelines using a two-dimensional finite element approach in Rocscience RS2, as it provides a geotechnical-focused finite-element environment that is well suited to modelling buried pipelines subjected to ground deformation. Unlike general structural FE codes, RS2/RS3 offers direct tools for simulating soil yielding, groundwater effects, staged construction, and slope-induced movements, allowing the pipeline to be analysed within a realistic ground profile rather than in isolation. The software’s built-in liner and interface elements enable efficient representation of soil–pipe interaction, while pseudostatic seismic loading can be applied directly to the soil domain to capture kinematic effects. These features make RS2/RS3 particularly effective for the parametric evaluation of burial depth, soil type, slope gradient, and groundwater level conducted in the present work.

The analysis investigates how variations in pipeline geometry and geotechnical conditions influence the structural response under seismic excitation. Specifically, parameters such as burial depth, pipe diameter, groundwater table, soil type, slope gradient, and earthquake peak ground acceleration (PGA) are systematically varied to assess their combined effects on shear forces and horizontal displacements along the pipeline wall. Unlike many previous studies that focused on a single variable or simplified assumptions about soil–pipe interaction, this work introduces an integrated parametric framework that captures the interdependence between geometric, seismic, and geotechnical parameters. The study’s findings aim to enhance the understanding of buried pipeline performance under realistic seismic loading conditions and provide design insights for improving the resilience of critical lifeline infrastructure in seismically active regions.

It is important to clarify that although earthquake hazards include dynamic effects such as wave propagation, cyclic degradation, pore-pressure buildup, and liquefaction, the present study does not attempt to model these phenomena. Instead, the analysis employs a pseudo-static horizontal acceleration, a common engineering approximation used to represent the lateral demand associated with permanent ground deformation (PGD). This approach captures relative trends in soil–pipe interaction but does not reproduce the full time-dependent dynamic response of buried pipelines. Accordingly, the results should be interpreted as simplified seismic demand rather than complete earthquake simulation.

2. Materials and Methods

2.1. Numerical Model Overview

A single continuous welded pipeline was analyzed using Rocscience RS2 (version 11.027; Rocscience Inc., Toronto, ON, Canada) to evaluate its behavior under seismic loading. The model represented a steel pipeline embedded in different types of soil and subjected to horizontal ground accelerations. The analysis aimed to determine the shear forces and horizontal displacements that develop in the buried pipeline when exposed to varying seismic and soil conditions.

2.2. Pipeline Geometry and Material Properties

The pipeline was idealized as a uniform continuous welded section; individual welds were not modeled explicitly. This approach is standard in pseudo-static soil–pipeline interaction analyses, where the focus is on global deformation and kinematic soil–pipe interaction rather than local stress concentrations. Fully welded joints in steel pipelines generally exhibit strength comparable to, or greater than, the pipe barrel, and therefore their omission does not affect the global displacement or shear trends investigated in this study. The diameters ranged from 8 in. (203.2 mm) to 56 in. (1422 mm), and wall thicknesses as listed in Appendix A. The pipeline material was steel (API 5L X60) with a modulus of elasticity of 2.06 × 10⁶ kPa. The pipeline was buried at depths ranging from 1 m to 5 m, with groundwater tables varying between 1.4 m and 20 m. Several PGA values (0.1–0.6 g) were applied, and different ground slopes were considered during the analysis.

The pipeline was represented in RS2 using a Standard Beam liner, whose mechanical properties were calibrated to reproduce the stiffness and inertia characteristics of a continuous welded steel pipe. The liner thickness was assigned as the actual wall thickness of the pipe (e.g., 9.525 mm for a 24-in API 5L X60 pipe), and the Young’s modulus was set to 2.06 × 10⁸ kPa, corresponding to the elastic modulus of steel. A Poisson’s ratio of 0.28 was used, consistent with steel’s elastic behavior. The unit weight (78.5 kN/m³) accounted for the steel density to correctly represent the gravitational body force and seismic inertial contribution. The Timoshenko beam formulation was selected to capture both bending and shear deformation, which is necessary for buried pipelines subjected to lateral soil loading. No sliding gap was assigned, ensuring continuous transfer of soil–pipe interaction forces. With these modifications, the liner element exhibits the correct axial, bending, and shear stiffness that govern the global deformation of a steel pipeline during seismic loading. This calibration allows the RS2 liner to replicate the pipe’s structural response under pseudo-static earthquake-induced ground deformation, even though local stresses at welds and circumferential ovalization are beyond the scope of a 2D beam representation.

2.3. Pipeline Geometry and Material Properties

The pipeline design geometry was defined based on a Typical Ditch Standard (STD-CS-02) commonly adopted in projects of the Iraqi State Company for Oil Projects (Baghdad, Iraq), as illustrated in Figure 1. The soil data were obtained from the Al-Kut Gas Factory Project, located in one of the most seismically active regions of Iraq, due to its position between the Tigris and Al-Gharraf Rivers, where groundwater is generally shallow (Figure 2).

Figure 1. Typical Ditch Standard (STD-CS-02).

Figure 2. Map of Iraq showing the location of the study area (Al-Kut).

2.4. Modeling Configuration and Boundary Conditions

Six scenarios were developed by varying one parameter at a time while keeping all other conditions constant. The analyzed parameters included burial depth, pipeline diameter, groundwater table, peak ground acceleration (PGA), slope, and soil type.

The numerical model had overall dimensions of 50 m × 109.655 m. The affected points of the pipeline along the section where shear forces and displacements were extracted are shown in Figure 3. Boundary conditions were applied as illustrated in Figure 4. The vertical boundaries on both sides were restrained in the X-direction to prevent horizontal movement, while the bottom boundary was fixed in the Y-direction, except at two corner nodes where both X and Y displacements were fixed to eliminate rigid-body motion. The top surface was left free of restraints to allow natural deformation under the applied loads.

Figure 3. The affected points of the pipeline.

Figure 4. The restraints in the model.

The pipeline was modeled using a liner whose properties were adjusted to represent the characteristics of the steel material. The stress analysis tolerance was set to 0.001 for high numerical accuracy, and the convergence criterion was defined based on absolute force and energy parameters. This setting ensures that stresses converge to within 0.1% between iterations, providing a stable and accurate representation of soil yielding and pipe deformation, while absolute force and energy are used to ensure the finite element solution accurately balances forces and displacements during analysis, especially with complex loads like seismic or excavation. Body force creates initial stresses (like overburden), and energy checks ensure the solution accurately reflects these self-weights and applied loads. Without body force included in the initial loading, the seismic force becomes zero despite the seismic coefficient being defined.

2.5. Soil Characterization

The surrounding soils were modeled as clayey, sandy, and heterogeneous materials to represent the range of site conditions encountered in the study area. The engineering parameters for these soils—such as unit weight, cohesion, friction angle, Young’s modulus, and Poisson’s ratio—were defined in accordance with the Al-Kut Gas Factory soil investigation data and incorporated directly into the RS2 model.

The Mohr–Coulomb (M–C) model was adopted to represent soil behavior in this study. The authors acknowledge that M–C is an elastic–perfectly plastic model that does not reproduce cyclic stiffness degradation, strain-dependent shear modulus variation, damping evolution, or pore-pressure generation during seismic loading. Because the present work employs a pseudo-static approximation of lateral seismic demand rather than dynamic time-domain loading, the M–C model was used only to capture relative differences in soil strength and stiffness for the parametric comparison. Therefore, the results should be interpreted as simplified estimates of soil–pipeline interaction rather than fully dynamic seismic response.

Regarding the soil properties in the first five scenarios, the initial element loading was defined as Field Stress and Body Force, and the moisture content was considered in unit weight. The dry, moist, and saturated unit weights were 16.2, 19.44, and 19.8 kN/m³, respectively, with a porosity value of 0.5.

The material stiffness was modeled as isotropic, having a Young’s Modulus (E) of 10,390 kPa and a Poisson’s Ratio (ν) of 0.35, extracted from field measurements. Strength behavior followed the Mohr–Coulomb failure criterion with plastic material type, where the cohesion (c) was 6.34 kPa, the friction angle (φ) was 31.25°, and the tensile strength reached 153.64 kPa. The dilation angle was set to 0°, and the residual parameters were taken equal to the peak values to maintain a consistent post-yield response.

Hydraulic behavior was defined as drained, with a fluid bulk modulus of 2.2 × 10⁶ kPa, a piezometric water mode, and a Hu value of 1. All options in the Datum Dependency and Stage Factors fields were left unselected to ensure uniform boundary and loading conditions for this layer throughout the analysis.

2.6. Seismic Loading and Analysis Scenarios

In Rocscience RS2, pseudo-static earthquake loading is applied using a seismic coefficient, which represents the peak ground acceleration (PGA) as a dimensionless fraction of gravitational acceleration. When this coefficient is defined, RS2 automatically applies an additional horizontal body force to every finite element. The seismic body force is computed as:

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

where $k_h$ is the seismic coefficient (equal to PGA), $\gamma$ is the unit weight of the soil or material, and $A$ is the element area. In essence, the seismic force corresponds to the horizontal inertial demand induced by ground shaking, while the body force due to gravity represents the element’s self-weight. RS2 combines these effects by vectorially adding the horizontal seismic body force to the existing vertical gravitational body force to obtain the total body force acting on each element. This formulation allows the pseudo-static method to approximate permanent-ground-deformation-type loading without performing full dynamic time–history analysis [20,21]. Table 1 shows the correlations of peak ground acceleration with the peak ground velocity, potential damage, the perceived shaking and instrumental intensity scale developed by USGS (USGS, 2011) [22,23].

Table 1. Correlation of PGA with the Instrumental Intensity scale and Richter scale [22,23].

Moment Magnitude (Richter)	Instrumental Intensity	Acceleration (g)	Velocity (cm/s)	Perceived Shacking	Potential Damage
Under 2.0	I	<0.0017	<0.1	Not felt	None
2.0–2.9	II–III	0.0017–0.014	0.1–1.1	Weak	None
3.0–3.9	IV	0.014–0.039	1.1–3.4	Light	None
4.0–4.9	V	0.039–0.092	3.4–8.1	Moderate	Very light
5.0–5.9	VI	0.092–0.18	8.1–16	Strong	Light
6.0–6.9	VII	0.18–0.34	16–31	Very strong	Moderate
7.0–7.9	VIII	0.34–0.65	31–60	Severe	Moderate to heavy
8.0 or higher	IX	0.65–1.24	60–116	Violent	Heavy
	X+	>1.24	>116	Extreme	Very heavy

A uniform finite-element mesh composed of 8-noded quadrilateral elements was used, with approximately 750 elements generated over the model domain. The seismic demand was represented using a pseudo-static horizontal body force applied to the soil domain. This method imposes a uniform horizontal acceleration and is widely used as a simplified representation of permanent ground deformation (PGD) rather than full dynamic wave propagation. The pseudo-static load does not capture cyclic loading, frequency content, or pore-pressure generation; instead, it provides a monotonic approximation suitable for parametric comparison of soil–pipe interaction trends. In most scenarios, the pseudo-static seismic load was applied in a single horizontal direction. Under this simplified framework, the analysis focuses on the direction producing the largest relative soil–pipe displacement. True seismic excitation, however, is multi-directional and involves simultaneous horizontal and vertical components. Scenario 5 incorporates bidirectional loading to illustrate directional effects, while full multi-directional shaking requires dynamic time–history analysis and is identified as a direction for future work.

After each model run, coordinate data were exported from Rocscience RS2 to Microsoft Excel. Using the auxiliary program SW-DTM, the coordinates were imported into AutoCAD (2025) as discrete points and connected using the CPT command (previously integrated into AutoCAD) to visualize the pipeline’s deformation profile under the imposed seismic loading. SW-DTM was used only to convert RS2-exported node coordinates into an AutoCAD-compatible format, and the CPT command generated a continuous polyline of the pipeline perimeter for precise visualization. These tools do not modify numerical results; they simply enable accurate geometric overlays of static and deformed shapes. The multi-software workflow (RS2 → Excel → SW-DTM → AutoCAD) was adopted because RS2 does not export continuous perimeter geometry for direct comparison across scenarios. To avoid transfer errors, coordinates were exported directly from RS2, processed using the same routine for all cases, and cross-checked against RS2 displacement plots to ensure that no scaling or distortion occurred.

2.7. Data Processing and Scenario Summary

The maximum shear forces and horizontal displacements obtained from each scenario were compiled and plotted in Microsoft Excel for comparison and trend evaluation. The six scenarios of the study and their corresponding variable and constant parameters are summarized in Table 2.

Table 2. The six scenarios of the research and their variable and constant characteristics.

Scenario	1	2	3	4	5	6	Typical Case
Variables	Depth	Diameter	GWT	PGA	Slope	Soil Type
Fixed values for control groups	*	Depth-1.5 m	Depth-1.5 m	Depth-1.5 m	Depth-1.5 m	Depth-1.5 m	Depth-1.5 m
	D-24 in.	**	D-24 in.	D-24 in.	D-24 in.	D-24 in.	D-24 in.
	GWT-10 m	GWT-10 m	***	GWT-10 m	GWT-10 m	GWT-10 m	GWT-10 m
	PGA-0.4 g	PGA-0.4 g	PGA-0.4 g	****	PGA-0.4 g	PGA-0.4 g	PGA-0.4 g
	Slope-0°	Slope-0°	Slope-0°	Slope-0°	*****	Slope-0°	Slope-0°
	Soil-Layered	Soil-Layered	Soil-Layered	Soil-Layered	Soil-Layered	******	Soil-Layered

* 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. ***** Slope Gradients are: (0, 5, 10)° in both directions. ****** Soil Types are: Homogeneous soil (Clayey soil and Sandy soil), Heterogeneous soil.

3. Results

For completeness, plots from the 2D FEA simulations in RS2 are provided in Appendix B (horizontal displacement) and Appendix C (shear force), while the detailed quantitative analysis for each scenario is presented in the following subsections.

3.1. Scenario 1—Effect of Burial Depth

Table 3 summarizes the maximum shear forces and horizontal displacements for burial depths from 1–5 m under a constant PGA of 0.4 g.

Table 3. Maximum shear forces (kN) and horizontal displacements (m) in single empty pipelines under a PGA of 0.4 g with variable burial depths (1–5 m).

Burial Depth (m)	Shear Force (kN)	Horizontal Displacement (m)
1	11.539	1.6133
1.5	12.983	1.6219
2	14.523	1.6321
3	15.383	1.6482
4	15.162	1.6636
5	15.993	1.6788

3.1.1. Shear Force Variation with Burial Depth

Figure 5 shows that when pipelines are buried between 1 m and 3 m, the maximum shear force increases noticeably with depth. This indicates that shallow pipelines are more vulnerable to seismic-induced forces, while deeper burial enhances confinement and resistance due to improved soil–pipe interaction. Beyond 3 m, the increase in shear force stabilizes, suggesting that the overburden soil provides greater lateral restraint. A slight secondary rise at depths beyond 4 m is attributed to increased confinement pressure or a contrast in soil stiffness at deeper levels, which affects stress redistribution around the pipe.

Figure 5. Relationship between depth of burial and the maximum shear force when the PGA = 0.4 g in single empty pipelines.

This pattern is consistent with geotechnical principles, where increased burial depth initially improves stability through confinement but later introduces nonlinear effects as overburden stiffness and confinement alter the dynamic response.

3.1.2. Horizontal Displacement Variation with Burial Depth

Figure 6 illustrates that horizontal displacement increased slightly with depth, following a nonlinear trend. Although deeper embedment provides stronger confinement, it also increases the mass of surrounding soil, transferring more seismic inertia to the pipeline. This explains the slight rise in displacement despite higher confinement. The results align with O’Rourke and Liu (1983) [24], who noted that deeper burial can, in some cases, amplify seismic demand on buried pipelines.

Figure 6. Relationship between burial depth and the maximum horizontal displacement when the PGA = 0.4 g in single empty pipelines.

As shown in Figure 7, AutoCAD plots of pipeline perimeter displacements reveal a consistent increase in movement of about 1.5 cm per additional meter of burial depth, confirming a gradual increase in deformation irrespective of soil variability.

Figure 7. Plotted coordinates showing the horizontal displacement of a single empty pipeline during both static and seismic conditions (PGA = 0.4 g) for burial depths (1–5) m.

3.2. Scenario 2—Effect of Pipeline Diameter

The results for varying pipeline diameters (8–56 in.) under a constant PGA of 0.4 g are summarized in Table 4.

Table 4. Maximum shear forces (kN) and horizontal displacements (m) in single empty pipelines under a PGA of 0.4 g with variable diameters (8–56 in.).

Pipeline’s Diameter (in.)	Shear Force (kN)	Horizontal Displacement (m)
8	6.6868	1.6168
16	11.384	1.6203
24	12.986	1.6225
48	14.154	1.6373
56	23.117	1.6382

3.2.1. Shear Force Variation with Diameter

The maximum shear force initially increases with pipe diameter, as larger pipelines mobilize greater passive soil resistance when subjected to seismic forces. This results in higher shear stresses at the pipe–soil interface. However, beyond a certain range (approximately 24 to 48 inches in diameter), the rate of increase in shear force gradually levels off. This behavior marks a transitional stage in the soil–structure interaction, where the increased stiffness of the larger pipe reduces the relative deformation between the pipe and surrounding soil, thereby leading to a stabilization in the net shear force (Figure 8).

Figure 8. Relationship between pipeline’s diameter and the maximum shear force in single empty pipelines when the PGA = 0.4 g.

Nevertheless, the maximum shear force increased sharply for pipelines with diameters exceeding 48 inches, as larger pipes experience greater inertial forces under pseudostatic loading. In addition, the expanded contact area between the pipe and surrounding soil enhances soil resistance, which in turn amplifies the internal shear force developed along the pipe wall. Consequently, the relationship between pipe diameter and maximum shear force exhibits a nonlinear, curved trend, reflecting the combined influence of inertia and soil–structure interaction effects.

3.2.2. Horizontal Displacement Variation with Pipeline’s Diameter

Figure 9 illustrates the gradual increase in maximum horizontal displacement, rising from approximately 1.615 m at 8 inches to about 1.639 m near 50 inches in diameter. Beyond this range, the curve flattens slightly, suggesting a plateau in displacement response. Smaller-diameter pipelines possess lower mass and, consequently, lower inertial forces during seismic excitation, allowing the surrounding soil to confine them more effectively. In contrast, as the pipe diameter increases, the associated mass and seismic inertia also increase, producing greater displacement. Furthermore, a larger contact area between the pipe and the soil enhances interaction, but this interaction may not fully restrain motion, particularly when a stiffness mismatch exists between the pipe and the surrounding soil. At even larger diameters, the flattening of the curve indicates that displacement tends to stabilize, as the combined effects of increased stiffness and inertial resistance counteract further movement, while soil resistance and pipe–soil contact likely reach their capacity limit in resisting additional displacement.

Figure 9. Relationship between pipeline’s diameter and the maximum horizontal displacement in single empty pipelines when the PGA = 0.4 g.

The AutoCAD plot (Figure 10) illustrated the horizontal spacing of single pipelines subjected to a pseudostatic seismic load with a peak ground acceleration (PGA) of 0.4 g. For small to moderate pipeline diameters, the horizontal displacement was directly proportional to pipe diameter, meaning that the horizontal displacement of the pipeline increased as the diameter enlarged. However, when the diameter exceeded 24 inches, the displacement trend fluctuated—it decreased for the 48-inch pipeline and increased again for the 56-inch pipeline. This variation was not distinctly visible in the plot, as the spacing measurements were taken from pipe center to center, which may have masked the subtle differences caused by deformation and soil–pipe interaction effects.

Figure 10. Plotted coordinates showing the horizontal displacement in (m) of a single empty pipeline during both static and seismic conditions (PGA = 0.4 g) for diameters (8–56) in.

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

Table 5 presents the results for groundwater table depths between 1.4 m and 20 m, including the case of no groundwater, under a PGA of 0.4 g.

Table 5. Maximum shear forces (kN) and horizontal displacements (m) in single empty pipelines under various groundwater table (GWT) depths.

GWT (m)	Shear Force (kN)	Horizontal Displacement (m)
1.4	9.2444	2.76
2	10.815	2.5762
3.6	12.368	2.1735
5	12.966	1.9843
10	12.986	1.622
15	12.343	1.4695
20	11.564	1.3465
NO GWT	10.983	1.1737

3.3.1. Shear Force Variation with GWT

The plotted curve in Figure 11 demonstrated that the maximum shear force increased significantly as the groundwater table (GWT) rose, reaching its highest values around 5 m below ground level. Beyond this point, the rate of increase nearly flattened before declining markedly at greater depths. At a very deep GWT of 20 m, the shear force dropped below 12 kN, whereas the dry condition (no groundwater) produced shear forces comparable to those observed when the water table was directly beneath the pipeline.

Figure 11. Relationship between GWT level and the maximum shear force in single empty pipelines when the PGA = 0.4 g.

This behavior reflects the influence of pore water pressure and effective stress on soil stiffness. A higher groundwater level increases pore pressure, thereby reducing effective stress and producing softer, more deformable soil. Consequently, the pipeline experiences greater relative movement and interaction force due to the weakened confining effect, resulting in higher shear forces. In contrast, as the water table deepens, the surrounding soil becomes denser and stiffer, enhancing stability and reducing shear demand. However, in a completely dry condition, the absence of buoyant forces increases the effective weight of the soil, which may slightly raise passive resistance and stress transfer, leading to moderately higher shear forces compared to saturated but stable cases.

3.3.2. Horizontal Displacement Variation with GWT

As shown in Figure 12, when the groundwater table (GWT) lies near the surface (1.4–5 m U/G), the maximum horizontal displacement is noticeably higher, peaking at approximately 2.7 m when the GWT fully covers the pipeline. This occurs due to the reduction in soil stiffness and the buoyant effect of water, which decrease effective stress and make the surrounding soil more deformable, thereby allowing the pipeline to move more freely under seismic excitation. As the GWT descends to about 5 m, the displacement reduces sharply, and beyond this depth, the decrease becomes more gradual. This trend reflects the increasing confinement and stiffness of the less saturated or drier soil, which better supports the pipeline and limits lateral movement during seismic loading.

Figure 12. Relationship between GWT level and the maximum horizontal displacement in single empty pipelines when the PGA = 0.4 g.

At a deep GWT of 20 m, the maximum horizontal displacement is around 1.35 m, while in dry conditions (no groundwater), the value drops further to approximately 1.2 m, indicating the highest soil resistance due to the absence of pore pressure and saturation effects.

The AutoCAD plot (Figure 13) revealed that, in general, higher groundwater table (GWT) levels corresponded to greater horizontal spacing or lateral movement of the pipeline. The largest horizontal displacement occurred when the GWT was at 1.4 m below ground, where the water level was above the pipeline, resulting in maximum soil softening and reduced confinement.

Figure 13. Plotted coordinates showing the horizontal displacement in (m) of a single empty pipeline during both static and seismic conditions (PGA = 0.4 g) for GWT levels (1.4–20) m, including the case when there is no GWT.

It is important to note that the large displacement values obtained for very shallow groundwater (e.g., >2 m) may not represent realistic field-scale pipeline movements. These values result from the combination of (i) monotonic pseudo-static loading, which lacks the confining effects of cyclic soil response; (ii) severe reduction of effective stress under near-saturated conditions; and (iii) the elastic–perfectly plastic Mohr–Coulomb soil model, which does not incorporate strain-dependent stiffness degradation or liquefaction onset. Accordingly, the absolute displacement magnitudes should be interpreted as a relative indicator of sensitivity to groundwater depth, not as a prediction of true seismic displacement.

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

The variation of PGA from 0.1 g to 0.6 g was analyzed under constant burial and geometric conditions, as shown in Table 6.

Table 6. Maximum shear forces (kN) and horizontal displacements (m) in single empty pipelines under PGAs of 0.1–0.6 g.

PGA (g)	Shear Force (kN)	Horizontal Displacement (m)
0.1	0.68649	0.20021
0.2	2.3816	0.4581
0.3	7.3031	0.8569
0.4	12.983	1.6219
0.5	13.715	2.764
0.6	15.919	4.7457

3.4.1. Shear Force Variation with PGA

As shown in Figure 14, the maximum shear force increased progressively with rising peak ground acceleration (PGA), which is a logical trend since stronger ground shaking generates greater inertial forces within the pipeline–soil system. The increase in shear force was moderate at PGA values below 0.2 g, as mild seismic excitation produces smaller inertial effects and limited shear demand. However, between 0.2 g and 0.4 g, the maximum shear force rose sharply, reflecting the greater relative displacement between the soil and pipeline, as well as the enhanced lateral pressures acting on the pipe and the mobilization of passive and active soil resistance during stronger shaking.

Figure 14. Relationship between PGA and the maximum shear force in single empty pipelines.

After this stage, the rate of increase in maximum shear force slowed down, forming a temporary plateau around a PGA of 0.5 g. This behavior can be attributed to soil yielding, energy dissipation, or localized soil failure, which temporarily limit further force development despite continued ground motion. However, at higher PGA levels, the maximum shear force began to rise again, as very strong seismic excitation surpasses the soil’s capacity, resulting in greater pipeline–soil interaction and internal force buildup within the pipeline structure.

The overall pattern indicates potential stiffness variations within the soil–structure system, likely arising from material yielding, nonlinear stress–strain behavior, or other energy-dissipating mechanisms that influence the seismic response at higher PGA levels

3.4.2. Horizontal Displacement Variation with PGA

The results in Figure 15 demonstrated that the maximum horizontal displacement increased nonlinearly with rising peak ground acceleration (PGA). At lower PGA values (0.1–0.3 g), displacement increased gradually and nearly linearly, indicating elastic behavior of both the soil and the pipeline. However, at higher PGA levels (0.4 g and above), the displacement grew more sharply, reflecting amplified soil and pipeline deformation under stronger shaking. This nonlinear response suggests that the soil–pipeline system begins to transition into a non-elastic or plastic state as seismic intensity increases, where stiffness degradation and potential localized failure start to dominate the deformation behavior.

Figure 15. Relationship between PGA and the maximum horizontal displacement in single empty pipelines.

The overall trend illustrates that soil–structure systems (such as slopes, buried pipelines, and retaining walls) exhibit minimal movement under low seismic loads, but experience rapidly increasing displacements once a critical acceleration threshold is exceeded, marking the onset of nonlinear or inelastic behavior. In our analyses, this threshold PGA value is obtained as 0.6–0.7 g. With a PGA value greater than 0.6 g, no logical values are obtained in the FEM analyses.

Plotting the results in AutoCAD (Figure 16) revealed that the pipelines experienced greater movement and deformation as the peak ground acceleration (PGA) increased. Such behavior signifies a higher risk of structural damage, since the pipeline network is composed of connected segments, fittings, and valves that are highly sensitive to differential movement. Excessive deformation can lead to joint loosening, misalignment, or leakage, potentially compromising the integrity and functionality of the entire pipeline system.

Figure 16. Plotted coordinates showing the horizontal displacement in (m) of a single empty pipeline during both static and seismic conditions of PGA (0.1–0.6) g.

3.5. Scenario 5—Effect of Slope Gradient

In this scenario, the influence of slope gradient was examined in both orientations: the negative slope, where the lowest ground level faces the seismic load (i.e., in the same direction as the seismic excitation), and the positive slope, where the highest ground level faces the earthquake (i.e., in the reverse direction of the seismic load). The results are shown in Table 7:

Table 7. Maximum shear forces (kN) and horizontal displacements (m) in single empty pipelines under PGA of 0.4 g in different slope gradients (0–10)°.

Slope Gradient	(−ve) Direction		(+ve) Direction
	Shear Force (kN)	Horizontal Displacement (m)	Shear Force (kN)	Horizontal Displacement (m)
0°	12.609	1.6018	12.583	1.6341
5°	10.319	1.0315	11.624	1.8164
10°	5.6495	0.85638	11.755	2.2406

3.5.1. Shear Force Variation with the Negative Slope Gradient

The trend illustrated in Figure 17 shows that at flat ground conditions (0° slope), the pipeline is uniformly confined and symmetrically supported by the surrounding soil. When the slope orientation opposes the direction of seismic loading, the seismic force drives the soil mass from right to left, while the gravitational component of the slope acts in the opposite direction, pushing the soil back toward the right. As the slope angle increases, this gravitational resistance becomes more significant, thereby reducing the net relative movement between the soil and the pipeline and lowering the mobilized shear force. Consequently, steeper slopes counteract a larger portion of the seismic force, resulting in a more pronounced decrease in the maximum shear force.

Figure 17. Relationship between slope gradient and the maximum shear force in single empty pipelines when the PGA = 0.4 g in the negative direction.

3.5.2. Horizontal Displacement Variation with the Negative Slope Gradient

The trend shown in Figure 18 indicates that under flat ground conditions (0° slope), the pipeline exhibits the maximum horizontal displacement. This occurs because the soil provides symmetrical and firm confinement, yet the lateral seismic forces act uniformly on both sides, resulting in greater overall deformation and movement. In contrast, as the slope gradient increases, the gravitational component opposing the seismic load becomes stronger, resulting in a reduction in net soil movement and, consequently, lower horizontal displacement of the pipeline. At steep slopes (around 10°), the counteracting gravitational effect becomes dominant, significantly limiting displacement and enhancing the overall stability of the soil–pipeline system.

Figure 18. Relationship between slope gradient and the maximum horizontal displacement in single empty pipelines when the PGA = 0.4 g in the negative direction.

After plotting the horizontal displacement values in AutoCAD (Figure 19), it was observed that the pipeline exhibited shorter horizontal movement as the ground inclination increased. This reduction in displacement is attributed to the horizontal component of pipeline motion acting in the opposite direction of the seismic load, which generates an additional resisting lateral force. This opposing action effectively diminishes the net impact of the seismic excitation, thereby reducing the overall horizontal displacement and enhancing the stability of the pipeline on steeper slopes.

Figure 19. Plotted coordinates showing the horizontal displacement in (m) of a single empty pipeline during both static and seismic conditions for slope gradients (0–10°) in the negative direction.

3.5.3. Shear Force Variation with the Positive Slope Gradient

The chart presented in Figure 20 illustrates that under flat ground conditions (0° slope), the maximum shear force reaches its highest value of approximately 12.6 kN. When the slope is aligned with the seismic direction, the soil mass is more easily mobilized by the ground motion. At this stage, the gravitational component along the slope is relatively small, meaning the shear force is primarily governed by seismic excitation and frictional resistance at the soil–pipe interface. As the slope angle increases, the maximum shear force decreases, reaching a minimum of about 11.6 kN at a 5° slope. This reduction occurs because the downslope gravitational component reduces the normal stress acting on the pipeline, thereby lowering frictional resistance and permitting slightly freer relative movement between the soil and the pipe. Beyond this point, the maximum shear force rises slightly again, to around 11.8 kN, as the combined effects of gravity and seismic acceleration increase the downslope soil movement, thereby reintroducing higher shear demand on the pipeline.

Figure 20. Relationship between slope gradient and the maximum shear force in single empty pipelines when the PGA = 0.4 g in the positive direction.

3.5.4. Horizontal Displacement Variation with the Positive Slope Gradient

Figure 21 illustrates that the maximum horizontal displacement increases nonlinearly with slope gradient, and as the slope becomes steeper, the rate of increase in displacement becomes more pronounced. This behavior results from the greater gravitational driving forces acting in the same direction as the seismic excitation, combined with a reduction in resisting forces such as soil shear strength and frictional resistance. Consequently, pipelines situated on steeper slopes become more susceptible to lateral deformation during strong ground shaking.

Figure 21. Relationship between slope gradient and the maximum horizontal displacement in single empty P/Ls when the PGA = 0.4 g in the positive direction.

Referring to the coordinate plots shown in Figure 22, it was observed that, unlike the reverse (negative) slope condition, the pipelines experienced greater horizontal movement as the slope inclination increased. This behavior results from the combined influence of the horizontal seismic load and the downslope weight of the overlying soil mass, both acting in the same direction. The gravitational component of the slope therefore amplifies the seismic effect, causing the pipeline to move longer distances and experience greater lateral deformation with increasing slope steepness.

Figure 22. Plotted coordinates showing the horizontal displacement in (m) of a single empty pipeline during both static and seismic conditions for slope gradients (0–10°) in the positive direction.

3.6. Scenario 6—Effect of Soil Type

The sixth scenario examined three soil types—clayey, sandy, and heterogeneous—under identical seismic and geometric conditions.

For each soil type, the initial conditions, stiffness, strength, and hydraulic properties were defined consistently throughout the analysis to ensure that variations in pipeline response resulted solely from the parameters set for each type. In all analyses, the Datum Dependency and Stage Factors options were left unselected to maintain uniform boundary and loading conditions across all runs. Moreover, all soil types were modeled under drained conditions, with a fluid bulk modulus of 2.2 × 10⁶ kPa, piezometric water mode, and Hu = 1.

In clayey soil, the Initial Element Loading was set to Field Stress and Body Force, with moisture content considered in the unit weight. The dry, moist, and saturated unit weights were 16.2, 19.44, and 19.8 kN/m³, respectively, with a porosity value of 0.5. The material exhibited isotropic stiffness, characterized by E = 6100 kPa and ν = 0.35.

The Mohr–Coulomb failure criterion was employed, assuming plastic behavior with c = 4.23 kPa, φ = 31°, tensile strength = 40.86 kPa, and dilation = 0°.

In sandy Soil, the Initial Element Loading was set to Field Stress and Body Force, and moisture content was included in the unit weight. The dry, moist, and saturated unit weights were 16.2, 20.25, and 21 kN/m³, respectively, with porosity = 0.5. The material was defined as isotropic, with E = 10,470 kPa and ν = 0.30.

The Mohr–Coulomb model governed the strength behavior, assuming plastic material response with c = 0 kPa, φ = 34.76°, and tensile strength = 64.98 kPa, while the dilation angle was 0°. The residual parameters were set equal to the peak values.

In heterogeneous Soil, the Initial Element Loading was also defined as Field Stress and Body Force, with moisture content accounted for in the unit weight. The dry, moist, and saturated unit weights were 16.2, 19.8, and 20.5 kN/m³, respectively, and the porosity was 0.5. The material exhibited isotropic stiffness, with E = 8000 kPa and ν = 0.32.

The Mohr–Coulomb failure criterion was used with plastic behavior, where c = 2.5 kPa, φ = 32°, and tensile strength = 52.75 kPa, with a dilation angle = 0°. The residual parameters were taken as equal to the peak values.

The results are summarized in Table 8.

Table 8. Maximum shear forces (kN) and horizontal displacements (m) in single empty pipelines under a PGA of 0.4 g in different soil types.

Soil Type	Shear Force (kN)	Horizontal Displacement (m)
Clayey soil	15.203	2.835
Sandy soil	13.836	1.7963
Heterogeneous	11.72	1.7687

3.6.1. Shear Force Variation with Soil Type

In general, the shear force distribution along the pipeline was nearly uniform across all soil types, with the highest maximum shear force recorded in clayey soil and the lowest observed in heterogeneous soil. Clayey soil, characterized by high cohesion but low frictional resistance, tends to deform more plastically under seismic loading, resulting in greater relative movement between the soil and the pipeline, which mobilizes larger shear forces along the pipe–soil interface. In contrast, heterogeneous soil (a mixture of clay, sand, and silt) exhibits better interlocking and variable stiffness, which dissipates seismic energy more effectively and reduces the concentration of shear stresses along the pipeline, leading to lower maximum shear force values.

3.6.2. Horizontal Displacement Variation with Soil Type

Figure 23 illustrates that the pipelines exhibited greater horizontal movement in clayey soil. This behavior can be attributed to the lower frictional resistance and higher plasticity of clay, which reduce the soil’s ability to restrain pipeline motion, leading to larger overall displacements under seismic loading.

Figure 23. Horizontal displacement that each point of a single empty pipeline section’s perimeter moves during a seismic event (PGA = 0.4 g) for different types of soil.

After plotting the pipeline movement outputs in AutoCAD (Figure 24), the results revealed that the pipelines embedded in clayey soil experienced greater horizontal displacement compared to those in sandy and heterogeneous soils. This indicates that the lower shear strength and higher deformability of clayey soil allow greater relative movement between the pipeline and the surrounding soil during seismic loading.

Figure 24. Results plot showing the single empty pipelines movement (m) in different types of soils for a PGA of 0.4 g.

4. Validations of RS2 Results Using Rocscience RS3

To evaluate the reliability of the two-dimensional (2D) finite element results, a representative model was re-simulated using the three-dimensional (3D) FEM platform Rocscience RS3. This verification step aims to determine whether the simplified 2D plane–strain formulation of RS2 adequately captures the essential soil–pipeline interaction mechanisms under seismic loading. Only one configuration was analyzed in RS3, as performing 3D analyses for all parametric cases would be computationally prohibitive.

The RS2 model geometry, soil profile, seismic loading, and pipeline properties were imported directly into RS3. Boundary conditions were applied to reproduce the RS2 setup: the two side faces were restrained in the horizontal directions corresponding to out-of-plane movement, the front and back faces were restrained in the Y direction, the base was fixed in the Z direction, and the top surface was free. This configuration is consistent with standard 3D approximations of 2D plane–strain conditions and minimizes artificial constraints (Figure 25).

Figure 25. Restraints in RS3 FEM Model (green: Y restraint, blue: Z restraint, red: X restraint).

After the analysis, a mid-length section cut was taken through the pipeline to extract shear force and displacement data along the pipe perimeter (Figure 26). RS3 presents sectional outputs in terms of global coordinates rather than along the arc length of the pipe; therefore, the X-coordinate was selected as the common reference axis for comparison with RS2.

Figure 26. Section-cut view of the RS3 model showing pipeline response.

4.1. Shear Force Response

Figure 27 and Figure 28 present the shear force distributions obtained from the RS2 and RS3 analyses. Although the RS3 output contains small oscillations resulting from 3D mesh sampling along the sectional cut, the global shape of the shear force curve closely follows that produced by RS2. Both models exhibit two pronounced shear peaks along the pipeline perimeter, corresponding to the locations of maximum bending curvature generated by the lateral seismic excitation.

Figure 27. Shear force distribution from the RS2 analysis.

Figure 28. Shear force distribution from the RS3 analysis.

The comparison reveals strong quantitative agreement between the two simulations. RS2 predicted shear forces ranging from approximately +20.95 to −20.49 kN/m, while the trend values from RS3 fell between +19.48 and −21.80 kN/m. The difference between the two results remains within roughly five to ten percent—an acceptable margin considering the inherent complexity of soil–pipeline interaction and the dimensional differences between 2D and 3D formulations. More importantly, the spatial distribution of the shear forces is nearly identical, indicating that RS2 accurately reproduces the primary load-transfer mechanism from soil to pipeline under the applied seismic loading.

This close correspondence is noteworthy because 3D models inherently allow deformation modes that are suppressed in 2D, such as torsional components and out-of-plane bending. The limited influence of these additional degrees of freedom in RS3 suggests that the seismic response in this configuration is governed predominantly by in-plane bending and shear transfer. These behaviors are well represented within the plane–strain framework used in RS2, reinforcing the reliability of the 2D approach for capturing the essential shear response of buried pipelines during seismic events.

4.2. Horizontal Displacement Response

Figure 29 and Figure 30 show the horizontal displacement distributions along the pipeline cross-section for both RS2 and RS3. Both models produce a smooth displacement profile with distinct maximum and minimum values, reflecting the characteristic pattern of lateral cross-sectional distortion under seismic loading. RS2 predicted horizontal displacements between 1.624 m and 1.635 m, while RS3 values ranged from 1.648 m to 1.661 m. Although the magnitudes obtained from RS3 are slightly higher—typically by 1–2%—the overall deformation pattern remains nearly identical. This difference is expected due to the 3D model’s ability to accommodate minor out-of-plane deformation components and represent three-dimensional stress redistribution within the surrounding soil more realistically.

Figure 29. Horizontal displacement profiles from RS2.

Figure 30. Horizontal displacement profiles from RS3.

The corresponding reduction in the pipe diameter was also consistent between the two models, with RS2 predicting 0.011 m of contraction and RS3 predicting 0.013 m. The small difference between these values confirms that RS2 captures the essential mechanism of seismic-induced cross-sectional distortion, reinforcing the suitability of the 2D approach for representing lateral deformation behavior in buried pipelines.

4.3. Interpretation and Implications

The comparison between the RS2 and RS3 simulations clearly demonstrates that the 2D plane–strain formulation provides a reliable approximation of the seismic response of buried pipelines. Although RS3 offers a higher-fidelity representation of the soil continuum and allows for fully three-dimensional deformation modes, the 2D model successfully reproduced the essential features of the system’s behavior. Both approaches yielded highly similar deformation mechanisms, consistent shear and bending trends, comparable displacement magnitudes, and nearly identical stress-transfer patterns along the pipeline perimeter. The close agreement is particularly meaningful given the substantial computational effort typically required for full 3D analyses, especially when conducting broad parametric studies involving multiple geometric and loading variables such as burial depth, diameter, slope angle, groundwater level, and permanent ground deformation. The results, therefore, indicate that RS2 can be confidently used as an efficient and accurate tool for evaluating a large number of scenarios, while 3D analyses may be reserved for critical configurations or verification purposes.

The similarity of the responses also suggests that out-of-plane effects and torsional components contribute minimally to the overall behavior under the loading configuration examined in this study. The governing mechanism appears to be dominated by in-plane bending and shear transfer between the soil and the pipeline, both of which are effectively captured by the plane–strain elements in RS2. This reinforces the suitability of 2D modeling for buried pipeline systems subjected to lateral seismic excitation and supports the methodological framework adopted for the broader analysis presented in this paper.

5. Relationship Between the Moment Magnitude and Surface Displacement

The classic set is from Wells & Coppersmith (1994); below for “all slip types” forms [25]:

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

$$Mw=6.69+0.74 {\log }_{10}\left(\mathit{MD}\right)$$

Inverted:

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

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: log₁₀(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.

It should be emphasized that the present validation is numerical rather than physical. Although the 2D–3D agreement demonstrates internal consistency and confirms that RS2 captures the dominant in-plane deformation mode, this comparison does not constitute validation against experimental data, analytical formulations, or empirical models. Classical analytical approaches—such as elastic foundation beam theory, Newmark sliding block formulations, Kennedy’s fault-crossing equations, and Wang & Yeh’s simplified soil–spring models—assume small strains, linear or bilinear soil springs, and prescribed PGD patterns. These assumptions differ fundamentally from the large-deformation pseudo-static loading applied in this study. Therefore, direct quantitative comparison with analytical solutions is not feasible, and the present validation should be interpreted as a numerical consistency check rather than full physical verification.

It should be noted that the RS2 liner captures bending stiffness but does not explicitly represent ring ovalization or hoop flexibility. The cross-sectional displacement profiles therefore represent in-plane deformation consistent with the 2D plane–strain assumption. RS3, which employs full 3D solid elements, provides a more realistic representation of circumferential deformation; however, the good agreement between the two models indicates that in-plane bending dominates the response for the simplified pseudo-static loading used in this study.

6. Discussion

The pseudo-static loading scheme adopted in this study provides an efficient means of exploring parametric trends, but it inherently omits dynamic soil behaviors such as inertial amplification, cyclic degradation, strain-rate effects, and pore-pressure evolution. As a result, the computed responses correspond to simplified lateral seismic demand, not the full earthquake response of buried pipelines. The findings should therefore be interpreted as indicative rather than predictive for dynamic conditions.

The results indicate that the pseudo-static seismic response of continuous buried pipelines is governed by a competition between confinement gains and inertial penalties, which are modulated by the soil state and local topography. In the depth series (Table 3; Figure 5, Figure 6 and Figure 7), shear demand increases distinctly from 1–3 m and then approaches a plateau, while horizontal displacement rises gently yet persistently with depth. This pattern is consistent with deeper cover mobilizing higher interface resistance while simultaneously engaging a larger overburden “inertia block” that transfers additional seismic demand to the pipe; such non-monotonic depth sensitivity and the emergence of a most-unfavorable embedment range have been reported for underground systems subjected to strong shaking [26]. Classic lifeline guidance also notes that continuous pipelines can shift between stiffness- and inertia-controlled regimes depending on embedment and input intensity, which explains the persistence of lateral displacement at greater cover [27].

Diameter effects (Table 4; Figure 8, Figure 9 and Figure 10) exhibit a stiffness–inertia crossover: interface shear scales nonlinearly with size (marked rise up to ~48 in, then a sharp increase at 56 in), whereas displacement grows only slightly and then flattens. Recent finite-element studies that explicitly model lateral soil–pipe interaction in sands report similar scaling—internal forces escalate nonlinearly with diameter and wall properties even where global displacements change modestly—mirroring the shear jump at the upper end of the diameter range and the near-plateau in displacement [28].

Groundwater conditions emerge as a first-order modifier (Table 5; Figure 11, Figure 12 and Figure 13). The largest displacements occur when the water table envelops the pipe, with sharp reductions as GWT drops below the springline and gradual decreases thereafter—an ordering compatible with effective-stress reasoning (reduced stiffness under elevated pore pressure, restored stiffness in deeper/dry states). Recent regional and mechanistic analyses document that rising groundwater markedly increases the extent and severity of earthquake-induced softening/liquefaction, thereby amplifying near-surface deformation potential; the observed peak under shallow GWT aligns with that evidence. Parametric work focused on buried pipelines likewise shows monotonic amplification of response as the water table approaches the pipe zone, reinforcing the generality of the mechanism across loading modalities [28].

The PGA series (Table 6; Figure 14, Figure 15 and Figure 16) reveals a clear elastic-to-nonlinear transition near 0.3–0.4 g, after which both shear and—more prominently—displacement increase rapidly. Shaking-table and numerical studies on buried oil/gas pipelines under bidirectional multi-point excitation identify comparable thresholds beyond which demands escalate sharply, attributing the jump to stiffness degradation, interface slip, and spatial variability of input—consistent with the curvature beyond ~0.4 g and the large displacement increment at 0.5–0.6 g [29]. In this study, the threshold value of PGA is determined to be 0.6–0.7 g.

Topography and loading direction materially influence response (Table 7; Figure 17, Figure 18 and Figure 19). When the slope’s gravitational component opposes input motion (negative slope), both shear and displacement decrease with inclination; when gravity aligns with shaking (positive slope), both increase, and the effect strengthens beyond ~5°. This directional asymmetry is consistent with topographic site effects, where slopes can either attenuate or amplify near-surface motion and strains depending on geometry and excitation orientation. Recent two- and three-dimensional analyses of pipelines traversing relief features report increased strains where topography favors downslope mobilization, corroborating the amplification observed for slopes aligned with seismic demand [30].

Soil type controls both the magnitude and partitioning of demands (Table 8; Figure 23 and Figure 24): clayey > sandy > heterogeneous for displacement and shear. The ordering is expected—clayey matrices permit larger cyclic strains and develop lower interface frictional restraint; well-graded or mixed deposits add internal interlocking and effective stiffness that cap movement. Emerging mitigation studies are consistent: engineered trench backfills (e.g., foam concrete) demonstrably reduce pipeline strains under deformation hazards, and micropile reinforcement in sandy deposits suppresses seismic settlement and pipe response—trends that align with the lower demands measured in the heterogeneous (stiffer/interlocked) condition [31].

Synthesizing across scenarios, three implications follow. First, hazard and state variables—PGA and GWT—are dominant amplifiers of lateral pipeline movement and warrant explicit upper-bound combinations in design checks, consistent with contemporary lifeline assessments [30]. Second, geometric choices tune the stiffness–inertia balance: increasing burial depth and diameter increases confinement and rigidity but also inertial participation, yielding a non-monotonic shear demand with a mild, persistent rise in displacement—precisely the crossover behavior reported in recent numerical literature. Third, terrain and backfill significantly alter the demand envelope: counter-slopes attenuate, and aligned slopes amplify, while stiffer/interlocked, or engineered fills reduce both lateral movement and internal forces, in line with current topography-aware pipeline studies and backfill innovations [28].

7. Limitations Regarding Analytical and Experimental Validation

A further limitation of this study is that the seismic demand was applied predominantly in one horizontal direction. The pseudo-static formulation commonly assumes a single controlling deformation direction, consistent with practice in PGD-based pipeline design. However, real earthquakes induce multi-directional shaking and vertical components that can influence soil–pipe interaction. Bidirectional effects were explored for the slope scenarios, but a comprehensive multi-directional assessment requires dynamic modeling, which is beyond the scope of the present study and will be addressed in future work.

Another limitation arises from the liner representation of the pipeline. While the liner includes axial, shear, and bending stiffness, it does not capture circumferential hoop behavior or explicit ovalization mechanisms. Therefore, the deformation patterns reflect 2D plane–strain in-plane distortion rather than true 3D ring ovalization. For loading conditions where cross-sectional collapse, wrinkling, or localized buckling are critical, shell or solid elements would be required. Future work will incorporate 3D modeling to investigate these deformation modes.

Another limitation is the absence of comparison with analytical solutions, empirical relationships, or physical model test data. While several classical formulations exist for buried pipelines—such as Winkler elastic foundation beam theory, Newmark-type PGD models, Kennedy’s strain-based criteria, and O’Rourke & Liu’s soil–spring representations [24,32,33,34,35]—these models are derived for small-strain, idealized loading and therefore do not directly correspond to the monotonic pseudo-static loading adopted here. Nevertheless, the qualitative trends observed in the present work (e.g., increased lateral resistance with burial depth, higher deformation under shallow groundwater, and larger soil compliance in clay) are broadly consistent with the mechanisms these analytical formulations describe.

Future studies will incorporate direct comparisons using closed-form foundation-beam solutions and available experimental results to establish quantitative validation beyond software-to-software agreement.

8. Conclusions

This study employed two-dimensional numerical modeling using Rocscience RS2 to evaluate the seismic response of buried steel pipelines subjected to fault-induced ground movements. The analysis investigated the influence of burial depth, pipeline diameter, groundwater table depth, peak ground acceleration (PGA), slope gradient, and soil type on the resulting shear forces and horizontal displacements. The results confirmed that each parameter exerts a distinct influence on the pipeline’s structural behavior during seismic loading.

The simulations revealed that deeper burial and larger pipe diameters increase both shear force and horizontal displacement due to higher confinement and inertial effects. Similarly, increasing PGA produced nonlinear amplification of both response measures, highlighting the sensitivity of buried pipelines to strong ground motion. In contrast, shallow groundwater tables reduced soil stiffness and increased buoyancy, leading to greater displacements. Clayey soil exhibited the highest shear and deformation responses because of its low frictional resistance and high plasticity, while sloped ground conditions affected the response depending on whether the seismic load acted upslope or downslope.

Among all the investigated factors, pipeline diameter was identified as the most influential parameter affecting shear force, followed by burial depth and soil type; slope gradient had the least impact. The computed shear forces were significantly lower than the theoretical yielding thresholds (ex. 4.378 MN for 24 in. pipeline), indicating that the modeled pipelines remain structurally safe against shear-induced failure. However, local deformations or strain concentrations may still occur under combined or cyclic loading, which warrants further investigation.

It is recommended that seismic design and maintenance strategies for lifeline pipeline networks prioritize regions with shallow groundwater levels, cohesive soils, or steep slopes. Particular attention should be given to joints, fittings, and connectors, which are most susceptible to failure under seismic displacement. Future studies should incorporate internal pressure effects, multiple parallel pipeline interactions, and surcharge loading to capture more realistic boundary conditions. Future work will incorporate fully dynamic time–history analyses using RS3 and real earthquake records to capture wave propagation, pore-pressure buildup, and liquefaction effects not represented by the pseudo-static approach used here. The validation can also be extended by comparing numerical results with analytical solutions (e.g., elastic foundation beam theory), empirical PGD relationships, and available experimental data to establish full physical verification. Ultimately, integrating parametric results into cost–benefit frameworks can support the development of effective mitigation and protection measures for buried pipeline infrastructure in seismically active regions.